A Trust-region Based Sequential Implicit Nonlinear Solver for Geothermal Simulations

Abstract

The so-called condensation problem poses numerical convergence challenges for nonlinear solvers. Herein, a single-cell condensation problem setup is used to demonstrate the conditional and restrictive nonlinear convergence of existing schemes, which include the fully coupled fully implicit method with standard Newton solver (FIM), the sequential fully implicit method (SEQ) with constant enthalpy during the Flow subproblem and constant total density during the Thermal subproblem and the modified sequentially preconditioned FIM (MSFIP-FIM). Then a new scheme, the trust-region based adaptive sequential fully implicit scheme (aSEQ), is proposed. aSEQ is composed of two parts; a trust-region relaxation based on prevention of crossing of kinks in the residual space for the Flow subproblem and adaptivity between constant pressure and constant total density constraint depending on phase conditions during the thermal subproblem. Using numerical examples, aSEQ is shown to have significantly superior nonlinear convergence compared to the mentioned existing methods in terms of the timestep size for which the convergence can be achieved and a larger convergence radius for a given timestep size. Modifications are proposed to generalize aSEQ to multiple cells and a one-dimensional example is used to demonstrate the significant computational savings of aSEQ.