A lattice Boltzmann method for Biot's consolidation model of linear poroelasticity

Abstract

Biot's consolidation model is a classical model for the evolution of deformable porous media saturated by a fluid and has various interdisciplinary applications. While numerical solution methods to solve poroelasticity by typical schemes such as finite differences, finite volumes or finite elements have been intensely studied, lattice Boltzmann methods for poroelasticity have not been developed yet. In this work, we propose a novel semi-implicit coupling of lattice Boltzmann methods to solve Biot's consolidation model in two dimensions. To this end, we use a single-relaxation-time lattice Boltzmann method for reaction-diffusion equations to solve the Darcy flow and combine it with a recent pseudo-time multi-relaxation-time lattice Boltzmann scheme for quasi-static linear elasticity. We employ a multi-grid method for the latter scheme to achieve quasi-optimal computational cost. For the coupling between the equations, we develop a centered update scheme, that incorporates both explicit and semi-implicit contributions. The numerical results demonstrate that naive (explicit or semi-implicit) coupling schemes lead to instabilities when the poroelastic system is strongly coupled. However, the newly developed centered coupling scheme is stable and accurate in all considered cases, even for the Biot--Willis coefficient being one. Furthermore, the numerical results for Terzaghi's consolidation problem and a two-dimensional extension thereof highlight that the scheme is even able to capture discontinuous solutions arising from instantaneous loading.