Modified flux-vector-based Green element method for problems in steady-state anisotropic media—Generalisation to triangular elements

doi:10.1016/j.enganabound.2010.09.004

Engineering Analysis with Boundary Elements

Volume 35, Issue 3, March 2011, Pages 495-498

https://doi.org/10.1016/j.enganabound.2010.09.004 Get rights and content

Abstract

This paper is concerned with the generalisation of a numerical technique for solving problems in steady-state anisotropic media, namely the ‘flux-vector-based’ Green element method ( $‘ q -based’$ GEM) for anisotropic media, to triangular elements. The generalisation of the method to triangular elements is based on the same concepts as for a rectangular grid, namely satisfying a nodal flux condition at each node of the mesh and the continuity of the tangential pressure gradient across the elements sharing a node.

Introduction

Anisotropic media are widely encountered in nature, for example in oil and gas reservoirs. In many reservoirs, the production of gas and oil is seriously affected by the highly anisotropic and/or heterogeneous structure of the media.

Steady-state problems in anisotropic media can be solved using the modified $‘ q -based’$ GEM for anisotropic media, introduced by Lorinczi et al. [1]. The approach introduced by Lorinczi et al. [1] has been implemented for non-uniform rectangular grids. Lorinczi et al. [2] used this in geological problems in faulted/fractured anisotropic media.

A similar approach has been previously introduced by Lorinczi et al. [3] for isotropic media, and it was applied to highly heterogeneous isotropic media. The two approaches are using the concept of the $‘ q -based’$ GEM [4], which maintains the high-order accuracy of the GEM, diminished by some approximations used in the GEM, see Taigbenu [5].

This paper introduces the $‘ q -based’$ GEM for anisotropic media for triangular elements. Lorinczi et al. [3] showed that the $q -based’$ GEM for isotropic media can be naturally extended to triangular finite element grids. The work presented here is based on satisfying similar conditions at nodes from a triangular mesh as in Lorinczi et al. [3], but considering the medium anisotropy.

Section snippets

Mathematical formulation

In this section we present the extension of the modified ‘flux-vector-based’ GEM for anisotropic media to triangular finite element grids. This is a technique suitable to solve problems in an anisotropic porous medium in which the equation governing the flow in a bounded domain $Λ \subset R^{2}$ is given by $\nabla \cdot (K \nabla p) = - F (x) in Λ$ where $K$ is the permeability tensor, p is the fluid pressure and $F (x)$ is an internal/external source forcing function (term) which incorporates the fluid viscosity. For simplicity, we

Conclusions

In this article we have presented the generalisation of a numerical technique for solving steady-state flow problems, namely the modified $‘ q -based’$ GEM for anisotropic media, to triangular elements. This is an extension of the modified $‘ q -based’$ GEM has been developed for rectangular grids [1], the two techniques being based on the same concepts—satisfying a nodal flux condition at each node and on the continuity of the tangential pressure gradient across the element boundaries.

The extension of

Acknowledgements

Piroska Lorinczi would like to acknowledge the financial support received from the ORS and the Rock Deformation Research Group, University of Leeds, UK.

References (5)

P. Lorinczi et al.
Modified flux-vector based GEM for problems in steady-state anisotropic media
Eng Anal Bound Elem
(2009)
P. Lorinczi et al.
Modelling of highly-heterogeneous media using a flux-vector-based Green element method
Eng Anal Bound Elem
(2006)

There are more references available in the full text version of this article.

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