Buildup Analysis for Interference Tests in Stratified Formations
Authors: Streltsova T.D. 
Published: December 2007

Summary
Interlayered crossflow modifies the responses of both the producing layer and the supporting less-permeable layer in such a manner that the conventional analysis based on the Ei type-curve graphical method becomes no longer valid. A forced match of pressure data from a stratified formation to the Ei type curve results in overestimation of both the formation storage and the permeability. Type curves that take into account the interlayered crossflow are developed and applied to analyze buildup data from interference tests at a water injectivity test site, Prudhoe Bay, AK.

Introduction
Vertical heterogeneity formed by a depositional sequence of interbedded, fine-grain sandstone and shale is a characteristic of the Prudhoe Bay reservoir. For example, Fig. 1 gives the shale correlation diagram developed from well logs for a Prudhoe Bay field water injectivity test site. 1 The cross sections show the presence of shaly sands of various thickness and continuity that overlie and underlie the field producing intervals. The extent and vertical permeability of these shaly sands will determine whether the reservoir permeable zones are in vertical hydraulic communication with the producing interval. If shales are continuous and have a negligible permeability, they form a barrier practically impermeable to flow. If, however, shaly sands that separate permeable reservoir layers have a measurable permeability or if they are discontinuous, then the production of an intermediate permeable zone will be enhanced by induced flow from the adjoining zones. The time-variant crossflow that augments the flow of a producing zone is dependent on the shaly sands' "leaking" properties-thickness, permeability, and the pressure difference across the sands-and on the amount of fluid that may be released from the reservoir volume extended by the contributing formations. Neither the producing zone with augmented flow nor the supporting zone with induced crossflow will, however, have the purely radial flow pattern implicit in any conventional analysis method. Customarily used buildup analyses-namely, those based on application of the Ei type curve for interference well testing and on the Homer analysis technique for single-well testing-are developed strictly for reservoirs with radial flow to a well. Therefore, if one applies the conventional analysis technique to pressure data measured in a producing zone "enlarged" by supporting flow formations, the formation permeability so determined will be overestimated by an amount proportional to the formation flow enhancement. Application of the Ei type-curve methods to a supporting zone with induced vertical flow impaired by low-permeability shaly sands is even less justified, since it is only radial flow that the exponential integral pressure curve describes.
Recognition of alterations produced by crossflow is important for interpretation of pressure data, expecially if one analyzes the responses of a reservoir during a limited time of well testing. Here, a transient flow behavior is considered for both producing, and supporting layers when the interlayered crossflow is taken into account. The analysis of crossflow is made for two different assumptions: when it is assumed to be permeability controlled only and when it results front a diffusion mechanism of fluid transfer through the lowpermeability supporting zone. The first, the permeability-controlled or lumped-parameter crossflow assumption, implies that the entire storage of the supporting zone contributes to the crossflow equally at any time, without adding any time dependency to the flux resulting from vertical position of the point of observation. The second the diffusivity-controlled or distributed-parameter crossflow, implies that the amount of storage contributing to the crossflow increases with time starting with that adjacent to the layer interface and including finally the entire storage of the lowpermeability layer at pseudosteady state. The resulting pressure behavior, therefore, will be dependent on position as well as time, the time variation at any given position as well as time, the time variation at any given position being different from that predicted by the lumped-parameter approach. Both assumptions are used extensively in hydrogeology literature in developments of leaky aquifers, confined two-aquifer system, unconfined flows, and aquifer-aquitard systems. Both approaches are in use as well in the flow description of naturally fractured reservoirs. Quantitative aspects of transient flow behavior under the two different assumptions on crossflow are discussed in detail elsewhere. Here, an extension is made to a stratified reservoir, the responses of which are affected by crossflow.

Pressure Behavior of a Producing Zone Augmented by Crossflow
Suppose that the shaly sections overlying or underlying the producing reservoir layers do not form absolute flow barriers but have a certain permeability. Selective reservoir production will then cause a differential pressure depletion and a fluid flow from (or through) the adjoining low-permeability beds.
JPT
P. 301^