POWERPOINT FILES FOR RECENT INVITED TALKS
5. Plenary talk at CSIRO Computational & Simulation Science Transformational Platform Conference, Melbourne, Australia, 18 March 2010.
Hydraulic Fractures: multiscale
phenomena, asymptotic and numerical solutions
A.P. Peirce
(pdf file, 3 Mb).

Abstract: 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 introduce models of HF and provide examples of natural HF and situations in which HF are used in industrial problems. I describe the governing equations in 12D as well as 23D models of HF, which involve a coupled system of degenerate nonlinear integropartial differential equations as well as a free boundary. I demonstrate, via rescaling the 12D 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 23D HF problem including: the rapid construction of Green's functions for cracks in layered elastic media, 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. 
4. Plenary talk at SANUM 2009, Stellenbosch, South Africa, April 2009.
Hydraulic Fractures: multiscale
phenomena, asymptotic and numerical solutions
A.P. Peirce
(PowerPoint with animations and
movies 27 Mb, pdf File 3 Mb).

Abstract: 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 introduce models of HF and 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 12D as well as 23D models of HF, which involve a coupled system of degenerate nonlinear integropartial differential equations as well as a free boundary. I demonstrate, via rescaling the 12D 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 23D HF problem including: the rapid construction of Green's functions for cracks in layered elastic media, 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. 
3. Talk at University of the Witwatersrand (WITS), Johannesburg, 1 April, 2008.
One dimensional models
of hydraulic fracture
A.P. Peirce
(PowerPoint
with animations and movies 9.8 Mb, .pdf
file 2.4 Mb).

Abstract:
Fluiddriven fractures are a class of tensile fractures that propagate
in compressively prestressed solid media due to the internal pressurization
by an injected viscous fluid. These hydraulic fractures occur both in
natural geological processes and in geoengineering applications. One
engineering application of particular interest is the deliberate generation
of hydraulic fractures in oil reservoirs to enhance the extraction of
oil. Typically, the oil is trapped in a sedimentary layer called the "pay zone" by the existence of ambient geological confining stresses in the neighbouring sedimentary layers that are much higher than that in the pay zone itself. This leads to the development of fractures that have a slender footprint, which can be exploited to derive simple one dimensional models. In this talk we will review the classical PerkinsKernNordgen (PKN) model which leads to a nonlinear diffusion equation that admits similarity solutions. However, the local elasticity assumption of the PKN model, in which the fluid pressure is assumed to be proportional to the fracture opening, precludes the incorporation of a fracture propagation condition which is consistent with Linear Elastic Fracture Mechanics (LEFM). In this talk we discuss recent work to include the nonlocal behaviour in a one dimensional model. We describe an asymptotic analysis of this one dimensional model, which not only reduces to the PKN model as an "outer solution" but also includes the expected tip asymptotic behaviour consistent with LEFM. We present some numerical results which explore the propagation of these slender fractures in two distinct regimes of propagation. In the first regime, the viscous dissipation is the dominant component of the energy required to propagate the fracture. In the second regime, the energy expended in breaking the rock dominates the energy required to propagate the fracture. We also introduce another class of one dimensional models which are also extensions of the PKN model  the socalled P3D models commonly used in the oil industry to model the penetration of the high stress barriers that surround the pay zone. 
2. Talk at the Workshop on Rock Mech. and Logistics in Mining, Chile, 8th March. 2007
Hydraulic Fracture: multiscale
processes and moving interfaces
A.P. Peirce
(PowerPoint with animations and movies 9.7 Mb, .pdf file 2.2 Mb).

Abstract: Hydraulic fracturing (HF) is a process that has recently been used to induce "goafing" and caving in mines. Other mining applications include fault preconditioning to reduce seismic hazards by releasing accumulated shear stresses on faults. We introduce the problem of Hydraulic Fracturing and provide examples of situations in which Hydraulic Fractures are used in industrial problems. We describe the governing equations in 12D as well as 23D models of HF, which involve a coupled system of degenerate nonlinear integropartial differential equations defined on a domain with a free boundary. We demonstrate, via rescaling the 12D model, how the active physical processes manifest themselves in the HF model. We also show how a balance between the dominant physical processes leads to special solutions. We discuss the challenges for efficient and robust numerical modeling of the 23D HF problem including novel multigrid and incomplete LU factorizations to precondition the stiff coupled system of equations. We demonstrate the efficacy of these techniques with numerical results. 
1. Talk at the IMS conference, Nanoscale material interfaces: experiment, theory and simulation, National University of Singapore, Singapore, 1014 Jan 2005.
Hydraulic Fracture: multiscale
processes and moving interfaces
A.P. Peirce
(PowerPoint
with animations and movies 9.5 Mb, .pdf
file 1.9 Mb)

Abstract: We introduce the problem of Hydraulic Fracture (HF) and provide examples of situations in which Hydraulic Fractures are used in industrial problems. We describe the governing equations in 12D as well as 23D models of HF, which involve a coupled system of degenerate nonlinear integropartial differential equations as well as a free boundary. We demonstrate, via rescaling the 12D model, how the active physical processes manifest themselves in the HF model. We also show how a balance between the dominant physical processes leads to special solutions. We discuss the challenges for efficient and robust numerical modeling of the 23D HF problem including: the rapid construction of Green's functions for cracks in layered elastic media and novel multigrid procedures to solve the coupled system of equations. We demonstrate the efficacy of these techniques with numerical results. 