Pressure Analysis of Infinite, Semi-infinite And Closed Rectangular Naturally Fractured Reservoirs
Authors: Aguilera R. , Ng M.G. 
Published: March 2010

Abstract
This paper presents methods for handling wellbore storage and skin in pressure drawdowns and buildups conducted in wells located off centre in bounded rectangular naturally fractured reservoirs. The methods integrate into a single technique many problems found in practice in well test analysis.
The principle of superposition via the method of images is used for the drawdown evaluation. The principle of desuperposition is utilized for the buildup evaluation. It is concluded that, in addition to bounded rectangular naturally fractured reservoirs, the equations permit proper design of drawdown, reservoir limit and buildup tests and the analysis of many cases found in practice, including a well in an infinite acting reservoir, a well affected by one, two or three intersecting linear discontinuities, and a well in a channel or bounded by parallel linear discontinuities. Drawdown and buildup applications are presented. It is shown that under favourable conditions it is possible to make estimates of hydrocarbons-in-place from drawdown and buildup data.

Introduction
Drawdown and buildup analysis of hydrocarbon reservoirs has been widely discussed in the petroleum engineering literature, as well as superposition and desuperposition techniques(4,5,8). Numerical techniques(3), specialized crossplots(5) and type curves(1–9) have proved very powerful for this type of interpretation in naturally fractured reservoirs.
This paper uses superposition and desuperposition techniques to generate drawdowns and buildups under different geologic scenarios, including infinite acting, channel, faulted and fully bounded naturally fractured reservoirs. A computer program has been generated to carry out the calculations. Under favourable conditions it is possible to make estimates of hydrocarbons-in-place from drawdown and buildup tests.

Theory
We use the LaPlace transform technique. We start off with a single well in an infinite double porosity reservoir. Then we use method of images for a rectangular reservoir. Finally, we use the principle of superposition in time to account for buildups. Bourdet and Gringarten(1) gave the dimensionless pressure drop in an infinite reservoir with double porosity behaviour:
Equation (1)(Available in full paper)
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The principle of superposition via image wells in a rectangular reservoir is defined following guidelines published by Earlougher(4). The location of the well can be at any point within the rectangular reservoir. The use of image wells for faulted naturally fractured reservoirs has been discussed by Aguilera(5). This paper extends that work to the use of image wells in fully bounded naturally fractured reservoirs. The approach involves the use of Equation (1) to determine a dimensionless pressure, Po in the LaPlace space at any point x, y within the system. Skin and wellbore storage are handled using techniques pub