August 24, 2023

Journal Article

## Newton trust-region methods with primary variable switching for simulating high temperature multiphase porous media flow

## Abstract

Numerical simulation of multiphase flow in large-scale engineered subsurface systems is widely recognized as computationally demanding due to strong non-linearity in the governing equations and the need to resolve a wide range of spatial and temporal scales. Coupling multiphase flow with energy transport due to high temperature heat sources introduces significant new challenges since boiling and condensation processes can lead to dry-out conditions with subsequent re-wetting. The transition between two-phase and single-phase behavior can require changes to the primary dependent variables (PDVs) adding discontinuities as well as extending constitutive nonlinear relations to extreme physical conditions. Discretization of the coupled system of conservation equations for liquid, gas and energy leads to a system of nonlinear algebraic equations for the solution at the new time level; this system is typically solved iteratively by the Newton-Raphson (NR) method, which entails solving a linear Jacobian system at each iteration. Practical simulations of large-scale engineered domains lead to Jacobian systems with a very large number of unknowns that must be solved efficiently using iterative methods in parallel on high-performance computers. Performance assessment (PA) of potential nuclear repositories, carbon sequestration sites and geothermal reservoirs can require numerous Monte-Carlo simulations to explore uncertainty in material properties, boundary conditions, and failure scenarios. Due to the numerical challenges, standard NR iteration may not converge over the range of required simulations. We use the open-source simulator PFLOTRAN for the important practical problem of the safety assessment of future nuclear waste repositories in the U.S. DOE geologic disposal safety assessment (GDSA) Framework. The simulator applies PETSc parallel framework and a backward Euler, finite volume discretization. We demonstrate failure of conventional NR method and the success of trust-region modifications to Newton's method for a series of test problems of increasing complexity. Trust-region methods essentially modify the Newton step size and direction under some circumstances where the standard NR iteration can cause solution to diverge or oscillate. We show how the Newton Trust-Region method can be adapted for Primary Variable Switching (PVS) when the multiphase state changes due to boiling or condensation. The simulations with high-temperature heat sources which led to extreme nonlinear processes with many state changes in the domain did not converge with NR, but they do complete successfully with the trust-region methods modified for PVS. This implementation effectively decreased weeks of simulation time needing manual adjustments to complete a simulation down to a day. Furthermore, we show the strong scalability of the methods on a single node and multiple nodes in HPC cluster.*
Published: August 24, 2023*

## Citation

Park H.D., M. Paul, G.E. Hammond, and A.J. Valocchi. 2022. Newton trust-region methods with primary variable switching for simulating high temperature multiphase porous media flow.*Advances in Water Resources*168. PNNL-SA-169602. doi:10.1016/j.advwatres.2022.104285