  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.
