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A new method for analysing productivity index (PI) on vertical wells is the main objective of this study. Well performance is often measured in terms of the well’s productivity which is dependent on a number of factors such as the reservoir’s configuration, the type of completion, petrophysical and fluid properties, formation damage, etc. The effect of partial completion is the main focus of making the productivity index analysis since almost all vertical wells are partially completed due to the reasons of water coning or gas cap issue, etc. It is also very expensive to fully complete a well especially when the formation thickness is so large.

Pressure behaviour solutions for both closed boundary and constant pressure boundary have been obtained, taking into consideration the effect of partial completion.

Productivity of a well is usually evaluated on the long time performance behaviour, thus the pseudo-steady state (late time) approach has been employed for calculation of the productivity index.

Several key factors have been tested on productivity index such as pseudo skin, shape factors, penetration ratio, reservoir drainage area and etc. The effects of these factors have been analysed on PI. Theoretical data were used in carrying out the analysis with results indicating that, productivity index increases with increasing completion interval and vice versa, whiles pressure drop due to skin as a result of restricted entry to fluid flow increases tremendously with decreasing completion interval.

Shape factors of various well positions in bounded reservoirs were computed and compared with results obtained by Dietz, and Babu and Odeh.




Well productivity is one of the major concerns in oil field development, and provides the means for oil field development strategy. Sometimes, well performance is measured in terms of productivity index. In order to arrive at the economic feasibility of drilling a well, petroleum engineers require proven and reliable methods to estimate the expected productivity of that well. Well productivity is often evaluated using the productivity index, defined as the production rate per unit pressure drawdown. Petroleum engineers often relate the well productivity evaluation to the long-time performance behaviour of a well, that is, the behaviour during pseudo-steady-state or/and steady-state flow of a closed system or/and constant pressure system respectively.

The long-term productivity of oil wells is influenced by many factors. Among these factors are petrophysical properties, fluid properties, degree of formation damage and/or stimulation, well geometry, well completions, number of fluid phases, and flow-velocity type (Darcy, non-Darcy) (Yildiz, 2003).

Depending upon the type of wellbore completion configuration, it is possible to have radial, spherical or hemispherical flow near the wellbore. A well with a limited perforated interval (partial completion) could result in spherical flow in the vicinity of the perforations as depicted in fig. 2.1. A well which only partially penetrates the pay zone, could result in hemispherical flow. These conditions could arise where coning of bottom water or gas cap becomes a serious issue (Ahmed, 2005). Figures 3.1 and 3.2 respectively depict the true picture of radial and spherical flow behaviour in a partially completed vertical well.

Partial completion is the completion of or flow from less than the entire producing interval. This situation causes a near-well flow constriction that result in a positive skin effect in a well-test analysis. The net result of partial completion yields extra pressure drop in the near wellbore region and reduces the well productivity.

The present analytical method of evaluating productivity index in vertical wells with partial completion does not account for the effect of pressure drop due to partial completion.

The purpose of this study is to develop analytical model for evaluating productivity index (P.I) of vertical wells with partial completion, where the effect of pressure drop due to partial completion is taken into account and compare results with conventional methods.

The partial differential equations were solved for both no-flow boundary and constant pressure boundary systems in Laplace and Fourier Transform domains before inversion to real time domain.

1.3         OBJECTIVES

The main objectives of this work are to:

·         Develop analytical model for pressure behaviour in closed and constant pressure boundary systems

·         Develop an analytical model for evaluating productivity index of vertical wells with partial completion for both closed-boundary and constant pressure boundary systems taking into account the effect of pressure drop due to partial completion

·         Calculate shape factors and compare with the existing ones and

·         Investigate the factors and parameters that influence or control productivity index.

1.4         METHODOLOGY

The partial form of the diffusivity equation in dimensionless terms is solved for both closed system and constant pressure boundary case employing Laplace and Fourier transforms. Gringarten and Ramey’s source functions (1974) for closed and bottom water drive (mixed boundaries system) have been used in conjunction with Babu and Odeh’s approach (1989) for obtaining pressure drawdown in terms of average reservoir pressure. Finally new productivity index equations are generated for both closed system and bottom water drive of a vertical well with partial completion....

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