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Abstract This paper discusses the use of quantitative production diagnostic analyses for unfractured vertical wells and finite-conductivity vertically fractured wells that have partial or entirely absent well flowing pressure records corresponding to the flow rate data. The production rate data are almost always available for a production performance analysis of a well, since the hydrocarbon production relates directly to the revenue from the well and often the water production data is available since it is a production or operating cost. However, the corresponding flowing pressure data are often incomplete, or in the case of artificially lifted wells, the flowing pressure data are generally not available. The diagnostic analyses considered in this paper are complimentary to the decline curve analysis matching procedure reported in an earlier publication that utilized an effective convolution analysis that does not require that the flowing pressures be known for each of the production data points in the production history of a well. The use of diagnostic analyses procedures in conjunction with the log-log production decline curve analyses can dramatically reduce the uniqueness problems often encountered in production decline analysis type curve matching procedures using the effective convolution analysis methodology.

Introduction Rate-transient based decline curve analyses have been in use in the petroleum industry for a long time1-3. Rate-transient solutions of the diffusivity equation describing fluid flow in porous media are those that are derived for a constant terminal pressure (Dirichlet) inner boundary condition, whereas pressure-transient solutions are those obtained for a specified flux or rate (Neumann) inner boundary condition. Arps1 reported that an empirical relationship could be used to match the boundary-dominated flow rate decline trend of flowing oil wells. Van Everdingen and Hurst2 also reported

pressure and rate-transient analytic solutions for unfractured wells in infinite-acting reservoirs and finite reservoirs with noflow and constant pressure outer boundary conditions. The paper by van Everdingen and Hurst2 also represented a couple of other significant firsts for the petroleum industry. One of these was the extensive use of the Laplace transformation technique to obtain transient solutions of the diffusivity equation governing fluid flow in porous media. The popularity of the Laplace transformation technique for solving reservoir fluid flow problems greatly increased after 1970 with the development and use of efficient numerical inverse Laplace transformation algorithms4 which utilize real space function evaluations, and improvements in modern computing capabilities. Another important result of van Everdingen and Hursts work2 is that the authors demonstrated the fundamental relationship between the pressure and rate-transient solutions of fluid flow in porous media, which utilized the convolution integral (Faltung Theorem) that accounts for the superposition-in-time effects of a varying pressure and flow rate production history. While both the Arps1 empirical decline analysis for boundary-dominated flow production performance and the van Everdingen and Hurst2 transient solutions are both useful in their own rights, matching with either set of solutions alone can often be quite cumbersome, suffer from uniqueness problems, or even result in unsuccessful decline analysis matches. The popularity of production rate decline analyses for evaluating well and reservoir parameters using the existing production decline behavior of flowing wells greatly increased in 1973 when Fetkovich3 introduced a composite decline curve that coupled the early time transient solutions of an unfractured vertical well (analogous to the van Everdingen and Hurst2 solutions) with the late-time (boundary-dominated flow) empirical decline stems developed by Arps1. The coupling of the transient and late-time (Arps1) stems on a single composite decline curve set was accomplished by Fetkovich3 using empirically derived scaling factors. The Fetkovich3 composite decline curve has subsequently been used extensively in the petroleum industry for the analysis of existing production data and the extrapolation of production rate decline trends of flowing oil wells to future time levels. Blasingame and coworkers5-9 have more recently reported composite production decline type curve sets and analyses for unfractured vertical, vertically fractured, and horizontal wells that utilize a theoretical basis for coupling of the transient and boundary-dominated flow decline stems. The theoretical basis for the coupling of the early and late time behavior on those

SPE 84224

Production Diagnostic Analyses With Incomplete or No Pressure Records B. D. Poe Jr., SPE, Schlumberger

2 B. D. Poe Jr. SPE 84224

composite decline curve sets was most clearly demonstrated in Ref. 9 for vertically fractured injection well analyses. Shih and Blasingame7 extended the composite decline curve solution methodology to applications for horizontal well analyses. In each of these publications5-9, as well as a later publication by Agarwal et al10, the Horner approximation of the pseudoproducing time (referred to as the material balance time function) has been utilized. As has been clearly demonstrated both numerically and analytically in Refs. 11 and 12, the Horner approximation of the pseudoproducing time (a.k.a. material balance time function) relationship to the superposition equivalent time of a vertically fractured well varies considerably over the transient life of the well. An illustration of this variation in the relationship between the superposition equivalent time and Horners approximation of the pseudoproducing time for a vertically fractured well in an infinite-acting reservoir is presented in Figs. 1 through 3. Figure 1 is a Cartesian graph of the dimensionless material balance time as a function of the corresponding superposition equivalent time function and fracture dimensionless conductivity. The same data is also presented on a log-log graph in Fig. 2. Both of these depictions of the relationship between the two time functions, while numerically correct, are very deceptive. The same numerical solution data for a vertically fractured well is also presented in Fig. 3 as the ratio of the material balance time function to the corresponding superposition equivalent time function, plotted as a function of the superposition equivalent time function scale. As has been reported in Refs. 11 and 12, the error associated with utilizing an uncorrected material balance time function in production decline curve analyses can be as much as 200% in the time scale for vertically fractured well analyses. Similar comparisons for unfractured vertical and horizontal wells are presented in Figs. 4 through 7. A Cartesian graph of the dimensionless material balance time function versus the superposition equivalent time function for an unfractured vertical well with a range of steady state skin effect values is presented in Fig. 4. The correlation between the two time functions does appear to be linear, though not a one-to-one relationship. A much clearer picture of the actual relationship between the two time functions is given in Fig. 5. Note that the only osteady state skin effect values for which there is actually a linear relationship between the two time functions, are for damage skin effect values of 100 or more. For the completeness of the discussion of the relationship between Horners approximation of the pseudoproducing time and the superposition-in-time equivalent time function for all well types, and since horizontal wells have been considered in Ref. 7, the corresponding comparisons of these two time functions for horizontal wells is presented in Figs. 6 and 7. The numerical results presented in these figures were generated using the uniform flux well solution reported by Ozkan13, evaluated at a wellbore location that results in an equivalent infinite conductivity wellbore in an infinite-acting reservoir response. The Cartesian graph of the dimensionless material balance time function versus the superposition equivalent time function indicates a general linear trend between the two time functions. However, properly depicting

the transient relationship between the two time functions as presented in Fig. 7 clearly demonstrates that the relationship is not linear at all.

These results are not surprising at all, since one would not expect an approximation (the Horner approximation of the pseudoproducing time) to accurately represent the rigorous superposition time function. At this point in the discussion, it is beneficial to introduce two fundamental inequalities that govern the pressure and rate-transient behavior of all well types and all flow regimes, as well as the relationship between the dimensionless material balance (Horner approximation of the pseudoproducing time) and the rigorous superposition equivalent time functions. These inequalities are offered without proof (left to the reader), since is well beyond the scope of this paper, but both can and have been readily proven in this research effort.

The first of these inequalities is the fundamental relationship between the reciprocal of the dimensionless wellbore rate-transient solution and the corresponding dimensionless wellbore pressure-transient solutions. In all flow regimes of all well types, the reciprocal of the wellbore rate-transient solution is greater than or equal to the corresponding pressure-transient solution at the same value of dimensionless time. This inequality is presented mathematically in Eq. 1. It is applicable to all well types and flow regimes, including steady state and boundary-dominated flow in closed finite reservoirs.

( ) ( )1

wD DwD D

p tq t

(1)

The second inequality that will be introduced in this

discussion involves the fundamental relationship between the dimensionless Horner approximation of the pseudoproducing (material balance) time and the dimensionless superposition equivalent time functions. This fundamental inequality is also applicable to all well and reservoir types, as well as all flow regimes. This inequality states that the dimensionless Horner approximation of the pseudoproducing time function is always greater than or equal to the corresponding superposition equivalent time function. This relationship is expressed mathematically as given in Eq. 2.

( ) ( )Dmb Dt t t t (2)

The second inequality (Eq. 2) results from the first

inequality given (Eq. 1) and the fundamental definition of the rate-transient analysis dimensionless material balance time function introduced in references 11 and 12. The definition of the dimensionless material balance time function for rate-transient analyses is defined mathematically as is given in Eq. 3.

( )( )

( )pD D

Dmb DwD D

Q tt t

q t= (3)

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 3

With these fundamental relationships (Eqs. 1-3), the dimensionless rate-transient solutions for the well and reservoir types of interest, and the appropriate analytic approximations for the rate-transient behavior of the various flow regimes of the well and reservoir types that we wish to consider, we can construct production analysis diagnostic plotting procedures using a corrected Horner approximation of the pseudoproducing (material balance) time function that are analogous to those reported in Refs. 14 and 15. The following section of this paper presents the development work for implementation of this type of production analysis plotting procedures that aid in production decline curve analysis matching procedures where a corrected material balance time function is used to approximate the superposition equivalent time function in the analyses.

The flow regime specific analytic expressions for the rate-transient behavior of a well intersected by a finite-conductrvity fracture have been previously presented in detail in Refs. 11, 14, and 15 and therefore will not be presented again in this paper. However, the rate-transient flow regime specific solutions or approximations for unfractured vertical wells has not been effectively reported in the literature and are therefore included in Appendix A. Appendix B details a systematic procedure for computing estimates of the sandface flowing pressure history from the decline curve analysis match, with the flow rate versus cumulative production decline curve analysis matching procedure for which some or all of the flowing sandface pressures are unknown that has been detailed in Ref. 11.

Development of Diagnostic Analyses Diagnostic plotting functions for unfractured vertical and vertically fractured wells in which an incomplete or absent well flowing pressure history is available, can be developed using the flow regime specific analytic solutions or approximations for each well type, a corrected material balance time function that is coupled with the appropriate rate-transient solution for the well and reservoir type under consideration, and the diagnostic plotting function techniques employed in Refs. 14 and 15. The only major difference between the resulting diagnostic plotting procedures of Ref. 14 and this paper is the use of the corrected material balance time function in place of the superposition time functions for each of the flow regimes of the well. In unfractured vertical wells, the infinite-acting radial flow regime can be characterized with a diagnostic graph of flow rate normailized pressure drop versus a corrected material balance time function. With the analytic expressions for the rate-transient behavior of an unfractured vertical well during the infinite-acting reservoir radial flow regime presented in Appendix A of this paper, a verification or validation diagnostic graph of the production decline analysis match can be directly computed. By coupling the two analyses, the flow rate versus cumulative production decline curve analysis matching and the radial flow verification diagnostic graph, the analyst can immediately visually determine the aptness of the decline analysis current match. Similarly, during the boundary-dominated flow regime of an unfractured vertical well, a verification or match validation diagnostic plot of the flwo rate normalized pressure drop

versus the corrected material balance time function will indicate whether the late time decline curve match selected is appropriate. Specialized diagnostic plotting procedures for vertically fractured wells are also constructed in a similar manner using the flow regime specific rate-transient analytic solutions presented in Refs. 11 and 14. Of particular interest are the diagnostic verification graphs for the bilinear, pseudolinear (or formation linear), pseudoradial, and boundary-dominated flow regimes. The corrected material balance time function resulting from the decline curve analysis match selection is used to generate the time scale used in the verification plotting procedures. Two verification plots are prepared for each flow regime analysis, a flow rate normalized pressure drop (or pseudopressure drop for gas reservoir analyses) versus the corrected material balance time function graph and a pressure drop normalized cumulative production versus corrected material balance time function graph. Each of these verification graphs must indicate acceptable agreement between the data plotting function and the computed characteristic straight line behavior of the flow regime that corresponds to the analytic solution for that flow regime and the match results derived from the flow rate versus cumulative production decline curve analysis match. Once an acceptable match has been obtained with the coupled flow rate versus cumulative production decline curve analysis and the corresponding flow regime specific verification diagnostic analyses, the production history data points for which the sandface flowing pressures are unknown can be evaluated using the decline analysis matched solution and the computational procedures detailed in Appendix B of this paper. The evaluation of the sandface flowing pressures in this manner permits a reconstruction of the sandface flowing pressure history that corresponds to the wells flow rate decline and cumulative production history. Examples of the diagnostic verification plotting proceduires that have been discussed in this paper are given in Figs. 8 through 11 for a vertically fractured well. The bilinear and pseudolinear verification graphs are presented in Figs. 8 and 9 for the bilinear flow regime and in Figs. 10 and 11 for pseudolinear flow. Similar match validation graphs for unfractured wells can also be constructed in a similar manner using the appropriate well specific rate-transient solutions and the corrected material balance time function. Conclusions The major conclusions that can be drawn from this and ongoing development work in this research area are the following: 1. An effective convolution analysis of a wells production

performance that has a partial or absent flowing pressure record can be accomplished using a corrected material balance time function and the well and reservoir specific rate-transient solutions.

2. Specialized flow regime diagnostic analyses developed using a corrected material balance time function in place of the superposition time function for wells with incomplete or absent flowing sandface pressure histories, aid in the decline curve analysis matching procedure by

4 B. D. Poe Jr. SPE 84224

providing a validation or verification of the decline curve analysis match selected.

3. The analysis technique presented in this paper for the reconstruction of a wells sandface flowing pressure history using the decline curve analysis match results and the wells available production history (flow rates and cumulative production) provides a means of effectively evaluating how the sandface flowing pressures of the well over its production history. The synthetic sandface flowing pressures generated in this manner can also be checked against any measured bottomhole pressures that have been recorded during the wells production history, offering even further validation of the decline curve analysis match obtained.

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.

Nomenclature A Well drainage area, ft2

DA Dimensionless drainage area, 2/D CA A L= oB Oil formation volume factor, rb/STB

tc Reservoir total system compressibility, 1/psia pG Cumulative gas production, MMscf

h Reservoir net pay thickness, ft 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

Dt Dimensionless time Dit ith dimensionless time in production history it ith time level in production history, hr nt Last or current time level in production history, hr SCT Standard condition temperature, deg R DX Dimensionless X direction spatial position eDX Dimensionless X direction drainage areal extent 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 Dimensionless parameter Reservoir effective porosity, fraction BV

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

o Oil viscosity, cp Functions exp Exponential function ln Natural logarithmic function l i Logarithmic integral Ei Exponential integral function

Acknowledgements The author would like to express his appreciation to Schlumberger for the permission to publish the results of this research effort. References 1. Arps, J.J.: Analysis of Decline Curves, Trans., AIME

(1945) 160, 228-247. 2. Fetkovich, M.J. Decline Curve Analysis Using Type

Curves, JPT (June 1980) 1065-1077. 3. van Everdingen, A.F. and Hurst, W.: The Application of

the Laplace Transformation to Flow Problems in Reservoirs, Trans., AIME (1949) 186, 305-324.

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

5. 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.

6. 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.

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 5

7. 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.

8. 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.

9. 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.

10. 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.

11. Poe, B.D. Jr.: Effective Well and Reservoir Evaluation Without the Need for Well Pressure History, paper SPE 77691 presented at the 2002 Annual Technical Conference and Exhibition, San Antonio, TX, Sept. 29-Oct. 2.

12. 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.

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

14. 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.

15. England, K.W., Poe, B.D. Jr., and Conger, J.G.: Comprehensive Evaluation of Fractured Gas Wells Utilizing Production Data, paper SPE 60285 presented at the 2000 SPE Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, Mar. 12-15.

16. Lee, W.J.: Well Testing, SPE Textbook Series, 1, SPE, Richardson, TX, (1982).

Appendices Appendix A Approximations of the Transient Behavior of Unfractured Vertical Wells Specialized production diagnostic analyses can be developed for unfractured vertical or slanted wells by considering the two prevalent flow regimes that may be exhibited by these types of wells in finite reservoirs, the infinite-acting radial flow and fully boundary-dominated flow regimes. Rather simple expressions can be utilized for each of these flow regimes of an unfractured vertical well. Infinite-Acting Radial Flow Regime To begin, we consider the commonly used logarithmic approximation for the pressure-transient behavior of an

unfractured vertical well exhibiting infinite-acting reservoir radial flow, given by Eq. A-1.

( ) ( )1 ln 4 22

wD D Dp t t S + (A-1)

The corresponding rate-transient approximation of the infinite-acting radial flow behavior of a vertical unfractured well may be written as given by Eq. A-2.

( ) ( )2

ln 4 2wD D

Dq t

t S

+ (A-2)

As noted in Ref. 11, this approximation can be obtained by either (1) assuming steady state conditions, or (2) by applying an approximate Laplace transform inversion procedure to the Laplace space pressure-transient approximation of this flow regime, obtained from Eq. A-1. The dimensionless cumulative production of an unfractured vertical well during the infinite-acting reservoir radial flow regime can be estimated using Eq. A-3 for

12.5Dt ,

( ) ( )02

ln 4 2D

pD Dt dQ t

S

+ (A-3) or more simply as:

( ) ( ) 22 li , 4D SpD D a tQ t a ea

= (A-4)

The logarithmic integral (li) appearing in Eq. A-4 is fundamentally related to the exponential integral function as defined in Eq. A-5. Therefore, readily available approximations of the exponential integral function can be used to evaluate function values of the logarithmic integral.

( ) ( )( )li Ei ln , 0x x x= > (A-5) Note also that Lee16 has reported a reasonably accurate approximation of the cumulative production behavior of an unfractured vertical well during the infinite-acting reservoir radial flow regime, except that it does not directly permit the consideration of a steady state skin effect. This approximation of the dimensionless cumulative production of an unfractured vertical well during the infinite-acting radial flow regime is given by Eq. A-6.

( ) ( )4.29881 2.02566 , 200

lnD

pD D DD

tQ t tt

+ > (A-6)

6 B. D. Poe Jr. SPE 84224

Fully Developed Boundary-Dominated Flow Regime The rate-transient behavior of an unfractured vertical well in a finite, closed -boundary drainage area during the fully developed boundary-dominated flow regime is given by Eq. A-7.

( ) 1 2exp DwD DD

tq tA

=

(A-7)

For a well centered in a closed cylindrical drainage area, the imaging function ( ) is defined by Eq. A-8. Similarly, the imaging function for fully developed boundary-dominated flow in a closed rectangular drainage area (in which the well can be arbitrarily located in the drainage area) is given by Eq. A-9. The corresponding late time approximations of the pressure-transient behavior of an unfractured vertical well in a finite closed cylindrical or rectangular reservoir have been previously reported by Ozkan13.

( ) 3ln4

eDr = (A-8)

2 2

2

123 2

eD D D WD

eD eD eD

Y YY YX Y Y

+

= +

1

12 cos coswD DeD eDm

m X m Xm X X

=

+

( )cosh cosh

sinh

eD D wDeD D wD

eD eD

eD

eD

Y Y YY Y Ym m

X Xm Y

X

+ +

(A-9) The dimensionless cumulative production of an unfractured vertical well in a finite, closed boundary reservoir is described by the relationship given in Eq. A-10. The imaging functions utilized in this relationship have been given in Eqs. A-8 and A-9 for cylindrically and rectangularly bounded reservoirs, respectively.

( ) 21 exp2

D DpD D

D

A tQ tA

=

(A-10)

Appendix B Evaluation of Missing Well Flowing Pressure History Values The convolution integral that relates the effects of a varying flow rate and flowing pressure history of a well with the wellbore rate-transient solution may be expressed in discrete time form as given in Eq. B-1.

( ) ( ) ( )1

1

11

n

wD Dn Di D Dn Di D Dn Di

in

q t p q t t q t t

=>

=

( )1D Dn Dnq t t + (B-1)

Similarly, the superposition of the effect of a varying flowing pressure history on the cumulative production of the well may be expressed in discrete form as given by Eq. B-2.

( ) ( ) ( )1

1

11

n

pwD Dn Di pD Dn Di pD Dn Di

in

Q t p Q t t Q t t

=>

=

( )1pD Dn DnQ t t + (B-2)

For liquid fluid flow, the dimensionless pressures appearing in Eqs. B-1 and B-2 are defined by Eq. B-3. Similarly, the dimensionless pseudopressures for gas reservoir analyses are given by Eq. B-4.

( )( )

i wfs iDi

i wfs n

p p tp

p p t

=

(B-3)

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

p i p wfs iDi

p i p wfs n

p p p p tp

p p p p t

=

(B-4)

At the initial production data point ( )1n = in the production history, the flowing pressure can be directly evaluated from the matched decline curve values. For liquid flow, substitution into the superposition relationships for the definition of the dimensionless variables permits the computation of the flowing pressure. In conventional oilfield units, the liquid flow flowing pressures are evaluated using the flow rate and cumulative production with Eqs. B-5 and B-6, respectively.

( ) ( )( )1

11

141.205 o o owfs i

o Dm

q t Bp t p

k h q t

= (B-5)

( ) ( )( )

11

211.11909

p owfs i

t pDmc

N t Bp t p

c h L Q t= (B-6)

Similarly, the flowing sandface pseudopressure at the initial production point can be evaluated using the flow rate and cumulative production relationships for gas reservoir analyses with Eqs. B-7 and B-8, respectively.

( )( ) ( ) ( )( )1

11

50300 sc gp wfs p i

g sc Dm

p T q tp p t p p

k h T q t= (B-7)

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 7

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

11

21 1

318313 sc pp wfs p i

g t sc pDmc

p T G tp p t p p

h c t T L Q t =

(B-8) For successive production data points ( )1n > , application of the discrete-time form of the rate-transient convolution relationships given in Eqs. B-1 and B-2, in a sequential and systematic manner, results in an effective and efficient means of directly evaluating the sandface flowing pressure history of the well. For liquid flow, the rate-transient convolution relationships can be readily rearranged to solve for the sandface flowing pressure at each point in the production history of the well ( )nt .

( ) ( )141.205 o n o owfs n io

q t Bp t p

k h

=

( )( ) ( ) ( )1

111

n

i wfs i Dm Dn Di Dm Dn Diin

p p t q t t q t t

=>

( )1/ Dm Dn Dnq t t (B-9) A similar expression for evaluating the liquid flow sandface flowing pressure history can be obtained with the cumulative production convolution relationship.

( ) ( ) 21.11909p n o

wfs n it c

N t Bp t p

c h L

=

( )( ) ( ) ( )1

111

n

i wfs i pDm Dn Di pDm Dn Diin

p p t Q t t Q t t

=>

( )1/ pDm Dn DnQ t t (B-10) The sandface flowing pressure (or pseudopressure) of a gas well can also be evaluated from the rate-transient decline analysis match using the flow rate and cumulative production convolution relationships, given by Eqs. B-11 and B-12, respectively.

( )( ) ( ) ( )50300 sc g np wfs n p ig

p T q tp p t p p

k h Tsc=

( ) ( )( )( ) ( ) ( )[ ]1

1

11

n

p i p wfs i Dm Dn Di Dm Dn Di

in

p p p p t q t t q t t

=>

( )1/ Dm Dn Dnq t t (B-11)

( )( ) ( ) ( )( ) 2318313 sc p n

p wfs n p ig t n sc c

p T G tp p t p p

h c t T L

=

( ) ( )( )( ) ( ) ( )[ ]1

1

11

n

p i p wfs i pDm Dn Di pDm Dn Di

in

p p p p t Q t t Q t t

=>

( )1/ pDm Dn DnQ t t (B-12)

8 B. D. Poe Jr. SPE 84224

Fig. 1 Cartesian graph of dimensionless material balance time function versus superposition equivalent time function for finite coinductivity fractured wells.

Fig. 2 Log-log graph of dimensionless material balance time function versus superposition equivalent time function for finite coinductivity fractured wells.

0

20000

40000

60000

80000

100000

120000

0 20000 40000 60000 80000 100000tD

t mbD

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

0.0000010.00001

0.00010.001

0.010.1

110

1001000

10000100000

1000000

0.000001 0.0001 0.01 1 100 10000 1000000

tD

t mbD

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

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 9

Fig. 3 Log-log graph of ratio of dimensionless material balance time and superposition equivalent time functions versus superposition equivalent time function for vertically fractured wells.

Fig. 4 Cartesian graph of dimensionless material balance time versus superposition equivalent time functions for unfractured vertical wells with various values of skin effect.

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

t Dm

b/tD

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

Vertical Well - Skin Effect

0.E+00

2.E+19

4.E+19

6.E+19

8.E+19

1.E+20

1.E+20

0.E+00 2.E+19 4.E+19 6.E+19 8.E+19 1.E+20 1.E+20

tD

t Dm

b

S=-7S=-5S=-3S=-2S=-1S=0S=1S=2S=3S=5S=7S=10S=20S=30S=50S=70S=100S=200S=300S=500

10 B. D. Poe Jr. SPE 84224

Fig. 5 Log-log graph of ratio of dimensionless material balance time and superposition equivalent time functions versus superposition equivalent time for unfractured vertical wells with skin effect.

Fig. 6 Cartesian graph of dimensionless material balance time versus superposition equivalent time functions for horizontal wells with various dimensionless well lengths.

Vertical Well - Skin Effect

1

10

1.E+00 1.E+04 1.E+08 1.E+12 1.E+16 1.E+20tD

tDm

b/t D

S=-7 S=-5 S=-3 S=-2S=-1 S=0 S=1 S=2S=3 S=5 S=7 S=10S=20 S=30 S=50 S=70S=100+

Horizontal Well - (ZwD=0.5, rwD = 0.003)

020406080

100120140

0 20 40 60 80 100 120

tD

t Dm

b

LD = 2 LD = 3LD = 5 LD = 7LD = 10 LD = 20LD = 30 LD = 50LD = 70 LD = 100LD = 200 LD = 300

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 11

Fig. 7 Log-log graph of ratio of dimensionless material balance time and superposition equivalent time functions versus superposition equivalent time for horizontal wells.

Fig. 8 Cumulative production verification graph of bilinear flow regime behavior from decline curve analysis match.

Horizontal Well - (ZwD=0.5, rwD=0.003)

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

tD

tDm

b/t D

LD = 2 LD = 3LD = 5 LD = 7LD = 10 LD = 20LD = 30 LD = 50LD = 70 LD = 100LD = 200 LD = 300

Pseudo Pressure Normalized Cumulative Production Analysis - Bilinear Flow Regime

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

6.00E-07

7.00E-07

8.00E-07

9.00E-07

1.00E-06

0.00E+00 5.00E+06 1.00E+07 1.50E+07 2.00E+07 2.50E+07 3.00E+07

Superposition-In-Time Function

Gp

/ Pp(

Pi) -

Pp(

Pwf)

12 B. D. Poe Jr. SPE 84224

Fig. 9 Flow rate verification graph of bilinear flow regime behavior from decline curve analysis match.

Fig. 10 Cumulative production verification graph of pseudolinear flow regime behavior from decline curve analysis match.

Rate Normalized Pseudo Pressure Analysis - Bilinear Flow Regime

0.00E+00

5.00E+05

1.00E+06

1.50E+06

2.00E+06

2.50E+06

3.00E+06

0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02 3.50E+02 4.00E+02

Superposition-In-Time Function

Pp(P

i) - P

p(Pw

f) / q

g

Pseudo Pressure Normalized Cumulative Production Analysis - Formation Linear Flow

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

6.00E-07

0.00E+00 2.00E+04 4.00E+04 6.00E+04 8.00E+04 1.00E+05 1.20E+05

Superposition-In-Time Function

Gp

/ Pp(

Pi) -

Pp(

Pwf)

SPE 84224 Production Diagnostic Analyses With Incomplete Or No Pressure Records 13

Fig. 11 Flow rate verification graph of pseudolinear flow regime behavior from decline curve analysis match.

Rate Normalized Pseudo Pressure Analysis - Formation Linear Flow Regime

0.00E+00

1.00E+06

2.00E+06

3.00E+06

4.00E+06

5.00E+06

6.00E+06

7.00E+06

8.00E+06

9.00E+06

1.00E+07

0.00E+00 2.00E+04 4.00E+04 6.00E+04 8.00E+04 1.00E+05 1.20E+05 1.40E+05 1.60E+05 1.80E+05 2.00E+05

Superposition-In-Time Function

Pp(P

i) - P

p(Pw

f) / q

g