international conference scaling up and modeling for transport and flow in porous media

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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: Random Quadrangular Grids

10000

01K

Mesh at refinement level 1

Smooth solution

Domain = (0,1)x(0,1)

Random refinement

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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

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