Abstract
We present a set of algorithms for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a two-phase system with rock compressibility and equal fluid compressibilities. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the time step. In the absence of gravity and capillary forces, the splitting is unconditionally stable and the resulting nonlinear system of discrete transport equations can be permuted to lower triangular form by using a simple topological-sorting method from graph theory. This gives an optimal nonlinear solver that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of nonlinear scalar equations that can be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in runtimes and increases the feasible time-step sizes. For cases with gravity, the same method can be applied as part of a nonlinear Gauss-Seidel method. Altogether, our results demonstrate that sequential splitting combined with standard upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.
