SPE 39931

. . -. .- . . -. ..-. .-. ,.- .- .,.Production Analysis of Linear l-low Into Fractureu IRobefi. A. Wattenbarger /SPE, Texas A&M University, and Ahmed H.Mauricio E. Villegas /SPEt Chevron Petroleum Technology Company,University

~@9ht Im, societyof P~leum Enginwm, tnc.

This paper w% p~~~ti~ori ti ~~ _ X Rocky Mountain RegionaWLow-%rmeab~ Resem-r$ SympTurn and ExhWMM held in m, Colorado, S-8 April 1998

This Pawr was sdected for prmontation by an = Pmgmm @mm”& Mng m“w cf-fin contained In ●n absbact submti by the ●uthor(s) Contont$ & the paper, aspresati, havenot been m“w8d by the sad~ d Potmleum Enginaam and am subject to~ by the author(s). The material, ●s pmwnted, & nd newssarily reflect anyposition of the -~ ot Petroleum Engineem, itc Mcers, or members. PaPrs presented atSPE meetings am subject bYpublicationtiw.by Ediial hmti ~ ~ ~~ dmum Engineers. EIeclrunic mpmduti, dishiution, or storage d ●ny part & Ws pap6rfor wmmerciaI purposes *out the_ ~w~ of the Sociely & -cum Engineers isPmhibti. P0fm18s10n to wmduce in print is Mctad to ●n ●b8M & nor more than ~words; illustrations may nd Im *. The •k~ mud contin conspicuousacknowl~gment & Mere snd by whom rho paper ws presented. Write Librarian, SPE, PO.Sox ~, Riihardwn, TX 7S0S3-=, USA,, h 01 -972-9S2-9433

AbstractSometimes decline curves for tight gas wells indicate thatlinear ftow may last for over 10 or 20 years. These declinemes may show outer boun~ry effects but no pseudo-radialflow. This paper presents d&line curve analysis methods for

rsuch wells. Values for k X1and drainage area can be

caltited. Stabilized (bounded) flow equations are alsopresented for forecasting. me solutions and type curvesdeveloped are for both itilnite-acting and stabilized flow witheither constant pwJ or constant rate. Several field cases areshown with example calculations.

IntroductionMany gas wells have been obsewed which stay in the linearflow regime for several years. These are usually wells in verytight gas reservoirs which may have hydraulic fracturesdesigned to extend to or n=ly to the drainage boundary ofthe welI. For these wells, no pseudo-radial flow is expectednor observed during the production period. Linear flow seemsto k the dominant flow regime throughout the well’sproduction life.

Fig 1 is a top view of a hydraulically fractured weIIwhose fracture extends all the way to the lateral boundaries.The weIt is in the center of a rectangular drainage area. Thedistanm to the outer boundary in the direction perpendicularto the fracture is y,. The fmcture is assumed to have ifilniteconductivity. This is a g~ assumption for largedimensionless fracture conductivity, FUI >50.

Dimensionless fracture conductivity is defined by:

Societyof Petroleum Engineers

lgnt Gas wellsE~-Banbi /SPE, Texas A&M University, andand J. Bryan. Maggard /SPE, Texas A&M

kwFCD= ~

k Xf............................................................(l)

With these conditions, the flow is linear and isperpendicular to the fracture.

The petroleum literature has many complicated cases forhydraulically fractured wells’-g. Solutions for these cases arepresented as type curves, tables, or sometimes as equations.

Usually the cases involve fractures in intinite reservoirs,which means that flow eventually leaves the linear regimeand becomes “pseudo-radial”. Almost all of the type curvesare for constant rate flow. There are very few solutions forconstant (wellbore) pressure, which is of interest for Iong-term production analysis.

Apparently, the only solution published for the problem ofFig. 1 is for a square with constant flow rateg. This is shownas a special case of a more general set of solutions.

Miller10 provided solutions for linear flow in aquifers.Miller’s solutions were for the infinite acting and boundedaquifers for both constant rate and constant pressure cases.Nabor and Barhamll generalized these solutions indimensionless variabIes. They also added the solutions forconstant pressure outer boundary case. The mathematics forthese solutions is also available from Carslaw and Jaeger{z forlinear heat conduction.

Wattenbarger, E1-Banbi, and Maggard*3 have adaptedthese linear reservoirs solutions to fractured wells in thegeometry of Fig. 1. They included the constint rate andconstant pressure cases for linear flow in a rectangle. Theequations and type curves in the foIlowing seetions are new,for the most part, and provide tools for analyzing long-termproduction petiormance of tight gas wells.

Solutions of Linear Flow Into Fractured WellsThe following equations apply for linear flow into a fracture.These equations strictly apply to the liquid case, but they canbe used for gas wells with good accuracy if dimensionless realgas pseudo-pressure14, mD, (fOr gaS) iS substituted fOrdimensionless pressure, pD, (for liquid).

265

2.. R.A. WATTENBARGER, A.H. EL-BANBI, M.E. VILLEGAS, J.B. MAGGARD ‘SPE 39931— —. —

. .:-—-.we *OW the solutions for two inner bounda~ conditions

(cofi~”h~f-ami cqgstant pwl) and closed reservoir outerboundary condition. Eq. 2 is the solution for constant rateptiuction ~om a cIosed Iifiear reservoirs.

~w.=:[:][:+(;)’,..f]

-+[$)~($)exP[-n’~’(~JtDx,]...(2)

Eq. 3 is the soIution for constant pwf production from acIosed linear reservoir. .,

““”[l.,~~—=___.__—_____;D...~ap[--.:(;j;h,l.................(3)where dimensionless variables are defined by:

‘fi(Pi ‘P~)~wD _=

141.2qBp.............................................(4)

I “=-zh~i-pti)-—--- ..............................................141.2qBp

(5]~D

0.00633k/t–‘f : #pc, x;

.................................................(6)

Note ‘tit-tie definitions of pw~ and llq~ appear to be thesame except that p.f varies in the PWDcase while q varies inthe I/qD ~.

‘Short-tetiW-Approximations. The solutions given by Eqs.2 and 3 have short-term approximations. Theseapp-tions are aIso the solutions for intinite resemoirouter boundary. ~ese approximations are given for constant

rate and constant pwf by the following equations, respectively:

PwD = ~n‘DxJ .......................................................(7) ‘

lxzt—= —r Dx, ....................................................(8)

qD 2

.“..-

UnIike ifilnite radial reservoir solutions’5, note that.constant rate and constant pressure solutions for ;Wlnitelinear reservoirs differ by a factor of d2.

hng-term Approximations. Closed resewoir solutions canbe approximated by the following equations for Iong’tirnel:,,, .- .,

pwD”=:[~)tDx,+:[$)......................(9.

;=;[:)exp[;(:]tDx,]....................(10)

These equations would result in ~fferent expressions forproductivity indexes. The “folIowing equations arc ‘theproductivity index expressions for constant rate and constantp~f production respectively:

. . .

Jm =kh

[[)1.,.....................-.m..

141,2Bp : :f

J,, =kh

[[)1......<..................<ti.....(12)

2 Y.141.2Bp – —

n Xf,, i -: :-*-–-–“’ ‘We see that the ratio of productivity index for con;~nt pwj

production to the productivity index for constant rateptiuction is x2/12. This difference may be important andthe right equation should be used for long-term productionforecast.

..

Type Curves. The most practical way to illustrate oursolutions is through Iog-log type curves, Fig. 2 is a type curvefor each of the two cases for alI times of interest. These curves

266

SPE W31 PRODUCTION ANALYSIS OF LINEAR FLOW INTO FRACTURED TIGHT GAS WELLS 3

have been plotted against tx rather than tD#. The “y axis”has been m~]ed to ( xj/y~ ) pm and ( Xfi ya ) (1/gD) ratherthan P.D and l/q~. ‘fhiS giVeS only one cuIve for each ~,for any rectangular geometxy, rather than families of curves.The following equations beeome the solutions for closedreservoirs for constimt rate and constant pwf respectively:

-+~(+)exP(-'J'~'~Q.).........................('3)

[1‘fl– zy, q~ “ ~ _~2~2t~,x PI 1

.....................A

The dimensionless time, tD>v,is defined by:

..(14)

[)2

t).00633kt _ ~ tt .......................!.,.“ ‘. ##cty: - y* “J

(15)

. ...=. ..: O.. — -

It- is ‘Ilgmficant that th& type cu~es (log-log plot of~erence in pressure or ~e~procal of production rate versustime) bend upward ._when the outer’ boundary is “reached.This is ~e-opposite of the downward bend which is seenwhen fractured wells tend toward pseudo-radial flow5’15.Thisshape difference may b ‘important in analyzing field data. (Itshotid’”be understood that this rule is reversed when we plot~----production tie, rather than reciprocal of rate, versus time)..

Figs. 3 and 4 are type curves drawn using Eqs. 2 and 3 fordifferent aspect ratios, ( XI/y, ). Fig. 5 is a deeline curvedrawn for the constant p~f solution given by Eq. 14.

End of Linear Flow [1/2 slope line). An important featureof the type curves is the end of the infinite acting linear flow(seen as the “half-slope”). This can be used to estimatereservoir size from field data, Once again, the two cases aredifferent. The end of the half slope is tLJp= 0.5 for theconstant rate case but is ?Pp+=0.25 for the constant P.fcase(seeFig. 2). These values, 0.5 and 0.25, are taken to be valueswhere the CUIVCSvisua/ly depart from straight lines, makingthem useful for the purpose of establishing minimumdrainage area from field data (discussed Iatcr).

Application to Gas We!lsThe solutions for liquid flow can be adapted to gas flow byusing the real gas pseudo-pressurel 4. In the liquid solutions,PwD is replacedby m.D which is defined as follows:

kh[m(pi)-m(pw)] “- “ ‘m=WD

1424 qg T............................(16)

l/q~ has the following definition:

1 k h[m(Pi) - m(pti)]= .....................i....w-...(l~

;. 1424qgT. . ..,-

where m@) is the real gas pseudo-pressurel 4defined by:

‘Pm(p) = 2~~p ....... ............. ........... ....... ........... ..(l8)

and the dimensionless time referenced to the distance to thereservoir boundary, ye, is defined in terms of initial fluidproperties as follows:

0.00633kt - ‘“t‘y= (+ PC,)lY.2

‘-- -~19)................................................

The liquid solutions, which have been previously shownand are pIotted in the type curves, apply very accurately forgas in transient flow. This has been determined for a numberof weI1 flow problems over the years. Thus, the use of Eqs.~.16-19 works very well for the infinite-acting t behavior.Fraim and Wattenbarger normalized timelc should be used tocorrect for changes in gas properties for boundary dominatedflow ‘data, if these solutions are used in forecasting. Analternative procedure for forecasting boundary dominatedflow production will be discussed later.

Analysis of Field DataWith the above solutions, there are several ways to analyzegas well field data. The following is a description of analysismethods. An example well will be used later for illustrativepurposes.

Log-Log Plot. The first step is to plot either pseudo-pressuredifference versus time (m@,)- m(pw~) vs. t),for constant “rate

267

4 R.A. WA~NBARGER. A.H. EL-BANBI, M.E. VILLEGAS, J.B. MAGGARD.-. ..—

SPE 3w3i

productio% or reciprocal of production rate versus time (l/qgvs. t), for constant p.f production, on a log-log plot. This is asimple plot tid wiII show indications of linear flow if the pIothas a “half-slope”.

Many weIls may have periods where the well pressurevaries or the well is shut in because of market curtailment,etc. For these reasons, the half-slope may not always beapparent or steady for the Iife of the well, even if the behaviorof the well is entirely linear flow. Therefore, the log-log plotshotid be considered to be a screening method rather than aquantitative analysis tool.

Square-Root of Time Plot. The next step is to make a more

definitive plot for Iinear flow - the ~ plot. The half-slopepart of a log-Iog pIot is equivalent to a straight line on the

pseudo-pressure difference vs.~plot, for the constant ~te

case, or l/q&vs. tiplot, forthe constant pressuremse.The slope of this line is given by the following equations

for constant rate and constant pwf, respectively:

200.8 T ggmm = — .. .......... ... ............ ...........(20)

hdm fixf

315.4T 1..........................(21)‘e ‘h ~m ~b~xf

where:

Am(p) =m(pi)-m(pti) .....................................(22)

Calculation of ~Xf. We can calculate ~xj from the

sIope of the fiploti mm or ma. From this slope, andknowing vaIues for other resemoir parameters, we can use thefoIlowing eqtitions for constant rate and constant ptip@uction, respectively:

&xf= 200.8T qg— .......................................(23)h~w mcR

This equation requires that permeability be known todetermine x} Unless k is known independently, it is verydi~cult to determine Xf

It should be emphasized that these two equations differ byd2.

Calculation of ‘Distance of Investigationn. The distance, y,,m be calcdated by identifying the end of the “~lf-slo~”line, t.h, and comparing this time, in days, to thecorresponding dimensionless time, tDP.This value of (tDy)#kis 0,5 for the constant rate case and 0.25 for the constant pwj-. The corresponding distance to the outer bounda~ isthen given by the following equations for both casesrespectively:

i

ktek

‘“=0”113 (@Wl)i

......... ...... ......... .. ....................(25)

/

k teb

‘= ‘0”159 ($x,),...... ........ ......... ......................(26)

:.i’; ?

This boundary distance can be considered to be “rninjnlunzvalue if all the history data is still on the half-sIope trend. Inthis case, the latest production time is used instead of (/),b inEqs. 25 and 26, This would be the “distance of investigation”at the current time. Agai& ‘these equations, require that@rmeability be known. Of course, this may be a weti pointin determining, by this method since k may ~,$~ncetiin..,

Calculation of drainage area. The good news, is that thedrainage area can be calculated directly, without knowingeither k or x} For the model of Fig. 1, the d~inage area isgiven by: .

A = 4xfy, ..................................................................(27)

Eq. 27 can be used to calculate drainage area from theslope, nzm or mm, and the end of haIf-sIope time, t,k,for bothCas

A=4xf yc=

{}

90.8T qg tr eh

........ ... ..........(28)h(@#t)i rnCR ~

fix = 315.4T

‘ --~da” ~i)mcp

............. ..... .......(24)

268

SPE 39931 PRODUCTtON ANALYSIS OF LINEAR FLOW INTO FRACTURED TIGHT GAS WELLS 5

‘=4x~yc=::::[{~cp~(P)}fi(2’)where the drainage ~ A, is in fi2 but can be changed toacres by dividing by 43,560. Once again, if the wellperformance is still in the half-slope period, the latest time is- in Eqs. 2$ and 29 and the drainage area is considered tobe minimum @rev@.

Calculation of Pore Volume, Now we can also calculate thepore volume, Vp We get a direet determination of Vp if wemtitipIy Eqs. 28 and 29 by + and h. This maybe a big help,since + and h are often not known in tight gas formations.‘fhe equation for Vpfor the constant rate is given by:

{}”~=90.8T’4g ~— — r‘ (~t)i ‘CR ‘b

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (30)

and for the constant pwfcase by:. . .

‘~=t::{~cp~(P)}&(31)We notice that the dculation of V’ does not depend on

having good estimates of k, h, and+.

Calculation of OGIP. Once the pore volume is calculated,then OGIP can be direct.Iy calculated by the followingquation

~G1p= v,(1 - Sw)R ..............................................(32)

If gas compressibility dominates c,, the calculation ofOHP becomes insensitive to the value used for Sw.Thus, wemay determine OGIP accurately without actually having goodknowledge of k, h, +, and SW

Example CalculationWe choose to appiy our linear flow analysis techniques toproduction data of a hydrauficdly fractured tight gas weIl in aSouth Texti ‘fieid. We &d’monthly rates for our plots. Therewere no recorded “presm”tis m the earIy years, but it wasassumed that pressure was constant throughout the producingtime of the well.

3?iE 6 shows a log-log decline curve for the example welI.The production rate foIIows a “half-slope” pattern for most ofits life. This is an indication of linear flow with constant

bottom-hole pressure. Actually, it is seen that the dab’-startsbending downward ~er about fifteen years. Applying ourlinear ftow analysis indicates that the well encountered theouter boundmy at y, at this time. Note that this we~”is riottendirig toward pseudo-radial flow since that would m&e thecurve bend upward from the half-slope, not downward.

For this well, we need fiber analysis and finallyconfirmation by reservoir simulation since the variation ofproduction rates and possible variation of pressures must havesome effect on the decline curve. Nonetheless, the log-logdecline cmve =ma to& a usefil and easy anaIysis=ttiI forour example well.

Fig. 7 shows the lfq~ vs. ~plot for the example well fora period of nearIy 23 years. The cumulative production is also

a straight line when plotted vs. ~ (not shown in the paper),The slope of the cumulative curve is 21mW. This tends tosmooth the data, but subtilties in the bounda~ effect may notbe seen as easily.

This figure was used to obtain both the slope, mm, andend of half-slope time, t,k. The values are 0.00009 and 5,625respectively. Other well data are given in Table 1. The resultsof the calculations from Eqs. 24, 29, 31, and 32 are asfollows: ,,

A = 2,872,000 ftz = 65.9 acres...

,.:,. .

Vp = 3.% x 107rcf .

OGIP = 6.67 X109SCf=6.67 Bcf

The above anaIysis should be accurate if aII the assumedconditions are perfeetly met. In the example \v~lI as”weII-asmany other wells in the same field, the reservoir/fracturesystem seems to cause linear flow, no pseudo-radial flow wasdeteeted. However, the conditions of constant rate or constantPwf were never perfect. In some cases, the ‘actual flowingconditions were not even close to either case. Therefore, itwas necessary to cotitrm or modi~ the analysis by -historymatching with GAS~17, a single phase gas simulator. Thesimulation results were close to the resuIts we obtained withthe simple hand calculation technique. “ “ -_, ‘‘

DiscussionThe analysis presented here may be usefil for many fbcturedtight gas wells. This method based on Iinear flow anaIysis hasnot previously appeared in the literature aIthough much of themathematics was previously avaiIable. We expect that therewill be increasing use of long hydraulic fractures in the futureand closer well spacing as well. These two wiII tend to filtillthe conditions of this linear flow analysis method.

....

269

e R.A. WAUENBARGER, A.H. EL-BANBI, M.E. VILLEGAS, .I.B. MAGGARD SPE 39931

~ analysis here is simple but may have wideapplication. It is worthwhile to discuss several aspects of itsapplication.

In a recent anaIysis of a field with about 60 wells, long-krm linear flow was observed in about one-third of the wells.Many of the other weIIs could not be analyzed because ofsevere rate fluctuation caused by gas market problems. Butnone of &. wells showed pseudo-radial flow theoretically_ in weIIs with hydraulic fractures.

Ftite Conductivity Fracture% This analysis has been forate fracture conductivity. When formation permeability is-mely low, the conductivity of the fracture will behave as. ..—tiough it is relatively high in “most cases. If the fracturecondti~-~ is significantly Iow (Fo less than 50) then thistid appear as a “skinw in long-term production behavior.This type of behavior was analyzed by Agarwal, et a17. ~Istid be recognizable by a non-zero intercept on the y axis of

the 1/q~vs. &plot.

~ intercept was not noticed on any of the well+ in thefield analyzed. It should. be recognized, however, that theearly ‘khavior was sometimes interrupted by the problems ofpu~ng a new well on production. It is interesting thatpressure buildup tests on the example well have shown earlyflow to be dominated by hi-linear flow, indicating that theticture conductivity was limited. Still, this was notnoticeable on the long-term production behavior.

Possible Causes of Linear Flow Behavior. In Fig. 1 theticture length extends all the way to the drainage boundary(xf = x.), This may not seem to be plausible in some cases,However, there may be other reasons for seeing this type oflinear behavior.

Stright and Gordon18 reported long-term linear behaviorin tight gas wel[s which did not have particularly largefracture treatments. They believed that this might be caused~“ natural fticturing in the formation caused by normaltectonic processes in a relatively hard formation, Thesetitures would tend to “beparallel to the fracture plane andwotid promote Iinear flow even if the fracture length werelimited.

Fig 8 shows how directional fractures in a reservoirwotid tend to change the reservoir into an equivalent “long,Skinnyw reservoir. The sketch on the left shows naturaltictures oriented in the x direction. Mer the transformationwhich accounts for the anisotropic permeability, thedimensions in the x direction are changed to:

.

x =xJ~ ...................................................... (3,3)

while the dimensions iri they direction are changed to:

.L

Y = yJ~ ......................................................(34)

It is likely that anisotropy is sometimes much more severethan this example.

The calculated fixf from either Eq. 23 or 24 would be

ye product of the square root of the equivalent &~;;bilitygiven by Eq. 35 and the transformed dimension Xl .....

. -“<

K=~~ ............................ ................................ (35)..—-—.... -. .- ;.”

However, the calculated drainage area, A, and consequentlythe pore volume, VP,and the OGIP are correctly determinedeven with the anisotropy.

...

KohIhaas and Abbott]g reported another possible reasonfor linear flow behavior other than the geometry of Fig. 1.They described massive tight formations which could beexpected to have layers of higher permeability. The pressuredrops in the higher permeability layers would then causevertical linear flow into the higher permeability layers. This,too, would result in long-term linear behavior.

Layered Reservoirs at Linear Flow. The analysis givenhere is for a single layer reservoir. If the well. produces frommdtiple layers, it would still appear as linear flow if flow is

,, .‘i~lnite acting= in alI layers. The calculated fix~ productwould be a thickness weighted average summation of thesevalues for the individual layers as shown by the following~tion:

fix . ;6”fn” ..................................j .........(36)

j=l

Determination of Fracture Wngth. This is a continualpuzzle for a fractured well. In our fractured well analysis, it is

seen that the A XJ product is determined from the wellbehavior. The conventional thought is that &rmcability mustW determined from a pre-fracture buildup test. This wouldwork in a simple reservoir, but it is eipected tl~at ~ny tightgas reservoirs will behave like rntitiple-layered resew.oirs, Inthese cases, the permeability averaging that takes place before.the fracture treatment and the ~k Xj averaging that takesplace after the fracture treatment may not bc related, Sometype curves of complicated cases may help this probIem butthe results may not be unique.

,;.L

270

SPE 39931 PRODUCTION ANALYSIS OF LINEAR FLOW INTO FRACTURED TIGHT GAS WELLS _. ..- 7

.

Forecasting. Once OGIP bs been determined, forecasting isatrai@tiorward. “Usethe m“ostrecent data which has a reliablevaIue o[itabilized rate and ~ttom-hole pressure. From this, aproductivity index can be determined directly, based on thefoIIowing equation:

J .[mb)-mbti)lg

..=. .. ......... ............ .............(37)9g

-4

If the boundary effects have not yet been observed, this will bea conservative forecast. The average pressure, ~, is

determined from material balance, using actual cumulativeproduction. Forecasting is then just a matter of selecting timeintervals and applying material balance and productivityindex equations at each fiture time interval.

Conclusions1. Many weIIs in tight gas reservoirs have long-term

production which exhibits only linear flow. These wells donot show pseudo-radial flow as sometimes expected withhydrmdic fmctures.

2. The equations for analyzing long-term (constant p.,)linear flow production are different than for buildup analysis(constant rate). Both sets of equations are presented here.

3. Drainage area can be directly determined if the outerboundary effect has been observed. (If the resemoir is stillinfinite-acting, this would be a minimum value). Knowledgeof permeability, k, is not required.

4. Pore vohune and OGIP can be directly determined ifthe outer boundary effect has been observed. (If the reservoiris stiIl intinite-acting, these would be minimum values).Knowledge of permeability, k, thickness, h, and porosity, @,are not required.

5. If gas compressibility dominates cl, the calculation ofOGIP becomes insensitive to the value used for Sw.Thus, wemay determine OG]P accurately without actually having goodkmowIedgeof k, h, +, and SW.

NomenclatureA = weI1drainage area, L2, flz.B = oil FVF, dimensionless, RB/STB

Bz =gas FVF, dimensionless, rcflsefBxi =gas FVF at initial pressure, dimensionless, rcflscfc~ “gas ~biIity, Lt2/m, psa-lc~= total compressl%ility, Lt2/m, psa”l

Fm = dimensionless fracture conductivity~ =formation volume factor, dimensionless

h =net formation thickness, L, fitJo = productivity index for constant p.f production,

LZ/mt3,ST13/D/psiJm = productivity index for constant rate production,

-....=- .*L2/mt3, STBID/psi

Jg =gas productivity index, L4t2/m, Mscf,cp~/psiz.k =permeability, L2, mdkj =fracture permeability, L2, md “k. =permeability in the Xdirection, L*, rndkY =permeability in the Ydirection, L2, mdK = uivaIent permeability for anisotropic reservoir,

?L,md

mcp = slope of l/q~ vs. &, D1’2/Mscf

ma = slope of ~ vs.&, psi2/cp D1nm(p) = real ‘gaspseudopressure, fits, p:iaz[~

m(~) = mo) at average reservoir pressure,” fit3,

psiazlepm@Wf) = m(p) at flowing wellbore pressure, fit3, psizlcp

n = number of layers in a linear layered reservoirOGIP = Original Gas in Place, m3, scf

_p = absolute pressure, tit’, psiap =average reservoir pressure, fif~, psia

PD = dimensionless pressure

PO =arbit~ry lower limit of nl@) integration, m/Lt2,psia

p~f ‘bottom-hole flowing pressure, m/Lt2, psiaqD = dimensionless flow rateq~ =gas flow rate, L3/t, Msc~q = oil flow rate, L3/t, STB/D

SW= water saturation, fractiont = producing time, days

?W = dimensionless time based on XftDY = dimensionless time based on y,

T = reservoir temperature, T, %VP= pore volume, L3, ft3w = fracture width, L, R.Xj= fracture half-length, L, ft. ~ ,..x= = reservoir half-width, L, ft.X. = transformed reservoir half-widtl; “because of

anisotropy, L, R. ,.y. = distance from fracture to outer boun@~, L, R,z = gas deviation factor, dimensionless~ =porosity, fractiong =viscosity, tit, cp

* =gas viscosity, mfLt, ep

Subscriptsehs = end of “half-slope” period

i = initial conditionsj = layer index

AcknowledgmentsWe thank Tai Pham of Coastal Oil and Gas Corp. for sl]aringfidd data with us. We also thank the Reservoir ModelingConsortium for providing funding for this project.

.,

27i

8 R.A. WATTENBARGER, A.H. EL-BANBI, M.E. VILLEGAS, J.B. MAGGARD .=SPE 39931

. .

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Problems-Case Studies; paper SPE 11088 presented atthe 57th Annual Fall Cotimence and Exhibition held jn

Behavior M“VerticaIIy Fractured Re-oirsfl JPT (Get. New GrIms, LA, Sept. 26-29, 1982. “““ .

1964), I 159-1170.Wattenbarger, R A., and Ramey, H. J.: “Well TestInterpretation of ~ertically Fractured Gas Wells: J. Pet.Tech. (May 1969) 625-32; Tmns., AIME, 246.Morse, R..A. and Von Gonten, D.: “productivity ofVerti~y Fractured Wells Prior to Stabilized Flowflpaper SPE 3631 presented at the 1971 Annual TechnicalC@~_a and Exhibition, New Grleans, Ott.3-6.-g= AIain C., Ramey, H. J. , Jr., and Raghavan,R: ‘Unsteady-State Pressure Distributions Created by aWeIl With a SingIe ktinite-conductivity VerticalFractnr~n Soc.Pet.~.J. (Aug. 1974) 347-360,Trans.,~.Cinco, H., Samaniego, F. and Dominguez, N..: ‘TransientPressure Bhvior for a Well With a Finite-ConductivityVerticalFracture,” ~’(Aug. 1978), 253-264.AgarwaI, R.G., ‘“Carter, R.D. and Pollock, C.B.:Evaluation and Pdormance Prediction of Low-Paability Gas We~s Stimulated by Massive HydraulicFracturing: JPT March, 1979) 362-372; Trans. AlME,267.Cinco-Ley, H., Smiego, F.: ‘Transient PressureAnalysis for Fractured Wells:’ JPT (Sept. 1981) 1749-1766.Gringarta A.G.:’=Xoir Limit Testing for FracturedWelIs~ paper SPE 7452 presented at the 53rd Annual FallTechnical Conference and Exhibition, Houston, Texas,Oct., 1978.Miller, F. G.: ‘Theory of Unsteady-State Influx of Waterin Linear Reservoir,” Joumai gj the Institute OSPetroleum, Vohune 48, Number 467- Nov. 1962,365-79.Nabor, G. W., and Barham, R. H.: ‘%inear AquiferBehavior: JPT, (May, 1964), 561-563,Carslaw, H. S., and Jaeger, J. C.: Conduction oJ Heat inSolids, Oxford University ~ess, Second Edition, 1959.W~barger, R. A., E1-Banbi, A., and Maggard, J. B.:-y wfi for the Reservoir Modeling Consortium,Texas A&MUniversity, May, 1996.A1-Hussainy, R., Ramey, H.J., Jr., and Crawford, P.B.:me Flow of ~ Gas Through Porous Media; JPT~y 1966) 624-636.Earlougher, R.C., Jr.: Advances in Well Test Analysis,-graph Vol. 5, Society of Petroleum Engineers ofAIME, New YorkR~lIas, 1977.Fra@ M.L. and” Wattenbarger, R.A.: “Gas Re~oirDecline-curve AnaIysis Using Type Curves With RealGas P=bpressures and Normalized Timefl SPEl?E(Dec. 1987) 671-6=.@ W. J. and Watibarger, Robert A.: “Gas Reservoirweeting, SPETextbook Series, Vol. 5, 19%.Stright, D. H., and Gordon, J, I,: “Decline Curve Analysisin Fracf.ured bw Permeability Gm Wells in the PiceanceBash” paper S-E 11640 prewted at the 1983SPEiDGE Symposium on Low Permeability held in

272

SPE M31 PRODUCTION ANALYSIS OF LINEAR FLOW INTO FRACTURED TIGHT GAS WELLS 9. . ..- .— ——

Table 1- Datafor example well.

Initialpressure, Pt 6600 pMom-hole flowing pressure, * 1600~udo-~ure at p,, rn(pl) 2.67 X 10° @~:ppseudo-preaaure at M, *) 1.69X108 pai’lcpgas specif~ gravity, y~ 0,717reservoir temperature, T 290 “F

formation net pay thickness, h 92 ft.

fornlatii ~oany, + 0.15

averagewater saturath, SW 0.47

totsl cornpreaaibilii et pj, c“ 3.s3 x 10+ w’

100

10

1

0.1

Xf●

Ye

XeFig. 1- A hydraulically fiacturad well in a rectangular resewoir.

aol 0.1

f;ye

10 100

Fig. 2- Constant rate ●nd constant pw solutions for closed linear reservoirs.

273

10 R.A. WATTENBARGER, A.H. EL-BANBI, ME. VILLEGAS, J.B. MAGGARD SPE 39931

10

—..‘.

“A

. .

ill

0.1 1 100 lmf&Fig.3- Constant production rate type curve for closed linear reservoirs. 9

,-

. .

’10

0.01 0.1 10 100

. ...- t;xf,.

Ha. 4- Constant ptitype curve for cIosed tinear reservoirs.

-.

.

---

.2. ”,..

;

.-.—..

SPE 39931 PRODUCTION ANALYSIS OF LINEAR FLOW INTO FRACTURED TIGHT GAS WELLS 11

0.1

0.01

O.wl 0.01tDye

Fig. 5- kfine curve for closed linear reservoirs,

Io,oou

1 10

,................

10 100 i,Ooo 10,OOO

t (days)

Fig. 6- iog-log decline curve for example well.

f2 R.A. WATTENBARGER, A.H. EL-BANBI, M.E. VILLEGAS, J.B. MAGGARD SPE 39931. *:_ ~_._, -

.&..=-

0.035

0.010

0.005

O.mo 10 m 30 40 80

,.

~ (::ys )’: 70

m. 7- Square rOOt Of time plot for the reciprocal of rate of the example well.

—. ———

— ——

— —

—— _——

—.

kx=16ky

1111

11112:1

-.

90 Im

1:2

..”.

Fig. 8- Effect of anisotropy from natural fractures, a square with kx = 16 kYtransforms to a 4:1 rectangle.