Comparison of fully-coupled and sequential solution methodologies for enhanced geothermal systems

Abstract

The simulation of enhanced geothermal systems (EGS) requires finding the solution to a highly coupled nonlinear set of partial differential equations. Verification of EGS modelling has been difficult due to a lack of analytical or semi-analytical solutions and is typically limited to a subset of processes. A comparison of alternative solution schemes is one method to numerically verify the coupling between all processes simultaneously and provide confidence in the solutions produced by the different solution schemes. This article presents the first such comparison of monolithic and sequential solution schemes for the modelling of EGS wells. A custom thermo-hydro-mechanical finite element model for an EGS well connected by planar fractures is developed along with two monolithic schemes; one using the Newton–Raphson iterative method and another using the Newton–Raphson iterative method modified by Aitken’s Δ2 relaxation method. The two monolithic schemes are compared against two sequential schemes and one loosely-coupled scheme in terms of accuracy and computational efficiency. The monolithic schemes are shown to have an optimal rate of convergence with respect to mesh and timestep refinement. The monolithic Newton–Raphson scheme is shown to require small timesteps that violate the minimum timestep size of the β-method of time integration, introducing spurious oscillations into the solution. Aitken’s Δ2 relaxation method is shown to improve the ability of the Newton–Raphson scheme to find a converged solution at larger timesteps, avoiding spurious oscillations and reducing overall computation time. The sequential solution schemes are shown to provide optimal rates of convergence with respect to mesh and timestep refinement and converge to the same solution as the monolithic solution schemes, verifying the results of both methodologies and previous modelling efforts. It is demonstrated that the sequential solution schemes are computationally more expensive than the monolithic schemes when seeking well-converged solutions. The loosely-coupled scheme demonstrates an optimal rate of convergence with mesh and timestep refinement, but is also shown to be much less accurate than the other schemes. The recommended solution scheme for the efficient simulation of EGSs is thus the Newton–Raphson scheme with Aitken’s Δ2 relaxation.