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

Abstract

Abstract We perform detailed stability and convergence analyses of iteratively coupled solution methods for coupled fluid flow and reservoir geomechanics. We analyse four different operator-split strategies: two schemes in which the mechanical problem is solved first (drained and undrained splits), and two schemes where the flow problem is solved first (fixed-strain and fixed-stress splits). We use the Van Neumann method to obtain sharp stability criteria. The drained and fixed-strain splits, which are commonly used, are only conditionally stable. Their stability limits are independent of time step size and depend only on the strength of the coupling between flow and mechanics. This implies that problems with strong coupling cannot be solved by the drained or fixed-strain split methods. Moreover, oscillations in the solution by the drained and fixed-strain split methods can occur even when the stability limit is honored. For problems where the deformation may be plastic in nature, the drained and fixed-strain sequential schemes suffer from severe stability problems when the system enters the plastic regime. However, the undrained and fixed-stress splitting methods are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. We also show that the drained and fixed-strain split methods with a fixed number of iterations are inconsistent. That is, they converge to the wrong solution as the time step size goes to zero. On the other hand, the fully coupled, undrained, and fixed-stress methods are consistent even in the case of a single iteration per time step. While both the undrained and fixed-stress schemes are stable, the fixed-stress method is more accurate for a given number of iterations than the undrained method. As a result, we strongly recommend the fixed-stress split. These results have immediate and widespread applicability in the design of reservoir flow simulators that account for geomechanics. 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)). However, the reservoir simulation community has traditionally emphasized flow modeling and oversimplified the mechanical response 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, 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 Longeumare, 2002; Minkoff et al., 2003; Thomas et al., 2003; Tran et al., 2004, 2005; Dean et al., 2006; Hua 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: