iSooietv of PetroleumEngineers

SPE 29891

Relative Productivities and Pressure Transient Modeling ofHorizontal Wells with Multiple FracturesR. N. Home, Stanford University,and K. O. Temeng, Saudj Aramoo

SPE Members

Copyright 1995, %}ety of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Middle East Oil Show held in Bahrain, 11-14 March 1995,

This paper waa selected for presentation by sn SPE Program Committee following review of information aontained in an abstract submined by the author(s), Contents of the papar,aa p~aanted, have not been reviewed by the .%ciety of Petro(em Engineers and are aubjecs to correction by the author(s), The material, as presented, does not naceaaarily reflect anyposition of the Society of Petroleum Engineers, iRaofficers, or rnambara. Papara presented at SPE metilngs are subjaot to publication revlaw by Editorial Committees of the Societyof Petroleum Engineers, Permission to copy ia restricted to an abstract of not mere than 300 words, Illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the papar is presented. Write Librarian, SPE, P.O. Box s33S3S, Richardaan, TX 75083-3S26, U. S.A., Telex, 163245 SPEUT,

Abstract

An analytical model has been developed to describe the. . .-A +. . . . ...nn+r..o. ,,1.ah.havinr of ainfiow performance iiUU u ~UalcllU ~,v-su..- “.- . . ..- -. _

horizontal well with multiple hydraulic fractures. Themodel has been used to compare the relative produc-tivities of multiple fractures, the objective being to de-termine the conditions under which multiple fracturesprovide significant improvement over a single fracture.

The approach used was to approximate the series offractures as fully penetrating, uniform flux, verticalfractures in a box-shaped reservoir of closed bound-aries. Interference between the multiple fractures wasaccounted for by the superposition of influence func-tions. The effect of wellbore storage and skin was in-corporated by numerically converting the solution intothe Laplace space.

Introduction

Hydraulic fracturing has been shown to be an effec-tive way of significantly enhancing the performance of

horizontal wells (1). It is especially beneficial in lowpermeability formations, and where low vertical per-meability reduces the effectiveness of horizontal wells.Two types of hydraulic fractures are possible with hor-

.>1–(2). :P AL.- -..;” -$ +L. n . ,all ;0 “nvlzon~a~ wel~ ‘-J j 11 bIIC @O U1 bLIC +=.’ 10 .- ~i%! b tk

minimum horizontal stress direction then a single largefracture is formed along the axis of the well. Fracturestransverse to the wellbore axis will be created when thewell is parallel to the minimum horizontal stress. Morecomplicated fracture geometries will result if the well-bore axis is not normal to either principal horizontalstress directions.

This paper describes an investigation into the preductivity enhancement achieved by creating multipletransverse hydraulic fractures on a horizontal well.The primary objective of the study was to determinewhether multiple fractures provide worthwhile improve-ment over a few or a single fracture. An essential fea-ture of multiple fractures is that the fractures will ul-timately interfere with each other, and this will resultin reduced effectiveness at later times. Thus, more hy-draulic fractures will not necessarily lead to proportion-ately greater productivity. This study addresses thefactors that control the effectiveness of multiple frac-tures, and seeks to provide the analytical tools for pre-

563

-2 Relative Productivities and Pressure Modeling ofllorizontal Wehk with Multiple Fract ures SPE 29891

dieting the performance of multi-fractured horizontalwells. Guo and Evans (3J have developed methods forthe performance prediction of horizontal well with mul-tiple fractures. Their work does not account for theeffects of interference between fractures, and is thusvalid for the cases where interference between fracturesis absent, such as during early time flow in tight gasform.ation~ with few widely-spaced fractures.

A secondary objective of the study was to understandthe pressure transient behavior of multi-fractured hor-izontal wells for the purpose of designing and inter-preting well tests. Guo and Evrms 131,and Larsen andHegre (4J have recently presented models for the hori-—–—*–1.-.11 - . . .. Al. —..14:-1,. +---,. +.,....” Iv’.; c .+1,,4., .rmm–zonLal wens wim UIUIL.IPICII CIUUUIC=. I M= SUUUJ WM-

plements the work of Guo and Evans (3J by accountingfor interference between fractures, and that of Larsenand Hegre (4) by incorporating the effects of wellborestorage and providing additional perspectives.

Mathematical Development

The horizontal well and the series of vertical fracturesthat intersect it are depicted in Figure 1. The hori-,ra..+-l ..,o11l;~o =Ifimrr +hn ,, =v; c amA the fradllrm ~~~fiuuucu WCL1 11-C- cb1wA15 UIsv y -.-, -.1.. .11” .--” ”-. ““

parallel to the plane of the z axis. The fractures areconsidered to fully penetrate the formation, and aretreated as uniform flux plane sources. Although theassumption of uniform flux is an approximation, it isuseful for the purpose of comparing the productivitiesof multiple fractures, and is much easier to formulateand compute.

The analysis treats only the flow into the fracturesthemselves; the horizontal well is not considered tocontribute to the production. This is a reasonable as-sumption since multiple fractures are typically createdin very low permeabilityy formations, and the wells arevery often cased unperforated.

The relative productivity of the completion can be eval-uated by investigating the values of the dimensionlesspressure drop, pD, which by definition is a ratio of thepressure drop to a given flow rate:

2xkh L@——‘D=Bp q

(1)

Dimensionless pressure functions for two different outerboundary conditions were derived, The first was for areservoir of infinite extent, and the second was for ahorizontal we!! located in a boxed-shaped reservoir of---. .. . .. .—.dimensions, x~, ye. The infinite reservoir case is ac-tually a special case of the closed-boundary case, butis derived and presented separately because it is sim-pler to compute, and easier for illustrating the basicconcepts.

Infinite-Acting System

The source function for the multiple fractures can begenerated using Newman’s Product Theorem by inter-secting a slab source aligned with the y axis with aseries of plane sources perpendicular to the y axis. Thesource function for the slab source is (5) :

[S=(x,t) = ~ erf

(Xj+x) + erf(zf - x)

2/57 2/7 1 (2)

while the source function for each of the multiple planesources, centered at y = YWwill be:

sy=J-2@j7 “P[-(Y221‘3)

The total source function for a single fracture would beS. x SV, while for n fractures it would be:

n

S(X, y,t) = s.~sdY!Yw! t); (4)i= 1

where qi is the flow rate from the individual fracture i,

and q is the total flow from all of the fractures. Thepressure drop due to the multiple fractures would thenbe given by:

tAp(z, y,t) = A

/#c ~q(r)s(ic, Y,t – ~)dr (5)

564

SPE 29891 R. N. Horneand K. O. Temeng 3

Writing these expressions in dimensionless terms, thetotal dimensionless pressure drop, pD~ at fracture i interms of the pressure drops, pDij at fracture i due toproduction at fracture j would be:

n

pDi = ~ qDjPDij(zD1 yD, ywDj) (6)j=l

@ ‘“

/‘D’j = 4 ~s=D(ZD, r)syD(vDi, !/wDj, T)dr (7)

SYD=[[

(YD - YwD)2‘eirp -

&D/~ 1

s.D=e’f (’k+’”’ (%)t

tD = -~‘f

(8)

(9)

(10)

(11)

(12)

(13)

(14)

[15)

Since all the fractures are connected through the hori-zontal wellbore (assumed to be of infinite conductivity),the pressure drops, pDi must all be the same (equal topLI). In addition, the sum of the individual flow ratesmust add up to the total flow rate, i.e. qD = 1. Hencewe can write a matrix equation (for an example withthree fractures):

[

PD1l PD12 PD13 -1

1[1[1qD 1 0

PD21 pD22 pD23 –1 qD2 o

PD31 PD32 PD33 ‘1 qD3 = o1110 PD 1

(16)

This matrix expression can be expanded to howevermany fractures are present in the system, and can bemodified to incorporate the effects of boundaries.

Closed Boundary System

The effects of a confining impermeable boundary of sizex~ by ye can be imposed by replacing the source func-tions S= and Sg by the corresponding solutions withinparallel boundaries (source functions number X andVII respectively in Gringarten and Ramey(5J). Thus,the dimensionless pressure drop for each fracture canbe evaluated from:-

/

t*PD = ~qDj SzDSYDdT

o

where the source functions S.D and SYD

the appendix.

Relative Productivity

(17)

are defined in

The dimensionless pressure drop is a function princi-pally of the following parameters: (1) the dimension-less time, tD, (2) the length of the horizontal wellrelative to the fracture length, L/xf, (3) the num-ber of fractures,n and (4) for a closed system, thedimensions of the system, Z,D and yeD.

By comparing values of pD at the same VdU6! of tD fOrdifferent numbers of fractures, the relative productivi-ties can be determined. Due to the definition of pD asa pressure drop for a given flow rate, it is sometimesnecessary to compare the reciprocal of pD rather thanpl) itself.

Figure 2 is a log-log plot of pL) versus tL) for n=l,3,5,7and 9 fractures, for L/rj = 10 and XeD and y@ = 21.

565

4 Relative Productivities and Pressure Modeling of Horizontal Wells with Multiple Fract ures SPE 29891

It is seen that the improvement in performance overa single fracture is limited once the fractures begin tointerfere with each other. lJltimately the performanceof multiple fractures deteriorate to the point where itapproximately equals that of a single fracture.

In Fig. 3 we have plotted the relative productivities(with respect to a single fracture) of multiple fracturesversus time for L/xf = 10 and xeD and y,~ = 50. Therelative productivities, shown on the y-axis, is definedas the ratio of the reciprocal ~D for n fractures to thereciprocal ~D fOr a single fracture, i.e. pDl /pD~. We

,.. , +h~ i .F. GP ifl ~e~efit Ofnotice mat at arly fiXed tilm$?, . . ..- .~.. -Q..ml,]t ip]e fractllres decre=es as the number of fractures.,Au... .- .. . .increases. The greatest advantage in multiple fracturesis obtained at early time, before fracture interferencebegins. In actual practice, what constitutes an optimalnumber of fractures will depend on the time period ofeconomic significance. Because the dimensionless timeis proportional to the permeability, and inversely pro-portional to compressibility, multiple fractures wouldbe expected to be most effective in tight gas reservoirs.

The effect of interference can also be illustrated in Table1 which shows the flow rates in each of three fractures,and the pressure drop at the well, for an infinite-actingsystem. The fractures initially each produce one-thirdof the total flow, but begin to interfere with each otherat a dimensionless time of around u.i. AS time goeson the outside fractures tend to produce a greater flowrate at the expense of the inner one.

Table 2 shows the pressure drop, the relative produc-tivities, and the relative cumulative productivities fordifferent numbers of fractures, for L = Xj and L = 10irfat a dimensionless time of 1000. The relative pro-ductivities and the relative cumulative productivities

.“, .,, . . . *l ---,. -r . “;” 1,3 fr=ct,, reare dehIK?U wm respec L LO LJIU3C U. a -lllg.L ..w.. -.v.

Mathematically, the relative cumulative productivity isdefined by the quantity s p~DdtD / f pnDdtD. Table 2shows that if the fracture length is similar to the hori-zontal well length, then the increased productivity formore than orie 1!~~.ULC,= ,,U,,..,,-..r-,. ,.+.,-a :. rn; nim.1 For ~h~rt fracture

leng$hs (or long wells) the increase in productivity ismore significant. This is because the onset of inter-ference is delayed as the length of the well increasesrelative to the length of the fractures.

Example Application

This example demonstrates an application of the re-sults of this study to a horizontal gas well. We wishto predict the performance of the well for a maximumof 4 fractures to help decide on an optimal number.Reservoir and well information is given below:

k = .015 rnd

9 = 0.075LW = 2500 ftrf = 300ftyg= 0.62T = 290” Fh = 280 ftpi = 8500 psiaXe = 15000 ftye = 15000 ft

The comparison is performed on the dynamic produc-tivity index, PI, defined as:

(18)

Figure 4 is a plot of PI versus time for n = 1,2,3,4f~Mtures, We see that going from 1 to 2 fractures re-

sults in a significant increase in PI over a significanttime period. There is an advantage, albeit reduced, ingoing to 3 fractures, and very small benefit in going to4 fractures. After about 10 years we notice that thereis practically no difference in performance between 3and 4 fractures. Although the ultimate decision wouldbe based on the economics (cost of fractures, gas price,etc. ) it appears that over a 20 year period, 3 fracturesm,ig!lt be Optir-na!

Pressure Transient Analysis

In this section we discuss the pressure transient be-havior of a multi-fractured horizontal well and the im-plications for test design and interpretation. The twoparameters of primary interest are the formation per-meability, k, and the fracture half-length, Zf. We beginthe discussion by describing the flow regimes that are

566

iPE 29891 R. N. Home and K. O. Temeng 5

expected with the system under consideration.

For an infinite-acting system without wellbore ~t~i~~~,

four flow regimes are possible with a multi-fracturedhorizontal well. These are: (1) First Linear, (2) FirstRadial, (3) Second Linear, and (4) Second Radial.These flow regimes are illustrated in the schematic ofFig. 5.

During the first linear period, flow occurs linearly anddirectly from the formation to the individual fractures.I–.11_ ..:___41.a C..”+l~n~s- nc.rin A and if the fractures arerOllO~lIl~ L1lC 111=. M1lQCU P-I...-, ------ ---- ---

short and widely spaced, a first radial flow period willensue during which each fracture will experience sepa-rate pseud~radial flow around it. The requirements of

. . ,, . 1.–- L --------- +km+ ;m+ovfawanmwide spacing ana snore Iengtn CIIWL c LUKJ. .,,.-. ,titti...-

L+..r .- i he f,9~t11rp~ dOeS not mask this early pseudo-U=..veul, ..1- .. . . . . . . ..-----radial period. The third flow regime, the late linearperiod, may occur if boundaries are very distant andthe fractures are relatively short and many. During thelate linear period, the flow lines to the fractures areparallel to each other, and are normal to the axis of thewellbore. Finally a late pseud~radial period will occurduring which, from the point of view of the reservoir aa

. . . 1 ._... -- ●I.A -.l+;l.o .* ‘311-a whole, the now appears radial wwtwds UK cli..,~ .J-..

fracture system The time required for this flow regimeto develop may be too long to be observable in actualfield tests.

Figure 6 is a log-log plot of dimensionless pressure andpressure derivative versus dimensionless time for a wellwith two fractures, L/zf = 10, and no wellbore stor-age. The plot shows a first linear flow indicated by ahalf slope on both the pressure and derivative curves.This is followed by the first radial period where thederivative stabilizes at a value of 1/4. Interference be-tween the fractures begins at a dimensionless time ofabout 10, and the derivative eventually stabilizes at a~,ahdeof I/Z during the second radial flow, without go-

ing through an observable second linear period.

Figure 7 is a log-log plot for 4 fractures, L= = 10, and‘ no wellbore storage. As in the previous case with two

fractures, there is an early linear period clearly indi-cated by a 1/2 slope on both the pressure and derivativecurves. The actual values of pressure and derivativeduring linear flow of the 4-fracture case are exactly 1/2these of the 2-fracture case. In Fig. 7 a short periodof first radial flow is seen to occur at a dimensionless

time of about 1.0. This is indicated by an attempt atstabilization by the derivative curve at a value of 1/8.in generai, during the first radiai period the derivativestabilizes at a value of l/(2n), where n is the number offractures. The second linear period follows the first ra-dial flow, and begins at a dimensionless time of about 1.Late radial flow occurs at iate time with the ckrivatiw..el-:l:-:_” ..* 1 IQStdulllLIIl& ah L/ A.

Specialized analysis procedures may be used for the in-dividual flow periods to infer reservoir permeability andfrac~~dreieng~~ ~Q de~~.ribed in the following subsec-

tions. All equations are presented in field units.

First Linear Period

As discussed previously, this period can be identified onthe iog-iog piot where ‘both the p~~ii~iii~ and derivativecurves both have 1/2 slope at early time.

During the linear flow period, a plot of bottomhole pres-sure versus the square root of time will yield a straightline of slope r-nfl. This may be used to estimate k~~ bymeans of the following equation:

(19)

First Radial Period

This period is recognized by the first stabilization onthe derivative plot after the first linear flow period.

A plot of pressure versus log of time of data from thisperiod will give a straight line of slope, mrl, from whichthe reservoir permeability can be computed as:

a=162,6qBp

(20)Tbrl??h

Second Linear Period

From the slope, mn, of a Cartesian plot of pressureversus square root of time of data from this period weobtain,

(21)

567

-6 Reiative Productivities and Pressure Modeling of Horizontal Wells with Multiple Fract ures SPE 29891

where L is the distance between the two extreme frac-tures.

Second Radial Period

With data from this period we plot pressure versus logof time to get a straight line of slope, mrz. Reservoirpermeability is obtained from the equation:

162.6qpBm= m,,~ (22)

We notice from Equations 19 to 22 that to obtain rj,k=, and kv we need data from the first and second linearperiods, and one of the two radial periods, If we assumethat the formation is isotropic (i.e. k. =kv = k), thenwe would require data from the first linear and at leastone of the other periods to compute both k and ~j.

Wellbore Storage

Figures 8, 9, and 10 are log-log plots of pressure andderivative for L/z j = 10, 4 fractures, and dimensionlesswellbore storage constants of 0.1, 0.5, and 2.0, respec-tively. The dimensionless iVeiiiXXe Siofzige COiWiiil$,

P. A-fined ~~~~ ~~~nect, to r # M foiiows:uf”~,” , is u.-!....-= ~.-- .- -,

We notice that even in the case of the unrealisticallylow CD=j of 0.1 (Figure 8), the first linear period isbarely noticeable. For higher values of wellbore stor-age, such as in Figures 9 and 10, the first linear periodis completely absent. This means that it would not bepossible to obtain values of Zj by means of the special-ized methods discussed previously, although nonlinearparameter estimation techniques may still be able toobtain an estimate.

T.. .11 +1. a +h aa e. a. ah W“ iII 17iu11rec A Q and ~~, only111 all ~llc .Ilrcc ~~vo ~llo.. .. ..l . .5---- -, -, -..the late radial period is evident. For low permeabilityreservoirs it may require an unrealistically long time forthe late radial to occur.

For test interpretation type-curve matching or nonlin-ear regression may offer the most practical way of ob-

taining the desired parameters.

Test Design

The preceding discussion suggests that eliminating orreducing the influence of wellbore storage to a minimumis essential to obtaining interpretable pressure transientdata. Unfortunately the horizontal section of the well-bore, and the fractures themselves contribute such asignificant fraction of the storage that standard procedures such as downhole shut-in during buildup canprovide only minimal relief.

All of this means that a conventional testing of a multi-fractured horizontal well may only provide limited in-formation, and that special testing methods may berequired to obtain interpretable data. The followingare some suggest ions:

1.) For buildup tests, it is essential to have down-holeshut-in.

2.) Obtain an independent estimate of permeabilityfrom a pre-frac test or preferably from an offset verticalwell. This would reduce the need to have data beyondthe first flow regime.

3.) Test the first fracture before adding additionalfractures, and assume that fracture properties will bethe same.

4.) It is necessary to plan for long tests in all cases.

5.) Ideally one should obtain downhole rate and pres-sure measurements across all the fractures.

Conclusions

An analytical model has been developed to describe theperformance and pressure behavior of horizontal wellswith multiple hydraulic fractures. The model assumesfuily penetrating, uniform flux fractures. The produc-tivity enhancement achieved by creating more than onefracture was investigated in the study. The model wasalso used to provide methods and ideas for designingand interpreting well tests,

568

SPE 29891

The study has shown that

multiple fractures along a

R. N. Home and K. O. Temeng

the effectiveness of creatinghorizontal well depends on

the relative lengths of the well and the fractures, as weiias the time at which the comparison is economicallysignificant. We observed that if the fracture length issimilar to the length of the horizontal well, then theincrease in productivity for more than one fracture issmall. For short fracture lengths (or long wells) theincrease in productivity is more significant. For smallvalues of dimensionless time, there is some advantagein using multiple fractures, even for fractures that aresimilar in length to the horizontal weli. Eiecause ofthe influence of interference, the time for which theproductivity enhancement is maintained is increased forlonger horizontal wells.

Four basic. flow regimes have been identified, and equa-tions have been provided for specialized analysis ofpressure transient data from each flow period. Becauseof the nature of the well-fracture configuration, and theinfluence of wellbore storage many of the flow regimesmay not be observable in actual tests. Special testingprocedures, and type-curve matching or nonlinear re-gression will be needed in most cases to obtain reliableresults from well tests.

Nomenclature

formation volume factorcompressibilitywellbore storage coefficientformation thicknesspermeabilityhorizontal well lengthnumber of fracturespressureinitial pressurewell pressureproductivity indexflow ratesource functiontimehalf-length of fracturediffusivityviscosityporosity

Subscripts

D=” dimensionlessXf = with respect to fracture half-lengtht = total

References

1. Yost, A. B. and Overby, W. K.: “Productionand Stimulation Analysis of Muitipie Hydrauiic Frac-turing of a 2000 ft Horizontal Well”, paper SPE 19090,presented at the SPE Gas Technology Symposium, Dal-las, TX, June 1989.

2. Soliman, M. Y., Hunt, .7. L. and El Rabaa,A. M.: “Fracturing Aspects of Horizontal Wells”, JPT

(Aug. 1990) 966-973.

3. Guo, G. and Evans, R. D.: “Pressure TransientBehavior and Inflow Performance of Horizontal WellsIntersecting Discrete Fractures”, paper SPE 26446, pre-sented at the 68th Annual Technical Conference andExhibition of SPE, Houston, TX, Oct. 1993.

4. Larsen, L. and Hegre, T. M.: “ Pressure Tran-sient Analysis of Multifractured Horizontal Wells”, pa-per SPE 28398, presented at the 69th Annual TechnicalConference and Exhibition of SPE, New Orleans, LA,Sep. 1994.

5. Gringarten, A. C. and Ramey, H. J., Jr.:“The Use of Source and Green’s Functions in SolvingUnsteady-Flow Problems in Reservoirs”, SPEJ (Ott.1973), 285-296.

Appendix

The dimensionless source functions for the closed sys-tem case is,

at eariy time (tD < .&j:

m

&D=; ~ [( 1 + (XD – XWD – 2mxeD) +erf

mm=-cc )

569

Relative Productivities and Pressure Modeling of Horizontal Wells with Multiple Fract uresSPE 29891

8

(1 – (XD + XWD – 2~~eD)t r j

= )1(24)

SYD= 1 E [=’(‘(YD - YwD - 2~yeD)2’ +

@z7L=-m ~4tD/ti )

(

‘(VD + YwD – ~~YeD)2exp )1 (25)

AtD/cl

and at late time (tD > ~~D):

s,D=+[,+2~,’.p(-m2fj/a)xmTywD m~yD

Cos — eos —YeD YeD 1

.,A

(26)

(27)

In these equation, tD and a are as defined in Equations10 and 13, and reD and yeD are both relative to rj.

570

tn ql qz q, Pnl.ooxlo~ 0.3333 03333 03333 9.35X1O-5

1.05X1o-5 03333 0.3333 0.3333 4 27x10-3

1 I7X104 0.3333 0.3333 0.3333 L80K10_2

1.O5X1O-3

1.O4X1OJ

0.102

1004

1088

10729599

0.33330.33340.3588041920.4338043530.4355

0.33330.33330.282401617013240.12940.1291

0.33330.33340.35880.4192~ 4338

0.435304355

5.73x10_20.38370.72352.808, -“A0 UU

9.40612.61

Table 1: Flow rates and pressure drops for 3 tlactures (L=x]

L=%{n PD “% /JP!A

jPd%3 4.311 1.134 1.162

I 4 4.308 1135 1164 I

L=IOX{n PD

“%. ‘P’tfkdb/

3 2932i,tii , 0,loi .0”.

4 2892 1691 1.896

6 2.828 1.729 1.971

Table 2 Comparative productivities of different numbers of flactures

z

A7illY

‘Ta--VFigure 1: Schematic of multiple fractures along a horizontal well

mo

0.1

1 10 100 lmo 1mo

to

Figure 2 Multiple fracture solution for closed rectsn~e J!/x~ 10 and xeD and yeD=2]

571

7

6. ‘, *,I

\

I ti

\

161116212631 36

td

I@rC 3: Multiple fracture sohmon for closed rectmde LxrfO ad XtD ~dY,D=50

prcaucivitv Index vs. R-e0.4

1 Frac

0.3 L,_ _ ~ ,crcc~

. 3 Fracs

\ -.– 4 Fracs

“\0.2 -.,

\

\~.0.1 - ~ - ------____ .-. —.- .. _,___---- ____ __— _-’___._-.___

~,. I .~OZ46Bl O12 14

20

Years

Figure 4: Compa.rkan of dynamic productiti~ index for the example gas well

–’+i–l”– Second Radisl

/ +-

FIS. s setumatle at Flow Rwglm-

572

I-J.’ -1010.000 I\

~ooe..o*00000

2 Fractures ~ooo.. ..s.0

1.000 -~o..

~o.o#..

., . . ...”. . . . . . . .

*.*~o

. . . .

.0”” . . . . . . . . . . . . . . . ...8 . . . .

..” . . .0.100 - #0 ●.

. .

# *O””*..*‘.O ●. .

●.*. .

0.0100.001 0.010 0.100 1.000 10.000 100.000 1Oocl.000

toxf

F@Ire 6: pressure and derivative for 2 fractures (No wellbo~ storage)

LJxf = 10

10.000

I

4 Fractures ..00. ...@”’.

~ooe&

1.000 - #o.# ●. . . . . . . . . . . . ..

# . . . . .

~o ~e.. . ..”..0 . ..”

*OO . . .

0.100 -& . . . . . .

.00”. .””””**O . . .

.00 ●.. .

.+”” .“‘.O /.

0.010 “00.001 0.010 0.100 1.000 10.000 100.0001owJ.oofJ

‘Dxf

F@e 7: pW.SUm and derivative for 4 tlactures Wo Weilbre st(orage)

OJJO, ~~O.DO1 0.010 0.100 1.000 10.000 100.000 1000.000

toxli

Figure 8: Pressure and derivative for 4 fractures, CDxf = 0.1

0.00, ~~

0.001 0.010 0.100 1.Oc)o 10.000 100.0001000.000toxf

Figure 9 Pressure and derivative for 4 hsctures, CDxf = 0.5

-0cu

10.000~ , ,

L

1.0001nlnnu. Ivw 1

--4- IU.u 1u

[

[ Lw/xf = 10

4 Fracs

c. .= 2.0

. . . .nn0.0011 ‘*” ““’”””’,

0.001 0.010 0.100 1.000 10.000 100.000 1 Uuu.uuu

‘Dxf

Figure 10: Pressure and derivative for 4 fractures, C~Xf = “2.0

574