Numerical Simulation Using Finite-Difference Schemes with Continuous Symmetries for Processes of Gas Flow in Porous Media

Markov Markov,Rodionov Rodionov

doi:10.3390/computation7030045

Markov Markov, Rodionov Rodionov

Open Access

https://doi.org/10.3390/computation7030045

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Abstract

This article presents the applications of continuous symmetry groups to the computational fluid dynamics simulation of gas flow in porous media. The family of equations for one-phase flow in porous media, such as equations of gas flow with the Klinkenberg effect, is considered. This consideration has been made in terms of difference scheme constructions with the preservation of continuous symmetries, which are presented in original parabolic differential equations. A new method of numerical solution generation using continuous symmetry groups has been developed for the equation of gas flow in porous media. Four classes of invariant difference schemes have been found by using known group classifications of parabolic differential equations with partial derivatives. Invariance of necessary conditions for stability has been shown for the difference schemes from the presented classes. Comparison with the classical approach for seeking numerical solutions for a particular case from the presented classes has shown that the calculation speed is greater by several orders than for the classical approach. Analysis of the accuracy for the presented method of numerical solution generation on the basis of continuous symmetries shows that the accuracy of generated numerical solutions depends on the accuracy of initial solutions for generations.

Highlights

Modern problems in modeling of natural oil and gas reservoirs require the use of complex coupled equations for physically different unsteady processes such as multiphase seepage, geomechanics, and multicomponent thermodynamical processes for three spatial dimensions
One type of partial differential equations is considered in this article—the equations of gas flow in one-dimensional porous media [10]
These equations represent an example of simple models for problems of gas flow in porous media, which are mentioned in the introduction

Summary

Introduction

Modern problems in modeling of natural oil and gas reservoirs require the use of complex coupled equations for physically different unsteady processes such as multiphase seepage, geomechanics, and multicomponent thermodynamical processes for three spatial dimensions. That is why one of the most popular topics in mathematical modeling of processes related to natural oil and gas reservoirs is how to simplify a model to a stage where it is still interesting and can be numerically solved with acceptable results [2]. All these approaches require reliable benchmark models, such as the mentioned simple equations, and a basis for their construction and numerical solving, for example, for a more effective choice of initial iterations. Another application of simple models is for fast calculations in the field of petroleum engineering, where time is more important than model complexity in some cases

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