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 CO$_2$ 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 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.
Relevant papers can be found at: http://www.math.ubc.ca/~peirce
Over 40 years ago, Orszag pointed out the importance of dealiasing in the pseudospectral method. However, the computational and storage cost of dealiasing, either by padding or phase-shift dealiasing, is so great that it is still sometimes neglected in strongly damped turbulence simulations: this is typically justified with a claim that high-wavenumber damping is sufficiently strong so that dealiasing error contributes negligibly to the larger energy scales.
On the other hand, Hou and Li demonstrated in 2007 that high-order Fourier smoothing captures nearly singular solutions of the 1D inviscid Burgers equations and the 3D Euler equations more accurately and efficiently than explicit dealiasing via 2/3 zero padding.
Given that high-Reynolds number turbulence, with well-resolved inertial ranges, falls midway between these two limiting cases of large viscosity vs. vanishing viscosity, it seems prudent to reconfirm the importance of properly dealiasing turbulence simulations. Moreover, the recent introduction of implicit dealiasing techniques, which in two and three dimensions are roughly twice as fast as explicit dealiasing, should more than offset the claim that smoothing via a Fourier filter is 20% more efficient than dealiasing.
In this talk, we revisit the issue of dealiasing, beginning with a review of recent advances in the pseudospectral method. We emphasize that implicit dealiasing outperforms zero padding by decoupling the data and temporary work arrays. We also discuss the parallelized implementations, for distributed and shared memory architectures, now available in our open-source library FFTW++.
Conventional seismic images are created by a standard methodology -or SM- which progressively refines raw data through a complex sequence of steps called a flow. Modern imaging flows have evolved from a mixture of practical needs, such as noise reduction, and more physical concepts such as reflectivity estimation. Seismic images are intended to estimate reflectivity, which is subsequently converted to an impedance image, typically using information from well control, by a process called “impedance inversion”. In recent years, more rigorous inversion methods have been proposed which, conceptually, can replace conventional flows and estimate impedance directly from raw data. Called full-waveform inversion -or FWI- these emerging methods are not yet practical for a variety of reasons. I will compare and contrast SM with FWI, illustrating where there are commonalities and essential differences. This will lead to the suggestion of modifications to FWI by borrowing essential techniques from SM. This may result in a more practical FWI.
We study the displacement flow of two Newtonian fluids in an inclined pipe. The fluids have the same viscosity but different densities. The displacing fluid is denser than the displaced fluid and is placed above the displaced fluid (i.e. a density-unstable configuration). Three dimensionless groups describe these flows: a densimetric Froude numberFr, a Reynolds number Re and the pipe inclination. A rich variety of flow phenomena can be observed in varying these parameters. At pipe inclinations close to horizontal and for moderate Re/Fr, the flows are laminar and viscous-dominated. A simple thin-film style modelling approach proves very effective in understanding these flows. As the inclination angles are reduced and the pipe becomes progressively vertical, we transition through buoyancy-dominated inertial exchange regimes to mixed and fully diffusive regimes. We give a leading order characterization of these flows. Joint work with K. Alba and S.M. Taghavi.