Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 907206, 35 pages
Research Article

Pseudo-Steady-State Productivity Formula for a Partially Penetrating Vertical Well in a Box-Shaped Reservoir

1Department of Petroleum Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, UAE
2Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, T-301 Sarkeys Energy Center, 100 East Boyd Street, Norman, OK 73019-1003, USA

Received 8 September 2009; Accepted 9 February 2010

Academic Editor: Alexander P. Seyranian

Copyright © 2010 Jing Lu and Djebbar Tiab. 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.


For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long-producing times. Taking a partially penetrating well as a uniform line sink in three dimensional space, by the orthogonal decomposition of Dirac function and using Green's function to three-dimensional Laplace equation with homogeneous Neumann boundary condition, this paper presents step-by-step derivations of a pseudo-steady-state productivity formula for a partially penetrating vertical well arbitrarily located in a closed anisotropic box-shaped drainage volume. A formula for calculating pseudo skin factor due to partial penetration is derived in detailed steps. A convenient expression is presented for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle, and for arbitrary position of the well within the rectangle.