An Unconditionally Stable Splitting Method Using Reordering for Simulating Polymer Injection

We present an unconditionally stable algorithm 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. The resulting nonlinear system of discrete transport equations can, in the absence of gravity and capillary forces, be permute to lower triangular form by using a simple topological sorting method from graph theory. This gives a nonlinear Gauss--Seidel method that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of scalar nonlinear equations that can easily 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. Hence, sequential splitting combined with standard upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.