The research of multiphysical thermal-hydraulic-mechanical (THM) simulation has achieved significant progress in the past decade. Currently, two general approaches for poromechanical simulation co-exist in the reservoir simulation community, namely the stress approach with stress as the primary variable for the mechanical governing equations and the displacement approach with displacement as the primary variable. In this work, we aim to provide a theoretical foundation and a practical semi-analytical solution for the stress approach based on the Navier-Beltrami-Michell Equations. Moreover, we will clarify the relationship (and equivalence) between the two approaches.
We have firstly proven the existence and uniqueness of the stress solution of Navier-Beltrami-Michell equation with given pressure and temperature field. Moreover, we have demonstrated the equivalence of the stress formulation to the displacement formulation. Based on Fourier's expansion, we have developed a general semi-analytical solution for thermal-hydraulic-mechanical process. The semi-analytical solution takes the pressure solution from the hydraulic simulation module (or a commercial reservoir simulator) and directly predicts the stress tensor of the multiphysical system. As such, the solution can be programmed fully coupled with the hydraulic simulation module to predict the stress field with varying pressure and temperature of homogeneous poroelastic rocks under given stress boundary conditions.
From the work above, we have laid a theoretical foundation for the stress approach. The derived semi-analytical solution of the stress field shows excellent accuracy. The solution has been used to predict the transient stress field of a dual-porosity system during primary depletion.
This paper is arguably the first trial to clarify the relationship between the stress approach and the displacement approach. Moreover, the derived semi-analytical solution provides a convenient yet precise way to obtain the stress field without time-consuming numerical simulation.