Design And Analysis Of A Fracture With Changing Conductivity
Authors: Soliman M.Y. 
Published: March 2010

Abstract
Most reservoirs fracture studies assume uniform fracture conductivity, however, proppant crushed by closure stress can reduce fracture conductivity near the wellbore. Sometimes, operators place a higher conductivity proppant near the wellbore. Sometimes, operators place a higher conductivity proppant near the wellbore to create a "tailed-in " fracture. The dual purposes of this paper are tostudy the effect of tail-in and/or crushed proppant on reservoir performance; andshow how to design a tailed-in fracture using graphs furnished in the paper.
Two models developed and analytically solved in this paper describe the performance of a well under either constant pressure or constant rate conditions. In these models the well intercepts a vertical fracture consisting of two segments of different conductivity. Solutions of the models are presented as graphs of dimensionless rate or dimensionless pressure plotted against dimensionless time. Five curves in each graph show the effect of the relative lengths and conductivities of two segments of the fracture.
The two analytical models are used to quantitatively evaluate and explain the behavior of fractured wells computer simulated by a numerical model. Reverse analysis of this procedure can be used to aid in fracture design. These examples demonstratewhy some fractures appear to have limited success while others cause sharp improvement in well performance; andthe adverse effect of proppant crushing on well productivity.

Introduction
Although most studies in the literature assume that a fracture is of uniform conductivity throughout its length, many actual field causes show otherwise. Most designs have a higher conductivity near the wellbore (coarser proppant, or higher concentration). Even if the fracture is designed with a uniform conductivity, the high closure pressure near the wellbore could reduce the fracturing conductivity.
Until now only numerical simulators could be used to evaluate the effect of changing fracture conductivity on reservoir performance. The purpose of this paper is to present a simple method and graph to be used in design of changing conductivity fracture.
Two analytical models dealing with a tailed-in fracture will be developed. The models are solved using Laplace transform and the final solution is inverted numerically. Because of the many variables involved in the problem, only selected figures are presented, however, similar graphs can be reproduced utilizing a numerical inverter for the appropriate Laplace transform.
Two samples, are discussed in this paper to illustrate the effect of variable conductivity. To verify the results, a numerical simulator is used to model the same examples.

Simplified Mathematical Model
The mathematical model is similar to the one proposed by Cinco, et al.(1). However, it is expanded to consider the effect of changing conductivity fracture for constant rate or constant pressure boundary conditions.
The solution presented by Cinco(1) only an approximate one and more accurate solutions are presented in the literature(1–5). The approximate solution presented in this paper and earlier by Cinco(1) agrees very well with the complete solution presented in literature(1–5).
The model considers a fracture in an finite reservoir that is horizontal, homogeneous and contains a slightly compressible fluid.