Three-dimensional finite element calculations are employed to study interactions in space and time between the creeping segment of the San Andreas fault in central California and the adjacent currently locked zones of the 1857 and 1906 great earthquakes. Vertically, the model consists of an elastic upper crust over a Maxwell viscoelastic region, representing the entire lower crust or a narrower horizontal detachment layer, and a stiffer and more viscous upper mantle. The crust has a single vertical fault extending to the top of the mantle at 25 km depth. In zones along strike corresponding to the 1857 and 1906 events, the top 12.5 km of the fault is locked against slip, except in great earthquakes. Below the locked zones and everywhere along the creeping region between them, the fault is freely slipping. The model parameters are compatible with seismological and geological observations, and with a ratio of Maxwell relaxation time to the relaxing layer thickness in the range 1 to 2 yr/km, as established by Li and Rice (1987) and Fares and Rice (1988) based on fits to geodetic data along the San Andreas fault. An imposed constant far field shear motion and periodic 1857- and 1906 - type earthquakes generate slip rates along the creeping fault segment that evolve in time throughout the entire earthquake cycle. Shortly after an adjacent great earthquake, slip rates in the creeping zone are higher than the far field velocity, while later in the cycle they are lower. Hence, time dependency should be accounted for when measurements of fault slip are used to estimate the plate motion. If Parkfield earthquakes are a response to a time dependent loading of the type simulated here, their recurrence interval would tend to lengthen with time since the 1857 event. Thus, the hypothesis of characteristic periodic earthquakes at Parkfield may not provide the best estimate of the occurrence time of the next event. Using, for example, the statistics of past events and assuming that Parkfield earthquakes ate a response to a slip deficit near Middle Mountain, and that the elastic crustal layer is 17.5 km thick, we find that the next event is predicted for about 1992 +/- 9 years if the lower crust is a 7.5 km thick layer having a material relaxation time of 15 years, and 1995 +/- 11 years if the 7.5 km thick lower crust is characterized by a relaxation time of 7.5 years. These values may be compared to the 1988 +/- 7 years estimate based on periodicity in time. The modeling results also indicate that the interaction between the 1857 and 1906 rupture zones is small.
