Modelling the development of fracture aperture distributions using a simple aperture growth law

Fractures in shallow sedimentary aquifers may become enlarged over relatively short geological time scales due to groundwater circulation. The aim of the present study is to develop a model to investigate general relationships between fracture aperture growth and the geometry of evolved fracture arrays using a simple aperture growth law and simple aperture geometries. The evolution in the conductivity of the system is also investigated. The model is formulated as an initial value problem and it has been used to study the growth of an array of orthogonal fractures with an initial, spatially uncorrelated, lognormal aperture distribution, where aperture growth-rate is an exponent of the flow rate through each fracture. The geometries of the evolved aperture arrays display a range of self-organised structures that are sensitive to the aperture growth-rate exponent, e, and to the standard deviation of the initial aperture distribution, a0. They show geometrical phase changes in the evolved structure as a function of changes in the boundary conditions. For example, low values of e and a0 lead to bi-modal aperture distributions, where apertures perpendicular to flow undergo limited growth and apertures parallel to flow are preferentially enlarged. At moderate values of e and  a0, there is a transition to a regime of more complex aperture geometries with anastomosing, channel-like, structures of preferentially enlarged apertures. At larger values of e, single continuous channel-like paths of preferentially enlarged apertures develop, where the tortuosity of the channel-like paths is a function of  a0. As they develop, the arrays show a power-law like increase in effective transmissivity, with transmissivity inversely proportional to e. Continuum percolation models provide a good description of the conductivity of the initial aperture arrays. However, because of selforganisation in the evolved aperture arrays, the conductivity of most of the evolved networks departs systematically from that predicted by continuum percolation theory. Future work is to be funded by the NERC Micro-to-Macro Thematic Programme. It will involve the development of a theoretical framework to describe the dynamic behaviour of evolving porous media, code development, model validation, investigation of scaling and self-organisation phenomena in simple systems, and case studies including the investigation of enlarged fractures in the Chalk.