The CALIBRE Source-Term Code: Technical Documentation for ...

SKI Report 95:13

SITE-94

The CALIBRE Source-Term Code:Technical Documentation for Version 2

K J WorganP C Robinson

March 1995

ISSN 1104-1374ISRNSKI-R--95/13--SE

STATf:Nr, KARNKRAFTINSPRKTIOM

^wectir.h Nuclear Pov^w lnr,pf;rlorntr

SKI Report 95:13

The CALIBRE Source-Term Code:Technical Documentation for Version 2

K J WorganP C Robinson

Intera Information Technologies,Chiltern House, 45 Station Road, Henley-on-Thames,

Oxfordshire, RG9 1AT, United Kingdom

March 1995

IM3583-3, Version 2

This report concerns a study which has been conducted for the Swedish NuclearPower Inspectorate (SKI). The conclusions and viewpoints presented in the report

are those of the author(s) and do not necessarily coincide with those of SKI.

13389 Stockholm 1995

PREFACE

This report concerns a study which is part of the SKI perfonnance assessmentproject SITE-94. SITE-94 is a performance assessment of a hypothetical repositoryat a real site. The main objective of the project is to determine how site specific datashould be assimilated into the performance assessment process and to evaluate howuncertainties inherent in site characterization will influence perfonnance assessmentresults. Other important elements of SITE-94 are the development of a practical anddefensible methodology for defining, constructing and analyzing scenarios, thedevelopment of approaches for treatment of uncertainties, evaluation of canisterintegrity, and the development and application of an appropriate Quality Assuranceplan for Performance Assessments.

Johan AnderssonProject Manager

The CALIBRE Source Term Code

Summary

The CALIBRE source term model has been updated, to make it a com-putationally more robust and faster code, and to incorporate some newfeatures. These include the addition of a canister pinhole release modeland a "tunnel" option. The tunnel is represented by a zero concentrationboundary condition at the interface of the backfilled tunnel with the ben-tonite and rock. The redox front model has been incorporated into theradionuclide transport code, so that a redox calculations may be run aloneor prior to a radionuclide migration calculation. Redox front results canalso be saved and re-used for radionuclide transport.

More generally, the code simulates the diffusion of radionuclides from ahigh-level waste canister, through a backfill region and into a fracturedrock matrix. The model includes chain decay and ingrowth, linear equi-librium sorption, solubility limiting and response to a redox front as itemerges from the canister and migrates through the near-field. Radial ad-vection (which approximates the advection downstream from the canisterand buffer) is applied in the fracture, in addition to diffusion. There isalso an option to allow for fixed total concentrations (ie concentrationssorbed on the solid and dissolved in the pore water) of naturally occurringisotopes in the fractured rock.

This document describes the mathematical model and numerical methodsused in developing CALIBRE, together with a number of verification testswhich compare the results with those computed using analytic solutions.

Contents

1 Introduction 1

2 Mathematical Model 3

2.1 Transport in the Buffer and Rock Matrix 3

2.2 Transport in the Fracture 8

2.3 Precipitate Formation and Dissolution 14

2.4 Pinhole Release 14

2.5 Tunnel Option 16

3 Numerical Methods 17

3.1 Spatial Discretization 17

3.2 Outputs 19

3.3 Time Stepping 20

4 Code Verification 22

4.1 One-Dimensional Radial Diffusion with a Delta-FunctionSource Term 22

4.2 One-Dimensional Radial Diffusion with a Constant Con-centration Source Term 25

4.3 Two-Dimensional Diffusion with a Delta-Function. Z-DependantSource Term 27

4.4 Two-Dimensional Diffusion with Constant ConcentrationSource Term 29

4.5 One-Dimensional Radial Diffusion with a 3-Nuclide DecayChain 31

4.6 Two-Dimensional Diffusion with a 3-Nuclide Decay Chain 35

4.7 Verification of the Radial Flow Approximation 40

4.8 Conclusions 41

References 42

List of Tables

Table 4.1 Comparison of numerical and analytic results for testcase 1 24



Table 4.4 Comparison of the flux from the numerical and analyticmodels for test case 3 29


Table 4.6 Comparison of numerical and analytic results for testcase 5. Np-237 32

Table 4.7 Comparison of numerical and analytic results for test-case 5, U-233 33

Table 4.8 Comparison of numerical and analytic results for testcase 5, Th-229 34

Table 4.9 Comparison of numerical and analytic results for testcase 6. Np-237 36

Table 4.10 Comparison of numerical and analytic results for testcase 6, U-233 37

Table 4.11 Comparison of numerical and analytic results for testcase 6, Th-229 38

Table 4.12 Comparison of flux for test case 6, Np-237 and U-233 39

Table 4.13 Comparison of flux for test case 6, Th-229 39

Table 4.14 Comparison of KBS-3 and CALIBRE Qeq Values . . 41

List of Figures

F i g u r e 1 . 1 T h e C o n c e p t u a l M o d e l G e o m e t r y i n C A L I B R E . . . 1

Figure 2.1 Schematic Representation of Channeling and Connec-tion Lengths between Channels 9

Figure 2.2 Schematic Representation of the Flow Field Arounda Cylindrical Container in the Fracture Plane 10

Figure 2.3 Cell Discretization in the Fracture Plane. ShowingRadial and Angular Dependence of Contaminant Concen-tration 11

Figure 2.4 Radial Concentration and Flow Approximation . . . 13

Figure 3.1 Schematic Representation of Spatial Discretization . 17

Figure 4.1 Schematic Representation of Test Case 1 23






Introduction

The CALIBRE computer program has been developed as a modelling toolfor the Swedish Nuclear Power Inspectorate. The original version was usedin SKIs Project-90. a safety assessment exercise for a reference repositoryof the disposal of spent nuclear fuel [1]. The conceptual model geome-try for CALIBRE is illustrated in figure 1.1. and is based on the KBS-3disposal concept. The model calculates the behaviour of radionuclides re-leased to the near-field environment after the degradation of a high-levelwaste canister. Processes modelled include leaching of the nuchdes fromthe solid waste matrix, diffusion through a bentonite backfill surroundingthe canister, diffusion through a fractured rock matrix and advection anddiffusion in the fracture water. As well as decay and ingrowth of chainsof radionuclides. the model includes linear equilibrium sorption and solu-bility limiting under reducing or oxidising conditions. Changes in redoxconditions occur in response to the movement of a redox front, which orig-inates in the canister and migrates through the near-field. The redox frontinformation is calculated first and may be stored for re-use. The theorybehind redox front motion is described in detail in separate documents.[2.3].

WATER FLOWTHROUGHCANISTERS

CRYSTALLINEROCK

Figure 1.1 The Conceptual Model Geometry in CALIBRE

The mathematical model of radionuclide transport in the near-field is setout in section 2. The original detailed mathematical description of the

1

system is given in the earlier technical document :4 and is not repeatedhere. The numerical methods used to solve the equations are described insection 3. This includes the finite difference space discretization and timestepping scheme employed, together with the approach used to handle theadvection. diffusion, decay and solubility limiting processes.

Section 4 describes a number of verification problems, where the CALI-BRE results are compared with those computed using analytic methods.A detailed description of these methods is given in reference [9j. Theyshow good agreement overall and also inaicate areas where care should betaken in using the model, to ensure the accuracy of the results.

2

2.1

Mathematical Model

Transport in the Buffer and Rock Matrix

In the earlier technical document [4] the equations for the release of ra-dionuclides from the canister and their migration through the bentoniteand rock matrix were set out. These have been followed with few modifi-cations.

The approach taken to calculating nuclide concentrations in the canisterregion has been simplified in the new version of CALIBRE- The quantitiessolved for are Afj(t). the total concentrations of released nuclides. Suffixij denotes isotope i of element j . Each A?(t) is partitioned into theconcentration dissolved in the pore water. C£(*). the concentration sorbedon solid material. Sf}(t) and the concentration precipitated.

(II

where zC is the porosity in the canister.

The initial inventory J° of any nuclide may be assigned for release viagap. grain boundary and/or matrix release modes, provided the fractionsapportioned to each mode sum to one.

Any gap inventories are assigned at canister failure time to the canisterregion- At any given time t. the amounts released from the waste formover a timestep interval At are given by

p- + I^(t)At^-lab »I

(2)

Where I^{t). I™(t) are the initial grain boundary and matrix inventoriesdecayed to time t. T9(, is the grain boundary release period and ^ is theslope of the function representing the fraction of nuclides remaining inthe matrix at time t. against time. These amounts are then decayed overthe time step and added to the amounts in the grid cells representing thecanister region. The matrix release curve is specified by the user in theinput data file, as a time series.

Once released from the solid waste form the nuclides in the pore waterdiffuse radially into the bentonite. and axially within the degraded canisterregion. The governing equation for Af}{t) is given by

id dcf.az-

(3!

where D0 is the effective pore water diffusion coefficient in the degradedcanister region, and IJ represents the parent isotope.

The partitioning between dissolved, sorbed and precipitated concentra-tions in the absence of solubility limits is given by:

Cj(f) = ASf,(t) = pc

= o.

14}

(5)

(6)

where pc is the density of material in the source region. Kf is the element-dependent distribution coefficient and Q^ is the capacity factor given by:

(7)

If at any time the solubility limit for element j . CJMsol) is exceeded insidethe container then equation 4 ceases to hold. Instead the dissolved con-centration of each isotope is proportional to the ratio of the total isotopeconcentration to total element concentration:

Cflt) =tt}[t)

(8)

where the summation over isotopei includes the stable species. The ex-pression for the sorbed concentration equation 5 is unchanged while theprecipitate concentration is given by:

(9)

Before the canister fails, the inventory is calculated using the Batemanequations, and the concentrations released in all states are set to zero.After canister failure, water enters the container so that nuclides start todissolve. This may occur under reducing conditions, before the canisterregion becomes fully oxidising.

In the bentonite and rock the governing equations for the Ay are similarto equation 3. In the bentonite equation 3 becomes:

§(r, z, t) = z

at

l B ~-(r-rdr\ dr dz2

and in the rock

dA*(r,z,t) =

dti*(r,z,t) + \uAfj{riZ,t)

+D IM'* dz2

• (10)

(11)

where DD, DR are the effective pore water diffusion coefficients in thebentonite and rock respectively.

The boundary conditions are as follows. At the innermost radius of thegrid, which corresponds to the axis of the cylinder, a zero flux conditionis imposed:

= 0 (12)r=0

Flux across the interface of the canister and bentonite is calculated byassigning an interface diffusion coefficient, DAV which is the weighted av-erage of the canister and bentonite diffusion coefficients. The weights aredetermined by the volumes Vc, V# of the cells adjoining the interface.

(13)ir=Tl

Where rt, is the radius of the canister/bentonite interface. This particularweighted average was chosen in preference to the more familiar harmonic

average as it was found to give better agreement with Calibre Version 1and also better agreement with the KBS-3 equivalent flow verification testsreported in Section 4.7. Fluxes across boundaries are calculated slightlydifferently in Calibre Version 2 owing to the more conventional choice ofaligning media boundaries with cell boundaries. In Calibre Version 1,media boundaries were aligned with all centres, forcing the boundary cellsto have mixed properties. The switch was made in Calibre Version 2 asit was felt to be more straightforward for the user, when designing thegeometry and discretisation of a problem.

At the upper and lower boundaries of the grid, zero flux conditions areagain imposed:

dz z=0

= 0 (14)z=Z

Here C represents the concentration in the waste, bentonite, rock or frac-ture and Z is the half-distance between fractures. The exception to thisis the tunnel boundary condition, where

(15)

in the bentonite and rock.

Flux across the interface of the bentonite and rock and the bentonite andfracture is again calculated by assigning an interface diffusion coefficientwhich is a weighted average of the two media diffusion coefficients. Hence

^Z.( r , z,t)\o<z<b dr

r=r2

b<z<Z

r=r2

o<z<b

(16)

(17)

Here TI is the radius of the interface and D-41, DA2 are the averaged diffu-sion coefficients and superscript F denotes the fracture.

At the interface of the fracture and rock, the flux is computed using therock diffusion coefficient, since it is this which is the limiting factor. Inaddition, terms are added to account for the possibility of channelling inthe fracture, where only a fraction of the fracture surface area is availablefor matrix diffusion. The boundary condition becomes

z=b

(18)2=6

Fs. the fraction of the fracture surface area available for diffusion, maybe expressed in terms of the specific wet surface area per unit volume ofrock, a, the fracture half-width, b, and the fractured rock porosity, 6:

e (19)

The far boundary is given by:

In the program, the boundary condition (20) is replaced by

(20)

Cg(r,z,t) = 0. (21)

where r3 is chosen to be far enough away from the canister for the conditionto be valid.

Partitioning between dissolved, sorbed and precipitated states in each ma-terial is treated in the same manner as in the canister (equations 4 - 9 )with appropriate choice of the material-dependent quantities of density,porosity and distribution coefficients. In the rock and fracture, allowancehas also been made for the inclusion of naturally occurring isotopes of theelement. A very simple approach to the inclusion of such nuclides hasbeen taken, at this stage. Their main effect on the waste nuclides will beto reduce the proportion of these nuclides dissolved in the rock and frac-ture pore-water, in the event that the elemental solubility limits in theseregions are exceeded. Hence all naturally occurring isotopes of an elementare lumped together and assigned (by the user) fixed, total concentra-tions (ie dissolved plus sorbed) in the rock and fracture. The nuclides aretherefore only accounted for in the calculation of total concentrations ofeach element, and not in the transport calculations. Consequently, theirpresence will have no influence on the results, unless the pore water be-comes saturated, so that individual isotopic concentrations are calculatedaccording to equation (8). Note that the concentrations of naturally oc-curring nuclides in the bentonite and canisters regions are always assumedto be zero.

2.2 Transport in the Fracture

The mass transfer in the fracture can be described in terms of the conti-nuity equation for the total concentration A of radionuclides. comprisingthe fractions dissolved in the water, sorbed on the fracture walls and pre-cipitated:

dt

(22)

The subscript ij denotes isotope i of element j , with IJ representingthe parent nuclide. A is the total concentration of the nuclide. C is theconcentration dissolved in the water, Dw is the diffusion coefficient infree water, vT and ve axe the radial and tangential components of the flowvelocity and Ay, Xu are the decay constants of the nuclide and its parent.

The partitioning between dissolved, sorbed and precipitated concentra-tions in the absence of solubility limits is given by:

OL-i

Stj(t)

Pa(t)

J \

= 0

(23)

(24)

(25)

where pR, Kj1, eR are the density, distribution coefficient (of element j)and porosity of the rock, 6 is the fractured rock porosity, 6 is the effectivedepth of surface sorption on the fracture walls and a is the channellingparameter, or specific wet surface area per unit volume of fractured rock.This definition corresponds to that used in the Project-90 geosphere trans-port code, CRYSTAL [5]. The capacity factor Q^ of the fracture is givenbv:

(26)

where the porosity of the fracture is 1. Hence if there is no surface sorption.the 8 parameter is simply set to zero.

plane-parallelfracture width

water-bearingchannels in thefracture plane

x typical "connection lengths"between channels and betweena channel and rock matrix

Figure 2.1 Schematic Representation of Channeling and Con-nection Lengths between Channels.

The results of the Project-90 near-field and far-field calculations indicatedthat using the same values of the specific wet surface area parameter inboth CALIBRE and CRYSTAL gave different results, in terms of theproportion of matrix diffusion that occurs. Matrix diffusion was moreeffective in CALIBRE because it occurs in two dimensions in the rock,whereas it occurs in only one-dimension in CRYSTAL [6]. Consequently anew parameter, called the channel connection length, has been introducedin CALIBRE, in an attempt to reflect more accurately the concept ofchannelling within the fracture. This concept is illustrated in Figure 2.1,which shows a number of water-bearing channels within a fracture plane.

A few large channels in the fracture will have a greater separation distancethan many smaller channels that are more densely packed, even if thetotal surface areas are equal. The opportunities for matrix diffusion willtherefore be less in the former case. In the code, the channel separationdistance is used in place of the cell separation between the fracture andadjacent rock, in equation 18. If this distance is greater than the physicalseparation then the concentration gradient between the fracture and rockis reduced compared with the case of an open fracture, and the effects ofmatrix diffusion are reduced. This new parameter can therefore be usedin conjunction with the specific wet surface area parameter, to restrict theeffects of matrix diffusion on a scale that is more comparable with thatimposed in CRYSTAL.

If the solubility limit Cj(sol) of element j is exceeded in the fracture thenthe dissolved concentration of each isotope is proportional to the ratio of

the total isotope concentration to the total element concentration:

Cij{t)=Cj(sol)- (27)

where the summation over isotopes includes the stable species. The ex-pression for the sorbed concentration (24) is unchanged whilst the precip-itate concentration is given by:

(28)

The radial and tangential flow velocity components i/P, v$ may be derivedfrom the solution to Laplace's equation for potential flow around a cylin-der. They may be written in the form:

vr = v cos 9 11 — \\ T2

vg = -vsin6 1 + —7

(29)

(30)

where v is the velocity far from the cylinder. The flow field is illustratedschematically in figure 2.2.

V

Upstream

Figure 2.2 Schematic Representation of the Flow Field Around aCylindrical Container in the Fracture Plane

The flow around the canister and bentonite destroys the cylindrical sym-metry of the system, resulting in the three-dimensional mass transfer equa-tion (22). Figure 2.3 illustrates the radial and angular dependence of the

10

contaminants in the cells of the discretized region in the fracture plane. Toavoid the complication and computational cost of a fully three-dimensionalmodel, an approximation to the flow has been derived which accounts forthe net advective mass transfer downstream from the canister, within atwo-dimensional model.

>• r

Cell i

Figure 2.3 Cell Discretization in the Fracture Plane, ShowingRadial and Angular Dependence of Contaminant Concentration

In the approximation, the angular dependence of the dissolved concentra-tion is assigned to a weighting function, f{6):

C(r,8) = C(r)f(0) (31)

where C(r) is the average concentration in the fracture water at radius r.For very thin fractures it is also assumed that there is no vertical variationin dissolved concentration within the fracture. The average concentrationC{r) is obtained by averaging C(r, 8) over 8:

\8)d8 (32)

This implies that the weighting function f{8) must satisfy the condition:

= 1 (33)

The continuity equation (22) may also be expressed in terms of the diffu-sive and advective flux components:

11

dAtj(t)dt

T drdJ?

dJz

rd§ldJf

dr rÖO

where the diffusive flux terms are given by:

or

DwddjJ6 = - ^ •

dl

(34)

(35)

(36)

(37)

and the advective flux terms are given by:

JaT = -VrCij,

Jg = -

(38)

(39)

Substituting the weighting function approximation into the angular-dependentflux terms gives:

C(T) df{6)ae Uw»

( rV\ C(r) T r3"/2 r*/2

« = —t/1 1 1 1 ̂ i i / f{6) cos Odd + / cos0f{6)dd\ rl J 2TT [A/2 J—K/2rl) 2TT

J% = v (1 + f̂ ^ / f(e)sined6+ / f(9)sm9dO\ / ^7T JTT/I J—TT/2

(40)

(41)

,(42)

where we have split the weighting function into an upstream term for 9ranging from n/2 to 3TT/2 and a downstream term for 9 ranging from —TT/2

to TT/2.

If f{9) = 1 for all 6, then all the above terms are zero. Non-trivial choicesfor the weighting function, for which J" is non-zero and J$ and Jfi are zeroinclude:

• f[9) = 2 on the downstream side and zero on the upstream side,

• f{9) = rrcosO on the downstream side and zero on the upstreamside.

12

The radial flux term mav be written:

-Tl^{3o- A) C(r).

where

and

1 [/= - — / cos9f{d)dft

2~ A/2

= — / cos6f(6)dd2iT J-TT/2

(43)

(44)

(45)

Hence, for /(Ö) = 2, 0O = 0 and & = 2/TT. and for /(0) = TTCOSÖ. ^o = 0and A = 7r/4. The choice made in CALIBRE Version 2 is f(0) = 7rcos0downstream, and /(0) = 0. upstream.

The radial flow and downstream weighted concentration approximationare illustrated schematically in figure 2.4.

Weighted downstreamaverage concentration

Figure 2.4 Radial Concentration and Flow Approximation

13

2.3 Precipitate Formation and Dissolution

In CALIBRE, the independent variable solved tor is the total concentra-tion Aij(t). where the subscript refers to isotope i of element j . The totalconcentration is partitioned into the concentration dissolved in the porewater. Cij(t). the concentration sorbed on solid material. Si3(t) and theconcentration precipitated. Pij{t):

= Q C, (t) + Pi (t) (46)

where e is the porosity of the material (canister, bentonite or rock), pis the material density. Kj is the distribution or sorption coefficient ofelement j . and a3 is the capacity factor. In the fracture, a3 is defined asin equation (26).

If the precipitate is zero, then the dissolved concentration is related to thetotal concentration via the capacity factor:

dj(t) = (47)

If the solubility limit Cj(sol) is exceeded, then the dissolved concentrationof each isotope is proportional to the ratio of the total isotope concentra-tion to the total element concentration:

Ctj{t) = Cj(sol)-

= M*)/*' (48)

where a' is an 'effective' capacity factor, which varies according to theextent to which the solubility limit is exceeded.

For cases where there is a migrating redox front, both the capacity factorsand solubility limits of an isotope may change abruptly when the frontarrives. In the code, this is handled by allowing the properties of a cell tostart changing as soon as any of the nearest cells become fully oxidised.Thus the properties may change from reducing to oxidising linearly overa number of timesteps, depending on how quickly the front is moving.

2.4 Pinhole Release

The pinhole release model employed in CALIBRE is taken from the paperby Chambre et. al. [7] and Rae [8]. This assumes instantaneous mixingin the canister (corresponding to a uniform concentration) with diffusion

14

through a small hole in a thin-walled container, into a porous medium. Atsteady-state, the mass flux of nuclide i. F \ entering the porous mediumthrough the pinhole is given by:

B,F = ADaCT{i)T (49)

DB is the effective diffusion coefficient of the porous medium (in this case,bentonite), r is the pinhole radius and C*(t) is the concentration of nuclidei within the container.

During the initial phase, the mass flux is given by:

r(t) = (50)

S(t) is a time dependent shape factor. For a circular hole, this is approx-imated by the expression

(51)

R, is the retardation of nuclide i in the bentonite. The shape factor ap-proaches the value 4 as time increases, giving back the expression for F1.in equation (49). The retardation R, is defined by the equation

4>B(52)

where Kf is the distribution coefficient of nuclide i in the bentonite, andPB, &B are the density and porosity of the bentonite.

Vertical symmetry no longer applies in this model, as it is assumed thatthere is only one pinhole per canister. Consequently, the final flux resultsare not scaled up, as they are in the standard release model.

In the code, the concentration within the waste container is calculated ina manner similar to that for whole-canister releases, except that the fluxlost to the bentonite is now much less. Release from the waste matrixvia gap, grain boundary and matrix dissolution is all handled within themodel. If an isotope becomes solubility limited then its solubility limit isimposed on the waste pore-water concentration. The pore-water concen-tration calculated in this way is used in equation (50), to calculate the loss

15

2.5

from the container. The cumulative loss over a given timestep is addedto the total amount in the neighbouring cell in the bentonite. positionednext to the pinhole.

Tunnel Option

The presence of the tunnel, located vertically above the canister and bufferis included as a model option in Version 2 of CALIBRE. It is representedconservatively by a zero concentration boundary condition. The modelledsection comprises that of the standard CALIBRE model (ie the verticalsection between horizontal planes bisecting the fracture and rock slabs,midway between the fractures) together with an additional vertical sec-tion of buffer and rock above the cansiter up to the tunnel. The zeroconcentration boundary condition at the interface with the tunnel impliesthat advection in the tunnel is sufficient to maintain the concentration ofradionuclides here at zero.

Flux lost at the upper boundary is calculated in this model, together withthat lost at the far radial boundary. Vertical symmetry no longer applies,so that the flux results are not scaled to represent loss from a full canister,as they are in the standard model. The canister may either fail completely,or via pinhole failure.

16

3.1

Numerical Methods

The basic approach to solving the equations used within CALIBRE in-volves a finite difference discretization in the spatial dimensions with atime-stepping scheme to handle the time dimension. Within each timestep the radioactive decay and ingrowth pan of the system is decoupledfrom the diffusion part. The solubility limits are used to derive relation-ships between concentration in the pore water and total concentration,which is held fixed for the duration of each time-step. With these approx-imations, the diffusion for each radionuclide is calculated separately. Afully implicit time-stepping scheme is used, allowing larger time-steps tobe taken with reasonable accuracy, compared with the ADI time-steppingscheme used in the previous version of CALIBRE. All these aspects of themethod are expanded in the following sections.

Spatial Discretization

The spatial discretization used is based on a rectangular array of cells inr-z co-ordinates. Figure 3.1 shows how these cells fit into the physicalstructure of canister, bentonite. rock and fracture. Each cell has a nodewhose position is used when gradient calculations are made. Where irreg-ular cell sizes are used the cell nodes are placed symmetrically between itsboundaries. Figure 3.1 shows a rather coarse grid for clarity: in practicethe grid would be much finer.

X

X

X

X

X

Bentonite

X

X

X

X

X

X •

X ;

x ;

x

X

x ;

X

x !

X

x ( •

X

X

X

X

X

Rock

X :ii

X I1

X

X

t " y • •

X

X

X

X

. X

X

X

X

X

X

r=0 Fracture

Canister

Figure 3.1 Schematic Representation of Spatial Discretization

The independent variable solved for is the average amount per unir volumewithin each cell. This is related to the pore-water concentration throughthe capacity factor a. and where relevant the solubility limit. In either

r

case we can denote the ratio of amount per unit volume to pore-waterconcentration by an effective capacity factor a', which is time dependentbecause of the solubility limits and because of the presence of a movingredox front. The movement of radionuclides between cells is determinedby the diffusive terms in the equations.

We label the columns by i and the rows by j and then the discrete versionof the basic equation 10 or 11 for cell ij in the bentonite or rock is

1 - 1 - ,

Är (53)

— l\

~J

where superscripts n. n + 1 represent the quantities .4 or C at time levelstn. fn^i respectively, superscript p represents the parent nuclide. the D'sare the diffusion coefficients at the interfaces between cells, the S's are thecell interface surface areas, the r ; 's and z/s are the radial and verticalcoordinates of neighbouring cell nodes, and V'y is the cell volume.

In the fracture, the above terms are supplimented by two additional terms:

1 -

l - Ar.2

?£3 - 3lC?jl)v (54)

v is the average water velocity in the fracture. Ar, is the radial width ofcell i. and 3o- 3\ are the weighting functions, as discussed in the previoussection.

If 3Q is zero then the approximation for the advective term takes the formof a single-point upstream weighting approximation, i.e. in computingthe advective flux across a cell boundary, we take the concentration in thecell from which the contaminants are flowing. Such approximations are

18

often preferred in the solution of difference equations for advective flow,as they help to reduce numerical dispersion.

In addition, the spatial cell separation distance between the fracture andadjacent rock. zJ+i — Zj is replaced by the channel connection length, auser input parameter.

3.2 Outputs

The main output quantity of interest is the total flux leaving the systemat the far boundary, and at the upper boundary, in the case of a tunnelcalculation. The condition imposed here is one of zero concentration, sothe flux F{t) is simply calculated by monitoring the total amounts lost tothe boundary cells over each time-step:

(55)

where the summation is taken over the complete row of cells at the farboundary and At is the time-step.

Additional outputs which are calculated include the concentrations at eachgridpoint, the flux from the canister to the bentonite and the flux acrossthe bentonite/rock/fracture interface. The latter are defined as follows:

r = r^Z-t\ (56).7=1 iyr

FB(t) = t ^ ( r 2 ) ö ^ ^ . (57)

where the D's and Sy are the interface diffusion coefficients and cross-sectional areas, respectively. The gradient of the concentration is calcu-lated using a first-order finite difference approximation across neighbour-ing cells.

Accounts are also kept of the total flux entering the system from the waste,the total mass in the grid system and the total flux lost, for each nuclide.The mass balance account is written to the output file at the end of arun, together with the mass total in the waste, if canister failure had notoccurred. Generally, the latter will be equal to the influx (or grid totalplus flux lost) unless the waste matrix is still dissolving, or the influx isgoverned by pinhole failure.

19

3.3 Time Stepping

The discretized equations can be solved by a variety of different time-stepping schemes. rnhere are however a number of characteristics of thecurrent problem which narrow the choice considerably.

Firstly, the moving redox front leads to sudden changes in properties. Thisrules out time-stepping schemes which use the history of the solution asguidance to the solution at the next time.

The next point to consider is the possibility of diverse timescales. Highlyabsorbed nuclides will move very slowly, while non-sorbed ones may moverapidly. Some nuclides have very short half-lives while others have very-long ones. This suggests that rather short time-steps may be required,which calls for the use of a computationally efficient and adaptable scheme.

Finally the solubility limitation requires that all isotopes of an element aresolved for simultaneously, again pointing to the need for a very efficientscheme.

In the first version of CALIBRE, the alternating direction implicit (ADI)scheme was employed. While this is generally accurate it does imposesmall timesteps for the kind of problems solved within Project-90, forexample. If the timesteps are too large then instabilities may arise in theform of negative concentrations. Consequently, it was decided to move toa fully implicit scheme in the new version, which is inherently stable andpotentially more efficient. The new code incorporates an automatic time-step choice based on comparisons of results over a full time-step, and twohalf-steps. If the comparison is satisfactory, within a pre-set tolerance,then the timestep is increased. It may also be decreased automatically atany time, to ensure consistent accuracy throughout a calculation.

As in the previous version of CALIBRE, the calculation of radionuclidedecay and ingrowth is decoupled from the transport calculation. Similarly,the ratio of total nuclide concentration to pore water concentration iskept fixed in each cell, over any given timestep. Consequently, to avoidinaccuracies that may be introduced by nuclide decay and ingrowth overdifferent timescales, the maximum timstep that can be taken is restrictedto be a fraction of the minimum half-life in the problem data set.

The actions performed over a timstep comprise the following:

• Calculate the effective capacity factor for each cell, taking accountof redox front motion. Interpolate between reducing and oxidisingsolubility limits and K'ds, if necessary.

20

• Calculate release from the waste into the degraded canister cells,over a full timestep.

• Calculate decay over the timestep of all species in all grid cells,including the canister.

• Solve the transport equations over the timestep.

• Calculate flux and cumulative flux lost from the system at C=0boundaries.

• Repeat the above steps over two half timesteps.

• Check accuracy and select a new timestep.

21

4.1

Code Verification

The code has been verified where possible against analytic solutions. Theanalytical model and methods used to solve the equations are described inreference [9]. Basically each of the following tests involve the diffusion ofmaterial placed at the inner radial boundary of a thick cylinder, composedof a single material only. In each case zero-flux conditions are imposedat the inner boundary and lower surface of the thick cylinder, togetherwith a zero concentration condition at the far radial boundary. The upperboundary condition is specified as zero-flux or zero concentration. Thelatter condition would represent a situation in which advective flow in thetunnel is sufficiently fast to maintain an effectively zero concentration ofcontaminants there.

In the analytical model, a source is introduced at the inner boundary asa line (or surface) source. This may be a pulse or delta function sourceat time zero, or a source of constant concentration. It may be distributedevenly across the height of the cylinder, or have a z-coordinate dependence.In the numerical model, the line source is approximated by specifying theinitial total concentration in each of the innermost cells, i.e. the source isactually dispersed in a thin, cylindrical region.

One-Dimensional Radial Diffusion with a Delta-FunctionSource Term

The physical system modelled is shown in figure 4.1. A thick cylinderof inner radius 0.5 (arbitrary) units and outer radius 5 units has zeroflux boundary conditions at its inner radius and at the upper and lowersurfaces, while the concentration at the outer boundary is fixed at zero. Adelta function source of strength 10 units is input at zero time at the innerboundary with zero concentration everywhere else. The cylinder materialhas a diffusion coefficient of 0.01 arbitrary units (L2T-1) and a capacityfactor of 0.1 (dimensionless).

22

Delta function input of total amount 10 zero-flux boundary condition

Ii

l-d radial diffusion, no radioactive decay

D = 0.01 a = 0.1

zeroconcentration

boundarycondition(c=0)

a = 0.5 b = 5.0

^ r

Figure 4.1 Schematic Representation of Test Case 1

The cylindrical region is discretized using a variable radial grid spacing,with 32 radial cells in each layer. Four rows of cells are used in the verticaldirection, to check that the results are truly 1-dimensional.

The source is distributed in the first layer of cells with a total concentrationin each cell given by 10/V, where V is the total volume of the innermostcells.

The results for the numerical and analytic models are compared at variousoutput times in taole 4.1. At early times, the numerical model results arehigher for points beyond a radius of 1.0, with good agreement in the innercells. This may be attributed to the dispersion introduced by the finitedifference and time-stepping approximations. At later times, excellentagreement is achieved throughout the grid.

23

Time

0.5

1.0

2.0

5.0

10.0

20.0

Radial

Coordinate

1.0

2.0

3.0

4.0

1.0

2.0

3.0

4.0

1.0

2.0

3.0

4.0

1.0

2.0

3.0

4.0

1.0

2.0

3.04.0

1.0

2.0

3.0

4.0

Vertical

Coordinate

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.50.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

Concentration(Numerical)

1.41xl0i

1.22xlO"2

1.66xlO-6

1.01 xlO-10

1.75X101

1.75X10-1

2.03xl0"4

7.37X10-8

1.51 xlO1

1.10

1.30xl0-2

4.40xl0"5

9.26

3.00

3.81X10-1

2.13xlO~2

5.673.14

1.06

2.25x10-1

3.26

2.38

1.33

5.28x10-1

(Analytic)

1.44 xlO1

4.97xlO-4

6.93 xlO-8

6.44xlO"8

1.75x10"!9.35 xlO"2

3.55 xlO"6

3.09xl0-8

l.SOxlO1

1.03

5.96 xlO-3

2.93xlO~6

9.24

3.01

3.65X10-1

1.65xlO-2

5.64

3.151.07

2.19x10"!

3.21

2.37

1.35

5.41X10-1

Table 4.1 Comparison of numerical and analytic results for test case 1

24

4.2 One-Dimensional Radial Diffusion with a Constant Con-centration Source Term

Constant pore-water concentration, c = 10.0

11-d radial diffusion with solubilitylimiting, no radioactive decay.

D = 0.01 a = 0.1

zeroconcentration

boundarycondition

a =1.0

r

b = 4.0


The physical system is illustrated in figure 4.2. The positions of the radialboundaries are changed and there is now a source of constant concentrationat the inner boundary. In the numerical model this boundary condition ismodelled as a solubility limited source term with solubility 10 mass unitsper unit volume. The region is discretized into 23 columns and 4 rows.

The results are presented in table 4.2. They display a similar pattern tothose obtained in example 1, with the early results less accurate than thelater ones, when the system is approaching steady-state.

25

Time

1.0

2.0

5.0

10.0

20.0

100.0

Radial

Coordinate

1.5

2.0

2.5

3.0

1.52.0

2.5

3.0

1.5

2.0

2.5

3.0

1.5

2.0

2.5

3.0

1.5

2.0

2.5

3.0

1.5

2.0

2.5

3.0

Vertical

Coordinate

0.7

0.7

0.7

0.7

0.7

0.7

Concentration

(Numerical)

2.10

2.06 xlO"1

1.09 xlO-2

3.61 xlO-4

3.49

8.12 xlO"1

1.25 xlO-1

1.31 xlO"2

5.10

2.288.62 xlO"1

2.73 xlO-1

6.033.48

1.87

9.05 xlO-1

6.704.45

2.83

1.64

7.07

4.99

3.39

2.07

(Analytic)

2.17

1.81 xlO"1

5.08 xlO"3

4.49 xlO-5

3.55

8.17 xlO"1

1.14 xlO"1

9.16 xlO-3

5.14

2.30

8.68 xlO"1

2.69 xlO"1

6.06

3.51

1.89

9.17X10"1

6.734.49

2.87

1.66

7.08

5.003.39

2.08


26

4.3 Two-Dimensional Diffusion with a Delta-Function, Z-DependantSource Term

Delta function concentration profile of3sinMs-zl/2s]

zero concentrationboundary condition

zA

2-d diffusion, no radioactive decay

0=0.01 a=5.0

zeroconcentration

boundarycondition

a=0.5 b=50


The physical system is illustrated in figure 4.3. The capacity factor isincreased from 0.1 to 5.0 and the upper boundary condition is one of zeroconcentration. The source has a profile of strength 3sin(7r(s — z)/2s) perunit length. In the numerical model, each source cell is assigned an initialconcentration of 3sin(7r(s — z)/2s).6z/V, where z denotes the z coordinateof a cell node, 6z is the cell height. 5 is the cylinder height and V is thecell volume. The region is discretized into 32 columns and 20 rows. Theresults are presented in table 4.3. Again, best agreement is obtained atlater times. Comparison is also made in this case of the flux into thetunnel, represented by the zero concentration condition along the z = 3.0boundary. Excellent agreement of within 3% of the analytic results isobtained, as shown in table 4.4.

27

Time

50.0

100.0

200.0

500.0

Radial

Coordinate

1.0

2.0

3.0

1.0

2.0

3.0

4.0

1.0

2.0

3.0

4.0

1.0

2.0

3.0

4.0

Vertical

Coordinate

1.0

2.01.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

10

2.0

1.0

2.01.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

1.0

2.0

Concentration

(Numerical)

5.19 xlO-2

9.00 xlO~2

4.39 xlO-4

7.61 xlO-4

1.85 xlO~7

3.2 xl0~7

4.27 xlO-2

7.40 xlO~2

3.14 xlO-3

5.44 xlO-3

3.01 xlO-5

5.21 xlO"5

5.87 xlO~8

1.02 xlO-7

2.84 xlO-2

4.93 xlO"2

7.17 xlO-3

1.24 xlO~2

5.67 xlO~4

9.83 xlO~4

1.61 xlO-5

2.80 xlO"5

1.28 xlO"2

2.23 xlO"2

7.13 xlO"3

1.24 xlO~2

2.41 xlO"3

4.18 xlO"3

5.07 xlO"4

8.80 xlO-4

(Analytic)

5.01 xlO"2

8.84 xlO"2

2.73 xlO"4

4.73 xlO-4

1.04 xlO"8

1.80 xlO-8

4.27 xlO-2

7.39 xlO"2

2.91 xlO-3

5.04 xlO~3

1.69 xlO~5

2.93 xlO-5

8.31 xlO"9

1.44 xlO~8

2.86 xlO-2

4.95 xlO-2

7.13 xlO"3

1.24 xlO"2

5.20 xHT4

9.00 xlO"4

l . l lx lO- 5

1.92 xlO~5

1.29 xlO-2

2.23 xlO"2

7.18 xlO"3

1.24 xlO-2

2.44 xlO~3

4.22 xlO"3

5.00xl0"4

8.67 xlO-4


28

Time

50.0

100.0

200.0

500.0

Flux into Tunnel

Numerical

3.15 xHT3

3.03 xlO"3

2.84 xlO"3

2.37 xlO"3

Analytic

3.06 xlO"3

2.97 xlO' 3

2.81 xlO~3

2.37 xHT3

Table 4.4 Comparison of the flux from the numerical andanalytic models for test case 3

4.4 Two-Dimensional Diffusion with Constant ConcentrationSource Term

Fixed concentration boundary condition of 4 (1 -cos (2rtz/s)) per unit length

zI

2-d diffusion, no radioactive decayzeroconcentration

boundarycondition(c = 0)

a = 5.0 b =10.0^ r


This case is illustrated in figure 4.4. The inner and outer radial boundariesare at 5.0 and 10.0 arbitrary units respectively and the cylinder height is5.0 units. The concentration at the inner boundary is constant, witha z-dependent profile of 4(1-COS(2TTZ/S)). In the numerical scheme thisis achieved by assigning the solubility of species in the source cells tothis function, and choosing sufficient inventory to maintain saturationthroughout the calculation. The region h discretized into 29 columns.A selection of results at various times is given in table 4.5. Excellentagreement is obtained at later times, with under 4% error at the earliertimes.

29

Time

1.0

2.0

10.0

20.0

Radial

Coordinate

6.0

7.0

8.0

6.0

7.0

8.0

6.0

7.0

8.0

6.0

7.0

8.0

Vertical

Coordinate

0.0

1.0

2.5

0.0

1.0

2.5

0.0

1.0

2.5

0.0

1.0

2.5

0.0

1.0

2.5

0.0

1.0

2.5

1.0

2.5

1.0

2.5

1.0

2.5

1.0

2.5

1.0

2.5

1.0

2.5

Concentration

(Numerical)

7.70 xlO"1

1.432.69

3.27X10-1

4.64 xlO"1

7.25 xlO"1

8.08 xlO-2

1.03 xlO"1

1.46 xlO"1

1.22

1.93

3.288.02 xlO"1

9.84 x 10"1

1.33

3.55 xlO"1

4.00 xlO-1

4.85 xlO"1

2.59

3.951.92

2.281.21

1.31

2.63

3.991.97

2.331.26

1.36

(Analytic)

7.79 xlO"1

1.45

2.73

3.28X10-1

4.70 xlO"1

7.40 xlO"1

7.42 xlO"2

9.73 xlO-2

1.41 xlO"1

1.23

1.94

3.30

8.13 xlO-1

9.97 xlO"1

1.35

3.58X1O-1

4.04 xlO"1

4.90 xlO"1

2.6

3.96

1.93

2.29

1.23

1.32

2.63

3.991.97

2.33

1.26

1.36


30

4.5 One-Dimensional Radial Diffusion with a 3-Nuclide De-cay Chain

Initial delta function concentrations for each nuclide

zs = 0.5/

1-d radial diffusion, 3 nuclide decay chain

D=1.3xlO"3m7yot(Np)=200,a(U)=40,a(fh)=1000

zeroconcentration

boundarycondition

a = 0.4 b = 2.0

^ r


This case is illustrated in figure 4.5. The inventory, diffusion coefficientand capacity factors are taken from reference [10]. A 3-nuclide chain (Xp-237, U-233, Th-229) with corresponding inventories of 2.44. 4.67 xKT4

and 0.0 GBq is used. The inventory figures are scaled down by a factor of 9.the ratio of the waste cylinder height (4.5 m) to the half-spacing betweenfractures (0.5 m). the latter being the section of the near-field systemmodelled by the CALIBRE code. The cylinder material is assigned theproperties of bentonite, under reducing conditions. Solubility limits are setartificially high, so that precipitation does not occur. Zero flux boundaryconditions are imposed at the upper and lower surfaces of the region, witha zero concentration far boundary condition. In the analytic model, thesource is introduced as a linearly distributed surface delta-function at theinner radial boundary. In the numerical model, the inventory in each ofthe source cells is specified so that their total concentrations are equal.The region is discretized into 30 columns and 3 rows.

The results are given in tables 4.6-4.8. Agreement between the analyticand numerical models is generally good (within 5%) over the timescaleconsidered.

31

Time

(y)lxlO4

2xlO4

5xlO4

lxlO5

2X105

5xlO5

Radial

Coord, (m)

0.8

1.21.6

0.8

1.2

1.6

0.81.2

1.6

0.81.2

1.6

0.81.2

1.6

0.8

1.2

1.6

Vertical

Coord, (m)

0.25

0.25

0.25

0.25

0.250.25

0.250.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

Np-237 Cone

(Numerical)

1.11 xlO"3

1.55 xlO~4

7.53 xlO~6

9.29 xlO~4

3.32 xlO"4

6.54 xl0~5

5.61 xlO~4

3.56 xlO~4

1.56 xlO"4

306 xlO~4

2.19 xlO"4

1.09 xHT4

1.04 xlO"4

7.28 xlO~5

3.66 xlO-5

3.66 xlO~6

2.64 xHT6

1.33 xlO-6

(moles/m3)

(Analytic)1.08 xlO"3

1.50 xlO"4

6.22 x 10"6

9.17 xlO"4

3.31 xlO"4

6.46 x 10"5

5.54 xlO"4

3.53 x 10"4

1.56 xlO"4

3.02 xlO"4

2.16 xlO"4

1.08 xlO"4

9.95 x 10-5

7.17 xlO~5

3.60 x 10-5

3.57 x 10-6

2.58 xlO"6

1.29 xlO"6

Table 4.6 Comparison of numerical and analytic results for test case 5. Np-237

32

Time

(y)lxlO4

2xlO4

5xlO4

lxlO5

2xlO5

5xlO5

Radial

Coord, (m)

0.8

1.2

1.6

0.8

1.21.6

0.81.2

1.6

0.8

1.2

1.6

0.81.2

1.6

0.81.2

1.6

Vertical

Coord, (m)

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

U-233 Cone.

(Numerical)

1.27 xlO-5

5.46 xl0~6

1.68 xl0~6

1.68 xlO~5

1.00 xlO"5

4.19 xlO-6

1.67 xlO"5

1.17 xHT5

5.75 xlO"6

1.04 xlO"5

7.51 xlO-6

3.77 xlO~6

3.48 xlO"6

2.51 xlO~6

1.26 xlO"6

1.26 xlO"7

9.11 xlO-8

4.58 xlO"8

(moles/m3)

(Analytic)

1.26 xlO~5

5.45 xlO-6

1.67 xlO~6

1.66 xlO"5

9.96 xlO-6

4.19 xlO-6

1.65 xlO"5

1.16 xlO"5

5.70 xlO-5

1.03 xlO-5

7.40 xlO"6

3.71 xlO-6

3.42 xlO-6

2.46 xlO~6

1.24 xlO~6

1.23 xlO-7

8.86 xlO-8

4.45 xlO"8

Table 4.7 Comparison of numerical and analytic results for test case 5. U-233

33

Time

(y)lxlO4

2xlO4

5xlO4

ixlO5

2xlO5

5xlO5

Radial

Coord, (m)

0.8

1.2

1.6

0.8

1.21.6

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

0.8

l.k

1.6

Vertical

Coord, (m)

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

Th-229 Cone

(Numerical)

9.53 xlO-9

3.03 xlO"9

7.31 xl0~10

2.10 xlO"8

1.07 xlO-8

3.98 xlO"9

3.10 xlO"8

2.12 xlO"8

1.01 xlO"8

2.13 xlO"8

1.53 xl0~a

7.64 xlO"9

7.17 xlO"9

5.17 xlO-9

2.60 xlO-9

2.60 xlO-10

1.88 xlO~10

9.42 xlO"11

. (moles/m3)

(Analytic)

9.28 xHT9

3.06 xlO~9

7.50 xlO-10

2.05 xlO~8

1.07 xlO~8

4.02 xlO~9

3.03 xlO~8

2.08 xlO~8

1.00 xlO~8

2.07 xlO~8

1.49 xlO~8

7.46 xlO"9

6.97 xlO"9

5.03 xlO-9

2.52 xlO-9

2.50 xlO~10

1.81 xlO~10

9.07 xlO~u

Table 4.8 Comparison r,f numerical and analytic results for test case 5, Th-229

34

4.6 Two-Dimensional Diffusion with a 3-Nuclide Decay Chain

|

Concentration profile=n/2s.sinfn(s-zl/2s] x initial inventory

zero concentrationboundary condition

'//////////////////////t///////.

2-d diffusion 3-nuclide decay chain

D=1.3x10*3m2/y

a(Np)=200, a(U)=40, a|Th)=1000

zeroconcentration

Iboundarycondition

V//////////////////////////////;


The final case is illustrated in figure 4.6. The same nuclides and physi-cal parameters as in the previous case are used, but the upper boundarycondition is changed to one of zero concentration (equivalent to high ad-vective flow in a tunnel) and the concentration profile along the innerradial boundary is set to the initial inventory of each nuclide, weighted bythe factor Trsin(n(s — z)/2s)/2s. The same radial discretization is used,but the discretization in the vertical direction is increased to 13 rows.

The results for the two models are shown in tables 4.9-4.11 for flux out ofthe rock, and in tables 4.12 and 4.13 for flux into the "tunnel". Overallthe agreement is reasonable, with between 10-20% error over all nuclidesand times. The results could probably be improved by further restrictingthe maximum time-step taken, or increasing the spatial discretization, atthe expense of computation time.

35

Time

(y)lxlO4

2xlO4

5xlO4

lxlO5

Radial

Coord, (m)

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

Vertical

Coord, (m)

0.45

0.25

0.45

0.250.45

0.25

0.45

0.250.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.450.25

0.45

0.25

0.45

0.25

0.45

0.25

Np-237 Cone

(Numerical)

8.59 xlO~4

6.15 xlO~4

1.20 xlO~4

8.61 xlO~4

6.13 xlO-6

4.39 xlO"6

3.94 xlO~4

2.82 xlO"4

1.39 xlO"4

9.98 xlO"5

2.73 xlO"5

1.96 xlO~5

3.55 xlO-5

2.55 xlO-5

2.25 xlO-5

1.61 xlO-5

9.99 xlO"5

7.16 xlO-6

8.12 xlO-7

5.81 xlO-7

5.83 xlO"7

4.18 xlO~7

2.97 xlO"7

2.13 xlO-7

(moles/m3)

(Analytic)

8.79 xlO"4

6.29 xlO-4

1.23 xlO-4

8.78 x 10"5

5.08 xlO"6

3.64 x 10-6

3.94 xlO-4

2.82 xlO~4

1.42 xlO-4

1.02 xlO-4

2.78 xlO"5

1.99 xlO~5

3.47 xlO"5

2.49 xlO'5

2.2] xlO"5

1.59 xlO-5

9.77 xlO~6

6.99 xlO-6

7.67 xlO~7

5.49 xlO~7

5.49 xlO"7

3.93 xlO"7

2.73 xlO"7

1.96 xlO"7

Table 4.9 Comparison of numerical and analytic results for test case 6, Np-237

36

Time

(y)lxlO4

2xlO4

5 xlO4

lxlO5

Radial

Coord, (m)

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

Vertical

Coord, (m)

0.45

0.25

0.450.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

U-233 Cone.(Numerical)

4.21 xlO~6

3.02 xlO~6

1.41 xlO~6

1.01 xl0~6

3.28 xl0~7

2.35 xlO~7

1.96 xlO~6

1.40 xlO~6

9.57 xlO~7

6.86 x 10~7

3.17 xlO~7

2.27 xlO~7

1.88 xlO~7

1.34 xlO~7

1.26 xlO"7

8.99 x 10~8

5.94 xlO"8

4.25 xlO"8

4.38 xl0~9

3.14 xlO-9

3.16 xlO"9

2.26 xlO-9

1.62 xlO"9

1.16 xlO"9

(moles/m3)

(Analytic)

4.22 xlO"6

3.02 xlO~6

1.44 xlO-6

1.03 xlO~6

3.32 xlO-7

2.37 xlO-7

1.93 xlO"6

1.38 xlO"6

9.57 xlO-7

6.85 xlO-7

3.16 xlO-7

2.26 xlO-7

1.81 xlO~7

1.30 xlO-7

1.22 xlO-7

8.70 xlO~8

5.67 xHT8

4.06 xlO-8

4.08 xlO"9

2.92 xlO"9

2.93 xlO~9

2.10 xlO"9

1.46 xlO"9

1.05 xlO~9

Table 4.10 Comparison of numerical and analytic results for test case 6, U-233

37

Time

(y)lxlO4

2xlO4

5xlO4

lxlO5

Radial

Coord, (m)

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

0.8

1.2

1.6

Vertical

Coord, (m)

0.45

0.25

0.45

0.250.45

0.25

0.45

0.250.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

0.45

0.25

Th-229 Cone

(Numerical)

4.68 xlO"9

3.35 xHT9

1.11 xlO~9

7.96 xl0~10

2.01 xl0-i°

1.44 xl0-i°

4.59 xlO"9

3.28 xlO"9

1.73 xlO"9

1.24 xlO"9

4.68 xlO-10

3.35 xlO"10

8.07 xlO"10

5.78 xlO-10

4.78 xlO"10

3.42 xlO-i°

2.01 xlO-10

1.44 xlO-10

2.19 xlO-11

1.57 xlO-n

1.54 xlO"11

1.10 xlO"11

7.70 xlO"12

5.51 xlO"12

. (moles/m3)

(Analytic)

4.66 xlO-9

3.34 xlO"9

1.14 xlO~9

8.13 xlO"10

2.04 xlO"10

1.46 xlO"10

4.48 xlO-9

3.21 xlO"9

1.73 xlO-9

1.24 xHT9

4.67 xlO-10

3.34 xlO"10

7.71 xlO"10

5.52 xlO"10

4.60 xlO-10

3.30 xlO"10

1.93 xlO"10

1.38 xlO-10

2.03 x H T n

1.45 xlO"11

1.42 xlO"11

1.02 xlO"11

6.96 xlO"12

4.98 xlO"12

Table 4.11 Comparison of numerical and analytic results for test case 6, Th-229

38

Time

(y)

lxlO4

2xlO4

5xlO4

lxlO5

2xlO5

Flux

Np-237Numerical

1.33

6.98

8.70

2.19

1.29

xlO~5

xlO~6

xlO~7

xlO~8

xlO-n

into tunnel, (moles/y)

Analytic

1.33

6.94

8.42

2.03

1.10

xlO"5

xlO-6

xlO-7

xlO~8

xlO-n

U-233

Numerical

7.55 xlO~8

4.07 xlO~8

4.80 xlO"9

1.19 xlO-i°

6.98 x IO-14

Analytic

7.48

3.98

4.56

1.08

5.86

xlO-8

xlO"8

xlO-9

xlO-i°

x 10-n

Table 4.12 Comparison of flux for test case 6, Np-237 and U-233

Time

(y)

lxlO4

2 xlO4

5xlO4

lxlO5

2xlO5

Flux into tunnel, (moles/y)

Th-229

Numerical

8.03 x IO-11

8.62 x 10-n1.89 xlO-n5.82 xlO"13

3.56 xlO"16

Analytic

7.87xlO-n

8.36x10-11

1.79xlO-n

5.29xlO"13

2.98 x 10"16

Table 4.13 Comparison of flux for test case 6, Th-229

39

4.7 Verification of the Radial Flow Approximation

A number of >,est cases were also performed to verify the radial flow ap-proximation employed in CALIBRE. In the KBS-3 report [11] the flowof nuclide i from the nearfield is calculated as a rate iVj defined by theequation

Ar, = QeqCai- (58)

Coi is the concentration of nuclides in the fuel pore-water and Qeq is theequivalent water flow that arrives at the canister with zero concentration ofnuclides and leaves it with a concentration of Coi- If the nuclide is solubilitylimited at a concentration of CiiSOi then the above equation becomes

Ni = QeqChSOl. (59)

In the CALIBRE test cases a single long-lived nuclide is considered, witha sufficient source inventory for it to remain solubility limited in the can-ister pore-water, throughout the simulations. To make the comparisonwith KBS-3, the effective diffusivity of the rock is set to zero to inhibitmatrix diffusion. The flow velocity of the water through a fracture ofwidth 0.1mm is varied to determine the effect of the radial flow approxi-mation at different flow rates. The simulations are run until the flux (F)leaving the system reaches an equilibrium value. From the calculated fluxand elemental solubility limit under reducing conditions values of Qeq arecalculated for comparison with KBS-3:

Qeq = -£— (60)

The results are listed in table 4.14. The ircosd weighting function is used.Results obtained using the alternative weighting function of 2 are notsignificantly different. At low Darcy velocities, the Qeq values calculatedusing CALIBRE agree reasonably well with the KBS-3 values [10]. Athigher flows, the radial flow approximation predicts lower fluxes than KBS-3.

Several calculations were performed to determine the sensitivity of theresults to the spatial discretisation. It was found necessary to use a mini-mum of 1 cm grid spacing in the region of the bentonite/fracture interface,to ensure that diffusion and flow at the fracture mouth were adequatelyrepresented. Further refinement produced practically no change in theresults, so the 1 cm grid-spacing was judged to be sufficiently accurate.

40

Darcy Velocity

l/m2/y0.1

0.3

1.0

3.0

CALIBRE Qeq

l/canister/y0.57

0.91

1.44

2.03

KBS-3 Qeq

l/canister/y0.57

0.94

1.57

2.41

Table 4.14 Comparison of KBS-3 and CALIBRE Qeq Values

4.8 Conclusions

Overall the agreement between the analytic and numerical models is verygood, verifying the correct behaviour of the numerical model for the casesconsidered above. The cases also illustrate the importance of choosing asufficiently fine mesh discretization, to achieve accurate results. Specialcare is also needed in choosing the position of the far boundary, to ensurethat a zero concentration boundary condition is valid at that position, anddoes not significantly influence the results. It is therefore advisable to dosome trial runs which vary (i) the discretization, and (ii) the far boundaryposition, to ensure that neither of these factors are significantly affectingresults.

41

References

[1] SKI Project-90 Summan- Report, Vols. I, II, SKI TR 91:23, SwedishNuclear Power Inspectorate, Stockholm, Sweden, 1991.

[2] Shaw, W.T., The Motion of a Redox Front in a System of Bentoniteand Rock, Incorporating Fracture Transport Effects, SKI TR 91:17,Swedish Nuclear Power Inspectorate, Stockholm, Sweden, 1992

[3] Shaw, W.T., The Oxidation State of the Near-field Environmentin the CALIBRE Source Term Model: Further Remarks, SKI TR91:20, Swedish Nuclear Power Inspectorate, Stockholm, Sweden,1992.

[4] Robinson, P. and Worgan, K, The CALIBRE Source-Term CodeTechnical Documentation for Project-90, SKI TR 91:18, SwedishNuclear Power Inspectorate, Stockholm, Sweden, 1991.

[5] Worgan, K., Robinson, P., CRYSTAL: A Model of a Fractured RockGeosphere for Performance Assessment within Project-90. SKI TR91:13, Swedish Nuclear Power Inspectorate, Stockholm, Sweden,1992.

[6] Worgan, K and Shaw, W., SKI Project-90 Geosphere CalculationsUsing CRYSTAL: Stand-alone and CALIBRE-CRYSTAL-BiosphereIntegrated Results. SKI TR 91:16, Swedish Nuclear Power Inspec-torate, Stockholm, Sweden. 1991.

[7] Chambré, P.L.. Lee, W. W-L, Kin, C.L. and Pigford, T.H.. Steady-State and Transient Radionuclide Transport through Penetrationsin Nuclear Waste Containers, LBL-21806, 1986.

[8] Rae, J., Leaks from Circular Holes in Intermediate-Level Waste Can-isters, AERE-R. 11631, 1985.

[9] Robinson, P.C., Verification Tests for Radionuclide Migration inCALIBRE, SKI TR 89:8, Swedish Nuclear Power Inspectorate,Stockholm, Sweden, 1989.

[10] Shaw, W., Smith, G., Worgan, K., Hodgkinson. D. and Anders-son, K., Source Term Modelling Parameters for Project-90, SKITR 91:27, Swedish Nuclear Power Inspectorate, Stockholm, Swe-den, 1992.

[11] SKBS/KBS, Final Storage of Spent Nuclear Fuel. KBS-3, SwedishNuclear Fuel Supply Co., Stockholm, Sweden, 1983.

42

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