Cell centered finite volume schemes for multiphase flow applications

Outline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Cell Centered Finite Volume Schemes for
Multiphase Flow Applications
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson .
1Institut Français du Pétrole
2 Université Paris Est
july 9th 2010
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
1 Applications and Motivations
2 Diffusion Model Problem
3 Cell Centered Finite Volume Discretizations
4 Numerical Experiments
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Applications
Basin Modeling
Reservoir simulation
CO2 geological storage
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Motivations of cell centered schemes for
compositional multiphase Darcy flow applications
Explicit linear fluxes
Pressure, Saturations, Compositions all defined at the cell centers
Existing Efficient Preconditioners like CPR-AMG
But cell centered VF schemes are non symmetric on general meshes
Possible lack of robustness due to mesh and permeability dependent
coercivity
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Model problem
dLetΩ⊂R be a bounded polygonal domain
2For f ∈ L (Ω), consider the following problem:

−div(ν∇u) = f inΩ,
u = 0 on ∂Ω
R
Let a(u, v)= ν∇u·∇v. The weak formulation reads
Ω
Z
1 1Find u ∈ H (Ω) such that a(u, v)= fv for all v ∈ H (Ω) (Π)0 0
Ω
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Model problem
Ω1
Ω3
Ω2
Let{Ω} be a partition ofΩ into bounded polygonal sub-domainsi 1≤i≤NΩ
ν| smooth and ν(x) is s.p.d. for a.e. x ∈ΩΩi
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Polyhedral admissible meshes
′K
′EK
xK
σdK,σ
K nK,σ
T : set of cells Kh
i bE =E ∪E : set of inner and boundary faces σh h h
m : surface of the face σσ
m : volume of the cell KK
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Discrete function space Vh
V : space of piecewise constant functions onTh h
v (x)= v for all x ∈ Kh K
1Equip V with the following discrete H norm:h 0
 1/2
X X mσ 2 ∀v ∈ V , kv k = |γ (v )− v |h h h V σ h Kh dK,σK∈T σ∈EKh
using the following trace reconstruction at the faces σ

v d +v dK L,σ L K,σ i γ (v ) = if σ = if σ =E ∩E ∈E , σ h K L hd +dL,σ K,σ
 bγ (v ) = 0 if σ ∈E .σ h h
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Finite Volume Scheme
Let F (u ) denote a conservative linear approximation ofK,σ hZ
ν∇u· nK,σ
σ
iconservativity: F (u )+ F (u )= 0, σ =E ∩E ∈E .K,σ h L,σ h K L h
The finite volume scheme reads
find u ∈ V such thath h
ZX
− F (u )= f for all K ∈T .K,σ h h
Kσ∈EK
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow ApplicationsOutline
Applications and Motivations
Diffusion Model Problem
Cell Centered Finite Volume Discretizations
Numerical Experiments
Discrete variational formulation
For all u , v ∈ V , leth h h
X X
a (u , v ) = F (u )(γ (v )− v )h h h K,σ h σ h K
K∈T σ∈Eh K
X X X
= F (u )(v − v )− F (u )vK,σ h L K K,σ h K
bσ=E ∩E ∈E K∈TK L h h σ∈E ∩EK h
The finite volume scheme is equivalent to:
find u ∈ V such thath h
Z
a (u , v )= fv for all v ∈ V .h h h h h h
Ω
1 1 2 1 1L. Agelas , D. Di Pietro , R. Eymard , C. Guichard , R. Masson . Cell Centered Finite Volume Schemes for Multiphase Flow Applications