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Finite Volume Schemes for multi-phase flow simulation on near well grids J. Brac ( 1 ) , R. Eymard ( 2 ) , C. Guichard ( 1 , 2 ) and R. Masson ( 1 ) ( 1 ) IFP and ( 2 ) Universit Paris-Est ECMOR XII - september 8th 2010 1 Objective Study of

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Finite Volume Schemes for multi-phase flow simulation on near well grids

  • J. Brac(1), R. Eymard(2), C. Guichard(1, 2) and R. Masson(1)

(1)IFP and (2)Université Paris-Est

ECMOR XII - september 8th 2010

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Objective

Study of Finite Volume Schemes on 3D near well multi-phase flow simulations.

Outline

  • Applications and Difficulties
  • 3D Near Well Grids
  • Finite Volume Schemes
  • Numerical Experiments
  • Conclusion

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Multi-phase Flow Simulation on Near Well Grids

Applications

Reservoir Simulation CO2 geological storage

Difficulties

Singular Pressure Distribution Well-Radius << Reservoir Dimension Deviated Well Anisotropy

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3D Near-well model

Exponentially refined radial mesh Unstructured mesh with

  • nly hexahedra

Hybrid mesh with hexahedra, tetrahedra and pyramids

Discretization on complex 3D general meshes = ⇒ MultiPoint Flux Approximation (MPFA) Finite Volume Schemes

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Finite Volume Scheme Model Problem

Model Equation

Find u (potential) in H1

0(Ω) |

−∇ · (Λ∇u) = f in Ω, u = 0 on ∂Ω Ω : bounded polygonal domain of Rd Λ : symmetric positive definite tensor field f : function of L2(Ω)

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Finite Volume Scheme Flux Formulation

Th : set of cells K Vh : space of piecewise constant functions

  • n Th
  • FK,σ(uh) ≈
  • σ

Λ∇u · nK,σ linearly

  • Conservativity : FK,σ(uh) + FL,σ(uh) = 0 , σ = K|L

Find uh ∈ Vh | −

σ∈EK FK,σ(uh) =

  • K f

∀K ∈ Th

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Finite Volume Scheme Discrete Variational Formulation

For all uh, vh ∈ Vh, let ah(uh, vh) =

  • σ=EK ∩EL∈Eh

FK,σ(uh)(vL − vK) −

  • K∈Th
  • σ∈EK ∩Eb

h

FK,σ(uh)vK The finite volume scheme is equivalent to Find uh ∈ Vh | ah(uh, vh) =

  • Ω fvh

∀vh ∈ Vh

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Sample of MPFA Finite Volume Schemes

O scheme

[Aavatsmark et al., 1996, Edwards and Rogers, 1998]

L scheme

[Aavatsmark, 2007]

G scheme inspired by the L scheme

[Agélas et al., 2010a] → Subcell gradients satisfying continuity conditions → Subfluxes F G

L,σ

→ Convex linear combination FK,σ = θG

σ F G K,σ

→ Choose the θG

σ to enhance the coercivity and remove singularities

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The GradCell scheme uses a discrete variational formulation

Non symmetric discrete variational formulation based on two cellwise constant gradients and residuals for stabilization

ah(uh, vh) =

  • K∈Th

mKΛK(∇huh)K · ( ∇hvh)K +

  • K∈Th
  • σ∈EK

mσ dK,σ RK,σ(uh)RK,σ(vh) (∇hvh)K =

1 mK

  • σ∈EK mσ(IK,σ(vh) − vK)nK,σ

( ∇hvh)K =

1 mK

  • σ∈EK mσ(γσ(vh) − vK)nK,σ

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The GradCell scheme has a compact stencil

Fluxes are derived from the bilinear form. ah(uh, vh) =

  • σ=EK ∩EL∈Eh

FK,σ(uh)(vL−vK)−

  • K∈Th
  • σ∈EK ∩Eb

h

FK,σ(uh)vK Fluxes FK,σ(uh) only between cells sharing a face The stencil is compact: neighbours of the neighbours For topologically cartesian grids : 13 cells in 2D, 21 cells in 3D [Agélas et al., 2010b]

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Outcome on Symmetry and Sparsity properties

Fact

Previous schemes : compact but non symmetric ⇒

conditionnal coercivity

If GradCell symmetric ⇒ large stencil (81 cells in 3D)

Difficult to combine both properties

Wish

Symmetric unconditionally coercive scheme Sparse stencil: 9 points in 2D and 27 points in 3D on

topologically Cartesian meshes

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SUSHI combining smart ideas... Scheme Using Stabilization and Harmonic Interfaces

combines...

O scheme ideas : subcell gradients (∇hu)s K and subfaces

unknowns us

σ A symmetric variational bilinear form for coercivity Weak and consistent subcell gradient for convergence Two point harmonic interpolation at the faces

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SUSHI Nice harmonic interpolation formula

Find a point yσ and a coefficient α with...

a linear two point interpolation... ... exact on piecewise linear functions, normal flux and potential continuity.

Harmonic point yσ Harmonic interpolation u(yσ) = α u(xK) + (1 − α) u(xL) [Agélas et al., 2009]

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How SUSHI uses the interpolation formula ?

At each face σ, choose the harmonic point yσ Subcell Ks around a vertex s Ks = (xK, yσ, s, yσ′, xK)

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SUSHI Discrete subcell gradient

(∇hu)Ks =

1 mKs ( ms σ(us σ − uK)nK,σ

+ ms

σ′(us σ′ − uK)nK,σ′

+ mˆ

σ(uˆ σ − uK)nKs,ˆ σ

+ mˆ

σ′(uˆ σ′ − uK)nKs,ˆ σ′)

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SUSHI A symmetric formulation

Symmetric discrete variational formulation

ah(uh, vh) =

  • K∈Th
  • s∈VK
  • mKsΛK(∇huh)s

K · (∇hvh)s K +

  • σ∈EK ∩Es

mKs (dK,σ)2 Rs

K,σ(uh)Rs K,σ(vh)

  • Subfluxes

ah(uh, vh) =

  • K∈Th
  • s∈VK
  • σ∈Es∩EK

F s

K,σ(uh)(vs σ − vK)

F s

K,σ(uh) + F s L,σ(uh) = 0

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Schemes comparison

  • n single-phase analytical solution

Problem studied

Single phase flow Anisotropy of the tensor permeability Λ Slanted well

⇒ Analytical solution [Aavatsmark and Klausen, 2003]

Numerical study

O, L, G, GradCell and SUSHI schemes Hexahedra and Hybrid mesh families Λ = diag(1,1, 1 20)

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Hexahedral mesh family l2 pressure error

mesh size h nonzero elements in the linear system

O and L schemes have the same behavior

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Hybrid mesh family l2 pressure error

mesh size h nonzero elements in the linear system

GradCell stencil ≈ 4 times smaller than O scheme L scheme fails but not the more flexible G scheme

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Two-phase flow (w-g) near well simulation

Injection of gaseous CO2 miscible in a reservoir full of water

Two-component

H2O (w) CO2 (w-g)

Thermodynamic equilibrium defined by the solubility ¯

C (w − g) : Cw

CO2 < ¯

C Sg = 0 (g) : Cw

CO2 = ¯

C Sg > 0 Test the O scheme on both types of meshes

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Total Mass of CO2 function of time

GOE = Grid Orientation Effect

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Mass of CO2 in phase gas function of time

Cell size affect the oscillations

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Conclusion

SUSHI scheme exhibits very promising results thanks to its

unconditional coercivity

Hybrid meshes show drawbacks of the schemes :

O scheme has a stencil ≈ 4 times bigger than GradCell L scheme fails but not the more flexible G scheme

Two-phase flow numerical solution is sensitive to GOE and

size of the cells

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References

Aavatsmark, I. (2007). Multipoint flux approximation methods for quadrilateral grids. In 9th International Forum on Reservoir Simulation, Abu Dhabi. Aavatsmark, I., Barkve, T., Boe, O., and Mannseth, T. (1996). Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media.

  • J. Comput. Phys., 127(1):2–14.

Aavatsmark, I. and Klausen, R. (2003). Well index in reservoir simulation for slanted and slightly curved wells in 3d grids. SPE Journal, 8(1):41–48. Agélas, L., Di Pietro, D., and Droniou, J. (2010a). The g method for heterogeneous anisotropic diffusion on general meshes. M2AN Math. Model.Numer. Anal. Agélas, L., Di Pietro, D., Eymard, R., and Masson, R. (2010b). An abstract analysis framework for nonconforming approximations of anisotropic heterogeneous diffusion. International Journal on Finite Volumes, 7(1):1–29. Agélas, L., Eymard, R., and Herbin, R. (2009). A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media.

  • C. R. Math. Acad. Sci. Paris, 347(11-12):673–676.

Edwards, M. and Rogers, C. (1998). Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Computational Geosciences, 2(4):259–290. 24