Numerical implementation of displacement discontinuity method and its application in hydraulic fracturing

A formulation of the quadratic displacement discontinuity method is presented to model two-dimensional elastic fracture problems. The new formulation is found by analytical integration of the fundamental solution along a straight-line crack within which three collocation points are taken. Each element has a quadratic distribution of displacement discontinuity values. For a curved crack, a general formulation of the displacement discontinuity method is proposed, based on the boundary integral equation. This formulation contains strongly singular and hypersingular integrals which can be evaluated in the sense of Cauchy principal value and Hadamard principal value, respectively. Special crack tip elements are adopted to show the √r variation of displacements near the crack tips. Some examples are used to illustrate the accuracy of two formulations. General formulation is also used to simulate hydraulic fracturing propagation and interaction between a crack and a fault. The fracture propagation path is obtained by using the maximum tensile stress criterion.