This work considers flow to a restricted-entry well in a single-or multilayer reservoir where crossflow occurs between reservoir layers. A simple algebraic expression for approximating the pseudoskin factor caused by restricted entry is presented. It is shown that, for practical purposes, this equation yields sufficiently accurate estimates of the pseudoskin factor regardless of the location of the open interval. This work also extends single-layer results of interest in well testing to the multilayer case and shows that if sufficient data are available, the results of this work can be used to prepare a complete analysis of well testing pressure data obtained at a restrictedentry well.

Summary This work presents new semilog type curves for analyzing data obtained at a restricted-entry well completed in a multilayer reservoir where crossflow occurs between reservoir layers. Use of the new type curves along with standard semilog analysis of pseudoradial-flow pressure data makes possible a complete analysis of pressure data. Specifically, standard semilog analysis of pseudoradial-flow pressure data yields estimates of the total flow capacity and the total skin factor. The new type curves can then be used to estimate the real mechanical skin factor resulting from damage or stimulation, the flow capacity adjacent to the open (perforated) interval, and the effective average vertical permeability. Introduction Wells are often completed over only a fraction of their productive zone to delay water and/or gas coning. A well completed in this way is referred to as a restricted-entry or partially penetrating well. This work considers the analysis of well test pressure data from a restricted-entry well completed in a multilayer reservoir where crossflow occurs between reservoir layers. The well is assumed to be unfractured, so flow occurs only in the radial and vertical directions. It is well known that pressure data obtained at a restricted-entry well may exhibit two semilog straight lines.1–7 The first (early-time) semilog straight line reflects plane radial flow in the part of the reservoir adjacent to the open (perforated) interval of the wellbore. The second semilog straight line reflects pseudoradial flow. Given a set of pressure data that exhibits both semilog straight lines, Bilhartz and Ramey1 showed that a complete analysis is possible. Ref. 1 considers only a single-layer reservoir, but Refs. 3 and 4 extended the results to the multilayer case. Unfortunately, the first semilog straight line seldom is observed in practice because it usually occurs too early to be clearly identified or is obscured by wellbore storage effects. Thus, in practice, the analysis of pressure data is usually based on pseudoradial-flow data. Standard semilog analysis of pseudoradial-flow data yields estimates of the total flow capacity, thickness-averaged permeability, and total skin factor,1–7 which is a linear combination of the real skin factor caused by damage or stimulation and the pseudoskin factor caused by restricted entry. Thus, to obtain an estimate of the real skin factor resulting from damage or stimulation, one must be able to obtain an independent estimate of the pseudoskin factor. Refs. 8 through 12 present procedures for estimating the pseudoskin factor for single-layer reservoirs, and Reynolds et al.7 presented a procedure for a special two-layer case. Refs. 3 and 4 recently presented a simple equation for estimating the pseudoskin factor caused by a rericted-entry well in a multilayer reservoir. Unfortunately, using any of the equations or graphs presented previously for estimating the pseudoskin factor requires an independent estimate of average vertical permeability, which is seldom available in practice. This work presents new semilog type curves for analyzing well test pressure data from a restricted-entry well draining a multilayer reservoir. By using the type curve in conjunction with standard semilog analysis, one can obtain estimates of all key parameters. Mathematical Models and Basic Definitions We consider flow to a restricted-entry well located at the center of a bounded, cylindrical, multilayer reservoir. The pressure is initially assumed to be uniform throughout the reservoir and equal to pi. Wellbore storage and gravity effects are assumed to be negligible. The well is produced at a constant total sandface rate. The reservoir consists of n horizontal layers. Each layer is assumed to be homogeneous and of uniform thickness. The top, bottom, and outer reservoir boundaries are sealed and thus represent no flow boundaries. The specific results presented here assume single-phase flow of a slightly compressible fluid of constant viscosity; however, the results of Ref. 7 suggest that our results may be applicable to multiphase-flow problems. This work presents new type curves for analyzing pressure data from restricted-entry wells. The solutions used for generating these type curves were obtained by solving the relevant initial-boundary problem using a finite-difference model. Tihs numerical model has been used extensively in previous well testing research and yields highly accurate numerical solutions.3,4,7,13–15 For the layered-reservoir problem considered here, it is appropriate to define the thickness-averaged horizontal permeability and the thickness-averaged porosity-compressibility product byEquation 1and Equation 2 respectively, where j=layer index (reservoir Layer j), n=number of layers, and h=total reservoir thickness. The dimensionless wellbore pressure drop and dimensionless time based on average properties are defined byEquation 3and Equation 4 respectively, where q is the total sandface rate of production in reservoir barrels per day. (All definitions used in this work are given in terms of customary units.) As is well known,1–7 well test pressure data obtained at a restricted-entry well may exhibit two semilog straight lines. The first semilog straight line reflects radial flow in the part of the reservoir adjacent to the open (perforated) interval of the wellbore. In considering this early-time semilog straight line, we find it appropriate to base our definitions on the properties in the part of the reservoir adjacent to the open interval. In this regard, we let kw denote the thickness-averaged horizontal permeability (based on hw) in the part of the reservoir adjacent to the open interval. The general definition of k is given byEquation 5 where hw is the length of the open interval and the sum is over all layers at least partly open to flow. For example, for both cases shown in Fig. 1, no=1, mo=2, and kw=(k1hw1+k2hw2)hw, where hw1 and hw2 are as shown in Fig. 1. In general, hwj denotes the length of the part of the open interval adjacent to reservoir Layer j. For example, for Case B of Fig. 1, hw1=h1 but hw2<h2. For cases where only one layer is open to flow with the thickness of the open layer equal to the length of the perforated interval of the wellbore, kw is equal to the permeability of the open layer. Such cases are shown in Figs. 2 and 3; for these two systems, kw=k1.

This paper presents new methods to analyze fractured well responses in heterogeneous reservoirs. We consider wells producing formations that are naturally fractured and use the idealizations proposed by Warren and Root (pseudosteady-state flow in the matrix-system) and by deSwaan-O (unsteady-state flow in the matrix-system) to model the naturally fractured reservoir. Pressure responses are correlated in a manner suitable for direct application of field data. Methods to determine fracture half-length are presented. Two field applications are discussed. The consequences of neglecting the heterogeneous character of the porous medium are also discussed.

Summary This work presents and illustrates a new general procedure for constructing type curves on the basis of a new pressure-derivative group that involves the dimensionless pressure change divided by its logarithmic derivative. When type curves based on this group are constructed, the vertical scale of the dimensionless derivative group is automatically aligned with the vertical scale of the field-data derivative group. This automatic alignment of the vertical scales simplifies type-curve matching and is advantageous for determining whether field data actually represent the solution assumed by a given type curve. It is also shown that the basic idea used in constructing the type curves can be used to ensure that proper semilog straight lines are chosen when well-test pressure data are analyzed by semilog methods. Introduction To the best of our knowledge, type curves based on pressure derivatives were first presented in the petroleum engineering literature by Tiab and Crichlow. Ref. 2 used the approach of Ref. 1 to present type curves based on the derivative of dimensionless pressure with respect to dimensionless time for analyzing well-test data obtained at an interference well. The most commonly used pressure-derivative type curves (e.g., Refs. 3 through 5), however are based on the ideas of Bourdet et al., who presented type curves based on the derivative of dimensionless pressure with respect to the logarithm of dimensionless time. This derivative is referred to throughout as the "logarithmic derivative of dimensionless pressure." The wellbore-storage and skin type curves of Ref. 3 represent log-log plots of both dimensionless pressure and its logarithmic derivative vs. dimensionless time divided by the dimensionless wellbore-storage constant. Type curves based on the Bourdet et al. idea have two main advantages over conventional type curves based on a log-log plot of only dimensionless pressure vs. dimensionless time. First, a plot of the logarithmic derivative of dimensionless pressure often has more character (curvature) than a plot of dimensionless pressure; hence, it is easier to obtain a unique type-curve match with derivative type curves. Second, such type curves usually involve a graph of both the dimensionless pressure and the derivative of dimensionless pressure; thus, in analyzing field data, one must match simultaneously both the measured pressure change and the derivative of the measured pressure change, which further enhances the likelihood of obtaining a unique type-curve match and a consistent analysis. An alternative method for constructing type curves on the basis of pressure derivatives was presented by Onur and Reynolds. This method involves plotting both dimensionless pressure and dimensionless pressure divided by the logarithmic derivative of dimensionless pressure vs. dimensionless time. With the new pressure/ pressure-derivative group, Ref. 6 generated new type curves for the classic wellbore-storage and skin problem and for uniform flux and infinite-conductivity vertically fractured wells. Duong also applied the same pressure/pressure-derivative group to prepare new wellbore storage and skin type curves, and Ozkan et al. used this group to construct type curves for horizontal wells. This work presents new type curves for finite-conductivity vertically fractured wells. These type curves are based on the same pressure/pressure-derivative group used by Onur and Reynolds. In addition, it is shown that the basic idea of plotting pressure change over its logarithmic derivative can be used to identify proper semilog straight lines when field pressure data are analyzed by semilog methods. Four example applications are presented. Basic Definitions The dimensionless pressure drop, dimensionless time based on wellbore radius, and dimensionless time based on fracture half-length are defined, respectively, by (1) (2) (3) Delta pwD' denotes throughout the logarithmic derivative of dimensionless pressure defined by (4) From the chain rule, it follows that (5) For finite-conductivity fractures, the dimensionless fracture conductivity is defined by (6) where kf and b denote the permeability in the fracture and the width of the fracture, respectively. For the classic wellbore-storage and skin problem, the dimensionless wellbore-storage constant is defined by (7) where W denotes the wellbore-storage coefficient in RB/psi. Basic Theory for New Type Curves From Eq. 5, it follows that the constants making time (in hours) dimensionless may be canceled from the logarithmic derivative; thus, (8) We let delta p (in psi) denote the pressure change during the test (for drawdown delta p=pi-pf) and define delta p' by (9) where the second equality of Eq. 9 follows from the chain rule. From Eqs. 1 through 5, 8 and 9, it follows easily that (10) i.e., the vertical scale on a plot of delta p/(2 delta p') vs. t is identical to the vertical scale on a plot of delta pwD/(2 delta pwD') vs. tD and thus, when type curves based on the derivative group on the left side of Eq. 10 are used, type-curve matching of the derivative data will require moving the field data only in the horizontal direction. This is one major advantage that the new type curves presented here have over conventional type curves based on pressure derivatives, which require both horizontal and vertical movement of the data plot to obtain a type-curve match. Because the vertical scale of derivative type curves based on the derivative group on the left side of Eq. 10 is always automatically aligned with the vertical scale on a plot of delta p/(2 delta p'), such type curves should also be advantageous in determining whether given field data are representative of the type curve chosen for analysis; i.e., a log-log plot of delta p/(2 delta p') vs. t should prove useful for determining which model (type curve) should be used for analysis. SPEFE P. 429^

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