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

Stephan B. Lunowa, Barbara Wohlmuth
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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.
线性孔弹性毕奥固结模型的格点玻尔兹曼法
比奥特固结模型是流体饱和可变形多孔介质演变的经典模型,具有多种跨学科应用。虽然通过有限差分、有限体积或有限元等典型方案求解孔隙弹性的数值求解方法已得到深入研究,但用于孔隙弹性的格点玻尔兹曼方法尚未开发出来。在这项工作中,我们提出了一种新颖的半隐式耦合晶格玻尔兹曼方法来求解二维的 Biot 固结模型。为此,我们使用反应扩散方程的单松弛时间晶格玻尔兹曼方法求解达西流,并将其与 Boolakee、Geier 和 De Lorenzis (2023, DOI:10.1016/j.cma.2022.115756) 最近提出的准静态线性弹性的伪时间多松弛时间晶格玻尔兹曼方案相结合。数值结果表明,当孔弹性系统强耦合时,天真的耦合方案会导致不稳定性。然而,新开发的使用全显和半隐式贡献的中心耦合方案在所有考虑的情况下都是稳定和精确的,即使 Biot--Willis 系数为 1。此外,对 Terzaghi 固结问题及其二维扩展的数值结果表明,该方案甚至能够捕捉瞬时加载产生的不连续解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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