## 1. Introduction

[2] Since the past decade, numerical experiments are commonly performed to determine and investigate transport properties of porous structures such as electrical conductivity, permeability, solute dispersion, relative displacement of a two-phase flow or NMR response. A better understanding of physical processes, and their complex interplay with the pore structure, is nowadays achieved by solving the governing equations directly in the pore space obtained from microtomographic (micro-CT) images [*Auzerais et al.*, 1996; *Arns et al.*, 2001, 2004; *Talabi et al.*, 2009; *Zhan et al.*, 2010; *Ovaysi and Piri*, 2010; *Bauer et al.*, 2011]. The improvement and development of such approach is important, for instance in geophysics, to understand the interaction between the rock and the fluids and the resulting consequences on the transport behavior. In contaminant hydrology and in petroleum engineering, the method permits the determination of the (relative) permeabilities of rocks and provides a fundamental understanding of the dispersion process at the pore level.

[3] Most of the past studies deal with the flow of incompressible viscous fluids and various numerical methods to solve the flow equations were proposed. Two distinct approaches should be mentioned: Pore network models [e.g., *Fatt*, 1956; *Joekar-Niasar and Hassanizadeh*, 2011] and direct numerical simulations (e.g., Finite Element [*Borne*, 1992; *Sun et al.*, 2010], Finite Volume [*Adler et al.*, 1990; *Mostaghimi et al.*, 2010], Lattice-Boltzmann [*Cancelliere*, 1990; *Zaretskiy et al.*, 2010]). The first is based on the modeling of the pore space by a network of interconnected pores and throats, reducing the governing equations to a system of linear equations, which can be easily solved. The second uses the voxel representation of the pore space obtained from the micro-CT images as numerical grid without major modification of the rock geometry and topology. Lately, Lattice-Boltzmann schemes became very popular. The reason for this lies mainly in the simplicity to solve the flow in complex geometries like porous media. In addition, Lattice-Boltzmann schemes are easily parallelized and therefore ideally suited to treat relatively large volumes. This technique was developed following the pioneering works of *Rothman* [1988] on Lattice Gas Automata (LGA) and *Succi et al.* [1989] on Lattice-Boltzmann Equation (LBE). Among the different schemes, the BGK model [*Qian et al.*, 1992] with a single-relaxation parameter in combination with the bounce-back boundary rule is the most popular. A major drawback of the method is that the fluid velocity is not proportional to the inverse of its viscosity. As a consequence, the permeability obtained with this method varies with the viscosity. This dependence is mainly due to the bounce-back boundary rule that imposes the location of the solid boundary which has a numerical error that depends on the viscosity. To overcome this problem, one solution is to use a more precise boundary scheme [*Ginzburg and d'Humières*, 2003; *Ginzburg et al.*, 2008a] or to increase the resolution of the numerical domain, e.g., the pore space. Yet, imaging techniques including micro-CT have a finite resolution that can be of the same order as the pore size, making this approach inappropriate. Another solution proposed by *Manwart et al.* [2002] is to use a “magic” viscosity that corresponds to the value of the relaxation time ( ) at which the effective permeability does not depend on the resolution of the numerical domain. Yet, this procedure has two disadvantages. First, the prescribed depends on the porous structure [*Ginzburg and d'Humières*, 2003], the value obtained for Fontainebleau sandstone might not be the same for other media. Second, since the viscosity is imposed, this procedure cannot be used in the case of two phase flow with different viscosities or non-Newtonian flow. Such situations are however encountered in many applications such as in Enhanced Oil Recovery [*Wu and Pruess*, 1996] as well as in soil remediation. A possible solution is to improve the BGK model by using a multiple-relaxation-time (MRT) model [*d'Humières et al.*, 2002] or a two-relaxation-time (TRT) model [*Ginzburg*, 2007]. The ability of the MRT and TRT schemes were tested on synthetic model structures like fiber materials [*Ginzburg and d'Humières*, 2003], body-centered cubic arrays of spheres and random-sized sphere packings [*Pan et al.*, 2006] and for several other regular structures using the TRT model [*Ginzburg et al.*, 2008b]. As demonstrated analytically in the work of *d'Humières and Ginzburg* [2009], MRT schemes lead to viscosity independent results only if the free relaxation parameters are correctly chosen. The proper choice should keep fixed the specific combinations (defined in section 2) for all symmetric and antisymmetric collision modes. The TRT scheme is the most efficient operator for low Reynolds number flow. However, improper choice of the free relaxation parameter in the MRT scheme leads to viscosity-dependent permeability, explaining the result of *Narvaez et al.* [2010]. It is important to remark that even if the MRT, in its more general form, might be less efficient in terms of computational cost, this is not the case for the TRT scheme where computational costs equal those of BGK.

[4] Many articles on porous media apply the Navier-Stokes equilibrium (and assuming low Reynolds number). We will use Stokes equilibrium that is simpler and sufficient for our purpose. The goal of the present paper is to assess the TRT scheme to solve Stokes flow for pressure *P* and momentum , prescribing kinematic viscosity and external body forcing :

in different types of porous media (straight channels, 2-D generated porous media and binarized 3-D images of Fontainebleau sandstone). The emphasis will on the evaluation of the scheme in terms of parameter dependence, errors and convergence time on the basis of permeability. The structure of the paper is as follows: the numerical implementation is presented in section 2. In section 3, we present the results obtained from the different configurations. Section 4 is dedicated to the summary and conclusions. In Appendix A, we show how to compute the effective permeability from the viscous energy dissipation.