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.

翻译：Biot固结模型是描述流体饱和可变形多孔介质演化的经典模型，在多个交叉学科领域具有广泛应用。虽然通过有限差分、有限体积或有限元等典型数值格式求解多孔弹性的方法已得到深入研究，但用于多孔弹性的格子Boltzmann方法尚未发展成熟。本研究提出一种新颖的半隐式耦合格子Boltzmann方法，用于求解二维Biot固结模型。为此，我们采用反应-扩散方程的单松弛时间格子Boltzmann方法求解达西流动，并将其与Boolakee、Geier和De Lorenzis（2023, DOI: 10.1016/j.cma.2022.115756）提出的准静态线性弹性伪时间多松弛时间格子Boltzmann格式相结合。数值结果表明，当多孔弹性系统强耦合时，简单耦合格式会导致数值失稳。然而，新发展的采用全显式和半隐式贡献的中心耦合格式在所有考虑情况下均保持稳定和精确，即使在Biot-Willis系数为1时亦然。此外，针对Terzaghi固结问题及其二维扩展的数值结果表明，该格式甚至能够捕捉瞬时加载产生的不连续解。