Fast and Slow Compaction in Sedimentary Basins

A mathematical model of compaction in sedimentary basins is presented and analyzed. Compaction occurs when accumulating sediments compact under their own weight, expelling pore water in the process. If sedimentation is rapid or the permeability is low, then high pore pressures can result, a phenomenon which is of importance in oil drilling operations. Here we show that one-dimensional compaction can be described in its simplest form by a nonlinear diffusion equation, controlled principally by a dimensionless parameter $\lambda$, which is the ratio of the hydraulic conductivity to the sedimentation rate. Large $\lambda$ corresponds to very permeable sediments, or slow sedimentation, a situation which we term "fast compaction," since the rapid pore water expulsion allows the pore water pressure to equilibrate to a hydrostatic value. On the other hand, small $\lambda$ corresponds to "slow compaction," and the pore pressure is in excess above the hydrostatic value and more nearly equal to the overburden value. We provide analytic and numerical results for both large and small $\lambda$, using also the assumption that the permeability is a strong function of porosity. In particular, we can derive Athy's law (that porosity decreases exponentially with depth) when $\lambda\gg 1$.