Sequential-implicit Newton's method for geothermal reservoir simulation

Numerical simulation of thermal multiphase fluid flow poses significant difficulties for nonlinear solvers. One approach to this problem is to solve the entire nonlinear system of equations simultaneously with a fully coupled method. However, due to the strong coupling and multiphysics interactions between equations, it is difficult to analyze and challenging to design solvers using this fully coupled method. Instead, a sequential-implicit method splits the multiphysics problem into different subproblems so that they can be each solved separately. The research described in this thesis developed and investigated sequential-implicit methods for geothermal simulation. The sequential-implicit method isolates each of the subproblems and enables the use of specialized solvers for each separate subproblem. Once each subproblem is solved in an efficient manner, the entire multiphysics problem is coupled through a sequential-implicit method. However, these sequential-implicit methods can face difficulties converging to the coupled solution. This thesis describes various sequential-implicit methods that reduce the computational cost of subsurface geothermal simulations by improving their nonlinear convergence. This split of the different physics allows for more specialized solvers such as multiscale finite volume or linear solver preconditioning methods to be built upon it. We demonstrated that for sequential-implicit geothermal simulations, a hybrid constraint strategy is necessary. This hybrid approach involves imposing a fixed pressure constraint for single-phase cells (control volumes) and a fixed density for two-phase cells. However, numerical comparisons showed that the outer loop convergence for the hybrid method on complex scenarios performed poorly in comparison to the fully coupled method. To improve on the outer loop convergence, a modified-sequential fully implicit method was investigated. Although the modified sequential-implicit method improves the outer loop convergence, the additional computational cost of the modified-sequential fully implicit method could diminish the gains from the improved convergence. One of the main contributions of this work is the development of a sequential-implicit Newton's method. Sequential-implicit methods often suffer from slow convergence when there is a strong coupling between the individual subproblems. This is due to the slow linear convergence rate of the fixed-point iteration that is used in current sequential-implicit methods. This new sequential-implicit Newton's method follows the same sequential scheme as the current sequential-implicit fixed-point method, but with a faster quadratic convergence rate compared to the current linear convergence rate. This method is not only applicable to geothermal simulation but also to all sequential-implicit multiphysics simulations. We demonstrated the effectiveness of this approach on two different multiphysics porous media problems: flow-thermal in geothermal simulation and flow-mechanics in geomechanics reservoir simulation. The numerical experiments show an improvement in outer loop convergence across all multiphysics problems and test cases considered. For some specific cases where there was a strong coupling, up to two orders of magnitude improvement was seen in the outer loop convergence for the sequential-implicit Newton's method. Following these investigations of the sequential-implicit method, we developed a sequential-implicit nonlinear solver to better solve a condensation problem in fully coupled geothermal simulation. This condensation problem is associated with the flow of cold water into a cell that is at saturated conditions that is also known as a ``negative compressibility'' problem. In order to deal with this problem, the nonlinear solver must be modified. We developed a sequential-implicit nonlinear solution strategy that overcomes this nonlinear convergence problem associated with the condensation front. For one-dimensional problems, this nonlinear solution strategy converged for all timesteps sizes, while the fully coupled strategy only converged for a limited sized timestep. Furthermore, for a two-dimensional heterogeneous problem, the largest Courant-Friedrichs-Lewy (CFL) number for this method is at least an order of magnitude larger than the CFL number that can be used with the fully coupled approach.

Keywords: Energy Resources Engineering.
Advisor: Horne Roland N. 
Tchelepi Hamdi 
Tomin Pavel