Well-test Analysis For a Wen In a Finite, Circular Reservoir
Authors: Ambastha Anil K. , Ramey H.J. 
Published: March 2010

Abstract
This study presents drawdown and buildup pressure derivative type-curves for a well producing at a constant rate from the centre of a finite, circular reservoir. Early time response (wellbore storage and skin effects) is correlated by C De2s, and late time response (outer boundary effects) by reD2/CD. The outer boundary may be closed or at a constant pressure. Design relations are developed for the time to the beginning and the end of infinite-acting radial flow. Producing time effects on buildup response are also discussed.

Introduction
Transient pressure response for a well producing from a finite Reservoir of a circular, square, and rectangular drainage shapes has been studied by many investigators(1–8).Mishra and Ramey(9) presented a buildup pressure derivative type-curve for a well with storage and skin, and producing from the centre of a closed, circular reservoir. Their type-curve applies for large producing times such that lpD &gt; lDpss. This work presents drawdown and buildup pressure derivative type-curves for a well producing at a constant rate from the centre of a finite, circular reservoir. The outerboundary may be closed or at a constant pressure. The differences between the responses for a well in a closed, circular reservoir (fully a developed field) and a well in a circular reservoir with a constant-pressure outer boundary (active edge-water drive system or developed five-spot, fluid injection pattern) are discussed. Design relations are developed to estimate the time period which correspond to infinite-acting radial flow regime. Producing time effects on buildup responses are studied using the slope of a dimensionless Agarwal(10) buildup graph.

Theory
The dimensionless wellbore pressure drop for a constant-rate well with storage and skin may be expressed as(1):
Equation (1) Available In Full Paper.
where L −1 is the inverse Laplace transform operator. In Equation (1), PD refers to the dimensionless wellbore pressure drop in Laplace transform without storage of skin. For the case of a constant-rate well producing from the centre of a closed circle, the expression for PD is(1).
TABLE 1: Dimensionless wellbore drawdown pressure and pressure derivative expressions for a well in a finite, circular reservoir. Illustrations available in full paper.
Equation (2) Available In Full Paper.
For the case of a constant-rate well producing from the centre of a circular reservoir with a constant-pressure outer boundary, The expression for P D is (1):
Equation (3) Available In Full Paper.
The dimensionless wellbore pressure drop from Equation (1) was obtained by inverting the Laplace space solution numerically with the Stehfest(11) algorithm.

Drawdown Responses
Table 1 shows the dimensionless wellbore pressure drop and the semi-log pressure derivative expressions for a well in a finite, circular reservoir during specific flow periods. All expressions in Table 1 may be written as combinations of lD/C D, CDe 2s, and le D2/CD. For example,
Equation (4) Available In Full Paper.
FIGURE 1: Verification or CDe 2s and re D2/CD as the correlating parameters for the drawdown responses. Illustrations at available in full paper.
FIGURE 2: Drawdown pressure derivative type-curve. Illustrations at available in full paper.
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