SPESociety of Petroleum Engineers

SPE 21492

Analysis of Slug Test Data From Hydraulically Fractured CoalbedMethane WellsJ.A. Rushing, T.A. Blasingame, B.D. Poe Jr., A.M. Brimhall, and W.J. Lee, Texas A&M U.SPE Members

Copyright 1991, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Gas Technology Symposium held in Houston, Texas, January 23-25, 1991-

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 reflectany 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 Societyof Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.

ABSTRACT

This paper presents new type curves for analyzing slug tests inhydraulically fractured coal seams. The type curves weredeveloped using a finite-eonductivity, vertical fracture model andare presented in terms of three parameters -- dimensionlesswellbore storage coefficient, dimensionless fracture conductivity,and fracture-face skin. With these new curves, we may estimatethe hydraulic fracture half-length, the formation permeability, andthe fracture conductivity. We also present a procedure for usingthe new curves and illustrate the procedure with an example.

INTRODUCTION

Slug testing has been proven to be an effective method forcharacterizing the production potential of coal seams. A slug testinvolves the imposition of an instantaneous change in pressure (orfluid head) in a well and the measurement of the resulting changein pressure as a function of time. This change in pressure iscreated by either injecting into or withdrawing from the well aspecific volume of fluid (Le., a slug). From this measuredpressure response, we may estimate the permeability and near­wellbore conditions.

Initially, slug testing methods and analysis techniques weredeveloped for estimating the transmissivity of shallow,underpressured aquifers l -3, but also have found applications inthe petroleum industry, especially for analyzing the flow periodduring drill stem tests4-6. Recently, slug testing has beenextended to the evaluation of the production potential of coalseams? Since most coal seams are saturated initially with water,slug testing provides a simple but effective method for estimatingflow properties early in the productive life before the initiation ofgas production. Reference 7 provides an overview ofconventional slug testing in coal seams.

Conventional slug test analysis techniques are based on radialflow models. However, many wells completed in coal seamsrequire hydraulic fracturing in order to become economically

References and illustrations at end of paper

105

viable producers. Therefore, these conventional slug test analysistechniques cannot be used to either assess the success of thefracture treatment or to evaluate the post-fracture potential of thesestimulated coal seams. Karsaki, et al.S studied the pressureresponse of slug tests in infinite-conductivity vertical fractures,but they did not investigate the behavior of finite-conductivityfractures. The purposes of this paper are to develop a model forslug testing in coal seams with [mite-conductivity vertical fracturesand to illustrate application of this model to the analysis of slugtests.

MATHEMATICAL MODEL

We developed our slug test model using Cinco-Ley, et al.'s9model which considers a well intersected by a fully-penetrating,finite-conductivity, vertical fracture. The reservoir is assumed tobe an isotropic, homogeneous, infinite medium having a uniformthickness, h, permeability, k, and porosity, l/J. In addition, the

reservoir contains a slightly compressible fluid of viscosity, j.l,and compressibility, c, that are independent of pressure.

Cinco-Ley, et al's9 model assumes the fracture to be ahomogeneous, uniform slab with height, h, width, bl' and halflength, Lf. Because the fracture width is much smaller thanfracture fength and height, the model assumes the flow in thefracture is linear and that fluid influx at the fracture tips isnegligible. In addition, the model assumes that fluid productionfrom the reservoir to the wellbore occurs only through thefracture. Further, since the fracture volume is small, the modelneglects the fracture compressibility and assumes flow within thefracture is relatively incompressible. Additional details concerningthe model formulation and problem solution may be obtained inRef. 9 and Appendix A.

Under these conditions, Cinco-Ley, et al.9 derived anexpression for the dimensionless pressure drop at the wellbore(Le., XD = 0) during constant rate production from a wellintersected by a finite-conductivity fracture as

2 Analysis of Slug Test Data From Hydraulically Fractured Coalbed Methane Wells SPE 21492

XD LX'-.1I.- Xli t dx" dx' + tT: XD- FCD L 0 qfD( ,Lp) FCD···········(I)

where qjD(x,t) is the ~ensionless flow rate, per unit of ~cturelength, of fluids entenng the fracture from the reservOIr. Inaddition, the dimensionless time based on fracture half-length, 4,is defmed as

tLp = Q.QQQ2637ktf/JJ.lCH (2)

and, the dimensionless fracture conductivity, FCD, is defmed as

FCD=1t (3)

If we assume the fracture is symmetric and homogeneous, thenqjD(XD,t) = qtp(-XD,t). Making this substitution into Eq. 1 andtiling the Laplace transform yields

PtD(s) =

1.t QfD(x',s) [KoOxD - x'fir) + KoOxD + x'I-/S)) dx'2}0

LXD

LX'_-.1I.- - Xli S dx"dx' + 1rXDFCD 0 0 qfD( ,) SFCD (4)

If we include a zone of altered permeability around the fractureface, Eq. 4 becomes

+ 1.t QfD(x',s) [KoOxD - x'/VS) + KoOxD + x'l-/S)) dx'2}0

XD x'-.1I.- ( (- (Xli s) dx"dx' +..!!:!JL

- FCD Jo Jo qfD , sFCD· .. · .... ··· .. ··(5)

where the damage to the fracture face is quantified by a positiveskin factor, Sf.

Recently, Cinco-Ley and MenglO and Meehan, et a1.1 1presented a method for solving Eqs. 4 and 5. The fracture hal~­

length is divided into n discrete elements, and each element ISmodeled as a uniform flux fracture. Since the flux is unknown,we write an equation describing the pressure and flux distributionin each element, and from this system of equations, we solve forthe wellbore pressure and flux in Laplace space. We thencompute the real space solutions using the Stehfest inversionalgorithm12. Details of this solution technique are presented inRefs. 10 and 11 and Appendix B.

Before inverting the system of equations and solving for thedimensionless pressure at the wellbore, we include wellborestorage as follows:

106

PwD(S) = PtD(s)1 + s2CLtDP/D(S) (6)

where the wellbore storage coefficient based on fracture half­length is

CLjD = ~,h;; , (7)

and C is the wellbore storage coefficient for a changing liquidlevel.

The conventional pressure plotting function for productionslug tests is defined as

p' - p(t)PDslug =~ (8)

and, for an injection slug test,

p(t) - p'PDslug =~ (9)

where Pi is the original reservoir pressure, Po is the wellborepressure prior to the beginning of the slug test, and pet) is thepressure measured during the test. According to Ramey andAgarwal13, Eqs. 8 and 9 also define the annulus unloading rate,

PDslug = CLP:: (10)

which may be computed in Laplace space as

PDslug(S) = SCLjD PwD(S) (11)

Following Ramey and Agarwal13, we also define a dimensionlesssandface rate as

qDslug = 1 - CLp:ri; (12)

or,

qDslug = 1 - PDslug (13)

The dimensionless sandface rate defmed by Eqs. 12 and 13 isuseful as a slug test plotting function to provide better resolutionof very early data.

SLUG TEST PRESSURE RESPONSE

In this section, we investigate the effects of various fractureand reservoir parameters on the slug test pressure response.Cinco-Ley, et al.9 have described four distinct flow periods thatmay occur in hydraulically fractured wells -- fracture linear,bilinear, formation linear, and pseudoradial flow. These periodsand their associated flow patterns are illustrated in Fig. 1.

Fracture linear flow occurs at very small values ofdimensionless time. During this period, most of the fluid flowcomes from expansion of the fluid contained in the fracture, andthe flow pattern is essentially linear. Bilinear flow occurs whenfluid flows linearly from the formation into the fracture. Duringthis flow period, the fracture-tip effects have not yet influenced thepressure response. For most practical values of dimensionlesstime, bilinear flow appears only in fractures with FCD < lQQtT:.Following a transition period, the next flow pattern is formation,

SPE 21492 J.A. Rushing, T.A. Blasingame, B.D. Poe, Jr., R.M. Brimhall, and W.J. Lee 3

linear flow, which occurs only in fractures with FCD ~ 100tr (Le.,essentially infinite-conductivity fractures). The fmal flow period,pseudoradial flow, occurs with fractures of all conductivities.However, the higher the conductivity, the later the flow patterncan be characterized as essentially radial.

Effects of Wellbore Storage Coefficient and FractureConductivity. For hydraulically fractured wells in which skineffects are negligible, the parameters controlling the pressureresponse are the dimensionless wellbore storage coefficient andfracture conductivity. The dimensionless wellbore storagecoefficient, defmed by Eq. 7, is based on fracture half-length. Inthis study, we investigate values of CLfD ranging from 10-4 to 1.Small values of CLfD represent small volumes of fluid in thewellbore and/or long hydraulic fractures. Conversely, largevalues of CLJI! are typical of large volumes of fluid in the wellboreand/or short fracture half-lengths. We also investigate the effectsof dimensionless fracture conductivities ranging from O.ltr to

100tr. As defined by Eq. 3, small values of FCD represent long,low-conductivity fractures and/or very permeable formations,while values ofFCD ~ 100tr are indicative of high or even infmiteconductivity fractures.

Similar to the type curves presented by Ramey, et a/.6, wedeveloped our type curves using tLfD/CLfD as the time plottingfunction. Figure 2 is a semilog plot of PDs/ug vs. tLfD/CLfD for

FCD =O.ltr and several values of CLfD. All of the curves havethe characteristic S-shape exhibited by radial flow models. Infact, the curves do not display any distinct characteristics whichwould allow us to distinguish them readily from curves generatedwith radial flow models. Similarly, semilog plots showing thepressure responses for values of F CD = tr, lOtr, and 100tr areshown in Figs. 3-5, respectively. Note that as FCD increases, thecurve shapes become more distinct and are more sensitive to thevalue of CLfD. In addition, we observe that for all values ofdimensionless fracture conductivity, the level of essentiallyconstant or static pressure response is maintained for longerperiods for larger values of CLfD, which suggests larger radii ofinvestigation are achieved.

The late-time pressure reponses shown in Figs. 6-9, which arelog-log plots of PDs/ug vs. tLfD/CLfD as a function of CLfD,illustrate that many of the curves form a unit-slope line at largevalues of tLJL!/CLfD. From a comparison with the behavior ofslug tests WIth radial flow, the appearance of the late-time unit­slope line suggests that a stabilized pseudoradial flow period isieached. As we would expect, this pseudoradial flow periodappears sooner for small values ofFCD. In addition, we observethat regardless of the value of FCD, the curves for CLfD ~ 0.1converge to form a unit-slope line by tLfD/CLfD = 100.Unfortunately, these same curves have similar shapes for therange of fracture conductivites studied, which indicates thatunique type curve matches may be difficult to obtain for largevalues of CLfD (Le., for short fracture half-lengths and/or largewellbore storage coefficients). Conversely, the curves for ClJD <0.1 exhibit unique shapes for different values ofFCD .

The early-time pressure responses are illustrated in Figs. lO­B, which are log-log plots of qvslug vs. tLfD/CLfD' For FCD =O.ltr and CLfD ~ 0.01, we observe a unit-slope line at very earlytimes (Le., tLfD/CLfD< 10-3) For radial mo~els Sageev16 .hasshown that in the absence of wellbore skm, the early-umeresponse also has a unit-slope prior to the final pressure buildup.Note that when we use log-log plots of the early pressureresponse, the eurves for values of CLfD ~ 0.1 exhibit slightlymore character than with semilog plots, especially at large valuesof FCD. Therefore, if the early data are available, these log-log

107

plots may help reduce the ambiguity in type-curve matching.Unfortunately, the small values ofPDslug and tLfD/CLfD shown inFigs. 10-13 correspond to very small pressures changes and timesthat may be impractical to obtain from field tests.

To estimate the time at which the slug test pressure reponsereflects bilinear flow, we compared the semianalytical solutiongiven by Eq. 4 to the approximate bilinear flow solution given byCinco-Ley, et aJ.l4,

~DOO- tr 4)w - S5/4f'IFCii + trs2CLp .. · (1

or, the slug test pressure response in Laplace space is

- trSCLpPDslug(S) = s5/4f'IFCii + trs2CLtD (15)

We then compute PDslug by inverting Eq. 15 with the Stehfest12algorithm. For the range of dimensionless times studied, weobserved bilinear flow patterns for all values of FCD and'CLfD,but especially at the lower values of CLf!?: In general, for CLjD ~

0.1, bilinear flow ends at tLfD/CLfD < 0.01 or much earlier for thelarger dimensionless fracture conductivities. At smaller values ofCLfD' a significant portion of the pressure response represents

bilinear flow. For example, for F CD =O.ltr and CLfD =10-4, theend of bilinear flow occurs at approximately tLfD/CLfD = 1, andas shown by Fig. 6, the pressure appears to be approaching aunit-slope line within three or four log cycles after this time.When F CD = 100tr and CLiP = 10-4, bilinear flow ends atapproximately tLfD/CLfD =0.01; however, as shown by Fig. 11,the subsequent pressure reponse does not form a unit-slope linefor the range of tL~/CLJI! investigated. We suspect that much ofthe pressure behavior after tLjD/CLfD = 0.01 reflects formationlinear flow. In general, we observe that the pressure responsesfor small values of CLfD and large values of F CD are moresensitive to the various flow patterns.

Effects of Fracture-Face Skin. Cinco-Ley andSamanieg017 have suggested that two types of fracture damagemay occur during the hydraulic fracturing process -- within thefracture adjacent to the wellbore and in the formation around thefracture face. The first type of damage, often described as achoked fracture, is thought to be caused by proppant crushing andembedding in the formation. The second type of damage,quantified as a fracture-face skin, is probably caused by fluidlosses into the formation. In this study, we address only theeffects of fracture-face damage on the slug test pressure response.Appendices A and B describe the manner in which damage to thefracture face is modeled as an infinitesimal skin with no storage.

In a slug test, a damaged zone around the wellbore reduces therate of fluid flow into the wellbore and, therefore, tends to reducethe rate at which the pressure changes. We would expect a similarreduction in fluid flow from the reservoir to the fracture whenfracture-face skin is present However, when compared to thecurve shape for no skin, Figs. 14 and 15 show that the presenceof fracture-face skin has little effect on the pressure response inlow-conductivity fractures, thus greatly reducing the character ofthe curve shapes. Specifically, fracture-face skin factors less than0.1 are indistinguishable from the zero skin case. In addition,note that these skin effects are less pronounced for small values ofCLfD and FCD (Fig. 15), which suggests that we cannot estimatethe fracture-face skin accurately in short, low-conductivityfractures. The early-time effects shown in Figs. 16 and 17 forFCD = O.ltr and CLfD = 10-4 and 1.0, respectively, helpdistinguish the pressure response at early times, but the press~e

differences may be too small to be measured accurately withconventional pressure gauges.

4 Analysis of Slug Test Data From Hydraulically Fractured Coalbed Methane Wells SPE 21492

For highly-conductive fractures with small values of CLjD(Fig. 18), we see that fracture-face skin affects the pressurebehavior significantly. We note that larger values of skin tend tomaintain the static pressure response longer (Le., skin reduces therate at which the pressure changes). The early-time pressureresponses for these same parameters are plotted in Fig. 19. As theskin increases, we observe the formation of the unit-slope lineearlier than when skin is not present. For high-conductivityfractures with large values of CLfD. (Fig. 20), we see that fracture­face skin, especially values less than 0.1, has little effect on thepressure response. Only the early-time pressure behavior isaffected by the presence of skin, as illustrated by Fig. 21. Fromthese plots, we may conclude again that fracture-face skin effectsare difficult to discern from slug tests in wells with short fractures(Le., large values of CLjD)' As we would expect, fracture faceskin affects the pressure response most when formation linearflow behavior predominates (Le., large values of F CD and smallvalues of CLjD).

SLUG TEST ANALYSIS PROCEDURE

In this section, we present a procedure for analyzing slug .testsin hydraulically fractured wells, and we illustrate the procedurewith a simulated injection slug test.

Analysis Procedure

1. Prepare semilog and log-log plots of the field data. Wesuggest plotting both PDslug and qDslug vs. time using thesame scales as the type curves. For prOduction slug tests,PDslug is defmed by Eq. 8, while for injection type tests,PDslug is defined by Eq. 9. In addition, the early-timeplotting variable, qDslug, is defined by Eq. 13. As wehave discussed previously, semilog and log-log plots ofPDslug are best suited for analyzing intermediate- and late­time data, while semilog and log-log plots of qDslugprovide resolution of early-time data, and under someconditions, help to reduce the ambiguity of type curvematching.

2. Next, we select the type curves for analysis of the slugtest. As we discussed previously, the slug test responsedepends on three parameters -- CLfD, F CD, and Sf.Therefore, we should use all available information toestimate these parameters and reduce the uncertainty in ouranalysis. For example, if previous fracture treatments inthe particular coal seam have not resulted in significantfluid losses into the formation, we may select type curvesfor no fracture-face skin.

or, the fracture half-length Ljis

LI=y :/~~;p

Note that we must have estimates of tP and Ct. Thewellbore storage coefficient (C, bbVpsi) for a changingliquid level in the wellbore is defined as

C=AwbPI

where Awb is the wellbore area in bbl/ft and Pf is thedensity of the fluid (psi/ft) used for the slug test.

6. Using the time match point obtained in Step 4, calculatethe formation permeability as follows:

k = 3,390 pC (tLtD/CLtD)h t MP

Ifpre-fracture estimates of permeability are available, wemay estimate the porosity-compressibility product usingthe time match point and the correlating parameter CLjDfrom the type curve match, or

t/>c - 0.0002637 k ( t )/ - J.LL.lCL,D tLp/CLp 'MP

7. From the dimensionless fracture conductivity obtained inStep 4, we may estimate fracture conductivity as

kjbf= (kLjJFCD

Example Problem

We illustrate the application of our type curves with asimulated injection slug test in a hydraulically fractured coal seam.The test data were generated with the coal properties summarizedin Table 1. We have assumed the slug test was conducted in awell completed with 7-inch casing (1.0. = 6.094 in.) and usingfresh water. The wellbore pressure at the beginning of the testwas Po = 600 psia. In addition, we note that previous fracturetreatments have not suffered significant fluid losses, so weassume fracture-face skin is negligible.

TABLE 1

Well and Coal Seam Parameters

The first step is to calculate the slug test plotting functions,PDslug and qDslug, and prepare sernilog and log-log plots of both.Next, we attempt to match the log-log plot of the field data withthe slug test type curves for zero skin. Note that we may obtainreasonable matches ofPDslug for several values ofFCD and CLjD;however, the best match of both plotting functions was obtained

3. Next, we find a match between the field data and typecurve plots. Because of the manner in which PDslug andqDslug have been defined, we simply align equal values ofthe functions on the vertical axes and slide the graphshorizontally until we obtain a match.

4. From the type curves, read values of the correlatingparamete~s, c.LjD, FeD, and Sf- In addition, obt~n a timematchpomt, I.e., tLjD/CLj!J and the correspondmg valueof t from the plot of the field data.

5 . Using the definition of dimensionless wellbore storagecoefficient, estimate the hydraulic fracture half-length.From Eq. 7,

CLP=~t/>c,hLl

108

B, RB/STBc/, psia-1h, ft

tPp, cp

pt, psi/ftrw, ftPi, psiaLf, ftk,mdFCD

1.03 x 10-3

100.025

1.0

0.43330.25500100

51r

SPE 21492 J.A. Rushing, T.A. Blasingame, B.D. Poe, Jr., R.M. Brimhall, and W.J. Lee 5

with type curves for FCD = 1C, CLfD =0.01, and Sf= 0 (Fig. 22).The time match point is

t =1 hr and tLfDlCLfD =0.19

Using this same match point, we also obtain a good match of thesemilog plot of the data, as shown by Fig. 23.

We estimate the wellbore area (Awb, bbVft) to be

_ 1rrw2 _ 1C(0.25)2 _Awb - 5.615 - 5.615 - 0.035 bbVft

and the wellbore stomge coefficient is

C = Awb = 0.035 bb?/ft = 0.081 bbl/psiPf 0.433 pSllft

The hydmulic fracture half-length is estimated to be

Lf= y:c~~~:~

L _ . / (0.894)(0.081)f- ~ (0.025)(3 x 10-3)(10)(0.01)

Lf= 98.3 ft

which agrees with the value used to simulate the slug test. Thepenneability of the coal seam is estimated to be

k = 3,390 Jl.C (tLtD/CLtD)h t MP

correlating pammeter that could be used for all flow regimes.Perhaps an alternative analysis technique would be to develop anautomatic type curve matching technique using our slug test modeland a multi-parameter estimation scheme.

We also used our new slug test model to investigate the effectsof various fracture and reservoir parameters on the slug testresponse in hydraulically fractured coal seams. In general, weobserved bilinear flow patterns for all values of CLfD and FCD,but especially at the smaller values of CLfD. However, this flowpattern ended at tLJPICLfD < 0.01 or much earlier for larger valuesof F CD. In addition, we suspect that formation linear flow isexhibited at late times for small values of CLfD in infinite­conductivity fractures. Further, we saw a unit-slope line at verylate times, which suggests a stabilized pseudoradial flow period.As we would expect, this pseudoradial flow period appearssooner in low-conductivity fractures. Finally, we investigated theeffects of fracture face skin and found that these effects are lesspronounced at large values of CLfD and FCD, which suggests thatwe cannot estimate the fracture face skin accurately from slug testsin short, low-conductivity fractures.

As we have discussed, our slug test model was developedusing Cinco-Ley, et al's9 finite-conductivity, vertical fracturemodel and is limited to the assumptions listed earlier in the paper.In addition, our model assumes a constant wellbore storagecoefficient during the test. Further, our model is valid only forsingle-phase flow conditions. Finally, we should note that weassume the fracture fully penetrates a single zone and that thissingle zone is the only zone in pressure communication with thewellbore.

NOMENCLATURE

k = (3,390)(1.0)(0.081) (Q..l2.)10 1.0

k= 5.2 md

which also agrees with the value given in Table 1. The fractureconductivity is estimated to be

kjbf= ("4)FCD

kjbf = (5.2)(98.3)(1C)

kjbf= 1,606 md-ft

SUMMARY AND CONCLUSIONS

We have developed new type curves for analyzing slug tests inhydmulically fractured reservoirs and have presented a procedure,which is illustrated with an example, for applying these new typecurves. With these type curves, we may estimate the hydraulicfracture half-length, formation permeability, and fractureconductivity. We have also investigated the effects of fracture­face skin, and have concluded that these skin effects are difficultto quantify in short, low-conductivity fractures.

The new type curves are based on Cinco-Ley, et al's9 finite­conductivity, vertical fracture model and are presented in terms ofthree pammeters -- CLfD' FCD, and Sf. Like Ramey, et a/.6 whodeveloped their slug-test type curves using the correlatingparameter CDe2S, we attempted to find a similar parameter whichwould reduce the number of curves. However, because of thevarious flow regimes that occur during production from finite­conductivity fractures (Le., fracture linear, bilinear, formationlinear, and pseudoradial flow), we were unable to develop a single

109

AwbB

~

C

Ct

feDhKo

k

4PPDppslugPDslug

Pf

l!JDPfD

PiPoPwDPwD

PwfqqDqfDqDslug

area of wellbore, bbl/ftformation volume factor, RB/STBfracture width, ftwellbore storage coefficient based on a changingliquid level, bbl/psidimensionless wellbore storage coefficient basedon fracture half-lengthfluid compressibility, psi- l

total system compressibility, psi-l

dimensionless fracture conductivityformation thickness, ftmodified Bessel function of the second kind, zeroorder

= formation permeability, mdfracture penneability, mdhydmulic fracture half-length, ftmeasured pressure during slug test, psiadimensionless pressure

= slug test pressure plotting functionLaplace transform of slug test pressure plottingfunctionpressure in fracture, psiadimensionless fracture pressure

Laplace transform of dimensionless fracturepressure

= initial pressure, psiameasured pressure at beginning of slug test, psiadimensionless pressure at the wellbore

= Laplace transform of dimensionless wellborepressurebottomhole flowing pressure, psiaflow rate, bbl/day

= dimensionless flow ratedimensionless fracture flow mteslug test mte plotting function

6 Analysis of Slug Test Data From Hydraulically Fractured Coalbed Methane Wells SPE21492

SfstLjDxXDYYD

tPJ1.

Pf

fracture face skin factor= Laplace transform variable= dimensionless time based on fracture half-length= distance along fracture, ft

dimensionless distance along the fracture= distance perpendicular to fracture= dimensionless distance perpendicular to fracture

porosity, fraction

viscosity, cpo= density of liquid in wellbore, psi/ft

14. Cinco-Ley, H. and Samaniego-V.,E: "Transient PressureAnalysis of Fractured Wells," I. Pet. Tech. (September1981) 1749-1766.

15. Cinco-Ley, H., Ramey, H.J., Jr., Samaniego-V., E, andRodriguez, F.: "Behavior of Wells with Low-ConductivityVertical Fractures," paper SPE 16776 presented at the 1987SPE Annual Meeting, Dallas, TX., September 27-30.

16. Sageev, A.: "Slug Test Analysis," Water ResourcesResearch I. (August 1986) 22, No.8 1323-1333.

REFERENCES

1. Ferris, J.G. and Knowles, D.B.: "The Slug Test forEstimating Transmissibility," U.S. Geological SurveyGround Water Note 26 (1954),1-7.

2. Cooper, H.H., Jr., Bredehoeft, J.D., and Papadopulos,I.S.: "Response of a Finite-Diameter Well to anInstantaneous Charge of Water," Water Resources ResearchI. (1967) 3, No.1, 263-269.

3. Papadopulos, S.S., Bredehoeft, J.D., and Cooper, H.H.,Jr.: "On the Analysis of Slug Test Data," Water ResourcesResearch I. (1973) 9, No.4, 1087-1089.

4. van Poollen, H.K. and Webber, J.D.: "Data Analysis forHigh Influx Wells," paper SPE 3017 presented at the 1970SPE Annual Meering, Houston, TX., Oct. 4-5.

5. Kohlhaas, c.A.: "A Method for Analyzing PressuresMeasured During Drillstem-Test Flow Periods," I.Pet.Tech. (October, 1972) 793-800.

6. Ramey, H.J., Jr., Agarwal, R.G., and Martin, I.:"Analysis of 'Slug Test' or DST Flow Period Data," I. Cdn.Pet. Tech. (July-September, 1975) 37-47.

7. Koenig, R.A. and Schraufnagel, R.A.: "Application of theSlug Test in Coalbed Methane Testing," paper 8743presented at the 1987 Coalbed Methane Symposium,Tuscaloosa, AL., Nov. 16-19.

8. Karsaki, K., Long, J.C.S., and Witherspoon, P.A.:"Analytical Models of Slug Tests," Water ResourcesResearch I. (1988) 24, No.1, 115-126.

9. Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.:"Transient Pressure Behavior for a Well with a Finite­Conductivity Vertical Fracture," Soc. Pet. Eng. I. (August1978) 253-264.

10. Cinco-Ley, H. and Meng, H.Z.: "Pressure TransientAnalysis of Wells with Finite Conductivity Vertical Fracturesin Double Porosity Reservoirs," paper SPE 18172 presentedat the 1988 SPE Annual Meeting, Houston, TX., Oct. 5-8.

11. Meehan, D.N., Horne, R.N., and Ramey, H.J., Jr.:"Interference Testing of Finite Conductivity HydraulicallyFractured Wells," paper SPE 19874 presented at the 1989Annual Meeting, San Antonio, TX., Oct. 8-11.

12. Stehfest, H.: "Numerical Inverstion of LaplaceTransforms," Communcations of the ACM (January 1970),13, No.1, 47-49.

13. Ramey, H.I., Jr. and Agarwal, R.G.: "Annulus UnloadingRates as Influenced by Wellbore Storage and Skin Effect,"Soc. Pet. En!!. I. (October 1972), 453-462.

110

17. Cinco-Ley, H. and Samaniego-V., E: "Transient PressureAnalysis: Finite Conductivity Fracture Case VersusDamaged Fracture Case," paper SPE 10179 presented at the1981 SPE Annual Meeting, San Antonio, TX., October 5-7.

Appendix A - Derivation of Finite ConductivityVertical Fracture Model

Following the model formulation and problem solution presentedby Cinco-Ley, et a/.9 , the partial differential equation indimensionless variables describing the transient flow in thefracture is

2~+-LoP(D=OOXD2 FCD OYD (A-l)

where the dimensionless variables are defined as follows:

(X t ) = k h(Pi - PAx,t»PfD D, ljD 141.2 qBJ1. (A-2)

tLjD = 0.0:£1jkt (A_3)

FCD = ~:; (A-4)

xD =t·.· · · · · (A-5)

YD =Ii· ····· ..····· ·..·..· ·..··· · (A-6)

The fracture is at initial reservoir pressure Pi, or

pjD(XD,tLjD=O) = 0, for 0 5X 5XD (A-7)

while the boundary condtion describing constant rate production atthe wellbore (Le., XD = 0) is

OP(D _ nOXD - - FCD (A-8)

The boundary condition describing the no flux condition at thefracture tips (Le., XD = ±1) is

OP(D =0......................................................... (A-9)OXD

After integrating Eq. A-I twice with respect to XD and applyingthe initial and boundary conditions, Cinco-Ley, et a1.9 derived thepressure drop between the wellbore (i.e., XD = 0) and any point ,-­the fracture as

SPE 21492 J.A. Rushing. T.A. Blasingame. B.D. Poe. Jr.• R.M. Brimhall. and W.J. Lee 7

LXD

LX'- -1I- q v(x" tL v) dx" dx'FCD 0 0 fi .,fi (A-tO)

where the dimensionless pressure at the wellbore for constant rateconditions is

(x =0 t ) = k h(Pi - Pwt<x=O.t»PfD D •Lp 141.2qBjl (A-ll)

and qfD is defined as the dimensionless flow rate per unit offracture length going from the reservoir to the fracture. or

2q,<x.t)LfqjD(x.t) = q (A-12)

Modeling the fracture as a plane source with dimensionless fluxdensity qD(X',tLjD)' the dimensionless pressure drop at any pointin the reservoir lS

'd-r ........ (A-B)

or. solving for the pressure drop at the face of the fracture (i.e.,YD =0).

PD(XD,yD=O.tLfD) =

{{(xD-x')2]}tf4D LqD(x'.-r) exp tL~ .J·'d-r (A-14)

If we equate Eqs. A-tO and A-14. we may solve for the pressurein the fracture at the wellbore.

L

XDLX " tI I nx--ltD 0 0 qjD(X .tLp)dx dx + Fc~ (A-15)

If we include the effects of fracture face skin (Sf), Eq. A-15becomes

+ ;~D + J: SfqfD(X',tLp) dx' (A-16)

111

The Laplace transform ofEq. A-15 is

PjD(S) = tIl 7jjD(X'.s) Ko(IxD - xlv'S) dx'-1

xD x'_--lL ( (7j (x" s) dx" dx' +E1L

FCD Jo Jo fD • SFCD .......... ·.. (A-17)

and. the Laplace transform ofEq. A-16 is

LXD x'

- --.lL (7jfD(X".S) dx" dx'FCD 0 Jo

+ S;~ +LSf7jfD(X'.s)dx' (A-18)

Appendix B • Derivation of the Semianalytical SOlutionfor Finite Conductivity Vertical Fracture

In this appendix. we summarize the solution technique for Eq. A­17 as presented by Cinco-Ley and MenglO and Meehan. et al. ll .A similar technique may be used to solve Eq. A-18. Thetechnique is a semianalytical approach in which the fracture isdivided into n discrete elements of equal length Llx. and eachelement is treated as a uniform flux fracture. If we assume thefracture is symmetric and homogeneous. then7jfD(XD.S) =7jfD(-XD.S) and Eq. A-17 may be rewritten as

PfD(s) =

t{ 7jfD(X'.S) [Ko(\xD - xlv'S) + KoOxD + xlv'S)] dx'

lXD LX'- --.lL 7j v(x" s) dx" dx' + 1rXDFCD 0 0 'f. SFCD (B-l)

The fIrst integral on the right side of Eq. B-1 may be approximatedas

n lXDi+1= ~ 7jjDi(S) XD; [Ko(lxD - xlv'S) +Ko(lxD + x'!VS)]dx' ... .(B-2)

where XDi and XDi+l are the beginning and end, respectively, ofthe ith element. We may also approximate the second integral onthe right side ofEq. B-1 as

+ Llx (XDj - iLlx~jDi(S) + (~f7jjD}s) (B-3)

where xDl is the midpoint of the ith element of the discretizedfracture. Substituting Eqs. B-2 and B-3 into B-1. we have

8 Analysis of Slug Test Data From Hydraulically Fractured Coalbed Methane Wells

PfD(S) =

n (XDi+l

t~ qfDi(s)JXDi [Ko(jxDj - xl'S)+ Ko(jxDj + xl'S)] dx' -

~~(~f'ltv,{,) +.1x ~Dj· i.1x)qflJ,{') + (,,:fqflJi')]

SPE 21492

1rxD'+ sFc~ (B-4)

In addition, we utilize the condition that the summation of the flowentering each fracture element is equal to the flow at the wellbore,or, in Laplace space,

n

AxL qfDi(S) = t '" , (B-5)i=l

We may write an equation for each fracture element and with Eq.B-5 develop a system of n+1 equations with n+1 unknowns, i.e.,qfDi(S), i = 1,..., nand PfD(s). The unknowns are obtained bysolving the system of equations. Once we have the dimensionlessflux and pressure in Laplace space, we may then use theStehfest12 algortihm to obtain these variables in real space. Asimilar methodology is used to solve Eq. A-18 when fracture faceskin is included. Additional details of the solution procedure areprovided in Refs. 10 and 11.

112

'SPE 21491

Fracture

(bl Bilinear Flow

10'

FCD = a.IxSf=a

QtJ)

10-4 10,3 10,2 10-1 1 10

0.4

0.8

0.2

l.°T-----iiiiiiii~~;:::::----------_,

0.6

~J~

(dl Pseudoradial Flow

Wellbore

Fracture

(al Fracture Linear Flow

~ ~ ~ ~ ~

iii i i(e) Formation Linear Flow

~==! E=~Fracture

Fig. I - Flow Periods for Hydraulically Fractured Wells lLtD'CLtD

Fig. 2 - Slug Test Pressure Response forFCD ;O.llt and Sf; 0

10-4 10,3 10,2 IO'!

0.8

0.2

0.0 J,--'-rTTnTn"---r-.-rTTm,--r-rTTTTnr--r-1n-nm,,-::;:::;:;:~~~~10. 4

l.°riiiiiii~~§§§~==---------=--1FCD = XSf= a

10,2 10,1 1 10

FCD = IOnSf= a

0.8

1.0~~~~;::=-------------1

0.6

~t;l."l"0"0

~0.4

0.2

tLtD'CLtD

Fig. 4 - Slug Test Pressure Response for Fen; 1011: and Sf; 0Fig. 3 - Slug Test Pressure Response for FeD; It and Sf; 0

113

1.0"""§::::::::==:::::=--------------iFCD = O.ln

Sf=O

Q,n

10-4 10-3 10-2 10-1 I 10

10or---_~~~~-------1

QeD

10-2 10-1 I 10

FCD = 100TtSf=O

0.0 -t--r-r-rrn.",.---r-T....."nn-.........~:::;::;::;:;;;;:;;=~~~~,...'III/1 0··

0.8

0.2

0.6

j'J 0.4

Fig. 5 - Slug Test Pressure Response for Fen = IOO1t and Sf = 0 Fig. 6 - Intennediate- and Late-Time Slug Test Response for Fm = O.ln and Sf = 0

FCD = IOnSf= 0

10-4 10-3 10-2 10-1

1OO~~!!!.!i!~§:===:=====_--------------_,FCD=nSf= 0

10-4 10-3 10-2 10-1 I 10

10°-r-_1I!!!!!!!i!!!!~~;;;:::::::=_-------------_,

1000100100.10.011 O··-;--r"T">.,..".............,..,~.............,..,~mr"""'T..,..,......,mr---.........rTTmr--,-rTTT..../

0.001

Fig. 7 - Intennediate-and Late-Time Slug Test Response for Fm =n and Sf = 0 Fig. 8 - Intennediate- and Late-Time Slug Test Response for Fm = 10ltand Sf= 0

114

10'·1!!!!!!:!~:::::::::::====:::=------------:---1 1O·T------------------=;:::::;~!!i!!!...FCD = 1001t

Sf=OFCD =O.l7t

Sf:; 0

CLjD

10-4 10-3 10-2 10"1 1 10

10'10·'+......""T"'n-nm,--......,...,n-nm,--...-.-rrrmr-...-.-rrrmr--.........T"TT......-....,,................/

1 0. 51000100100.10.011O··+..............-rnm--..""T"'n-nm--..""T"'n-nm,--......,...,n-nm,--...-.-rrrmr--.........T"TTTTT/

0.001

Fig. 9 - Intennediate- and Late-Time Slug Test Response for Fco =lOO1t and Sf =0 Fig. 10 - Early-Time Slug Test Response for FCO =0.11t and Sf =0

FCD =101tSf= 0

10' 10'

FCD = 7tSf=O

10. 110"

C!<o 1 10

~ Q 10·'10-4 10.3 10-2 10-1 1 10 10. 2

j' ~~""-0 ""-0

§ §~

10. 3 10. 3

10·'+-..""T"'n-nm,--...-.-rrrmr--.........T"TTmr--.........T"TT"'!"....,-........Tnf-,.....,..,.,..,=/10. 5 10. 4 10. 3 10. 2 10" 10

010

1

Fig. 11 - Early-Time Slug Test Response for Fco = 1< and Sf = 0 Fig. 12 - Early-Time Slug Test Response for Fco =10lt and Sf =0

115

FCD = O.l1t

C4D = 10-4

Sr0.01 0.1o

1.01---_r:::iii~=::::--------------,

0.6

~,~

U 0.4

0.8

0.2Fcn = 1OO7t

Sf= 0

1 O· '+-....,...,..,...,.....---..,...,...,.-..nmr---...,...,.-..nmr---..........,.mr--.-.-TTTTnf----,c-r-TTl-m/10. 5

Fig. 13 - Early-Time Slug Test Response forFCD= 100lt and Sf= 0 Fig. 14 - Slug Test Response forFm = O.ht and CLtD= 104

10'1 o· '....,..,:.....,...,...,crnmr---.........,.."mr---.........,.."mr---..........,.mr--.-.-TTTTnf----,c-r-TTT-m/

1 o's1 000100100.10.01

0.2

o. 0 +--r-T'TT'"""I----."""T'"'......"T"""""T"""........""I"---r....,..,........~....-.:::;:;;n:nr=9'......-/0.001

1.0 10'

FCD = O.l1t FCD = O.l1tC4D=1 C4D= 10-4

0.8 10. 1

0.8 ~ Cl 10"

~~j' Sr

.,,00"'00 0 0.01 d 0 0.01 0.1

d 0.4 - 10. 3

Fig. 15 - Slug Test Response for Fm = O.17t and CLjI) = 1

tLrJ>'CLtD

Fig. 16 - Early-Time Slug Test Response for FCD = O.lx and CLtD = 104

116

se.E. 21492

10""'1"""-----------------------:> 1.0-.:::~--~:=_-....=:::~====----------__,

10'

FCD = 1OO7tCLrD = 10-4

~

0.01 0.1

O.B

0.2

O.B

~,:l- 0

U 0.4 -'---'----'-----'-~r_-_+_--~----\

10''0'

0.10.01o

FCD= O.htCLrD= 1

1 o' ·-t":...,......._rTTn,...-__.__1"T"l"'"',._..............."""-.....................--.-......,.,.,mr--r..,...,,..,.,n!1 O·S

Fig. 17 - Early-Time Slug Test Response for FCO = O.llt and CLjD = 1

tLp'C4D

Fig. 18 - Slug Test Response for Fco = lOOltandCLjD= 1()4

Fig. 19 - Early-Time Slug Test Response for Fco = lOOlt and CLjD = 10-4

1.0

Fco = lOO7tCLrD= 1

O.B

~

0 0.01 0.1O.B

~~€a 0.4

0.2

0.010. 4 10. 3 10. 2 10. 1 10' 10' 10'

lJ.PC4D

Fig. 20 - Slug Test Response for FCO = lOOlt and CLiO = I

'0', 0'

FCD = 1OO7tCLrD = 10-4

'o'-r------------:====?;;;=--::::"--:::::--j

1 o··~_........_rTTn...___.__.,..,.,rTm,._.......~TTn1r-.,.....,..,.." .......-,_....rrrmr__r..,...,n-nn/1 0. 5

16' ~

0.01 0.1

117

Se.E. 214q 2

t, hours

1O· ·+..J.,-rTTr.,.,.,.,-r-o..,..,.Tmr-"'T""T""T",...,.,....---.-~"""mr-,,...,.TnrTn1r_.....,,.,..,CTTTr/__...J1 0'4

10°....-----------------------==FeD = lOO1t

CLrD = 1

10. 1

Sr0 0.01 0.1 be

~... 10. 20be

"O!0

""

1 0. 3

1O···-¥:......,-rTT'l.....-r-o...,..,Tmrr-"'T""T""T",...,.,....---.-~"""mr_,,...,.TrrTn1r_...-.....,......../1 0. 5

0.001 0.01 0.1 100U

Fig. 21 - Early-Time Slug Test Response for Fen = 100lt and CLtD = 1

t, hours

Fig. 22 - Log-log Match of the Example Slug Test

0.6

I 0.6

l5be

"~ 0.4

0.2

0.001 0.1 10 100 1000

Fig. 23 - Semilog Match of the Example Slug Test

118