hydro-chemical modelling of in situ behaviour of …intermediate- and high-level radioactive waste....

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X. SillenN. Mokni, S. Olivella, E. Valcke, N. Bleyen, S. Smets, X. Li and bituminized radioactive waste in Boom Clay

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Hydro-chemical modelling of in situ behaviour of bituminized

radioactive waste in Boom Clay

N. MOKNI1*, S. OLIVELLA1, E. VALCKE2, N. BLEYEN2,

S. SMETS2, X. LI3 & X. SILLEN4

1Department of Geotechnical Engineering and Geosciences, Universitat

Politecnica de Catalunya, Barcelona, Spain2Waste and Disposal Expert Group, The Belgian Nuclear Research

Centre (SCK†CEN), Boeretang 200, 2400 Mol, Belgium3EIG EURIDICE, Boeretang 200, 2400 Mol, Belgium

4ONDRAF/NIRAS, Kunstlaan 14, 1210 Brussel, Belgium

*Corresponding author (e-mail: [email protected])

Abstract: The hydro-chemical (CH) interaction between swelling Eurobitum bituminized radio-active waste (BW) and Boom Clay was investigated to assess the feasibility of geological disposalfor the long-term management of this waste. First, the long-term behaviour of BW in contact withwater was studied. A CH formulation of chemically and hydraulically coupled flow processes inporous materials containing salt crystals is discussed. The formulation incorporates the strongdependence of the osmotic efficiency of the bitumen membrane on porosity and assumes the exist-ence of high salt concentration gradients that are maintained for a long time and that influence thedensity and motion of the fluid. The impacts of temporal and spatial variations of key transport par-ameters (i.e. osmotic efficiency (s), intrinsic permeability (k), diffusion, etc.) were investigated.Porosity was considered the basic variable. For BW porosity varies in time because of the wateruptake and subsequent processes (i.e. dissolution of salt crystals, swelling of hydrating layers, com-pression of highly leached layers). New expressions of s and k describing the dependence of theseparameters on porosity are proposed. Several cases were analysed. The numerical analysis wasproven to be able to furnish a satisfactory representation of the main observed patterns of the be-haviour in terms of osmotic-induced swelling, leached mass of NaNO3 and progression of thehydration front when heterogeneous porosity and crystal distributions have been assumed.Second, the long-term behaviour of real Eurobitum drums in disposal conditions, and in particularits interaction with the surrounding clay, was investigated. Results of a CH analysis are presented.

Final disposal in deep geologically stable for-mations is considered worldwide as the preferredoption for the long-term management of long-livedintermediate- and high-level radioactive waste. Thedisposal concepts proposed by many countriesconsist of a system of natural and engineered bar-riers to separate the waste from the biosphere. TheBelgian Agency for the Management of RadioactiveWaste and Fissile Materials (ONDRAF/NIRAS)envisages geological disposal of Eurobitum bitumi-nized radioactive waste (BW) (ONDRAF/NIRAS2009). Eurobitum is an intermediate-level long-lived radioactive waste form that consists of c.60 w% of hard bitumen Mexphalt R85/40 and c.40 wt% of waste. The waste originates from thechemical reprocessing of spent nuclear fuel andfrom the cleaning of high-level waste storage tanks,and contains mainly NaNO3 and CaSO4. These saltsare present with a concentration of, respectively,20–30 and 4–6 wt% of Eurobitum. In Belgium,

the Boom Clay, which is a 30–35 million yearsold and c. 100 m thick marine sediment, isbeing studied as a potential host formationbecause of its favourable properties to limit anddelay the migration of any leached radionuclidesto the biosphere over extended periods of time.The current disposal concept foresees that several220 l drums of Eurobitum would be grouped in athick-walled cement-based secondary container,which in turn would be placed in concrete-lined dis-posal galleries that are excavated at mid-depth in theclay layer (Marien et al. 2013). Only 80–90% of thetotal volume of the drum would be filled with Euro-bitum. The remaining voids between the drumswould be backfilled with a cement-based material.The emplacement of Eurobitum BW in the BoomClay is expected to result in a geomechanical anda physico-chemical perturbation of the clay(Valcke et al. 2010). The large amounts of dehy-drated and hygroscopic salts in the BW, with

From: Norris, S., Bruno, J., Cathelineau, M., Delage, P., Fairhurst, C., Gaucher, E. C., Hohn, E. H.,Kalinichev, A., Lalieux, P. & Sellin, P. (eds) Clays in Natural and Engineered Barriers for Radioactive WasteConfinement. Geological Society, London, Special Publications, 400, http://dx.doi.org/10.1144/SP400.40# The Geological Society of London 2014. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics

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bitumen acting as a highly efficient membrane, willprovoke an osmosis-driven uptake of pore water,resulting in swelling of the BW and/or increasedstress on the clay (i.e. a geomechanical perturbationof the clay). In addition, diffusion of large amountsof salts, mainly NaNO3, will result in physico-chemical perturbation of the clay. After closure ofthe disposal gallery, two different phases can be dis-tinguished (Valcke et al. 2010; Marien et al. 2013).In the first, free swelling phase, an osmoticallyinduced swelling of the BW will take place untilall free volume is filled with swollen BW. After-wards, the continued water uptake and swelling ofthe BW will induce an increasing stress on theclay. The resultant deformation of the clay createsadditional space for the BW to swell, but comparedwith the free swelling phase, the swelling rate islimited by the increasing stress exerted by theclay. This second phase is the restricted swellingphase. The number of drums per gallery cross-section must be limited to prevent unacceptable geo-mechanical perturbations of the clay.

An experimental programme aiming at under-standing the swelling and leaching behaviour ofBW samples in contact with alkaline solutions is inprogress at the Belgian Nuclear Research CentreSCK†CEN. The set of tests that has been performedconsists of water uptake tests under restricted (i.e.constant stress and constant volume) and free swel-ling conditions (Valcke et al. 2010; Marien et al.2013). In a previous paper (Mokni et al. 2011) ahydro-chemical (CH) formulation of chemicallyand hydraulically coupled flow processes in porousmaterials containing salt crystals was discussed.The formulation incorporates the strong dependenceof the osmotic efficiency of the membrane on poros-ity, and assumes the existence of high concentra-tion gradients that are maintained for a long timeand that influence the density and motion of thefluid. The CH formulation was used to analyse theinfluence of osmosis on the swelling of EurobitumBW owing to water uptake under laboratory condi-tions. The experimental results of pressure increase,swelling deformation and formation of low per-meability compressed outer layers were reproducedsuccessfully by Mokni et al. (2011) and Marienet al. (2013). However, some differences wereobserved between the model results and the exper-imental observations in terms of progression of thehydration front and leached mass of NaNO3.Indeed, the modelled time to dissolve all salt crys-tals was shorter than observed in the experiments.According to the model, all crystals in the sampleare dissolved after about 2000 days, while inreality only 12–21% of the initial NaNO3 contentwas leached after about 4 years of hydration andonly the outer layers with a thickness of 1–2 mmwere hydrated after this time period (Marien et al.

2013). Marien et al. (2013) stated that the differenceis related to the simple relationship between poros-ity and membrane efficiency proposed by Mokniet al. (2011), which shows a too rapid decrease inefficiency with porosity increase (see Fig. 2further in the text).

In order to better reproduce the experimentalobservations in terms of progression of the hydra-tion front and leached mass of NaNO3, a newexpression of the osmotic efficiency coefficient(s), including the dependence of s on porosityand on crystal volume fraction, is proposed in thispaper. Because of the interdependency of most ofthe key transport functions (i.e. permeability, diffu-sion coefficient and osmotic efficiency) that con-trol the magnitude of coupled fluxes (principallychemical osmosis and ultrafiltration), a new expres-sion of the intrinsic permeability (k), including thedependence of k on porosity, is proposed in thispaper. The water uptake tests under constant verti-cal stress performed on BW were modelled usingthe proposed expressions and several cases wereanalysed. A sensitivity analysis was performed inorder to study the effect of the variation rates of sand k with porosity on swelling, leaching and pro-gression of the hydration front of BW. The CHformulation was used to model the in situ behav-iour of BW, to obtain insights into the kinetics ofwater uptake by BW, dissolution of the embeddedNaNO3 crystals, solute leaching and maximumgenerated pressure under disposal conditions.

Long-term swelling behaviour of BW:

phenomenology and main processes

observed in laboratory experiments

A research programme involving the testing of theswelling behaviour of specimens of BW under con-stant volume and constant stress conditions was per-formed at the Belgian Nuclear Research CentreSCK†CEN. The experimental results are reportedin Valcke et al. (2010) and Marien et al. (2013).Swelling under various constant stresses, osmosis-induced pressure in constant volume conditionsand leached mass of NaNO3 were measured duringthe tests. After dismantling, the hydrated sampleswere characterized with micro-focus X-ray compu-ter tomography (mCT) and environmental scanningelectron microscopy (ESEM). Modelling results ofthose tests were presented by Mokni et al. (2011);Mokni (2011) and Marien et al. (2013). The phe-nomenology of the water uptake, swelling and saltleaching processes in restricted swelling conditionsis now reasonably well understood (Valcke et al.2010; Mokni et al. 2011; Marien et al. 2013). Thereported results and observations show that theuptake of water by and the subsequent swelling

N. MOKNI ET AL.



and/or pressure increase of BW (composed of40 wt% of waste containing 28.5 wt% soluble saltsNaNO3) are largely controlled by osmosis. Thebitumen surrounding the soluble salts and otherinorganic compounds behaves as a highly efficientsemipermeable membrane, especially when testedunder restricted swelling conditions. After almostfour years of hydration of small BW samples(38 mm diameter, 10 mm thickness), osmostic-induced pressures of up to 20 MPa and swellingdeformations ranging between 10 and 20% arebeing measured in constant volume and constantstress water uptake tests, respectively. A majorcharacteristic of the medium, which plays an impor-tant role in the process of water uptake, is thepresence of salt crystals. In fact, the large amountof hygroscopic salt incorporated into the bitumenmatrix will attract water from the surrounding reser-voir, resulting in dissolution of the soluble crystals,diffusion of the dissolved salts and volume increaseof the leaching BW. When the external appliedstress is lower than the osmotic pressure, water isdriven into the pores of the medium because of thegradient of concentration existing between thereservoir and the solution initially filling the pores.The pore water concentration will decrease until itreaches a value corresponding to an osmotic pres-sure equal to the external applied stress. The dif-fusion of NaNO3 out of the BW, combined withthe external stress and the osmosis-driven trans-port of water from the outermost pores with alower NaNO3 concentration (higher water activity)towards deeper pores with a higher NaNO3 concen-tration (lower water activity), contributes to a localconsolidation and the formation of a re-compressed,low-porosity layer at the outer boundary of theleached material (Fig. 1b). Compression of the out-ermost layers is considered here as a limiting orcontrolling process for the water uptake and therelease of NaNO3. Valcke et al. (2010) characterizedsome leached samples with mCT. This techniquepermitted the hydrated area of the leached sampleto be visualized. Figure 1a shows the mCT imageof samples S0 and S4 after 887 and 1472 days ofhydration, respectively. The red parts represent thestainless steel filters. The thin green–yellow layersnear the filters are the hydrated layers. The centralpart in red–yellow corresponds to the dry Eurobi-tum. For both samples, only a thin layer had beenhydrated. The central part of the samples appearedto be not hydrated, which suggested that it wouldtake several years before the hydration frontreached the centre of the samples, and it wouldrequire even more time before all NaNO3 had dis-solved. During that time, the sample was expectedto continue to swell. In fact, size reduction of theouter pores induces a decrease in the permeabilityand the diffusion coefficient, and an increase in

the osmotic efficiency in these compressed layers,while deeper in the sample these parameters varyin the opposite way. Therefore, the outer layers actas a highly efficient semipermeable membrane sur-rounding the swollen material with less efficientmembrane properties. This maintains a very slowinflow of water towards the sample and outflow ofthe solute over a long time.

Coupled CH formulation

Mokni et al. (2010a, 2011) presented a CH formu-lation of chemically and hydraulically coupledflow processes in porous materials containing saltcrystals, which incorporates the strong dependenceof the efficiency of the membrane on porosity. Theformulation assumes the existence of high con-centration gradients that are maintained for a longtime and that influence the density and motionof the fluid. The CH formulation is presentedbriefly below.

Balance equations

The medium is composed of three phases:

† solid phase – solid matrix and salt crystals;† liquid phase (l ) – dissolved salt in water, dis-

solved air;† gas phase (g) – mixture of dry air and water

vapour.

It also comprises four species:

† solid matrix (m) – matrix embedding thecrystals;

† salts – as crystal (c) and as dissolved salt (s)resulting from the dissolution of the crystals;

† water (w) – as liquid or evaporated in the gasphase;

† air (a) – dry air as gas or dissolved in theliquid phase.

The total volume of the medium (Vt) can be decom-posed into the volume occupied by the crystals(Vc), the pores (voids, Vv) and the bitumen (Vb),that is, Vt ¼ Vc + Vv + Vb. The volume of thepores varies as a consequence of dissolution of thecrystals and may be occupied by water or air.

These volumes permit the volume fraction of thedifferent components (fc and fb) and the volumefraction of pores, that is, the porosity (ff), to bedefined. These variables are defined as:

fc =Vc

Vt

; ff =Vv

Vt

; fb =Vb

Vt

;

Vt = Vc + Vv + Vb; 1 = fc + ff + fb = fc + fm

fm = ff + fb (1)

MODELLING BEHAVIOUR OF BW IN BOOM CLAY



The sum of the porosity (ff) plus the volumefraction of the bitumen (fb) is defined as thevolume fraction of the porous bitumen matrixfm, that is, fm ¼ f+ fb. In this way, 1 2 fm ¼fc is the volumetric fraction occupied by thecrystals.

An initial porosity ff0is assumed, correspond-

ing to the volume initially occupied by the gas andwater in contact with salt. As a consequence of thedissolution of the crystals, the pore volume avail-able for the fluids (ff) will increase, resulting in adeformation of the sample.

To establish the balance equations, the com-positional approach is adopted, consisting of balan-cing the species rather than the phases. The massbalance equations for water, dissolved salts, crystalsand solid phase, including the coupled flows,namely osmotic flow and ultrafiltration, and thedissolution/precipitation of salts, are described indetail in Mokni et al. (2011). The formulationoutlined below has been discretized in space andtime in order to be used for numerical analysisand has been included in the computer codeCODE_BRIGHT (Olivella et al. 1996).

Fig. 1. (a) Micro-focus X-ray computer tomography (mCT) images of two Eurobitum samples: visualization of thehydration front in Eurobitum samples with 28.5 wt% NaNO3 after 887 and 1472 days (Valcke et al. 2010). (b) ESEMphotos of the leached layer of an Eurobitum sample (sample S0) after 887 days of contact with 0.1 m KOH in constantstress conditions (Valcke et al. 2010).

N. MOKNI ET AL.



Constitutive equations

The following assumptions are made in the CHformulation:

† The medium constitutes a single soluble salt(NaNO3) plus an inert solid phase.

† Multiphase flow including phase change is con-sidered (dissolution/precipitation of the saltcrystals).

† Only chemical and hydraulic gradients areconsidered.

† The medium is assumed not to be saturatedwith water.

† Binary diffusion of salt and water at high con-centration is assumed.

† Permeability, diffusion and osmotic efficiencycoefficients are considered to be variablewith porosity.

The constitutive equations that relate the mass-average flow of fluid and the mass flow of soluteto chemical and pressure gradients are as follows(Mokni et al. 2011):

(1) Flux of liquid phase (Jl) induced by bothpressure and concentration gradients:

Jl = − kkrl

m(∇p + rlg∇z) + as

mks∇(ws

l );

as =krlRTrl

Ms

(2)

This is an advective flux in the sense that ittransports both the solute and the water.

(2) Nonadvective flux of solute:

Jsl = −sws

lrlJl − Drl(1 − s)(1 + gwsl )∇ws

l ;

g = 1

rl

∂rl

∂wsl

(3)

(3) Nonadvective flux of water including ultrafil-tration is written as:

Jwl = b

asws

lrlJl − Drl(1 − s)(1 − gwwl )∇ww

l

a = 1 + 1

rl

∂rl

∂wsl

wsl = 1 + gws

l

b = 1 + 1

rl

∂rl

∂wwl

wwl = 1 − gww

l

(4)

where k (m2) denotes the intrinsic per-meability tensor, krl the relative permeabilityof the liquid, g the gravity vector, m (MPa s)the dynamic viscosity, p (MPa) the hydraulicpressure, ws

l the mass fraction of solute inthe liquid phase, ww

l the mass fraction ofwater in the liquid phase, R (J (mol K)21)the gas constant, T (K), the temperature, rl

(kg m23) the liquid density, Ms (kg mol21)

the solute molar mass, s the osmotic effi-ciency coefficient, and D (m2 s21) the dif-fusion coefficient.

The total fluxes of water and solutes in the liquidphase are expressed as:

jwl = −ww

l rl

k

m∇p + D(1 − s)rl(1 − gww

l )∇wsl

+ ks

m(ww

l rl)∇p+ ks2

m

b

aww

l rl

( )∇p

− ks

m

b

aww

l rl

( )∇p + ww

l rlff

du

dt(5a)

jsl = −ws

lrl

k

m∇p − D(1 − s)rl(1 + gws

l )∇wsl

+ ks

m(ws

lrl)∇p − ks2

m(ws

lrl)∇p

+ ks

m(ws

lrl)∇p+ wslrlff

du

dt(5b)

The first and second terms in equations (5a) and(5b) represent Darcy’s and Fick’s laws, respect-ively. The solute diffusion law is dependent uponthe osmotic efficiency coefficient (s). This is anadditional coupling effect included in order tomodel diffusion for semipermeable membranematerials. For the total flux of water (equation 5a)the third term represents chemical osmosis. Thefourth and the fifth terms are coupled fluxes repre-senting respectively the flow of water induced byan osmotic (∇p) and a hydraulic (∇p) pressure gra-dient. For the total flux of the solute, the coupledfluxes terms include ultrafiltration (third term).The solute flow is also driven by an osmotic pres-sure gradient (fourth and fifth terms in equation5b). The sixth terms in both equations describe theadvective flux owing to solid motion.

Constitutive relationships

Important parameters that affect the water uptakeand salt release rates and hydration front are thebitumen membrane efficiency, the permeabilityof the BW, and the diffusion coefficient of dis-solved salt in the BW. These parameters dependon the porosity of the BW, which varies in timebecause of the water uptake and subsequent pro-cesses, that is, dissolution of salt crystals, swellingof hydrating layers and compression of highlyleached layers.

The diffusivity of solutes and water (m2 s21). Thediffusion coefficient controls the diffusion of dis-solved salt and water. The parameter D is con-sidered as an effective diffusion coefficient equalto D ¼ t ff D0 where t is the tortuosity and D0 is




a Fickian diffusion coefficient for a solute in freewater. The effective diffusivity for semipermeablemembranes is defined as D* ¼ (1 2 s)D. There-fore, for media presenting ideal membrane pro-perties, the diffusion of the solute and water iscompletely hindered.

The osmotic efficiency s. The osmotic efficiencycoefficient controls the flow of water and solutesowing to osmotic effects. This parameter describesthe nonideality of membrane behaviour. It rangesfrom 1 for an ideal membrane to 0 for porousmedia having no membrane properties. For clays,it has been shown that the efficiency of the claymembrane depends on several factors such ascation exchange capacity, solute concentration orporosity. For a composite porous medium such asBW, composed of a porous bituminized matrixembedding crystals, Mokni et al. (2011) proposeda simple relationship that describes the decrease ofs when porosity increases:

s = s0

f0

ff

( ); f0 ≤ ff (6a)

This equation shows a rapid decrease in osmoticefficiency with porosity increase (Fig. 2a).

In the present paper, the osmotic efficiency coef-ficient is assumed to be dependent on porosity andon crystal content. The latter dependency hasbeen reported by Valcke et al. (2010), who men-tioned the existence of a threshold salt content atwhich BW loses its membrane properties. Indeed,for high crystal contents, the bitumen membranearound the salt crystals becomes too thin to induceany osmotically driven swelling.

The osmotic efficiency coefficient s is assumedto be a nonlinear function of the porosity (ff) andof the crystal volume fraction (fc):

s = s0

1 − ff

1 − f0

( )n

(1 − fc − ff ); f0 ≤ ff (6b)

where s0 is the reference osmotic efficiency for thereference porosity f0, and n is a material parameterthat controls the rate of decrease of s with increaseof ff and fc. Figure 2a shows the evolution ofthe osmotic efficiency coefficient as a function ofporosity for several values of n. The choice of thisexpression for the membrane efficiency is basedon experimental observations. Indeed, there isexperimental evidence that, even when the layersare partially leached, the membrane efficiency isstill very high (Valcke et al. 2010).

The intrinsic permeability k. The intrinsic permea-bility controls the flow of water owing to hydraulicand osmotic pressure gradients. It also controls theflow of solutes by advection. Mokni et al. (2011)used a Kozeny–Carman-type equation (Fig. 2b) todescribe the evolution of permeability with porosityexpressed as:

k = k0

f3f

(1 − ff )2

(1 − f0)2

f30

(7a)

This equation is useful for soil materials. Never-theless, for a composite porous media such asBW, experiencing a structure evolution induced bydissolution and swelling, an expression that isspecifically applicable to that particular material isneeded. Mokni et al. (2011) performed a sensitivityanalysis on the effect of the initial values of intrin-sic permeability, osmotic efficiency, and diffusioncoefficients on the magnitude and duration ofswelling deformation of BW. It was demonstratedthat these three parameters, dependent on poro-sity, evolve in a coupled way. Indeed, a materialhaving a high value of permeability and diffusionhas a low value of the efficiency coefficient (lessefficient semi permeable membrane).

In this paper, in order to investigate the effectof the rate of change of permeability with porosity,a generalized form of equation (7a) is proposedfor BW.

(a)

(b)

Fig. 2. Evolution of (a) osmotic efficiency coefficientfor different values of n in equation (6b) and for previousfunction used in Mokni et al. (2011) and (b) intrinsicpermeability used in the different analysed cases andKozeny–Carman function used in Mokni et al. (2011).

N. MOKNI ET AL.



k = k0

faf

(1 − ff )2b+2

(1 − f0)2b+2

fa0

(7b)

where k0 is the reference permeability at the refer-ence porosity f0, and a and b are material para-meters between 1 and 2 that control the rate ofincrease of k with ff. Figure 2b shows the evolutionof the intrinsic permeability as a function of porosityfor several values of a and b.

Sensitivity analysis: effect of the rate of

variation of s and k with porosity on

swelling, leached mass of NaNO3 and on

leached thickness of BW

In this section, results from the modelling work ofthe water uptake tests under constant stress con-ditions described in Valcke et al. (2010) using theproposed expressions of s and k (equations 6b &7b) are presented. Several cases are analysed con-sidering different values of n, a and b, parameterscontrolling the rate of change of osmotic efficiencyand intrinsic permeability with porosity, swelling,leached mass of NaNO3 and leached thickness.

For the experimental aspects of these wateruptake tests, the reader is referred to Valcke et al.(2010). Specifically for this paper, it is importantto mention that, prior to contact with water, thesamples in the water uptake cells were submitted torepeated compression–decompression steps (‘com-pressibility test’) to determine the mechanicalproperties of BW. Mokni et al. (2008, 2010a, b)proposed an elasto-viscoplastic constitutive modelto simulate these compression tests in oedometricconditions. The CH formulation, combined withthe elasto-viscoplastic mechanical constitutivemodel (Mokni et al. 2010b) was used to simulatethe water uptake tests. The result of the differentanalyses performed will be compared with theresult of the Base case described in Mokni et al.(2011).

Model domain, boundary conditions and

(initial) parameter values

The model domain consists of two regions (Fig. 3).One region corresponds to a reservoir representing afilter that is saturated with a low NaNO3 concen-tration solution. The initial NaNO3 mass fractionfor the reservoir is w ¼ 0.001 kg NaNO3 kg21 solu-tion (c. 0.01 m). Within the reservoir, the perme-ability is set to high values (k ¼ 10216 m2). Theother region corresponds to the BW containingNaNO3 crystals occupying 16% of the totalvolume of the BW (corresponding to 28 wt%).

Pure water inflow, brine outflow and salt diffusionthrough the reservoir are allowed at the upper andlower sides of the sample. The medium is con-sidered to be saturated (Sr ¼ 1) with NaNO3 brine(initial mass fraction w ¼ 0.47 kg NaNO3 kg21 sol-ution (c. 7.46 m)).

The different cases were simulated under aconstant vertical stress. Along the vertical bound-aries of the domain, horizontal displacements arerestricted to represent the oedometric conditions.Material properties are summarized in Table 1.The initial values for permeability, osmotic effi-ciency and diffusion coefficient are the samereference values used by Mokni et al. (2011) andMarien et al. (2013). The mechanical parameterswere derived from the calibration of oedometercompressibility tests that are presented in Mokniet al. (2010b).

Effect of a and b on swelling and leached

mass of NaNO3

Table 2 summarizes the different cases analysed.The transport functions (permeability and osmoticefficiency) used for the different cases are displayedin Figure 2b.

Fig. 3. Schematic representation of the model domain.




Case 1. In this case the Kozeny–Carman type per-meability function (equation 7a) used by Mokniet al. (2011) is maintained, while for the osmoticefficiency coefficient the new expression (equation

6b) that describes the dependency of s on ff andfc is used. Several values of n are considered.The evolution of volumetric deformation (i.e. swel-ling) is displayed in Figure 4. The plot shows

Table 1. Material properties

Parameter (initial values) Symbol Value

Crystal volume fraction fc00.16

Porosity f f00.01

Intrinsic permeability k0 (m2)k (m2)

BW 3 × 10226

Filter 10216

Efficiency coefficient s0 0.95Diffusion coefficient D* (m2 s21) 1.6 × 10216

Dissolution rate constant k (kg s21 m23) 1025

Solute mass fraction wsl (kg NaNO3 kg21 solution) BW 0.47

Filter 0.001

Initial values of transport parameters used for the different cases.BW, Eurobitum bituminized radioactive waste.

Table 2. Transport functions used for the different cases

Analysed cases Constitutive relationships

Case 1*k = k0

ff3

(1 − ff )2

(1 − f0)2

f03

; s = s0

1 − ff

1 − f0

)n(; n ¼ 2,3,5

Case 2 k = k0

ff

(1 − ff )6

(1 − f0)6

f0

; s = s0

1 − ff

1 − f0

( )n

; n ¼ 2,3,8

Case 3 k = k0

ff2

(1 − ff )4

(1 − f0)4

f02

; s = s0

1 − ff

1 − f0

( )2

; n ¼ 2

*Kozeny–Carman function used in Mokni et al. (2011).

Fig. 4. Case 1: swelling (or volumetric deformation) for different values of n (equation 6b).

N. MOKNI ET AL.



comparisons between the experimental data and thenumerical simulations (at constant vertical appliedstress, 2.2 MPa). It is observed that the model pre-dicts lower swelling rates in comparison to theexperimental results. Indeed, in the CH formulation,water uptake, swelling and leaching are consid-ered to be the result of several transient fully cou-pled processes (diffusion, advection, dissolution,involving a low permeability material with evolvingthickness and properties) that are strongly nonlinear.The overall time-dependence of the swelling andleaching rates and rate of progression of the hydra-tion front depend on the relative contribution of allthese processes and on their time constants. Infact, the variation of the intrinsic permeability andosmotic coefficient plays an important role in thecoupled migration of water and solute. While

the direct advective fluxes are proportional to k,the coupled fluxes are proportional to the productks (equation 5). It is convenient to define at thispoint the following transport functions: kp/m,kps/m and kps2/m. These functions will be com-pared with the diffusion coefficient D* to identifythe major processes that control the transport ofwater and solute. Figure 5 shows the evolutionwith porosity of the above-defined functionstogether with the diffusion coefficient D* forcase 1 (Fig. 5a) and for the base case presented byMokni et al. (2011) (Fig. 5b). The plots indicatethat, for case 1, the transport functions controllingthe swelling and leaching rates are significantlydifferent from the base case. Indeed, for case 1 thetransport functions are similar and evolve with por-osity at similar rates. In addition, higher values and

Fig. 5. Variation with porosity of the transport functions kp/m, kps/m, kps2/m and D* for (a) case 1 and (b) base case(Mokni et al. 2011).




evolution rate are observed for the transport func-tion kps2/m in comparison to the base case. As aresult there is rapid leaching of the outermostlayers of the BW sample and a rapid decrease inNaNO3 concentration and therefore in osmoticpressure in these layers owing to the higher waterand solute flow rates. The transport of NaNO3

solutes out of the BW, combined with the externalstress and the osmosis-driven transport of waterfrom the outermost pores with a lower NaNO3 con-centration (higher water activity) towards deeperpores with a higher NaNO3 concentration (lowerwater activity), contributes to a local consolidationof the hydrated layers. Swelling of the deeperlayer is then counteracted by compression of theleached layers leading to a fast levelling off of theleaching and swelling rates.

Case 2. In this case the new permeability (equation7b) and osmotic efficiency (equation 6b) func-tions are used (Table 2). The used permeabilityfunction predicts a slower increase of the per-meability with increasing porosity compared withthe base case (Fig. 2b). Several values of the para-meter n, which controls the rate of decrease of theosmotic efficiency coefficient with porosity, areconsidered. The same initial values of the transportfunction as for the previous case have been used(Table 1). The evolution of volumetric deformationis displayed in Figure 6. Experimental and numeri-cal results concerning the sodium nitrate releaseare shown in Figure 7. In this case the model repro-duces well the swelling and leaching rates. Severalobservations can be made. First, it is seen that,after more than 4 years of hydration, the samples

Fig. 6. Case 2: swelling (or volumetric deformation) for (a) different applied stresses and (b) different values of n.Experimental results and model predictions.

N. MOKNI ET AL.



continued to swell with no tendency to level off.Both modelling and experiments were performedunder various vertical applied stresses (2.2, 3.3,and 4.4 MPa). For this range of stresses, the effecton swelling and leaching rates is moderate accord-ing to the model. The effect of stress level on swel-ling was discussed by Mokni et al. (2011) in regardto the maximum swelling pressure achievabletaking into account the high osmotic pressure asso-ciated with saturated sodium nitrate pore solution.Second, no significant effect of the variation of thevalue of n is observed on the swelling and leachingrates (Figs 6b & 7). Figure 8 shows the evolution

with porosity of the above-defined transport func-tions together with the diffusion coefficient D* forseveral values of the parameter n. The plot showsthat, in both cases (i.e. cases of n ¼ 8 and n ¼ 2)for low porosity, diffusion is the dominant processfor the transport of water and solute. As porosityincreases, permeability increases and the directand coupled advective fluxes become the dominantprocesses. For both cases, the transport parameterscontrolling the swelling and leaching rates are sig-nificantly different at high porosity values, whilemore similar values are observed for low porosi-ties. The fact that no effect of n on the swelling

Fig. 7. Case 2: leached mass of NaNO3. Experimental results and model predictions.

Fig. 8. Case 2: variation with porosity of the transport functions kp/m, kps/m and kps2/m and D*, which control themagnitude of the direct and coupled fluxes of water and solute (for a given osmotic and hydraulic pressure gradients;p ¼ 42 MPa and m ¼ 1029 MPa s).




and leaching rates is observed demonstrates thatboth are controlled by the fluxes in the outermostlow porosity layer. Compression of the outermostlayers is then a limiting or controlling process forthe water uptake and the release of NaNO3.

Case 3. In this case, the rate of increase of per-meability with porosity is higher than in case 2 butlower than in case 1 (Fig. 2a; Table 2). The samematerial parameters as for the previous cases havebeen used (Table 1). Modelling results of swell-ing and NaNO3 leaching of the samples in the con-stant stress test (sv ¼ 2.2 MPa) leads to lowerswelling and leaching rates in comparison to theexperimental results (Fig. 9). In fact, in this casethe outermost layers are predicted to leach quiterapidly owing to the higher water and solute flow

rates inducing a fast decrease in NaNO3 concen-tration and in osmotic pressure in these layers. Theosmotic pressure would then rapidly approach thevalue of the externally applied stress, which wouldlead to compression of the hydrated layers.

Effect of a and b on leached thickness

of BW and hydration front

The progression of the water front is the result of therelative contribution of several transient processes(diffusion, advection and dissolution) involving alow porosity and low permeability material withevolving thickness and properties and that arestrongly coupled. Valcke et al. (2010) and Marienet al. (2013) observed that almost no pores can beseen on ESEM images of the dry BW. Indeed,

Fig. 9. Case 3. (a) Volumetric deformation v. time. (b) Leached mass of NaNO3. Experimental results andmodel predictions.

N. MOKNI ET AL.



in contact with water the NaNO3 salt crystals at theouter boundary of the material will dissolve rapidly.This results in the formation of pores filled with ahighly saturated NaNO3 solution. As a consequencepermeability and diffusivity increase but most of thewater that is taken up further by the BW stays in thefirst layer of pores to dissolve the remaining crystalsand in a next strep to dilute the saturated NaNO3 sol-ution. Profiles of crystal volume fraction (Fig. 10a)allow the thickness of the leached zones to be calcu-lated. Table 3 summarizes the calculated andmeasured leached thicknesses (obtained by mCTand ESEM analyses; Valcke et al. 2010) at differenttimes for the different analysed cases. For case 2, themodel predicts a slower progress of the water frontin comparison to the base case presented in Mokniet al. (2011). However, even for this slower progressof the water front, there is still a difference betweenthe modelled results and the mCT analyses, whichshow that after about c. 1500 days of hydration thewater front had not yet reached the centre of thesample (Table 3). For case 3, the model predicts a

slower dissolution of the crystals in the centre ofthe sample and a slower progression of the hydra-tion front compared with the base case and case 2(Table 3). Indeed, except for the outermost layers,the largest part of the sample is not hydrated after2000 days. This is a direct consequence of thesize reduction of the outer pores, which induces adecrease in the permeability and the diffusion coef-ficient, and an increase of the osmotic efficiencyin these compressed layers, while deeper in thesample these parameters are varying in the oppositeway. This maintains a very slow inflow of watertowards the sample and outflow of the solute overa long time.

Analysis of the effect of heterogeneous

distribution of porosity on leached thickness

of BW and hydration front

Visual examination of the mCT images after dif-ferent hydration periods shows irregular shape of

00.020.040.060.08

0.10.120.140.16

0 0.005 0.01

Cry

stal

vol

ume

frac

tion

(-)

Thickness (m)

Base case (Mokni et al.,2011)Case 2Heterogenous case

(a) (b)

Fig. 10. (a) Profiles of crystal volume fraction after 1500 days of hydration for case 2, heterogeneous case and base case(Mokni et al. 2011). (b) Heterogeneous distribution of porosity (each finite element is characterized by a single value ofporosity).

Table 3. Calculated and measured leached thickness (note that these data relate to the total thickness, that is,the sum of the thicknesses of the two leached layers per sample)

Duration(days)

Leachedthickness

(mm)mCT

Leachedthicknesscalculated

(mm) (case 2)

Leachedthicknesscalculated

(mm) (case 3)

Leached thicknesscalculated (mm)(heterogenous

case)

Leached thicknesscalculated (base

case) (mm)(Mokni et al. 2011)

c. 1500 6.29 9 6.4 c. 6.62 *

*Hydration front has already reached the centre of the sample.




the hydration front (Fig. 1a), which can be attributedto pre-existing heterogeneities inside the BW lead-ing to local variation of permeability, diffusivityand osmotic efficiency of the material. This hetero-geneity may originate from the distribution of crys-tals, which are often grouped together in largercrystal clusters, generating zones with variable saltcontent and porosity (Fig. 1b).

In an additional effort to more realisticallysimulate the progression of the wetting front,some heterogeneity of the transport properties hasbeen considered. The same transport function asin case 2 are used (Table 2, n ¼ 2) since they

lead to good estimates of swelling and leachingrates. Figure 10b shows the initial porosity fieldconsidered. The mean porosity is fmean ¼ 0.01(same initial porosity as the homogenous case:case 2) and the variance is in the order of 1025.This random field has no spatial correlation. Therange of initial porosity considered is justified bythe existence of only a few isolated pores withdifferent sizes observed in ESEM images of dryBW (Valcke et al. 2010; Marien et al. 2013).Note that, owing to the small variations in porosityconsidered, initial values of permeability, osmoticefficiency and diffusivity are different for each

Fig. 11. Heterogeneous case. (a) Volumetric deformation v. time. (b) Leached mass of NaNO3. Experimental resultsand model predictions.

N. MOKNI ET AL.



finite element. The profile of crystal volume frac-tion drawn after 1500 days for the heterogeneouscase is shown in Figure 10a. When comparing thethickness of the hydrated layer with the experi-mental results, the heterogeneous case shows acomparable result (Table 3). Figure 11 shows theevolution of swelling and leached mass of NaNO3

with time. Although the model predicts a slowerswelling and leaching rates as compared withcase 2, induced by the marked heterogeneity inthe initial values of the transport properties, theresults are still in the range of the experimentalmeasurements.

CH modelling of the in-situ disposal of BW

According to the present disposal design ofONDRAF/NIRAS, ten 220 l drums, each filledwith on the average 170 l of Eurobitum, would begrouped in two arrays of five drums in cement-basedsecondary containers, which are to be placed inconcrete lined disposal galleries that are exca-vated at mid depth in the host formation (Fig. 12a;Wacquier & Van Humbeeck 2009). The voidspace between the primary waste packages and thesecondary concrete container, and between the sec-ondary concrete container and the gallery liner, will

Fig. 12. (a) Axial cross section of the monolith for Eurobitum (Wacquier & Van Humbeeck, 2009). (b) 2D Finiteelement model.

Table 4. Initial values of transport functions for the different materials considered in the in-situ case

Parameters (initial values) Symbols Values

BW Crystal volume fraction fc0 0.16Porosity ff0 0.01Intrinsic permeability k0 (m2)

ab

3 × 10226

12

Efficiency coefficient s0 0.95n 2

Diffusion coefficient D (m2 s21) 1.6 × 10216

Dissolution rate constant k (kg s21 m23) 1025

Solute mass fraction wsl 0.47

Boom Clay Intrinsic permeability K (m2) 10219


Concrete container+liner Intrinsic permeability K (m2) 10218


Backfill Intrinsic permeability K (m2) 10216





be backfilled with cement-based materials. It isimportant to recognize that several aspects of theEurobitum disposal system are not yet finalized, inparticular the composition of the concrete secondarycontainer, the presence or the absence of rein-forcement bars in the secondary concrete containedand the composition of the cement based backfillmaterial. The design provided in Figure 12a isbased on the latest information and assumptions.

In the frame of the study of the long-term behav-iour of Eurobitum bituminized radioactive wastein final disposal conditions, a preliminary hydro-chemical analysis is presented here. The objective

is to obtain first insights into the kinetics of wateruptake, dissolution, solute leaching and maximumgenerated swelling and pressure under disposalconditions. A 2D finite element model is used con-sidering a section of the gallery taken perpen-dicularly to the axial direction (Fig. 12b). Thecementitious backfill, the concrete container, theconcrete liner and the Boom Clay are consideredto be saturated with water with low NaNO3 concen-tration (initial mass fraction is w ¼ 0.001 kg kg21).The BW filling the drums contains NaNO3 crystalsoccupying 16% of the total volume of the BW (cor-responding to 28% in weight) and is considered to

Fig. 13. Contours of crystal volume fraction (a) and porosity in BW drums (b) after 0.01 days, and 5 and 100 years

Fig. 14. Contour field of liquid pressure in BW drums after 5, 20 and 100 years.

N. MOKNI ET AL.



be saturated with NaNO3 brine (initial mass frac-tion: w ¼ 0.47 kg kg21). The same initial valuesfor permeability, osmotic efficiency and diffusioncoefficient for BW as for case 2 are used for thisanalysis. Detailed materials parameters are sum-marized in Table 4.

Figure 13 shows the contour field of crystalvolume fraction after, 0.01 days, and 5 and 100years together with contours of porosity. At earlytime, owing to high concentration gradient, wateris attracted from the backfill to the BW. After 5years, the crystal volume fraction (fc) decreasesand tends to zero at the outer boundary of thedrums, while no change in fc is observed in themiddle of the drum after 100 years (Fig. 13a). Thissuggests that it will take several hundred yearsbefore the hydration front has reached the centreof the drums. As a consequence of crystal dissol-ution, porosity increases at the hydrated layers(Fig. 13b). No compression of the outer hydratedboundary is observed since no mechanical coupl-ing is considered. Figure 14 shows contour fieldof liquid pressure after 5, 20 and 100 years. Thefigure shows that there is a pressure front that pro-gresses slowly towards the centre of the drums.After 5 years the maximum value is attained at theouter boundary of the BW drums and is around31 MPa. A decrease of the maximum value of liq-uid pressure is also observed after 100 year at theouter hydrated layers. In fact, the increase of poros-ity at this zone induces an increase of permeabilityand diffusion coefficient and a decrease of effi-ciency of the membrane. Drainage and solute diffu-sion out of the BW are favoured, resulting in adecrease in osmotic pressure and therefore in porepressure. Figure 15a shows contours field of con-centration after 100 years in BW. Except for the

outer boundary where the crystals have dissolved,the concentration of the NaNO3 solution insidethe BW drums remains constant and equal to thatof a saturated NaNO3 solution after 100 years. Infact, the crystals provide a source term of solutesthat prevent the decrease of the concentration. Con-tours field of concentration through the backfillmaterial, liner, concrete container and Boom Clayafter 100 years are shown in Figures 15b and c.Because the BW is not a perfect semipermeablemembrane, the solute is allowed to diffuse out ofBW and the NaNO3 concentration increases in thesurrounding materials.

Conclusion

In this paper, equations of the key transport vari-ables in the CHM formulation proposed by Mokniet al. (2011) are improved in order to better repro-duce some of the experimental results. In thiscontext a new expression of the osmotic efficiencycoefficient (s), including the dependence of s onporosity and on crystal volume fraction, has beenproposed. A new expression describing BW per-meability evolution with porosity has been pre-sented as well. The modelling results illustrate thata and b, parameters controlling the rate of changeof intrinsic permeability with porosity, have a sig-nificant influence on swelling and leaching ratesand on progression of the hydration front. The pro-posed functions are interdependent and controlthe magnitude of the coupled fluxes, namely chemi-cal osmosis and ultrafiltration. In addition, it wasdemonstrated that assuming a simple realizationof random fields of porosity that is of transportproperties (permeability, osmotic efficiency and

Fig. 15. Contour field of NaNO3 mass fraction in the liquid phase after 100 years in (a) BW, (b) backfill, concretecontainer and liner and (c) Boom Clay.




diffusivity) improves the modelling results in termsof progression of the hydration front. Indeed, in thiscase, the calculated and the measured thicknesses ofthe leached layers, after 1500 days of hydration,were quite similar. This approach implies differentmagnitude of direct and coupled fluxes at eachfinite element in the domain. Accounting for spatialand temporal variations of the key transport par-ameters was essential to reproduce all the aspectsthat were observed in the experiments (swelling,leaching of sodium nitrate and thickness of theleached layer). Finally, the hydro-chemical analysisof the behaviour of Eurobitum drums under dis-posal conditions provided first insights into the kin-etics of water uptake, dissolution, solute leachingand maximum generated pressure under disposalconditions.

This work was undertaken in close cooperation with, andwith the financial support of ONDRAF/NIRAS, the Bel-gian Agency for the Management of Radioactive Wasteand Enriched Fissile Materials, as part of its programmeon geological disposal of medium-level long-lived waste.

References

Marien, A., Mokni, N., Valcke, E., Olivella, S.,Smets, S. & Li, X. 2013. Osmosis-induced wateruptake by Eurobitum bituminized radioactive wasteand pressure development in constant volume con-ditions. Journal of Nuclear Materials, 432, 348–365.

Mokni, N. 2011. Deformation and Flow Induced byOsmotic Processes in Porous Materials. PhD thesis,Universitat Politecnica de Catalunya, Espana.

Mokni, N., Olivella, S., Li, X., Smets, S. & Valcke, E.2008. Deformation induced by dissolution of salts inporous media. Physics and Chemistry Earth, PartsA/B/C, 33(Supplement 1), S436–S443.

Mokni, N., Olivella, S. & Alonso, E. E. 2010a.Swelling in clayey soils induced by the presence ofsalt crystals. Applied Clay Science, 47, 105–112.

Mokni, N., Olivella, S., Li, X., Smets, S., Valcke, E. &Marien, A. 2010b. Deformation of bitumen basedporous material: experimental and numerical analysis.Journal of Nuclear Materials, 404, 144–153.

Mokni, N., Olivella, S., Valcke, E., Marien, A.,Smets, S. & Li, X. 2011. Deformation and flowdriven by osmotic processes in porous materials: appli-cation to bituminized waste materials. Transport inPorous Media, 86, 665–692.

Olivella, S., Gens, A., Carrera, J. & Alonso, E. E.1996. Numerical formulation for a simulator(CODE_BRIGHT) for the coupled analysis of salinemedia. Engineering with Computers, 13, 87–112.

ONDRAF/NIRAS 2009. The Long-term Safety Assess-ment Methodology for the Geological Disposalof Radioactive waste. ONDRAF/NIRAS reportNIROND TR-2009-14 E, www.nirond.be

Valcke, E., Marien, A., Smets, S., Li, X., Mokni, N. &Olivella, S. 2010. Osmosis-induced swelling ofEurobitum bituminized radioactive waste in constantstress conditions. Journal of Nuclear Materials, 406,304–316.

Wacquier, W. & Van Humbeeck, H. 2009. B&C Con-cepts and Open Questions. ONDRAF/NIRAS Techni-cal note, 2009-0146.

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