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 by Boolakee, Geier and De Lorenzis (2023, DOI: 10.1016/j.cma.2022.115756). The numerical results demonstrate that naive coupling schemes lead to instabilities when the poroelastic system is strongly coupled. However, the newly developed centered coupling scheme using fully explicit and semi-implicit contributions 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.

Figures (11)

• Figure 1: D2Q9 velocity set $\boldsymbol{c}_i$ with linear indices (left) and D2Q8 velocity set $\boldsymbol{c}_{ij}$ with 2D Miller indices, i.e., $\bar{1} = -1$ (right).
• Figure 2: Missing incoming distribution functions (blue) at the bottom boundary of the rectangular domain $\mathcal{D}$ and known incoming distribution functions (black) from inside the domain. Black dots: interior nodes, blue dots: boundary nodes, empty gray nodes: ghost nodes.
• Figure 3: Convergence results for the periodic problem with $\alpha = 0.5$ using $N_E = N_x$ and the explicit, centered and implicit scheme (left to right).
• Figure 4: Convergence results for the periodic problem with $\alpha = 0.8$ using $N_E = N_x$ and the explicit, centered and implicit scheme (left to right).
• Figure 5: Convergence results for the periodic problem with $\alpha = 0.8$ using the explicit explicit, centered and implicit scheme (top to bottom). The orange dashed line $N_E = 0.01 N_x^2$ indicates the transition between dominant error due to discretization of pseudo-timesteping.