A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory

The exploitation of unconventional reservoirs complements the practice of hydraulic fracturing, and with an ever-increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture, and the problem has been approached from different directions in the past 3 decades starting with the work of Gringarten et al. (1974) for an infinite-conductivity case, followed by contributions from Cinco-Ley et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991), and Blasingame and Poe (1993) for a finite-conductivity case. This topic remains an active area of research and, for the more-complicated physical scenarios such as multiple transverse fractures in ultratight reservoirs, answers are currently being sought. Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure-transient testing, although with an emphasis on the effects of natural fractures in pressure/rate behavior. In this paper, we begin by performing a rigorous analytical and numerical study of the fractal diffusivity equation (FDE), and we show that it is more fundamental than the classic linear and radial diffusivity equations. Thus, we combine the FDE with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semianalytic solution for flow in a finite-conductivity vertical fracture that we name the "fractal-fracture solution (FFS)." This new solution is instantaneous and comparable in accuracy with the Blasingame and Poe solution (1993). In addition, this is the first time that fractal theory is used in fluid flow in a porous medium to address a problem not related to reservoir heterogeneity. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is flexible, and we believe that the same methodology could be extended to different applications. One objective of this work is to develop a fast and accurate semianalytical solution for flow in a single vertical fracture that fully penetrates a homogeneous infinite-acting reservoir. This would be the first time that fractal theory is used to study a problem that is not related to naturally fractured reservoirs or reservoir heterogeneity. In addition, as part of the development process, we revisit the fundamentals of fractals in reservoir engineering and show that the underlying FDE possesses some interesting qualities that have not yet been comprehensively addressed in the literature.