Existence analysis of a single-phase flow mixture model with van der Waals pressure

The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the mass densities. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. The model consists of parabolic equations with cross diffusion with a hypocoercive diffusion operator. The global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, extending the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium mass density is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a minimum principle for the mass fractions. Without mass diffusion, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.

[1]  Eva Farkas,et al.  General Purpose Compositional Model , 1985 .

[2]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[3]  D. H. Cuong,et al.  Existence of traveling waves in van der Waals fluids with viscosity and capillarity effects , 2014 .

[4]  Raimund Bürger,et al.  Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression , 2003, SIAM J. Appl. Math..

[5]  Limit thermodynamic model for compositional gas–liquid systems moving in a porous medium , 2007 .

[6]  Zhangxin Chen,et al.  Analysis of a Compositional Model for Fluid Flow in Porous Media , 2000, SIAM J. Appl. Math..

[7]  Herbert Amann,et al.  Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems , 1993 .

[8]  K. Zumbrun,et al.  Navier–Stokes regularization of multidimensional Euler shocks , 2006 .

[9]  D. Johnston,et al.  Advances in Thermodynamics of the van der Waals Fluid , 2014 .

[10]  D. Peng,et al.  A New Two-Constant Equation of State , 1976 .

[11]  J. Mikyška,et al.  Energy inequalities in compositional simulation , 2016 .

[12]  J. Mikyška,et al.  Compositional Modeling of Two-Phase Flow in Porous Media Using Semi-Implicit Scheme , 2015 .

[13]  Martin Burger,et al.  Nonlinear Cross-Diffusion with Size Exclusion , 2010, SIAM J. Math. Anal..

[14]  Andrea Lamorgese,et al.  Phase Field Approach to Multiphase Flow Modeling , 2011 .

[15]  D. Gao.  Duality Principles in Nonconvex Systems , 2000 .

[16]  M. Mercier.  Global smooth solutions of Euler equations for Van der Waals gases , 2010, 1003.4903.

[17]  Nicola Zamponi,et al.  Analysis of degenerate cross-diffusion population models with volume filling , 2015, 1502.05617.

[18]  Ansgar Jüngel,et al.  Compact families of piecewise constant functions in Lp(0,T;B) , 2012 .

[19]  Dieter Bothe,et al.  Continuum thermodynamics of chemically reacting fluid mixtures , 2013, 1401.5991.

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

[21]  G. Soave.  Equilibrium constants from a modified Redlich-Kwong equation of state , 1972 .

[22]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[23]  Ansgar Jüngel,et al.  The boundedness-by-entropy method for cross-diffusion systems , 2015 .

[24]  Shu-Yi Zhang.  Existence of multidimensional phase transitions in a steady Van Der Waals flow , 2013 .

[25]  Hussein Hoteit,et al.  Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media , 2005 .