Abstract
Quite a lot has been published about pressure transient testing of hydraulically fractured wells since the pioneering work of Heber Cinco-Ley in 1976. A common approach of the problem since 1981 has been to solve the ordinary differential equations (ODEs) of rate and pressure versus time in the Laplace domain and to invert the Laplace solution back into the time domain. This method has led to exact asymptotic solutions for the so-called early time bilinear and formation linear flow regimes. Unfortunately, the inversion of the general Laplace solution was shown to be too complex. Therefore, the pressure and pressure derivative type curves of the transition period between the bilinear and linear regimes have always been numerically generated, most often by making use of Stehfest's inversion algorithm, sometimes by discretization methods in the time domain.
This paper shows that a single, simple and exact analytical solution of the ODEs can be obtained directly in the time domain, providing that the flow within the fracture is assumed to be incompressible (no fracture storage effects). This general solution has two asymptotes for early and late times that agree perfectly with the previously published solutions for bilinear and formation linear flow regimes.
The contribution of the present paper is that it provides exact pressure and pressure derivative type curves throughout the long (six log cycles) transition period between the bilinear and linear asymptotes. It therefore allows for rigorous definitions of transition times, which had been so far numerically approximated. In addition, the proposed solution handles without any difficulty the presence of skin at fracture walls (fracture face skin).
