article i nfo We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power-law basis functions, with arbitrary index 0b αb1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h 4 ) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.

/pdf/computer-methods-in-applied-mechanics-and-engineering-53g5581srr.pdf

Computer Methods in Applied Mechanics and Engineering

https://www.math.ubc.ca/~peirce/Computer%20simulation%20of%20HF_Adachi_Siebrits_Peirce_Desroaches_IJRM_MS_V44_p739_2007.pdf

Computer simulation of hydraulic fractures

The past decade has seen a substantial amount of research on air-sea gas exchange and its environmental controls. These studies have significantly advanced the understanding of processes that control gas transfer, led to higher quality field measurements, and improved estimates of the flux of climate-relevant gases between the ocean and atmosphere. This review discusses the fundamental principles of air-sea gas transfer and recent developments in gas transfer theory, parameterizations, and measurement techniques in the context of the exchange of carbon dioxide. However, much of this discussion is applicable to any sparingly soluble, non-reactive gas. We show how the use of global variables of environmental forcing that have recently become available and gas exchange relationships that incorporate the main forcing factors will lead to improved estimates of global and regional air-sea gas fluxes based on better fundamental physical, chemical, and biological foundations.

/pdf/advances-in-quantifying-air-sea-gas-exchange-and-4tk362q7zi.pdf

Advances in Quantifying Air-Sea Gas Exchange and Environmental Forcing*

This paper reviews recent results of a research program aimed at developing a theoretical framework to understand and predict the different modes of propagation of a fluid-driven fracture. The research effort involves constructing detailed solutions of the crack tip region, developing global models of hydraulic fractures for plane strain and radial geometry, and identifying the parameters controlling the fracture growth. The paper focuses on the propagation of hydraulic fractures in impermeable rocks. The controlling parameters are identified from scaling laws that recognize the existence of two dissipative processes: fracturing of the rock (toughness) and dissipation in the fracturing fluid (viscosity). It is shown that the two limit solutions (corresponding to zero toughness and zero viscosity) are characterized by a power law dependence on time and that the transition between these two asymptotic solutions depends on a single number, which can be chosen to be either a dimensionless toughness or a dimen...

Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks

Most concentrated colloidal suspensions such as cement, drilling fluids, paints, muds, etc., have been considered until now thixotropic fluids with a flow curve of an ideal yield stress fluid. We start by showing from inclined plane tests, intended to determine the yield stress, that these systems in fact exhibit peculiar properties. Unlike ideal yield stress fluids, they stop flowing abruptly below a critical stress, and start flowing at a high velocity beyond a critical stress, which in addition increases with the time of preliminary rest. In order to clarify these features we carried out a complete set of rheometrical tests with a model fluid, a bentonite suspension. Our results show that under controlled stress, in some cases after significant flow, there is bifurcation of the behavior towards either stoppage or rapid shear, depending on the relative values of the imposed and critical stresses. As an immediate consequence, we find that no (homogeneous) steady state flows at a shear rate below a critical value can be obtained. These results can be qualitatively predicted by a simple theoretical model that assumes that the viscosity of the material results from the competition between aging and shear rejuvenation, associated to, respectively, the organization or disorganization of the network of particle interactions. This shows that the flow curve in the steady state of concentrated colloidal suspensions and, more generally, of structured fluids, is strongly affected by their thixotropy.

/pdf/viscosity-bifurcation-in-thixotropic-yielding-fluids-8kwttfun10.pdf

Viscosity bifurcation in thixotropic, yielding fluids

We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

The crack tip region in hydraulic fracturing

On gas absorption into a turbulent liquid

This article first provides a brief and simple account of continuum models for transport in porous media, and of the role of length scales in passing from pore-scale phenomena to “Darcy” continuum scale representations using averaged variables. It then examines the influence of non-Newtonian rheology on the single- and multi-phase transport parameters, i.e. Darcy viscosity, dispersion lengths and relative permeabilities. The aim is to deduce functional forms and values for these parameters given the rheological properties of the fluid or fluids in question, and the porosity, permeability, dispersion lengths and relative permeabilities (based on Newtonian fluids and equivalent capillary pressures) of the porous medium. It is concluded that micro-models, typically composed of capillary networks, applied at a sub-Darcy-scale, parameterised using data for flows of a well-characterised set of non-Newtonian fluids, are likely to provide the most reliable means.

J. R. A. Pearson

Papers

The crack tip region in hydraulic fracturing

On gas absorption into a turbulent liquid

Models for flow of non-Newtonian and complex fluids through porous media

On the Stability of Nonisothermal Fiber Spinning

Melting in single screw extruders