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Summary Field studies have shown that, if an inclined fracture has a significant inclination angle from the vertical direction or the fracture has a poor growth along the inclined direction, this fracture probably cannot fully penetrate the formation, resulting in a partially penetrating inclined fracture (PPIF) in these formations. It is necessary for the petroleum industry to conduct a pressure-transient analysis on such fractures to properly understand the major mechanisms governing the oil production from them. In this work, we develop a semianalytical model to characterize the pressure-transient behavior of a finite-conductivity PPIF. We discretize the fracture into small panels, and each of these panels is treated as a plane source. The fluid flow in the fracture system is numerically characterized with a finite-difference method, whereas the fluid flow in the matrix system is analytically characterized on the basis of the Green's-function method. As such, a semianalytical model for characterizing the transient-flow behavior of a PPIF can be readily constructed by coupling the transient flow in the fracture and that in the matrix. With the aid of the proposed model, we conduct a detailed study on the transient-flow behavior of the PPIFs. Our calculation results show that a PPIF with a finite conductivity in a bounded reservoir can exhibit the following flow regimes: wellbore afterflow, fracture radial flow, bilinear flow, inclined-formation linear flow, vertical elliptical flow, vertical pseudoradial flow, inclined pseudoradial flow, horizontal-formation linear flow, horizontal elliptical flow, horizontal pseudoradial flow, and boundary-dominated flow. A negative-slope period can appear on the pressure-derivative curve, which is attributed to a converging flow near the wellbore. Even with a small dimensionless fracture conductivity, a PPIF can exhibit a horizontal-formation linear flow. In addition to PPIFs, the proposed model also can be used to simulate the pressure-transient behavior of fully penetrating vertical fractures (FPVFs), partially penetrating vertical fractures (PPVFs), fully penetrating inclined fractures (FPIFs), and horizontal fractures (HFs).

Summary For an empty fracture, the fracture permeability (kf) is mainly influenced by the effect of viscous shear from fracture walls and can be analytically estimated if the fracture width (wf) is known a priori (i.e., kf=β2wf2/12, where β2 is the unit-conversion factor). For an adequately propped fracture, the fracture permeability is mainly influenced by the proppant-pack properties and can be approximated with the proppant-pack permeability (kf=kp, where kp is proppant-pack permeability). It can be readily inferred that as the effect of viscous shear fades (or the proppant-pack effect becomes pronounced), there should be a regime within which both the viscous shear and the proppant-pack properties exert significant influences on the fracture permeability. However, the functional relationship between fracture permeability, viscous shear (or fracture width), and proppant-pack properties is still elusive. In this work, we propose a new fracture-permeability model to account for the influences of the proppant-pack permeability, proppant-pack porosity (ϕp), and fracture width on the fracture permeability. This new fracture-permeability model is derived from a modified Brinkman equation. The results calculated with the fracture-permeability model show that with different values of the Darcy parameter, the fluid flow can be divided into viscous-shear-dominated (VSD) regime, transition regime, and Darcy-flow-dominated (DFD) regime. If the Darcy parameter is sufficiently large, the effect of proppant-pack permeability on fracture permeability can be neglected and the fracture permeability can be calculated with the VSD fracture-permeability (FP) (VSD-FP) equation (i.e., kf=β2ϕpwf2/12). If the Darcy parameter is sufficiently small, the effect of viscous shear on fracture permeability can be neglected and the fracture permeability can be calculated with the DFD-FP equation (i.e., kf=kp). Both the VSD-FP and DFD-FP equations are special forms of the proposed fracture-permeability model. For the existing empirical/analytical fracture-conductivity models that neglect the effect of viscous shear, one can multiply these models by the coefficient of viscous shear to make these models capable of estimating the fracture conductivity with large values of Darcy parameter.

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