Analysis of Pressure Data From a Restricted-Entry Well in a Multilayer Reservoir

Abstract

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.

Computation of the Pseudoskin Factor Caused by a Restricted-Entry Well Completed in a Multilayer Reservoir

“…Although the results of Ref. 8 show that cases exist where Eq. 37 gives slightly more accurate results than Eq.…”

“…We letkw denote the thickness-averaged horizontal ~rmeability (based on hw) adjacent to the open interval. In general, kw is given by The dimensionless flow capacity of the open interval is denoted by II and is defined by (9) where Sb =pseudoskin factor caused by restricted entry and s=real skin factor reSUlting from damage or stimulation. The validity and interpretation of Eq.…”

“…Here, only the final approximations for sb are presented; derivations of these solutions are given in Ref. 8.…”

A New Approach for Constructing Derivative Type Curves for Well Test Analysis

“…During the time that the wellbore pressure drop is controlled by wellbore storage effects, it is well known (see Refs. 4,9,10, and 17) that Similarly, it is also well known that when wellbore storage effects become negligible, the dimensiunless pressure drop is given by the standard semilog equation; i.e., (8) where s is the skin factor. From fundamental results of Earlougher and Kersch 17 and Refs.…”

“…7 presented conceptually similar type curves for finite-conductivity fractures. Yeh and Reynolds 8 presented both pressure and pressure-derivative type curves for analyzing pressure data obtained at a restricted-entry well. Typically, these type curves represent a graph (usually log-log) of dimensionless pressure (or a dimensionless pressure group) and the derivative of dimensionless pressure (or a constant times the derivative of dimensionless pressure) vs. dimensionless time (or a dimensionless time group).…”