The mechanics of a near-surface crack under uniform pressure or shear in a transversely isotropic medium : with applications to hydraulic fracture

Abstract

The influence of material anisotropy (transverse isotropy) on half-plane surface tilt and displacement fields induced by a near-surface crack under uniform pressure or shear is investigated by solving a series of boundary value problems using (1) Dual Integral Methods, (2) Fourier Transform/Green's Function Methods, and (3) Complex Variable Methods. Lekhnitskii's (1981) Complex Variable Method is most general insofar as both the crack and principal anisotropy directions are arbitrarily oriented relative to the half-plane surface. The solutions obtained by the various methods are compared and are found to be in excellent agreement. The complex variable solutions for (a) the vertical displacements induced by a uniformly pressurized crack oriented parallel to the free-surface of a nearly isotropic half-plane, and (b) the surface tilts induced by a uniformly pressurized crack oriented perpendicular to the free-surface of a nearly isotropic half-plane concur with the solutions of Pollard and Holzhausen (1979), obtained using the Schwarz-Neumann Alternating Methods. It is shown that free-surface effects become negligible for depth to half-crack length ratios greater than four, d/a > 4. Surface tilts are highly non-unique, even in a nearly isotropic medium. A vertical shear crack generates the same surface tilt as a pressurized crack that dips at 60(DEGREES), and a 45(DEGREES) dipping shear crack generates the same surface tilt as a horizontal pressurized crack. The crack boundary value problem solutions are used to interpret certain "anomalous" tilt measurements made during a hydraulic fracture experiment conducted by Oak Ridge National Labs, in Oak Ridge, Tennessee. A series of hydraulic fractures propagated sub-parallel to bedding at a depth of 305 meters in the Cambrian Conasauga shale which regionally dips about 20(DEGREES) to the SSE. The surface tilts are anomalous since present isotropic theoretical models of fractures under uniform pressure in two-dimensional (Pollard and Holzhausen, 1979), and three-dimensional, mode I dislocation fracture models (Davis, 1983), predict that the position of "zero-tilt" is located down-dip of the fracture, and SSE of the wellbore at Oak Ridge. Observations indicate that the "zero-tilt" position is located up-dip of the fracture, and NNW of the wellbore. A best-fit analysis involving observed and theoretical tilts indicates that a component of shear is present whose relative magnitude is twice that of the normal fracture plane pressure.