Convergence and Error Analysis of Fully Discrete Iterative Coupling Schemes for Coupling Flow with Geomechanics

Abstract

The modeling of fluid flow coupled with mechanical deformations is needed for reliable production forecasts, especially in stress sensitive, and structurally weak reservoirs. It also plays a critical role in accurately predicting surface subsidence, well stability, CO2 sequestration, and sand production. Traditionally, changes in mechanical deformations are visible to fluid flow through a pore compressibility factor, which is insufficient for purposes described above. In fact, it is only through the coupling between flow and mechanics that reliable reservoir models can be obtained. In this work, we consider the multirate fixed-stress split iterative coupling scheme for the Biot system modeling coupled flow with geomechanics in a poro-elastic medium. Due to its physical nature, the geomechanics problem can cope with a coarser time step compared to the flow problem. This makes the multirate coupling scheme, in which several flow finer time steps are solved within one coarser mechanics time step, a natural candidate in this setting. For the multirate iterative scheme proposed here, our contributions are: 1. We establish the contracting behavior of the two successive iterates leading to geometric speed of convergence, 2. We derive error estimates for quantifying the error between any iterate and the true solution. Error estimates of the fully discrete fixed-stress split iterative coupling are derived for the first time in this work. 3. We show the sharpness of our derived theoretical estimates by implementing the proposed scheme numerically for field scale problems. Our approach is based on studying the equations satisfied by the difference of iterates and utilizing a Banach contraction argument to show that the corresponding scheme is a fixed point iteration. Obtained contraction results are then used to derive theoretical convergence error estimates for the iterative coupling scheme. Moreover, by comparing theoretical contraction estimates against numerical computations, we conclude that theoretical estimates can predict the contracting behavior, and subsequently, the rate of convergence of the iterative scheme with high accuracy.