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.