Effects of Normal Stress Changes on Porosity in Rate and State Friction

Many recent applications of fluid injection, for example, fluids from hydraulic fracture, geothermal stimulation, sequestration of CO2, and waste disposal, have resulted in seismic activity. A key element of the interaction between mechanical deformation and pore fluids is changes in porosity associated with slip. Based on experiments of Marone et al. (JGR, 1990), Segall and Rice (SR, JGR, 1995) proposed an expression for the rate of change of porosity that depends on the sliding velocity and a parameter ɛ that describes the magnitude of the change. In general, ɛ might be expected to depend on the effective normal stress, but there is evidence, although scant, that it does not (Samuelson et al., JGR, 2009). If ɛ is constant, then the SR relation predicts that the porosity is constant for a given sliding velocity. Sleep (JGR, 1995) proposed an alternative expression for the rate of porosity change. It is similar to the SR form, but depends on the state parameter. Consequently, predictions for the aging law and the slip law for the evolution of state differ, although they do not for the SR form. Calculations show that predictions of the two expressions are similar for sudden velocity changes at constant effective normal stress. But observations by Linker and Dieterich (LD, JGR, 1992) showed that changes in effective normal stress change the state at constant slip velocity and formulated a description of these changes. Thus, the Sleep form does predict that the porosity changes with changes in effective normal stress (even if ɛ is constant), whereas the SR form does not. Calculations for jumps in normal stress at constant slip velocity demonstrate that the effect can significant. Consequently, it will be important for applications when the effective normal stress is changed by pore pressure changes. Although calculations for the aging law and the slip law differ, the difference is small. Simulations with the spring block model of SR are used to illustrate the effect of these different porosity formulations. Using the aging law for the variation of the state (with the LD effect), the simulations indicate that the magnitude of rapid slip events is similar for the two models, but that the predicted time between the events can be significantly different.