DYNAS Workshop 6,7,8/12/04 Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous Media E. Mouche, C. Le Potier, P. Maugis, L.V. Benet. Commissariat

DYNAS Workshop 6,7,8/12/04

Mixed Hybrid Finite Element and Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous Iterative Methods for Flow in Porous

MediaMedia

E. Mouche, C. Le Potier, P. Maugis, L.V. Benet.E. Mouche, C. Le Potier, P. Maugis, L.V. Benet.

Commissariat à l'Energie Atomique, C.E. de Saclay, Gif sur Yvette Commissariat à l'Energie Atomique, C.E. de Saclay, Gif sur Yvette Cedex, FranceCedex, France

Email: [email protected]: [email protected]


Summary

Cast3M Code Richard’s equation & MHFE Formulation Iterative resolution Illustrations Air Water migration + Temperature Conclusion


Cast3M (CASTEM2000)http: //www-cast3m.cea.fr)

Finite Element Code (CEA) PDE : Structural mechanics, Fluid Mechanics,

Thermics, … Object oriented code :

Object2 = Operator Object1 (options) « Toll Box » (500 operators) 2 languages : user’s (Gibiane) & developer’s

(Esope) Pre & Post processors FE, MHFE & FV


Cast3M (CASTEM2000)

Porous media : Darcy (2d & 3d), Transport equations (adv. disp. diff.), …(See recent paper in computational

geosciences 2004, Gilles Bernard Michel et al, about the COUPLEX test case)

Basic brick for multiphase flow :

tstanconsC,B,A

FCJ

)t,x(SJ.BFt

FA

Iterative Solutions For multiphase flow F, G, …


Richard’s Equation, Iterative Resolution

Mixed Hybrid Finite Element formulation for water flow in unsaturated porous media », C.Le Potier et al., CMWR XII, 1998

)zh)((KU

U.t

h)h(C

dvU.dvt

h)h(C

dn.whdw.hdw.UK 1

MHFE

h : head pressure, U : Darcy velocity, C : Capillary capacity, K : permeability,Θ water content

FE Mesh(QUA4)

Face center Velocity, head

MHFE MeshEFMH (QUAF)

Element center, head



1i,1ni,1n1i,1n

1i,1nn1i,1n

i,1n

H)h(KU

U.t

HH)h(C

andzhH

Implicit time discretization & Picard algorithm

Time step : nIterative step : i



FN k

kFaceFace

Faces )(volume

)(surfacen.U

N

1

t

1

Time step strategy :

Parameter (X= C & K) homogeneization : Different types of means : 1) functions of X values on the faces, X=F(X(Hface)) , XA (arithmetic), XG (geometric), XH (harmonic) or centered value X = X(Hcenter)

MHFE MeshMHFE (QUAF)

Element center; X = X(Hcenter)

Face center;X =F(X(Hface))


Infiltration in a Heterogeneous Soil

Infiltration

Water

content

30mn

1 day

Overflow

Not at Scale

Clay lens

Infiltration and

Recharge of an aquifer

in a heterogeneous soil

Sand


Rainfall on a Slope (Cf DYNAS)

Localized RainfallUniform Rainfall

Runoff

T0

T1

Rainfall on a slope (Runoff, Infiltration, Recharge, Overflow),

Aquifer

Satu

ra tion



1. Importance of first time step in the convergence process. If well selected, convergence is achieved in 5-10 iterations.

2. Parameter homogeneization : Depending on the situation : Arithmetic mean or centered value. Arithmetic seems to lead to a better precision ?!.

3. No problem with the convective (gravity) term.

4. Heterogeneous media may lead to tough situations : flow from an impervious medium (low hydraulic diffusivity, D=K/C) to a pervious medium (high D)

5. No problem with the unsaturated - saturated transition (see 4.)


Air Water Migration

wa

wawc

wcaawcaa

ww

PP)(P

)w,a(gP.k

U

0UHU.t

))H((

0U.t

w : water , a : air , Hc : Henry’s constant , ρ : density , ω : porosity


Air Water Migration

zh)(KH.h)(KH)(K.

t

h)h(C)H1(

t

h)H(

zhh)(K.t

h)h(C

cwwcaawwcawaa

ccca

awcw

cawwc

c

w : water , a : air , Hc : Henry’s constant , ρ : density , ω : porosity

Rewritten in terms of capillary head & air head pressures


Air Water Migration

)UU.(t

hA

t

hA

)UU.(t

hA

2,21,2a

2,2c

1,2

2,11,1c

1,1

t

hhAU.U.

t

hhA

U.U.t

hhA

nc

1i,1nci,1n

1,21i,1n

1,21i,1n

2,2

na

1i,1nai,1n

2,2

i,1n2,1

1i,1n1,1

nc

1i,1nci,1n

1,1

And solved the same way as for Richard’s equation

It looks like the matrix system


Touma and Vauclin experiment Exp. and num. Analysis of two-phase infiltration in a partially saturated soil, TIPM (1) 1986

« Vertical infiltration in a sandy column with no lateral air flow and with air flow »

Air flowNo Air flow. The water infiltration is drastically slowed down by air


Air mass conservation problem THM and Geoch. Behaviour of clay barrier in radioactive waste repositories , Volckaert et al,

CCE Report EUR 16744 en 1996

« Vertical infiltration in a sandy column with no air flow and soluble air »

Air mass conservation Air Pressure


EVEGAS European Project Cannot find the reference !

«Production of a Hydrogen Bubble in a saturated porous media »

Does your code see the Cat’s ears ?

H2 H2


+ Temperature : Thermo Hydraulics

0)UUUU.(t

TA

t

hA

t

hA

0)UUUU.(t

TA

t

hA

t

hA

0)UUUU.(t

TA

t

hA

4,33,32,31,33,3A

2,3C

1,3

4,23,22,21,23,2A

2,2C

1,2

4,13,12,11,12,1C

1,1

Ah

tU U U U A

T

t

Ah

tU U U U U A

h

tA

T

t

A

N I CI

N I N N N N II

N I AI

N I N N N N N I CI

N II

N

1 11

1

1 11 1

1 2 1 3 1 4 1 21

2 21

1

2 21 1

2 1 2 1 2 3 2 4 2 11

2 31

3 31

,,

,,

, , , ,,

,,

,,

, , , , ,,

,,

,,

. .( )

. .( )

II

N I N N N N I CI

N I AIT

tU U U U A

h

tA

h

t

1

3 31 1

3 1 3 2 3 4 3 11

3 21

. .( ),

,, , , ,

,,

,

Physics

Numerics


Pollock D.W., WRR (22) 1986

« Simulation of Fluid Flow & Energy Transport Processes associated with HLW Disposal in Unsaturated Alluvium »

Saturation 10 -100 y.

Saturation 100 -1000 y.

Pollock’s results


Conclusion

Iterative methods work quite well. Not time consuming, as compared to « global »

methods (benchmarking with THM codes). Sometimes grandma’s tricks (choice of good variables)

must be introduced May become very tough with some media such as

unsaturated flow in fractured media, geothermy, … If you do it, have fun !

A paper is in preparation !

Documents

DYNAS Workshop 6,7,8/12/04 Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous Media E. Mouche, C. Le Potier, P. Maugis, L.V. Benet. Commissariat