On the Derivation of Closed‐Form Expressions for Displacements, Strains, and Stresses Inside Poroelastic Reservoirs

We critically review the derivation of closed‐form analytical expressions for elastic displacements, strains, and stresses inside a subsurface reservoir undergoing pore pressure changes using inclusion theory. Although developed decades ago, inclusion theory has been used recently by various authors to obtain fast estimates of depletion‐induced and injection‐induced fault stresses in relation to induced seismicity. We therefore briefly address the current geomechanical relevance of this method, and provide a numerical example to demonstrate its use to compute induced fault stresses. However, the main goal of our paper is to correct some erroneous assumptions that were made in earlier publications. While the final expressions for the poroelastic stresses in these publications were correct, their derivation contained conceptual mistakes due to the mathematical subtleties that arise because of singularities in the Green's functions. The aim of our paper is therefore to present the correct derivation of expressions for the strains and stresses inside an inclusion and to clarify some of the results of the aforementioned studies. Furthermore, we present two conditions that the strain field must satisfy, which can be used to verify the analytical expressions. Plain Language Summary Previous authors derived mathematical expressions to aid in understanding the nature of earthquakes as a result of the production of fluids from the subsurface or the injection of fluids into it. These expressions represent relationships between stresses and pore pressures in subsurface reservoirs (i.e., sealed bodies of porous rock containing fluids). Although the expressions are correct, the reasoning behind their derivation contained flaws related to mathematical subtleties in the underlying theory. In the present paper we explain those errors and present the correct derivation. Moreover we present two additional results that can be of help in the derivation of similar expressions for different configurations or purposes, and we give an example to demonstrate the use of these expressions in computing stresses caused by fluid production from a faulted reservoir. Key Points We clarify and correct expressions for poroelastic strains and stresses obtained with inclusion theory in previous studies Green's functions for strains and stresses can be integrated to obtain analytical expressions outside, but not inside poroelastic reservoirs We provide conditions to verify analytical solutions