Subsidence diffusion–convection: I. The direct problem

Abstract

Compaction of a collapsible substratum due to effective stress increase may give rise to the formation of the well-known trap-door mechanism. According to Litwiniszyn's theory, large-scale subsidence over a yielding underground geo-structure is seen as a stochastic (Markov) process. In simple terms we assume here that the displacement of a particle in vertical direction is diffused to both horizontal and vertical displacement for the particles lying above it. This process may be called kinematic diffusion, since it leads to the Einstein–Kolmogorov (E-K) integral equation. Moreover, following the observations in small-scale model experiments, we assume that the process is always confined between a set of steeply inclined shear bands, whose inclination is evolving. Under certain mathematical transformations the E-K integral equation reduces to a partial differential equation of parabolic type. The solution depends on a diffusivity coefficient, which determines the formation of the subsidence trough inside the body as well as in the surface.