Numerical Simulation of Thermoelastic Nonlinear Waves in Fluid Saturated Porous Media with Non-local Darcy Law

In this work we present a numerical study of a non-linear space-fractional parabolic system, describing thermoelastic waves propagation in fluid saturated porous media with non-local Darcy law. We use implicit cell-centered finite difference method for numerical discretizations, combined with \(L1-2\) formula for the fractional derivative approximation. We apply Newton’s method to compute the finite difference solution. We solve a linear system of algebraic equation in block-matrix form on each iteration, using Schur complement. Numerical experiments attest good attributes of the proposed numerical method.

[1]  D. F. McTigue,et al.  Thermoelastic response of fluid‐saturated porous rock , 1986 .

[2]  Monica Moroni,et al.  Flux in Porous Media with Memory: Models and Experiments , 2010 .

[3]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[4]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[5]  Roberto Garra,et al.  Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  M. Fellah,et al.  Transient wave propagation in inhomogeneous porous materials: Application of fractional derivatives , 2006, Signal Process..

[7]  M. Caputo.  Models of flux in porous media with memory , 2000 .

[8]  Eduardo Ramos,et al.  A Spatially Non-Local Model for Flow in Porous Media , 2012, Transport in Porous Media.

[9]  Jose Alvarez-Ramirez,et al.  A fractional-order Darcy's law , 2007 .

[10]  Yoshio Takane,et al.  The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems , 2009, Comput. Math. Appl..

[11]  Fuzhen Zhang.  The Schur complement and its applications , 2005 .

[12]  R. Garra,et al.  Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media , 2013 .

[13]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[14]  I. Podlubny.  Fractional differential equations , 1998 .

[15]  M. Bonafede,et al.  Hot fluid migration: an efficient source of ground deformation: application to the 1982–1985 crisis at Campi Flegrei-Italy , 1991 .

[16]  R. Horn,et al.  Basic Properties of the Schur Complement , 2005 .

[17]  Ettore Salusti,et al.  Transient solutions for temperature and pressure waves in fluid-saturated porous rocks , 1996 .

[18]  Zhi-Zhong Sun,et al.  A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications , 2014, J. Comput. Phys..