Fractured-Well-Test Design and Analysis in the Presence of Non-Darcy Flow

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Fractured-Well-Test Design and Analysis inthe Presence of Non-Darcy Flow

J.A. Gil, SPE, and E. Ozkan, SPE, Colorado School of Mines, and R. Raghavan, SPE, Phillips Petroleum Co.

Summary

The objective of this paper is to discuss the design and analysis of

fractured-gas-well tests to account for non-Darcy flow within thefracture. The results and discussions are based on a semianalyticalmodel. Guidelines and correlations are presented to enable engi-neers to design fractured-well tests whereby the magnitude of non-Darcy flow may be estimated and its effect minimized, if neces-sary. The effect of non-Darcy flow skin (or rate-dependent skin) isdiscussed in detail. A method is also presented to analyze frac-tured-well tests under non-Darcy flow conditions to estimate thenon-Darcy flow coefficient, fracture conductivity, and reservoirproperties. Unlike the conventional methods, this method does notrequire two tests at different production rates.

Introduction

For unfractured vertical wells, it is possible to ascertain conditionsfor which the effects of non-Darcy flow should be negligibly

small. Such information does not exist for fractured wells. Also, inthe latter case, we need to contend with multiple flow regimes.Should the effects of non-Darcy flow be ignored, then it has beenshown in the literature16 that lower estimates of fracture conduc-tivity and/or half-length will result. The conventional recommen-dation to conduct tests at two different rates to estimate fractureproperties (length and conductivity)4,5 is often not a practicalproposition. The adverse effect of non-Darcy flow on our ability toanalyze tests is further exacerbated by the influence of producingtime on buildup analysis.5,6

In this study, we use the semianalytical model discussed inRefs. 4 through 6 to investigate the effect of non-Darcy flow in thefracture. We incorporate the effect of wellbore storage along thelines suggested by Cinco-L. and Sameniego-V.7 It is assumed thatfor the ranges of the fluid and reservoir properties considered in

this study, flow obeys Darcys law in the reservoir or, as discussedby Wattenbarger and Ramey,2 the effect of non-Darcy flow in thereservoir is negligibly small when compared with that within thefracture. The ranges of the reservoir and fracture properties con-sidered in this work are similar to those used in Refs. 8 and 9 andare summarized in Table 1.

One of the contributions of this work is to improve fractured-well-test design and analysis under non-Darcy flow conditions. Wecalculated the magnitude of the skin factor caused by non-Darcyflow and determined the ranges of production rate and fracturehalf-length for which the estimates of fracture conductivity andhalf-length will be unaffected by non-Darcy flow. We assume thatif the pressure drop caused by non-Darcy flow is less than or equalto 10% of the total drawdown, then the effects of non-Darcy flowwill be negligibly small. We also present guidelines and correla-tions that can be used to design fractures in which non-Darcy skinfactor (rate-dependent skin) would not exceed this limit. Our cal-culations may be used to obtain similar information for other limitsor to obtain an alternate design of the fracture variables.

Another contribution of this work is to present an analyticaltechnique for fractured-well tests under the effect of fracture non-Darcy flow. This technique requires transient-pressure data during

the bilinear- and linear-flow periods. The technique is applicablefor the ranges of parameters required by the technique proposed byGuppy et al.,4,5 but unlike their method, our technique does notrequire multiple tests at different flow rates. As suggested by Lin-gen,10 we use sandface flow rates to determine the non-Darcy floweffects; thus, the effect of wellbore storage does not hinder theapplication of our technique.

We start this communication with the definition of the variablesused and the ranges of parameters investigated. We then presentthe results to delineate the effect of non-Darcy flow in the fractureand to quantify the non-Darcy flow skin. These discussions lead tothe guidelines and correlations to design fractures and well testswith acceptable effects of non-Darcy flow. Introduction of the newanalytical technique for fractured wells under non-Darcy flow condi-tions follows. We conclude this presentation with general comments.

Definitions and Ranges of the Data Used

Here, we introduce the system examined, define the variables used

in the discussions, and present the ranges of the data investigated.

Definitions. Let us first define the system investigated in thiswork. We consider a laterally infinite reservoir of uniform thick-ness, h, and a vertical well intercepted by a fully penetrating hy-draulic fracture. The hydraulic fracture has a height of h, half-length of xf, width of wf, and permeability of kf. A real gas flowsin the system, and the reservoir permeability is assumed to beconstant and uniform.

Because we deal with real-gas flow, we present our results interms of pseudopressure, m(p), defined by

m p = 2

0

p

p

Z p pdp, . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where p is the pressure, Z is the gas compressibility factor, and is the gas viscosity. The dimensionless pseudopressure is defined,in field units, by

mD tD =kh

1,422qT m pi m pwf , . . . . . . . . . . . . . . . . . . . . . . (2)

where pi and pwf stand, respectively, for the initial and flowingwellbore pressures; q is the constant surface production rate; and Tis the reservoir temperature. In Eq. 2, the dimensionless time, tD,is defined by

tD =2.637 10

4k

ct ixf2

t, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

for t in hours. The term (ct)i in Eq. 3 is the viscosity-compressibility product at initial conditions.

As discussed in Refs. 4 through 6, the transient-pressure re-sponses of finite-conductivity fractures with non-Darcy flow aregoverned by the dimensionless fracture conductivity (CfD) and thedimensionless flow rate (qDND) defined, respectively, by

CfD =kfwf

kxf, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where kfwf is the fracture conductivity, and

qDND =4.64 10

16 kf M q

wfhi, . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where Mis the molecular weight of the gas and is the non-Darcyflow coefficient. All the equations and results presented in this

Copyright 2003 Society of Petroleum Engineers

This paper (SPE 84846) was revised for publication from paper SPE 71573, first presentedat the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, 30 Septem-ber3 October. Original manuscript received for review 29 November 2001. Revised manu-script received 2 April 2003. Paper peer approved 4 April 2003.

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work are in field units. In this unit system, is measured in ft1.Sometimes, is expressed in terms of atm-s2/g. For conversion, 1atm-s2/g 30.8917959106 ft1.

Reservoir, Fluid, and Fracture Properties. To develop guide-lines and correlations for fractured-well tests under non-Darcyflow conditions, we consider the ranges of reservoir, fluid, andfracture properties shown in Table 1. These ranges of data weretaken from the works of Vincent et al.8 and Settari et al. 9 andshould cover most practical ranges of gas reservoirs and fracturedwells. Note that the work of Settari et al.9 concentrates on theanalysis of fractured-well performance in high-permeability reser-voirs in which fracture half-lengths are usually short (< 100 ft),whereas our attention is not restricted to a specific range of prop-erties. It should be emphasized, however, that in high-permeabilityreservoirs, fracture non-Darcy skin may be large even for smallfracture half-lengths. Because our main objective in this study is todelineate the conditions (ranges of properties and flow rates) underwhich the effect of non-Darcy flow can be held within acceptable

(arbitrarily defined) limits, high-permeability reservoirs normallyfall outside the scope of our interest. In other words, as we discussbelow, our results indicate that the production rates and fracturehalf-lengths required to limit the non-Darcy flow pressure drop to10% of the total drawdown are usually outside the practical ranges.Therefore, considering the reasonable ranges of production rateand fracture half-length in high-permeability reservoirs, it is prac-tically not possible to limit the non-Darcy flow effect to acceptableranges by design considerations. Under these conditions, oneshould expect non-Darcy flow effects to be significant.

Effect of Fracture Non-Darcy Flow andRate-Dependent Skin

Our objective in this section is to present guidelines and correla-tions for the design of fractures and fractured-well tests to estimatethe effect of fracture non-Darcy flow. To set the stage for discus-sion, we first briefly review the effect of non-Darcy flow on tran-sient-pressure responses of finite-conductivity fractures. We thenconsider the skin effect attributable to non-Darcy flow for theranges of the properties shown in Table 1, and we present chartsthat may be used to determine the ranges of properties for whichnon-Darcy flow effects should not be of concern.

Fig. 1 presents an example for the transient pseudopressureresponses of fractured wells with and without fracture non-Darcyflow in the fracture. Although the responses shown in Fig. 1 arepresented in terms of dimensionless variables, they were generatedwith the data shown in Table 2. We will use this data set as thebase case in our following developments and discussions. For the

particular case shown in Fig. 1, we chose the fracture half-length,

xf 200 ft, and considered three values of production rate, q

1,000, 5,000, and 10,000 Mscf/D.

As mentioned previously, the dimensionless pseudopressureresponses of fractured wells (dimensionless pseudopressure, mwD,

vs. dimensionless time, tD) under non-Darcy flow conditions are

governed by the dimensionless fracture conductivity, CfD, and di-

mensionless flow rate, qDND. For the case shown in Fig. 1, CfD375 (note that fractures with CfD 300 display infinite-

conductivity characteristics). The three unbroken lines in Fig. 1 are

for the non-Darcy flow cases ( 0.032 atm-s2/g), with qDND12.62, 63.12, and 126.24 corresponding to q 1,000, 5,000, and

10,000 Mscf/D, respectively. The dashed line in Fig. 1 represents

the responses for the Darcy flow case ( 0, qDND 0). Notethat for the Darcy flow case, the dimensionless pseudopressures

are not a function of the production rate. The pseudopressure dif-

ference between the Darcy and non-Darcy flow cases is a function

of qDND (or, because everything else is constant, the difference is

a function of the flow rate). This difference is called non-Darcyflow or rate-dependent skin. For a given value of qDND, the non-

Darcy flow skin is a function of time (and, thus, the flow regime)until the onset of pseudoradial flow, when it becomes a constant.In practice, the term non-Darcy flow skin is used in reference tothe constant value attained after the onset of pseudoradial flow.Hereafter, we will use the same terminology and denote the non-Darcy flow skin by SND.

Because non-Darcy flow skin is an indicator of the reduction inproductivity caused by non-Darcy flow in the fracture, it may beused to assess the influence of non-Darcy flow effects on welldeliverability and on the analysis of fractured-well tests. In thisstudy, we used a threshold value of 0.1 for the non-Darcy flowskin. For the ranges of the properties considered (see Table 1), SND 0.1 corresponds to a non-Darcy flow pressure drop that is lessthan 10% of the total drawdown. We must note, based on ourresults in this study, that as the reservoir permeability increases,SND 0.1 represents a much smaller proportion of the total draw-down than 10%. This result may not be readily understandable ormay even be contrary to the intuitive expectation. Therefore, anexplanation may be appropriate.

As discussed in Refs. 11 and 12, when the reservoir perme-ability increases, dimensionless fracture conductivity decreases,and more fluid enters the fracture from the regions closer to thewellbore. This decreases the effectiveness of the fracture or, asstated in common terminology, reduces the effective fracturelength. The effect of reduced conductivity alone has been shown toincrease the pressure drop required for a given production rate. Italso should be realized that when more fluid enters the fracture

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around the wellbore, flow rates and (thus) velocities in these sec-

tions of the fracture increase. This increases the non-Darcy floweffects and thus leads to a further increase in the drawdown. There-fore, when the reservoir permeability increases, unless the designof the fracture is changed (that is, the fracture permeability and/orwidth is increased, or the fracture length is decreased), both thedrawdown and the pressure drop caused by non-Darcy flow in-crease. As we show later, if the reservoir permeability is high,unless the fracture conductivity (kfwf) is increased, the non-Darcyskin values corresponding to a 10% contribution in the total draw-down usually cannot be accomplished unless very low flow ratesor extremely short fractures are considered.

Fig. 2 shows the non-Darcy skin factors as a function of frac-ture half-length and production rate for the base case (Table 2);results for xf = 200 ft were shown in Fig. 1. As expected, thenon-Darcy skin increases as the production rate and fracture half-

length increase. For example, if we want to increase the fracturehalf-length but limit the non-Darcy skin at the threshold value ofSND 0.1, the production rate should be decreased. Similarly, ifwe want to use higher production rates, then shorter fractureswould be needed to remain below the threshold value of SND

0.1. For the particular case shown in Fig. 2, when the production

rates were 5,000, 7,000, 10,000, and 15,000 Mscf/D, the fracturehalf-lengths would have to be shorter than 220, 160, 110, and 75ft, respectively, if no more than 10% reduction in productivitybecause of non-Darcy flow was desired. Because these values ofxfare rather small to maintain a viable production rate, we may nowseek an alternate design from the information given later. Plotssimilar to Fig. 2 for other cases are given in Ref. 13.

In this study, we considered over 1,200 combinations of theproperties shown in Table 1, generated transient-pseudopressureresponses, and examined the non-Darcy skin effects in the formshown in Fig. 2. As a result of these investigations, we determinedthe pairs of fracture half-length and production rate at the thresholdvalue of the non-Darcy skin. Our attempts to correlate the relationbetween the production rate and the fracture half-length yieldedthe result shown in Fig. 3 (the details of our investigation and the

development of the correlation may be found in Ref. 13).Fig. 3 presents the conditions for which non-Darcy flow effectis negligibly small as defined in this paper (SND 0.1). Strictlyspeaking, the correlation given in Fig. 3 is valid for the fracturepermeabilities, kf 50,000 md, and the initial pressures, pi 2,000 psia. As shown in Fig. 4, however, for pi < 2,000, thecorrelation developed in this study (Fig. 3) provides conservativeestimates of the properties for which non-Darcy flow should not beof concern. Similarly, Fig. 5 indicates that, even for fracture per-meabilities as low as kf 10,000 md, the correlation shown in Fig.3 (represented by the unbroken line in Fig. 5) is a good approxi-mation. Furthermore, as the fracture permeability decreases, theboundary of the acceptable region in Fig. 3 moves upward and tothe right. That is, the correlation presented in Fig. 3 always pro-vides conservative estimates. One final remark is on the effect ofspecific gravity, g. We have tested the effect of specific gravityon the correlation given in Fig. 3 and, as shown in Fig. 6, foundthat the correlation takes into account the variations in specificgravity in the range of 0.6 g 0.9.

The correlation presented in Fig. 3 is a convenient and practicaltool for the design of fractures and fractured-well tests. Using Fig.3, it is possible to design fractured-well tests and determine theflow rates to avoid the complexity of non-Darcy flow effects in theanalysis. Examination of the correlation shown in Fig. 3 also re-veals that sometimes, the non-Darcy flow effect may not beavoided in practice. For example, assuming the base-case proper-ties shown in Table 2, except for a higher permeability of k 10md and a production rate of q 3,000 Mscf/D, the fracturehalf-length should be as short as 4 ft to accomplish SND 0.1.

Fig. 1Effect of non-Darcy flow on transient-pressure responses of fractured wells for the base case (Table 2).



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As we mentioned earlier, as the reservoir permeability in-creases, both dimensionless pseudopressure and non-Darcy skinincrease. In these cases, SND 0.1 may correspond to pressuredrops that are much smaller than 10% of the total drawdown.Tables 3 through 5 present dimensionless pseudopressures andnon-Darcy skin values as a function of time for three reservoirpermeabilities, k 0.1, 1, and 10, and the values of the productionrate shown in the tables. Other properties are the same as those forthe base case (Table 2). The last value of the dimensionlesspseudopressure for each value ofxfcorresponds to the onset of thepseudoradial flow, and the value of the non-Darcy skin becomesconstant after this time. For example, for the k 0.1 and xf 50ft case (Table 3), even when the flow rate is as high as 5,000Mscf/D, the non-Darcy skin, SND, is approximately 0.8% of thedimensionless pseudopressure, mwD, at the onset of pseudoradialflow (t 53.8 hours). Table 5, however, indicates that if k 10

md, for the same fracture half-length (xf 50 ft) and a lower

production rate of q 3,000 Mscf/D, SND

at the onset of pseu-doradial flow (t 0.896 hours) is approximately 19% of mwD. Asmentioned before, when the reservoir permeability increases, non-Darcy skin may be the norm for practical ranges of fracture half-length and production rate; that is, the threshold value of 10% maybe too conservative. Fig. 7 is a visual representation of the pre-ceding comments for the data considered in Table 5. Fig. 7 indi-cates that, even for a threshold value of SND 0.5 and a produc-tion rate of 3,000 Mscf/D, the fracture half-length should not ex-ceed 25 ft . F or a production rate of 5,000 M s cf/D , thecorresponding fracture half-length is as low as a few feet. Theseresults indicate that when the reservoir permeability becomeslarge, the effect of non-Darcy flow may not be avoidable.

At this point, it may be useful to recapitulate the results of ourdiscussions so far. The preceding results and discussions empha-size that under certain conditions, the effect of non-Darcy flow in

the fracture may be minimized by careful design. In these cases, it

Fig. 3Envelope of SND 0.1 for pi 2,000 psia and kf 50,000 md.

Fig. 2Non-Darcy skin as a function of fracture half-length and production rate for k= 0.1.



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may be possible to conduct well tests with a minimum effect ofnon-Darcy flow (for a given reservoir and fracture, this normallymeans the selection of the appropriate flow rates). The resultspresented above and the correlation presented in Fig. 3 may beused to design fractures and fractured-well tests when fracturenon-Darcy flow is of concern. It may also be used to determineconditions for which non-Darcy flow would be of concern. (Notethat in developing the correlation for the envelope, we have usedthe criterion of SND 0.1. These results may be readily extendedto different tolerances by using Figs. 2, 3, and 7.) Our results alsohave indicated that under some conditions, the effect of non-Darcyflow may not be avoided by simple design considerations. In thesecases, considerable reductions in the productivity of the fracturemay result from non-Darcy flow effects, and it may not be possibleto estimate fracture length by well tests because of non-Darcyflow effects.

Before completing this section, we demonstrate the reduction inwell productivity caused by non-Darcy flow. The non-Darcy flow

skin values estimated in the preceding discussions were used tocompute the productivity indices of wells producing by finite-conductivity fractures and non-Darcy flow. We assume that theflux distribution has stabilized, and we define the productivityratio, PR, as the ratio of the productivity indices for the cases withand without non-Darcy flow in the fracture. For simplicity, assum-ing radial reservoir geometry and using the effective wellboreradius (rw ) concept, we may write the following expression for PR.

PR =ln rerw 0.75

ln rerw 0.75 + SND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

In Eq. 6, re is the reservoir radius, rw is the effective radius of thefracture given as a function of dimensionless conductivity, CfD(see Cinco-L and Sameniego-V.12), and SND is the non-Darcy flowskin. We have assumed that the mechanical skin is zero.

Fig. 8 shows PR (or the productivity reduction because ofnon-Darcy flow) as a function of fracture half-length for the six

Fig. 5Envelope of SND 0.1. Effect of fracture permeability, kf, for pi 2,000 psia.

Fig. 4Envelope of SND 0.1. Effect of initial pressure, pi.



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cases noted above the figure. The base-case properties given in

Table 2 were used to generate the results in Fig. 8, except for thosenoted on the figure. An 80-acre spacing was assumed. As ex-pected, Fig. 8 indicates that the PR decreases (or the productivityloss because of non-Darcy flow increases) as the fracture half-length (xf) and/or dimensionless flow rate (qDND) increase. Anincrease in the fracture width (wf), however, decreases the flowvelocities in the fracture and reduces the non-Darcy flow pressuredrop, leading to an increase in PR. Also, as we discussed earlier,as the reservoir permeability increases, the PR for a given fracturehalf-length decreases. For example, for Case 6 in Fig. 8, for k10 md, the productivity loss because of non-Darcy flow becomesas high as 60% as the fracture half-length approaches 300 ft. Onceagain, the results shown in Fig. 8 indicate that especially for mod-erate- to high-permeability reservoirs, non-Darcy flow effects maybe so significant that no practical design can reduce them to ac-

ceptable ranges. In such cases, special analytical techniques, suchas the one suggested next, are to be used.

Analysis of Fractured-Well-Test Data in thePresence of Non-Darcy Flow

The transient-pressure responses of finite-conductivity fracturesunder the influence of fracture non-Darcy flow are a function ofthe dimensionless conductivity, CfD (Eq. 4), and dimensionlessflow rate, qDND (Eq. 5). An examination of the definitions of thesedimensionless variables and the results presented in the litera-ture16 indicate that the non-Darcy flow coefficient, , should be

determined from well tests and used in the evaluation of the per-

formances of fractured wells. This, however, is not a simple task.As we mentioned in the Introduction, the conventional recom-

mendation to conduct two tests at different flow rates to estimatefracture properties (length and conductivity)4,5 is often not a prac-tical proposition. On the other hand, because reservoir and fractureflow rates should be lower during a buildup test, the effect ofnon-Darcy flow may be expected to be less significant on buildupanalysis. This, however, has been shown not to be true in Refs. 5and 6. The buildup tests follow production periods, and if non-Darcy flow influences the reservoir performance during the pro-ducing period, the conditions in the reservoir at the instant ofshut-in may dominate the buildup responses. The analysis of pres-sure-buildup tests in fractured wells is further complicated by theeffect of short producing time.5,6 Umnuayponwiwat et al.6 haveshown that the buildup analysis of a fractured-well test under

non-Darcy flow conditions would be equivalent to the correspond-ing drawdown analysis if the producing time was sufficiently long.For short producing times, however, the results of buildup analysesusually do not match those of the drawdown analyses, and thebuildup results include more error.

Another difficulty in the analysis of fractured-well responses inthe presence of non-Darcy flow has been addressed by Ref. 6. Thisdifficulty is a result of the fact that the non-Darcy flow responsescan be matched with the Darcy flow responses for a lower (ap-parent) conductivity. It can be shown that the apparent conductiv-ity is a function of the dimensionless flow rate, qDND (or the

Fig. 6Envelope of SN- 0.1. Effect of specific gravity, g, for kf 50,000 md and pi 2,000 psia.



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non-Darcy flow coefficient, ), and the true fracture conductivity,CfD, defined in Eq. 4. Therefore, provided that they yield the sameapparent conductivity value, infinitely many combinations ofqDNDand CfD may be found to match a Darcy flow type curve. Thismakes the regression analysis (or type-curve matching) a fruitlessexercise because of the non-uniqueness problem.

This discussion indicates that if non-Darcy flow affects theresponses, then the transient-test analyses (drawdown and buildup)should take this effect into account and determine the non-Darcyflow coefficient, . It is also clear that new techniques are required

to analyze fractured-well-test responses in the presence of non-Darcy flow if only one test is available. Here, we present ananalytical technique for fractured wells that does not require twotests at different rates to determine the non-Darcy flow coefficient, .

Theoretical Basis of the New Analytical Technique. The tech-nique presented here is based on the analytical expressions ofbilinear- and linear-flow regimes in finite-conductivity fracturesand the correlation of the apparent fracture conductivity undernon-Darcy flow conditions in the fracture.

Bilinear Flow. It has been shown by Cinco-L. and Sameniego-V.12 that during bilinear flow, the dimensionless pseudopressureresponses may be expressed in the following form:

mwD Darcy =

54

2CfD

tD14

. . . . . . . . . . . . . . . . . . . . . . . . (7)

Guppy et al.4 have shown that Eq. 7 could also be used underfracture non-Darcy flow conditions provided that the fracture con-ductivity, CfD, is replaced by an apparent fracture conductivity,CfD,app; that is,

mwD non-Darcy =

54 2CfD,apptD

14. . . . . . . . . . . . . . . . . (8)

They also presented a correlation for the apparent fracture con-ductivity, CfD,app, that is given here:

CfD,app =CfD

1 + 0.31 qDND, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

where CfD represents the true conductivity of the fracture as de-fined by Eq. 4. The ranges for which this correlation is applicableare given in Ref. 4 and cited below for convenience:

For all values of qDND that yield CfD,app 2 and CfD < 10. For 1 qDND 20 and 10 < CfD


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By analogy to bilinear flow and by using Eq. 11, we may writethe following relation for the dimensionless pseudopressure undernon-Darcy flow conditions:

mwD non-Darcy = tD +a

CfD,app. . . . . . . . . . . . . . . . . . . . (13)

Further assuming that the correlation given in Eq. 9 also holdsduring the linear-flow period, we may obtain

mwD Corrected, L = mwD non-Darcy 0.31 a qDN

CfD

= tD +a

CfD= mwD Darcy . . . . . . . . . (14)

Similar to the bilinear-flow case discussed previously, Eq. 14 in-dicates that if qDND and CfD are known, then, during the linear-flow period, the dimensionless non-Darcy flow responses may becorrected to match the pseudopressure responses of the Darcy flow

case with the true conductivity value.

It is important to note the following feature of the corrections

to the non-Darcy flow responses during bilinear- and linear-flow

periods. Because the form of the correction is different for linear-

and bilinear-flow periods and the correction requires that the in-dividual values ofqDND and CfD (not their combination in the form

of an apparent conductivity) be known, unless the correct values of

qDND and CfD are used, corrected pseudopressures cannot be

matched with a Darcy flow type curve during both bilinear- and

linear-flow regimes. This technique overcomes the nonuniqueness

problem encountered in the conventional regression analysis of the

non-Darcy flow responses of fractured wells mentioned earlier.

Furthermore, because it uses the responses of two different flow

regimes at the same flow rate (instead of the responses of the same

flow regime at two different rates), it provides a convenient and

practical method to compute the non-Darcy flow coefficient andthe true fracture conductivity from one single rate test. We dem-onstrate these features of the new analytical technique on the ex-ample cases considered next. Before discussing the examples,

however, we should make two comments.

Fig. 7Non-Darcy skin as a function of fracture half-length for k= 10 md.

Fig. 8Reduction in productivity caused by non-Darcy flow in the fracture.



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We have also tested the apparent fracture-conductivity equationgiven by Gidley,15 but we did not obtain good results in our

correction technique. Also, as will be demonstrated by the ex-amples below, the correction term used for the linear-flow re-sponses appears to be a good correction to the pseudoradial-flowresponses. This, however, is an empirical observation, and we donot have a theoretical justification for this observation.

Application Examples. Here, we consider two examples. The firstexample intends to demonstrate the application of the new ana-lytical technique. The second example concentrates on the effect ofwellbore storage and skin.

Example 1. Here, we consider the example shown in Fig. 9.The dashed line is the dimensionless pseudopressure responsesrepresenting the test data. The unbroken line is the Darcy flowresponses for the same case and represents the type curve to bematched. The circular and triangular data points correspond to the

corrected pseudopressure test data for the bilinear- and linear-flowregimes, respectively.

As expected from the preceding theoretical developments, Fig.9 shows that both bilinear- and linear-flow corrections yield ex-

cellent matches with the Darcy flow type curve during the corre-sponding flow regimes. The legend on the figure shows the matchproperties of and CfD, which are in excellent agreement with theinput data used to generate the test case denoted by the dashed line.

To demonstrate the uniqueness of the match shown in Fig. 9,we consider another combination of the and CfD values to gen-erate the corrected responses shown in Fig. 10. Although the ap-parent conductivities (computed by Eq. 9) are the same for thecombinations of and CfD used in Figs. 9 and 10, for the caseshown in Fig. 10, it is not possible to match both of the correctedresponses with the Darcy flow type curve. (Fig. 10 shows thematch of the corrected responses during the linear-flow period, butthe corrected responses for the bilinear flow do not match the typecurve. If the corrected bilinear-flow data were matched, then therewould be no match for the corrected linear-flow data.) This result

supports our claim on the uniqueness of the match obtained by ouranalytical technique.

Fig. 9Example application of the new analytical technique (Example 1).

Fig. 10Demonstration of the uniqueness of the analysis, Example 1.



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An important observation from Figs. 9 and 10 is that our cor-rection to the linear-flow responses also works during the pseudo-radial-flow period; that is, the linear-flow correction matchesthe non-Darcy flow responses with the Darcy flow type curve forall times after the beginning of linear flow. Note, however, thatour correction technique requires the existence of two flowperiods, and the bilinear-flow period has to be one of them. Thesecond flow period may be linear, pseudoradial, or late-time transitional.

Example 2. The objective of this example is to discuss theapplication of the new analytical technique under the influence ofwellbore storage. We have generated the data for this example withand without wellbore storage. The data shown in Fig. 11 are with-out the effect of wellbore storage and are intended to highlight theflow regimes that might be masked by wellbore storage effects.According to Fig. 11, at early times, we should have the bilinear-flow period. At later times, however, no apparent linear-flow be-

havior is observed. Our correction technique, however, still worksat late times.

Fig. 12 shows the example for the wellbore storage case. Here,the dimensionless wellbore storage coefficient is C

D0.01;

comparing Figs. 11 and 12, we can see that most of the bilin-ear-flow data are masked by wellbore storage effects. Fig. 13shows the results of the application of our correction technique.Despite the fact that most of the bilinear-flow period was lostbecause of wellbore storage and no well-defined linear flowexisted, our technique yields an excellent match with the corre-sponding Darcy flow type curve during the bilinear- and late-timeflow periods.

Here, two remarks are in order. First, as suggested by Lingen,10

in this case we computed the qDND terms to be used in the cor-rection by using the sandface rates. Although we obtained excel-lent results, we should note the difficulties in computing thesandface rates accurately from the pressure-derivative data. Thefollowing equation provides a means of computing the sandfaceflow rates:

qsf

q= 1 CD

dmwD

dtD, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

Fig. 11Data for Example 2 without wellbore storage effects.

Fig. 12Data (non-Darcy flow) and type curve (Darcy flow) for Example 2.



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where qsf

stands for the sandface rate and q is the constant surfaceproduction rate. The accuracy of the computations by Eq. 15 isusually not sufficient because of the accuracy or oscillations of thederivative data. An alternate approach is to first generate the sand-face flow-rate data from Eq. 15 and then match the data by usingthe exponential model given next:

qsf

q= 1 exp atD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

Once a satisfactory match is obtained and the constant a is fixed inEq. 16, the sandface flow rates can be generated by using Eq. 16,and they are used in the corrections.

The second remark is on the uniqueness issue. The results notshown here confirmed that the other combinations of and CfD,different from that used in Fig. 13, failed to correct the non-Darcy

flow responses to match the Darcy flow type curve during bothbilinear- and late-time flow periods.

Finally, for completeness, we include the derivative responsesin our analysis. Fig. 14 shows a composite pseudopressure andderivative plot for the example under consideration. It can be seenfrom this plot that the derivative responses for the corrected non-Darcy flow data also yield an excellent match with the Darcy flowderivative type curve.

Conclusions

In this study, we investigated the effect of non-Darcy flow on theperformance of fractured wells. We have shown that the non-Darcy flow effect may or may not be reduced by design consid-erations. In moderate- to high-permeability reservoirs, the effect ofnon-Darcy flow may be much too large for it to be avoided. Inother cases, the correlation and the results presented in this paper

Fig. 13Match of the corrected responses with Darcy flow type curve. Pseudopressure responses.

Fig. 14Match of the corrected responses with Darcy flow type curve. Pseudopressure and derivative responses.



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may be used to design fractures and fractured-well tests to mini-mize the influence of non-Darcy flow. Another important result ofthis work is a new analytical technique to account for non-Darcyflow in finite-conductivity fractures. Unlike the conventional tech-niques, the method presented in this paper does not require mul-tiple tests at different flow rates.

Nomenclature

CD dimensionless wellbore storage coefficient

CfD dimensionless fracture conductivity

ct total compressibility, psi1

h formation thickness, ft

k permeability, mdm(p) pseudopressure, psi2/cp

M molecular weight of the gas, lbm/lb-mol

p pressure, psi

q production rate, Mscf/D

qsf sandface production rate, Mscf/D

qDND dimensionless flow-rate constant

re reservoir radius, ft

rw effective wellbore radius, ft

SND non-Darcy flow skin

t time, hours

T reservoir temperature, R

wf fracture width, in., ft

xf fracture half-length, ft

Z gas compressibility factor non-Darcy flow coefficient, ft1

porosity

g specific gravity of gas

viscosity, cp

Subscripts and Superscripts

D dimensionless

f fracture

i initial

w wellbore

wf flowing wellbore

References

1. Millheim, K.K. and Cichowicz, L.: Testing and Analyzing Low-

Permeability Fractured Gas Wells, JPT (February 1968) 193; Trans.,

AIME, 243.

2. Wattenbarger, R.A. and Ramey, H.J. Jr.: Well Test Interpretation of

Vertically Fractured Gas Wells, JPT (May 1969) 625; Trans., AIME,

246.

3. Holditch, S.A. and Morse, R.A.: The Effects of Non-Darcy Flow on

the Behavior of Hydraulically Fractured Gas Wells, JPT (October

1976) 1169.

4. Guppy, K.H. et al.: Non-Darcy Flow in Wells With Finite-

Conductivity Vertical Fractures SPEJ (October 1982) 681.

5. Guppy, K.H., Cinco-Ley, H., and Ramey, H.J. Jr.: Pressure Buildup

Analysis of Fractured Wells Producing at High Flow Rates, JPT (No-

vember 1982) 2656.

6. Umnuayponwiwat, S. et al.: Effect of Non-Darcy Flow on the Inter-

pretation of Transient Pressure Responses of Hydraulically Fractured

Wells, paper SPE 63176 presented at the 2000 SPE Annual Technical

Conference and Exhibition, Dallas, 14 October.

7. Cinco-Ley, H. and Sameniego-V., F.: Effect of Wellbore Storage and

Damage on the Transient Pressure Behavior of Vertically Fractured

Wells, paper SPE 6752 presented at the 1977 SPE Annual Technical

Conference and Exhibition, Denver, 912 October.

8. Vincent, M.C., Pearson, C.M., and Kullman, J.: Non-Darcy and Mul-

tiphase Flow in Propped Fractures: Case Studies Illustrate the Dramatic

Effect on Well Productivity, paper SPE 54630 presented at the 1999

SPE Western Regional Meeting, Anchorage, 2628 May.

9. Settari, A., Stark, A.J., and Jones, J.R.: Analysis of Hydraulic Frac-

turing of High Permeability Gas Wells to Reduce Non-Darcy Skin

Effects, JCPT (May 2000) 39, No. 5, 56.

10. Lingen, P.L.: Rate-Dependent Skin From Afterflow, paper SPE

28832 presented at the 1994 SPE European Petroleum Conference,

London, 2527 October.

11. Cinco-L., H., Samaniego-V., F., and Dominguez-A., N.: Transient

Pressure Behavior for a Well With a Finite-Conductivity Vertical Frac-

ture, SPEJ (August 1978) 253.

12. Cinco-Ley, H. and Samaniego-V., F.: Transient Pressure Analysis for

Fractured Wells, JPT (September 1981) 1749.

13. Gil, J.A.: Fractured-Well-Test Design and Analysis in the Presence of

Non-Darcy Flow, MS thesis, Colorado School of Mines, Golden, Col-

orado (2001).

14. Camacho-V., R.G.: Response of Wells Producing Commingled Res-

ervoirs: Unequal Fracture Length, MS thesis, The U. of Tulsa, Tulsa

(1984).

15. Gidley, J.L.: A Method for Correcting Dimensionless Fracture Con-

ductivity for Non-Darcy Flow Effects, SPEPE(November 1991) 391.

SI Metric Conversion Factors

acre 4.046 873 E+03 m2

atm 1.013 250* E+05 Pa

cp 1.0* E03 Pas

ft 3.048* E01 m

ft3 2.831 685 E02 m3

in. 2.54* E+00 cm

lbm 4.535 924 E01 kg

psi 6.894 757 E+00 kPa

*Conversion factor is exact.

Jamir A. Gil is an engineer working for PDVSA, Venezuela.

e-mail: [email protected]. Gil holds BS and MS degrees inpetroleum engineering from U. Central de Venezuela and theColorado School of Mines, respectively. Erdal Ozkan is a pro-fessor in the Dept. of Petroleum Engineering at the ColoradoSchool of Mines. e-mail: [email protected]. Previously, he wason the faculty at Istanbul Technical U. Ozkan holds BS and MSdegrees from Istanbul Technical U. and a PhD degree from theU. of Tulsa, all in petroleum engineering. He has served on thereview committees of the SPE journals, was a Review Chairper-son for SPEREE, and was a steering committee member for anSPE Forum. Presently, he is a member of the JPTEditorial Com-mittee. Rajagopal Raghavan has worked at Stanford U.,Amoco Production Co., the U. of Tulsa, Texas A&M U., andPhillips Petroleum Co. e-mail: [email protected]. Heholds a BS degree in electrical engineering from the Birla Inst.of Technology and MS and PhD degrees from the U. of Birming-ham and Stanford U., both in petroleum engineering. Ragha-

van is an honorary member of SPE and has served on severalSPE committees and forums; he presently serves as the SeniorTechnical Editor.


Fractured-Well-Test Design and Analysis in the Presence of Non-Darcy Flow

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