Numerical simulation of gas migration through...

Computing and Visualization in Science manuscript No.(will be inserted by the editor)

Numerical simulation of gas migration through engineered andgeological barriers for a deep repository for radioactive waste

Brahim Amaziane · Mustapha El Ossmani · Mladen Jurak

Received: March 28, 2012 / Accepted: date

Abstract In this paper a finite volume method ap-

proach is used to model the 2D compressible and im-

miscible two-phase flow of water and gas in heteroge-

neous porous media. We consider a model describing

water-gas flow through engineered and geological barri-

ers for a deep repository of radioactive waste. We con-

sider a domain made up of several zones with differ-

ent characteristics: porosity, absolute permeability, rel-

ative permeabilities and capillary pressure curves. This

process can be formulated as a coupled system of par-

tial differential equations (PDEs) which includes a non-

linear parabolic gas-pressure equation and a nonlinear

degenerated parabolic water-saturation equation. Both

equations are of diffusion-convection types. An implicit

vertex-centred finite volume method is adopted to dis-

cretize the coupled system. A Godunov-type method isused to treat the convection terms and a conforming

finite element method with piecewise linear elements

is used for the discretization of the diffusion terms.

An upscaling technique is developed to obtain an ef-

fective capillary pressure curve at the interface of two

media. Our numerical model is verified with 1D semi-

analytical solutions in heterogeneous media. We also

present 2D numerical results to demonstrate the signif-

B. AmazianeUNIV PAU & PAYS ADOUR, IPRA–LMA, CNRS-UMR5142, Av. de l’universite, 64000 Pau, France.E-mail: [email protected]

M. El OssmaniUniversite Moulay Ismaıl, LM2I-ENSAM, Marjane II, B.P.4024, Meknes, 50000, Maroc.E-mail: [email protected]

M. JurakUNIV PAU & PAYS ADOUR, IPRA–LMA, CNRS-UMR5142, Av. de l’universite, 64000 Pau, France.E-mail: [email protected]

icance of capillary heterogeneity in flow and to illus-

trate the performance of the method for the FORGE

cell scale benchmark.

Keywords Finite volume · Heterogeneous porous

media · Hydrogen migration · Immiscible compressible ·Nuclear waste · Two-phase flow · Vertex-centred

PACS 47.40.-x · 07.05.Tp · 02.30.Jr · 47.11.Df ·47.56.+r · 47.55.Ca · 28.41.Kw · 47.55.-t

Mathematics Subject Classification (2000) 74Q99 ·76E19 · 76T10 · 76M12 · 76S05

1 Introduction

The modeling of multiphase flow in porous formations isimportant for both the management of petroleum reser-

voirs and environmental remediation. More recently,

modeling multiphase flow received an increasing atten-

tion in connection with the disposal of radioactive waste

and sequestration of CO2.

The long-term safety of the disposal of nuclear waste

is an important issue in all countries with a significant

nuclear program. Repositories for the disposal of high-

level and long-lived radioactive waste generally rely on

a multi-barrier system to isolate the waste from the bio-

sphere. The multi-barrier system typically comprises

the natural geological barrier provided by the reposi-

tory host rock and its surroundings and an engineered

barrier system, i.e. engineered materials placed within

a repository, including the waste form, waste canisters,

buffer materials, backfill and seals.

In this paper, we focus our attention on the numeri-

cal modeling of immiscible compressible two-phase flow

in heterogeneous porous media, in the framework of the

geological disposal of radioactive waste. As a matter of

2 Brahim Amaziane et al.

fact, one of the solutions envisaged for managing waste

produced by nuclear industry is to dispose it in deep

geological formations chosen for their ability to prevent

and attenuate possible releases of radionuclides in the

geosphere. In the frame of designing nuclear waste ge-

ological repositories, a problem of possible two-phase

flow of water and gas appears, for more details see

for instance [22,23]. Multiple recent studies have estab-

lished that in such installations important amounts of

gases are expected to be produced in particular due

to the corrosion of metallic components used in the

repository design. The creation and transport of a gas

phase is an issue of concern with regard to capability

of the engineered and natural barriers to evacuate the

gas phase and avoid overpressure, thus preventing me-

chanical damages. It has become necessary to carefully

evaluate those issues while assessing the performance of

a geological repository. As mentioned above, the most

important source of gas is the corrosion phenomena of

metallic components (e.g. steel lines, waste containers).

The second source, generally less important depending

on the type of waste, is the water radiolysis by radi-

ation issued from nuclear waste. Both processes would

produce mainly hydrogen. Hydrogen is expected to rep-

resent more than 90 % of the total mass of produced

gases, see e.g. [10,25] and the references therein.

Significant effort has been and continues to be ex-

pended in numerous national and international pro-

grams in attempts to understands the potential im-

pact of gas migration, and of two-phase gas-water pro-

cesses, on the performance of underground radioactive

waste repositories, and to provide modeling tools or ap-

proaches that will allow these impacts to be assessed.

The French Agency for the Management of Radioactive

Waste (Andra) [1] is currently investigating the feasibil-

ity of a deep geological disposal of radioactive waste in

an argillaceous formation. Recently, the Couplex-Gaz

benchmark was proposed by ANDRA, and the French

research group MOMAS [20] to improve the simulation

of the migration of hydrogen produced by the corro-

sion of nuclear waste packages in an underground stor-

age. This is a system of two-phase (water-hydrogen)

flow in a porous medium. This benchmark generated

some interest and engineers encountered difficulties in

handling numerical simulations for this model. One of

the purposes of this test case is to model the resatura-

tion of a disposal cell for intermediate- level long- lived

waste (known as Type-B waste) as soon as the cell is

closed until the end of the gas-production period, for

more details, see for instance [26]. Another benchmark

is currently being studied in the framework of the Euro-

pean Project FORGE: Fate Of Repository Gases [15].

The aim of this test case is to better understand the

mechanisms of the gas migration at the cell scale and

in particular to analyze the effect of the presence of

different material and interfaces on such mechanisms.

Verification of numerical models for immiscible com-

pressible flow in porous media by the means of appro-

priate benchmark problems is a very important step in

developing and using these models.

The usual set of equations describing this type of

flow is given by the mass balance law and Darcy-Muskat’s

law for each phase, which leads to a system of strongly

coupled nonlinear PDEs. In the subsurface, these pro-

cesses are complicated by the effects of heterogeneity

on the flow and transport. Simulation models, if they

are intended to provide realistic predictions, must ac-

curately account on these effects. Here, we will assume

that the porous medium is composed of multiple rock

types, i.e. porosity, absolute permeability, relative per-

meabilities and capillary pressure curves being differ-

ent in each type of porous media. Contrast in capillary

pressure of heterogeneous permeable media can have

a significant effect on the flow path in two-phase im-

miscible flow, see for instance [17] and the references

therein. Such heterogeneous porous media lead to a

possibly discontinuous solution at medium interfaces,

which is a consequence of the transmission conditions

at the interfaces. This should be taken into account in

the discretization.

The numerical modeling and analysis of two-phase

flow in porous media has been a problem of interest for

many years and many methods have been developed.

There is an extensive literature on this subject. We will

not attempt a literature review here, but merely men-

tion a few references. We refer to the books [8,9,16] and

the references therein. Closely related to our work, we

refer for instance to [3,7,13,18,21,24] and the references

therein, where finite volume methods which employ in-

terface conditions were studied in the incompressible

case. In the area of multicomponent models numerical

simulations were presented in [2,6].

The purposes of this paper are to derive a robust

and accurate scheme on unstructured grids by using

a vertex centered finite volume method for the coupled

system modeling immiscible compressible two-phase flow

in highly heterogeneous porous media with different

capillary pressures and present numerical simulations

for an academic test and a benchmark test in the con-

text of hydrogen migration in a nuclear waste reposi-

tory.

In this paper, the wetting and nonwetting phases are

treated to be incompressible and compressible, respec-

tively. This treatment is indeed necessary when a com-

pressible nonwetting phase, such as hydrogen, is sub-

jected to compression during confinement. The problem

Numerical simulation of gas migration for radioactive waste 3

is written in terms of the phase formulation, i.e. where

the phase pressures and the phase saturations are pri-

mary unknowns. We will consider a domain made up

of several zones with different characteristics: porosity,

absolute permeability, relative permeabilities and cap-

illary pressure curves. This process can be formulated

as a coupled system of partial differential equations

which includes a nonlinear parabolic pressure equation

and a nonlinear degenerate diffusion-convection satura-

tion equation. Moreover the transmission conditions are

nonlinear and the saturation is discontinuous at the in-

terfaces separating different media. There are two kinds

of degeneracy in the studied system: the first one is the

degeneracy of the capillary diffusion term in the satura-

tion equation, and the second one appears in the evolu-

tion term of the pressure equation. In this work, we ad-

vance the applicability of the vertex centred finite vol-

ume method in heterogeneous media on unstructured

grids, where we show how to account for the discon-

tinuity in saturation from different capillary pressure

functions. An upscaling technique is developed to ob-

tain an effective capillary pressure curve at the interface

of two media. Our aim is to study a fully implicit finite

volume scheme for the 2D problem where a special dis-

cretization at the interfaces is developed and present

numerical simulations.

The rest of the paper is organized as follows. In sec-

tion 2 we give a short description of the mathematical

and physical model used in this study. In section 3 we

describe a fully implicit finite volume scheme obtained

by combining a Godunov method for the convection

terms and a conforming finite element method with

piecewise linear elements for the diffusion terms. The

space discretization is performed using a vertex-centred

finite volume method and an implicit Euler approach

is used for time discretization, and the nonlinear sys-

tem is solved by Newton-Krylov’s method at each time

step. Furthermore, we formulate an upscaling method

used to treat the nonlinear interface conditions in het-

erogeneous porous media consisting of different rock

types. Namely, the continuity of the capillary pressure

at the interfaces separating different media. Our nu-

merical model is verified with 1D semi-analytical solu-

tions from [14] for incompressible immiscible two-phase

flow in heterogeneous media. Section 4 is devoted to

the presentation of the results of this test. Computa-

tional studies which illustrate the robustness of our fi-

nite volume approach are discussed for the FORGE cell

scale benchmark in section 5. Lastly, some concluding

remarks are forwarded.

2 Mathematical model

In this section we briefly recall the conservation equa-

tions and the constitutive laws for the two-phase (gas

and water) immiscible flow model and introduce the

fractional flow formulation.

Let Ω ⊂ R2 be a bounded domain representing the

porous medium and ]0, T [ the time interval of interest.

The porous medium is characterized by the porosity Φ

and the permeability tensor K, which are spatially vary-

ing and constant in time fields. Other porous medium

properties involve relative permeability and capillary

pressure relationships, which are given functions of sat-

urations and also of position in the case of different rock

types. Let the lower case subscripts w and g denote the

water and gas phases respectively. The mass conserva-

tion equations for each phase form a coupled system of

PDEs (see, e.g., [4,8,9,16]): for α ∈ w, g, we have

Φ(x)∂

∂t(ραSα) + div(ραqα) = Fα, (1)

where the phase velocity qα is defined by the Darcy-

Muskat law:

qα = −λα(Sα)K(x)(∇Pα − ραg

), (2)

where ρα, Sα, Pα and µα are, respectively, the α phase

density, saturation, pressure and viscosity; kr,α is the

relative permeability, and λα = kr,α/µα is the mobility

of the α-phase; Fα is the source/sink term and g is the

gravity vector.

Additionally, the system satisfies a volume constraint:

Sw + Sg = 1, (3)

and the capillary pressure law:

Pc(Sw) = Pg − Pw, (4)

where Pc is a given capillary pressure function.

In the sequel we will use a fractional flow formula-

tion with the total velocity, and the gas pressure and

the water saturation as primary unknowns. We assume

that the water is incompressible and the gas density is

given by the ideal gas law; the viscosities are supposed

to be constant. We set P = Pg, S = Sw, ρg(P ) = cgP

where the compressibility cg is a given positive con-

stant. Furthermore, for notational convenience, we will

neglect the gravity force in the model and set the source

terms to zero.

Using the total velocity q = qw + qg and the coef-

ficients λ(S) = λw(S) + λg(S), fw(S) = λw(S)/λ(S),

a(S) = −λw(S)λg(S)P ′c(S)/λ(S) and α(S) as the prim-

itive function of a(S), we have

qw = fw(S)q−K∇α(S),


and the water phase equation takes the form

Φ∂S

∂t+ div(fw(S)q−K∇α(S)) = 0. (5)

We will refer to equation (5) as the saturation equa-

tion. It will be used in the finite volume discretization

together with the pressure equation (6) which is defined

as a sum of equations (1) with certain weighting factor

R that will be defined later on, leading to:

Φ∂

∂t(ρg(P )(1− S)) +RρwΦ

∂S

∂t

− div([ρg(P )λg(S) +Rρwλw(S)]K∇P ) (6)

+ div(Rρwλw(S)K∇Pc(S)) = 0.

We assume the boundary ∂Ω to be divided in the

Dirichlet and the Neumann parts: ∂Ω = ΓN ∪ ΓD,

where ΓN ∩ΓD = ∅. Then the following boundary con-

ditions are imposed

ραqα · n = Qα, for α= w,g on ΓN ; (7)

S = Sbdw , P = P bdg on ΓD, (8)

where Qα are phase mass fluxes and Sbdw , P bdg are im-

posed values of the water saturation and the gas pres-

sure. To complete the system (5)–(8), we need to add

initial conditions for S and P .

We will consider heterogeneous porous domains com-

posed of several subdomains with different material prop-

erties, including relative permeability and capillary pres-

sure curves. Each subdomain, or a rock type, has a

unique set of relative permeability curves and capillary

pressure curve. In a such domain the PDEs (5), (6), are

valid only in each subdomain separately, while on the

boundary of different subdomains one has to impose the

interface conditions. These conditions are composed by

the continuity of phase fluxes and the continuity of the

phase pressures on the boundary of different materials.

The pressure continuity can be unfulfilled if the phase

is not present at the both sides of the interface and

positive entry pressure exists, leading to the extended

capillary pressure condition [11].

The main difficulties related to the approximation

of the solution of system (5), (6) is the coupling of the

equations, the degeneracy of the diffusion term in the

saturation equation, the degeneracy of the temporal

term in the pressure equation and the nonlinear in-

terface conditions appearing in heterogeneous porous

media containing different rock types.

3 Finite volume discretization

For the spatial discretization of the two-phase flow equa-

tions we use the vertex centered finite volume method.

The method is based on two spatial grids: a primary

grid, which is a conforming finite element grid, and a

secondary grid composed of control volumes centered in

the vertices of the primary grid. All model data, perme-

ability, porosity, sources, capillary pressures and rela-

tive permeabilities are constant element-wise on the pri-

mary grid. The material interfaces are therefore aligned

with the edges of the primary grid and that the satura-

tion functions are associated to elements of the primary

grid. The primary unknowns are represented as P1/Q1

finite element functions over the primary grid.

The control volume of the secondary grid, centered

at the vertex xi, is constructed by joining the centers

of all elements having vertex xi with the centers of cor-

responding edges emanating from xi (see Figure 1). El-

ements of the primary mesh will be denoted by symbol

E, while the volume of the secondary mesh, centered at

xi, will be denoted Vi.

xi

xj

xk

xi

xj

xk

γEi,j

E E

Vi

nEi,k

Fig. 1 The control volume Vi centered at the vertex xi ofthe primary grid composed of elements E.

Note also that the boundary ∂Vi of the control vol-

ume Vi is composed of the segments γEi,j ⊂ E which

connect the center of element E to the edge of E given

by the vertices xi and xj . A unit normal to γEi,j , directed

out of Vi, is denoted nEi,j .

At the interface of two materials, with different cap-

illary pressure curves, the phase pressure is continuous

if the phase is on the both side of the interface. If the

material capillary pressure curves end at zero at the

water saturation equal to one (entry pressure equal to

zero), and the convention that non existent of the non-

wetting phase has the “pressure” equal to the wetting

phase pressure is used, then the phase pressures are al-

ways continuous across the material interface, even in

the case where the non-wetting phase is not present on

the both sides of the interface. The capillary pressure

is then also continuous across the material interface.

In the case of materials with different entry pressures,


a discontinuity of the non-wetting phase pressure and

the capillary pressure is possible until the entry pres-

sure is not reached (for more details, see for instance

[11]). Since in our application we use the van Genuchten

capillary pressure curves, for simplicity of presentation

we will not consider the possibility of non continuous

non-wetting phase pressure. Our discretization adapts

naturally to this general situation.

Before presenting our numerical scheme, first we will

describe the treatment of the material interfaces in the

numerical method since this treatment influence our

choice of the primary variables in the discrete system.

The continuity of the capillary pressure implies the

discontinuity of the saturation across the material inter-

face (see Figure 3). The capillary pressure will be there-

fore represented on the primary mesh by its values in

the vertices and interpolated as linear or bilinear func-

tion in the elements (P1/Q1 finite elements). The satu-

ration, in the other hand, will be defined from the cap-

illary pressure on each element E as SE = (PEc )−1(Pc),

where Pc is the capillary pressure variable, and PEc is

the capillary pressure curve associated to the element

E. In the vertices that are on the material interface, as

xi and xj in Figure 2, we obtain different values of the

saturations SEi , SEj in different elements E.

mat. I mat. II

xi

xj

SE1i SE2

i

SE1j SE2

j

E1 E2

Fig. 2 An example of the primary grid in the case of two ma-terial subdomains showing the discontinuity of the saturationat the interface.

For example, the values SE1i and SE2

i in the vertex

xi are calculated from the capillary pressure value Pc,iin xi by inverting the corresponding capillary pressure

functions PE1c (S) and PE2

c (S) as shown in Figure 3.

Using the capillary pressure and one phase pressure

as primary variables in the discretized equations and

calculating the saturation, as described above, clearly

implies the continuity of both phase pressures over the

material interfaces. But, we can also replace the cap-

illary pressure by the saturation as a primary variable

and preserve the phase pressure continuity. It is suffi-

Sw

Pc

PE1c PE2

c

P ic

Pc,i

SE1i SE2

iSi 1

Fig. 3 Material capillary pressure curves (full lines) and theeffective capillary pressure curve (dotted line). Calculation of

the local saturations SE1

i , SE2

i and the effective saturationSi, for a given capillary pressure Pc,i.

cient to choose in each vertex xi one of the local satura-

tions SEi , for some neighboring element E, and use it as

a primary unknown in xi. If the vertex xi is not on the

material interface, then all surrounding elements have

the same capillary pressure curve and the saturations

SEi are equal for all elements E containing xi. If not,

we have to select one neighboring element E = Ei as

the one defining the global saturation unknown in the

vertex xi; from SEi,ii we can then calculate the capillary

pressure Pc,i = PEic (SEi,i

i ) in xi and then also all local

saturations in all neighboring elements E. Therefore,

by selecting one of the local saturation as the primary

unknown we can restore the capillary pressure and cal-

culate corresponding saturation everywhere, preserving

the continuity of the capillary pressure. There is only

one restriction to the choice of defining the element Ei:

in presence of the capillary pressure curves with posi-

tive entry pressures we have to choose the element cor-

responding to the curve with the smallest entry pres-

sure in order to satisfy the extended capillary pressure

condition [11]. This implementation of the continuity of

phase pressures is used in [21].

As described above, the treatment of the material

interface has a disadvantage of introducing the deriva-

tive of the function S 7→ (PEc )−1(PEic (S)) into the Ja-

cobian matrix. This derivative can be very large in the

case of heterogeneities with strong differences in the

capillary pressure curves. As this is the case in our ex-

ample (see Figure 7), we prefer to use as unknown the

effective saturation instead of the local saturation on

such interfaces. The effective saturation Si at the ver-

tex xi that corresponds to the capillary pressure Pc,i is


given as a solution of the nonlinear system

(

∫Vi

Φdx)Si =∑E

(

∫Vi∩E

Φdx)SEi . (9)

Pc,i = PEc (SEi ), ∀E, E ∩ Vi 6= ∅. (10)

Note that for given Pc,i one can find local saturations

SEi from (10) by inverting the capillary pressure func-

tions, and then the effective saturation Si is given as

a mean value of the local saturations in (9). More pre-

cisely, the effective saturation gives the same quantity

of the water in the control volume as the local satu-

rations if the mass lumping is used in the calculation.

By representing the capillary pressure value Pc,i as a

function of the effective saturation value Si we obtain

the effective capillary pressure curve P ic , associ-

ated to the vertex xi: Pc,i = P ic(Si). From (9) it follows

that the effective capillary pressure curve depends on

the geometry of the primary grid. An example of the

effective capillary pressure curve is given in Figure 3

(see also Figures 15 and 16).

An advantage of the effective saturation/capillary

pressure is that the derivative of the function

S 7→ (PEc )−1(P ic(S)) can be much smaller than the

derivative of S 7→ (PEc )−1(PEic (S)) in presence of ex-

tremely different curves, as one in Figure 7. More im-

portant is that in combination with the mass lumping

this term completely disappear from the accumulations

terms of the residual since, due to (9), we can approxi-

mate: ∫Vi

ΦSE dx ≈∫Vi

ΦdxSi,∫Vi

Φρg(P )(1− SE) dx ≈∫Vi

Φdxρg(Pi)(1− Si),

where only the effective saturation is used.

In order to avoid solving the nonlinear system (9)–

(10), we can tabulate, before starting the simulation,

the effective capillary pressure curves in each vertex of

the material interface. Then, the capillary pressure and

the local saturations are easily computable from the

effective saturation.

Finally, we describe our finite volume discretization.

Since all functions of saturation variable entering into

differential equations are given by elements, we will gen-

erally use the notation f |E(S) = fE(S) for an element

E of the primary mesh. First, we integrate the equation

(5) over the control volume Vi, we apply the divergence

theorem, a fully implicit time derivative discretization,

the mass lumping in the accumulation term and a full

upwind stabilization in the convective term, resulting

in the discrete saturation equation:1

1

∆t

∫Vi

Φdx(Sn+1i − Sni )

−∑E

∫∂Vi∩E

K∇αE(SE,n+1) · ndσ

+∑E

∑j∈C(i)

∫γEi,j

[fEw (SE,n+1i )(qn+1 · n)+ (11)

+ fEw (SE,n+1j )(qn+1 · n)−]dσ

+∑E

∫∂Vi∩ΓN

w ∩EQn+1w /ρw dσ = 0.

Here C(i) is the set of all indices j 6= i such that xi and

xj are the vertices of the same element E. For j ∈ C(i),γEi,j is the straight part of ∂Vi ∩ E which lays between

xi and xj (see Figure 1). Note that in the accumulation

term we use the effective saturations Sn+1i , Sni , while

in all other integrals we use the local saturation SE,n+1

obtained by inverting locally the capillary pressure. The

diffusion term can be further written as (the time level

(n+ 1) is omitted)∑E

∫∂Vi∩E

K∇αE(SE) · ndσ

=∑E

∑xk∈Ek 6=i

(αE(SEk )− αE(SEi ))

∫∂Vi∩E

K∇ϕk · ndσ.

where ϕk is the P1 (or Q1) finite element base function

on the triangle (quadrilateral) associated to the vertex

xk. All integrals are calculated by the midpoint numer-

ical integration rule applied to each segment γEi,j ⊂ ∂Vi.The second equation is obtained by integrating equa-

tions (1) for α = w, g over the control volume Vi. Water

equation is multiplied by ρg(Pn+1i )/ρw and added to

the gas equation. After mass lumping and cancellation

in the accumulation terms we get the discrete pressure

equation:

1

∆t

∑E

∫Vi∩E

Φdx(ρg(Pn+1i )− ρg(Pni ))(1− Sni )

−∑E

∫∂Vi∩E

mE,n+1i K∇Pn+1 · ndσ (12)

+∑E

∫∂Vi∩E

ρg(Pn+1i )λEw(SE,n+1)K∇PEc (SE,n+1) · ndσ

+∑E

∫∂Vi∩ΓN∩E

(ρg(Pn+1i )Qn+1

w /ρw +Qn+1g ) dσ = 0,

where

mEi = ρg(Pi)λ

Ew(SE) + ρg(P )λEg (SE).

1 x+ = max(x, 0), x− = min(x, 0).


Note that the equation (12) can be interpreted as a

discretization of the pressure equation (6) in which the

factor R is chosen in i-th discrete equation as R =

ρg(Pn+1i )/ρw. This choice, together with the mass lump-

ing, leads to a cancellation in the accumulation terms

and stabilisation of the method. Again, we have ex-

pressed the accumulation terms by using the effective

saturation Sni , rather than the local saturations.

All integrals in discrete pressure equation are ap-

proximated in similar way as the diffusion term in the

saturation equation. For example,∑E

∫∂Vi∩E

λEw(SE)K∇PEc (SE) · ndσ ≈

∑k∈C(i)

(P kc (Sk)− P ic(Si))∫∂Vi

λEw(SE)K∇ϕk · ndσ,

where we have left out the time level index n + 1 to

shorten the notation. The integrals are calculated by

applying the midpoint numerical integration rule to

each segment γEi,j composing ∂Vi. Note that, due to

(10), we could use the effective saturations in the cal-

culation of the capillary pressure through the effective

capillary pressure curves P kc , P ic .

Finally, the total volumetric velocity is each element

E is given by:

q(SE , P ) =− λE(SE)K∇P + λEw(SE)K∇PEc (SE).

It is calculated always in the interior of an element and

therefore can be calculated simply using linear/bilinear

interpolation of P and PEc (SE) in the element E.

The control volumes Vi are possibly heterogeneous

since the material properties are associated to the el-

ements of the primary mesh, and the material inter-

faces pass through edges of the primary mesh elements.

When applying the divergence theorem in the control

volume Vi we have apparently neglected the heterogene-

ity of the domain Vi and obtained only integrals over

∂Vi. This calculation is correct if the phase fluxes over

the material interface are continuous, leading to cancel-

lation of the integrals over the material interface inside

Vi. This is precisely the way how the flux continuity

over the material interface is taken into account. While

the phase pressure continuity is satisfied explicitly, the

flux continuity is taken into account only implicitly.

4 Verification on semi-analytical solution

In order to verify our treatment of heterogeneities we

have tested the code on semi-analytical solutions exist-

ing in the case of incompressible fluids. A 1D semi-

analytical solution of incompressible immiscible two-

phase flow in homogeneous domain, with capillary ef-

fects included, is obtained in [19]. The extensions to

heterogeneous domains are given in [12], including only

diffusion, and in [14], including both convection and

diffusion terms. We will present here only one example

from [14] consisting of a simple heterogeneous medium

composed of two porous domains separated by a sharp

interface at x = 0. Extension of the porous domain is

from x = −0.1 m to x = 0.1 m. The Brooks and Corey

model for the relative permeability and capillary pres-

sure functions is considered with entry pressure Pd and

pore size distribution index λ. The properties of the left

and the right porous sub-domains are given in Table 1.

Left domain Right domainPermeability 7 · 10−12 [m2] 5.3 · 10−12 [m2]Porosity 0.34 0.34Brooks-Corey λ 2.48 2.48

Brooks-Corey Pd 2218 [Pa] 2550 [Pa]

Table 1 Data used for comparison with semi-analytical so-lution.

The wetting and nonwetting phase viscosities are

respectively µw = 1 cP and µg = 0.9 cP.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1 -0.05 0 0.05 0.1

x [m]

Wetting phase saturation at t=1000 s

exactnumerical

Fig. 4 Comparison with a semi-analytical solution.

The irreducible saturations in the both subdomains

are equal to zero and the initial conditions for the wetting-

phase saturation in the left and right subdomains are

taken as Sw = 1 and Sw = 0 respectively. To fix the

semi-analytical solution one has to fix also the ratio R

of the total and the non-wetting flux at the discontinu-

ity (see [14]) which is taken here to be R = 0.5. Com-

parison of the wetting-phase saturation given by the

semi-analytical and the numerical solution at t = 1000

sec is given in Figure 4. The numerical solution is cal-

culated with 200 points in x-direction and we observe

numerical convergence on refined meshes.


5 FORGE cell scale benchmark

Long-term radioactive waste management usually con-

siders final disposal in a deep geological repository. This

includes a series of passive barriers, both engineered and

natural, in order to minimize the migration of radioac-

tivity and achieve the required level of safety for the

environment. As the repository system evolves, gases

that may be produced, such as hydrogen from the corro-

sion of metals and from the radiolysis of water, become

an important safety issue. Multi-disciplinary European

Commission project FORGE [15] is focused on gener-

ation and migration of waste-derived repository gases

and its influences on repository system performance.

For understanding the repository-scale gas migra-

tion and developing methodology for dealing with het-

erogeneities, the FORGE project Work Package 1 in-

cludes benchmark studies on numerical simulation of

gas migration through the underground nuclear waste

repository on different spatial scales. The smallest scale

considered contains only one canister and surrounding

materials, while larger scales include a module of hun-

dred canister and the whole repository. We will present

here only the cell scale benchmark.

5.1 Benchmark description

The domain of the FORGE cell scale benchmark is ax-

isymetric and composed of one waste canister, a ben-

tonite plug which isolates the canister from the access

drift, a zone damaged by excavation (EDZ) surrounding

the canister, the plug and the access drift. Finally, there

is the access drift with its backfill and the geological

medium (unperturbed rock). Axial cross-section of the

domain is presented in Figure 5. The axis of the domain

is horizontal, but the gravity force is neglected, and the

whole problem can be treated as two-dimensional by

means of axial symmetry.

Fig. 5 Axial cross-section of the computational domain.

The contact between the waste canister and the

EDZ, and equally the bentonite plug and the EDZ, is

not perfect. It is assumed that these materials are sep-

arated by a thin layer of a porous medium of very dif-

ferent characteristics, called interface (general retention

behaviour similar to a sand). The materials to be taken

into account include, therefore, the EDZ (of both the

cell and the access drift, which have the same charac-

teristics), the bentonite plug, the backfill of the access

drift, the geological medium and the interfaces (both

canister-EDZ and plug-EDZ interfaces). The waste can-

ister is constituted of a material impermeable to both

water and gas, and is not explicitly represented in the

model. It does not make part of the computational do-

main and it serves only as a source of the gas.

The gas-production term for the disposal cell is im-

posed on the external surface of the cylinder that rep-

resents the canisters.

Dimensions of the computational domain are given

on Figure 6 and in Table 2. Physical parameters of the

materials2 are given in Table 3.

Fig. 6 Dimensions in the computational domain. Values aregiven in Table 2.

Parameter Value Parameter Value

Lx 60 m Rx 20 mRd 3 m Ed 1 mLp 5 m Lc 40 mEc 0.5 m Rc 0.5 mEi 0.01 m Lr 11.5 m

Table 2 Dimensions in the Figure 6.

2 Originally, horizontal and vertical permeability Kh, Kv

were given, which is not consistent with supposed radial sym-metry of the problem. As a consequence we use an isotropicpermeability K, given by K = 3

√K2

hKv.


ParametersMaterials K[10−18m2] Φ n Pr [MPa]Access drift 50 0.4 1.5 2

Bentonite plug 10−2 0.35 1.6 16Geological mat. 7.9 10−3 0.15 1.5 15

EDZ 7.9 0.15 1.5 1.5Interface can. 1 1.0 4 10−2

Interface plug 7.9 0.3 4 10−2

Table 3 Material parameters: Permeability (K), porosity (Φ)and van Genuchten’s parameters, n and Pr.

The saturation functions are given by the van Ge-

nuchten law, where the irreducible saturations of the

gas and water are set to zero (S = Sw):

Pc(S) = Pr(S−1/m − 1)1/n, m = 1− 1/n,

kr,w(S) =√S[1−

(1− S1/m

)m]2,

kr,g(S) =√

1− S[1− S1/m

]2m.

The van Genuchten parameters for all materials are

given in Table 3, and in Figure 7 we show the capillary

pressure curves of all materials. The Interface curve is

so small compared to the other capillary pressure curves

that is practically invisible in Figure 7.

Figure 8 is an enlarged part of Figure 7, which shows

drastic difference between the curve in the interface and

the curves in other materials. This difference is a major

problem for numerical simulation.

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pc [M

Pa]

Water saturation

Capillary pressure curves

Bentonite plugGeological medium

Access driftEDZ

Interface

Fig. 7 Capillary pressure curves for all materials.

Note that the benchmark deals with water-gas flow

in a porous medium under high capillary pressures which

leads to a degeneracy in the system.

The initial conditions. In the initial moment the Ge-

ological medium and the EDZ are saturated by water.

The Access drift and the Bentonite plug have water

0

0.5

1

1.5

2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pc

[MP

a]

Water saturation

Bentonite plugGeological medium

Access driftEDZ

Interface

Fig. 8 A zoom of the capillary pressure curves for all mate-rials.

saturation Sw = 0.7 and in the Interfaces we have

Sw = 0.05. Water saturated media (Geological medium

and EDZ) have the initial water pressure of 5 MPa. All

unsaturated media (Access drift, Bentonite plug, Inter-

faces) have the initial gas pressure of 0.1 MPa, and the

water pressure is deduced from the capillary pressure

law. Note that in the initial moment the phase pres-

sures are discontinuous.

The gas source term is active the first 10,000 years

and for one canister it produces a quantity of the gas

of 100 mol/year. After 10,000 years the gas production

stops. The gas source is implemented as the gas flux

boundary condition imposed on the boundary of the

waste canister.

The boundary conditions are the following: at the

outer radius of the calculation domain r = 20 m we have

Dirichlet’s boundary conditions Pw = 5 MPa, Sw = 1.

At the boundary of the canister the water flux is

zero and the gas flux is given by the gas source term.

At x = 0 m, for r ≤ 3 m, the boundary conditions

are of the Dirichlet’s type variable in time for water and

gas. The representation of these variations are given in

Figure 10. The gas pressure is deduced from the capil-

lary pressure law.

At all other parts of the boundary no flow boundary

condition is applied.

5.2 Output results

The benchmark requests different outputs from the sim-

ulation.

5.2.1 Evolution with time of fluxes through surfaces

Central concern of this benchmark is migration of the

gas produced by the canister corrosion. A good measure

of that migration is given by evolution in time of the


Fig. 9 Boundary conditions in 2D computational domain.The waste canister is not a part of the computational domain.On its boundary the gas flux corresponding to the gas sourceterm is imposed.

phase fluxes through certain characteristic surfaces in

the model, which are given below.

Outer boundary of the model at r = 20 m (Sout in

Figure 11), fluxes counted positively out of the model.

Drift wall (Sdrift in Figure 11), fluxes counted pos-

itively toward the drift. Outside surface of the EDZ,

separated in 3 sections (see Figure 11): SEDZ1 (around

canister), SEDZ2 (around plug) and SEDZ3 (drift EDZ).

Fluxes counted positively out of the EDZ toward the

undisturbed rock. Inner cell surfaces (see Figure 11) :

Scell (section including interface and EDZ at canister-

plug junction), Sint1 (interface at canister-plug junc-

tion), Sint2 (interface at the drift wall). Fluxes counted

positively toward the drift.

5.2.2 Evolution with time along lines

Time evolution of all model variables are requested over

several lines. Lines at constant radius (see Figure 12):

Lint (passes through the interface), LEDZ (just outside

the cell EDZ), Lrock (inside the rock at a 5 m radius)

Lines at constant x (see Figure 12): Lx=0 m and

Lx=60 m (boundaries of the model), Lplug (in the mid-

dle of the plug), Lcell (in the middle of the canister).

5.2.3 Evolution with time at given points

Time evolution of all model variables are requested in

12 points (see Figure 13). Points 1 to 4, are at the same

radius as the centre of the interface: P1 and P4 (at the

boundaries), P2 (in the middle of the canister), P3 (at

canister-plug junction). Points 5 and 6 are at the same

radius as the centre of the cell EDZ: P5 (in the mid-

dle of the canister), P6 (at the canister-plug junction).

Point 7 is in the middle of the drift EDZ on x = 0

-4

-2

0

2

4

0.1 1 10 100 1000 10000 100000

Wate

r pre

ssure

[M

Pa]

Time [years]

Outer radius constant pressure

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.1 1 10 100 1000 10000 100000

Wate

r satu

ration

Time [years]

Outer radius constant saturation

Fig. 10 The Dirichlet boundary conditions on the Accessdrift boundary, for x = 0 and 0 ≤ r ≤ 3 m.

boundary. Points 8 to 12 are at 5 m radius: P8 and P12

(at the boundaries), P9 (at the same x as the middle ofthe canister), P10 (at the same x as the canister-plug

junction), P11 (at the same x as the intersection of the

drift and the interface).

5.3 Main difficulties

The geometry of the domain present the first difficulty

because of small diameter of the Interface (1 cm), com-

pared to other dimensions of the computational do-

main. Large computational time (100,000 years) force

us to use relatively coarse grid which then leads to a

grid with large difference in the elements size.

Second difficulty is the large difference in the capil-

lary pressure curves in the Interfaces and the curves in

the EDZ and the Bentonite plug. The continuity of the

capillary pressure through the material interface makes

the saturation in the Interface very sensible to varia-

tions of the saturations in surrounding materials, the

EDZ and the Bentonite plug. This instability is very


Fig. 11 Schematic representation of the surfaces troughwhich fluxes will be calculated.

Fig. 12 Schematic representation of the lines along whichresults should be calculated.

Fig. 13 Schematic representation of the points where resultsshould be calculated.

strong in the Interface near the plug and to avoid nu-

merical oscillations in the Interface, which can prevent

the Newton solver from converging, the time step must

often be reduced. This is serious difficulty because of

large final time of simulation.

Remark. Benchmark test case presented here is sim-

plified with respect to the original benchmark in two

aspects. First, we did not model gas dissolution in wa-

ter and dissolved gas transport by the convection and

the diffusion. We assume that all generated gas will be

present in the gas phase and we study its migration.

Second, we have changed the permeability of the In-

terface near the Canister, from originally 10−12 m2 to

10−18 m2. Original, very large permeability in the In-

terface near the Canister prevented the Newton solver

from converging, except for small time steps. No other

changes to the original data were done (see also foot-

note on page 8).

6 Simulation results

6.1 Simulation code and the grid

We have used simulation code developed in house based

on the numerical scheme presented in Section 3. The

non linear system of discretized equations is solved by

the inexact Newton method, using ILU preconditioned

GMRES linear solver. For linear and non linear solvers

we have used PETSc package, [5]. Fully implicit time

stepping was controlled by the convergence of the New-

ton solver.

Fig. 14 Simulation grid.

The grid used in the simulation, shown in Figure 14,

is composed of 1932 elements in the primary grid and it

is mostly uniform rectangular grid, except in the Access

drift and the EDZ, where general quadrilaterals and

triangles are used to fit the given geometry. In the EDZ

we have used five layers of elements, and four layers

in the Interfaces. Dimensions of the elements in radial

direction go from 25 mm in the Interface, to 1.6 m in

the Geological medium.

The effective capillary pressure curves were calcu-

lated as tables in all interface points before beginning

of the simulation; 9000 points are used in each table.

Since the effective capillary pressure curves depend on

local mesh geometry, we obtain one curve in each ver-

tex on the material interface. As an illustration, in Fig-

ure 15 we show one effective capillary pressure curve at


the material interface between the EDZ and the Geo-

logical medium. Corresponding material curves are also

shown. From Figure 15 we see, as well as from the def-

inition (9), (10), that the effective capillary pressure

curves have the same monotonicity properties as the

material curves and represent a mesh dependent mean

value of the material capillary pressure curves. In Fig-

ure 16 we show one effective capillary pressure curve

on the boundary of the EDZ and the Interface near the

Canister. Since the geometry of the primary grid near

the Canister is very regular all effective curves are clus-

tered in a few groups of almost identical curves. We

can also note in Figure 16 that the material curve of

the Interface is too small to be represented at the scale

of the figure.

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

Capill

ary

pre

ssure

[M

Pa]

Water saturation

effectiveEDZ

Geol. med.

Fig. 15 The effective capillary pressure between the EDZand the Geological medium.

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Capill

ary

pre

ssure

[M

Pa]

Water saturation

effectiveInterface

EDZ

Fig. 16 The effective capillary pressure between the EDZand the Interface.

6.2 Gas pressure evolution

The gas pressure is a primary variable in our model.

It is taken to be equal to the water pressure in wa-

ter saturated region. From imposed initial conditions

the gas pressure is initially discontinuous. It is assumed

that the fully saturated materials (EDZ and Geological

medium) are initially put into contact with unsaturated

materials (Interfaces and Access drift) which results in

discontinuity of water and gas pressures in the initial

moment. After the start of the simulation the gas pres-

sure becomes immediately continuous, as it can be seen

from Figure 17, for t = 1 day. Then, in the EDZ and the

Access drift, it stays relatively small and uniform until

sufficient quantity of gas is generated, when it starts to

grow (see Figure 17 for t = 2, 000 years). The maximum

gas pressure is attained at one corner of the Canister

at the end of the gas production interval (t = 10, 000

years).

6.3 Evolution with time of fluxes through surfaces

In the first year of the simulation the initial water pres-

sure discontinuity produce very strong flow of water

from the Geological medium to the EDZ. The water

flow through SEDZ1, after initial very strong flow into

EDZ, shortly changes direction and between 0.001 an

0.1 year flows to the Geological medium. After that

stays directed towards the EDZ (Figure 18). The water

flow through SEDZ3 (Figure 19) stays directed to the

EDZ until extinction of the gas source when its goes to

zero, but shows some small oscillations at later times,

obviously of numerical origin. The water flow through

SEDZ2 (Figure 18) is slightly positive (from the EDZ to

the Geological medium) which is a consequence of much

stronger water flow through the EDZ (surface Scell,

Figure 18) towards the Access drift. The flow of water

from the EDZ to the Access drift (surface Sdrift, Fig-

ure 19) stays very strong even after extinction of the gas

source. Water flow through the interfaces is negligible.

The EDZ is the main path for the water flow, which

goes from the Geological medium to the Access drift.

The Interfaces do not participate to the water flow since

they are mostly field with the gas (see Figure 26) keep-

ing water mobility very low. They prevent the flow of

water from the EDZ to the Bentonite plug, which resat-

urates primary from the Access drift. This slows down

the plug resaturations, which take approximately 2,000

years and explains small water flux from the EDZ to the

Geological medium through SEDZ2. Finally, the water

flux through the EDZ-Acess drift interface stops ap-

proximately at 23,000 years, when the hole domain, ex-

cept the Interfaces, is resaturated.


Fig. 17 Gas pressure (in MPa) in the domain at different times: 1 day, 1 year, 2,000 years and 100,000 years.

-150

-100

-50

0

50

100

0.1 1 10 100 1000 10000 100000

Wate

r flux [kg/y

ear

]

Time [years]

S_EDZ_2S_EDZ_1

S_cellS_int_1S_int_2

S_out

Fig. 18 Water fluxes through SEDZ1, SEDZ2, Scell, Sint1

and Sint2.

The gas flow in Figures 20 and 21 shows that the

gas is flowing through the EDZ in direction of the Ac-

cess drift (surface Scell, Figure 21) and and form the

EDZ to the Access drift, through Sdrift (Figure 21).

These two fluxes are the strongest ones. The flow of

the gas through the Interfaces (Sint1 and Sint2 in

Figure 20) is approximately 10-15 times smaller than

the flow through the EDZ in the direction of the Ac-

cess drift (Scell in Figure 21), and it is smaller than

the gas flow from EDZ into the undisturbed rock. We

may conclude, therefore, that importance of the inter-

-400

-200

0

200

400

0.1 1 10 100 1000 10000 100000

Wate

r flux [kg/y

ear]

Time [years]

S_driftS_EDZ_3

Fig. 19 Water fluxes through Sdrift and SEDZ3.

face for the gas migration is weak. The most of the

gas flows through the EDZ, whose properties has the

largest influence on behavior of the system. However,

the Interfaces keep their importance as a capillary bar-

rier that slows down the resaturation of the Canister

and the Bentonite plug.

Let us also note that in the first year of the simula-

tion the gas is flowing from the Access drift to the EDZ

(Sdrift in Figure 21) under the influence of the bound-

ary condition imposed on the boundary of the Access

drift and because the gas is replacing the water that is

entering into the Access drift.


The flow of the gas from the EDZ to the Geologi-

cal medium (surfaces SEDZ1 and SEDZ2 in Figure 20) is

approximately 5-8 times smaller than the flow through

the EDZ and the Interface (surface Scell in Figure 21),

and is a consequence of the gas replacing the water that

is entering into the EDZ. Similarly, we have the flow of

the gas from the Access drift EDZ into undisturbed

rock (SEDZ3 in Figure 21) which is induced by a strong

flow of the water in the opposite direction. After ap-

proximately 23,000 years the gas is finally evacuated to

the Access drift and from there, through the boundary

with imposed pressure, out of the system.

6.4 Evolution with time at given points

In this subsection we show the water saturation and

the water and the gas pressures in some of prescribed

points.

In Figure 22 we show water saturation in all points

outside of the EDZ and the Interface, and in Figure 23

we show the points in the Interface and the EDZ. We

see from Figure 22 that desaturation of the Geologi-

cal medium is very weak and that it is stronger in the

points closer to the Access drift, due to strog water flow

into the Access drift and some counter flow of the gas.

This weak desaturation is to be expected due to weak

permeability and strong capillary pressure curve of the

undisturbed rock.

In the points P2, P3 and P4 (Figure 23) which lay

in the Interface we observe a low water saturation un-

til 27,000 years when the resaturation of the Interface

starts. This behaviour is a consequence of very low

capillary pressure curve in the Interface (see Figure 7)

which keeps the Canister and the Bentonite plug iso-

lated of the EDZ and the Geological medium and slows

-0.01

0

0.01

0.02

0.03

0.04

1 10 100 1000 10000 100000

Gas flu

x [kg/y

ear]

Time [years]

S_EDZ_2S_EDZ_1

S_int_1S_int_2

S_out

Fig. 20 Gas fluxes through SEDZ1, SEDZ2, Sint1, Sint2 andSout.

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.1 1 10 100 1000 10000 100000

Gas flu

x [kg/y

ear]

Time [years]

S_driftS_cell

S_EDZ_3

Fig. 21 Gas fluxes through SEDZ3, Scell and Sdrift.

0.994

0.995

0.996

0.997

0.998

0.999

1

1.001

1.002

0.1 1 10 100 1000 10000 100000

Wate

r satu

ration

Time [years]

P1P8P9

P10P11P12

Fig. 22 Time evolution of the water saturation in points P1

and P8-P12.

down the resaturation of the Plug (and the Canister

in reality). Almost all gas must be evacuated from the

EDZ for resaturation of the Interface to start.

In the points P5 and P6 that lay in the center of

the EDZ a desaturation is significant only after 2,000

years (see Figure 23). This means that the generated

gas is efficiently evacuated to the Access drift until ap-

proximately 2,000 years, when the gas starts to accu-

mulate in the EDZ. In the point P7 the desaturation is

more significant (Figure 23), but this is influenced by

the boundary condition imposed on boundary of the

Access drift (compare to imposed water saturation in

Figure 10).

The gas pressure is shown in Figure 24 for points

P2-P7 which lay in the EDZ and in the Interfaces. In

the remaining points, P1 and P8-P12, we present the

water pressure in Figure 25.

In Figure 24 we show also the boundary condition

on the gas pressure imposed on the boundary of the


0

0.2

0.4

0.6

0.8

1

0.1 1 10 100 1000 10000 100000

Wate

r satu

ration

Time [years]

P2P3P4P5P6P7

Fig. 23 Time evolution of the water saturation in pointsP2-P7.

Access drift. This pressure is supposed to model inter-

action of the computational domain with the rest of the

repository. In this figure the gas pressure in the points

P2 and P5 are almost the same; similarly, the pressure

in the points P3 and P6 are almost the same, and the

pressures in P4 and P7 are almost equal to the gas pres-

sure imposed on the Access drift boundary. Since the

points P2, P5 and P3, P6 are close, this is to be ex-

pected. In the other hand, in the points P4 and P7 we

see strong influence of the Dirichlet boundary condition

at the boundary of the Access drift. The maximum gas

pressure achieved in the simulation is 5.43 MPa. This

results show that the pressurisation at the cell scale, due

to the gas generation by the canister corrosion, strongly

depends on interaction of the Canister with the rest of

the repository, which is represented here by time vary-

ing boundary condition on the Access drift.

0

1

2

3

4

5

6

0.1 1 10 100 1000 10000 100000

Gas p

ressure

[M

Pa]

Time [years]

P2P3P4P5P6P7

Fig. 24 Time evolution of the gas pressure in points P2-P7

and the gas pressure imposed on the boundary of the Accessdrift.

Evolution in time of the water pressures in the points

P1 and P8-P12 are presented in Figure 25. The initial

discontinuity of the water pressure produce immediate

pressure drop in the Geological medium. This pressure

drop attains it minimum before 100 years due to strong

initial water flux into the EDZ and the Access drift.

When these fluxes stabilize (see Figures 18 and 19) the

water pressure starts to increase, until reaching its ini-

tial value of 5 MPa after 10,000 years.

0

1

2

3

4

5

0.1 1 10 100 1000 10000 100000

Wate

r pre

ssure

[M

Pa]

Time [years]

P1P8P9

P10P11P12

Fig. 25 Time evolution of the water pressure in points P1

and P8-P12.

One of the main concerns in the nuclear waste repos-

itory safety assessment is the raise of the pressure of the

gas produced by the canister. In our simulation we find

that the pressure raise is quite small (0.43 MPa) and

therefore it does not present a threat to the safety of therepository. However, we have seen that the gas pressure

is strongly influenced by the gas pressure imposed on

the boundary of the Access drift, which is by itself an

assumption on interaction between one repository cell

with the rest of the repository. The other parameters

that influence the gas pressure buildup are the proper-

ties of the EDZ which determine the ability of the EDZ

to conduct the gas into the Acces drift, and thus reduce

the pressure buildup.

6.5 Evolution with time along lines

From solution over different lines we present only the

water saturation over line Sint which goes through the

Interfaces (Figure 26) which is the most interesting one,

since the behavior of the solution is the most complex

in the Interfaces. The Interfaces have a very small cap-

illary pressure curve which keeps it desaturated if there

is some quantity of the gas in the EDZ. Therefore we


see that after initial partial resaturation at 2,000 years,

water saturation diminish and stays approximately con-

stant up to 25,000 years. Around 27,000 years the water

saturation decrease to its minimum value (see also Fig-

ure 23) – probably due to local accumulation of the gas

that is evacuated from the Geological medium and the

EDZ to the area close to the Interfaces – and then starts

slowly to increase again. The resaturation starts from

the Access drift side.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

t=0.0 yearst=2000 years

t=10000 yearst=27000 yearst=60000 years

t=100000 years

Fig. 26 Water saturation over the line Sint given at severaltime instances.

It is of great importance to evaluate the role of the

Interfaces in the gas migration in the repository since

the meshing of the Interfaces in 3D model, on the scale

of whole repository, produces prohibitively large grids.

From our simulations we can conclude that the role ofthe Interface in the gas migration in minor compared

to the role of the EDZ. The main impact of the Inter-

face on the gas migration is to slow down resaturation

of the Bentonite plug (and the Canister). In the other

hand, modeling of the Interfaces as a porous medium

with very low capillary pressure curve is not an accurate

description of physical reality. We conclude, therefore,

that an upscaling of the Interfaces with the EDZ, that

is forming a new EDZ with slightly different properties

and eliminating the Interfaces, could produce a simpler

model of the repository, without perturbing important

gas migration characteristics.

7 Conclusion

We have presented a vertex centered finite volume nu-

merical scheme for the water-gas flow through highly

heterogeneous porous media. The method is fully im-

plicit and it includes a treatment of heterogeneities by

using an upscaling technique and the concept of the ef-

fective saturation. Our numerical model is verified with

1D semi-analytical solutions in heterogeneous media.

We also apply the method to the FORGE cell scale

benchmark developed for the evaluation of the hydro-

gen migration in an underground nuclear waste repos-

itory. Note that the benchmark deals with water-gas

flow in a porous medium under high and discontinuous

capillary pressures which leads to a degeneracy in the

system. The benchmark is presented in all details fol-

lowed by the numerical results obtained by our C++

homemade code based on the numerical scheme pre-

sented in this paper. The simulator uses the open source

library PETSc for solving discrete nonlinear systems.

The simulation shows that the most of the gas is

evacuated through the EDZ towards the Access drift.

Only a small quantity of the gas enters into the Geo-

logical medium and then slowly returns into the EDZ

to be evacuated from the system. The Interfaces have

only a secondary role in the gas transport, but they

slow down the process of re-saturation of the Bentonite

plug by keeping it isolated from the EDZ. The whole

system is strongly influenced by the Dirichlet boundary

condition imposed on the Access drift boundary.

Acknowledgements The research leading to these resultshas received funding from the European Atomic Energy Com-munitys Seventh Framework Programme (FP7/2007-2011) un-der Grant Agreement no230357, the FORGE project. Thiswork was partially supported by the GnR MoMaS (PACEN/CNRS,ANDRA, BRGM,CEA, EDF, IRSN) France, their supportsare gratefully acknowledged.

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