[Blasingame] SPE 107967
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SPE 107967 Application of the β -Integral -Integral Derivative Function to Production Analysis D. Ilk, SPE, N. Hosseinpour-Zonoozi, SPE, S. Amini, SPE, and T.A. Blasingame, SPE, Texas A&M U. Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & Gas Technology Symposium held in Denver, Colorado, U.S.A., 16–18 April 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. 01-972-952-9435.
Ab st rac t In this work we present the application of the β -integral -integral derivative function for the interpretation and analysis of production data. The β -derivative -derivative function was recently proposed for the analysis and interpretation of pressure transient data [Hosseinpour-Zonoozi, et al (2006)], (2006)], and we demonstrate that the β -integral -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 β -derivative -derivative function (i.e., i.e., the β -integral -integral derivative) for the purpose of estimating reservoir 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.
prefer the "pressure" analysis format because of the similarity with pressure transient analysis, while others are more comfortable with "rate decline" analysis. The β -integral -integral derivative functions are derived in complete detail in Appendix A, and the primary definitions are summarized as follows: q (t ) .......................................... .................... ...... (1) β [q Ddi (t Dd )] = Ddid Dd ............................ q Ddi (t Dd ) p (t ) .......................................... .................. .... (2) β [ p Ddi (t Dd )] = Ddid Dd ............................ p Ddi (t Dd )
The definitions of the component functions used in Eqs. 1 and 2) are given as follows: Function
Definition 1
t Dd
∫0
Rate Integral
q Ddi (t Dd ) =
Rate Integral Derivative
q Ddid (t Dd ) = t Dd
Pressure Integral
p Ddi (t Dd ) =
Pressure Integral Derivative
t Dd
1 t Dd
d dt Dd
∫
p Ddid (t Dd ) = t Dd
q Dd (τ ) d τ .........(3)
q Ddi (t Dd ) .....(4)
t Dd p Dd (τ ) d τ ........(5) 0 d
dt Dd
p Ddi (t Dd ) .......(6)
This paper provides the following contributions for the analysis and interpretation of gas production data using the β integral derivative function:
The associated definitions of these functions are provided in Appendix B and are referenced as appropriate in the Nomenclature.
Schematic diagrams of various production data functions using the β -integral -integral derivative formulation (type curves). Analysis/interpretation of production data using the β integral derivative formulation.
In addition to the definitions of the the β -integral -integral derivative functions, we have created an "inventory" of "type curve" solutions for unfractured and fractured wells — this inventory is provided in Appendix C.
●
●
Introduction
Orientation
This work introduces the new β -integral -integral derivative functions ( β [q Ddi(t Dd )] and β [ p p Ddi(t Dd )]) — where these functions are defined to identify the transient, transition, and boundarydominated flow flow regimes from production production data analysis. analysis. We have utilized two different formulations — β [q Ddi(t Dd )] is used for "rate decline" analysis (based on q/Δ p p functions) and p p ( t ) is used for "pressure" analysis (based on Δ p/ p/q β [ Ddi Dd ] functions).
As noted above, our inventory of solutions is provided in Ap( i.e.,, 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. ( i.e.,, the resolution of flow regime(s)).
The application (i.e. (i.e.,, the use of β [q Ddi(t Dd )] or β [ p p Ddi(t Dd )]) is essentially a matter of preference — there is no substantive difference in the application application of these functions. functions. Some analysts
We first consider the "decline rate" case β [q Ddi(t Dd )] and associated 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 elliptical flow geometry solution for a fractured well [Amini et al (2007)] — where we note that these are high fracture conduct-
2
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
ivity cases, and fractured well solutions are very similar (nearly identical) in this circumstance. Schematic of Dimensionless Rate Integral Derivative Functions Various Reservoir Models and Well Configurations (as noted) DIAGNOSTIC plot for Production Data (q Ddid and [qDdi ] )
SPE 107967
derivative function are compared with the results from conventional (i.e., established) production-analysis methods. Example 1: Southeast Asia — Oil Well
3
10
] i
d D
( (
q [
, n 2 t 10 n q , u n F o e i v t t c i n a u v 1 r F i e e 10 v D i t l a a v r g i r t e e D n l I e 0 a t r a 10 g R e t s n s I e e l t n a o R i s s n s e -1 e l m10 n i o D i s " n e m — i D w a L -2 r 10 e w o P "
Fractured Well in a Bounded Circular Reservoir (Finite C onductivity Vertical Fracture)
) )
d o i i d D c
( (
) )
Unfractured Well in a Bounded Circular Reservoir
[q Ddi ] ~ 1.0 (boundary dominated flow)
1
Transient Flow Region
In this case we have the measured rate and pressure data for an oil well — daily rates and bottomhole flowing pressures are available and are used. Fig. 3 shows the time-pressure-rate (TPR) data for this case. We note that the data are wellcorrelated except for an abrupt decline in rates at late times — which we believe indicates the evolution of wellbore damage.
NO Wellbore Storage or Skin Effects
2
For this analysis, we have chosen to use the rate decline integral functions to overcome the data-quality issues and the material balance time function to eliminate (at least to some extent) the variable-rate/variable pressure drop effects. In Fig. 4 we present the field data and model matches for the q Dd , q Ddi, β [q Ddi(t Dd )] "decline" functions in dimensionless (decline) format where the "data" functions are given by symbols.
[q Ddi ] = 0.5 (linear flow) 1 ( (
Legend: (q Ddid
Fractured Well i n a Bounded Elliptical Reservoir (Finite C onductivity Vertical Fracture)
) )
1
) ( [qDdi ] ) Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivi ty) Fractured Well (Elliptical Reservoir)
BoundaryDominated Flow Region
-3
10
-5
-4
10
-3
10
-2
10
-1
10
0
10
10
1
10
2
3
10
10
Example 1 — Exploration Well (Southeast Asia)
Dimensionless Material Balance Decline Time, t Dd,bar =NpDd /q Dd
4
5000
10
Figure 1
— Schematic of q Ddi (t Dd)] vs. t Dd — Unfractured and fractured well configurations.
Next we consider the pressure transient analysis analog case ( β [ p Ddi(t Dd )] and associated functions) as shown in Fig. 2. The major difference in Fig. 2 compared to Fig. 1 (other than the functions being inverted) is that we can clearly diagnose transient radial and linear flow (fracture cases). In addition, the boundary-dominated flow portion of the data is clearly evident as viewed from β [ p Ddi(t Dd )] (or the p Dd (t Dd ) and p Ddi(t Dd ) functions). As we noted earlier, the use of the q Dd (t Dd ) or p Dd (t Dd )format functions is a matter of preference, either (or preferably both) sets of functions can be used at the same time.
Legend: q o Data Function p wf Data Function
a 4500 i 4000
D / B T S , o
s p , f
w
p
3500 3000
q
, e 103 t a r w o l F l i O
2500
Oil Flowrate
2000 1500 Wellbore Flowing Pressure
1000 500
2
10
0
0 0 5
Schematic of Dimensionless Pressure Integral Derivative Functions Various Reservoir Models and Well Configurations (as noted) DIAGNOSTIC plot for Production Data (p Ddid and [pDdi ] )
0 0 0 1
0 0 5 1
0 0 0 2
0 0 5 2
0 0 0 3
0 0 5 3
0 0 0 4
0 0 5 4
, e r u s s e r P g n i w o l F e r o b l l e W
0
0 0 0 5
Production Time, t, hours
3
10
] i
Legend: (p Ddid
d D
p [
, n
d i o i d t D c
) ( [p Ddi] ) Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conducti vity) Fractured Well (Elliptical Reservoir)
NO Wellbore Storage or Skin Effects
Figu re 3 — Example 1: Time-Pressur e-Rate (TPR) history plot. Southeast Asia oil well — very good correlation of rates and pressures.
BoundaryDominated Flow Region
2
10
n , u n F o e i v t i c t n a u v 1 r F i e e 10 v D i t l a a r v g i r t e e D n l I e a r r 100 g u s e s t n e I r P e r s u s s e s l e n r o -1 P i 10 s s s n e e l m n i o D i s " n e m —10-2 i w D a L r e w o P " p
1
Transient Flow Region
Fetkovich-McCray Rate Function Type Curve 1 ( (
4
Unfractured Well Centered in a Bou nded Circular Reservoir (r eD = 1x10 ) Example 1 — Southeast Asia Oil Well
) )
Fractured Well in a Bounded Elliptical Reservoir (Finite C onductivity Vertical Fracture)
[p Ddi ] = 0.5 (linear flow)
10
10 [pDdi ] = 1.0 (boundary dominated flow)
2 1
( ) ( ) Fractured Well in a Bounded Circular Reservoir (Finite C onductivity Vertical Fracture)
Unfractured Well in a Bounded Circular Reservoir ( ) ( )
-3
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
Dimensionless Material Balance Decline Time, t Dd,bar =NpDd /q Dd
Figure 2
— Schematic of p Ddi (t Dd)] vs. t Dd — Unfractured and fractured well configurations (pressure transient analog for mat).
Ap pl ic ati on of th e -Int egr al Deri vat iv e Fun ct io n t o Production Analysis — Field Examples In this section we provide field examples to demonstrate/ illustrate the diagnostic value of the β -integral derivative function and its applications in production analysis. The main pur pose of this exercise is to provide the diagnostic value of the β -integral derivative function rather than focusing on it as a direct solution mechanism. Our results using the β -integral
s ) n ] o ) i t d D c t n ( i u d F D q e [ n i l c d e n D a , e ) t a d D R t ( i s d s D e q l , n ) o d i D s t n ( e d D q m ( i D
-4
10
2
-3
10
-2
10
-1
10
0
10
1
2
10
10
2
Model Legend: Fetkovich-McCray Rate Function Transient "Stems" Type Curve - Unfractured Well Centered in a Bounded (Transient Flow Region 4 4 Analy tical Solut ions : r eD = 1x10 ) Circular Reservoir (Dimensionless Radius: r eD = 1x10 )
10
Legend: q Dd (t Dd ), q Ddi (t Dd), and [ q Ddi (t Dd )] versus t Dd,bar q Dd(t Dd) Rate q Ddi (t Dd ) Rate Integral
1
10
1
[q Ddi (t Dd)] Rate Integral -Derivative
10
0
10
[q Ddi (t Dd)]
0
q Dd(t Dd ) Data Function q Ddi (t Dd) Data Function
10
q Ddi (t Dd )
4 r eD=1x10
-1
10 q Dd(t Dd )
[q Ddi (t Dd)] Data Function
10
10
-2
10
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior) -3
10
-4
10
-3
10
-2
10
-1
10
0
10
-1
1
10
-2
-3
2
10
t Dd,bar =NpDd (t Dd )/q Dd (t Dd )
Figure 4 — Example 1: Diagnostic log-log plot (dimensionless rate decline integral functions) excellent diagnostic performance of [q Ddi (t Dd)] data function.
The diagnostic log-log plot shown in Fig. 4 is excellent — we
SPE 107967
Application of the β -Integral Derivative Function to Production Analysis
obtained excellent data matches using the model for an unfractured well in a homogenous reservoir model. In this case we obtained a match of r eD = 1x104 — which, in isolation, does suggest well damage effects. The only discrepancy in the β [q Ddi(t Dd )] model and data functions occurs at relatively "early" values of the material balance time function, at times where we believe the data are transitioning from transient radial flow to a transitional flow regime prior to evidence of boundary effects. From the β [q Ddi(t Dd )] data function, it is clear that the boundaries of the drainage area have not yet established — i.e., the β [q Ddi(t Dd )] 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 take more 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 such as: k , s, r eD, N , pi (where we note that pi is imposed in this and all of our examples), we proceed and generate model-based pressures and rates using superposition in time. This "analysis" 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. Example 1 — Exploration Well (Southeast Asia) 4
Legend: q o Data Function p wf Data Function
D / B T S , o
Analys is Resu lts: South east Asi a Oil Wel l (Bounded Circular Reservoir Case)
(
) q o Model Function
(
) p wf Model Function
k r eD N
r e pi
= = = = =
130 md 4 1x10 (dimensionless) 24.1 MMSTB 3430 ft 2900 p si a (f or ced )
q
, e 103 t a r w o l F l i O
10
5
10000 Legend: East Texas Gas Well (SPE 84287) q g Data Function
4000
f w
7000 p ,
q
5000 4000 3000 2000 1000
2
0 0 0 5
3000
2000 1500 Wellbore Flowing Pressure
1000 500
0
0 0 5
0 0 0 1
0 0 5 1
0 0 0 2
0 0 5 2
0 0 0 3
0 0 5 3
0 0 0 4
0 0 5 4
0
Production Time, t, hours
Figure 5 — Example 1: Analysis by modeling, excellent performance of the model obtained from the log-log diagnostic plot.
0 0 5 1
0 0 0 2
0 0 5 2
0 0 0 3
0 0 5 3
0 0 0 4
0 0 5 4
0 0 0 5
0 0 5 5
0 0 0 6
0 0 5 6
0 0 0 7
0
0 0 5 7
0 0 0 8
Figu re 6 — Example 2: Time-Pressur e-Rate (TPR) his tor y plo t. East Tx gas well. Very good correlation of rate and pressure data — indicates likelihood of good analysis.
Since this well is hydraulically fractured, we use fractured well models for analysis/interpretation. Since this is a gas 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 rate integral decline functions — in particular, the β [q Ddi(t Dd )] data function indicates that the flow is in transition to the boundary-dominated flow regime (evolving trend in the β [q Ddi(t Dd )] data function approaches 1). Fetkovich McCray Rate Function Type Curve Fractured Well Centered in a Boun ded Circular Reservoir (FcD = 10) Example 2 — East TX Gas Well (Tigh t Gas Sand)
w
0 0 0 5
0 0 0 1
Production Time, t, hr
s p , f
, e r u s s e r P g n i w o l F e r o b l l e W
10
10
-5
3
-4
10
10
-3
-2
10
10
-1
10
0
10
Transient "Stems" (Transient Flow Region Analy tic al Sol utio ns: FcD = 10)
s ) n ] o ) i t d 2 D c t 10 n ( i u d F D q e [ n i l c d e n 101 D a , e ) t d a D t R ( i s d s D e q 100 l , n ) o d i D t s ( n d e D q m ( i -1 10 D
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 for this case.
10
10
1
10
3
Model Legend: Elliptical Flow Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 10)
Legend: q Dd(t Dd), q Ddi (t Dd), and [q Ddi (t Dd)] versus t Dd,bar Rate q Dd(tDd) Rate Integral q Ddi (t Dd)
10
2
[q Ddi (t Dd)] Rate Integral -Derivative q Ddi (t Dd)
q Ddi (t Dd) Data Function
10
1
q Dd (t Dd)
10
q Dd(t Dd) Data Function
0
[q Di(t Dd)] [q Di (t Dd)] Data Function
10
-1
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
-2
In Fig. 5 we 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.
e r u s e r P g n i w o l F e r o b l l e W
6000 s
, e t a r w o l F 3 s 10 a G
10
a i s
8000 p ,
D / F C 4 S10 M , g
p
3500
9000
p wf Data Function
a 4500 i
2500
Oil Flowrate
2
10
Example 2 — East Texas Gas Well (SPE 84287) (Tight Gas Sand)
5000
10
3
-5
-4
10
10
-3
-2
10
10
-1
10
0
10
10
-2
1
t Dd,bar =GpDd (t Dd )/q Dd(t Dd)
Figure 7 — Example 2: Diagnostic log-log plot (dimensionless rate decline integral functions) outstanding diagnostic performance of [ q Ddi (t Dd)] data function.
However, as we observe from the fractured well model, this case is in transition and requires approximately two more logcycles to reach complete boundary-dominated flow. Such an observation is neither unusual nor unexpected for a well in a low to very-low permeability gas reservoir. As in the previous case, we proceed from the analysis and generate the pressure and rate responses using the defined reservoir model and the estimated reservoir parameters (k , r eD, G, pi — where, again, pi is imposed all cases).
4
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
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 erratic behavior in the rate data. We note that our analysis results are very close to original results provided for this case [Pratikno et al (2003)]. Example 2 — East Texas Gas Well (SPE 84287) (Tight Gas Sand) 10
5
12000 Legend: East Texas Gas Well qg Data Function p wf Data Function qg Model Function p wf Model Function
D / F C 4 S10 M , g
Analysi s Resul ts: Eas t Tx Gas Wel l
11000
(Bounded Circular Reservoir Case) k x f FcD G r e pi
= = = = = =
a i p , f
10000 s
0 .0 55 4 m d 2 90 f t 10 (d im en si on les s) 1. 58 6 B SC F 339 ft 9330 psia (forced)
9000 8000 7000
q
, e t a r w o l F 3 s 10 a G
6000 5000 4000 3000 2000
w
p
, e r u s s e r P g n i w o l F e r o b l l e W
1000 2
10 0
0 0 5
0 0 0 1
0 0 5 1
0 0 0 2
0 0 5 2
0 0 0 3
0 0 5 3
0 0 0 4
0 0 5 4
0 0 0 5
0 0 5 5
0 0 0 6
0 0 5 6
0 0 0 7
0 0 5 7
0
0 0 0 8
Production Time, t, hr
Figure 8 — Example 2: Analysis by modeling, very good performance of the model obtained from the log-log diagnostic plot.
Example 3: Mexico — Very Tight Gas (long production) This example was recently evaluated using an elliptical flow model [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 field has only one well. The long production history and high quality data yield "near textbook" quality diagnostic plots (Figs. 9 and 10). Example 3 — Mexico Gas Well (Tight Gas Sand — Very Low Reservoir Permeability, Very Long Prod uction History) 4
1200
10
1100
a i p , f
1000 s
D / F C S M , g
900 800 700
q
, e 103 t a r w o l F s a G
600 500 400 300 Legend: qg Data Function pwf Data Function
200
w
p
, e r u s s e r P g n i w o l F e r o b l l e W
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., q Dd and the auxiliary functions q Ddi and β [q Ddi(t Dd )] versus t DA). We obtained an excellent match using the elliptical flow parameters — F E = 100 and ξ 0 = 0.25. We note that these are the 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 β [q Ddi(t Dd )] data function rather than q Ddid (t Dd ) — which indicates the transition to boundary-dominated flow uniquely. 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) -4
0 0 , 3 1
0 0 , 4 1
0 0 , 5 1
0 0 , 6 1
Figur e 9
We note as comment that the data scatter seen in the rate is not clearly reflected in the pressure data — but we also acknowledge 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 that these data are accurate and correlated — and we anticipate a consistent analysis/interpretation. The objective of this example is to apply and validate the elliptical boundary β -integral derivative type curves. For this purpose we have used the elliptical boundary model type
-2
10
-1
0
10
10
1
2
3
10
10
10
2
Model Legend: Elliptical Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 100) Legend: q Dd(t DA), q Ddi (t DA), and [q Ddi (t DA)] versus t DA q Dd(t DA) Rate q Ddi (t DA) Rate Integral
1
10
1
[q Ddi (t DA)] Rate Integral -Derivative q Dd(t DA) Data Function
10
[q Di (t DA)]
10 q Dd(t DA)
Transient "Stems" (Transient Flow Region Analyti cal Solu tion s: FE = 100) y
-1
q Ddi (t DA)
10
closed reservoir boundary (ellipse)
wellbore
0
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
[q Di (t DA)] Data Function
-2
fracture
-3
10
b
10
x
x f
-3
a
-4
10
-4
10
-3
10
10
-2
10
-1
0
10
10
1
10
2
-4
3
10
10
Dimensionless Decline Time Based on Drainage Area (t DA )
Figure 10 — Example 3: Diagnostic log-log plot (dimensionless rate decline integral functions) very good match of the [ q Ddi (t Dd)] function (excellent diagnostic).
The final step in our analysis is to generate the pressure and rate responses using the (elliptical) reservoir model that we deduced from the diagnostic plot (see Fig. 11). We note that for this case, the computed rates match the raw d ata extremely well — but the calculated bottomhole pressure response does show 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. Example 3 — Mexico Gas Well (Tight Gas Sand — Very Low Reservoir Permeability, Very Long Production History) 5
0 0 , 7 1
— Example 3: Time-Pressur e-Rate (TPR) his tor y plo t. Mexico gas well. Good quality data (bottomh ole pressures are given con stant).
10
0 = 0.25
s ) 10 n ] o ) i A t D c t n ( i u d F D 0 q 10 e [ n i l c d e n D a , e ) t -1 A a t D10 R ( i s d s D e q l , n ) o A i D -2 s t ( n d10 e D m ( q i D
0
Production Time, t, days
10
q Ddi(t DA) Data Function
10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 2 3 4 5 6 7 8 9 0 1 2 1 1 1
-3
10
2
10
100
2
SPE 107967
1600
10
Analys is Resu lts: Mexic o Gas Well (Bounded Elliptical Reservoir Case)
D / F C 4 S10 M , g
k x f FE
= = =
0.001 826 100
md ft (dimensionless)
G r e pi
= = =
9.6 BSCF 871 ft 5463 ps ia (f or ced )
Legend:
2
0 0 0 0 , 1
0 0 0 , 2
0 0 0 , 3
0 0 0 , 4
q g Model Function p wf Model Function
0 0 0 , 5
0 0 0 , 6
0 0 0 , 7
0 0 0 , 8
0 0 0 , 9
0 0 0 , 0 1
0 0 0 , 1 1
0 0 0 , 2 1
0 0 0 , 3 1
0 0 0 , 4 1
0 0 0 , 5 1
s p , f
w
p
q
10
1200
, e 1000 r u s s e r 800 P g n i w 600 o l F e r 400 o b l l e 200 W
, e t a r w o l F 3 s 10 a G q g Data Function p wf Data Function
1400 a i
0 0 0 , 6 1
0
0 0 0 , 7 1
Production Time, t, days
Figure 11 — Example 3: Analysis by modeling, very good rate match by the mod el, generated pressures fail to honor the given constant bottomhole pressures.
As closure in this section, we present the "average" analysis
SPE 107967
Application of the β -Integral Derivative Function to Production Analysis
results for these examples considered in this work (see Table 1). Table 1
— " Average" analysis results for this work.
Example 1 (oil) 2 (gas) 3 (gas)
k (md) 130 0.0055 0.0010
x f (ft) N/A 290 825
G (or N ) (BSCF or MMSTB) 24.1 1.6 23.0
Summary and Conclusion s Summary: The primary purpose of this paper is the presentation of the β -integral derivative function as a diagnostic tool for production data analysis. Two different (dimensionless) formulations of the β -integral derivative function are proposed for use in production analysis applications. β [q Ddi(t Dd )] formulation for "rate decline" analysis p Ddi(t Dd )] formulation for "pressure" analysis ● β [ ●
The β -integral derivative function can be computed directly using rate/pressure integral and rate/pressure integral derivative functions or rate/pressure and rate/pressure integral functions (the relevant derivations are provided in Appendix A). We provide a schematic "diagnosis worksheet" for the interpretation of the β -integral derivative function for rate integral and pressure integral cases (see Appendix C) as well as an inventory of type curves ( β -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 — Centered in a bounded elliptical reservoir ● Fractured well ● ●
We have applied and validated the application of β -integral derivative function for production analysis using various field cases. Conclusions: 1. The β -integral derivative function has the potential to become a significant diagnostic tool in production analysis as the β -integral derivative function exhibits unique character for several flow regimes. 2. The diagnostic matches of the production data obtained using the β -integral derivative function presented in this work are excellent. It is very likely that similar diagnostic matches would have been obtained using the rate integral derivative function. But we have shown that the β -integral derivative function provides more resolution — in particular, the β -integral derivative function yields the following behavior for the cases used in this work. Case ● Reservoir boundaries: — Closed reservoir (circle, rectangle, etc.) Fractured wells: — Infinite conductivity vertical fracture. — Finite conductivity vertical fracture.
β [q Ddi(t Dd )] 1
●
1/2 1/4
3. The incorporation of the β -integral derivative function in the modern production analysis tools will help to distinguish individual flow regimes, as well as help to differentiate transitional character — this may be the source o f most value for the β -integral derivative functions.
Recommendations/Comment :
Future work on this topic
5
should focus on the additional β -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 BSCF G p = Gas production, MSCF or BSCF h = Pay thickness, ft k = Permeability, md k f = Fracture permeability, md k R = Reservoir permeability, md N = Oil-in-place, STB N p = Cumulative oil production, STB p = Pressure, psia pi = Initial reservoir pressure, psia p p = Pseudopressure function, psia p R = Reservoir pressure, psia pwf = Flowing bottomhole pressure, psia q = Flowrate, STB/D q g
=
Gas flowrate, MSCF/D
qo r e r w r wa t t a x f
= = = = = = =
Oil flowrate, STB/D Drainage radius, ft Wellbore radius, ft Apparent wellbore radius, ft Time, hr Pseudo-time (adjusted time), hr Fracture half-length, ft
Dimensionless Variables b Dpss = Dimensionless pseudosteady-state constant F cD = Dimensionless fracture conductivity F E = Elliptical fracture conductivity p D = Dimensionless pressure p Dd
=
p Di
=
Dimensionless pressure derivative Dimensionless pressure integral
p Did
=
Dimensionless pressure integral derivative
β [ p Ddi]
=
Dimensionless β -pressure integral derivative
q D
=
q Di
=
Dimensionless flowrate Dimensionless rate integral
q Did
=
Dimensionless rate integral derivative
β [q Ddi]
=
Dimensionless β -rate integral derivative
r eD t D t Dd t DA t Dxf
= = = = =
Dimensionless outer reservoir boundary radius Dimensionless time (wellbore radius) Dimensionless decline time Dimensionless time (drainage area) Dimensionless time (fracture half-length)
Mathematical Functions and Variables a = Regression coefficient A = Auxiliary function b = Regression coefficient B = Auxiliary function Greek Symbols β = Beta-derivative φ = porosity, fraction μ = Viscosity, cp ξ 0 = Elliptical boundary characteristic variable Subscripts a = d = D =
Pseudotime Derivative or decline parameter Dimensionless
6
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
Dd f g i id mb pss r R
= = = = = = = = =
Dimensionless decline variable Fracture Gas Integral function or initial value Integral derivative function Material balance Pseudosteady-state Positive integer Reservoir
Superscripts — =
Material balance time
Constants π = γ =
Circumference to diameter ratio, 3.1415926… Euler’s constant, 0.577216…
Gas Pseudofunctions: p μ z i p p p = i i dp pi pbase μ z t a
= μ gic gi
t mba, gas
=
∫ ∫
t
1
dt μ ( p) c g ( p ) 0 g μ gi c gi t q(t ) q(t )
∫
0
μ g ( p) c g ( p )
dt
References Amini, S., Ilk, D., and Blasingame, T.A.: "Evaluation of the Elliptical Flow Period for Hydraulically-Fractured Wells in Tight Gas Sands — Theoretical Aspects and Practical Considerations," paper SPE 106308 presented at the 2007 SPE Hydraulic Fracturing Technology Conference held in College Station, Texas, U.S.A., 29–31 January 2007. Blasingame, T.A., Johnston, J.L., and Lee, W.J.: "Type Curve Analysis 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 of Oil 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 in Veracruz, 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 and Applications," paper SPE 103204 presented at the 2006 Annual SPE Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006. Palacio, J.C. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves — Analysis of Gas Well Production Data," paper SPE 25909 presented at the 1993 Joint Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 26-28 April 1993. Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "De-cline Curve Analysis Using Type Curves — Fractured Wells," paper SPE 84287 presented at the SPE annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October 2003.
SPE 107967
SPE 107967
Application of the β -Integral Derivative Function to Production Analysis
Ap pen di x A: Deri vat io n of Rate Int egr al Formulation
-Derivat iv e
In this Appendix we derive the β -derivative integral functions ( β [q Ddi(t Dd )] and β [ p Ddi(t Dd )]) where these functions are defined to identify the transient, transition, and boundary-dominated flow regimes from production data analysis.
Before we begin to derive the formulation for the β -integral derivative 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 rateintegral function is defined as: q Ddi (t Dd ) =
t Dd
t Dd
∫0
q Dd (τ ) d τ ..................................(A-1)
Where q Dd (t Dd ) is the dimensionless rate decline function [Fetkovich, 1980]. The dimensionless rate-integral derivative function (using the Bourdet derivative formulation) is: q Ddid (t Dd ) = t Dd
d dt Dd
q Ddi (t Dd ) ..............................(A-2)
The derivative of Eq. A-1 with respect to the dimensionless decline time, t Dd is: d dt Dd
(t Dd )
=
t Dd
1
1 t Dd
2
∫0
q Dd (τ ) d τ +
1 t Dd
β [q Ddi (t Dd )]
t Dd
d dt Dd
Pressure Integral Functions For reference, the dimensionless pressure-integral function is defined as [Blasingame, et al 1989]: (modified to "decline" variable format) p Ddi (t Dd ) =
...................................................................................... (A-4) Substituting Eq. A-3 into Eq. A-4, we obtain,
Where p Dd (t Dd ) is the dimensionless pressure decline function. The dimensionless rate-integral derivative function (using the Bourdet derivative formulation) is given as: p Ddid (t Dd ) = t Dd
=
d dt Dd
p Ddi (t Dd ) ................................ (A-8)
p Ddi (t Dd ) 1 (t Dd ) 2 1 t Dd
∫
t Dd 1 p Dd (τ ) d τ + p Dd (t Dd ) t Dd 0
[ p Dd (t Dd ) − p Ddi (t Dd ) ]
...................................................................................... (A-9) Multiplying through Eq. A-9 by the dimensionless decline time, t Dd yields: d dt Dd
p Ddi (t Dd )
The power-law derivative formulation (i.e., β -derivative formulation) for the dimensionless pressure-integral function is defined as:
⎡ 1
t Dd ⎢
⎣ t Dd
⎤
[ q Dd (t Dd ) − q Ddi (t Dd ) ]⎥
=
⎦ =
q Ddi (t Dd )
∫
β [ p Ddi (t Dd )]
β [q Ddi (t Dd )]
q Dd (t Dd )
t Dd
t Dd p Dd (τ ) d τ ................................. (A-7) 0
.................................................................................... (A-10) Where Eq. A-10 is a fundamental definition of the "pressure integral" given by [Blasingame, et al 1989].
q (t ) β [q Ddi (t Dd )] = Ddid Dd q Ddi (t Dd )
=
1
= p Dd (t Dd ) − p Ddi (t Dd )
q Ddi (t Dd )
Where this result reduces to:
= q Ddi (t Dd )
q Ddi (t Dd )
Where Eq. A-6 is exactly the definition given by [Doublet, et al 1994], and thus, confirms our definition of the β [q Ddi(t Dd )] function.
= t Dd
d ln[t Dd ]
1
q Dd (t Dd )
p Ddid (t Dd )
d ln[q Ddi (t Dd )]
q Ddi (t Dd )
= 1−
Solving for q Ddid (t Dd ) yields
=−
[ q Dd (t Dd ) − q Ddi (t Dd ) ]
1
q Ddi (t Dd )
dt Dd
The power-law derivative formulation (i.e., the β -derivative formulation) for the dimensionless rate-integral function is defined as:
=
q Ddid (t Dd )
d
q Dd (t Dd )
....................................................................................... (A-3)
=
Equating Eqs. A-5 and A-6 gives us:
The derivative of Eq. A-7 with respect to dimensionless decline time, t Dd is:
q Ddi (t Dd )
= −
q (t ) β [q Ddi (t Dd )] = 1 − Dd Dd ......................................... (A-5) q Ddi (t Dd )
q Ddid (t Dd ) = q Ddi (t Dd ) − q Dd (t Dd ) ............................. (A-6)
Rate Integral Functions
1
7
−1
Or, finally, we obtain:
=
d ln[ p Ddi (t Dd )] d ln[t Dd ] 1 p Ddi (t Dd ) 1 p Ddi (t Dd )
t Dd
d dt Dd
p Ddi (t Dd )
⎡ 1
t Dd ⎢
⎣ t Dd
Where this result reduces to:
⎤
[ p Dd (t Dd ) − p Ddi (t Dd )]⎥ ⎦
8
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
β [ p Ddi (t Dd )]
= =
1
[ p Dd (t Dd ) − p Ddi (t Dd )]
p Ddi (t Dd )
−1
p Ddi (t Dd )
....................................................................................(A-11) Where we note an alternate form of Eq. A-11 is obtained using β [ p Ddi (t Dd )]
= =
1
[ p Dd (t Dd ) − p Ddi (t Dd )]
p Ddi (t Dd ) p Ddid (t Dd )
⎡ r ⎤ 1 = ln ⎢ e ⎥ − ⎣ r wa ⎦ 2
p Dd (t Dd ) p Ddi (t Dd )
−1 =
p Ddid (t Dd ) p Ddi (t Dd )
p Ddid (t Dd ) = p Dd (t Dd ) − p Ddi (t Dd ) ...........................(A-13)
Where Eq. A-13 is exactly (as expected) the definition given by [Blasingame, et al 1989], and thus, confirms our definition of the β [ p Ddi(t Dd )] function.
Ap pen di x B: Dimen si on les s Variab les The most straightforward approach to defining dimensionless variables for this application is to use the approach of Fetkovich [Fetkovich, 1980] and reduce all cases to a single set of unified variables. This process is fairly easy for a given case, but will require knowledge of the reservoir model for each specific case. To simplify (somewhat) this exercise, we will use the approach of [Pratikno, et al 2003], which states the following relations for the dimensionless decline variables: ("decline" time).......................... (B-1)
t D A
b Dpss
q Dd = q D b Dpss p Dd =
1 b Dpss
("decline" rate)........................... (B-2) ("decline" pressure) ................... (B-3)
p D
+
= 0.00633
q D
= 141.2
p D
=
1
k φμ ct A
qBμ
(t in days)................................... (B-4)
t 1
kh ( pi − pwf ) kh
141.2 qBμ
............................................ (B-5)
( pi − p wf ) ............................................ (B-6)
The remaining task is to address the b Dpss variable for the unfractured well, fractured well, and elliptical flow cases — these results are: Unfractured Well : [Fetkovich, 1980] b Dpss
⎡ r ⎤ 3 = ln ⎢ e ⎥ − ⎣ r wa ⎦ 4
+ a 4 u 3 + a5 u 4 1 + b1 u + b2 u 2 + b3 u 3 + b4 u 4
...................................................................................... (B-8) Where, a1 = 0.93626800 b1 = -0.38553900 a2 = -1.00489000 b2 = -0.06988650 a3 = 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 more than sufficient for all applications. Elliptical Flow/Fractured Well : [Amini, et al 2007] Given a particular reservoir/fracture case — formulated in the elliptical flow geometry (i.e., ξ 0 and F E values), then b Dpss(ξ 0 ,F E ) can be estimated using : b = 1.00146ξ + 0.0794849e −ξ 0 − 0.16703u Dpss
+
(exact definition).................. (B-7a)
A B
0
− 0.754772
.................................................................................... (B-10) Where the auxiliary functions are: u = ln( F E )
Where (obviously) the b Dpss variable given in Eqs. B-1 to B-3 is model-dependent. For reference, the base or "universal" definitions of t DA, q D, and p D are: t D
a1 + a2 u + a3 u 2
u = ln ( F cD )
Solving for p Ddid (t Dd ) yields
2π
(Fetkovich definition) .......... (B-7b)
The difference in Eqs. B-7a and B-7b, is essentially irrelevant, and from a historical perspective, the Fetkovich definition is most widely accepted. We use Eq. B-7b in this work. Fractured Well : [Pratikno, et al 2003] Given a particular reservoir/fracture case (i.e., r eD and F cD values), then b Dpss(r eD, F cD) can be estimated using : −2 b Dpss = ln ( r eD ) − 0.049298 + 0.43464 r eD
p Ddi (t Dd )
....................................................................................(A-12) Equating Eqs. A-11 and A-12 gives us:
t Dd =
But we note that Fetkovich [Fetkovich, 1980] defined this variable as: b Dpss
p Dd (t Dd )
SPE 107967
A = a1 + a2u + a3u 2
+ a4u 3 + a5u 4
B = b1 + b2u + b3u 2
+ b4u3 + b5u 4
.................................................................................... (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 al 1994] — i.e., the "material balance time" concept. Using this approach, we "convert" variable-rate/variable pressure drop data into an equivalent constant rate case (analog to well test analysis). As such, we will always work in terms of the material balance time variable which is defined as: − N p (or G p ) t =
qo (or q g )
(liquid case)............................. (B-12)
SPE 107967
−
t a
=
Application of the β -Integral Derivative Function to Production Analysis
μ gi c gi q g (t )
∫
t
q g (τ )
0 μ g (τ ) c g (τ )
d τ (gas case)................... (B-13)
In practice, we will use the "decline" time variables based on the 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: −
t Dd
=
N pDd
.............................................................. (B-14)
q Dd
Where the dimensionless "decline" cumulative production is defined as: N pDd
=
∫
q Dd 0
q Dd (τ ) d τ ............................................. (B-15)
As a final comment, we want to state that for the unfractured reservoir case we have used (exactly) the Fetkovich definitions for the "decline" variables. Specifically, these definitions are: t Dd =
1
⎡
⎤ 2 ⎤⎥ ⎡ ⎡ r e ⎤ 1 ⎤ ⎢ ⎥ − 1⎥ ⎢ln ⎢ r ⎥ − 2 ⎥ 2 ⎢ ⎣ r w ⎦ ⎣ ⎦ ⎣⎢ ⎣ wa ⎦ ⎦⎥
1 ⎢ ⎡ r e
t D ........................... (B-16)
⎡ ⎡ r ⎤ 1 ⎤ q Dd = ⎢ln ⎢ e ⎥ − ⎥ q D ............................................... (B-17) ⎢⎣ ⎣ r wa ⎦ 2 ⎥⎦ p Dd =
1
⎡ ⎡ r e ⎤ 1 ⎤ ⎢ln ⎢ ⎥− ⎥ ⎣⎢ ⎣ r wa ⎦ 2 ⎦⎥
p D ............................................. (B-18)
9
Where for this case, the "ordinary" dimensionless time function is given as: t D
= 0.00633
k φμ ct r w2
t ................................................... (B-19)
Ap pen di x C: Dimens io nl ess " Typ e Cur ve" Repr esentations of the -Pressure Derivative and Various other Pressure Functions (selected reservoir/well configurations) In this appendix we present the "inventory" of type curve solutions for the proposed β -derivative integral functions (i.e., the β [q Ddi(t Dd )] and β [ p Ddi(t Dd )]). We use the dimensionless decline "material balance time" function given as: (i.e., the equivalent constant rate case) −
t Dd
=
=
1 q Dd
∫
q Dd 0
q Dd (τ ) d τ
N pDd q Dd
...................................................................................... (C-1) For the case of the elliptical flow geometry we elected not to use the t Dd -format due to certain early-time artifacts (some trends overlap in a non-uniform manner). We believe that this effect is not an error or flaw in the use of the t Dd function, but rather just an artifact of the formulation for this particular case. As an alternative, we use the t DA format as proposed by Amini et al [Amini et al (2007)] — this format works very well and yields no visible artifacts.
10
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
SPE 107967
Schematic of Dimensionless Rate Integral Derivative Functions Various Reservoir Models and Well Configurations (as noted) DIAGNOSTIC plot for Production Data (q Ddid and [q Ddi ] ) 3
10
] i
d D
( (
q [
, n
o d i i d t D c n
Fractured Well in a Bounded Circular Reservoir (Finite Conductivity Vertical Fracture)
) )
2
NO Wellbore Storage or Skin Effects
10
( (
q
, u n F o e i v t i c t n a v u i 1 F r e 10 e D v i t l a a r v g i r t e e n D I l e a 0 r t g a 10 e R t s n s I e e l t a n o R i s s s n e e 10-1 l m n i o D i s " n e m — i D w a L -2 r 10 e w o P "
) )
Unfractured Well in a Bounded Circular Reservoir
[q Ddi ] ~ 1.0 (boundary dominated flow)
1
Transient Flow Region
2
[q Ddi ] = 0.5 (linear flow ) 1 ( (
Legend: (q Ddid
Fractured Well in a Bounded Elliptical Reservoir (Finite Conductivity Vertical Fracture)
) )
1
) ( [q Ddi ] ) Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivity ) Fractured Well (Elliptical Reservoir)
BoundaryDominated Flow Region
-3
10
-5
10
-4
10
10
-3
-2
-1
10
0
10
10
10
1
2
3
10
10
Dimensionless Material Balance Decline Time, t Dd,bar =NpDd /q Dd
Figur e C.1 — Schematic of q Ddi (t Dd)] vs. t Dd — Unfractured and fractured well configurations (note the distinction of the " transition" flow regimes that the q Ddi (t Dd) function pr ovides) (analog of decli ne type curve analysis).
Schematic of Dimensionless Pressure Integral Derivative Functions Various Reservoir Models and Well Configurations (as noted) DIAGNOSTIC plot for Production Data (p Ddid and [p Ddi ] ) 3
10
] i
Legend: (p Ddid
d D
p [
, n o 2 d i i 10 t d D c p n u , n F o e i v t i c t n a v u i 1 F r e 10 e D v l i t a a r v g i r t e e n D I l e a r 0 r 10 g u e s t s e n r I P e r s u s s e s l e n r o -1 P i 10 s s s n e e l m n i o D i s " n e m — -2 i 10 D w a L r e w o P "
) ( [p Ddi ] ) Unfractured Well (Radial Flow) Fractured Well (Finite Fracture Conductivity ) Fractured Well (Elliptical Reservoir)
NO Wellbore Storage or Skin Effects
BoundaryDominated Flow Region
1
Transient Flow Region 1 ( (
) )
Fractured Well in a Bounded Elliptical Reservoir (Finite C onductivity Vertical Fracture)
[p Ddi ] = 0.5 (linear flow)
[p Ddi ] = 1.0 (boundary dominated flow) 2 1
( (
) )
Fractured Well in a Bounded Circular Reservoir (Finite Conductivity Vertical Fracture)
Unfractured Well in a Bounded Circular Reservoir ( ) ( )
-3
10
-5
10
-4
10
10
-3
-2
10
-1
10
0
10
10
1
2
10
3
10
Dimensionless Material Balance Decline Time, t Dd,bar =NpDd /q Dd
Figur e C.2 — Schematic o f p Ddi (t Dd)] vs. t Dd — Unfractured and fractured well configurations — good transition and strong indicator of the boundary-domin ated flow regime (analog of well test analysis).
SPE 107967
Application of the β -Integral Derivative Function to Production Analysis
11
Fetkovich-McCray Rate Function Type Curve— t Dd,bar Format (Unfractured Well Centered in a Bounded Circular Reservoir) -4
-3
10
-2
10
3
-1
10
10 ] ) d
0
10
10
1
2
10
10
3
10
3
10
D
t ( i
Model Legend: Fetkovich-McCray Rate Function Type Curve - Unfractured Well Centered in a Bounded Circular Reservoir
Transient "Stems" (Transient Flow Region)
d n q a [ , ) 2 d , , 10 D ) t e v d ( i i D d t t ( D a d q v D , i r q l e , a r D e t g a e - 101 R t n l a r e I e g n i t e l a t c R n e I D e e n t s i a 0 s l 10 e c e R l e n D n o s i i s s l c e e n l e n D m i o s i s D s e 10-1 n l e n o m i i D s n e m i D -2 d D
Legend: q Dd(t Dd), qDdi (t Dd) and [q Ddi (t Dd)] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
r eD=r e/r wa =5
2
10
10 20
q Dd(t Dd)
30 50
1
10
[q Ddi (t Dd)] Depletion "Stems" (Boundary-Dominated Flow Region)
100 500 1000 4 r eD=1x10
0
10 5
20
30
10
50 500 4 r eD=1x10
100
-1
10
1000
[q Ddi (t Dd)]
-2
10
-4
-3
10
10
-2
-1
10
0
10
10
1
2
10
10
10
3
10
t Dd,bar =NpDd (t Dd)/q Dd (t Dd)
Figur e C.3 —
q Ddi (t Dd)] vs. t Dd — Unfractured well configuration — also plotted with q Dd and q Ddi for comparison — very good resolution of transient and transition regimes using the q Ddi (t Dd)] functions.
Pressure Function Type Curve— t Dd,bar Format (Unfractured Well Centered in a Bounded Circular Reservoir) -4
10
] )
d D
t ( i d d D n p a [ , ) , , d D e ) t d ( v i i D d t t ( D a d p v D , i r p l e , a r D e g r e u t s n l s I a r e r e r g e P u t n s s I s s e e l r e r n P u o s s i s s s e e n l r e n P m i o s i D s s e n l e n o m i i s D n e m i D
10
10
3
-3
-2
-1
10
0
10
10
1
10
2
10
3
10
3
10 Model Legend: Fetkovich-McCray Rate Function Type Curve - Unfractured Well Centered in a Bounded Circular Reservoir
10
10
Legend: p Dd (t Dd ), p Ddi (t Dd) and [p Ddi (t Dd)] vs. t Dd,bar Pressure Function Curves Pressure Integral Function Curves Pressure Integral- -Derivative Function Curves
2
2
10
1
1
Transient "Stems" (Transient Flow Region)
10
p Dd(t Dd) p Ddi (t Dd)
10
4 r eD=1x10
0
0
10
1000 5
500
-1
Depletion "Stems" (Boundary-Dominated Flow Region)
100
500
30 20
10
10
30 20 50
100 50
-1
10
1000 4 r eD=1x10
10
[p Ddi (t Dd)]
r eD=r e/r wa=5
-2
10
-2
-4
10
10
-3
-2
10
-1
10
0
10
1
10
2
10
10
3
10
t Dd,bar =NpDd (t Dd)/q Dd (t Dd)
Figur e C.4 —
p Ddi (t Dd)] vs. t Dd — Unfractured well configuration — also plotted with p Dd and p Ddi for comparison, similar form as the q Ddi (t Dd)] functions — excellent resolution of all flow regimes.
Fetkovich-McCray Rate Function Type Curve- t Dd,bar Format ( Vertical Well with a Finit e Conductivity Vertical Fracture — FcD = 1 ) -4
-3
10
] )
-2
10
3
-1
10
0
10
10
1
2
10
10
Fetkovich-McCray Rate Function Type Curve- t Dd,bar Format ( Vertical Well with a Finit e Conductivity Vertical Fracture — FcD = 100 ) 3
-4
10
10
3
10
d
D t
( i d d D n q a [ , ) 2 d , , D e 10 ) t v d ( i i D t d t ( D a d q v D , i r q l e , a r D e t g 1 e a t R n l 10 a e I r n e g i e l t a t c R n e I D e e n t s i a 0 s l c R 10 e e l n D e n o i i s l s s c n l e e e n D m i o s i -1 D s s e 10 n l e n o m i i D s n e m i D -2
2
10
1
q Dd(t Dd )
10 Depletion "Stems" (Boundary-Dominated Flow Region)
0
10 5
20
30
10
50
500
-1
100
10
3
r eD=1x10
[ q Ddi (t Dd)]
-2
10
-4
-3
10
-2
10
-1
10
0
10
10
1
2
10
10
10
( i d d D n q a [ , ) 2 d , , D e 10 ) t v d ( i i D t d t ( D a d q v D , i r q l e , a r D e t g 1 e a t R n l 10 a e I r n e g i e l t a t c R n e I D e e n t s i a 0 s l c R 10 e e l n D e n o i i s l s s c n l e e e n D m i o s i -1 D s s e 10 n l e n o m i i D s n e m i D -2
10
3
-1
10
Transient "Stems" (Transient Flow Region)
-4
-2
10
-1
10
0
10
10
i
d d D n q a [ , ) 2 d , , D e 10 ) t v d ( i i D d t t ( D a d q v D , i r q l e a , r e g D t 1 e a t R n l 10 a e I r e g n i t e l a t c R n e I D e e n t s l i 0 s c a e e R 10 l e n D n o s i i l s s c e e n l e n D m i o s i -1 D s s e 10 n l e n o m i i D s n e m i D -2
1
2
10
10
1
100 1000
0
10 5
20
30
10
50 500 r eD=1x10
100
10
-2
-3
-2
10
-1
10
10
2
10
10
0
10 5
50
500
-1
100
10
r eD=1x10
[q Ddi (t Dd)]
-2
-4
-3
10
-2
10
-1
10
0
10
10
1
2
10
10
10
10
-1
10
10
] )
-2
10
3
-1
10
0
10
10
1
2
10
10
10
10
50 500
3
2
10
1
q Ddi (t Dd )
Depletion "Stems" (Boundary-Dominated Flow Region)
100 1000
0
10 5
20
30
10
50 500 r eD=1x10
100
-1
10
3
[ q Ddi (t Dd)]
-2
-4
10
-3
10
-2
10
-1
10
0
10
3
2
10
1
10
30 50
Depletion "Stems" (Boundary-Dominated Flow Region)
q Ddi (t Dd ) 100
500
1000
0
10 5
20
30
10
50 500
100
-1
10
3
r eD=1x10
[q Ddi (t Dd)]
-2
-3
-2
10
-1
10
0
10
10
1
2
10
10
10
3
10
1
10
2
10
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=500).
10
3
10
-3
10
] )
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=10).
-2
10
3
-1
10
0
10
10
1
2
10
10
3
10
10
t ( i d d D n q a [ , ) 2 d , , D e 10 ) t v d ( i i D t d t a ( D d q v D , i r q l e , a r D e t g 1 a t e R n l 10 a e I r n e g i e l t a t c R n e I D e e n t s i a 0 s l c R 10 e e l n D e n o i i s l s s c e e n l e n D m i o s i D s s e10-1 n l e n o m i i s D n e m i D -2
10
3
10 Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 1000)
Transient "Stems" (Transient Flow Region)
Legend: q Dd (t Dd), qDdi (t Dd) and [q Ddi (t Dd)] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
2
10
5 10
1
q Dd (t Dd)
20
10
30
Depletion "Stems" (Boundary-Dominated Flow Region)
q Ddi (t Dd )
50 500
100 1000
0
10 5
20
30
10
50
500
100
-1
10
3
r eD=1x10
[ q Ddi (t Dd )]
-2
-4
10
-3
10
t Dd,bar =N pDd (t Dd )/q Dd(t Dd )
Figur e C.7 —
3
10
q Dd(t Dd)
20
-4
10
30
2
10
5
d D
q Dd (t Dd )
20 10
1
10
Fetkovich-McCray Rate Functi on Type Curve- t Dd,bar Format ( Vertical Well with a Fini te Conductivity Vertical Fracture — FcD = 1000 )
Legend: q Dd (t Dd), q Ddi (t Dd) and [q Ddi (t Dd)] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
5
10
10
Figure C.9 —
3
Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 10)
Transient "Stems" (Transient Flow Region)
0
10
Legend: q Dd(t Dd ), q Ddi (t Dd ) and [q Ddi (t Dd )] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
10
10
t ( i d d D n q a [ , ) 2 d , , D e 10 ) t v d ( i i D t d t ( D a d q v D , i r q l e , a D e r t g 1 e a t R n l 10 a e I r n e g i e l t c a t n e R I D e e n t s i a 0 s l R 10 e c e l n D e n o i i s l s s c n l e e e n D m i o s i -1 D s s e 10 n l e n o m i i D s n e m i D -2
10
3
10
t Dd,bar =NpDd (t Dd)/q Dd (t Dd)
10
d D
2
10
Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 500)
Transient "Stems" (Transient Flow Region)
-4
10
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=5).
-3
1
10
10
( i d d D n q a [ , ) 2 d , , D 10 ) t e v d ( i i D t d t ( D a d q v D , i r q l e , a D e r g t 1 e a t R n l 10 a e I r e g n i t e l a t c R n e I D e e n t i s l a 0 s c R 10 e e l n D e n o s l i i s s c e e n l e n D m i o s i D s s e 10-1 n l e n o m i i D s n e m i D -2
3
-2
10
3
Fetkovich-McCray Rate Function Type Curve- t Dd,bar Format ( Vertical Well with a Fini te Conductivity Vertical Fracture — FcD = 10 ) -4
10
10
t Dd,bar =NpDd (t Dd)/q Dd (t Dd)
Figur e C.6 —
0
10
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=100).
-3
10
] )
d D
Depletion "Stems" (Boundary-Dominated Flow Region)
q Ddi (t Dd )
3
-1
3
[ q Ddi (t Dd )]
t
1
10
Depletion "Stems" (Boundary-Dominated Flow Region)
q Ddi (t Dd )
50 500
-4
3
qDd (t Dd)
20
10
q Dd (t Dd)
20 30
Figure C.8 —
3
10
Legend: q Dd(t Dd), q Ddi (t Dd ) and [q Ddi (t Dd )] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
30
2
10
Fetkovich-McCray Rate Functi on Type Curve- t Dd,bar Format ( Vertical Well with a Finit e Conductivity Vertical Fracture — FcD = 500 )
Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 5)
Transient "Stems" (Transient Flow Region)
3
5
10
10
d D
3
10
t Dd,bar =NpDd (t Dd)/q Dd (t Dd)
10
t (
2
10
10
-4
10
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=1).
-3
10
3
1
10
Legend: q Dd (t Dd), q Ddi (t Dd) and [q Ddi (t Dd)] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
Fetkovich-McCray Rate Functio n Type Curve- t Dd,bar Format ( Vertical Well with a Finite Conductivity Vertical Fracture — FcD = 5 ) ] )
10
Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 100)
t Dd,bar =N pDd (t Dd )/q Dd(t Dd )
Figur e C.5 —
0
10
10
D t
Legend: qDd (t Dd), q Ddi (t Dd) and [q Ddi (t Dd)] vs. t Dd,bar Rate Function Curves Rate Integral Function Curves Rate Integral- -Derivative Function Curves
q Ddi (t Dd)
-2
10
d
Model Legend: Fetkovich-McCray Rate Function Type Curve - Fractured Well Centered in a Bounded Circular Reservoir (Finite Conductivity: FcD = 1)
Transient "Stems" (Transient Flow Region)
-3
10
3
10
] )
-2
10
-1
10
0
10
1
10
2
10
10
t Dd,bar =NpDd (t Dd )/q Dd (t Dd )
Figur e C.10 —
3
10
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration (FcD=1000).
SPE 107967
Application of the β -Integral Derivative Function to Production Analysis
13
Ellipti cal Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 1) -4
-3
10
-2
10
2
-1
10
0
10
10
1
2
10
10
3
10
2
10
10 Model Legend: Elliptical Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 1)
Transient "Stems" (Transient Flow Region Analy tic al Solu tion s: FE = 1) 1
s ) n ] o ) i t A D c t n ( i u d 0 F D 10 q e [ n i l c d e n D a , 10-1 e ) t A a t D R ( i s d s D e q l , n ) -2 o A 10 i D s t n ( e d D q m ( i D
0 = 0.25
q Ddi (t DA)
0.50
0.75 1.50
1
Legend: q Dd (t DA), q Ddi (t DA), and [q Ddi (t DA)] versus t DA q Dd (t DA) Rate q Ddi (t DA) Rate Integral
10
[q Ddi (t DA)] Rate Integral
1.0
10
-Derivative
1.75 2.0
0
3.0
10
0 = 0.25
4.0
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
0 = 5.0
-1
10
0 = 5.0
[q Di (t DA)] -2
10
closed reservoir boundary (ellipse)
y wellbore
fracture
q Dd (t DA)
-3
10
-3
10
x
x f
b
a
-4
-4
10
-4
-3
10
-2
10
-1
10
0
10
10
1
2
10
10
10
3
10
Dimensionless Time Based on Drainage Area (t DA)
Figur e C.11 —
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration — elliptical flow model (FE=1).
Ellipti cal Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 10) -4
-3
10
-2
10
2
-1
10
0
10
10
1
2
10
10
3
10
10
2
10 Transient "Stems" (Transient Flow Region Analy tic al Solu tio ns: FE = 10) 0 = 0.25
1
0.75 0.50 1.50 1.0
s ) n ] o ) i t A D c ( t 0 n i u d 10 F D q e [ n i l c d e n D a , 10-1 e ) t A a D R t ( i s d s D e q l , -2 n ) 10 o A i D s t n ( e d D q m ( i D -3
1.75
10
Model Legend: Elliptical Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 10) q Ddi (t DA)
Legend: q Dd(t DA), q Ddi (t DA), and [q Ddi (t DA)] versus t DA q Dd(t DA) Rate q Ddi (t DA) Rate Integral
2.0
[ q Ddi (t DA)] Rate Integral
3.0
1
10
-Derivative
4.0
0
10
0 = 5.0
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
0 = 5.0
-1
10
[q Di(t DA)]
-2
10
closed reservoir boundary (ellipse)
y wellbore
fracture
q Dd(t DA)
10
-3
10
x
x f
b
a
-4
10
-4
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
10
3
10
Dimensionless Time Based on Drainage Area (t DA )
Figur e C.12 —
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration — elliptical flow model (FE=10).
14
D. Ilk, N. Hosseinpour-Zonoozi, S. Amini, and T.A. Blasingame
SPE 107967
Ellipti cal Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 100) -4
-3
10
-2
10
2
-1
10
0
10
1
10
2
10
3
10
10
10
2
10 Transient "Stems" Model Legend: Elliptical Flow Type Curve - Fractured (Transient Flow Region Well Centered in a Bounded Elliptical Reservoir Analy tic al Solu tio ns: FE = 100) (Finite Conductivity: FE = 100) q Ddi (t DA) Legend: q (t ), q (t ), and [q (t )] versus t
0 = 0.25
0.75 0.50 1.0 1.75 1.50
1
10
s ) n ) o ] i t A D c t n ( i u d 0 F D 10 q e [ n i l c d e n D a , -1 e ) t A10 a t D R ( i s d s D e q l , n ) -2 o A i D10 s t ( n d e D q m ( i D
Dd DA
q Dd (t DA) q Ddi (t DA)
2.0
Ddi DA
Ddi DA
1
10
[q Ddi (t DA)] Rate Integral
3.0
DA
Rate Rate Integral -Derivative
4.0
0
10
0 = 5.0
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
0 = 5.0
-1
10 [q Di (t DA)]
-2
10
closed reservoir boundary (ellipse)
y wellbore
fracture
q Dd(t DA)
-3
10
b
-3
10
x
x f
a
-4
10
-4
-4
-3
10
-2
10
-1
10
0
10
10
1
2
10
10
10
3
10
Dimensionless Time B ased on Drainage Area (t DA)
Figur e C.13 —
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration — elliptical flow model (FE=100). Ellipti cal Flow Type Curve - Fractured Well Centered in a Bounded Elliptical Reservoir (Finite Conductivity: FE = 1000) -4 0.25
-3
10
10
0.50
1.50
1
-1
10 q Ddi (t DA)
0 = 0.25
0.75
-2
10
2
1.0
10
1
2
10
10
3
10
2
10 Transient "Stems" Model Legend: Elliptical Flow Type Curve - Fractured (Transient Flow Well Centered in a Bounded Elliptical Reservoir Region - Analytical (Finite Conductivity: FE = 1000) Solutions: FE = 1000) Legend: q Dd (t DA), q Ddi (t DA), and
1.75
s ) 10 n ] o ) i t A D c t n ( i u d F D 0 q 10 e [ n i l c d e n D a , e ) t -1 A a t D 10 R ( i s d s D e q l , n ) o A i -2 D s t 10 n ( d e D q m ( i D
0
10
q Dd (t DA) q Ddi (t DA)
2.0 3.0
[q Ddi (t DA)] versus t DA
1
10
Rate Rate Integral
[q Ddi (t DA)] Rate Integral
-Derivative
4.0
0
10
0 = 5.0
5.0
Depletion "Stems" (Boundary-Dominated Flow Region-Volumetric Reservoir Behavior)
0 = 0.25
[q Di (t DA)]
-1
10
-2
10
closed reservoir boundary (ellipse)
y wellbore
fracture
q Dd(t DA)
-3
-3
10
b
10
x
xf
a
-4
10
-4
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
10
3
10
Dimensionless Time B ased on Drainage Area (t DA)
Figur e C.14 —
q Ddi (t Dd)], q Dd, and q Ddi vs. t Dd — Fractured well configuration — elliptical flow model (FE=1000).
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