Productivity Computations for Multilateral, Branched and Other Generalized and Extended Well Concepts
Authors: Larsen Leif 
Published: December 2007

Abstract
Simple methods are presented for productivity computations for multilateral, branched and other generalized and extended well concepts, mostly based on computations of pseudoradial skin factors for systems of connected wells, intervals or fractures with a common bottomhole pressure (or potential). Also included are applications to perforation size situations. For symmetric cases, either with respect to formation boundaries or with respect to a rotational surface around the well-system axis, simple closed-form expressions are presented. Similar, although more complex, expressions are also given for linear symmetric cases with 3 or 4 wells or branches. For other cases a numerical approach is outlined. The accuracy of the direct results depends on the vertical component and horizontal distance between well elements, and on methods used to compute skin values of individual well elements, but numerical schemes to improve the results have also been outlined.

Introduction
Although several methods have been introduced in the literature to determine productivity indices or skin values for multilateral and multifractured wells (see for instance Refs. 1–4), the cases covered are mostly restricted to highly symmetric well configurations. The present paper, on the other hand, considers a more general approach that can be used to determine productivity indices or skin values for quite arbitrary well configurations in homogeneous reservoirs of constant thickness.
The approach used assumes a certain distance between the midpoints of the well elements, e.g., branches, drainholes, and fractures, of the well or well system in question, and then assumes the pressure development to depend on the distance only. Even relatively close to a horizontal well, it is for instance possible to estimate the pressure development with negligible error by using a fully penetrating vertical line well. Moreover, since the line-source solution can be approximated with a simple logarithmic expression at sufficiently late times, it is possible to combine the effect of for instance several wells or branches by elementary methods. This is especially simple if the rate of each well or branch is known, e.g., if they can all be assumed identical from symmetry considerations. Otherwise the rates must be determined by solving a system of linear equations as part of the computation of the productivity index or skin value.
For cases where simple line-source solutions cannot be assumed to accurately represent the pressure development at a given observation point, it might be necessary to use more accurate analytical solutions to compute adjustment factors. These adjustments can be applied to the distance between well elements (e.g., midpoint of branches) as has been outlined in Appendix B.

Skin Factors and Productivity Indices
For the developments of this paper it suffices to consider pseudosteady state productivity indices of the form
(1)
in practical SI units, where y =0.57722156… denotes Euler's constant, CA is the Dietz form factor for the shape of the drainage area and well location, and S is the skin factor of the well. For a circular drainage area with the well at the center, Eq. 1 takes the familiar form
(2)
where re is the outer radius of an equivalent circular drainage area.
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