An approximate Jacobian nonlinear solver for multiphase flow and transport

We present an approximate Jacobian approach for solving nonlinear, multiphase flow and transport problems in porous media. A backward Euler time discretization scheme is used prior to spatial discretization with a lowest order mixed finite element method (MFEM). This results in a fully implicit nonlinear algebraic system of equations. Conventionally, an exact Jacobian construction is employed during the Newton linearization to obtain a linear system of equations after spatial and temporal discretization. This fully coupled, monolithic linear system, usually in pressure and saturation (or concentration) unknowns, requires specialized preconditioners such as constrained pressure residual (CPR) or two stage preconditioner. These preconditioners operate on the linear system to decouple pressure and saturation (or concentration) degrees of freedom (DOF) in order to use existing linear solvers for positive definite (PD) matrices such as GMRES or AMG, to name a few. In this work, we present an alternative to two-stage preconditioning (or CPR) for solving the aforementioned monolithic system after Newton linearization. This approach relies upon a decoupling approximation for the pressure-saturation (or concentration) block sub-matrices, during Newton linearization, to obtain block diagonal sub-matrices. The resulting linear system is easily reduced, trivially eliminating these diagonal sub-matrices, to obtain a system in pressure DOF circumventing the need for specialized preconditioners. Further, the linear system has lesser DOF owing to the elimination of saturation (or concentration) unknowns. This nonlinear solver is demonstrated to be as accurate as the exact Jacobian approach, measured in terms of convergence of nonlinear residual to a desired tolerance for both methods. Our numerical results indicate a consistent computational speedup by a factor of approximately 1.32 to 4.0 for the two-phase flow model formulation under consideration. This is related to the DOF of the linear systems for the approximate and exact Jacobian approaches. For multicomponent flow and transport, this speedup is expected to be directly proportional to the number of concentration degrees of freedom. A number of field scale numerical simulations are also presented to demonstrate the efficacy of this approach for realistic problems.