Adaptive nonlinear solver for a discrete fracture model in operator-based linearization framework

Abstract

Simulation of compositional problems in hydrocarbon reservoirs with complex heterogeneous structure requires adopting stable numerical methods that rely on an implicit treatment of the flux term in the conservation equation. The discrete approximation of convection term in governing equations is highly nonlinear due to the complex properties complemented with a multiphase flash solution. Consequently, robust and efficient techniques are needed to solve the resulting nonlinear system of algebraic equations. The solution of the compositional problem often requires the propagation of the displacement front to multiple control volumes at simulation timestep. Coping with this issue is particularly challenging in complex subsurface formations such as fractured reservoirs. In this study, we present a robust nonlinear solver based on a generalization of the trust-region technique to compositional multiphase flows. The approach is designed to embed a newly introduced Operator-Based Linearization technique and is grounded on the analysis of multi-dimensional tables related to parameterized convection operators. We segment the parameter-space of the nonlinear problem into a set of trust regions where the convection operators maintain the second-order behaviour (i.e., they remain positive or negative definite). We approximate these trust regions in the solution process by detecting the boundary of convex regions via analysis of the directional derivative. This analysis is performed adaptively while tracking the nonlinear update trajectory in the parameter-space. The proposed nonlinear solver locally constraints the updating of the overall compositions across the boundaries of convex regions. Besides, we enhance the performance of the nonlinear solver by exploring diverse preconditioning strategies for compositional problems. The proposed nonlinear solution strategies have been validated for both miscible and immiscible gas injection problems of practical interest.