In this paper, we develop a multiphysics finite element method for solving the quasi-static thermo-poroelasticity model with nonlinear permeability. The model involves multiple physical processes such as deformation, pressure, diffusion and heat transfer. To reveal the multi-physical processes of deformation, diffusion and heat transfer, we reformulate the original model into a fluid coupled problem that is general Stokes equation coupled with two reaction-diffusion equations. Then, we prove the existence and uniqueness of weak solution for the original problem by the $B$-operator technique and by sequence approximation for the reformulated problem. As for the reformulated problem we propose a fully discrete finite element method which can use arbitrary finite element pairs to solve the displacement $\bu$ pressure $\tau $ and variable $\varpi,\varsigma$, and the backward Euler method for time discretization. Finally, we give the stability analysis of the above proposed method, also we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Numerical experiments show that the proposed method can achieve good results under different finite element pairs and are consistent with the theoretical analysis.

翻译：本文针对具有非线性渗透率的准静态热-孔隙弹性模型，开发了一种多物理场有限元方法。该模型涉及变形、压力、扩散和热传导等多个物理过程。为揭示变形、扩散和热传导的多物理过程，我们将原始模型重构成一个流体耦合问题，即一般斯托克斯方程与两个反应扩散方程的耦合。然后，通过$B$-算子技术以及对重构问题的序列逼近方法，证明了原始问题弱解的存在性和唯一性。针对重构问题，我们提出了一种全离散有限元方法，该方法可使用任意有限元单元对求解位移$\bu$、压力$\tau$以及变量$\varpi$、$\varsigma$，并采用向后欧拉方法进行时间离散。最后，我们给出了上述方法的稳定性分析，同时证明了该全离散多物理场有限元方法具有最优收敛阶。数值实验表明，所提出的方法在不同有限元单元对下均能取得良好结果，且与理论分析一致。