# Numerical Solutions of a Hypersingular Integral Equation with Application to Productivity Formulae of Horizontal Wells Producing At Constant Wellbore Pressure

## Abstract

The performance of horizontal wells producing at constant wellbore pressure is a critical problem in petroleum engineering. But few articles on the well performance under constant wellbore pressure can be found in the literature due to the difficulty of hypersingular integral equations, which are needed for this problem. This article proposes and studies a new model using a hypersingular integral equation for the productivity of horizontal wells producing at constant wellbore pressure. An efficient numerical method is developed for this hypersingular integral equation based on a new expansion with respect to the singularity at arbitrary points. And numerical examples are provided to illustrate the convergence of the numerical methods. By using fluid potential superposition principle, productivity equations for a line sink model are derived from the point sink solution to the diffusivity equation. By solving the hypersingular integral equation, the authors obtain the productivity formulae of a horizontal well producing at constant wellbore pressure, which provide fast analytical tools to evaluate production performance of horizontal wells. Numerical examples are provided to illustrate the features of the model and the numerical method.

## Recommended Citation

C.
Hu
et al.,
"Numerical Solutions of a Hypersingular Integral Equation with Application to Productivity Formulae of Horizontal Wells Producing At Constant Wellbore Pressure," *International Journal of Numerical Analysis and Modeling, Series B*, vol. 5, no. 3, pp. 269 - 288, Institute for Scientific Computing and Information, Jan 2014.

## Department(s)

Mathematics and Statistics

## Research Center/Lab(s)

Center for High Performance Computing Research

## Keywords and Phrases

Hypersingular Integral Equation; Quadrature method; Horizontal Well; Constant Wellbore Pressure

## Document Type

Article - Journal

## Document Version

Citation

## File Type

text

## Language(s)

English

## Rights

© 2014 Institute for Scientific Computing and Information, All rights reserved.

## Publication Date

01 Jan 2014