A Semi-Analytic Solution for Flow in Finite-Conductivity Vertical Fractures Using Fractal Theory

Abstract

Abstract The exploitation of unconventional reservoirs goes hand in hand with 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 et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991) and Blasingame and Poe (1993) for a finite-conductivity case. This topic is still an active area of research and, for the more complicated physical scenarios such as multiple transverse fractures in ultra-tight reservoirs, answers are presently being sought. Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure transient testing, albeit 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 and show that it is more fundamental than the classic linear and radial diffusivity equations. Subsequently, we combine the Fractal Diffusivity Equation with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semi-analytic solution for flow in a finite-conductivity vertical fracture which we name the "Fractal Fracture Solution". This new solution is very fast and its accuracy is comparable to that of the Blasingame and Poe solution (1993). In its final, closed form, it is valid for a dimensionless fracture conductivity FcD ≥ 3. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is very flexible and we are optimistic about extending the same methodology to develop new solutions for pressing problems that the industry currently faces, the only caveat being that it must calibrated to a known solution or production history.