Bonded Half-Planes Containing Two Arbitrarily Oriented Cracks: A Study of Containment of Hydraulically Induced Fractures

In the first part of this paper, the title subject is studied by introducing two modified singular integrals. The problem is reduced to a set of singular integral equations; and problem is reduced to a set of singular integral equations; and it is solved numerically by employing the Lobatto-Chebyshev method. The stress intensity factor at the fracture tips of a hydraulically induced fracture in a layered medium is calculated in the second part of the paper. The effect of the fluid pressure and the in-situ stress gradient as well as the effect of the relative layer material properties on the magnitude of the stress intensity factors are properties on the magnitude of the stress intensity factors are studied numerically. It has been shown that the relative magnitude of the stress intensity factor at the fracture tips can he used to indicate the direction of fracture movement. Introduction Extensive analyses of bonded half-planes containing cracks have been conducted by many authors. In general, there are two approaches to this problem. In the first approach, the Mellin transform is applied to the field equations. This leads to a set of integral equations, which, in turn, are solved numerically. Erdogan and Biricikoglu, Cook and Erdogan, Ashbaugh, and Erdogan and Aksogan have used this method in their analyses of stresses in the bonded planes containing straight cracks. In the other approach, the same problem is studied by employing the complex potential function of Kolosov and Muskhelishvili. For a general discussion of this method, see Refs. 5 through 9. By using this method, the general problem of a half-plane containing a system of curvilinear problem of a half-plane containing a system of curvilinear cracks is solved by Ioakimidis and Theocaris. The associated boundary value problem is deduced to a system of complex singular integral equations, which then are solved numerically by applying the Lobatto-Chebyshev method. In this paper, the problem is studied by using a method very similar to that developed in Ref. 10, However, since we are studying the propagation of a hydraulically induced fracture in a layered rock medium, the loading condition of our problem is different from that previously cited. In our formulation, the cracks are subject to different distributions of internal loadings. As demonstrated later, although our method is, in principle, similar to that reported in Refs. 10 and 11, it differs in many ways. Our method is suitable forsolving the problems of two cracks situated in two different half-planes and oriented at an arbitrary angle with respect to one another andstudying the problems pertaining to the environment of hydraulic fracturing. It also should be mentioned here that the method used in this study is an extension of the method developed by Lu in his study of a plane problem of many cracks and the problem of a partially bonded plate. In our analysis of the problem, the plane of the fracture is assumed to be in a condition of plane strain. In view of the order of magnitude differences between the fracture length, height, and width of a hydraulically induced fracture, we believe that this assumption is acceptable except, perhaps, at a very early stage of fracturing. The general problem of two bonded half-planes containing many cracks of arbitrary shapes is considered first. The problem then is reduced to a case of two arbitrarily oriented straight cracks. The solution is carried out in full. Numerical values of the stress intensity factor at the fracture tips pertaining to the containment of a hydraulically induced fracture are presented and discussed at the end of the paper. Formulation of the Problem In the following derivations, we follow the notations in Ref. 15; for completeness and clarity, some obvious results are listed without further referencing. Throughout the paper, we use the superscripts phi (x) and psi (x) for x epsilon X (along the interface); the subscripts phi (s) and psi (s) for s epsilon L (along cracks); and zeta epsilon L+X; x, xi epsilon X; and s, L in integrals. Consider an elastic plane (under either plane stress or plane strain condition) made by bonding together two plane strain condition) made by bonding together two planes of different materials, where k +, G+ and k -, planes of different materials, where k +, G+ and k -, G - are the material constants for the upper (Z + ) and the lower (Z-) plane, respectively. Let there be p nonintersecting smooth cracks. Lj =ajbj (j=1..... p) on both these half-planes. Let the intensity (force/unit length) of the external load applied on the surface of crack Lj be Xj (s) + i Yj (s), where s is the complex coordinate of a point on Lj. SPEJ P. 55