Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle
materials by the injection of a pressurized viscous fluid. In this talk I provide examples of natural HF and situations in which HF are used in industrial problems. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy.
I describe the governing equations in 1-2D as well as 2-3D models of HF, which involve a coupled system of degenerate nonlinear integro-partial differential equations as well as a free boundary. I demonstrate, via re-scaling the 1-2D model, how the active physical processes manifest themselves in the HF model and show how a balance between the dominant physical processes leads to special solutions. I discuss the challenges for efficient and robust numerical modeling of the 2-3D HF problem and some techniques recently developed to resolve these problems: including robust iterative techniques to solve the extremely stiff coupled equations and a novel Implicit Level Set Algorithm (ILSA) to resolve the free boundary problem. The efficacy of these techniques is demonstrated with numerical results.
Relevant papers can be found at: http://www.math.ubc.ca/~peirce

In this talk, I consider the existence of local-in-time strong solutions to a well established coupled system of partial differential equations arising in Fluid-Structure interactions. The system consisting of an incompressible Navier-Stokes equation and an elasticity equation with velocity and stress matching boundary conditions at the interface in between the two domains where each of the two equations is defined. I discuss new existence results for a range of regularity in the initial data and the differences in the exsitence results when domains with non-flat boundaries are considered.