PhD Dissertation Defense: Sohail Waziri

Trust-region based fully implicit nonlinear solver for condensation-dominated geothermal systems 

Abstract: Thermal multiphase flow simulation is central to the sustainable management of subsurface energy systems. The fully coupled, fully implicit method (FIM) is the standard method for such problems, but it suffers from stability and convergence issues in geothermal systems undergoing steam condensation. FIM is unconditionally stable only when the underlying PDE is stable, yet the coupled mass and energy conservation equations for flow with condensation are inherently unstable, owing to the negative apparent compressibility of the steam–water mixture. FIM thus resolves the unstable mode only below a critical time-step size, and its convergence is further restricted by the highly nonlinear residual space arising from the orders-of-magnitude contrast between water and steam properties. We recast the governing equations in dimensionless form, derive the apparent compressibility—showing it takes opposite signs in the two-phase and liquid regions—and obtain the stability and convergence limits of FIM. We then introduce a multi-pass preconditioned fully implicit method (mppFIM), which combines nonlinear preconditioning of the condensation problem with physics-guided trust-region corrections to the post-Newton updates, applied adaptively—most tightly at the front—until convergence. Unlike FIM, mppFIM converges for arbitrarily large time steps on challenging problems, yielding substantial computational savings. Negative apparent compressibility also degrades linear solver performance. Using the true-IMPES constrained pressure residual (CPR) operator to extract the pressure system, we assess its suitability for M-matrix-based solvers such as AMG. The operator's departure from the M-matrix sign structure is confined to liquid/two-phase interfaces and stems directly from the contrasting compressibility signs of the two regions. The effects of rock energy accumulation and thermal conduction are also examined.