Pre-existing fluid-driven fracture: mathematical models and solution

Nchabeleng, Mathibele Willy
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The problem of a two-dimensional, pre-existing, fluid-driven fracture propagating in a permeable rock is considered. The flow of fluid in the fracture is laminar and the fracture is driven by a viscous incompress ible Newtonian fluid. Lubrication theory is applied to the fracturing fluid and the Cauchy principal value integral derived from linear elas tic fracture mechanics is used to describe the elasticity equation relat ing the fluid pressure to the fracture half-width. The fluid leak-off at the fracture interface into the rock formation is modelled in two ways, namely, using a leak-off velocity term and by using Darcy’s law. Ap propriate initial and boundary conditions for the model are stated and discussed. Similarity solutions are derived for the fracture half-width, length, leak-off velocity and leak-off depth. Numerical results are ob tained for a nonlinear diffusion equation with leak-off velocity term and for a nonlinear diffusion equation coupled with Darcy’s model. The results are illustrated using computer generated graphs
A thesis submitted in partial fulfilment for the degree of Doctor of Philosophy to the Faculty of Science, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, 2021