Efficient Simulation of Geothermal Processes in Heterogeneous Porous Media based on the Exponential Rosenbrock-Euler and Rosenbrock-type Methods

TL;DR

This paper introduces efficient numerical methods based on exponential Rosenbrock-Euler and Rosenbrock-type schemes for simulating geothermal processes in heterogeneous porous media, improving accuracy and computational efficiency over traditional methods.

Contribution

The paper develops and demonstrates the effectiveness of exponential Rosenbrock-Euler and Rosenbrock-type methods for geothermal system simulation, addressing limitations of standard time discretization schemes.

Findings

Exponential Rosenbrock--Euler method outperforms standard schemes in efficiency and accuracy.

Krylov subspace and Leja points techniques enable efficient matrix exponential computations.

Numerical examples confirm the advantages of the proposed methods.

Abstract

Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock--Euler and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock--Euler time integrator has advantages over standard time-dicretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Leja points techniques make these computations efficient. The Rosenbrock-type methods use the appropriate rational functions of the Jacobian from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples.