An Advanced Method for Predicting the Productivity Ratio of a Perforated Well
Authors: Locke Stanley 
Published: December 2007

Summary
This study extends and, in most details, corroborates the work of previous investigators in the area of relating productivity ratios to perforations. A more refined model, called the finite-element method, permits a significant advance in the precision of the general solution. The principal findings of this study include (1) corroboration of previous conclusions that perforator penetration is substantially more important than perforation diameter, within practical ranges of values, (2) confirmation that productivity continues to increase with increasing shot density, and (3) an analysis of the effect of angular phasing of successive shots, which, for the first time, compares the phasings used in real perforating guns. This analysis confirms earlier findings that 90° phasing is the best of those tested (0, 90, 120, and 180°) but produces more meaningful quantitative comparisons than were available before.

Introduction
An advanced mathematical modeling technique brings new light to the problem of predicting the performance of perforations under actual producing conditions.
The principal contributions of this study over previous work arise from the use of a model that more nearly approaches the characteristics of a real perforation. For the first time, a three-dimensional flow study takes into account the crushed zone around the perforation as well as a damaged zone around the wellbore. Moreover, this study considers the perforation a true cylinder rather than one of the mathematically simpler radial configurations used in most previous studies.
As in earlier studies, the flow problem is solved for various angular phasings of adjoining perforations: 0,90, 120, and 180°.
The more significant results of this analysis include strong corroboration of the previous findings that (1) within realistic ranges, the length of a perforation is considerably more critical to productivity than its diameter, (2) of the four angular phasings studied, 90° phasing is best, and (3) that under all conditions, productivity continues to increase with increasing shot density up to at least a level of eight shots per foot (26 shots per meter). In addition, a new nomograph is presented for estimating the productivity ratio* of downhole perforations.
A general solution of the flow-prediction problem requires knowledge of or assumption about many parameters, some of which are not determined readily. Although this study offers no new techniques for evaluating these variables, it focuses attention on areas that are in most urgent need of further investigation. Such studies are encouraged strongly.

The Problem
Effective planning of the perforating program has come to be recognized as a crucial factor in the full-term economic success of any well. Planning for the general case always has been impeded by the limitations of existing mathematical modeling studies, which have not succeeded in providing reliable predictions of productivity ratios. Such planning, therefore, has had to fall back. on experience, which is often lacking or of limited applicability.
Much work is needed to further the industry's ability to plan optimal well completions under a wide range of conditions. This study provides a new basis on which this work can be built.