MATHEMATICAL MODELING AND ANALYSIS OF MECHANISMS ASSOCIATED WITH THE PROPAGATION OF HYDRAULICALLY INDUCED FRACTURES IN PORO-ELASTIC MEDIA (POROUS MEDIA, ELLIPTICAL, POWER LAW FLUIDS, RADIAL, VARIATIONAL FORMULATION).

摘要

The mechanics of hydraulic fracturing generally entails the controlled pumping of a suitable proppant laden fracture fluid in a borehole and subsequent vertical fracture initiation and extension in a geological formation of interest. This technique of inducing artificial underground fractures has been used successfully by the petroleum industry in the enhancement of oil and gas production rates from low permeability reservoirs. Since its introduction to the industry, hydraulic fracturing has found widespread geotechnical applications, including geothermal energy extraction, underground coal conversion, nuclear waste disposal, and in-situ stress state determination.;The mechanism of fluid leak-off is analyzed from the vantage point of a pistonlike displacement of a compressible reservoir fluid by a penetrating fracture fluid. The associated moving boundary problem is solved for the case of a Newtonian fracturing fluid as well as the more general case of a non-Newtonian fluid of power law type penetrating a permeable reservoir. Various expressions for the leak-off coefficient characterizing the rate of fluid loss are derived, along with several conventional forms deduced as special cases. These solutions illustrate the pressure profile characteristics and provide guidelines for fluid loss minimization.;The developed elliptical and radial models for the propagation of hydraulically induced fractures furnish an excellent capability for parametric sensitivity comparisons and provide response calibration of sophisticated finite element model simulations. The Lagrangian method of analysis can be generalized to incorporate complicating effects such as multi-layering and in-situ stress variation, however the resulting differential equations must be solved numerically.;The propagation of hydraulically induced fractures is formulated from a variational point of view, based on Lagrange's equations of motion for nonconservative systems. The complex problem involving a coupling of elasticity, fracture mechanics, and viscous fluid flow is separated into its various parts for analysis and then recombined in the deduced equations of motion. Governing equations for the evolution of assumed elliptical and circular fractures in a uniformly confined, linear, elastic, homogeneous medium are derived. Explicit solutions of power law form for the generalized coordinates designating the time dependent fracture dimensions are obtained with and without the consideration of leak-off effects.