A New Vertical Interference Test Method in a Three Layer Reservoir With an Unstable Impermeable Interlayer
Authors: Xijian Tang , Shaowu Liu , Yanfeng Wang 
Published: December 2007

Abstract
This paper introduces a vertical interference test in a specific three layered reservoir. The upper and lower layers are permeable and middle layer is impermeable within the near wellbore and further more has a pinch-out extending from it or an "open window" in some regions. The numerical solution curves for this general model have been presented and are approximately equivalent to two basic models: disk pinch-out and ring window. For these two models, the paper determines the approximate analytical solutions at an early and a later time period. Based on these analytical solutions, a vertical interference test method has been designed for interpreting formation parameters. A field example is then presented. The data is analyzed. It demonstrates that this type of well testing method is applicable.

Introduction
The earliest method for determining vertical permeability using the vertical well testing method taken in the same well was presented by Burns in 1969. Later, there were a few articles published presenting various theoretical models and interpretation methods for developing a three layered reservoir from a single zone to determine the vertical permeability and cross-flow ability. Several employed the vertical interference or pulse well testing method.
For a three layer reservoir model, some provide better results, such as Bremer, Ehlig-Economides, Lee and Wijesinghe. Their models assume the interlayer is permeable and steady. Based on the hypothesis that cross-flow of interlayer is linear and two permeable zones have the same physical characteristics. Bremer and Ehlig-Economides gave an analytical solution for their model. Although the solution is reasonable, the conditions are not easily met, Ehlig-Economides extended Bremer's results and obtained an analytical solution in a Laplace space under general conditions. Likewise, they discussed the early and late approximate solutions, Lee's model was similar to theirs, but took into account effects of wellbore storage and skin. In the process of getting the analytical solution in Laplace space, he applied the supposition of linear source. However, this supposition does not produce a significant error in the results. Lee's other study was to employ a pulse test and present the pulse test curves, Wijesinghe obtained analytical solution in Laplace space on the condition that interlayer cross-flow is linear. He also took into account the effects of wellbore storage and skin of layers. Hence, his results are best to date.
This paper's model is a specific three layer reservoir in which the interlayer is unsteady. Prior studies examined situations where the interlayer is permeable. This study's interlayer is assumed to be impermeable surrounding the near wellbore and has an "'open window" in some regions or is completely pinch-out in some directions extending from the near wellbore (Fig. 1). This model comes from the separated flood development of an oil field. Realizing the separate flood (or production) is necessary to guarantee that the interlayers are impermeable and steady. During the deposit of the formation, the layer sometimes is produced pinch-out or has an "open window" in some regions. Identifying them while meeting the needs of separated flood and production is very important for improving an oilfield's development level.

Theoretical Model
let us examine the following reservoir's physical characteristics. Assume the first and third layers are homogeneous and horizontal. The second layer is impermeable, but has a vertical cross-flow within region D. The reservoir is fully saturated with a slightly compressible fluid with constant properties. The height effect is not being considered.
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