| | | 9.3.2 Lithostatic Stresses with a Uniform Friction Model: Scenario sslithouni |
In this case we assume that little or no water resides in the interstices of the grains, so that the pore pressures are negligible, and the effective normal stresses are lithostatic. We use the same slip-weakening friction model that we use in the base case. The parameters in the friction model do not vary with depth.
With negligible pore pressures the effective normal stresses equal the lithostatic stresses. In the case of our homogeneous half-space, the effective normal stresses increase linearly with depth. The shear tractions at failure increase linearly with depth, because they come from the product of the normal tractions, which increase linearly with depth, and the maximum coefficient of friction (µmax), which is uniform with depth. To create a propagating rupture with a uniform speed, we employ a uniform distance from failure. This implies that the initial shear tractions should increase linearly with depth. Maintaining a uniform distance from failure over the entire depth of the fault would require negative initial shear tractions near the ground surface, so we limit the shear tractions to positive values and taper the distance from failure near the ground surface. Because the effective normal tractions increase with depth, the change in the shear tractions will be larger at depth. Figure 9.9 summarizes the variations of these tractions with depth. We do not know the final shear stresses on the fault surface, but we expect the final shear stresses to generally follow the variations of the minimum sliding shear stresses. Figure 9.10 shows the initial shear and normal tractions applied to the fault surface. The asperity used to start the rupture requires only small relative increases in the shear tractions, so unlike the base case, it blends in with the surrounding shear tractions. Due to the large increase in the shear tractions with depth, we choose not to taper the shear tractions along the lateral edges and bottom of the fault.
The rupture propagates rapidly across the bottom of the fault in response to the increase in the dynamic stress drop with depth. Recall from section 8.3.1 that the size of the dynamic stress drop influences the rupture speed. Figure 9.11 shows the distributions of final slip and maximum slip rate on the fault. In contrast to scenario ssbase, both the final slip and maximum slip rate show a clear trend with depth with extraordinarily large values near the bottom of the fault. We see that a dynamic stress drop that increases with depth leads to unreasonable behavior, and, in particular, the final slips and maximum slip rates increase with depth.
We may scale the initial tractions to create reasonable values of final slip and slip rate near the bottom of the fault, but we cannot change the trend with depth as long as the parameters in the friction model remain uniform with depth. Similarly, if the pore pressures remain well below the lithostatic pressures, increasing the pore pressures will only reduce the rate at which the slip and slip rate increase with depth; it cannot change the trend. Additionally, any other friction model with uniform parameters with depth will produce the same trends with depth. This suggests that either the effective normal stresses are uniform with depth or the friction model parameters vary with depth. As we noted in section 8.2.1, the existence of topography and density variations imply the pore pressures do not generally approach the lithostatic normal stresses. Consequently, we will attempt to adjust the parameters in the friction model to compensate for effective normal stresses that increase with depth.
| | | 9.3.2 Lithostatic Stresses with a Uniform Friction Model: Scenario sslithouni |
