Homogeneous and Membrane Pore Fluid Diffusion in Spring Block Simulations of Fault Slip With Rate and State Friction

Spring block, and sometimes continuum, models of the effects of the coupling of fluid flow and frictional slip often employ the membrane approximation. This approximation assumes that the fluid flux between the slipping layer and a remote pore fluid reservoir is proportional to the difference between the value of pore pressure in the reservoir and in the slipping zone. In contrast, Darcy's law states that the fluid flux is proportional to the gradient of the pore pressure. We analyze and compare these two formulations by asymptotic analysis of the fluid flow equations and numerical simulations of a spring–block model using rate and state friction. This analysis shows that membrane diffusion agrees with homogeneous diffusion in the limit of undrained conditions and for nearly drained conditions. For homogeneous diffusion and essentially undrained conditions, both the asymptotic analysis and numerical simulations indicate the formation of a boundary layer near the shear zone where gradients of pore pressure are large. Outside this layer the pore pressure rapidly approaches the drained solution. A linearized stability analysis derives the dependence of the non‐dimensional critical stiffness () on fluid diffusivity, dilatancy factor (), and shear zone thickness. The homogeneous and membrane diffusion models exhibit nearly identical in the limits of drained and undrained conditions, but differ between the two limits. In this intermediate range, numerical simulations show that the two models produce similar slip behavior for but significant differences in slip velocity and recurrence patterns for .