A Semi-Analytical Model for Characterizing the Pressure Transient Behavior of Finite-Conductivity Horizontal Fractures

Abstract

Hydraulic fracturing may induce horizontal fractures in shallow, tectonically active or high-reservoir-pressure formations. Studying the pressure transient behavior of the horizontal fractures can provide insights into the size and shape of the formations as well as their productivities. However, our knowledge about the pressure transient behavior of a horizontal fracture is far from adequate. In this study, a semi-analytical model is proposed to characterize the pressure transient behavior of finite-conductivity horizontal fractures in bounded reservoirs. Specifically, we discretize the horizontal fracture into rectangle plane elements, each of which is treated as a plane source. (In this work, a plane source indicates a rectangle plane, and the oil flows from the matrix to this plane.) The transient flow in the fracture systems can be numerically characterized with the finite difference method, whereas the transient flow in the matrix system can be analytically simulated with the Green function method; as such, the flow behavior in both the fracture and matrix can be modeled. Subsequently, we construct the mathematical model by coupling the finite difference formulations for the fracture system and the analytical functions for the matrix system. This semi-analytical model is arranged into a matrix format and can be readily solved with the Gaussian elimination method. The key features of the proposed approach can be summarized as: it can model an irregular horizontal fracture by discretizing the horizontal fracture into small elements, such that the real fracture configuration can be better captured; the fracture conductivity can be taken into consideration; and the non-uniform influx distribution along the fracture can be modeled to better honor the actual flow behavior from the matrix to the fracture. We also apply the semi-analytical method to analyze the flow regimes of single-phase oil flow from a circular horizontal fracture and an elliptical horizontal fracture in a bounded reservoir; the flow regimes such as wellbore after flow, bilinear flow, formation linear flow, early pseudo-radial flow, late pseudo-radial flow, elliptical flow, and boundary dominated flow can be observed during oil production. In addition, we examine the influences of formation thickness, fracture’s vertical position, fracture conductivity, and wellbore storage on the pressure dynamics of a circular horizontal fracture. It is found that the early pseudo-radial flow results from two different flow scenarios: one is near the edge of the horizontal fracture when the formation thickness is small, while the other one is surrounding the horizontal fracture in the vertical direction when the formation thickness is sufficiently large. The wellbore storage mainly exerts influences on the early production period, and its duration tends to be longer as the fracture conductivity decreases. Furthermore, the flow regimes and pseudo-skin factor are comprehensively investigated for an elliptical horizontal fracture and an irregular-shaped horizontal fracture that may be induced by the stress heterogeneity in the reservoirs.