numerical simulation by homogenization of …heterogeneous porous media. computational geosciences,...

Introduction Mathematical Model FORGE Benchmark Numerical scheme Numerical results Conclusion

Numerical Simulation by Homogenization ofMultiphase Flow in Heterogenous Porous Media

E. Ahusborde1, B. Amaziane1, M. El Ossmani2, M. Jurak3

1 IPRA-LMA, UMR CNRS 5142, University of Pau, France.2 University Moulay Ismail, ENSAM, Meknes, Morocco.

3 University of Zagreb, Department of Mathematics, Croatia.

Conference in Honor of Abderrahmane BendaliDecember 12th−14th, 2017, Pau.


Outline

1 Introduction

2 Mathematical Model

3 FORGE Benchmark

4 Numerical scheme

5 Numerical results

6 Conclusion


Introduction


Introduction

Numerical modelling of gas migration and two-phase flowthrough engineered and geological barriers for a deep repositoryfor radioactive waste.French Research Group MoMaS (PACEN/CNRS ANDRA,BRGM, CEA, EDF, IRSN): http://www.gdrmomas.org/Euratom FP7 Project FORGE (Fate Of Repository Gases):http://www.bgs.ac.uk/forge/home.html

Goal:

Couple DuMuX to solve the coupled and non-linear system describingmiscible compressible two phase flow in heterogeneous porous mediaand an upscaling strategy.

[1] DuMuX , DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media. DuMuX

web-page : http://www.dumux.org.


Introduction

References

B. Amaziane, M. El Ossmani, M. Jurak, Numerical simulation of gas migration through engineered and geologicalbarriers for a deep repository for radioactive waste, Computat. and Visualiz. in Science 15:1, (2012), 3–20.

E. Ahusborde, B. Amaziane, M. Jurak, Three-dimensional numerical simulation by upscaling of gas migration throughengineered and geological barriers for a deep repository for radioactive waste, Geological Society Special Publication,415:1 (2015) 123–141.

B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski, Homogenization of immiscible compressible two-phase flow inporous media : application to gas migration in a nuclear waste repository, SIAM J. Multiscale Model. Simul., 8:5(2010) 2023-2047.

B. Amaziane, L. Pankratov, A. Piatnitski, An improved homogenization result for immiscible compressible two–phaseflow in porous media, Networks and Heterogeneous Media (NHM), 12:1 (2017), 147–171.


Two-phase miscible compressible flow equations

Notation: The index α ∈ {l,g} refers to the phase (liquid and gas),while the superscript c ∈ {H2O,H2} refers to the component.

Generalized Darcy-Muskat’s law

Uα =−krα(Sα)

µα

K (∇Pα −ραg) .

Mass conservation law for each component

∂

∂ t(Φ∑

α

ραXcαSα)+∑

α

∇ · (ραXcαUα +Jc

α)−∑α

Qcα = 0. (1)

Diffusive fluxes

Jcα =−ΦραSα

1τ2 Dc

α∇Xcα . (2)


Two-phase miscible compressible flow equations

Capillary pressure law : Van Genuchten-Mualem model

Pg−Pl = Pc(Sl), (3)

with

Sle =Sl−Slr

1−Slr−Sgr, (4)

Pc(Sl) = Pr(S−1/mle −1)1/n, (5)

krl(Sl) =√

Sle[1− (1−S1/mle )m]2, (6)

krg(Sl) =√

1−Sle(1−S1/mle )2m. (7)

Henry law

CH2l = HH2(T)P

H2g , CH2

l =XH2

l ρl

MH2

. (8)


Description of the benchmark

Figure: Schematic representation of a repository for HLW.



Figure: Schematic representation of the module to be simulated. Definition of theA-A’, B-B’ cross sections.



Figure: Schematic representation of the A-A’ vertical cross section.



Figure: Schematic representation of the module to be simulated. Definition of theA-A’, B-B’ cross sections.



Figure: Schematic representation of the B-B’ vertical cross section.


Physical parameters

MaterialsParameter Interface facing Interface facing Interface facing

plug waste backfill and edzK [m2] 5 ·10−18 10−12 10−15

Porosity [%] 30 100 40Van Genuchten parameters

n [-] 4 4 4Pr [Pa] 104 104 104

Tortuosity 1 1 1Materials

Parameter Backfills Bentonite EDZ COXK [m2] 5 ·10−17 10−20 5 ·10−18 10−20

Porosity [%] 40 35 15 15Van Genuchten parameters

n [-] 1.5 1.6 1.5 1.5Pr [Pa] 2 ·106 1.6 ·107 1.5 ·106 1.5 ·107

Tortuosity 2 4.5 2 2

Table: Physical parameters.


Initial conditions and source term

At the initial moment the pressures are discontinuous.

Variables MaterialsInterfaces Backfills Bentonite EDZ Geological

plugs MediumSl Sl = 0.05 Sl = 0.7 Sl = 0.7 Sl = 1 Sl = 1

Pg,Pl Pg = 0.1MPa, Pl = Pg−Pc(Sl) Pl = Pg = 5MPa

The source term:

Implemented as Neumann boundary condition.

Q = 100 mol/year for t < 10000 years; Q = 0 for t > 10000years.


Finite volume scheme & Upscaling

Home-made code in C++ for the upscaling.

Simulator : DuMuX , DUNE for Multi-{Phase, Component,Scale, Physics, ...} flow and transport in porous media(http://www.dumux.org).

Model used : 2p2c module which implements two-phase flowwith two components.Coupled fully-implicit approach.Spatial discretization: vertex-centred finite volume approach.Time discretization: implicit Euler scheme.Mesh generator used: Gmsh + conversion into Dune Grid Format(DGF).


Scale up of the benchmark

Complete benchmark model can be represented only on a very finegrid. By means of upscaling we need to represent all model data on acoarser simulation grid. We upscale:

Porosity Φ(x) and absolutepermeability K(x);Capillary pressure Pc(x,S) andrelative permeability curveskrα(x,S);Tortuosity coefficient τ(x).

The upscaling is based on a localcapillary equilibrium hypothesis.Heterogeneous quantities are replacedby homogeneous, effective ones.

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� � � � � � � � � � � � � � � ��

Numerical simulation model

Finely gridded geological model


Upscaling of permeability and porosity

Let the upscaling REV be denoted by V .We have to solve in V the following steady-state Darcy’s flow problem for i = 1,2,3:

−∇ · (K(x)∇Pi) = 0 in V,

Pi = xi on ∂V.

where xi is the i-th coordinate. Then we calculate the mean flux

< K∇Pi >=1

vol(V)

∫V

K(x)∇Pi(x)dx,

and the effective absolute permeability Kh is given by Khei =< K∇Pi >.The effective porosity Φh is computed such that pore volume is exactly conservedbetween the fine and coarses scales:

Φh =

1vol(V)

∫V

Φ(x)dx.

Amaziane, B., Koebbe, J. (2006). JHomogenizer: A computational tool for upscaling permeability for flow inheterogeneous porous media. Computational Geosciences, 10(4), 343-359.

Bourgeat, A., Jurak, M. (2010). A two level scaling-up method for multiphase flow in porous media; numericalvalidation and comparison with other methods. Computational Geosciences, 14(1), 1-14.


Effective capillary pressure

For given capillary pressure level u, we calculate the localsaturation distribution S0(x) given by the local capillaryequilibrium:

u = P1c(S1) = · · ·= Pn

c(Sn), Sh =∫

VΦ(x)S0(x)dx/Φ

h.

Effective capillary pressure is then Phc(S

h) = u.

Sl

Pc

P 1c P 2

c

P hc

u

S1 S2Sh 1


Effective relative permeability

For given effective Sh, we calculate Phc(S

h) and correspondinglocal saturation distribution S0(x).The effective relative permeability function krh

α(Sh)i, in the

space direction i, is defined by the formula:

krhα(S

h)i =1

vol(V)

∫V

K(x)krα(x,S0(x))∇Pi · eidx/Ki,i,

where Pi satisfies

−∇ · (K(x)krα(x,S(x))∇Pi) = 0 in V,

Pi = xi on ∂V.


Effective tortuosity

For given effective Sh calculate Phc(S

h) and corresponding localsaturation distribution S0(x).The effective tortuosity function τh

α(Sh)i, in the space direction i,

is defined for the effective saturation Sh and the phase α by:

ΦhShα

τhα(Sh)2

i=

1vol(V)

∫V

Φ(x)Sα(x)τ(x)2 ∇ξ

αi · eidx,

where ξi satisfies

−∇ · (Φ(x)Sα(x)τ(x)2 ∇ξ

αi ) = 0 in V,

ξαi = xi on ∂V.


Upscaling for the benchmark

U1. Elimination of the Interfaces:EDZ+Interface → homogeneous blockBackfill+Interface→ homogeneous blockMainDriftPlug+Interface→ homogeneous block

U2. Intermediate Upscaling:Canister+Interface+EDZ→ homogeneous blockPlug+Interface+EDZ→ homogeneous blockBackfill+Interface+EDZ → homogeneous blockMainDriftPlug+Interface→ homogeneous block

U3. Full Upscaling:Canister+Plug+Interface+EDZ+GM→ homogeneous blockBackfill+Interface+EDZ→ homogeneous blockMainDriftPlug+Interface→ homogeneous block



Figure: U1 grid: 510 112 elements.





Upscaling near the drifts

U1. Elimination of Interfaces:Backfill+Interface→ homogeneousblock

U2. Intermediate upscaling:Backfill+Interface+EDZ→homogeneous block

U3. Full upscaling:The same.

EDZInterface

Backfill

EDZEffective

U1

Effective

U2


Upscaling near the canisters

Geological medium

Interface

EDZ

Canister

Geological medium

Upscaled

Canister

U1

Geological medium

Upscaled

U2

Upscaled

U3

U1. EDZ+Interface→ homogeneous block

U2. Canister+Interface+EDZ → homogeneous block

U3. Canister+Plug+Interface+EDZ+GM → homogeneous block


Output results

Figure: Definition of the points where results have been extracted, of the lineL-MD and of the surfaces F-C25 and F-Pd.


Results for model U2 : Fluxes

Figure: Dissolved hydrogen fluxes (Left) and Gaseous hydrogen fluxes (Right).

Gaseous hydrogen total fluxes are three orders of magnitude larger than thedissolved hydrogen total fluxes.

In gaseous phase, the advective fluxes dominate diffusive fluxes by severalorder of magnitudes, while in the liquid phase they are comparable.


Results for model U2 : Data in points

0 1 2 3 4 5 6 7 8

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]

Gas pressure, Cell 50

C50-1 PgC50-2 PgC50-3 Pg

0 1 2 3 4 5 6 7 8

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]


C25-1 PgC25-2 PgC25-3 Pg

0 1 2 3 4 5 6 7 8

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]


C1-1 PgC1-2 PgC1-3 Pg

0

1

2

3

4

5

6

7

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]

Gas pressure, Main Drift

Pd-1 PgPu-1 Pg

Figure: Gas pressure in points.

The pressure maximum is about 7MPa.


Results for model U2 : Data on line L-MD

Figure: Gas pressure (left) and gas saturation (right) in the Main drift.

Pressurisation of the Main Drift, which does not exceed 7 MPa, and return tothe equilibrium which is not completely done even after 100,000 years.

The Main Drift plugs are almost fully resaturated after 1,000 years.


Comparison between U2 and U3 models

0 1 2 3 4 5 6 7 8

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]

Gas pressure, C25-3

U3U2

-2-1 0 1 2 3 4 5 6 7

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]

Water pressure, C25-3

U3U2

0.82 0.84 0.86 0.88

0.9 0.92 0.94 0.96 0.98

1

100 101 102 103 104 105

time [years]

Water saturation, C25-3

U3U2

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

100 101 102 103 104 105

pre

ssure

[M

Pa]

time [years]

Capillary pressure, C25-3

U3U2

Figure: Solution in the point P-C25-3 formodels U3 and U2.

CPU timeModel U3 4 hours (8 proc)Model U2 1 month (24 proc)

Table: CPU time for models U3and U2.


Gas pressure and saturation

Figure: Gas pressure and saturation at t=1000y (top) and t=10000y (bottom) for theU2 model.


Comparison with other participants

10-2

10-1

100

101

102

103

104

105

time (years)

0

1

2

3

4

5

6

7

8

9

Gas

Pre

ssu

re (

Mp

a)

Present model U3Present model U2LEINWMOANDRANDA

10-2

10-1

100

101

102

103

104

105

time (years)

0

1

2

3

4

5

6

7

8

Gas

Pre

ssu

re (

Mp

a)

Present model U3Present model U2LEINWMOANDRANDA

Figure: Comparison of the gas pressure for the points P-C50-3 (left) and P-Pd-1(right).


Conclusion and acknowledgements

The maximum pressure in the module will be about 7 MPa.

The advection is the main way of hydrogen transport.

Transport of hydrogen dissolved in water is about three orders ofmagnitude less significant than the transport of gaseoushydrogen.

This work was partially supported by the Euratom FP7 ProjectFORGE under grant agreement 230357 and the GnR MoMaS(PACEN/CNRS, ANDRA, BRGM,CEA, EDF, IRSN) France,their supports are gratefully acknowledged.

numerical simulation by homogenization of …heterogeneous porous media. computational geosciences,...

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