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Cell centered finite volume scheme for multiphase porous media flows with applications
in the oil industry
International Conference Scaling Up and Modeling
for transport and flow in porous media
Dubrovnik, Croatia
October 13rd-16th 2008
Léo Agelas, Daniele di Pietro, Roland Masson (IFP)
Robert Eymard (Paris East University)
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Outline
Finite volume discretization of compositional models
Cell Centered FV discretization of diffusion fluxes on general meshes
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Applications
Basin simulation
Reservoir simulation
C02 geological storage simulation
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Compositional Models
Pore
ii Vol
VolS
m
mCfixedTP
,),(,
),,(),,,(),,,(
CTPfCTPCTP
),,(),,,( ,, xSPxSk cr
,0
,,
,1
1
)( ,,
S
ff
C
S
QVCdivCS
gPPKk
V
ii
ii
iiit
cr
Phases: = water, oil, gas
Components i=1,...N (H2O, HydroCarbon species, C02, ...)
Unknowns
Thermodynamics laws (EOS):
Hydrodynamics laws:
present phases
present phases
absent phases
mass conservation of each component
pore volume conservation
thermodynamic equilibrium
Darcy phase velocities
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Discretization of compositional models
Main constraints Must account for a large range of physics Robustness and CPU time efficiency Avoid strong time step reduction
Cell centered FV discretization in space
Euler fully or semi implicit schemes in time Thermodynamic equilibrium and pore volume conservation are implicit
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Finite Volume Scheme
Discretization
Discrete conservation law
0)(1
1
1
dtdxQPKSMdivStt
n
n
t
t K
tnn
K
nK
nKnn
nK
nK dxQdsnPKSMm
tt
SS
K
11*1
1
.)(
TM
MMKLKK PTSSMdsnPKSM ),(.)(
0 ML
MK TT
LK
Kx
Lx
Kn
K
K
Kx
LK
K
,
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Discretization of compositional models
KnKi
KLKLLKKLiK
nKi
nKi mQGXXMCm
t
XmXm
K
1,
**,
1
),()()( ,*
TMMLKKLMc
nM
MKKL gZXXSPPTG ),()( **
,*
,1
ii CSPCXm ),()(
),(),(
1
1
11,11,
1,,
1,
nK
nKi
nK
nKi
i
nKi
nK
PCfPCf
C
S
CSPX ,, present phases
present phases
present phases
),(
)()(
PC
SkrXM
Component mass conservations
Pore volume conservation and thermodynamics equilibrium
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Finite Volume discretization of diffusion fluxes
Cell centered schemes Linear approximation of the fluxes Consistent on general meshes Cellwise constant diffusion tensors Cheap and robust
Compact stencil Coercivity Monotonicity
TCSPPu ic ,),(, ,
TM
MMKK uTdsnuK ,,
LK
Ku
Lu
Kn
LGR
Fault
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Reservoir and basin simulation meshesThe mesh follows the directions of anisotropy using hexahedra but is locally non orthogonal due to
- Faults
- Erosions (pinchout)
- Wells
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CPG faults
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Corner Point GeometriesStratigraphic grids with erosions
Examples of degenerate cells (erosions)
• Hexahedra
• Topologicaly Cartesian
• Dead cells
• Erosions
• Local Grid Refinement (LGR)
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Near well discretizations
Multi-branch well
Hybrid mesh using Voronoi cells
Hybrid mesh using pyramids and tetraedra
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Cell centered finite volume schemes on general meshes
O and L MPFA type schemes Piecewise constant gradient on a subgrid
Cellwise constant gradient construction Success (Eymard et al.): symmetric coercive but not compact
Ku
' su '
ssu
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Discrete cellwise constant gradient
x
K
KKK
K nuum
mu
,)()(
dKK
K
vallforvvxxnm
m
K
)(,
Cellwise constant linear exact gradient
KKu
u
center of gravity of the facex
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Hybrib bilinear form
KKK
KK
KKKKK
K
vRuRd
m
vumvua
)()(
)()(),(
,,,
)()()()(, KKKK xxuuuuR
with
HFV (Eymard et al.) or MFD (Shashkov et al.)
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Elimination of the face unknowns using interpolationsuccess scheme (Eymard et al)
ext
MMM
for
foruuu
0
)(int
)()())(()(, KKKK xxuuuuR
K
KKK
K nuum
mu
,))(()(
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Success scheme: discrete variational form
onu
onfuKdiv
0
hh VvdxvfvuatsVu ),(..
KKKh uuuV ,)(
KKK
KK
KKKKK
K
vRuRd
m
vumvua
)()(
)()(),(
,,,
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Success scheme: fluxes
0)()( uFuF LKKL
Stencil FKL :
LK
Fluxes in a general sense between K and L s.t.
LK
KorL
K
L
with
K
KKKL
LKKL
Kext
vuFvvuFvua
)())((),( ,
int
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Success scheme
Advantages Cell centered symmetric coercive scheme on general meshes Increased robustness on challenging anisotropic test cases
Drawbacks Discontinuous diffusion coefficients Fluxes between cells sharing e.g. only a vertex Large stencil
Non symmetric formulation with two gradients
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Consistent gradient
)(, uK
K
KKKK
K nuum
mu
,, ))(()(
LK
Kx
Lx
Kn
x,Kd
interpolation using only neighbors of K
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Interpolation
• Potential u linear in each cell K, L, M
• Flux continuity at the edges
• Potential continuity at the edges
',
',
s
Ku
Lu
Mu'
x
)(, UK
The scheme reproduces cellwise linear solutions for cellwise constant diffusion tensor
Use an L type interpolation (Aavatsmark et al.) using only neighbouring cells of K
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A "weak" gradient
ext
KL
LKKL
KKK
K
dd
ududu
nuum
mu
K
,0
,)(
))(()~
(
int,,
,,
,
LK
Kx
Lx
Kn
x,Kd
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Compact cell centered FV scheme: bilinear form
K KKK
KKKKKK
K
vRuRd
mvumvua
)()()~
()(),( ,,,
)()())(()( ,, KKKKK xxuuuuR with
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Compact cell centered FV scheme: discrete variational formulation
hh VvdxvfvuatsVu ),(..
KKKh uuuV ,)(
onu
onfuKdiv
0
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Compact cell centered FV scheme: fluxes
K
KKKL
LKKL
Kext
vuFvvuFvua
)())((),( ,
int
Stencil of the scheme: neighbors of the neighbors
13 points for 2D topologicaly cartesian grids
19 points for 3D topologicaly cartesian gridsK
L
0)()( uFuF LKKLwith
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Convergence analysis
2/1
2,
,
))((
TK KKK
KV
uud
mu
h
hdd VLL
uCuu ))(())(( 22
~
2),(
hVuuua
Stability of the gradients
Coercivity (mesh and K dependent assumption)
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Weak convergence property of the weak gradient
dxudivdxuhh )(.~
lim 0
thenMuhVhHh supandLinuuif hh )(lim 2
0
K
KKK
K nuum
mu
,))(()~
(
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Test case CPG 2DCPG meshes of a 2D basin with erosions
2 km
20 km
Mesh at refinement level 3
100
01K
Smooth solution
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Test case CPG 2D
Solver iterations (AMG preconditioner)
L2 error
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Test case: Random Quadrangular Grids
10000
01K
Mesh at refinement level 1
Smooth solution
Domain = (0,1)x(0,1)
Random refinement
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Test case Random Grid
L2 error Solver iterations (AMG preconditioner)
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Test case: random 3D
Diffusion tensor Smooth solution
2000
010
001
K
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Test case random 3D
L2 error
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Test case random 3D
Solver iterations using AMG preconditioner
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Test case random 3D
L2 error on fluxes
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Test case: random 3D aspect ratio 20
zoom
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Test case random 3D with aspect ratio 20
L2 error
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Conclusions
There exists so far no compact and coercive (symmetric) cell centered FV schemes consistent on general meshes
Among conditionaly coercive cell centered FV schemes GradCell Scheme exhibits a good robustness with respect to the anisotropy of K
and to deformation of the mesh Compact stencil 2 layers of communication in parallel
To be tested for multiphase Darcy flow