The Viscoelastic Solution to Geertsma’s Subsidence Problem

Abstract

This paper presents the viscoelastic analytical solution for stress and displacement due to prescribed time-varying changes in the pore fluid pressure of a disk-shaped inclusion embedded within a semi-infinite, viscoelastic medium. The correspondence principle of viscoelasticity, along with Hankel–Fourier and Laplace transforms, is used to derive the solution. The instantaneous viscoelastic solution, corresponding to the response immediately after the inclusion pore pressure change, recovers the elastic solution to the same problem (Geertsma 1973). Results are presented for fractional Maxwell and Burgers models of viscoelasticity after being applied to a set of experimental data from creep tests on shale. Solution results are demonstrated and discussed for the cases of constant inclusion depletion, as well as delayed injection of fluid into a previously depleted inclusion.