Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics

We perform detailed stability and convergence analyses of sequential-implicit solution methods for coupled fluid flow and reservoir geomechanics. We analyze four different sequential-implicit solution strategies, where each sub-problem (flow and mechanics) is solved implicitly. Two schemes in which the mechanical problem is solved first, namely, the drained and undrained splits, and two schemes, where the flow problem is solved first, namely, the fixed-strain and fixed-stress splits. The Von Neumann method is used to obtain the linear-stability criteria of the four sequential schemes, and numerical simulations are used to test the validity and sharpness of these criteria for representative problems. The analysis indicates that the drained and fixed-strain splits, which are commonly used, are conditionally stable, and that the stability limits depend only on the strength of coupling between flow and mechanics and are independent of the timestep size. So, the drained and fixed-strain schemes cannot be used when the coupling between flow and mechanics is strong. Moreover, numerical solutions obtained using the drained and fixed-strain sequential suffer from oscillations, even when the stability limit is honored. For problems where the deformation may be plastic (nonlinear) in nature, the drained and fixed-strain sequential schemes become unstable when the system enters the plastic regime. On the other hand, the undrained and fixed-stress sequential schemes are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. While both the undrained and fixed-stress schemes are unconditionally stable, for the cases investigated we found that the fixed-stress split converges more rapidly than the undrained split. Based on these findings, we strongly recommend the fixed-stress sequential-implicit method for modeling coupled flow and geomechanics in reservoirs. Introduction Reservoir geomechanics is concerned with the study of fluid flow and the mechanical response of the reservoir. Reservoir geomechanical behavior plays a critical role in compaction drive, subsidence, well failure, stress dependent permeability, as well as tar sand and heavy oil production (see, e.g., Lewis and Sukirman (1993); Settari and Mourits (1998); Settari and Walters (2001); Thomas et al. (2003); Li and Chalaturnyk (2005); Dean et al. (2006); Jha and Juanes (2007)). The reservoir simulation community has traditionally emphasized flow modeling and oversimplified the mechanical response of the formation through the use of the rock compressibility, taken as a constant coefficient or a simple function of porosity. In order to quantify the deformation and stress fields due to changes in the fluid pressure field and to account for stress dependent permeability effects, rigorous and efficient modeling of the coupling between flow and geomechanics is required. In recent years, the interactions between flow and geomechanics have been modeled using various coupling schemes (Settari and Mourits, 1998; Settari and Walters, 2001; Mainguy and Longuemare, 2002; Minkoff et al., 2003; Thomas et al., 2003; Tran et al., 2004, 2005; Dean et al., 2006; Jha and Juanes, 2007). Coupling methods are classified into four types: fully coupled, iteratively coupled, explicitly coupled, and loosely coupled (Settari and Walters, 2001; Dean et al., 2006). The characteristics of the coupling methods are: 1. Fully coupled. The coupled governing equations of flow and geomechanics are solved simultaneously at every time step. (Lewis and Sukirman, 1993; Wan et al., 2003; Gai, 2004; Phillips and Wheeler, 2007a,b; Jean et al., 2007). For nonlinear problems, an iterative (e.g., Newton–Raphson) scheme is usually employed to compute the numerical solution. The fully coupled method is unconditionally stable, but it is computationally very expensive. Development of a fully coupled flow-mechanics reservoir simulator, which is needed for this approach, is quite costly. 2. Iteratively coupled. These are sequential (staggered) solution schemes. Either the flow, or mechanical, problem is solved first, then the other problem is solved using the intermediate solution information (Prevost, 1997; Settari and Mourits,