co 2 matris | carburants diversifis | vhicules conomes | raffinage propre | rserves prolonges ifp...

CO2 maîtrisé | Carburants diversifiés | Véhicules économes | Raffinage propre | Réserves prolongées

Écrire ici dans le masque le nom de votre Direction – Écrire ici dans le masque le titre de la présentation – Date de la présentation

© IF

P© IF

P

Cell centered finite volume scheme for multiphase porous media flows with applications

in the oil industry

International Conference Scaling Up and Modelingfor 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)

2

© IF

P©

IFP

Outline

Finite volume discretization of compositional models

Cell Centered FV discretization of diffusion fluxes on general meshes

3

© IF

P©

IFP

Applications

Basin simulation

Reservoir simulation

C02 geological storage simulation

4

© IF

P©

IFP

Compositional Models

Pore

ii Vol

VolSmmCfixedTP

,),(,

),,(),,,(),,,(

CTPfCTPCTP

),,(),,,( ,, xSPxSk cr

,0,,

,1

1

)( ,,

Sff

C

S

QVCdivCS

gPPKk

V

ii

ii

iiit

cr

Phases: = water, oil, gasComponents 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

5

© IF

P©

IFP

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

6

© IF

P©

IFP

Finite Volume Scheme

Discretization

Discrete conservation law

0)(11

1

dtdxQPKSMdivStt

n

n

t

t Ktnn

K

nK

nKnn

nK

nK dxQdsnPKSMm

ttSS

K

11*1

1

.)(

TM

MMKLKK PTSSMdsnPKSM ),(.)(

0 ML

MK TT

LK

Kx

LxKn

K

KKx

LK

K

,

7

© IF

P©

IFP

Discretization of compositional models

KnKi

KLKLLKKLiK

nKi

nKi mQGXXMCm

tXmXm

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

8

© IF

P©

IFP

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

LuKn

LGR

Fault

9

© IF

P©

IFP

Reservoir and basin simulation meshesThe mesh follows the directions of anisotropy using hexahedra but is locally non orthogonal due to

- Faults

- Erosions (pinchout)

- Wells

10

© IF

P©

IFP

CPG faults

11

© IF

P©

IFP

Corner Point GeometriesStratigraphic grids with erosions

Examples of degenerate cells (erosions)

• Hexahedra• Topologicaly Cartesian• Dead cells• Erosions• Local Grid Refinement (LGR)

12

© IF

P©

IFP

Near well discretizations

Multi-branch well

Hybrid mesh using Voronoi cells

Hybrid mesh using pyramids and tetraedra

13

© IF

P©

IFP

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

14

© IF

P©

IFP

Discrete cellwise constant gradient

x

K

KKK

K nuummu

,)()(

dKK

K

vallforvvxxnmm

K

)(,

Cellwise constant linear exact gradient

KKu

u

center of gravity of the facex

15

© IF

P©

IFP

Hybrib bilinear form

KKK

KK

KKKKK

K

vRuRdm

vumvua

)()(

)()(),(

,,,

)()()()(, KKKK xxuuuuR

with

HFV (Eymard et al.) or MFD (Shashkov et al.)

16

© IF

P©

IFP

Elimination of the face unknowns using interpolationsuccess scheme (Eymard et al)

ext

MMM

for

foruuu

0

)(int

)()())(()(, KKKK xxuuuuR

K

KKK

K nuummu

,))(()(

17

© IF

P©

IFP

Success scheme: discrete variational form

onuonfuKdiv

0

hh VvdxvfvuatsVu ),(..

KKKh uuuV ,)(

KKK

KK

KKKKK

K

vRuRdm

vumvua

)()(

)()(),(

,,,

18

© IF

P©

IFP

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

LKKLKext

vuFvvuFvua

)())((),( ,int

19

© IF

P©

IFP

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

20

© IF

P©

IFP

Consistent gradient

)(, uK

K

KKKK

K nuummu

,, ))(()(

LK

Kx

LxKn

x,Kd

interpolation using only neighbors of K

21

© IF

P©

IFP

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

22

© IF

P©

IFP

A "weak" gradient

ext

KL

LKKL

KKK

K

ddudud

u

nuummu

K

,0

,)(

))(()~(

int,,

,,

,

LK

Kx

LxKn

x,Kd

23

© IF

P©

IFP

Compact cell centered FV scheme: bilinear form

K KKK

KKKKKK

K

vRuRdmvumvua

)()()~()(),( ,,,

)()())(()( ,, KKKKK xxuuuuR with

24

© IF

P©

IFP

Compact cell centered FV scheme: discrete variational formulation

hh VvdxvfvuatsVu ),(..

KKKh uuuV ,)(

onuonfuKdiv

0

25

© IF

P©

IFP

Compact cell centered FV scheme: fluxes

K

KKKL

LKKLKext

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

26

© IF

P©

IFP

Convergence analysis

2/1

2,

,

))((

TK KKK

KV

uudmu

h

hdd VLLuCuu

))(())(( 22

~

2),(hV

uuua

Stability of the gradients

Coercivity (mesh and K dependent assumption)

27

© IF

P©

IFP

Weak convergence property of the weak gradient

dxudivdxuhh )(.~lim 0

thenMuhVhHh supandLinuuif hh )(lim 2

0

K

KKK

K nuummu

,))(()~(

28

© IF

P©

IFP

Test case CPG 2DCPG meshes of a 2D basin with erosions

2 km

20 km

Mesh at refinement level 3

10001

K

Smooth solution

30

© IF

P©

IFP

Test case: Random Quadrangular Grids

1000001

K

Mesh at refinement level 1

Smooth solution

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

Random refinement

38

© IF

P©

IFP

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

co 2 matris | carburants diversifis | vhicules conomes | raffinage propre | rserves prolonges ifp...

Documents