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Abstract

This paper presents an analytical study of the wellbore pressure behavior for a vertically fractured well subject to limited entry and bottom water drive. The well is producing at a constant rate from (1) an infinitely large reservoir and (2) the center of a rectangular drainage region. The upper boundary of the formation is impermeable but the lower boundary is maintained at constant pressure equal to the initial pressure of the reservoir. Two fracture locations (top pressure of the reservoir. Two fracture locations (top and center) within the producing interval are considered. Both infinite-conductivity and uniform-flux fractures are examined. Analysis of drawdown and buildup pressure data by conventional semi-logarithmic techniques are conducted to determine the combined effect of limited entry and bottom water drive on well test analysis.

The study indicates that the use of conventional Horner and Miller-Dyes-Hutchinson semilog methods would give inaccurate results. Consequently, the type curves presented in this paper may be used to estimate presented in this paper may be used to estimate reservoir and fracture characteristics in oil, gas and geothermal reservoirs.

Introduction

The ability to determine reservoir performance due to production or injection through a fractured well, with a reasonable degree of accuracy, is of utmost importance in the petroleum industry since many wells have been hydraulically fractured. The analysis of pressure data in fractured wells has however received considerable attention.

These authors assume that the fracture extends over the entire vertical extent of the formation. But field observations have shown that this assumption is not always valid. The first to study the effect of vertical fracture height on transient flow behavior was Raghavan, et al. But their study was only applicable to an infinitely large reservoir with an impermeable upper and lower boundaries.

A more recent study by Cinco-Ley, et al. used a parallelepiped model to analyze the pressure behavior parallelepiped model to analyze the pressure behavior of geothermal steam wells with a vertical fracture. The type curves developed with that model can only be used for dry steam wells. In another study Economides, et al. also used the parallelepiped model to analyze pressure buildup of a geothermal steam well.

The model developed in this paper is similar to the parallelepiped model except that here the results are not restricted to dry steam wells. As a matter of fact the results obtained are for oil wells but they can be used for gas and geothermal wells. This paper presents an elaborate treatment to show the effect of presents an elaborate treatment to show the effect of vertical fracture height on the pressure behavior in an infinite and bounded reservoir subject to bottom water drive.

MATHEMATICAL MODEL

The Schematic diagrams of the models under study are shown in Figures 1 and 2. The following assumptions are made.

  1. In Figure 1 the reservoir is infinite in x and y directions and consists of a single layer with uniform thickness, the upper boundary is impermeable and the lower boundary is at constant pressure equal to the initial pressure of the system.

  2. In Figure 2 the reservoir is in the form of a rectangular drainage region with uniform thickness, the upper boundary is impermeable and the lower boundary is at constant pressure equal to the initial pressure of the system. pressure of the system.

  3. The reservoir is uniform and homogeneous but anisotropic (that is, the vertical permeability kz, is different from the horizontal permeability, k).

  4. Initially, the pressure is constant everywhere in the reservoir; i.e., p(x, y, z, 0) = pi.

  5. A single-phase slightly compressible liquid with compressibility ct and viscosity p flows from the reservoir into the wellbore a constant reservoir rate qB.

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