A mathematical model is developed in order to evaluate the productivity ofmultilateral completions. The model may also be used to understand and discernthe pressure behaviour of multilateral completions. The ability to incorporatethe location of each lateral rigorously is the unique feature of the model. Asa result of the development of this model, it is now possible to calculaterapidly the efficacy of various options and to incorporate this completionscheme in numerical models to evaluate the possible influences of interference, boundaries, changes in reservoir properties, etc.


Even a perfunctory survey of the literature indicates that, althoughmultiple-lateral completions are quite common, there is a dearth of informationon methods to compute productivity, to devise procedures to evaluate thecompletion efficacy, and to forecast performance. The goal of this paper is todevelop a mathematical model to attaint these general objectives. To ourknowledge, the results presented in this paper are not available in theliterature. The principal feature of the analytical model described here isthat it is possible to examine the influence of the location of the laterals ofthe multilateral system rigorously and completely. This option cannot bereadily incorporated into conventional numerical schemes. The specificcontributions of this work are as follows: (i) the model may be used to designfor specific configurations of horizontal sections and evaluate theconsequences that follow from a specific choice, (ii) to screen candidates forthis ompletion scheme, and (iii) to discuss transient behavior and utline amethodology to analyze pressure tests with a view to determine reservoirproperties and completion efficiency. Before considering the specific detailsof this work, we notethat myriad completion schemes are referred to asmultilateral wells. In this paper, we refer to a system with one or moreorizontal sections in a single zone that produce against a ommon pressure andproducing string as a multilateral completion. Methods are outlined in the textto extend the scheme discussed here to multiple, non-communicating, producingzones.

Mathematical Model

We consider the flow of a single-phase, slightly-compressible fluid or constantviscosity in a uniform porous medium in the form of a slab that is infinite inarea extent. The upper and bottom boundaries (z=o and z=ze = h) areassumed to be mpermeable. The permeabilities in the x, y, and z directions arc, respectively, kx.ky, and kz. Initially, thepressure is uniform throughout the reservoir.

Each lateral is assumed to be parallel to the plane Z= 0 and its properties(length, radius, compass orientation, elevation, skin factor, etc.) arespecified independently. All laterals produce against a common pressure and themultilateral system (herein after referred to as the System) may be produced ata specified rate. For convenience, most of the results given here are for thecase when the System is produced at a constant (surface) rate. The contributionof each radial or lateral to the total production rate is determined bycalculations.

For convenience and ease of comparison, we present results in dimensionlessform. Dimensionless pressure and dimensionless time are defined, respectively,as:

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