Generalized three-dimensional lattice Boltzmann color-gradient method for immiscible two-phase pore-scale imbibition and drainage in porous media

Abstract

This article presents a three-dimensional numerical framework for the simulation of fluid-fluid immiscible compounds in complex geometries, based on the multiple-relaxation-time lattice Boltzmann method to model the fluid dynamics and the color-gradient approach to model multicomponent flow interaction. New lattice weights for the lattices D3Q15, D3Q19, and D3Q27 that improve the Galilean invariance of the color-gradient model as well as for modeling the interfacial tension are derived and provided in the Appendix. The presented method proposes in particular an approach to model the interaction between the fluid compound and the solid, and to maintain a precise contact angle between the two-component interface and the wall. Contrarily to previous approaches proposed in the literature, this method yields accurate solutions even in complex geometries and does not suffer from numerical artifacts like nonphysical mass transfer along the solid wall, which is crucial for modeling imbibition-type problems. The article also proposes an approach to model inflow and outflow boundaries with the color-gradient method by generalizing the regularized boundary conditions. The numerical framework is first validated for three-dimensional (3D) stationary state (Jurin's law) and time-dependent (Washburn's law and capillary waves) problems. Then, the usefulness of the method for practical problems of pore-scale flow imbibition and drainage in porous media is demonstrated. Through the simulation of nonwetting displacement in two-dimensional random porous media networks, we show that the model properly reproduces three main invasion regimes (stable displacement, capillary fingering, and viscous fingering) as well as the saturating zone transition between these regimes. Finally, the ability to simulate immiscible two-component flow imbibition and drainage is validated, with excellent results, by numerical simulations in a Berea sandstone, a frequently used benchmark case used in this field, using a complex geometry that originates from a 3D scan of a porous sandstone. The methods presented in this article were implemented in the open-source PALABOS library, a general C++ matrix-based library well adapted for massive fluid flow parallel computation.

Three-dimensional lattice Boltzmann method benchmarks between color-gradient and pseudo-potential immiscible multi-component models

“…[41][42][43]. The CGM and PPM used for the benchmarks are, respectively, from Leclaire et al 35 and Porter et al 13 These models are not described in this article and the reader may consult the previously mention references for a better understanding. It should be noted, however, that additional consideration for the PPM can be found in the appendix.…”

“…[30][31][32] There are also approaches to model the static and dynamic contact angles accurately at a solid boundary. [33][34][35] Recently, a generalized CGM with enhanced equilibrium for the D2Q9, D3Q15, D3Q19 and D3Q27 have been proposed. This model also generalized the modeling of static contact angle for 3D simulations as well as adapting the regularized pressure and velocity boundary condition for the CGM.…”

Wetting boundary condition for the color-gradient lattice Boltzmann method: Validation with analytical and experimental data

“…This method accurately simulates wetting phenomena on a flat surface ( Huang et al, 2014;Li et al, 2016;Liu et al, 2015 ) and can be extended to 3D ( Yu et al, 2017 ), but the implementation for arbitrary surfaces is not obvious. Leclaire et al (2017) proposed a way to find the proper direction of the color gradient ∇ N based on the recurrence relation for the secant method. However, the recurrence relation would in principle require many iterations, which could increase computational costs.…”

“…To compute multiphase flow properties at the pore scale several direct numerical simulation methods have been developed using the volume-of-fluid method ( Raeini et al, 2015;Shams et al, 2018 ), the level set method ( Jettestuen et al, 2013;Prodanovi ć and Bryant, 2006 ) and the lattice Boltzmann method (LBM) ( Boek et al, 2017;Leclaire et al, 2017;Ramstad et al, 2012;Tölke et al, 2006 ). Several studies have used the LBM to compute flow through porespace images ( Ahrenholz et al, 2008;Boek et al, 2017;Leclaire et al, 2017;Ramstad et al, 2012;.…”

“…Several studies have used the LBM to compute flow through porespace images ( Ahrenholz et al, 2008;Boek et al, 2017;Leclaire et al, 2017;Ramstad et al, 2012;. In these cases, a color gradient (CG) LBM was used.…”