Breakdown of Chapman-Enskog expansion and the anisotropic effect for lattice-Boltzmann models of porous flow

Lattice-Boltzmann based models are frequently used to study fluid flow in porous media. The macroscopic governing equations as well as the viscosity and the permeability for such models are usually determined using a Chapman-Enskog expansion approach. However, this approach has been validated numerically only for a narrow range of the viscosity and the permeability. We have analytically derived the exact macroscopic governing equations of the lattice-Boltzmann model for the case of simple shear flows in porous media. We find that the effective viscosity in the governing equations is different from the one obtained from the Chapman-Enskog expansion. These findings have been validated numerically. We also find that the effective viscosity is anisotropic. Implications for modeling flow in porous media are discussed.

[1]  H. Brinkman.  A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles , 1949 .

[2]  Raphael Aronson,et al.  Theory and application of the Boltzmann equation , 1976 .

[3]  F. Dullien.  Porous Media: Fluid Transport and Pore Structure , 1979 .

[4]  Sauro Succi,et al.  The permeability of a random medium: Comparison of simulation with theory , 1990 .

[5]  S. Zaleski,et al.  Lattice Boltzmann model of immiscible fluids. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[6]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[7]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[8]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[9]  Shan,et al.  Lattice Boltzmann model for simulating flows with multiple phases and components. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Nicos Martys,et al.  Computer simulation study of the effective viscosity in Brinkman’s equation , 1994 .

[11]  George Em Karniadakis,et al.  Rarefaction and Compressibility Effects in Gas Microflows , 1996 .

[12]  Chen,et al.  Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  F. Phelan,et al.  Lattice Boltzmann methods for modeling microscale flow in fibrous porous media , 1997 .

[14]  L. Luo,et al.  Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model , 1997 .

[15]  Shiyi Chen,et al.  Lattice-Boltzmann Simulations of Fluid Flows in MEMS , 1998, comp-gas/9806001.

[16]  David Freed,et al.  Lattice-Boltzmann Method for Macroscopic Porous Media Modeling , 1998 .

[17]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[18]  Xiaobo Nie,et al.  Lattice Boltzmann simulation of the two-dimensional Rayleigh-Taylor instability , 1998 .

[19]  M. Kollmann,et al.  Colloidal electrolyte friction and sedimentation: A mode-coupling approach , 2000 .

[20]  J. Buick,et al.  Gravity in a lattice Boltzmann model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  N. Martys.  Improved approximation of the Brinkman equation using a lattice Boltzmann method , 2001 .

[22]  B. Shi,et al.  Discrete lattice effects on the forcing term in the lattice Boltzmann method. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Zhaoli Guo,et al.  Lattice Boltzmann model for incompressible flows through porous media. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Dongxiao Zhang,et al.  Unified lattice Boltzmann method for flow in multiscale porous media. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Sauro Succi,et al.  Lattice Boltzmann method at finite Knudsen numbers , 2005 .