Copyright 2002, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 29 September–2 October 2002. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract This paper presents a practical and useful analysis technique for effectively accomplishing a rigorous superposition-in-time (convolution) analysis of the transient drawdown performance of oil and gas wells that does not require that the flowing pressures (wellhead or bottom hole) be known for every flow rate data point in the production history. If at least one flowing pressure point is available at any point in time during the production history of the well, a unique analysis of the production data for the evaluation of the reservoir effective permeability, well drainage area, and well parameters such as unfractured well radial flow steady-state skin effect, fractured well effective fracture half-length and average fracture conductivity, or horizontal well effective length in the pay zone may be obtained. Even if no flowing pressures are available at all during the production history of the well, estimates of these well and reservoir parameters can still be obtained with reasonable accuracy using the analysis techniques presented in this paper. The general evaluation procedures presented in this paper are applicable for all well types, including flowing wells and wells with artificial lift systems (i.e., conventional beam pumping units and electrical submersible pumps (ESP)). The theoretical basis for a rate-transient analysis of the production performance of a vertically fractured well is presented in detail in this paper. Application of the proposed general analysis procedure to other well types (unfractured vertical/slanted wells, horizontal wells, etc.) may also be found in a similar manner as presented in this paper. The theoretical basis for a pressure-transient based solution for a vertically fractured well is also available in the references. In the relatively short period of time that this new analysis technique has been available, it has radically changed the way

that engineers may now rigorously analyze the production performance of oil and gas wells to evaluate the well and reservoir properties from the well’s production performance, as well as greatly expanding the number and types of wells that can now be considered. The new analysis techniques presented in this paper therefore represent a step-change in the petroleum industry’s technical capabilities for the analysis of the production performance of oil and gas wells over that which existed prior to this work. Introduction One of the most common problems encountered by a production or reservoir engineer when analyzing an oil or gas well’s production history to evaluate the formation and completion system properties (by inversion of the production history record of the well), is the lack of a complete data record with which to employ a conventional convolution analysis. The variable that is most commonly not recorded that is required for a conventional convolution analysis of the production history of the well is the well flowing pressure. An analysis technique is presented in the following sections of this paper that will remove this limitation in a convolution analysis of the production data record of a vertically fractured well. Similar analysis techniques have also been developed for other types of wells1. The flow rates of the hydrocarbon phases (oil and gas) of a well are generally known with a reasonably high degree of accuracy, since the oil and gas production directly correspond to revenue for the operator and royalty owners. Custody transfers also dictate that the production of these fluids be monitored as accurately as possible. The exception to this general statement is the case where the production of several wells are processed at a single separation facility and the actual production of any single well in the system may be estimated from the combined system production with periodic tests of the individual wells in the system. Often known with a much lower degree of certainty is the water production from the well, since this variable generally represents an operating cost of the well, it is commonly not recorded or reported with a great deal of precision. Often the most reliable way of ascertaining a representative water production history for a producing well is through the use of run tickets for the loads of water transported to disposal wells. The effect of a varying flow rate and sandface flowing pressure history of a well on its dimensionless wellbore

SPE 77691

Effective Well and Reservoir Evaluation Without the Need for Well Pressure History B. D. Poe Jr., SPE, Schlumberger

2 B. POE SPE 77691

pressure solution at a point in time of interest has been established with the Faltung Theorem as employed by van Everdingen and Hurst2. The general form of the well-known convolution relationship that accounts for the superposition-in-time effects of a varying sandface pressure and flow rate on the dimensionless wellbore pressure transient behavior of a well is given by Eq. 1.

( ) ( ) ( )'

0

D

wD D D D D

t

p t q p t dτ τ τ= −∫ (1)

The pressure transient behavior of a well with a varying flow rate and flowing pressure history can be readily evaluated using Eq. 1 for specified terminal flow rate (Neumann) inner boundary condition transients such as constant flow rate drawdown or injection transients, or shut-in well sequences such as pressure buildup or falloff transients. The development of a pressure-transient based analysis that is analogous to the rate-transient analyses presented in the remainder of this paper may be found in Ref. 1. As fully discussed in Ref. 3, the most appropriate inner boundary condition solution to use for the analysis of the production history of a well corresponds to that of a specified terminal pressure (Dirichlet) inner boundary condition. Therefore, further discussion of the pressure-transient solution based analyses will be abbreviated at this point and an example of its application for the analysis of a vertically fractured well is provided in Ref. 1 for the sake of completeness only. It has been included in this discussion simply due to its historical significance and to demonstrate the similarity of the form of the two relationships. More importantly, one thing that the practicing engineer should be cognizant of is the fact that the dimensionless flow rate and pressure appearing in Eq. 1 do not equate to the dimensionless flow rate and pressure of the rate-transient based analyses presented in the balance of this paper. Equation 2 gives the corresponding dimensionless rate-transient behavior of a well with a varying flow rate and sandface flowing pressure history3.

( ) ( ) ( )'

0

D

wD D D D D

t

q t q t p dτ τ τ= − −∫ (2)

With a substitution of variables, the rate-transient convolution integral can be written in the more readily amenable form presented in Eq. 3.

( ) ( ) ( )'

0

D

wD D D D D

t

q t p q t dτ τ τ= − −∫ (3)

Using either the pressure-transient (Eq. 1) or rate-transient (Eq. 3) convolution integrals to account for the varying flow rate and sandface pressure history of a well, a discrete time approximation of the convolution integral can be derived to permit the analysis of a varying flow rate and sandface pressure production history. The corresponding rate-transient convolution integral approximation of the dimensionless well flow rate is given in Eq. 4.

( ) ( ) ( )1

111

n

wD Dn Di D Dn Di D Dn Diin

q t p q t t q t t−

−

=>

= − − − ∑

( )1D Dn Dnq t t −+ − (4) The corresponding rate-transient solution dimensionless cumulative production of a well with a varying flow rate and sandface pressure production history can also be evaluated using a discrete time approximation3 with Eq. 5.

( ) ( ) ( )1

111

n

pD Dn Di pD Dn Di pD Dn Diin

Q t p Q t t Q t t−

−

=>

= − − − ∑

( )1pD Dn DnQ t t −+ − (5) The dimensionless pressure appearing in the superposition-in-time relationships of Eqs. 4 and 5 is defined for oil and gas reservoirs with Eqs. 6 and 7, respectively.

( )( )

i wf iDi

i wf n

p p tp

p p t−

=−

(6)

( ) ( )( )( ) ( )( )

p i p wf iDi

p i p wf n

p p p p tp

p p p p t−

=−

(7)

The well bore dimensionless flow rate is defined for oil and gas reservoirs in conventional oilfield units with Eqs. 8 and 9, respectively.

( )( )

141.205 o o owD

o i wf

q t Bq

k h p pµ

=−

(8)

( )( ) ( )( )

50299.5 sc gwD

g sc p i p wf

p T q tq

k h T p p p p=

− (9)

The dimensionless cumulative production of oil and gas reservoirs is defined in conventional oilfield units as given in Eqs. 10 and 11, respectively.

( ) ( )( )( )21.11909

p n opD n

t c i wf n

N t BQ t

c h L p p tφ=

− (10)

( ) ( )( ) ( ) ( )( )( )2

318313 sc p npD n

g t n sc c p i p wf n

p T G tQ t

h c t T L p p p p tφ µ=

− (11) The dimensionless time that corresponds to a given value of dimensional time ( )nt for oil and gas reservoir analyses is defined with Eqs. 12 and 13, respectively.

( ) 2

0.000263679 o nD n

o t c

k tt tc Lφ µ

= (12)

( ) ( )2

0.000263679 g a nD n

c

k t tt t

Lφ= (13)

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 3

The system characteristic length ( )cL appearing in Eqs. 10 through 13 is dependent upon the system under consideration. In an unfractured vertical well analysis it is equal to the wellbore radius (half the wellbore diameter, not necessarily or generally equal to the hole size). An apparent (or effective) well bore radius is also commonly used as the system characteristic length in unfractured vertical well decline analyses, particularly in cases where the well has been stimulated to improve its productivity, resulting in a negative steady state skin effect. In this case, the apparent well bore radius (or the system characteristic length) is simply the well bore radius multiplied by the exponential function value of the negative of the steady state skin effect. In a vertically fractured well analysis, the system characteristic length is the fracture half-length, or half of the total effective fracture length in the system. Similarly, in a horizontal well analysis the system characteristic length is equal to half of the total effective wellbore length in the pay zone. The evaluation of the pseudotime integral transformation is addressed in detail in Ref. 4 and a detailed discussion of this topic need not be given in this paper. However, suffice it to say that significant care must be employed in low-permeability gas reservoir analyses so that this integral transformation is accurately and properly evaluated. With these rate-transient analysis fundamental relationships established, we turn our attention next to the development of a practical means of estimating the superposition-in-time function values of production history data points for which the flowing sandface (or wellhead) pressure are not available. For a production history data point that has the flowing wellhead pressure and well flow rates recorded, the corresponding bottom hole wellbore and sandface flowing pressures can be readily be estimated using the industry-accepted wellbore pressure traverse and completion pressure loss models documented in Ref. 5. When the wellhead flowing pressure is not available at a production data point, and bottom hole pressure measurements are also not available, a conventional convolution analysis of the type prescribed by Eqs. 4 and 5 is not possible without guessing (or in some way roughly estimating) what the missing sandface flowing pressure should have been at that point in time in the production history. Palacio and Blasingame6, using the “material balance” time function that was derived and first used by McCray7, proposed an alternative solution to this problem. The “material balance” equivalent time function casually appears similar to the Horner approximation commonly used for the evaluation of the pseudo-producing time of a smoothly varying flow rate history in pressure buildup analyses. From pressure-transient theory, Palacio and Blasingame6 established that during the pseudo-steady state flow regime (fully boundary dominated flow in a closed system), that the “material balance” time function is exactly equal to the rigorous superposition-in-time relationship for the pressure-transient solution of a varying flow rate history.

For rate-transient analyses, we can also define a “material balance” time approximation for oil reservoir analyses that is identical in general form as the “material balance” time function reported by Palacio and Blasingame6. This relationship is expressed mathematically in Eq. 14 for rate-transient oil reservoir analyses. In the rate-transient case, the exact relationship between the flow rate and cumulative production functions change with each flow regime as a function of time.

( ) ( )( )

24 p nmb n

o n

N tt t

q t= (14)

Employing an equivalent “material balance” time function analogous to that presented in Ref. 6 for pressure-transient analyses (instead developed for rate-transient analyses of the production performance of gas reservoirs), we define a “material balance” time function for gas reservoir analyses that is given by Eq. 15.

( ) ( )( )

24000 p namb n

g n

G tt t

q t= (15)

While the “material balance” time function has been shown to have a theoretical basis for the pressure-transient behavior of a well during the pseudosteady state flow regime, it is not correct for any other pressure-transient flow regime, and is not correct for any rate-transient flow regime1. Other authors8,9

have also mistakenly adopted and used the “material balance” time function, with only a cursory review of its applicability and with no correction or modifications, for the analysis of the production performance of all other flow regimes in wells besides the pseudosteady-state flow regime. One very important fundamental inconsistency exists in all of the work reported in Refs. 8 through 13. The inconsistency exhibited in those references is that they all use the “material balance” time function (derived from pressure-transient theory for only the pseudosteady state flow regime) to evaluate the rate-transient performance of wells. All of the referenced analyses utilize the conventional flow rate decline type curve (rate-transient) solutions in some form to evaluate the production behavior of oil and gas wells. As clearly demonstrated in Ref. 1, the uncorrected “material balance” time function is not correct for any rate-transient solution flow regime, not even for fully boundary-dominated flow. The analyses presented in this paper are internally consistent in that they use a “material balance” time function derived directly from rate-transient theory and also use the appropriate rate-transient solutions for all of the analyses. No other previous publication has provided a truly completely consistent and proper analysis methodology for the analysis of production performance data of oil and gas wells using the “material balance” time function. Agarwal et al8 have also erroneously reported that the rate-transient and pressure-transient solutions are equivalent. They are not equivalent in any flow regime, and the proof of this fact is documented in Ref. 1. Agarwal et al8 reported numerical simulation results of a comparison between the

4 B. POE SPE 77691

“material balance” time function and the equivalent superposition-in-time function. An example of this comparison is given for a vertically fractured well in Fig. 1. Figure 1 appears rather benign and innocuous at first sight, and does appear to follow a generally linear trend. However, replotting the same data as the ratio of “material balance” time to the equivalent superposition time function, as a function of the equivalent superposition time as in Fig. 2 gives the true picture of what is actually happening. The results presented in Figs. 1 and 2 were generated using a reservoir simulator constructed with the complete, rigorous, Laplace domain, rate-transient, analytic solution of a finite-conductivity vertical fracture in an infinite-acting reservoir14. Bounded reservoir solutions have also been generated in this study to verify these results and findings. These results have also been identically duplicated with a commercial finite-difference reservoir simulator15. As clearly presented in Ref. 1, the bounding limits of each of the flow regimes are easily identified. The “material balance” to superposition time ratio has a constant value during the bilinear flow regime of 4/3. During the formation linear flow regime, the ratio of the “material balance” time to the superposition time reaches a constant value (which is a maximum on the graph) of 2. Not only are these two time functions not equivalent, the ratio between the two functions also varies continuously over the transient history of the well. An earlier flow regime (fracture storage or fracture linear flow regime) also exists in the transient behavior of a vertically fractured well but is not depicted in Figs. 1 and 2 since it (1) ends very quickly (in much less time than is generally recorded as the first data point in production data records), and (2) is commonly “masked” or distorted by wellbore storage (only applicable for pressure-transient solutions) if it is present. During the fracture linear flow regime, the ratio of the “material balance” to the equivalent superposition time also has a constant value of 2. Even though this flow regime is generally not observed in production data, the theoretical basis for the solution of the rate-transient behavior of this flow regime is presented in the Appendix of this paper for completeness. A late time flow regime may also exist for all types of wells (unfractured vertical, vertically fractured, and horizontal wells) in closed (no flow outer boundary condition) systems, which is also not depicted in Figs. 1 and 2. In rate-transient analyses this flow regime is simply referred to as the fully boundary-dominated flow regime. It occurs during the same interval in time as the pseudo-steady state flow regime of pressure-transient solutions, but the pressure distributions in the reservoir during the boundary-dominated flow regime of rate-transient solutions are completely different from those exhibited in pressure-transient solutions. The rate-transient behavior of oil and gas wells during the boundary-dominated flow regime is fully presented in the Appendix of this paper for a vertically fractured well in cylindrically and rectangularly bounded reservoirs. Even during the radial flow regime of unfractured vertical wells (analogous to the pseudoradial flow regime of vertically

fractured wells), the ratio of the “material balance” time function to the equivalent superposition time function achieves a stabilized numerically determined value of about 1.08, as clearly demonstrated in Fig. 2. Therefore, for radial (or pseudoradial) flow analyses, an error in the time function of 8 % may be acceptable and within engineering accuracy, but errors in the time function of as much as 200 % during the formation linear (or pseudolinear) flow regime of vertically fractured wells are not. The rate-transient (flow rate versus time or cumulative production) decline curve solutions that have been ubiquitously used in production data analyses are the appropriate solutions to use in most all practical cases. At least as early as 194516, rate decline curves have been commonly used to characterize the production performance of oil and gas wells. Fetkovich and co-workers17,18 have greatly expanded the use and applicability of decline curve analyses for characterizing formation and well properties from the production performance of oil and gas wells. Blasingame and co-workers10-13 have also reported the development of production analyses using decline curves that also incorporate the use of the “material balance” time function. If the proper corrections are made to the “material balance” time function, a modified “material balance” time function can be constructed and used to obtain an “effective” time function value that is equivalent in magnitude to the rigorous superposition time function. This type of equivalent time function would permit the analysis of production history data points for which the flowing pressures are not known. Therefore, a convolution analysis of all of the production history is performed, using the known pressure data points where they exist in a conventional convolution analysis, and using the modified “material balance” time function to evaluate the equivalent superposition time function values that correspond to the data points at which the pressures are not known. This is essentially what is done in the model construction that is described in the following section of this paper. Model Description The construction of a production analysis model that incorporates the flexibility of combining a conventional rate-transient convolution analysis for production data points with known pressures, and the modified “material balance” time concept for data points at which the pressure is unknown, into a robust and accurate production analysis system, requires tremendous planning and care. A production analysis system of this type has been constructed and used, and is referred to in this paper simply as the Pressure Optional Effective Well And Reservoir Evaluation production analysis system. The construction of the production analysis system presented in this paper is most readily accomplished by generating and storing the rate-transient decline curve solutions for a family of well types, outer boundary conditions, and for a range of parameter values that relate to the model under consideration. The dependent variables that are required in the solution are the dimensionless well flow

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 5

rate and cumulative production as a function of time. Rate decline curves of this type are generated and stored for a practical range of the independent variable values. For unfractured vertical well rate-transient type curves, the independent variables considered were dependent on the outer boundary condition specified. In a closed cylindrically bounded reservoir, the dimensionless well drainage radius ( )eDr , referenced to the apparent wellbore radius, is the independent variable used to generate a family of rate-transient decline type curves. In an infinite-acting reservoir system, the radial flow steady-state skin effect is the independent variable used to construct the family of type curves. The latter set is of particular importance for the case of all well types (unfractured, fractured, and horizontal) where no sand face flowing pressures are available at all for the convolution analysis. The details of this procedure will be discussed in the following section of this paper. For vertically fractured wells in infinite-acting reservoirs, the independent variable of interest is the dimensionless fracture conductivity ( )fDC . In closed reservoirs, the fractured well decline curves are also constructed with the dimensionless well drainage area ( )DA as an independent variable. For horizontal well decline curves, a larger number of independent parameter values must be considered. In infinite-acting systems, the dimensionless wellbore length ( )DL ,

vertical location in the pay zone ( )wDZ , and well bore radius

( )wDr are all considered. The effect of the wellbore location has been demonstrated by Ozkan19 to have a lesser impact on the wellbore transient behavior than the dimensionless wellbore length and wellbore radius and may be fixed at a constant average value (equal to approximately one half) if limitations of array storage and interpolation are encountered. In a finite closed reservoir there is also the dimensionless well drainage area ( )DA that must be included in the independent variables considered when generating that family of decline curves. While the production analysis model described in this paper presently only considers the common well types and outer boundary conditions, the analysis methodology is entirely general. A numerical simulation model can be used to consider any type of well and reservoir configuration desired, and the resulting generated rate-transient decline curves are then used in the analysis. This is the only requirement of the Pressure Optional Effective Well And Reservoir Evaluation production analysis methodology described in this paper, that the dimensionless flow rate and cumulative production transient behavior of the particular well and reservoir configuration under consideration can be accurately generated and stored to be used in the decline curve analysis.

The evaluation of the ratio of the “material balance” time function to the rigorous equivalent superposition-in-time function, as a function of the equivalent superposition time, is defined in its most fundamental form for rate-transient analyses in Eq. 16. Note that Eq. 16 directly provides the necessary correction for the conventional “material balance” time function.

( )( )

( )( )

( )( ) ( )

mb n Dmb n pD n

e n D n wD n D n

t t t t Q tt t t t q t t t

= = (16)

Therefore, the dimensionless time, flow rate, and cumulative production, obtained for whatever well type and reservoir configuration considered, are used to directly compute the correction for the “material balance” time function over the entire transient history of the well. The modified “material balance” equivalent time function that is used to effectively perform the convolution for production data points for which the sandface pressures are unknown is obtained by simply dividing the appropriate uncorrected “material balance” time function value (given by Eqs. 14 or 15) by the correction defined with Eq. 16. Therefore, the superposition time function value can be effectively (and internally consistently) estimated using the “material balance” time function (computed from a well’s production data) and the decline curve analysis matched well and reservoir model dimensionless rate-transient behavior. The actual implementation and application of this new technology in the model is discussed in the following section of the paper. Implementation and Application The implementation and application of the Pressure Optional Effective Well And Reservoir Evaluation production analysis methodology requires the consideration of two specific cases. Each case must be considered separately, since each requires a different solution procedure. In the first case, a fully determined system results that can be directly solved at each of the production data time levels with known sandface flowing pressures. The second case involves a two-step or iterative evaluation procedure in order to estimate the well and reservoir properties using both (a) an unfractured vertical well and infinite-acting reservoir decline curve analysis and (b) a decline curve analysis corresponding to the actual well and reservoir configuration of the system. The first case considered is one in which at least one production data point (at any point in time during the entire production history of the well) has a known flowing sandface pressure that corresponds to the flow rate data at that point in time. A factor as to whether or not this case applies is also dependent upon whether or not significant completion pressure losses are present in the system. Since typically the wellhead flowing pressures (or possibly bottom hole flowing wellbore pressure measurements from permanent downhole gauges) are the well flowing pressure data that are available, and because the completion losses are in general formation effective permeability (and skin effect in some models) dependent, simultaneous solution of the sandface flowing

6 B. POE SPE 77691

pressure and the formation effective permeability and skin effect generally require an iterative solution procedure. The first case discussed requires that the sandface flowing pressure for at least one point in time in the production history be known (or that the completion losses can be assumed to be negligible in which case the sandface flowing pressures would be assumed to be known). If the production data set and the well conditions do not meet these requirements, then the analysis given in the second case that follows applies. The second case is that in which (1) none of the sandface flowing pressures are available for any of the production data flow rate points, (2) one in which the sandface flowing pressures can not be estimated directly from the bottomhole wellbore flowing pressures, such as in the case of non-negligible completion pressure losses, or (3) the case where you actually have an unfractured vertical well in an infinite-acting reservoir. For any of these three conditions, an initial analysis of the early transient (infinite-acting reservoir response) production data on an unfractured vertical well infinite-acting reservoir decline curve set is required. This must be done, regardless of the actual well type. With the first two conditions listed for the second case, this initial step is required in order to reduce the number of unknowns in the problem by one, typically the reservoir effective permeability is chosen as the parameter of interest. For the first condition of the second case, none of the necessary sandface flowing pressures are available for a convolution analysis. Therefore, by plotting the well’s flow rate versus its cumulative production against a dimensionless flow rate versus dimensionless cumulative production type curve of this type, the only unknowns in the relationships between the dimensionless and dimensional plotting functions on the graph is the formation effective permeability, related to the ordinate scale dimensionless flow rate function. In this type of analysis, only the early transient (infinite-acting reservoir behavior) is used in determining the appropriate decline curve match. It is important to note that for any point on the matched decline curve, the pressure drop (or pseudopressure drop for gas reservoir analyses) appears in the denominator of the dimensionless flow rate and cumulative production, the ordinate and abscissa values, respectively. Therefore, for any point on the decline curve, the abscissa and ordinate scale values can be used to resolve the remaining unknown of the problem directly related to the scales of the plotting functions since the pressure drop term cancels out of the evaluation. This principle is applicable for the initial infinite-acting reservoir unfractured vertical well decline curve analysis of all of the second case conditions. It is also important to note that the abscissa variable (dimensionless cumulative production) in this particular analysis is referenced to the actual wellbore radius ( )wr that is known, not the apparent or effective well bore radius. The only other match variable (radial flow steady-state skin effect) in the analysis is obtained directly from the decline curve stem matched on the graph.

For the second case and first condition, the formation effective permeability is actually generally the only parameter estimate that is used in subsequent computations. The transient behavior of vertically fractured or horizontal wells are best characterized using the specific dimensionless parameters associated with those well types (i.e.,

, , ,fD D wD wDC L r Z ) and the steady state skin effect is generally not a good way to characterize that behavior unless the well is actually an unfractured vertical well. The second condition of the second case also requires an initial analysis of the production data with a set of infinite-acting reservoir unfractured vertical well decline curves to obtain an initial estimate of the reservoir effective permeability so that the completion pressure losses and corresponding sandface flowing pressures may be computed. Once again, the reservoir effective permeability is generally the only parameter from this analysis step that is used in the subsequent calculations. For the last condition of the second case (unfractured vertical well in an infinite-acting reservoir), all of the analysis results are used. The reservoir effective permeability and the matched radial flow steady-state skin effect values resulting from the analysis represent the final results for those parameters. Once this graphical analysis step is completed, the production data analysis is also completed for the unfractured vertical well and infinite-acting reservoir case. CASE 1: The production analysis procedure that is used for the first case discussed in this paper is accomplished in a very straightforward manner. The dimensional flow rates of the well are plotted against the corresponding dimensional cumulative production at each of the production data time levels on a log-log dimensionless flow rate versus dimensionless cumulative production decline curve set that corresponds to the actual reservoir type, outer boundary condition, and well type of interest. For each of the production data points that have known sandface flowing pressure values, the reservoir effective permeability is directly determined from the matched decline curve values, the production data, and the relationship between the dimensional and dimensionless well flow rates (ordinate values). The system characteristic length is directly computed from the relationship between the dimensional and dimensionless cumulative production (abscissa values). This means that independent estimates of these parameters can be determined for each and every production data point for which the sandface flowing pressures are known. While one may be tempted to think that they may be able to evaluate how each of these parameters change with respect to time, this is not the case. There are two reasons why this is not possible; (1) the convolution integral as employed in this analysis does not permit the use of a non-linear function (reservoir model) which would be implied if either of these parameters change with respect to time, and (2) the rate-transient decline curve solutions employed in the analysis have been generated for

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 7

constant system properties. Therefore, these are just independent estimates of the two parameters determined for each of the known sandface flowing pressure data points and should simply be averaged to report representative values for these parameters. Statistical analysis techniques have also been employed in the averaging process to minimize the effects of outliers in the computed results for these parameters. With the reservoir effective permeability and system characteristic length known from the analysis just described, the other well and reservoir properties are subsequently determined from the dimensionless parameters associated with the matched dimensionless solution decline curve stem. For example, an unfractured well in a closed cylindrically bounded reservoir has decline curve stems that are associated with the dimensionless well drainage radius, referenced to the system characteristic length. Therefore, the well’s effective drainage radius and drainage area can be readily computed from the match result. The radial flow steady-state skin effect is also directly obtained from the matched system characteristic length and the wellbore radius using the effective wellbore radius concept. It should be noted that for closed finite reservoir decline curve analyses, the decline curve sets displayed on the graphs that are used for matching purposes should be modified using the appropriate pseudosteady state coupling relationship for the well model of interest, analogous to the systematic methodology proposed in Ref. 13. This results in all of the boundary-dominated flow regime decline data of the decline curves in the set collapsing to a single decline stem on the displayed graph, which greatly aids in graphical matching efforts. Similarly, for vertically fractured wells in closed rectangularly bounded reservoirs the decline curve stems correspond to specific values of the dimensionless fracture conductivity and the dimensionless drainage area of the well. The dimensional fracture conductivity is computed from the matched dimensionless fracture conductivity, and the average estimates of the reservoir effective permeability and fracture half-length (equal to the matched system characteristic length). The well’s drainage area is directly computed from the matched dimensionless well drainage area ( )DA and the system characteristic length A similar scenario exists for the production analysis of a horizontal well in a closed finite reservoir. In this case the decline stems correspond to values of the dimensionless wellbore length in the pay zone (referenced to the net pay thickness), the dimensionless well effective drainage area, the dimensionless well vertical location in the pay zone (if this parameter is considered as variable in the analysis), and the dimensionless wellbore radius. The total effective length of the wellbore in the pay zone may be computed as an average of twice the matched system characteristic length and the value of effective wellbore length derived from the matched dimensionless wellbore length and the net pay thickness. The effective wellbore radius is computed from the matched dimensionless wellbore radius and the net pay thickness. The

well’s effective drainage area is readily obtained from the matched dimensionless drainage area and the system characteristic length. CASE 2: The analysis procedure that is used for the second case discussed in this paper requires a two-step or iterative solution procedure. As previously mentioned, the initial analysis step for the second case involves matching the early transient data (infinite-acting reservoir behavior) of the actual well on an infinite-acting reservoir unfractured vertical well decline curve set. This step is used to determine an initial estimate of the formation effective permeability. For the first condition of the second case discussed, where none of the flowing pressures are known in the production history, this is the only practical way of reliably estimating the reservoir effective permeability independently from the effects of all other parameters governing the rate-transient response of the system. If this condition of the second case is applicable in the production analysis, only estimates of the well and reservoir properties can be obtained from the analysis since all of the subsequent computations for the other parameter estimates are dependent on the accuracy of the reservoir effective permeability estimate obtained in this step. This point may appear to be of minor significance, however, consider the case of a vertically fractured well that has only exhibited bilinear or pseudolinear flow in the production data record. In both of these flow regimes (in fact in all of the transient behavior prior to the onset of pseudoradial flow), the apparent radial flow skin effect exhibited by the system is transient, i.e. changes continuously with respect to time. Only once the pseudoradial flow regime has been exhibited in the transient behavior of the well has the flux distribution in the fracture stabilized such that the transient behavior of the vertically fractured well can be characterized by a meaningful and constant steady-state radial flow apparent skin effect. Prior to that point in time, the production data decline on the graph may not exactly follow a single transient decline stem, which has an associated constant radial flow skin effect value. However, in spite of this limitation, it has been found empirically by matching numerous sets of numerical simulation transient production results of fractured wells that the error resulting from inverting the production data record in this manner to estimate the reservoir effective permeability is generally small, typically less than 5 %. Since the early transient behavior of low dimensionless conductivity (CfD < 10) vertical fractures may not generally follow a single constant skin effect decline stem on the unfractured vertical well and infinite-acting reservoir decline analysis graph, it would also not be expected that the skin effect derived from the analysis would be appropriate for characterizing the transient behavior of the well. For higher dimensionless conductivity (CfD > 50) fractures, the early transient production decline data do generally tend to follow a single decline stem. However, in general only the estimate of the reservoir effective permeability is used in the subsequent

8 B. POE SPE 77691

analyses of the production data and the remaining well and reservoir specific parameters of interest are obtained using a decline curve analysis that corresponds to those particular well and reservoir conditions. A similar discussion could also be given for the early transient behavior of horizontal wells, with their model specific early transient flow regimes, in which case the reservoir effective permeability is also the only parameter estimate retained from the initial unfractured vertical well and infinite-acting reservoir decline curve analysis. Once the reservoir effective permeability has been estimated from the initial analysis step previously discussed, the production data are then plotted on a decline curve set for the actual well and reservoir conditions of interest. With the previously determined reservoir effective permeability estimate, the only unknown remaining unresolved between the dimensionless parameter scales of the reference decline curve set and the dimensional production data is the system characteristic length, which is associated with the abscissa scale of each of the matched production data points. As previously discussed in this paper, at each production data point on the matched decline curve stem of the graph, the pressure (or pseudopressure) drop terms present in the definitions of both the dimensionless flow rate and cumulative production variables cancel out when resolving the ordinate and abscissa match points of the dimensionless and dimensional scales for each of the matched points. Therefore, independent estimates of the system characteristic are directly evaluated for each of the well’s actual production data flow rate points. Also, as previously discussed in this paper, a statistical analysis of the independent estimates of the system characteristic length is employed to obtain a representative average value for this parameter. With estimates of the reservoir effective permeability and system characteristic length obtained in the manner just described, the remaining unknowns of the decline curve production analysis are obtained in the same manner as previously given for the first case. For the third condition of the second case, where the well is actually an unfractured vertical well and the reservoir is still infinite-acting at the end of the historical production data record, the analysis may be repeated using the infinite-acting reservoir unfractured well decline curve set to improve the estimates of the reservoir effective permeability and steady state skin effect. For the first and second conditions of the second case discussed, an iterative procedure is used to update the parameter estimates used in the completion loss and sand face pressure calculations, whether these are measured values (condition 2) or synthetic values (conditions 1 and 2) as detailed in the following discussion. The iterative matching process for this case and these conditions uses a reference dimensionless decline curve set that corresponds to the actual well and reservoir type being considered. The iterative matching and analysis process are continued until convergence and a satisfactory decline analysis match are achieved. Concurrently with the graphical analysis matching, the sand face flowing pressure history of the well is synthetically

generated in a systematic point-by-point manner (beginning with the initial production data point) by resolution of the matched dimensionless decline curve stem solution (and the corresponding dimensionless time scale associated with that curve) and the superposition relationships given in Eqs. 4 and 5. Definitions of the dimensionless variables used in these relationships have been given previously in Eqs. 6 through 13. Note that this solution procedure for estimating the sand face flowing pressures at each of the production data flow rate points is applicable to all well and reservoir types, and can be performed whether any historical measured well flowing pressures are available or not. If some sand face pressures are known (such as in the first case discussed), a direct comparison of the actual and synthetic sand face flowing pressure values can be used to verify the quality of the decline curve analysis match obtained for the production data set. The well bore bottom hole flowing pressures can also be back-calculated from the synthetic sand face flowing pressure history by including the completion losses of the system5. Field Examples and Discussion The use of the production analysis model described in this paper is best demonstrated with some illustrative field examples. Numerous synthetic examples have been evaluated in the testing and validation of the model, however, field examples have been chosen for the paper since they provide an additional complexity in the analysis due to the fact that the production performance of the wells are often not recorded under the most ideal of conditions in the field. In this paper, two examples have been chosen, which demonstrate some of the advantages and capabilities of this production analysis technique, for which independent estimates of the well and reservoir properties are available. The independent estimates of these properties are derived from conventional production analyses or geophysical measurements such as core analyses. The first example selected is that of a vertically fractured gas well located in South Texas for which a complete flowing tubing pressure record is available, which permits a conventional convolution analysis of the production performance of the well to evaluate the well and reservoir properties. The second example is that of an unfractured vertical well completed in a heavy oil reservoir in South America (produced with an electrical submersible pump (ESP) for which there are no pump intake pressures recorded) that has a fairly complete set of laboratory core analyses from whole cores. Figure 3 is the decline curve match of the first example well that had been previously analyzed with a production analysis history matching model to obtain estimates of the reservoir effective permeability, fracture half-length, and conductivity of 0.05 md, 80 ft, and 0.5 md-ft, respectively. It can be seen that the production analysis model discussed in this paper provided essentially the same results (kg=0.049 md, Xf=83 ft, kfbf=0.41md-ft) as the production analysis that used the conventional rate-transient convolution analysis. The oil well example production analysis required the two-step decline analysis of the production data. Figure 4 is the

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 9

decline curve analysis of the early transient (infinite-acting reservoir) production performance of the well used to determine the estimate of the reservoir effective permeability. The production analysis resulted in an estimate of the average reservoir effective permeability for the 54 ft thick sand of 1.28 md that is in excellent agreement with the average permeability reported from the core analyses of 1.4 md. Therefore, the production data analysis methodology reported in this paper makes it both possible and practical to reliably estimate the insitu reservoir effective permeability from the production behavior of a well with absolutely no measured well flowing pressures with which to perform a conventional convolution analysis of the production performance of the well. The final step in the oil well example decline curve analysis is depicted in Fig. 5. This graph is an illustration of the decline analysis matching performed to evaluate the radial flow steady state skin effect and obtain an estimate of the effective well drainage area. There is not an independent estimate of the steady state skin effect available for the well with which to compare but the inverted estimate of skin effect is consistent with the well’s completion type and performance. The effective well drainage area estimate obtained from the analysis of 194 acres is also in good agreement with the well spacing of about 200 acres on which the wells in this field have been drilled. Additional examples that demonstrate the application of this powerful production data analysis methodology could also be readily presented. However, some of the more subtle advantages of this production analysis technique are probably best presented in a discussion format. As an example, the production analysis technique presented in this paper of matching the well flow rate versus the well’s cumulative production, against a log-log graph of the decline curve data set consisting of the dimensionless well flow rate versus the dimensionless cumulative production, besides having the advantage of not requiring that the sand face flowing pressures be known for each of the production data points plotted on the graph, also directly eliminates most all of the problems encountered in conventional convolution analyses related to partial day or partial month production in the production data record. If the well is only on production for part of a day (or month if monthly production data are used), it is often not readily apparent how to choose an average flowing pressure to assign to that production data point and time value in the conventional convolution analysis. Most importantly of all is that with this production analysis technique, values of the well flowing pressure need not be guessed or estimated for the missing pressure values to complete the convolution analysis of the production data. It is also readily apparent from the theory provided in the Appendix of this paper and in Ref. 1, as well as the oil well ESP example in this paper, that this production analysis technique results in an effectively rigorous convolution analysis of the production data, even with no sand face flowing pressures to use in the production data analysis.

Another example of a commonly encountered production analysis problem is that of the case where an operator has acquired the producing well from another operator that did not record the flowing tubing pressure history of the well. The original operator may not have even made it a practice to maintain their own database of the production flow rate and cumulative production history of their wells and simply rely on a commercial production data service to keep track of the production records of the well. Thus the current operator is left with the daunting task of trying to evaluate the current well condition and reservoir properties with no practical or mathematically correct way of accomplishing that task using the conventional convolution techniques currently employed in valid production data analysis models. A similar scenario arises in the case of an oil well that would initially produce naturally on its own, but later required the use of an artificial lift system in order to continue producing. While the well was initially flowing under its own reservoir drive energy, the flowing wellhead pressures could readily be recorded and used in the convolution analysis. However, once the well was switched over to a pump (either ESP or beam pump) the wellhead flowing pressure would not generally be recorded or would be meaningless in the analysis if it were. In this case, if an evaluation of the current well condition is required, the analyst is faced with the problem of how to perform the convolution analysis of the production history while the well was being artificially lifted. The production analysis techniques reported in this paper provide for the first time a truly mathematically correct, internally-consistent, and practical means of effectively performing a convolution analysis of these types of production analysis problems to permit the estimation of the well and reservoir properties. Conclusions The major conclusions that can be drawn from this work are the following: 1. An effective convolution analysis of a well’s production

performance that has a partial or absent flowing pressure record can be accomplished using a corrected “material balance” time function and the rate-transient solutions using the analysis procedures presented in this paper.

2. The uncorrected “material balance” time function is only truly applicable for the pressure-transient analysis of a well’s production behavior during the pseudosteady state flow regime. It is not truly correct for any other pressure-transient flow regime and is incorrect for all rate-transient flow regimes of all well types.

3. The production analysis techniques presented in this paper permit for the first time a mathematically correct means of evaluating the production performance of artificially lifted wells to obtain estimates of their well and reservoir properties.

4. The effective convolution analysis reported in this paper is entirely general and is applicable to all well conditions and reservoir types as long as the appropriate dimensionless rate-transient solution can be generated for those cases.

10 B. POE SPE 77691

Nomenclature A Well drainage area, ft2

DA Dimensionless drainage area, 2/D CA A L=

fb Fracture width, ft oB Oil formation volume factor, rb/STB fDC Dimensionless fracture conductivity, /fD f f fC k b kX=

tc Reservoir total system compressibility, 1/psia tfc Fracture total system compressibility, 1/psia BFf Cumulative production bilinear flow superposition time

function 1BFf Flow rate bilinear flow superposition time function

FLf Cumulative production formation linear flow superposition time function

1FLf Flow rate formation linear flow superposition time function

FSf Cumulative production fracture storage linear flow superposition time function

1FSf Flow rate fracture storage linear flow superposition time function

pG Cumulative gas production, MMscf h Reservoir net pay thickness, ft

fk Fracture permeability, md gk Reservoir effective permeability to gas, md ok Reservoir effective permeability to oil, md CL System characteristic length, ft DL Dimensionless horizontal well length in pay zone,

/ 2D hL L h= hL Effective horizontal well length in pay zone, ft

m Summation index n Index of current or last data point

pN Cumulative oil production, STB Dp Dimensionless pressure solution Dip Dimensionless pressure at the ith time level ip Initial reservoir pressure, psia pp Real gas pseudopressure potential, psia2/cp scp Standard condition pressure, psia wDp Dimensionless well bore pressure wfp Sand face flowing pressure, psia Dq Dimensionless flow rate gq Gas flow rate, Mscf/D oq Oil flow rate, STB/D pDQ Dimensionless cumulative production

wDq Dimensionless well flow rate er Effective well drainage radius, ft eDr Dimensionless well drainage radius, /eD e Cr r L= wr Well bore radius, ft

wDr Dimensionless well bore radius, /wD wr r h= T Reservoir temperature, deg R

at Pseudotime integral transformation, hr-psia/cp aet Equivalent pseudotime superposition function,

hr-psia/cp ambt Gas reservoir “material balance” time, hr Dt Dimensionless time Dit ith dimensionless time in production history et Equivalent time superposition function, hr it ith time level in production history, hr mbt Oil reservoir “material balance” time, hr nt Last or current time level in production history, hr

SCT Standard condition temperature, deg R

DX Dimensionless X direction spatial position *

DX Dimensionless fracture spatial position eDX Dimensionless X direction drainage areal extent fX Effective fracture half-length, ft wDX Dimensionless X direction well spatial position

DY Dimensionless Y direction spatial position eDY Dimensionless Y direction drainage areal extent wDY Dimensionless Y direction well spatial position wDZ Dimensionless well vertical spatial position

Greek β Dimensionless parameter ξ Dimensionless parameter φ Reservoir effective porosity, fraction BV

fφ Fracture effective porosity, fraction BV σ Pseudoskin due to dimensionless fracture conductivity δ Pseudoskin due to bounded nature of reservoir

fDη Dimensionless fracture hydraulic diffusivity

g tcµ Mean value gas viscosity-total system compressibility, cp/psia

oµ Oil viscosity, cp Functions erfc Complimentary error function exp Exponential function ln Natural logarithmic function Acknowledgements The author would like to express his appreciation to Schlumberger for the permission to publish the results of this research effort.

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 11

References 1. Poe, B.D. Jr.: “Oil and Gas Reservoir Production Analysis

Apparatus and Method With an Effective Convolution Analysis That Does Not Require Well Pressure History”, U.S. Patent pending.

2. van Everdingen, A.F. and Hurst, W.: “The Application of the Laplace Transformation to Flow Problems in Reservoirs,” Trans., AIME (1949) 186, 305-324.

3. Poe, B.D. Jr., Conger, J.G., Farkas, R., Jones, B., Lee, K.K., and Boney, C.L.: “Advanced Fractured Well Diagnostics for Production Data Analysis,” paper SPE 56750 presented at the 1999 Annual Technical Conference and Exhibition, Houston, TX, Oct. 3-6.

4. Poe, B.D. Jr. and Marhaendrajana, T.: “Investigation of the Relationship Between the Dimensionless and Dimensional Analytic Transient Well Performance Solutions in Low-Permeability Gas Reservoirs,” paper SPE 77467 presented at the 2002 SPE Annual Technical Conference and Exhibition, San Antonio, TX, Sept. 29 – Oct. 2.

5. The Technology of Artificial Lift Methods, Brown, K.E. (ed.), 4 PennWell Publishing Co., Tulsa, OK (1984).

6. Palacio, J.C. and Blasingame, T.A.: “Decline-Curve Analysis Using Type Curves – Analysis of Gas Well Production Data,” paper SPE 25909 presented at the 1993 SPE Rocky Mountain Regional / Low Permeability Reservoirs Symposium, Denver, CO, Apr. 12-14.

7. McCray, T.L.: Reservoir Analysis Using Production Decline Data and Adjusted Time, M.S. Thesis, Texas A&M University, College Station, TX (1990).

8. Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., and Fussell, D.D.: “Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts,” SPE Res. Eval. and Eng., (Oct. 1999) Vol. 2, No. 5, 478-486.

9. Crafton, J.W.: “Oil and Gas Well Evaluation Using the Reciprocal Productivity Index Method,” paper SPE 37409 presented at the 1997 Production Operations Symposium, Oklahoma City, OK, Mar. 9-11.

10. Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.: “Decline Curve Analysis Using Type Curves – Analysis of Oil Well Production Data Using Material Balance Time: Application to Field Cases,” paper SPE 28688 presented at the 1994 SPE Petroleum Conference and Exhibition of Mexico, Veracruz, MX, Oct. 10-13.

11. Shih, M.Y. and Blasingame, T.A.: “Decline Curve Analysis Using Type Curves: Horizontal Wells,” paper SPE 29572 presented at the 1995 SPE Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Denver, CO, Mar. 19-22.

12. Doublet, L.E. and Blasingame, T.A.: “Decline Curve Analysis Using Type Curves: Water Influx/Waterflood Cases,” paper SPE 30774 presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct. 22-25.

13. Doublet, L.E. and Blasingame, T.A.: “Evaluation of Injection Well Performance Using Decline Type Curves,”

paper SPE 35205 presented at the 1995 SPE Permian Basin Oil and Gas Recovery Conference, Midland, TX, Mar. 27-29.

14. Poe, B.D. Jr., Shah, P.C., and Elbel, J.L.: “Pressure Transient Behavior of a Finite-Conductivity Fractured Well With Spatially Varying Fracture Properties,” paper SPE 24707 presented at the 1992 SPE Annual Technical Conference and Exhibition, Washington D.C., Oct. 4-7.

15. SABRETM - A General Purpose Petroleum Reservoir Simulator, S.A. Holditch & Associates, Inc., College Station, TX, Oct. 1993.

16. Arps, J.J.: “Analysis of Decline Curves,” Trans., AIME (1945) 160, 228-247.

17. Fetkovich, M.J. “Decline Curve Analysis Using Type Curves,” JPT (June 1980) 1065-1077.

18. Fetkovich, M.J. et al: “Decline Curve Analysis Using Type Curves – Case Histories,” SPEFE (Dec. 1987) 637-656.

19. Ozkan, E.: Performance of Horizontal Wells, Ph.D. dissertation, University of Tulsa, Tulsa, OK (1988).

20. Stehfest, H.: “Numerical Inversion of Laplace Transforms,” Communications of the ACM (Jan. 1970), 13, No. 1, 47-49 (Algorithm 368 with corrections).

Appendix Relationship Between “Material Balance” Time Function and Superposition Time Function for Vertically Fractured Wells Specialized analyses can be derived for oil and gas reservoirs for each of the flow regimes of a vertically fractured well. The flow regimes that will be considered in this discussion are the rate-transient flow regimes of a fractured well that correspond to the fracture storage linear, bilinear, formation linear or pseudolinear, pseudoradial, and boundary-dominated flow regimes. Fracture Storage Linear Flow Regime The wellbore dimensionless rate transient behavior of a fractured well during the fracture storage linear flow regime is given3 by Eq. A-1.

( ) 0.179587fD fDwD D

fD DfD D

C Cq ttt ηπ π η

= = (A-1)

The dimensionless cumulative production of a fractured well during the fracture storage linear flow regime is given3 by Eq. A-2.

( )1/ 2 1/ 2

3/ 2 0.3591742 fD D D

pD D fDfD fD

C t tQ t Cπ η η

= =

(A-2)

The dimensional well bore rate-transient behavior of a fractured oil well during the fracture storage linear flow regime is presented in Eq. A-3 and the corresponding superposition-in-time function for this flow regime is given in Eq. A-4.

( )( ) ( )1

12.7677o n f f f tf

FS ni wf n o o

q t h b k cf t

p p t Bφ

µ=

− (A-3)

12 B. POE SPE 77691

( ) ( )( )

1

1

1 111

1 1 1ni wf i

FS n

i wf n n i n i n nin

p p tf t

p p t t t t t t t

−

− −=>

−= − +

− − − −

∑

(A-4) The pressure drop normalized cumulative production of a fractured oil well during the fracture storage linear flow regime is given by Eq. A-5 and the corresponding superposition-in-time function for this flow regime is given in Eq. A-6.

( )( ) ( )

153.212p n f f f tf

FS ni wf n o o

N t h b k cf t

p p t Bφ

µ=

− (A-5)

( ) ( )( )

1

1 1

11

ni wf n

FS n n i n i n n

i wf nin

p p tf t t t t t t t

p p t

−

− −

=>

−= − − − + −

− ∑

(A-6) The superposition-in-time functions are directly related to one another during the fracture storage linear flow regime of a fractured oil well as expressed in Eq. A-7.

( ) ( )( )

2

21

1e n FS n

FS n

t t f tf t

= =

(A-7)

The relationship between the “material balance” time function and the rigorous superposition time function for the fracture storage linear flow regime behavior of fractured oil wells is given by Eq. A-8.

( ) ( )2mb n e nt t t t= (A-8) The dimensional well bore rate-transient behavior of a fractured gas well during the fracture storage linear flow regime is presented in Eq. A-9.

( )( ) ( )( ) ( )

( )( )1

4548.06

g n sc f f f tfFS a n

p i p wf n sc t n

q t hT b k cf t t

p p p p t p T c t

φ=

−

(A-9)

( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )

1

1

11

1

1np i p wf i

FS a n

p i p wf nin

a n a i

p p p p tf t t

p p p p t t t t t

−

=>

−

−=

−

−

∑

( ) ( ) ( ) ( )1

11a n a na n a i t t t tt t t t −

+−

−−

(A-10) Eq. A-11 gives the corresponding pseudopressure drop normalized cumulative production behavior of a fractured well during the fracture storage linear flow regime.

( )( ) ( )( )

( ) ( )( )( )

54577300

sc f g n g t n f f fp n

FS a n

p i p wf n sc

hT b t c t k c tG tf t t

p p p p t p T

µ µ φ=

−

(A-11)

( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )

1

111

n p i p wf iFS a n a n a i

p i p wf nin

p p p p tf t t t t t t

p p p p t

−

−

=>

− = −−∑

( ) ( ) ( ) ( )1a n a i a n a nt t t t t t t t −− − + −

(A-12) The relationship between the flow rate and cumulative production superposition-in-time functions for a fractured gas well during the fracture storage linear flow regime is given by Eq. A-13 and is equal to the equivalent pseudotime function for a varying sandface pressure history.

( ) ( )( )( )( )

2

21

1ae n FS a n

FS a n

t t f t tf t t

= = (A-13)

The proof of the identities established in Eqs. A-7 and A-13 for fractured oil and gas well behaviors during the fracture storage linear flow regime can be accomplished by at least one of four possible means: (1) by simple numerical evaluation, (2) function (series) analysis, (3) considering a simple case such as a single inner boundary condition value production history, or (4) simply proven heuristically. The first two options are rather tedious and will not be presented in this discussion. The third method of proof is trivial, since for n=1 the two functions are easily proven to be exactly equal. Heuristically we know that for any given value of dimensional time during the fracture storage linear flow regime that the flow rate and cumulative production superposition-in-time functions must relate directly to the same value of dimensionless time. This general and fundamental heuristic principle applies to all flow regimes of all types of wells. Therefore, the relationship between the “material balance” time and superposition time functions for the fracture storage linear flow regime of a fractured gas well is given by Eq. A-14.

( ) ( ) ( )2amb n g t n ae nt t c t t tµ= (A-14) Bilinear Flow Regime The wellbore dimensionless rate-transient behavior of a fractured well during the bilinear flow regime is given3 by Eq. A-15.

( ) 1/ 41/ 4

2 0.36735134

fD fDwD D

DD

C Cq tttπ

= = Γ

(A-15)

The corresponding dimensionless cumulative production of a fractured well during the bilinear flow regime is given by Eq. A-16.

( ) 3/ 4 3/ 44 2 0.489801334

fDpD D D fD D

CQ t t C tπ

= = Γ

(A-16)

The dimensional rate-transient behavior of a fractured oil well during the bilinear flow regime is given by Eq. A-17.

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 13

( )( ) ( )

1/ 4

1348.9821

o n f f o tBF n

i wf n o o

q t h k b k c f tp p t B

φµ

= −

(A-17)

( ) ( )( ) ( ) ( )

1

1 1/ 4 1/ 411

1

1 1n i wf iBF n

i wf n n i n iin

p p tf t

p p t t t t t

−

−=>

−= −

− − − ∑

( )1/ 4

1

1n nt t −

+−

(A-18)

The pressure drop normalized cumulative production behavior of a fractured oil well during the bilinear flow regime is given by Eq. A-19.

( )( ) ( )

1/ 4

3881.678

p n f f o tBF n

i wf n o o

N t h k b k c f tp p t B

φµ

= −

(A-19)

( ) ( )( ) ( ) ( )

13/ 4 3/ 4

111

n i wf iBF n n i n i

i wf nin

p p tf t t t t t

p p t

−

−

=>

− = − − − −∑

( )3/ 41n nt t −+ − (A-20)

The equivalent time function that can be used for the rate-transient behavior of a fractured oil well during the bilinear flow regime is given by Eq. A-21.

( ) ( )( )

4/3

41

1e n BF n

BF n

t t f tf t

= =

(A-21)

The rate-transient equivalent “material balance” time approximation for the bilinear flow regime of a fractured oil well is therefore given by Eq. A-22.

( ) ( )43

mb n e nt t t t= (A-22)

The dimensional rate-transient behavior of a fractured gas well during the bilinear flow regime is given by Eq. A-23.

( )( ) ( )( )

( ) ( )( )1/ 4

17448.2g n sc g f f

BF a nscp i p wf n

q t hT k k bf t t

p Tp p p p tφ

=−

(A-23)

( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )

1

1 1/ 411

1

1n p i p wf iBF a n

p i p wf n a n a iin

p p p p tf t t

p p p p t t t t t

−

−−>

−=

− −

∑

( ) ( )( ) ( ) ( )( )1/ 4 1/ 4

1

1 1

a n a i a n a nt t t t t t t t −

− +− −

(A-24) The dimensional pseudopressure drop normalized cumulative production behavior of a fractured gas well during the bilinear flow regime is given by Eq. A-25.

( )( ) ( )( )

( ) ( )( )1/ 4

314071000p n sc g f f

BF a nscp i p wf n

G t hT k k bf t t

p Tp p p p tφ

=−

(A-25)

( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )

1 3/ 41

11

n p i p wf iBF a n a n a i

p i p wf nin

p p p p tf t t t t t t

p p p p t

−

−

=>

− = −−∑

( ) ( )( ) ( ) ( )( )3/ 4 3/ 41a n a i a n a nt t t t t t t t −− − + −

(A-26) The relationship between the flow rate and cumulative production superposition time functions for a fractured gas well during the bilinear flow regime is given by Eq. A-27. This relationship also constitutes the equivalent pseudotime that can be used in the rate-transient analysis of the behavior of a fractured gas well during the bilinear flow regime.

( ) ( )( )( )( )

4/3

41

1ae n BF a n

BF a n

t t f t tf t t

= = (A-27)

Therefore, the relationship between the “material balance” and the superposition time function for the bilinear flow regime of a fractured gas well is given by Eq. A-28.

( ) ( ) ( )43

amb n g t n ae nt t c t t tµ= (A-28)

Formation Linear Flow Regime The dimensionless rate-transient behavior of a high conductivity (CfD > 300) fractured well during the formation linear flow regime is given3 by Eq. A-29.

( ) 2 0.359174wD D

DD

q tttπ π

= = (A-29)

The dimensionless cumulative production of a fractured well during the formation linear flow regime is given3 by Eq. A-30.

( ) 3/ 2

4 0.718348DpD D D

tQ t tπ

= = (A-30)

Eq. A-31 gives the dimensional rate-transient behavior of a fractured oil well in the formation linear flow regime.

( )( ) ( )1

6.38385o n f o t

FL ni wf n o o

q t h X k cf t

p p t Bφµ

=−

(A-31)

( ) ( )( )

1

1

1 111

1 1 1n i wf iFL n

i wf n n i n i n nin

p p tf t

p p t t t t t t t

−

− −=>

−= − +

− − − −

∑

(A-32) The dimensional pressure drop normalized cumulative production behavior of a fractured oil well during the formation linear flow regime is given by Eq. A-33.

14 B. POE SPE 77691

( )( ) ( )

76.6062p n f o t

FL ni wf n o o

N t h X k cf t

p p t Bφµ

=−

(A-33)

( ) ( )( )

1

1 1

11

n i wf iFL n n i n i n n

i wf nin

p p tf t t t t t t t

p p t

−

− −

=>

−= − − − + −

− ∑

(A-34) The relationship between the flow rate and cumulative production superposition-in-time functions of a fractured oil well during the formation linear flow regime is given by Eq. A-35. This relationship also represents the equivalent time that can be used in rate-transient analyses of fractured oil wells during the formation linear flow regime.

( ) ( )( )

2

21

1e n FL n

FL n

t t f tf t

= =

(A-35)

The equivalent rate-transient “material balance” time function for a fractured oil well during the formation linear flow regime is given by Eq. A-36.

( ) ( )2mb n e nt t t t= (A-36) The dimensional rate-transient behavior of a fractured gas well during the formation linear flow regime is defined as given in Eq. A-37.

( )( ) ( )( ) ( )( )1

2274.03g n sc f g

FL a nscp i p wf n

q t h T X kf t t

p Tp p p p tφ

=−

(A-37)

( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )

1

1

111

1n p i p wf iFL a n

p i p wf n a n a iin

p p p p tf t t

p p p p t t t t t

−

−=>

−=

− −

∑

1

1 1a n a i a n a nt t t t t t t t −

− +− −

(A-38) The pseudopressure drop normalized cumulative production behavior of a fractured gas well during the formation linear flow regime is given by Eq. A-39.

( )( ) ( )( )

( ) ( )( )27288600

p n sc f g t n gFL a n

p i p wf n sc

G t h T X c t kf t t

p p p p t p Tµ φ

=−

(A-39)

( )( ) ( ) ( )( )( ) ( )( )

( ) ( )1

1

11

n p i p wf iFL a n a n a i

p i p wf nin

p p p p tf t t t t t t

p p p p t

−

−

=>

−= −

−∑

( ) ( ) ( ) ( )1a n a i a n a nt t t t t t t t −− −− +

(A-40) The effective pseudotime that can be used in rate-transient analyses of a fractured gas well during the formation linear flow regime is presented in Eq. A-41. This expression also

provides the relationship between the flow rate and cumulative production superposition time functions for a fractured gas well during the formation linear flow regime.

( ) ( )( )( )( )

2

21

1ae n FL a n

FL a n

t t f t tf t t

= = (A-41)

The equivalent “material balance” time function that can be derived for rate-transient analyses of fractured gas wells during the formation linear flow regime is given by Eq. A-42.

( ) ( ) ( )2amb n g t n ae nt t c t t tµ= (A-42) Pseudolinear Flow Regime The dimensionless rate-transient behavior of low to moderate dimensionless conductivity fractured wells during the pseudolinear flow regime is given by Eq. A-43. The pseudolinear linear flow rate-transient solution given in Eq. A-43 can be readily shown to reduce to that of Eq. A-29 for infinite conductivity fractures.

( )23 9 3exp

4 2fD fD D fD D

wD DC C t C tq t erfcπ

=

(A-43) The rate-transient flow rate and cumulative production behavior of vertically fractured oil and gas wells during the pseudolinear flow regime are presented in detail in Ref. 1 and rather lengthy. Of greater importance for this discussion however is the resulting relationship between the “material balance” time function and the equivalent superposition time function during this flow regime for a vertically fractured well. This result is given in Eq. A-44 for a fractured oil well analysis.

( ) ( )( ) ( )

2 11

exp

e nmb n

t tt t

erfc

βπ

β β β

−

= +

(A-44)

where:

( ) ( )2

41685.55f f e n

o o t f

k b t tk c X

βφ µ

= (A-45)

The corresponding relationship between the “material balance” and superposition time functions during the pseudolinear flow regime of a fractured gas well is given in Eq. A-46.

( ) ( ) ( )( ) ( )

2 11

exp

g t n ae namb n

c t t tt t

erfc

βµ π

β β β

−

= +

(A-46) where:

SPE 77691 EFFECTIVE WELL AND RESERVOIR EVALUATION WITHOUT THE NEED FOR WELL PRESSURE HISTORY 15

( ) ( )2

41685.55f f ae n

g f

k b t tk X

βφ

= (A-47)

Pseudoradial Flow Regime The dimensionless rate-transient behavior of a fractured well during the infinite-acting pseudoradial flow regime is given3 by Eq. A-48.

( )( ) ( ) ( )

1

*

22 ln 4 2 2 ,0 ln

wD D

D

q t Ls X sγ σ

−=+ − + −

(A-48) Similarly, the dimensionless cumulative production of a fractured well during the pseudoradial flow regime can be evaluated in real space using the Laplace space rate-transient solution. The pseudoskin term ( )*,0DXσ appearing in Eqs.

A-48 and A-49 is defined in Ref. 1 and is a function of the fracture dimensionless conductivity.

( )( ) ( ) ( )

1

2 *

2

2 ln 4 2 2 , 0 lnpD D

D

Q t Ls X sγ σ

−=+ − + −

(A-49) The relationship between the “material balance” time and the equivalent superposition time functions during the infinite-acting pseudoradial flow regime of a vertically fractured well can be obtained using the fundamental relationship between the two functions, given in Eq. A-50.

( )( )

( )( )

( )( ) ( )

mb n Dmb n pD n

e n D n wD n D n

t t t t Q tt t t t q t t t

= = (A-50)

Generally, the pseudoradial flow regime rate-transient Laplace domain solutions (Eqs. A-48 and A-49) are inverted into the real space domain using a numerical inversion technique such as the Stehfest Algorithm20. The results of this application may be found in detail in Ref. 1 and will not be included in this discussion due to length constraints. However, the ratio of the “material balance” to the equivalent superposition time functions during the late pseudoradial flow regime does tend to stabilize at a value of approximately equal to 1.06 for CfD = 0.1 and 1.084 for CfD = 10000, as can be seen in Fig. 2 of this paper. Boundary-Dominated Flow Regime The dimensionless rate-transient behavior of a fractured well that is centrally located in a closed cylindrically bounded reservoir may be characterized3 with Eq. A-51.

( ) ( ) ( ) ( )[ ]( ) ( ) ( )

2 * *

* *

*

2exp

ln 0.25 , 0 ,

ln 0.25 , 0 ,, ,

D

eD eD D D eD

eD D D eD

wD D eD D

t

r r X X r

r X X rq X r t

σ δ

σ δ

−

+ + +=

+ + +

(A-51) The dimensionless cumulative production behavior of a fractured well that is centrally located in a closed cylindrically

bounded reservoir during the fully developed boundary-dominated flow regime may also be evaluated using Eq. A-52. The imaging pseudoskin term due to the bounded nature of the reservoir ( )*,D eDX rδ is defined in Ref. 1.

( )* , ,pD D eD DQ X r t =

( ) ( ) ( )2

2 * *221 exp

ln 0.25 ,0 ,eD D

eD eD D D eD

r tr r X X rσ δ

−−+ + +

(A-52) Similarly, the boundary-dominated flow regime rate-transient behavior of a fractured well that is located in a closed rectangularly bounded reservoir is given by Eq. A-53.

( ) 1 2, , , , , , exp DwD wD wD eD eD D D D

eD eD

tq X Y X Y X Y tX Y

πξ ξ

−=

(A-53) where:

2 2

2

123 2

eD D D wD

eD eD eD

Y Y Y YX Y Y

ξ π +

= − +

21

2 1 sin cos coseD wD D

eD eD eDm

X m m X m Xm X X X

π π ππ

∞

=

+

∑ i

( )cosh cosh

sinh

eD D wDeD D wD

eD eD

eD

eD

Y Y YY Y Ym m

X Xm Y

X

π π

π

− +− −+

(A-54) The dimensionless cumulative production of a fractured well located in a closed rectangularly bounded reservoir during the boundary-dominated flow regime is given by Eq. A-55.

( ) 2, , , , , , 1 exp

2

eD eD DpD wD wD eD eD D D D

eD eD

X Y tQ X Y X Y X Y t

X Y

π

π ξ

−= −

(A-55) For both of these drainage area shapes, the relationship between the “material balance” time function and the equivalent superposition time function is given in general form by Eq. A-56.

( )( )

( )( )

1 expexp

mb n

e n

t t ut t u u

− −=

− (A-56)

where: 2 D

D

tuAπξ

= (A-57)

For rectangularly bounded reservoir cases, the parameter ( )ξ has been defined in Eq. A-54. For a cylindrically bounded reservoir, the appropriate value of this parameter is given by Eq. A-58.

( ) ( ) ( )* *ln 0.25 ,0 ,eD D D eDr X X rξ σ δ= + + + (A-58)

16 B. POE SPE 77691

Fig. 1 – Cartesian graph of tmbD versus tD for vertically

fractured well in infinite-acting reservoir.

Fig. 2 – Semi-log graph of tmbD/tD versus tD for vertically

fractured well in infinite-acting reservoir.

Fig. 3 – Example vertically fractured gas well with known flowing pressures.

Fig. 4 – Vertical oil well example with ESP (no pressures), match for oil effective permeability.

Fig. 5 – Vertical oil well example with ESP, evaluation of skin effect and drainage area.

1

1.5

2

2.5

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06tD

tmbD

/t D

CfD = 0.1CfD = 1CfD = 10CfD = 100CfD = 1000CfD = 10000

0

20000

40000

60000

80000

100000

120000

0 20000 40000 60000 80000 100000 120000tD

tmbD

CfD = 0.1CfD = 1CfD = 10CfD = 100CfD = 1000CfD = 10000