Pre-existing fluid-driven fracture: mathematical models and solution
Date
2021
Authors
Nchabeleng, Mathibele Willy
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Abstract
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
Description
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