Recognition and Location of Pinchout Boundaries by Pressure Transient Analysis
Authors: Horne Roland N. , Temeng Kwaku O. 
Published: December 2007

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In the analysis of well pressure transient tests, only two types of boundaries are recognized currently-i.e., impermeable and constant pressure. Furthermore, from a sectional viewpoint, boundaries of either type frequently are considered vertical. However (to the best of our knowledge) there is no consideration of a reservoir boundary that results from a tapering of the reservoir to an eventual pinch out boundary. In such a case, the boundary is clearly impermeable, and yet would not behave in the same manner as a perpendicular sealing boundary. Since many oil and gas reservoirs are supposed to contain pinch out boundaries, there appears to be a need to determine the effect of such boundaries on the reservoir pressure performance and, if possible, to determine their pressure performance and, if possible, to determine their properties and position. Both proved possible in this properties and position. Both proved possible in this investigation.
In the case of a well fully penetrating the formation, the pinch out configuration is illustrated in Fig. 1, where the upper and lower confining beds meet at an angle, theta, at the pinch out. The reservoir is semi-infinite in the direction away from the pinch out (i.e., other boundaries of the system are too distant to influence the solution), and the reservoir is infinite in the dimension out of the plane of Fig. 1. plane of Fig. 1. Solution to the problem hinges on accomplishing representation of the unusual wedge-shaped boundaries. Superposition may be performed in the usual manner, so that "image" wells form a polygon of line source segments. The superposition is complex in all cases except where theta is an integral division of 180 deg. -i.e., except when n is an even integer, where(1)
Ref. 1 gives details of the derivation of the solution.
Briefly stated, the total dimensionless pressure drop, P (with usual definition), caused by superposition of the pressure drops due to all the single line source segments pressure drops due to all the single line source segments is given as a function of dimensionless time, t (with usual definition), by
(2)
This equation holds for either isotropic or non isotropic cases. The only difference lies in the definition of the non dimensional variables (Ref. 1). Here, r and z are the dimensionless radius and distance, respectively, from the lower end of the i th image (nondimensionalized using the wellbore radius, r ). These radii and distances may be calculated recursively from
(3) and(4)
by using simple geometry, where h = h/r and h is the formation thickness.
The integrals in Eq. 2 were calculated with a small computer program using Simpson's rule with 16 time divisions per log cycle. The short-time behavior follows the line-source solution.
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