Effect of Relative Permeability and Mobility Ratio on Pressure Falloff Behavior
Authors: Sosa A. , Raghavan R. , Limon T.J. 
Published: December 2007

Summary
This study examines the influence of saturation gradients on pressure falloff data by rigorously accounting for the relative permeability characteristics of the porous medium. The study examines the significance of the various straight-line segments that appear when results are analyzed on semilogarithmic paper as a function of the relative permeability characteristics of the porous medium, injection rate, injection time, and distance to the external boundary. Conditions necessary to obtain the mobilities of the completely flooded region, the transitional (oil/water) zone, and the undamaged region are prescribed. The applicability of the methods suggested in the literature to locate the radii of fluid banks was investigated.

Introduction
Virtually all studies pertaining to well test analysis for multiphase flow through porous media have considered only the simultaneous flow of oil and gas. The flow of oil, gas, and water, or oil and water, on the pressure behavior of a well has not been investigated. This paper examines and documents the pressure behavior at a well due to the simultaneous flow of oil and water in the reservoir. Specifically, we examine the pressure behavior during falloff tests. Falloff tests are the counterpart of pressure buildup tests and are used for the same purposes. Since the fluid injected prior to a falloff test is different from the displaced fluid, a saturation gradient is established.The first objective of this work is to examine the effect of saturation gradients on pressure falloff tests. The second objective is to investigate the applicability of models which have been proposed in the literature to describe the movement of fluid banks. These models assume that the various fluid banks, which develop as a result of fluid injection, can be represented by composite zones. Such models implicitly assume that no saturation gradients exist within each zone and that an abrupt change in mobility occurs at the interface of each zone. In this study, these assumptions are avoided since the fluid distribution in the reservoir is governed by the relative permeability characteristics of the reservoir rock. All of the results presented here were obtained with the aid of a two-phase, numerical (finite difference) model.

Background
Fig. 1 presents a schematic diagram of a model that has been considered in the literature. As a result of water injection, three zones containing oil, gas, and water have formed within the reservoir. This figure describes conditions prior to fillup. (For conditions after fillup, the gas saturation would be zero.) The effect of the unflooded region on the pressure response at the well is neglected in most studies.All works cited previously assume that the diffusivity equation is obeyed in each zone and that a constant saturation exists within each zone which changes abruptly at the interface. Also, none of the studies specify how one should approximate the transitional region that would exist between the oil and water banks. Finally, the definition of the distance from the well to a bank - i.e., whether it includes or excludes the transitional region - always has been open to speculation.More recently, Weinstein, using a numerical model, analyzed the effect of "cold" waterflooding a "hot" reservoir.
JPT
P. 1125^