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Sequential Refinement Solver using Space-Time Domain Decomposition for Non-linear Multiphase Flow Problems

Hanyu Li, Mary F. Wheeler

TL;DR

The paper tackles convergence failures in solving nonlinear multiphase flow equations by introducing a sequential local refinement strategy in space-time domain, enabling larger time steps without sacrificing accuracy. It employs a space-time mixed finite element formulation and a multi-level refinement procedure guided by saturation-change indicators to provide accurate initial guesses for Newton iterations. Numerical tests on SPE10-based problems demonstrate approximately 5x speedup with good accuracy for Gaussian-like media, while channel-like permeability cases reveal limitations of isotropic refinement and motivate separating time and space refinement. The approach offers a robust, scalable pathway to accelerate large-scale nonlinear multiphase flow simulations in subsurface porous media, with potential extensions to more complex models.

Abstract

Convergence failure and slow convergence rate are among the biggest challenges with solving the system of non-linear equations numerically. While using strictly small time steps sizes and unconditionally stable fully implicit scheme mitigate the problem, the computational load becomes enormous. We introduce a sequential local refinement scheme in space-time domain that improves convergence rate and prevents convergence failure while not restricting to small time step, thus boosting computational efficiency. We rely on the non-linear two-phase flow model. The algorithm starts by solving the coarsest mesh. Then regions with certain features such as saturation front is refined to the finest resolution sequentially. Such process prevents convergence failure. After each refinement, the solution from the previous mesh is used to estimate initial guess of the current mesh for faster convergence. Numerical results are presented to confirm accuracy of our algorithm as compared to the traditional fine time step approach. We also observe 5 times speedup in the runtime by using our algorithm.

Sequential Refinement Solver using Space-Time Domain Decomposition for Non-linear Multiphase Flow Problems

TL;DR

The paper tackles convergence failures in solving nonlinear multiphase flow equations by introducing a sequential local refinement strategy in space-time domain, enabling larger time steps without sacrificing accuracy. It employs a space-time mixed finite element formulation and a multi-level refinement procedure guided by saturation-change indicators to provide accurate initial guesses for Newton iterations. Numerical tests on SPE10-based problems demonstrate approximately 5x speedup with good accuracy for Gaussian-like media, while channel-like permeability cases reveal limitations of isotropic refinement and motivate separating time and space refinement. The approach offers a robust, scalable pathway to accelerate large-scale nonlinear multiphase flow simulations in subsurface porous media, with potential extensions to more complex models.

Abstract

Convergence failure and slow convergence rate are among the biggest challenges with solving the system of non-linear equations numerically. While using strictly small time steps sizes and unconditionally stable fully implicit scheme mitigate the problem, the computational load becomes enormous. We introduce a sequential local refinement scheme in space-time domain that improves convergence rate and prevents convergence failure while not restricting to small time step, thus boosting computational efficiency. We rely on the non-linear two-phase flow model. The algorithm starts by solving the coarsest mesh. Then regions with certain features such as saturation front is refined to the finest resolution sequentially. Such process prevents convergence failure. After each refinement, the solution from the previous mesh is used to estimate initial guess of the current mesh for faster convergence. Numerical results are presented to confirm accuracy of our algorithm as compared to the traditional fine time step approach. We also observe 5 times speedup in the runtime by using our algorithm.

Paper Structure

This paper contains 9 sections, 34 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Two phase flow formulation
  3. Governing equations
  4. Fully discrete formulation
  5. Solution algorithm
  6. Sequential local refinement
  7. Data structure
  8. Numerical results
  9. Conclusions

Figures (15)

  • Figure 1: Saturation and normalized initial non-linear residual at 100 and 800 days
  • Figure 2: Sequential local mesh refinement in space-time domain from coarsest (level 0) to finest (level 1) resolution
  • Figure 3: Normalized initial non-linear residual at each refinement level
  • Figure 4: Normalized error indicator at each refinement level
  • Figure 5: Cumulative distribution function fitted to initial residual data at each refinement level
  • ...and 10 more figures