Multi-component Multiphase Porous Flow

An axiomatic foundation for models of multi-component multiphase porous flow appearing ubiquitously in the engineering literature is developed. This unifies and extends various disparate and empirical formulations appearing in the literature. Constitutive restrictions are derived from an appropriate statement of the second law of thermodynamics, and the corresponding dissipation inequalities establish stability of solutions. The convexity properties and variational structure of these models are elucidated.

[1]  Grégoire Allaire,et al.  Homogenization of the stokes flow in a connected porous medium , 1989 .

[2]  F. Otto,et al.  Thermodynamically driven incompressible fluid mixtures , 1997 .

[3]  D. S. Drumheller,et al.  Theories of immiscible and structured mixtures , 1983 .

[4]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[5]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[6]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[7]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[8]  Morteza Zadimoghaddam,et al.  Minimizing movement , 2007, SODA '07.

[9]  M. Peszynska,et al.  Pore-to-core simulations of flow with large velocities using continuum models and imaging data , 2013, Computational Geosciences.

[10]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[11]  S. Flügge.  Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie , 1960 .

[12]  William G. Gray,et al.  High velocity flow in porous media , 1987 .

[13]  T. Hashida,et al.  The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  M. Gurtin,et al.  The Mechanics and Thermodynamics of Continua , 2010 .

[15]  L. Ambrosio,et al.  Infinite-dimensional porous media equations and optimal transportation , 2010 .

[16]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[17]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[18]  Yuanle Ma,et al.  Computational methods for multiphase flows in porous media , 2007, Math. Comput..

[19]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[20]  Massimiliano Zingales,et al.  A mechanical picture of fractional-order Darcy equation , 2015, Commun. Nonlinear Sci. Numer. Simul..

[21]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[22]  W. Noll.  Lectures on the foundations of continuum mechanics and thermodynamics , 1973 .

[23]  R. Showalter,et al.  Micro-Structure Models of Diffusion in Fissured Media* , 1991 .

[24]  A. Mielke.  Deriving Effective Models for Multiscale Systems via Evolutionary $$\varGamma $$-Convergence , 2016 .

[25]  Amir Beck,et al.  Introduction to Nonlinear Optimization - Theory, Algorithms, and Applications with MATLAB , 2014, MOS-SIAM Series on Optimization.

[26]  M. Peszynska,et al.  Upscaling Non-Darcy Flow , 2009 .

[27]  Ralph E. Showalter,et al.  Distributed Microstructure Models of Porous Media , 1993 .