A New Model of a Pressure-Dependent-Conductivity Hydraulic Fracture in a Finite Reservoir: Constant Rate Production Case
Authors: Pedroso C. A. , Corrêa A. C. F. 
Published: December 2007

Abstract
A new model of a symmetrical, fully penetrating vertical fracture, intercepting a well locate in a finite reservoir is developed. This model, in addition to the two-parameter used in conventional fracture modeling, that is, fracture conductivity and diffusivity ratio, uses a third parameter: the permeability modulus. This additional parameter represents the permeability variation for a fracture with a pressure-dependent permeability.
The three-parameter model is numerically solved by finite difference schemes in both fracture and reservoir. The constant rate solution presents, at very early time, the influence of the permeability modulus in well pressure response. The conventional pressure analysis of bilinear and linear formation flow can be modified to consider the permeability modulus.
We also investigate the effects of the decline in fracture conductivity on the pressure profile along the fracture, on the flux density (from the matrix to fracture) distribution, and on the reservoir drainage. Great differences with respect to the two parameter fracture model are observed.

Introduction
Hydraulic fracturing is a worldwide technique for either producing damaged wells or increasing production from low permeability reservoirs. For this reason, a great amount of work has been directed to determine the flow behavior of fractured wells, to analyze pressure testing results, to design fracture geometry and to select project parameters. Besides the research of Prats, Raghavan, Cinco-Ley and Gringarten, which depict de general theory of fracture flow and well test analysis, additional work has been done in specific fracture topics, such as partial penetrating and asymmetrical positioning of fractures, and fractures in which conductivity varies with height. However an specific topic, has been poorly discussed: the flow behavior of a fracture in which permeability and conductivity dependent on the effective stresses.
Hydraulic fracturing increases well productivity or injectivity by changing the radial flow pattern, characterized by a large concentration of flux lines nearby the well, to a linear flow pattern, in which fluid flows towards a high conductivity fracture. For this reason, the effectiveness of a fracture depends on its conductivity (kf,w) and on the ratio of fracture to reservoir permeability, expressed by the fracture dimensionless conductivity (kf,w/lf,k).
The fracture conductivity is dependent on the effective stress tension acting over the fracture propping agent. The effective stress tension increases as the porous pressure decreases. Hence, production from a fractured well increases the effective stress, causing a reduction in the fracture permeability. Thus fracture productivity decreases and pressure drop increases with production.
The purpose of this work is to establish a model which permits to compute the changes in fracture conductivity with the fracture porous pressure. This requires numerical solution of the non-linear flow equation obtained from the consideration of variable fracture permeability.