Injection/Falloff Testing in Heterogeneous Reservoirs
Authors: Banerjee R. , Thompson L.G. , Reynolds A.C. 
Published: December 2007

Abstract
In this work we present a new theoretical understanding of pressure data obtained at a water injection well. The theory provides new physical insight on how permeability heterogeneity, saturation gradients and mechanical skin factor combine to influence the pressure response at the well. Based on the theoretical equations, methods for analyzing injection/falloff pressure data in both homogeneous and radially heterogeneous reservoirs are presented. For homogeneous reservoirs, we present procedures for estimating the mechanical skin factor from injection or falloff pressure data. Our theory provides a procedure for analyzing the pressure response during the second injection period of a two-rate test. It is shown that the information that can be obtained from a two-rate test is similar to that obtained from a falloff test.

Introduction
Injection testing is pressure transient testing during injection of a fluid into a well. It is analogous to drawdown testing for both constant and variable rates. Shutting in an injection well results in a pressure falloff which is similar to pressure buildup in a production well. However, the distinction between injection/falloff and conventional drawdown/buildup testing is that the flow characteristics of the injected fluid are different from those of the original reservoir fluids so that multiphase reservoir flow has to be considered in order to understand these tests.
A novel insight into the theory of multiphase flow pressure transient testing was presented by Thompson and Reynolds. Their theory describes the averaging process that occurs during multiphase flow drawdown and buildup and explains pressure transient behavior of both single and multiphase flow in radially heterogeneous reservoirs. Although the focus was on gas condensate reservoirs, their studies included injection/falloff testing. In summary, the theory stated that well test mobilities reflect weighted average mobilities in those regions of the reservoir where rate is changing with time and where mobility is changing with time, i.e.,
(1)
where, C1 is a units conversion constant, (see Nomenclature), KR is a "rate kernel", defined as, and KM is a "mobility kernel" defined as.
In the case of injection/falloff testing, they argued that if changing mobility were the dominant factor occurring during the injection phase, it could explain the common belief that it is impossible to see beyond the "flood front" during injection. However, further numerical experiments on injection testing indicated that there were some cases where permeability beyond the flood front could affect injection-well pressure data.
In this work, we investigate this apparently anomalous behavior carefully, and show that for injection/falloff testing, multiphase flow pressure derivative data yield information both about the reservoir region close to the moving flood front and the unflooded zone ahead of the front.

Approximate Analytical Injection Solution
In this section, we derive an approximate analytical solution for water injection in a radially heterogeneous reservoir. We assume an infinite cylindrical reservoir with a fully penetrating injection well of radius rw at the center of the reservoir. Wellbore storage effects are neglected. Water is injected into the reservoir at a constant rate, qwBw RB/D. Except for connate water, the reservoir is initially assumed to be filled with oil of a small and constant compressibility. The reservoir is made up of N+1 concentric cylinders having radii of r1, r2, … rN, with corresponding permeability values of k1, k2,…,kN, kN+1 respectively.
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