Coupling-Strength Criteria for Sequential Implicit Formulations

Sequential Fully Implicit (SFI) schemes have been proposed as an alternative to the Fully Implicit Method (FIM). A significant advantage of SFI is that one can employ scalable strategies to the flow and transport problems. However, the primary disadvantage of using SFI compared with FIM is the fact that the splitting errors induced by the decoupling operator, which separates the pressure from the saturation(s), can lead to serious convergence difficulties of the overall nonlinear problem. Thus, it is important to quantify the coupling strength in an adaptive manner in both space and time. We present criteria that localize the computational cells where the pressure and saturation solutions are tightly coupled. The approach is using terms in the FIM Jacobian matrix, we quantify the sensitivity of the mass and volume-balance equations to changes in the pressure and the saturations. We identify three criteria that provide a measure of the coupling strength across the equations and variables. The standard CFL stability criteria, which are based entirely on the saturation equations, are a subset of the new criteria. Here, the pressure equation is solved using Algebraic MultiGrid (AMG), or a multiscale solver, such as the Multiscale Restricted-Smooth Basis (MsRSB) approach. The transport equations are then solved using a fixed total-velocity. These ‘coupling strength’ criteria are used to identify the cells where the pressure-saturation coupling is strong. The applicability of the derived coupling-strength criteria is tested using several test cases. The first test is using a gravitational immiscible dead-oil lock-exchange under a unit mobility ratio and large differences in density. For this case, the SFI algorithm fails to converge to the fully coupled solution due to the large splitting errors. Introducing a fully coupled solution stage on the local subdomains as an additional correction step restores nonlinear convergence. Detailed analysis of the ‘coupling strength’ criteria indicates that the criteria related to the sensitivity of the mass balance to changes in the pressure and the sensitivity of the volume balance to changes in the saturations are the most important ones to satisfy. Other test cases include an alternate gas-water-gas injection in a top layer of the SPE 10 test case and an injection-production scenario in a three-dimensional reservoir with layered lognormally distributed permeability. We propose novel criteria to estimate the strength of coupling between pressure and saturation. These CFL-like numbers are used to identify the cells that require fully implicit treatment in the nonlinear solution strategy. These criteria can also be used to improve the nonlinear convergence rates of Adaptive Implicit Methods (AIM).