Coupling-strength criteria for sequential implicit formulations

Abstract

The Fully Implicit Method (FIM) is used widely in the reservoir simulation community, especially for the so-called black-oil model. The primary reason for resorting to FIM is its unconditional stability, but this comes at the high computational cost of solving coupled nonlinear systems of all the conservation equations simultaneously. In addition, having a scalable FIM simulator relies on specialized multi-stage linear solvers. In the last few years, the Sequential Fully Implicit (SFI) formulation has received a great deal of attention. SFI decomposes the fully coupled nonlinear problem into two nonlinear sub-problems: flow and transport. The flow problem is solved implicitly for pressure, and the transport problem is solved implicitly for the saturation(s). The splitting usually assumes that the total-velocity is fixed during the transport computations. Because the pressure has a near-elliptic behavior and the saturation(s) have near-hyperbolic behavior, the flow-transport splitting of SFI allows for the use of dedicated scalable nonlinear and linear solvers for each sub-problem.At convergence, SFI and FIM lead to the same numerical solution within the specified tolerances on the residuals. SFI, however, can suffer from convergence difficulties of the overall nonlinear iterations between flow and transport. The convergence difficulties of SFI are intimately related to the splitting, and they depend on the type and strength of coupling between the nonlinear flow and transport problems. In this paper, we develop indicators that quantify the nature and strength of coupling across the systems of equations and solution variables. The indicators quantify the coupling in both space and time. Our strategy employs terms in the FIM Jacobian matrix to quantifying the sensitivity of the mass conservation equations and the global volume-balance to changes in pressure and saturation.These ‘coupling strength’ indicators are used in a domain-decomposition strategy to define local sub-domains for which a fully coupled correction is performed instead of the usual fixed-point iteration update. We demonstrate the accuracy and utility of these indicators for several challenging problems, including buoyancy driven lock-exchange - with or without gas solubility, as well as, gas and water injection problems in heterogeneous reservoir models.