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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.
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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
Reservoir(FiniteConductivityVertical Fracture)
( )( )
( )( )
( )( )
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)
Transient FlowRegion
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
Unfractured Well ina Bounded Circular
Reservoir
Fractured Well ina Bounded Elliptical
Reservoir(FiniteConductivityVertical Fracture)
Fractured Well ina Bounded Circular
Reservoir(FiniteConductivityVertical Fracture)
( )( )
( )( )
( )( )
NO Wellbore Storageor Skin Effects
[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)
DimensionlessRateDeclineFunctions
(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
Depletion "Stems"(Boundary-Dominated Flow
Region-VolumetricReservoir Behavior)
Transient "Stems"(Transient Flow Region -
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).
7/25/2019 [Blasingame] SPE 107967
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4 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967
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
Depletion "Stems"(Boundary-Dominated Flow
Region-VolumetricReservoir Behavior)
Transient "Stems"(Transient Flow Region -
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
DimensionlessRateDeclineFunctions
(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
7/25/2019 [Blasingame] SPE 107967
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SPE 107967 Application of the -Integral Derivative Function to Production Analysis 5
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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6 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967
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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SPE 107967 Application of the -Integral Derivative Function to Production Analysis 7
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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8 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967
[ ]
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
GNt=
(liquid case)............................. (B-12
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SPE 107967 Application of the -Integral Derivative Function to Production Analysis 9
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 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967
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)
Transient FlowRegion
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
(boundarydominated flow)
1
1
1
2
Unfractured Well ina Bounded Circular
Reservoir
Fractured Well ina Bounded Elliptical
Reservoir(Finite ConductivityVertical Fracture)
Fractured Well ina Bounded Circular
Reservoir(Finite ConductivityVertical Fracture)
( )( )
( )( )
( )( )
NO Wellbore Storageor Skin Effects
[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)
Transient FlowRegion
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
(boundarydominated flow)
1
1
1
2
Unfractured Well ina Bounded Circular
Reservoir
Fractured Well ina Bounded Elliptical
Reservoir(FiniteConductivityVertical Fracture)
Fractured Well ina Bounded Circular
Reservoir(Finite ConductivityVertical Fracture)
( )( )
( )( )
( )( )
NO Wellbore Storageor Skin Effects
[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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SPE 107967 Application of the -Integral Derivative Function to Production Analysis 11
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
)]
tDd,bar=NpDd(tDd)/qDd(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)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"(Transient 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
)]
tDd,bar=NpDd(tDd)/qDd(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)]
Depletion "Stems"(Boundary-Dominated
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
),
DimensionlessDeclineRateIntegral,qDdi(
tDd
),an
d
DimensionlessDeclineRateIntegral-
Derivative,
[qDdi(
tDd
)]
tDd,bar=NpDd(tDd)/qDd(tDd)
Fetkovich-McCray Rate Functio n Type Curve-tDd,barFormat
( Vertical Well with a Finite Conductivity Vertical Fracture FcD= 5 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir
(Finite Conductivity: FcD= 5)Legend: qDd(tDd), qDdi(tDd) and [qDdi(tDd)] vs. tDd,bar
Rate Function CurvesRate Integral Function Curves
Rate Integral- -Derivative Function Curves
qDd(tDd)
qDdi(tDd)
[qDdi(tDd)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"(Transient 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
DimensionlessDeclineRate,qDd
(tDd
),
Dimension
lessDeclineRateIntegral,qDdi(
tDd
),and
DimensionlessD
eclineRateIntegral-
Derivative,
[qDdi(
tDd
)]
tDd,bar=NpDd(tDd)/qDd(tDd)
Fetkovich-McCray Rate Function Type Curve-tDd,barFormat
( Vertical Well with a Fini te Conductivity Vertical Fracture FcD= 10 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir
(Finite Conductivity: FcD= 10)
Legend: qDd(tDd), qDdi (tDd) and [qDdi (tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves
Rate Integral- -Derivative Function Curves
5
1020
3050
100500
1000
qDd(tDd)
qDdi(tDd)
[qDdi(tDd)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"
(Transient Flow Region)
reD=1x103
500 100
50
3020
10
5
Figure C.7 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well
configuration (FcD=10).
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
)]
tDd,bar=NpDd(tDd)/qDd(tDd)
Fetkovich-McCray Rate Function Type Curve-tDd,barFormat
( Vertical Well with a Finit e Conductivity Vertical Fracture FcD= 100 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir
(Finite Conductivity: FcD= 100)
Legend: qDd(tDd), qDdi (tDd) and [qDdi (tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves
Rate Integral- -Derivative Function Curves
5
10
20
30
50
100500
1000
qDd(tDd)
qDdi(tDd)
[qDdi(tDd)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"
(Transient Flow Region)
reD=1x103
500100
5030
2010
5
Figure C.8 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well
configuration (FcD=100).
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
),
DimensionlessDeclineRateIntegral,qDdi(
tDd
),an
d
DimensionlessDeclineRateIntegral-
Derivative,
[q
Ddi(
tDd
)]
tDd,bar=NpDd(tDd)/qDd(tDd)
Fetkovich-McCray Rate Functi on Type Curve-tDd,barFormat
( Vertical Well with a Finit e Conductivity Vertical Fracture FcD= 500 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir
(Finite Conductivity: FcD
= 500)
Legend: qDd(tDd), qDdi(tDd) and [qDdi(tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves
Rate Integral- -Derivative Function Curves
5
10
20
30
50
100500
1000
qDd(tDd)
qDdi(tDd)
[qDdi(tDd)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"(Transient 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
DimensionlessDeclineRate,qDd
(tDd
),
Dimension
lessDeclineRateIntegral,qDdi(
tDd
),and
DimensionlessD
eclineRateIntegral-
Derivative,
[qDdi(
tDd
)]
tDd,bar=NpDd(tDd)/qDd(tDd)
Fetkovich-McCray Rate Functi on Type Curve-tDd,barFormat
( Vertical Well with a Fini te Conductivity Vertical Fracture FcD= 1000 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - FracturedWell Centered in a Bounded Circular Reservoir
(Finite Conductivity: FcD= 1000)
Legend: qDd(tDd), qDdi (tDd) and [qDdi(tDd)] vs. tDd,barRate Function CurvesRate Integral Function Curves
Rate Integral- -Derivative Function Curves
5
10
20
30
50
100500
1000
qDd(tDd)
qDdi(tDd)
[qDdi(tDd)]
Depletion "Stems"(Boundary-Dominated
Flow Region)
Transient "Stems"
(Transient Flow Region)
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
SPE 107967 Application of the -Integral Derivative Function to Production Analysis 13
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)]
Model Legend: Elliptical Flow Type Curve - FracturedWell Centered in a Bounded Elliptical Reservoir
(Finite Conductivity: FE= 1)
Legend: qDd(tDA),qDdi (tDA), and [qDdi(tDA)] versus tDA q
Dd(t
DA
) Rate
qDdi (tDA) Rate Integral
[qDdi (tDA)] Rate Integral -Derivative
Depletion "Stems"(Boundary-Dominated Flow
Region-VolumetricReservoir Behavior)
Transient "Stems"(Transient Flow Region -
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
closed reservoirboundary (ellipse)
fracturewellbore
b
a
xf
x
y
DimensionlessRateDeclineFunctions
(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)
Legend: qDd(tDA),qDdi(tDA), and [qDdi(tDA)] versus tDA qDd(tDA) Rate
qDdi(tDA) Rate Integral
[qDdi(tDA)] Rate Integral -Derivative
Depletion "Stems"
(Boundary-Dominated FlowRegion-VolumetricReservoir Behavior)
Transient "Stems"
(Transient Flow Region -Analy tic al Solu tio ns: FE= 10)
Ellipti cal Flow Type Curve - Fractured Well Centered in aBounded Elliptical Reservoir (Finite Conductivity: FE= 10)
0.500.75
1.0
0= 5.0
closed reservoirboundary (ellipse)
b
fracture
a
xf
wellbore
x
y
Dime
nsionlessRateDeclineFunctions
(qDd
(tDA
),qDdi(tDA
),and
[qDdi(tDA
)])
Figure C.12 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well configuration elliptical flow model (FE=10).
7/25/2019 [Blasingame] SPE 107967
14/14
14 D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame SPE 107967
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)]
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
Depletion "Stems"(Boundary-Dominated Flow
Region-Volumetric
Reservoir Behavior)
Transient "Stems"(Transient Flow Region -
Analy tic al Solu tio ns: FE= 100)
Ellipti cal Flow Type Curve - Fractured Well Centered in aBounded Elliptical Reservoir (Finite Conductivity: FE= 100)
0= 0.25
0.50
0= 5.0
closed reservoirboundary (ellipse)
fracturewellbore
b
a
xfx
y
DimensionlessRateDeclineFunction
s
(qDd
(tDA
),qDdi(tDA
),and
[qDdi(tDA)])
Figure C.13 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well configuration elliptical flow model (FE=100).
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)]
Model Legend: Elliptical Flow Type Curve - FracturedWell Centered in a Bounded Elliptical Reservoir
(Finite Conductivity: FE= 1000)
Legend: qDd(tDA),qDdi(tDA), and [qDdi(tDA)] versus tDA qDd(tDA) Rate
qDdi(tDA) Rate Integral
[qDdi(tDA)] Rate Integral -Derivative
Depletion "Stems"(Boundary-Dominated Flow
Region-Volumetric
Reservoir Behavior)
Transient "Stems"(Transient Flow
Region - AnalyticalSolutions: FE= 1000)
Ellipti cal Flow Type Curve - Fractured Well Centered in aBounded Elliptical Reservoir (Finite Conductivity: FE= 1000)
0= 0.25
0.50
0= 0.25
5.0
closed reservoirboundary (ellipse)
fracturewellbore
b
a
xfx
y
DimensionlessRateDeclineFunctions
(qDd
(tDA
),qDdi(tDA
),and
[qDdi(tDA
)])
Figure C.14 qDdi(tDd)], qDd, and qDdivs. tDd Fractured well configuration elliptical flow model (FE=1000).