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Ž .The Science of the Total Environment 272 2001 3342

Numerical modelling of radon-222 entry into houses:an outline of techniques and results

Claus E. Andersen

Risø National Laboratory, DK-4000 Roskilde, Denmark

Abstract

Numerical modelling is a powerful tool for studies of soil gas and radon-222 entry into houses. It is the purpose ofthis paper to review some main techniques and results. In the past, modelling has focused on Darcy flow of soil gasŽ .driven by indooroutdoor pressure differences and combined diffusive and advective transport of radon. Models ofdifferent complexity have been used. The simpler ones are finite-difference models with one or two spatialdimensions. The more complex models allow for full three-dimensional and time dependency. Advanced featuresinclude: soil heterogeneity, anisotropy, fractures, moisture, non-uniform soil temperature, non-Darcy flow of gas, andflow caused by changes in the atmospheric pressure. Numerical models can be used to estimate the importance ofspecific factors for radon entry. Models are also helpful when results obtained in special laboratory or test structure

Žexperiments need to be extrapolated to more general situations e.g. to real houses or even to other soilgas.pollutants . Finally, models provide a cost-effective test bench for improved designs of radon prevention systems. The

paper includes a summary of transport equations and boundary conditions. As an illustrative example, radon entry iscalculated for a standard slab-on-grade house. 2001 Elsevier Science B.V. All rights reserved.

Keywords: Advection; Diffusion; Houses; Modelling; Radon-222; Soils; Transport

1. Introduction

Because of the increased lung-cancer riskŽbelieved to be associated with radon UNSCEAR,

.1993 , it is of interest to identify houses with high

Tel.: 45-4677-4912; fax: 45-4677-4959.Ž .E-mail address: [email protected] C.E. Andersen .

levels and to minimize radon entry from the soil.Many resources have therefore been devoted tothis task, and much has been learned about this

Žsoilhouse interaction Nazaroff et al., 1988;.Nazaroff, 1992 .

Essentially, radon entry is controlled by fourŽ . Ž .processes: 1 generation of radon in the soil; 2

Ž .transport of radon through soil; 3 transportŽthrough the building shell e.g. through cracks in

0048-969701$ - see front matter 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 0 4 8 - 9 6 9 7 0 1 0 0 6 6 2 - 3

( )C.E. Andersen The Science of the Total Enironment 272 2001 334234

. Ž .the slab ; and 4 driving forces such asindooroutdoor pressure differences. With themany factors in play, radon entry is a relativelycomplex problem. One approach to learn moreabout it is to use numerical modelling as a stand-alone tool or to use modelling in conjunction withspecial experiments. The purpose of this paper isto outline some modelling techniques and results.

2. Equations

Consider a reference element V of soil. Thisvolume may be split into three parts: V for thegvolume of grains, V for the volume of water,wand V for the volume of air:a

Ž .VV V V 1g w a

Ž .Hence, the total porosity , the water porosity , and the air porosity can be expressed as:w a

V Vw a Ž . 2V

Vw Ž . 3w V

Va Ž . 4a V

Likewise, the total activity A of radon-222Žsimply referred to as ‘radon’ in all of the fol-

.lowing in V may be split into three parts:

Ž . A A A A 5g w a

where the indices have the same meaning asbefore. We now define the concentration of radonin the air-filled parts of the pores as:

Aa Ž .c 6a Va

and the radon concentration in the water-filledparts of the pores as:

Aw Ž .c 7w Vw

Part of the grain activity A is available forgtransport in the pore system. This is the radonadsorbed to soil-grain surfaces: A . The im-g,s

Ž .mobile part A A is radon produced byg g,sthe ‘non-emanating’ part of the grain radium. Inline with the framework presented by Rogers and

Ž .Nielson 1991a , we introduce the sorbed radonŽ 1 .concentration per kg dry mass Bq kg as:

Ag,s Ž .c 8s Mg

where M is the grain mass within V.gŽWe assume rapid sorption kinetics Wong et

.al., 1992 such that the partitioning of radonbetween air, water and soil grains is permanentlyin equilibrium everywhere in the soil:

Ž .c Lc 9w a

Ž .c Kc 10s a

where L is the Ostwald partitioning coefficient L equals approx. 0.36 at 10C and 0.23 at 25CŽ .Clever, 1979 and K is the radon surface sorp-

Žtion coefficient Rogers and Nielson, 1991a;.Nazaroff, 1992 . The equilibrium assumption sim-

plify the problem considerably as we can thenexpress the total mobile radon activity by refer-ence to the concentration in just one phase. Nor-mally, the radon concentration in the air phase cais selected as ‘reference concentration’, and themobile activity in V is hence given as:

Ž . A A A c V 11a w g,s a

where

Ž . L K 12a w ds

is sometimes called the partition-correctedporosity. If the medium is dry and without grainsorption, we have: . The equilibrium as-sumption is widely used in models of pollutant

Žtransport, but is not universally correct Thomson

( )C.E. Andersen The Science of the Total Enironment 272 2001 3342 35

.et al., 1997 . Support of the assumption can beŽ .found in Nazaroff et al. 1988 and Nazaroff

Ž . Ž .1992 . Finally, we note that in Eq. 12 is thedsdry soil density:

Ž . Ž . 1 13ds g

Ž 3where is the grain density approx. 2.6510g3 .kg m for a wide range of soils .

If radium is present only in soil grains, wedefine the radon generation rate per pore volumeŽ 1 3 .Bq s per m -pore as:

Eds Ž .G 14

Žwhere is the decay constant of radon 2.098386 1.10 s , and E is the emanation rate of radon

Žto the soil pores i.e. the number of atoms thatemanates into water and air per second per kg

.dry mass . We can write the emanation rate asE fA , where f is the fraction of emanationRa

Ž 1 .and A is the activity concentration Bq kgRaof radium-226 per dry mass.

A mass-conservation equation for the mobileradon activity in V is:

c a Ž .Gc j 15at 1Žwhere j is the bulk flux density in units of Bq s

2 .per m at time t. The term ‘bulk’ means that thedensity is measured per total cross-sectional area

1Ž .perpendicular to j. Hence, a flux J Bq sŽacross some plane with geometric area A e.g. a

2 .120-m crawl-space floor and uniform bulk flux

density j gives: J j Aa, where a is a unit vectorˆ ˆperpendicular to the plane.

The bulk flux density is divided into two:

Ž .j j j 16a d

Ignoring water movement, the advective fluxdensity is given by:

Ž .j c q 17a a

Žwhere q is the bulk flux density of soil gas in

3 1 2 .units of m s per m discussed later. Weassume, that the diffusive flux can be written as:

Ž .j Dc 18d a

such that the bulk diffusivity D accounts forradon diffusion through air and water in thepores. D is a function of temperature and pres-

Ž .sure Washington et al., 1994 and may thereforeŽ .if not for other reasons change in time andspace. We assume, that the soilgas flow is so low

Žthat mechanical dispersion can be ignored i.e. D. Žis independent of q Domenico and Schwartz,

.1990 .The flow of soil gas is usually calculated from

Darcy’s law:

k Ž .q p 19

Ž . Ž 2 .where k is the isotropic gas permeability m , Ž 6 1is the dynamic viscosity 17.510 Pa s at

.10C , and p is the disturbance pressure fieldŽ . ŽLoureiro, 1987; Loureiro et al., 1990 i.e. the

.deviation from the aerostatic absolute pressure .Under natural conditions, p is at the order of afew Pa. The disturbance pressure field is found asa solution to the equation of mass conservation:

pa Ž .q 20P t0

where P is the mean absolute atmospheric pres-0Ž 5 .sure approx. 10 Pa . Sometimes more refined

Ž .treatments are considered: 1 If the soil isanisotropic, k is substituted with a tensor. Withlayered soil, horizontal and vertical permeabilities

Ž .are introduced. 2 In cases of fan-driven flows ofŽsoil gas e.g. in connection with sub-slab depres-

.surization systems the flow cannot be well de-Ž .scribed by Darcy’s law so-called non-Darcy flow .

Ž .Eq. 19 is then extended with a Forchheimerterm to account for the non-linear relation

Žbetween flow velocity and pressure gradient Bon-. Ž .nefous et al., 1992 . 3 If the soil does not have a

uniform temperature, natural convection can oc-Ž .cur and Eq. 19 is extended with a term to

account for thermal expansion of the gas when itflows between regions of different temperatures


Ž .Revzan et al., 1991 . A separate equation may beneeded to account for the temperature field inthe soil.

2.1. Boundary conditions

Ž .Eq. 15 is solved for given boundary condi-tions. In most cases, c is set to some fixed valueaŽ .e.g. zero at the soilatmosphere interface , orthe flux density of radon is set to some fixed valueŽe.g. zero at the boundary between soil and some

.low-diffusivity material . Some situations, how-ever, require more elaborate boundary condi-tions. For example, the boundary condition at theinterface between soil and indoor environmentcan be based on a ‘room accumulation model’Ž .Loureiro, 1987; Andersen, 1992 . Imagine astraight, smooth-walled crack in a concrete slab.The top of the crack ends in a well-mixed room

Žwith concentration c . The total flux of radon inin1 . Ž .Bq s into the room not just from the crack is

J. The air-exchange rate of the room is . Hence,vthe boundary condition needed to model radonentry through the crack is that the radon concen-tration at the top of the crack equals c whereinc in turn fulfils:in

Ž .JV c 21v in

where V is the volume of the room. J is calcu-lated by the numerical model and is a function of

Žc for example, diffusion of radon through thein.crack will diminish as c increases . The solutionin

may be found iteratively: Guess a value of c ,incalculate J with the numerical model, and find animproved value of c . This process is continuedin

Ž .until Eq. 21 is fulfilled. It may be necessary tounderrelax the process.

In situations when soil interfaces with a free-airŽcompartment that is not well mixed e.g. by turbu-

.lence from fans it may be inappropriate to im-pose a fixed radon concentration at the boundary.A more refined treatment is to introduce a stag-nant air layer between soil and open air, and toextent the computational domain to include thefilm, or to derive special boundary condition for

Žexample based on diffusion through the film Bird

et al., 1960; Jury et al., 1984; Petersen et al., 1996;.He et al., 1998 .

In the normal situation, the pressure field isŽ .found from Eq. 20 based on the requirement

that the pressure should be fixed at a certainvalue at the boundary and that there should befixed flows of soil gas at others. Sometimes, thecalculation of boundary pressures is linked toexperimental data or other models. For example,

Ž .Riley et al. 1996 used windtunnel data for thegroundsurface pressure field around a houseexposed to wind, and a computational fluid dy-namics code for determining house ventilationrates in the presence of wind. Subsequently, a soilmodel was used for the calculation of sub-soil

Ž .flows soil venting and radon entry into the house.In a Dutch study, indoor airflows and pressuredifferences were calculated with a house infiltra-tion model for given temperature and wind sce-narios. These results were used as boundary con-ditions for a numerical model of radon entryŽ .Janssen et al., 1998 .

2.2. Sample calculation

It is instructive to perform a sample calculationŽAndersen, 1992; Nazaroff, 1992; van der Spoel,

.1998 . We consider, a semi-infinite homogeneoussoil column dominated by diffusion. Homogeneityhere means that all material characteristics are

Ž .uniform and constant in all parts of the soil. Forexample, the soil is assumed to have a uniformdistribution of moisture and a uniform tempera-ture. The radon concentration at the atmosphericsurface is zero. For steady-state conditions, we

Ž .obtain the solution to Eq. 15 :

zŽ . Ž .c z c 1exp 22a a , ž /ž /Ld

where z is the depth and the deep-soil radonconcentration is:

G Ž .c 23a,

Observe that at 10C, c changes by a factora,of 1L2.8 if we consider c for a fully water-a,


saturated medium compared to a dry mediumŽ . Ž .Rose et al., 1990 . For this reason and others itis therefore important to take moisture into ac-count when modelling soilgas radon. The diffu-sion length is:

D D e Ž .L 24((d

where we have introduced the effective diffusivityD . The relation to the bulk diffusivity is:e

Ž .DD 25e

Observe, that under dry conditions, we haveD D . In the literature it is sometimes un-eclear what type of diffusion constant that is ap-

Ž .plied see e.g. Rogers et al., 1994 , and terms like‘effective’, ‘bulk’, and ‘pore averaged’ diffusivitiesmean different things to different people. Fortu-nately, it appears that there is general agreementon the meaning of ‘diffusion length’: it is the

Ž .inverse of the z factor in Eq. 22 . Hence, Eq.Ž .24 may sometimes be used to identify the mean-ing of particular types of diffusivity.

Ž .Finally, we state that the exhalation rate j 0Ž 1 2 .in Bq s m from the surface at z0 is:

ca ,Ž . Ž .j 0 D 26Ld

2.3. Special considerations

Ž .Numerical solution of Eq. 15 in medias thatare not homogeneous requires special considera-tion as and will change from point to pointŽ .Andersen, 1992; van der Spoel, 1998 . It is there-

Ž .fore not advisable to divide Eq. 15 by or toŽobtain ‘effective equations’ compare Rogers and

.Nielson, 1991a corresponding to writing fluxequations on the basis of flux per air-filled porearea or the like. When media of different porosity,moisture content or temperature meet it shouldbe required that bulk fluxes are the same at eachside of the interface between the media and thatradon concentrations ‘phase-by-phase’ are contin-uous:

Ž . Ž . Ž .j j 27

Ž . Ž . Ž .c c 28a a

Ž . Ž . Ž .c c 29w w

Ž . Ž . Ž .c c 30s s

where and designate the two sides of theinterface. Observe, that these requirements are

Ž .automatically fulfilled by Eq. 15 . A benchmarktest relating to these problems is described inŽ .Andersen et al., 1999 .

Many expressions exist for gas diffusivity insoils. For radon diffusion, the most popular ex-pression is probably that of Rogers and NielsonŽ .1991a,b . It appears that their empirical expres-sion is an effective diffusivity in the sense given

Ž .by Eq. 25 :

Ž 14 . Ž .D D exp 6m6m 31e 0

where D 1.1105 m2 s1 and m is the0water saturation . This interpretation hasw

Ž .been tested by van der Spoel 1998 for wet sand.All the equations given here treat soil as a

porous medium. However, some studies indicatethat soil properties of interest for gas flow may bedominated by bioporosity created by roots or

Žworms Garbesi, 1993; Holford et al., 1993; Hoff,.1997 . This can lead to preferential flow. This

does not necessarily mean that it is inappropriateto model such soils as ‘porous media equivalents’.

Ž .For example, Hoff 1997 studied a sample ofclayey-till and demonstrated that although thebioprosity controlled the effective gas permeabil-

Žity and provided fast-flow paths for radon-222 as.well as radon-220 it was a fine approximation to

model the column as if it was homogeneous.

2.4. Numerical procedures

Ž . Ž .To solve Eqs. 15 and 20 numerically, a rangeof procedures are available as described by GadgilŽ .1992 . A particularly simple and robust proce-dure is the finite-difference method known as the


Žcontrol-volume approach sometimes also called. Ž .the finite-volume method Patankar, 1980 . First,

a computational grid is defined for the problem inquestion. Depending on the selected numericalscheme the equation is then discretized into oneŽ .large matrix equation. The unknowns are field

Ž .values pressures and radon concentrations . Thematrix coefficients depend on the grid, materialproperties and boundary conditions. The mainproblem with programming numerical models isthe bookkeeping of coefficients and materialproperties, etc. The model output should be eval-uated carefully. In particular, it is important toverify that the solution is insensitive to furthergrid refinements.

3. Standard house example

As a specific example of how numerical mod-elling can be used to better understand the im-portance of various factors on combined diffusiveand advective entry into houses, we consider theslab-on-grade house sketched in Fig. 1. This houseis an extension of the standard house used in

Ž .UNSCEAR 1993 . The house is cylindrical andhas a floor area of 100 m2. There is a 3-mm gapof air between the slab and the footer along thefull 35-m perimeter of the house. Below the slab,a highly permeable layer of gravel exists. Thehouse is located on a 10-m thick soil block of 20m radius. Parameters of the various componentsare given in Table 1. These parameters have beenselected in collaboration with H. Arvela, STUK,

Ž .Finland personal communication, 1997 . Observe,that the diffusivity of the slab is relatively high

Ž .the diffusion length amounts to approx. 22 cmin line with recent results for concrete from

Ž .Florida, USA Rogers et al., 1994 .All components are assumed to be homoge-

neous. We consider gas permeabilities of the soilin the range from 1014 to 109 m2. The house isconstantly depressurized 1 or 2 Pa relative to theoutdoors. The air-exchange rate of the 250 m3

house is set to 1 h1. Other constants are: 18106 Pa s1, L0.3, and 2.7103 kggm3. The model RnMod3d developed at Risø is

Žused for the computations Andersen, 1992; An-.dersen et al., 1999 .

The results of model calculations are shown inFig. 2. The full-line curves are for 1 Pa depressur-ization, the dashed-line curves are for 2 Pa. Theplot shows the total radon entry rate as well ashow much that comes through the concrete slaband the 3-mm air gap, respectively.

We make the following observations from thefigure: radon enters the house from the slab at a

1 Ž .rate of approximately 1 Bq s almost indepen-dently of soil permeability and house depressur-ization. Mobile radon is generated in the slab at arate of approximately 0.23 Bq s1, so the mainpart of the slab entry must be radon from the soilthat diffuses through the slab. There is littlepressure-driven soil gas entry through the slabbecause of its low permeability. The entry ratefrom the slab corresponds to an indoor radonlevel of approximately 15 Bq m3 for an air-ex-change rate of 1 h1 or 30 Bq m3 at 0.5 h1.Radon also diffuses through the gap. For low soilpermeabilities, it can be seen that this type ofentry amounts to less than 0.2 Bq s1. For higherpermeabilities, the flow of soil gas increases the

Table 1aParameters used in the calculations for the standard slab-on-grade house

A f D kRa w1 2 1 2Bq kg m s m

8 15Slab 50 0.1 0.20 0 2.010 106 9Gravel 40 0.2 0.40 0 1.810 5107Soil 40 0.2 0.25 0.2 4.310 Variable1 0 15Footer 0 0 0 0 1.010 1.0105 7Gap 0 0 1.00 0 1.210 7.510

a The symbols are as defined in the text.


Fig. 1. Sketch of standard slab-on-grade house.

advective radon entry. For permeabilities in theŽ . 11 Žrange 0.51 10 depending on the depres-

.surization of the house , radon enters through theslab and through the air gap in equal amounts.For larger permeabilities, advection through thegap is the main mode of entry. For 1 Pa depres-surization and 1010 m2 gas permeability, theentry rate is approximately 10 Bq s1. This corre-sponds to 150 Bq m3 for an air-exchange rate of1 h1 or 300 Bq m3 at 0.5 h1. To betterunderstand the relative importance of diffusionand advection similar calculations of entry into ahouse with a bare soil floor is compared with

Ž .entry into a slab-on-grade house Andersen, 1992 .

4. Result highlights

4.1. General simulation studies

Radon entry modelling has been undertaken by

Ž .many groups Gadgil, 1992 . One of the firststudies is from 1983, when a Canadian group setup a finite-element model of Darcy flow andassessed soilgas and radon entry as a result ofindooroutdoor pressure differences created by

Ž .stack effects and wind DSMA, 1983 . In 1987,Loureiro published a Ph.D. thesis on modelling ofsoilgas and radon entry into basements under

Žconstant negative pressure Loureiro, 1987;.Loureiro et al., 1990 . This work demonstrated

the usefulness of the control-volume approachŽ .Patankar, 1980 , and dealt with problems such ascombined advective and diffusive transport andtransport through the crack. Furthermore, thethree-dimensional model allowed for a relativelyrealistic account of building components. Later,this work was continued by Revzan and col-

Žleagues Revzan et al., 1991; Revzan and Fisk,.1992; Revzan et al., 1993 . For example, they

found that the non-uniform temperature field inthe soil around a heated basement could increasethe radon entry rate by 40% compared to theisothermal situation. They also found that one ofthe most important building factors controllingradon entry is the presence or absence of a highlypermeable gravel layer below the slab. Such agravel layer can increase the entry by a factor of5. This was later verified experimentally by

Ž .Robinson and Sextro 1995 .

4.2. Entry by atmospheric pressure

Ž .Robinson et al. 1997 also set up a numericalmodel of soil gas transport and demonstrated howentry caused by changes in atmospheric pressuredepend on other reservoir characteristics thanentry by indooroutdoor pressure differences. For

Ž .example, they found that the effective soilgasentry rate caused by atmospheric pressure changeswas not directly proportional to soil permeabilityŽit will, however, decrease with decreasing perme-

.ability . Although a steady 1 Pa indooroutdoorpressure difference would be the main drivingforce for houses on soils of high permeability,changes in atmospheric pressure become increas-

Ž .ingly important relatively speaking for houseson lower permeabilities. For a permeability below1013 m2, changes in atmospheric pressure would


Fig. 2. Radon entry rate into standard slab-on-grade house vs. gas permeability of the soil.

Žbe the main driving force for soilgas entry radon.entry may be dominated by diffusion .

4.3. Gas permeability measurements

Comparing detailed experiments with numeri-Ž .cal modelling results based on measured soil

parameters has been a source of improved under-standing and advancement in site characterizationtechniques. For example, Garbesi and colleaguesobserved that numerical models based on mea-sured gas permeability consistently and signifi-cantly underestimate soilgas entry into housesŽ .Garbesi, 1993 . They identified the conventionalsingle-probe permeability technique to be an im-portant source for the underestimation. The prob-lem is scale dependency: the single-probe perme-ability measurement technique interacts with soilat the 20 cm scale, whereas a house will interactat the scale of several meters. This is unimportant

Ž .for homogeneous situations e.g. a house on sand ,

but very important when soil inhomogeneitiessuch as root or worm holes come into play. Onthe basis of the model-measurement comparison,a new dual probe for gas permeability measure-ments was developed.

4.4. Radon entry reduction

Numerical modelling has also been used inconnection with design considerations of radonprevention and remediation. For example, Bonne-

Ž .fous et al. 1993 used numerical modelling toexplore the effect of placing a low-diffusivitymembrane under the sub-slab gravel layer. An-

Ž .dersen et al. 1996 and van der Spoel et al.Ž .1998b used numerical modelling to investigatemembranes as a mitigation technique in crawl-space houses. Imperfect membrane placement is

Ž .included in the modelling. Bonnefous et al. 1992 ,Ž .Gadgil et al. 1994 have used numerical mod-

elling to investigate the importance of various


factors on the radon efficiency of active sub-slabventilation systems.

5. Model limitations

To test the limitations of numerical radon mod-els, a range of tests have been done. In thelaboratory, van der Spoel and colleagues at theKVI have conducted very detailed experiments

Ž .with a sand vessel 2 m diameter and 2 m heightŽvan der Spoel et al., 1997; van der Spoel, 1998;

.van der Spoel et al., 1998a . They have demon-strated that for room-dry homogeneous sand,numerical models can reproduce virtually all as-pects relating to diffusive and advective transport:model-measurement differences are less than10%. Particularly noteworthy is that their inputparameters to the model are independent of theactual experiments. For example, gas diffusivity inthe sand is derived on the basis of electricalconductivity measurements of tortuosity. Tests formoisturized sand are less favorable. For certainaspects, model-measurement differences are as

Žlarge as 40% which is much larger than the.uncertainties associated with the results . It ap-

pears, however, that the main reason for thediscrepancies is that it is difficult to obtain correctinput parameters. The prime problem is to knowthe distribution of moisture in the column. In thefield, experiments have been undertaken withŽ . Žsmall test structures Andersen, 1992; Garbesi,1993; Nielson et al., 1994; Andersen et al., 1994;

.Andersen, 1995 . These results also indicate thatthe main problem is to obtain correct parameters.As already mentioned, this has lead to improvedmethods for finding effective soil parameters.Nielson and colleagues have compared entry into

Žreal houses with model estimates Nielson et al.,.1994 . They found relatively good agreement for

50 slab-on-grade houses in Florida.

6. Conclusion

Although radon can be measured relatively eas-ily in the indoor environment, it is much moredifficult to measure entry rates of soil gas and

soilgas radon. One problem is that theindooroutdoor pressure difference controls bothhouse air-exchange rate and radon entry. For thisreason, numerical modelling of radon entry hasbeen a valuable and comparatively inexpensivetool for assessments of the relative importance ofthe many factors controlling radon entry. In thefuture, numerical modelling will probably be usedto better understand the effect of moisture andbioporosity on radon entry.

References

Andersen CE. Entry of soil gas and radon into houses. Risø-Ž .R-623 EN , Risø National Laboratory, DK-4000 Roskilde,

Denmark, 1992.Andersen CE, Søgaard-Hansen J, Majborn B. Soil gas and

radon entry into a simple test structure: comparison ofexperimental and modelling results. Radiat Prot Dosim1994;56:151155.

Andersen CE. Summary of project 3. In: Radon sourcesmodels and countermeasures, Final report, 19921995, EUcontract F13P-CT920064d, 1995.

Andersen CE, Koopmans M, de Meijer RJ. Identification ofadvective entry of soilgas radon into a crawl-space cov-

Ž .ered with sheets of polyethylene foil. Risø-R-876 EN ,Risø National Laboratory, DK-4000 Roskilde, Denmark,1996.

Andersen CE, Albarracın D, Csige I, van der Graaf ER,´Jiranek M, Rehs B, Svoboda Z, Toro L. ERRICCA radon´

Ž .model intercomparison exercise. Risø-R-1120 EN , RisøŽNational Laboratory, DK-4000 Roskilde, Denmark availa-.ble as internet publication at http:www.risoe.dk , 1999.

Bird RB, Stewart WE, Lightfoot EN. Transport phenomena.John Wiley & Sons, 1960.

Bonnefous YC, Gadgil AJ, Fisk WJ, Prill RJ, NematollahlAR. Field study and numerical simulation of subslab venti-

Ž .lation systems. Environ Sci Technol 1992;26 9 :17521759.Bonnefous YC, Gadgil AJ, Revzan KL, Fisk WJ, Riley WR.

Impact of a sub-slab aggregate layer and a sub-aggregatemembrane on radon entry rate: a numerical study.Proceedings of indoor air ‘93. The 6th International confer-ence on indoor air quality and climate, 4. Helsinki, Finland:Indoor air ‘93, 1993:569574.

Ž .Clever HL ed. . Solubility data series. Volume 2. Krypton,xenon and radon-gas solubilities. Pergamon Press, 1979.

DSMA Atcon Ltd. Review of existing instrumentation andevaluation of possibilities for research and development ofinstrumentation to determine future levels of radon at aproposed building site. Research report INFO-0096, AtomicEnergy Control Board, Ottawa, Canada, 1983.

Domenico PA, Schwartz FW. Physical and chemical hydroge-ology. John Wiley and Sons, 1990.

Gadgil AJ. Models of radon entry. Radiat Prot DosimŽ .1992;45 14 :375380.


Gadgil AJ, Bonnefous YC, Fisk WJ. Relative effectiveness ofsub-slab pressurization and depressurization systems forindoor radon mitigation: studies with an experimentallyverified numerical model. Indoor Air 1994;4:265275.

Garbesi K. Toward resolving model-measurement discrepan-cies of radon entry into houses. LBL-34244, LawrenceBerkeley Laboratory, CA 94720, USA, 1993.

He B, Shang A-G, Guo H-P. Discussion about surfaceboundary conditions of radon concentration and surfaceexhalation rate calculations in indoor concrete slab. Health

Ž .Phys 1998;74 3 :366369.Hoff A. Radon transport in fractures soil: Laboratory experi-

Ž .ments and modelling, Ph.D. dissertation. Risø-R-975 EN ,Risø National Laboratory, DK-4000 Roskilde, Denmark,1997.

Holford DJ, Schery SD, Wilson JL, Philips FM. Modelingradon transport in dry, cracked soil. J Geophys Res

Ž .1993;98 B1 :567580.Janssen MPM, de Vries L, Phaff JC, van der Graaf ER,

Blaauboer RO, Stoop P, Lembrechts J. Modelling radontransport in Dutch dwellings.RIVM report no. 610050005,Bilthoven, the Netherlands, 1998.

Jury WA, Spencer WF, Farmer WJ. Behavior assessmentmodel for trace organics in soil: IV. Review of experimen-

Ž .tal evidence. J Environ Qual 1984;13 4 :580586.Loureiro CO. Simulation of the steady-state transport of

radon from soil into houses with basements under constantnegative pressure. LBL-24378, Lawrence Berkeley Labora-tory, CA 94720, USA, 1987.

Loureiro CO, Abriola LM, Martin JE, Sextro RG. Three-di-mensional simulation of radon transport into houses withbasements under constant negative pressure. Environ Sci

Ž .Technol 1990;24 9 :13381348.Nazaroff WW, Moed BA, Sextro RG. Soil as a source of

indoor radon: generation, migration, and entry. In: NazaroffWW, Nero AV, editors. Radon and its decay products inindoor air. Wiley-Interscience, 1988.

Nazaroff WW. Radon transport from soil to air. Rev GeophysŽ .1992;30 2 :137160.

Nielson KK, Rogers VC, Rogers V, Holt RB. The RAETRADmodel of radon generation and transport from soils into

Ž .slab-on-grade houses. Health Phys 1994;67 4 :363377.Patankar SV. Numerical heat transfer and fluid flow. Hemi-

sphere Publishing Corporation, New York, 1980.Petersen LW, El-Farhan YH, Moldrup P, Rolston DE, Ya-

maguchi T. Transient diffusion, adsorption, and emission ofvolatile organic vapors in soils with fluctuating low water

Ž .contents. J Environ Qual 1996;25 5 :10541063.Revzan KL, Fisk WJ, Gadgil AJ. Modelling radon entry into

houses with basements: model description and verification.Indoor Air 1991;2:173189.

Revzan KL, Fisk WJ. Modelling radon entry into houses withbasements: the influence of structual factors. Indoor Air1992;2:4048.

Revzan KL, Fisk WJ, Sextro RG. Modeling radon entry intoFlorida slab-on-grade houses. Health Phys 1993;65Ž .4 :375385.

Riley WJ, Gadgil AJ, Bonnefous YC, Nazaroff WW. Theeffect of steady winds on radon-222 entry from soil into

Ž .houses. Atmos Environ 1996;30 7 :11671176.Robinson AL, Sextro RG. The influence of a subslab gravel

layer and open area on soilgas and radon entry into twoŽ .experimental basements. Health Phys 1995;69 3 :367377.

Robinson AL, Sextro RG, Riley WJ. Soilgas entry intohouses driven by atmospheric pressure fluctuations theinfluence of soil properties. A tm os E nviron

Ž .1997;31 10 :14871495.Rogers VC, Nielson KK. Multiphase radon generation and

transport in porous material. Health Phys 1991a;Ž .60 6 :807815.

Rogers VC, Nielson KK. Correlations for predicting airpermeabilities and 222 Rn diffusion coefficients of soils.

Ž .Health Phys 1991b;61 2 :225230.Rogers VC, Nielson KK, Holt RB, Snoddy R. Radon diffusion

coefficients for residential concretes. Health PhysŽ .1994;67 3 :261265.

Rose AR, Ciolkosz EJ, Washington JW. Effects of regionaland seasonal variations in soil moisture and temperatureon soil gas transport. The International Symposium onRadon and Radon Reduction Technology: Volume III.Preprints. EPA6009-90005c. Paper number: C-VI-5.January, 1990:1990.

van der Spoel WH, van der Graaf ER, de Meijer RJ. Diffusivetransport of radon in a homogeneous column of dry sand.

Ž .Health Phys 1997;72 5 :766778.van der Spoel WH. Radon transport in sand: A laboratory

study. Ph.D. dissertation, Technical University Eindhoven,the Netherlands, ISBN 90-386-0647-8, 1998.

van der Spoel WH, van der Graaf ER, de Meijer RJ. Com-bined diffusive and advective transport of radon in a homo-geneous column of dry sand. Health Phys 1998a;Ž .74 1 :4863.

van der Spoel WH, van der Graaf ER, de Meijer RJ. Foilcoverage of a crawl-space floor: measurements and mod-

Ž .elling of radon entry. Health Phys 1998b;74 5 :581593.Thomson NR, Sykes JF, van Vliet D. A numerical investiga-

tion into factors affecting gas and aqueous phase plumes inthe subsurface. J Contam Hydrol 1997;28:3970.

ŽUNSCEAR United Nations Scientific Committee on the ef-.fects of Atomic Radiation . Sources and effects of ionizing

radiation, United Nations, New York, 1993.Washington JW, Rose AR, Ciolkosz EJ, Dobos RR. Gaseous

diffusion and permeability in four soil profiles in centralŽ .Pennsylvania. Soil Sci 1994;157 2 :6576.

Wong CS, Chin Y-P, Gschwend PM. Sorption of radon-222 tonatural sediments. Geochim Cosmochim Acta 1992;56:39233932.

Radon-222 in soil, water and building materials: Presentationof laboratory measurement methods in use at Risø

Claus E. AndersenRisø National Laboratory, Build. NUK-125, DK-4000 Roskilde Denmark

Abstract

Three methods for measurements of radon-222 in soil, water and building materials are de-scribed, and sample results are given. The methods are used both in connection with the healthproblem of radon-222 in houses and with radon-222 as a tracer of environmental transport.

1 Introduction

Radon-222 constitutes a dominant part of the life-time radiation dose for most persons in theNordic countries. This is, however, not the only reason why radon-222 is an interesting objectof research: radon-222 is also an excellent tracer of certain transport processes in parts of theenvironment. At Risø, we are engaged in studies of both of these aspects. To this end wehave adapted laboratory methods relating to radon measurements in soil, water and buildingmaterials. The purpose of this contribution is to describe three such methods, and to outlinesome results.

2 Emanation from soil samples

The emanation rate of soil is the number of radon atoms that (effectively) escape the soil grainsinto the pore system per kg dry mass per second. The emanation rate partly controls the soil-gasradon concentration. It is measured as follows: The sample is mildly disaggregated by forcingit through a brass plate with a 22 mm hole. The sample (typically about 300 g) is placed ina 6 L steel chamber. The chamber is flushed to near-zero radon concentration with about 30L of aged nitrogen from a pressurized cylinder. Thereafter the chamber is closed, and radonstarts to build up inside the chamber. Over the following days (or weeks) the chamber radonconcentration is determined at selected times. Evacuated 200 mL scintillation cells are used forthe purpose. An airbag with aged nitrogen is used to balance the pressure in the chamber aftersampling. The scintillation cells are counted on a computer-controlled sample-changer with onephotomultiplier tube. Samples are weighted before and after analysis. A moisture determina-tion is also carried out. The analysis of the data includes corrections for dilution of the chamberbecause of sampling and pressure and temperature dependent transfer coefficients. Analysis ofprecision is carried out both for each chamber concentration determination (typically each cellis counted 3 to 4 times) and for the overall emanation-rate analysis (typically each analysis isbased on 3 or more scintillation cell samples). The analysis-of-precision is a chi-squared testwhere the a priori uncertainty of the quantity of interest is compared with the experimental stan-dard deviation. The final result of the analysis is reported on standardized computer-generatedmeasurement sheets. Normally, batches of 12 samples are analyzed. In total, 480 analyses havebeen carried out. About 250 of these come from the Geological Survey of Denmark: To helpunderstand differences in indoor radon potential for different geologies, samples from different

A

BC

D

E

F

Structure 1

Structure 2

North5 m

0

0

0

0

0

0

0

20

20

20

20

20

20

1

2

3

ABCDEF

Emanation rate [atoms s−1 kg−1]

Dep

th[m

]

Fig. 1. Emanation rate results for six soil cores (named A to F) taken at the clayey till field siteof Risø’s radon test structures. The site is sketched above the graph. The dashed reference lineat 5.8 atoms s−1 kg−1 is the mean of all results below 1.4 m. Uncertainties (expressed as singlestandard deviations) are in most cases smaller than the plotted data points.

parts of Denmark have been analyzed. A small number of analyses have been done for engi-neering companies to help assess the risk for high indoor radon for larger construction works.About 200 emanation-rate determinations are from the site of Risø’s radon test structures (thefirst structure, which is no longer in existence, is described in (Andersen, 1992)). Part of theseanalyses have been used to investigate the effect of disaggregation and moisture content (Ander-sen, 1998). Another purpose has been to provide soil parameters needed for the comparison ofnumerical models of radon transport in soil and entry into houses. Figure 1 shows 95 emanationrate results for six 3 m soil cores from the test structure site. Below 1.4 m depth, there is littlevariability with depth and from core to core. The pooled mean and standard deviation of theseresults amount to 5.8 and 1.4 atoms s−1 kg−1, respectively (N=48). For the top (0–1.4 m) layer,the mean and standard deviation are 10.1 and 4.4 atoms s−1 kg−1, respectively (N=47). It canbe seen from the figure, that the emanation rate peaks in the layer between 0.5 to 0.85 m belowthe surface. The maximum value of 25 atoms s−1 kg−1 occurs at 0.83 m depth for profile A.The high-emanation layer may result because the layer is more rich in radium than the otherparts of the profile. Another possibility is that the fraction of emanation for the layer is high.The detailed mapping based on soil cores seems to allow for a much more clear understandingof the emanation-rate conditions of the site than previous measurements (cf. results given in(Andersen, 1992)).

000 1 2 3 10 2020 40

atoms s−1 kg−1 Bq kg−1 %

LAC (M1) . . . . .

LAC (M2) . . . . .LAC (M3) . . . . .

Concrete (M7) .

AAC (M4) . . . .AAC (M5) . . . .

AAC (M6) . . . .

Gypsum (M8) .Bricks (M9) . . .LECA (M10) . .

Rn-222 exhalation rate Ra-226 activity Fraction of exhalation

Fig. 2. Exhalation rate results. The meaning of abbreviations is given in the text. M1 meansmaterial 1 etc. The fraction of exhalation is the exhalation rate divided by the radium-226 concen-tration. Uncertainties (expressed as single standard deviations) are in many cases smaller than theplotted data points.

3 Exhalation from building materials

The mass-specific exhalation rate of radon-222 from a building material sample (such as a brickor a concrete slab) is the amount of radon-222 that escapes the sample per kg per second. Stud-ies carried out by Jonassen, Ulbak and coworkers in the 1970’ies and 1980’ies showed that theradon-222 exhalation rate of ordinary Danish building materials is low. Special building ma-terials with large radon-222 exhalation rates do however exist (at least in other countries). Forthis reason, it is of interest for Danish producers of building materials to be able to quantify thisaspect of their products. Risø has therefore set up a method for exhalation rate measurements.It is a closed-chamber method. The sample (typically 30 x 30 x 5 cm3) is placed in a 55 Lstainless steel chamber together with a radon monitor that measures the radon-222 concentra-tion every hour. Also temperature, humidity and pressure in the chamber is registered. Thesample is conditioned for 24 h with a flow of aged nitrogen. The flow has a relative humidityof about 50 %. After conditioning, the chamber is closed and the radon concentration starts tobuild up. The measurement extends from 3 to 10 days. The sample is weighted before and aftermeasurement. The method is documented in a report (Andersen, 1999) together with measure-ment results for 10 Danish building materials. All materials were found to have exhalation ratesbelow 2.7 atoms s−1 kg−1. The highest value were for ordinary concrete, lightweight aggre-gate concrete (LAC) and autoclaced areated concrete (AAC). Bricks, gypsum and lightweightexpanded clay aggregate (LECA) had values below about 0.3 atoms s−1 kg−1. Under consid-eration of the application of the mateials in a typical Danish single-family house, it was foundthat such materials cannot increase the indoor radon concentration by more than 10 Bq m−3.The Danish Institute for Radiation Hygiene (SIS) has measured the radium-226 concentrationof the samples. The results are shown in Figure 2. The figure shows, for example, that some ofthe materials have a fraction of exhalation above 20 %. Other materials (such as bricks) havea very low fraction of exhalation. Although bricks have the largest radium concentration, verylittle radon exhale from the surface (the fraction of exhalation is less than 1 %).

Step 1: Trapping of activity

Flow control

Carrier gas (N2)

Moisture trap

Radon-222

Cold trap−78 C

Step 2: Degassing to scint. cell

200 mLscintillation cell

Carrier gas

5 L water sample

Oven400 C

Fig. 3. Radon-222 degassing of seawater samples. The water sample is stripped for radonby flushing it with 30 L of nitrogen. The activity is collected on activated charcoal cooledto dry-ice temperature. In step 2, the trap is heated to 400C, and the activity is transferredto an evacuated scintillation cell.

4 Radon in seawater

Groundwater tends to have a high radon-222 concentration compared with seawater. Radon-222 can therefore be used to trace the sub-marine supply of ground water to the sea. In 1998Risø engaged in an EU-project called sub-Gate which deals with submarine ground water fluxesand transport processes from methane rich coastal sedimentary environments. In this project,an area of Eckernforde Bay in the western Baltic Sea is investigated. In the area many so-calledpockmarks exist, and ground water (containing radon) is believed to seep into the sea. Becauseof the relative close distance from the study area to Risø, a method has been adopted wheresamples are transported back to the laboratory for radon analysis. The method is sketched inFigure 3. It has been modified from that described by Mathieu et al. (1988). Cruises have beenmade in December 1998 and in July 1999. Preliminary results indicate that excess radon-222exist close to the sea floor. In December 1998, near sea-floor radon-222 concentrations as highas 30 mBq L−1 were observed. In contrast, near sea-surface radon-222 concentration were onlyabout 3 mBq L−1. The radium-226 concentration was found to be less variable. The average of19 measurements was 3.3 mBq L−1.

ReferencesAndersen, C.E. Entry of soil gas and radon into houses. Risø-R-623(EN), Risø National

Laboratory, DK-4000 Roskilde, Denmark (1992).Andersen, C.E. Radon-222 emanation rate measurements of disaggregated soil samples. In:

Radon investigations in the Czech Republic 7 and the 4. International workshop on thegeological aspects of radon risk mapping. Radon investigations in the Czech Republic 7;Barnet, I.; Neznal, M. (eds.), (Czech Geological Survey, Prague, 1998) p. 63–66.

Andersen, C.E. Radon-222 exhalation from Danish building materials: H+H Industri A/S re-sults. Risø-R-1135(EN), Risø National Laboratory, DK-4000 Roskilde, Denmark (1999)(available electronically at www.risoe.dk).

Mathieu, G.G, Biscaye, P.E., Lupton, R.A., and Hammond, D.E.: System for measurementof 222Rn at low levels in natural waters. Health Physics, vol. 55, no. 6, 989–992, 1988.

Reference:

Andersen, C.E.: Radon-222 in soil, water and building materials: Presentation of laboratorymeasurement methods in use at Risø. In: Proceedings. Nordic Society for RadiationProtection 12. Ordinary meeting, Skagen (DK), 23-27 Aug 1999. Søgaard-Hansen, J.;Damkjær, A. (eds.), ISBN 87-550-2617-6, (Risø National Laboratory, Roskilde, 1999) p. 105-108.

Risø-R-1201(EN) report errata

Claus E. Andersen, Risø National Laboratory

November 10, 2008

Corrections

Equation 9 in Risø-R-1201(EN) Radon transport modelling: User’s quide to RnMod3d shouldbe changed to:

ρws =δM

δV= (1− ε)ρg + εθvρw (1)

Equation 11 should be changed to:

θg ≡ δMw

δMg(2)

=ρwδVw

ρgδVg(3)

=ρwεwδV

ρg(1− ε)δV(4)

=ρwεw

ρds(5)

=ρwθvε

ρds(6)

= ερw

ρdsθv (7)

1

Abstract Numerical models based on nite-dierence or nite-element meth-

ods are used by various research groups in studies of radon-222 transport through

soil and building materials. Applications range from design of radon remediation

systems to more fundamental studies of radon transport. To ascertain that results

obtained with these models are of good quality, it is necessary that such models are

tested. This document reports on a benchmark test organized by the EU project

ERRICCA: European Research into Radon in Construction Concerted Action.

The test comprises the following cases: (1) Steady-state diusive radon proles in

dry and wet soils, (2) steady-state entry of soil gas and radon into a house, (3)

time-dependent radon exhalation from a building-material sample. These cases

cover features such as: soil heterogeneity, anisotropy, 3D-eects, time dependency,

combined advective and diusive transport of radon, ux calculations, and par-

titioning of radon between air and water in soil pores. Seven groups participated

in the intercomparison. All groups submitted results without knowing the results

of others. For these results, relatively large group-to-group discrepancies were ob-

served. Because of this, all groups scrutinized their computations (once more) and

engaged in follow-up discussions with others. During this debugging process, prob-

lems were indeed identied and eliminated. The accordingly revised results were in

better agreement than those reported initially. Some discrepancies, however, still

remain. All in all, it seems that the exercise has served its purpose and stimulated

improvements relating to the quality of numerical modelling of radon transport. To

maintain a high quality of modelling, it is recommended that additional exercises

are carried out.

This document has been typeset with the document preparation system LATEX.

This document can be obtained electronically at: www.risoe.dk/nuk/

ISBN 87-550-2557-9

ISBN 87-550-2558-7 (Internet)

ISSN 0106-2840

Information Service Department Ris 1999

Contents

1 Introduction 1

2 Basic denitions 1

3 Problem denitions 3

3.1 Assumptions and basic parameters 3

3.2 Case 0: Prole in dry soil 4

3.3 Case 1: Prole in soil 5

3.4 Case 2: Entry into house 6

3.5 Case 3: Building-material exhalation 8

4 Results 9

5 Discussion 9

6 Conclusion 13

Acknowledgements 13

References 14

A Addresses and models 15

Ris-R-1120(EN) iii

1 Introduction

Numerical models are used by a number of research groups in studies of radon-222

transport through soil and building materials. Some apply models to aid in the

design of mitigation systems. Others use models in conjunction with experiments

to help better understand the mechanism for radon transport. Regardless of the

application it seems, however, always to be important that modelling results are

of good quality. To demonstrate this, it is necessary that models are tested. The

present report describes a model-model intercomparison exercise organized by the

modelling group of ERRICCA (European Research into Radon in Construction

Concerted Action1) to help modellers identify (gross) errors in model calculations,

for example, as a result of incorrect programming or incorrect model use.

In the exercise, the task is to compare modelling results for problems for which

exact analytical solutions are (probably) not known. This provides an opportunity

for the participating groups to test their models for relatively complex problems.

Also, the test is realistic in the sense that models are normally applied to solve

problems for which answers are unknown. In case of model-model discrepancies

it is, however, not possible to say who is right and who is not. To resolve such

problems, additional tests are needed.

The exercise includes the following cases:

Case 0: Steady-state diusive radon prole in dry soil

Case 1: Steady-state diusive radon prole in wet soil

Case 2: Steady-state entry of soil gas and radon into a house

Case 3: Time-dependent radon exhalation from a building-material sample

These problems cover features such as: soil heterogeneity, anisotropy, 3D-eects,

time dependency, combined advective and diusive transport of radon, and par-

titioning of radon between air and water in soil pores.

History

An exercise proposal was sent to 21 potential participants February 12, 1998. After

some minor changes, problem denitions were xed April 14, 1998. The initial

deadline for submission of results was June 10, 1998, but that was extended to July

1, 1998. After the deadline, results were circulated among the participating groups.

One group joined the exercise after the deadline, and some groups submitted

revised results. The results listed in section 4 are the latest results reported. In

total, seven European groups participated in the exercise.

2 Basic denitions

Soil structure

In this exercise, soil can be viewed as a porous medium consisting of grains and

pores. Radon (222Rn) exists in trace amounts in the pores along with water and soil

gas. Beyond quantities such as porosity and bulk transport properties, there will

be no detailed descriptions of the materials involved. This exercise therefore favors

models that treat transport through porous media using a continuum approach.

1http://arcas.nuclear.ntua.gr/~erricca/

Ris-R-1120(EN) 1

A small reference element of volume ÆV located at the point (x; y; z) can be

divided into grains of volume ÆVg, air-lled pores of volume ÆVa, and a water-

lled pores of volume ÆVw such that:

ÆV = ÆVg + ÆVp = ÆVg + ÆVa + ÆVw (1)

where ÆVp is the total pore volume. From this, we dene porosity (), air porosity

(a), and water porosity (w) as:

=ÆVp

ÆV(2)

a =ÆVa

ÆV(3)

w =ÆVw

ÆV(4)

We also dene the fraction of water saturation of the pores:

m =ÆVw

ÆVp=

w

(5)

and the density of soil grain material g (kgm3):

g =Æmg

ÆVg(6)

where Æmg is the mass of grains in ÆVg.

Radon concentrations

The activity of radon in ÆVa is called ÆAa, and we dene the radon activity con-

centration of the air-lled pores (Bq m3) as:

ca =ÆAa

ÆVa(7)

Likewise, the activity of radon in ÆVw is called ÆAw, and we dene the radon

activity concentration of the water-lled pores (Bq m3) as:

cw =ÆAw

ÆVw(8)

Observe, that the above denitions imply that ca gives the radon activity per

geometric volume of gas-lled pore space. An alternative denition could have

been to relate radon activity to the total number of gas molecules in the air-lled

pores expressed as the volume those molecules occupy at some standard conditions

of pressure and temperature.

Phase partitioning

In this exercise, it can be assumed that radon will not adsorb to any solid surfaces

(e.g. soil grain surfaces). Furthermore, it can be assumed that the partition of

radon between gas and water (locally) is instantaneous and permanently in equi-

librium such that the concentration ratio in the pore space is well described by

the solubility coeÆcient, L:

cw = Lca (9)

Radon uxes

For any point (x; y; z), we characterize the net transport of radon by the bulk

ux density ~j(x; y; z) in units of Bqm2 s1. The term 'bulk' means that density

is measured per total cross-sectional area perpendicular to ~j. Hence, a ux J

2 Ris-R-1120(EN)

(Bq s1) across some plane with geometric area A and uniform bulk ux density~j gives: J = ~j Aa, where a is a unit vector perpendicular to the plane. In this

calculation, it is unimportant if A is void, or partially blocked by soil grains or

water.

Radon emanation rate

Soil grains contain radium (226Ra) and radon emanates into air-lled and water-

lled pores. The emanation rate, E, gives the number of atoms that emanate

into the pores per unit time per kg dry mass of the soil (atoms kg1 s1). In this

exercise, it is not specied what fraction that ends up in the water-lled pores and

what fraction that goes into the air-lled pores.

Diusion coeÆcient

Diusion through soil will in this exercise be described by a bulk diusion constant

D (m2 s1) dened by:~jd = Drca (10)

which states that the bulk ux density of radon is proportional to the gradient

of the radon concentration in the air-lled pores ca. This denition is in common

use, but it is certainly not the only possible denition. Observe, that D accounts

for diusion through both air and water lled pores such that D can account for

diusion even if the medium is fully water saturated!

Pressure eld

We dene the disturbance pressure eld p(x; y; z) (Pa) at a given location of the

soil as the dierence between the absolute pressure P (x; y; z) (Pa) at that point

and the 'aerostatic' pressure at that depth PA(z) (Pa):

P (x; y; z) = PA(z) + p(x; y; z) (11)

where

PA(z) = Patm agz (12)

where the z-axis is oriented vertically (pointing upwards), Patm is the absolute

pressure at the atmospheric surface (z = 0), a the soil-gas density (kgm3), and

g the gravitational acceleration (m s2).

Permeability

The transport of soil gas is assumed to be well described by Darcy's law:

~q = k

rp (13)

where ~q is the bulk ux density of soil gas (m3 m2 s1), k is the gas permeability

of the soil (m2), and the viscosity (Pa s). If the soil is not isotropic k may be

specied for each direction of interest.

3 Problem denitions

3.1 Assumptions and basic parameters

This section lists assumptions, constants etc. that are common for all cases con-

sidered in this exercise:

Ris-R-1120(EN) 3

All experiments are conducted at the same temperature.

Although anisotropic gas permeabilities are considered in case 2, all other soil

properties are in all cases set to be isotropic.

Table 1 lists parameters that are common for the problem cases dened in

this exercise.

When an air chamber is said to be 'well mixed' with a certain radon concen-

tration, it means that the chamber has a uniform radon concentration. It is

ignored, that a thin stagnant layer of air may exist, for example, where the

chamber meets the soil surface.

The emanation rate E is assumed to be independent of moisture conditions

and other soil parameters.

The bulk diusion constantD is assumed to be well described by the equation:

D = D0

e(14)

where = a + Lw = (1 m + Lm) which can be called the partition-

corrected porosity, and D0

eis set to be the eective diusion constant found

by Rogers and Nielson [Rog91A, Rog91B]:

D0

e= D0 exp(6m 6m14) (15)

and where D0 is the diusion constant of radon in free air.

Table 1. Parameter values common for all cases in this exercise. The values of

solubility L and viscosity are characteristic for a temperature about 10 ÆC.

Quantity Value Unit Remark

E 10 atoms kg1 s1 Radon emanation rate

g 2:65 103 kgm3 Density of soil grain material

2:09838 106 s1 Decay constant of 222Rn

17:4 106 Pa s Air viscosity

L 0.3565 Radon solubility in water

D0 1:1 105 m2 s1 Radon diusivity in free air

3.2 Case 0: Prole in dry soil

Case 0 (zero) concerns radon transport in a column of dry soil. The geometry

is identical to that described in case 1 (see Figure 1(A) and (B)). The column

extends from z = 0 m to 3 m depth (z = 3 m). In the x and y directions, the

column measures 1 m by 1 m. At z = 0 m, radon diuses into a well-mixed air

chamber. The system is assumed to be in steady state.

Boundary conditions

The air in the chamber is maintained at a uniform radon concentration of 1000

Bq m3. Other boundaries of the soil column than that at z = 0 are closed o for

radon transport. The disturbance pressure eld p(x; y; x) is zero at all places of

the column.

4 Ris-R-1120(EN)

Figure 1. Case 1 geometry. (A) is the soil column viewed from the top, (B) is

a side view of the column. (C) and (D) are plots of porosity () and moisture

saturation (m), respectively.

Parameters

The soil is set to be homogeneous with porosity (z)=0.3 and moisture saturation

m(z) =0.0 (dry soil). Other parameters are given in Section 3.1

Model output to be reported

The task is to calculate the ux of radon J (Bq s1) in the z-direction at z = 0.

3.3 Case 1: Prole in soil

Case 1 concerns radon transport in a column of soil. As shown in Figure 1, the

column extends from z = 0 m to 3 m depth (z = 3 m). In the x and y directions,

the column measures 1 m by 1 m. At z = 0 m, radon diuses into a well-mixed

air chamber. The system is assumed to be in steady state.

Boundary conditions

The air in the chamber is maintained at a uniform radon concentration of 1000

Bq m3. Other boundaries of the soil column than that at z = 0 are closed o for

Ris-R-1120(EN) 5

radon transport. The disturbance pressure eld p(x; y; x) is zero at all places of

the column.

Parameters

As shown in Figure 1(C), the porosity (z) is set to 0.5 above z = 1 m, and

0.3 below. The moisture saturation m = m(z) increases linearly from m = 0:2 at

z = 0 to m = 1 at z = 2 m. Below z = 2 m, the moisture saturation is 1. This

is shown in Figure 1(D). Other parameters are given in Section 3.1


The task is to calculate the radon concentration in the air-lled pore space ca(Bq m3) and the ux of radon J (Bq s1) in the z-direction at z = 0, z = 1 m,

and z = 2 m.

3.4 Case 2: Entry into house

Case 2 concerns the slab-on-grade house sketched in Figure 2. Domains of impor-

tance for the problem are: slab, sub-slab layer, footer, air gap, and soil.

Radon can enter the house (only) through a smooth-walled air gap that sepa-

rates the slab from the footer. The gap width is 1 mm. The thickness of slab and

sub-slab layer are both equal to 10 cm. The footer reaches 1 m below the atmo-

spheric surface and has a thickness of 30 cm. As indicated in Figure 2, the soil

extends 10 m vertically, and 30 m by 30 m, horizontally. The system is assumed

to be in steady state.

Flow through the gap is described by the Navier-Stokes equation. In this ex-

ercise, it can be assumed, that a ow rate Q (m3 s1) through the gap causes a

pressure dierence pc across the gap equal to:

pc =12 z

Lc w3Q (16)

where z = 0:1 m is the slab thickness, w = 0:001 m is the gap width, and Lc

is the total gap length. To account for diusion of radon through the gap, the

diusion constant of radon in free air D0 can be assumed (see Table 1).

Boundary conditions

The disturbance pressure is (1) zero at the atmospheric surface, and (2) p at

the top of the air gap, i.e. where soil and indoor atmosphere meet (i.e. at z = 0).

As given in Table 2, we consider p equal to 0 and 3 Pa. The open atmosphere

is set to have a radon concentration equal to 0 Bq m3. The radon concentration

cin at the top of the air gap (i.e. at z = 0) is given by the equation:

J = v V cin (17)

where J (Bq s1) is the entry rate of radon into the house through the air gap,

v is 0.5 h1, and V is 250 m3. Other boundaries than those mentioned above are

closed o for gas transport.

Parameters

Slab and footer are made of radium-free solid materials that are impermeable to

any kind of gas transport. Soil and sub-slab gravel layer are both assumed to have

porosity and moisture saturation equal to =0.40 and m=0.20, respectively. The

6 Ris-R-1120(EN)

Figure 2. Case 2 geometry (not drawn to scale). (A) is the house viewed from the

top, and (B) to (D) are dierent side views of the house.

Table 2. Parameter sets considered in case 2.

ID Dist. pressure Gas permeability, m2

in house, Pa Sub-slab layer Soil

p ksub kh kv

Case 2-A 0 1011 1011 1011

Case 2-B 3 1011 1011 1011

Case 2-C 3 109 1011 1011

Case 2-D 3 109 1010 1011

Case 2-E 3 109 1011 1010

sub-slab layer is set to have the (isotropic) gas permeability ksub. The soil is set

to have the gas permeability kv in the vertical direction and kh in the horizontal

direction. Five combinations of the parameters p, ksub, kv, and kh are considered

in this exercise. These are identied as case 2-A to case 2-E as shown in Table 2.

Other parameters are given in Section 3.1.

Ris-R-1120(EN) 7

Figure 3. Case 3 geometry. (A) and (B) show the sample as viewed from top and

side, respectively. (C) is sketch of the sample placed in a 12.8 L air chamber (not

drawn to scale).


The task is to calculate the entry rates of soil gas Q (m3 s1) and radon J (Bq s1)

into the house for the ve parameter sets given in Table 2.

Notes

The boundary condition specied by equation 17 simulates the accumulation of

radon indoors when the air-exchange rate v is taken into account. This type of

boundary condition can be a problem for some models. Another complication will

probably be to design a suÆciently ne computational grid close to the gap.

3.5 Case 3: Building-material exhalation

Case 3 concerns a sample of building material placed in an 12.8 L air chamber

as sketched in Figure 3. Radon diuses from the sample into the chamber. The

task is to calculate the exhalation rate J (Bq s1) at selected times. The sample

is cylindrical with the dimensions: 15 cm diameter and 30 cm height. Although

Figure 3(C) shows the chamber as a box of dimensions 20 cm by 20 cm by 32 cm,

the chamber is well mixed and its shape can be considered to be unimportant for

the problem in question.

In this exercise, no formal distinction is made between soil and building materi-

als. Therefore, all denitions and assumptions presented in section 2 and 3.1 also

apply to the building-material sample considered in this case.

8 Ris-R-1120(EN)

Initial and boundary conditions

For time t = 1 to 0, the chamber is maintained at a radon concentration equal

to 0. At t = 0, the chamber is closed o and there is no transport of radon through

the walls of chamber.

Parameters

The building-material sample is set to have porosity and moisture saturation equal

to =0.2 and m=0.8, respectively. Other parameters are identical to the soil pa-

rameters given in section 3.1.


The task is to calculate the exhalation rate J (Bq s1) at times t = 0, t = 1 hour,

t = 12 hours, and at t =1.

4 Results

The results reported by the seven participating groups are listed in the following.

With one exception, all groups solved only part of the problems. Group 2 used

two dierent models (A and B), and their results are therefore given as Group 2A

and Group 2B, respectively.

The rst part of the exercise was conducted as a blind test: All groups submitted

results without knowing the results of others. Relatively large group-to-group dis-

crepancies were observed. All groups therefore scrutinized their calculations once

more. Some groups discovered misunderstandings relating to problem denitions,

or found that it was necessary to repeat calculations with a ner grid size. Also

programming errors were identied. A mixture of original and revised results are

listed in Table 3 to 6. Each result is assigned notes as follows:

Note 1 Original results (submitted without knowing results of other groups).

Note 2 Revised results after correction for (trivial) misunderstandings relating

to input parameters.

Note 3 Revised results based on a ner grid.

Note 4 Revised results after changes of the model.

The appendix (page 15) contains information on the models used by the various

groups.

5 Discussion

Case 0

Case 0 can be solved analytically, and it is observed from Table 3 that all groups

are consistent with the following solution based on [Co81]2:

J =pD (

G

c0) tanh(L0

r

D) = 4:722824 : : : 102 Bq s1 (18)

2One potential problem with the use of results from [Co81] is that that publication is notbased on bulk diusivity in the same sense that this exercise is. To be useful, the results in [Co81]therefore have to be "translated" into quantities consistent with those used here.

Ris-R-1120(EN) 9

where = 0:3, = 2:09838 106 s1, D = D0

e= 2D0 = 9:9 107m2 s1,

D0 = 1:1 105 m2 s1, c0 = 1000 Bq m3, L0 = 3 m, and

G = g1

E = 0:129749 : : : Bqm3s1 (19)

where g = 2:65 103 kgm3 and E = 10 atoms kg1 s1.

Case 1

Table 4 shows the results for case 1. For each of the six reported quantities (cal-

culated concentrations and uxes at three depths) there are always four or more

groups in favor of a particular value. The agreement does, however, not comprise

the same four (or more) groups. Only group 1, 2A, and 2B agree on all results

(within about 4 %). In turn group 3, 4 or 5 constitutes the fourth or fth "partner

of agreement". For example, at z = 0, Group 1, 2A, and 2B found a radon ux of

2:8 102 Bq s1. This is in agreement with group 4 and 5, but much smaller that

group 3 and 6. At z = 1, group 1, 2A and 2B found a ux of 1:8 102 Bq s1.Now the result is in agreement with group 3 and 4, but too low compared with

group 5 and 6.

All groups agree that the radon concentration (ca) in the air-lled pore parts

should be 1000 Bq m3 (i.e. the same as in the well-mixed box). This is consistent

with the requirement that the radon concentration fullls continuity phase-by-

phase (see later) at media interfaces. Also, all groups agree that the radon con-

centration at z = 2 m is about 170 kBq m3. As there is little transport at that

depth, this probably means that there good agreement on how the source term

should be calculated.

Case 2

Only group 1 and 7 attempted to solve case 2. In the pure diusion case (case 2-A)

the two models agree within 15 %. For the other cases, discrepancies between 20

to 70 % are observed. Part of the discrepancy could be that group 1 used a full

3D model for the problem whereas group 7 used a 2D model. It is observed, that

Q and J increase from 2-B to 2-C. This is the eect of adding a sub- oor gravel

layer to the house. The results of group 1 show that Q and J increase by a factor

of 3.9. For group 7, the increase amounts to a factor of 3.0 for the ow Q and 2.4

for the radon entry J .

Table 3. Results reported for Case 0: Prole in dry soil. The meaning of the notes

is given in the text.

J at z = 0 Note

Bq s1

Group 1 4:72 102 1

Group 2A 4:72 102 1

Group 2B 4:72 102 1

Group 3 4:63 102 1

Group 4 4:71 102 4

Group 5 4:70 102 1

Group 6 - -

Group 7 4:72 102 2

10 Ris-R-1120(EN)

Table 4. Results reported for Case 1: Prole in wet soil. The meaning of the notes

is given in the text.

z = 0 z = 1 m z = 2 m Note

ca J ca J ca J

Bqm3 Bq s1 Bqm3 Bq s1 Bqm3 Bq s1

Group 1 1:000 103 2:83 102 3:08 104 1:08 102 1:71 105 1:22 105 1

Group 2A 1:000 103 2:82 102 3:07 104 1:08 102 1:71 105 1:27 105 2,3

Group 2B 1:000 103 2:82 102 3:07 104 1:07 102 1:71 105 1:19 105 2,3

Group 3 1:000 103 3:73 102 3:59 104 1:03 102 1:68 105 1:25 105 1

Group 4 1:000 103 2:80 102 3:03 104 1:04 102 1:70 105 0:93 105 4

Group 5 1:000 103 2:76 102 2:85 104 1:26 102 1:71 105 3:75 105 4

Group 6 1:000 103 8:46 102 3:62 104 1:76 102 1:73 105 9:42 105 1

Group 7 - - - - - - -

Table 5. Results reported for Case 2: Entry into house. Group 7 modelled case 3,

with a 2D-model. The meaning of the notes is given in the text.

Case 2-A Case 2-B Case 2-C Note

Q J Q J Q J

m3 s1 Bq s1 m3 s1 Bq s1 m3 s1 Bq s1

Group 1 0 0.111 1:04 105 0.45 4:1 105 1.74 1

Group 7 0 0.094 1:8 105 0.59 5:4 105 1.42 2

Case 2-D Case 2-E Note

Q J Q J

m3 s1 Bq s1 m3 s1 Bq s1

Group 1 7:29 105 3.00 13:5 105 5.21 1

Group 7 - - - - 2

Table 6. Results reported for Case 3: Building-material exhalation. The meaning

of the notes is given in the text.

J Note

t = 0 t = 1 h t = 12 h t =1Bq s1 Bq s1 Bq s1 Bq s1

Group 1 2:28 104 2:22 104 2:15 104 2:14 104 1

Group 2A - - - - -

Group 2B 2:15 104 2:10 104 2:03 104 2:02 104 3

Group 3 2:20 104 2:15 104 2:08 104 2:07 104 1

Group 4 2:77 104 2:73 104 2:65 104 2:56 104 4

Group 5 - - - - -

Group 6 - - - - -

Group 7 - - - - -

Ris-R-1120(EN) 11

Figure 4. Plot of results reported for case 3. Group 4 used a one-dimensional

approximation in their computations.

Case 3

The results of case 3 are shown graphically in Figure 4. The calculated concen-

trations at t = 0, span over a range that corresponds to about 25 % of the mean

result. The four groups involved seem to agree on the (relative) time development

during the rst 12 hours.

The reason why group 4 nds higher values than the other three groups could

well be that group 4 applied certain simplifying assumptions to simulate the three-

dimensional problem (dened in the exercise) with a one-dimensional model.

Sources of discrepancies

The observed discrepancies initiated follow-up discussions, and some groups took

action to change their models or to subject them to further tests. The follow-up

discussions were carried at a group-to-group level. To stimulate further improve-

ment of models, it is probably useful to highlight some of the technical issues

discussed:

Diusivity. The terms bulk diusivity and eective diusivity are widely used

in the radon literature, however, not always with the same meaning! Misun-

derstandings relating to these terms therefore is an excellent source of model

discrepancies. Although some eort had been given (see the section Basic def-

initions), to dene what in this exercise it meant by bulk diusivity, follow-up

discussion indicated that the stated diusivities had caused some confusion.

Interface conditions. One interesting feature of case 1 is that it includes media

of dierent porosity and moisture content. This exercise therefore tests the

treatment of radon in dierent phases (radon is partitioned between the air-

lled parts of the pores and the water-lled parts of the pores). One aspect

of all this is the question of continuity of radon concentration and radon ux

at interfaces between regions of dierent porosities and moisture contents3.

3It is interesting to note, that Rogers and Nielson [Rog91A] discuss this problem: "Interface

12 Ris-R-1120(EN)

In this exercise, some groups advocate the following conditions of continuity:

(1) that radon concentrations phase-by-phase should be the same at the two

sides of the interface (e.g. ca(+) = ca() and cw(+) = cw() where + and

designate locations in the two media) and (2) that the bulk ux should be

the same at the two sides of the interface (e.g. ~J(+) = ~J()). This implies

that bulk concentrations of radon as well as uxes of radon evaluated on a

pore-area basis in general will change (abruptly) across such interfaces (i.e.

the concentration gradient will be undened). This in turn has consequences

for the treatment of diusion: the diusive ux must be related to the concen-

tration of radon in one single phase. In the exercise the radon concentration in

the air-lled parts of the pores (ca) is used for that purpose (see equation 10).

Computational ux "measurements". Another question of discussion was the

following: "My model outputs radon concentrations at all these locations and

it also gives the ow rate of soil gas, but how can I calculate the radon ux at

z = 2 m?" It appears that the solution to this problem will vary from model

to model. One easy-to-understand approach is given by Patankar [Pa80].

6 Conclusion

A model-model intercomparison exercise involving seven groups has been com-

pleted. All groups submitted results without knowing the results of others. For

these results, relatively large group-to-group discrepancies were observed. Because

of this, all groups scrutinized their computations (once more) and engaged in

follow-up discussions with other groups. In this process, problems were indeed

identied and eliminated. In some cases trivial errors had been made in the trans-

lation of input parameters from problem denitions (as stated in this document)

to the framework used in the models. Others had underestimated the need for

further grid renement at critical locations. Also, some computer code errors were

identied. The accordingly revised results were in better agreement than those

reported initially. Some discrepancies, however, still remain. All in all, it seems

that the exercise has served its purpose and stimulated improvements relating to

the quality of numerical modelling of radon transport. To maintain a high quality

of modelling, it is recommended that additional exercises are carried out.

Acknowledgements

This work was supported nancially by ERRICCA (European Research into Radon

in Construction Concerted Action, EU contract no. F14P-CT96-0064 (DG12-

WSMN). ERRICCA is carried out in the Nuclear Fission Safety research and

training programme.

conditions for adjacent soil regions are: continuity of the pore air Rn concentration, C0, andcontinuity of the pore-average ux, F=p, across the interface. It is incorrect to assume continuityof the pore-average Rn concentration across an interface. This condition generally does not occur.Continuity of pore air ux or of bulk Rn ux (F ) may cause errors."

Ris-R-1120(EN) 13

References

[Co81] R. Colle, R.J. Rubin, L.I. Knab, and J.M.R. Hutchinson: Radon

transport through and exhalation from building materials: A Re-

view and assessment. NBS Technical Note 1139. National Bureau

of Standards, U.S. Department of Commerce, 1981.

[Pa80] S.V. Patankar: Numerical heat transfer and uid ow. Hemisphere

Publishing Corporation, New York, 1980.

[Rog91A] V.C. Rogers and K.K. Nielson: Multiphase radon generation and

transport in porous material. Health Physics, vol. 60, no. 6 (June),

pp. 807815, 1991.

[Rog91B] V.C. Rogers and K.K. Nielson: Correlations for predicting air per-

meabilities and 222Rn diusion coeÆcients of soils. Health Physics,

vol. 61, no. 2 (August), pp. 225230, 1991.

14 Ris-R-1120(EN)

A Addresses and models

Group 1

Claus E. Andersen

Ris National Laboratory

Building NUK-125

DK-4000 Roskilde

Denmark

Telephone: +45 4677 4677 (main laboratory number)

Telephone: +45 4677 4912 (direct)

Fax: +45 4677 4959

E-mail: [email protected]

Web page: www.risoe.dk/nuk/nuk-clan.htm

All calculations were carried out with the model RnMod3d. This is a three-

dimensional, time-dependent model based on the control-volume approach. The

basis of the model is described in C.E. Andersen: Entry of soil gas and radon

into houses, Report Ris-R-623(EN) (1992), but the model interface is not docu-

mented. The model has been programmed in Pascal and is run on a PC from the

compiler/editor environment of Borland Pascal 7.0 or Delphi. The model is rela-

tively exible, and all exercise cases could be set up without troubles. The model

has not been prepared for external use. The model is primarily used in connection

with studies of radon in soil.

Group 2A & 2B

Emiel R. van der Graaf

Kernfysisch Versneller Instituut

Zernikelaan 25

NL-9747 AA Groningen

The Netherlands

Telephone: +31 50363 3562

Fax: +31 50363 4003


Model 2A: RAETRAP is based on the code RAETRAN developed by Rogers and

Associates Engineering Corporation, Salt Lake City, USA. KVI added a (more)

user friendly interface. The model solves the one-dimensional steady-state radon

transport equation (diusion and advection) by setting up the exact simultaneous

equations for a multi-layer system that approximates the real situations. These

exact equations are then solved by matrix methods. The model is documented in:

Stapel, C.: Manual for RAETRAP, a 1D code for calculation of diusion and ow

of radon through porous media. KVI. Technical Document, RT-01, 1992. Main use

of the model is in the Dutch Integrated Radon Model, that consists of a intercon-

nected series of computer models to study the in uence of various parameters on

exposure to radon of inhabitants of dierent dwelling types. Reference: Janssen,

M.P.M. de Vries, L., Pha, J.C., van der Graaf, E.R., Blaauboer, R.O., Stoop, P.,

Lembrechts, J.: Modelling radon transport in Dutch dwellings, RIVM report no.

610050005, RIVM, Bilthoven, The Netherlands, 1998.

Model 2B: KVI-1D and KVI-2D are based on the multi-phase radon transport

formalism introduced by Nielson and Rogers [Rog91A] (see page 14) and are im-

plemented by W.H. van der Spoel. Both models use the control volume approach

Ris-R-1120(EN) 15

(and thus fall under the method of nite dierences) and numerically solve the

time dependent radon transport equation including generation, decay, adsorption,

advection and diusion. The models are documented in: Van der Spoel, W.H.,

Radon transport in sand: A laboratory study. PhD Thesis, Eindhoven University

of Technology, 1998. Main use of the models has been in studying radon transport

in soil. Currently, the model is extensively used in studies on radon release from

concrete and cellular concrete.

Lessons learned from the intercomparison were that one has to be extremely

careful in translating parameters given in an exercise into the input parameters of a

specic model. Furthermore, we observed that both 1D-models are very sensitive to

the choice of degree of discretization at interfaces were rapid changes in properties

take place.

Group 3

Istvan Csige

Institute of Nuclear Research

H-4001 Debrecen

POB 51

Hungary

Telephone: +36 52 417 266

Fax: +36 52 416 181


Web page: www.atomki.hu/atomki/Radon

The model is named RnFlow. It has been programmed by Istvan Csige. The

model is a three-dimensional, time-dependent model based on the nite-dierence

approach. The geometry can be given in cartesian, polar or spherical coordinates.

One special feature of the model is that it includes heat transport in soil. The

model is not yet available on a commercial basis. The model documentation is

in preparation. The model is used mainly in connection with basic research. It is

planned to use the model in a procedure to determine the radon source potential

of building sites.

The model was not been modied as a result of the exercise, but the exercise

revealed the importance of the grid size eects on the accuracy of calculations.

Group 4

Laszlo Toro

Radiation Hygiene Dept.

Institute of Public Health "prof. dr. Leonida Georgescu"

16-18 V. Babes av.

RO 1900 Timisoara

Romania

Telephone: +40 56 192106

Fax: +40 56 192101


The model has been programmed at the Radiation Hygiene Dept., Institute of

Public Health "prof. dr. Leonida Georgescu", Romania. The model is based on

nite elements: rst order Lagrange trial functions (a variant with higher order

functions is under development). The model is one dimensional in space (con-

stant/variable grid), and has Crank-Nicolson time stepping with variable length

of the time step. All input parameter values are given from the console, and the

16 Ris-R-1120(EN)

output is in the form of an ASCII le. The transport model includes ow of water

and air (the air part is in progress). The model will be available for free download

from the web page of the laboratory given above. Detailed documentation is in

preparation. The main use of the model is to predict migration in soil of radon

and other contaminants dissolved in air or water.

As a result of the exercise (and further tests) the following problems were identi-

ed (and eliminated): certain programming errors, wrong use of given parameters,

mathematical errors (illegal move of space variable parameters), treatment of the

saturated-unsaturated boundary.

Group 5

Bernd Rehs

Isotopenlaboratorium fur biologische und medizinische Forschung

der Georg-August-Universitat Gottingen

Burckhardtweg 2

D-37077 Gottingen

Germany

Telephone: +49 0551 39 8112

Fax: +49 0551 39 8110


The model has been programmed at the Isotopenlaboratorium in Gottingen.

The model is based on one-dimensional explicit nite dierences and the FTCS

scheme. Model documentation is in preparation. The main use of the model is

to study the in uence of meteorological parameters on radon transport in soil.

The model was changed as a result of the exercise (treatment of the diusion

coeÆcient).

Group 6

Diego Albarracn and Llus Font Guiteras

Grup de Fisica de les Radiacions

Edici Cc

Universitat Autonoma de Barcelona

E-08193 Bellaterra (Barcelona)

Spain

Telephone: +34 3 5811530

Fax: +34 3 5812155



The model is named transrad. It has been developed at the Grup de Fisica de

les Radiacions, Universitat Autonoma de Barcelona. The model works with time

and two space dimensions. The geometry consists of a vertical cut of a house,

including indoor and outdoor air and soil. There is a crack around the oor of

the building, simulating the oor-wall joint. Equations for pressure and radon

concentration elds are solved using nite dierences. There is a regular square

grid with a subnested one of ner mesh size immediately under the house. This is

where the largest radon concentration gradients occur. The model gives contour

levels for both elds, including their time evolution, when some of their transport

parameters change in time. The model is object of a Ph.D. thesis, and is not yet

available in the public domain. This exercise promoted the inclusion of the eect

of the water saturation in the model. Some problems were identied in case 1 at

Ris-R-1120(EN) 17

z=1 m, where the porosity (and related parameters) change abruptly.

Group 7

Martin Jiranek and Zbynek Svoboda

Czech Technical University

Faculty of Civil Engineering

Department of Building Structure

Thakurova 7

CS-166 29 Praha 6

Czech Republic

Telephone: +420 2 24354806

Fax: +420 2 3119987


The name of the model is RADON2D for Windows. The program authors are M.

Jiranek and Z. Svoboda. The model is based on nite elements. It is a steady-

state, two-dimensional model. One special feature of the model is that it has a

graphical input and output user interface. The model is not available in the public

domain, but it is in use by a couple of research institutes in the Czech Republic.

The model has been described in the following papers:

Jiranek, Svoboda: The verication of radon protective measures by means of

a computer model. In: Proc. of 5th IBPSA Conference, Building Simulation

97, Prague 1997, pp. 165171.

Svoboda: The numerical solution of convective-diusive transport, PhD thesis,

Prague 1997.

Manual for the program RADON2D, Prague 1997.

The main use of the program relates to the verication of radon protective mea-

sures in the eld of civil engineering. The model was changed as a result of the

exercise. The problem was in the denition of the radon diusion coeÆcient De,

which we had dened in a dierent way from the other participants.

18 Ris-R-1120(EN)

Ris-R-1120(EN) 19

Bibliographic Data Sheet Ris-R-1120(EN)

Title and author(s)

ERRICCA radon model intercomparison exercise

Claus E. Andersen, Diego Albarracn, Istvan Csige, Emiel R. van der Graaf,

Martin Jiranek, Bernd Rehs, Zbynek Svoboda, and Laszlo Toro

ISBN

87-550-2557-9

87-550-2558-7 (Internet)

ISSN

0106-2840

Dept. or group

Nuclear Safety Research and Facilities Department

Date

April 1999

Groups own reg. number(s) Project/contract No.

EU DG12-WSMN no. F14P-CT96-0064

Pages

20

Tables

6

Illustrations

4

References

5

Abstract (Max. 2000 char.)

Numerical models based on nite-dierence or nite-element methods are used by

various research groups in studies of radon-222 transport through soil and building

materials. Applications range from design of radon remediation systems to more

fundamental studies of radon transport. To ascertain that results obtained with

these models are of good quality, it is necessary that such models are tested. This

document reports on a benchmark test organized by the EU project ERRICCA:

European Research into Radon in Construction Concerted Action. The test com-

prises the following cases: (1) Steady-state diusive radon proles in dry and wet

soils, (2) steady-state entry of soil gas and radon into a house, (3) time-dependent

radon exhalation from a building-material sample. These cases cover features such

as: soil heterogeneity, anisotropy, 3D-eects, time dependency, combined advec-

tive and diusive transport of radon, ux calculations, and partitioning of radon

between air and water in soil pores. Seven groups participated in the intercompari-

son. All groups submitted results without knowing the results of others. For these

results, relatively large group-to-group discrepancies were observed. Because of

this, all groups scrutinized their computations (once more) and engaged in follow-

up discussions with others. During this debugging process, problems were indeed

identied (and eliminated). The accordingly revised results were in better agree-

ment than those reported initially. Some discrepancies, however, still remain. All in

all, it seems that the exercise has served its purpose and stimulated improvements

relating to the quality of numerical modelling of radon transport. To maintain a

high quality of modelling, it is recommended that additional exercises are carried

out.

Descriptors INIS/EDB

ADVECTION; BENCHMARKS; BUILDING MATERIALS; DIFFUSION; EN-

VIRONMENTAL TRANSPORT; FINITE DIFFERENCE METHOD; FINITE

ELEMENT METHOD; GAS FLOW; HOUSES; RADON 222; SOILS

Available on request from:Information Service Department, Ris National Laboratory(Afdelingen for Informationsservice, Forskningscenter Ris)P.O. Box 49, DK4000 Roskilde, DenmarkPhone (+45) 46 77 46 77, ext. 4004/4005 Fax (+45) 46 77 40 13

Risø-R-1135(EN)

Radon-222 Exhalation fromDanish Building Materials:H + H Industri A/S Results

Claus E. Andersen

Risø National Laboratory, Roskilde, DenmarkAugust 1999

Risø-R-1135(EN)

Radon-222 Exhalation fromDanish Building Materials:H + H Industri A/S Results

Claus E. Andersen


Abstract This report describes a closed-chamber method for laboratory mea-

surements of the rate at which radon-222 degasses (exhales) from small building

material samples. The chamber is 55 L in volume and the main sample geometry

is a slab of dimensions 5x30x30 cm3. Numerical modelling is used to assess (and

partly remove) the bias of the method relative to an ideal measurement of the

free exhalation rate. Experimental results obtained with the method are found to

be in agreement with the results of an open-chamber method (which is subject to

dierent sources of error).

Results of radon-222 exhalation rate measurements for 10 samples of Danish

building materials are reported. Samples include ordinary concrete, lightweight

aggregate concrete, autoclaved aerated concrete, bricks, and gypsum board. The

maximum mass-specic exhalation rate is about 20 mBq h1 kg1. Under con-

sideration of the specic applications of the investigated building materials, the

contribution to the indoor radon-222 concentration in a single-family reference

house is calculated. Numerical modelling is used to help extrapolate the labo-

ratory measurements on small samples to full scale walls. Application of typical

materials will increase the indoor concentration by less than 10 Bq m3.

Claus E. Andersen

Ris National Laboratory

Department of Nuclear Safety Research

Build. NUK-125

DK-4000 Roskilde

Denmark

Phone, main Ris number: (+45) 4677 4677

Phone, direct: (+45) 4677 4912

Fax: (+45) 4675 4959

Internet: www.risoe.dk/nuk/

E-mail: <[email protected]>

This document can be obtained electronically at: www.risoe.dk/nuk/

ISBN 87-550-2594-3

ISBN 87-550-2595-1 (Internet)

ISSN 0106-2840

Information Service Department Ris 1999

Contents

1 Introduction 1

1.1 Background 1

1.2 Organization of the report 2

2 Theoretical framework 2

2.1 Radiometric quantities 2

2.2 Measurement procedures 6

3 Materials 8

3.1 Samples 8

3.2 Equipment 8

4 Experimental procedures and data analysis 10

4.1 Experimental procedures 10

4.2 Data and error analysis 11

4.3 Radium-226 measurements 13

5 Experimental results 13

6 Modelling results 16

6.1 Simulation of the closed-chamber method 17

6.2 Bound-to-free exhalation rate ratio 19

6.3 g for laboratory samples 19

6.4 g for walls 20

6.5 From laboratory measurements to full-scale walls 22

7 Reference house calculations 24

8 Discussion 28

8.1 Chamber leakage and other sources of error 28

8.2 Comparison with previous measurements 31

9 Conclusions 34

Acknowledgements 35

References 35

A Guide to measurement sheets 37

B Measurement sheets 39

Ris-R-1135(EN) iii

1 Introduction

The main objectives of this report are:

to describe a closed-chamber method used at Ris for laboratory measure-

ments of the radon-222 1 exhalation rate of building materials,

to investigate the various sources of errors characteristic for this method,

to report results for 10 Danish building material samples, and

to extrapolate the results to a typical Danish single-family house.

The report includes a brief review of other methods for exhalation rate measure-

ments, as well as results of previous exhalation-rate measurements conducted in

Denmark. The company H+H Industri A/S, lsted, Denmark, has supported the H+H Industri A/S

present work nancially. Furthermore, all building-material samples have been se-

lected and supplied by that company, and most (but not all) of the materials were

produced there.

1.1 Background

Danish homes without direct ground contact (for example, apartments in multi-

story buildings) typically have a low indoor radon level of about 20 Bq m3

[SIS87b]. In comparison, the average indoor radon level in Danish single-familiy

houses is normally about 70 Bq m3, and the level can be very dierent from

house to house. For example, single-family houses with annual averages from 10

to 1000 Bq m3 have been found [An97b]. The reason for the pronounced dier-

ence between houses with and without direct ground contact is that soil gas has

a high concentration of radon (typically about 10 000 to 100 000 Bq m3). Even

a minute entry rate of soil gas therefore can have a large impact on the indoor Soil-gas entry

radon concentration. Such gas entry is possible because most houses (apparently)

do not have a gas-tight oor construction and because houses are normally at

a slight underpressure (as it is normally warmer indoors than outdoors). Soil is

therefore the main source of indoor radon in most Danish single-family houses

[SIS87b, An97a].

In line with the above, studies carried out by Jonassen, Ulbak and co-workers in

the 1970'ies and 1980'ies (see Section 8.2), showed that radon exhalation rates of

ordinary Danish building materials are low. This is conrmed by the present inves- Building materials

tigation. Special building materials with large radon exhalation rates do however

exist (at least in other countries): alum-shale concrete, granite, Italian volcanic

tu, and by-product gypsum2 [UN93]. For example, Sweden has about 300 000

houses with alum-shale building materials sometimes referred to as "blue con-

crete" a lightweight aerated concrete used in blocks [SSI93]. This building ma-

terial can raise the indoor radon level above 1000 Bq m3. Alum-shale concrete

is no longer produced, and its application in Denmark has been limited [Ul80].

Finally, it should be observed that soil can have a large emanation rate (e.g.

above 10 atoms s1 kg1 which is about ve times the mass-specic exhalation

1The most abundant radon isotope is radon-222. It ordinates from radium-226 and is part ofthe Uranium Series (U-238). The half life of radon-222 is 3.83 days. Because of the importance ofthis particular isotope, it is often referred to as "radon". As this report concerns only radon-222(and not radon-220 or any other isotope of radon) we will adopt this practice here.

2Phosphogypsum is used in some countries as a substitute for natural gypsum in the manufac-ture of cement, wallboards and plaster [UN93]. Phoshogypsum is a by-product from the fertilizerindustry. The elevated levels of radium-226 in phospogypsum comes from the phosphate rockthat tends to have elevated concentration of Uranium-238 decay products.

Ris-R-1135(EN) 1

rate for ordinary concrete). Therefore, so-called "ecological" houses build in Den-

mark from clay excavated directly on the building site may have building materials

with above-average radon exhalation rates.

The Danish radon budget

In an investigation of radon in 117 newer Danish single-family houses [An97a],Budget 1

it was found that radon entered at a (geometric) mean rate of 9.6 kBq h1 and

that 80 % of the houses had radon entry rates above 6 kBq h1. In the typical

Danish single-family house built of clay bricks and/or aerated concrete, the radon

entry rate resulting from building materials has been estimated to be 13 kBqm3

[Jo80, Ul84]. Entry from the outdoors amounts to about 2 kBq h1 or less. Hence,

this calculation suggests that building materials and outdoor air cannot account

for observed indoor radon levels in 80 % of the investigated houses.

Another "radon budget" comes from the results of the 198586 national sur-Budget 2

vey [SIS87b]: The average radon level in multi-family houses was found to be

19 Bq m3. Since the outdoor radon level is about 8 Bq m3 [Ma86], and since

entry from the soil is probably marginal for most of the multi-family houses in-

cluded in the survey, it seems reasonable to attribute 198 = 11 Bq m3 to radon

from building materials.

1.2 Organization of the report

Section 2 gives a presentation of the theoretical framework for radon exhalation.

Quantities such as the mass-specic exhalation rate are dened, and it also out-

lines the characteristics of various methods for exhalation rate measurements.

Section 3, includes descriptions of samples as well as the experimental appara-

tus. In Section 4, the experimental procedure is described, and it is shown how

the nal exhalation rate results are found from raw measurements. The results of

the exhalation rate measurements are given in Section 5. Section 6 contains the

results of numerical model calculations. Based on solution of the 3-dimensional

time-dependent diusion equation, a number of issues related to the measure-

ment procedure are investigated. A reference house is then dened in Section 7,

and under consideration of the various applications of the building materials, the

concentration in such a house is estimated. The nal two sections of the report

contain discussion and conclusions. An appendix contains measurement sheets for

all exhalation-rate measurements.

2 Theoretical framework

2.1 Radiometric quantities

Exhalation rates, J

Consider a certain sample of building material placed in some well-dened envi-

ronment of given temperature, humidity, pressure, stress, radon concentration etc.

Under the given conditions, we dene the sample specic exhalation rate J to be

the net amount of radon that escapes the sample per time unit.

In this report 'the amount of radon that escapes per time unit' is expressed asMain units

the number of radon atoms that escapes per second (atoms s1), or as the amount

of radon activity measured in Bq that escapes per hour (Bq h1). From the basic

2 Ris-R-1135(EN)

Figure 1. Illustration of the main processes involved: 1) radium inside a grain

decays to radon, 2) some of the radon atoms reach the pores of the material (this

is called emanation), 3) radon diuses through the pore system, and 4) part of the

pore radon degasses from the surface of the material (this is called exhalation).

law of radioactive decay, we have that:

[J in units of Bq h1] = 3600 s

1 h[J in units of atoms s1] (1)

where =2:09838 106 s1 is the decay constant of radon. Hence, the statement

that a sample has an exhalation rate of 1.0 atoms s1 is equivalent to the statement

that the sample has an exhalation rate of 7:6 103 Bq h1.

If A is the total geometric surface area of the sample and M is the mass of the

sample, we then calculate the area specic (JA) and the mass specic (JM) radon

exhalation rates as:

JA =J

A(2)

JM =J

M(3)

In this report JA is expressed in units of atoms s1 m2 or Bq h1 m2. Likewise,

JM is expressed in units of atoms s1 kg1 or Bq h1 kg1.

As already indicated, the exhalation rate of a given sample depends on the

environment in which the sample is placed. The situation when the environment

has zero radon concentration is of special interest. We refer to this situation as

"free", and add the letter 'f' as a subscript to exhalation rate quantities obtained Free exhalation

under this condition (JM;f and JA;f). Likewise, so-called bound exhalation rates

(dened page 5) are given the letter 'b' as subscript (e.g. JM;b and JA;b). Bound exhalation

Radium concentration, ARa

Radium-226 is transformed to radon by radioactive decay. Therefore, radon is

produced in all materials containing radium-226. The concentration of radium

ARa in units of Bq radium-226 per kg dry mass directly gives the production rate

of radon. For example, if a sample contains 23 Bq radium-226 then it means that

radon is produced at a rate of 23 atoms per second. The radium content of the

building materials depend solely on the selected raw materials.

Ris-R-1135(EN) 3

Radon emanation rate, E

In porous materials, radium is situated in solid grains. Not all radon produced

in the grains actually escape to the pores in between grains. We dene the radon

emanation rate E to be the number of radon atoms per second per kg dry material

(atoms s1 kg1) that escape the solid parts of the material and are available for

transport at a scale larger than the characteristic pore diameter of the material.

Essentially, the emanation rate is the rate at which radon is supplied to the pores

of the material.

Fraction of radon emanation, f

The fraction of emanation, f , is here dened as the ratio between the rate of

emanation, E, and the rate of radon production inside the sample (i.e. the radium-

226 concentration, ARa):

f =E

ARa

(4)

The fraction of emanation depends on the distribution of radium-226 in the grains,

the grain size distribution and the presence of moisture in between grains. The-

oretically, f may take values from 0 to 100 %. For example, a large fraction of

emanation can be expected if radium exists as a surface coating on the grains, if

the grains have a large inner porosity, or if the grains are very small. Presence

of water in between grains can also moderate the emanation process [Ta80]. For

soils, the typical maximum value of the emanation fraction is about 20 %. For aIn uence of moisture

given material (with xed grain size distribution etc.) the emanation fraction is

essentially only a function of the moisture in the sample. It is assumed, that f is

independent, for example, of the radon concentration in the pores.

Fraction of exhalation-to-emanation, g

Radon can move through the sample by diusion and advection. Because of the

nite half-life of radon, only a fraction of the pore-space radon escapes the sample

before decay. We introduce the fraction of emanation-to-exhalation, g, as:

g =JME

(5)

where JM is the mass-specic exhalation rate and E is the emanation rate. g takes

values from 0 to 100 %. If the sample is very large, only a small fraction of the

radon generated inside the sample will reach the surface. In that case, g will be

close to zero. If the sample is very small, all radon generated in the sample will

probably reach the environment and g is unity.

Pressure dierences across the sample can induce ows of air which in turn

transport radon advectively. The main source of bulk air movement through intactAdvective transport

samples (i.e. samples without macroscopic cracks) are probably changes in the

absolute atmospheric pressure.

Fraction of radon exhalation, h

We dene the fraction of exhalation, h, as the ratio between the rate of radon

exhalation from the sample and the rate of radon production inside the sample

(i.e. the radium concentration):

h =JMARa

(6)

4 Ris-R-1135(EN)

h takes values from 0 to 100 %. As already described, the process of exhalation

can be split into two parts as described by f and g, and we have:

h =JME

E

ARa

= f g (7)

Diusivity, D

Exhalation of radon from building materials such as concrete mainly results from

molecular diusion [St88, Rog94, Re95]. The bulk diusivity, D, of building ma-

terials is therefore an important parameter. D is normally believed to be in the

range from 1010 m2 s1 to 106 m2 s1, and this is also the range considered in

the model calculations in Section 6.

The diusive ux is proportional to the gradient of the radon concentration

eld. To clarify what this means, we consider the following example (a more for-

mal denition of bulk disuivity can be found elsewheresee e.g. [An92, An99]):

Imagine a A=120 m2 house positioned on soil. The house has an intact slab of

L =0.1 m in thickness. The slab has a bulk diusivity D=106 m2 s1. The radon Diusion through a

concrete slabconcentration below the slab is set to be cA=50 000 Bq m3. The indoor envi-

ronment has a near-zero radon concentration cB . Ignoring radioactive decay, the

diusive entry J of radon to the house is:

J = ADcA cB

L(8)

= 120 m2 106 m2 s1 50 000 Bq m3

0:1 m(9)

= 21 600 Bq h1 (10)

If the house has an air-exchange rate of v=0.5 h1 and a volume of V=300 m3,

then the diusive entry rate can increase the indoor radon concentration by as

much as 173 Bq m3 (see mass-balance model described page 25). If the bulk

diusivity of the slab is 1010 m2 s1, the diusive entry through the slab can not

even increase the indoor radon concentration by 1 Bq m3.

Concentration eld reshaping

The diusive exhalation rate from a sample is always at a maximum when the

sample is placed in a zero-concentration environment. This is referred to as free

exhalation (Jf). If the sample is placed in a closed chamber (with no other sources Free exhalation

of radon), and if the chamber is initially at zero concentration, then initially

(i.e. at t = 0) radon will exhale from the sample at a rate corresponding to

the free exhalation rate, J(0) = Jf . Because the chamber is closed, the radon

concentration will inevitable increase as a result of this. This leads to a new (less

steep) radon concentration prole in the samplei.e. the eld is reshaped [Sa84]

and the exhalation rate decreases (i.e. J < Jf for t > 0). If the chamber is small

compared with the sample, the change in exhalation rate can be large. If no changes

are made to the system, the radon concentration in the chamber will approach

some equilibrium value, c1. At that point, the net exhalation from the sample

is balanced by radioactive decay of radon in the air volume of the chamber and

leakage of radon out of the chamber. The exhalation rate at this point is called

'the bound exhalation rate', and we use the subscript b to mark this condition. Bound exhalation

Hence, we use Jb for the bound sample-specic exhalation rate, JA;b for the bound

area-specic exhalation rate, and JM;b for the bound mass-specic exhalation rate.

Ris-R-1135(EN) 5

From exhalation rate measurements to full-scale walls

An important application of laboratory measurements of the exhalation rate of

(small) samples of building materials is to assess the contribution of those materials

when applied in specic house-construction parts. This is discussed in Section 6.5.

2.2 Measurement procedures

Measurement of radon exhalation rates can be performed in a multitude of ways.

The most important ones are outlined below.

Gamma measurements of the radium-226 content

Radium-226 is the source of radon. A crude (but robust) measure of 'potential

radon exhalation' is to obtain the radium-226 concentration ARa (Bq kg1) of the

material. Such a measurement can be performed by gamma spectroscopy. From

the conservative assumption, that all radon generated inside the building material

gets out (i.e. assuming h=1.0 in equation 7), we have:

JM = ARa (11)

With further assumptions, it is even possible to put an upper bound on the indoor

radon level. For example, building materials complying with the Swedish radiumRadium index

index requirement:

ARa < 200 Bq kg1 (12)

cannot raise the indoor radon level by more than about 150 Bq m3 even if the

oor, ceiling, and all house walls are made with that material [Cl92, p. 102]. The

house is set to have an air-exchange rate of 0.5 h1.

Gamma measurements of the above type are relatively easy to conduct, but do

not give the actual rate of radon exhalation.

Laboratory measurements of radon exhalation

Laboratory measurements are conducted by placing the sample under investiga-

tion in a chamber from which the radon concentration can be measured. The main

two measurement procedures can be outlined as follows:

Open-chamber method (Method A) A ow of air (typically about 1 L min1)

is established through the chamber. This provides the sample with a well-

dened environment with respect to humidity and radon. A near-zero radon

concentration is preferable for measurements of the free exhalation rate (see

page 5). After a selected time of conditioning (e.g. 12 h), the sample is as-

sumed to be in equilibrium with the chamber environment, and radon mea-

surements are conducted. There are now two ways to proceed:

Activity collection (Method A1) A device able to trap radon is placed

at the outlet of the chamber. This can for example be a cold trap of ac-

tivated charcoal placed in a dewar with dry ice (temperature 78 ÆC).

Such a trap will eectively collect all radon leaving the chamber. The

trapped activity subsequently can be determined by gamma spectroscopy.

Alternatively, radon may be released into a scintillation cell [Ma88]. If

the activity A (Bq) is trapped over a period of time T (s), then the

exhalation rate J (Bq s1) from the sample is:

J =A

T(13)

6 Ris-R-1135(EN)

The main feature of this approach is that only the determination of A

is a subject to uncertainty. The experimenter even has the opportunity

to diminish the counting error relating to the A-determination by se-

lecting a suÆciently long time of integration. This method therefore is

(potentially) very accurate and highly sensitive.

Air concentration measurement (Method A2) A sample of air is taken

from the chamber outlet or the radon concentration of the chamber is

monitored continuously. From the measured concentration c (Bq m3)

and the ow rate Q (m3 s1) through the chamber, the exhalation rate

J (Bq s1) can then be found as:

J = cQ (14)

Since the radon concentration is low (typically below 5 Bq m3), this

method is only useful with a sensitive method for radon concentration

determination. In most cases, counting error will be an important source

of uncertainty. Another source of error is in the ow rate determination.

The open-chamber method A1 follows the general recommendations given

by the Danish Standards Association regarding degassing measurements for

building products [DS94]. Another reason to consider method A1 a good ref- Standard methods

erence method is that it follows the principles in the proposed Dutch norm for

exhalation rate measurements (the pre-standard is identied as NVN5699).

It is used for example by the KVI [Gr97].

Closed-chamber method (Method B) First the sample is conditioned as de-

scribed with the previous method. If the ow rate of air through the chamber

is suÆciently large (and is without radon), the chamber will quickly approach

a near-zero level. At time t = 0, the chamber is closed in the condition:

c(0) 0 (15)

As a result of exhalation from the sample, the radon concentration starts to

build up inside the chamber for t > 0. Monitoring of the radon concentration

c(t) in the chamber over a certain period of time (e.g. 530 days) is done

by grab sampling or by a continuous radon monitor. The analysis of the so

called 'growth curve' c(t) is conducted as follows: If there are no leaks in the

chamber (i.e. if the chamber is truly closed), mass balance requires that:

Vdc

dt= J(t) V c (16)

where we have assumed that the chamber is well mixed. V is the volume of

the chamber. If the chamber is suÆciently large (compared with the sample)

it is a good approximation (see section 6) to assume that the exhalation rate

is constant:

J(t) J (17)

such that equation 16 has the solution:

c(t) = c1(1 et) (18)

where

c1 =J

V(19)

is the radon concentration (Bq m3) of the chamber as t!1. Fitting equa-

tion 18 to the measured growth curve c(t) provides an estimate of the param-

eter c1 from which the exhalation rate can be found as:

J = V c1 (20)

Ris-R-1135(EN) 7

This method is critically dependent on the assumption that the chamber is

leak free. If this is not the case, in equation 20 needs to be substituted

by some eective 'decay constant' e that incorporates radioactive decay as

well as the leakage. e can be made part of the tting procedure described

above. The correction is only valid if the (leaky) chamber is placed in a room

with a radon concentration much lower than that of the chamber.

3 Materials

This section describes the investigated samples and the experimental apparatus.

3.1 Samples

Two batches of each 10 building material samples were supplied by H+H Industry

A/S. Most (but not all) samples were produced by that company. The rst batch

was delivered to Ris on September 29, 1997. The second batch was delivered on

November 10, 1997. Both batches were produced 12 months prior to the dates

of delivery. At H+H Industry A/S, all samples had been conditioned to be in

equilibrium with air at 23 ÆC and 43 % relative humidity. This means that the

moisture contentW in all samples were less than 3 %.W is the mass of (removable)

moisture divided per dry mass. At Ris, the samples were located in a basement

laboratory room in building 125. The typical conditions of that room were 24 ÆC

and 40 % relative humidity. The average radon concentration in the room was

about 30 Bq m3.

The samples in batch 1 are identied as M1 to M10 (i.e. material no. 1, material

no. 2 etc.). Measurements were not performed for batch 2.

For the samples in batch 1, Table 1 gives linear dimensions, masses (M), surface

areas (A), volumes (V ), area-to-volume ratios (A/V ), and densities (m =M=V ).

Sample M10 is an aggregate (single grains), and the volume has been calculated

for a relatively loose packing. The corresponding surface area has been calculated

from an assumed area-to-volume ratio of 0.533 m1. This ratio is identical to that

obtained for a slab of dimensions 30 x 30 x 5 cm3.

3.2 Equipment

Figure 2 shows the experimental set-up.

Chamber

All measurements were performed in a cylindrical stainless-steel chamber. The

volume of the chamber is 55.76 L (about 34.4 cm diameter and 60 cm depth). The

lid of the chamber is sealed with an o-ring and is closed by 16 bolts. The chamber

is equipped with two fans of the type used for cooling in personal computers.

Flow control

The ow system consists of a dry line and a wet line. The dry-line ow comes from

a 4 m3 nitrogen (pressurized) gas cylinder. The ow rate is regulated manually

by use of the pressure reduction valve. The ow has a relative humidity of 0 %.

The wet-line ow comes from another 4 m3 nitrogen gas cylinder. This ow is

controlled by a mass- ow controller (Brooks Instrument B.V., the Netherlands)

8 Ris-R-1135(EN)

Table 1. Dimensions of the samples in batch 1. The densities given in the second

column are nominal factory densities in units of kgm3. Lightweight aggregate

concrete and autoclaved aerated concrete are abbreviated as LAC and AAC, re-

spectively.

ID Description Dimensions Mass Area Volume A/V Density

kg m2 L m1 kgm3

M1 LAC, 1 slab 30.0 x 30.0 x 4.9 cm3 2.89 0.239 4.41 54 656

density 600

M2 LAC type 1, 1 slab 29.8 x 30.0 x 5.4 cm3 7.32 0.243 4.83 50 1516

density 1500

M3 LAC type 2, 1 slab 29.8 x 29.9 x 5.0 cm3 7.04 0.238 4.46 53 1579

density 1500

M4 AAC, 1 slab 30.1 x 30.0 x 5.5 cm3 2.56 0.247 4.97 50 515

density 450

M5 AAC, 1 slab 30.0 x 30.1 x 5.1 cm3 3.12 0.242 4.61 53 677

density 650

M6 AAC, 1 slab 29.9 x 30.1 x 5.1 cm3 3.66 0.241 4.59 53 797

density 735

M7 Ordinary concrete, 1 slab 29.9 x 30.0 x 5.0 cm3 10.08 0.239 4.49 53 2248

density 2300

M8 Gypsum board 5 boards 3.96 0.963 5.55 173 713

29.8 x 29.8 x 1.3 cm3

M9 Bricks 3 bricks 10 x 20 x 5.4 cm3 8.56 0.330 4.84 68 1768

2 bricks 10 x 14.8 x 5.4 cm3

M10 Lightweight expanded Single grains (no packing) 1.51 (0.277)a 5.19 (53)a 291

clay aggregatea See text

Figure 2. Experimental set-up.

Ris-R-1135(EN) 9

set to 500 mLn min1 (nL = normal liter). The ow is led through a humidier

such that the ow thereafter can be assumed to have a relative humidity of 100 %.

The two ow lines are mixed and the ow is supplied to the chamber. The

total ow rate from the chamber was measured with a Gilian bubble ow meter

(20 mLmin16 L min1; Gilian Instruments Corp. USA).

Radon instrument

Radon concentration measurements were conducted with an ionization chamber

placed inside the chamber (AlphaGuard, PQ-2000 from Genitron, Germany). The

monitor has sensors for relative humidity, temperature and absolute pressure. It

is assessed that about 1.55 L of the monitor is rigid. This value is used for the

calculation of the free air volume in chamber.

Numerical model

A numerical nite-dierence model called RnMod3d developed at Ris was used in

certain parts of the error analysis. RnMod3d is a 3D time-dependent model of gas

and radon transport through porous media. The principles behind the model are

outlined in [An92]. The model has been compared with other models [An99], and

it has also successfully been tested against the analytical steady-state solutions

given by Berkvens et al. [Be88].

4 Experimental procedures anddata analysis

An experimental procedure corresponding to the closed-chamber method (Method

B) described page 7 is adopted as primary method in this work. A (less accurate)

version of the open-chamber method (Method A2) is used to check for gross errors.

To distinguish between results obtained with the two methods, all open-chamber

results are marked with an OC (Open Chamber) as in Jf;OC.

4.1 Experimental procedures

1. The sample was weighted and positioned in the chamber.

2. From the computer, the continuous radon monitor was set to store results in

a new data le. The cycle time of the monitor was set to 1 hour (preferable)

or 10 min.

3. The continuous radon monitor was placed in the chamber.

4. The lid was put on the chamber.

5. The ow was started, and the time was noted in the log book as the start of

conditioning. The mass- ow controller was set to 0.5 L min1. The reduction

valve of the dry- ow line was adjusted such that the total ow Q leaving

the chamber was about 1 L min1. The ow was maintained at this level for

1224 hours. The total ow rate was manually measured at selected times

with the bubble ow meter.

6. The tubing was removed from the chamber, and the chamber was closed. The

time was noted in the log book as the time the conditioning stopped (and the

build-up started).

10 Ris-R-1135(EN)

7. After 214 days of build-up, the chamber was opened. The time was noted in

the log book as the time the build-up stopped.

8. The sample was removed from the chamber and was (re)weighted.

9. The data from the experiment was stored in the database.

4.2 Data and error analysis

Radon monitor bias

The raw radon concentrations reported by the monitor are corrected for instru-

ment bias by subtraction of 14:120:72 Bq m3. Hence, a (raw) instrument read-

ing of 15:12 0:5 Bq m3 is corrected to 1:0 0:9 Bq m3, where all indicated

(statistical) uncertainties are expressed as one standard deviation, and where the

uncertainty of the corrected result is found by quadrature summation. This cor-

rection was deduced from one single blank experiment conducted from August 11

to August 13, 1998. The radon monitor was left in the chamber (without any sam-

ple), and the chamber was ushed with nitrogen let through an activated charcoal

cold trap [Ma88]. Such a trap is known to eectively remove any radon in the

nitrogen. To allow for desorption of radon from chamber walls, the chamber was

ushed on two occasions.

The radon monitor is calibrated against three local standards all traceable to

NIST (see [An97b]). The uncertainty of the bias of the results (expressed as one

relative standard deviation) is judged to be about 5 %.

Closed-chamber method (Method B)

Equation 18 was extended with a constant term c0 and an eective decay constant

e :

c(t) =

(c0 for t < 0 (conditioning)

c0 + c1(1 ee t) for t 0 (build-up)(21)

c0 re ects the (potential) o-set of the radon monitor and the fact that the radon

concentration of the air in the chamber during conditioning was only near zero

but not exactly zero. e was set to xed values (see later) and was not made

part of the tting procedure. A value of e greater than the radioactive decay

constant of radon (=2:09838 106 s1) means that the chamber is leaky.

Equation 21 was tted to the measured radon concentration in the chamber Non-linear tting

during conditioning and build up. Non-linear curve tting was conducted with

the Marquard method as described in Bevington and Robinson [Be92, p. 161].

Essentially, the tting procedure determines the values of c0 and c1 such the

sum-of-squares:

2 =

NXi=1

ci c(ti)

i

2(22)

becomes minimal. ci is the radon concentration measured at regular time intervals

ti (every hour or every 10 minutes), N is the number of measurement points and

i is the uncertainty associated with each ci as estimated by the radon monitor.

The reduced-2 (2) is calculated as:

2 =2

(23)

where = N 2. A value of 2 very dierent from unity indicates that either the

tting function or the error estimates i are inappropriate.

The free exhalation rate was calculated using a modied version of equation 20: Corrections

Ris-R-1135(EN) 11

Jf = eV c1 (24)

where V is the air volume of the chamber3 and c1 is the tted equilibrium

radon concentration. The factor converts the measured bound exhalation rate

to the free exhalation rate Jf . Based on the results of model calculations presented

page 20, was in all cases set to 1=0:987 = 1:013. As discussed page 28, e was

set to 1:037 for measurement 103 to 120, and 1:0 for measurement 121,

where is the (true) decay constant of radon (2:09838 106 s1). The statistical

variability ufJfg of any Jf-determination is found by quadrature summation ofUncertainty analysis

the following contributions:

The statistical error associated with the tted parameter c1 (the inverse of

the diagonal element in the error matrix).

Although measurements numbers 103 to 120 are corrected for leakage from the

chamber, this leakage was probably dierent from experiment to experiment.

The variability from this source on the nal result (Jf) is judged to be about

2 % 4.

The correction from bound to free exhalation is set to be about 0.4 %

(half the maximum range of the results shown in Figure 9). Hence: =

1:0130.004, where the uncertainty is expressed as one standard deviation.

Other (random) sources of errors (such as sink eects, interference of radon-

220 and errors connected to the determination of the air volume in the cham-

ber) are judged to be at the order of 1 %

The combined uncertainty UcfJfg of a Jf-determination is found by quadrature

summation of the 5 % uncertainty of the radon instrument and the value for ufJfg

just discussed.

Open-chamber method (Method A2)

In the open-chamber method, the free exhalation rate is calculated from a modied

version of equation 14):

Jf;OC = (ccond cgas)Q (25)

where ccond is the average radon concentration in the chamber from 4 hours after

start of conditioning to the time the chamber is closed. The rst 4 hours are

excluded from the analysis because initially the chamber is lled with room air.

cgas is the radon concentration of the nitrogen gas source. In this investigation,

cgas was set to zero because a set of four newly purchased cylinders of nitrogen

was found to have radon concentrations below 0.05 Bq m3. In most cases, gas

cylinders were stored for days or weeks before use in the experiment. However,

with the exception just given, there was no systematic control of the specic radon

concentration of the carrier gas in the actual experiments, and therefore this source

of error may bias some of the Jf;OC-results.

The statistical variability ufJf;OCg of any Jf;OC-determination is found byUncertainty analysis

quadrature summation of the following contributions:

The statistical error associated with the measurement of the radon concen-

tration. This source is large as the concentrations are close to zero.

3V equals the chamber volume (55.76 L) minus the dead volume of the radon monitor (1.55 L)and the geometric volume of the sample. For a 30 x 30 x 5 cm3 sample of concrete (4.5 L), Vequals 55:76 1:55 4:5 = 49:71 L.

4Observe, that a change of e will cause a change of both the tted estimate of c1 and thecalculation of Jf as given in equation 24. For example, a typical exhalation rate measurement, achange of 5 % of e will lead to a change of only about 2 % on the nal result (i.e. Jf).

12 Ris-R-1135(EN)

The ow rate Q is judged to be subject to an uncertainty (expressed as one

relative standard deviation) of about 10 %. This source of variability result

from imperfections of the (manual) ow control (e.g. changes of gas cylinders

during experiments).

The possible in uence of large cgas-values is not included in the uncertainty esti-

mate.

4.3 Radium-226 measurements

Radium-226 concentration determinations were conducted by Danish Institute for

Radiation Hygiene.

5 Experimental results

16 exhalation rate measurements were conducted (identication numbers 103,

105: : :117, 120 and 121). M7 was measured 7 times, whereas the other 9 ma-

terials were measured only once. The measurements were conducted during the

period October 22, 1997 to August 10, 1998. Appendix B contains measurement

sheets for all measurements.

Figure 3 shows the radon concentration in the chamber during a typical ex-

periment. Initially, the sample is conditioned with a ow of about 1 L min1 for

1 day. Observe, that the concentration has a non-zero value. This is used in the

so-called open-chamber method. Then at day 0, the chamber is closed, and the

radon concentration increases towards some equilibrium value. This part of the

curve is used in the closed-chamber method.

-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5 6 7

Time, days

Rn-222, Bq m-3

GEXH0110.dat

-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5 6 7

Time, days

Rn-222, Bq m-3

GEXH0110.dat

Figure 3. Typical build-up curve. This is measurement no. 110. Day zero on the

x-axis is January 21, 1998. The tted curve: c(t) = c0+ c1(1 exp(t)) has the

parameters c0=2.20.9 Bqm3, and c1=3504 Bqm3. The free mass-specic

exhalation rate is calculated to be JM;f=2.600.07 atoms s1 kg1. The indicated

uncertainties are expressed as standard deviations of the given results and include

all (known) sources of errors except bias of the radon instrument.

Ris-R-1135(EN) 13

Table 2. Main results obtained with the closed-chamber method. The indicated un-

certainties include all (known) sources of error except the uncertainty of the cali-

bration of the radon monitor. All uncertainties are expressed as one standard de-

viation of the given results. Lightweight aggregate concrete and autoclaved aerated

concrete are abbreviated as LAC and AAC, respectively. The fraction of exhalation

(in the last column) is the quantity h dened page 4.

ID Description Exhalation rate Radium Fraction

Meas. JA;fufJA;fg JM;fufJM;fg ARa of exhal-

no. Bq h1 m2 mBq h1 kg1 atoms s1 kg1 Bq kg1 ation, %

M1 LAC, 112 0.2390.009 19.80.7 2.620.10 32.6a 8

density 600 kgm3

M2 LAC type 1, 103 0.5840.015 19.40.5 2.570.07 11.8a 22

density 1500 kgm3

M3 LAC type 2, 110 0.5800.016 19.60.5 2.600.07 17.6a 15

density 1500 kgm3

M4 AAC, 111 0.0870.006 9.50.6 1.260.08 10.1b 12

density 450 kgm3

M5 AAC, 109 0.1030.006 8.00.5 1.060.06 16.6a 6

density 650 kgm3

M6 AAC, 108 0.2860.010 18.90.7 2.500.09 11.9a 21

density 735 kgm3

M7 Ordinary concrete, 105 1.0120.026 23.90.6 3.170.08 13.8a 23

density 2300 kgm3 114 0.8610.024 20.40.6 2.700.07 - 20

115 0.8490.021 20.20.5 2.670.07 - 19

116 0.8590.022 20.40.5 2.700.07 - 20

117 0.8340.022 19.80.5 2.620.07 - 19

120 0.8080.021 19.20.5 2.540.07 - 18

121 0.8140.021 19.30.5 2.560.06 - 19

M8 Gypsum board, 106 0.0100.001 2.40.3 0.320.04 2.1c 15

density 710 kgm3

M9 Bricks, 107 0.0200.003 0.80.1 0.100.01 39.8a 0.26

density 1800 kgm3

M10 Lightweight expanded 113 0.0010.004 0.10.7 0.020.10 32.5b 0.05

clay aggregate,

density 290 kgm3

Relative standard uncertainties: a 5 %, b 10 %, c 15 %.

Closed-chamber method

The main results obtained with the closed-chamber method are given in Table 2

together with the results of radium-226 measurements provided by the Danish

Institute for Radiation Hygiene. It is seen from the table that concrete-based ma-

terials (M1 to M7) have mass-specic exhalation rates in the range from about 1

to 3 atoms s1 kg1. The other materials have values below 0.3 atoms s1 kg1.

Radium-226 concentrations are in the range from about 2 Bq kg1 for gypsum

(M8) to 40 Bq kg1 for bricks (M9). Also the fraction of exhalation, h (see de-

nition page 5) varies over a wide range. The lowest value is for M10, where less

than 0.1 % of the radon atoms generated in the material escapes the material.

The largest fraction of exhalation is for the concrete-based samples M2, M6 and

M7, where h amounts to 20 %.

14 Ris-R-1135(EN)

Figure 4. Results for sample M7: Free mass-specic exhalation rate (JM;f) and

sample mass versus the time since November 11, 1997.

Figure 5. Free mass-specic exhalation rates obtained with the closed-chamber

method (JM;f) and the open-chamber method (JM;f;OC). The indicated uncertain-

ties ufJM;f;OCg include all (known) sources of error except the uncertainty of the

bias of the radon instrument.

Ris-R-1135(EN) 15

Figure 4 shows the results of 7 (repeated) exhalation rate measurements for

sample M7 versus time since the rst measurement on November 11, 1997. The

total mass of the sample is also shown. As can be seen from the gure, the exhala-

tion rate decreases signicantly (i.e. the variability is larger than can be explained

by the uncertainty associated with the results). The decrease amounts to about

20 % over a period of 250 days. During the same period, the mass decreases about

0.5 %. The main reason for the change in mass is probably that the samples dries

out.

Open-chamber method

The results JM;f;OC of the open-chamber methods are shown in Figure 5 along

with the results for the closed-chamber method JM;f . There is good agreement

between the results of the two methods. Observe, that the uncertainty associated

with the open-chamber method is large.

Chamber leakage

On ve occasions during the period November 7, 1997 to March 20, 1998 special

experiments were conducted to test the chamber for leakage. These experiments

are presented and discussed page 28.

6 Modelling results

The main cause of exhalation of radon from building materials is believed to

be molecular diusion [St88, Rog94, Re95]. The following investigation therefore

focuses on this aspect whereas other mechanisms (for example, exhalation resulting

from pressure changes) are ignored. The following issues are investigated:

Test if an equation of the form c(t) = c1(1 exp(t)) (see page 7 and

11) gives a good description of the radon concentration build-up during an

exhalation rate measurement with the closed-chamber method. This is a test

of the goodness of the approximation that the exhalation rate J is constant

throughout the measurement period.

Quantify the bias caused by the closed-chamber method (see page 7) com-

pared to an ideal measurement of the free exhalation rate. This is a calculation

of the so-called bound-to-free exhalation rate ratio.

Quantify how sample geometry in uences on the measurement result. This

aspect is useful for example when results in this investigation are compared

with results obtained previously using other sample geometries.

Quantify how laboratory measurement on small samples (slabs or cylinders)

may be extrapolated to full-scale construction parts in real houses.

Although few (if any) published investigations have dealt with the above problems

in full 3D and time dependency, many have certainly considered similar situations.

The main results can therefore also be extracted from those publicationssee for

example: [Jo80, Co81, Sa84, Be88, Al94].

Model

All simulations have been performed with the numerical model called RnMod3d

(see page 10). The model solves the relevant diusion equation [An92].

16 Ris-R-1135(EN)

Diusivity, D

It is assumed, that exhalation occurs solely as a result of molecular diusion, and

the main parameter of the hypothetical building material is therefore D, the bulk

diusivity (see page 5). To cover what seems to be the relevant range of diusivity

values, most calculations are performed for values in the range from 1010 m2 s1

to 106 m2 s1.

Other model parameters

Other parameters of the building materials have (arbitrarily) been set to the fol-

lowing constant values: ARa=14 Bq kg1 (radium-226 concentration), f=0.2 (frac-

tion of emanation), g=2:7 103 kgm3 (density of grains), =0.2 (total porosity),

and W=0.0 (water content by mass).

Observe, that with the above values for grain density and porosity, the material

has a bulk density of

m = (1 )g = 2:16 103 kgm3 (26)

Also, observe that the emanation rate is:

E = fARa = 2:8 atoms s1 kg1 (27)

The main case under consideration (see next section): a 5 x 30 x 30 cm3 slab hence

has a total mass of M=9.72 kg and a maximum rate of exhalation of M E =

27.2 atoms s1 or 0.206 Bq h1.

Model geometry

Several geometries are explored. The main set of calculations concern a slab with

dimensions 5 x 30 x 30 cm3 (4.5 L) placed in a chamber of 58.5 L volume. The free

air in the chamber is assumed to be well mixed. Observe, that the free air volume

(54.0 L) in the model chamber volume is a bit larger than that of the actual

experiment (49.71 L after correction for the rigid volume of the radon monitor,

see footnote page 12).

Other geometries that are also considered: cylindric samples and (house) walls

with one or two sides facing the indoors. All calculations are performed in full 3D.

Boundary conditions and initial conditions

The chamber is assumed to the leak tight, such that the eective decay constant

of the chamber is equal to the decay constant of radon: =2:098 106 s1. In

some of the calculations the radon concentration of the chamber (or house) is,

however, maintained at 0 (free exhalation condition).

Model output

The main output from the model calculations is the total exhalation rate J .

6.1 Simulation of the closed-chamber method

Figure 6 to 8 show simulations of exhalation rate measurements: A sample with

dimensions 5 x 30 x 30 cm3 is placed in a closed chamber initially at zero radon

concentration and the radon concentration in the chamber starts to build up.

In Figure 6, the sample has a diusivity D of 1010 m2 s1. In Figure 7, D is

108 m2 s1, and in Figure 8, it is as high as 106 m2 s1. The gures show

Ris-R-1135(EN) 17

Figure 6. Model simulation of exhalation rate measurement with D=1010 m2 s1.



18 Ris-R-1135(EN)

calculated exhalation rates J as well as calculated radon concentrations c in the

chamber. The right plots show the time development over a period of 30 days. The

left plots focus on the time immediately after the start of the build-up period.

As marked in the top-right plot of Figure 6, the exhalation rate J is at a maxi-

mum at t = 0. This is the free exhalation rate Jf . Here, it amounts to 0.126 Bq h1.

As time progresses, the exhalation rate decreases towards some equilibrium value

Jbthe so-called bound exhalation rate. Here, it amounts to 0.125 Bq h1. Al-

though the plots show a distinctive change in J over time, the relative change is

marginal (less than 1 %), and it is a good approximation to treat J as a constant

as suggested page 7. As a result of this, the build-up of radon in the chamber c(t)

is well described by an equation of the form c(t) = c1(1 exp(t)) (see page 7

and 11).

The results for D=108 m2 s1 (Figure 7) are similar to those just discussed for

D=1010 m2 s1. Two dierences do however exist: the exhalation is much larger

and the time it takes to reach the state of bound exhalation is smaller.

For D=106 m2 s1 (Figure 8), the exhalation rate is only slightly higher than

that obtained for D=108 m2 s1. This is because these diusivities are so high

that almost all radon will anyway exhale from the sample, regardless of the exact

value of D. For D=106 m2 s1 the free exhalation rate Jf amounts to a value

just below 0.206 Bq h1 which is the rate radon is produced in the sample (i.e.

the maximum exhalation ratesee page 17).

6.2 Bound-to-free exhalation rate ratio

Simulations like the ones just described are conducted for a range of diusivity

values. In each case, the values of the free exhalation rate Jf and the bound

exhalation rate Jb are calculated. The bound-to-free exhalation rate ratios:

JbJf

(28)

are shown as function of D in Figure 9 for two geometries: a 5 x 30 x 30 cm3 slab

and a 10 x 30 x 30 cm3 slab.

For the 5 cm slab (i.e. the main geometry used in this study), the bound-

to-free exhalation rate ratio is always in the range from about 0.983 to 0.99.

Eectively this means that the result obtained with the current closed-chamber

method (regardless of the diusivity of the sample) are biased by less than 2 %

(as a result of the eect in question) in comparison with an ideal measurement of

Jf . For the measurements with the 10 x 30 x 30 cm3 slab, the potential bias is a

little bit larger. The results are specic for the 58.8 L chamber considered in the

calculation.

6.3 g for laboratory samples

The fraction of (mass specic) exhalation-to-emanation g was dened on page 4

as the rate radon exhale from the sample relative to the rate radon is supplied to

the pores of the sample (i.e. the rate of emanation, E). Figure 10 shows calculated

values of g for a range of sample geometries. The calculations assume equilibrium

conditions and that the samples are in an environment with zero radon concen-

tration.

Geometry A is a slab of dimensions 1 x 30 x 30cm3. The thickness of only 1 cm

means that most radon supplied to the pores will escape from the sample regardless

of the diusivity of the material. In contrast, the exhalation rate for a sample

formed as a cylinder with 15 cm diameter and 30 cm height will depend somewhat

on the diusivity of the material. For example, imagine a material of diusivity

Ris-R-1135(EN) 19

Figure 9. Calculated bound-to-free exhalation rate ratio versus bulk diusivity for

closed-chamber measurements with a 5x30x30 cm3 slab placed in a 58.5 L chamber.

D=109 m2 s1. Two samples are prepared for exhalation rate measurements: One

of geometry A and another of geometry E. Although the materials are the same,

Figure 10 shows that the results for geometry E will be about 20 % lower than

those obtained with geometry A. For D=1010 m2 s1, the dierence is about

40 %.

The present investigation mainly applies the 5 x 30 x 30 cm3 slab geometry

(geometry B). It is seen from the gure that even if the diusivity is a low as

1010 m2 s1, this geometry can impede the exhalation only by 30 %.

6.4 g for walls

The fraction of exhalation-to-emanation, g, has been calculated for a range of wall

geometries. As shown in Figure 11, both walls having two sides facing the indoors

(internal walls) and one side facing the indoors (external walls) are considered. In

the one-sided situation, it is assumed that the other side is sealed o, such that

no radon can escape from that side.

As can be seen from the gure, there are large dierences among the g-values

for the dierent walls. For example, imagine two walls made of the same material:

The rst wall is 5 cm thick with both sides facing the interior (this is geometry

F in Figure 11). The second is a 20 cm thick wall with only one side facing the

interior (this is geometry D in Figure 11). We assume that the diusivity of the

material is 1010 m2 s1. As can be seen from the gure, g amounts to about 0.56

for the rst wall and 0.08 for the second. Hence, as much as 56 % of the radon

atoms supplied to the pores of the material degas from the surface of the rst

wall. The "eectiveness" of degassing is only 8 % for the second wall.

20 Ris-R-1135(EN)

Figure 10. Fraction of exhalation-to-emanation g for a range of sample geometries.

Figure 11. Fraction of exhalation-to-emanation g for dierent wall geometries.

Ris-R-1135(EN) 21

6.5 From laboratory measurements to full-scalewalls

Exact solution

By denition, the exact total exhalation rate J(wall) (Bq h1) from a wall can be

calculated as:

J(wall) =M(wall) g(wall) E (29)

where M(wall) is the mass of the wall, g(wall) is the exhalation-to-emanation

ratio for the given wall geometry and diusivity, and E is the emanation rate

of the material. Normally the latter is unknown, and is therefore estimated from

laboratory measurement of the mass-specic exhalation rate JM(sample) of a small

sample:

E =JM(sample)

g(sample)(30)

such that we obtain:

J(wall) =M(wall)g(wall)

g(sample)JM(sample) (31)

which is the formula by which a laboratory measurement can be extrapolated to

a full-scale wall. The problem with the extrapolation is, however, that it is only

accurate, if the diusivity of the material is known. In that special case, g(sample)

and g(wall) can simply be read from Figure 10 and 11.

Fortunately, two straightforward approximate solutions exist: One based on a

high-diusivity assumption and another based on a low-diusivity assumption.

The true diusivity of the material will be in between these two limiting cases.

High-diusivity limit

In the high-diusivity limit, we assume that the diusivity of the material is so

large that all radon escape both the laboratory sample and the wall (i.e. both are

considered to be small compared to the diusion length of radon). This means

that:

g(wall) = 1 (32)

g(sample) = 1 (33)

such that from equation 31 , we obtain:

J(wall) M(wall) JM(sample) (34)

For real materials, equation 32 and 33 are never fullled exactly. However, as long

as g(wall) g(sample), equation 34 provides an overestimate of the wall exha-

lation rate. Essentially this means that the laboratory sample should be thinner

than the wall in question.

Low-diusivity limit

In the low-diusivity limit, we assume that the diusivity is so small that radon

escape only from a very thin surface layer of the material. Hence, we assume that

the area-specic exhalation rate is the same for the sample and the wall:

JA(wall) = JA(sample) (35)

We can therefore write the fractions of exhalation-to-emanation as:

g(wall) =A(wall) JA(sample)

M(wall) E(36)

g(sample) =A(sample) JA(sample)

M(sample) E(37)

22 Ris-R-1135(EN)

Figure 12. Calculation of the total exhalation rate J(wall) for an internal 10 cm

wall. The calculations are based on laboratory measurements of a sample slab of

dimensions 5 x 30 x 30 cm3. The exact value of J(wall) for each diusivity D is

calculated with equation 31. The horizontal lines are results for the simplied mod-

els in equations 34 (the high-diusivity limit) and equation 38 (the low-diusivity

limit) based on mass-specic and area-specic exhalation rates, respectively.

Inserting into equation 31 gives:

J(wall) A(wall) JA(sample) (38)

where A(wall) is the surface area of the wall and JA(sample) is the area-specic

exhalation rate of the sample. This is an underestimate of the true J(wall) as the

diusivity of real materials is not innitely low.

Internal wall example

To illustrate how good (or bad) these approximations are we will consider a specic

example and compare the approximate solutions with the exact value calculated

from equation 31.

Imagine the following situation: Laboratory measurements are conducted on a

sample slab of dimensions 5 x 30 x 30 cm3. Assume, the mass-specic exhalation

rate is found to be JM(sample)=19.6 mBq h1 kg1, and that the area-specic

exhalation rate is found to be JA(sample)=0.58 Bq h1 m2. The material is to

be used as an internal house wall of 10 cm thickness. The wall has a mass M(wall)

of 14 400 kg, and a total (exposed) surface area of 196 m2. The question, is what

the total exhalation rate will be from the wall? The diusivity of the material is

unknown.

First, we will consider the exact (true) value. The 10 cm double sided wall Exact value

is identied as geometry is G in Figure 11. If the material has a diusivity

D=109 m2 s1, the gure gives g(wall) equal to about 75 %. For the same diu-

sivity, Figure 10 shows that g(sample) equals 95 %. Hence, from equation 31, the

Ris-R-1135(EN) 23

total exhalation rate from the wall J(wall) can be calculated to be: 225 Bq h1.

Likewise, exact results can be obtained for any other hypothetical value of the

diusivity. Results for values in the range from 1010 to 106 m2 s1 are shown

in Figure 12.

Now, in the high-diusivity limit, the total wall exhalation rate is calculated ashigh-D limit

the product of the wall mass and the mass-specic exhalation rate of the sample.

In the example from above, the estimate of J(wall) amounts to 282 Bq h1. This

value is plotted as the top horizontal line in Figure 12. It is observed, that the

approximation is indeed good for high diusivities.

In the low-diusivity limit, the total wall exhalation rate is calculated as thelow-D limit

product of the exposed area of the wall and the area-specic exhalation rate of the

sample. This results in the estimate: J(wall)=114 Bq h1. This value is the bottom

horizontal line in Figure 12. It is observed, that the approximation is indeed good

for low diusivities.

This example shows that the extrapolation of laboratory measurements to a

full wall geometry is subject to considerable uncertainty (a factor of two or such)

when the diusivity of the material is unknown.

7 Reference house calculations

The impact of exhalation from a given building material on the radon concentra-

tion in the house where it is used depends on the following factors:

Mass-specic exhalation rate of the material

Mass of material in use

Location of material (e.g. internal wall or external wall)

House volume

Air-exchange rate of the house

The building materials investigated in this report have a range of applications.

Each application imply that a certain amount of material is used in a certain

Figure 13. Layout of the reference house used in the assessment of how much

the indoor radon concentration increases as a result of specic applications of the

investigated building materials.

24 Ris-R-1135(EN)

Table 3. Dimensions of the reference house. The areas of the walls have been

corrected for doors (1.8 m2 each) and windows (1.4 m2 each). The projected area

is simply the volume of material in use divided by the thickness (i.e. the cross-

sectinal area). The exposed area (Ae) is the area facing the house interior.

Room height 2.5 m

Floor area 123 m2

House volume (V ) 308 m3

Internal wall with 7 doors

Projected area 96 m2

Exposed indoor area (Ae) 192 m2

External wall with 3 doors and 9 windows



Wall (internal or external) with 10 doors and 9 windows



part of the house construction. In the following, we assess the impact of these

applications for the reference house sketched in Figure 13. The house is a typical

Danish slab-on-grade house from the 1970'ies. House dimensions are summarized Slab-on-grade houese

in Table 3.

The diusivity of the materials has not been measured, and we therefore are Unknown diusivity

bound to estimate exhalation rates in the house on the basis of the approximations

described in section 6.5. Those approximations concern the total wall exhalation

rate (as if radon from all sides of the wall would exhale into the house), and it

is necessary to apply some modications to account for applications other than

internal walls.

In the the high-diusivity limit, the exhalation rate is calculated from the mass-

specic exhalation rate (JM) as measured in the laboratory for a small sample

(see equation 34). We apply a geometry factor G to account for how much of the G factor

radon exhaling from the building material that reaches the indoor environment.

For contruction parts with one side facing the free atmosphere, G is set to 0.5. For

all other locations (internal walls, sub-slab, ceiling construction etc.) we set G to 1.

These values for G will probably lead to an overestimation of the exhalation rate.

In the the low-diusivity limit, the exhalation rate is calculated from the area-

specic exhalation rate (JA) as measured in the laboratory for a small sample (see

equation 38). The wall area is now the exposed area (i.e. the area facing the house Exposed area

interior).

House concentration

Under steady-state conditions, the rate J (Bq h1) radon is supplied to the house

(all sources considered) equals the rate radon is removed:

J = vV c (39)

where v is the air-exchange rate (normally assumed to be 0.5 h1), V is the

volume of the house, and c is the indoor radon concentration (Bq m3). Thus, c

can be calculated as:

c =J

vV(40)

In the calculations, we assume that the house air is well mixed, that is has an

air exhange og 0.5 h1 and that it can be treated as one single zone.

Ris-R-1135(EN) 25

Building-material applications

In the following the term external wall is used to designate a wall that faces the

outdoors. In line with this, the term internal wall designates a wall that faces only

the indoors. The following applications are considered:

Front wall (Danish: formur) and back wall (Danish: bagmur) are the main

components of a cavity wall. The front wall faces the outdoors, and the back

wall faces the interior of the house.

Solid load-bearing external wall (Danish: massiv mur) An external con-

crete wall constructed as one single layer.

Internal wall (Danish: skillevg) Walls separating rooms in the house.

Ceiling

Sub-slab layer A layer of large porosity positioned below the slab to prevent

ingress of moisture into the oor construction (i.e. a capillary-breaking layer).

Results

The results of the calculations are shown in Table 4. The top part of the table

lists the contribution of each individual material. As an example consider the

application of material M1 as a 10 cm internal wall: After correction for doors, the

reference house contains internal walls corresponding to an exposed area of 192 m2Internal wall example

and a mass of 5800 kg. In the laboratory measurements (see Table 2) it was found

that M1 has an area-specic exhalation rate (JA) of 0.239 Bq h1 m2, and a mass-

specic exhalation rate (JM) of 19.8 mBq h1 kg1. In the low-diusivity limit, the

internal wall will increase the indoor radon concentration by 0.30 Bq m3. This

result is given in the second last column of the table. In the high-diusivity limit,

the house concentration will increase by 0.74 Bq m3. This is given in the last

column of the table. The true contribution of the wall to the radon concentration

in the house will be somewhere between 0.30 and 0.74 Bq m3.

The bottom part of Table 4 gives an example of a full (reference) house con-Full house example

structed with a set of the investigated materials. The calculation shows that the

selected building materials could increase the indoor radon by up to 4.3 Bq m3.

Observe, that the ordinary concrete in the slab accounts for most of the radon.

If there is no radon entry from the soil, and if the outdoor air has a radon con-

centration of 8 Bq m3 (as is typical for Denmark), the house will reach a radon

concentration of about 12 Bq m3 or less.

The results given above apply to the reference house with the air-exchange rateAir-exchange rate

set to 0.5 h1. Although this value is universally used in such calculations (see

e.g. [DS94]), actual measurements [Be94] show that newer Danish houses have an

average air-exchange rate of 0.35 h1. With this air-exchange rate in the reference

house, all concentration estimates in Table 4 increase by 43 %. Hence for the

full-house example, the contribution of the building materials to the indoor radon

concentration would be 6.1 Bq m3 or less (and not 4.3 Bq m3 or less). Observe,

that this value is somewhat less than the value of 11 Bq m3 deduced from the

measurement of radon levels in multi-family houses given on page 2.

26 Ris-R-1135(EN)

ID

Description

Application

Projected

Exposed

Thickness

Density

Mass

G

JA

JM

c(fromJA)

c(fromJM)

area,m2

area,m2

mm

kgm

3

103

kg

Bqh

1m

2

mBqh

1kg

1

Bqm

3

Bqm

3

M1

LAC,density600

Backwall

97

97

100

600

5.8

1

0.239

19.8

0.15

0.75

Internalwall

96

192

100

600

5.8

1

0.239

19.8

0.30

0.74

M2

LACtype1,density1500

Backwall

97

97

100

1500

14.6

1

0.584

19.4

0.37

1.83

Internalwall

96

192

100

1500

14.4

1

0.584

19.4

0.73

1.81

M3

LACtype2,density1500

Backwall

97

97

100

1500

14.6

1

0.58

19.6

0.37

1.85

Internalwall

96

192

100

1500

14.4

1

0.58

19.6

0.72

1.83

M4

AAC,density450

Fullexternalwall

97

97

400

450

17.5

0.5

0.087

9.5

0.05

0.54

M5

AAC,density650

Backwall

97

97

100

650

6.3

1

0.103

8.0

0.06

0.33

Internalwall

96

192

100

650

6.2

1

0.103

8.0

0.13

0.32

M6

AAC,density735

Backwall

97

97

100

735

7.1

1

0.286

18.9

0.18

0.87

Internalwall

96

192

100

735

7.1

1

0.286

18.9

0.36

0.87

M7

Ordinaryconcrete

Backwall

97

97

100

2300

22.3

1

0.814

19.3

0.51

2.80

Slab

123

123

100

2300

28.3

1

0.814

19.3

0.65

3.55

Internalwall

96

192

100

2300

22.1

1

0.814

19.3

1.01

2.77

M8

Gypsonboard

Ceiling

123

123

13

718

1.1

0.5

0.01

2.4

0.01

0.01

Walls

289

289

13

718

2.7

1

0.01

2.4

0.02

0.04

M9

Bricks

Frontwall

97

97

110

1768

18.9

0.5

0.02

0.8

0.01

0.05

Backwall

97

97

100

1768

17.1

1

0.02

0.8

0.01

0.09

Internalwall

96

192

100

1768

17

1

0.02

0.8

0.02

0.09

M10

Lightw.expand.clayagg.

Floorinsulation

123

123

280

291

10

1

0.001

0.1

0.00

0.01

Full-houseexample(seetext):

M7

Ordinaryconcrete

Slab

123

123

100

2300

28.3

1

0.814

19.3

0.65

3.55

M8

Gypsonboard

Ceiling

123

123

13

718

1.1

0.5

0.01

2.4

0.01

0.01

M9

Bricks

Frontwall

97

97

110

1768

18.9

0.5

0.02

0.8

0.01

0.05

M5

AAC,density650

Backwall

97

97

100

650

6.3

1

0.103

8.0

0.06

0.33

M5

AAC,density650

Internalwall

96

192

100

650

6.2

1

0.103

8.0

0.13

0.32

M10

Lightw.expand.clayagg.

Floorinsulation

123

123

280

291

10

1

0.001

0.1

0.00

0.01

Total(full-houseexample):

0.86

4.27

Table4.CalculatedimpactofspecicapplicationsofthematerialsM1toM10inthereferencehouse.Inthebottompartofthetable,afull-houseexampleis

shown.LightweightaggregateconcreteandautoclavedaeratedconcreteareabbreviatedasLACandAAC,respectively.

Ris-R-1135(EN) 27

8 Discussion

8.1 Chamber leakage and other sources of error

A critical assumption linked to the closed-chamber method is that there is no leak-

age from the chamber. Some eorts was therefore devoted to check this particular

aspect of the measurements.

Radon tests

Five radon tests were conducted to identify potential leakage. In each test, a rel-

atively large radon activity was injected into the chamber, and the decay was

followed over time. The results are shown in Figure 14. Part (A) of the gure

suggests that the decrease of the radon concentration in the chamber is well de-

scribed by an exponential decay function (i.e. the concentration vs. time curves

are linear in a semi-log coordinate system). Part (B) of the gure shows the es-

timated decay constants normalized with the (true) decay constant for radon

(=2:09838 106 s1). Values larger than 1 indicate that radon is removed from

the chamber at a rate faster than can be accounted for by radioactive decay. The

indicated uncertainties are one standard deviation of the tted slopes. Experi-

ment (3) indicates a normalized decay constant which is about 9 % larger than

unity. The other experiments either are more uncertain or give a normalized decay

constant lower than or equal to unity. The weighted mean of the ve results is:

0:9965 0:0039. This value is not signicantly dierent from unity.

Pressure tests and one additional radon test

Although the radon tests suggest that there is no signicant leakage from the

chamber, the measured absolute pressure in the chamber (see measurement sheets

in Appendix B) reveals that the pressure in the chamber does not remain con-Leaky chamber

stant when the chamber is closed! Unfortunately, this was rst realized after com-

pletion of most of the measurements. A pressure test performed on August 3,

1998 demonstrated that the chamber could not maintain any reasonable room-to-

chamber pressure dierence for periods much longer than about 10 minutes. The

cause of the trouble was found to be a non-standard cable plug mounted on the

chamber lid installed in order maintain communication between the continuous

radon monitor and a computer. The exhalation rate measurements identied as

number 103 to 120 had been conducted with the leaky plug! The nal measure-

ment (no. 120) was for material M7. That measurement ended July 28, 1998. The

plug was removed on July 29, 1998, and the chamber was pressurized to about

1500 hPa (i.e. to about 500 hPa above the atmospheric pressure). Over a period

of 24 h, it was not possible to measure any change in the pressure in the chamber.

Sample M7 was then measured one nal time. This is measurement no. 121. The

dierence between the results for measurement 120 and 121 is less than 1 %. This

is insignicant when measurement errors are considered.

Correction for leakage

In each individual exhalation-rate measurement, it is (in principle) possible to

test if the chamber is subject to leakage: If the eective decay constant is much

dierent from the xed value assumed in the analysis, it will not be possible to

make the theoretical curve t the experimental data, and the reduced-2 (see

equation 23) will reach a value which is statistically dierent from 1.0. Such a

28 Ris-R-1135(EN)

Figure 14. Results of ve tests of the tightness of the chamber. Slopes steeper than

that indicated in part (A) by the dashed line indicate that radon is removed from

the chamber faster than can be explained by radioactive decay. Normalized decay

constants are shown in part (B).

Figure 15. Plot of the sum of all values of the reduced-2 obtained in the exper-

iments number 103 to 120 for a range of (xed) eective decay constants e=.

test, however, requires that the errors of the radon-concentration measurements

are known accurately, which is not the case in this investigation. Here, the results

are simply used to nd the average leakage during the closed-chamber experiment,

such that the results can be corrected for the problem by use of an eective

decay constant as described page 7 and 12. Figure 15 shows (grand) sums of the

reduced-2 values for the 15 exhalation rate results obtained with the leaky plug

Ris-R-1135(EN) 29

(measurement numbers 103 to 120). Each sum is calculated for a xed value of the

eective decay constant e . The value that best describes the (average) leakage

present during the experiments is where the curve has its minimum. The value

amounts to e = 1:037 which is therefore used in the calculation of Jf (see

equation 24, page 12).

Constant J assumption

Another important assumption behind the closed-chamber method is that the

exhalation rate remains constant during the build-up period (see page 7). This

problem was addressed by numerical modelling (see page 19). For conditions sim-

ilar to those of the present experimental set up, it was found that the exhalation

rate can be considered to be constant (within about 1.5 %) for materials with diu-

sivities D in the range from 1010 m2 s1to 106 m2 s1. This range of diusivity

values probably covers all materials of interest in this context.

Bound-to-free exhalation ratio

The numerical model was also used to assess the decrease of the exhalation rate

as the radon concentration increases in the chamber during the build-up period

(see page 19). Calculation of so-called bound-to-free exhalation rate ratios showed

that the bound exhalation rate could not be biased (for this reason) by more

than about 2 %. A small correction factor equal to 1.013 was introduced in the

calculation of the nal result to correct for this problem. See equation 24, page 12.

Comparison with open-chamber method

The best test of bias of a particular measurement procedure is probably to com-

pare its results with results obtained by other means. In this investigation, results

of the closed-chamber method can be compared with results of the open-chamber

method. The comparison is shown graphically in Figure 5, page 15. It can be seen

from the gure that the open-chamber method is subject to considerable uncer-

tainty, but that there seems to be no signicant dierence between the results ob-

tained with the two methods. The seven measurements of sample M7 are of special

interest. The mean of the open-chamber results is 2.96 atoms s1 kg1, whereas

the mean for the closed-chamber method is about 10 % lower: 2.71 atoms s1 kg1.

A t-test of paired samples shows that the dierence among results obtained with

the two methods is insignicant (p = 46 %). This suggests that the closed-chamber

method is not strongly biased for example, as a result of chamber leakage or other

sources of errors.

In uence of the age of concrete and moisture

It is known from previous investigations [Ul84, St88, Di91], that the exhalation

rate of concrete may change over time. For example, Roeloft and Scholten [Roe94]

found the exhalation rate of their concrete samples to vary by as much as a

factor of 1.5 during the rst 6 to 12 months after pouring of the samples. 1 year

after pouring, the variability was much less. During the following 6 to 8 years,

the exhalation rate decreased monotonously to 0.30.6 of the maximum value. In

addition, van Dijk and de Jong [Di91] have shown that gypsum as well as concrete

samples conditioned to dierent moisture contents have dierent exhalation rates.

The reason for these observations could be that the emanation rate change with

moisture. Changes in diusivity (for example, as concrete degrade with time) could

also be part of the explanation [Rog94]. Furthermore, it has been speculated if

30 Ris-R-1135(EN)

vapor transport or change in adsorption characteristics could play a role.

For the above reasons, a 24 h conditioning period at room-like values of tem-

perature and humidity was adopted as part of the measurement protocol. The

in uence of this particular protocol (or deviations from this5) on the nal mea-

surement results was not investigated directly. Only the problem of ageing of

concrete was studied: Sample M7 was measured 7 times over a period of 250 days

(using the same protocol each time). The results in Figure 4 on page 15 show that

the exhalation rate decreases by 20 % over the 250 days. The decreases correlates

with the decrease in mass. This probably means that the exhalation rate decreases

as the sample dries out.

Extrapolation of laboratory results to full walls

As demonstrated by the example page 23, the extrapolation of laboratory exhala-

tion rate measurements to full walls is subject to considerable uncertaintywhen

the diusivity of the material is unknown. The true result, however, is bound to

be within the so-called low- and high-diusivity limits. To make conservative es-

timates, it is best to use the high-diusivity limit, where the exhalation rate is

estimated as the mass of the wall multiplied by the mass-specic radon exhalation

rate of the material as measured in the laboratory (see equation 34).

Selected methodology

The main measurements reported here are based on the closed-chamber method

identied as Method B on page 7. This method was selected because previous

exhalation rate measurements in Denmark had been carried out with this method

(see the following section). It also played a role that the instruments needed for the

method were readily available at Ris. This report demonstrates that the sources

of errors related to the aging of concrete and to the extrapolation of results from

laboratory measurements to full-scale houses are generally much larger than the

uncertainty associated with the individual laboratory measurements. From this

perspective there is probably little need to develop more accurate measurement

procedures than the closed-chamber method presented in this work. From the per-

spective of harmonizing (standard) methods for radon exhalation rate measure-

ments it may, however, be of interest to adopt an open-chamber method equivalent

to Method A1 described page 6. This would be in better line with the ongoing

standardization work in the Netherlands and Denmark (see page 7).

8.2 Comparison with previous measurements

A number of investigations of radon exhalation rates and radioactivity of Danish

building materials were carried out by Jonassen, Ulbak and co-workers in the

1970'ies and 1980'ies. The key references are:

Ulbak, 1980 [Ul80] In 1980, a survey of radioactivity in Danish building ma-

terials was carried out by the Danish Institute for Radiation Hygiene [Ul80].

The survey included gamma measurements of potassium-40, thorium-232 and

radium-226, but none of radon exhalation rates. As radium-226 is the source

of radon, those results can, however, be used to identify candidates for build-

ing materials with the highest radon exhalation rates. The emphasis of the

5Observe that with the adopted protocol there is no control of humidity in the chamber duringthe build-up period: if the sample is not in moisture equilibrium with the chamber air when thechamber is closed, then the humidity will change in time. This on the other hand can be seen asa feature of the method: Such deviations from equilibrium are easy to detect with this method.

Ris-R-1135(EN) 31

survey was on bricks and concrete aggregates such as sand, stones and gran-

ite. 257 samples were investigated. Few measurements were conducted on real

concrete samples. The main results for radium-226 are reproduced in Table 5.

Samples believed to be representative for Denmark are marked in the table.

The highest radium values were found for aerated alum-shale concrete (of

Swedish origin), y ash, tiles (from various countries), concrete aggregates of

granite, and bricks from mo-clay (Danish: moler) from Mors. Comparison of

Table 2 and 5 shows that there is little dierence between the results of the

present investigation and the 1980 survey. For example, the radium concen-

tration of brick sample (material M9) in the present investigation is virtually

identical to the national average value found in the 1980 survey.

Jonassen and McLaughlin (1976,1980) [Jo76, Jo80] reported radon exha-

lation rates for 14 building material samples. The results for those of the

samples believed to have a Danish origin are reproduced in Table 6. Jonassen

and McLaughlin used a closed-chamber method for the investigation. The

main dierence between their procedure and the present work probably is

that they used larger chambers (120 L and 200 L) and larger samples (typi-

cally 50100 kg slabs). The linear dimensions of samples are unclear, but the

volume-to-surface ratios are given: samples of concrete had volume-to-surface

ratios of about 30 m1 which is smaller than the values of about 53 m1 in

this investigation (see Table 1, page 9). This probably means that the samples

used by Jonassen and McLaughlin were a good deal thicker than the 5 cm

used here. This can be of importance for comparison of the results as shown

in Figure 10, page 21. The fraction of the chamber volume taken up by the

sample ranged from 10 to 55 % with a typical value around 40 %. The cham-

ber used in this investigation is in all cases lled less than 10 %. It is unclear

how the samples were conditioned in the chamber.The values reported by Jonassen and McLaughlin and those found in this in-

vestigation are surprisingly similar. For example, the values for mass-specic

exhalation (JM) for ordinary concrete was found by Jonassen and McLaugh-

lin to be about 2.02.4 atoms s1 kg1 (sample J1 and J2) compared to

2.6 atoms s1 kg1 found here for sample M7. The mass-specic exhalation

rate for bricks (solid type) of 0.08 atoms s1 kg1 (J11) is identical to the

value of 0.10 atoms s1 kg1 found here for sample M9. It is probably rea-

sonable to compare the lightweight concrete samples J7 (density 750 kgm3)

and J8 (density 780 kgm3) with sample M1 (density 600 kgm3) of the

present investigation. It is seen from the tables, that the area-specic exha-

lation rates are almost identical (about 0.2 Bq h1 m2 in both cases). The

mass-specic exhalation rates, do however, deviate by a factor of 1.9: J7 has

a mass-specic exhalation rate of 1.4 atoms s1 kg1, while the value of M1 is

2.6 atoms s1 kg1. The reason the area-specic exhalation rates agree while

the mass-specic values do no, could be that radon exhale only from a rel-

atively thin surface layer of the sample (corresponding to the low-diusivity

limit discussed page 23). The larger surface-to-volume ratios of the samples

used by Jonassen and McLaughlin will produce lower mass-specic exhalation

rates compared with this investigation. This is supported by the observation

that M1 has a fraction of exhalation of only 8 %.

Ulbak, Jonassen and Bkmark (1984) [Ul84] investigated radon exhalation

rates for cylindrical concrete samples (15 cm in diameter and 30 cm in height).

This is geometry E in Figure 10. Among other things, it was found:

that radon exhalation rate measurements conducted during the rst year

after production were very variable. In the present investigation, no such

variability was observed. It was, however, observed that the exhalation

32 Ris-R-1135(EN)

Table 5. Selected radium-226 results (ARa) from the 1980-survey of radioactivity

in Danish building materials reported by Ulbak [Ul80]. The results are sorted in

order of decreasing average radium concentration.

Group Number of ARa (Bq kg1)

samples min average max

Alum-shale aerated concrete 2 670

Fly asha 10 110 150 210

Tiles from various counties 13 22 66 108

Bricksa 79 23 42 86

Leca 3 36 40 43

Concrete aggregatesa (e.g. sand, stones, granite) 107 < 4 19 95

Concrete 6 13 16 24

Aerated concrete 3 9 15 25

Natural gypsuma 12 < 4 8 13a Representative value for Denmark.

Table 6. Results of Danish exhalation rate measurements as reported by Jonassen

and McLaughlin [Jo76, Jo80].

ID Description Exhalation rate

JA JM

Bq h1 m2 mBq h1 kg1 atoms s1 kg1

J1 Ordinary concrete, gravel and sand from 1.3 18 2.38

the sea, Danish deposits, density 2300 kgm3

J2 Ordinary concrete, gravel and sand from 1.0 15 1.98

pits, Danish deposits, density 2200 kgm3

J7 Lightweight concrete, Danish origin, 0.24 11 1.42

clay based, density 750 kgm3

J8 Lightweight concrete, Danish origin, 0.16 11 1.42

clay based, density 780 kgm3

J9 Expanded clay concrete, Leca, 0.16 9 1.19

density 650 kgm3

J10 Bricks, solid type, 0.017 0.6 0.08

density 1900 kgm3

J11 Bricks, cavity type, 0.007 0.6 0.08

density 1900 kgm3

J14 Gypsum board, 0.005 0.8 0.11

density 980 kgm3

rate of sample M7 changed over time. For a 250 day period, the exhala-

tion rate decreased 20 %.

that the use of y ash as a substitute for cement in ordinary concrete

had no signicant impact on the radon exhalation rates (when compared

with ordinary concretes).

and that ordinary Danish concrete has a mass-specic exhalation rate

of 17 mBq h1 kg1 (i.e. 2.25 atoms s1 kg1). This is only slightly less

than the value of about 20 mBq h1 kg1 found in this investigation for

sample M7. This result is interesting as the ordinary concrete studied

Ris-R-1135(EN) 33

by Ulbak et al. has a radium-226 contents of about 34 Bq kg1 com-

pared with the value of only 13.8 Bq kg1 for sample M7. This means

that radon exhales more "eÆciently" from M7 compared to the concrete

studied by Ulbak et al. In other words, the fraction of exhalation h are

dierent: 6.6 % (= 2:25=34) versus about 20 % for sample M7. The dif-

ference could be related to dierences in radium content and transport

properties of the aggregates used for the two concretes. Also part of the

explanation could be measurement errors related to the geometry of the

samples. From Figure 10, it can be seen that the cylindrical geometry

used by Ulbak et al. (for any given diusivity) impedes the radon exha-

lation more than the 5 x 30 x 30 cm3 geometry used in the present study.

If the diusivity of both materials happens to be as low as 1010 m2 s1,

the slab should have an exhalation rate about 0:70=0:42 = 67 % larger

than that of the cylinder (g(sample) = 0.70 for the slab and 0.42 for the

cylinder, see Figure 10).

9 Conclusions

A procedure for measurement of radon exhalation rates from building material

samples has been established at Ris National Laboratory, and the method has

been applied to 10 samples supplied by H+H Industri A/S, lsted, Denmark. The

applied closed-chamber method is similar to the one previously used in Denmark

by Jonassen and co-workers.

With respect to the analytical aspects of the method, the following conclusions

were reached :

The applied closed-chamber method was found to be in need of minor cor-

rections relating to leakage and the problem of bound exhalation. With these

corrections, it was not possible to demonstrate any signicant dierence be-

tween the results obtained with the closed-chamber method and an open-

chamber method. Since these two methods are sensitive to dierent types of

errors this suggests that the closed-chamber method (as applied in this work)

provides results of good quality.

It was demonstrated that sources of errors related to the aging of concrete

and to the extrapolation of results from laboratory measurements to full-scale

houses are generally much larger than the assessed accuracy of any of the

laboratory measurements. From this perspective, there is probably little need

to develop more accurate measurement procedures than the closed-chamber

method presented in this work. Only from the perspective of harmonizing

(standard) methods for radon exhalation rate measurements it could be of

interest to adopt a more accurate open-chamber method than was used in

this work.

The following sample specic conclusions were reached:

The exhalation rates were in the range from about 1 to 3 atoms s1 kg1.

For a typical single-family house, the contribution of any single application of

the materials was 3.6 Bq m3 or below. A reference house build exclusively

from a set of the investigated materials was found to have an indoor concen-

tration of 4.3 Bq m3 or below (all other sources neglected). This value is at

the same level as the outdoor radon concentration.

The results reported here are essentially identical to those reported in by

Jonnassen, Ulbak and co-workers.

34 Ris-R-1135(EN)

Acknowledgements

This work was supported nancially by H+H Industri A/S, lsted, Denmark.

Emiel van der Graaf of the Nuclear Geophysics Division, Kernfysisch Versneller

Instituut, Groningen, the Netherlands is thanked for providing detailed informa-

tion about the procedures in use at the KVI and the ongoing Dutch standard-

ization work. Kaare Ulbak and colleagues at the Danish Institute for Radiation

Hygiene (SIS) are thanked for the radium-226 determinations.

References

[Al94] Aldenkamp, F.J. and Stoop, P.: Sources and transport of indoor

radonMeasurements and mechanisms. Ph.d. thesis. University of

Groningen, the Netherlands (1994).

[An92] Andersen C.E.: Entry of Soil gas and radon into houses. Ris-R-

623(EN), Ris National Laboratory (1992).

[An97a] Andersen, C.E., Bergse, N.C., Majborn, B. and Ulbak, K.: Radon

and natural ventilation in newer Danish single-family houses. In-

door Air, vol. 7, 278286 (1997).

[An97b] Andersen, C.E., Bergse, N.C., Brendstrup, J., Damkjr,

A., Gravesen, P., and Ulbak, K.: Radon-95: En undersgelse

af radonkoncentrationen i danske enfamiliehuse. Ris-R-

979(DA), Ris National Laboratory, (available electronically

at http:nnwww.risoe.dk) (1997).

[An99] Andersen, C.E., Albarracn, D., Csige, I., van der Graaf, E.R.,

Jiranek, M., Rehs, B., Svoboda, Z. and Toro, L.: ERRICCA radon

model intercomparison exercise, Ris-R-1120(EN), Ris National

Laboratory, (available electronically at http:nnwww.risoe.dk)

(1999).

[Be88] Berkvens, P., Kerkhove,E., and Vanmarcke., H.: Three-

dimensional treatment of steady-state 222Rn diusion in building

materials: Introducing a practical modied one-dimensional

approach. Health Physics vol. 55(5), 793799 (1988).

[Be94] Bergse, N.C.: Ventilationsforhold i nyere, naturligt ventilerede

enfamiliehuse. SBI-rapport 236. Statens Byggeforskningsinstitut

(1994).

[Be92] Bevington, P.R. and Robinson, D.K.: Data reduction and error

analysis for the physical sciences (second edition). McGraw-Hill,

Inc., New York (1992).

[Cl92] Clavensjo, B. and Akerblom, G.: Radonboken. Atgarder mot

radon. Byggforskningsradet. Stockholm (1992).

[Co81] Colle, R., Rubin, R.J., Knab, L.I. and Hutchinson, J.M.R.: Radon

transport through and exhalation from building materials: A Re-

view and assessment. NBS Technical Note 1139. National Bureau

of Standards, U.S. Department of Commerce (1981).

Ris-R-1135(EN) 35

[DS94] Danish Standards Association (Dansk Standard): Anvisning for

bestemmelse og vurdering af afgasning fra byggevarer (in Danish)

(Directions for the determination and evaluation of the emission

from building products) DS/INF 90 (1994).

[Di91] van Dijk, W. and de Jong, P.: Determining the 222Rn exhala-

tion rate of building materials using liquid scintillation counting.

Health Physics, vol. 61, no. 4, pp. 501509 (1991).

[Gr97] van der Graaf, E.R, Cozmuta, I., and van der Spoel, W.H: Cali-

bration of the KVI instruments to measure radon exhalation rates

from building materials under controlled conditions. Report R99,

Environmental Radioactivity Research and Consultancy Group,

Kernfysisch Versneller Instituut (KVI), Groningen, the Nether-

lands (1997).

[Jo76] Jonassen, N. and McLaughlin, J.P.: Radon in Indoor air I & II.

Research report 6 & 7. Internal report, Laboratory of Applied

Physics I, Technical University of Denmark, Lyngby (1976).

[Jo80] Jonassen, N. and McLaughlin, J.P.: Exhalation of radon-222 from

building materials and walls. In: Gesell, T.F. and Lowder, W.M.

(eds.): Natural Radiation Environment III, US department of En-

ergy, CONF-780422, 12111224 (1980).

[Jo96] de Jong, P., van Dijk, W., van Hulst, J.G.A., and van Heijnin-

gen, R.J.J.: The eect of the composition and production process

of concrete on the 222Rn exhalation rate. Environmental Interna-

tional, vol. 22, suppl. 1, pp. S287S293 (1996).

[Ma86] Majborn, B. (Unpublished results), Ris National Laboratory,

DK-4000 Roskilde, Denmark (1986).

[Ma88] Mathieu, G.G, Biscaye, P.E., Lupton, R.A., and Hammond, D.E.:

System for measurement of 222Rn at low levels in natural waters.

Health Physics, vol. 55, no. 6, 989992 (1988).

[Re95] Renken, K.J., and Rosenberg, T.: Laboratory measurements of the

transport of radon gas through concrete samples. Health Physics,

vol. 68, no. 6, pp. 800808 (1995).

[Roe94] Roelofts, L.M.M. and Scholten, L.C.: The eect of aging, humidity,

y-ash additives on the radon exhalation from concrete. Health

Physics, vol. 67, no. 3, pp. 266271 (1994).

[Rog94] Rogers V.C., Nielson, K.K., Rodger, B.H., and Snoddy, R.: Radon

diusion coeÆcients for residential concretes. Health Physics, vol.

67, no. 3, pp. 261265 (1994).

[Sa84] Samuelson, C. and Pettersson, H.: Exhalation of 222Rn from

porous materials. Radiation Protection Dosimetry, vol. 7, no. 1-4,

95100 (1984).

[SIS87b] National Institute of Radiation Hygiene (SIS) and Ris National

Laboratory: Naturlig straling i danske boliger (1987).

[SSI93] Swedish Radiation Protection Institute: Radon 1993: En rapport

over laget. SSI-rapport 93-10, Stockholm, Sweden.

[St88] Standen E.: Building materials as a source of indoor radon. In:

Nazaro, W.W. og Nero, A.V. (eds.): Radon and its decay prod-

ucts in indoor air. John Wiley & Sons, 113-130 (1988).

36 Ris-R-1135(EN)

[Ta80] Tanner, A.B.: Radon migration in the gropund: A supplementary

review. In: Gesell, T.F. and Lowder, W.M. (eds.): Natural Radi-

ation Environment III, US department of Energy, CONF-780422,

556 (1980).

[UN93] United Nations Scientic Committee on the Eects of Atomic Ra-

diation (UNSCREAR): Sources and eects of ionizing radiation.

1993 report to the General Assembly. United Nations.

[Ul80] Ulbak, K.: Radioaktive stoer i danske byggematerialer. National

Institute of Radiation Hygiene (SIS)(1980).

[Ul84] Ulbak, K., Jonassen, N. and Bkmark, K.: Radon exhalation from

samples of concrete with dierent porosities and y ash additives.

Radiation Protection Dosimetry, vol. 7, no. 14, 4548 (1984).

A Guide to measurement sheets

The following list describes the type of information listed on the measurement

sheets (see next appendix):

Method (primary) This line of text tells that the closed-chambermethod (method

B, page 11) is used as primary method.

Method (secondary) This line of text tells that the measurement sheet also

contains data based on the open-chamber method identied as Method A2

(see page 12). Results obtained with this method are relatively uncertain in

comparison with the primary method. The open-chamber results are given

mainly to help identify gross errors.

Measurement procedure This line of text describes the measurement proce-

dure.

Measurement no. / Series / Sample ID Each measurement has a unique iden-

tication number. The series and the sample are identied in free text format.

Sample descriptor Description of the sample.

Chamber / Carrier gas Sample and carrier gas identication.

Radon instrument / Cycle time Identication of the radon monitor in use

and the selected cycle time (10 m or 1 h).

Analysis program / Datale / Graphle Identication of the data analysis

program and data les.

Date of reporting The date the results were reported. If the results have been

revised, then it is the date of the last revision.

Experimenter Name of the person who carried out the experimental work.

JM;f The free mass-specic exhalation rate of the sample. This is the main result

of the measurement. It is obtained with the closed-chamber method. The

term free means that the exhalation rate given corresponds to the exhalation

rate of the sample when it is placed in an environment with air maintained

at zero radon concentration (see page 3 and 5).

UcfJM;fg The combined uncertainty (expressed as one standard deviation) of the

measurement result JM;f . All known sources of errors are included.

Ris-R-1135(EN) 37

ufJM;fg This is the same as UcfJM;fg except that the uncertainty of the radon

monitor bias is not included.

JA;f The free area-specic exhalation rate of the sample.

UcfJA;fg The combined uncertainty (expressed as one standard deviation) of the

measurement results JA;f . All known sources of errors are included.

ufJA;fg This is the same as UcfJA;fg except that the uncertainty of the radon

monitor bias is not included.

JM;f;OC An estimate of the free mass-specic exhalation rate as found with the

open-chamber method. The used open-chamber method is considered to be

much less accurate than the closed-chamber method, and the open chamber

results are given only to help identify gross errors.

ufJM;f;OCg The combined uncertainty of the measurement results JM;f;OC. All

known sources or errors (except the uncertainty of the bias of the radon

monitor) are included.

Sample dimensions This gives the overall shape and dimension of the sample.

Mass (before, after and lost) The sample is weighted before it is put in the

chamber and after it is taken out (i.e. after completion of the exhalation

rate measurement). The mass lost is the dierence between the two masses.

Typically mass is lost by the drying out of the sample or because sample

grains break o.

Volume, Area, Density This is the bulk volume of the sample, the total surface

area, and the density of the sample.

Empty chamber vol. etc. This eld gives the total volume of the chamber, the

volume of rigid things inside the chamber such as the radon monitor (but not

the sample), and the air volume in the chamber during measurements after

correction for dead space and sample.

Model equation This is the main equation used in the non-linear tting of the

radon concentration in the chamber. The full equation is given as equation 21,

page 11.

Chi-2 reduced, N This is a measure of the non-linear t (see page 11). N is the

number of individual measurements involved.

Fitted parameter c0, sfc0g This is the result of tting the equation to the data.

sfc0g is the standard deviation of the tted parameter c0 (sometimes called

the standard error).

Fitted parameter c1, sfc1g This is the result of tting the equation to the

data. sfc1g is the standard deviation of the tted parameter c1 (sometimes

called the standard error).

Fixed parameter e The eective decay constant is set to a xed value in the

tting equation. e is expressed as a factor multiplied by the decay constant

of radon-222 (2:09838 106 s1). e > means that the chamber is leaky.

Bound-to-free exhalation correction This factor attempts to convert the mea-

sured bound exhalation rate to a free exhalation rate (see page 12).

Start and Stop This gives the dates and times for the conditioning period (rst

column) and the build-up period (second column).

Period This gives the total duration in days of the conditioning period (rst

column) and the build-up period (second column).

38 Ris-R-1135(EN)

Qwet This is the setting of the mass- ow controller in the "wet ow line".

Q=Qwet+Qdry This is the total ow rate of air leaving the chamber as measured

with the ow meter at the pressure and temperature given under notes (see

Figure 2, page 9).

c sfcg This is the mean and standard deviation of the radon concentration mea-

surements in the chamber made during the conditioning period (rst column)

and the build-up period (second column).

T sfTg This is the mean and standard deviation of the temperature measure-

ments made in the chamber during the conditioning period (rst column)

and the build-up period (second column).

RH sfRHg This is the mean and standard deviation of the relative humidity

measurements made in the chamber during the conditioning period (rst col-

umn) and the build-up period (second column).

Patm sfPatmg This is the mean and standard deviation of the pressure measure-

ments made in the chamber during the conditioning period (rst column) and

the build-up period (second column).

N This is the number of measurements (of c, T , RH, or Patm) made in the chamber

during the conditioning period (rst column) and the build-up period (second

column).

Figures The gures show the radon concentration, pressure, relative humidity,

and temperature in the chamber during the conditioning and the build-up

period. The chamber is closed at time t=0.

B Measurement sheets

This appendix contains measurement sheets with detailed results for all exhalationrate determinations.

Ris-R-1135(EN) 39

222Rn exhalation rate measurementMethod (primary) Closed-chamber method w. continuous radon monitor

Method (secondary) Open-chamber method w. continuous radon monitor

Measurement procedure Conditioning and build-up (June 1998 procedure)

Measurement no. / Series / Sample ID #0103 / H+H Industri A/S / M2

Sample descriptor LAC type 1, density 1500

Chamber / Carrier gas RnChamber2 / nitrogen

Radon instrument / Cycle time AlphaGuard PQ2000 EF-231 / 10 min

Analysis program / Datale / Graphle ExhBas02.pas / ExhData4.dat / GEXH0103.dat

Date of reporting August 20, 1998

Experimenter Claus E. Andersen, Ris National Lab., Denmark

Free mass-specic exhalation rate JM;f 2.57 atoms s1 kg1

Combined uncertainty (k = 1) UcfJM;fg 0.15 atoms s1 kg1

Type A + B uncert. except Rn monitor bias ufJM;fg 0.07 atoms s1 kg1

Free area-specic exhalation rate, JA;f 0.584 Bq h1 m2

Combined uncertainty (k = 1) UcfJA;fg 0.034 Bq h1 m2

Type A + B uncert. except Rn monitor bias ufJA;fg 0.015 Bq h1 m2

Open-chamber result JM;f;OC / ufJM;f;OCg 2:71 atoms s1 kg1/ 0.83 atoms s1 kg1

Sample dimension Slab 29.8 cm x 30.0 cm x 5.4 cm

Mass 7.3170 kg

Volume (V ) / Area (A) / Density (m) 4.83 L / 0.243 m2/ 1516 kgm3

Empty chamber vol. / dead space (sample) / air 55.76 L / 1.55 L / 49.38 L

Model equation c(t) = c0 + c1(1 exp(e t))

Chi-2 reduced (2) / N 0.775 / 125 + 1478 = 1603

Fitted parameter c0 / sfc0g 4:83 Bq m3 / 0.51 Bq m3


Fixed parameter e 1:037 , where = 2:09838 106 s1

Bound-to-free exhalation correction (JM;f/JM) 1.013

Conditioning Build-up Unit Notes

Start 221097-17:20 231097-14:01

Stop 231097-14:01 021197-20:30

Period 0.86 10.27 d

Qwet 500 0 mLn min1 Mass- ow controlled

Q=Qwet+Qdry 988.9 0 mLmin1 Measured at 1005 hPa and 25 ÆC

c sfcg 3:7 1:0 203:0 2:5 Bq m3

T sfTg 24:3 0:0 23:2 0:0 ÆC

RH sfRHg 46:9 0:2 54:8 0:1 %

Patm sfPatmg 1006:3 0:1 1022:5 0:2 hPa

N 125 1478 - Number of measurements

-50

0

50

100

150

200

250

300

350

400

450

500

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Rn-222, Bq m-3

GEXH0103.dat

-50

0

50

100

150

200

250

300

350

400

450

500

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Rn-222, Bq m-3

GEXH0103.dat

950

960

970

980

990

1000

1010

1020

1030

1040

-3-2-10 1 2 3 4 5 6 7 8 910

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

60

-3-2-10 1 2 3 4 5 6 7 8 910

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

25

-3-2-10 1 2 3 4 5 6 7 8 910

Time, days

Temperature, degC

Notes: -

40 Ris-R-1135(EN)





Sample descriptor Ordinary concrete, density 2300


Radon instrument / Cycle time AlphaGuard PQ2000 EF-231 / 1 h












Mass (before) / Mass (after) / Mass (lost) 10.1264 kg / 10.1253 kg / 1:1 g




Chi-2 reduced (2) / N 0.612 / 73 + 124 = 197






Start 111197-14:34 141197-16:11

Stop 141197-16:11 191197-20:18

Period 3.07 5.17 d



c sfcg 4:6 0:5 226:3 10:2 Bq m3

T sfTg 23:2 0:0 22:1 0:1 ÆC

RH sfRHg 46:8 0:2 51:5 0:2 %

Patm sfPatmg 1003:0 0:5 1019:7 0:4 hPa


-50

0

50

100

150

200

250

300

350

400

450

-3 -2 -1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0105.dat

-50

0

50

100

150

200

250

300

350

400

450

-3 -2 -1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0105.dat

950

960

970

980

990

1000

1010

1020

1030

-3 -2 -1 0 1 2 3 4 5

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

-3 -2 -1 0 1 2 3 4 5

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

-3 -2 -1 0 1 2 3 4 5

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 41





Sample descriptor Gypsum board













Sample dimension 5 plates of 29.8 cm x 29.8 cm x 1.3 cm





Chi-2 reduced (2) / N 0.614 / 27 + 114 = 141






Start 201197-14:41 211197-17:53

Stop 211197-17:53 261197-11:49

Period 1.13 4.75 d



c sfcg 0:3 0:9 7:4 0:6 Bq m3

T sfTg 20:3 0:1 21:7 0:0 ÆC

RH sfRHg 37:2 0:8 42:0 0:0 %

Patm sfPatmg 1012:0 0:5 1018:6 0:2 hPa


-10

-5

0

5

10

15

20

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0106.dat

-10

-5

0

5

10

15

20

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0106.dat

950

960

970

980

990

1000

1010

1020

1030

-1 0 1 2 3 4

Time, days

Pressure, hPa

20

25

30

35

40

45

-1 0 1 2 3 4

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

-1 0 1 2 3 4

Time, days

Temperature, degC

Notes: -

42 Ris-R-1135(EN)





Sample descriptor Bricks













Sample dimension 5 bricks, 3:10x20x5.4cm, 2:10x14.8x5.4cm





Chi-2 reduced (2) / N 0.661 / 30 + 260 = 290






Start 171297-11:15 181297-17:32

Stop 181297-17:32 291297-14:00

Period 1.26 10.85 d



c sfcg 0:0 0:7 10:0 0:4 Bq m3

T sfTg 19:7 0:1 22:4 0:0 ÆC

RH sfRHg 43:5 0:6 41:1 0:1 %

Patm sfPatmg 1023:2 0:8 1005:2 0:6 hPa


-10

-5

0

5

10

15

20

25

30

35

40

-1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Rn-222, Bq m-3

GEXH0107.dat

-10

-5

0

5

10

15

20

25

30

35

40

-1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Rn-222, Bq m-3

GEXH0107.dat

950

960

970

980

990

1000

1010

1020

1030

1040

-1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Pressure, hPa

20

25

30

35

40

45

50

-1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

-1 0 1 2 3 4 5 6 7 8 9 10

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 43





Sample descriptor AAC, density 735


















Chi-2 reduced (2) / N 0.487 / 19 + 141 = 160






Start 050198-08:01 060198-12:43

Stop 060198-12:43 120198-09:05

Period 1.20 5.85 d



c sfcg 1:5 1:1 70:5 2:9 Bq m3

T sfTg 23:6 0:0 23:6 0:0 ÆC

RH sfRHg 40:4 0:3 40:4 0:0 %

Patm sfPatmg 991:8 1:0 1011:2 0:7 hPa


-20

0

20

40

60

80

100

120

140

0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0108.dat

-20

0

20

40

60

80

100

120

140

0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0108.dat

950

960

970

980

990

1000

1010

1020

1030

0 1 2 3 4 5

Time, days

Pressure, hPa

20

25

30

35

40

45

0 1 2 3 4 5

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

25

0 1 2 3 4 5

Time, days

Temperature, degC

Notes: -

44 Ris-R-1135(EN)


















Sample dimension Slab 30 cm x 30.1 cm x 5.1 cm





Chi-2 reduced (2) / N 0.734 / 25 + 134 = 159






Start 120198-18:00 130198-18:27

Stop 130198-18:27 190198-08:59

Period 1.02 5.61 d



c sfcg 1:0 1:2 24:1 1:2 Bq m3

T sfTg 23:5 0:0 23:4 0:0 ÆC

RH sfRHg 38:0 0:5 39:7 0:0 %

Patm sfPatmg 1011:0 0:3 1003:0 0:4 hPa


-10

0

10

20

30

40

50

60

-1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0109.dat

-10

0

10

20

30

40

50

60

-1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0109.dat

950

960

970

980

990

1000

1010

1020

-1 0 1 2 3 4 5

Time, days

Pressure, hPa

20

25

30

35

40

45

-1 0 1 2 3 4 5

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

-1 0 1 2 3 4 5

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 45





Sample descriptor LAC type 2, density 1500


















Chi-2 reduced (2) / N 0.496 / 26 + 185 = 211






Start 200198-16:40 210198-19:20

Stop 210198-19:20 290198-12:47

Period 1.11 7.73 d



c sfcg 2:6 1:0 168:5 5:6 Bq m3

T sfTg 22:5 0:0 21:9 0:0 ÆC

RH sfRHg 37:6 0:3 38:4 0:0 %

Patm sfPatmg 1030:9 1:0 1025:4 0:7 hPa


-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5 6 7

Time, days

Rn-222, Bq m-3

GEXH0110.dat

-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5 6 7

Time, days

Rn-222, Bq m-3

GEXH0110.dat

950

960

970

980

990

1000

1010

1020

1030

1040

-1 0 1 2 3 4 5 6 7

Time, days

Pressure, hPa

20

22

24

26

28

30

32

34

36

38

40

-1 0 1 2 3 4 5 6 7

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

-1 0 1 2 3 4 5 6 7

Time, days

Temperature, degC

Notes: -

46 Ris-R-1135(EN)























Chi-2 reduced (2) / N 0.505 / 19 + 151 = 170






Start 020298-14:25 030298-15:30

Stop 030298-15:30 090298-22:28

Period 1.05 6.29 d



c sfcg 0:1 1:2 22:9 1:0 Bq m3

T sfTg 21:3 0:0 20:7 0:0 ÆC

RH sfRHg 28:1 0:5 29:5 0:0 %

Patm sfPatmg 992:1 1:1 1009:7 0:4 hPa


-10

0

10

20

30

40

50

60

0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0111.dat

-10

0

10

20

30

40

50

60

0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0111.dat

950

960

970

980

990

1000

1010

1020

0 1 2 3 4 5 6

Time, days

Pressure, hPa

20

22

24

26

28

30

32

0 1 2 3 4 5 6

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

0 1 2 3 4 5 6

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 47





Sample descriptor LAC, density 600


















Chi-2 reduced (2) / N 0.609 / 30 + 146 = 176






Start 170298-20:10 190298-17:06

Stop 190298-17:06 250298-19:30

Period 1.87 6.10 d



c sfcg 1:4 1:0 60:0 2:4 Bq m3

T sfTg 22:3 0:0 22:7 0:0 ÆC

RH sfRHg 44:5 0:5 45:1 0:0 %

Patm sfPatmg 1023:4 0:6 1017:1 0:5 hPa


-20

0

20

40

60

80

100

120

140

-1 0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0112.dat

-20

0

20

40

60

80

100

120

140

-1 0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0112.dat

950

960

970

980

990

1000

1010

1020

1030

-1 0 1 2 3 4 5 6

Time, days

Pressure, hPa

20

25

30

35

40

45

50

-1 0 1 2 3 4 5 6

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

-1 0 1 2 3 4 5 6

Time, days

Temperature, degC

Notes: Initial build-up period (180298-21:15 to 190298-09:49) abandoned because of ill-closed chamber .

48 Ris-R-1135(EN)





Sample descriptor Lightw. expand. clay agg.













Sample dimension Single grains (no packing)





Chi-2 reduced (2) / N 0.638 / 28 + 116 = 144






Start 260298-10:20 270298-20:14

Stop 270298-20:14 040398-16:52

Period 1.41 4.86 d



c sfcg 0:6 0:8 0:5 0:4 Bq m3

T sfTg 22:9 0:0 22:4 0:0 ÆC

RH sfRHg 50:4 1:1 51:2 0:1 %

Patm sfPatmg 1007:9 1:4 995:3 0:6 hPa


-10

-5

0

5

10

15

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0113.dat

-10

-5

0

5

10

15

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0113.dat

950

960

970

980

990

1000

1010

1020

-1 0 1 2 3 4

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

60

65

-1 0 1 2 3 4

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

-1 0 1 2 3 4

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 49























Chi-2 reduced (2) / N 0.456 / 26 + 115 = 141






Start 050398-19:42 060398-21:16

Stop 060398-21:16 110398-16:08

Period 1.07 4.79 d



c sfcg 2:8 0:8 181:1 8:6 Bq m3

T sfTg 22:0 0:0 21:0 0:0 ÆC

RH sfRHg 44:2 0:6 43:7 0:0 %

Patm sfPatmg 1006:8 0:5 1011:4 0:9 hPa


-50

0

50

100

150

200

250

300

350

400

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0114.dat

-50

0

50

100

150

200

250

300

350

400

-1 0 1 2 3 4

Time, days

Rn-222, Bq m-3

GEXH0114.dat

950

960

970

980

990

1000

1010

1020

1030

-1 0 1 2 3 4

Time, days

Pressure, hPa

20

25

30

35

40

45

50

-1 0 1 2 3 4

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

-1 0 1 2 3 4

Time, days

Temperature, degC

Notes: -

50 Ris-R-1135(EN)























Chi-2 reduced (2) / N 0.551 / 24 + 270 = 294






Start 020498-10:00 030498-14:07

Stop 030498-14:07 140498-21:00

Period 1.17 11.29 d



c sfcg 3:8 1:0 304:1 7:9 Bq m3

T sfTg 20:8 0:0 21:9 0:1 ÆC

RH sfRHg 53:3 1:0 46:2 0:1 %

Patm sfPatmg 1007:8 0:4 998:6 0:3 hPa


-100

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 9 10 11

Time, days

Rn-222, Bq m-3

GEXH0115.dat

-100

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 9 10 11

Time, days

Rn-222, Bq m-3

GEXH0115.dat

950

960

970

980

990

1000

1010

1020

0 1 2 3 4 5 6 7 8 91011

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

60

0 1 2 3 4 5 6 7 8 91011

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

0 1 2 3 4 5 6 7 8 91011

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 51























Chi-2 reduced (2) / N 0.420 / 19 + 276 = 295






Start 160498-18:12 170498-13:12

Stop 170498-13:12 290498-09:50

Period 0.79 11.86 d



c sfcg 2:9 1:0 310:5 7:8 Bq m3

T sfTg 23:1 0:0 23:4 0:0 ÆC

RH sfRHg 43:8 0:4 46:4 0:1 %

Patm sfPatmg 995:4 0:7 1009:3 0:4 hPa


-100

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10 11

Time, days

Rn-222, Bq m-3

GEXH0116.dat

-100

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10 11

Time, days

Rn-222, Bq m-3

GEXH0116.dat

950

960

970

980

990

1000

1010

1020

1030

0 1 2 3 4 5 6 7 8 91011

Time, days

Pressure, hPa

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 91011

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

25

0 1 2 3 4 5 6 7 8 91011

Time, days

Temperature, degC

Notes: -

52 Ris-R-1135(EN)























Chi-2 reduced (2) / N 0.456 / 28 + 131 = 159






Start 020698-16:15 030698-21:40

Stop 030698-21:40 090698-09:00

Period 1.23 5.47 d



c sfcg 3:6 0:6 192:9 8:3 Bq m3

T sfTg 24:6 0:0 24:7 0:0 ÆC

RH sfRHg 49:2 0:5 48:1 0:1 %

Patm sfPatmg 1012:1 2:3 1012:8 0:4 hPa


-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0117.dat

-50

0

50

100

150

200

250

300

350

-1 0 1 2 3 4 5

Time, days

Rn-222, Bq m-3

GEXH0117.dat960

980

1000

1020

1040

1060

-1 0 1 2 3 4 5

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

-1 0 1 2 3 4 5

Time, days

Rel. humidity, %

16

18

20

22

24

26

-1 0 1 2 3 4 5

Time, days

Temperature, degC

Notes: -

Ris-R-1135(EN) 53























Chi-2 reduced (2) / N 0.453 / 25 + 165 = 190






Start 200798-17:30 210798-18:37

Stop 210798-18:37 280798-15:33

Period 1.05 6.87 d



c sfcg 4:0 1:0 217:7 8:0 Bq m3

T sfTg 23:4 0:0 23:8 0:0 ÆC

RH sfRHg 47:6 0:0 48:8 0:0 %

Patm sfPatmg 1009:7 0:4 1013:6 0:2 hPa


-50

0

50

100

150

200

250

300

350

400

-1 0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0120.dat

-50

0

50

100

150

200

250

300

350

400

-1 0 1 2 3 4 5 6

Time, days

Rn-222, Bq m-3

GEXH0120.dat

950

960

970

980

990

1000

1010

1020

-1 0 1 2 3 4 5 6

Time, days

Pressure, hPa

20

25

30

35

40

45

50

-1 0 1 2 3 4 5 6

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

25

-1 0 1 2 3 4 5 6

Time, days

Temperature, degC

Notes: -

54 Ris-R-1135(EN)























Chi-2 reduced (2) / N 0.476 / 25 + 237 = 262






Start 300798-10:55 310798-11:20

Stop 310798-11:20 100898-08:56

Period 1.02 9.90 d



c sfcg 4:6 0:9 277:1 8:0 Bq m3

T sfTg 24:1 0:0 24:2 0:0 ÆC

RH sfRHg 53:6 0:2 53:2 0:0 %

Patm sfPatmg 1001:4 0:3 1003:6 0:1 hPa


-50

0

50

100

150

200

250

300

350

400

450

500

-1 0 1 2 3 4 5 6 7 8 9

Time, days

Rn-222, Bq m-3

GEXH0121.dat

-50

0

50

100

150

200

250

300

350

400

450

500

-1 0 1 2 3 4 5 6 7 8 9

Time, days

Rn-222, Bq m-3

GEXH0121.dat

950

960

970

980

990

1000

1010

1020

1030

-1 0 1 2 3 4 5 6 7 8 9

Time, days

Pressure, hPa

20

25

30

35

40

45

50

55

60

-1 0 1 2 3 4 5 6 7 8 9

Time, days

Rel. humidity, %

15

16

17

18

19

20

21

22

23

24

25

-1 0 1 2 3 4 5 6 7 8 9

Time, days

Temperature, degC

Notes: Chamber pressure test is ok.

Ris-R-1135(EN) 55

Bibliographic Data Sheet Ris-R-1135(EN)

Title and author(s)

Radon-222 exhalation from Danish building materials: H+H Industri A/S results

Claus E. Andersen

ISBN

87-550-2594-3

87-550-2595-1 (Internet)

ISSN

0106-2840

Dept. or group

Nuclear Safety Research and Facilities Department

Date

August 1999


Pages

56

Tables

6

Illustrations

15

References

29


This report describes a closed-chamber method for laboratory measurements of the

rate at which radon-222 degasses (exhales) from small building material samples.

The chamber is 55 L in volume and the main sample geometry is a slab of dimen-

sions 5x30x30 cm3. Numerical modelling is used to assess (and partly remove) the

bias of the method relative to an ideal measurement of the free exhalation rate.

Experimental results obtained with the method are found to be in agreement with

the results of an open-chamber method (which is subject to dierent sources of

error).

Results of radon-222 exhalation rate measurements for 10 samples of Danish

building materials are reported. Samples include ordinary concrete, lightweight

aggregate concrete, autoclaved aerated concrete, bricks, and gypsum board. The

maximum mass-specic exhalation rate is about 20 mBq h1 kg1. Under con-

sideration of the specic applications of the investigated building materials, the

contribution to the indoor radon-222 concentration in a single-family reference

house is calculated. Numerical modelling is used to help extrapolate the labo-

ratory measurements on small samples to full scale walls. Application of typical

materials will increase the indoor concentration by less than 10 Bq m3.


BRICKS; CONCRETES; DEGASSING; DIFFUSION; FINITE DIFFERENCE

METHOD; GYPSUM CEMENTS; HOUSES; MEASURING METHODS; RA-

DIOECOLOGICAL CONCENTRATIONS; RADIUM 226; RADON 222; RISOE

NATIONAL LABORATORY

Available on request from:Information Service Department, Ris National Laboratory(Afdelingen for Informationsservice, Forskningscenter Ris)P.O. Box 49, DK4000 Roskilde, DenmarkPhone (+45) 46 77 46 77, ext. 4004/4005 Fax (+45) 46 77 40 13

Risø National Laboratory carries out research within science and technology,providing Danish society with new opportunities for technological development.Theresearch aims at strengthening Danish industry and reducing the adverse impactonthe environment of the industrial, energy and agricultural sectors.

Risø advises government bodies on nuclear affairs.

This research is part of a range of Danish and international researchprogrammes andsimilar collaborative ventures. The main emphasis is on basic research andparticipation in strategic collaborative research ventures and market driventasks.

Research is carried out within the following programme areas:

• Industrial materials• New functional materials• Optics and sensor systems• Plant production and circulation of matter• Systems analysis• Wind energy and atmospheric processes• Nuclear safety

Universities, research institutes, institutes of technology and businesses areimportant research partners to Risø.

A strong emphasis is placed on the education of young researchers through Ph.D.and post-doctoral programmes.

ISBN 87-550-2594-3ISBN 87-550-2595-1 (Internet)ISSN 0106-2840

Copies of this publicationare available from

Risø National LaboratoryInformation Service DepartmentP.O. Box 49DK-4000 RoskildeDenmarkTelephone +45 4677 [email protected] +45 4677 4013Website www.risoe.dk

Risø-R-1201(EN)

Radon Transport Modelling:User’s Guide to RnMod3d

Claus E. Andersen


Risø-R-1201(EN)

Radon Transport Modelling:User’s Guide to RnMod3d

Claus E. Andersen


Abstract RnMod3d is a numerical computer model of soil-gas and radon trans-port in porous media. It can be used, for example, to study radon entry fromsoil into houses in response to indoor-outdoor pressure differences or changes inatmospheric pressure. It can also be used for flux calculations of radon from thesoil surface or to model radon exhalation from building materials such as concrete.The finite-volume model is a technical research tool, and it cannot be used

meaningfully without good understanding of the involved physical equations. Someunderstanding of numerical mathematics and the programming language Pascalis also required. Originally, the code was developed for internal use at Risø only.With this guide, however, it should be possible for others to use the model.Three-dimensional steady-state or transient problems with Darcy flow of soil gas

and combined generation, radioactive decay, diffusion and advection of radon canbe solved. Moisture is included in the model, and partitioning of radon betweenair, water and soil grains (adsorption) is taken into account. Most parameters canchange in time and space, and transport parameters (diffusivity and permeability)may be anisotropic.This guide includes benchmark tests based on simple problems with known

solutions. RnMod3d has also been part of an international model intercomparisonexercise based on more complicated problems without known solutions. All testsshow that RnMod3d gives results of good quality.

Copyright The copyright to the model code called RnMod3d (described in thisguide) belongs to Risø National Laboratory, Denmark.

Disclaimer Although great care has been taken in preparing RnMod3d, and al-though many of the features implemented in the model have been tested by com-parison with exact solutions, it cannot in any way be guaranteed that the programis free of errors. The model is provided ”as is” without warranty of any kind. Inno event shall Risø be liable for any damages whatsoever arising out of the use of,inability to use, or malfunctioning of RnMod3d.

Version This guide concerns RnMod3d, version 0.8 (Sep. 15, 1997 – July 18, 2000).

Claus E. AndersenRisø National LaboratoryNuclear Safety Research DepartmentBuilding NUK-125DK-4000 Roskilde, DenmarkPhone: +45− 4677 4677 (main lab.)Phone: +45− 4677 4912 (direct)Fax: +45− 4677 4959E-mail: [email protected]: www.risoe.dk/nuk

Printed August 21, 2000

ISBN 87-550-2734-2 (printed edition)

ISBN 87-550-2733-4 (internet edition)

ISSN 0106-2840

Information Service Department · Risø · 2000

Contents

1 Introduction 11.1 What problems can be solved? 11.2 What problems can not be solved? 11.3 How to get a copy of RnMod3d 11.4 Structure of this guide 21.5 How to use RnMod3d 21.6 Making a job file 2

2 Installation 52.1 Pascal compiler (Delphi) 52.2 Pascal compiler (Borland Pascal 7) 62.3 Source files 62.4 Test case: F0000prg.dpr 6

3 Method 73.1 Basic definitions 73.2 Radon transport equation 83.3 Soil-gas transport equation 93.4 RnMod3d treatment of radon and soil gas 113.5 Finite-volume method 12

4 Control variables 144.1 runid 154.2 runtitle 154.3 solution 154.4 geometry 154.5 Ly 15

4.6 grid def 154.7 force new grid in every run 154.8 boundary conditions def 164.9 flux def 164.10 probe def 164.11 materials def 164.12 e def 164.13 beta def 174.14 G def 174.15 lambda def 174.16 D def 174.17 initialfield def 174.18 import initialfield 174.19 import finalfield guess 174.20 export field 184.21 use fieldbuffer 184.22 flowfield 184.23 flowfactor 184.24 import field name 194.25 export field name 194.26 flowfield name 194.27 plotfiles def 19

Risø-R-1201(EN) iii

4.28 user procedure each iteration def 194.29 wr details 204.30 wr main procedure id 204.31 wr all procedure id 204.32 wr iteration line log 204.33 wr iteration line screen 204.34 wr residual during calc log 204.35 wr residual during calc screen 214.36 wr flux during calc log 214.37 wr flux during calc screen 214.38 wr probes during calc log 214.39 wr probes during calc screen 224.40 wr final results log 224.41 wr final results screen 224.42 wr axes 224.43 wr nodes 224.44 wr nodes numbers 224.45 wr node sizes 234.46 wr coefficients 234.47 wr material volumes 234.48 warning priority log 244.49 warning priority screen 244.50 solver def 254.51 scheme 254.52 relax factor 254.53 flux convset 254.54 probe convset 254.55 conv evaluation period 264.56 min iterations 264.57 max iterations 264.58 max time 264.59 max change 264.60 max residual sum 274.61 dtim 274.62 BC running 274.63 BC running update of cBCs def 274.64 BC running min iterations 274.65 BC running max residual sum before new BC 274.66 BC running convergence def 274.67 wr BC running messages log 284.68 wr BC running messages screen 284.69 press enter wanted 28

5 Geometry 285.1 Grid size (memory issues) 295.2 geometry 305.3 set FixVal 305.4 wFixVal 315.5 Node spacing 315.6 set axis single 315.7 set axis double 335.8 set axis triple 335.9 Location and size of specific control volumes 345.10 Grid inspection: wr axes 36

iv Risø-R-1201(EN)

5.11 Grid evaluation: dcdx and dcdxnorm 37

6 Nodes and connectors 386.1 Node types 396.2 Connector types 396.3 Default nodes and connectors 396.4 Inspection of nodes and connectors 406.5 set node 416.6 change node 416.7 boundary conditions def 416.8 in cube 436.9 in plane 446.10 in region 446.11 in interval 45

7 Materials 457.1 materials def (mat1, mat2 etc.) 467.2 Porosity, e def 467.3 Partition-corrected porosity, beta def 487.4 Generation rate, G def 487.5 Decay constant, lambda def 497.6 Diffusivity, D def 497.7 Moisture 49

8 Flux probes (Flx1, Flx2 etc.) 518.1 Fluxes between individual pairs of control volumes 518.2 update flxval 528.3 FlxVal 548.4 Standard flux probe output 55

9 Field probes (Obs1, Obs2 etc.) 559.1 ObsVal 559.2 Standard field probe output 569.3 fieldvalue 569.4 fieldvalue2D 579.5 get fieldvalue 579.6 get fieldvalue2D 579.7 get avgfield 579.8 get avgfield2D 58

10 Solution procedure 5810.1 First guess 5810.2 Relaxation 5810.3 Iterative solution procedures 5910.4 Criteria for convergence and residuals 5910.5 Scheme (space) 6010.6 Scheme (time) 61

11 Time dependency 6111.1 solution := steady 6111.2 solution := unsteady 6111.3 Initial conditions 6211.4 Time-dependent boundary conditions 6311.5 Time-dependent material properties 64

Risø-R-1201(EN) v

11.6 Time-dependent flow field of soil gas 6411.7 Full time dependency (cBUF1, cBUF2 and qBUF) 65

12 Special boundary conditions 6712.1 Trial-and-error by hand 6812.2 BC running 68

13 Output and debugging 7313.1 Standard files 7313.2 Other file output 7313.3 Contour plots: update plotfile 7413.4 Stream lines 7513.5 Warnings 7613.6 Error messages 7713.7 Critical evaluation of results 77

14 RnMod3d inside 7814.1 Index coordinates: i, j, and k 7814.2 The main data structure: GP 7814.3 Other variables 7914.4 datatype 8014.5 Memory 8014.6 Enumerated types 8014.7 Sequence of actions in run model 81

15 Benchmark tests 8315.1 F0100prg: Steady flow of soil gas 8315.2 F0101prg: Steady diffusion of radon 8415.3 F0102prg: Diffusion and advection of radon 8415.4 F0103prg: Time-dependent flow of soil gas 87

16 House simulation example 90

A F0100prg.dpr 94

B Output: F0100LOG.dat 96

C F0101prg.dpr 98

D F0102prg.dpr 100

E F0103prg.dpr 105

F F0130prg.dpr 108

References 114

vi Risø-R-1201(EN)

1 Introduction

RnMod3d is a computer model of radon transport in porous media. It can beenused for:

• flux calculations of radon from the soil surface into the atmosphere

• simulations of entry of soil gas and radon into houses in response to indoor-outdoor pressure differences or changes in atmospheric pressure

• calculation of radon exhalation from building materials

• error analysis of measurement procedures related to radon

The model is a technical research tool, and it cannot be used meaningfully withoutgood understanding of the involved physical equations. Some understanding of nu-merical mathematics and the programming language Pascal is also required. Orig-inally, the model was developed for internal use at Risø only. With this guide, how-ever, it should be possible for others to use the model. The design has emphasizedflexibility, robustness, programming transparency, and the ability to documentand verify computations. Features such as speed, use of memory, and portabilityhave been given a lower priority. The model can be run on personal computers PC modelwith an Intel processor DX486 or above.

1.1 What problems can be solved?

Three-dimensional steady-state or transient problems with Darcy flow of soil gasand combined generation, radioactive decay, diffusion and advection of radon canbe solved. Moisture is included in the model, and partitioning of radon betweenair, water and soil grains (adsorption) is taken into account. Most parameters canchange in time and space, and transport parameters (diffusivity and permeability)may be anisotropic.The model can treat problems where both the soil-gas and the radon problem are

time-dependent. For example, the model can calculate time-dependent combineddiffusive and advective entry into a house when the flow of soil gas is created bychanges in the atmospheric pressure.

1.2 What problems can not be solved?

Clearly it is difficult to list all the things RnMod3d can not do. Here are, how-ever, the most important ones: The model cannot treat non-Darcy flow of soil gasor soil-gas flow in non-isothermal soil. In transient soil-gas simulations there aretwo restrictions: the air-filled porosity must be constant in time, and the pressurevariations must be small (compared with the absolute pressure). The numericalprocedures implemented in RnMod3d are relatively simple and the model is notparticularly fast. Although the model in principle can treat time-dependent prob-lems in full 3D, the computational time required to solve such problems can betoo large to be of practical use. Finally, it is mentioned that RnMod3d is basedon orthogonal grids. Hence, it is not possible to perform accurate calculations forcomplex geometries.

1.3 How to get a copy of RnMod3d

RnMod3d can be obtained from the author of this report.

Risø-R-1201(EN) 1

1.4 Structure of this guide

The remaining part of this section tries to give an overview of what it takes to setup a problem. Section 2 tells how the model can be ”installed” on a PC, and howthe test case can be run. Section 3 presents the equations solved by RnMod3d. Thenumerical method is also described. After these ”introductory” sections, Section 4then describes all the so-called control variables used in RnMod3d. This is thereference section of the guide. Then from Section 5 and onwards specific issues aretreated one by one. First it is shown how a grid is set up, then the idea of nodesand connectors are introduced etc. The final part of the guide gives examples ofcomputations performed with RnMod3d.

1.5 How to use RnMod3d

To do calculations with RnMod3d it is necessary for the user to write, compile,and run a Pascal program. The program is here called a job file. An example isJob fileshown in the appendix starting page 94. The job file contains a link to the RnMod3dcode plus all information about the problem in question: the computational grid,boundary and initial conditions, soil parameters, what output to calculate etc. Toset up a job, the user needs to make proper assignments to what is here calledcontrol variables. Many of the control variables are Pascal pointers to user-definedControl variablesfunctions or procedures. This design makes the model highly flexible.The most difficult part of setting up a job is probably to define the geometry

of the problem. However, when the geometrical model has first been established(and verified) it is an easy task to change parameters, and to get the output ofinterest. The ”geometrical model” can be saved and used (directly or in modifiedform) in other computations.

1.6 Making a job file

The structure of a job file is very simple: First, values are assigned to control vari-ables. Then, RnMod3d is run by calling the predefined procedure run model. There-run modelafter, it is possible to redefine one or more of the control variables, and additionalsimulations can be done with run model. Each time run model is called, RnMod3dperforms a simulation corresponding to the settings of the control variables. Theprimary result of a simulation is pressures and flows of soil gas and/or concen-trations and fluxes of radon. After the simulations, the procedure close model isclose modelcalled (once) to close output files etc.The structure of a prototype job file is shown in example 1. Only a few dec-

larations and control-variable assignments are shown. Most code lines have beenleft out as indicated by the dots. The example includes two runs. In the first run,RnMod3d solves the problem on the basis of the coarse grid specified in the proce-dure my coarse grid. In the second run, a finer grid is used. Which grid is usedis controlled by the pointer: grid def. The difference in results from run 1 to 2will show how sensitive the solution is to the selected grid.

Example 1 Prototype structure of a job file.

program F001prg;

$I R3dirs03

uses R3Defi03,R3Main03,R3Writ03; (* Links to RnMod3d *)

procedure my_coarse_grid;

begin

set_FixVal(xFix1,0.0); (* xFix1 = 0.0 m *)

set_FixVal(xFix2,3.3); (* xFix2 = 3.3 m *)

set_axis_single(xFix1,xFix2,5,FocusA,1.0) (* Allocate 5 nodes between xFix1 and xFix2 *)

2 Risø-R-1201(EN)

...

end;

procedure my_fine_grid;

begin

set_FixVal(xFix1,0.0);


set_axis_single(xFix1,xFix2,33,FocusA,1.0) (* Allocate 33 nodes between xFix1 and xFix2 *)

...

end;

...

begin (* main *)

runid:=’001’;

...

grid_def:=my_coarse_grid;

run_model; (* first run *)

grid_def:=my_fine_grid;

run_model; (* second run *)

close_model;

end.

grid def is just one single control variable (out of more than 60). The differencefrom run to run could have related to almost any other aspect of the computation:boundary conditions, material properties, requirement for convergence, numericalscheme, relaxation etc. RnMod3d is therefore particularly well suited for sensitivityanalyses. As another example, imagine that entry into a house has to be calcu- Sensitivity analyseslated for a range of 10 permeabilities and 10 indoor-outdoor pressure differencescontained in two arrays: perm and press defined by the user. Such a sensitivityanalysis could be programmed as follows:

Example 2 Prototype sensitivity analysis.

program F002prg;

$I R3dirs03

uses R3Defi03,R3Main03,R3Writ03;

var ii,jj:1..10;

perm,pres:array[1..10] of real;

...

begin (* main *)

runid:=’002’;

...

perm[1]:=1e-14; (* m2 *)

perm[2]:=2e-14;

perm[3]:=1e-13;

...

pres[1]:=-10; (* Pa *)

pres[2]:=-8;

pres[3]:=-3;

...

for ii:=1 to 10 do

for jj:=1 to 10 do

begin

... (* set permeability to perm[ii] and pressure to pres[jj] *)

run_model;

end;

close_model;

end.

The steps needed to set up a job file are described in the following.

Risø-R-1201(EN) 3

Run identification

Before anything else, assign the job an identification tag with the runid controlvariable. If the job file is saved under the name: F0997prg.dpr then it would beconvenient to set runid := ’0997’ because then standard output from RnMod3druns will go to files such as F0997LOG.dat and F0997FLW.dat. This makes it easyto find out what files belong to what jobs.

Geometry

Then the geometry of the problem should be considered. All dimensions (for ex-ample, of building components) of importance for the problem should be identifiedand formally set up as so-called fix points called xFix1, xFix2 etc. A link mustFix pointsthen be established between the physical (x, y, z) world in meters and a three-dimensional grid of control volumes with index coordinates (i,j,k). Initially (i.e.Control-volume gridbefore the model set-up has been fully verified), it is best to use only a very coarsegrid. Fortunately, the control-volume approach guarantees that even results ob-tained with coarse grids are physically meaningful (for example, the solution willnot become unstable and the radon concentrations will not become negative forthis reason). In the end, the grid must, however, have a sufficiently high resolution,otherwise the results will be too inaccurate. The use of fix points means that thesepoints do not move as grids with more control volumes are used.

Nodes and connectors

Then it must be defined how each control volume should ”work”. Most controlvolumes will be under the control of the transport equations for radon or soil gas,but some others may be fixed at certain external values (boundary conditions) ormay not be part of the computations at all. In RnMod3d, each control volume hasa property called node type which reflects these aspects. Likewise, some (adjacent)control volumes will be connected and some others will be disconnected. Theseaspects are specified in control-volume properties called connectors. Each controlvolume has six connectors (one for each neighbor). The user can set all nodes andconnectors in accordance with the problem in question.

Materials

The next step is to define material properties. For example, in a simulation ofradon transport, it is of course necessary to specify values for porosity and diffu-sivity etc. The assignment of material properties can be based on physical (x, y, z)coordinates (for example, it can be specified that the radon generation rate shouldchange with soil depth in some user-defined way or that the top-soil moisture con-tent should change in time because it rains). It is also possible to divide thecomputational plane into blocks of materials (called mat1, mat2 etc.) and to setthe material properties to be block-wise constant.

Output

RnMod3d solves the specified equations and returns field values at all nodes in the(i,j,k) computational field. This type of output is, however, seldom the endFlux ”probes” etc.result from the user’s point of view. Often the prime output will be field valuesor fluxes at a few selected locations (given in physical (x, y, z) coordinates). Toget this type of output without troubles, RnMod3d is equipped with special ”fieldmeasurement probes” (called Obs1, Obs2 etc.) and ”flux measurement probes”

4 Risø-R-1201(EN)

(called Flx1, Flx2 etc.). In radon simulations, this framework provides the userwith the ability to monitor radon concentrations and fluxes of radon. In soil-gassimulations the probe values correspond to pressures and soil-gas flow rates. The”field measurement probes” are normally placed at given points defined by physical(x, y, z)-coordinates. In contrast, ”flux measurement probes” are normally definedin relation to plane surfaces defined by reference to fix points.

2 Installation

To use RnMod3d, a Pascal compiler must be installed and the five source files (listedin Section 2.3) must be copied to a directory ”visible” for the compiler togetherwith the test job file: F0000prg.dpr.

2.1 Pascal compiler (Delphi)

It is best to run RnMod3d from the editor/compiler environment of Delphi (onlyDelphi 3 has been tested, but other versions are probably all right). Observe, Delphithat RnMod3d is a pure console application. No use is made of the Windows user-interface in Delphi. Here is what to do to make an old-fashioned hello-world pro-gram. A job file for RnMod3d can be made in the same way.

1. Open Delphi.

2. Create a new application using File | New Application.

3. Go to the Project Manager (View | Project Manager).

4. Remove the default form from the project (highlight the unit and hit delete).Do not save changes.

5. Go to Project Source (View | Project Source).6. Edit the project source file:

• Remove code inside begin end.

• Replace the Forms unit in the uses section with SysUtils.

• Remove $R *.RES.• Place $apptype console in a line by itself right after the programstatement.

7. Add whatever statements needed in the body of the program. The programcan look like this:

program test;

$apptype console

begin

writeln(’Hi there’);

readln;

end.

8. Compile the program with (Project | Compile).

9. Run the program with Run | Run.10. When the program is saved, it is best to use the default extension for Delphi

projects: .dpr. Any units that are created should be saved with extension:.pas.

Risø-R-1201(EN) 5

2.2 Pascal compiler (Borland Pascal 7)

RnMod3d can also be run from Borland Pascal 7. The only change is that thefollowing two lines must be removed from the code file R3Dirs03.pas:

$DEFINE Delphi$apptype console

There are three reasons why it is best to run RnMod3d from Delphi:

• RnMod3d runs much slower under Borland Pascal v. 7 compared with Delphi(a factor of 2 or such).

• The memory model is better in Delphi.

• The max time control variable does not work under Borland Pascal (see Sec-tion 4.58)

2.3 Source files

RnMod3d consists of more than 5500 code lines. The code is placed in the followingfive files:

R3Dirs03.pas Compiler directives

R3Defi03.pas Global declarations

R3Main03.pas The main program

R3Writ03.pas Additional procedures (mainly output routines)

R3Delp03.pas Special code for Delphi and Borland Pascal 7

These files should be placed in the working directory or better in a separate codeCode directorydirectory. In the latter case, Delphi needs to know about this directory. This isdone by adding the path (e.g. d:\data\pascal\rnmod3d\code) in the Libary Path(Environmantal Options | Library). It is advisable to make the code files read only.

2.4 Test case: F0000prg.dpr

F0000prg.dpr is a test job file where everything is defined by default. The (hid-den) problem that is solved is a simple heat-conduction problem of no inter-est here. It just serves as a simple test. When the model is run, the followingtwo (output) files are created: f0000LOG.dat and f0000RES.dat. The first is alog file with all sorts of output. The second is a general purpose result file. Inthe default case, no output goes to the RES file. If everything works, runningF0000prg.dpr should give: Flx1 : J = -1.0000000E-0006 (this is a flux) andObs1 : c = 1.5000000E+0000 (this is a concentration).

Example 3 Test case: F0000prg.dpr

program F0000prg;

$I R3dirs03


begin

default_problem;

run_model;

close_model;

end.

6 Risø-R-1201(EN)

3 Method

The purpose of this section, is to present the basic transport equations solved byRnMod3d. The framework is consistent with that used in [An92] and [An99c]. Anoutline of the finite-volume approach is also given.

3.1 Basic definitions

Consider a reference element δV of soil. This volume may be split into three parts:δVg for the volume of grains, δVw for the volume of water, and δVa for the volumeof air:

δV = δVg + δVw + δVa (1)

Hence the (total) porosity ε, the water porosity εw, and the air porosity εa can beexpressed as:

ε =δVw + δVa

δV(2)

εw =δVw

δV(3)

εa =δVa

δV(4)

We define the fraction of water saturation of the pore volume (i.e. the volumetricwater content) as:

θv =δVw

δVa + δVw=

εwε

(5)

Hence θv = 1 means that the pores are completely filled with water whereas θv = 0means that the soil is dry. The total mass of the reference element is:

δM = δMg + δMw (6)

where δMg is the mass of grain material and δMw is the mass of water. The massof air is neglected. The density of the grain material is:

ρg =δMg

δVg(7)

For a wide range of soils ρg is in the (narrow) range from 2.65 to 2.75 · 103 kgm−3.The density for water:

ρw =δMw

δVw(8)

is about 1.0 · 103 kgm−3. The wet-soil density for given porosity and water contentcan be calculated as:

ρws =δM

δV= (1− ε)ρg + θvρw (9)

The dry-soil density is:

ρds =δMg

δV= (1− ε)ρg (10)

We define the amount of water per dry mass of soil (i.e. the gravimetric watercontent) as:

θg =δMw

δMg=

ρwδVw

ρgδVg=

εw1− ε

ρw

ρds=

ε

1− ε

ρw

ρdsθv (11)

Hence if the porosity of the soil is ε = 0.3, then full water saturation (θv = 100 %)means that the amount of water per dry mass is normally about θg = 16 %.

Risø-R-1201(EN) 7

3.2 Radon transport equation

The total activity δA of radon-222 (simply referred to as ”radon” in all of thefollowing) in the reference element δV may be split into three parts:

δA = δAg + δAw + δAa (12)

where the indices have the same meaning as in equation 1. We now define theconcentration of radon in the air-filled parts of the pores as:

ca =δAa

δVa(13)

and the radon concentration in the water-filled parts of the pores as:

cw =δAw

δVw(14)

Part of the grain activity δAg is available for transport in the pore system. This isthe radon adsorbed to soil-grain surfaces: δAg,s. The immobile part (δAg − δAg,s)is radon produced by the ”non-emanating” part of the grain radium. In line withthe framework presented by Rogers and Nielson [Rog91A], we introduce the sorbedradon concentration per kg dry mass (Bq kg−1) as:

cs =δAg,s

δMg(15)

where δMg is the grain mass within δV .We assume rapid sorption kinetics [Wo92] such that the partitioning of radon

between air, water and soil grains is permanently in equilibrium at any point ofthe soil:

cw = Lca (16)

cs = Kca (17)

where L is the Ostwald partitioning coefficient given in Table 1, and K is theradon surface sorption coefficient [Rog91A, Na92]. The equilibrium assumptionsimplify the problem considerably as we can then express the total mobile radonactivity by reference to the concentration in just one phase. Normally, the radonconcentration in the air phase ca is selected as ”reference concentration”. Thisapproach is also taken in RnMod3d. The mobile activity in δV is hence given as:

δAg,s + δAw + δAa = βcaδV (18)

whereβ = εa + Lεw +Kρds (19)

is sometimes called the partition-corrected porosity. If the medium is dry andwithout grain sorption, we have: β = ε. The equilibrium assumption is widely usedin models of pollutant transport, but is not universally correct [Th97]. Supportfor the assumption can be found in [Na88, Na92]If radium is present only in soil grains, we define the radon generation rate per

pore volume (Bq s−1 per m3-pore) as:

G =λρdsE

ε= λE

1− ε

ερg (20)

where λ is the decay constant of radon (2.09838 · 10−6 s−1), andE is the emanationrate of radon to the soil pores (i.e. the number of atoms that emanates intowater and air per second per kg dry mass). We can write the emanation rateas E = fARa, where f is the fraction of emanation and ARa is the activityconcentration (Bq kg−1) of radium-226 per dry mass.

8 Risø-R-1201(EN)

Table 1. Radon solubility L in water as function of temperature (from [Cl79], p.228).

Temperature L

K -273.15 0.5249278.15 0.4286283.15 0.3565288.15 0.3016293.15 0.2593298.15 0.2263303.15 0.2003308.15 0.1797

A mass-conservation equation for the mobile radon activity in δV is:

∂βca∂t

= εG− λβca −∇ ·j (21)

where j is the bulk flux density (in units of Bq s−1 per m2) at time t. The term’bulk’ means that the density is measured per total cross-sectional area perpen-dicular to j. Hence, a flux J (Bq s−1) across some plane with geometric area A

(e.g. a 120 m2 crawl-space floor) and uniform bulk flux density j gives: J = j ·Aa,where a is a unit vector perpendicular to the plane.The bulk flux density is divided into two:

j = ja +jd (22)

Ignoring water movement, the advective flux density is given by:

ja = caq (23)

where q is the bulk flux density of soil gas (in units of m3 s−1 per m2) discusedlater. We assume, that the diffusive flux can be written as:

jd = −D∇ca (24)

such that the bulk diffusivity D accounts for radon diffusion through air and waterin the pores.D is a function of temperature and pressure [Wa94] and may therefore(if not for other reasons) change in time and space. We assume, that the soil-gasflow is so low that mechanical dispersion can be ignored (i.e. D is independent ofq) [Do92].

3.3 Soil-gas transport equation

It is assumed that the flow is of the Darcy type, that the soil has a uniformtemperature (natural convection in the soil is ignored), and that εa is constant intime. Also, it is assumed that pressure variations are small in comparison withthe absolute pressure. The equation can be derived as given next.The equation of continuity for soil gas transport is [Bi60]:

∂εaρa

∂t= −∇ · (ρaq) (25)

where ρa is the density of the gas (in kgm−3). For an ideal gas under isothermalconditions, ρa is proportional to the absolute pressure P (x, y, z, t) (in Pa). Hence,we have:

∂εaP

∂t= −∇ · (P q) (26)

Risø-R-1201(EN) 9

We can split the absolute pressure into three parts:

P (x, y, z, t) = P0 − ρa,0 g z + p(x, y, z, t) (27)

where P0 is the mean pressure at the atmospheric surface, and where p is thedisturbance pressure field. The middle term consists of: the average air density atthe given temperature ρa,0 (about 1.3 kgm−3), the acceleration due to gravity g

(about 9.8 m s−2), and the depth −z below the atmospheric surface (located atz = 0). The z-axis points upwards. The ”aerostatic” pressure:

PH(z) = P0 − ρa,0 g z (28)

increases about 13 Pa per m depth.The left-hand side of equation 26 can be evaluated as follows:

∂εaP

∂t=

∂εa(PH(z) + p(x, y, z, t))∂t

(29)

= PH(z)∂εa∂t

+∂εap

∂t(30)

We limit the treatment to the situation when εa is constant in time, and wetherefore have:

∂εaP

∂t=

∂εap

∂t(31)

On the right-hand side of equation 26, we assume that the disturbance pressureis small in comparison with PH(z) such that:

P q = (PH(z) + p) q (32)

≈ P0 q (33)

From this, we can approximate equation 26 as:

∂εap

∂t= −∇ · (P0q) (34)

orεaP0

∂p

∂t= −∇ · q (35)

where q is given by Darcy’s law:

q = −k

µ∇p (36)

In the special case of homogeneous soil, we can reduce equation 35 and 36 to:

∂p

∂t= Dp∇2p (37)

which is a usual diffusion equation, where

Dp =kP0

µεa(38)

is the diffusivity. Observe, that without the important simplification in equa-tion 33, we would had obtained a transport equation with the term: ∇2p2. In-stead, only ∇2p is part of the final equation1. Hence, equation 33 has lead to alinearization of the problem.

1Observe, that in one dimension,

∂

∂x

(p

∂p

∂x

)=

1

2

∂2

∂x2p2 (39)

10 Risø-R-1201(EN)

RnMod3d solves the following equation for radon transport:

∂βca∂t

= εG− λβca −∇ ·j (40)

wherej = caq −D∇ca (41)

is the bulk flux density of radon (in Bq s−1 per m2) and where

ca is the radon concentration in the air-filled parts of the pores (Bq m−3)

t is the time (s)

β = εa + Lεw +Kρds is the partition-corrected porosity (dimensionless)

ε is the porosity (dimensionless)

G is the radon generation rate per pore volume (Bq s−1 per m3)

λ is the decay constant for radon (2.09838 · 10−6 s−1 for radon-222)

D is the bulk diffusivity (m2 s−1)

q is a known bulk flux density of soil gas (m3 s−1 per m2)

Box 1: Radon transport equations.

RnMod3d solves the linearized equation for soil-gas transport:

εaP0

∂p

∂t= −∇ · q (42)

where:q = −k

µ∇p (43)

is the bulk flux density of soil gas (in m3 s−1 per m2), and where

p is the disturbance pressure (Pa)

t is the time (s)

εa is the air porosity (dimensionless)

P0 is the mean absolute pressure (about 105 Pa)

k is the gas permeability (m2)

µ is the dynamic viscosity (about 17.5 · 10−6 Pa s at 10 C)

Box 2: Soil-gas transport equations.

3.4 RnMod3d treatment of radon and soil gas

RnMod3d is programmed to solve equations 40 and 41 in Box 1 and the formalismused to define problems in RnMod3d closely follows that used in these equations.For example, the control variable relating to the radon-decay constant (λ) is calledlambda def in RnMod3d. Hence, equation 40 and 41 relate to RnMod3d in a straight-forward manner. For the soil-gas problem, the situation is a bit more complicated.First, we observe, that equations 42 and 43 in Box 2 are also of the form given inequations 40 and 41. We just have to substitute ca with p, D with k

µ and β withεa/P0. The rest of the ”radon-equation coefficients” must be set to zero: λ = 0,G = 0, ε = 0 and q = 0. If we do that, RnMod3d solves the soil-gas problem asdefined by equations 42 and 43.Table 2 should ease the translation of quantities used in RnMod3d and the two Translation table

Risø-R-1201(EN) 11

Table 2. Quantities etc. used by RnMod3d in radon and soil-gas problems.

RnMod3d Radon problem Soil-gas problemEquation 40 + 41 Equation 42 + 43

Basic field value ca [Bq m−3] p [Pa]

D def D [m2 s−1] kµ [m2 Pa−1 s−1]

e def ε [-] 0

beta def β [-] εaP0

[Pa−1]

G def G [Bq s−1 m−3] 0

lambda def λ [s−1] 0

flowfield import export

J∫Ωj · da [Bq s−1]

∫Ωq · da [m3 s−1]

Q∫Ωq · da [m3 s−1] 0

sets of transport equations. The first line of the table concerns the ”field values”used by RnMod3d. For radon problems, these field values represent the radon con-centration in the air-filled pore parts (ca). For soil-gas problems, they correspondto the disturbance pressure (p). Therefore these quantities should be used whenspecifying fixed-value boundary conditions. Furthermore, it should be observedthat model output of field values are based on these quantities.The next lines concern control variables D def to Lambda def. These controlDual meaning of material

properties variables are Pascal pointers to user-defined functions as described in Section 7.Here we just state that their meaning relates directly to the coefficients of equa-tions 40 and 41. Hence, for radon problems, D def should point to the user-definedfunction where the bulk diffusivity D of the material is defined and e def shouldpoint to the user-defined function of (total) porosity. For soil-gas problems, thesituation is different as already stated: D def should point to the function wherethe gas permeability divided by the dynamic viscosity is defined, e def shouldpoint to a function which always return zero etc.The next line of the table concerns the soil-gas flow field q. It links the soil-Flow field

gas problem and the radon problem. It must be observed, that in the soil-gasequation, q results from the calculation. Its relation to the pressure field is givenin equation 43. In advective radon problems, q is a known flow field of soil gas.RnMod3d has a control variable called flowfield. In a soil-gas simulation, wemay set this to export meaning that the calculated flow field q should be output(exported) to a file. Later, in a radon simulation, we may want to import thissoil-gas flow field. We can do that by setting flowfield to import. Other settingsare also possible.The two final lines of the table concern RnMod3d ”probes” for flux measurements.

More details can be found in Section 8. Here we just mention, that in radonOutput ”probes”problems J and Q give the flux of radon and soil gas, respectively. In problemswith pure diffusion, the soil-gas flow will be zero. In soil-gas problems, J is thecalculated flow of soil gas whereas Q is without meaning (the model returns Q =0). Flux measurements are done over some surface Ω as indicated in the table.

3.5 Finite-volume method

RnMod3d is based on a finite-volume (also called control-volume) method. Thismethod is closely related to the finite-difference method. Information about these

12 Risø-R-1201(EN)

S

N

W E

B

T

P

x

y

z

Figure 1. Sketch of the control volume located around node P . The six adjacentnodes are called W for west, E for east, S for south, N for north, B for bottom,and T for top.

S

N

W EP

x

y

Figure 2. Two-dimensional projection of a grid of control volumes. Observe, thatcontrol volumes need not have the same size, and that nodes are always locatedmidway between control-volume interfaces.

Risø-R-1201(EN) 13

numerical techniques can be found in [Pa80, Ve95, He96, Fe99]. The finite-volumeapproach has been used also in other models of radon transport [Lo87, An92,Sp98].The computational grid is divided into control volumes as sketched in Figure 1

and 2. Each control volume is a box with one (center) node and six faces. Theprime variable is the value of the field at the nodes. Soil-gas problems are based onthe disturbance pressure field p(x, y, z) whereas radon problems are based on theradon concentration field in the air-filled parts of the pores ca(x, y, z). Transportfrom one control volume to another is approximated by linear flux expressions.These expressions involve field values at pairs of adjacent nodes (e.g. P and E inFigure 1). Fluxes are calculated for each of the six control-volume faces. Consid-ering sources and sinks and that soil-gas and radon may accumulate in the controlvolume, we then require strict conservation of mass. This gives one algebraic equa-tion for each control volume P :

aP cP = aEcE + aW cW + aNcN + aScS + aT cT + aBcB + b (44)

where the a’s and the b are coefficients, the c’s are the unknown field values, andwhere indices E, W , N , S, T , and B refer to the adjacent control volumes on theeast, west, north, south, top, and bottom sides of P . The coefficients are calculatedfrom material properties and control volume sizes.We do the same thing for all control volumes in the grid and therefore obtain

a traditional matrix equation with N equations and N unknown field values. Nis typically 10 000 or more, so it is virtually impossible to solve the equation byordinary matrix inversion (the main matrix would be of size: N by N). RnMod3dtherefore uses iterative methods for finding the solution.In summary, RnMod3d is based on field values at control-volume nodes and fluxes

at interfaces between pairs of adjacent nodes. Section 5.9 contains a few moredetails about these matters.

4 Control variables

RnMod3d is controlled by more than 60 control variables. Some control variablescan be assigned simple types of data: strings, floating-point numbers, integers,or booleans (i.e. true or false). Other variables are enumerated types of data.Enumerated typesThese are used to help clarify the meaning of variable assignments and to restrictassignments to what is actually meaningful. For example, the control variablegeometry is declared as an enumerated type, and it can only be assigned thevalues: cartesian3D, cartesian2D or cylindrical2D. Otherwise an error willoccur.Some of the control variables are pointers to pre-defined or user-defined proce-

dures and functions. These variables have names ending with def. For example,Pointer variables end withdef grid def is a pointer to the user-defined procedure where the grid is defined. If

the user has defined a procedure called mygrid, then we can tell RnMod3d about itwith the assignment: grid def := mygrid. Sometimes it makes sense to set suchvariables to nil. For example, during debugging it may be of interest to avoidcalling the solver. This can be done with solver def:=nil.In the following, all global control variables in RnMod3d are presented one by one.

The order of presentation follows the likely order of assignment during creation ofa job file.

14 Risø-R-1201(EN)

Default values

When RnMod3d starts, all control variables are set to their default values. If moreruns are conducted within the same job file, it is sometimes desirable to reset allcontrol variables to these values. This can be done with the procedure:

set_control_variables_to_defaults;

4.1 runid

Type: String with four characters Default: ’0000’ Description: Each modelcalculation is assigned an identification tag called runid. If we set runid :=’0997’ and run the model, then standard output goes to the files f0997LOG.datand f0997RES.dat. It is a good idea to name that job file f0997PRG.dpr, be-cause then all files relating to the run can be found as f0997*.*. Additionalinformation: See Section 13.1

4.2 runtitle

Type: String Default: ’Default control variables’Description: This vari-able is used to assign a descriptive line of text to the model calculation. For ex-ample, runtitle := ’My first job’. This line of text is output to the screenand to the LOG-file.

4.3 solution

Type: Enumerated variable with two possible assignments: steady or unsteadyDefault: steady Description: The assignment: solution := steady impliesthat the model calculation corresponds to steady-state conditions. The assign-ment: solution := unsteady is used for time-dependent problems. Additionalinformation: See Section 11.

4.4 geometry

Type: Enumerated variable with three possible assignments: cartesian3d, cartesian2d,or cylindrical2d Default: cartesian3d Description: Selection of coordinatesystem. Additional information: See Section 5.

4.5 Ly

Type: Floating-point number larger than 0 Default: 1.0 Description: Ly givesthe thickness of the grid (in m) in the y-direction. The value of Ly is only ofimportance when geometry := cartesian2d.

4.6 grid def

Type: Pointer Default: nil Description: This is a pointer to the user-definedprocedure with the grid. Additional information: See Section 5.

4.7 force new grid in every run

Type: BooleanDefault: falseDescription: The assignment force new grid in every run:= true forces a re-calculation of the grid pointed to by grid def every timerun model is issued. If the variable is false, the grid will only be recalculated

Risø-R-1201(EN) 15

if the procedure pointed to by grid def changes from one run to another. Forexample, it will do so in:...

grid_def:=mygrid;

run_model; (* Run 1 *)

grid_def:=myothergrid;

run_model; (* Run 2 *)

...

regardless of the setting of force new grid in every run. In the following exam-ple, it is, however, important that force new grid in every run is set to true:...

procedure mygrid;

begin



set_axis_single(xFix1,xFix2,NN,FocusA,1.0);

...

end;

...

grid_def:=mygrid;

for NN:=10 to 100 do

run_model;

...

otherwise all model runs will be performed for the grid size corresponding toNN:=10.

4.8 boundary conditions def

Type: Pointer Default: nil Description: This is a pointer to the user-definedprocedure where the boundary conditions are defined. Other information aboutnodes and connectors located within the grid boundary can also be specified here.Additional information: See Section 6.

4.9 flux def

Type: Pointer Default: nil Description: This is a pointer to the user-definedprocedure where the ”flux measurement probes” (Flx1, Flx2 etc.) are defined.Additional information: See Section 8.

4.10 probe def

Type: Pointer Default: nil Description: This is a pointer to the user-definedprocedure where pressure or radon-concentration probes (Obs1, Obs3 etc.) aredefined. Additional information: See Section 9.

4.11 materials def

Type: Pointer Default: nil Description: This is a pointer to the user-definedprocedure where the materials mat1, mat2 etc. are defined. Additional informa-tion: See Section 7.1.

4.12 e def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction where the (total) porosity (ε) is defined. Additional information: SeeSection 7.2.

16 Risø-R-1201(EN)

4.13 beta def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction where the partition-corrected porosity (β) is defined. Additional infor-mation: See Section 7.3.

4.14 G def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction where the radon generation rate per pore volume (G) is defined. Addi-tional information: See Section 7.4.

4.15 lambda def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction where the decay constant of radon (λ) is defined. Additional informa-tion: See Section 7.5.

4.16 D def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction where the bulk diffusivity (D) is defined. Additional information: SeeSection 7.6.

4.17 initialfield def

Type: Pointer Default: nil Description: This is a pointer to a user-definedfunction. In time-dependent problems, it is possible to specify that the initial fieldshould be given by some function (e.g. equal to a non-zero constant). The variableinitialfield def can be used to tell RnMod3d about such a function. A nil-assignment: initialfield def := nil is required if the initial field is specified byother means (see import initialfield).Warning: Only nodes of the type free(see Section 6.1 page 39) will be initialized by initialfield def. For example,nodes that are fixed to certain values (such as boundary conditions) are unaffectedby the initial field read through initialfield def.

4.18 import initialfield

Type: Boolean Default: false Description: In time-dependent problems, it ispossible to specify that the initial field should be read from a file with: import initialfield:= true. The name of the file is given by import file name. Observe, import initialfieldshould be set to false after the first time step has been taken.

4.19 import finalfield guess

Type: Boolean Default: false Description: The solver in RnMod3d solvesthe field equations iteratively until some requirements of convergence are met.If import finalfield guess := false, then the first ”guess” is the field alreadyin the main data structure GP. In the very first run in a job file, the field in GPis always 0 everywhere. If import finalfield guess := true, then the solverimports a field from the file given by import file name and uses that as an ini-tial guess of the final solution. It is important to distinguish between the initialfield for a time-dependent problem and the initial guess for the iterative solutionprocedure.

Risø-R-1201(EN) 17

4.20 export field

Type: Boolean Default: false Description: If export field := true thenthe final field is exported to the file given by export file name. If the file al-ready exists, then it will be overwritten without warning. It is particularly usefulto export a ”field” if the computations have not converged. The field can thenbe imported with import finalfield guess := true and used as a good start-ing guess for more iterations. If export field := false, then no such field isexported.

4.21 use fieldbuffer

Type: Enumerated variable with three possible assignments: cBUF1, cBUF2 orno cBUF Default: no cBUF Description: If use fieldbuffer has been set tocBUF1 then the state of RnMod3d in the next run model call will be reset to thatstored in the buffer called cBUF1. Likewise, the results of the new computationswill be stored in the same buffer. Buffers are needed in problems when both thesoil gas and the radon problems are time-dependent. If use fieldbuffer hasbeen set to cBUF2 then the state of RnMod3d is encapsulated in the buffer calledcBUF2. If use fieldbuffer has been set to no cBUF, then no such buffer is used.Additional information: Section 11.7.

4.22 flowfield

Type: Enumerated variable with three possible assignments: none, export, import,export to qBUF, or import from qBUF Default: none Description: The termflowfield refers to the flow of soil gas (q in Box 1 and 2, page 11). In a cal-culation of soil-gas transport, the assignment flowfield := export, forces theflow field of soil gas between control volumes to be exported to the file given byflowfield name. In a later calculation with advective radon transport, it is possi-ble to import this flow field with flowfield := import. The flow field is read fromthe file given by flow field name. In a calculation of radon transport with purediffusion, we set flowfield := none. With flowfield set to export to qBUF,the flow field is stored in the flow field buffer called qBUF. This is a dynamic vari-able created for the purpose only when needed. It can be used only within a singlejob file. With flowfield set to import from qBUF, a flow field already stored inqBUF can be restored. An example of the use of qBUF can be found in Section 11.7.Warning: Observe, that it is meaningful to use flow fields in other calculationsonly if these calculations are performed with grids identical to that used in theoriginal flow calculation. For example, if the flow of soil gas is calculated with avery fine grid of 20 000 nodes, then this grid cannot be used in a later radon calcu-lation based on a coarser grid with only 5 000 nodes. RnMod3d tests if the numberof nodes are identical in the two situations. If they are not, an error message willappear. The model does, however, not test if the two grids are truly identical.

4.23 flowfactor

Type: Floating-point number Default: 1.0 Description: During import of aflow field of soil gas, all flows between control volumes are multiplied by theflowfactor. For example, imagine a problem with diffusive and advective radonentry into a house depressurized 1 Pa relative to the outdoors. If the soil-gas flowfield has been calculated (in a previous model run) with a steady depressurizationof 1 Pa, then we use flowfactor := 1.0 in the radon calculation (no scaling).However, if we want to know the radon entry in the situation of a 5 Pa depressur-

18 Risø-R-1201(EN)

ization, we set flowfactor := 5.0. Thereby all flows between control volumesare multiplied by a factor of 5. If the radon entry during house pressurization isneeded, we set flowfactor to be negative (all flows are reversed). The flow fieldcan be scaled meaningfully in this fashion because of the linearity of the transportequation for the soil-gas flow. Observe, that the procedure is not directly applica-ble if the house has entry points with different depressurizations etc. flowfactoraffects both flow fields imported from files and from the buffer called qBUF, seeSection 4.22.

4.24 import field name

Type: String with a valid file name or ’’ Default: ’’ Description: This variableis used to specify the name of a file from which a field may be imported (seeinitialfield def and import finalfield guess). If import field name = ’’then the import file takes the standard name fxxxx 00.dat where xxxx is givenby the runid. For example, if runid := ’0997’ then import will be done from thefile f0997 00.dat. If the field should be imported from a file called myfile.datthen use the assignment: import field name := ’myfile.dat’.

4.25 export field name

Type: String with a valid file name or ’’ Default: ’’ Description: This vari-able is used to specify the name of a file to which the final field should be ex-ported (see export field). If export field name := ’’ then the export filetakes the standard name fxxxx 00.dat where xxxx is given by the runid (seeimport field name).

4.26 flowfield name

Type: String with a valid file name or ’’ Default: ’’ Description: This variableis used to specify the name of a file to which or from which the flow field of soilgas should be imported or exported (see flowfield). If flowfield name =: ’’then the flowfield file takes the standard name fxxxxflw.dat where xxxx is givenby the runid (see import field name).

4.27 plotfiles def

Type: Pointer Default: nil Description: This is a pointer to the user-definedfunction that makes data files for plotting purposes. Additional information:See Section 13.3.

4.28 user procedure each iteration def

Type: Pointer Default: nil Description: This is a pointer to a user-definedprocedure that is called once every iteration by the solver. This procedure is calledfrom within the find field-procedure. See Section 14.7. Normally, this variable isset to nil. The procedure can be used to monitor the convergence of the iterativesolution procedure:

procedure monitor_convergence;beginwriteln(’Iteration = ’,iter,’c = ’,GP[5]^[2]^[5].c);end;

where

Risø-R-1201(EN) 19

user_procedure_each_iter_def := monitor_convergence;

In principle it is also possible to let the procedure change the boundary condi-tions. For example, the cBC[fixed1] can be changed. The methods described inSection 12 are, however, more suited for that purpose.

4.29 wr details

Type: BooleanDefault: falseDescription: wr details := true forces RnMod3dto output detailed information about the progress of the computations. The out-put goes to the screen during run time. This can be used to debug problematicjob files.

4.30 wr main procedure id

Type: Boolean Default: false Description: wr main procedure id := trueforces RnMod3d to output identification headers every time the main proceduresare called. This can be used to debug problematic job files.

4.31 wr all procedure id

Type: Boolean Default: false Description: wr all procedure id := trueforces RnMod3d to output identification headers whenever procedures are called.This can be used to debug problematic job files.

4.32 wr iteration line log

Type: Boolean Default: false Description: wr iteration line log := truemakes RnMod3d output information to the LOG-file about the number of iterationconducted:

Iteration = 501 (1000) Time = 13.02 min (60.00) Residual = 3.40E+0002

The meaning is a follows: The line was written after completion of iteration number501. The number in parentheses shows that a maximum of 1000 iterations isallowed. 13.02 minutes have passed since the run model was issued. The maximumtime allowed for the computations is 60.00 minutes. The sum of residuals amountsto 3.4 · 102. The concept of residuals is described page 60. The ”iteration line” isoutput whenever the iteration number divided by conv evaluation period givesan integer.

4.33 wr iteration line screen

Type: Boolean Default: true Description: This variable has the same meaningas wr all procedure id, except that the line of information goes to the screen.

4.34 wr residual during calc log

Type: Boolean Default: false Description: wr residual during calc log:= true forces RnMod3d to write information in the LOG-file about residuals. Theoutput comes during the computations, and it is useful for monitoring if the so-lution converges. The output comes whenever the iteration number divided byconv evaluation period gives an integer. An example is shown here:

* Abs. sum of bs = 0.00000E+0000

* Abs. sum of residuals = 2.63340E+0002 (change = -5.31864E-0003)

20 Risø-R-1201(EN)

* Max residual = 1.02542E+0001 (change = 7.43469E-0003)

* Max residual at (i,j,k) = ( 2, 1, 3)

* Max residual at (x,y,z) = ( 7.500E-0001, 0.000E+0000, 2.250E+0000)

The quantities involved are defined in Section 10.4. The output includes the sumof absolute values of residuals, the max residual, and the location of the maxresidual (both as control-volume coordinates (i,j,k) and physical coordinates(x, y, z)). The ”relative change per iteration” is also given. In the example, thesum of absolute residuals decreases about 0.53 % per iteration. Observe: To ob-tain information about the number of iterations reached, wr iteration line logshould be set to true.

4.35 wr residual during calc screen

Type: Boolean Default: false Description: This variable has the same mean-ing as wr residual during calc log, except that the output goes to the screen.

4.36 wr flux during calc log

Type: Boolean Default: false Description: If this variable is set to true,RnMod3d will output results of ”flux measurements” with the probes Flx1, Flx2etc. The output comes during the computations and can therefore be used tomonitor the convergence of the run. The output does not come for every singleiteration, unless conv evaluation period := 1. An example of output is shownhere:

Flx1 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx2 : J = -3.3263329E+0000 ( change = -1.3561772E-0002 ) Q = 1.2035000E-0007

Flx3 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx4 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx5 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

If this output concerns a radon problem, then the meaning of the numbers is asfollows: Flux probe no. 2 (Flx2) reports a flux (J) of about -3.326 Bq s−1. Therelative change per iteration is about -1.35 %. The flow of soil gas (Q) is about1.2 · 10−7 m3 s−1. Observe: To obtain information about the number of iterationsreached, wr iteration line log should be set to true.

4.37 wr flux during calc screen

Type: Boolean Default: false Description: This variable has the same mean-ing as wr flux during calc log, except that the output goes to the screen.

4.38 wr probes during calc log

Type: Boolean Default: false Description: If this variable is set to true,RnMod3d will output results of ”concentration measurements” with the probesObs1, Obs2 etc. The output comes during the computations and can therefore beused to monitor the convergence of the run. The output does not come for everysingle iteration, unless conv evaluation period := 1. An example of output isshown here:

Obs1 : c = 2.7050633E-0001 ( change = 1.2874671E-0001 )

Obs2 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs3 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs4 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs5 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Risø-R-1201(EN) 21

If this output concerns a radon problem, then the meaning of the numbers isas follows: Concentration probe no. 1 (Obs1) reports a field value (c) of about0.27 Bq m−3. The relative change per iteration is about -12.9 %. Observe: To ob-tain information about the number of iterations reached, wr iteration line logshould be set to true.

4.39 wr probes during calc screen

Type: Boolean Default: false Description: This variable has the same mean-ing as wr probes during calc log, except that the output goes to the screen.

4.40 wr final results log

Type: Boolean Default: true Description: If this variable is set to true,output will be written to the LOG-file after the solver has ended its computa-tions. The output includes: (1) the title of the run as specified by runtitle,(2) a statement of why the computations stopped (e.g. because the computa-tions converged), (3) a line showing the number of iterations and time used forthe computations (see wr iteration line log), (4) results about residuals (seewr residual during calc log), (5) results of ”flux measurements” (see wr fluxduring calc log), and (6) results of ”concentrationmeasurements” (see wr probesduring calc log).

4.41 wr final results screen

Type: Boolean Default: true Description: This variable has the same meaningas wr final results log, except that the output goes to the screen.

4.42 wr axes

Type: Boolean Default: true Description: If this variable is set to true, theinformation about the grid is output to the LOG-file. Additional information:See Section 5.10.

4.43 wr nodes

Type: Boolean Default: false Description: If this variable is set to true,information about each individual control volume is written to the LOG-file. Anexample is shown here:

wr_nodedata

---------------------------------------------------------------------------

i j k c nodetyp west east south north bottom top

---------------------------------------------------------------------------

1 1 1 0.000 fixed1 nill noflow nill noflow nill noflow

---------------------------------------------------------------------------

The output includes: (1) The index coordinates of the control volume (i,j,k),(2) the field value (c) in Bq m−3 if a radon problem is solved, (3) the type of node,and (4) the connectors for each of the six faces of the control volume.

4.44 wr nodes numbers

Type: BooleanDefault: trueDescription: If this variable is set to true, outputwill be written to the LOG-file about the number of each type of control volumesin the grid. For example:

22 Risø-R-1201(EN)

wr_count_nodes

* Type and number of nodes incl. boundary conditions :

* NOP 328

* free 640

* fixed1 16 value = 1.00000000000E+0000

* fixed2 16 value = 0.00000000000E+0000

* fixed3 0 value = 0.00000000000E+0000

* unchanged 0

* Total 1000

tells that there are 328 ”no operation” control volumes, 640 control volumes thatare ”free floating” (i.e. controlled by the transport equation), 16 control volumesof the type fixed1, and 16 control volumes of type fixed2. In total there are 1000control volumes in the grid. When the computations ended the ”fixed values” atfixed1 and fixed2 were 1.0 and 0.0, respectively. The item ”unchanged” has nomeaning here.

4.45 wr node sizes

Type: Boolean Default: false Description: If this variable is set to true,output will be written to the LOG-file about the sizes of each individual controlvolume in the grid. For example:

wr_cvsize

2 5 8 ArW= 3.000E+0000 ArE= 3.000E+0000

ArS= 3.000E+0000 ArN= 3.000E+0000

ArB= 2.250E+0000 ArT= 2.250E+0000

dV= 4.500E+0000

tells that the area of the west face (ArW) of control volume (i,j,k) = (2,5,8)amounts to 3 m2. The areas of the east, south, north, bottom, and top faces arealso given. The volume dV of the control volume is 4.5 m3.

4.46 wr coefficients

Type: Boolean Default: false Description: If this variable is set to true,output will be written to the LOG-file about the coefficients of each individualcontrol volume in the grid. For example:

wr_all_coefficients

1 1 1 mat1 ap= 1.000E+0000 b= 1.000E+0000

aw= 0.000E+0000 ae= 0.000E+0000

as= 0.000E+0000 an= 0.000E+0000

ab= 0.000E+0000 at= 0.000E+0000

tells that the material of control volume (i,j,k) = (1,1,1) is mat1, and thatthe ap coefficient amounts to 1.0 etc.

4.47 wr material volumes

Type: BooleanDefault: trueDescription: If this variable is set to true, outputrelating to the materials mat1, mat2 etc. (see Section 7.1) will be written to theLOG-file. This information can be particularly useful for testing if the problem hasbeen set up correctly. For example, in 3D simulations with building componentsof different materials, it is useful to test if the volumes of these components areas intended. For example,

wr_material_volumes_etc (volume-averaged field values)

mat Avg(conc) Activity Volume N N_invalid

mat1 1.197706031E+0004 9.581648252E+0002 4.000000000E-0001 36 89

Risø-R-1201(EN) 23

mat2 2.638484718E+0004 6.860060266E+0003 1.300000000E+0000 36 89

mat3 2.221443068E+0004 5.775751977E+0003 1.300000000E+0000 63 162

mat Min(conc) i j k x y z

mat1 2.999521243E+0003 2 4 2 1.852E-0002 9.815E-0001 -2.950E+0000

mat2 2.434101159E+0004 2 4 10 1.852E-0002 9.815E-0001 -1.500E+0000

mat3 2.156342193E+0004 2 4 18 1.852E-0002 9.815E-0001 -5.000E-0002

mat Max(conc) i j k x y z

mat1 2.095874291E+0004 2 2 5 1.852E-0002 3.519E-0001 -2.650E+0000

mat2 2.851119095E+0004 4 2 8 6.481E-0001 3.519E-0001 -2.300E+0000

mat3 2.300846592E+0004 4 2 11 6.481E-0001 3.519E-0001 -1.100E+0000

Total geometric volume = 3.00000000000E+0000

Total activity = 1.35939770688E+0004

Overall mean concentration = 2.26566284481E+0004

gives the following information for the control volumes set to mat1: (1) the averageair concentration (ca) is 11.97 kBq m−3, (2) the total activity considering all phases(∑

βcaδV ) is 958 Bq, (3) the total geometric volume taken up by mat1 (∑

δV ) is0.4 m3, (4) there are 36 control volumes with valid field values and 89 with invalidfield values (invalid field values could be from control volumes of zero volume orNOP’s–see Section 6.1 page 39), (5) the minimum concentration is 3.0 kBq m−3 andthis value occurs at control volume (i,j,k) = (2,4,2) with physical coordinates(x,y,z) = (0.019 m, 0.98 m, −2.95 m), and (6) the maximum concentration is21 kBq m−3. Similar information is given for the two other materials: mat2 andmat3.The item ”Total geometric volume” gives the total geometric volume included

in the grid without consideration for porosity. The item ”Total activity” is thetotal (mobile) activity covered by the grid regardless of phase (

∑βcaδV ). The

”Overall mean concentration” corresponds to the average air concentrations (ca)listed for the individual materials. The only difference is that this value includesresults from all materials. Observe: all average air concentrations use the volumeof the included control volumes as weight.

4.48 warning priority log

Type: Enumerated variable with the following possible assignments:war interpolation,war other, war fileimport, war convergence, war residual,or war none Default: war other Description: It is used to control what type ofwarnings that should be written to the LOG-file: only warnings of sufficient impor-tance will be output. If warning priority log := war none, then no warningswhatsoever will be output. If warning priority log := war other, then warn-ings of this type plus those lower on the list (convergence warnings, file importwarnings, and residual warnings) will be output. Warnings from the (less impor-tant) field interpolation functions will not be output. Normally, warning priority logwill be set to war other because in some circumstances the output can be floodedwith warnings from the field interpolation procedure.

4.49 warning priority screen

Type: Enumerated variable with the following possible assignments:war interpolation,war other, war fileimport, war convergence, war residual,or war none Default: war other Description: This variable has the same mean-ing as warning priority log, except that the output goes to the screen.

24 Risø-R-1201(EN)

4.50 solver def

Type: Pointer to nil, find better field Thomas, or find better field Gauss SeidelDefault: nil Description: This is a pointer to the RnMod3d procedure thatshould be used as solver. (1) nil means that no solver is defined. This is use-ful during debugging. For example, it can be tested if the grid is set up cor-rectly or if there are memory problems etc. without really solving the prob-lem. (2) find better field Thomas means that the Thomas algorithm is usedas solver. This procedure sweeps the grid line by line in alternating directions.(3) find better field Gauss Seidel means that the Gauss-Seidel procedure isused as solver. This is a point-iterative procedure. The procedure pointed to bysolver def is called once in each iteration. Additional information: See Sec-tion 10.3.

4.51 scheme

Type: Enumerated variable with the following possible assignments: powerlaw,central, upwind, hybrid, or exact. Default: exact Description: This variableis used to control how the coefficients are calculated. Additional information:See the function Apower in the file R3Main03.pas and [Pa80].

4.52 relax factor

Type: Positive floating-point number Default: 1.0 Description: This variablecontrols the relaxation of the iterative solution procedure:

c_new:=c_old+relax_factor*(c_now-c_old)

where c old is the field value reached in the previous iteration, c now is the re-sult reached in this iteration, and c new is the new result after relaxation. Withrelax factor := 1.0 there is no relaxation. Relaxation factors above 1 can beused to obtain quicker convergence (over-relaxation). Relaxation factors below 1(under-relaxation) can be used to ”tame” unstable problems. Relaxation factorsabove 2.0 are unstable. Optimal relaxation factors for soil-gas problems are oftenaround 1.98.

4.53 flux convset

Type: Set of the enumerated values: Flx1, Flx2 etc. Default: [ ] (i.e. anempty set) Description: The variable specifies which ”flux measurement probes”that should be used by the solver for for convergence tests. For example, ifflux convset := [Flx3], then only the convergence of Flx3 is used by the solver.Examples of other assignments are: Calculation with Pascal

sets!flux_convset := [Flx1,Flx2]; (* Flx1 and Flx2 *)

flux_convset := [Flx1,Flx3..Flx5]; (* Flx1, Flx3, Flx4, and Flx5 *)

flux_convset := [Flx1..Flx5]-[Flx2]; (* Flx1, Flx3, Flx4, and Flx5 *)

flux_convset := []; (* Empty set, no flux probes *)

Warning: Flux probes with end results close to zero should not be included influx convset. Additional information: See Section 8.

4.54 probe convset

Type: Set of the enumerated values: Obs1, Obs2 etc. Default: [ ] (i.e. an emptyset) Description: This variable has the same meaning as flux convset (see

Risø-R-1201(EN) 25

the previous section) except that this one concerns ”the probes for concentra-tion measurements”: Obs1, Obs2 etc. Warning: Probes with end results close tozero should not be included in probe convset. Additional information: SeeSection 9.

4.55 conv evaluation period

Type: Integer number Default: 50 Description: This variable is used to controlwhen convergence tests should be conducted. This is of interest because the solverworks iteratively, and because it costs computational time to evaluate if conver-gence has been reached. In particular, calculation of fluxes Flx1, Flx2 etc. can besomewhat time consuming. If conv evaluation period := 1 then a convergencetest is performed after every single iteration. If conv evaluation period := 100then a convergence test is performed only after every 100 iterations. There is oneside effect to the setting of conv evaluation period: The procedures associatedwith the variables:

wr_iteration_line_logwr_iteration_line_screenwr_residual_during_calc_logwr_residual_during_calc_screenwr_flux_during_calc_logwr_flux_during_calc_screenwr_probes_during_calc_logwr_probes_during_calc_screen

will generate output only for those of the iterations with convergence tests.

4.56 min iterations

Type: Integer number Default: 5 Description: This variable sets the minimumnumber of iterations that the solver should use. For example: min iterations:=50will force the solver to do at least 50 iterations regardless of all other settings.

4.57 max iterations

Type: Integer number Default: 500 Description: This variable sets the maxi-mum number of iterations that the solver can use. For example: max iterations:=1000will force the solver to stop after a maximum of 1000 iterations. The iterationsmay stop before that (e.g. if convergence is reached).

4.58 max time

Type: Floating-point number Default: 180 Description: This variable sets themaximum computational time (wall-clock time in seconds) that the solver can use.For example: max time:=12*3600 will force the solver to stop after 12 hours ofcomputations. The iterations may stop before that (e.g. if convergence is reached).Warning: This variable has no meaning if RnMod3d is complied with BorlandPascal v. 7.

4.59 max change

Type: Floating-point number Default: 1e-6 Description: This variable setspart of the criteria for ”convergence”. When all of the probes included in thesets flux convset and probe convset changes less (per iteration) than the value

26 Risø-R-1201(EN)

given by max change, we consider this part of the requirement for convergenceto have been met (but there are others). Hence max change := 1e-4 means thatconvergence is not reached before the monitored values change by less than 0.01 %per iteration.

4.60 max residual sum

Type: Floating-point number Default: 1e-4 Description: In addition to ”fluxmeasurements” and ”concentration measurements” (see max change), RnMod3dalso uses the sum of residuals as a criteria for convergence. If the sum of residualsis larger than the value assigned to max residual sum, the model does not considerthe solution to have converged. Additional information: See Section 10.4.

4.61 dtim

Type: Non-negative floating-point number Default: 0.0 Description: dtim isthe time step given in seconds. Hence, in a time-dependent problem (i.e. whensolution has been set to unsteady) the call run model will advance the field ofpressures or radon concentrations by dtim. For example, if each time step shouldbe 1 hour, then we set dtim := 3600 Additional information: See Section 11.2.

4.62 BC running

Type: Boolean Default: false Description: If this variable is set to falsethen no adjustment of boundary conditions are carried out. Hence this value mustbe set to true when ”running boundary conditions” are needed. Additionalinformation: See Section 12.

4.63 BC running update of cBCs def

Type: PointerDefault: nil Description: This is a pointer to a user-defined pro-cedure that controls how the boundary conditions (e.g. cBC[fixed1]) are changed.To prevent unstable solutions the process is normally under-relaxed. Additionalinformation: See Section 12.

4.64 BC running min iterations

Type: Interger number Default: 100 Description: This variable is of type in-teger. It sets the minimum number of iterations that RnMod3d needs to carryout before it attempts to change the boundary conditions. If the value is set toolow, the solution procedure can become unstable. Additional information: SeeSection 12.

4.65 BC running max residual sum before new BC

Type: Floating-point number Default: 1e-9 Description: This variable givesthe maximum sum-of-residuals before RnMod3d attempts to change the boundaryconditions. If the value is set too high, the solution procedure can become unstable.Additional information: See Section 12.

4.66 BC running convergence def

Type: Pointer Default: nil Description: This is a pointer to a user-definedfunction that returns the value true if some user-defined criteria for convergence

Risø-R-1201(EN) 27

has been met. Otherwise it should return the value false. For example, in asimulation of exhalation from concrete into a chamber it can be tested if thereis consistency between the assumed fixed-concentration and the calculated flux.Additional information: See Section 12.

4.67 wr BC running messages log

Type: Boolean Default: false Description: This variable that controls ifRnMod3d should output information about the running boundary conditions tothe LOG-file. Additional information: See Section 12.

4.68 wr BC running messages screen

Type: Boolean Default: false Description: This variable controls if RnMod3dshould output information about running boundary conditions to the screen. Ad-ditional information: See Section 12.

4.69 press enter wanted

Type: Boolean Default: true Description: If the variable is set to true, thenthe console (window) where RnMod3d runs does not close before the user haspressed enter. If the variable is set to false RnMod3d can be run in batch mode.

5 Geometry

RnMod3d uses a grid of control volumes as the basis for all computations. This sec-tion tells how the grid spacing is controlled. Essentially, the user needs to write aprocedure that maps the computational (i,j,k) space onto the physical (x, y, z)world in meters. Only orthogonal grids are possible in RnMod3d. This means thatcontrol volumes with the same i-coordinate have identical x coordinates (regard-less of j and k). The same is true for the other dimensions. Hence the mappingcan be done independently for each axis:

i ↔ x

j ↔ y

k ↔ z

The mapping involves three steps:

• Selection of coordinate system The type of coordinate system is se-lected with the control variable geometry. Three possibilities are available:cartesian2d, cylindrical2d and cartesian3d.

• Declaration of fix points All physical dimensions of importance for theproblem should be associated formally with fix points called xFix1, xFix2etc. For example, in a house simulation there may be a crack in the floor atx = 3 m. We can associate this location with xFix2 with the call:set FixVal(xFix2,3.0). In other parts of the job file, reference should bemade to this physical location through xFix2. Reference directly to x = 3.0 mshould be avoided.

• Node spacing To achieve a sufficiently accurate numerical solution it isimportant that grid points are closely spaced in regions with large field gradi-ents. In RnMod3d, the grid is generated ”by hand”. Essentially, each axis can

28 Risø-R-1201(EN)

be subdivided as specified by the user. If each of the three axes are dividedinto 50 pieces, then the grid will consist of 503 = 125 000 control volumes.

When the grid has been set up in this way, each control volume has a certain loca-tion and size. Section 5.9 provides more details about this. However, geometricalinformation for individual control volumes is normally not needed because of theuse of fix points.An example of a user-defined grid is shown next. The grid is from a three-

dimensional problem. The full meaning of the statements is explained in the fol-lowing.

Example 4 Three-dimensional grid, where geometry:=cartesian3d .

procedure mygrid;

begin

set_FixVal(xFix1,0.0); (* x-axis *)


set_axis_single(xFix1,xFix2,10,FocusA,2.0);

set_FixVal(yFix1,0.0); (* y-axis *)

set_FixVal(yFix2,1.0);

set_axis_single(yFix1,yFix2,10,FocusB,2.0);

set_FixVal(zFix1,-3.0); (* z-axis *)

set_FixVal(zFix2,-2.5);


set_FixVal(zFix4, 0.0);

set_axis_single(zfix1,zfix2,1,FocusA,1.0);

set_axis_single(zfix2,zfix3,5,focusA,1.0);

set_axis_single(zfix3,zfix4,1,FocusA,1.0);

end;

To force RnMod3d to use the grid defined in the example, we need set the controlvariable called grid def as follows:

grid_def := mygrid;

5.1 Grid size (memory issues)

The only limitation for the size of the grid is the available computer memory.Grids with as many as 250 000 nodes have been used on a pc with 128 Mb of ram.By default RnMod3d, however, is limited to the grid size given by the compilerdirectives in the file R3DIRS03.pas. Typical settings are:

$DEFINE imax100$DEFINE jmax100$DEFINE kmax200

This means that a maximum of 100 nodes can be located on the x-axis (index i)and the y-axis (index j), whereas 200 nodes are allowed on the z-axis (index j).The following DEFINE-directives are possible for the x-axis:

imax3imax10imax50imax100imax150imax200imax250imax300

Risø-R-1201(EN) 29

imax350imax400imax450imax500

Similar directives can be used for the other axes. The maximum number of nodesthat can be allocated on the x-, y-, and z-axes are given by imaxTot, jmaxTot,and kmaxTot. The values of these variables are output to the LOG-file and thescreen when a job is run. In case too many nodes are specified in a job file, thecomputations will end with an error message.

5.2 geometry

The coordinate system used by RnMod3d is set by the control variable geometry.As shown in Figure 3 there are three possibilities.The assignment geometry := cartesian3d implies that an ordinary carte-

sian (x, y, z)-coordinate system is used for the computations. With geometry :=cartesian2d, only the cartesian (x, z)-coordinates are used. The thickness of thegrid in the y-direction can be set with the control variable Ly (see Section 4.5).The default thickness is 1 meter. With geometry := cylindrical2d, the (x, z)-coordinates refer to the (r, z) cylindrical coordinates. In this case, the y coordinatehas no meaning. One-dimensional problems are solved with either of the three co-Grids for 1D-problemsordinate systems. To minimize the use of memory, only one node should be devotedto each of the ”passive” dimensions.

x

x x

y

z

z

z

cartesian2d cylindrical2d cartesian3d

Figure 3. Types of grid geometries that can be selected with the control variablegeometry. The idea for this figure comes from [Ho94, p. 30].

5.3 set FixVal

As already stated, physical dimensions of importance for the problem should beassociated with fix points called xFix1, xFix2 etc. for the x-axis, yFix1, yFix2etc. for the y-axis, and zFix1, zFix2 etc. for the z-axis. Fix points are associatedwith physical dimensions by calls such as:

set_FixVal(xFix2,3.0)

that links xFix2 to the physical dimension x = 3.0 m. Fix points should be set upin accordance with the following rules:

30 Risø-R-1201(EN)

• Fix points should be defined starting from xFix1 for the x-axis, yFix1 forthe y-axis, and zFix1 for the z-axis.

• The physical dimensions associated with fix points should be given in ascend-ing order: ”xFix1” < ”xFix2” < ”xFix3” etc.

• There can be ”no gaps” in the series of fix points used. For example, it is notpossible to define xFix2 and xFix4 without also defining xFix3.

5.4 wFixVal

RnMod3d stores all information about fix points in the array wFixVal. For each fixpoint, the array contains a record of two items called defined and w. The firstelement is a boolean that tells if the fix point has been defined or not. The secondelement is the physical (meter) coordinate of the fix point. This type of informationis useful in some special situations. For example, imagine that a basement slab isset to span vertically from zFix2 to zFix3. To ascertain that this slab thicknesshas been correctly implemented in the job file, we could make the following callin the job file:

Example 5 Reference to fix points.

if (wFixVal[zFix2].defined) and (wFixVal[zFix3].defined) thenwriteln(’The slab thickness is = ’,wFixVal[zFix3].w-wFixVal[zFix2].w,’ m’);

5.5 Node spacing

The grid of control volumes is created by subdividing each of the three axes. InRnMod3d, this subdivision is always done between single pairs of fix points. Fixpoints never move (regardless of the grid spacing). This means, for example, thatcontrol volumes will never ”cross” a fix point. This has the following importantimplication: If, for example, a concrete slab in a house simulation is defined byreference to fix points then the ”concrete” is always filled up completely by controlvolumes. There will be no control volumes which are partly in the soil and partly inthe concrete. To say it differently: All ”cuts” are made along fix points. Subdividingan axis between pairs of fix points can be done with the three procedures:

set_axis_single(wFixA,wFixB,hAB,f,pow);set_axis_double(wFixA,wFixB,hAM,hMB,fA,fB,powA,powB,wdiv);set_axis_triple(wFixA,wFixB,hAM,hMM,hMB,fA,fM,fB,powA,powM,powB,wdivA,wdivB);

The fix points ”FixA” and ”FixB” must be adjacent. Hence, we cannot makea call such as set axis single(xFix2,xFix5...). To help read the set axis-procedure headers it is probably useful to know that w is used for x, y and z.Likewise h is a generic reference to the index variables: i, j and k.

5.6 set axis single

set_axis_single(wFixA,wFixB,hAB,f,pow);

where

wFixA and wFixB are fix points such as xFix1 and xFix2.

hAB is the wanted number of subdivisions between ”A” and ”B”.

f is focusA or focusB.

pow is a real number (e.g. 1.0).

Risø-R-1201(EN) 31

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 4. Z-axis generated with: set axis single(zFix1,zFix2,5,FocusA,1.0);Five nodes are placed at uniform distances between the fix points at -3 and 0.0.

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 5. Z-axis generated with: set axis single(zFix1,zFix2,25,FocusA,1.0);Now 25 nodes are located between the two fix points.

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 6. Z-axis generated with: set axis single(zFix1,zFix2,25,FocusA,2.0);Now the 25 nodes are not distributed uniformly. The density is highest in the leftpart of the axis (FocusA).

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 7. Z-axis generated: set axis single(zFix1,zFix2,25,FocusB,2.0);Now the focus is to the right side (FocusB).

32 Risø-R-1201(EN)

This procedure divides the axis between the fix point pair: wFixA and wFixB intohAB subdivisions. Power functions of the type wpow are used for the purpose.pow equal to 1 makes the node-spacing uniform. If a non-uniform distribution iswanted, we set pow to values different from 1. For example, pow set to 2, will givea spacing that increases as a square-function. The parameter f is used to controlwhere the density of divisions should be highest. If the density should be highestclose to fix point A, then set f=focusA. If the density should be highest in theother end, use f=focusB. The term ’single’ in the name of the procedure refersto the fact that the grid point density changes monotonically (i.e. in one singleinterval) as specified by the distribution function.Sample calls with set axis single are given in Figure 4 to 7. The two fix

points zFix1 and zFix2 are set as follows:

set_FixVal(zFix1,-3.0)set_FixVal(zFix2, 0.0)

5.7 set axis double

set_axis_double(wFixA,wFixB,hAM,hMB,fA,fB,powA,powB,wdiv);

where


hAM and hMB are the wanted numbers of subdivisions between ”A” to ”M” and”M” to ”B”, respectively.

fA and fB are assigned the values focusA or focusB.

powA and powB are real numbers.

wdiw is a real number (e.g. 0.5).

In this procedure, the axis between the two fix points (A and B) is split intotwo. The physical location of the middle point (M) is given by wdiv. wdiv givesthe location of M as a fraction of the total physical distance between A and B.wdiv=0.5 means that M is half way between A and B. wdiv=0.01 means that Mis located very close to A: The distance A–M is 1 % of the total distance from A toB. wdiv=0.99 locates M very close to B. The number of subdivisions allocated tocover the interval A to M is specified by hAM. Likewise, the number of subdivisionsbetween M and B is hMB. Within the two intervals: AM and MB, subdivisions aredistributed as described for the set axis single procedure. Each interval has itsown pow-parameter. So for example, it is possible to have uniform spacing betweenA and M and highly non-uniform spacing from M to B.Sample calls with set axis double are given in Figure 8 and 9. The two fix

points zFix1 and zFix2 are set to -3 and 0.0, respectively.

5.8 set axis triple

set_axis_triple(wFixA,wFixB,hAM,hMM,hMB,fA,fM,fB,powA,powM,powB,wdivA,wdivB);

where


hAM, hMM, and hMB are the wanted numbers of subdivisions between ”A” to ”M”,”M” to ”M”, and ”M” to ”B”, respectively.

fA, fM, and fB are assigned the values focusA or focusB.

Risø-R-1201(EN) 33

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 8. Z-axis generated with: set axis double(zFix1,zFix2,5,20,FocusA,FocusB,1.0,2.0,0.5); 5 Nodes are devoted to the left interval and 20 to theright. The middle point (that separates left from right) cuts the z-interval into twoparts of equal parts (f=0.5). The node distribution is uniform in the left interval(from −3 to −1.5 m and non-uniform (the exponent equals 2.0) to the right (from−1.5 to 0 m).

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 9. Z-axis generated with: set axis double(zFix1,zFix2,5,20,FocusA,FocusB,1.0,2.0,0.2); Now the two intervals are not of equal size. The left in-terval covers 20 % of the distance between the two fix points. The right intervalcovers the remaining 80 %.

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = 0

Figure 10. Z-axis generated with: set axis triple(zFix1,zFix2,5,20,5,FocusA,FocusA,FocusB,3.0,1.0,3.0,0.2,0.8); The distance between the twofix points is now divided into three. Cuts are made after 20 % and 80 %.

powA, powM, and powB are real numbers.

wdiwA and wdiwB are real numbers (e.g. 0.5).

This procedure is a natural extension of set axis double. The only difference isthat now the interval between node A and B is split into three subintervals. Asample call with set axis triple is given in Figure 10. The two fix points zFix1and zFix2 are set to -3 and 0.0, respectively.

5.9 Location and size of specific control volumes

Figure 11 shows a generic control volume. It is located at (i,j,k). The controlvolume is represented by the gray region. Any material property assigned to thecontrol volume (porosity, diffusivity etc.) applies to that region. Hence, material

34 Risø-R-1201(EN)

x[i] xnod[i] x[i+1]

dx[i]

x-axis

W P Ejw je

Figure 11. Geometry and x-coordinates for control volume (i,j,k).

properties are assumed to be constant within each control volume. RnMod3d findsthe field value (e.g. the radon concentration) exactly at the node P in the centerof the control volume, and fluxes are calculated exactly at the interfaces betweenadjacent control volumes (je and jw in the figure).The x-coordinates for the control volume (i,j,k) are shown in Figure 11.

The west-side interface is located at x[i] and the east-side interface is locatedat x[i+1]. The node P is located at xnod[i] midway between the interfaces.Hence xnod[i] = x[i] +0.5 · dx[i]. The length of the control volume is dx[i] =x[i+ 1]− x[i].In RnMod3d the location of control-volume interfaces are stored in the arrays

x[i], y[j], and z[k]. Likewise the ”height, width and depth” are stored in thearrays dx[i], dy[j], and dz[k]. Hence such information can be accessed directlyby the user as given in the following example. Alternatively, a standard list canbe written to the LOG-file as shown in Section 5.10.

Example 6 How to print out the coordinates of a specific control volume.

...

run_model;

writeln(’Control volume (5,3,6) has its west interface at: x = ’,x[5]);

writeln(’Control volume (5,3,6) has its south interface at: y = ’,y[3]);

writeln(’Control volume (5,3,6) has its bottom interface at: z = ’,k[6]);

...

xnod(i), ynod(j) and znod(k)

Often it is necessary only to get the (x, y, z) coordinate of the node (not theinterfaces). This information is most easily obtained with the functions xnod(i),ynod(j) and znod(k). For example, the physical x-coordinate of the center-nodeof the i’th control-volume is xnod(i). An illustrative example of the use of thesefunctions is given page 47.

Areas and volume of a given control volume

As shown in Figure 1 page 13 each control volume is a ”box” with six sides. Thearea of each side are output to the LOG-file if the control variable wr node sizesis set to true (see Section 4.45). It is also possible to get the information for just

Risø-R-1201(EN) 35

one single control volume. This can be done with the procedure set cvsize. Thefollowing example shows what to do.

Example 7 Application of set cvsize .

procedure wr_interface_areas(i:itype; j:jtype; k:ktype);

var ArW,ArE,ArS,ArN,ArB,ArT,dV:datatype;

begin

set_cvsize(i,j,k,ArW,ArE,ArS,ArN,ArB,ArT,dV);

writeln(’The area of the west-side interface of control volume: ’);

writeln(i:4,j:4,k:4,’ is: ’,ArW,’ m2’);

writeln(’The volume of control volume : ’);

writeln(i:4,j:4,k:4,’ is: ’,dV,’ m3’);

end;

5.10 Grid inspection: wr axes

The location of control volumes can be inspected if the control variable wr axesis set to true. A table like the one given next will appear in the LOG-file (seepage 74).

Example 8 Grid output created with mygrid in the example page 29.

axis i x[i] x[i+1] dx[i] dcdx dcdxnorm Fixpts

x 1 0.00000 0.00000 0.0000000 8.883E-0005 0.00005922 xFix1

x 2 0.00000 0.01000 0.0100000 4.254E-0004 0.00028357 -

x 3 0.01000 0.04000 0.0300000 6.003E-0004 0.00040023 -

x 4 0.04000 0.09000 0.0500000 6.801E-0004 0.00045337 -

x 5 0.09000 0.16000 0.0700000 7.312E-0004 0.00048749 -

x 6 0.16000 0.25000 0.0900000 7.477E-0004 0.00049845 -

x 7 0.25000 0.36000 0.1100000 7.191E-0004 0.00047942 -

x 8 0.36000 0.49000 0.1300000 6.325E-0004 0.00042165 -

x 9 0.49000 0.64000 0.1500000 4.695E-0004 0.00031299 -

x 10 0.64000 0.81000 0.1700000 1.878E-0004 0.00012520 -

x 11 0.81000 1.00000 0.1900000 2.168E-0019 0.00000000 -

x 12 1.00000 1.00000 0.0000000 0.000E+0000 0.00000000 xFix2

axis j y[j] y[j+1] dy[j] dcdy dcdynorm Fixpts

y 1 0.00000 0.00000 0.0000000 8.181E-0005 0.00005454 yFix1

y 2 0.00000 0.19000 0.1900000 8.392E-0005 0.00005594 -

y 3 0.19000 0.36000 0.1700000 3.420E-0004 0.00022799 -

y 4 0.36000 0.51000 0.1500000 5.414E-0004 0.00036095 -

y 5 0.51000 0.64000 0.1300000 6.500E-0004 0.00043333 -

y 6 0.64000 0.75000 0.1100000 6.927E-0004 0.00046177 -

y 7 0.75000 0.84000 0.0900000 6.845E-0004 0.00045634 -

y 8 0.84000 0.91000 0.0700000 6.361E-0004 0.00042405 -

y 9 0.91000 0.96000 0.0500000 5.471E-0004 0.00036470 -

y 10 0.96000 0.99000 0.0300000 3.096E-0004 0.00020641 -

y 11 0.99000 1.00000 0.0100000 2.168E-0019 0.00000000 -

y 12 1.00000 1.00000 0.0000000 0.000E+0000 0.00000000 yFix2

axis k z[k] z[k+1] dz[k] dcdz dcdznorm Fixpts

z 1 -3.00000 -3.00000 0.0000000 0.000E+0000 0.00000000 zFix1

z 2 -3.00000 -2.50000 0.5000000 1.500E+0000 1.00000000 -

z 3 -2.50000 -2.50000 0.0000000 0.000E+0000 0.00000000 zFix2

z 4 -2.50000 -2.10000 0.4000000 1.127E-0002 0.00751807 -

z 5 -2.10000 -1.70000 0.4000000 1.021E-0002 0.00681031 -

z 6 -1.70000 -1.30000 0.4000000 8.553E-0003 0.00570229 -

z 7 -1.30000 -0.90000 0.4000000 6.397E-0003 0.00426476 -

z 8 -0.90000 -0.50000 0.4000000 1.824E-0003 0.00121574 -

z 9 -0.50000 -0.50000 0.0000000 2.280E-0003 0.00151967 zFix3

z 10 -0.50000 0.00000 0.5000000 4.337E-0019 0.00000000 -

z 11 0.00000 0.00000 0.0000000 0.000E+0000 0.00000000 zFix4

36 Risø-R-1201(EN)

The example corresponds to the mygrid procedure given page 29. First informationabout the x-axis is given. Nodes on this axis are indexed by the variable i. Thereare 12 nodes on the axis, so i goes from 1 to 12. x[i] gives the physical coordinatefor the left interface of control volume i. Likewise x[i+1] is the coordinate of theright interface. dx[i] is the thickness of the control volume i (see Figure 11,page 35). The columns dcdx and dcdxnorm will be described in the followingsection. They contain information about maximum gradients of the field (c) inthe direction of the x-axis, so this information can be used to identify where moregrid points should be located. The final column Fixpts marks fix-point locations.The same type of information is given for the other two axis. Here, j and k areused as index variables for the y and z-axeses, respectively. Observe that for eachfix point, there will be a ”ghost” node of zero thickness.

x [m]

0.0 0.2 0.4 0.6 0.8 1.0

xFix1 = 0xFix2 = 1

y [m]

0.0 0.2 0.4 0.6 0.8 1.0

yFix1 = 0yFix2 = 1

z [m]

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

zFix1 = -3zFix2 = -2.5

zFix3 = -0.5zFix4 = 0

Figure 12. Plot of axes generated with mygrid defined page 29. The circles rep-resent nodes. Fixpoints are named xFix1 etc. Interfaces of control volumes areindicated on the line above the nodes.

5.11 Grid evaluation: dcdx and dcdxnorm

The output from wr axes (see the previous section) also indicates where to addmore grid points in the grid: For all nodes at i, RnMod3d calculates the fielddifferences (”∆c”) to the adjacent nodes at i+1. The largest field difference iscalled dcdx. In a soil-gas simulation, dcdx is measured in Pa. In a radon simulation,dcdx is measured in Bq m−3. What does this quantity tell? If dcdx is found to be1 Pa at i=10 whereas dcdx is much smaller than 1 Pa, for all other i’s in the grid,then it would probably be good to add some more grid points between i=10 andi=11. Observe, dcdx is the field ”gradient” per node in the x-direction.Similar calculations are carried out for the two other axes, and the results are

stored in dcdy and dcdz. To find out which axis that may be in most need of moregrid points, the maximum of dcdx, dcdy, and dcdz is calculated. From this globalmaximum, we normalize all the dcdx-values etc. The results are called dcdxnorm,dcdynorm, and dcdznorm.

Risø-R-1201(EN) 37

x

y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

z

0.0 0.2 0.4 0.6 0.8 1.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

y

z

0.0 0.2 0.4 0.6 0.8 1.0

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Figure 13. Two-dimensional projection plots of the grid generated with mygriddefined page 29. The circles represent nodes. Fix points are drawn with thick lines.Interfaces of control volumes are drawn with thin lines.

As shown in the previous section, dcdx and dcdxnorm etc. are output to theLOG-file by setting wr axes to true.

6 Nodes and connectors

Having created a (i,j,k)-grid of control volumes and linked them to the physical(x, y, z)-world in meters (see the previous section), it is now time to define howeach control volume should ”behave” and how each control volume should beconnected with its nearest neighbors. This section tells how to do that. Like thegeometry of the grid is contained in the procedure pointed to by the controlvariable grid def, the control variable boundary conditions def2 points to theprocedure where the ”grid behavior” is defined. In other words, this section tellshow the boundary conditions def procedure should be programmed.The section is divided in two. First, the different types of nodes and connectors

are presented, and it is shown how these may be set with the procedures set nodeand change node. Then we describe the so-called in-functions: in cube, in plane,in region, and in interval, which are used to pin point ”collections” of controlvolumes (for example an entire boundary) by reference to fix points (xFix1, xFix2etc.) defined in the grid procedure. As described in subsequent sections, the in-functions are used also for setting up material properties and flux measurements.

2The name of this control variable is a little bit misleading. The procedure pointed to byboundary conditions def controls not only boundary conditions, but all nodes and connectorsin the grid.

38 Risø-R-1201(EN)

6.1 Node types

Three types of nodes are used in RnMod3d:

the standard node free,

the no-operation node nop, and

the fixed-value nodes fixed1, fixed2 etc.

Control volumes with a node type set to free are controlled by the transportequation for radon or soil gas given in Section 3. Of course, this is normallythe majority of control volumes. The no-operation node type, nop, is used forcontrol volumes which are not part of the problem (they just happen to be inthe grid because the grid is always a regular box). For example, imagine a 3D-simulation of soil-gas entry into a basement house: That part of the grid that is”in the basement” is not controlled by Darcy’s law and control volumes in thisregion should be set to nop. The fixed-value node types: fixed1, fixed2 etc. areused for control volumes where the field is held fixed at certain constant values(regardless of transport equations). The fixed values are set up in the array cBC.For example, in the previous basement example, the pressure at the interfacebetween concrete and basement (or soil and basement) may be set to some fixedvalue (e.g. −3.0 Pa). If we set these control volumes to be of node type fixed1,then we set cBC[fixed1] := -3.0. Likewise, the collection of control volumeslocated at the interface between soil and atmosphere may be assigned the nodetype fixed2. If this boundary is maintained at zero Pa, then we set cBC[fixed2]:= 0.0. The control volumes in the soil (or concrete) are of the type free.

6.2 Connector types

Each control volume is like a box (see Figure 1, page 13): it represents a certainvolume and has six faces. For control volumes (deep) inside the grid, each controlvolume interfaces with six other control volumes. Radon (or soil gas) thereforenormally can flow from one control volume to six others (nearest neighbors). Theability of having transport between neighboring control volumes is handled byconnectors. All control volumes have six connectors. These are called: Econ, Wcon,Ncon, Scon, Tcon, and Bcon for the east, west, north, south, top, and bottomof the control volume faces, respectively. There are three types of connectors inRnMod3d:

the standard connector std,

the no-flow connector noFlow, and

the connector to ”nowhere” called nill.

The standard connector type, std, is used when radon or soil gas can flow freely(as given by the governing transport equations) between the two control volumeslinked by the connector. The no-flow connector called noFlow is used when suchtransport is explicitly set to be zero. This represents a no-flow boundary condition.Clearly, connectors are only meaningful between control volumes. Control volumeslocated at the boundary of the grid will have one or more faces pointing to nowhere.These connectors are set to be of the type called nill.

6.3 Default nodes and connectors

When run model is called the first time and the grid geometry has been set inaccordance with the procedure pointed to by grid def, RnMod3d assigns the fol-lowing default values to grid nodes and connectors:

Risø-R-1201(EN) 39

All nodes are set to be of type free.

Connectors between control volumes are set to be of type std.

Connectors at the boundary of the grid (pointing to nowhere) are set to nill.

This means that the default computational grid behaves as a closed box. Forexample, if we add some radon activity, it cannot leave the box: All boundariesare closed off. However, radon can move around inside the box and decay asspecified by the radon transport equation.If we make a list of nodes in the grid (e.g. by setting the control variable

wr nodes to true) it will be found that the above description is not entirelytrue. A glance at such a table will reveal that (all) grids contain a number of nopnodes and noFlow connectors. These occur for control volumes that have all threecoordinates at fix points (for example, the x-coordinate could be at xFix3, they-coordinate could be at yFix2, and the z-coordinate could be at zFix5). Suchcontrol volumes have zero volume, and we refer to them as being ”dead”. The”dead” control volumes plays absolutely no role for the user when a problem is setup. They can be ignored completely3. Only, they may cause a little confusion inthe situation (already mentioned) where the node types in the grid are inspected:The user may think that something is wrong with the grid because it contains abunch of nop’s that he or she had not explicitly defined.The type of nodes and connectors should be changed from the default with

the procedures set node and change node. These procedures are ”clever” in thesense that if for example the connector at the top face of control volume (i,j,k)is set to noFlow, then the program automatically updates the connector of thebottom face of the control volume (i,j,k+1). Also, these procedures take care ofthe ”dead” control volumes.

6.4 Inspection of nodes and connectors

Often it is useful to be able to verify that the correct nodes and connectors areset up. An easy way to inspect the grid is to set the control variable wr nodesto true. Then a complete listing of alle nodes and connectors are output to theLOG-file (see Section 4.43). Another way is to interact with the main data structureGP directly (see Section 14). A list of specific nodes or connectors can be made asshown in the following examples.

Example 9 Print list of control volumes with specific nodes. The procedure canbe called as wr nodelist(NOP).

procedure wr_nodelist(what:nodetyptype);

var i:itype;

j:jtype;

k:ktype;

begin

for i:=1 to imax do

for j:=1 to jmax do

for k:=1 to kmax do

if GP[i]^[j]^[k].nodetyp=what then

writeln(i,’ ’,j,’ ’,k,’ Found one node = ’,nodetyp_string(what))

end;

Example 10 Print list of control columes with specific connectors. The procedurecan be called as: wr connectorlist(noFlow).

3Well, they can almost be ignored completely. If the user makes direct access to the fieldvalues of the grid make sure to test if the grid values are valid. See the examples in Section 9 forhow to do that.

40 Risø-R-1201(EN)

procedure wr_connectorlist(what:nodecontype);

var i:itype;

j:jtype;

k:ktype;

begin

for i:=1 to imax do

for j:=1 to jmax do

for k:=1 to kmax do

if GP[i]^[j]^[k].Wcon=what then

writeln(i,’ ’,j,’ ’,k,’ Found one connector = ’,nodecon_string(what))

end;

6.5 set node

The procedure set node is used to set the node type of a single control volume.For example, if the control volume (i,j,k) should be set to type fixed2, thenwe simply make the call:

set_node(i,j,k,fixed2)

The connectors of the control volume remain unaffected by set node unless thenode type is set to nop. In that case, all connectors to other control volumes areset to noFlow.

6.6 change node

The procedure change node can be used to change any feature of nodes andconnectors. The procedure is called as follows:

change_node(i,j,k,nodetypNew,wconNew,econNew,sconNew,nconNew,bconNew,tconNew)

where (i,j,k) are the index coordinates of the control volume to be affected,nodetypNew is the new node type that should be assigned to the control volume,and wconNew is the new connector type that should be assigned to the west face ofthe control volume. Similarly, the other parameters concern connectors at the east,south, north, bottom, and top faces of the control volume. Hence, the followingcall sets the node type of control volume (i,j,k) to free and all connectors(except the one at the top) to std. The top connector is set to noFlow:

change_node(i,j,k, free, std,std, std,std, std,noFlow)

Often, we want to change only the node type (or a single connector) and leave nodX and conXeverything else unchanged. To do that, we use nodX for ”unchanged node type”and conX for ”unchanged connector”. Hence, if we want to set the top connectorof control volume (i,j,k) to noFlow and leave everything else unchanged we usethe call:

change_node(i,j,k, nodX, conX,conX, conX,conX, conX,noFlow);

6.7 boundary conditions def

All node types and connectors (deep inside the grid or on the true boundary) canbe set by the procedure pointed to by the control variable: boundary conditions def.In the simple situation where we want to use the default settings (i.e. to modeltransport in a closed box as explained previously) and nothing more, we ”pro-gramme” a procedure with no changes of nodes or connectors:

Risø-R-1201(EN) 41

Example 11 Default equations and boundary conditions.

procedure my_closed_box(i:itype;j:jtype;k:ktype);beginend;

With the assignment:

boundary_conditions_def := my_closed_box;

RnMod3d is told that this is the procedure with all our changes.Normally, changes of nodes and connectors are not really made for individual

control volumes. In the typical situation we make changes for ”collections” ofcontrol volumes. One example of such a ”collection” for a house simulation isthe collection of nodes that are located at the atmospheric soil-air interface. In asoil-gas calculation we may want to set the pressure to zero with a fixed nodetype. To pinpoint such ”collections” of control volumes by reference to fix points(xFix1, xFix2 etc.), special in-functions have been developed. This means thatthe user need not (explicitly) know the index coordinates (i,j,k) of the controlvolumes in the ”collection”. The in-functions are: in cube, in plane, in region,and in interval. These functions are look-up tables: It can be tested if anycontrol volume (i,j,k) is within, outside or at the ”edge” of a certain regiondefined by fix points (xFix1 xFix2 etc.). The functions return the value true,if the control volume belongs to the region, and false if it does not. Beforedescribing how the in-functions are used, we will present a simple example.Imagine, a 30 m high column of sand with cross-sectional area of 2 x 2 m. The

sand is placed in some container with walls impermeable to gas flow. The top ofthe container is maintained at −3 Pa relative to the bottom. To treat this problemwe define the following two procedures:

Example 12 Sand column example.

procedure grid_column;

begin



set_axis_single(xFix1,xFix2,5,Focus,1.0);



set_axis_single(yFix1,yFix2,5,Focus,1.0);

set_FixVal(zFix1, 0.0); (* z-axis *)

set_FixVal(zFix2,30.0);

set_axis_single(zFix1,zFix2,10,Focus,1.0);

end;

procedure BC_column(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=0;

cBC[fixed2]:=-3.0;

if in_plane([eqAB,inside],i,xFix1,xFix2,yFix1,yFix2,zFix1,zFix1) then

set_node(i,j,k,fixed1); (* Boundary at zFix1 *)

if in_plane([eqAB,inside],i,xFix1,xFix2,yFix1,yFix2,zFix2,zFix2) then

set_node(i,j,k,fixed2); (* Boundary at zFix2 *)

end;

and we make the assignments:

geometry := cartesian3D;grid_def := grid_column;boundary_conditions_def := BC_column;

42 Risø-R-1201(EN)

Observe the following: (1) control-volume sizes etc. are in grid column and controlvolume ”behaviors” are as defined by default with the changes given in BC column.(2) All geometrical features are assigned to fix points (column height and cross-sectional dimensions). The procedure BC column contains references only to fixpoints. Hence, BC column remains valid even if we (later) change the column di-mensions or if we add more control volumes to the grid (i.e. if we make a finergrid).

6.8 in cube

The boolean function in cube is called as:

in_cube(reg, i,xFixA,xFixB, j,yFixA,yFixB, k,zFixA,zFixB)

where

reg is a Pascal set of inside, outside, and eqAB.

i, j and k are index coordinates of control volumes.

xFixA and xFixB are adjacent fix points on the x-axis.

yFixA and yFixB are adjacent fix points on the y-axis.

zFixA and zFixB are adjacent fix points on the z-axis.

The function concerns the location of the control volume with index coordinates(i,j,k) in relation to the ”cubic”4 region defined by the six planes defined sym-bolically by x = xFixA, x = xFixB, y = yFixA etc. where x = xFixA is the planeof all control volumes with physical x-coordinates equal to the fix point xFixAetc. If the control volume (i,j,k) belongs to the region, then in cube returnsthe value true. Otherwise it returns the value false.The parameter reg is a Pascal set of the elements: inside, outside, and eqAB5.

The meaning will become clear after the example given next. An example call is:

Example 13 Test procedure for the function in cube.

procedure test;

var i:itype;

j:jtype;

k:ktype;

begin

for i:=1 to imax do

for j:=1 to jmax do

for k:=1 to kmax do

if in_cube([inside],

i,xFix1,xFix4,

j,yFix3,yFix5,

k,zFix1,zFix2) then

writeln(’Inside the cube : ’,i:4,j:4,k:4);

end;

This example prints a list of all control volumes that have physical coordinateswithin the cube given by (xFix1 < x < xFix4) and (yFix3 < y < yFix5) and(zFix1 < z < zFix2). To refer to the control volumes that are not inside thecube, [inside] should be substituted with [outside]. To refer to those controlvolumes exactly on the faces of the cube, [inside] should be substituted with[eqAB]. To refer to those control volumes, that are inside or on the face of the

4The name ”cube” is misleading in the sense that the sides of the region need not be of equalsize. It had probably been better to call the function: in box.

5For the other in-functions the values eqA and eqB can also be used.

Risø-R-1201(EN) 43

cube, write [inside,eqAB]. To refer to those control volumes, that are inside oroutside (but not on the face of the cube), write [inside,outside]. To refer toall control volumes, write [inside,outside,eqAB]. To specify a an empty regionuse [ ],

6.9 in plane

The boolean function in plane is called as:

in_plane(reg, i,xFixA,xFixB, j,yFixA,yFixB, k,zFixA,zFixB)

where

reg is a Pascal set of inside, outside and eqAB.





The function concerns the location of the control volume with index coordinates(i,j,k) in relation to the plane defined by whichever (single) pair of fix pointsthat are identical. If xFixA=xFixB, then the function concerns the plane of controlvolumes with physical x-coordinates equal to the fix point xFixA. Planes in theother directions can be specified by yFixA=yFixB or zFixA=zFixB. Only one pairof fix points can be identical.If xFixA=xFixB, then the fix points for the y- and z-axis are used to limit the

region to be some rectangular part of the plane. An example will be given below.The parameter reg is a Pascal set of the elements: inside, outside, and eqAB.The meaning is identical to that described for the function in cube.The following example shows three sample calls of in plane. The first case con-

cerns the region defined symbolically as: (x = xFix1) and (yFix3 < y < yFix5)and (zFix1 < z < zFix2). So this is a rectangle in the yz-plane through x =xFix1. The function returns true if (i,j,k) is inside the region.

Example 14 Sample calls of in plane.

if in_plane([inside] ,i,xFix1,xFix1, j,yFix3,yFix5, k,zFix1,zFix2) then

writeln(’A ’,i:4,j:4,k:4); (* case A *)

if in_plane([eqAB] ,i,xFix1,xFix2, j,yFix2,yFix2, k,zFix1,zFix2) then

writeln(’B ’,i:4,j:4,k:4); (* case B *)

if in_plane([inside,eqAB],i,xFix1,xFix1, j,yFix3,yFix5, k,zFix3,zFix3) then

writeln(’C ’,i:4,j:4,k:4); (* case C *)

6.10 in region

The boolean function in region is called as:

in_region(i,xFixA,xFixB,xreg, j,yFixA,yFixB,yreg, k,zFixA,zFixB,zreg)

where

xreg, yreg, and zreg are Pascal sets of inside, outside, eqA, eqB and eqAB.




44 Risø-R-1201(EN)


This is a more general function than in cube and in plane. Six planes are defined:two yz-planes at x = xFixA and x = xFixB, two xz-planes at y = yFixA and y =yFixB, and two xy-planes at z = zFixA and z = zFixB. For each pair of planes, itcan be specified if it is the region inside, outside etc. that is of interest. The functionwill return true if the control volume (i,j,k) is within the ”x-region” and the”y-region” and the ”z-region”. Otherwise, it will return the value false. Theparameters xreg, yreg, and zreg are sets of the elements: inside, outside, eqA,eqB, eqAB. The meaning of the elements inside, outside, and eqAB is identicalto that described for the function in cube. The values eqA and eqB can be usedto include only fix point A or B. This is demonstrated by the following samplecall. Here, the in region is true for all control volumes that fulfill: (xFix2 ≤ x< xFix3) and (yFix2 < y < yFix3) and (zFix2 ≤ z ≤ zFix3).

Example 15 Sample call of in region.

if in_region(i,xFix2,xFix3,[inside,eqA],

j,yFix2,yFix3,[inside],

k,zFix2,zFix3,[inside,eqA,eqB]) then writeln(i:4,j:4,k:4);

6.11 in interval

The boolean function in interval is called as:

in_interval(h,wFixA,wFixB,wreg):boolean;

h is an index coordinate (i, j or k) of a control volume.

wFixA and wFixB is a pair of adjacent fix points on the x-, y- or the z-axis.

wreg, yreg, and zreg is a Pascal set of inside, outside, eqA, eqB and eqAB.

This function concerns only one coordinate. h is a generic index variable i, j, andk. Likewise, w is a generic physical coordinate x, y, and z. The parameter wregis a Pascal set of the elements inside, outside, eqA, eqB, or eqAB. The meaningof these elements is identical to that described for the function in region. Thesample call shown next, concerns all control volumes that have x-coordinates inthe interval: xFix2 ≤ x < xFix3.

Example 16 Sample call of in interval.

if in_interval(i,xFix2,xFix3,[inside,eqA]) then writeln(i:4,j:4,k:4);

7 Materials

RnMod3d solves transport equations of the form given in Box 1, page 11. Theseequations involve five material properties: β, ε, G, D and λ. The physical interpre-tation of the coefficients is clear in the case of radon transport. When problems ofsoil-gas transport are considered, the same ”coefficients” are used, however, witha different physical interpretation. This is discussed in Section 3.4.This section outlines how RnMod3d is linked to user-defined functions of mate-

rial properties through the control variables: beta def, e def, G def, D def andlambda def. The technique is flexible as it allows material properties to change inspace and time.To ease the assignment of material properties, it is useful to divide the compu-

tational grid into different types of materials. RnMod3d has a tool for that. This isdescribed next.

Risø-R-1201(EN) 45

7.1 materials def (mat1, mat2 etc.)

Most computations involve materials of different types. For example, calculationsof entry into houses almost always involve concrete, fill and undisturbed soil. InRnMod3d, each control volume can be set to a given material. These materials arenamed: mat1, mat2 etc. The assignment of control volumes to materials takes placethrough the user-defined procedure pointed to by materials def. The simplestexample is if all control volumes are set to be of the same material:

Example 17 Homogeneous problem.

function mymaterials(i:itype;j:jtype;k:ktype):mattype;

begin

mymaterials:=mat2;

end;

where the control variable materials def must be set to mymaterials. An exam-ple involving four materials is given below. The materials mat1, mat2, mat3 andmat4 could be layers of soil in a laboratory column experiment. The in-proceduredescribed in Section 6 are useful for the task.

Example 18 Inhomogeneous problem.

function mymaterials(i:itype;j:jtype;k:ktype):mattype;

var mat:mattype;

begin

mat:=mat1;

if in_cube([inside,eqAB],i,xFix1,xFix2,j,yFix1,yFix2,k,zFix1,zFix2) then

mat:=mat2;


mat:=mat3;


mat:=mat4;

mymaterials:=mat;

end;

A simple way to verify that the geometrical extension of the involved materialshas been defined correctly, is to set the control variable wr material volume totrue. This will make RnMod3d output a list of the total volume occupied by eachof the defined materials (see page 23). Other material-specific information is alsooutput.It should be observed that the use of materials mat1, mat2 etc. is just a ”book-

keeping tool”. As will be described in the following, this tool is useful when ma-terial properties are defined, however, it is perfectly all right not to use the tool.This can be done by setting all control volumes to be of the same type (e.g. mat1).

7.2 Porosity, e def

The control variable e def has to point to the user-defined procedure where theporosity is defined. The following example shows how to set all control volumesto have a porosity equal to 0.5:

Example 19 Homogeneous porosity.

function e_test(i:itype;j:jtype;k:ktype):datatype;

begin

e_test:=0.5;

end;

where e def must be set to e test. In principle we can assign an individualporosity for each control volume (i,j,k):

46 Risø-R-1201(EN)

Example 20 Porosity specified by index variables.


var ee:datatype;

begin

ee:=0.5;

if i=10 then ee:=0.3;

if j=5 the ee:=0.4;

if znod(k)>5.33 then ee:=0.2;

e_test:=ee;

end;

If the grid has been split into four materials called mat1, mat2, mat3 and mat4, withporosities 0.5, 0.4, 0.3 and 0.3, respectively, we would define e test as follows:

Example 21 Blockwise (in)homogeneous porosity.


var ee:datatype;

begin

case materials_def(i,j,k) of

mat1: ee:=0.5;

mat2: ee:=0.4;

mat3: ee:=0.3;

mat4: ee:=0.3

else

error_std(’e_test’,’Unknown material’);

end; (* case *)

e_test:=ee;

end;

Observe, the following: (1) the function pointed to by the control variable materials defis used to look up the type of material assigned to each individual control volume.(2) It was said that only material mat1, mat2, mat3 and mat4 were used in the ap-plication. Hence, porosities are defined only for these materials. If materials defreturns some other material, an error has occurred. We therefore stop the com-putations by calling error std. The use of error procedures is described in Sec-tion 13.6. (3) Imagine that mat2 represents the soil layer from the atmosphericsurface down to 0.3 m. If we at some point discover, that in fact this soil layergoes down to a depth of only 0.2 m, then we make changes only in the ”materials”function. The porosity of mat2 is unchanged.In the examples above, each individual material were assumed to be homoge-

neous. Material properties can, however, easily be non-constant. A simple example,is if the porosity of mat3 changes with depth as described in the example page ??:

Example 22 Depth dependent porosity.


var ee:datatype;

begin

case materials_def(i,j,k) of

mat1: ee:=0.5;

mat2: ee:=0.4;

mat3: ee:=0.50+0.125*znod(k);

mat4: ee:=0.3

else

error_std(’e_test’,’Unknown material’);

end; (* case *)

e_test:=ee;

end;

The function znod(k) is used to get the physical z-coordinate (in meters) of controlvolume (i,j,k) (see Section 5.9).

Risø-R-1201(EN) 47

In some cases, the porosity has been measured in the field for a number of depths(e.g. from 0 to 3 m at 10 cm intervals). Such results can easily be used by RnMod3din the following way: First, the data are gathered in a file. Then a Pascal functionis written that can read the file and perform (e.g. linear) interpolation betweenmeasurement points. Here we imagine a function called look up etot(depth). Itsimply returns the estimated porosity at any given depth within the measurementinterval. Finally, the function is used by the porosity procedure in RnMod3d:

Example 23 Depth dependent porosity read from a file.


var depth:datatype;

begin

depth:=-znod(k);

e_test:=look_up_etot(depth);

end;

Parameters can also change in time. See Section 11.5.

7.3 Partition-corrected porosity, beta def

The control variable beta def links RnMod3d to the user-defined function wherethe partition-corrected porosity β is defined. For example, consider a homogeneousmedium with air porosity (εa) equal to 0.2, water porosity (εw) equal to 0.2, anda partition coefficient L equal to 0.36, we would write:

Example 24 Homogeneous β.

function my_beta(i:itype;j:jtype;k:ktype):datatype;

var ea,ew,L:datatype;

begin

ea:=0.2;

ew:=0.2;

L:=0.36;

my_beta:=ea+L*ew;

end;

and set beta def to my beta.The physical meaning of the beta def-procedure is different in radon problems

and in problems of soil-gas transport. This is discussed Section 3.4.

7.4 Generation rate, G def

The gereration rate of radon per pore volume is defined by the function pointedto by G def. For example, consider a homogeneous medium with generation rateequal to 0.209838 Bq s−1 per m3. In dry soil this gives a deep-soil radon concen-tration equal to G/λ ≈ 100 kBq m−3. In that case we would write:

Example 25 Homogeneous G.

function my_G(i:itype;j:jtype;k:ktype):datatype;

begin

my_G:=0.209838;

end;

and set G def to my G.The physical meaning of the G def-procedure is different in radon problems and

in problems of soil-gas transport. This is discussed in Section 3.4.

48 Risø-R-1201(EN)

7.5 Decay constant, lambda def

The decay constant of radon is defined by the function pointed to by lambda def.Normally, the decay constant is set to the same value in all parts of the computa-tional plane:

Example 26 Decay constant λ.

function my_lambda(i:itype;j:jtype;k:ktype):datatype;

begin

my_lambda:=2.09838e-6;

end;

and set lambda def to my lambda. If the soil-gas pollutant in question is not radon,but some trace chemical being removed from the soil by a first-order process, ”λ”could indeed change from place to place. Ventilation can also be lumped into ”λ”.The physical meaning of the lambda def-procedure is different in radon prob-

lems and in problems of soil-gas transport. This is discussed in Section 3.4.

7.6 Diffusivity, D def

The bulk diffusivity of radon is defined by the function pointed to by D def. Thetechnique is identical to that described for β, ε, G, and λ. Only one thing is differ-ent: Diffusitivity may be anisotropic. The header of the D def-function thereforeincludes a directional parameter. We return to this shortly. In the situation withhomogeneous isotropic soil and a bulk diffusivity equal to 10−6 m2 s−1, we define:

Example 27 Homogeneous isotropic D.

function my_D(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

begin

my_D:=1e-6;

end;

with D def set to my D. The dir parameter in the header of the diffusion functioncan take the values: xdir, ydir and zdir. If the diffusivity is homogeneous butanisotropic with D =10−6 m2 s−1 in the x and y directions (i.e. horizontally) and0.2 · 10−6 m2 s−1 in the z direction (i.e. vertically), we define:

Example 28 Homogeneous anisotropic D.

function my_D(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

var dd:datatype;

begin

case dir of

xdir,ydir: dd:=1e-6;

zdir: dd:=0.2e-6;

else

error_std(’my_D’,’Unknown direction’);

end;

my_D:=dd;

end;

Example 29 shows how the diffusion constant found by Rogers and Nielson canbe implemented in RnMod3d.The physical meaning of the D def-procedure is different in radon problems and

in problems of soil-gas transport. This is discussed in Section 3.4.

7.7 Moisture

RnMod3d uses only the material properties defined by the functions pointed toby beta def, e def, G def and lambda def. When these parameters are derived

Risø-R-1201(EN) 49

Figure 14. Sketch of the geometry used in case 1 of the ERRICCA model intercom-parison exercise [An99a]. (A) is the soil column viewed from the top, (B) is a sideview of the column. (C) and (D) are plots of porosity (ε) and moisture saturation(m = θv), respectively.

from (or related to) some common quantities such as soil moisture content or soiltemperature, it is often helpful to introduce such quantities explicitly in the job file.The example below shows how case 1 in the ERRICCA model intercomparisonexercise was modelled with RnMod3d (see [An99a]). The problem is sketched inFigure 14. z goes from 0 at the atmospheric surface to −3.0 m. The function mdescribes the moisture profile (i.e. θv as defined page 7). Rogers and Nielson’sformula [Rog91A, Rog91B] is used for the calculation of diffusivity. pw(x,y) is anin-built power function that returns xy.

Example 29 Material properties for ERRICCA case 1.

function e(i:itype;j:jtype;k:ktype):datatype;

begin (* Porosity *)

if znod(k)>-1 then e:=0.5 else e:=0.3;

end;

function m(i:itype;j:jtype;k:ktype):datatype;

var mres,depth:datatype;

begin (* Moisture saturation, m=ew/e *)

depth:=znod(k);

mres:=0.20-0.4*depth;

50 Risø-R-1201(EN)

if mres>1 then mres:=1;

if mres<0 then error_std(’m’,’m<0!’);

m:=mres;

end;

function beta(i:itype;j:jtype;k:ktype):datatype;

var ea,ew,L:datatype;

begin (* Partition-corrected porosity *)

ew:=m(i,j,k)*e(i,j,k); (* Water porosity *)

ea:=e(i,j,k)-ew; (* Air porosity *)

L := 0.3565; (* L Ostwald *)

beta:=ea+L*ew;

end;

function D(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

const Da=1.1e-5;

var e1,b1,m1:datatype;

begin (* Bulk diffusivity *)

e1:=e(i,j,k);

m1:=m(i,j,k);

b1:=beta(i,j,k);

D:=b1*Da*e1*exp(-6*m1*e1-6*pw(m1,14*e1));

end;

function G(i:itype;j:jtype;k:ktype):datatype;

var Ema,rhog,etot:datatype;

begin (* Radon generation rate *)

Ema:=10.0; (* Emanation rate, atoms/kg/s *)

rhog:=2.65e3; (* Grain density, kq/m3 *)

etot:=e(i,j,k);

if etot<=0 then error_std(’G’,’etot<=0!’);

G:=rhog*(1-etot)/etot*lambda_use*Ema;

end;

function lambda(i:itype;j:jtype;k:ktype):datatype;

begin (* Decay constant *)

Lambda:=lambda_use;;

end;

where we have set:

e_def := e;

beta_def := beta;

G_def := G;

D_def := D;

lambda_def := lambda;

and where lambda use is a user-defined constant set to 2.09838 · 10−6 s−1.

8 Flux probes (Flx1, Flx2 etc.)

The primary output of many simulations is the total flux across some plane surface.For example, in house simulations the primary output is the radon entry rate orthe soil-gas entry rate into the house.

8.1 Fluxes between individual pairs of control vol-umes

The basic fluxes in RnMod3d are those that go between individual pairs of (ad-jacent) control volumes. An example is the flux called je in Figure 11, page 35.

Risø-R-1201(EN) 51

This is the flux in the x-direction between control volume P and E. Such fluxescan be found with the function called node flux(dir,i,j,k). The parameter dirtells which side of control volume (i,j,k) that is considered: west, east, south,north, bottom, or top. For example, to find the east-side flux (je) of controlvolume (3,70,4) simply call the function as:

writeln(’je = ’,node_flux(east,3,70,4));

The same value be found if the west-side flux of the adjacent control volume at(i.e. at (4,70,4)) is looked up:

writeln(’jw = ’,node_flux(west,4,70,4));

If a soil-gas problem is considered, then node flux returns a flow of soil gas inUnitsunits of m3 s−1. If a radon problem is considered, then the result is in units ofBq s−1. If the flux density is needed, then the flux should be divided by the areaof the interface between the control volumes. This area can be found as describedin Section 5.9.

8.2 update flxval

RnMod3d has a way for keeping track of fluxes involving many control volumes.Essentially it is possible to ask RnMod3d to integrate fluxes over specific areas (notjust between single pairs of control volumes). These ”flux measurement probes”can be used to monitor fluxes wherever the user wants.The flux probes are called Flx1, Flx2 etc. These probes are positioned by

the user through the user-defined procedure pointed to by the control variableflux def. The idea can be explained with the example given below:

Example 30 Simple flux measurements.

procedure myfluxes(i:itype;j:jtype;k:ktype);

begin

if in_plane([inside,eqAB],

i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix1,zFix1) then update_flxval(Flx1,top,i,j,k,plus);

end;

where we have set flux def equal to myfluxes. This procedure defines Flx1 asthe grand sum of the fluxes through the ”top” faces of all single control volumesthat are part of the xy-plane pin-pointed by the in plane function (Ω).

Flx1 =∑Ω

jtop(i, j, k) (45)

Essentially, the measurements occur along individual connectors. In the case ofnoFlow or nill connectors, there will be no contribution to the flux measurement.For connectors of the type std, the flux will be assessed using an approximateversions of equation 41 for radon and equation 43 for soil gas.It is possible to use any of the control-volume faces for flux measurements–not

just the ”top” as in the above example. To do that simply call to update flxvalwith the second parameter set to bottom, east, west, north or south.Fluxes are taken to be positive if they are in the direction of the x, y or z axis.plus and minus

The last parameter in the update flxval-call can be used to change the sign whenthe control-volume fluxes are added. plus means no change of sign, minus meansthat the sign should be changed. In example 30, Flx1 will therefore be positive ifthe flux is in the (positive) direction of the z-axis. A more complicated example,demonstrates the use of plus and minus

52 Risø-R-1201(EN)

Example 31 More complex flux measurements.

procedure myfluxes(i:itype;j:jtype;k:ktype);

begin


i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix1,zFix1) then

begin (* zFix1 plane *)

update_flxval(Flx1,top,i,j,k,plus);

update_flxval(Flx3,top,i,j,k,plus);

end;


i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix2,zFix2) then

begin (* zFix2 plane *)

update_flxval(Flx2,bottom,i,j,k,plus);

update_flxval(Flx3,bottom,i,j,k,minus);

end;

end;

In this example, Flx1 is the flux across the zFix1 plane as before. The newfluxes are Flx2 and Flx3. Flx2 is the flux across the zFix2 plane, and Flx3 isthe difference between Flx1 and Flx2. Adding fluxes in this fashion is useful forexample, in house simulations with more than one entry point.Sometimes it is important to use the ”correct” control-volume face for the flux Flux probes at nill

connectors always returnzero

measurements. In the example below, we consider a cubic grid (it could be a cubicsample of concrete). We want to measure the flux through each of the six faces(for example, it could be the total radon exhalation from the sample). If we usemyfluxes not ok for the flux measurements the model will report all fluxes Flx1to Flx6 to be zero! The problem is that we conduct flux measurements on theouter rim of the cube (i.e. between the outer control-volume nodes and nowhere;these connectors are of the type called nill). The fluxes defined by myfluxes okare the correct ones.

Example 32 Flux measurements at grid boundaries.

procedure mygrid;

begin



set_axis_single(xFix1,xFix2,5,Focus,1.0);



set_axis_single(yFix1,yFix2,5,Focus,1.0);

set_FixVal(zFix1,0.0); (* z-axis *)


set_axis_single(zFix1,zFix2,5,Focus,1.0);

end;

procedure myfluxes_not_ok(i:itype;j:jtype;k:ktype);

begin


i,xFix1,xFix1,

j,yFix1,yFix2, (* xFix1 plane *)

k,zFix1,zFix2) then update_flxval(Flx1,west,i,j,k,plus);


i,xFix2,xFix2,


k,zFix1,zFix2) then update_flxval(Flx2,east,i,j,k,plus);


i,xFix1,xFix2,

Risø-R-1201(EN) 53

j,yFix1,yFix1, (* yFix1 plane *)

k,zFix1,zFix2) then update_flxval(Flx3,south,i,j,k,plus);


i,xFix1,xFix2,


k,zFix1,zFix2) then update_flxval(Flx4,north,i,j,k,plus);


i,xFix1,xFix1,

j,yFix1,yFix2, (* zFix1 plane *)

k,zFix1,zFix1) then update_flxval(Flx5,bottom,i,j,k,plus);


i,xFix1,xFix2,



end;

procedure myfluxes_ok(i:itype;j:jtype;k:ktype);

begin


i,xFix1,xFix1,


k,zFix1,zFix2) then update_flxval(Flx1,east,i,j,k,plus);


i,xFix2,xFix2,


k,zFix1,zFix2) then update_flxval(Flx2,west,i,j,k,plus);


i,xFix1,xFix2,


k,zFix1,zFix2) then update_flxval(Flx3,north,i,j,k,plus);


i,xFix1,xFix2,


k,zFix1,zFix2) then update_flxval(Flx4,south,i,j,k,plus);


i,xFix1,xFix1,




i,xFix1,xFix2,



end;

8.3 FlxVal

After each run model call, the results of flux measurements are stored in an arraycalled FlxVal. This array contains two components: J and Q. In radon simulations,J is the calculated flux of radon in Bq s−1 and Q is the (imported) soil-gas flow ratein m3 s−1. For example, we can write the result of the Flx2 probe measurementsas follows:

writeln(’The results are: ’,FlxVal[Flx2].J,’ Bq/s ’,FlxVal[Flx2].Q,’ m3/s ’);

In a soil-gas simulation FlxVal[Flx2].J is the calculated soil-gas flow rate inm3 s−1, and FlxVal[Flx2].Q has no meaning.This is discussed in Section 3.

54 Risø-R-1201(EN)

8.4 Standard flux probe output

The results of flux measurements are available as standard output in the LOG-fileand on the screen. The output is discussed in Section 4.36. The output can beturned on and off with the control variables:

wr_final_results_logwr_final_results_screen

Preliminary flux estimates can also be monitored as RnMod3d solves the problemiteratively. This is done with:

wr_flux_during_calc_logwr_flux_during_calc_screen

In fact, the flux measurements should normally be part of the requirement for con-vergence. This is done with the control variable flux convset which can containa Pascal set of flux probes. For example, Flx2, Flx4 and Flx6 can be included inthe convergence test with:

flux_concset:=[Flx2,Flx4,Flx6];

See page 25 and Section 10.4 for further details.

9 Field probes (Obs1, Obs2 etc.)

RnMod3d is also equipped with a framework for doing radon concentration measure-ments (in radon simulations) and pressure measurements (in soil-gas simulations).The probes are called Obs1, Obs2 etc.

9.1 ObsVal

Probe Obs2 can be set to monitor the radon concentration in control volume(i,j,k) = (3,4,2) as follows:

Example 33 Simple radon concentration probe.

procedure myprobes;

begin

obsval[obs2]:=GP[3]^[4]^[5].c;

end;

The probe definition must be linked to RnMod3d with the control variable assign-ment:

probe_def:=myprobes;

The example makes direct access to the main data structure GP, where

GP[i][j][k].c

is the field value. GP also has a record that tells if the field value is valid or not:

GP[i][j][k].valid_fieldvalue

This can be used as follows:

Example 34 Simple radon concentration probe with test of valid fieldvalue.

Risø-R-1201(EN) 55

procedure myprobes;

var i:itype;j:jtype;k:ktype;

begin

i:=3; j:=4; k:=2;

obsval[obs2]:=-999;

if GP[i]^[j]^[k].valid_fieldvalue then

obsval[obs2]:=GP[i]^[j]^[k].c;

end;

This is one way to avoid problems with ”dead nodes” (see page 40). Further detailsabout GP are given in Section 14.After each run model call, the results of field-value measurements can be found

in the array called ObsVal. This array is similar to the one used for flux measure-ments (FlxVal, see Section 8.3). In radon simulations ObsVal is the concentrationUnitsof radon in Bq m−3. In soil-gas simulations, ObsVal is the pressure in Pa. Forexample, we can write the result of the Obs4 probe measurements as follows:

writeln(’The result is: ’,ObsVal[Obs4].c,’ Bq/m3 ’)

Normally, we are not interested in field values for specific control volumes(i,j,k) since their significance change with the grid resolution. Instead we need tofind field values for physical (x, y, z) locations. RnMod3d has four procedures/functionfor this purpose. These are described in the Section 9.3 to 9.8.

9.2 Standard field probe output

The results of field measurements are available as standard output in the LOG-fileand on the screen. The output is discussed in Section 4.38. The output can beturned on and off with the control variables:

wr_final_results_logwr_final_results_screen

Preliminary field probe estimates can also be monitored as RnMod3d solves theproblem iteratively. This is done with:

wr_probes_during_calc_logwr_probes_during_calc_screen

In fact, the field probe measurements should normally be part of the requirementfor convergence. This is done with the control variable probe convset which cancontain a Pascal set of field probes. For example, Obs2, Obs4 and Obs6 can beincluded in the convergence test with:

probe_concset:=[Obs2,Obs4,Obs6];

See page 25 and Section 10.4 for further details.

9.3 fieldvalue

Assume we need to estimate the radon concentration at some physical location(x, y, z). We can use the function:

fieldvalue(xp,yp,zp,valid)

for this purpose. The parameter valid is a boolean return variable that tells if thecall was successful or not. The function finds the field value by linear interpolationamong the nearest control volumes. To monitor the radon concentration at (x, y, z)= (2.3 m,−0.2 m,10 m) with probe Obs4 we could define a procedure as follows:

56 Risø-R-1201(EN)

Example 35 Radon concentration probe at physical location (x, y, z).

procedure myprobes;

var cc:datatype; valid:boolean;

begin

cc:=fieldvalue(2.3,-0.2,10,valid);

if valid then obsval[obs4]:=cc else obsval[obs4]:=0

end;

end;

9.4 fieldvalue2D

In two-dimensional simulations (set by the control variable geometry), there isno y-coordinate. A special two-dimensional version of fieldvalue therefore isavailable:

fieldvalue2D(xp,zp,valid)

9.5 get fieldvalue

In simulations of actual soil-gas radon measurements the physical probe locationsmay not be known exactly. For example, we may not know the exact depth fromwhich some specific soil-gas radon is taken. For example, we may assess that thesampling depth is ”1 m ± 5 cm”, where the 5 cm is one standard uncertainty.In a model simulation of the sampling, we may want to assess the influence ofthe uncertainty of the sampling depth on the radon concentration determination.Clearly, the answer depend on the gradient of the radon concentration field at thesampling location. RnMod3d has a (very) simple procedure which can be used forthe assessment:

get_fieldvalue(xp,dxp,yp,dyp,zp,dzp,c,dc,valid)

(xp,yp,zp) is the (x, y, z)-coordinate of the field location of interest. dxp is theuncertainty of the xp-coordinate. dyp and dzp are the uncertainties of the twoother coordinates. The estimated field value is returned in the variable c and theassociated uncertainty is in dc. The variable valid tells if the estimated result isvalid or not. The uncertainty is estimated as:

dc =

√(∂c

∂xdx)2

+(∂c

∂ydy)2

+(∂c

∂zdz)2

(46)

9.6 get fieldvalue2D

The two-dimensional version of the previous procedure is:

get_fieldvalue2D(xp,dx,zp,dz,c,dc,valid)

9.7 get avgfield

To get the average field over an entire region, the procedure:

get_avgfield(x1,x2,y1,y2,z1,z2,ddd,c,dc)

can be used. The region is a box with the physical coordinates given by x1, x2, y1,y2, z1 and z2. ddd is the resolution (e.g. 0.01 meter). The main result is returnedin the variable c. The variable dc returns the variability of the result.

Risø-R-1201(EN) 57

9.8 get avgfield2D

The two-dimensional version of the previous procedure is:

get_avgfield2D(x1,x2,z1,z2,ddd,c,dc)

10 Solution procedure

Essentially, RnMod3d solves a matrix equation of the form:

Ac = b (47)

This equation is set up on the basis of equation 44, page 14. A is a matrix ofcoefficients. These tell how the field quantity (i.e. pressure or radon) moves fromone control volume to another. Hence, matrix elements reflect material propertieslike diffusivity, the size of control volumes etc. Luckily most of the elements are zeroas transport can take place only between adjacent control volumes. c represents afield of radon concentrations or pressures. If there are 10 000 nodes in the grid thenc is a column vector with 10 000 elements. Likewise, A is a matrix with 10 000 by10 000 elements. Finally, b is a vector with coefficients that relate to the sourceterm. In radon problems, b reflects the radon generation rate. In time-dependentproblems, b also include information about the field at the previous time step.Because of the shear size of a typical matrix A, this equation cannot be solved

by simple matrix inversion. Instead iterative solution procedures are used. Theiterative solution procedures work as follows: First, a solution c0 is guessed. Thenon the basis of the procedure, an improved guess c1 is found. From this, a newfield c2 is found etc. This is continued until convergence is met.

10.1 First guess

There are two possible initial field guesses:

• If the control variable import final field guess is set to true, then themodel imports the initial field guess from the file with the name given byimport field name. The field could come from a file saved after an earliercalculation, where the computation was not carried out all the way to con-vergence. The earlier calculation could also have been subject to less strictcriteria for convergence. See Section 4.19, page 17.

• If the control variable import final field guess is set to false, then theinitial field guess equals whatever field is stored in the main data structureGP. In the very first model run this field is zero all over: c0 = 0.

10.2 Relaxation

To minimize the time it takes to reach convergence, computations are often over-relaxed. The idea is quite simple. Take a look at one particular node in the grid.After the i’th iteration, the field value at this node is ci. After the next iteration anew value called ci+1 is obtained. Each iteration leads to an improved estimate ofthe true value. In the beginning, relatively large steps are taken (i.e. the differencebetween ci and ci+1 is large), but eventually step sizes get smaller. Now, if weknow in what ”direction” the true value can be found, why not take a larger step?With relaxation, we multiply the step size by a factor α:

ci+1,R = ci + α(ci+1 − ci) (48)

58 Risø-R-1201(EN)

If the relaxation factor α is too large, unstability will result. The relaxation factorcan be set by the user with the control variable relax factor, see Section 4.52,page 25.

10.3 Iterative solution procedures

The two solution procedures available in RnMod3d are both iterative:

• Gauss-Seidel: This is a point-iterative solution procedure. The grid is sweptpoint by point. For each point, we calculate an improved estimate of the fieldvalue directly from equation 49 as:

cP,i+1 =aEcE,i + aW cW,i + aNcN,i + aScS,i + aT cT,i + aBcB,i + b

aP(49)

where cP,i+1 is the new improved estimate and all other field values: cE,i, cW,i

etc. are from the previous iteration.

• Thomas: The Thomas algorithm is similar to that of Gauss-Seidel. The onlydifference is that the Thomas procedure works line by line. This means fasterconvergence. The reason is that e.g. the impact of boundary conditions canreach all the way to the other side of the computational plane in one singleiteration. To further speed up convergence, the direction of lines is alternatedfrom one iteration to the next: First, a line parallel to the x-axis is selected,then one parallel to the y-axis and finally one parallel to the z-axis.

The solution procedure is selected with the control variable solver def (see Sec-tion 4.50).

10.4 Criteria for convergence and residuals

In RnMod3d, the convergence criterion consists of three elements:

• The first criteria for convergence is that all flux probes included in flux convsetchange by less than the value given by max change (see Section 4.53 and 4.59).For example, imagine that flux convset := [flx1,flx4] and max change:= 1e-4, then convergence is not met before the results for flux probe flx1and probe flx4 change by less than 0.01 % per iteration. The values of otherflux probes (e.g. flx2 and flx3) play no role for the convergence. Observe,that if the final value of one of the flux probes is close to zero, then this canbe a problematic requirement. It is best to avoid flux probes with values closeto zero in flux convset. Flux probes can be located anywhere in the com-putational plane as described in Section 8. If none of the flux probes shouldbe part of the convergence criteria, then simply use: flux convset := [ ].The convergence of flux measurements can be monitored during the iterativeprocedure as described Section 4.36, page 21.

• The second criteria for convergence is that all field value probes includedin probe convset change by less than the value given by max change. Forexample, imagine that probe convset := [obs3] and max change := 1e-4,then convergence is not met before the results for probe obs3 change by lessthan 0.01 % per iteration. It is best to avoid probes with values close to zeroin probe convset. The probes can be located anywhere in the computationalplane as described in Section 9. It seems best to place probes close to regionsof main interest. Probes can also be placed in ”corners” of the computationalplane where the field (by experience) takes a long time to settle down. Theconvergence of field-value measurements can be monitored during the iterativeprocedure as described Section 4.38, page 21.

Risø-R-1201(EN) 59

• The final requirement for convergence is that the sum of residuals is lessthan max residual sum (i.e. sufficiently small). This criterion is based on therecommendations given by Patankar [Pa88, p. 236]. After the i’th iteration,the guessed solution of the matrix equation is ci. To evaluate how close thissolution is to the right one, we insert ci into equation 47 and calculate theresidual vector ri:

ri = Aci −b (50)

We then define the absolute sum of residuals Ri (after the i’th iteration) as:

Ri =∑

|ri| (51)

where (as before) the sum is over all nodes in the grid. In the end, Ri shouldapproach zero. However, as already mentioned, we consider the problem to besolved when Ri < max residual sum. To better understand the significanceof Ri, it is sometime of interest to know the value:

R0 =∑

|b| (52)

where the sum is over all nodes in the grid. This is the value of Ri thatis obtained when ci = 0. In the end, Ri should reach a value that is lowcompared with R0. In fact, RnMod3d gives a warning if Ri multiplied bythe constant residual sum warning limit is not less than R0. By defaultresidual sum warning limit is set to 100. The results of Ri, R0 and themaximum value of ri as well as its location in the computational plane canbe output from RnMod3d during and after the iteration solution procedure,see Section 4.34. The value of R0 is output as Abs. sum of bs.

It takes time to test for convergence. Therefore it is best not to do so in everysingle iteration. How often the convergence is tested can be set by the controlvariable conv evaluation period, see Section 4.55.Convergence is not the only thing that controls when the iterative solution pro-

cedure stops. See min iterations (Section 4.56), max iterations (Section 4.57),and max time (Section 4.58).

10.5 Scheme (space)

The coefficients aE , aW etc. in equation 44 can be calculated in a number of ways.Essentially, the different possibilities relate to the assumed field profile betweenadjacent nodes. In other words there are different interpolation schemes available.For example, the so-called central scheme is based on the assumption of a linearprofile. This is a good approximation if diffusion dominates in the region betweenthe two nodes. On the other hand, if the profile is dominated by advection, thenthe profile will be shifted to one side. This is used in the so-called up-wind scheme.In real problems, the best profile is somewhere between these two extremes. InRnMod3d the following schemes are available:

powerlawcentralupwindhybridexact

For example, to use the scheme based on the exact solution of the diffusive-advection equation, simple set the control variable scheme to exact. See Sec-tion 4.51, page 25.

60 Risø-R-1201(EN)

10.6 Scheme (time)

The fully implicit scheme is used (see [Pa80, p. 56]). No alternatives have beenimplemented.An important feature of the fully implicit scheme is that steady-state fields can

be calculated in one single (large) time step. Another feature is that solutions areunconditionally stable. However, the accuracy is only first order in time, so smalltime steps are needed to ensure good accuracy [Ve95, p. 173].

11 Time dependency

11.1 solution := steady

If the control variable solution is set to steady, then RnMod3d performs a calcu-lation as if the conditions defined by the coefficient functions (i.e. D def, beta defetc.) and the boundary conditions (i.e. boundary conditions def) have existedsince t = −∞. When run model is called, the solution will reflect these condi-tions. The final solution does not depend on the initial field. This is a so-calledsteady-state solution.

11.2 solution := unsteady

If the control variable solution is set to unsteady, then RnMod3d performs a time-dependent calculation. Each time run model is called, the solution is progressedby one single time step dtim. Normally it is necessary to split the simulation intomany (small) time steps. Hence run model is called many times.The ”global” time is given by the variable tim. Both dtim and tim are measured

in seconds. Calculation of time-dependent problems are simple to set up. In thefollowing example, we first calculate a steady-state field. Then we perform a time-dependent calculation where each time step is given by dtim. Initially, dtim isonly 10 seconds, but we let dtim expand by 20 % in each step. After 12 hours (i.e.when tim > 12 ·3600 seconds) we perform one additional steady-state calculation.

Example 36 Prototype time-dependent problem.

solution := steady;

tim := 0;

dtim := 0;

run_model; (* Initial field at t=0 *)

solution := unsteady;

dtim := 10;

repeat

dtim:=dtim*1.2; (* Take larger time steps *)

tim:=tim+dtim; (* Update tim *)

run_model; (* Advance the field by dtim *)

writeln(’’Results for time = ’,tim/3600,’ hr’,’ Flux = ’,FlxVal[Flx1].j,’Bq/s’)

until (tim>12*3600);

solution:= steady;

run_model;

close_model;

The only thing that binds two consecutive model runs together is the calculatedfield: The ”old” field (in GP) tells how much radon (or soil gas pressure) is stored inthe computational grid. The new model run simply updates the field in accordancewith the problem specification. In fact almost everything is set up from scratch

Risø-R-1201(EN) 61

before each time step. Hence everything that controls coefficients and boundaryconditions can be time-dependent. The sole purpose of the variable tim is to havea global time that can be referred to in the procedures that change in time. Inother words, tim is not used explicitly by the model itself.Observe, that if a control variable such as wr axes is set to true (see Section 4.42,

page 22) then the grid is output every time run model is issued. In a time depen-dent problem, it is therefore best to set such control variables to false after thefirst run. Otherwise the LOG-file will be flooded.

Order of statements

In example 36, the order of the statements:

tim:=tim+dtim;

and

run_model;

is important. The reason is as follows:

1. tim is used to control changes in boundary conditions etc. as described in thefollowing (see e.g. Section 11.4).

2. When the statement run model is issued, the boundary conditions etc. mustbe those that prevail at tim := tim + dtim.

Problems may occur if the order of tim:=tim+dtim and run model is reversed.

11.3 Initial conditions

There are three possible ways to specify initial conditions:

• The initial field may be read from a file. This is accomplished by setting thecontrol variable: import initialfield to true as described in Section 4.18,page 17. This is, however, only meaningful if the initial field has been cal-culated on the basis of a grid identical to that used in the (new) computation.Also observe, that after the first time step has been taken, import initialfieldshould be set to false. A typical example of this type of initial condition isgiven next. The initial field is assumed to be in the file called c0.dat.

Example 37 Initial field in a file.

import_field_name := ’c0.dat’

import_initialfield := true;


dtim := 200;

tim := 0;

repeat

tim:=tim+dtim;

run_model;

import_initialfield := false; (* No further imports *)


To store any field (for reuse as an initial field in some later calculation) simplyuse the control variable export field (see Section 4.20, page 18).

• The initial field is specified in a function. Imagine that the initial field shouldequal 3000 Bq m−3 at all grid points. This can be done as follows. First, wedefine a function that describes the initial field:

Example 38 Initial field by function (part 1).

62 Risø-R-1201(EN)

function myfunction(i:itype; j:jtype; k:ktype):datatype;

begin

myfunction := 3000

end;

Then we make the appropriate reference in the body of the program with thecontrol variable initialfield def. For example, we may write:

Example 39 Initial field by a function (part 2).

initialfield_def := myfunction; (* Initial cond. by function *)


dtim := 200;

tim := 0;

repeat

tim:=tim+dtim;

run_model;

initialfield_def:= nil; (* No further initial fields *)


It is easy to define more complicated fields. The same methods as given in theexample page 47 can be used. Additional details can be found in Section 4.17,page 17.

• The initial field is ”calculated on the fly”. For example, we may start amodel simulation by calculation of some steady-state field. This is the methoddemonstrated in example 36. The point is that all model runs (steady-stateor time-dependent) end up with a field that can be used as initial conditionfor further computations.

11.4 Time-dependent boundary conditions

The first example shows how a boundary condition can change in time. We considerthe problem when the pressure at the boundary (e.g. the atmospheric surface)changes periodically in time as:

p = cos(2πT0

t) (53)

where T0 is a period time (e.g. 12 hours). If the pressure at the boundary is calledfixed1, then we can implement the problem as follows:

Example 40 Time-dependent change of boundary conditions.

procedure boundary_conditions(i:itype;j:jtype;k:ktype);

const T0=12*3600;

begin

cBC[fixed1]:=cos(2*pi/T0*tim);


i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix3,zFix3) then set_node(i,j,k,fixed1);

end;

As described in Section 6.7, page 41, the fixed-value nodes are controlled bycBC[fixed1], cBC[fixed2] etc. It is possible also to change the types of nodesin time. For example, imagine that at tim equal to 200 seconds, the boundary atzFix1 should change from being fixed at 0 to being closed off for transport. Wecould implement this as follows:

Example 41 Time-dependent change of type of boundary conditions.

Risø-R-1201(EN) 63


begin

cBC[fixed1]:=0;


i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix1,zFix1) then

begin

if (tim<200) then

set_node(i,j,k,fixed1)

else

set_node(i,j,k,free)

end;

end;

The use of set node is described in Section 6.5, page 41.

11.5 Time-dependent material properties

Changes of coefficients in time, can be implemented as follows:

Example 42 Time-dependent diffusivity.


var DD:datatype;

begin

DD:=1e-5;

if tim>6*3600 then DD:=1e-8;

D:=DD;

end;

In this example, the diffusivity changes from 10−5 to 10−8 m2 s−1 as tim equals6 hours. Other coefficients like porosity, radon generation rate etc. can be madetime dependent in a similar fashion.

11.6 Time-dependent flow field of soil gas

In radon problems, the imposed flow field of soil gas may change in time. Forexample assume, that a flow field has been calculated previously, and that it isimported into the model run by setting flowfield := import (see Section 4.22,page 18). For the time period from 0 to 2 hours, we may want to use this flow fielddirectly in the radon calculation. Then we may want to decrease the flow field to30 % of the original value (see Section 4.23, page 18). When tim equals 12 hours,we may want to turn the flow field off. This can all be done as follows:

Example 43 Time-dependent adjustment of flow field in a radon problem.


dtim := 300;

tim := 0;

flowfactor := 1.0;

repeat

tim:=tim+dtim;

run_model;

wr_result_line; (* some user-defined procedure *)

if (tim>2*3600) then flowfactor:=0.3;

if (tim>12*3600) then flowfactor:=0.0;


close_model;

Another possibility, is that the flow field changes altogether. For example, imag-ine two flow fields have been calculated and stored in the files: Nwind.dat and

64 Risø-R-1201(EN)

Wwind.dat. The first could correspond to soil-gas flow created as a result of windfrom the north. The other could reflect wind from the west. We may want to seethe change in the radon field if the wind changes abruptly from north to westwhen tim equals 12 hours:

Example 44 Time-dependent shift in flow field in a radon problem.


dtim := 300;

tim := 0;

flowfactor := 1.0;

flowfield_name := ’Nwind.dat’

repeat

tim:=tim+dtim;

run_model;

if (tim>12*3600) then flowfield_name:=’Wwind.dat’;


close_model;

11.7 Full time dependency (cBUF1, cBUF2 and qBUF)

In time-dependent problems, RnMod3d simply updates the main data structure GPby one time step dtim each time run model is called. This procedure works wellif the problem concerns only time-dependent soil-gas transport or if it concernsonly time-dependent radon transport. In the general case, however, when bothproblems are time dependent, the radon simulation will destroy the state of thepressure field (in GP) and likewise, the pressure field simulation will destroy thestate of the radon concentration field. The model cannot ”remember” more thanone field at a time. To treat such problems, it is therefore necessary to be ableto store the state of all calculations in some other variable than the main datastructure GP. RnMod3d can use two buffers called cBUF1 and cBUF2 for the purpose.These buffers are dynamic variables that are created only when needed. There isalso a buffer called qBUF where the flow of soil gas can be stored. With these threebuffers, RnMod3d can keep track of two time-dependent problems concurrently.The use of buffers is controlled by the control variable use fieldbuffer. With

use fieldbuffer set to cBUF1 the next run model calculation is encapsulatedby the field buffer cBUF1. This means that the first thing that happens afterrun model has been called is that the main data structure GP is reset to the statein cBUF1 (the list of actions undertaken in run model is described in Section 14.7).If there is no such state in cBUF1 (which is always the case in the first run in a jobfile), GP is not affected by this. Then the computations are performed by RnMod3din the usual fashion. The last thing that happens before the run model procedureends is that the full state of the computed field is stored in the buffer cBUF1.Hence, the next time run model is called with use fieldbuffer set to cBUF1, thecomputations can resume from the state of this field. If the soil-gas problem isencapsulated by the buffer cBUF1, then the radon problem can be encapsulatedby cBUF2.The soil-gas and the radon problems are coupled to each other only by the flow

field of soil gas q (see equation 40). Luckily radon is present only in trace levels, sothe pressure field does not change with the radon concentration. Hence, there is nocoupling from the radon field back to the soil-gas problem: The soil-gas problemis completely independent of the radon problem.There are two methods with which the field of soil-gas flows can be transferred qBUF

from an ”ongoing” soil-gas simulation to an ”ongoing” radon problem:

• A file is used. This means that flowfield should be set to export in thesoil-gas problem, and to import in the radon problem.

Risø-R-1201(EN) 65

• The flow-field buffer (called qBUF) is used. In the soil-gas problem, flowfieldshould be set to export to qBUF. In the radon problem, flowfield shouldbe set to import from qBUF.

A prototype job file with full time dependency is shown in the following example.Observe how control variables have been split into three groups:

• Those control variables that are common for both the soil-gas and the radonproblem. These variables are given at the beginning of the main body of thejob file, and as they will be not overwritten in the following, these settings re-main valid throughout the job file. For example, both problems are calculatedwith the same grid: grid def := grid.

• Those control variables that are specific for the soil-gas problem. These vari-ables are collected in the procedure called define soilgas problem. For ex-ample, here the permeability of the soil is defined.

• Those control variables that are specific for the radon problem. These vari-ables are collected in the procedure called define radon problem. For exam-ple, here the radon generation rate is defined.

Example 45 Full time dependency.

program fxxxxprg;

...

procedure define_soilgas_problem;

begin

use_fieldbuffer := cBUF1;

flowfield := export_to_qbuf;

boundary_conditions_def := boundary_conditions_soilgas;

D_def := D_soilgas;

e_def := e_soilgas;

beta_def := beta_soilgas;

G_def := G_soilgas;

lambda_def := lambda_soilgas;

...

end;

procedure define_radon_problem;

begin

use_fieldbuffer := cBUF2;

flowfield := import_from_qbuf;

boundary_conditions_def := boundary_conditions_radon;

D_def := D_Rn;

e_def := e_Rn;

beta_def := beta_Rn;

G_def := G_Rn;

lambda_def := lambda_Rn;

...

end;

begin (* main *)

runid := ’xxxx’;

runtitle := ’Buffer test’;


geometry := cartesian3d;

grid_def := grid;

materials_def := materials;

...

tim :=0;

dtim:=200;

repeat

tim:=tim+dtim;

66 Risø-R-1201(EN)

define_soilgas_problem;

run_model;

define_radon_problem;

run_model;

until (tim>1000);

close_model;

end.

If more problems of the above nature are conducted within the same job file, dispose fieldbufferit may be necessary to reset the buffers. This can be done with the proceduredispose fieldbuffer. An example shows what to do.

Example 46 Use of dispose fieldbuffer.

...

begin (* main *)

runid := ’xxxx’;

runtitle := ’Buffer test’;



grid_def := grid;


...

tim :=0;

dtim:=200;

repeat

tim:=tim+dtim;


run_model;


run_model;

until (tim>1000);

grid_def := some_new_grid;

dispose_fieldbuffer(cBUF1); (* Reset buffers *)

dispose_fieldbuffer(cBUF2);

tim :=0;

dtim:=200;

repeat

tim:=tim+dtim;


run_model;


run_model;

until (tim>1000);

close_model;

end.

12 Special boundary conditions

The standard boundary conditions in RnMod3d are (as described in Section 6.1and 6.2):

fixed-value conditions where the field is fixed at a given level regardless of thetransport equations. For example, in a simulation of radon exhalation from

Risø-R-1201(EN) 67

the soil surface into open atmospheric air, we may want to set up a transportsimulation for the soil where the concentration at the soil surface is alwaysequal to 5 Bq m−3. Such a condition is modelled by setting all control-volumesat the boundary to be of type fixed1 where cBC[fixed1]:=5.

No-flow conditions where the flux is set to zero. For example, in a simulationof radon transport in soil, we may assume that at the ground-water level,there is no transport. This is accomplished by setting all bottom connectorsof the control-volumes next to the ground water to be of type noFlow.

In some radon simulations it is necessary to enforce other boundary conditions.The most important case is when part of the porous medium is in direct contactwith open air where radon may accumulate. This occurs, for example, in closed-chamber exhalation measurements. For example, a sample of concrete may belocated in a small closed chamber where the air is well mixed by fans [An99a,An99b]. This section tells how to do treat such problems.

12.1 Trial-and-error by hand

Clearly the radon concentration in the chamber depends on the flux out of thesample. However, the opposite is normally also true: the flux depends on the radonconcentration in the chamber. For example, the maximum flux out of the sampleis when the chamber concentration is zero. If there are no other sources than theconcrete sample and if the chamber is closed then in steady-state, the followingmass balance is fulfilled:

J = λV c (54)

where J is the total exhalation rate out of the sample (Bq s−1), λ is the decayconstant (s−1), V is the chamber volume (m3), and c is the concentration of radonin the chamber (Bq m−3). Simulation of this type of a problem with RnMod3d canbe done as follows:

• The computational grid should only include the concrete. The chamber shouldnot be made part of the grid because here the air is well mixed and thetransport is not really covered by the transport equation solved by RnMod3d.

• Impose fixed-concentration nodes at the concrete-air boundary. If fixed1 isused, then set cBC[fixed1]:=0.

• Perform a run with the model and calculate the flux of radon into the cham-ber. The calculated flux is then inserted into equation 54 and the correspond-ing chamber radon concentration is found. Observe the difference between theassumed chamber concentration (0 in the first run) and the calculated value.

• Now increase the imposed chamber concentration cBC[fixed1] by trial-and-error until there is consistency between flux and chamber concentration asgiven in equation 54.

12.2 BC running

As described in the previous subsection, special boundary conditions can be han-dled by manual change of the value of a fixed concentration at the boundary. Itis, however, sometimes better to let RnMod3d do the trial-and-error part of theproblem. In particular, it is virtually impossible to solve time-dependent problems”by hand”.In the lack of a better name, the RnMod3d system for changing the boundary

conditions during the iterative solution procedure is here called running boundaryconditions. The following control variables are used for the purpose:

68 Risø-R-1201(EN)

BC_runningBC_running_update_of_cBCs_defBC_running_min_iterationsBC_running_max_residual_sum_before_new_BCBC_running_convergence_defwr_BC_running_messages_logwr_BC_running_messages_screen

BC running

This is a boolean variable. If it is set to false then no adjustment of boundaryconditions are carried out. Hence this value must be set to true when ”runningboundary conditions” are needed.

BC running update of cBCs def

This is a pointer to a user-defined procedure that controls how the boundaryconditions (e.g. cBC[fixed1]) are changed. To prevent unstable solutions theprocess is normally under-relaxed.

BC running min iterations

This variable is of type integer. It sets the minimum number of iterations thatRnMod3d needs to carry out before it attempts to change the boundary conditions.If the value is set too low, the solution procedure can become unstable.

BC running max residual sum before new BC

This floating-point variable gives the maximum sum-of-residuals before RnMod3dattempts to change the boundary conditions. If the value is set too high, thesolution procedure can become unstable.

BC running convergence def

This is a pointer to a user-defined function that returns the value true if some user-defined criteria for convergence has been met. Otherwise it should return the valuefalse. For example, in a simulation of exhalation from concrete into a chamberit can be tested if there is consistency between the assumed fixed-concentrationand the calculated flux.

wr BC running messages log

This is a boolean variable that controls if RnMod3d outputs information about theproblem to the LOG-file.

wr BC running messages screen

This is a boolean variable that controls if RnMod3d outputs information about theproblem to the screen.

Example

An application of running boundary conditions will now be demonstrated. Imaginethat a sample of concrete is placed in a chamber. The chamber volume is V and

Risø-R-1201(EN) 69

the total flux of radon from the sample into the chamber is called J . There are noother sources of radon in the chamber. The chamber is ventilated with radon-freeair. The ventilation rate is λv in units of s−1 (i.e. the number of air-changes persecond). The first task is to write a boundary condition for the chamber. Thereare three obvious possibilities:

• If the ventilation rate (or the chamber volume) is very large then the chamberradon concentration cch can be maintained at a near-zero level:

cch ≈ 0 (55)

• If system is in steady-state, then the chamber radon concentration must fulfill:

J = (λ + λv)V cch (56)

where λ is the decay constant for radon.

• It the system is not in steady state then some initial condition must be de-scribed for cch = cch(t) at time zero. For example, the concentration mayinitially be zero:

cch(t = 0) ≈ 0 (57)

For t > 0 the following condition applies:

Vdcchdt

= J − (λ+ λv)V cch (58)

To simulate such conditions with RnMod3d, we first write a function that returnsthe value for cch:

function c_chamber:datatype;

const chamber_open=true; (* Open or close the chamber *)

vol=0.050; (* Volume is 50 L *)

lamv=2/3600; (* Air exchange rate is 2 times per hour *)

lamd=2.098e-6; (* Decay constant for radon-222 *)

lam = lamd+lamv;

var J,dc_dt:datatype;

begin

J:=FlxVal[Flx1].J; (* Read flux from probe Flx1 *)

if chamber_open then

c_chamber:=0 (* free exhalation *)

else

begin (* bound exhalation *)

if solution=steady then

c_chamber:=J/(lam*vol)

else

begin (* unsteady *)

dc_dt:=(J-(lam*vol)*c_chamber_old)/vol;

c_chamber:=c_chamber_old+dc_dt*dtim;

end;

end; (* bound exhalation *)

end;

where we assume that flux probe Flx1 monitors the total flux of radon out of thesample, and where

c_chamber_old

is a floating-point variable declared in the job file which is initially set to zero (i.e.before RnMod3d is called the first time).The nodes at the boundary of the concrete is set to be of type fixed1 and the

chamber radon concentration is hence imposed with cBC[fixed1]. For examplethe (standard) boundary conditions can be programmed as:

70 Risø-R-1201(EN)


begin

cBC[fixed1]:=0;


i,xFix1,xFix2,

j,yFix1,yFix2,


...

end;

A procedure is then needed that can adjust cBC[fixed1] in such a way that thedesired boundary condition is fulfilled. The following procedure could be used:

procedure BC_running_update_of_cBCs;

const relax=0.7;

var cBC_old:datatype;

begin

cBC_old:=cBC[fixed1];

cBC[fixed1]:=cBC_old+relax*(c_chamber-cBC_old);

end;

Observe, that we under-relax the update of cBC[fixed1] compared to the situa-tion where:

cBC[fixed1]:=c_chamber;

We also need a procedure that measures if there is consistency between the im-posed chamber concentration (cBC[fixed1]) and the value that can be calculatedfrom the boundary condition and the measured flux (c chamber). For example,we could use the following function:

function BC_running_convergence:boolean;

const maxchange=1e-7;

begin

if (cBC[fixed1]>0) and

(abs((cBC[fixed1]-c_chamber)/cBC[fixed1])<maxchange)

then

BC_running_convergence:=true

else

BC_running_convergence:=false;

end;

Then we just need to set the control variables to use the above procedures. Forexample the job file could look like this:

program F0027prg;

...

var chamber_open:boolean;

c_chamber_old:datatype;

...

function c_chamber:datatype;

...

end;

function BC_running_convergence:boolean;

...

end;

procedure BC_running_update_of_cBCs;

...

end;


...

end;

Risø-R-1201(EN) 71

function initialfield(i:itype;j:jtype;k:ktype):datatype;

...

end;

...

begin (* main *)

runid := ’0027’;

runtitle := ’Test case’;

geometry := cylindrical2d;

grid_def := grid;

force_new_grid_in_every_run := false;

boundary_conditions_def := boundary_conditions;

...

flux_convset := [flx1,flx2];

probe_convset := [obs1..obs3];

conv_evaluation_period := 50;

BC_running := false;

BC_running_convergence_def := nil;

BC_running_update_of_cBCs_def := nil;

BC_running_min_iterations := 0;

BC_running_max_residual_sum_before_new_BC := 1e-5;

wr_BC_running_messages_log := false;

wr_BC_running_messages_screen := false;

min_iterations := 60;

max_iterations := 20000;

max_time := 15*60;

max_change := 1e-9;

max_residual_sum := 1e-19;

solution := steady;

tim := 0;

dtim := 0;

c_chamber_old := 0;

cBC[fixed1] := 0;

run_model; (* Initial field at t= 0*)


dtim := 1800;

c_chamber_old:=cBC[fixed1];

BC_running := true;

BC_running_convergence_def := BC_running_convergence;

BC_running_update_of_cBCs_def := BC_running_update_of_cBCs;;





repeat

tim:=tim+dtim;

run_model;

c_chamber_old:=cBC[fixed1];


close_model;

end.

72 Risø-R-1201(EN)

13 Output and debugging

RnMod3d can be set to generate various types of output. This is mainly controlledby those of the control variables in Section 4 that start with wr . Output may bedirected to the screen or to the LOG-file.

13.1 Standard files

Each model calculation is assigned an identification tag through the control vari-able called runid. If we set runid := ’0997’ and run the model, then standardoutput goes to the files listed in Table 3. The column named file variable showsthe identification that can be used in Pascal write-statements to write to the files.For example, to write something to the standard result file, simply use:

writeln(RES,’Hi there’);

The standard output files are assigned and opened during the call run model. Thismeans that the user cannot write to the files before run model has been called.For example, the following sequence will give run-time error 103: File not open.

Example 47 The following job file gives a run-time error.

program F003prg;

...

begin (* main *)


run_model;

close_model;

end.

Example 48 Correct use of RES-file.

program F003prg;

...

begin (* main *)

run_model;


close_model;

end.

The standard RnMod3d files are closed again by close model. Some of the defaultfilenames in Table 3 may be changed with the control variables:

import_field_nameexport_field_nameflowfield_name

This is explained in Section 4.

13.2 Other file output

Sometimes it is desirable to output results to non-standard files. This can be doneeasily. First, a text file variable must be declared, and then a file name should beassigned to it. Finally, the file should be opened and closed. For example:

Example 49 User-defined output.

program F0997prg;

...

var MyF:text;

Risø-R-1201(EN) 73

Table 3. Standard RnMod3d files if runid := ’0997’. All these files are ASCIIfiles.File name File variable Type of output Purposef0997LOG.dat LOG Log file User readable filef0997RES.dat RES Generic result file User readable file

f0997 01.dat PLT1 Plot file User readable filef0997 02.dat PLT2 Plot file User readable filef0997 03.dat PLT3 Plot file User readable file

f0997 00.dat File with field (c) Used by RnMod3df0997FLW.dat File with soil-gas flow field (q) Used by RnMod3df0997TMP.dat Reserved for later use Used by RnMod3d

...

begin (* main *)

assign(MyF,’myfile.dat’);

rewrite(MyF);

writeln(MyF,’Hi there’);

run_model;

...

close_model;

close(MyF);

end.

13.3 Contour plots: update plotfile

RnMod3d has a system for creating 2D plot files. These files can be used by softwaresuch as Surfer (Golden Software) to create contour plots of the calculated pressureor radon concentration fields. An example is shown in Figure 15. The user has to

0 5 10 15 20

x [m]

-10

-5

0

z[m

]

Figure 15. Example of calculated pressure field and streamlines of steady soil-gasentry into a slab-on-grade house. The pressure field is also shown. The contourplot was created with Surfer ver. 7 from Golden Software.

write a procedure with specifications about what should be output. For example,it may be desirable to avoid output for control volumes of the type NOP or tolimit the output in other ways. An example will be shown in the following. In thiscase the name of the procedure is myplots. RnMod3d will use this procedure if thecontrol variable plotfiles def is set as follows:

74 Risø-R-1201(EN)

plotfiles_def := myplots;

The default value for plotfiles def is nil. In this case no plot files will begenerated during a model run. To specify what should be output, the user needsto use the procedure:

update_plotfile(plt,dir)

where

plt is one of the standard file variables: PLT1, PLT2 or PLT3.

dir is one of the ”directions”: xdir, ydir or zdir.

The first parameter tells where the output should go. One of the three plottingfile variables given in Table 3 can be used (e.g. PLT1). The second parameter givesthe ”direction” of the plot. For example, if xdir is selected, then the 2D plot willbe perpendicular to the x-axis. Hence, a (y, z)-plot will be generated. An exampleof a plot file procedure is shown next:

Example 50 A plot file procedure.

procedure myplots(i:itype;j:jtype;k:ktype);

begin

if (j=2) and (GP[i]^[j]^[k].nodetyp<>NOP) then update_plotfile(plt1,ydir);

if (j=2) and (GP[i]^[j]^[k].mat=mat2) then update_plotfile(plt2,ydir);

if (k=5) then update_plotfile(plt3,zdir);

end;

The meaning is as follows: The output directed to the PLT1-file includes all non-NOP control volumes with j-index equal to 2. The PLT2-file gets the same typeof output except that now only control volumes of material mat2 are included.Finally, the PLT3-file gets output for all control volumes with k-index equal to 5.The PLT-output includes index and physical coordinates, field values at the

control-volume nodes, coded node type (where 1=free, 2=fixed1 etc.), codedmaterial type (where 2=mat1, 3=mat2 etc.), names of the node type, and namesof material. The coded numbers for node type and materials are included to helpcreate plots (e.g. mat1 can be colored in one color and mat2 in another).

Example 51 Content of a PLT-file created with update plotfile(plt,dir)wheredir has been set to ydir.

i k x z c nodeno matno node mat

1 124 0.000000E+0000 0.000000E+0000 -3.00000000E+0000 2 3 fixed1 mat2

2 1 1.879347E-0001 -1.000000E+0001 -7.80006595E-0001 1 2 free mat1

2 2 1.879347E-0001 -9.984201E+0000 -7.80006595E-0001 1 2 free mat1

...

2 73 1.879347E-0001 -2.500000E-0001 -2.92318620E+0000 1 4 free mat3

2 74 1.879347E-0001 -2.453292E-0001 -2.92319051E+0000 1 4 free mat3

...

4 116 5.641896E+0000 -8.349609E-0002 -2.99910051E+0000 1 6 free mat5

...

133 122 2.000000E+0001 -8.789062E-0004 -1.81602884E-0005 1 2 free mat1

133 123 2.000000E+0001 -9.765625E-0005 -2.01780972E-0006 1 2 free mat1

133 124 2.000000E+0001 0.000000E+0000 0.00000000E+0000 3 2 fixed2 mat1

13.4 Stream lines

In simulations of steady soil-gas transport it is useful to create a plot of thepressure field in the soil. This can be done with the procedure update plotfileas described earlier. Often it is desirable also to calculate stream lines as this is agood way to visualize the flow. RnMod3d does not include a general procedure forthis task. However, it is not difficult to write a procedure for this purpose. Thestreamlines in Figure 15 have been calculated with the procedure in Example 52.

Risø-R-1201(EN) 75

Example 52 Calculation of stream lines in the xz-plane.

procedure wr_streamlines;

const streamlinefactor=1e8;

var i:itype;

j:jtype;

k:ktype;

psi,psi_temp:datatype;

QB,QW:datatype;

LM:text;

begin

writeln(’Write streamlines ....’);

assign(LM,’caflow02.dat’);

rewrite(LM);

psi_temp:=0;

i:=2;

j:=2;

k:=2;

repeat

psi:=psi_temp;

k:=1;

repeat

writeln(LM,xnod(i):20,’ ’,znod(k):20,’ ’,streamlinefactor*psi:20);

QW:=GP[i]^[j]^[k].aW*(GP[i-1]^[j]^[k].c-GP[i]^[j]^[k].c);

psi:=psi+QW;

k:=k+1;

until (k=kmax);

QB:=GP[i]^[j]^[2].aB*(GP[i]^[j]^[1].c-GP[i]^[j]^[2].c);

psi_temp:=psi_temp-QB;

i:=i+1;

until (i=imax);

close(LM);

end;

13.5 Warnings

After computations with warnings, RnMod3d will show a little table indicating thenumber of warning flags raised during the run. Further information about whereand why the warnings were given can normally be found in the LOG-file (searchfor warning).

Example 53 Table of warnings that were issued during the execution of the jobfile.

---------------------------------------------------------------------------

OBSERVE : Warnings were issued during this session.

Warning: war_interpolation was issued 20 times

Warning: war_other was issued 1 times

Warning: war_convergence was issued 1 times

Warning: war_residual was issued 1 times

---------------------------------------------------------------------------

RnMod3d uses the following types of warnings. The warnings are listed in orderof importance.

war interpolation This warning can almost always be ignored completely. Thewarning is issued by the procedures fieldvalue and fieldvalue2D. To findthe field value at any location (x, y, z) these procedures perform interpolationbetween adjacent nodes. If one or more nodes are of the type NOP (and there-fore without a valid field value) this warning is issued. The available (valid)nodes are used for the interpolation.

76 Risø-R-1201(EN)

war other This category contains warnings for problems not covered by the othergroups. Examples include warnings from the solver that the maximum numberof iterations was reached or that the grid has been redefined.

war fileimport This warning is issued if attempts are made to import a file thatdoes not exist or if the field in the imported file does not match the currentgrid.

war convergence This warning tells that the run was stopped before the solverreached convergence. This may or may not be a real problem.

war residual This warning signals that the sum of absolute residuals seems tobe too large compared to the source term (b). See Section 10.4 for furtherdetails.

In jobs with many model runs, it may be useful to include the number of warningsin the result file. The user has access to the warning table through the array:

warning_table:array[warningtype] of longint;

For example, warning table[war convergence] equals 0 if all runs have reachedconvergence. The user can invoke warnings with the procedure:

warning_std(idst,message,war)

where idst and message are descriptive strings that tells where and why thewarning was created. The type of warning is set by war. User-generated warningsshould use war other.

13.6 Error messages

Errors will halt RnMod3d. Normally, the error message includes the name of the pro-cedure that generated the error and a brief message. If the error cannot be identi-fied from this try to set wr details, wr main procedure id, or wr all procedure idto true, and run the job file again. The user can invoke errors messages with theprocedure:

error_std(idst,message)

where idst and message are descriptive strings that tells where and why the erroroccurred.

13.7 Critical evaluation of results

It is important to evaluate the output from RnMod3d critically. In particular, it isimportant to ascertain that the problem solved by RnMod3d is actually the problemwanted by the user. Here is a list of things to do:

• Try to start with a very simple version of the problem. Test each level ofcomplexity, and do not add more complexity before tests have been passed.

• Test that the model agrees with simple calculations. For example, in radonsimulations try to compare the deep-soil radon concentration with the analyt-ical solution. If the soil is not very deep, then make it so or set the diffusivityto a low value.

• Test if the geometry has been set up correctly. For example, test that the vol-ume of materials is as wanted. This can be done through the control variablewr material volume (see Section 4.47). This procedure also output mate-rial specific minimum and maximum radon concentrations (or pressures in asoil-gas simulation). Are these results as expected?

Risø-R-1201(EN) 77

• Test that the boundary conditions have been set up correctly. For example tryto change values and see if the system responds as intended. For example, ina soil-gas simulation involving constant pressures at one or more boundaries,try to set the pressure to zero or change the sign of pressures. In a steadyflow situation, the flow of gas into the system must equal the flow out of thesystem at the other boundaries. Is this fulfilled in the model?

• If a flux measurement give zero result when it should not, then test if theprobes are really located correctly (see page 53).

• Test that the solution is not sensitive to further grid refinement. For example,see what results are reached if the number of nodes in the grid is doubled (orhalved). Make sure the grid is sufficiently fine in regions where large gradientsoccur. The technique describe in Section 5.11 may be useful for this purpose.

• In time-dependent problems, test that the solution is not sensitive to theselected time step dtim. Try to see what happens if dtim is doubled (orhalved).

• Make plots of the calculated fields.

14 RnMod3d inside

This section explains a little about the inside part of RnMod3d. This informationmay be helpful during debugging. Fortunately, most variable names are long, de-scriptive and easy to read. For example, in the code file, the variable that flags ifthe buffer cBUF1 has been created or not is called:

cBUF1_has_been_created

It is a boolean variable, and can take only the values true or false. Likewiseenumerated data types have been used for many variables. This is discussed inSection 14.6.

14.1 Index coordinates: i, j, and k

The index coordinates i, j, and k are used in RnMod3d to refer to specific controlvolumes. These variables are restricted in range by the associated types definedas follows:

itype = 1..imaxTot;jtype = 1..jmaxTot;ktype = 1..kmaxTot;

The constants imaxTot etc. are set by the user as described in Section 5.1.

14.2 The main data structure: GP

The main data structure in RnMod3d is GP (”grid pointer”). This structure containsinformation about all control volumes. Each node in GP point to data which hasbeen declared as follows:

nodedatatype=recordc:datatype;aE,aW,aN,aSS,aT,aB,b,ap,ap_old_dt:datatype;qE,qW,qN,qS,qT,qB:datatype;

78 Risø-R-1201(EN)

nodetyp:nodetyptype;Wcon,Econ,Scon,Ncon,Bcon,Tcon:nodecontype;mat:mattype;valid_fieldvalue:boolean;

end;

The meaning is as follows:

• Field values. The field value of control volume (i,j,k) is stored as:

GP[i]^[j]^[k].c

• a-coefficients. The aE-coefficient of control volume (i,j,k) (see equation 44,page 14) is stored as:

GP[i]^[j]^[k].aE

The other a-coefficients are stored in a similar fashion. Observe, that thecoefficient ap old dt is not part of equation 44. This coefficient has beenintroduced to maintain conservation of mass even if β changes from one timestep to another. Without going into details, we observe, that aP old (see[Pa80]) should correspond to the time when the last field was calculated. If weignore transport, generation and decay we have (symbolically) : β(1) · ca(1) =β(0) · ca(0), where 0 and 1 represent old and new, respectively. In terms ofcoefficients this means, that ap new*ca = ap old*ca old.

• Material type (i.e. mat1, mat2 etc.) is stored as:

GP[i]^[j]^[k].mat

• Soil-gas fluxes. The soil-gas flux through the east interface of the controlvolume (i,j,k) is:

GP[i]^[j]^[k].qE

Fluxes through the other interfaces are stored as qW, qN etc.

• Node type. The type of node (e.g. free or fixed1) of the control volume(i,j,k) is given by:

GP[i]^[j]^[k].nodtyp

• Connector type. The type of connector (e.g. std or NoFlow) through the eastinterface of the control volume (i,j,k) is given by:

GP[i]^[j]^[k].Econ

Connectors for the other interfaces are stored as Wcon, Ncon etc.

• Valid field value. GP also contains a flag that tells if the field value is valid ornot. That is given by:

GP[i]^[j]^[k].valid_fieldvalue

14.3 Other variables

This is a list of some other variables that sometimes are needed in job files:

cBC: The fixed values used in fixed-value boundary conditions fixed1 etc. SeeSection 6.7.

FlxVal: The results of flux measurements with Flx1 etc. See Section 8.3.

Obsval: The results of field measurements with Obs1 etc. See Section 9.1.

wFixVal: The values of fix points xFix1 etc. See Section 5.4.

x[i], y[j], z[k], dx[i], dy[j], and dz[k]. The location and size of individualcontrol volumes. See Section 5.9

Risø-R-1201(EN) 79

14.4 datatype

All floating-point computations are done with variables of the type called datatype.By default datatype is set to equal extended. To decrease the use of memory or totest the sensitivity of the results to the internal number representation datatypeshould be set to double, real or single.

14.5 Memory

Memory is allocated dynamically (during runtime) for the main data structure GP.Maximum grid sizeSection 5.1 tells how the maximum grid size can be changed. Other data structuresare static variables.

14.6 Enumerated types

Wherever meaningful, enumerated data types have been used in RnMod3d. Forexample, variables that hold the type of node of a control volume are declared tobe of type nodetyptype, which in turn is declared as:

nodetyptype = (nop,

free,

fixed1,

fixed2,

fixed3,

fixed4,

fixed5,

NodX);

The use of enumerated types has four implications: (1) job-file assignmentslike: geometry := cartesian3D are readable, (2) it is easy to find the possiblegeometries implemented in RnMod3d (just look up the declaration of geometry inthe source file), (3) should there be any problems with one of the geometries, itis relatively easy to identify those places in the source code where that geometryis treated, and finally (4) enumerated types can be used in Pascal set calculations(see example page 25). Most enumerated types have predefined functions thatcan convert variables to descriptive strings. For example, the main program filecontains a function that can be used to print out the value of a variable of thetype nodetyptype:

function nodetyp_string(x:nodetyptype):string;

var st:string;

begin

case x of

NOP: st:=’NOP ’;

free: st:=’free ’;

fixed1: st:=’fixed1 ’;





NodX: st:=’unchanged ’;

else

st:=’Unknown !!’;

end; (* case *)

nodetyp_string:=st;

end;

These functions can be useful during debugging.

80 Risø-R-1201(EN)

14.7 Sequence of actions in run model

RnMod3d starts to do computations when the procedure run model is called. Themodel may find a steady-state field (if solution has been set to steady) or itmay advance the field by one single time step dtim (if solution has been set tounsteady). To use RnMod3d with confidence, it is important to know the sequenceof actions in the procedure run model. The details can be read from the exactprogramming of run model in the file R3Main03.pas. To learn more, it may alsobe useful to let RnMod3d echo procedure names etc. during runtime. This can bedone with the control variables:

wr_detailswr_main_procedure_idwr_all_procedure_id

The following gives a summary of the actions in run model:

1. Initially two things can happen:

• If this is the first run model call in a job file then all variables will beinitialized. For example the entire field in the main data structure GP isset to zero:

GP[i]^[j]^[k].c:=0

• If this is not the first run in a session, then two situations can occur:

– If the control variable use fieldbuffer is set to cBUF1, then thestate of RnMod3d is set back to the state RnMod3d ended up in thelast time run model was called with use fieldbuffer set to cBUF1.Likewise, if use fieldbuffer is set to cBUF2, RnMod3d is restoredto that found in the field buffer cBUF2. If nothing has yet been savedin the buffer, then nothing is restored. Hence GP will not be changed(for this reason). This situation is identical to than discussed nextwhere use fieldbuffer has been set to no cBUF.

– If the control variable use fieldbuffer is set to no cBUF, then nochange of the main data structure GP is performed at this stage.Hence the results of the previous run (still present in GP) are usedas a starting point in the current computation (unless changed inone of the steps listed below).

2. If needed, then a grid is generated. Three situations can occur:

• If this is the first run in a job file, then a grid is generated in accordancewith the procedure pointed to be grid def.

• If the control variable grid def has changed from a previous call ofrun model, then a new grid is generated (as specified by the procedurenow pointed to by grid def).

• If the control variable force new grid in every run has been set totrue, then a new grid is always generated with the procedure now pointedto by grid def. This may or may not be a truly new grid (see Sec-tion 4.7).

3. A flow field of soil gas may be imported. Clearly this is only meaningful if aradon problem is solved. This field corresponds to q in Box 1, page 11. Threesituations exist:

• If the control variable flowfield has been set to import, then a flowfield is imported from the file given by flowfield name.

Risø-R-1201(EN) 81

• If the control variable flowfield has been set to import from qbuf thena flow field is imported from the flow-field buffer called qBUF. Of coursethis is possible only if a flow field has been calculated (and saved in thebuffer) earlier in the job file.

• For all other settings of flowfield, the flow field will be set to zero allover.

4. All nodes and connectors are always (even in time-dependent problems) setback to the default configuration. As described in Section 6.3, the computa-tional domain now simulates a ”closed box”.

5. Nodes and connectors are then changed (from the default settings) in accor-dance with the procedure pointed to by boundary conditions def.

6. If this run is the first in a time-dependent problem (i.e. if solution is set tounsteady) then an initial field may be set up as follows:

• If import initialfield is set to true then an initial field is importedfrom the file given by import field name.

• If initial field def is not set to nil then an initial field is set up fromthe procedure pointed to by initial field def.

• In all other cases, the field in the grid pointer GP is taken to be the initialfield. The following situations occur:

– If this is the first run in the job file, then the initial field is zero at allnodes except those fixed to certain values as given by the procedurepointed to by boundary conditions def.

– If this is not the first run in the job file, then the result of theprevious calculation is now used as initial field in this run. Observethat the previous run could have been a steady-state calculation aswell as a time step in a time-dependent calculation.

7. The procedure materials def is called. This means that the control volumesin the computational domain are set to be of materials mat1, mat2 etc.

8. The coefficient matrix is then set up. This involves calling the user-definedprocedures:

• beta def

• e def

• G def

• D def

• lambda def

If any of the nodes are of the type fixed1, fixed2 etc. then the values ofthese nodes are set to equal the values in cBC.

9. The problem has now been fully specified. Before it is solved it is possible tomake a guess of the solution. If import finalfield guess is set to true thensuch a guess is imported from the file given by import field name. If such aguess is not imported, then the field present in GP is used as initial guess. Forexample, in time-dependent problems the result of the previous time step isusually a good guess of the next one.

10. The procedure find field is then called. This procedure in turn calls thesolver pointed to by solver def. When convergence has been reached (orthe solver was stopped for other reasons), the final field is returned in GP.During the iterative solution procedure it is possible to revise the boundaryconditions (e.g. cBC[fixed1]) as RnMod3d calls the procedure pointed to by

82 Risø-R-1201(EN)

BC_running_update_of_cBCs_def

Such running boundary conditions are described in Section 12. Furthermore,the procure pointed to by:

user_procedure_each_iter_def

is called during each iteration (see Section 4.28).

11. Various types of output are generated.

12. In a soil-gas problem, it is meaningful to store the resulting flow from nodeto node as a flow field. It can then be used in a later radon simulation. Threesituations occur:

• If the control variable flowfield has been set to export then a flowfield is exported to the file given by flowfield name.

• If the control variable flowfield has been set to export to qbuf thena flow field is exported to the flow-field buffer. It can then be used laterin the current session.

• For all other settings of flowfield, the flow field will not be stored.

13. Finally the state of RnMod3d may be stored for later use:

• If the control variable use fieldbuffer is set to cBUF1 or cBUF2, thenthe present state of RnMod3d is stored in buffer cBUF1 or cBUF2, respec-tively.

• If use fieldbuffer is set to no cBUF, then the state of RnMod3d is notstored in a buffer.

The results of the computations (GP), will however, in all cases remain intactand can be used in additional run model calls within the same job file. How-ever, observe, that all information (in bufferes or GP) are always lost when thejob file ends. To transfer information from one job to another, results have tobe stored in files.

15 Benchmark tests

To verify that RnMod3d gives accurate results, it is necessary to perform benchmarktests on the basis of problems with known solutions. This section contains somesimple examples. RnMod3d has also been compared with other radon models formore complicated problems (without known solutions) [An99a].

15.1 F0100prg: Steady flow of soil gas

This problem concerns steady Darcy flow of soil gas in a 1 m x 1 m x 3 m columnwith homogeneous sand. The gas permeability k of the sand is 2 · 10−10 m2 andthe dynamic viscosity µ is 17.5 · 10−6 Pa s. The disturbance pressure is 0.0 Pa atthe bottom of the column and −3.0 Pa at the top. Other column sides are closedoff for transport.

Risø-R-1201(EN) 83

Analytical solution

The exact flow through the column is:

Q = Ak

µ

∆p

L(59)

= 1 m2 2 · 10−10 m2

17.5 · 10−6 Pa s3 Pa3 m

(60)

= 1.14285714 . . . · 10−5 m3 s−1 (61)

The pressure in the center of the column is −1.5 Pa.

Model implementation

The full job file can be found in Appendix A. Observe, that the ”diffusivity” isset to k

µ . This is in accordance with the formalism presented in Table 2, page 12.Also, observe, that other soil parameters are set to zero. Since the flow is steady,it is not necessary to assign any value to the ”partition-corrected porosity” β.In unsteady flow simulations β must be set to εa/P0. A flux-measurement probecalled Flx1 is placed at the bottom of the column, and another probe called Flx2is placed at the top. A pressure probe called obs1 is placed in the center of thecolumn (i.e. at (x, y, z) = (0.5 m,0.5 m,−1.5 m)).

Results

The model gives the following results:

Flx1: J = 1.1428571 · 10−5 m3 s−1

Flx2: J = 1.1428571 · 10−5 m3 s−1

Obs1: c = −1.500 Pawhich is in perfect agreement with the true result. Additional results can be foundin the LOG-file, which is listed in Appendix B.

15.2 F0101prg: Steady diffusion of radon

This problem is case 0 of the ERRICCA model intercomparison exercise described[An99a]. The problem concerns steady diffusion of radon in a sand column. Thejob file F0101prg.dpr is listed in Appendix C. The following results are obtained:

Flx1: J = 0.000 Bq s−1

Flx2: J = 4.72197 · 10−2 Bq s−1

Obs1: c = 4.19221 · 104 Bq m−3

The Flx1 flux measurement verify that the no-flow boundary condition is fulfilledat the bottom. The Flx2 result is in good agreement with the true result: J =4.722828 · 10−2.

15.3 F0102prg: Diffusion and advection of radon

This problem concerns a column of homogeneous sand of height L = 5 m and cross-sectional area 1 m2. Both steady Darcy flow of soil gas and combined diffusionand advection of radon are considered. The problem is sketched in Figure 16. Thesand has the following properties:

The sand is homogeneous and dry

84 Risø-R-1201(EN)

0

p(0) = ∆p

c(0) = cs

p(L) = 0

c(L) = 0

z

q c(z)

L = 5 m

Figure 16. Sketch of the problem treated by F0102prg.

The gas permeability k is 10−11 m2

The radon generation rate G equals λ · 10000 Bq m−3

The porosity ε is 0.3

The bulk diffusivity D is 10−6 m2 s−1

The following boundary conditions apply for the flow of soil gas:

At z = L, the disturbance pressure is 0 Pa.

At z = 0, the disturbance pressure is ∆p. Three cases will be investigated:

• ∆p = −100 Pa• ∆p = 0 Pa

• ∆p = 100 Pa

Other boundaries are closed for transport.

The following boundary conditions apply for the transport of radon:

At z = L, the radon concentration is set to 0.

At z = 0, the radon concentration is cs = 5000 Bq m−3.

Other boundaries are closed for transport.

The decay constant (λ) is set to 2.09838 · 10−6 s−1 and the dynamic viscosity (µ)is set to 17.5 · 10−6 Pa s.

Risø-R-1201(EN) 85

0 1 2 3 4 5

010

0020

0030

0040

0050

00

z

c

Flow

0 1 2 3 4 5

020

0040

0060

0080

00

z

c

0 1 2 3 4 5

010

0020

0030

0040

0050

0060

00

z

c

Flow

Figure 17. Radon concentration profiles in the sand column in F0102prg. z ismeasured in m and c in Bq m−3. The three plots correspond to ∆p equal to −100 Pa(top), 0 Pa (middle), and 100 Pa (bottom). The exact results are shown as lines.RnMod3d results are shown as circles.

86 Risø-R-1201(EN)

Analytical solution

The analytical solution to the above problem can be found in [Co81, p. 26]. Theradon concentration in the column (0 ≤ z ≤ L) is:

c(z) =G

λ

(1− exp( qz

2D ) sinh L−zΛ + exp −q(L−z)

2D sinh zΛ

sinh LΛ

)

+cs exp(qz

2D)sinh L−z

Λ

sinh LΛ

(62)

where the soil-gas flow rate in the direction of the z-axis is:

q =k

µ

∆p

L(63)

and

Λ =(

q2

4D2+ L−2

d

)−0.5

(64)

and where Ld is the diffusion lenght:

Ld =

√D

ελ(65)

The flux at z = L is:

j =(−D

dcdz

+ qc

)∣∣∣∣z=L

(66)

=G

λ

(q

2+

D

Λcosh L

Λ − exp qL2D

sinh LΛ

)+ cs

D

Λexp qL

2D

sinh LΛ


Appendix D shows how the problem with ∆p = −100 Pa has been implementedin RnMod3d. The job file also contains the exact solution.

Comparison of results

Table 4 and Figure 17 show that there is good agreement between the results ofRnMod3d and the analytical solution.

Table 4. Results for the radon flux at z = L.

P RnMod3d Exact DeviationPa Bq s−1 Bq s−1 %

−100 5.4545 · 10−4 5.4820 · 10−4 −0.50 7.7855 · 10−3 7.7896 · 10−3 −0.05

100 7.0965 · 10−2 7.0973 · 10−2 −0.01

15.4 F0103prg: Time-dependent flow of soil gas

This case concerns unsteady Darcy flow of gas in a finite soil column of height 3.Initially at t = 0, the disturbance-pressure field in the column equals zero. Thenat t = 0, the disturbance pressure at one end of the column (z = 3) starts tooscillate as:

patm(t) = p1 sin(ωt+ ϕ) (67)

Risø-R-1201(EN) 87

where p1 is the amplitude of the variations (e.g. 1 Pa), and where the period timeis:

T =2πω

(68)

The disturbance pressure field at the other end of the column (z = 0) remains atzero. Inside the soil column, the disturbance-pressure field p(z, t) is governed byequations 42 and 43. The problem geometry is sketched in Figure 18.

z

0

3

pamt(t)

p(z, t)

p = 0

Figure 18. Sketch of problem treated by F0103prg: The disturbance pressure at theboundary at z = 3 starts to oscillate at t = 0. After some transient period, this inturn starts oscillations of the same frequency in the soil. The phase and amplitudechange with z (and soil parameters).

Analytical solution

The exact solution to the problem (0 ≤ z ≤ 3 and t ≥ 0) can be found in Carslawand Jaeger [Car59, p. 105]:

p(z, t) = p1A sin(ωt+ ϕ+ φ) + (69)

2p1πDp

∞∑n=1

n(−1)n(Dpn2π2 sin ε− ω32 cos ε)

D2pn

4π4 + ω234sin(nπz

3

)exp

(−Dpn2π2t/32

)where

A =∣∣∣∣ sinh θz(1 + i)sinh θ3(1 + i)

∣∣∣∣ (70)

φ = arg(sinh θz(1 + i)sinh θ3(1 + i)

)(71)

and where

θ =(

ω

2Dp

) 12

(72)

Dp is defined in Section 3.3.

Parameters

We consider the following parameters:

88 Risø-R-1201(EN)

3 = 5.0 m

k = 10−14 m2

P0 = 1.0 · 105 Pa

εa= 0.2

µ = 17.5 · 10−6 Pa s

p1 = 3.0 Pa

T = 10 hours (period time)

ϕ = π/2

Observe, that with ϕ = π/2, the pressure at z = 3 will ”jump” from 0 to p = 3 Paat t = 0+.


The model implementation is shown in Appendix E. The properties of the soil areset in accordance with Table 2. In particular observe, that the ”diffusivity” is setto k

µ , and that beta is set to εaP0.

The conditions at tim:=0 are calculated with solution:=steady. For the sub-sequent runs, solution is set to unsteady. Only the boundary condition fixed2at the atmospheric surface changes in time. The main results of the computationsare written to the file F0103RES.dat. The output includes pressures in the centerof the column:

• obs1 at z = 0.2 m

• obs2 at z = 1.0 m

• obs3 at z = 2.5 m

• obs4 at z = 4.0 m

• obs5 at z = 4.8 m

The computations are stopped after 4 cycles (i.e. when tim equals 40 hours).

Evaluation

As can be seen from Figure 19, the results obtained with RnMod3d agree well withthose of the exact analytical solution in equation 69.

Additional comments

Riley et al. use the same test as just discussed to verify their model code calledRapidSTART [Ri99].RnMod3d has been tested against also the case described in Carslaw and Jaeger

as 2.6 Semi-infinite solid. Surface temperature a harmonic function of the time[Car59, p. 64]. Again near-perfect agreement between RnMod3d results and thoseof the exact analytical solution was obtained. However, the analytical solution wasrelatively difficult to integrate numerically6.

6Peter Kirkegaard at Risø is thanked for performing the integration.

Risø-R-1201(EN) 89

0 10 20 30 40

-2

-1

01

2

0 10 20 30 40

-0.06

-0.02

0.02

0.06

t [hr]

t [hr]p(z

=0.2,t)[Pa]

p(z

=4.8,t)[Pa]

Figure 19. Comparison of results obtained with F0103prg (circles) and the analyt-ical solution given in equation 69 (line). The top plot is the disturbance pressureat z = 4.8 m. The bottom plot is for z = 0.2 m.

16 House simulation example

This section demonstrates how RnMod3d can be set up to do calculations of soil gasand radon entry into a house. The house sketched in Figure 20(A) is considered.It is a 100 m2 slab-on-grade house. For simplicity, the house is chosen to becylindrical. There is a 3 mm gap of air between the slab and the footer along the full35 m perimeter of the house. This is clearly an important route of entry, however,transport can also take place through the concrete slab. A highly permeable layerof gravel exists below the slab. The house is located on a 10 m thick soil block of20 m radius. The house is constantly depressurized 1 Pa relative to the outdoors.Further details about parameters etc. will be given in the following.

Geometry

Figure 20(A) shows the geometry of the house. The house is cylindrical, so we set

geometry := cylindrical2D;

With the fix points xFix1 to xFix5 and zFix1 to zFix5, it is possible to give anaccurate representation of all geometrical features of the house. Table 5 gives thedimensions of the different components. For example, the width (i.e. the radius) ofthe slab is 5.6419 m, such that in fact the area of the floor is the required 100 m2.Accordingly, we assign the fix points as follows:

set_FixVal(xFix1,0.000); (* x-axis *)set_FixVal(xFix2,Lx_slab-Lx_gap);set_FixVal(xFix3,Lx_slab);set_FixVal(xFix4,Lx_slab+Lx_footer);set_FixVal(xFix5,Lx_soil);

90 Risø-R-1201(EN)

xFix1 xFix5

zFix5Fixed1

No flow

Fixed2

zFix1(B)

xFix1 xFix5

zFix5

Flx1

Obs4

Obs5

Obs3

Obs2

Obs1

Flx3Flx2

zFix1(C)

xFix3

Slab

Gravel

Footer

Soil

Gap

House interior

xFix1 xFix2 xFix5

X-axis

z-a

xis

xFix4

zFix2

zFix3zFix4

zFix5

zFix1(A)

Figure 20. Sketch of slab-on-grade house.

Risø-R-1201(EN) 91

Table 5. Geometry for the slab-on-grade house. All dimensions are in meters.

Width (x) Thickness (z)

Soil Lx soil = 20.0 Lz soil = 10.0Slab Lx slab = 5.6419 Lz slab = 0.10Gravel (as Lx slab) Lz gravel = 0.15Footer Lx footer = 0.3 Lz footer = 0.80Gap Lx gap = 0.003 (as Lz slab)

set_FixVal(zFix1,-Lz_soil); (* z-axis *)set_FixVal(zFix2,-Lz_footer);set_FixVal(zFix3,-Lz_slab-Lz_gravel);set_FixVal(zFix4,-Lz_slab);set_FixVal(zFix5, 0.00);

Observe, that the x-axis goes from 0 to 20 m, and that the z-axis goes from −10 mat the bottom of the soil to 0 at the atmospheric surface.

Boundary conditions

The soil-gas and the radon problems are based on the same types of boundary con-ditions. As sketched in Figure 20(B) the interface between the house and the slabis set to the fixed-value boundary condition: fixed1. Likewise, the atmosphericsurface is set to fixed2. The rest of the boundary is set to no-flow conditions.In the soil-gas problem we force the house to be depressurized 1 Pa relative to

the atmospheric surface. This is programmed with the assignments:

cBC[fixed1]:=-1; (* Pa *)cBC[fixed2]:= 0;

For the radon problem, we set both indoor and outdoor radon concentrations tozero:

cBC[fixed1]:= 0; (* Bq/m3 *)cBC[fixed2]:= 0;

Flux probes: Flx1 etc.

We are interested in the fluxes indicated in Figure 20(C). Flx1 gives the fluxthrough the slab (i.e. through the concrete). Flx2 is the flux through the gap.Flx3 is the flux into the atmosphere. The total entry rate into the house is givenby Flx4. This is the sum of Flx1 and Flx2. Observe, that all fluxes are taken tobe positive if they are in the direction of the z-axis.

Field probes: Obs1 etc.

To monitor the pressure and radon concentration fields, the probes Obs1 to Obs5are placed as indicated in Figure 20(C). For example, Obs2 is located just belowthe footing. The exact positions of the probes can be read from the job file. Noticethat wFixVal is used to find the location of fix points (see Section 5.4).

92 Risø-R-1201(EN)

Materials

Table 6 lists the materials used in the computations and their parameters. Otherconstants are: µ = 18 · 10−6 Pa s, λ = 2.09838 · 10−6s−1, L = 0.3 (see equation 19),and ρg=2.7 · 103 kgm−3.

Table 6. Parameters used in the calculations for the slab-on-grade house: gas per-meability (k), radium-226 concentration (ARa), fraction of emanation f , totalporosity (ε), volumetric water content (θv), and bulk diffusivity (D).

Name of k ARa f ε θv D

materials m2 Bq kg−1 m2 s−1

Soil mat1 10−11 40 0.2 0.25 0.2 4.3 · 10−7

Slab mat2 10−15 50 0.1 0.20 0 2.0 · 10−8

Gravel mat3 5 · 10−9 40 0.2 0.40 0 1.8 · 10−6

Footer mat4 10−15 0 0 0.20 0 10−10

Gap mat5 7.5 · 10−7 0 0 1.00 0 1.2 · 10−5

Names are assigned to the different materials. For example, the slab materialis called mat2. In the job file, the procedure materials defines exactly what partof the computational grid that contains mat2. This information is used in otherparts of the job file. For example, the function where the porosity is defined lookslike this:

function e_radon(i:itype;j:jtype;k:ktype):datatype;

var ee:datatype;

begin

ee:=0;

case materials(i,j,k) of

mat1: ee:=0.25;

mat2: ee:=0.20;

mat3: ee:=0.40;

mat4: ee:=0;

mat5: ee:=1.0;

else

error_std(’e_radon’,’Unknown material’);

end;

e_radon:=ee;

end;

The radon generation rate (G) is calculated from equation 20 on the basis ofthe given radium concentrations and fractions of emanation. The permeabilityassigned to the gap is calculated from [An92]:

k =d2

12(73)

where d is the width of the gap. For a 3 mm gap, this corresponds to k =7.5 · 10−7 m2. The other parameters for the gap corresponds to free air.

Job file

The complete job file is shown in Appendix F.

Results

The main results of the computations are output to the LOG-file as:

Risø-R-1201(EN) 93

The total soil-gas entry into the house is (Flx4) =1.6532545E-0005 m3/s

The total radon entry into the house is (Flx4) =1.9368863E+0000 Bq/s

An extended version of this house simulation can be found in [An99c]. Figure 15,page 74 shows pressure contours and streamlines.

A F0100prg.dpr

program F0100prg;

(* --------------------- RnMod3d jobfile ---------------------- *)

(* Project: User guide example: Steady soil-gas flow, 1D *)

(* Created: May 24, 1999 *)

(* Revised: July 17, 2000 *)

$I R3dirs03


procedure grid;

begin






set_axis_single(yFix1,yFix2,1,FocusA,1.0);



set_axis_double(zFix1,zFix2,30,30,FocusA,FocusB,1.1,1.1,0.5);

end;


begin

cBC[fixed1]:=0;

cBC[fixed2]:=-3.0;


i,xFix1,xFix2,

j,yFix1,yFix2,



i,xFix1,xFix2,

j,yFix1,yFix2,


end;

procedure fluxes(i:itype;j:jtype;k:ktype);

begin


i,xFix1,xFix2,

j,yFix1,yFix2,



i,xFix1,xFix2,

j,yFix1,yFix2,


end;

94 Risø-R-1201(EN)

procedure probes;


begin

cc:=fieldvalue(0.5,0.5,-1.5,valid);

if not valid then cc:=0.0;

obsval[obs1]:=cc;

end;

function materials(i:itype;j:jtype;k:ktype):mattype;

begin

materials:=mat1;

end;


begin

e:=0;

end;


begin

beta:=0;

end;


var mu:datatype;

begin

mu:=17.5e-6;

D:=2e-10/mu;

end;


begin

G:=0;

end;


begin

lambda:=0;

end;

begin (* main *)

runid := ’0100’;

runtitle := ’User guide example: Steady soil-gas flow, 1D’;

solution := steady;


Ly := 1.0;

grid_def := grid;



flux_def := fluxes;

probe_def := probes;


e_def := e;

beta_def := beta;

G_def := G;


D_def := D;

initialfield_def := nil;

import_initialfield := false;

import_finalfield_guess := false;

export_field := false;

use_fieldbuffer := no_cBUF;

flowfield := none;

flowfactor := 1.0;

Risø-R-1201(EN) 95

import_field_name := ’’;

export_field_name := ’’;

flowfield_name := ’’;

plotfiles_def := nil;

user_procedure_each_iter_def := nil;

wr_details := false;

wr_main_procedure_id := false;

wr_all_procedure_id := false;

wr_iteration_line_log := false;

wr_iteration_line_screen := true;

wr_residual_during_calc_log := false;

wr_residual_during_calc_screen := false;

wr_flux_during_calc_log := false;

wr_flux_during_calc_screen := false;

wr_probes_during_calc_log := false;

wr_probes_during_calc_screen := false;

wr_final_results_log := true;

wr_final_results_screen := true;

wr_axes := true;

wr_nodes := false;

wr_node_numbers := true;

wr_node_sizes := false;

wr_coefficients := false;

wr_materials_volumes := true;

warning_priority_log := war_other;

warning_priority_screen := war_other;

solver_def := Find_better_field_thomas;

scheme := exact;

relax_factor := 1.0;


probe_convset := [obs1];




max_time := 5*60;

max_change := 1e-9;


dtim := 0;








press_enter_wanted := true;

run_model;

close_model;

end.

B Output: F0100LOG.dat

---------------------------------------------------------------------------

Description : Radon and soil gas transport model

Program name : RnMod3d (Copyright, Risoe National Laboratory, Denmark)

Version : Version 0.8 (Sep. 15, 1997 - July 18, 2000)

Documentation : User’s Guide to RnMod3d, Risoe-R-1201(EN)

---------------------------------------------------------------------------

* Time = 19-07-2000 09:55:04

**** LOG File : f0100LOG.dat

96 Risø-R-1201(EN)

**** RES File : f0100RES.dat

**** RUN ID : 0100

**** RUN TITLE : User guide example: Steady soil-gas flow, 1D

---------------------------------------------------------------------------

wr_memory_status

imax = 1 jmax = 1 kmax = 1

imaxTot = 100 jmaxTot= 100 kmaxTot = 200

---------------------------------------------------------------------------

wr_memory_status

imax = 3 jmax = 3 kmax = 63

imaxTot = 100 jmaxTot= 100 kmaxTot = 200

---------------------------------------------------------------------------

wr_count_nodes

* Type and number of nodes incl. boundary conditions :

* NOP 244

* free 305

* fixed1 9 value = 0.00000000000E+0000

* fixed2 9 value = -3.00000000000E+0000

* fixed3 0 value = 0.00000000000E+0000

* fixed4 0 value = 0.00000000000E+0000

* fixed5 0 value = 0.00000000000E+0000

* unchanged 0

* Total 567

---------------------------------------------------------------------------

**** RUN ID : 0100

**** RUN TITLE : User guide example: Steady soil-gas flow, 1D

***** Converged

---------------------------------------------------------------------------

Iteration = 301 (5000) Time = 0.01 min ( 5.00) Residual = 2.07E-0021

* Abs. sum of bs = 0.00000E+0000

* Abs. sum of residuals = 2.07079E-0021 (change = -9.99979E-0001)

* Max residual = 2.11758E-0022 (change = -9.99851E-0001)

* Max residual at (i,j,k) = ( 2, 2, 62)

* Max residual at (x,y,z) = ( 5.000E-0001, 5.000E-0001,-1.716E-0002)

Flx1 : J = 1.1428571E-0005 ( change = 9.4091795E-0019 ) Q = 0.0000000E+0000

Flx2 : J = 1.1428571E-0005 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx3 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx4 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Flx5 : J = 0.0000000E+0000 ( change = 0.0000000E+0000 ) Q = 0.0000000E+0000

Obs1 : c = -1.5000000E+0000 ( change = 5.7824116E-0019 )

Obs2 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs3 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs4 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

Obs5 : c = 0.0000000E+0000 ( change = 0.0000000E+0000 )

---------------------------------------------------------------------------

wr_material_volumes_etc (volume-averaged field values)

mat Avg(conc) Activity Volume N N_invalid

mat1 -1.500000000E+0000 0.000000000E+0000 3.000000000E+0000 61 506

mat Min(conc) i j k x y z

mat1 -2.982838178E+0000 2 2 62 5.000E-0001 5.000E-0001 -1.716E-0002

mat Max(conc) i j k x y z

mat1 -1.779212754E-0002 2 2 2 5.000E-0001 5.000E-0001 -2.982E+0000

Total geometric volume = 3.00000000000E+0000

Total activity = 0.00000000000E+0000

Overall mean concentration = -1.50000000000E+0000

---------------------------------------------------------------------------

wr_axes_proc

axis i x[i] x[i+1] dx[i] dcdx dcdxnorm Fixpts

x 1 0.00000 0.00000 0.0000000 9.758E-0019 0.00000000 xFix1

x 2 0.00000 1.00000 1.0000000 1.084E-0019 0.00000000 -

x 3 1.00000 1.00000 0.0000000 0.000E+0000 0.00000000 xFix2

axis j y[j] y[j+1] dy[j] dcdy dcdynorm Fixpts

y 1 0.00000 0.00000 0.0000000 9.758E-0019 0.00000000 yFix1

Risø-R-1201(EN) 97

y 2 0.00000 1.00000 1.0000000 1.084E-0019 0.00000000 -

y 3 1.00000 1.00000 0.0000000 0.000E+0000 0.00000000 yFix2

axis k z[k] z[k+1] dz[k] dcdz dcdznorm Fixpts

z 1 -3.00000 -3.00000 0.0000000 0.000E+0000 0.00000000 zFix1

z 2 -3.00000 -2.96442 0.0355843 3.814E-0002 0.69579013 -

z 3 -2.96442 -2.92372 0.0406923 4.178E-0002 0.76227505 -

z 4 -2.92372 -2.88085 0.0428727 4.361E-0002 0.79566866 -

z 5 -2.88085 -2.83650 0.0443531 4.492E-0002 0.81951967 -

z 6 -2.83650 -2.79101 0.0454874 4.595E-0002 0.83830367 -

...

z 60 -0.11493 -0.07357 0.0413539 4.030E-0002 0.73527063 -

z 61 -0.07357 -0.03432 0.0392507 3.679E-0002 0.67114101 -

z 62 -0.03432 0.00000 0.0343236 1.716E-0002 0.31309835 -

z 63 0.00000 0.00000 0.0000000 0.000E+0000 0.00000000 zFix2

---------------------------------------------------------------------------

* Time = 19-07-2000 09:55:05

C F0101prg.dpr

program F0101prg;

(* --------------------- RnMod3d jobfile ---------------------- *)

(* Project: User guide example: Steady radon diffusion, 1D *)

(* Created: May 24, 1999 *)

(* Revised: July 17, 2000 *)

$I R3dirs03


procedure grid;

begin










end;


begin

cBC[fixed2]:=1000;


i,xFix1,xFix2,

j,yFix1,yFix2,


end;


begin


i,xFix1,xFix2,

j,yFix1,yFix2,



i,xFix1,xFix2,

98 Risø-R-1201(EN)

j,yFix1,yFix2,


end;

procedure probes;


begin

cc:=fieldvalue(0.5,0.5,-1.5,valid);


obsval[obs1]:=cc;

end;


begin

materials:=mat1;

end;


begin

e:=0.3;

end;


begin

beta:=e(i,j,k);

end;


begin

D:=9.9e-7;

end;


begin

G:=0.12974983

end;


begin

lambda:=2.09838e-6;

end;

begin (* main *)

runid := ’0101’;

runtitle := ’User guide example: Steady radon diffusion, 1D’;

solution := steady;


Ly := 1.0;

grid_def := grid;



flux_def := fluxes;



e_def := e;

beta_def := beta;

G_def := G;


D_def := D;

initialfield_def := nil;

import_initialfield := false;

import_finalfield_guess := false;

export_field := false;

use_fieldbuffer := no_cBUF;

Risø-R-1201(EN) 99

flowfield := none;

flowfactor := 1.0;

import_field_name := ’’;

export_field_name := ’’;

flowfield_name := ’’;

plotfiles_def := nil;

user_procedure_each_iter_def := nil;

wr_details := false;

wr_main_procedure_id := false;

wr_all_procedure_id := false;

wr_iteration_line_log := false;


wr_residual_during_calc_log := false;

wr_residual_during_calc_screen := false;

wr_flux_during_calc_log := false;

wr_flux_during_calc_screen := true;

wr_probes_during_calc_log := false;

wr_probes_during_calc_screen := false;

wr_final_results_log := true;

wr_final_results_screen := true;

wr_axes := true;

wr_nodes := false;

wr_node_numbers := true;

wr_node_sizes := false;

wr_coefficients := false;

wr_materials_volumes := false;

warning_priority_log := war_other;

warning_priority_screen := war_other;

solver_def := Find_better_field_thomas;

scheme := exact;


flux_convset := [flx2];





max_time := 5*60;

max_change := 1e-9;


dtim := 0;








press_enter_wanted := true;

run_model;

close_model;

end.

D F0102prg.dpr

program f0102prg;

(* --------------------- RnMod3d jobfile ---------------------- *)

(* Project: User guide example: *)

(* Steady radon diffusion + advection, 1D. *)

(* Created: May 24, 1999 *)

(* Revised: July 17, 2000 *)

100 Risø-R-1201(EN)

$I R3dirs03


const lambda_use = 2.09838e-6;

mu = 17.5e-6;

var ksoil,cS,dP,velocity,Lz,Dsoil,esoil,Gsoil:datatype;

procedure grid;

begin








set_FixVal(zFix2, Lz);

set_axis_double(zFix1,zFix2,30,30,FocusA,FocusB,2,2,0.5);

end;

procedure boundary_conditions_Soilgas(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=dP;


i,xFix1,xFix2,

j,yFix1,yFix2,


cBC[fixed2]:=0.0;


i,xFix1,xFix2,

j,yFix1,yFix2,


end;

procedure boundary_conditions_Rn(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=cS;


i,xFix1,xFix2,

j,yFix1,yFix2,


cBC[fixed2]:=0.0;


i,xFix1,xFix2,

j,yFix1,yFix2,


end;


begin


i,xFix1,xFix2,

j,yFix1,yFix2,



i,xFix1,xFix2,

j,yFix1,yFix2,


end;

Risø-R-1201(EN) 101

procedure probes;


begin

cc:=fieldvalue(0.5,0.5,Lz/2,valid);


obsval[obs1]:=cc;

end;


begin

materials:=mat1;

end;

function e_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

e_soilgas:=0;

end;

function beta_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

beta_soilgas:=0;

end;

function D_soilgas(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

begin

D_soilgas:=ksoil/mu;

end;

function G_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

G_soilgas:=0;

end;

function lambda_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

lambda_soilgas:=0;

end;

function e_Rn(i:itype;j:jtype;k:ktype):datatype;

begin

e_Rn:=esoil;

end;

function beta_Rn(i:itype;j:jtype;k:ktype):datatype;

begin

beta_Rn:=e_Rn(i,j,k);

end;

function D_Rn(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

begin

D_Rn:=Dsoil;

end;

function G_Rn(i:itype;j:jtype;k:ktype):datatype;

begin

G_Rn:=Gsoil;

end;

function lambda_Rn(i:itype;j:jtype;k:ktype):datatype;

begin

lambda_Rn:=lambda_use;

end;

function sinh(x:datatype):datatype;

102 Risø-R-1201(EN)

var z:datatype;

begin

z:=exp(x);

sinh:=(z-1/z)/2

end;

function cosh(x:datatype):datatype;

var z:datatype;

begin

z:=exp(x);

cosh:=(z+1/z)/2

end;

function c_exact(z:datatype):datatype;

var v,D,cinf,s,alpha,Ld:datatype;

(* See NBS technical note 1139, p. 26 *)

begin

D:=Dsoil;

v:=velocity;

alpha:=v/2/D;

Ld:=sqrt(D/esoil/lambda_use);

s:=1/sqrt(sqr(alpha) + sqr(1/Ld));

cinf:=Gsoil/lambda_use;

c_exact:=cinf*(1-1/sinh(Lz/s)*(exp(v*z/2/D)*sinh((Lz-z)/s) + exp(-v*(Lz-z)/2/D)*sinh(z/s)))+

cS*exp(v*z/2/D)*sinh((Lz-z)/s) / sinh(Lz/s);

end;

function j_exact:datatype;

var v,D,cinf,s,alpha,Ld:datatype;

(* See NBS technical note 1139, p. 26 *)

begin

D:=Dsoil;

v:=velocity;

alpha:=v/2/D;

Ld:=sqrt(D/esoil/lambda_use);

s:=1/sqrt(sqr(alpha) + sqr(1/Ld));

cinf:=Gsoil/lambda_use;

j_exact:=cinf*(V/2 + D/s/sinh(Lz/s)*(cosh(Lz/s)-exp(v*Lz/2/D)))+

cS*D/s*exp(v*Lz/2/D)/sinh(Lz/s);

end;

procedure wr_flux;

begin

writeln(LOG,’ dP = ’,dP:6:2);

writeln(LOG,’RnMod3d Rn flux at z=0: ’,FlxVal[flx2].j:16,’ Bq/m2/s’);

writeln(LOG,’Exact Rn flux at z=0: ’,j_exact:16,’ Bq/m2/s’);

writeln(LOG,’Deviation: ’,100*(FlxVal[flx2].j-j_exact)/j_exact:16:4,’ %’);

end;

procedure wr_profile;

var Nsteps,zstart,zstop,dzz,zz,cc:datatype;

valid:boolean;

begin

(* This procedure finds the field at (x,y,z) where x=0.5m *)

(* and y=0.5m, and z is looped through the values from top to *)

(* bottom. *)

Nsteps:=800;

if not (wFixVal[zFix1].defined and wFixVal[zFix2].defined) then

error_std(’wr_profile’,’Undefined fixpoints!’);

zstart:=wFixVal[zFix1].w;

zstop :=wFixVal[zFix2].w;

dzz:=(zstop-zstart)/Nsteps;

zz :=zstart;

Risø-R-1201(EN) 103

writeln(RES,’z’:12,’,’,’c’:12,’,’,’cexact’:12);

while (zz<zstop) do

begin

cc:=fieldvalue(0.5,0.5,zz,valid);

if valid then

writeln(RES,zz:12:6,’,’,cc:12:6,’,’,c_exact(zz):12:6);

zz:=zz+dzz;

end;

end;

begin (* main *)

runid := ’0102’;

runtitle := ’User guide example: Steady Rn diff. and adv.’;

solution := steady;


Ly := 1.0;

grid_def := grid;

flux_def := fluxes;








wr_axes := false;

wr_node_numbers := false;


(* User-defined constants *)

Lz := 5; (* Column depth *)

ksoil := 1e-11; (* Soil permeability *)

cS := 5000; (* Radon conc. at z=0 *)

dP := -100; (* Pressure difference *)

Dsoil := 1e-6; (* Diffusivity *)

esoil := 0.3; (* Porosity *)

Gsoil := 10000*lambda_use; (* Generation rate *)

velocity:=ksoil/mu*dP/Lz;

(* First, the soil gas problem *)


D_def := D_soilgas;

e_def := e_soilgas;


G_def := G_soilgas;


flowfield := export_to_qBUF;


max_change := 1e-12;


run_model;

(* Second, the radon problem *)

flowfield := import_from_qBUF;

boundary_conditions_def := boundary_conditions_Rn;

D_def := D_Rn;

e_def := e_Rn;

beta_def := beta_Rn;

G_def := G_Rn;

lambda_def := lambda_Rn;




104 Risø-R-1201(EN)

run_model;

wr_flux;

wr_profile;

close_model;

end.

E F0103prg.dpr

program f0103prg;

(* --------------------- RnMod3d jobfile ---------------------- *)

(* Project: User guide example: Transient gas flow (1D) *)

(* Created: October 12, 1999 *)

(* Revised: July 17, 2000 *)

$I R3dirs03


const mu = 17.5e-6;

easoil = 0.2;

ksoil = 1e-14;

Tper = 10*3600;

phi = pi/2;

p1 = 3;

P0 = 100000;

omega = 2*pi/Tper;

Dp = ksoil*P0/easoil/mu;

Lz = 5.0;

z_obs1 = 0.2; (* probe locations *)

z_obs2 = 1.0;

z_obs3 = 2.5;

z_obs4 = 4.0;

z_obs5 = 4.8;

procedure grid;

begin








set_FixVal(zFix2,Lz);

set_axis_double(zFix1,zFix2,10,10,FocusA,FocusB,2,2,0.5);

end;

procedure boundary_conditions_Soilgas(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=0;


i,xFix1,xFix2,

j,yFix1,yFix2,


cBC[fixed2]:=0;

if tim>0 then

cBC[fixed2]:=p1 * sin(omega*tim + phi);

Risø-R-1201(EN) 105


i,xFix1,xFix2,

j,yFix1,yFix2,


end;


begin


i,xFix1,xFix2,

j,yFix1,yFix2,



i,xFix1,xFix2,

j,yFix1,yFix2,


end;

procedure probes;

var valid:boolean;

begin

obsval[obs1]:=fieldvalue(0.5,0.5, z_obs1,valid);





end;


begin

materials:=mat1;

end;


begin

e_soilgas:=0;

end;


begin

beta_soilgas:=easoil/P0;

end;


begin

D_soilgas:=ksoil/mu;

end;


begin

G_soilgas:=0;

end;

function lambda_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

lambda_soilgas:=0;

end;

begin (* main *)

runid := ’0103’;

runtitle := ’User guide example: Transient gas flow in slab’;

solution := steady;


grid_def := grid;

106 Risø-R-1201(EN)



flux_def := fluxes;



D_def := D_soilgas;

e_def := e_soilgas;


G_def := G_soilgas;


flux_convset := [flx2];

probe_convset := [obs1,obs2,obs3,obs4,obs5];




max_time := 5*60;



wr_iteration_line_screen := false;

wr_final_results_screen := false;

wr_axes := false;

wr_node_numbers := false;


(* First do steady-state for t=0 *)

tim:=0;

run_model;

(* Then do the unsteady part *)

solution:=unsteady;

dtim:=Tper/500;

(* Write header w. labels *)

writeln(’tim/Tper’:16,’ ’,’dtim/Tper’:16,’ ’,’cBC[fixed2]’:16,’ ’,’obsval[obs5]’:16);

writeln(RES,’tim’:16,’, ’,

’hr’:16,’, ’,

’Patm’:16,’, ’,

’Q1’:16,’, ’,

’Q*2’:16,’, ’,

’P1’:16,’, ’,

’P2’:16,’, ’,

’P3’:16,’, ’,

’P4’:16,’, ’,

’P5’:16);

repeat

writeln(tim/Tper:16:4,’ ’,dtim/Tper:16:4,’ ’,cBC[fixed2]:16:4,’ ’,obsval[obs5]:16:4);

writeln(RES,tim:16,’, ’,

tim/3600:16,’, ’,

cBC[fixed1]:16,’, ’,

FlxVal[Flx1].j:16,’, ’,

FlxVal[Flx2].j:16,’, ’,

obsval[obs1]:16,’, ’,




obsval[obs5]:16);

tim:=tim+dtim;

run_model;

until tim>4*Tper;

Risø-R-1201(EN) 107

wr_gridfiles;

close_model;

end.

F F0130prg.dpr

program F0130prg;

(* --------------------- RnMod3d jobfile ---------------------- *)

(* Project: User guide example: *)

(* House simulation (slab-on-grade) *)

(* Created: September 26, 1998 *)

(* Revised: July 18, 2000 *)

$I R3dirs03


const

LambdaRn222 = 2.09838e-6; (* 1/s *)

mu = 18.0e-6; (* Pa s *)

rho_g = 2.7e3; (* kg/m3 *)

LOstwald = 0.30; (* water/gas partitioning *)

deltaP = -1.0; (* Pa *)

(* Horizontal (x) dimensions, m *)

Lx_soil = 20.00;

Lx_slab = 5.6419;

Lx_footer = 0.300;

Lx_gap = 0.003;

(* Vertical (z) dimensions, m *)

Lz_soil = 10.00;

Lz_slab = 0.10;

Lz_gravel = 0.15;

Lz_footer = 0.80;

(* Radium-226 concentration, Bq/kg *)

ARa_soil = 40;

ARa_slab = 50;

ARa_gravel = 40;

ARa_footing = 0;

ARa_gap = 0;

(* Fraction of emanation, - *)

f_soil = 0.2;

f_slab = 0.1;

f_gravel = 0.2;

f_footing = 0;

f_gap = 0;

(* Porosity, - *)

etot_soil = 0.25;

etot_slab = 0.20;

etot_gravel = 0.40;

etot_footing = 0.20;

etot_gap = 1.00;

(* Volumetric water content, - *)

msat_soil = 0.20;

msat_slab = 0.0;

msat_gravel = 0.0;

108 Risø-R-1201(EN)

msat_footing = 0.0;

msat_gap = 0.0;

(* Bulk diffusivity, m2/s *)

D_soil = 4.3e-7;

D_slab = 2.0e-8;

D_gravel = 1.8e-6;

D_footing = 1.0e-10;

D_gap = 1.2e-5;

(* Gas permeability, m2 *)

k_soil = 1e-11;

k_slab = 1e-15;

k_gravel = 5e-9;

k_footing = 1e-15;

k_gap = 7.5e-7;

procedure grid;

begin

(* x-axis *)


set_FixVal(xFix2,Lx_slab-Lx_gap);

set_FixVal(xFix3,Lx_slab);

set_FixVal(xFix4,Lx_slab+Lx_footer);

set_FixVal(xFix5,Lx_soil);

set_axis_double(xFix1,xFix2,15,15,FocusB,FocusB,2.1,3.0,0.97);


set_axis_double(xFix3,xFix4,4,4,FocusA,FocusB,2.0,2.0,0.5);


(* z-axis *)

set_FixVal(zFix1,-Lz_soil);

set_FixVal(zFix2,-Lz_footer);

set_FixVal(zFix3,-Lz_slab-Lz_gravel);

set_FixVal(zFix4,-Lz_slab);




set_axis_double(zFix3,zFix4,15,5,FocusB,FocusB,2.0,2,0.95);

set_axis_single(zFix4,zFix5,4,FocusB,3.0);

end;

procedure boundary_conditions_soilgas(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=deltaP;

cBC[fixed2]:=0;

if in_plane([inside,eqAB], (* Observe: Full slab, not just the gap *)

i,xFix1,xFix3,

j,yFix1,yFix2,

k,zFix5,zFix5) then

change_node(i,j,k,fixed1,ConX,ConX,ConX,ConX,ConX,ConX);

if in_plane([inside,eqAB], (* Atmospheric surface *)

i,xFix4,xFix5,

j,yFix1,yFix2,

k,zFix5,zFix5) then


end;

procedure boundary_conditions_radon(i:itype;j:jtype;k:ktype);

begin

cBC[fixed1]:=0;

cBC[fixed2]:=0;

if in_plane([inside,eqAB], (* Observe: Full slab, not just the gap *)

i,xFix1,xFix3,

Risø-R-1201(EN) 109

j,yFix1,yFix2,

k,zFix5,zFix5) then


if in_plane([inside,eqAB], (* Atmospheric surface *)

i,xFix4,xFix5,

j,yFix1,yFix2,

k,zFix5,zFix5) then


end;


begin


i,xFix1,xFix2,

j,yFix1,yFix2,

k,zFix5,zFix5) then

begin

update_flxval(Flx1,bottom,i,j,k,plus); (* slab *)

update_flxval(Flx4,bottom,i,j,k,plus); (* add to total house entry *)

end;


i,xFix2,xFix3,

j,yFix1,yFix2,

k,zFix5,zFix5) then

begin

update_flxval(Flx2,bottom,i,j,k,plus); (* gap *)

update_flxval(Flx4,bottom,i,j,k,plus); (* add to total house entry *)

end;


i,xFix4,xFix5,

j,yFix1,yFix2,

k,zFix5,zFix5) then

update_flxval(Flx3,bottom,i,j,k,plus); (* atm. surface *)

end; (* fluxes *)

procedure probes;

var c1,dc1:datatype;

valid1:boolean;

begin

get_fieldvalue2d((wFixVal[xfix2].w+wFixVal[xfix3].w)/2,0.0005,

wFixVal[zfix4].w,0.000001,c1,dc1,valid1);

obsval[obs1]:=c1;

get_fieldvalue2d((wFixVal[xfix3].w+wFixVal[xfix4].w)/2,0.001,

wFixVal[zfix2].w-0.05,0.02,c1,dc1,valid1);

obsval[obs2]:=c1;

get_fieldvalue2d(wFixVal[xfix1].w+0.3,0.001,

wFixVal[zfix1].w+0.3,0.02,c1,dc1,valid1);

obsval[obs3]:=c1;

get_fieldvalue2d(wFixVal[xfix5].w-0.3,0.001,

wFixVal[zfix1].w+0.3,0.02,c1,dc1,valid1);

obsval[obs4]:=c1;

get_fieldvalue2d(wFixVal[xfix5].w-0.3,0.001,

wFixVal[zfix5].w-0.3,0.02,c1,dc1,valid1);

obsval[obs5]:=c1;

end; (* probes *)


var mat:mattype;

begin

mat:=mat1; (* soil *)

if in_region(i,xFix1,xFix2,[inside,eqab],

j,yFix1,yFix2,[inside,eqab],

k,zFix4,zFix5,[inside,eqab]) then mat:=mat2; (* slab *)


110 Risø-R-1201(EN)


k,zFix3,zFix4,[inside,eqab]) then mat:=mat3; (* gravel *)



k,zFix2,zFix5,[inside,eqab]) then mat:=mat4; (* footing *)



k,zFix4,zFix5,[inside,eqab]) then mat:=mat5; (* gap *)

materials:=mat;

end; (* materials *)

function m(i:itype;j:jtype;k:ktype):datatype;

var mm:datatype;

begin

mm:=0;


mat1: mm:=msat_soil;

mat2: mm:=msat_slab;

mat3: mm:=msat_gravel;

mat4: mm:=msat_footing;

mat5: mm:=msat_gap;

else

error_std(’m’,’Unknown material’);

end;

m:=mm;

end;


begin

e_soilgas:=0;

end;


begin

beta_soilgas:=0;

end;


begin

G_soilgas:=0;

end;

function Lambda_soilgas(i:itype;j:jtype;k:ktype):datatype;

begin

Lambda_soilgas:=0;

end;


var kk:datatype;

begin

kk:=0;


mat1: kk:=k_soil;

mat2: kk:=k_slab;

mat3: kk:=k_gravel;

mat4: kk:=k_footing;

mat5: kk:=k_gap;

else

error_std(’D_soilgas’,’Unknown material’);

end;

D_soilgas:=kk/mu

end;

function e_radon(i:itype;j:jtype;k:ktype):datatype;

Risø-R-1201(EN) 111

var ee:datatype;

begin

ee:=0;


mat1: ee:=etot_soil;

mat2: ee:=etot_slab;

mat3: ee:=etot_gravel;

mat4: ee:=etot_footing;

mat5: ee:=etot_gap;

else

error_std(’e’,’Unknown material’);

end;

e_radon:=ee;

end;

function beta_radon(i:itype;j:jtype;k:ktype):datatype;

var ea,ew:datatype;

begin

ew:=m(i,j,k)*e_radon(i,j,k);

ea:=e_radon(i,j,k)-ew;

beta_radon:=ea+LOstwald*ew;

end;

function G_radon(i:itype;j:jtype;k:ktype):datatype;

var GG,ee,lam:datatype;

begin

GG:=0;

ee:=e_radon(i,j,k);

lam:=lambdaRn222;


mat1: GG:=rho_g*(1-ee)/ee*lam*f_soil * ARa_soil;

mat2: GG:=rho_g*(1-ee)/ee*lam*f_slab * ARa_slab;

mat3: GG:=rho_g*(1-ee)/ee*lam*f_gravel * ARa_gravel;

mat4: GG:=rho_g*(1-ee)/ee*lam*f_footing * ARa_footing;

mat5: GG:=rho_g*(1-ee)/ee*lam*f_gap * ARa_gap;

else

error_std(’G’,’Unknown material’);

end;

G_radon:=GG;

end;

function Lambda_radon(i:itype;j:jtype;k:ktype):datatype;

begin

Lambda_radon:=LambdaRn222;

end;

function D_radon(dir:dirtype;i:itype;j:jtype;k:ktype):datatype;

var DD:datatype;

begin

DD:=0;


mat1: DD:=D_soil;

mat2: DD:=D_slab;

mat3: DD:=D_gravel;

mat4: DD:=D_footing;

mat5: DD:=D_gap;

else

error_std(’D_radon’,’Unknown material’);

end;

D_radon:=DD;

end;

begin (* main *)

runid := ’0130’;

112 Risø-R-1201(EN)

solution := steady;

geometry := cylindrical2d;

grid_def := grid;

flux_def := fluxes;




wr_flux_during_calc_screen := true;

wr_axes := false;

(* First do the soil-gas simulation *)

runtitle := ’Slab-on-grade house (pressure)’;


e_def := e_soilgas;


G_def := G_soilgas;


D_def := D_soilgas;

import_finalfield_guess := true;

export_field := true;

flowfield := export;

import_field_name := ’PRES00.dat’;

export_field_name := import_field_name;


flux_convset := [flx1..flx3];





max_time := 60*60;



run_model; (* Soil gas run *)

(* Then do the radon simulation *)

runtitle := ’Slab-on-grade house (radon)’;

boundary_conditions_def := boundary_conditions_radon;

e_def := e_radon;

beta_def := beta_radon;

G_def := G_radon;

lambda_def := lambda_radon;

D_def := D_radon;

import_finalfield_guess := true;

export_field := true;

flowfield := import;

import_field_name := ’Rn0000.dat’;

export_field_name := import_field_name;


flux_convset := [flx1..flx3];





max_time := 60*60;



run_model; (* Radon run *)

writeln(LOG,’The total soil-gas entry into the house is (Flx4) = ’,FlxVal[Flx4].Q:16,’ m3/s’);

writeln(LOG,’The total radon entry into the house is (Flx4) = ’,FlxVal[Flx4].J:16,’ Bq/s’);

close_model;

end.

Risø-R-1201(EN) 113

References

[An92] C.E. Andersen: Entry of soil gas and radon into houses. Risø-R-623(EN), Risø National Laboratory, DK-4000 Roskilde, Denmark,1992.

[An99a] C.E. Andersen, D. Albarracın, I. Csige, E.R. van der Graaf, M.Jiranek, B. Rehs, Z. Svoboda, and L. Toro: ERRICCA radon modelintercomparison exercise, Risø-R-1120(EN), Risø National Labo-ratory, DK-4000 Roskilde, Denmark, 1999 (This document can bedownloaded from Risø’s web-site: www.risoe.dk).

[An99b] C.E. Andersen: Radon-222 exhalation from Danish building mate-rials: H+H Industri A/S results. Risø-R-1135(EN), Risø NationalLaboratory, DK-4000 Roskilde, Denmark, 1999 (This documentcan be downloaded from Risø’s web-site: www.risoe.dk).

[An99c] C.E. Andersen: Numerical modelling of radon-222 entry intohouses: An outline of techniques and results. Presented at Radonin the living environment, April 19–23, 1999, Athens, Greece asabstract no. 64. Submitted for publication in the workshop pro-ceedings.

[Bi60] R.B. Bird, W.E. Stewart, and E.N. Lightfoot: Transport phenom-ena. John Wiley & Sons, 1960.

[Car59] H.S. Carslaw and J.C. Jaeger: Conduction of Heat in Solids. Secondedition. Oxford Science Publications, Oxford, 1959.

[Cl79] H.L. Clever (ed.): Solubility data series. Volume 2. Krypton, xenonand radon - gas solubilities. Pergamon Press, 1979.

[Co81] R. Colle, R.J. Rubin, L.I. Knab, and J.M.R. Hutchinson: Radontransport through and exhalation from building materials: A Re-view and assessment. NBS Technical Note 1139. National Bureauof Standards, U.S. Department of Commerce, 1981.

[Do92] P.A. Domenico and F.W. Schwartz: Physical and Chemical Hydro-geology. John Wiley and Sons, 1990.

[Fe99] J.H. Ferziger and M. Peric: Computational methods for fluid dy-namics. Second edition. Springer, Berlin, Germany, 1999.

[He96] R. Helmig: Einfuhrung in die numerischen Methoden der Umwelt-stromungsmechanik. Institut fur Computer Anwendungen imBauingenieurwesen, Techniche Universitat Braunschweig, Ger-many, 1996.

[Ho94] D.J. Holford: Rn3D: A finite element code for simulating gas flowand radon transport in variably saturated, nonisothermal, porousmedia: User’s manual, version 1.0. Pacific Northwest Laboratory,USA, PNL-8943, 1994.

[Lo87] C.O. Loureiro: Simulation of the steady-state transport of radonfrom soil into houses with basements under constant negative pres-sure. LBL-24378, Lawrence Berkeley Laboratory, CA 94720, USA,1987.

[Na92] W.W. Nazaroff: Radon transport from soil to air. Review of Geo-physics, vol. 30(2), pp. 137–160, 1992.

114 Risø-R-1201(EN)

[Na88] W.W. Nazaroff, B.A. Moed, and R.G. Sextro: Soil as a source of In-door Radon: Generation, Migration, and Entry. IN: W.W. Nazaroffand A.V. Nero (eds.). Radon and its Decay Products in Indoor Air.Wiley-Interscience, 1988.

[Pa80] S.V. Patankar: Numerical heat transfer and fluid flow. HemispherePublishing Corporation, New York, 1980.

[Pa88] S.V. Patankar: Elliptic systems: Finite-difference method I. IN:W.J. Minkowycz, E.M. Sparrow, G.E. Schneider, and R.H.Pletcher: Handbook of numerical heat transfer. John Wiley & SonsInc., New York, 1988.

[Ri99] W.J. Riley, A.L. Robinson, A.J. Gadgil, and W.W. Nazaroff: Ef-fects of variable wind speed and direction on radon transport fromsoil into buildings: Model developement and exploratory results.Atmospheric Environment, vol. 33, pp. 2157–2168, 1999.

[Rog91A] V.C. Rogers and K.K. Nielson: Multiphase radon generation andtransport in porous material. Health Physics, vol. 60, no. 6 (June),pp. 807–815, 1991.

[Rog91B] V.C. Rogers and K.K. Nielson: Correlations for predicting airpermeabilities and 222radon diffusion coefficients of soils. HealthPhysics, vol. 61, no. 2 (August), pp. 225–230, 1991.

[Sp98] W.H. van der Spoel: Radon transport in sand: A laboratory study.Ph.D. dissertation, Technical University Eindhoven, the Nether-lands, ISBN 90-386-0647-8, 1998.

[Th97] N.R. Thomson, J.F. Sykes, and D. van Vliet: A numerical investi-gation into factors affecting gas and aqueous phase plumes in thesubsurface. Journal of Contaminant Hydrology, vol. 28, pp. 39–70,1997.

[Ve95] H.K. Versteeg and W. Malalasekera: An introduction to computa-tional fluid dynamics. The finite volume method. Longman, Edin-burg, England, 1995.

[Wa94] J.W. Washington, A.R. Rose, E.J. Ciolkosz, and R.R. Dobos:Gaseous diffusion and permeability in four soil profiles in centralPennsylvania. Soil Science, vol. 157(2), pp. 65–76, 1994.

[Wo92] C.S. Wong, Y-P. Chin, and P.M. Gschwend: Sorption of radon-222to natural sediments. Geochimica et Cosmochimica Acta, vol. 56,pp. 3923–3932, 1992.

Risø-R-1201(EN) 115

Bibliographic Data Sheet Risø-R-1201(EN)

Title and author(s)

Radon transport modelling: User’s guide to RnMod3d

Claus E. Andersen

ISBN

87-550-2734-2 (printed edition)87-550-2733-4 (internet edition)

ISSN

0106-2840

Dept. or group

Nuclear Safety Research Department

Date

August, 2000


Pages

116

Tables

6

Illustrations

20

References

25


RnMod3d is a numerical computer model of soil-gas and radon transport in porousmedia. It can be used, for example, to study radon entry from soil into houses inresponse to indoor-outdoor pressure differences or changes in atmospheric pres-sure. It can also be used for flux calculations of radon from the soil surface or tomodel radon exhalation from building materials such as concrete.The finite-volume model is a technical research tool, and it cannot be used mean-ingfully without good understanding of the involved physical equations. Someunderstanding of numerical mathematics and the programming language Pascalis also required. Originally, the code was developed for internal use at Risø only.With this guide, however, it should be possible for others to use the model.Three-dimensional steady-state or transient problems with Darcy flow of soil gasand combined generation, radioactive decay, diffusion and advection of radon canbe solved. Moisture is included in the model, and partitioning of radon betweenair, water and soil grains (adsorption) is taken into account. Most parameters canchange in time and space, and transport parameters (diffusivity and permeability)may be anisotropic.This guide includes benchmark tests based on simple problems with known so-lutions. RnMod3d has also been part of an international model intercomparisonexercise based on more complicated problems without known solutions. All testsshow that RnMod3d gives results of good quality.


ADVECTION; BUILDINGMATERIALS; COMPUTERIZED SIMULATION; COM-PUTER PROGRAM DOCUMENTATION; DIFFUSION; ENVIRONMENTALTRANSPORT; FINITE DIFFERENCEMETHOD; GAS FLOW; HOUSES; RADON222; R CODES; SOILS

Available on request from:Information Service Department, Risø National Laboratory(Afdelingen for Informationsservice, Forskningscenter Risø)P.O. Box 49, DK–4000 Roskilde, DenmarkPhone (+45) 46 77 46 77, ext. 4004/4005 · Fax (+45) 46 77 40 13E-mail: [email protected]

Risø National Laboratory carries out research within science and technology,providing Danish society with new opportunities for technological development. Theresearch aims at strengthening Danish industry and reducing the adverse impact onthe environment of the industrial, energy and agricultural sectors.

Risø advises government bodies on nuclear affairs.

This research is part of a range of Danish and international research programmes andsimilar collaborative ventures. The main emphasis is on basic research andparticipation in strategic collaborative research ventures and market driven tasks.

Research is carried out within the following programme areas:

• Industrial materials• New functional materials• Optics and sensor systems• Plant production and circulation of matter• Systems analysis• Wind energy and atmospheric processes• Nuclear safety

Universities, research institutes, institutes of technology and businesses areimportant research partners to Risø.

A strong emphasis is placed on the education of young researchers through Ph.D.and post-doctoral programmes.

ISBN 87-550-2734-2ISBN 87-550-2733-4 (Internet)ISSN 0106-2840

Copies of this publicationare available from

Risø National LaboratoryInformation Service DepartmentP.O. Box 49DK-4000 RoskildeDenmarkTelephone +45 4677 [email protected] +45 4677 4013Website www.risoe.dk

PDF

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