`Mathematical Problems in EngineeringVolume 2012, Article ID 428596, 18 pageshttp://dx.doi.org/10.1155/2012/428596`
Research Article

## Productivity Formulae of an Infinite-Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

1College of Mathematics, Sichuan University, Sichuan, Chengdu 610041, China
2Department of Petroleum Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, UAE
3Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401, USA

Received 13 April 2012; Accepted 29 May 2012

Academic Editor: Kue-Hong Chen

Copyright © 2012 Chaolang Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.