A finite element approach for modelling single-phase compressible flow in dual porosity systems

Abstract Fluid flow in a rock formation that contains a network of fractures occurs through two coupled systems, the fracture network and the blocks of porous and permeable rock matrix. Dynamic modelling is challenging when low permeability of the rock matrix creates transient effects that can persist for long periods. In this paper, we describe a finite element (FE) based algorithm developed to model the flow of highly compressible gas through a fractured reservoir. The fracture network is represented by a two-dimensional (2D) FE mesh, coupled to a number of customised strings of one-dimensional (1D) elements representing individual matrix blocks. The use of multiple FEs to model each matrix block replaces the transfer function and facilitates accurate modelling of transient effects in the matrix blocks while still honouring the fully compressible nature of gas. We use the algorithms to study some of the peculiarities of gas flow through fractured reservoirs with low matrix permeability. Matrix block geometry is shown to be an important parameter requiring accurate representation, particularly under transient flow conditions. We show how different geometric shapes can be reduced to 1D representations while still retaining much of the essential geometric information. It is shown that where transient effects occur, neglecting to capture them in the model can result in large errors. A conceptual system is suggested for classifying fractured reservoirs into six categories, each with a preferred modelling approach determined by the values of matrix permeability and fracture density. Finally, a case study is presented of a gas field producing from a low permeability fractured carbonate via two wells. A history match is successfully accomplished due to the flexible meshing capability of the finite element method and the transient modelling capabilities of the algorithms.