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Implicit-Explicit schemes for decoupling multicontinuum problems in porous media

Maria Vasilyeva

TL;DR

This work addresses efficient simulation of flow in fractured multicontinuum porous media by formulating a general implicit-Explicit (ImEx) time discretization that decouples the continua and pairing it with NLMC nonlocal multicontinua upscaling for accurate coarse-grid models in place of costly fine-grid solves. The authors develop a unified framework for both fine-scale (FVM) and coarse-scale (NLMC) spaces, establish stability for two- and three-level implicit schemes, and propagate the decoupling to two-level and three-level ImEx schemes with D/L/U variants. Numerical experiments on two- and three-continuum tests demonstrate that ImEx decoupling maintains accuracy while dramatically reducing computational effort, with coarse NLMC solutions achieving comparable errors at a fraction of the cost. The result is a scalable, robust solver that leverages multiscale upscaling and operator splitting to enable fast simulations of high-contrast multicontinuum flows in fractured reservoirs, with substantial practical impact for reservoir simulation and related geoscience applications.

Abstract

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct an accurate and physically meaningful coarse-scale approximation. The NLMC method is an accurate technique to develop a physically meaningful coarse scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.

Implicit-Explicit schemes for decoupling multicontinuum problems in porous media

TL;DR

This work addresses efficient simulation of flow in fractured multicontinuum porous media by formulating a general implicit-Explicit (ImEx) time discretization that decouples the continua and pairing it with NLMC nonlocal multicontinua upscaling for accurate coarse-grid models in place of costly fine-grid solves. The authors develop a unified framework for both fine-scale (FVM) and coarse-scale (NLMC) spaces, establish stability for two- and three-level implicit schemes, and propagate the decoupling to two-level and three-level ImEx schemes with D/L/U variants. Numerical experiments on two- and three-continuum tests demonstrate that ImEx decoupling maintains accuracy while dramatically reducing computational effort, with coarse NLMC solutions achieving comparable errors at a fraction of the cost. The result is a scalable, robust solver that leverages multiscale upscaling and operator splitting to enable fast simulations of high-contrast multicontinuum flows in fractured reservoirs, with substantial practical impact for reservoir simulation and related geoscience applications.

Abstract

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct an accurate and physically meaningful coarse-scale approximation. The NLMC method is an accurate technique to develop a physically meaningful coarse scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.
Paper Structure (14 sections, 4 theorems, 67 equations, 7 figures, 9 tables)

This paper contains 14 sections, 4 theorems, 67 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Fine-scale approximation by space (FVM)
  4. Multiscale approximation by space (NLMC)
  5. Implicit approximation by time (coupled scheme)
  6. Two-level scheme
  7. Three-level scheme
  8. Implicit-Explicit scheme (decoupled scheme)
  9. Two-level scheme
  10. Three-level scheme
  11. Numerical results
  12. Implicit-Explicit for fine-scale system
  13. Implicit-Explicit for coarse-scale system
  14. Conclusion

Key Result

Theorem 1

The solution of the discrete problem mm4 is stable with $\theta \geq 1/2$ and satisfies the following estimate

Figures (7)

  • Figure 1: Computational domain $\Omega = [0,2] \times [0,1]$ with 10 fractures $\gamma_l$, $\gamma = \cup_{l=1}^{10} \gamma_l$
  • Figure 2: Two-continuum media (2C). Reference solution at $t = T_{max}/4, T_{max}/2$ and $T_{max}$ (from left to right)
  • Figure 3: Three-continuum media (3C). Reference solution at $t = T_{max}/4, T_{max}/2$ and $T_{max}$ (from left to right). First row: fist continuum. Second row: second continuum
  • Figure 4: The dynamic of the error with $N_t = 16$ (first row) and $64$ (second row). The label is given with the error at the final time
  • Figure 5: Two-continuum media (2C). Multiscale method solution (coarse grid, coupled scheme Im1 with $N_t = 128$) at final time for different number of oversampling layers $K_i^{+,l}$ for $l=2,3,4$ and $5$ (from left to right)
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof