Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries

We present results of lattice-Boltzmann simulations to calculate flow in realistic porous media. Two examples are given for lattice-Boltzmann simulations in twoand threedimensional (2D and 3D) rock samples. First, we show lattice-Boltzmann simulation results of the flow in quasi-two-dimensional micromodels. The third dimension was taken into account using an effective viscous drag force. In this case, we consider a 2D micromodel of Berea sandstone. We calculate the flow field and permeability of the micromodel and find excellent agreement withMicroparticle Image Velocimetry (μ-PIV) experiments. Then, we use a particle tracking algorithm to calculate the dispersion of tracer particles in the Berea geometry, using the lattice-Boltzmann flow field. Second, we use lattice-Boltzmann simulations to calculate the flow in Bentheimer sandstone. The data set used in this study was obtained using X-ray microtomography (XMT). First, we consider a single phase flow. We systematically study the effect of system size and validate Darcy’s law from the linear dependence of the flux on the body force exerted. We observe that the values of the permeability measurements as a function of porosity tend to concentrate in a narrower region of the porosity, as the system size of the computational sub-sample increases. Finally, we compute relative permeabilities for binary immiscible fluids in the XMT rock sample. © 2009 Elsevier Ltd. All rights reserved. 1. Lattice-Boltzmann simulation of flow in 2D porous media A fundamental understanding of flow in porousmedia is of crucial importance inmany applications, such as the recovery of hydrocarbons from oil reservoirs. To characterise flow at the pore scale, it is convenient to use simplified representations of porous media, such as physical micromodels, which can be constructed in the form of pseudo-two-dimensional (pseudo2D) capillary networks. Because the experimental analysis is often very difficult, simulation can be a useful complementary method. In particular, the lattice-Boltzmann technique is very well suited for solving the flow in complex geometries, and it has been successfully used in the study of flow in porous media at the pore scale [1–3]. For a general introduction to the lattice-Boltzmann theory, we refer to [4]. In two dimensions, lattice-Boltzmann simulations have been used to investigate viscous fingering of binary immiscible fluids using the Shan–Chen model [5] and the flow of non-Newtonian fluids in porous media [6,7]. In studies of viscous fingering in a Hele-Shaw cell, the third dimension was taken into account using an effective viscous drag force [8,9]. Recently, we used the effective viscous drag method to compare lattice-Boltzmann simulations of a single phase flow in micromodels with experimental results in the same geometry [10]. Here, we extend these calculations to flow in a 2D micromodel of Berea sandstone. We calculate the permeability of the micromodel and ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (E.S. Boek), [email protected] (M. Venturoli). 1 Current address: Department of Chemical Engineering, Imperial College London, SW7 2AZ, UK. 0898-1221/$ – see front matter© 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.08.063 2306 E.S. Boek, M. Venturoli / Computers and Mathematics with Applications 59 (2010) 2305–2314 Fig. 1. Berea pattern used in the lattice-Boltzmann 2D simulations. The obstacles are represented in black, and the pore space inwhite. The size is 1418μm by 1774 μm. The etch depth of the physical Berea unit is 24.54 μm. compare with experimental data. Then, we use a particle tracking algorithm to calculate the dispersion of tracer particles in the Berea geometry, using the lattice-Boltzmann velocity field. 1.1. Berea sandstone We study the flow in a 2D micromodel of Berea sandstone. This micromodel has been engineered at Schlumberger Cambridge Research, based on a thin section of a 3D Berea sandstone rock sample. The micromodel has been discretised on a lattice, and bit-mapped (0 pore, 1 obstacle) to create the matrix for the lattice-Boltzmann simulations. A picture of the Berea micromodel is presented in Fig. 1. 1.1.1. Single phase flow We consider the lattice-Boltzmann method for flow in two dimensions, which has been described in detail in [10]. In that work, we have shown that we can make successful use of the Hele-Shaw viscous drag approximation when the third dimension of the micromodel is small in comparison with the lateral two dimensions. The micromodel described here is pseudo-2D, and, for symmetry reasons, the fluid velocity is zero in the z direction (depth). We adopt an approach used in the literature [11,8] to solve the flow field in such a geometry using 2D lattice-Boltzmann simulations. This approach consists in introducing a drag force, acting on the fluid, that represents the (approximate) effect of the bounding walls in the third dimension. This viscous drag force depends on the fluid kinematic viscosity, ν, on the fluid velocity, u, and on a depth parameter, h, which represents the distance between the walls bounding the implicit third dimension. If the average velocity of the Poiseuille profile is used, the drag force takes the expression used by Boon et al. [8,9] fdrag = − 12ν h2 u. (1) This is the expression that we will use here. 1.1.2. Calculation of permeability The permeability of a porous medium can be calculated from the empirical Darcy’s law. This well known relation states that the flow rate is proportional to the force driving the fluid, the coefficient of proportionality being the permeability of the medium divided by the dynamic viscosity of the fluid. Darcy’s law can be written as J = − K μ (∇P − ρg), (2) where J is the flow rate per unit area of cross section (flux), K is the permeability, ∇P is the pressure drop between inlet and outlet, ρ is the fluid density, g is a body force (for example gravity), andμ is the dynamic viscosity of the fluid (with the E.S. Boek, M. Venturoli / Computers and Mathematics with Applications 59 (2010) 2305–2314 2307 Fig. 2. Flux versus force and linear fit (dashed line) of the data points. kinematic viscosity given by ν = μ/ρ). By measuring (or calculating) the flux for different pressure drops (or body force values), and using Eq. (2), the permeability K can be derived. The permeability has dimensions of an area, and it is measured in units of Darcy. 1.1.3. Single phase permeability We have carried out lattice-Boltzmann simulations to calculate the single phase permeability of the micromodel using Darcy’s law. To estimate the single phase permeability of the sample, we impose a flow in the positive y direction of the rock. The flow is driven only by a body force g (and no pressure drop is explicitly present). A correspondence between the body force, g , and the pressure drop, ∇P , can be defined using the following equation, ∇P = (Pi − Po) L = gρ (3) where Pi and Po are the pressures at the inlet and outlet respectively, L is the distance between inlet and outlet, and ρ is the fluid density. Eq. (3) can be used to compare the simulations with experiments, in which usually a pressure drop is used to drive the fluid flow. The precise form of the forcing term used, generalised for a multi-component fluid system, is specified as follows: F = g m ∑