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Reduced Basis method for finite volume simulations of parabolic PDEs applied to porous media flows

Jana Tarhini, Sébastien Boyaval, Guillaume Enchéry, Quang Huy Tran

TL;DR

This paper addresses the computational burden of multi-query, finite-volume simulations of parabolic Darcy flows in porous media under uncertain permeability. It develops a reduced-basis framework built on POD-Greedy basis construction guided by a novel, parameter-independent space-time energy-norm a posteriori estimator, complemented by EIM for non-affine dependence and SCM for coercivity bounds. The approach yields reliable primal and QoI error certification and achieves substantial online speedups (e.g., ~90% reduction) while maintaining high accuracy for interior-boundary flux QoIs relevant to CO2 storage risk assessment. The results demonstrate the method’s practicality for rapid uncertainty quantification and decision-support in underground gas storage scenarios and similar reservoir-scale problems.

Abstract

Numerical simulations are a highly valuable tool to evaluate the impact of the uncertainties of various modelparameters, and to optimize e.g. injection-production scenarios in the context of underground storage (of CO2typically). Finite volume approximations of Darcy's parabolic model for flows in porous media are typically runmany times, for many values of parameters like permeability and porosity, at costly computational efforts.We study the relevance of reduced basis methods as a way to lower the overall simulation cost of finite volumeapproximations to Darcy's parabolic model for flows in porous media for different values of the parameters suchas permeability. In the context of underground gas storage (of CO2 typically) in saline aquifers, our aim isto evaluate quickly, for many parameter values, the flux along some interior boundaries near the well injectionarea-regarded as a quantity of interest-. To this end, we construct reduced bases by a standard POD-Greedyalgorithm. Our POD-Greedy algorithm uses a new goal-oriented error estimator designed from a discrete space-time energy norm independent of the parameter. We provide some numerical experiments that validate theefficiency of the proposed estimator.

Reduced Basis method for finite volume simulations of parabolic PDEs applied to porous media flows

TL;DR

This paper addresses the computational burden of multi-query, finite-volume simulations of parabolic Darcy flows in porous media under uncertain permeability. It develops a reduced-basis framework built on POD-Greedy basis construction guided by a novel, parameter-independent space-time energy-norm a posteriori estimator, complemented by EIM for non-affine dependence and SCM for coercivity bounds. The approach yields reliable primal and QoI error certification and achieves substantial online speedups (e.g., ~90% reduction) while maintaining high accuracy for interior-boundary flux QoIs relevant to CO2 storage risk assessment. The results demonstrate the method’s practicality for rapid uncertainty quantification and decision-support in underground gas storage scenarios and similar reservoir-scale problems.

Abstract

Numerical simulations are a highly valuable tool to evaluate the impact of the uncertainties of various modelparameters, and to optimize e.g. injection-production scenarios in the context of underground storage (of CO2typically). Finite volume approximations of Darcy's parabolic model for flows in porous media are typically runmany times, for many values of parameters like permeability and porosity, at costly computational efforts.We study the relevance of reduced basis methods as a way to lower the overall simulation cost of finite volumeapproximations to Darcy's parabolic model for flows in porous media for different values of the parameters suchas permeability. In the context of underground gas storage (of CO2 typically) in saline aquifers, our aim isto evaluate quickly, for many parameter values, the flux along some interior boundaries near the well injectionarea-regarded as a quantity of interest-. To this end, we construct reduced bases by a standard POD-Greedyalgorithm. Our POD-Greedy algorithm uses a new goal-oriented error estimator designed from a discrete space-time energy norm independent of the parameter. We provide some numerical experiments that validate theefficiency of the proposed estimator.
Paper Structure (27 sections, 3 theorems, 132 equations, 13 figures, 3 tables, 5 algorithms)

This paper contains 27 sections, 3 theorems, 132 equations, 13 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. A parametrized High-Fidelity model
  3. A Darcy model of single phase porous media flows
  4. Finite volume discretization
  5. Reduced Basis technique in time-dependent setting
  6. Low-fidelity model
  7. POD-Greedy method
  8. A posteriori estimation of the primal error
  9. A posteriori estimation of the QOI error
  10. Computational aspects
  11. Affine decomposition.
  12. Residual norm evaluation.
  13. Coercivity constant computation.
  14. Numerical results
  15. Affine decomposition of the scheme coefficients
  16. ...and 12 more sections

Key Result

Proposition 3.1

Denote $\boldsymbol{A}= \frac{1}{2} (\boldsymbol{A}+\boldsymbol{A}^T )+\frac{1}{2} (\boldsymbol{A}-\boldsymbol{A}^T) := \boldsymbol{A}_{\rm sym}+\boldsymbol{A}_{\rm skew}$ the symmetric and skew-symmetric of matrices $\boldsymbol{A}$. For any $\xi$, given lower bounds there holds where the upper bound is taken as and where $\boldsymbol{A}_{\rm sym}^*$ is the symmetric part of $\boldsymbol{A}^*$

Figures (13)

  • Figure 2.1: Typical domain configuration.
  • Figure 2.2: Harmonic averaging point.
  • Figure 4.1: Permeabilities' distribution used for the offline stage
  • Figure 4.2: Spatial repartition of the permeabilities within $\Omega$: the yellow zone includes cells having the high permeability value $\kappa_1$ and the low permeability value $\kappa_2$ is used in the blue zone.
  • Figure 4.3: Evolution of the EIM interpolation error \ref{['maxInterpError']} with respect to the number of selected parameters $\rm M$.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Proposition 3.1: Energy a posteriori error estimate for the primal problem
  • proof
  • Proposition 3.2: Energy a posteriori error estimate for the dual problem
  • proof
  • Proposition 3.3: Output error evaluation
  • proof
  • proof
  • proof
  • proof