Design Of Well Tests To Determine The Properties Of Stratified Reservoirs
Authors: Dogru Ali H. , Seinfeld John H. 
Published: December 2007

Abstract
A general framework for the optimal design of well tests to determine the properties of stratified reservoirs is presented. In this framework an analytical method for calculating the sensitivity coefficients, pw/ bi, where pw is the wellbore pressure, and bi is any reservoir parameter, is pressure, and bi is any reservoir parameter, is presented. The only approximation involved is spatial presented. The only approximation involved is spatial discretization of the reservoir. The method is applied to the computation of the effect of radial and vertical permeability distributions on the wellbore pressure for permeability distributions on the wellbore pressure for a cylindrically symmetric formation with a centrally located well. Design variables are taken to be the vertical location of the pressure measurement in the wellbore, the fractional flow time, tf/T in a drawdown/ build-up test, and the flow configuration at the well. It is demonstrated that situations with restricted flow entry or injection and production provide the most information on reservoir properties, and that the optimum values of tf/T range from 0.5 to 0.75 depending on the nature of the test.

Introduction
Vertical reservoir heterogeneities play an influential role in the effectiveness of enhanced recovery operations. Since coring and logging information does not generally provide sufficient detail concerning the properties of the provide sufficient detail concerning the properties of the drainage area, drillstem tests are often used to estimate reservoir properties in the vicinity of the wellbore. There exist several prior studies on the use of drillstem tests to determine the structure of stratified reservoirs. Vertical reservoir heterogeneities have traditionally been represented by horizontal layers or strata within which flow may be strictly horizontal or horizontal and vertical, the two situations being referred to as non-communicating or communicating layers, respectively.
Most of the prior studies aimed at estimating the properties of stratified reservoirs have been based on properties of stratified reservoirs have been based on situations for which analytical solutions for the wellbore pressure can be obtained. On the basis of an analytical pressure can be obtained. On the basis of an analytical solution for the wellbore pressure, Lefkowitz et al. showed that the properties of non-communicating, stratified reservoirs could not be determined from the results of pressure build-up tests alone. Russell and Prats considered a stratified, communicating reservoir and showed that its early transient behavior was the same as that of the non-communicating system. An analytical solution of this problem was also investigated by Katz and Tek. On the basis of a numerical study, Pendergrass and Berry concluded that, except for the very early flow period, the transient performance of a stratified reservoir period, the transient performance of a stratified reservoir was essentially the same as that of a homogeneous reservoir having the same permeability-thickness product. Several authors have considered the estimation of radial and vertical permeabilities for cylindrical reservoirs. Prats and Hirasaki permeabilities for cylindrical reservoirs. Prats and Hirasaki suggested injection tests to estimate vertical permeability. Burns employed both analytical and numerical solutions to estimate vertical permeability. He designed a test procedure based on injection and partial well shut in and investigated the effects of casing leaks and damage zone on the estimates derived from the tests. Kazemi and Seth used numerical solutions to examine the effect of restricted flow entry into the well on the pressure behavior at the wellbore.
The object of this work is to develop a general theory for the design of well tests for determining the properties of stratified reservoirs. We confine our attention in this work to reservoirs that can be represented as containing a single fluid of small and constant compressibility. However, we seek a method that is applicable to reservoirs of arbitrary geometry and distribution of properties. We begin by defining the model of a radially symmetric, cylindrical reservoir with a centrally located well that has an arbitrary spatial distribution of permeability and porosity. By discretizing the spatial domain of the reservoir and solving the resulting set of ordinary differential equations for the grid block pressures we obtain an analytical solution for the grid block pressures we obtain an analytical solution for the grid block pressures under arbitrary flow conditions. Sensitivity pressures under arbitrary flow conditions. Sensitivity coefficients relating grid block pressures and reservoir properties are introduced and calculated from the analytical properties are introduced and calculated from the analytical pressure solution. pressure solution.