Study of Full Implicit Petroleum Engineering Finite-Volume Scheme for Compressible Two-Phase Flow in Porous Media

An industrial scheme, to simulate the compressible two-phase flow in porous media, consists of a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness since the proposed scheme discretizes the equations with gravity and capillary terms. We show that the proposed scheme satisfies the maximum principle for the saturation, a discrete-energy estimate on the pressures, and a function of the saturation that denotes capillary terms. These stability results allow us to derive the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. To our knowledge, this is the first convergence result of a finite volume scheme in the case of two-phase compressible flow in several space dimensions. The proof is given for the complete system when the density of each phase depends on its own pressure.

[1]  Mazen Saad,et al.  Convergence of a Finite Volume Scheme for Gas Water Flow in a Multi-Dimensional Porous Media , 2011, 1102.0582.

[2]  F. Brezzi,et al.  Mathematical models and finite elements for reservoir simulation : single phase, multiphase and multicomponent flows through porous media , 1988 .

[3]  Jérôme Jaffré,et al.  Upstream differencing for multiphase flow in reservoir simulation , 1991 .

[4]  Mazen Saad,et al.  On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media , 2004 .

[5]  T. Arbogast.  Two-phase incompressible flow in a porous medium with various nonhomogeneous boundary conditions , 1990 .

[6]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[7]  Raphaèle Herbin,et al.  Mathematical study of a petroleum-engineering scheme , 2003 .

[8]  Brahim Amaziane,et al.  Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods , 2008 .

[9]  Mazen Saad,et al.  Two compressible immiscible fluids in porous media , 2008 .

[10]  S. Kružkov,et al.  BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF EQUATIONS OF TWO-PHASE POROUS FLOW TYPE; STATEMENT OF THE PROBLEMS, QUESTIONS OF SOLVABILITY, JUSTIFICATION OF APPROXIMATE METHODS , 1977 .

[11]  C. Marle,et al.  Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière , 1996 .

[12]  Zhangxin Chen.  Degenerate Two-Phase Incompressible Flow: I. Existence, Uniqueness and Regularity of a Weak Solution , 2001 .

[13]  Mazen Saad,et al.  Two-Component Two-Compressible Flow in a Porous Medium , 2012 .

[14]  Mazen Saad.  SOLUTIONS TO A MODEL FOR COMPRESSIBLE IMMISCIBLE TWO PHASE FLOW IN POROUS MEDIA , 2010 .

[15]  Mazen Saad,et al.  On a fully nonlinear degenerate parabolic system modeling immiscible gas–water displacement in porous media , 2011 .

[16]  Stephan Luckhaus,et al.  Flow of Oil and Water in a Porous Medium , 1984 .

[17]  Stephan Luckhaus,et al.  Quasilinear elliptic-parabolic differential equations , 1983 .

[18]  D. W. Peaceman.  Fundamentals of numerical reservoir simulation , 1977 .

[19]  Zhangxin Chen,et al.  Degenerate two-phase incompressible flow II: regularity, stability and stabilization☆ , 2002 .

[20]  K. Aziz,et al.  Petroleum Reservoir Simulation , 1979 .