Finite element simulations of coupled solid-deformation/fluid-diffusion occurring in earthquake fault zones often require high-fidelity descriptions of the spatial and temporal variations of excess pore water pressure. Large-scale calculation of the coupled fault zone process is often inhibited by the high-order interpolation of the displacement field required to overcome unstable tendencies of the finite elements in the incompressible and nearly incompressible limit. In this work we utilize a stabilized formulation in which the balance of mass is augmented with an additional term representing a stabilization to the incremental change in the pressure field. The stabilized formulation permits equal-order interpolation for the displacement and pore pressure fields and suppresses pore pressure oscillations in the incompressible and nearly incompressible limit. The technique is implemented with a recently developed critical state plasticity model to investigate transient fluid-flow/solid-deformation processes arising from slip weakening of a fault segment. The accompanying transient pore pressure development and dissipation can be used to predict fault rupture and directivity where fluid flow is an important driving force.