Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part I: formulation and properties of the mathematical model

This paper is a prequel to that of Marchand et al. (Comput Geosci 16:691–708, 2012), where an efficient and accurate hybrid-mixed finite element approximation for a system of time-dependent nonlinear conservation equations has been formulated, implemented, and tested, which are general enough to represent most of the existing formulations for two-component liquid–gas flow in porous medium with phase exchange, also allowing for any (dis)appearance of one of the phases. Temperature variation is neglected, but capillary effects are included by extended Darcy’s law, and Fickian diffusion is taken into account. The efficiency and stability of the numerical method of Lake (1989) relies on an equivalent reformulation of the otherwise commonly used model in terms of new principal variables and subsequent static (flash) equations allowing more generally for any (dis)appearance of one of the phases without the need of variable switching or unphysical quantities. In particular, the formulation in terms of complementarity conditions allows for an efficient and stable solution by the semismooth Newton’s method.

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

[2]  Farid Smaï,et al.  Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository , 2008, 0802.4389.

[3]  M. Michelsen.  The isothermal flash problem. Part II. Phase-split calculation , 1982 .

[4]  Felix Otto,et al.  L1-Contraction and Uniqueness for Quasilinear Elliptic–Parabolic Equations , 1996 .

[5]  Karsten Pruess,et al.  CO2-H2O mixtures in the geological sequestration of CO2. I. Assessment and calculation of mutual solubilities from 12 to 100°C and up to 600 bar , 2003 .

[6]  Farid Smaï,et al.  Développement d’outils mathématiques et numériques pour l’évaluation du concept de stockage géologique , 2009 .

[7]  Jérôme Jaffré,et al.  Henry’ Law and Gas Phase Disappearance , 2009, 0904.1195.

[8]  Robert Eymard,et al.  Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage , 2011, Math. Comput. Simul..

[9]  Hamdi A. Tchelepi,et al.  A New Method for Thermodynamic Equilibrium Computation of Systems With an Arbitrary Number of Phases , 2009 .

[10]  Jeoung Yoon.  Underground Storage of CO2 and Energy , 2010 .

[11]  Christian Kanzow,et al.  The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions , 2011, Comput. Optim. Appl..

[12]  Mikhail Panfilov,et al.  Interfaces of Phase Transition and Disappearance and Method of Negative Saturation for Compositional Flow with Diffusion and Capillarity in Porous Media , 2010 .

[13]  Curtis H. Whitson,et al.  The negative flash , 1989 .

[14]  Olaf Ippisch,et al.  Modeling and simulation of two-phase two-component flow with disappearing nonwetting phase , 2012, Computational Geosciences.

[15]  Jérôme Jaffré,et al.  Gas phase appearance and disappearance as a problem with complementarity constraints , 2011, Math. Comput. Simul..

[16]  Peter Knabner,et al.  Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results , 2012, Computational Geosciences.

[17]  Ismael Herrera,et al.  Enhanced Oil Recovery , 2012 .

[18]  Igor Mozolevski,et al.  Discontinuous Galerkin method for two-component liquid–gas porous media flows , 2012, Computational Geosciences.

[19]  L. Landau,et al.  statistical-physics-part-1 , 1958 .

[20]  N. W. Lanfredi,et al.  HP 67/97 calculator waves application programs , 1987 .

[21]  A. Bourgeat,et al.  Modelling and Numerical Simulation of gas migration in a nuclear waste repository , 2010, 1006.2914.

[22]  Franklin M. Orr,et al.  An analysis of the vanishing interfacial tension technique for determination of minimum miscibility pressure , 2007 .

[23]  Rainer Helmig,et al.  Numerical simulation of non-isothermal multiphase multicomponent processes in porous media.: 1. An efficient solution technique , 2002 .

[24]  S. Kräutle,et al.  The semismooth Newton method for multicomponent reactive transport with minerals , 2011 .

[25]  Leonard F. Konikow,et al.  Modeling Groundwater Flow and Pollution , 1988 .

[26]  G. Chavent.  Mathematical models and finite elements for reservoir simulation , 1986 .

[27]  Stephen W. Webb,et al.  Gas transport in porous media , 2006 .

[28]  E. Stenby,et al.  Thermodynamics of the multicomponent vapor-liquid equilibrium under capillary pressure difference , 2001 .

[29]  Ibtihel Ben Gharbia,et al.  Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix , 2012, Math. Program..

[30]  SUMMARY AND DISCUSSION OF THE RESULTS , 1966 .

[31]  R. Krishna,et al.  The Maxwell-Stefan approach to mass transfer , 1997 .

[32]  Farid Smaï,et al.  Compositional Two-Phase Flow in Saturated-Unsaturated Porous Media: Benchmarks for Phase Appearance/Disappearance , 2013 .

[33]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[34]  Rainer Helmig,et al.  A new approach for phase transitions in miscible multi-phase flow in porous media , 2011 .

[35]  R. Pierotti,et al.  THE SOLUBILITY OF GASES IN LIQUIDS , 1963 .

[36]  V. S. Vaidhyanathan,et al.  Transport phenomena , 2005, Experientia.

[37]  Kazufumi Ito,et al.  On a semi-smooth Newton method and its globalization , 2009, Math. Program..

[38]  M. D. Siegel,et al.  A benchmark in portable FORTRAN: Speeds of CPU and in-memory data-transfer operation for hydrogeochemical models , 1987 .

[39]  Laurent Trenty,et al.  A benchmark study on problems related to CO2 storage in geologic formations , 2009 .

[40]  Anahita Abadpour,et al.  Method of Negative Saturations for Modeling Two-phase Compositional Flow with Oversaturated Zones , 2009 .