Self-similar solution of a fluid-driven fracture

The research described in this thesis has focused on the development of a self-similar solution for the problem of a plane-strain fluid-driven fracture in an impermeable elastic medium, in the limiting case of zero toughness. The problem is solved semi-analytically, taking into account recent results that describe the near-tip behavior of a hydraulic fracture.;The theory of elasticity provides a singular integral equation for the relation between the pressure and the fracture aperture. The fluid flow is modeled in accordance with lubrication theory. A similarity solution is derived for the coupled system, which allows the initial boundary value problem to be transformed into a self-similar form where time does not appears explicity. The lubrication equation is reduced to a non-linear algebraic equation by expressing the crack opening and the fracture pressure in terms of Jacobi polynomials, such that the elasticity equation is satisfied. Solution for crack opening, pressure within the crack, flow rate and fracture length are computed with any given degree of accuracy.;Comparison between this semi-analytical solution and results obtained with a numerical model (which can account for toughness and fluid lag) indicates a very good agreement between the two solutions when the rate of energy dissipated in the rock to propagate the fracture is less than 1% the energy dissipated in the fluid by viscous losses.