Connectivity of fracture networks: The effects of anisotropy and spatial correlation

Abstract

In fractured formations of very low matrix permeability connectivity of fractures is a crucial parameter which may have a significant impact on the overall flow. The connectivity behaviour of fracture networks can be analysed by using percolation theory. The scaling law within this theory is used to predict the connectivity and its associated uncertainty very rapidly. Although the effects of some geometrical parameters of fractures (e.g. size distribution) on the universal connectivity curves are extensively investigated, the effects of anisotropy or spatial correlation of fractures has not yet been fully addressed. In this paper we present a framework to derive and verify numerically the scaling including the effects of anisotropy in two and three dimensions. Anisotropy arises as a consequence of restricting the orientational disorder of fractures in two dimensions and using fractures with different aspect ratios in three. The main effect of anisotropy is to create an ‘easy’ direction for connectivity and a ‘difficult’ direction. This leads to the concept of an ‘apparent’ percolation threshold defined as the value which collapses all the anisotropic mean connected fraction curves onto the same isotropic curve. This is changed as a function both of the system size and the aspect ratio/s involved. We then used certain symmetry relations involved and numerical results to formulate the aspect ratio/s dependency of the apparent threshold. The curves for the fluctuations about this mean connected fraction have to use this apparent threshold as well as a change in magnitude which can be accounted for by rescaling with the geometric mean length. Classical percolation usually used to study the connectivity assumes spatially uncorrelated fractures. However, the same basic methodology still applies for correlated systems with small modifications which depend on the nature of correlation. A modelling technique based on the idea that the elastic free energy due to the fracture density follows a Boltzmann distribution is presented to generate realizations of correlated fracture networks. A simulated annealing algorithm is used with an objective function based on the derived expression for the spatial correlation in the elastic displacement. The connectivity aspects of these networks are then analysed. As a result this has extended the applicability of percolation concepts to the anisotropic and correlated fracture systems which may be useful for practical engineering purposes when a very rapid risk assessment is necessary.