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

TL;DR

This paper develops a reduced basis method using a goal-oriented error estimator to efficiently simulate porous media flows governed by Darcy's parabolic PDE, significantly reducing computational costs for multiple parameter scenarios.

Contribution

It introduces a novel goal-oriented error estimator for reduced basis methods applied to finite volume discretizations of Darcy's model, enhancing simulation efficiency.

Findings

The new estimator effectively guides basis construction.

Reduced basis methods significantly decrease simulation time.

Numerical experiments validate estimator accuracy and efficiency.

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.