Stanford Mechanics and Computation
Home
People
Research
Teaching
Seminar Schedule
FAQ
for
The tip region of a near-surface hydraulic fracture
Title: The Tip Region of a Near- Surface Hydraulic Fracture By Emmanuel Detournay Department of Civil, Environmental, and Geo- Engineering University of Minnesota Abstract: The talk investigates the stationary problem of a semi-infinite hydraulic fracture propagating near a free- surface. The mathematical model, which is based on the Euler- Bernoulli beam theory and on Reynolds lubrication theory, leads to the formulation of a strongly nonlinear diffusion equation. Because this governing equation does not tolerate a pressure singularity at the advancing front of the hydraulic crack, the fluid front is lagging behind the crack edge except in the limiting case of an inviscid fluid. The lag is a priori unknown, and is part of the solution. For a finite dimensionless toughness, the asymptotic forms of the solution in the regions near and away from the tip are discussed. The apertures in these two regions have similar square power-laws dependence on the distance from the tip. In other words, the curvature in these two regions is uniform with their values depending on the dimensionless toughness. The transition zone between these two regions, which is near the fluid front, shrinks as the dimensionless toughness increases. For large dimensionless toughness, the solution can be expressed in terms of an asymptotic expansion. A numerical method for solving the nonlinear diffusion equation is outlined, and the numerical results are shown to agree well with the asymptotic results. The asymptotic tip solution can be used to develop efficient numerical algorithms of near-surface hydraulic fracturing problems. As an illustration of the application of the tip solution, we solve the problem of a steadily moving bubble of viscous fluid trapped beneath an elastic sheet and the problem of a near-surface penny-shaped hydraulic fracture propagating as the result of injection of fluid at a constant rate.
Return to
The tip region of a near-surface hydraulic fracture
.