New rheological and porosity equations for steady-state compaction

Diagenetic theory, as it is often stated, is formally incomplete in the sense that it contains more dependent variables than the number of equations in the theory. Heretofore, this situation has been resolved ordinarily by introducing an empirical equation for porosity, phi, or equivalently the solid volume fraction, phi(s) = 1 - phi, as a function of depth. In contrast, the theory of compaction, that is combined momentum and stress balances, leads to a differential equation that governs the behavior of phi(s), thus completing standard diagenetic theory. Based on recently acquired in situ data, we advance that the steady state change in solid volume fraction, d phi(s), during compaction is web described by a function of the change in effective stress on the solids, d sigma'; specifically, d phi(s) = A exp (-b sigma')d sigma' where A and b are parameters that specify the initial compressibility and the attenuation of compressibility, respectively. From this rheology and the justifiable assumption that the Darcian contribution to the stress can be neglected, the steady-state distribution of phi(s) is governed by the equation phi(s) = (phi(s))(o)(phi(s))(infinity)/(phi(s))(o) + ((phi(s))(infinity) - (phi(s))(o)) exp (-beta x) where x is depth, beta is a depth-attenuation constant, and the subscripts o and infinity indicate values at the sediment-water interface and the asymptotic value at great depth, respectively. Fits of phi(s) data with this new equation are similar in quality to those obtained with the classical exponential and power-law functions.