Pressure-Transient Behavior of Wells Producing From Complex Patterns of Interconnected Linear Segments
Authors: Larsen Leif 
Published: December 2007

Abstract
The pressure-transient-behavior of wells producing from reservoirs with complex patterns of interconnected linear segments has been investigated by case specific and general methods. Proper understanding of reservoirs of this type is critical in situations where extrapolation of short-time test data to possible late-time production characteristics is attempted. This is shown by theoretical considerations and illustrated with examples of wells with one or two flow entries in simple and more complex patterns of interconnected linear segments.

Introduction
The results and methods of the paper extend those presented by Larsen on the pressure-transient behavior of wells producing from a single well in a two-dimensional pattern of interconnected linear reservoirs. Except for added complexity because of an increased number of segments and flow entries, there is no fundamental difference between systems with a single well in a 2D pattern and multiple wells, or flow entries, in a 3D pattern. Unfortunately, it is very difficult to work with general models and solutions, except for highly regular patterns. It is much easier to work with case specific models and solutions, e.g., models with one or two flow entries in a reservoir made up of two intersecting bounded or unbounded linear segments.
Although single wells with single or multiple flow entries in patterns of linear segments are the primary objectives, it is more convenient to consider systems of single and multiple wells in such patterns. One can then work with different boundary conditions, such as constant individual rates or constant total rate with common bottomhole pressure. The latter is equivalent to having a single well with multiple flow entries and infinite conductivity.
Since the mathematical approach depends strongly on the type of pattern being studied, the basic patterns with few segments studied in Ref. 1 have also been reviewed in the present paper.

Generalized Linear Reservoirs
Fig. 1 shows one linear and two generalized linear reservoirs, i.e., reservoirs that can be modelled as simple linear systems by introducing parallel no-flow boundaries. By measuring length along the center line of Cases b and c, it should be clear that all three cases have the same reservoir volume affected by distance of influence along the center line, and hence the same pressure-transient response, at least for wells away from possible sharp bends and corners.
For Cases b and c in Fig. 1 a horizontal well can possibly intersect the reservoir more than once. This would be just a special case of multiwell production from a linear reservoir with constant total rate and common bottomhole pressures.

Self-Intersecting Linear Reservoirs
A self-intersecting linear reservoir is shown in Fig. 2, with four well locations indicated for reference. With production at just one of these locations, the situation is quite different for Locations 1 and 2 on the loop, and Locations 3 and 4 outside. For wells at the outside locations the loop can be replaced by a branch of length equal to half the distance around the loop centerline with doubled width, i.e., by a "Y-shaped" model with one segment of finite length. For wells at Locations 1 and 2 a transient period will exist in the data with a more complex response before the loop again will affect the response as a branch of finite length.
For a reservoir of the type indicated in Fig. 2, and also for the "Y-shaped" case, it is again realistic to consider the possibility of multiple intercepts by a horizontal well. Both single-well and two-well cases have been included in the paper to illustrate possible pressure-transient behavior of reservoirs of this type.
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