Inverse modeling of tracer experiments in FEBEX compacted Ca-bentonite

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Physics and Chemistry of the Earth 31 (2006) 640–648

Inverse modeling of tracer experiments in FEBEXcompacted Ca-bentonite

Javier Samper a,*, Zhenxue Dai a,1, Jorge Molinero b, M. Garcıa-Gutierrez c,T. Missana c, M. Mingarro c

a Escuela Tecnica Superior de Ingenieros de Caminos, Canales y Puertos, Campus de Elvina s/n, 15192 La Coruna, Spainb Escola Politecnica Superior, Universidad de Santiago de Compostela, 27002 Lugo, Spain

c Dpto. de Impacto Ambiental de la Energıa, Caracterizacion Hidrogeoquımica de Emplazamientos, CIEMAT, Edificio 20-A,

Avda. Complutense, 22, 28040 Madrid, Spain

Received 18 September 2005; received in revised form 28 January 2006Available online 22 June 2006

Abstract

Solute transport parameters of compacted Ca-bentonite used in the FEBEX Project were derived by Garcıa-Gutierrez et al. (2001)from through- and in-diffusion experiments using analytical solutions for their interpretation. Here we expand their work and presentthe numerical interpretation of diffusion and permeation experiments by solving the inverse transport problem which is formulatedas the minimization of a weighted least squares criterion measuring the differences between computed and measured concentration values.The inverse problem is solved with INVERSE-CORE2D�, a finite element code which accounts for both dissolved and sorbed concentra-tion data, uses either the Golden section search or Gauss–Newton–Marquardt methods for minimizing the objective function and allowsthe estimation of transport and retardation parameters such as diffusion coefficient, total and kinematic porosity and distribution coef-ficients. Diffusion and permeation experiments performed on FEBEX compacted bentonite using tritium, cesium, selenium, and stron-tium have been effectively interpreted by inverse modeling. Estimated parameters are within the range of reported values for thesetracers in bentonites. It has been found that failing to account for the role of sinters may lead to erroneous diffusion coefficients by a factorof 1.4. Possible ways to improve the design of in-diffusion and permeation experiments have been identified. The interpretation of thetritium permeation experiment requires the use of a double-porosity model with mobile porosity of 0.14 for a dry density of 1.18 g/cm3.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Inverse modeling; Solute transport parameters; FEBEX; Bentonite; Diffusion experiments; Sorption

1. Introduction

Predictions of reactive solute transport through com-pacted bentonites are needed for the performance assess-ment of the disposal of hazardous wastes such as high-level nuclear waste (HLW). The FEBEX bentonite wasextracted from the Cortijo de Archidona deposit, exploited

1474-7065/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pce.2006.04.013

* Corresponding author. Address: Escuela de Caminos, Campus deElvina s/n, Universidad de La Coruna, 15192 La Coruna, Spain. Tel.: +34981 16 70 00x1433; fax: +34 981 16 71 70.

E-mail address: [email protected] (J. Samper).1 Present address: Earth and Environmental Sciences Division,

Los Alamos National Laboratory, New Mexico, USA.

by Minas de Gador, S.A. in Serrata de Nıjar (Almerıa,Spain). This bentonite was selected by ENRESA (Empresa

Nacional de Residuos Radioactivos, S.A.) prior to theFEBEX project (Huertas et al., 2000) as a suitable materialfor backfilling and sealing of a HLW repository because ithas a very large montmorillonite content, large swellingpressure and sorption capacity, extremely low permeabil-ity, acceptable thermal conductivity and ease of compac-tion for the fabrication of blocks. Bentonite used in theFEBEX project was homogenized to reduce the uncertain-ties in parameter variability (Huertas et al., 2000).

The solution of the inverse problem provides a waywhereby measurements of state are used to determineunknown flow and transport parameters by fitting model

J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648 641

outputs to measurements. A general framework anddescription of the coupled inverse problem of water flowand solute transport is given by Mishra and Parker(1989) and Sun (1994). More work on inverse flow and sol-ute and transport in aquifers has been done by Samperet al. (1990), Gailey et al. (1991), Wagner (1992), Xianget al. (1993), Medina and Carrera (1996) and Poeter andHill (1997).

However, the ability to fully characterize transportparameters of compacted bentonite has not kept pace withthe numerical and modeling expertise. Deriving reliableestimates of transport and sorption parameters of theFEBEX bentonite is one of the objectives of the FEBEXProject. Experimental characterization of diffusive trans-port of radionuclides in FEBEX bentonite was carriedout by means of through- and in-diffusion experiments(Garcıa-Gutierrez et al., 2001). Application of the inverseapproach to estimate various flow and transport parame-ters of the FEBEX bentonite are presented by Dai andSamper (1999). Solute transport parameters of compactedCa-bentonite used in the FEBEX Project were presentedby Garcıa-Gutierrez et al. (2001). These parameters werederived from through- and in-diffusion experiments whichwere interpreted by means of analytical solutions. Herewe expand on the latter approach and provide results notonly for diffusion but also for flow-through or permeationexperiments. Numerical interpretation of laboratory exper-iments is achieved by solving the inverse problem of solutetransport. The inverse approach overcomes the limitationsof analytical solutions by accounting for: (1) Measurementerrors; (2) The role of the sinters; (3) Time-varying bound-ary conditions in permeation experiments; and (4) Tran-sient concentrations in the cell and total concentrationdata at the end of in-diffusion experiments. The inverseanalysis has been applied to the numerical interpretationof through-diffusion, in-diffusion and permeation experi-ments performed on FEBEX Ca-bentonite using tritium,strontium, cesium and selenium. The following solutetransport parameters are estimated: molecular diffusioncoefficient, dispersivity, distribution coefficient, Kd, andkinematic as well total porosities.

2. Direct problem of solute transport and sorption

The equation for the transient transport of a sorbing sol-ute through a saturated porous medium under steady-stateflow is given by (Bear, 1972):

r � ð/DrCÞ � qrC þ rðC� � CÞ ¼ /RoCot

ð1Þ

where / is porosity, C is solute concentration; q is volumet-ric water flux; r is a fluid sink/source term having a concen-tration C*, t is time, and R is the retardation coefficientdefined as

R ¼ 1þ qKd

/ð2Þ

where Kd is the distribution coefficient and q the bulk den-sity. Tracer concentrations are extremely small. The largestconcentrations are 2.69 · 10�9 M for cesium, 5.4 · 10�9 Mfor selenium and 10�9 M for strontium. For such low con-centrations all sorption isotherms are linear. The dispersiontensor, D, in (1) is given by

D ¼ DeIþDh ð3Þand includes the hydrodynamic or mechanical dispersiontensor, Dh, and the molecular diffusion term. Since claysamples were prepared by compacting powder FEBEXbentonite which had been previously homogenized, molec-ular diffusion is expected to be isotropic. Therefore, the dif-fusion term can be written as DeI, where De is the effectivemolecular diffusion coefficient and I is the identity tensor.In one-dimensional solute transport the mechanical disper-sion, Dh, is equal to aq where a is dispersivity. The effectivediffusion coefficient in porous media, De, is usually ex-pressed as:

De ¼ /Dp ð4Þ

where Dp is the molecular diffusion coefficient in the porespace which in turn is given by

Dp ¼ D0s ð5Þwhere D0 is the molecular diffusion coefficient in pure waterand s is the tortuosity of the medium which is assumed hereto be equal to /1/3 where / is porosity. The apparent diffu-sion coefficient, Da, takes into account the combined effectof diffusion and sorption and is defined as

Da ¼De

/þ qKd

ð6Þ

Solution of Eq. (1) requires appropriate initial conditions:

Cjt¼0 ¼ C0 ð7Þ

and boundary conditions:

�/DrC � nC1¼ bðC1 � CÞ þ F 0D ð8Þ

where n is a unit vector normal to the boundary C1 point-ing outwards, C0 is initial concentration, C1 is a prescribedconcentration; b is a parameter controlling the type ofboundary condition. For a Neumann condition, b = 0and solute flux through the boundary is equal to the disper-sive flux, F 0D. For a Dirichlet condition, b =1, F 0D ¼0 andthe concentration is prescribed to be equal to C1. At waterinflow boundaries b = q Æ n and F 0D ¼0.

3. Inverse problem of solute transport and sorption

The essence of the inverse problem lies on deriving opti-mum parameter estimates from known concentration data.Optimum parameters are those which minimize an objec-tive function measuring the difference between measuredand computed concentrations.

Our formulation of the inverse problem is based ongeneralized least squares criterion (Sun, 1994; Dai and

TRACER

Pre

ssu

re

TRACERIN

PERMEATION

THROUGH DIFFUSION

IN DIFFUSION

CLAY

CLAY

CLAY

C

TRACER OUT

TRACERIN

TRACERIN

Fig. 1. Methods used for determining sorption and diffusion parameters(from Garcıa-Gutierrez et al., 2001).

642 J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648

Samper, 2004). Let P = (p1,p2,p3, . . . ,pM) be the vector ofM unknown parameters. The least squares criterion, E(p),can be expressed as,

EðpÞ ¼XNE

i¼1

W iEiðpÞ ð9Þ

where i ¼ 1;NE denotes different types of data, i = 1 fordissolved concentration; i = 2 for total concentration;i = 3 for prior information on model parameters, Wi isthe weighting coefficient of the ith generalized least-squarescriterion Ei(p) which is defined as:

EiðpÞ ¼XLi

l¼1

w2lir

2liðpÞ

rliðpÞ ¼ uilðpÞ � F i

l

where uilðpÞ is the computed value of the ith variable at the

lth observation point; F il are measured values; Li is the

number of observations, either in space or in time, forthe ith type of data; and rli is the residual or difference be-tween model outcome and measurement; wli is the weight-ing coefficient for the lth measurement of the ith type ofdata. Its values depend on the accuracy of observations.If some data are judged to be unreliable, they are assignedsmall weights in order to prevent their pernicious effect onthe optimization process.

Eq. (9) is a weighted multiobjective optimization crite-rion. Weighting coefficients Wi are computed from (Car-rera and Neuman, 1986; Dai and Samper, 2004):

W i ¼W 0i

EiðpÞLi

i ¼ 1;NE ð10Þ

where W0i are user-defined dimensionless initial weightingcoefficients for different types of observation data. Whilecoefficients W0i are dimensionless, Wi have dimensionswhich are reciprocal of those of Ei(p). Generally Ei(p) hasunits equal to the square of the unit used for each typeof data (i.e., (mol/l)2 for dissolved concentrations). WeightsWi are updated automatically during the iterative optimiza-tion process according to Eq. (10) whenever NE > 1.

Weights wli for different observation points in (6) mustbe specified in advance. In a statistical framework, thesecoefficients are inverse standard deviations expressing theuncertainty of measured data. Such standard deviationscan be derived from measurement or analytical errors,rm, or in the case of time series from random fluctuationsabout a trend. For a given type of data (i.e., for a fixedi), wli for all l = 1,2, . . . ,Li are all equal to r�1

m wheneverall data values are equally reliable.

Minimization of the expression in Eq. (9) can achievedby different methods such as: golden section search (Sun,1994), Newton, Gauss–Newton, Gauss–Newton–Leven-berg–Marquardt (Yeh, 1986; Sun, 1994), conjugate gradi-ent (Carrera and Neuman, 1986), quasi-Newton, andsimulated annealing. In this paper, golden section searchand Gauss–Newton–Levenberg–Marquardt methods have

been used to solve the inverse problem. Combining thesetwo inverse methods with the forward modeling codeCORE2D (Samper et al., 2000; Xu et al., 1999), an inversecode INVERSE-CORE2D (Dai and Samper, 1999) hasbeen developed. The code can estimate any flow and trans-port parameter in the code CORE2D and provides the sta-tistical measures of goodness-of-fit as well as parameteruncertainty by computing the covariance and correlationmatrices, eigenvalues and approximate confidence intervals(see Dai and Samper, 2004).

4. Description of transport experiments

Several types of diffusion experiments (Fig. 1) were car-ried out by Garcıa-Gutierrez et al. (2001) to estimate trans-port and sorption parameters of FEBEX bentonite,including total and kinematic porosity, diffusion coeffi-cients (effective and apparent) and distribution coefficient.These experiments were interpreted by means of availableanalytical solutions by Garcıa-Gutierrez et al. (2001). Theconditions of some experiments are not taken into accountby such solutions. The use of numerical models for exper-iment interpretation overcomes these limitations of analyt-ical solutions.

4.1. Through-diffusion experiments

Through-diffusion experiments were performed usingtritium (experiments TD-HTO1, TD-HTO2 and TD-HTO3) and strontium (experiments TD-Sr5, TD-Sr6,TD-Sr7 and TD-Sr8).

Experiments were performed on bentonite plugs of50 mm of diameter, with lengths ranging from 5.3 mm for

0.0 0.5 1.0 1.5 2.0

0

200

400

600

800

1000

Act

ivity

(M

coun

t/l)

Time (days)

Permeation 1 (P-1d) (Double-porosity)

Time function 1 Obj.=185.6Time function 2 Obj.=72.0Time function 3 Obj.=12.2

Fig. 2. Time functions used to define the tracer pulse of the inflow water.Also shown are the values of the objective function for each time function.One can see the large sensitivity of the fit to the selected time function.

J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648 643

tritium to 8.3 mm for strontium (Fig. 1). They wereimmersed in water for 4 weeks achieving a final densityof 1.18 g/cm3. The clay plug was located in contact withtwo cells. To ensure a uniform diffusion through the wholecross-section of the plug, porous stainless-steel sinters man-ufactured by Mott were located at both sides of the plug.According to technical specifications sinters are 3.35 mmlong and have a porosity of 0.5. A known mass of tracerwas added initially to one of the cells. The evolution of tra-cer activity was measured in both cells at regular intervals.Experiments lasted around 54 days for tritium and nearly200 days for strontium.

The model domain of through-diffusion experimentsincludes the following subdomains: the upstream anddownstream cells, two sinters and the clay plug. Finite ele-ments representing the cells were assigned a porosity of 1and a large-enough diffusion coefficient in order to ensurethat all nodes within the cell have the same activity. Forthe sinters the effective diffusion was computed using Eqs.(4) and (5) with a porosity of 0.5.

The initial activity in the upstream cell was taken equalto the initial prescribed tracer activity. Initial activity isequal to zero in the rest of the subdomains.

Two finite element grids were used for the numerical sim-ulation of through diffusion experiments. That of tritium(TD-HTO1, TD-HTO2 and TD-HTO3) includes 200 trian-gular elements and 202 nodes. The grid for strontium exper-iments (TD-Sr5, TD-Sr6, TD-Sr7 and TD-Sr8) has 114triangular elements and 116 nodes. The simulation timehorizon covers the expected duration of the experiments.For TD-HTO1, TD-HTO2 and TD-HTO3 the total timeis 54 days, which was discretized with 800 time increments.For strontium experiments the total experiment time of 193days was divided into 1800 time increments. Before runningthe inverse model, some forward runs were made to test andensure the numerical stability of the solutions.

4.2. In-diffusion experiments

In-diffusion (ID) experiments consist on the immersionof a clay sample of 38 mm of diameter and 25 or 50 mmof length into a cell containing granite synthetic groundwa-ter until saturation. Then a tracer, which is initially notpresent either on the clay or on the sinters, is added tothe solution and allowed to diffuse into clay samples fromboth sides. The gradient in tracer activities induces a diffu-sive flux from the cell into clay samples (Fig. 1). Thedecrease in tracer activity in the immersion cell was moni-tored continuously. At the end of the experiment clay plugswere sliced to obtain the activity profile along clay samples.In-diffusion experiments were carried using cesium andselenium. Se(IV) was added as a sodium selenite SeO3Na2.Tests were performed under oxic conditions.

The model domain for in-diffusion experiments containsfive parts: two cells, two sinters and a clay plug. Initial activ-ities in the two cells are equal to the initial tracer activitywhile in the sinters and the clay plug they are equal to zero.

Two kinds of numerical models were used in theseexperiments, one for cesium (In-Cs-9 and In-Cs-10) with600 triangular elements and 602 nodes, and the other forselenium (In-Se-17, In-Se-18) with 200 triangular elementsand 202 nodes. The simulation time for both tracers(cesium and selenium) is 439 days and it is discretized with630 time increments.

4.3. Permeation experiments

A high-pressure stainless steel cell was used for perme-ation experiments. The clay plug, 5.3 mm in length and50 mm of diameter was sealed between two 3.35 mm thickstainless steel sinters (Fig. 1). The granite synthetic ground-water was injected into the sample with a low driving pres-sure (6 bar) ensured by a HPLC pump with pressure andflow regulation (the flow rate in the cells ranged from 2.7to 4.0 ml/day). The final density is 1.18 g/cm3. All theexperiments were performed under oxic conditions. After4 weeks of sample saturation, a small pulse of tritium tracer(500 ll or 183,000 counts) was injected into the fluid path-way. The breakthrough curve of the tracer at the outlet wasmonitored continuously by a fraction collector.

Single and double porosity models were used to modelpermeation experiments. The former is simulated with a 1-D grid while the latter requires a 2-D finite element grid con-taining a mobile part where water can flow and an immobilepart where only molecular diffusion can take place.

Since the tracer was injected as a non-instantaneouspulse, the duration and distribution of the pulse is uncer-tain. Only the flow rate and the total tracer mass injectedare known. Tracer injection was modeled by using atime-dependent pulse function. A trial and error methodwas used to estimate such pulse function (Fig. 2) by keep-ing constant the total mass of tracer injected. Initial activ-ities in all of the nodes are zero.

Fig. 3. Finite element mesh for the double-porosity model.

644 J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648

The 1-D single porosity model (P-1) has 200 triangularelements and 202 nodes. This model proved to be unableto match observed data and then a 2-D double porositymodel (P-1d) was used which was discretized with 560 tri-angular elements and 324 nodes. Fig. 3 illustrates the finiteelement mesh used for the double-porosity model. Thetotal length of the system is 1.2 cm, including two sintersat both sides (2 · 0.335 cm long) and the bentonite samplein the middle (0.53 cm long). The width is 0.2 cm. Based ona prior estimate of mobile porosity of about 20%, the widthof the mobile part (bottom row) is 0.04 cm while that of theimmobile part (the upper seven rows) is 0.16 cm.

The simulation time is 3.78 days. It is discretized with 88time periods with gradually increasing time increments.

5. Interpretation by inverse modeling

Diffusion experiments were interpreted numericallyusing the inverse code INVERSE-CORE2D (Dai and Sam-

Table 1Summary of interpretation of CIEMAT diffusion experiments performed on s

Tracer Experiments andbentonite density (g/cm3)

Numerical

/ De (m2/s) Da (

Tritium TD-HTO1 1.18 0.56b 1.41 · 10�10

0.5 1.41 · 10�10

TD-HTO2 1.18 0.56b 1.24 · 10�10

0.45 1.24 · 10�10

TD-HTO3 1.18 0.56b 1.18 · 10�10

0.44 1.17 · 10�10

P-1 1.18 0.52/0.14 3.5 · 10�9

Strontium TD-Sr5 1.39 0.48b 1.10 · 10�9 6.74TD-Sr6 1.39 0.48b 1.81 · 10�9 7.13TD-Sr7 1.65 0.38b 1.25 · 10�9 5.47TD-Sr8 1.65 0.38b 9.70 · 10�10 5.28

Cesium ID-Cs-9 1.65 0.42b 8.03 · 10�10 3.340.49 8.05 · 10�10 3.32

ID-Cs-10A 1.57 0.42b 3.3 · 10�10 1.370.49 3.44 · 10�10 1.43

ID-Cs-10B 1.57 0.42b 7.57 · 10�10 3.210.45 8.81 · 10�10 3.39

Selenium ID-Se-17 1.57 0.42b 4.35 · 10�13 6.510.21 1.73 · 10�13 3.4

ID-Se-18 1.57 0.42b 1.39 · 10�13 2.290.25 1.23 · 10�13 2.23

a Analytical results were obtained by Garcıa-Gutierrez et al. (2001). Here / deapparent diffusion coefficient and Kd is distribution coefficient in ml/g.

b Parameter is fixed and therefore not estimated.

per, 1999). Results of the numerical interpretation of TD,IN and P experiments are shown in Table 1. This table con-tains also the parameters obtained by Garcıa-Gutierrezet al. (2001) by using analytical methods. It should benoticed that not all experiments could be interpretedanalytically.

5.1. Through-diffusion experiments

Several sets of through-diffusion (TD) experiments wereperformed on clay samples at dry densities ranging from1.18 to 1.65 g/cm3 using tritiated water (tritium) and stron-tium. Tritium was used in three TD experiments whichlasted about 50 days, a time long enough for the experi-ment to attain steady activities along the samples.

Tritium experiments were used to test the estimationalgorithms and to explore the role of the sinters. Ourresults indicate that the sinters should be taken intoaccount for a proper interpretation of the experiments.The results in Table 1 clearly demonstrate that bentonitediffusion coefficients obtained from analytical solutionsunderestimate the true diffusion coefficient by a factor of1.4. To demonstrate the accuracy of the numerical results,experimental data were interpreted numerically with amodel without sinters. In this case, inverse estimates of dif-fusion coefficients coincide with those obtained with ana-lytical methods. Therefore, failing to account for thesinters can lead to the underestimation of diffusion coeffi-

amples of compacted FEBEX bentonitea

Analytical

m2/s) Kd (ml/g) De (m2/s) Da (m2/s) Kd (ml/g)

1.03 · 10�10

8.89 · 10�11

9.80 · 10�11

· 10�13 1176 8.81 · 10�12 1410· 10�13 1823 3.44 · 10�12 1200· 10�13 1383 3.99 · 10�12 1550· 10�13 1114 5.61 · 10�12 1300

· 10�13 925.1 3.07 · 10�13 823 ± 121· 10�13 931.8· 10�13 924.1 2.51 · 10�13

· 10�13 924.4· 10�13 907.9 3.51 · 10�13

· 10�13 999.6

· 10�14 2.41 6.25 · 10�14 3.35 ± 1.5· 10�14 1.87· 10�14 2.17· 10�14 2.02

notes diffusion-accessible porosity; De is effective diffusion coefficient; Da is

J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648 645

cients. Based on this finding, the sinters were considered forthe interpretation of all the diffusion experiments.

In general, the numerical solution matches observed tra-cer data in both cells (Fig. 4). Effective diffusion coefficientsfor tritium range from 1.17 to 1.41 · 10�10 m2/s which arewithin the range of published values for this tracer in bent-onites (Yu and Neretnieks, 1996). The 95% confidenceinterval of the estimated effective diffusion coefficient ofTD-HTO1 is rather small: (1.34 · 10�10, 1.48 · 10�10) m2/s. The best fit to TD tritium data is obtained with diffusiveporosities ranging from 0.44 to 0.5 (see Table 1) which areslightly smaller than the total porosity. Tortuosity factorranges from 0.76 to 0.79.

Estimates of porosity and diffusion (D0), however, arestrongly correlated as attested by their large computed cor-relation coefficients which are greater than 0.95. Given thelarge uncertainties in porosity and diffusion coefficient, thevalue of porosity was fixed. Several runs were performedusing porosities ranging from 0.45 to 0.56. These runs ledto similar fits but different estimates of D0. Estimates of

0 10 20 30 40 50 60

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Act

ivity

(M

coun

t/L)

Time (days)

Through-Diffusion 1 (TD-HTO1) Observations (IN) Numerical results (IN) Observations (OUT) Numerical results (OUT)

0 50 100 150 200

0

5

10

15

20

25

Act

ivit

y (M

coun

t/L

)

Time (days)

Through-Diffusion 5 (TD-Sr5) Observations (IN) Numerical results (IN) Observations (OUT) Numerical results (OUT)

Fig. 4. Numerical interpretation of through diffusion experiments: tritiumTD-HTO1 (top) and strontium TD-Sr5 (bottom). Upstream (IN cell) anddownstream (OUT cell) cell activities are shown.

porosity and D0 always provide a similar values of theeffective diffusion coefficient. This means that tritium TDexperiments only allow the unambiguous estimation ofthe effective diffusion coefficient.

Four TD experiments were performed using strontium:TD-Sr5 and TD-Sr6 with a dry density of 1.18 and TD-Sr7 and TD-Sr8 with a density of 1.39 g/cm3. This tracerhas a large distribution coefficient. In fact, the tracer doesnot reach the downstream cell after almost 200 days whileit has been fully flushed from the upstream cell after thattime. These experiments were interpreted in two groups.For TD-Sr5 and TD-Sr6 porosity was fixed to a value of0.48 that corresponds to a dry density of 1.39 g/cm3. Forthe second group (TD-Sr7 and TD-Sr8) porosity was fixedto 0.38 that corresponds to a dry density of 1.65 g/cm3. Kd

and D0 were estimated in all four experiments. The best fitis shown in Fig. 4. Effective diffusion coefficients De rangefrom 9.7 · 10�10 to 1.81 · 10�9 m2/s with a mean value of1.28 · 10�9 m2/s. Estimated values of Kd show also awide range (from 1114 to 1823 cm3/g) with a mean of1374 cm3/g. Estimated results of the apparent diffusionDa show much less scatter (from 5.28 · 10�13 to7.13 · 10�13 m2/s) with a mean value of 6.15 · 10�13 m2/s.Estimation errors and confidence intervals are similar forall four experiments. For experiment TD-Sr5, the 95%confidence intervals for the estimates of effective diffusionand Kd are (5.47 · 10�10, 1.65 · 10�9) m2/s and (786.6,1566) cm3/g, respectively. These values are within the rangeof published values for this tracer in bentonites. It shouldbe noticed that in this case the estimates of effective diffu-sion and Kd show consistently a significant negative corre-lation of about �0.5. The inverse analysis indicates that Kd

is the parameter having the largest uncertainty (estimationerror).

5.2. In-diffusion experiments

Several sets of ID experiments were performed on claysamples having a total porosity of 0.42 which amounts toa dry density of 1.57 g/cm3 using radioactive cesium(137Cs) and selenium SeO2þ

3 . Tracer activity in the cellwas monitored for full duration of the experiment. At theend of the experiment, clay samples were sliced and totaltracer activity was measured in each slice. This activityincludes the activity in the liquid and solid phases, inMcount/per gram of sample (clay + water).

In order to fit the transient liquid activity in the cell andthe final total activity (including the liquid and solidphases) in the clay at the end of experiments, it is necessaryto translate modeling results from liquid activity to totalactivity by means of

CT ¼ CL

/þ qdKd

/qw þ qd

ð11Þ

where CT = total activity (Mcount/g), CL = liquid activity(Mcount/cm3), qd = bulk dry density of the porous med-ium (g/cm3), and qw = density of water (g/cm3). Therefore,

646 J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648

inverse modeling of ID experiments uses two types of data:transient liquid activity in the cell and final total activity inthe clay.

Two cesium experiments were available (ID-Cs-9 andID-Cs-10). In one of them, the behavior of the sample isnot symmetric, i.e., the tracer diffuses on one side fasterthan on the other. In this case, data from each side ofthe sample were interpreted separately (tests ID-Cs-10Aand ID-Cs-10B). The experiments were interpreted in twostages. First, the porosity was fixed to a value of 0.42 whileKd and D0 were estimated. In the second stage, porositywas also estimated, resulting in an excellent fit to measuredfinal concentration data. Some discrepancies are observedin the time evolution of tracer activity in the cell whichare attributed to uncertainties in the initial activity andthe lack of well-mixed conditions in the cell. A detailed sen-sitivity analysis was performed to evaluate the effects ofseveral sources of uncertainty including: (1) the initial con-centration in the sinters, (2) the appropriate boundary con-dition for the cell, (3) the initial concentration in the cell,and (4) the effective volume of the cell. The results of thisanalysis indicate that estimated parameters are very sensi-tive to the properties and conditions prevailing at the cell.Therefore, for the proper interpretation of this type ofexperiment we suggest that: (1) only one clay sampleshould be located in each cell, (2) the initial concentrationshould be measured as accurately as possible, and (3) spe-

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

0.20

0.25

Act

ivity

(M

coun

t/g)

Distance (dm)

In-Diffusion-Cs-9 Measurements Model fit

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.000

0.001

0.002

0.003

0.004

Act

ivity

(M

coun

t/g)

Distance (dm)

In-Diffusion-Se-18 Measurements Model fit

Fig. 5. Numerical interpretation of in diffusion cesium ID-Cs-9 (top) and seactivities in the clay sample and figures on the right show the concentration e

cial care should be taken in measuring the activities of theslices located near the faces of the clay samples.

The best fit is obtained with diffusive porosities slightlylarger (0.49) than total porosity (0.42). This result couldbe caused by Cs diffusion through clay interlayers. Diffusiveporosities range from 0.45 to 0.49. Effective diffusion coef-ficients De range from 3.44 · 10�10 to 8.81 · 10�10 m2/s.Estimated values of Kd range from 924.4 to 999.6 cm3/g.Apparent diffusion coefficients range from 1.43 · 10�13 to3.39 · 10�13 m2/s. These values are within the range of pub-lished values for this tracer in bentonites. Numerical solu-tions fit perfectly measured data (Fig. 5).

Two selenium (in the form of SeO2þ3 ) experiments were

performed on clay samples of 2.5 cm, half the length ofthe cesium cells. Again, the experiments were interpretedin two stages. First, the porosity was fixed to a value of0.42 while Kd and D0 were estimated. In the second stage,porosity was also estimated, resulting in an excellent fitto both final concentration data and time evolution oftracer activity in the cell. The best fit is obtained withdiffusive porosities smaller than total porosity (Fig. 5).Diffusive porosities are equal to 0.21 and 0.25. Effectivediffusion coefficients De range from 1.23 · 10�10 to 1.72 ·10�10 m2/s. Estimated values of Kd range from 1.87 to2.02 cm3/g. Apparent diffusion coefficients range from2.23 · 10�13 to 3.40 · 10�13 m2/s. Garcıa-Gutierrez et al.(2001) reported difficulties in obtaining a unique fit to sele-

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lenium ID-Se-18 (bottom) experiments. (Left figures show the final totalvolution in the cell.)

J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648 647

nium IN diffusion experiments that were attributed tochemical reactions. Selenium could be present in severaloxidation states and form different complexes with differentproperties. Although Se(IV) oxidation into Se(VI) seemsunlikely, Se(IV) could be reduced by Fe (II) or organic mat-ter inside the clay sample.

5.3. Permeation experiments

Contrary to in- and through-diffusion experiments, per-meation experiments involve advective and dispersivetransport. These experiments are intended to provide infor-mation on kinematic porosity which measures the volumeof well-interconnected water-flowing pores. Permeationexperiments were performed using tritiated water. Follow-ing the parsimony principle, the classic single-porositymodel was tested first. For the permeation experiment aconvergent but suboptimal solution is obtained with aporosity of 0.35 (Fig. 6). Given the limitations of the sin-gle-porosity model, a more complex double-porosity modelwas considered.

In our double-porosity model the domain is divided intotwo domains. Let fm and fim be the fractions of the total

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Fig. 6. Numerical interpretation of tritium permeation experiment.Computed activities (lines) and measured data (symbols) are shown forsingle-porosity (top) and double-porosity (bottom) models.

volume of the sample occupied by the mobile and immobiledomains, respectively. From prior information we setfm = 0.2 and fim = 0.8. In our formulation of double poros-ity, model porosities for mobile and immobile zones are notglobal porosities, but relative porosities. Let /m and /im bethe relative porosities of the mobile and immobile zoneswhich are defined as the ratio of void volume to total vol-ume of each domain. The total porosity of the sample isgiven by / = /mfm + /im fim where /mfm is the overallporosity of the mobile domain and /imfim represents theoverall porosity of the immobile domain. According toour experience, this way of parameterizing double porosityis not affected by the geometry adopted in the model.Therefore, the values /mfm and /imfim are independent ofthe model geometry.

The inverse double-porosity model leads to an excellentfit of the activity breakthrough curves (Fig. 6) with relativeporosities /m = 0.73 and /im = 0.65 for which the overallporosity of the mobile domain is 0.14 and that of theimmobile domain is 0.52. It should be noticed that the esti-mated total porosity (0.66) is slightly greater than total vol-umetric porosity (0.56). This could be caused by tritiumaccumulation in clay minerals which according to Kalini-chenko et al. (2002) takes place under room conditionsby a two-stage isotope exchange. Tritium ions first migratefrom the solution to the bound layers and then substitutethe protons of the structural hydroxyls.

The effective diffusion coefficient in the mobile phase is3.5 · 10�9 m2/s while that of the immobile phase is almostthree orders of magnitude smaller. The longitudinal disper-sivity for the mobile phase was set equal to one-tenth of thelength of the clay sample and was not estimated.

5.4. Comparison of numerical estimations with analytical

solutions

Standard analytical methods (see Garcıa-Gutierrez et al.,2001) for the interpretation of TD and ID experiments donot take into account the sinters. Such a limitation of analyt-ical methods can lead to erroneous estimates of diffusioncoefficients. Inverse modeling of tritium TD experimentshas revealed that for the experimental conditions of FEBEXbentonite failing to account for the sinters can lead to under-estimation of effective diffusion coefficients by a factor of 1.4.

Analytical methods for the interpretation of ID experi-ments rely entirely on activity data measured at the endof the experiment. They make no use of data measuredat the immersion cell and allow only the estimation ofapparent diffusion coefficient and /R (see Eq. (2)). Inversemodeling, on the contrary, allows the estimation of trans-port and sorption parameters by using simultaneously bothfinal total activities and time evolution of activities in theimmersion cells.

Numerical methods are better suited than analyticalmethods especially for the interpretation of permeationexperiments which may require time-varying pulse func-tions and double porosity models for bentonite.

648 J. Samper et al. / Physics and Chemistry of the Earth 31 (2006) 640–648

Limitations of analytical methods may lead to biasedestimates of diffusion coefficients. According to the com-parison shown in Table 1, such a bias is about a factorof 1.4 for the effective diffusion coefficient of tritium andan order of magnitude for the apparent diffusion coefficientof strontium. However, the biasness is rather small for theapparent diffusion and the distribution coefficient of Csand Se (Table 1).

6. Conclusions

Diffusion and permeation experiments performed onFEBEX compacted bentonite using tritium, cesium, sele-nium, and strontium have been effectively interpreted byinverse modeling. Contrary to standard analytical meth-ods, numerical inverse modeling allows an efficient, fastand optimum interpretation of these experiments. It hasbeen found that failing to account for some experimentalconditions may lead to erroneous diffusion coefficientswhich for the experimental conditions of FEBEX bentonitecould deviate from true values by a factor of 1.4.

Inverse modeling of permeation experiments allows theuse of time-varying pulse functions and double porositymodels for bentonite. Numerical methods have proved use-ful for indicating identifiability problems and suggestingpossible improvements in experimental conditions such asmeasuring directly the effective diffusion coefficient of por-ous sinters.

Possible improvements for in-diffusion experimentsinclude: (1) Determining accurately the initial tracer con-centration and the effective volume of the cells; and (2)Obtaining detailed and accurate concentration profiles,especially near the faces of the clay samples, and using onlyone clay sample in each cell.

In our work the porosity and the effective diffusion ofthe sinters were taken from manufacturer technical specifi-cations. However, it is recommended to determine theseparameters experimentally by means of diffusion experi-ments to reduce unneeded uncertainties.

The duration of the tracer pulse should be much smallerthan the duration of the permeation experiment. Other-wise, the concentration of the inflow water should be care-fully measured. Similar to other clays, FEBEX bentoniteexhibits a complex porosity structure. Two types of poros-ities can be distinguished: a mobile porosity along whichporewater flows and an immobile porosity where waterdoes not flow while only molecular diffusion takes place.The results of the tritium permeation experiment per-formed on FEBEX bentonite indicate that mobile porosityis 0.14 for a dry density of 1.18 g/cm3.

Acknowledgement

This work was supported by the Spanish Nuclear WasteCompany (ENRESA) within the framework of the FEBEXResearch Project through Research Grants signed with theUniversity of La Coruna (Contracts 70323 and 770045).

The FEBEX Project as a whole is funded by the EC (Pro-jects FI4W-CT95-0008 and FIKW-CT-2000-0016) withinthe Nuclear Fission Safety Programme. We gratefullyacknowledge the assistance of John Apps of LBNL. Wethank the two anonymous reviewers for their thoughtful re-views of the paper which contributed to improve it.

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