[Blasingame] SPE 107967

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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & GasTechnology Symposium held in Denver, Colorado, U.S.A., 1618 April 2007.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than300 words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment 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.

Abst ract

In this work we present the application of the -integral

derivative function for the interpretation and analysis of pro-duction data. The -derivative function was recently proposed

for the analysis and interpretation of pressure transient data

[Hosseinpour-Zonoozi, et al(2006)], and we demonstrate thatthe -integral derivative and its auxiliary functions can be used

to provide the characteristic signatures for unfractured and

fractured wells.

The purpose of this paper is to demonstrate the application of

the "production data" formulation of the -derivative function

(i.e., the -integral derivative) for the purpose of estimatingreservoir properties, contacted in-place fluid, and reserves.Our main objective is to introduce a new practical tool for the

analysis/interpretation of the production data using a new

diagnostic rate and pressure drop diagnostic function.

This paper provides the following contributions for theanalysis and interpretation of gas production data using the -

integral derivative function:

Schematic diagrams of various production data functionsusing the -integral derivative formulation (type curves).

Analysis/interpretation of production data using the -

integral derivative formulation.

IntroductionThis work introduces the new -integral derivative functions([qDdi(tDd)]and [pDdi(tDd)]) where these functions aredefined to identify the transient, transition, and boundary-

dominated flow regimes from production data analysis. Wehave utilized two different formulations [qDdi(tDd)]is used

for "rate decline" analysis (based on q/p functions) and[pDdi(tDd)] is used for "pressure" analysis (based on p/qfunctions).

The application (i.e., the use of [qDdi(tDd)]or[pDdi(tDd)]) isessentially a matter of preference there is no substantive

difference in the application of these functions. Some analysts

prefer the "pressure" analysis format because of the similaritywith pressure transient analysis, while others are more com-

fortable with "rate decline" analysis.

The -integral derivative functions are derived in completedetail in Appendix A, and the primary definitions are sum-

marized as follows:

)(

)()]([

DdDdi

DdDdidDdDdi

tq

tqtq = ................................................ (1

)(

)(

)]([ DdDdi

DdDdidDdDdi tp

tp

tp = .............................................. (2

The definitions of the component functions used in Eqs. 1 and2) are given as follows:

Function Definition

Rate Integral dqt

ttq Dd

Dd

DdDdDdi )(

0

1

)( = .........(3)

Rate Integral-Derivative )()( DdDdi

DdDdDdDdid tq

dt

dttq = .....(4)

Pressure

Integral dp

t

ttp Dd

Dd

DdDdDdi )(0

1

)(= ........(5)

PressureIntegral-

Derivative

)()( DdDdiDd

DdDdDdid tpdt

dttp = .......(6)

The associated definitions of these functions are provided in

Appendix B and are referenced as appropriate in the

Nomenclature.

In addition to the definitions of the the-integral derivative

functions, we have created an "inventory" of "type curve"

solutions for unfractured and fractured wells this inventory

is provided in Appendix C.

Orientation

As noted above, our inventory of solutions is provided in Ap

pendix C these solutions were selected for relevance (i.e.the likelihood of a practical need), but also for the value of

each case as schematic example (i.e., the resolution of flow

regime(s)).

We first consider the "decline rate" case [qDdi(tDd)] and as

sociated functions) as shown in schematic form in Fig. 1. This

schematic plot (or "type curve") consists of unfractured and

fractured well cases for comparison including the ellipticalflow geometry solution for a fractured well [Amini et a(2007)] where we note that these are high fracture conduct

SPE 107967

Application of the -Integral Derivative Function to Production AnalysisD. Ilk, SPE, N. Hosseinpour-Zonoozi, SPE, S. Amini, SPE, and T.A. Blasingame, SPE, Texas A&M U.

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2 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967

ivity cases, and fractured well solutions are very similar (near-

ly identical) in this circumstance.

10-3

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Dimensionless Material Balance Decline Time, tDd,bar=NpDd/qDd

Legend: (qDdid ) ( [qDdi] )

Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivi ty) Fractured Well (Elliptical Reservoir)

Transient FlowRegion

Schematic of Dimensionless Rate Integral Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)

DIAGNOSTIC plot for Production Data(qDdidand [qDdi] )

DimensionlessRateIntegralDerivativeFunction,qDdid

"PowerLaw

"DimensionlessRateIntegralD

erivativeFunction,

[qDdi]

Boundary-Dominated

Flow Region

[qDdi] ~ 1.0(boundary

dominated flow)

1

1

1

2

Unfractured Well ina Bounded Circular

Reservoir

Fractured Well i na Bounded Elliptical

Reservoir(FiniteConductivityVertical Fracture)

Fractured Well ina Bounded Circular


( )( )

( )( )

( )( )

NO Wellbore Storageor Skin Effects

[qDdi] = 0.5

(linear flow)

Figure 1 Schematic of qDdi(tDd)] vs. tDd Unfractured and

fractured well configurations.

Next we consider the pressure transient analysis analog case([pDdi(tDd)]and associated functions) as shown in Fig. 2. Themajor difference in Fig. 2compared to Fig. 1(other than the

functions being inverted) is that we can clearly diagnose tran-

sient radial and linear flow (fracture cases). In addition, the

boundary-dominated flow portion of the data is clearly evident

as viewed from [pDdi(tDd)](or the pDd(tDd)and pDdi(tDd)func-tions). As we noted earlier, the use of the qDd(tDd)or pDd(tDd)-

format functions is a matter of preference, either (or preferably

both) sets of functions can be used at the same time.

10-3

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Dimensionless Material Balance Decline Time, tDd,bar=NpDd/qDd

Legend: (pDdid ) ( [pDdi] )

Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conducti vity) Fractured Well (Elliptical Reservoir)


Schematic of Dimensionless Pressure Integral Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)

DIAGNOSTIC plot for Production Data(pDdidand [pDdi] )

DimensionlessPressureIntegralDerivativeFunction,pDdid

"PowerLaw

"DimensionlessPressureIntegralDerivativeFunction

,

[pDdi]

Boundary-Dominated

Flow Region

[pDdi] = 1.0

(boundarydominated flow)

1

1

1

2


Reservoir

Fractured Well ina Bounded Elliptical




( )( )

( )( )

( )( )


[pDdi] = 0.5

(linear flow)

Figure 2 Schematic of pDdi(tDd)] vs. tDd Unfractured and

fractured well configurations (pressure transientanalog format).

Appl ication of the -Integral Deri vat ive Funct ion toProduction Analysis Field Examples

In this section we provide field examples to demonstrate/illustrate the diagnostic value of the -integral derivative func-

tion and its applications in production analysis. The main pur-

pose of this exercise is to provide the diagnostic value of the

-integral derivative function rather than focusing on it as adirect solution mechanism. Our results using the -integral

derivative function are compared with the results from

conventional (i.e., established) production-analysis methods.

Example 1: Southeast Asia Oil Well

In this case we have the measured rate and pressure data for an

oil well daily rates and bottomhole flowing pressures areavailable and are used. Fig. 3 shows the time-pressure-rate

(TPR) data for this case. We note that the data are well

correlated except for an abrupt decline in rates at late times which we believe indicates the evolution of wellbore damage.

For this analysis, we have chosen to use the rate decline

integral functionsto overcome the data-quality issues and the

material balance timefunction to eliminate (at least to someextent) the variable-rate/variable pressure drop effects. In Fig

4we present the field data and model matches for the qDd, qDdi[qDdi(tDd)] "decline" functions in dimensionless (decline

format where the "data" functions are given by symbols.

102

103

104

OilFlowra

te,qo,

STB/D

5000

4500

4000

3500

3000

2500

2000

1500

1000

5000

Production Time, t, hours

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

0

We

llbore

Flow

ing

Pressure,pwf,ps

ia

Wellbore FlowingPressure

Oil Flowrate

Example 1 Exploration Well (Southeast Asia)

Legend:qo Data Function

pwfData Function

Figure 3 Example 1: Time-Pressure-Rate (TPR) history plotSoutheast Asia oil well very good correlation o

rates and pressures.

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

DimensionlessRateDeclineFunctions

(qDd

(tDd),qDdi(tDd

),and

[qDdi(tDd

)])

tDd,bar=NpDd (tDd)/qDd(tDd)

Fetkovich-McCray Rate Function Type Curve

Unfractured Well Centered in a Bou nded Circular Reservoir (reD= 1x104)

Example 1 Southeast Asia Oil Well

Model Legend: Fetkovich-McCray Rate FunctionType Curve - Unfractured Well Centered in a Bounded

Circular Reservoir (Dimensionless Radius: reD= 1x104)

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]

Depletion "Stems"(Boundary-Dominated Flow

Region-VolumetricReservoir Behavior)

Transient "Stems"(Transient Flow Region -

Analy tical Solut ions : reD

= 1x104)

reD=1x104

Legend: qDd(tDd),qDdi(tDd), and [qDdi(tDd)] versus tDd,bar qDd(tDd) Rate

qDdi(tDd) Rate Integral

[qDdi(tDd)] Rate Integral -Derivative

qDd(tDd) Data Function

qDdi(tDd) Data Function

[qDdi(tDd)] Data Function

Figure 4 Example 1: Diagnostic log-log plot (dimensionlessrate decline integral functions) excellent diagnos

tic performance of [qDdi(tDd)] data function.

The diagnostic log-log plot shown in Fig. 4is excellent we

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SPE 107967 Application of the -Integral Derivative Function to Production Analysis 3

obtained excellent data matches using the model for an

unfractured well in a homogenous reservoir model. In this

case we obtained a match of reD = 1x104 which, in

isolation, does suggest well damage effects. The only dis-crepancy in the [qDdi(tDd)] model and data functions occurs at

relatively "early" values of the material balance time function,

at times where we believe the data are transitioning fromtransient radial flow to a transitional flow regime prior to

evidence of boundary effects.

From the [qDdi(tDd)] data function, it is clear that the boun-daries of the drainage area have not yet established i.e.,the

[qDdi(tDd)] values have not yet stabilized at 1, nor is this

function approaching 1 at that time. Specifically using the

model match for diagnosis, it can be concluded that it will takemore than another log-cycle for the response function to

exhibit full boundary-dominated flow.

Once we have identified the appropriate (i.e., likely) reservoir

model and we have estimated reservoir model parameters suchas: k,s, reD,N,pi(where we note thatpiis imposed in this and

all of our examples), we proceed and generate model-based

pressures and rates using superposition in time. This "analy-sis" procedure is performed to validate the diagnosis (obtained

from the log-log plot) in terms of history matching, to confirm

the reservoir model, and finally to check the data consistency.The summary plot for this case is shown in Fig. 5.

102

103

104

OilFlo

wra

te,qo,

STB/D

5000

4500

4000

3500

3000

2500

2000

1500

1000

5000

Production Time, t, hours

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

0

We

llbore

Flow

ing

Pressure,pwf,ps

ia

Wellbore FlowingPressure

Oil Flowrate

Example 1 Exploration Well (Southeast Asia)

Legend:qo Data Function

pwfData Function

( ) qo Model Function

( ) pwf Model Function

Analys is Resu lts: South east Asi a Oil Wel l

(Bounded Circular Reservoir Case)

k = 130 md

reD = 1x104

(dimensionless)

N = 24.1 MMSTBre = 3430 ft

pi = 2900 p si a (f or ced )

Figure 5 Example 1: Analysis by modeling, excellent perfor-

mance of the model obtained from the log-logdiagnostic plot.

In Fig. 5we find excellent agreement between the data and the

pressures and rates generated by the reservoir model. For

reference, the reservoir model does not honor the data at late

times where we suspect that well damage is evolving.

Example 2:East Texas(US) Tight Gas

This case is taken from Pratikno et al[Pratikno et al(2003)],

and all of the relevant data and the analysis results for this

case can be found in that reference. The time-pressure-rate(TPR) plot for this case is shown in Fig. 6. We note that the

production data for this example case are of very good quality

(although only given on a daily basis). We advocate that most

gas wells in low permeability formations should have data

acquisition programs which are comparable to those used forthis case.

102

103

104

105

8000

7500

7000

6500

6000

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000

5000

Production Time, t, hr

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

We

llbore

Flo

wing

Pressure,pwf,ps

ia

Legend: East Texas Gas Well (SPE 84287) qgData Function

pwfData Function

GasF

lowra

te,qg,

MSCF/D

Example 2 East Texas Gas Well (SPE 84287)(Tight Gas Sand)

Figure 6 Example 2: Time-Pressure-Rate (TPR) his tor y plo tEast Tx gas well. Very good correlation of rateand pressure data indicates likelihood of goodanalysis.

Since this well is hydraulically fractured, we use fracturedwell models for analysis/interpretation. Since this is a ga

case (i.e., flowing fluid is compressible), we use pseudo

pressure and pseudotime functions. The diagnostic log-log

plot (Fig. 7) shows outstanding matches for all of the rateintegral decline functions in particular, the [qDdi(tDd)] data

function indicates that the flow is in transition to the boun-

dary-dominated flow regime (evolving trend in the [qDdi(tDd)

data function approaches 1).

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

tDd,bar=GpDd(tDd)/qDd(tDd)

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

Fetkovich McCray Rate Function Type CurveFractured Well Centered in a Boun ded Circular Reservoir (FcD= 10)

Example 2 East TX Gas Well (Tigh t Gas Sand)


(qDd(tDd

),qDdi(tDd

),and

[qDdi(tDd

)])

Model Legend: Elliptical Flow Type Curve - FracturedWell Centered in a Bounded Circular Reservoir

(Finite Conductivity: FcD= 10)

Legend: qDd(tDd),qDdi(tDd), and [qDdi(tDd)] versus tDd,bar qDd(tDd) Rate

qDdi(tDd) Rate Integral [qDdi(tDd)] Rate Integral -Derivative

[qDi(tDd)] Data Function

qDdi(tDd)

Data Function

qDdi(tDd)

qDd(tDd)

[qDi(tDd)]

qDd(tDd) Data Function




Analy tic al Sol utio ns: FcD

= 10)

Figure 7 Example 2: Diagnostic log-log plot (dimensionlessrate decline integral functions) outstanding diag

nostic performance of [qDdi(tDd)] data function.

However, as we observe from the fractured well model, this

case is in transition and requires approximately two more log-

cycles to reach complete boundary-dominated flow. Such anobservation is neither unusual nor unexpected for a well in a

low to very-low permeability gas reservoir. As in the previou

case, we proceed from the analysis and generate the pressure

and rate responses using the defined reservoir model and theestimated reservoir parameters (k, reD, G,pi where, again,pis imposed all cases).

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As seen in Fig. 8, the overall match of the generated responses

(rates and pressures) and the raw data are very good to

excellent for this case even taking into account the erraticbehavior in the rate data. We note that our analysis results are

very close to original results provided for this case [Pratikno et

al(2003)].

102

103

104

105

8000

7500

7000

6500

6000

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000

5000

Production Time, t, hr

12000

11000

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

We

llbore

Flow

ing

Pressure,pwf,

ps

ia

Legend: East Texas Gas Wellqg Data Function

pwf Data Function

qg Model Function

pwf Model Function

Gas

Flowra

te,qg,

MSCF/D

Analysi s Resul ts: Eas t Tx Gas Wel l(Bounded Circular Reservoir Case)

k = 0 .0 55 4 m dxf = 2 90 f t

FcD = 10 (d im en si on les s)

G = 1. 58 6 B SC Fre = 339 ft

pi = 9330 psia (forced)

Example 2 East Texas Gas Well (SPE 84287)(Tight Gas Sand)

Figure 8 Example 2: Analysis by modeling, very good per-

formance of the model obtained from the log-logdiagnostic plot.

Example 3:Mexico Very Tight Gas(long production)

This example was recently evaluated using an elliptical flowmodel [Amini et al (2007)] and it was concluded that the

reservoir has a permeability of < 0.001 md (estimated by

several analyses). In addition, it is worth noting that this fieldhas only one well. The long production history and high

quality data yield "near textbook" quality diagnostic plots

(Figs. 9 and 10).

1200

1100

1000

900

800

700

600

500

400

300

200

100

0

We

llbore

Flow

ing

Pressure,pwf,ps

ia

17

,000

16

,000

15

,000

14

,000

13

,000

12

,000

11

,000

10

,000

9,0

00

8,0

00

7,0

00

6,0

00

5,0

00

4,0

00

3,0

00

2,0

00

1,0

000

Production Time, t, days

102

103

104

Legend:qgData Function

pwfData Function

Gas

Flowra

te,qg,

MSCF/D

Example 3 Mexico Gas Well(Tight Gas Sand Very Low Reservoir Permeability, Very Long Prod uction History)

Figure 9 Example 3: Time-Pressure-Rate (TPR) his tory plo t.

Mexico gas well. Good quality data (bottomholepressures are given constant).

We note as comment that the data scatter seen in the rate is notclearly reflected in the pressure data but we also acknow-

ledge that this scenario could be one of data scaling, as the

pressure data are certainly not measured at the same accuracy

as the rate data. Even given this comment, we believe thatthese data are accurate and correlated and we anticipate a

consistent analysis/interpretation.

The objective of this example is to apply and validate theelliptical boundary -integral derivative type curves. For this

purpose we have used the elliptical boundary model type

curves in the matching process in the diagnostic log-log plot

(Fig. 10). In this example we utilize type curve solutions in

terms of the equivalent constant rate case in "decline" form(i.e., qDdand the auxiliary functions qDdiand [qDdi(tDd)] versustDA). We obtained an excellent match using the elliptical flow

parameters FE= 100 and 0= 0.25. We note that these arethe same results as obtained by the original reference for this

case [Amini et al(2007)]. The only substantive difference in

this analysis is that we employed the [qDdi(tDd)] data functionrather than qDdid(tDd) which indicates the transition to

boundary-dominated flow uniquely.

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionless Decline Time Based on Drainage Area (tDA)

qDd(tDA) Data Function

qDdi(tDA) Data Function

[qDi(tDA)] Data Function

Model Legend: Elliptical Flow Type Curve - FracturedWell Centered in a Bounded Elliptical Reservoir

(Finite Conductivity: FE= 100)

Legend: qDd(tDA),qDdi(tDA), and [qDdi(tDA)] versus tDA qDd(tDA) Rate

qDdi(tDA) Rate Integral

[qDdi(tDA)] Rate Integral -Derivative




Analyti cal Solu tion s: FE= 100)

Ellipti cal Flow Type Curve - Fractured Well Centered in a

Bounded Elliptical Reservoir (Finite Conductivity: FE= 100, 0= 0.25)

Example 3 Mexico Gas Well (Tight Gas Sand Very Low Reservoir Permeability)

0= 0.25

fracture

closed reservoirboundary (ellipse)

wellbore

xf

a

b

y

x


(qDd

(tDA

),qDdi(

tDA

),and

[qDdi(tDA

)])

qDdi(tDA)qDd(tDA)

[qDi(tDA)]

Figure 10 Example 3: Diagnostic log-log plot (dimensionlessrate decline integral functions) very good match o

the [qDdi(tDd)] function (excellent diagnostic).

The final step in our analysis is to generate the pressure andrate responses using the (elliptical) reservoir model that we

deduced from the diagnostic plot (see Fig. 11). We note tha

for this case, the computed rates match the raw data extremely

well but the calculated bottomhole pressure response doesshow some disagreement with the raw pressure data. In

fairness, the pressures are the "weakest" data, and are likely

affected by phenomena such as liquid-loading.

1600

1400

1200

1000

800

600

400

200

0

We

llbore

Flow

ing

Press

ure,pwf,ps

ia

17

,000

16

,000

15

,000

14

,000

13

,000

12

,000

11

,000

10

,000

9,0

00

8,0

00

7,0

00

6,0

00

5,0

00

4,0

00

3,0

00

2,0

00

1,0

000

Production Time, t, days

102

103

104

105

Legend:qgData Function

pwfData Function

Gas

Flowra

te,qg,

MSCF/D

Example 3 Mexico Gas Well(Tight Gas Sand Very Low Reservoir Permeability, Very Long Production History)

Analys is Resu lts: Mexic o Gas Well

(Bounded Elliptical Reservoir Case)

k = 0.001 mdxf = 826 ft

FE = 100 (dimensionless)

G = 9.6 BSCFre = 871 ft

pi = 5463 ps ia (f or ced )

qg Model Function

pwfModel Function

Figure 11 Example 3: Analysis by modeling, very good ratematch by the model, generated pressures fail tohonor the given constant bottomhole pressures.

As closure in this section, we present the "average" analysis

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results for these examples considered in this work (see Table

1).

Table 1 "Average" analysis results for this work.

Example

k(md)

xf(ft)

G (or N)(BSCF or

MMSTB)

1 (oil) 130 N/A 24.1

2 (gas) 0.0055 290 1.6

3 (gas) 0.0010 825 23.0

Summary and Conclusions

Summary: The primary purpose of this paper is the presen-

tation of the -integral derivative function as a diagnostic tool

for production data analysis. Two different (dimensionless)formulations of the -integral derivative function are proposed

for use in production analysis applications.

[qDdi(tDd)] formulation for "rate decline" analysis

[pDdi(tDd)] formulation for "pressure" analysis

The -integral derivative function can be computed directly

using rate/pressure integral and rate/pressure integral deri-

vative functions or rate/pressure and rate/pressure integralfunctions (the relevant derivations are provided in Appendix

A). We provide a schematic "diagnosis worksheet" for the

interpretation of the -integral derivative function for rate

integral and pressure integral cases (see Appendix C) as well

as an inventory of type curves (-integral derivative solutions)

for specified reservoir models having closed boundaries.

Unfractured well Centered in a bounded circular reservoir

Fractured well Centered in a bounded circular reservoir

Fractured well Centered in a bounded elliptical reservoir

We have applied and validated the application of -integral

derivative function for production analysis using various field

cases.

Conclusions:1. The -integral derivative function has the potential to

become a significant diagnostic tool in production analysis

as the -integral derivative function exhibits uniquecharacter for several flow regimes.

2. The diagnostic matches of the production data obtained

using the -integral derivative function presented in thiswork are excellent. It is very likely that similar diagnosticmatches would have been obtained using the rate integral

derivative function. But we have shown that the -integralderivative function provides more resolution in parti-

cular, the -integral derivative function yields the following

behavior for the cases used in this work.

Case [qDdi(tDd)]Reservoir boundaries:

Closed reservoir (circle, rectangle, etc.) 1

Fractured wells:

Infinite conductivity vertical fracture. Finite conductivity vertical fracture.

1/21/4

3. The incorporation of the -integral derivative function inthe modern production analysis tools will help to distin-guish individual flow regimes, as well as help to different-

iate transitional character this may be the source of most

value for the -integral derivative functions.

Recommendations/Comment: Future work on this topic

should focus on the additional -integral derivative solutions

for various (preferably complicated) reservoir models and

configurations which were not described in this work as

well as more applications of the functions in practice.

Nomenclature

Field Variables

ct = Total system compressibility, psi-1

G = Gas-in-place, MSCF or BSCFGp = Gas production, MSCF or BSCF

h = Pay thickness, ft

k = Permeability, md

kf = Fracture permeability, mdkR = Reservoir permeability, mdN = Oil-in-place, STB

Np = Cumulative oil production, STBp = Pressure, psiapi = Initial reservoir pressure, psiapp = Pseudopressure function, psia

pR = Reservoir pressure, psiapwf = Flowing bottomhole pressure, psia

q = Flowrate, STB/D

qg = Gas flowrate, MSCF/Dqo = Oil flowrate, STB/D

re = Drainage radius, ftrw = Wellbore radius, ft

rwa = Apparent wellbore radius, ftt = Time, hr

ta = Pseudo-time (adjusted time), hrxf = Fracture half-length, ft

Dimensionless Variables

bDpss = Dimensionless pseudosteady-state constantFcD = Dimensionless fracture conductivity

FE = Elliptical fracture conductivity

pD = Dimensionless pressure

pDd = Dimensionless pressure derivative

pDi = Dimensionless pressure integral

pDid = Dimensionless pressure integral derivative

[pDdi] = Dimensionless-pressure integral derivative

qD = Dimensionless flowrate

qDi = Dimensionless rate integral

qDid = Dimensionless rate integral derivative

[qDdi] = Dimensionless-rate integral derivative

reD = Dimensionless outer reservoir boundary radius

tD = Dimensionless time (wellbore radius)tDd = Dimensionless decline timetDA = Dimensionless time (drainage area)tDxf = Dimensionless time (fracture half-length)

Mathematical Functions and Variablesa = Regression coefficientA = Auxiliary functionb = Regression coefficient

B = Auxiliary function

Greek Symbols = Beta-derivative

= porosity, fraction

= Viscosity, cp

0 = Elliptical boundary characteristic variable

Subscriptsa = Pseudotime

d = Derivative or decline parameter

D = Dimensionless

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Dd = Dimensionless decline variable

f = Fractureg = Gasi = Integral function or initial value

id = Integral derivative function

mb = Material balancepss = Pseudosteady-state

r = Positive integer

R = Reservoir

Superscripts = Material balance time

Constants

= Circumference to diameter ratio, 3.1415926

= Eulers constant, 0.577216

Gas Pseudofunctions:

dpz

pp

pp

zp

basei

iip

i

=

dtpcp

tct

gggigia

)()(

1

0

=

dtpcp

tqt

tq

ct

gg

gigigasmba

)()(

)(

0

)( ,

= References

Amini, S., Ilk, D., and Blasingame, T.A.: "Evaluation of theElliptical Flow Period for Hydraulically-Fractured Wells in TightGas Sands Theoretical Aspects and Practical Considerations,"

paper SPE 106308 presented at the 2007 SPE Hydraulic FracturingTechnology Conference held in College Station, Texas, U.S.A.,2931 January 2007.

Blasingame, T.A., Johnston, J.L., and Lee, W.J.: "Type CurveAnalysis Using the Pressure Integral Method," paper SPE 18799

presented at the 1989 SPE California Regional Meeting,Bakersfield, CA, 05-07 April 1989.

Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame,T.A.: "Decline Curve Analysis Using Type Curves Analysis ofOil Well Production Data Using Material Balance Time:Application to Field Cases," paper SPE 28688 presented at the

1994 Petroleum Conference and Exhibition of Mexico held inVeracruz, MEXICO, 10-13 October 1994.

Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves,"

JPT (March 1980) 1065-1077.

Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The

Pressure Derivative Revisited Improved Formulations andApplications," paper SPE 103204 presented at the 2006 AnnualSPE Technical Conference and Exhibition, Dallas, TX, 23-27September 2006.

Palacio, J.C. and Blasingame, T.A.: "Decline Curve AnalysisUsing Type Curves Analysis of Gas Well Production Data,"

paper SPE 25909 presented at the 1993 Joint Rocky Mountain

Regional/Low Permeability Reservoirs Symposium, Denver, CO,26-28 April 1993.

Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "De-clineCurve Analysis Using Type Curves Fractured Wells," paper

SPE 84287 presented at the SPE annual Technical Conference andExhibition, Denver, Colorado, 5-8 October 2003.

7/25/2019 [Blasingame] SPE 107967

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Appendix A: Derivat ion of Rate Integral -Derivat iveFormulation

In this Appendix we derive the -derivative integral functions([qDdi(tDd)] and[pDdi(tDd)]) where these functions are definedto identify the transient, transition, and boundary-dominated

flow regimes from production data analysis.

Rate Integral FunctionsBefore we begin to derive the formulation for the -integralderivative rate function, we start with the definitions of the so

called "rate-integral" functions [Palacio and Blasingame 1993;

Doublet, et al 1994]. For reference, the dimensionless rate-

integral function is defined as:

dqt

ttq Dd

Dd

DdDdDdi )(

0

1

)( = ..................................(A-1)Where qDd(tDd) is the dimensionless rate decline function[Fetkovich, 1980]. The dimensionless rate-integral derivative

function (using the Bourdet derivative formulation) is:

)()( DdDdiDd

DdDdDdid tqdt

d

ttq = ..............................(A-2)

The derivative of Eq. A-1 with respect to the dimensionless

decline time, tDdis:

[ ])()(1

)(1

)(

0

)(

1

)(

2

DdDdiDdDdDd

DdDdDd

DdDd

Dd

DdDdiDd

tqtqt

tqt

dqt

t

tqdt

d

=

+=

....................................................................................... (A-3)

The power-law derivative formulation (i.e., the -derivativeformulation) for the dimensionless rate-integral function isdefined as:

)()(

1

]ln[

)](ln[

)]([

DdDdiDd

DdDdDdi

Dd

DdDdi

DdDdi

tqdt

dt

tq

td

tqd

tq

=

=

Where this result reduces to:

)(

)()]([

DdDdi

DdDdidDdDdi

tq

tqtq =

...................................................................................... (A-4)

Substituting Eq. A-3 into Eq. A-4, we obtain,

[ ]

1)(

)(

)()(1

)(

1

)]([

=

=

DdDdi

DdDd

DdDdiDdDdDd

DdDdDdi

DdDdi

tq

tq

tqtqt

ttq

tq

Or, finally, we obtain:

)(

)(1)]([

DdDdi

DdDdDdDdi

tq

tqtq = ......................................... (A-5

Equating Eqs. A-5 and A-6 gives us:

)(

)(1

)(

)(

DdDdi

DdDd

DdDdi

DdDdid

tq

tq

tq

tq=

Solving for qDdid(tDd) yields

)()()( DdDdDdDdiDdDdid tqtqtq =

............................. (A-6Where Eq. A-6 is exactly the definition given by [Doublet, e

al1994], and thus, confirms our definition of the [qDdi(tDd)function.

Pressure Integral Functions

For reference, the dimensionless pressure-integral function is

defined as [Blasingame, et al 1989]: (modified to "decline"variable format)

dpt

ttp Dd

Dd

DdDdDdi )(

0

1

)( = ................................. (A-7WherepDd(tDd) is the dimensionless pressure decline function

The dimensionless rate-integral derivative function (using theBourdet derivative formulation) is given as:

)()( DdDdiDd

DdDdDdid tpdt

dttp = ................................ (A-8

The derivative of Eq. A-7 with respect to dimensionless de

cline time, tDdis:

[ ])()(1

)(1

)(

0

)(

1

)(

2

DdDdiDdDd

Dd

DdDdDd

DdDd

Dd

DdDdiDd

tptp

t

tpt

dpt

t

tpdt

d

=

+=

...................................................................................... (A-9

Multiplying through Eq. A-9 by the dimensionless decline

time, tDdyields:

)()(

)(

)(

DdDdiDdDd

DdDdiDd

Dd

DdDdid

tptp

tpdt

dt

tp

=

=

.................................................................................... (A-10

Where Eq. A-10 is a fundamental definition of the "pressure

integral" given by [Blasingame, et al1989].

The power-law derivative formulation (i.e., -derivative for

mulation) for the dimensionless pressure-integral function isdefined as:

[ ]

=

=

=

)()(1

)(

1

)()(

1

]ln[

)](ln[

)]([

DdDdiDdDdDd

DdDdDdi

DdDdiDd

DdDdDdi

Dd

DdDdi

DdDdi

tptpt

ttp

tpdt

dt

tp

td

tpd

tp

Where this result reduces to:

7/25/2019 [Blasingame] SPE 107967

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[ ]

1)(

)(

)()()(

1

)]([

=

=

DdDdi

DdDd

DdDdiDdDdDdDdi

DdDdi

tp

tp

tptptp

tp

....................................................................................(A-11)

Where we note an alternate form of Eq. A-11 is obtained using

[ ]

)(

)(

)()()(

1

)]([

DdDdi

DdDdid

DdDdiDdDdDdDdi

DdDdi

tp

tp

tptptp

tp

=

=

....................................................................................(A-12)

Equating Eqs. A-11 and A-12 gives us:

)(

)(1

)(

)(

DdDdi

DdDdid

DdDdi

DdDd

tp

tp

tp

tp=

Solving forpDdid(tDd) yields

)()()( DdDdiDdDdDdDdid tptptp = ...........................(A-13)

Where Eq. A-13 is exactly (as expected) the definition given

by [Blasingame, et al1989], and thus, confirms our definition

of the [pDdi(tDd)]function.

Appendix B: Dimensionless Variables

The most straightforward approach to defining dimensionless

variables for this application is to use the approach of Fet-

kovich [Fetkovich, 1980] and reduce all cases to a single set ofunified variables.

This process is fairly easy for a given case, but will require

knowledge of the reservoir model for each specific case. Tosimplify (somewhat) this exercise, we will use the approach of

[Pratikno, et al2003], which states the following relations for

the dimensionless decline variables:

ADpss

Dd Dtb

t 2

= ("decline" time).......................... (B-1)

DpssDDd bqq = ("decline" rate)........................... (B-2)

1D

DpssDd p

bp = ("decline" pressure) ................... (B-3)

Where (obviously) the bDpssvariable given in Eqs. B-1 to B-3

is model-dependent. For reference, the base or "universal" de-finitions of tDA, qD, andpDare:

tAc

kt

tD 00633.0

= (tin days)................................... (B-4)

)(

12.141

wfiD

ppkh

qBq

=

............................................ (B-5)

)(2.141

1wfiD pp

qB

khp =

............................................ (B-6)

The remaining task is to address the bDpss variable for the

unfractured well, fractured well, and elliptical flow cases

these results are:

Unfractured Well: [Fetkovich, 1980]

4

3ln

=

wa

eDpss

r

rb (exact definition).................. (B-7a)

But we note that Fetkovich [Fetkovich, 1980] defined this

variable as:

2

1ln

=

wa

eDpss

r

rb (Fetkovich definition) .......... (B-7b

The difference in Eqs. B-7a and B-7b, is essentially irrele

vant, and from a historical perspective, the Fetkovich defini

tion is most widely accepted. We use Eq. B-7b in this work

Fractured Well: [Pratikno, et al2003]Given a particular reservoir/fracture case (i.e., reD and FcDvalues), then bDpss(reD,FcD) can be estimated using :

44

33

221

45

34

2321

2

1

43464.0049298.0)(ln

ubububub

uauauauaa

rr eDeDDpssb

++++

+++++

+=

...................................................................................... (B-8

Where,

)(ln cDFu=

a1 = 0.93626800 b1 = -0.38553900

a2 = -1.00489000 b2 = -0.06988650a3 = 0.31973300 b3 = -0.04846530

a4 = -0.04235320 b4 = -0.00813558

a5 = 0.00221799

...................................................................................... (B-9

The correlation given by Eq. B-8 is an approximation of the

exact values for this case, but this result should be morethan sufficient for all applications.

Elliptical Flow/Fractured Well: [Amini, et al2007]

Given a particular reservoir/fracture case formulated in

the elliptical flow geometry (i.e., 0 and FE values), then

bDpss(0,FE) can be estimated using :

754772.0

16703.00794849.000146.1 00

+

+=

B

A

uebDpss

.................................................................................... (B-10

Where the auxiliary functions are:

)ln( EFu=

45

34

2321 uauauauaaA ++++=

45

34

2321 ububububbB ++++=

.................................................................................... (B-11

The correlation given by Eq. B-10 is sufficiently accurate

for all practical applications.

In addition to the "decline" variables, we also employ the"equivalent constant rate" concept proposed by [Doublet, et a1994] i.e., the "material balance time" concept. Using thi

approach, we "convert" variable-rate/variable pressure drop

data into an equivalent constant rate case (analog to well testanalysis). As such, we will always work in terms of the

material balance time variable which is defined as:

)(or

)(or

go

pp

qq

GNt=

(liquid case)............................. (B-12

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d

c

q

tq

ct

gg

gt

g

gigia

)()(

)(

)(

0=

(gas case)................... (B-13)

In practice, we will use the "decline" time variables based onthe appropriate material balance time functions (liquid or gas),

and we will also present the "type curve" solutions in terms of

the (dimensionless) "decline" material balance time, given as:

Dd

pDd

Dd q

N

t =

.............................................................. (B-14)

Where the dimensionless "decline" cumulative production is

defined as:

dqN Dd

Ddq

pDd )(0= ............................................. (B-15)

As a final comment, we want to state that for the unfractured

reservoir case we have used (exactly) the Fetkovich defini-

tions for the "decline" variables. Specifically, these defini-tions are:

D

wa

e

w

e

Dd t

rr

rr

t

21ln1

21

1

2

= ........................... (B-16)

Dwa

eDd q

r

rq

2

1ln

= ............................................... (B-17)

D

wa

eDd p

r

rp

2

1ln

1

= ............................................. (B-18)

Where for this case, the "ordinary" dimensionless time func-

tion is given as:

trc

kt

wtD 00633.0 2

= ................................................... (B-19

Appendix C: Dimensionless "Type Curve" Representations of the -Pressure Derivative and Variousother Pressure Functions (selected reservoir/welconfigurations)

In this appendix we present the "inventory" of type curve

solutions for the proposed -derivative integral functions (i.e.

the [qDdi(tDd)] and [pDdi(tDd)]). We use the dimensionlesdecline "material balance time" function given as: (i.e., the

equivalent constant rate case)

Dd

pDd

Dd

Ddq

Dd

Dd

q

N

dqq

t

=

=

)(1

0

...................................................................................... (C-1

For the case of the elliptical flow geometry we elected not touse the tDd-format due to certain early-time artifacts (sometrends overlap in a non-uniform manner). We believe that this

effect is not an error or flaw in the use of the tDdfunction, bu

rather just an artifact of the formulation for this particular

case. As an alternative, we use the tDAformat as proposed byAmini et al [Amini et al (2007)] this format works very

well and yields no visible artifacts.

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10-3

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Dimensionless Material Balance Decline Time, tDd,bar=NpDd /qDd

Legend: (qDdid ) ( [qDdi ] )

Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivity ) Fractured Well (Elliptical Reservoir)


Schematic of Dimensionless Rate Integral Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)

DIAGNOSTIC plot for Production Data(qDdidand [qDdi ] )

DimensionlessRateIntegralDerivativeFunction,qD

did

"PowerLaw

"DimensionlessRateIntegralDerivativeFunction,

[qDdi]

Boundary-

DominatedFlow Region

[qDdi ] ~ 1.0


1

1

1

2


Reservoir


Reservoir(Finite ConductivityVertical Fracture)



( )( )

( )( )

( )( )


[qDdi ] = 0.5

(linear flow )

Figure C.1 Schematic of qDdi(tDd)] vs. tDd Unfractured and fractured well configurations (note the distinction of the " transition" flow

regimes that the qDdi(tDd) function provides) (analog of decli ne type curve analysis).

10-3

10-2

10-1

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Dimensionless Material Balance Decline Time, tDd,bar=NpDd /qDd

Legend: (pDdid ) ( [pDdi ] )

Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivity ) Fractured Well (Elliptical Reservoir)


Schematic of Dimensionless Pressure Integral Derivative FunctionsVarious Reservoir Models and Well Configurations (as noted)

DIAGNOSTIC plot for Production Data(pDdidand [pDdi ] )

Dime

nsionlessPressureIntegralDerivativeFunction,pDdid

"PowerLaw

"DimensionlessPressureIntegralDerivativeFunction,

[pDdi]

Boundary-Dominated

Flow Region

[pDdi ] = 1.0


1

1

1

2


Reservoir





( )( )

( )( )

( )( )


[pDdi ] = 0.5

(linear flow)

Figure C.2 Schematic o f pDdi(tDd)] vs. tDd Unfractured and fractured well configurations good transition and strong indicator ofthe boundary-domin ated flow regime (analog of well test analysis).

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10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimens

ion

less

Dec

line

Ra

te,qDd

(tDd

),

Dime

ns

ion

less

Dec

line

Ra

teIntegra

l,qDdi(

tDd

),an

d

Dimens

ion

less

Dec

line

Ra

teIntegra

l-

Deriva

tive,

[qD

di(

tDd

)]

tDd,bar=NpDd(tDd)/qDd(tDd)

Fetkovich-McCray Rate Function Type CurvetDd,barFormat

(Unfractured Well Centered in a Bounded Circular Reservoir)

Model Legend: Fetkovich-McCray Rate Function Type Curve - UnfracturedWell Centered in a Bounded Circular Reservoir

Legend: qDd(tDd), qDdi(tDd) and [qDdi(tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves

Rate Integral- -Derivative Function CurvesreD=re/rwa=5

10

20

30

50

100

500

1000

qDd(tDd) [qDdi(tDd)]

[qDdi(tDd)]

Depletion "Stems"(Boundary-Dominated

Flow Region)

Transient "Stems"(Transient Flow Region)

reD=1x104

500 100

5030 20 10

5

1000

reD=1x104

Figure C.3 qDdi(tDd)] vs. tDd Unfractured well configuration also plotted with qDdand qDdi for comparison very good resolution

of transient and transition regimes using the qDdi(tDd)] functions.

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dime

nsionlessPressure,pDd

(tDd

),

DimensionlessPressureIntegral,pDdi(

tDd

),and

DimensionlessP

ressureIntegral-

Derivative,

[pDdi(

tDd

)]


Pressure Function Type CurvetDd,barFormat

(Unfractured Well Centered in a Bounded Circular Reservoir)

Model Legend: Fetkovich-McCray Rate Function Type Curve - UnfracturedWell Centered in a Bounded Circular Reservoir

Legend: pDd(tDd), pDdi(tDd) and [pDdi(tDd)] vs. tDd,barPressure Function CurvesPressure Integral Function Curves

Pressure Integral- -Derivative Function Curves

reD=re/rwa=5

10

20

30

50100

5001000

pDd(tDd)

pDdi(tDd)

[pDdi(tDd)]


Flow Region)


reD=1x104

500 100

5030 20

105

1000

reD=1x104

Figure C.4 pDdi(tDd)] vs. tDd Unfractured well configuration also plotted with pDd and pDdi for comparison, similar form as the

qDdi(tDd)] functions excellent resolution of all flow regimes.

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10-2

10-2

10-1

10-1

10

0

10

0

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionle

ssDeclineRate,qDd

(tDd

),

DimensionlessDeclineRateIntegral,qDdi(

tDd

),and

DimensionlessDeclineRateIntegral-

Derivative,

[qDdi(

tDd

)]


Fetkovich-McCray Rate Function Type Curve-tDd,barFormat

( Vertical Well with a Finit e Conductivity Vertical Fracture FcD= 1 )

Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir

(Finite Conductivity: FcD

= 1)

Legend: qDd(tDd), qDdi (tDd) and [qDdi (tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves

Rate Integral- -Derivative Function Curves

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)

Transient "Stems"

(Transient Flow Region)

reD=1x103

500100

5030

2010

5

Figure C.5 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well

configuration (FcD=1).

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

DimensionlessDeclineRate,qDd

(tDd

),


tDd

),an

d


Derivative,

[qDdi(

tDd

)]


Fetkovich-McCray Rate Functio n Type Curve-tDd,barFormat

( Vertical Well with a Finite Conductivity Vertical Fracture FcD= 5 )


(Finite Conductivity: FcD= 5)Legend: qDd(tDd), qDdi(tDd) and [qDdi(tDd)] vs. tDd,bar

Rate Function CurvesRate Integral Function Curves


qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)


reD=1x103

500100

50

30 2010

5

Figure C.6 qDdi(tDd)], qDd, and qDdivs. tDd Fractured wellconfiguration (FcD=5).

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103


(tDd

),

Dimension

lessDeclineRateIntegral,qDdi(

tDd

),and

DimensionlessD

eclineRateIntegral-

Derivative,

[qDdi(

tDd

)]



( Vertical Well with a Fini te Conductivity Vertical Fracture FcD= 10 )





5

1020

3050

100500

1000

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)

Transient "Stems"


reD=1x103

500 100

50

3020

10

5



10-2

10-2

10-1

10-1

10

0

10

0

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionle

ssDeclineRate,qDd

(tDd

),


tDd

),and


Derivative,

[qDdi(

tDd

)]








5

10

20

30

50

100500

1000

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)

Transient "Stems"


reD=1x103

500100

5030

2010

5



10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103


(tDd

),


tDd

),an

d


Derivative,

[q

Ddi(

tDd

)]


Fetkovich-McCray Rate Functi on Type Curve-tDd,barFormat



(Finite Conductivity: FcD

= 500)

Legend: qDd(tDd), qDdi(tDd) and [qDdi(tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves


5

10

20

30

50

100500

1000

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)


reD=1x103

500 100

5030

2010

5

Figure C.9 qDdi(tDd)], qDd, and qDdivs. tDd Fractured wellconfiguration (FcD=500).

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103


(tDd

),

Dimension

lessDeclineRateIntegral,qDdi(

tDd

),and

DimensionlessD

eclineRateIntegral-

Derivative,

[qDdi(

tDd

)]


Fetkovich-McCray Rate Functi on Type Curve-tDd,barFormat

( Vertical Well with a Fini te Conductivity Vertical Fracture FcD= 1000 )



Legend: qDd(tDd), qDdi (tDd) and [qDdi(tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves


5

10

20

30

50

100500

1000

qDd(tDd)

qDdi(tDd)

[qDdi(tDd)]


Flow Region)

Transient "Stems"


reD=1x103

500100

50

3020

105

Figure C.10 qDdi(tDd)], qDd, and qDdivs. tDd Fractured

well configuration (FcD=1000).

7/25/2019 [Blasingame] SPE 107967

13/14


10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionless Time Based on Drainage Area (tDA)

4.0

0= 5.0

qDd(tDA)

qDdi (tDA)

[qDi(tDA)]



Legend: qDd(tDA),qDdi (tDA), and [qDdi(tDA)] versus tDA q

Dd(t

DA

) Rate

qDdi (tDA) Rate Integral

[qDdi (tDA)] Rate Integral -Derivative




Analy tic al Solu tion s: FE= 1)

Ellipti cal Flow Type Curve - Fractured Well Centered in aBounded Elliptical Reservoir (Finite Conductivity: FE= 1)

3.02.0

1.75

1.50

0.500.75

1.0

0= 0.25

0= 5.0

0= 0.25


fracturewellbore

b

a

xf

x

y


(qDd

(tDA

),qDdi(tDA

),and

[qDdi(tDA

)])

Figure C.11 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well configuration elliptical flow model (FE=1).

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionless Time Based on Drainage Area (tDA)

0= 0.25

1.50

1.75

2.0

3.0

4.0

0= 5.0

qDd(tDA)

qDdi(tDA)

[qDi(tDA)]

Model Legend: Elliptical Flow Type Curve - Fractured

Well Centered in a Bounded Elliptical Reservoir(Finite Conductivity: F

E= 10)




Depletion "Stems"

(Boundary-Dominated FlowRegion-VolumetricReservoir Behavior)

Transient "Stems"

(Transient Flow Region -Analy tic al Solu tio ns: FE= 10)


0.500.75

1.0

0= 5.0


b

fracture

a

xf

wellbore

x

y

Dime

nsionlessRateDeclineFunctions

(qDd

(tDA

),qDdi(tDA

),and

[qDdi(tDA

)])


7/25/2019 [Blasingame] SPE 107967

14/14


10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionless Time B ased on Drainage Area (tDA)

1.0

1.501.75

2.0

3.0

4.0

0= 5.0

0.75

qDdi(tDA)

qDd(tDA)

[qDi(tDA)]







Region-Volumetric

Reservoir Behavior)


Analy tic al Solu tio ns: FE= 100)


0= 0.25

0.50

0= 5.0


fracturewellbore

b

a

xfx

y

DimensionlessRateDeclineFunction

s

(qDd

(tDA

),qDdi(tDA

),and

[qDdi(tDA)])


10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

Dimensionless Time B ased on Drainage Area (tDA)

0.25

1.01.50

1.75

2.0

3.0

4.0

0= 5.0

0.75

qDd(tDA)

qDdi(tDA)

[qDi(tDA)]







Region-Volumetric

Reservoir Behavior)

Transient "Stems"(Transient Flow

Region - AnalyticalSolutions: FE= 1000)


0= 0.25

0.50

0= 0.25

5.0


fracturewellbore

b

a

xfx

y


(qDd

(tDA

),qDdi(tDA

),and

[qDdi(tDA

)])


[Blasingame] SPE 107967

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Transcript of [Blasingame] SPE 107967