Closed-form solution for vertical fracture width in anisotropic elastic formations

A closed-form analytical solution for estimating the hydraulic fracture width in anisotropic tight shales is presented, as a function of the anisotropic elastic properties and fluid pressure. The suggested formula is a generalization of the well-known Sneddon solution for fracture width in isotropic formations that is widely used by petroleum engineers for hydraulic fracture applications. The classical isotropic formula becomes inappropriate for applications in unconventional tight shale formations, where the ratio of moduli in orthogonal directions may reach 4.0. This new fracture width closed form solution was derived for infinite, homogeneous, anisotropic elastic medium, and for the particular case of a vertical fracture of given height, aligned with the axes of elastic anisotropy. This solution is less general than the known 2D analytic solutions for the infinite elliptic cavity arbitrarily oriented to the anisotropy axes a medium. These general analytic solutions require multiple complex steps to compute fracture width, including the selection of roots of high-order algebraic equations, and conformal coordinate transformations. In contrast, the presented closed-form solution allows one to quickly and easily understand the influence of rock elastic properties and fluid pressure on fracture width, without extensive numerical simulations. Thus, it is well suited for parametric studies and for handling large amounts of input parameter combinations.