Frictional heating on a fault plane causes pore fluids to pressurize. When the permeability of the fault zone materials is sufficiently low, fluid pressure on the fault can approach the normal stress, though it can never be exceeded due to the feedback between pore pressure and frictional strength. However, if slip occurs at a boundary between materials of different permeabilities, such as fault gouge and a damage zone, the highest pressure develops within a few millimeters of the fault in the lower permeability material, rather than at the fault surface. The pressure increase off the fault can reach or exceed the normal stress given a large enough permeability contrast, because there is no direct feedback between this pressure and the frictional heating at the fault surface. High fluid pressures off the fault might result in the slip shifting into the lower permeability material, where the frictional strength has been reduced.
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 Pore fluid pressurization is a possible mechanism for dynamic weakening of faults [Andrews, 2002; Rice, 2006; Mase and Smith, 1987; Suzuki and Yamashita, 2006]. If there are fluids present at the fault surface, frictional heat released during the slip process will cause these fluids to expand. If fluids are unable to escape from the fault surface over the time scale of the slip due to low permeability in the surrounding rock, then the fluid pressure will increase, lowering the effective normal stress across the fault. This effect in turn will lower the frictional resistance, according to the equation:
where τ is the shear stress due to friction, μ the coefficient of friction, σ the normal stress, and P the pore pressure. Assuming a constant slip rate, this lower shear stress will reduce the amount of heat generation on the fault.
 To investigate how properties of the fault zone affect pore fluid pressurization and frictional resistance, we develop a numerical model for the fluid flow perpendicular to an infinitesimally thin planar fault undergoing coseismic slip. Previous research has focused on faults with either uniform materials adjacent to the fault surface [e.g., Andrews, 2002; Mase and Smith, 1987; Rice, 2006], or with material properties varying symmetrically on both sides of the fault [Noda and Shimamoto, 2005]. Others have looked at fluid pressure changes induced by the asymmetric wave propagation at a bimaterial interface [Rudnicki and Rice, 2006]. In contrast, we focus our models on the fluid pressure and stress response when the slip surface is at a boundary between regions of different permeability, where the pressure change is a result of the frictional heating. The San Andreas fault in Parkfield is an example of such a configuration. It is located at the boundary between Salinian basement to the west and Great Valley sediments and the Franciscan assemblage to the east [e.g., Hickman et al., 2004]. Electrical resistivity profiles [e.g., Becken et al., 2005] show that the Salinian basement is electrically resistive (and therefore less permeable) and rocks to the east are electrically conductive (and therefore more permeable). A permeability contrast could also occur between fault gouge and the surrounding damage zone [e.g., Chester and Chester, 1998; Noda and Shimamoto, 2005].
 We solve the following equations using a 1D Galerkin Finite Element Model, starting from the analytical development of Mase and Smith :
The definitions of the symbols in equations (2) to (4) are given in Table 1. Equation (2) describes how pressure changes in response to fluid pressurization, and it accounts for convection, advection, and differential thermal expansion and compressibilities of the matrix and fluid. Equation (3) for heat diffusion includes terms for conduction, convection, advection and heat storage. Equation (4) describes how porosity changes in response to differential compression and thermal expansion of the matrix and fluid. We solve these three equations simultaneously using a Newton-Raphson integration approach. The 1D heat and pressurization model is solved for a line perpendicular to the fault because the pressure and temperature gradients are strongest in that direction. In the fault parallel direction, nearby parts of the fault will experience similar amounts of slip at nearly the same times, and will have similar pressure. Thus, any fluid flow parallel to the fault will likely be much lower than perpendicular flow (during fault slip). As we will show, the effects of fluid pressurization are observed to distances of only millimeters from the slipping surface and the assumption of much smaller flow parallel to the fault is valid over these distances. Heat is released only at the node in the middle of our model space corresponding to the infinitesimally thin fault. Optimal grid spacing and time steps were determined by reducing them to the point at which the model results were no longer dependent on grid spacing or time step size, i.e., 0.1 mm for spacing and 0.01 s or smaller for time steps, depending on the permeability.
Temperature, change in temperature, initial temperature
n′, n, n0
(n′ = no + n)
Porosity, change in porosity, and initial porosity
2 × 10−4 Pa s
Viscosity of water
4.2 × 10−10 Pa−1
Compressibility of water
1 × 10−11 Pa−1
Compressibility of the solid
1 × 10−10 Pa−1
Compressibility of the matrix
6.24 × 10−4 K−1
Thermal expansivity of water
2.5 × 10−5 K−1
Thermal expansivity of the solid
2.5 × 10−5 K−1
Thermal expansivity of the matrix
4.196 × 106 K−1
Heat capacity of water
2.62 × 106 K−1
Heat capacity of the matrix
Fluid volume expelled/volumetric dilation
Slip rate (=0 except at the fault surface)
Coefficient of friction
 We specify a step change in permeability across the fault in order to simulate slip at a boundary between different materials, or at a fault gouge/damage zone interface. All other properties of the material are assumed to be homogenous in order to isolate the effect of a permeability contrast. We compare results for permeability steps of different magnitude, and compare the results to those from models with uniform permeability. In all of our models we use a constant slip rate of 0.5 m/s, and run the simulations for two seconds of slip. We assume a constant friction coefficient in order to make fluid pressurization the only source of fault weakening in the model; in real faults there are likely multiple sources of reduced friction [e.g., Brodsky and Kanamori, 2001; Goldsby and Tullis, 2002; Lachenbruch, 1980; Rice, 2006]. Our model faults initially have an effective normal stress of 24 MPa. In some cases, the calculated pore-fluid pressure exceeds the normal stress. Pressures in excess of the lithostatic pressure would likely result in hydraulic fracturing, opening new pore space and increasing the permeability. For the sake of simplicity our model does not account for this process, so when the fluid pressure exceeds the lithostatic pressure, the model loses accuracy.
3. Results and Discussion
 The three curves in Figure 1 show results for a model with a step change in permeability across the fault, from k = 10−18 (on the left) to 10−20 m2 (on the right). In addition, results are shown for models with uniform permeabilities of 10−18 and 10−20 m2. Results are shown for 1 and 2 s of simulated slip. In all cases, frictional heat causes the temperature on the fault to increase, and heat slowly conducts outward. Frictional heating leads to increased fluid pressure on the fault, which diffuses outward. Due to the low permeability, fluid flow advects an insignificant amount of heat.
 When the permeability is a uniform 10−20 m2, the fluid pressure on the fault increases rapidly, nearly reaching 24 MPa, at which point the effective normal stress would be zero. Because of the low effective normal stress, there is little frictional resistance on the fault, so that little heat is generated during subsequent slip. This is why the temperature remains low in this case. In contrast, when the permeability is a uniform 10−18 m2, fluid is able to flow more freely away from the fault, so the pressure increase is smaller. The frictional heating is thus higher, resulting in the higher temperature increase on the fault. For the case of the permeability contrast, values of pressure and temperature on the fault lie between those of the two uniform models, but are closer to those of the uniform 10−18 m2 permeability model. This result is because the pressure on the fault is determined by the total fluid flow, and flow to the left is much greater than flow to the right.
 An important feature of these results is that the peak in pressure in the permeability contrast case is offset from the fault surface to the low permeability side, and the peak pressure is higher than it is for either of the homogeneous cases. This result is due to a combination of the less permeable right hand side, which is more prone to pressurization, and the more permeable left hand side. The more permeable side allows the pressure on the fault to be relieved more effectively, increasing the frictional heating, which leads to higher temperatures on the fault. It is useful to compare the temperature distribution in Figure 1 for the model with a low uniform permeability of 10−20 m2 to the temperature distribution for the permeability-contrast model. There is a much higher temperature in the contrast model because the pressure on the fault plane is lower. This high temperature leads to greater heat conduction into the low permeability material on the right hand side, which increases the pressure there more than in the uniform case with low permeability. In other words the combination of greater heating on the fault, allowed by the more permeable left side, and slow fluid flow due to low permeability on the right side, results in pressure on the right side that is higher than the lithostatic pressure. This effect can never occur in a model with a uniform permeability, because in such a case the highest pressures are always on the fault, where the tradeoff between pressurization and frictional heat production prevents the fluid pressure from reaching lithostatic values. Figure 2 shows that the off-fault peak in pressure moves away from the fault surface with time, as heat conducts outward. Note that since the peak is off the fault, fluid flows toward the fault surface from the peak, following the pressure gradient. Also note that the temperature distributions between the uniform and contrast models are nearly the same, suggesting that heat transfer by advection is relatively unimportant.
Figure 3 compares the results for models employing two different permeability contrasts. The solid curve is for the case in which permeability changes by two orders of magnitude (from 10−18 to 10−20 m2), while the dashed line is for the case in which permeability changes by one order of magnitude (from 10−18 to 10−19 m2). The model with k = 10−20 m2 on the right side has a higher peak pressure off the fault, since fluids there are more effectively trapped in this case. However, temperature and left hand pressure profiles are nearly the same in both cases, because the higher permeability on the left side of the fault controls the amount of fluid pressurization and hence heat generation on the fault. Since permeability on the right side of the fault is much smaller than on the left side in both cases, fluid flow is dominated by the leftward flow. Thus, the fluid pressure on the fault surface and to the left of the fault is quite similar for both models, and is controlled by the higher permeability side.
 Pore fluid pressurization is one mechanism of dynamic fault weakening during an earthquake. When slip occurs along a fluid saturated interface separating materials of different permeabilities, fluid pressurization, and thus frictional heating of the fault is controlled by the more permeable side. The accompanying increased pressure on the less permeable side of the fault may reach the lithostatic pressure if the permeability contrast is high enough. Hydraulic fracture is a possible result of this effect, although we do not model this process in the current methodology. Nonetheless, one may postulate that this mechanism might cause the slip in an earthquake to shift from the original fault to a surface that is inside the less permeable material, since the strength there will be less than the strength on the original slip surface. We also note that pore fluid weakening occurs within the order of tens of centimeters of slip. For small earthquakes, fluid pressurization may not be a very important factor, as the weakening distance is a large fraction of the total slip. Our model does not take into account the influence of fluid pressurization on the slip rate. We plan to combine our fluid pressurization model with a dynamic rupture model in order to explore the feedback between fluid pressurization and fault slip.
 The authors gratefully acknowledge a number of helpful conversations with J. Rice, T. Shimamoto, H. Noda, and B. Duan. We also thank two anonymous reviewers for their helpful comments and our editor Eric Calais, for his help with the publication process. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship, and NSF Award 0229391.