Development and Application of New Type Curves for Pressure Transient Analysis
Abstract: Use of type curves based on the dimensionless pressure and its logarithmic derivative has become a standard method for analyzing well tests data. Recently, a new procedure for constructing type curves using a dimensionless pressure group which involves the ratio of the dimensionless pressure and its derivative was developed and reported in the literature. This paper extends the preceding concept not only to the pressure variable but also to the time variable. That is a new dimensionless time group is introduce… Show more
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Cited by 3 publications
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Summary New pressure and pressure-derivative functions for analyzing well-test data are presented. The new pressure function represents the integral of pressure with respect to time divided by time. Thus, this new pressure function represents a moving or cumulative average of the pressure over time. It is shown that the logarithmic derivative of this pressure function is equal to a moving or cumulative average of the standard logarithmic pressure derivative. It also is shown that this new derivative function is useful for analyzing well-test pressure data in cases where the point wise pressure-derivative data are difficult to interpret point wise pressure-derivative data are difficult to interpret because of noise. Introduction In recent years, the pressure derivative has become a highly popular method for interpretation of well-test pressure data. The popular method for interpretation of well-test pressure data. The main disadvantage of the pressure derivative is that one must construct derivative data by numerical differentiation of measured pressure data. The resulting pressure derivative data often are pressure data. The resulting pressure derivative data often are noisy and difficult to interpret. While various "smoothing" technique shave been used to reduce noise, some concern exists that the smoothing procedures may alter the basic character of the data. In this work, we present new pressure functions based on the cumulative average of pressure and its derivatives. This averaged pressure-derivative function is advantageous in practice because it frequently is much smoother than the standard pressure-derivative data. pressure-derivative data. The cumulative average of dimensionless pressure is given by the integral of dimensionless pressure overtime divided by dimensionless time and is referred to throughout as the integral-pressure function. Similarly, the cumulative average of the logarithmic derivative of dimensionless pressure is referred to throughout as the integral-pressure-derivative function, and the analogous average of the second derivative is referred to as the integral-pressure second-derivative function. The pressure-integral function used here differs in definition and concept from the integral used in Ref. 1 to analyze slug-test data. Specifically, in Ref. 1, we integrated the slug-test pressure change to obtain the pressure drop one would obtain for the equivalent wellbore storage and skin problem. In this paper and its supplement, we use the integral functions to obtain the same type of analyses procedures (type curve and semilog) that commonly are applied to standard well-testing problems when one can generate smooth pressure-derivative data. problems when one can generate smooth pressure-derivative data. The objectives of this work are (1) to present new type curves based on the integral-pressure function and its derivatives, (2) to illustrate the utility of these new pressure functions for identifying proper semilog straight lines, and (3) to illustrate application of the new pressure functions by applying them to two typical field examples. Ref. 2 presents two additional field examples. Similar work was done independently and concurrently by Blasingame et al., who presented a variety of type curves based on the integral-pressure and integral-pressure-derivative functions. Duong and McLauchlin also used the basic relationship between the integral-pressure function and the integral-pressure-derivative function to develop aregression-analysis procedure to estimate reservoir parameters. However, they did not pursue the use of pressure-integral and pressure-integral-derivative functions in pressure-integral and pressure-integral-derivative functions intype-curve analysis. Assumptions and Definitions Throughout this work, we consider the single-phase flow of as lightly-compressible fluid of constant viscosity to a well that is produced at a constant rate. All type curves presented in this work are based on these drawdown solutions. However, these type curves can be used to analyze buildup data provided that producing time effects are negligible or can be eliminated with producing time effects are negligible or can be eliminated with Agarwal'sequivalent time. Dimensionless pressure, dimensionless time based on wellbore radius, and dimensionless time based on fracture half-length are defined by (1) (2) and (3) respectively. Let P'wD denote the logarithmic derivative of the dimensionless pressure; i.e, pressure; i.e, (4) Let I(PwD) and I(P'wD) be the integrals of the dimensionless pressure and the logarithmic derivative of the dimensionless pressure and the logarithmic derivative of the dimensionless pressure with respect to dimensionless time, respectively, pressure with respect to dimensionless time, respectively,(5) and (6) For the classic wellbore-storage and skin problem, the dimensionless wellbore-storage constant is defined by (7) For finite-conductivity fractured wells, the dimensionless fracture conductivity is defined by (8) where kf and b are the permeability in the fracture and the width of the fracture, respectively. Basic Theoretical Equations Dividing both sides of Eqs. 5 and 6 by dimensionless time, tD, gives (9) and (10) respectively. Note that Eqs. 9 and 10 represent the correct mathematical definition of the average value of PwD and P'wD, respectively, over the time interval [0, tD]. Throughout this work, we refer to I(PwD)/tD and I(P'wD)/tD as the integral-pressure function and the integral-pressure-derivative function, respectively. SPEFE P. 135
Summary New pressure and pressure-derivative functions for analyzing well-test data are presented. The new pressure function represents the integral of pressure with respect to time divided by time. Thus, this new pressure function represents a moving or cumulative average of the pressure over time. It is shown that the logarithmic derivative of this pressure function is equal to a moving or cumulative average of the standard logarithmic pressure derivative. It also is shown that this new derivative function is useful for analyzing well-test pressure data in cases where the point wise pressure-derivative data are difficult to interpret point wise pressure-derivative data are difficult to interpret because of noise. Introduction In recent years, the pressure derivative has become a highly popular method for interpretation of well-test pressure data. The popular method for interpretation of well-test pressure data. The main disadvantage of the pressure derivative is that one must construct derivative data by numerical differentiation of measured pressure data. The resulting pressure derivative data often are pressure data. The resulting pressure derivative data often are noisy and difficult to interpret. While various "smoothing" technique shave been used to reduce noise, some concern exists that the smoothing procedures may alter the basic character of the data. In this work, we present new pressure functions based on the cumulative average of pressure and its derivatives. This averaged pressure-derivative function is advantageous in practice because it frequently is much smoother than the standard pressure-derivative data. pressure-derivative data. The cumulative average of dimensionless pressure is given by the integral of dimensionless pressure overtime divided by dimensionless time and is referred to throughout as the integral-pressure function. Similarly, the cumulative average of the logarithmic derivative of dimensionless pressure is referred to throughout as the integral-pressure-derivative function, and the analogous average of the second derivative is referred to as the integral-pressure second-derivative function. The pressure-integral function used here differs in definition and concept from the integral used in Ref. 1 to analyze slug-test data. Specifically, in Ref. 1, we integrated the slug-test pressure change to obtain the pressure drop one would obtain for the equivalent wellbore storage and skin problem. In this paper and its supplement, we use the integral functions to obtain the same type of analyses procedures (type curve and semilog) that commonly are applied to standard well-testing problems when one can generate smooth pressure-derivative data. problems when one can generate smooth pressure-derivative data. The objectives of this work are (1) to present new type curves based on the integral-pressure function and its derivatives, (2) to illustrate the utility of these new pressure functions for identifying proper semilog straight lines, and (3) to illustrate application of the new pressure functions by applying them to two typical field examples. Ref. 2 presents two additional field examples. Similar work was done independently and concurrently by Blasingame et al., who presented a variety of type curves based on the integral-pressure and integral-pressure-derivative functions. Duong and McLauchlin also used the basic relationship between the integral-pressure function and the integral-pressure-derivative function to develop aregression-analysis procedure to estimate reservoir parameters. However, they did not pursue the use of pressure-integral and pressure-integral-derivative functions in pressure-integral and pressure-integral-derivative functions intype-curve analysis. Assumptions and Definitions Throughout this work, we consider the single-phase flow of as lightly-compressible fluid of constant viscosity to a well that is produced at a constant rate. All type curves presented in this work are based on these drawdown solutions. However, these type curves can be used to analyze buildup data provided that producing time effects are negligible or can be eliminated with producing time effects are negligible or can be eliminated with Agarwal'sequivalent time. Dimensionless pressure, dimensionless time based on wellbore radius, and dimensionless time based on fracture half-length are defined by (1) (2) and (3) respectively. Let P'wD denote the logarithmic derivative of the dimensionless pressure; i.e, pressure; i.e, (4) Let I(PwD) and I(P'wD) be the integrals of the dimensionless pressure and the logarithmic derivative of the dimensionless pressure and the logarithmic derivative of the dimensionless pressure with respect to dimensionless time, respectively, pressure with respect to dimensionless time, respectively,(5) and (6) For the classic wellbore-storage and skin problem, the dimensionless wellbore-storage constant is defined by (7) For finite-conductivity fractured wells, the dimensionless fracture conductivity is defined by (8) where kf and b are the permeability in the fracture and the width of the fracture, respectively. Basic Theoretical Equations Dividing both sides of Eqs. 5 and 6 by dimensionless time, tD, gives (9) and (10) respectively. Note that Eqs. 9 and 10 represent the correct mathematical definition of the average value of PwD and P'wD, respectively, over the time interval [0, tD]. Throughout this work, we refer to I(PwD)/tD and I(P'wD)/tD as the integral-pressure function and the integral-pressure-derivative function, respectively. SPEFE P. 135
The type curves using pressure/pressure-derivative ratio (PDR) was recently introduced so that the type-curve matching requires the movement of field data only in the horizontal direction. This improvement not only facilities the type-curve matching, but also increases the reliability of analysis results. However, the uniqueness of the results from a type-curve match is still the major concern.This paper presents a new plotting technique for type curves and field data plot that will eliminate the movement of field data in both directions. The horizontal scales on both type curve and field data plots are fixed by the use of the time/pressure-derivative ratio (TOR) as a parameter in constr~cting the type curves and field data plot.The paper concludes that using the proposed type curves with PDR and TOR as parameters willgive the automatic alignments on both vertical and horizontal scales. Thus, the movement of data for type-curve matching is not needed. TheReferences and illustrations at end of paper.
SPE Members Abstract In recent years, the pressure derivative has become a highly popular method for analyzing well test data. The main disadvantage of the pressure derivative is that one must construct derivative data by numerical differentiation of measured pressure data. The resulting pressure derivative data is often noisy and difficult to interpret. While various "smoothing techniques" have been used to reduce the noise in pressure data, there exists some concern that the smoothing procedures may alter the basic character of the data. In this work, we present new pressure functions based on the moving or cumulative average of pressure and its first and second derivatives. Using the average value of the pressure derivative is advantageous in that corresponding field data is much smoother than the standard pressure derivative data. It is shown that various combinations of these functions can be used to construct new type curves and to identify proper semilog straight lines. Introduction For well over a year, we have been using analysis methods based on the integration of field data. Some of our work involving applications of integration has been reported previously in Refs. 1 and 2. Specifically in Ref. 2, we noted one could use various combinations of the moving or cumulative average value of the pressure derivative to construct type curves for all well testing problems; see, for example, Refs. 3–11. The cumulative average of dimensionless pressure is given by the integral of dimensionless pressure (with respect to dimensionless time) divided by pressure (with respect to dimensionless time) divided by dimensionless time and is referred to throughout as the integral-pressure function. Similarly, the cumulative average of logarithmic derivative of dimensionless pressure is referred to throughout as the integral-pressure-derivative function and the average of the second derivative is referred to as the integral-pressure second derivative function. As noted by Ref. 2, the new integral-pressure and integral-pressure-derivative functions, respectively, have the same basic character (curvature) as the dimensionless pressure and its logarithmic time derivative. The main advantage of using the integral functions over the the pressure and the standard pressure derivative in data analysis is that the integral functions are essentially free of the noise normally exhibited by pressure and pressure derivative data, which in turn, simplifies the interpretation of the well-test data. Ref. 1 focused on the analysis of slug test data by using integral pressure function and Ref. 2 focused on the analysis of interference test data by using type curves based on various combinations of integral pressure functions. Recently, Blasingame et al. have presented a variety of type curves based on the integral-pressure and integral-pressure-derivative functions. Ref. 12 also presented a new numerical integration scheme which is presented a new numerical integration scheme which is highly accurate especially for data from vertically fractured wells. Duong and McLauchlin also used the basic relation between the pressure-integral and the pressure-integral derivative function to develop a pressure-integral derivative function to develop a multiple regression analysis procedure for estimating reservoir parameters. While Ref. 13 did not pursue the use of pressure-integral and pressure-integral-derivative function in type curve analysis, it is clear that the authors are aware of this possibility; see Fig. 5 of Ref. 13. The objectives of this work are as follows:to present new type curves based on -integral-pressure, present new type curves based on -integral-pressure, integral-pressure derivative and integral-pressure second derivative;to illustrate the utility of these new pressure functions in identifying proper semilog straight pressure functions in identifying proper semilog straight lines;to present a method based on the integral-pressure-derivative which can be used to determine a well's drainage area from buildup data; andto illustrate the application of the new pressure functions for analyzing field data. As noted previously, Blasingame et al. have presented type curves for several problems based on the new presented type curves for several problems based on the new integral-pressure and pressure-derivative functions. In this work, type curves which have the word "new" as part of their figure captions represent type curves which were not presented in Ref. 12. presented in Ref. 12. P. 497
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