Rate Decline Curves Analysis of a Vertical Fractured Well ...Ž‹磊.pdf · Rate Decline Curves Analysis of a Vertical Fractured Well With Fracture Face Damage Rate decline analysis

Wang Lei

Wang Xiao-dong

Ding Xu-min

Zhang Li

Li Chen

School of Energy Resources,

China University of Geosciences,

Beijing 100083, China

Rate Decline Curves Analysis ofa Vertical Fractured Well WithFracture Face DamageRate decline analysis is a significant method for predicting well performance. Previousstudies on rate decline analysis of fractured wells are all based on homogeneous reser-voirs rather than homogeneous ones considering fracture face damage. In this article, awell model intercepted by a finite conductivity vertical fracture with fracture face damageis established to investigate how face damage factor affects the productivity of fracturedwell. Calculative results show that in transient flow, dimensionless rate decreases withthe increase of fracture face damage and in pseudo steady-state flow, all curves underdifferent face damage factors coincide with each other. Then, a new pseudo steady-stateanalytic formula and its validation are presented. Finally, new Blasingame type curvesare established. It is shown that the existence of fracture damage would decrease the ratewhen time is relatively small, so fracture damage is an essential factor that we shouldconsider for type curves analysis. Compared with traditional type curves, new type curvescould solve the problem of both variable rate and variable pressure drop for fracturedwells with fracture face damage factor. A gas reservoir example is performed to demon-strate the methodology of new type curves analysis and its validation for calculating im-portant formation parameters. [DOI: 10.1115/1.4006865]

Keywords: fractured wells, fracture conductivity, type curves, fracture face damage,formation parameters

1 Introduction

Hydraulic fracturing is an effective technique for productivityenhancement of wells with damaged zones or producing from lowpermeability formations. During the last few decades, there hasbeen a continuous increasing interest [1–4] in the determination offormation properties from transient pressure test or flow rate dataanalysis. Gringarten had made an extraordinary contribution tothe development of transient pressure data analysis and typecurves analysis of fractured wells [5]. Three basic solutions werepresented: the infinite fracture conductivity solution, the uniformflux solution for vertical fractures, and the uniform flux solutionfor horizontal fractures. Since fracture conductivity could not beignored, a semi-analytical solution and an analytical solution forfinite conductivity vertical fractured well were presented [6,7].These solutions were quite significant to the later analysis of pro-ductivity and well test data for fractured wells.

It has been shown that the increase in the productivity of afractured well depends on fracture characteristics [1,2], such asfracture conductivity, length, penetration [8,9] and also dependson a possible damage to the formation surrounding the fracture.Skin damage concept for fractured wells was first presented byEvans [10]. He assumed that the flow from formation to fracturewas linear, passing through two porous media in series, of whichone is damaged zone around the fracture and the other is undam-aged formation. Seven years later, dimensionless skin damage fac-tor was redefined [11] and then they established a finiteconductivity model including fracture damage.

In terms of productivity analysis and prediction, the most repre-sentative masterpiece is Arps decline equations [12]. He general-ized the production decline law into three types: exponentialdecline, harmonic decline, and hyperbolic decline. Subsequentdecades, there was no progress in the aspect of productivity

analysis. Later, Fetkovich [13] provided the nature of rate declinein theory and thus laid the foundation of modern rate decline anal-ysis. He perfectly combined theoretical decline curves with Arpsdecline equations to further analyze rate decline law [13]. How-ever, Fetkovich type curves could not solve the problem of bothvariable rate and variable bottom pressure drop. Blasingame et al.[14,15] solved this problem skillfully by introducing the conceptsof material balance time and integral average. Later, many schol-ars extended Fetkovich and Blasingame type curves into otherwell types, such as fractured wells and horizontal wells [16–26].

However, at present, there is still no literature in which theeffect of fracture face damage on type curves is analyzed and cor-responding type curves are presented. First, this paper establishesa mathematical model for a fractured well of finite conductivitywith face damage. Then, we acquire a numerical solution by usingboundary element method and a new pseudo steady-state analyticformula is obtained by means of asymptotic analysis and multipleregression methods. Finally, new Blasingame type curves are pre-sented and then applied to solve a gas reservoir example.

2 Fracture Flow Model

Figure 1 shows the physical model assumed and the assump-tions of this model are as follows [6].

(1) A homogeneous, isotropic, horizontal, slap reservoir isbounded by an upper and a lower impermeable stratum.The reservoir has uniform thickness, h, permeability, k, andporosity, /, which do not change with pressure.

(2) The reservoir contains a slightly compressible fluid of com-pressibility, c, and viscosity, l, which are constant.

(3) Fluid is produced through a vertically fractured well inter-sected by a fully penetrating, finite conductivity fracture ofhalf-length, xf, width, w, permeability, kf, and porosity, /f,which are also constant.

For the fracture, we also consider that no flow is allowed intothe fracture through the fracture tips and the flow of the fluid inthe fracture is linear flow (Fig. 2). Finally, we assume that fluid

Contributed by the Petroleum Division of ASME for publication in the JOURNAL

OF ENERGY RESOURCES TECHNOLOGY. Manuscript received November 28, 2011; finalmanuscript received April 23, 2012; published online June 21, 2012. Assoc. Editor:Desheng Zhou.

Journal of Energy Resources Technology SEPTEMBER 2012, Vol. 134 / 032803-1Copyright VC 2012 by ASME

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flow is steady because the fracture is very small compared withthe reservoir.

Under these conditions assumed, the fracture flow model basedon dimensionless variables can be described by

@2pfD

@xD2þ 2

CfD

@pD

@yD

��yD¼wD

2

¼ 0 0 < xD < 1 (1)

Inner boundary condition

@pfD

@xD

��xD¼0

¼ � pCfD

(2)

Outer boundary condition

@pfD

@xD

��xD¼1

¼ 0 (3)

The flow correlation formula for surface of the fracture

qfD ¼ �2

p@pD

@yD

��yD¼wD

2

(4)

We can obtain solutions in Laplace domain through combiningthe Eqs. (1)–(4), that is

~pwDðsÞ � ~pfDðxDÞ ¼p

sCfDxD � s

ðxD

0

ðv

0

~qðuÞdudv

� �(5)

where

xD ¼x

xf; yD ¼

y

xf; wD ¼

w

xf; CfD ¼

kf w

kxf

pD ¼2pkhðpi � pÞ

qu; pfD ¼

2pkhðpi � pf Þqu

; qfD ¼2qðx; tÞ

q

3 Formation Flow Model

As is assumed above, we consider formation flow model as aplane source in a circular bounded reservoir (Fig. 3), so pointsource integral method must be used in Laplace domain. Themathematical model of point source for formation flow can bedescribed as follows:

@2pD

@r2D

þ 1

rD

@pD

@rD¼ @pD

@tDtD > 0 (6)

Initial condition

pD rD; 0ð Þ ¼ 0 (7)

Inner boundary condition

limrD!0

rD@pD

@rD

� �¼ �1 (8)

Outer boundary condition

@pD

@rD

��rD¼reD

¼ 0 (9)

Combining Eqs. (6)–(9) can obtain point source solution. Further,in order to gain the solution of plane source, point source solutionmust be written in form of integral expressions of Bessel functions[27]. The pressure distribution of this system can be eventuallygiven by

~pDðxD; 0; sÞ ¼1

2

ð1

�1

~qfDða; sÞ½K0f½ðxD � aÞ2�1=2 ffiffispg

þ K1ðreDffiffispÞ

I1ðreDffiffispÞ I0f½ðxD � aÞ2�1=2 ffiffi

spg�da (10)

where

tD ¼k � t

ulctx2f

; red ¼re

xf; rD ¼

r

xf

4 Skin Damage Model

As is shown in Fig. 4, the fracture is surrounded by a skin dam-aged zone. It is easy to know that the total pressure drop of thisfracture is equal to the sum value of normal pressure drop in thereservoir and the additional pressure drop caused by the damagedzone, that is

Fig. 1 Finite conductivity vertical fracture in a bounded slapreservoir

Fig. 2 Fracture flow model

Fig. 3 The reservoir flow model

032803-2 / Vol. 134, SEPTEMBER 2012 Transactions of the ASME


~pfDðxDÞ ¼ ~pDðxD; yD ¼ 0; sÞ þ ~qfDðxD; sÞsfD (11)

Introducing dimensionless skin factor defined by Cinco-Ley andSamaniego [11], that is

sfD ¼p2

ws

xf

k

ks� 1

� �

Now putting Eqs. (10) and (11) into Eq. (5), we obtain

~pwDðsÞ �1

2

ð1

�1

~qða; sÞ½K0f½ðxD � aÞ2�1=2 ffiffispg

þ K1ðreDffiffisp Þ


spg�

� ~qfDðxD; sÞsfD ¼p

sCfDxD � s

ðxD

0

ðv

0

~qðuÞdudv

� �(12)

From Eq. (12), we can find that only ~pwD and ~qfDðxD; sÞ areunknown. But the problem is that it is hard to gain analytical solu-tion from Eq. (12), so we must employ numerical algorithm tosolve it.

5 Solution

5.1 Numerical Solution. Assuming the fracture can bedivided into n segments, the right side of Eq. (10) would have thefollowing transformation,

1

2

ð1

�1

~qða; sÞ½K0f½ðxD � aÞ2�1=2 ffiffispg

þ K1ðreDffiffispÞ


spg� ¼

Xn

i¼1

qiðsÞ

�ðxDiþ1

xDi

K0ð xD � x0j jffiffispÞ þ K0½ðxD þ x0Þ

ffiffisp�

�

þ K1ðreDffiffisp Þ

I1ðreDffiffispÞ ðI0ð xD � x0j j

ffiffispÞ þ I0½ðxD þ x0Þ

ffiffisp�Þ�

dx0 (13)

And the second-order integral of Eq. (12) can be also transformedas

ðxDj

0

ðx0

0

~qðx00;sÞdx00dx0 ¼Xj

i¼1

~qiðsÞDx2

2þDxðxDj� iDxÞ

� �þ Dx2

8~qjðsÞ

(14)

In addition to the above expressions, by virtue of steady flow, wealso have

DxXn

i¼1

~qiðsÞ ¼1

s(15)

The unknowns qi(s) and pwD(s) can be obtained through combingEqs. (12)–(15). Then, qi(t) and pwD(t) for any time given t and anygiven sfD can be figured out by Stehfest numerical algorithm [28].

5.2 Validation of Solution. Riley [29] gave an analytical so-lution for elliptical finite conductivity fractures without fracturedamage. To validate the solution presented in this paper, wecompared our solution with Riley’s results. In our model, SfD isconsidered to be equal to zero because Riley’s results did not con-sider fracture damage. Figure 5 shows the comparison of the twosolution under different fracture conductivity CfD, the good agree-ment validates the solution obtained in this work.

5.3 Pseudo Steady-State Analytic Solution. Especially, wefurther study pseudo steady-state by other method. Since the ana-lytic formula of pseudo steady-state directly obtained from Eqs.(12)–(15) is very complex, we managed to gain a new pseudosteady-state analytic formula by means of asymptotic analysis andmultiple regression methods [27]. A pseudoradial formula couldbe obtained as follows by combining Eqs. (37)–(39) in Ref. [29]

PwDðtDf Þ ¼1

2ln tDf þ 3� cþ 2Sf

þ f ðCfDÞ (16)

By comparing the pseudo steady-state formula of a vertical well[28] with Eq. (16), a new pseudo steady-state formula of fracturedwells could be obtained as follows:

pwDðtDÞ ¼2tD

reD2 � 1

þ lnreD

2þ 3

4þ f ðCfDÞ þ SfD (17)

where

f ðCfDÞ ¼X1n¼1

pCfD

n 2nþ pCfDðnþ 1Þ� �� pCfD

pCfD þ 2(18)

Equation (18) is not convenient to solve practical problemsbecause it involves a sum of an infinite series. Therefore, accord-ing to Eq. (18), we manage to get a simple regression equation asfollows:

f ðCfDÞ

¼ 0:95� 0:56wþ 0:16w2 � 0:028w3 þ 0:0028w4 � 0:00011w5

1þ 0:094wþ 0:093w2 þ 0:0084w3 þ 0:001w4 þ 0:00036w5

(19)

w ¼ lnðCfDÞ

Equation (19) is a fitted function, whose results are comparedwith the results of analytic function of Eq. (18) in Fig. 6. Theexcellent matching condition fully indicates the validity of thefitted function. It can also be seen from Fig. 6 that the changing

Fig. 4 Fractured well with a damaged zone around the fracture

Fig. 5 The comparison for the results of this paper and Riley [29]

Journal of Energy Resources Technology SEPTEMBER 2012, Vol. 134 / 032803-3


scope of the conductivity could meet the need of solving real engi-neering problems.

In particular, we define bDpss as dimensionless pseudo steady-state constant, which is given by,

bDpss ¼ lnreD

2

� þ 3

4þ f ðCfDÞ þ SfD (20)

Figures 7 and 8 show the comparison of numerical solution andpseudo steady-state solution. We note excellent agreementbetween these two solutions. Thus, the validation of the new for-mula presented is reached.

6 Discussions and Application of Type Curves

6.1 Decline Analysis. According to Duhamel principle [28],the pressure solution under constant rate and the rate solutionunder constant pressure have the following correlation in Laplacedomain

~qDðsÞ ¼1

s2 ~PwDðsÞ(21)

Combining Eqs. (12)–(15) and Eq. (21), we can obtain the correla-tions between dimensionless rate qD(tD) and dimensionless timetD for any given SfD and any given reD (Figs. 9 and 10).

In Fig. 9, it can be seen that in transient flow, dimensionlessrate decreases with the increase of SfD at the same time point,which shows that the more serious the fracture damage, thesmaller the rate. In pseudo steady-state flow, all curves under dif-ferent skin damage factors coincide with each other. It also dem-onstrates in Fig. 10 that time of transient flow become longer with

the increase of drainage radius. Besides, transient flow may extendto infinity when drainage radius reD becomes large enough. Inpseudo steady-state flow, productivity declines fast but all curvesunder different skin damage factors still coincide. This behaviorcan be inferred from Eq. (17). When the flow reaches thepseudo steady-state, the value of [2tD/(reD

2� 1)þ ln(reD/2)þ 3=4þ f(CfD)] stem is far more greater than the value of fracture dam-age factor SfD in Eq. (17). Besides, when it is plotted in log–logcoordinate, the influence of fracture damage factor on dimension-less pressure and dimensionless rate (Eq. (21)) becomes smaller,so it seems that all rate curves in pseudo steady-state flow in bothFigs. 9 and 10 are almost normalized. In a strict sense, the ratecurves have some tiny difference.

Fig. 7 Comparison of numerical solution and pseudo steady-state solution for a well with a finite conductivity vertical frac-ture (CfD 5 0.1, SfD = 0)

Fig. 8 Comparison of numerical solution and pseudo steady-state solution for a well with a finite conductivity vertical frac-ture (CfD 5 0.1, SfD = 0.5)

Fig. 9 Rate decline curves for a well with a finite conductivityvertical fracture at different values of SfD (reD 5 4, CfD 5 0.5)

Fig. 10 Rate decline curves for a well with a finite conductivityvertical fracture at different values of reD (SfD 5 0.5 and SfD 5 0,CfD 5 0.5)

Fig. 6 The comparison for analytic function and fitted functionof f(CfD)



6.2 Blasingame Type Curves and Discussions. The generaldefinitions of the base decline for type curve variables can begiven by

qDd ¼ qDbDpss (22)

tDd ¼2

bDpssðreD2 � 1Þ tD (23)

The definitions of Eqs. (22) and (23) were presented by Fetkovich[13]. However, to introduce material balance time, we useEq. (12) as a constant rate solution [14] and then use 1/pwD resultas qD. To eliminate multiple solutions and errors, integral averagemethod of rate was created by Blasingame et al. [15]. The auxil-iary variables in this method typically used for decline type curveanalysis are given by

(1) Rate integral function: qDdi

qDdi ¼NpDd

tDd¼ 1

tDd

ðtDd

0

qDdðsÞds (24)

(2) Rate integral derivative function: qDdid

qDdid ¼ �dqDdi

d lnðtDdÞ¼ �tDd

dqDdi

dtDd¼ qDdi � qDd (25)

Now incorporating Eqs. (22)–(25) with the flow model assumedpreviously, we can consequently establish new Blasingame typecurves for a finite conductivity vertical fractured well with frac-ture face damage, which are shown in Figs. 11–14.

From Figs. 11 and 12, we can find that in transient flow, underthe same fracture conductivity CfD, the values of qDdi and qDdid

both decrease with the increase of skin damage factor SfD. Inpseudo steady-state flow, curves of both groups normalize, respec-tively. Contrast Fig. 13 with Fig. 14, we note that under the sameskin damage factor SfD, the value of qDdi and qDdid both increasewith conductivity CfD in transient flow and the curves of twogroups also normalize, respectively, in pseudo steady-state flow.

Figures 15–17 show the effect of fracture damage factor SfD

(SfD¼ 0 and 1) on dimensionless rate integral qDdi and rate inte-gral derivative qDdid for the same dimensionless drainage radiusreD but different fracture conductivity CfD values of 0.5, 5, and 50,respectively. It can be seen from all of these three plots that, whendimensionless decline time is relatively smaller (tDd � 1) thecurves of qDdi and qDdid with fracture damage are always lowerthan the ones without fracture damage, which shows the existenceof the fracture damage decreases the rate. When the values oftime become larger (tDd> 1), the two kinds of curves all reachgood agreement and the effect of fracture damage becomes notobvious. These features are similar to the results presented inFig. 9. Besides, by comparing Figs. 15–17, we find that the largerthe value of CfD, the greater the difference between curves withfracture damage and curves without fracture damage. This isbecause for the case of low conductivity, fracture flow is dominantand the effect of damage region on flow is weak while for the caseof high conductivity, reservoir flow is dominant, and thus, theeffect of damage region on flow is relatively strong.

Figures 18–20 show the effect of fracture damage factor SfD

(SfD¼ 0 and 1) on dimensionless rate integral qDdi and rate inte-gral derivative qDdid for the same fracture conductivity CfD butdifferent dimensionless drainage radius reD values of 5, 25, and

Fig. 11 New Blasingame type curves for fractured wells withfracture face damage (CfD 5 10, SfD 5 0)

Fig. 12 New Blasingame type curves for fractured wells withfracture face damage (CfD= 10, SfD 5 0.5)

Fig. 13 New Blasingame type curves for fractured wells withfracture face damage (CfD 5 0.5, SfD 5 0.1)

Fig. 14 New Blasingame type curves for fractured wells withfracture face damage (CfD 5 5, SfD 5 0.1)



100, respectively. From these three plots, we can get the sameconclusion with Figs. 15–17 that the existence of the fracturedamage decreases the rate when time is relatively smaller.Besides, by comparing Figs. 18–20, we find that the smaller thevalue of reD, the larger the difference between curves with fracturedamage and curves without fracture damage. This is because thesmaller the dimensionless drainage radius, the larger the ratio ofarea of fracture damage region and area of reservoir drainageregion, and thus, fracture damage can exert more serious effect onthe rate of smaller dimensionless drainage radius. From the studyabove, we can conclude that fracture damage is an important fac-tor for type curves analysis and it is essential for us to consider it.

6.3 Application

6.3.1 Analysis Procedures for Rate Decline Curves. Takinggas reservoir as an example, type curve matching procedures aregiven below.

(1) For data of gas well, we should adopt the next four specialfunctions presented by Blasingame et al. [14,15], the mate-rial balance pseudotime function, the pseudopressure dropnormalized rate function, the pseudopressure drop normal-ized rate integral function and the pseudopressure drop nor-malized rate integral derivative function, which are,respectively, given by

�ta ¼lgicgi

qg

ðt

0

qgðsÞlgðp�Þcgðp

�Þds (26)

Fig. 15 The effect of fracture damage factor on type curves forreD 5 15 and CfD 5 0.5

Fig. 16 The effect of fracture damage factor on type curves forreD 5 15 and CfD 5 5







qg

Dpp¼ qg

ppi � pp(27)

qg

Dpp

� �i

¼ 1

�ta

ð�ta

0

qg

Dppds (28)

qg

Dpp

� �id

¼ d

d lnð�taÞqg

Dpp

� �i

� �¼ � 1

�ta

d

d�ta

qg

Dpp

� �i

� �(29)

where

pp ¼lgicgi

qg

ðp

pref

p

lgzdp

(2) Based on Eqs. (26)–(29), the following data functionsshould be plotted on a scaled log–log coordinate for typecurve matching:

ðaÞ qg

�Dpp

i

versus �ta

ðbÞ qg

�Dpp

id

versus �ta

(3) We now match the depletion data trends onto the Arps b¼ 1stem for each of the new Blashingame type curves beingused: qDdi, qDdid. Once the best matching is obtained, we re-cord the time and rate axis match points as well as the red

transient flow stage.

(a) Rate axis matching point

qg

�Dpp

i

h iMP� qDdi½ �MP

qg

�Dpp

id

h iMP� qDdid½ �MP

(b) Time axis match point

�tað ÞMP � tDdð ÞMP

(c) Transient flow stem (red)

Select the best matching points between practical data and typecurves to gain the transient flow stem.

(d) Calculate the bDpss value using Eq. (20).

6.3.2 A Gas Reservoir Example. In this section, we presentgas field data analysis using the procedures of previous section.Table 1 shows reservoir fluid properties and production data ofthe chosen gas field.

Figure 21 shows the matching result of field data and new typecurves. We get the best match of the data when SfD¼ 0.1,CfD¼ 20 and reD¼ 20. Using the obtained match points, we thencalculate estimates of gas in place, reservoir drainage area, effec-tive permeability to gas, and fracture half-length. Table 2 showsthe summary of results.

Matching results: SfD¼ 0.1, CfD¼ 20, reD¼ 20

tM ¼ �tað ÞMP= tDdð ÞMP ¼ 245 days

pM ¼ qg=Dpp

MP= qDdð ÞMP ¼ 1:25� 103m3=day=Mpa

Calculations(1) Gas in place

G ¼ 1

cgi

�tað ÞMP

tDdð ÞMP

qg=Dpp

MP

qDdð ÞMP

(30)

G ¼ 245 days� 1:25� 103m3=day=Mpa

8:2� 10�3=MPa

¼ 3:73� 108m3

(2) Gas reservoir drainage area and equivalent drainage radiusThe drainage area is estimated using Eq. (31)

A ¼ 100GBgi

/hð1� SwiÞ(31)

Therefore

A ¼ 100� 3:73� 108m3 � 1:45� 10�4m3=m3

0:15� 9:7m� ð1� 0:23Þ¼ 4:82� 106m2

We then calculate the equivalent drainage radius, re, usingEq. (32)

re ¼ffiffiffiffiffiffiffiffiA=p

p(32)

re ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4:82� 106m2=3:14159

q¼ 1239 m

Table 1 Gas reservoir, fluid properties, and production data ofa gas well

Name of parameters Basic data

Estimated net pay thickness, h 9.7 mAverage porosity, / 0.15Water saturation, Swi 0.23Volume factor at pi, Bgi 1.45� 10�4 m3/m3

Gas viscosity at pi, lgi 0.03 cpGas compressibility at pi, cgi 8.2� 10�3/MPaInitial reservoir pressure, pi 24 MPaInitial reservoir pseudopressure, ppi 17.17 MPa

Fig. 21 Match of production data for an example on the newBlasingame decline type curve for a well of a finite conductivityvertical fracture with face damage (SfD 5 0.1, CfD = 20)

Table 2 Summary of calculative results

Name of parameters Results

The match point, tM 245 daysThe match point, pM 1.25� 103 m3/day/MPaThe conductivity factor, CfD 20Fracture skin factor, SfD 0.1Gas in place, G 3.73� 108 m3

Equivalent drainage radius, re 1293 mEffective gas permeability, kg 0.98 mDFracture half-length, xf 61.95 m



(3) Effective gas permeability

kg ¼542:8Bgilgi

hbDpss

ðqg=DppÞMP

ðqDdÞMP

� �(33)

From Eq. (33), we note that before calculating effective gas per-meability, we should first calculate bDpss. According to Eq. (20),we have

bDpss ¼ ln20

2

� �þ 3

4þ ½0:95� 0:56� lnð20Þ þ 0:16� ðlnð20ÞÞ2 � 0:028� ðlnð20ÞÞ3 þ 0:0028� ðlnð20ÞÞ4 � 0:00011� ðlnð20ÞÞ5�½1þ 0:094� lnð20Þ þ 0:093� ðlnð20ÞÞ2 þ 0:0084� ðlnð20ÞÞ3 þ 0:001� ðlnð20ÞÞ4 þ 0:00036� ðlnð20ÞÞ5�

þ 0:1 ¼ 3:22

kg ¼ 542:8� 1:45� 10�4 � 0:03cp� 3:47

� 1:25� 103m3=day=MPa=9:7m ¼ 0:98mD

(4) Facture half-length

xf ¼re

red(34)

xf ¼1239 m

20¼ 61:95 m

7 Conclusions

The following conclusions are derived from this study:We have successfully constructed fracture flow model, forma-

tion flow model and skin damage model for a vertical fracturedwell with fracture face damage centered in a closed, circularreservoir.

Based on all established models, we manage to obtain numeri-cal solution of the whole flow process and analytic solution forpseudo steady-state. Fortunately, the two solutions exhibit excel-lent agreement, which shows the correctness of the two solutions.

According to the relationship between pressure solution andrate solution in Laplace domain, a set of rate decline curves arepresented. From the curves, we note that rate of the well decreaseswith the increase of fracture face damage.

New Blasingame type curves are established for different frac-ture skin damage factors and different fracture conductivity. It isshown that rate integral function decreases with the increase ofskin damage factor. The effect of fracture damage on type curvesis also discussed in details in this paper. We conclude that fracturedamage is an important consideration for type curves analysis.This set of decline type curves are then applied to interpret gasproduction data which contains both variable rate and variablepressure. We successfully get the relevant parameters.

As the practitioner we require additional cases, the proceduresand governing relations presented above are completely generaland should be readily reproducible. In other words, the theoreticalmethods presented in this paper could be extended and we cangain more decline type curves to meet the needs of practicalproblems.

Acknowledgment

This article was supported by Important National Science andTechnology Specific Projects of the Twelfth Five Year PlanPeriod (Grant No. 2011ZX05013-002) and the National BasicResearch Program of China (Grant No. 2011ZX05009-004).

Nomenclature

Dimensionless Variables: Real DomainbDpss ¼ dimensionless pseudo steady-state constant

CfD ¼ dimensionless fracture conductivitypwD ¼ dimensionless well bottom pressurepfD ¼ dimensionless fracture pressure

qDd ¼ dimensionless decline rateqDdi ¼ dimensionless decline rate integral

qDdid ¼ dimensionless decline rate integral derivative.sfD ¼ dimensionless fracture damage skintDd ¼ dimensionless decline timexDj ¼ midpoint of the j segment

c ¼ Euler canstangt, 0.5771

Dimensionless Variables: Laplace Domain~pD ¼ the pressure pD in Lapace domain

~pwD ¼ bottom pressure pwD in Lapace domain~pfD ¼ fracture pressure pfD in Lapace domain

~qðuÞ ¼ fracture rate q(x,t) in Lapace domain~qfD ¼ the fracture rate qfD in Lapace domain

s ¼ time variable in Lapace domain, dimensionless

Field VariablesA ¼ reservoir drainage area, m2

c ¼ compressibility, 1/Mpacgi ¼ initial compressibility for gas, 1/MPaG ¼ gas in place, m3

k ¼ effective permeability, mDks ¼ permeability of the damaged zone, mDkg ¼ effective gas permeability, mDp ¼ formation pressure, MPapi ¼ initial formation pressure, MPapp ¼ reservoir pseudopressure for gas, Mpappi ¼ initial reservoir pseudopressure, MPa

Dpp ¼ pseudopressure drop, MPapref ¼ reference pressure, Mpa

q ¼ rate of per unit fracture length from formation, m3/dqg ¼ production rate for gas, m3/dl ¼ fluid viscosity, cp

lgi ¼ initial viscosity for gas, cp/ ¼ porosity, fractionr ¼ reservoir radius, m

re ¼ equivalent drainage radius, mt ¼ time variable, days

~ta ¼ normalized pseudotime function, daysxf ¼ fracture half-length, m

ws ¼ width of the damaged zone, ma ¼ integral variable

Special FunctionsK0(x) ¼ Modified Bessel function (2nd kind, zero order)K1(x) ¼ Modified Bessel function (2nd kind, first order)I0(x) ¼ Modified Bessel function (1st kind, zero order)I1(x) ¼ Modified Bessel function (1st kind, first order)

Special SubscriptsDd ¼ dimensionless decline variable

i ¼ integral function (or initial value)id ¼ integral derivative function

pss ¼ pseudo steady-state



References

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http://dx.doi.org/10.2118/3013-PA





http://dx.doi.org/10.2118/945228-G

http://dx.doi.org/10.2118/49222-MS

http://dx.doi.org/10.1016/S1001-6058(06)60005-X

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http://dx.doi.org/10.1115/1.4000700

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