ITERATIVE SOLUTION OF LINEAR EQUATIONS IN FINITE ELEMENT ANALYSIS
Mixed Hybrid Finite Element and Iterative Methods for Flow in Porous … · 2013-11-06 · DYNAS...
Transcript of 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]
DYNAS Workshop 6,7,8/12/04
Summary
Cast3M Code
Richard’s equation & MHFE Formulation
Iterative resolution
Illustrations
Air Water migration
+ Temperature
Conclusion
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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, …
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
Richard’s Equation, Iterative Resolution
∇−=
∇−=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
DYNAS Workshop 6,7,8/12/04
Richard’s Equation, Iterative Resolution
∑ 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))
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
Rainfall on a Slope (Cf DYNAS)
Localized RainfallUniform Rainfall
Runoff
T0
T1
Rainfall on a slope (Runoff, Infiltration, Recharge, Overflow),
Aquifer
Satu
ration
DYNAS Workshop 6,7,8/12/04
Richard’s Equation, Iterative Resolution
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.)
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
+ 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
DYNAS Workshop 6,7,8/12/04
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
DYNAS Workshop 6,7,8/12/04
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 !