mixed hybrid finite element and iterative methods for flow in porous … · 2013-11-06 · dynas...

DYNAS Workshop 6,7,8/12/04

Mixed Hybrid Finite Element andMixed Hybrid Finite Element andIterative Methods for Flow in PorousIterative Methods for Flow in Porous

MediaMedia

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

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

EmailEmail: emmanuel.mouche@cea.: [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), Transportequations (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

∇−=

=∇++∂

∂

rr

rrrIterative SolutionsFor multiphaseflow 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

∇+∇θ−=

∇−=∂∂

)zh)((KU

U.th

)h(Crrr

rr

Ω∇−=Ω∂

∂

Γ−Ω∇=Ω

∫ ∫

∫∫∫

Ω Ω

ΩΩΩ

−

dvU.dvt

h)h(C

dn.whdw.hdw.UK 1

rr

rrrrrr

MHFE

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

FE Mesh(QUA4)

Face centerVelocity, head

MHFE MeshEFMH (QUAF)

Element center,head



∇−=

∇−=Δ

−+=

+++++

++++

+

1i,1ni,1n1i,1n

1i,1nn1i,1n

i,1n

H)h(KU

U.t

HH)h(C

andzhHrr

rrImplicit timediscretization &Picard algorithm

Time step : nIterative step : i



∑ Ω

Ω×=

ΔFN k

kFaceFace

Faces )(volume

)(surfacen.U

N

1

t

1rr

Time step strategy :

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

ration



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

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

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

4. Heterogeneous media may lead to tough situations : flowfrom 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

( )( )

( )

θ+θ=ω

−=θ

=αρ−∇µ

−=

=+ρ∇+∂

θ+θρ∂

=∇+∂

∂θ

ααα

αα

wa

wawc

wcaawcaa

ww

PP)(P

)w,a(gP.k

U

0UHU.t

))H((

0U.t

rr

rrr

rr

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


Air Water Migration

( )( )

( )( ) ( )( )

∇+∇θρ∇+∇θρ+θρ∇

=∂

∂−ρ−

∂

∂θ+θ−ω

∇+∇+∇θ∇=∂

∂

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

t

h)h(C)H1(

t

h)H(

zhh)(K.t

h)h(C

cwwcaawwcawaa

ccca

awcw

cawwc

c

rrrrr

rrrr

w : water , a : air , Hc :Henry’s constant , _ : density ,_ : 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

rrr

rrr

Δ

−−∇−=∇+

Δ

−

∇−=∇+Δ

−

+++++++

+++

+++++

+

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

rrrr

rrrrAnd solvedthe same wayas forRichard’sequation

It looks likethe matrixsystem


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 andwith air flow »

Air flowNo Air flow. The water infiltrationis 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 andsoluble air »

Air mass conservation Air Pressure


EVEGAS European ProjectCannot find the reference !

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

Does your code seethe 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

rrrrr

rrrrr

rrrrr

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

,,

,,

, , , ,,

,,

,,

, , , , ,,

,,

,,

. .( )

. .( )

++

+ + +

++

+ + + +

+

+∇ = −∇ + + −

+∇ = −∇ + + + − −

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

r r r r r r

r r r r r r r

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

r r r r r r. .( ),

,, , , ,

,,,

Physics

Numerics


Pollock D.W., WRR (22) 1986

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

Saturation10 -100 y.

Saturation100 -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 asunsaturated flow in fractured media, geothermy, …

If you do it, have fun !

A paper is in preparation !

mixed hybrid finite element and iterative methods for flow in porous … · 2013-11-06 · dynas...

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