  In this paper, we study the numerical algorithm for a nonlinear
poroelasticity model with nonlinear stress-strain relations. By using variable
substitution, the original problem can be reformulated to a new coupled
fluid-fluid system, that is, a generalized nonlinear Stokes problem of
displacement vector field related to pseudo pressure and a diffusion problem of
other pseudo pressure fields. A new technique is used to get the existence and
uniqueness of the solution of the reformulated model and a fully discrete
nonlinear finite element method is proposed to solve the model numerically. The
multiphysics finite element is used to get the discretization of the space
variable and the backward Euler method is taken as the time-stepping method in
the fully discrete case. Stability analysis and the error estimation are given
for the fully discrete case and numerical test are taken to verify the
theoretical results.
