Nonlinear Acceleration of the Iterative-Sequentially Coupled Flow and Geomechanics Fixed Stress Scheme
Authors: Waziri S. , Ren G. , Tomin P. , Guyaguler B. 
Published: August 2024

ABSTRACT:
The iterative sequential fixed stress coupling scheme (SC) for coupled flow and geomechanics has proven to be unconditionally stable. However, SC suffers from poor outer loop convergence, particularly evident in tightly coupled poromechanics problems. This work addresses the issue of slow sequential convergence by applying nonlinear acceleration to the outer loop. First, a coupling strength indicator, coupling strength ratio (CSR), is introduced. CSR is a function of fluid and rock properties, and reservoir models with stronger coupling are characterized by higher CSR values. Numerical examples prove CSR to be a suitable metric of the coupling strength, and the linear convergence rate of the fixed point SC scheme renders it ill-suited for such tightly coupled models. To that end, Aitken relaxation is applied to the porosity update after the mechanics subproblem to improve the outer loop convergence rate. Numerical examples show significant computational gains in terms of fewer outer loops needed to converge to the same residual tolerance.

1 INTRODUCTION
Conventional subsurface simulation typically captures the mechanical effects associated with flow in subsurface porous media using a rock compressibility term. The rock compressibility is either taken to be a constant value or a function of pressure (Aziz, 1979). However, accurate modeling of rock deformations and the changes in permeability and porosity due to stress changes requires proper coupling of flow and mechanics (Settari and Mounts, 1998) which entails solving a coupled system of mass conservation equations and momentum balance for flow and mechanics variables respectively. Carbon capture and sequestration (CCS) (Xu et al., 2021; Pawar et al., 2015), well stability (Wang and Lu, 2001), land subsidence studies (Kidambi et al., 2017), sand production (Wang and Lu, 2001), and unconventional subsurface resources (Settgast et al., 2017; Ren and Younis, 2021) are some of the applications where accurate and efficient simulation of coupled flow and mechanics is paramount.
The widely used coupling scheme for solving coupled flow and mechanics is the monolithic fully coupled fully implicit method (FC). FC solves the coupled system of mass and momentum conservation equations simultaneously. The attractive feature of FC is its unconditional stability. However, linear solver performance can be a computational bottleneck especially for large, tightly coupled poromechanics problems. Alternative to the FC scheme, explicit and iterative sequential methods have been proposed (Settari and Walters, 2001) and compared to FC (Dean et al., 2006). The order in which the subproblems are solved and the constraints imposed during each subsolve have been researched. For the iterative sequential method where mechanics is solved, the undrained split is unconditionally stable for α ≥ 0.5 where α is the parameter of time discretization (Kim et al., 2011c), whereas the fixed stress split method, where flow subproblem is solved first, is unconditionally stable for same α condition (Kim et al., 2011b). However, the fixed stress split is found to be more accurate for a given number of iterations than the undrained split indicating a faster convergence rate (Kim et al., 2011a). Nonetheless, all sequential methods, by design, allows the use of specialized preconditioners and linear solver that target the individual physics and permits the extension of existing single physics simulators into a coupled simulator with relatively less code development (Rin et al., 2017; Garipov et al., 2018).