Linear and nonlinear solvers for simulating multiphase flow within large-scale engineered subsurface systems

Heeho D. Park, Glenn E. Hammond, Albert J. Valocchi, Tara LaForce

Research output: Contribution to journalArticlepeer-review


Multiphase flow simulation is well-known to be computationally demanding, and modeling large-scale engineered subsurface systems entails significant additional numerical challenges. These challenges arise from: (a) the presence of small-scale discrete features like shafts, tunnels, waste packages, and barriers; (b) the need to accurately represent both the waste form processes at the small spatial scale of the repository and the large-scale transport processes throughout heterogeneous geological formations; (c) the strong contrast in material properties such as porosity and permeability, as well as the nonlinear constitutive relations for multiphase flow. Numerical solution entails discretization of the coupled system of nonlinear governing equations and solving a linear system of equations at each Newton–Raphson iteration. Practical problems require a very large number of unknowns that must be solved efficiently using iterative methods in parallel on high-performance computers. The unique challenges noted above can lead to an ill-conditioned Jacobian matrix and non-convergence with Newton's method due to discontinuous nonlinearity in constitutive models. Moreover, practical applications can require numerous Monte-Carlo simulations to explore uncertainly in material properties, geological heterogeneity, failure scenarios, or other factors; governmental regulatory agencies can mandate these as part of Performance Assessments. Hence there is a need for flexible, robust, and computationally efficient methods for multiphase flow in large-scale engineered subsurface systems. We apply the open-source simulator PFLOTRAN to the practical problem of performance assessment of the US DOE Waste Isolation Pilot Plant (WIPP) site. The simulator employs a finite volume discretization and uses the PETSc parallel framework. We evaluate the performance of several preconditioners for the iterative solution of the linearized Jacobian system; these range from stabilized-biconjugate-gradient with block-Jacobi preconditioning (BCGS) to methods adopted from reservoir modeling, such as the constrained pressure residual (CPR) two-stage preconditioner and flexible generalized residual solver (FGMRES). We also implement within PETSc the general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), which truncates the Newton update or modifies the update with a Cauchy solution that is within the quadratic model trust-region of the objective function. We demonstrate the effectiveness of each method for a series of test problems with increasing difficulty. We find that the NTRDC and FGMRES-CPR-ABF (FCA) preconditioners generally perform best for the test problem having the extreme nonlinear processes, achieving a 50x speed-up compared with BCGS. The most ill-conditioned and extreme nonlinear simulations do not converge with BCGS (as one may expect), but they do complete the simulation with NTRDC and FCA. We also investigate the strong scalability of each method and demonstrate the impact of node-packing upon parallel performance on modern processor architectures.

Original languageEnglish (US)
Article number104029
JournalAdvances in Water Resources
StatePublished - Oct 2021


  • Multiphase flow
  • Nonlinear
  • Porous media
  • Preconditioner
  • Subsurface system
  • Trust-region

ASJC Scopus subject areas

  • Water Science and Technology


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