A Novel Analytical Fracture-Permeability Model Dependent on Both Fracture Width and Proppant-Pack Properties

Abstract

SummaryFor 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.