Modelling and Numerical Simulation of Gas …web.math.pmf.unizg.hr/~jurak/BJS-CG-final.pdfModelling and Numerical Simulation of Gas Migration in a Nuclear Waste Repository 3 From deﬁnition

Computational Geoscience manuscript No.(will be inserted by the editor)

Modelling and Numerical Simulation of Gas Migration in a NuclearWaste Repository

Alain Bourgeat · Mladen Jurak · Farid Smaı

January 19, 2012

Abstract We present a compositional compressible two-phase,liquid and gas, flow model for numerical simulations of hy-drogen migration in deep geological radioactive waste repos-itory. This model includes capillary effects and gas diffu-sivity. The choice of the main variables in this model, totalor dissolved hydrogen mass concentration and liquid pres-sure, leads to a unique and consistent formulation, no matterwhether there is a gas phase or not. After introducing andexplaining our model, we show computational evidences ofits adequacy to simulate gas phase appearance and disap-pearance in different but typical situations for gas migrationin an underground radioactive waste repository.

Keywords Two-phase flow, compositional flow, porousmedium, underground nuclear waste management, gasmigration

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Modeling Physical Assumptions . . . . . . . . . . . . . . . 23 Liquid Saturated/Unsaturated state; a general formulation . 44 Numerical experiments . . . . . . . . . . . . . . . . . . . . 75 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . 15

Alain BourgeatUniversite de Lyon, Universite Lyon1, CNRS UMR 5208 InstitutCamille Jordan, F - 69200 Villeurbanne Cedex, France

Mladen JurakFaculty of Science, University of Zagreb, Bijenicka 30, Zagreb, Croa-tia,E-mail: [email protected]

Farid SmaıUniversite de Lyon, Universite Lyon1, CNRS UMR 5208 InstitutCamille Jordan, F - 69200 Villeurbanne Cedex, France

1 Introduction

The simultaneous flow of immiscible fluids in porous me-dia occurs in a wide variety of applications. The most con-centrated research in the field of multiphase flows over thepast four decades has focused on unsaturated groundwaterflows, and flows in underground petroleum reservoirs. Mostrecently, multiphase flows have generated serious interestamong engineers concerned with deep geological repositoryfor radioactive waste, and for CO2 capture and storage simu-lations (CSS). For radioactive waste geological repositoriesin Europe, which are in Argillaneous or Granitic rocks, thereis a growing awareness that the effect of hydrogen gas gen-eration, due to anaerobic corrosion of the radioactive wastepackages steel engineered barriers (carbon steel overpacksand stainless steel envelopes), can affect all the functionsallocated to the canisters or to the buffers and even to thebackfill (see [18], [31], [35]). The host rock safety functionmay even be threaten by overpressurisation, [7], leading toopening fractures in the host rock and inducing groundwa-ter flow and transport of radionuclides outside of the wastesite boundaries. Our ability to understand and predict un-derground gas migration is crucial in the designing and theperformance assessment of any reliable geological nuclearwaste storage.

In nuclear waste management, the migration of gas throughthe near field environment and the host rock, involves twocomponents, water and pure hydrogen H2; and two phases:“liquid” and “gas”. Due to the inherent complexity of thephysics, equations governing this type of flow in porous me-dia are nonlinear and coupled. Moreover the geometries andmaterial properties characterizing many applications can bequite irregular and contrasted. As a result of all these diffi-culties, numerical simulation often offers the only viable ap-proach to modeling transport and multiphase flows in porousmedia.

2 Alain Bourgeat et al.

An important consideration, in the modeling of fluid flowwith mass exchange between phases, is the choice of theprimary variables that define the thermodynamic state ofthe system. When a phase appears or disappears, the setof appropriate thermodynamic variables may change. Thereare two different approaches to that problem. The first one,widely used in simulators such as TOUGH2, [34], relies on aprimary variable substitution algorithm. This algorithm usesin two-phase conditions the appropriate variables like pres-sure and saturation, and, when a transition to single-phaseconditions occurs, it switches to new variables adapted to theone-phase conditions, like pressure and concentration. Thisvariable substitution is done after each Newton iteration ac-cording to some “switching criteria”, see [13,19,32,36]. Adifferent presentation of this approach was done recently in[22,24], where the solubility conditions are formulated ascomplementary conditions which complement the conser-vation law equations. The whole system is then solved by asemismooth Newton method as in [16,37], which consistsin working on an intermediate active node set (see [24]).The second possibility is to use a set of primary “persistent”variables, such as pressure and component density, whichwill remain well defined when phase conditions change, sothat they can be used throughout the single and two-phaseregions, like in [8,3].

This paper addresses the problem of the phase appear-ance/disappearance through a single set of persistent vari-ables, well adapted to heterogeneous porous media, whichdoes not degenerate and hence could be used, without re-quiring switching the primary variables, as an unique formu-lation for both situations: liquid saturated and unsaturated.We will demonstrate through four numerical tests, the abilityof this new formulation to actually cope with the appearanceor/and disappearance of one phase in simple, but typical andchallenging situations like the ones we met in undergroundradioactive waste repository simulations.

Although the application we had in view for this modelwas the gas migration in geological radioactive waste repos-itories; we are aware that the very same problem of phaseappearance and disappearance is also crucial in modellingthe recently discussed technology of Carbon Capture andStorage (see for instance [14,15]).

2 Modeling Physical Assumptions

We consider herein a porous medium saturated with a fluidcomposed of 2 phases, liquid and gas, and according to theapplication we have in mind, we consider the fluid as a mix-ture of two components: water (only liquid) and hydrogen(H2, mostly gas) or any gas with similar thermodynamicalproperties. In the following, for sake of simplicity we willcall hydrogen the non-water component and use indices wand h for the water and the hydrogen components.

According to our goal, which was to focus on the phaseappearance and disappearance phenomena, we have doneseveral simplifying assumptions which are not essential forunderstanding our approach. Not doing these assumptionswould not have affected neither our choices of primary vari-ables nor the conclusions, but they would have considerablycomplicated the present paper.

– The porous medium is assumed to be in thermal equi-librium. This hypothesis can be questionable in the caseof application to nuclear waste repository where heat isgenerated by the nuclear waste, but, as argued in [7], thenear field thermal characteristic time is usually smallerthan the corrosion time, so that the most of the hydrogenproduction take place when the system is close to ther-mal equilibrium. Hence, although thermal flux and en-ergy conservation could be taken easily in account, forsimplicity, they will not be discussed herein and we willconsider only isothermal flows.

– After restoring thermal equilibrium in the repository andresaturation of the clay engineered barriers (in severalhundred years, [7]), the water pressure will be sufficientlyhigh compared to the water vapour pressure and the wa-ter vaporization can be safely neglected.

– Although at the depth of some storages, the water den-sity could be affected by the pressure, we suppose forsimplicity in our presentation that the water componentis incompressible. For the very same reasons, the porousmedium is supposed rigid, meaning that the porosity Φ

is only a function of the space variable Φ = Φ(x).– We are assuming that the gas flow can be described by

the generalized two-phase Darcy’s law, and we are nottaking in account the possibility, in clayey rocks of hav-ing the gas transported by other mechanism, see [29].

The two phases are denoted by indices, l for liquid, and gfor gas. Associated to each phase α ∈ l,g, we have, in theporous medium, the phase pressures pα , the phase satura-tions Sα , the phase mass densities ρα and the phase volu-metric fluxes qα . The phase volumetric fluxes are given bythe Darcy-Muskat law (see [33,4])

ql =−K(x)λl(Sl)(∇pl−ρlg) ,qg =−K(x)λg(Sg)(∇pg−ρgg) ,

(1)

where K(x) is the absolute permeability tensor, λα(Sα) isthe α−phase relative mobility function, and g is the grav-ity acceleration; Sα is the effective α−phase saturation andthen satisfies:

Sl +Sg = 1. (2)

Pressures are connected through a given capillary pressurelaw (see [25,5])

pc(Sg) = pg− pl . (3)

Modelling and Numerical Simulation of Gas Migration in a Nuclear Waste Repository 3

From definition (3) we notice that pc is a strictly increasingfunction of gas saturation, p′c(Sg)> 0, leading to a capillaryconstraint:

pg > pl + pc(0), (4)

where pc(0) ≥ 0 is the capillary curve entry pressure (seeFigure 2).

The water component and the gas component which arenaturally in liquid state and in gas state at standard condi-tions are also denoted respectively solvent and solute. Wewill assume herein, for simplicity that the mixture containsonly one solvent, the water and one gas component, the hy-drogen.

Writing all the quantities relative to one component withthe superscript i ∈ w,h, we define then Mi as the molarmass of the i−component and ρ i

α , ciα , X i

α , as, respectively:the dissolved mass and the dissolved molar densities, themolar fraction, of the i−component in the α−phase, α ∈l,g. All these quantities satisfy :

ρiα = Mici

α , X iα =

ciα

cα

,

ρα =h

∑k=w

ρkα , cα =

h

∑k=w

ckα .

(5)

As said before, in the gas phase, we neglect the water vapor-ization and we use the ideal gas law (see [17]):

ρg =Cv pg, (6)

with Cv = Mh/(RT ), where T is the temperature, R the uni-versal gas constant.

Mass conservation for each component leads to the fol-lowing differential equations:

Φ∂

∂ t(Slρ

wl )+div

(ρ

wl ql + jw

l

)= F w, (7)

Φ∂

∂ t

(Slρ

hl +Sgρg

)+div

(ρ

hl ql +ρgqg + jh

l

)= F h, (8)

where the phase flow velocities, ql and qg, are given by theDarcy-Muskat law (1), F k and jk

l , k ∈ w,h, are respec-tively the k−component source terms and the diffusive fluxin the liquid phase (see eq. (13)).

Assuming water incompressibility and independance ofthe liquid volume from the dissolved hydrogen concentra-tion, we may assume the water component concentration inthe liquid phase to be constant, i.e.:

ρwl = ρ

stdw , (9)

where ρstdw is the standard water mass density. The assump-

tion of hydrogen thermodynamical equilibrium in both phasesleads to equal chemical potentials in each phase: µh

g (T, pg,Xhl )=

µhl (T, pl ,Xh

l ). Assuming that in the gas phase there is onlythe hydrogen component and no water, leads to Xh

g = 1,

ρhl

pl

ρhl=

C h(p l

+pc(0))

Sg = 0

ρhl ≤ Ch(pl + pc(0))

Sg > 0

ρhl = Chpg

≥ pl + pc(0)

pg = pl + pc(Sg)

Fig. 1 Phase diagram: Henry’s law; localization of the liquid saturatedSg = 0 and unsaturated Sg > 0 states

and then, from the above chemical potentials equality, wehave a relationship pg = F(T, pl ,Xh

l ). Assuming that the liq-uid pressure influence could be neglected in the pressurerange considered herein and using the hydrogen low sol-ubility, ρh

l ρwl = ρstd

w , we may then linearise the solu-bility relation between pg and Xh

l , and obtain the Henry’slaw pg = KhXh

l , where Kh is a constant specific to the mix-ture water/hydrogen and depends on the temperature T (see[17]). Furthermore, using (9) and the hydrogen low solubil-

ity, the molar fraction, Xhl , reduces to ρh

l Mw

ρstdw Mh (see eqs. (9)-

(11) in [8]) and the Henry’s law can be written as

ρhl =Ch pg, (10)

where Ch = HMh = ρstdw Mh/(MwKh); with H, the Henry’s

law constant.

Remark 1 On the one hand the gas pressure obey the Capil-lary pressure law (3) with the constraint (4), but on the otherhand it should also satisfy the local thermodynamical equi-librium and obey a solubiliiy equation like the Henry’s law(10). More precisely if there are two phases, i.e. if the dis-solved hydrogen mass density, ρh

l , is sufficiently high to leadto the appearance of a gas phase (Sg > 0), we have from (10)and (3) :

ρhl =Ch(pl + pc(Sg)). (11)

Moreover, Sg > 0 with the capillary constraint (4) and theHenry’s law (10), give the solubility constraint:

ρhl >Ch(pl + pc(0)). (12)

But if the dissolved hydrogen mass density, ρhl , is smaller

than the concentration threshold (see Figure 1), then thereis only a liquid phase (Sg = 0) and none of all the relation-ships (3) or (12), connected to capillary equilibrium, appliesanymore; we have only Sg = 0, with ρh

l ≤Ch pg.


The concentration threshold line, ρhl =Ch(pl + pc(0)) in

the phase diagram, is then separating the one phase (liquidsaturated) region from the two phase (unsaturated) region.

The existence of a concentration threshold line can alsobe written as unilateral conditions:

0≤ Sg ≤ 1, 0≤ ρhl ≤Ch pg, Sg(Ch pg−ρ

hl ) = 0;

which could be added to the conservation laws (7)-(8), andsolved conjointly at each time step by means of a semi-smooth Newton’s method, as explained in the Introductionand in [22] or [24].

Since hydrogen is highly diffusive we include the dis-solved hydrogen diffusion in the liquid phase. The diffu-sive fluxes in the liquid phase are given by the Fick’s lawapplied to Xw

l and to Xhl , the water component and the hy-

drogen component molar fractions (see eqs. (12) and (13) in[8]). Using the same kind of approximation as in the Henry’slaw, based on the hydrogen low solubility, we obtain, for thediffusive fluxes in this binary mixture (see Remark 2 andRemark 3 in [8] and see [6]):

jhl =−ΦSlD∇ρ

hl , jw

l =−jhl , (13)

where D is the hydrogen molecular diffusion coefficient inthe liquid phase, possibly corrected by the tortuosity factorof the porous medium (see [23,34]).

If both liquid and gas phases exist, (Sg 6= 0), the porousmedia is said unsaturated, then the transport model for theliquid-gas system can be obtained from (1), (7)-(8), usingequations (5), (6), (9), and (13),

Φρstdw

∂Sl

∂ t+div

(ρ

stdw ql− jh

l

)= F w, (14)

Φ∂

∂ t(Slρ

hl +Cv pgSg) (15)

+div(

ρhl ql +Cv pgqg + jh

l

)= F h,

ql =−Kλl(Sl)(

∇pl− (ρstdw +ρ

hl )g), (16)

qg =−Kλg(Sg)(∇pg−Cv pgg) , (17)

jhl =−ΦSlD∇ρ

hl . (18)

But in the liquid saturated regions, where the gas phasedoesn’t appear, Sl = 1, the system (14)–(18) degenerates to:

div(

ρstdw ql− jh

l

)= F w, (19)

Φ∂ρh

l∂ t

+div(

ρhl ql + jh

l

)= F h, (20)

ql =−Kλl(1)(

∇pl− (ρstdw +ρ

hl )g), (21)

jhl =−ΦD∇ρ

hl . (22)

3 Liquid Saturated/Unsaturated state; a generalformulation

As recalled in the Introduction a traditional choice for theprimary unknowns, in modeling two phase flow and trans-port process, is the saturation and one of the phases pres-sure, for example Sg and pl . But as seen above, in (19)–(22),saturation is no longer a consistent variable in saturated re-gions and this set of unknowns cannot describe the flow ina region where there is only one phase (see [36]). In thissection, we recall and compare two possible choices of pri-mary variables presented in [8] for circumvent this difficultproblem.

3.1 Modeling based on the total hydrogen mass density, ρhtot

To solve this problem, instead of using the gas saturation Sgwe have proposed, in [8], to use ρh

tot , the total hydrogen massdensity, defined as:

ρhtot = Slρ

hl +Sgρ

hg . (23)

Then, defining

a(Sg) =Ch(1−Sg)+CvSg ∈ [Ch,Cv]; (24)

with

a′(Sg) =Cv−Ch =C∆ > 0, (25)

since Cv > Ch, from the assumption of weak solubility; wemay rewrite the total hydrogen mass density, ρh

tot , defined in(23), as:

ρhtot =

a(Sg)(pl + pc(Sg)) if Sg > 0ρh

l if Sg = 0.(26)

As noticed in the previous section in Remark 1, using themonotonicity of functions pc(Sg) and a(Sg), we see a con-centration threshold corresponding to Ch(pl + pc(0)) sepa-rating the liquid saturated zone, ρh

tot ≤Ch(pl + pc(0)), fromthe unsaturated zone, ρh

tot >Ch(pl + pc(0)).With this choice of primary variables, ρh

tot and pl , thetwo systems of equations (14)–(18) and (19)–(22) reduce toa single system of equations:

Φρstdw

∂Sl

∂ t−div

(ρ

stdw Kλl(Sl)

(∇pl− (ρstd

w +ρhl )g))

+div(

ΦSlD∇ρhl

)= F w,

(27)

Φ∂ρh

tot

∂ t−div

(ρ

hl Kλl(Sl)

(∇pl− (ρstd

w +ρhl )g))

−div(

Cv pgKλg(Sg)(∇pl +∇pc(Sg)−Cv pgg))

−div(

ΦSlD∇ρhl

)= F h.

(28)


If we want to study the mathematical properties of the opera-tors in this system of equations, we should develop the abovesystem of equations using first the dependency of the sec-ondary variables Sg = Sg(pl ,ρ

htot), Sl = 1−Sg = Sl(pl , ρh

tot),ρh

l = ρhl (pl ,ρ

htot), and secondly computing the derivatives of

the saturations, from equation (26),

∂Sg

∂ pl=−

a(Sg)21lρh

tot>Ch(pl+pc(0))

C∆ ρhtot +a(Sg)2 p′c(Sg)

, (29)

∂Sg

∂ρhtot

=a(Sg)1lρh

tot>Ch(pl+pc(0))


, (30)

where 1lρhtot>Ch(pl+pc(0)) is the characteristic function of the

set ρhtot >Ch(pl + pc(0)).

As noted in section 2.5 in [8], we have ∂Sg/∂ pl ≤ 0and ∂Sg/∂ρh

tot > 0, when the gas phase is present. Then thesystem (14)–(15) can be written :

−Φρstdw

∂Sg

∂ pl

∂ pl

∂ t−div

(A1,1

∇pl +A1,2∇ρ

htot +B1Kg

)−Φρ

stdw

∂Sg

∂ρhtot

∂ρhtot

∂ t= F w

(31)

Φ∂ρh

tot

∂ t−div

(A2,1

∇pl +A2,2∇ρ

htot +B2Kg

)= F h; (32)

where the coefficients are defined by:

A1,1(pl ,ρhtot) =λl(Sg)ρ

stdw K−Φ(1−Sg)DChNI, (33)

A1,2(pl ,ρhtot) =−Φ(1−Sg)

1−Na(Sg)

DChI, (34)

A2,1(pl ,ρhtot) = (λl(Sg)ρ

hl +λg(Sg)Cv pgN)K+Φ(1−Sg)DChNI,

(35)

A2,2(pl ,ρhtot) = λg(Sg)

1−Na(Sg)

Cv pgK

+Φ(1−Sg)1−Na(Sg)

DChI,(36)

B1(pl ,ρhtot) =−λl(Sg)ρ

stdw [ρstd

w +ρhl ], (37)

B2(pl ,ρhtot) =− (λl(Sg)ρ

hl [ρ

stdw +ρ

hl ]+λg(Sg)C2

v p2g); (38)

with I denoting the identity matrix and with the auxiliaryfunctions

N(pl ,ρhtot) =

C∆ ρhtot


1l ∈ [0,1),

1l = 1lρhtot>Ch(pl+pc(0))

(39)

ρhl (pl ,ρ

htot) = min(Ch pg(pl ,ρ

htot),ρ

htot), (40)

pg(pl ,ρhtot) = pl + pc(Sg(pl ,ρ

htot)). (41)

We should notice first that equation (32) is uniformlyparabolic in the presence of capillarity and diffusion; but ifcapillarity and diffusion are neglected, this same equationbecomes a pure hyperbolic transport equation (see sec. 2.6in [8]). Then, if we sum equations (31) and (32) we obtain auniformly parabolic/elliptic equation, which is parabolic inthe unsaturated (two-phases) region and elliptic in the liquidsaturated (one-phase) region.

Remark 2 Simulations presented in sec. 3.2 in [8] show thatthis last choice of primary variables, ρh

tot , and pl , could eas-ily handle phase transitions (appearance/disappearance ofthe gas phase, saturated zones, ...) in two-phase partiallymiscible flows. However, the discontinuity of the charac-teristic function with respect to the main variable ρh

tot , onthe concentration threshold line, 1lρh

tot>Ch(pl+pc(0)), in (29),(30) and in (39), have some effect on the conditioning ofthe Jacobian matrix and hence on the number of Newton it-erations and the number of iterations required to solve theJacobian system. Except if the fraction in front of the char-acteristic function in (39) tends to zero as Sg → 0, whichis the case when the van Genuchten’s capillary curves areused.

An other variant is presented in [1], where using the totalhydrogen concentration,

Chtot =

(1−Sg)ρhl +Sgρg

(1−Sg)ρl +Sgρg; (42)

an extended saturation can be defined from the inverse of(42):

Sg =Ch

totρl−ρhl

Chtotρl−ρh

l +(1−C)ρg. (43)

This saturation which was initially defined in the two-phaseregion is then extended outside this region by doing ρg = ρlin (42). Since, no matter in what region we are, there existsalways an “extended“ saturation (Sg ≤ 0, outside the two-phase region), which can be chosen as primary variable. Itis then possible to model both the one-phase flow and thetwo-phase flow with the same system of equations writtenwith this extended saturation as main unknown, and the gasappearance and disappearance is actually treated through thetotal hydrogen concentration Ch

tot expression (see [28]).

3.2 Modeling based on the dissolved hydrogen massdensity in the liquid phase, ρh

l

We have seen that the variables pl and ρhtot , introduced in

the last section, describe perfectly the flow system, both inthe one-phase and in the two-phase regions, independently


Sg

pc(Sg)

0 10

π =ρhlCh

− pl

f(π)

1

0

0

Fig. 2 Capillary pressure curve, pc = pg− pl , and inverse function

of the presence of diffusion or capillary forces. But if we as-sume moreover that the effects of the capillary forces are notnegligible we can choose an other set of primary variables.

Namely, using the retention curve (inverse of the capil-lary pressure curve), we may define the phase saturation asfunction of the dissolved hydrogen mass density in the liq-uid, ρh

l , and of the liquid pressure, pl ; and hence use themas main unknowns. With these two variables, ρh

l and pl , thetwo systems (14)–(18) and (19)–(22) are transformed in asingle system of equations able to describe both liquid satu-rated and unsaturated flow.

Since the capillary pressure curve Sg 7→ pc(Sg) is a strictlyincreasing function we can define an inverse function (reten-tion curve) f : R→ [0,1], (see Fig. 2), by

f (π) =

p−1c (π) if π ≥ pc(0)

0 otherwise.(44)

By definition of the retention curve f , using (10) and (12),we have:

f

(ρh

lCh− pl

)= Sg, (45)

and it is then possible to compute the gas saturations, Sg,from pl and ρh

l . These two variables being well defined inboth the one and two-phase regimes, we will now use themas principal unknowns.

Equations (14)-(18) with unknowns pl and ρhl can be

written as:

−Φρstdw

∂

∂ t

(f

(ρh

lCh− pl

))−div

(A1,1

∇pl + A1,2∇ρ

hl +B1Kg

)= F w,

(46)

Φ∂

∂ t

(a∗ f

(ρh

lCh− pl

)ρ

hl)

−div(A2,1

∇pl + A2,2∇ρ

hl +B2Kg

)= F h,

(47)

where the coefficients are given by the following formulas:

A1,1 = λl(Sg)ρstdw K, A1,2 =−Φ(1−Sg)DI, (48)

A2,1 = λl(Sg)ρhl K, (49)

A2,2 = λg(Sg)Cv

C2h

ρhl K+Φ(1−Sg)DI, (50)

with B1 and B2 defined as in (37), (38), and

a∗(Sg) =a(Sg)

Ch= 1+(

Cv

Ch−1)Sg. (51)

If we consider first, equation (47), we may write it as

Φ(a∗(Sg)+ρ

hl

∂a∗(Sg)

∂ρhl

)∂ρhl

∂ t

−div(A2,1

∇pl + A2,2∇ρ

hl +B2Kg

)+Φρ

hl

∂a∗(Sg)

∂ pl

∂ pl

∂ t= F h .

Moreover, from (51) and because f and f ′ are positive, wehave

a∗(Sg)+ρhl

∂a∗(Sg)

∂ρhl

=

1+(Cv

Ch−1)

(f

(ρh

lCh− pl

)+

ρhl

Chf ′(

ρhl

Ch− pl

))≥ 1;

and if the diffusion is not neglected, we have definite posi-tiveness of the quadratic form A2,2 , in equation (47); i.e. forany ξ 6= 0,

(A2,2ξ ·ξ ) = λg(Sg)

Cv

C2h

ρhl Kξ ·ξ +Φ(1−Sg)D|ξ |2 > 0,

and therefore equation (47) is strictly parabolic in ρhl .

If we develop, equation (46) as follows:

Φρstdw f ′

(ρh

lCh− pl

)∂ pl

∂ t−div

(A1,1

∇pl + A1,2∇ρ

hl +B1Kg

)− ρstd

w

ChΦ f ′

(ρh

lCh− pl

)∂ρh

l∂ t

= F w;

we have, for any ξ ,

λl(Sg)ρstdw Kξ ·ξ ≥ 0,

and then positiveness of (A1,1ξ ·ξ ) and of (A2,1ξ ·ξ ).Moreover,

Φρstdw f ′

(ρh

lCh− pl

)≥ 0.

However, equations in system (46)-(47) are not uniformlyparabolic/elliptic for the pressure pl , because the coefficients,A1,1, A2,1, in front of ∇pl in (46)– (47) tend to zero asSg→ 1.

Remark 3 It is worth noticing that this system (46)-(47),with variables pl and ρh

l , has interesting properties for nu-merical simulations in strongly heterogeneous porous me-dia. These two variables are continuous through interfacesseparating different porous media with different rock types


(different absolute permeability, different capillary and per-meability curves), as we will see in 4.3; which is not thecase for the total hydrogen density ρh

tot . An other advantageis the continuity, in the neighbourhood of the concentrationthreshold line, of all the coefficients Ai, j, in (46)–(47) andof f in (47). However, the choice of total hydrogen massdensity, ρh

tot , for primary variable does not require capillaryeffects, making it useful when the capillary effects are negli-gible; which is not the case with the choice of the dissolvedhydrogen density, ρh

l , as primary variable, which is relyingon an invertible capillary pressure curve (see equation (45)).Moreover, with this choice ρh

l , as primary variable, steep-est or infinite slope in the capillary curve or in the retentioncurve have effect on the conditioning of the Jacobian andmake problem for computing back the secondary variables.

Remark 4 From the solubility equation given by the Henry’slaw (10), it is possible to define an “extended” gas pressureby pg = ρh

l /Ch even inside the liquid saturated region. Ob-viously, this “extended” gas pressure coincide with the truegas pressure pg in the two-phase region. In [3] and [30], this“extended” gas pressure and the liquid pressure are chosenfor primary variables and the gas appearance and disappear-ance is treated through the retention curve.

4 Numerical experiments

In this last section, we present four numerical tests speciallydesigned for illustrating the ability of the model describedby equations (46)-(47) to deal with gas phase appearanceand disappearance. All the computations were done usingthe variables, pl and ρh

l , we are also displaying, for eachtest, the saturation and pressure level curves. These two lastquantities are obtained after a post processing step using theCapillary Pressure law (3), equations (45), Henry’s law (10),and the constraints (4) and (12) (see Figure 1).

The first test focuses on the gas phase appearance pro-duced by injecting pure hydrogen in a 2-D homogeneousporous domain Ω (see Figure 3), which is initially liquidsaturated by pure water.

Because the main goal of all these numerical experi-ments is to test the model efficiency, for describing the phaseappearance or disappearance, the porous domain geometrydoes not really matter and we will use a porous domainwith a simple geometry. Consequently, we choose a simple,quasi-1D, porous domain (see Figure 4) for the followingthree tests .

The test case number 2 is more complex, it shows localdisappearance of the gas phase created by injecting pure hy-drogen in a homogeneous unsaturated porous medium (ini-tially both phases, liquid and gas, are present everywhere).

The two last tests aim is to focus on the main challengesin simulating the flow crossing the engineered barriers, lo-

cated around the waste packages. In the test case number3, the porous medium domain is split in two parts with dif-ferent and highly contrasted rock types, and like in the firstone, the gas phase appearance is produced by injecting purehydrogen in an initially water saturated porous domain. Thetest case number 4 addresses the evolution of the phases,from an initial phase disequilibrium to a stabilized station-ary state, in a closed porous domain (no flux boundary con-ditions).

Fluid parameters: phases and components characteristicsParameter Value

θ 303 KDh

l 3 10−9 m2/sµl 1 10−3 Pa.sµg 9 10−6 Pa.s

H(θ = 303K) 7.65 10−6 mol/Pa/m3

Mw 18 g/molMh 2 10−3 kg/molρstd

w 103 kg/m3

Table 1 Fluid parameters: phases and components characteristics.

In all these four test cases, for simplicity, the porousmedium is assumed to be isotropic, such that K = kI withk a positive scalar; and the source terms are assumed tobe null: Fw = 0 and Fh = 0. As usual in hydrogeology,the van Genuchten-Mualem model for the capillary pressurelaw and the relative permeability functions (see [20,27] ) areused for underground nuclear waste modeling, i.e. :

pc = Pr

(S−1/m

le −1)1/n

, λl =1µl

√Sle

(1− (1−S1/m

le )m)2

and λg =1µg

√1−Sle

(1−S1/m

le

)2m

with Sle =Sl−Sl,res

1−Sl,res−Sg,resand m = 1− 1

n.

(52)

Note that in the van Genuchten-Mualem model, there is noCapillary pressure jump at 0, pc(0)= 0, but the presence of ajump, like in the Brooks-Corey model (see [10]), would notlead to any difficulty, neither from the mathematical point

Mesh sizes and time steps, used in the different Numerical TestMesh size range Time step range

Test number 1 2 m – 6 m (∗) 102 years – 5 104 yearsTest number 2 1 m (∗∗) 102 years – 5 103 yearsTest number 3 1 m (∗∗) 102 years – 2 104 yearsTest number 4 s

2 10−3 m (∗∗) 0.33 s – 16.7 103 s(*) Unstructured triangular mesh(**) Regular quadrangular mesh

Table 2 Mesh sizes and time steps used in the different numerical tests.


Γimp

Ω

Γout

Γimp

Γimp

Γin

Γimp

Ld

Ld

Ls

Ls

Ls

Ls

Fig. 3 Test case number 1: Geometry a the 2-D porous domain, Ω .

Γimp

Γimp

ΓinΩ1 Γout

Ω2

Lx

L1

Ly

Fig. 4 Test cases number 2, 3 and 4: Geometry of the quasi-1D porousdomain, Ω = Ω1∪Ω2.

of view, nor for the numerical simulations. Concerning theother fluid characteristics, the values of the physical parame-ters specific to the phases (liquid and gas) and to the compo-nents (water and hydrogen) are given in Table 1. All the sim-ulations, presented herein, were performed using the mod-ular code Cast3m, [11]. The differential equations systemwas first linearized by a quasi-Newton method and then dis-cretized by a finite volume, implicit in time, scheme; withthe discretization parameters (mesh size and time step) givenin Table 2.

4.1 Numerical Test number 1

The geometry of this test case is given in Figure 3; and therelated data are given in Table 3. A constant flux of hydro-gen is imposed on the input boundary, Γin, while Dirichletconditions pl = pl,out , ρh

l = 0 are given on Γout in order tohave only the water component on this part of the boundary.The initial conditions, pl = pl,out and ρh

l = 0, are uniform onall the domain, and correspond to a porous domain initiallysaturated with pure water.

The main steps of the corresponding simulation are pre-sented in Figure 5.

Boundary conditions Porous mediumInitial condition Param. Value

φ w ·ν = 0 on Γimp k 5 10−20 m2

φ h ·ν = 0 on Γimp Φ 0.15φ w ·ν = 0 on Γin Pr 2 106 Pa

φ h ·ν = Qh on Γin n 1.49pl = pl,out on Γout Sl,res 0.4

ρhl = 0 on Γout Sg,res 0

pl(t = 0) = pl,out in Ω Othersρh

l (t = 0) = 0 in Ω Param. Valuepl,out = 106 Pa Ld 200 m

Ls 20 mQh 9.28 mg/m2/year

Table 3 Numerical Test case number 1: Boundary and Initial Condi-tions; porous medium characteristics and domain geometry; φ w and φ h

are denoting respectively the water and hydrogen flux.

We observe in the beginning (see time t = 1200years inFigure 5) that all the injected hydrogen through Γin is totallydissolved in the liquid phase, the gas saturation stay null onall the domain (there is no gas phase). During that same pe-riod of time: the increasing in liquid pressure is relativelysmall and the liquid phase flux originates slowly (they areboth hard to see on the figures); then the hydrogen is trans-ported mainly by diffusion of the dissolved hydrogen in theliquid phase.

Later on, the dissolved hydrogen accumulates around Γinuntil the dissolved hydrogen mass density ρh

l reaches thethreshold ρh

l = Ch pl (according to Figure 1 and pc(0) = 0in 1), at time t = 1600 years, when the gas phase appearsin the vicinity of Γin. Then this unsaturated region progres-sively expands. The gas phase volume expansion creates agradient of the liquid pressure in the porous domain, caus-ing the liquid phase to flow from Γin to Γout . Consequently,after this time, t = 1600 years: the hydrogen is transportedby convection in the gas phase and the dissolved hydrogenis transported by both convection and diffusion in the liquidphase. The liquid phase pressure increases globally in thewhole domain until time t = 260 000 years (see Figure 5),and it starts to decrease in the whole domain until reachinga uniform and stationary state at t = 106 years, in which thewater component flux is null everywhere.


The geometry and the data of this numerical test are givenin Figure 4 and Table 4. The porous medium is homoge-neous and the initial conditions uniform; there is no needfor defining two parts of the porous domain, Ω1 and Ω2; theparameter L1 will be considered as null.

In this second test a constant flux of hydrogen is im-posed on the input boundary Γin, while Dirichlet conditionspl = pl,out , pg = pg,out are chosen, on Γout , such that ρh

l >

Ch pl , in order to keep the gas phase (according to the phase


ρhl pl Sg

ρhl at t = 1200 years pl at t = 1200 years Sg at t = 1200 years

ρhl at t = 4 104 years pl at t = 4 104 years Sg at t = 4 104 years

ρhl at t = 2 105 years pl at t = 2 105 years Sg at t = 2 105 years

ρhl at t = 106 years pl at t = 106 years Sg at t = 106 years

Fig. 5 Numerical Test case number 1: Evolution of ρhl ,the hydrogen density in the liquid phase; pl the liquid phase pressure; and Sg the gas

saturation ; at times t = 1200,4 104,2 105 and 106 years (from the top to the bottom).

diagram in Figure 1) present on this part of the boundary.The initial conditions pl = pl,out and ρh

l =Ch pg,out are uni-form and imply the presence of the gas phase in the wholedomain.

The main steps of the corresponding simulation are pre-sented in Figures 6 and 7 where we show the liquid pres-sure pl , the dissolved hydrogen molar density ch

l (equal toρh

l /Mh) and the gas saturation Sg profiles, at different times.At the beginning, up to t < 1400 years, the two phases

are present in the whole domain (see time t = 500 years on

Figure 6). The permanent injection of hydrogen increasesthe gas saturation in the vicinity of Γin. The local gas sat-uration drop is due to the difference in relative mobilitiesλα(Sα) between the two phases: the lower liquid mobilityleads to a bigger liquid pressure increase, compared to thegas pressure increase; which is finally producing a capillarypressure drop (according to definitions (1) and (3); see Fig-ure 2), and creating a liquid saturated zone. At time t = 1400years, the gas phase starts to disappear in some region of theporous domain (see time t = 1500 years, in Figure 7).


Boundary conditions Porous mediumInitial condition Param. Value

φ w ·ν = 0 on Γimp k 5 10−20 m2

φ h ·ν = 0 on Γimp Φ 0.15φ w ·ν = 0 on Γin Pr 2 106 Pa

φ h ·ν = Qh on Γin n 1.49pl = pl,out on Γout Sl,res 0.4

ρhl =Ch pg,out on Γout Sg,res 0

pl(t = 0) = pl,out in Ω Othersρh

l (t = 0) =Ch pg,out in Ω Param. Valuepl,out = 106 Pa Lx 200 m

pg,out = 1.1 106 Pa Ly 20 mQh = 55.7 mg/m2/year L1 0 m

Table 4 Numerical Test case number 2: Boundary and Initial Condi-tions; porous medium characteristics and domain geometry. φ w and φ h

are denoting respectively the water and hydrogen flux.

Then, a saturated liquid region (Sg = 0)will exist untiltime t = 17 000 years (see Figure 6); and during this pe-riod of time, the saturated region is pushed by the injectedHydrogen, from Γin to Γout .

After the time t = 17 000 years, due to the Dirichletconditions imposed on Γout , the liquid saturated region dis-appears and all together the phases pressure and the gassaturation are growing in the whole domain (see the timet = 20 000 years in Figure 7).

Finally the liquid pressure reaches its maximum at timet = 20 000 years and then decreases in the whole domain(see the Figure 7). This is caused, like in the numerical testcase number 1, by the evolution of the system towards a sta-tionary state which is characterized by a zero water compo-nent flow.


The geometry and the data of this numerical test are givenin Figure 4, Table 5 and Table 6. Like in the Numerical Testnumber 2, a constant flux of hydrogen is imposed on theinput boundary, Γin, while Dirichlet conditions pl = pl,out ,ρh

l = 0 are given on Γout , in order to have only the liquidphase on this part of the boundary. The initial conditions,pl = pl,out and ρh

l = 0, are uniform on all the domain, andcorrespond to a porous domain initially saturated with purewater. Contrary to the two first numerical tests, the porousdomain is non homogeneous, there are two different poroussubdomains Ω1 and Ω2; Lx = 200 m, L1 = 20 m and L2 =

180 m.The simulation time of this test case is T = 106 years;

the discretization space mesh is 1 meter; the time step is 102

years at the beginning and grows up to 2 · 104 years in theend of the simulation (see Table 2).

Figures 9 and 10 represent the liquid pressure pl , thedissolved hydrogen molar density (equal to ρh

l /Mh) and thegas saturation Sg profiles at different times.

Boundary conditions Otherinitial condition Param. Value

φ w ·ν = 0 on Γimp Lx 200 mφ h ·ν = 0 on Γimp Ly 20 mφ w ·ν = 0 on Γin L1 20 m

φ h ·ν = Qh on Γin pl,out 106 Papl = pl,out on Γout Qh 5.57 mg/m2/year

ρhl = 0 on Γout

pl(t = 0) = pl,out on Ω

ρhl (t = 0) = 0 on Ω

Table 5 Numerical Test case number 3: Boundary and Initial Condi-tions, domain geometry. φ w and φ h are denoting respectively the waterand hydrogen flux.

Porous mediaParam. Value on Ω1 Value on Ω2

k 10−18 m2 5 10−20 m2

Φ 0.3 0.15Pr 2 106 Pa 15 106 Pan 1.54 1.49

Sl,res 0.01 0.4Sg,res 0 0

Table 6 Numerical Test case number 3: Porous medium characteris-tics.

Sg

pcp(1)c (Sg)

p(2)c (Sg)

p(1)c (S(1)g ) = p(2)c (S(2)

g )

S(1)g S(2)

g

Fig. 8 Saturation discontinuity at the interface of two materials withdifferent capillary pressure curves; test case number 3.

The main difference from the previous simulations (whichwere in a homogeneous porous domain) is the gas satura-tion discontinuity, staying on the porous domain interfacex = 20 m. Due to the height of this saturation jump , we hadto use a logarithm scale for presenting the gas saturation Sgprofiles; but as a consequence although all the Sg curves goto zero this cannot be seen with a logarithmic scale in Fig-ures 9 and 10

There are four main steps :

– From 0 to 3.8 · 104 years the increasing of both the gassaturation and the liquid pressure are small and slow inthe whole domain and hard to see on the figures; whilethe hydrogen injection on the left side Γin of the domainincreases notably the hydrogen density level .


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t=0.00E+00t=5.00E+02t=1.50E+03t=3.00E+03t=8.00E+03t=1.40E+04t=1.70E+04

Fig. 6 Test case number 2; Lx = L2 = 200 m: Time evolution, in years, of chl the dissolved hydrogen molar density (top right), pl (top left) and Sg

(bottom) profiles; during the first time steps.

– From 3.8 ·104 to 5.4 ·104 years both the liquid pressureand the hydrogen density are increasing in the whole do-main. The gas start to expanding from the left side ofthe domain Γin. The saturation front is moved towardsthe porous media discontinuity, at x = 20 m, which isreached at t = 5.4 ·104 years; see Figures 9.

– From 5.4 ·104 years to 1.3 ·105 years, see Figures 10, thesaturation front has crossed the medium discontinuity atx = 20 m and, from now, all the saturation profiles willhave a discontinuity at x = 20 m.

– From 1.3 · 105 years to 106 years, see Figures 10, boththe hydrogen density and the gas saturation keep grow-ing while the liquid pressure decreases towards zero onthe entire domain. The gas saturation front keeps movingto the right, pushed by the injected gas, up to x≈ 150 mat 106 years.

Until the saturation front reaches the interface betweenthe two porous media, for (t = 5.4 · 104 years), appearanceand evolution of both the gas phase and the unsaturated zoneare identical to what was happening in the test case 1 (with

a homogeneous porous domain) during the period of gas in-jection: the dissolved hydrogen is accumulating at the en-trance until the liquid phase becomes saturated, at time (t >3.8 104 years), letting the gas phase to appear.

When the saturation front crosses the interface betweenthe two porous subdomains (at x = 20 m and t = 5.4 · 104

years), the gas saturation is strictly positive on both sides ofthis interface and the capillary pressure curves is differenton each side (see Table 6). The capillary pressure continuityat the interface imposes to p(1)c , the Capillary Pressure inΩ1, and to p(2)c , the Capillary Pressure in Ω2, to be equalon this interface. Then p(1)c = p(2)c is satisfied only if thereare two different saturations,on each interface side S(1)g , andS(2)g : p(1)c (S(1)g ) = p(2)c (S(2)g ) ; see Figure 8.

In the same way as in the numerical test case number 1,the system tends to a stationary state


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Fig. 7 Test case number 2; Lx = L2 = 200 m: Time evolution, in years, of chl the dissolved hydrogen molar density (top right), pl(top left) and Sg

(bottom) profiles ; during the six last time steps.


This last numerical test is different from all the precedentones; it intends to be a simplified representation of what hap-pens when an unsaturated porous block is placed within awater saturated porous structure. The challenge is then: howthe mechanical balance will be restored in a homogeneousporous domain, which was initially out of equilibrium, i.e.with a jump in the initial phase pressures?

The initial liquid pressure is the same in the entire porousdomain; Ω , pl,1 = pl,2, and in the subdomain Ω1 the ini-tial condition, pl,1 = pg,1 in Table 7, corresponds to a liquidfully saturated state with a hydrogen concentration reach-ing the gas appearance concentration threshold (pg = pl andρh

l = Ch pg; Figure 1). In the subdomain Ω2 the initial con-dition (pl,2 6= pg,2 and pg,2 6= pg,1) corresponds to a non sat-urated state (see Table 7).

The porous block initial state is said out of equilibrium,because: if this initial state was in equilibrium, in the twosubdomains Ω1 and Ω2, the local mechanical balance would

have made the pressures, of both the liquid and the gas phase,continuous in the entire domain Ω .

For simplicity, we assume the porous medium domainΩ is homogeneous and all the porous medium characteris-tics are the same in the two subdomains Ω1 and Ω2, andcorresponding to concrete. The system is then expected toevolve from this initial out of equilibrium state towards astationary state.

We should notice that, in order to see appearing the fi-nal stationary state, in a reasonable period of time, we haveshortened the domain Ω ( Lx = 1m), taken the porous mediacharacteristics, and set the final time of this simulation Tf inat Tf in = 106 s ≈ 11.6 ) days. The complete set of data ofthis test case is given in Table 7.

The space discretization step was taken constant equal to2 ·10−3 m and the time step was variable, going from 0.33 sin the beginning of the simulation to 16.7 · 103 s at the endof the simulation (see Table 2). Figures 11 and 12 representthe liquid pressure pl , the dissolved hydrogen molar con-


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Fig. 9 Test case number 3; Lx = 200 m, L1 = 20m: Time evolution,in years, of chl the dissolved hydrogen molar density (top right), pl(top left)

and Sg (bottom) profiles ; during the first time steps.

Boundary conditions Porous mediuminitial condition Param. Value

φ w ·ν = 0 on ∂Ω k 10−18 m2

φ h ·ν = 0 on ∂Ω Φ 0.3pl(t = 0) = pl,1 on Ω1 Pr 2 106 Pa

ρhl (t = 0) =Ch pg,1 on Ω1 n 1.54pl(t = 0) = pl,2 on Ω2 Sl,res 0.01

ρhl (t = 0) =Ch pg,2 on Ω2 Sg,res 0

pl,1 = 106 Pa Otherpg,1 = 106 Pa Param. Valuepl,2 = 106 Pa Lx 1 m

pg,2 = 2.5 106 Pa Ly 0.1 mL1 0.5 m

Table 7 Data of the numerical test number 4 : boundary and initialconditions;domain geometry. The porous medium domain Ω is homo-geneous, all the porous medium parameters are the same in the twosubdomains Ω1 and Ω2; φ w and φ h are denoting respectively the waterand hydrogen flux .

centration chl and the gas saturation Sg profiles at different

times.

There are essentially two steps:

– For 0 < t < 1.92 · 105 s (see Figure 11), the initial gassaturation jump moves from x = 0.5 m, at t = 0 andreaches Γin, the left domain boundary, at t = 1.92 ·105 s.During this movement, the saturation jump height (ini-tially≈ 0.16 ) decreases, until approximately 0.03, whenit reaches the left boundary Γin. In front of this disconti-nuity there is a liquid saturated zone, Sg = 0, and in thiszone both the liquid pressure and the hydrogen densityare spatially uniform (see Figure 11, top ). But, while thehydrogen density remains constant and equal to its ini-tial value, the liquid pressure becomes immediately con-tinuous and starts growing quickly (for instance, pl(t =103 s) ≈ 1.6 · 106 Pa), and then more slowly until t =1.3 ·105 s, when it starts to slightly decrease.In Figure 11, located on the gas saturation discontinuity,there are both a high contrast in the dissolved hydrogenconcentration (this concentration stays however contin-uous, but with a strong gradient, as seen in the top rightof Figure 11), and a discontinuity in the liquid pressuregradient (see the top left of Figure 11).


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Fig. 10 Test case number 3; Lx = 200 m, L1 = 20m: Time evolution,in years, of chl the dissolved hydrogen molar density (top right), pl(top left)

and Sg (bottom) profiles ; during the last seven time steps.

– For 1.92 · 105 s < t < 106 s = Tf in (see Figure 12), allthe entire domain is now unsaturated (Sg >0). The liq-uid pressure, the hydrogen density and the gas saturationprofiles are all strictly monotonous and continuous, go-ing towards a spatially uniform distribution, correspond-ing to the stationary state (see Figure 12).

As expected, the system initially out of equilibrium (dis-continuity of the gas pressure), comes back immediately toequilibrium (the gas pressure is continuous) and evolves to-wards a uniform stationary state (due to the no mass inflowand outflow boundary conditions). Although the liquid pres-sure and the dissolved hydrogen density are immediatelyagain continuous for t > 0 , the hydrogen density still havea locally very strong gradient until t = 1.92 ·105 s.

At first, and at the very beginning (≈ 102 s), see topleft of Figure 11, only the liquid pressure evolves in theliquid saturated zone. Due to a gas pressure in the unsatu-rated zone higher than in the liquid saturated zone (Sg = 0;pg = 2.5MPa > pl = 1MPa, for the initial state in Table 7),and due to the no flow condition imposed on Γin, the liquid in

the saturated zone is compressed by the gas from the unsatu-rated zone. Then, a liquid gradient pressure appears aroundthe saturation front and makes the liquid to flow from theliquid saturated zone towards the unsaturated one, and thenthe gas saturation front to move in the opposite direction.

The very strong hydrogen density gradient (until t =1.92 · 105 s), located on the saturation front, is due to thecompetition between the diffusion and the convective fluxof the dissolved hydrogen around the saturation front: thewater flow convecting the dissolved hydrogen, from left toright, cancels the smoothing effect of the gas diffusion prop-agation in the opposite direction. On the one hand the diffu-sion is supposed to reduce the hydrogen concentration con-trast, by creating a flux going from strong concentrations (inthe unsaturated zone) towards the low concentrations (in theliquid saturated zone), and on the other hand the flow of theliquid phase goes in the opposite direction (left to right, fromSg = 0 to Sg > 0). Once the disequilibrium has disappeared,the system tends to reach a uniform stationary state deter-mined by the mass conservation of each component present


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Fig. 11 Numerical test case number 4, Lx = 1 m, L1 = 0.5 m: Time evolution, in seconds, of chl the dissolved hydrogen molar density (top right),

pl(top left) and Sg (bottom) profiles; during the six first time steps.

in the initial state (the system is isolated, with no flow onany of the boundaries).

5 Concluding remarks

From balance equations, constitutive relations and equationsof state, assuming thermodynamical equilibrium, we havederived a model for describing underground gas migrationin water saturated or unsaturated porous media, includingdiffusion of components in phases and capillary effects. Inthe second part, we have presented a group of numericaltest cases synthesizing the main challenges concerning thegas migration in a deep geological repository. These numer-ical simulations, are based on simplified but typical situa-tions in underground nuclear waste management; they showevidence of the model ability to describe the gas (hydro-gen) migration, and to treat the difficult problem of cor-rectly following the saturated and unsaturated regions cre-ated by the gas generation. The selection of primary vari-ables depends in general on the ability to chose a set of

variables to represent the system state in saturated as wellas in unsaturated regime. The optimal selection of primaryvariables depends, in general, on the particular problem be-ing simulated. In some circumstances, the selection of pri-mary variables has a large effect on the conditioning of theJacobian matrix and hence on the number of Newton itera-tions and the number of iterations required to solve the Ja-cobian system. The simplicity in evaluating other secondaryvariables may have also large impact on efficiency of thechosen set of primary variables, and may require furthersolving of nonlinear algebraic equation as for the methodin Section 3.1. In heterogeneous media it would be prefer-able to chose the primary variables which are continuousthrough material interfaces (as a phase pressures, and dis-solved hydrogen mass ρh

l as in Section 3.2), than variableslike saturations or total hydrogen mass which are discontin-uous through the interface. We have shown in the numericaltests that the primary variables chosen in Section 3.2 can ef-ficiently treat the gas appearance/disappearance in presenceof heterogeneities; although,the formulation based on total


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10

12

14

abscissa (m)

gas

satu

ratio

n (%

)

t=1.80E+05t=2.00E+05t=3.00E+05t=6.00E+05t=1.00E+06

Fig. 12 Numerical test case number 4, Lx = 1 m, L1 = 0.5 m: Time evolution, in seconds, of of chl the dissolved hydrogen molar density (top

right), pl(top left) and Sg (bottom) profiles; during the five last time steps.

hydrogen mass is more suitable when the capillary effectsare negligible.

Acknowledgements This work was partially supported by the GNRMoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN). Mostof the work on this paper was done when Mladen Jurak was visiting, atUniversite Lyon 1, the CNRS-UMR 5208 ICJ.

References

1. Abadpour A., Panfilov M.: Method of Negative Saturationsfor two-phase Compositional Flow with Oversaturated Zones.Transp. Porous Media 79, 197-214 (2009).

2. Myron B. Allen III: Numerical Modelling of Multiphase Flowin Porous Media, Adv. Water Resources, 8, 162-187 (1985).

3. Angelini, O., Chavant, C., Chenier, E., Eymard, R., Granet, S.:Finite Volume Approximation of a Diffusion-dissolution modeland application to nuclear waste storage, Mathematics & Com-puters in simulation, 81, 2001-2017 (2011).

4. Bear, J.: Dynamics of Fluids in Porous Media. American Else-vier (1979).

5. Bear, J., Bachmat, Y.: Introduction to Modeling of TransportPhenomena in Porous Media. Kluwer, Dordrecht (1991).

6. Bear, J., Bensabat, J., Nir, A.: Heat and mass transfer in un-saturated porous media at a hot boundary: I. One-dimensionalanalytical model. Transp. Porous Media 6, 281-298 (1991).

7. Bonina, B., Colina, M., Dutfoy, A.: Pressure building during theearly stages of gas production in a radioactive waste repository.Journal of Nuclear Materials, 281(1), 1-14 (2000).

8. Bourgeat, A., Jurak, M., Smaı, F.: Two partially miscible flowand transport modeling in porous media; application to gas mi-gration in a nuclear waste repository. Comput. Geosci. 13(1),29-42 (2009).

9. Bourgeat, A., Jurak, M., Smaı, F.: Modelling and NumericalSimulation of Gas Migration in a Nuclear Waste Repository.June 2010; http://arxiv.org/abs/1006.2914.

10. Brooks, R.J., Corey, A.T.: Properties of porous media affectingfluid flow. J. Irrigation Drainage Div., in: Proc. Am Soc CivilEng (IR2) 92, 61-88 (1966).

11. CEA, Cast3m, http://www-cast3m.cea.fr. Accessed 19 Jan-uary 2012

12. Chavent, G., Jaffre, J.: Mathematical Models and Finite Ele-ments for Reservoir Simulation. North-Holland (1986).

13. Class, H., Helmig, R., Bastian, P.: Numerical simulation of non-isothermal multiphase multicomponent processes in porous me-dia 1. An efficient solution technique. Adv. Water Resour. 25,533-550 (2002).


14. Class, H., Dahle, H.K., Helmig,R.: Guest Editors, Special Issue:Numerical Models for Carbon-Dioxide Storage in GeologicalFormations. Computational Geosciences, 13(4) (2009).

15. Delshad, M., Thomas, S.G., Wheeler, M.F.: Parallel NumericalReservoir Simulations of Non-Isothermal Compositional Flowand Chemistry. SPE-118847-PP (2009).

16. Facchinei, F., Pang, J.S.: Finite Dimensional Variational In-equalities and Complementarity Problems. Springer, New York(2003).

17. Firoozabadi, A.: Thermodynamics of Hydrocarbon Reservoirs.McGraw-Hill, New York (1999).

18. FORGE project, http://www.bgs.ac.uk/forge. Accessed 19January 2012

19. FORSYTHA, P.A., UNGER, A.J.A., SUDICKY, E.A.: Nonlin-ear iteration methods for nonequilibrium multiphase subsurfaceflow. Adv. Water Resour. 21, 433-449 (1998).

20. Van Genuchten, M.: A closed form equation for predicting thehydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am.J. 44, 892-898 (1980).

21. Horseman,S.T., Higgo,J.J.W., Alexander, J., Harrington,J.F.:Water, gas and solute movement through argillaceous media.NEA Report CC-96/1, Paris, OECD (1996).

22. Jaffre, J. and Sboui, A.: Henry’s Law and Gas Phase Disappear-ance. Transp. Porous Media 82, 521-526 (2010).

23. Jin Y., Jury W.A.: Characterizing the dependence of gas dif-fusion coefficient on soil properties. Soil Sci. Soc. Am. J., 60,66-71 (1996).

24. Lauser, A., Hager, C., Helmig, R., Wohlmuth, B.: A new ap-proach for phase transitions in miscible multi-phase flow inporous media. Adv. Water Resour. 34, 957-966 (2011).

25. Marle, C.M.: Multiphase Flow in Porous Media. Edition Tech-nip (1981).

26. Millington R.J., Shearer R.C.: Diffusion in aggregated porousmedia. Soil Science 3(6), 372-378 (1971).

27. Mualem, Y.: A New Model for Predicting the Hydraulic Con-ductivity of Unsaturated Porous Media. Water Resour. Res.12(3), 513-522 (1976).

28. Oladyshkin, S., Panfilov, M.: Hydrogen penetration in waterthrough porous medium: application to a radioactive waste stor-age site. Environ Earth Sci. (2011) DOI 10.1007/s12665-011-0916-0.

29. Olivella, S., Alonso, E.E.: Gas flow through clay barriers.Geotechnique 58, No. 3, 157-176 (2008)

30. Olivella, S., Carrera, J., Gens, A., Alonso E. E.: Non-isothermalmultiphase flow of brine and gas through saline media. Transp.Porous Media, 15, 271-293 (1994).

31. PAMINA project: http://www.ip-pamina.eu. Accessed 19January 2012

32. Panday, S., Forsyth, P.A., Falta, R.W., Yu-Shu Wu, Huyakorn,P.S.: Considerations for robust compositional simulations ofsubsurface nonaqueous phase liquid contamination and reme-diation. Water Resour. Res., 31, 1273-1289 (1995).

33. Peaceman, D.W.: Fundamentals of numerical reservoir simula-tion. Elsevier, Amsterdam (1977).

34. Pruess K., Oldenburg C. and Moridis G.: Tough2 User’s Guide,version 2.0. Lawrence Berkeley National Laboratory, Berkeley(1999).

35. Talandier, J.: Synthese du benchmark couplex-gaz. Journeesscientifiques du GNR MoMaS, Lyon, 4-5 septembre 2008,http://http://momas.univ-lyon1.fr/presentations/

couplex\_lyon2008\_andra.ppt. Accessed 19 January 201236. Yu-Shu Wua, Forsyth, P.A.: On the selection of primary vari-

ables in numerical formulation for modeling multiphase flow inporous media. J. Cont. Hydr. 48, 277-304 (2001).

37. Hintermuller M., Ito K., Kunisch K.: The primal-dual active setstrategy as a semismooth Newton method. SIAM J. Optim., 13,865-888 (2002).

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