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PRESSURE BEHAVIOR OF WELLS INTERCEPTING FRACTURES

R. RaghavanThe University of Tulsa

Tulsa, Oklahoma

Abstract

The pressure behavior of wells interceptingfractures is of considerable interest to thepetroleum industry due to the large number ofwells that have been hydraulically fractured .toimprove well productivity. Hydraulic fracturing isrecognized as one of the'major developments inpetroleum production technology within the last30 years. As a result of extensive research toresolve differences between field results andexpectations based on analytical studies, a con­siderable body of knowledge on the performance offractured wells has been accumulated. This paperis a bEief survey of the current level of under­standing of this aspect of pressure transientanalysis by petroleum engineers.

The topics considered here include the fol­lowing: (i) the effect of vertical, horizontal,and inclined fractures on pressure behavior at thewell, (ii) the influence of fracture flow capacityon pressure vs. time data, (iii) the effect ofwellbore storage and damage on pressure response,(iv) the influence of closed (depletion or zerorecharge) or constant pressure (complete recharge)boundaries. Both flowing and shut-in pressurebehaviors are discussed.

This survey also indicates some of theproblems that should be solved to improve ourunderstanding of fractured well behavior.

Introduction

Virtually every commercial oil and gas wellhas been stimulated either at the start of pro­duction or during its productive life. The mainobjective of well stimulation is to bring pro­ductive capacity to commercial levels. Initiallystimulation treatments consisted of acidizingwells being produced from limestone reservoirs.The first acid treatment job was performed onFebruary 11, 1932. By 1934 acidizing had becomean accepted practice for stimulating wells pro­ducing from intervals containing substantialamounts of acid-soluble components in the reser­voir rock matter. 1 The acidizing process usuallyconsists of injecting hydrochloric acid (normally15 percent by weight) along with surface activeagents and inhibitors (to protect casing and otherequipment) •

It was soon realized that pressure parting orformation lifting also played an important partin the ease with which the acidization is per­formed. 2 For example, at pressures below thoserequired to lift the overburden, very littlefluid could be injected; however, when the

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pressure became high enough to part or fracturethe formation, the injection rate could be raisedsignificantly with little or no additional increasein pressure. Similar observations were made byDickey and Anderson 3 and Yuster and Ca1houn4 intheir studies of injection rates in water injectionwells. These authors concluded that formationscould be parted by excessive injection pressures.Similar observations were reported by other inves­tigators studying the use of squeeze cementing. 5- 7

This process normally involves the injection of aslurry into a porous formation.

The realization that formations could bebroken down or fractured during acidizing, squeezecementing, and water injection operations, servedas a precursor to hydraulic fracturing. }~draulic

fracturing was introduced to the petroleum indus­try in the Hugoton gas field in western Kansas.This method of increasing well productivity wasconceived and patented by Farris of the PanAmerican Petroleum Corporation,8 and has beendefined as "the process of creating a fracture orfracture system in a porous medium by injectinga fluid under pressure through a wellbore in orderto overcome native stresses and to cause materialfailure of the porous medium. "9

The fluid used in hydraulic fracturingdepends on the physical and chemical nature of thereservoir fluids and rock. Generally a proppant(Ottawa sand, glass beads, nutshells, or plasticparticles) is also injected along with the fluidsince hydraulically formed fractures tend to heal,that is, they lose their fluid carrying capacityafter the parting pressure is released. lO

Over the past twenty-eight years hydraulicfracturing has served as an inexpensive way ofincreasing the productive or injection capacityof wells. The success of many marginal wells andnear-depleted fields can be directly attributed tothis procedure. It is estimated that over500,000 wells have been hydraulically fractured.The method has been used to accomplish four tasks:(i) to overcome wellbore damage, (ii) to improvewell productivity by creating highly conductivepaths to the wellbore, (iii) to aid in fluidinjection operations and (iv) to assist in thedisposal of brines and industrial waste material.

The principal objective of this paper is toprovide a summary of the state of the art on thepressure analysis of wells intercepted by frac­tures. It shall examine wells that are inten­tionally fractured as well as those that intersectnatural fractures. This paper is restricted onlyto the examination of a single fracture existingin a uniform, homogeneous porous formation; thatiS,naturally fractured reservoirs consisting of a

system of interconnected cracks or failure surfacescoupled to a matrix of different porosity and per­meability in a random fashion are not examined.

This critique assumes that the reader is fami­liar with some of the developments in the petro­leum engineering literature. Three developmentswhich would be extremely useful for the reader tounderstand in following the subject matter of thispaper are: (i) the concept of wellbore storage,(ii) the infinitesimally thin skin concept, and(iii) the pressure behavior of an unfractured well(plane radial flow) producing at a constant rateand located at the center of a square drainageregion with the outer boundary closed (depletionor zero recharge) or at constant pressure (fullrecharge). Details regarding all of the aboveaspects are discussed in three monographs. Two ofthese have been published by the Society of Pet­roleum Engineersll ,12 and another by the AmericanGas Association. 13

Prior to considering various aspects offractured wells I shall enumerate the purposes ofpressure transient testing and also provide abrief historical sketch of pressure transientanalysis. Only those papers which have a directbearing to this review are mentioned. This sketchis intended to provide those in the audience notfamiliar with the petroleum engineering-literaturesome idea of the parallel developments that tookplace in the ground water hydrology and petroleumengineering literature pertaining to pressuretransient behavior in the 1940's and 1950's.

Objectives of Pressure Transient Analysis

It is well established in the petroleum indus­try that p~assure transient analysis is the mostpowerful tool existing to enable an engineer todetermine the characteristics of a given reservoir,and then prepare a long-range forecast of produc­tion performance.

Questions which a petroleum engineer normallyencounters include the following: (i) Is the lowproductivity of a well due to 'low formation flowcapacity, to a low driving force for moving fluidto the wellbore or to well damage? (ii) Is itlikely to be worthwhile to perform a stimulationtreatment? and (iii) was a stimulation treatment,which was conducted to eliminate formation damage,successful? Answers to the above questions canenable an engineer to make decisions regardingoperating practices and/or stimulation programs.Pressure transient tests can be used to providethese answers.

Today pressure transient tests are used forthe following purposes: (i) A quantitativeestimate of the formation flow capacity (permea­bility - thickness product) of the volume drainedby a well, (ii) quantitative information on theshape and size of the drainage volume, (iii) anestimate of the mean or average reservoir pressure(this is necessary for material balance calcula­tions), (iv) determination of reservoir hetero­geneity, and (v) diagnosis of the well condition(whether the region near the sandface has beendamaged or plugged, or whether it has been

stimulated). It is not unusual to conduct a testfor the sole purpose of determining the well con­dition.

Pressure Transient Analysis:

A Brief Historical Review

One of the earliest measurem~nts of bottom­hole pressures was for the estimate of the averageor "static" reservoir pressure. This measurementis useful in material balance calculations toestimate the quantity of oil and/or gas in thereservoir. To obtain it a producing well wasusually shut in for a period of 24 to 72 hours.The measured pressure after this period wasassumed to be the static pressure. However, itwas soon realized that estimates of static pres­sure were dependent on the time for which a wellhad been shut in and that the lower the permeabi­lity, the longer the time required for the well tobe shut in. This immediately led to the importantrealization that the formation permeability can bedetermined from a well test. To my best knowledgethe first determination of formation permeabilityvia this T~thod was presented by Moore, Schilthuisand Hurst in 1933.

15In 1935, Theis presented a classic studyon pump testing of water wells and discussed theanalysis of pressure recovery data. Pressurerecovery data are known as pressure build-up datain the petroleum engineering literature. The formof graphing and analysis suggested by Theisremains one of the basic techniques used in petro­leum engineering today. In 1951, Horner16 summar­izing the important contributions of the researchpersonnel of the Shell companies, presented thesame method of analysis. (It should be mentionedthat Horner and co-workers arrived at theirapproach independently.) At approximately thesame time, Miller, Dyes and Hutchinson (MDH)17presented an alternate method of analysis.Although the methods of analysis of the Horner andMDH procedures were different, both methodsreported that the formation permeability can bedetermined from wellbore pressures. The relation­ship between these two methgds was shown onlyrecently by Ramey and Cobb. I8

In 1937, Muskat19 presented a method fordetermining ultimate static pressure from pressuretransient data. This method is especially usefulin situations where early time data are unavail~

able or are dominated by wellbore storage effects.

Many studies appeared during the 1940's. Ofnote in the ground water literature were the worksof Wenze120 , Cooper and Jacob 2l , and Jacob. 22Jacob 23 was also the first to recommend semi-loggraphical analys~s for pressure draw-down data.In 1946, Elkins 2 presented graphs for analysisof interference test data. This information formsthe basis for analyzing many of the interferencetests conducted today. In,1949, vanEverdingen and Hurst 25 presented a study on theapplication of Lap lace Transforms to transient flowproblems. Much of the work that has followed inthe petroleum engineering literature is a direct

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result of the van Everdingen-Hurst study.

All of the studies mentioned above concernpressure behavior when the well is flowing at aconstant rate. Though studies of wells producingat a constant wellbore pressure have been examined(Hurst26 , Jacob and Lohman27 , van Everdingen andHurst 25 ) these solutions have not been used inwell test analysis,probably due to the fact thatit is not readily apparent how this condi-tion would affect the important case of pressurebuild-up after the well had been shut-in at thesandface.

As mentioned earlier the results of twostudies by Horner16 and Miller, Dyes and Hutchin­son17 were presented in the early 1950's. Thesetwo methods formed the backbone of pressure tran­sient analysis in the petroleum industry. As aresult of the success of the above studies indescribing pressure behavior of wells, a tremendouswealth of information pertaining to pressurebehavior under a variety of conditions has beenaccumulated. These included studies on the effectof damage and stimulation (van Everdingen28 andHurst29)~ on the effect of partial penetration(Hantush JO , Nisle3l , Burns 32 , and Prats 33)*, onwellbore storage phenomena (Agarwal, Al-Hussainyand Ramey 34 , Ramey 35 , Ramey and Agarwa136 , Ramey,Agarwal and Martin37 , and Cooper, Bredehoeft andPapadopulos 38) and on the effect of heterogen­eities. ll ,12 Most of these studies are summarizedin Refs. 11, 12, 13, and 39.

Flowing Pressure Behavior of Fractured Wells

With the advent of hydraulic fracturing as astimulation technique in low permeability reser­voirs, it soon became obvious that standard radialflow solutions considered by earlier works wereinadequate for the analysis of fractured wells.The main problem was that a fracture is a plane ofhigh conductivity extending into the formation forsome distance and the radial flow idealization didnot include this aspect.

Before proceeding further, a discussion ofthe azimuthul orientation of fractures intercept­ing wellbores is warranted. Howard and Fast statethat the azimuthul orientation depends on whetherthe porous medium acts as an elastic, brittle,ductile or plastic material. 40 According toHubbert and Willis 4l , the general state of stressunderground is one in which the three principalstresses are unequal and the plane of the hydrau­lic fracture would be perpendicular to the axis ofthe least stress. In tectonically relaxed areasthe least stress is horizontal. Thus verticalfractures would result. But if orogenic forcesare active, the direction of the least stresscould be vertical (in this case it would equal theeffective overburden stress) and could result inhorizontal fractures. This then implies that, atleast theoretically, the injection pressure dur­ing hydraulic fracturing must be equal to orgreater than the effective overburden pressure for

*This list only includes those of directconsequence to this paper.

a horizontal fracture to result.

Today it is generally believed that hydraulicfracturing normally results in a single verticalfracture, the plane of which includes the well­bore. 42 But it is also agreed that if formationsare shallow then horizontal fractures can result.The specific orientation of a fracture with res­pect to the vertical axis may be unidentifiableif it is a naturally occurring fracture. However,vertical fractures are by far the most common.Thus, most of the attention in the literaturehas been directed towards vertical fractures.

Much of the early work on fractured wellsconcerned the study of steady state behaviorusing potentiometric or analytical models(Muskat 43 , Howard and Fast 44 , McGuire andSikora45 Prats46 , van Poollen, Tinsley andSaunders 47 , Craft, Holden and Graves 48 , Dyes, Kempand Caudle49 , Tinsley, Williams, Tiner andMalone50 ). Most of these papers were primarilyinterested in the productivity increase thatwould result due to a fracture treatment. Unfor­tunately, as shown in Fig. 1 the results are notin agreement. 5l Here J act represents the produc-

tivity of the well following a fracture treatment,J theo is the productivity prior to fracture

treatment, kf is the fracture permeability in md.,and w is the fracture width in feet.

As far as pressure transient analysis isconcerned Dyes, Kemp and Caudle49 were the firstto investigate the effect of a vertical fractureon the straight line that results on a Horner orMiller-Dyes-Hutchinson graph. In the limitednumber of cases they examined, they concluded thatfractures which extend over 15 percent of thedrainage radius alter the position and slope ofthe straight line on the pressure build-up curve.Others also studied the production response andpressure behavior of a closed cylindrical reser­voir producing an incompressible46 or a slightlycompressible52 fluid through a single, verticalfracture located at the center of the cylinder.They found that the production rate declineincreases as the fracture length increases. Thus,they suggested that lateral extent of the frac­ture can be determined from a comparison of theproduction rate declines before and after frac­turing, or it can be determined from the ratedecline if the fluid and formation properties areknown. Prats 46 also found that if the ratio ofreservoir radius to fracture radius was greaterthan two, then the production behavior of such afractured system can be represented by an equi­valent radial-flow system having an effectivewell radius equal to one-fourth of the totalfracture length. (Muskat43 had arrived at asimilar conclusion earlier when he examined afractured well in an infinite reservoir.) In thepetroleum engineering literature this observationis known as the "effective wellbore radius con­cept." Scott53 developed curves of wellborepressure versus time for a fractured well in aclosed circular reservoir using this concept.

Russell and Truitt54 , in a comprehensivetreatment of the subject, studied the pressurebehavior of infinite-conductivity fractured wellsin a square reservoir using a finite difference

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The Infinite-Conductivity Vertical Fracture in

a Closed Square Drainage Region

Gringarten, et al. 55 , have presented draw­down data for an infinite-conductivity verticallyfractured well located at the center of a closedsquare drainage region and producing a slightlycompressible constant viscosity fluid at a con­stant rate (see Fig. 2). The solution for theproducing pressure in psi, PWf' at time, t,expressed in hours is

model. An infinite-conductivity fracture impliesthat there is no pressure drop along the fractureplane at any instant in time. They considered ahomogeneous, isotropic reservoir in the form of aclosed square completely filled with a slightlycompressible liquid of constant viscosity. Pres­sure gradients were assumed to be small everywhe£eand gravity effects were neglected. The plane ofthe fracture was located symmetrically within thereservoir and parallel to one of the sides of thesquare boundary (Fig. 2). The fracture extendedthroughout the vertical extent of the formationand production at a constant rate was assumed tocome only through the fracture. Russell andTruitt computed the pressure at the wellbore as afunction of time and fracture penetration ratio.Here the term fracture penetration ratio will bedefined as the ratio xe/xf and will be used con-sistently in all of the foll~ing. They demon­strated the effect of fracture length on the draw­down and build-up behavior of a vertically frac­tured well for a wide variety of conditions.

In 1974, Gringarten, Ramey and Raghavan55

found it necessary to re-examine the solutionspresented by Russell and Truitt as the Russell­Truitt study was not intended for short timeanalysis. They examined the problem analyticallyby the use of Green's functions 56 and the Newman57product method. which had been discussed earlier byGringarten and Ramey.58 Gringarten,~ al., werealso the first to present a complete and compre­hensive view of the pressure behavior of aninfinite-conductivity vertical fracture. Thework of Gringarten, et al., will serve as astarting point for our discussion.

kh )141.2 qB~ (Pi - Pwf (1)

Here, p D (tD ,xe/xf) represents the dimension-w xf

less wellbore pressure drop for a particularfracture penetration ratio, x Ixf , and t D ise x

fdimensionless time. The formation permeabilityis denoted by kin md., the thickness by h in feetthe porosity by <1>, the system compressibility by ,ct in psi-l and the initial pressure by Pi' Theflow rate is ~measured in Stock Tank Barrels/Day,the formation volume factor is B, which is Reser­voir Barrels per Stock Tank Barrel, and the fluidviscosity in cp is~. The distance from the cen­ter of the well to the external boundary is xefeet, and the fracture length end-to-end is 2xf •

All equations here are expressed in oil fieldengineering units. Figures 3 and 4 are graphs ofPwD vs. t

Dxffor the system under examination on

log-log and semi-log coordinates. The fracturepenetration ratio, xe/xf, is the parameter ofinterest.

Prior to considering pressure behavior in thebounded system, let us for the present examine avertically fractured well in an infinite reservoi~

This corresponds to the xe/xf = 00 line on Figs. 3

and 4. Note that we are examining a fracturedwell in an infinite reservoir and not an unfrac­tured well in an infinite reservoir. For aninfinite-conductivity fractured well in an infi­nite reservoir Gringarten, et al., have shown thatthe wellbore pressure drop I; given by the follow­ing expression:

PWD(tDx ) = 1:~ (erf 0.134 + erfO.866 )f

2 DXf

r-- r--vtD vtDxx f f

_ 0.067 Ei (- 0.018) - 0.433 Ei (_ 0.750) (3)

~xf t Dxf

where erf (x) is the error function of x, and-Ei (-x) is the exponential integral. 59 At largevalues of time (tDx ~ 3) it can be shown that

fEq. (3) can be written as

where In tDx + 1. 100f

(4)

0.000264kt

<l>Ct~X~

*In ground water hydrology PwD = 2TITs/(q~)where s is the head draw-down and T is the trans-missivity; dimensionless time t Dx Tt/(~Sx~)

fwhere S is the storage coefficient.

where In refers to natural logarithms. The abovetime limit of tDx = 3 was obtained empirically by

fexamining the xe/xf = 00 line on Fig. 4 and deter-

mining the time at which a straight line with aslope of 1.l5l/log", starts. This may be done by

10placing a triangle with the proper slope on Fig. 4and checking the xe/xf = 00 curve for the start of

the straight line with the proper slope.

For small values of time (tDx ~ 0.016)f

-120-

(5)

Equation (5) contains two unknowns, k andxf • But if k can be determined then xf can becalculated from a Cartesian ~h of~p = (Pi - Pwf) or Pwf vs. Itime, since Eq. (5)indicates that

(10)

This early time period is generally referred to asthe linear flow period. As shown in Fig. 3 onlog-log coordinates this period is characterizedby a straight line of slope of 0.5. The reasonfor this may be seen if the logarithm of eachside of Eq. 5 is considered. Taking theselogarithms we obtain

~p a: It

The relevant formula is

log [PWD (tDx )]f

log lIT + 1 log t2 DX

f(6) (11)

Then the reason for the "half slope line" isclear. Here the abbreviation "log" refers to"logarithm to the base 10." The time limit oft Dx = 0.016 was also obtained empirically. The

flog-log graph was used for this purpose and theend of the linear flow period was determined byplacing a triangle with the correct slope onFig. 3 and then searching for the end of the linearflow period.

From a practical viewpoint Eq. (4) impliesthat if data are obtained for a long enough per­iod, then the permeability-thickness product maybe calculated from the slope of the draw-downcurve. The equation to be used would be the wellknown radial flow formula:

where m is the slope of the semi-log straight lineand m' is the slope of the straight line onCartesian coordinates in psi/Ihr. The aboverepresents the approach presented by Russell andTruitt 54 and Clark. 60

Gringarten, et al., also proposed that log­log type curve matching (actually Ramey35suggested this approach in 1970) be used tocalculate permeability and fracLure length. Thebasis for the type curve matching procedure iswell-known. It will be repeated here only forcontinuity. Taking the logarithm of both sidesof Eqs. (1) and (2\ respectivel~we have:

kh l62.6qB)lm

(7)

log kh141. 2 qB)l

Once the semi-log straight line has beenidentified and Eq. (7) has been used to estimateformation permeability the skin factor, s, can bedetermined from the expression12

where m is the slope of the straight line portionof the draw-down curve in psi/logo on semi-logpaper. It is important to note, however, that thestart of the semi-log straight line cannot bedetermined a priori. This can be a problem inanalyzing pressure data by this approach.

(12)

(13)

+ log (p -p f)i w

log 0.000264 k + log t

<PCt)lX~log t DX

f

If actual draw-down data are plotted as the log­arithm of the absolute difference between initialpressure at the start of the test and pressureafter the rate change versus the logarithm oftime, the actual field data should be similar to alog-log graph of PwD vs. t Dx • The difference

between the two graphs is onty a linear transla­tion of both coordinates, represented by the firstterms on the right-hand side of Eqs. (12) and(13). If a proper match of the field curve withthe dimensionless curve is obtained then kh/)l canbe determined from the vertical displacement ofthe horizontal axes and k/(<PCt)lX1) from the hori­zontal displacement of the vertical axes. Thusthe permeability-thickness product and fracturehalf-length can be determined. The advantage ofthe type curve matching procedure is that theentire data obtained during a test can be used.

(8)

(9)

m

s = _-"k:.:;:h=--_l41.2qB)l

s = 1.151 [ _

In Eq. (8), Plhr is the pressure on the correct

semi-log straight line at one hour or on theextrapolation of the correct semi-log straightline to one hour12 and rw is the wellbore radius infeet. Note that the skin factor, s, is related tothe skin pressure drop, ~Pskin' by the expression:

-121-

In addition it can be shown that the duration oftesting can be greatly reduced if this procedure isfollowed.

At long times, t ~ 2, Eq. (14) maybe writtenDXf

as

The pressure response shown in Figs. 3 and 5. may also be displayed on a different type curve

as PwD vs. tDA where tDA is the dimensionless time

based on the drainage area, A, and is given by

On the basis of Eqs. (5) and (15), we can saythat Eqs. (7) and (11) can be used to calculateformation permeability and fracture length,respectively provided that the test is run for along enough period,

This expression is similar to that for an unfrac­tured well in an infinite reservoir. For smalltimes, t D ~ 0.16, Eq. (5) applies. The

xf

time limits mentioned above were obtained alongthe same lines as for the infinite-conductivitycase.

(15)

(16)

1 (In t + 2.80907)2 DXf

0..000264kt~ct\lA

P D (tD

)w xf

Figure 7 displays the same information as thatshown in Fig. 3 as PwD vs. t DA• It is convenient

for long time analysis when bends due to the outerboundary become evident. As shown in Fig. 7, thetime for the start of pseudo-steady state isapproximately t DA of 0.12 for all xe/xf • Thus the

vertically-fractured well reaches pseudo-steadystate in about the same time as an unfracturedwell,in a closed square. 6l Figure 5 may also be

Figures (5) and (6) are log-log and semi-loggraphs, respectively, for the pressure behavior ata fractured well for the uniform-flux case.Again three different flow periods can be charac­terized. A linear flow period occurs at earlytimes. This corresponds to a straight line with aslope of one-half on log-log coordinates (Fig. 5).After a period of transition, there is a pseudo­radial flow period corresponding to the semi-logstraight line (Fig. 6). After a second periodof transition, pseudo-steady state flow occurs,which is characterized by an approximate unit slopestraight line on log-log coordinates. This flowperiod results because fluid is produced at aconstant rate from a closed reservoir. Dependingupon xe/xf' one or more of these flow periods maybe missing: in the total fracture penetration case(xe/xf = 1), for instance, the first transitionperiod and the pseudo-radial period do not appear,whereas only the pseudo-radial period is missingfor values of xe/xf.between 1 and 3. Figures 3and 5 can be used for type curve matching toobtain estimates of formation permeability frac­ture length, and distance to a drainage limit. 55

(14)

During the course of their investigation oninfinite-conductivity vertical fractures, Grin­garten, et al., also arrived at another solutioncalled the "uniform-flux" solution. This solutiongave the appearance of a high, but not infinite,conductivity fracture. Thus unlike the infinite­conductivity case the pressure varies along thefracture length at any given instant in time.Application of these solutions to field dataindicates that the uniform-flux solution matcheswells intersecting naturally occurring fracturesbetter than the infinite-conductivity solution.On the other hand the infinite-conductivity solu­tion matches the pressure behavior of hydraulicallyfractured weels better than does the uniform-fluxsolution. More recent experience has indicatedthat the uniform-flux solution matches injectionwell pressure data, and wells that are acid­fractured much better than the infinite­conductivity solution. The exact nature of thesesolutions will be discussed in the section onfinite-capacity fractures.

As already pointed out, the uniform-fluxsolution is useful in analyzing data obtained fromwells intersecting natural fractures. For a wellin an infinite reservoir it can be shown that thepressure drop at the wellbore is given by:

Square Drainage Region

The Uniform-Flux Vertical Fracture in a Closed

( _1_)_ 1:. Ei(- _1_ )2~ 2 4tDDx xf

f

All of the above discussion pertains to afractured well in an infinite reservoir. Let usnow consider the effect of outer boundaries.Returning to Fig. 3 we note that at early timesthe solutions for the bounded case are identicalto that for a fractured well in an infinite reser­voir. They possess an initial period controlledby linear flow to or from the vertical fracturesurface. During this period, pressure is a func­tion of the square root of time. On l@g-logcoordinates the pressure behavior during thisperiod is characterized by a straight line ofslope of 0.5. Following the linear flow period apseudo-radial flow period (slope = 1.15l/logcycle on a semi-log graph) exists for fracturepenetration ratios, xe/Xf' greater than 5. Thepseudo-radial flow period begins at tDx = 3. This

fcan be demonstrated by plotting the graph ofPwD vs. tDx on semi-log coordinates (see Fig. 4).

fFinally, all solutions reach pseudo-steady statebecause fluid is produced at a constant rate froma closed system}2 The advantages of identifyingthe various flow regimes are discussed laterunder "Some Practical Considerations." The timelimits mentioned above can also be used in thedesign of field tests.

-122-

udisplayed in a similar manner.

61In 1968, Earlougher, et al., showed that

for an unfractured well in-a closed square thetime for onset of pseudo-steady state is greaterfor the well point than for any other point in thesystem. Thus pseudo-steady state flow at the wellguarantees that all points in the drainage areaare at pseudo-steady state. The same is also truefor a vertically fractured well in a closed squaredrainage region. 62

Comparison of Infinite-Conductivity

and Uniform-Flux Solutions

Though the shapes of the infinite-conductiv­ity and uniform-flux solutions are similar, someof the differences are worth mention. Comparisonof the two solutions indicates that the pseudo­radial flow period begins somewhat earlier for theuniform-flux case (tDx = 2 for uniform-flux,

ft Dx = 3 for infinite-conductivity). Furthermore,

fif xe/xf ~ I, the linear flow period for a

uniform-flux fracture exists for a much longerperiod than for the infinite-conductivity case(tD = 0.16 for uniform-flux, t Dx = 0.016 for~ f

infinite-conductivity). In conclusion, it shouldbe noted that distinctions between the two casesvanish if Xe/xf = 1.

The Vertically Fractured Well in a

Constant Pressure (Full Recharge) Square

Recently Raghavan and Hadinot063 extended thesolutions presented by Gringarten, et al.,55 byconsidering that the pressure at the-outer bound­ary was maintained at a constant value equal tothe initial pressure (full recharge). Figures 8and 9 are log-log graphs of PwD vs. t Dx for the

finfinite-conductivity and uniform-flux casesrespectively. On both of these figures theresults shown in Figs. 3 and 5 are also presentedfor purposes of comparison. Again the linecorresponding to xe/Xf = 00 represents a verticallyfractured well in an infinite reservoir.

The results in Figs. 8 and 9 indicate threeCharacteristic flow periods. A linear flow periodoccurs at early times--the oae-half slope line.After a period of transition a pseudo-radial flowperiod exi.sts. Like the closed case this flow per­iod exists only for certain values of xe/xf. Aftera ser 'nd period of transition steady state flowcondil IS are reach,. ~ for all xe/xf. This flow

period ~s analogous to pseudo-steady state flowbehavior for wells in closed systems. Duringsteady state the pressure at each point within thedrainage region is independent of time and thereis no decline in pressure. Steady state conditionsresult when t DA = 0.4 for all xe/xf.

..;."'

For practical purposes, Figs. 8 and 9 may beused for type curve matching for the appropriatefracture type. If a drainage limit should becomeevident during the test, then data points wouldfollow the appropriate xe/xf line. If the system

under study is located in a constant pressuresquare then field data would fall below thexe/xf = 00 curve, and follow the appropriate xe/xfline. On the other hand if the system boundariesare closed, then data would rise above the xe/xf

= 00 curve and follow the corresponding xe/xf line.Figures 7 and 8 may also be used for analyzingfall-off or build-up data. This aspect of pres­sure analysis will be considered in the section on"Shut-in Pressure Behavior."

For unfractured wells Hurst, Haynie andWalker64 , remarked that the system boundaries(closed or constant pressure) affect pressurebehavior at the same time, i.e., curves influencedby outer boundary conditions will depart simultan­eously from the infinite reservoir curve, regard­less of the nature of the outer boundary. Theresults in Figs. 8 and 9 indicate that this obser­vation also applies to fractured wells for allcases except xe/xf = 1. This then implies that alimiting statement can be made concerning thedrainage volume for a fractured well which doesnot indicate a drainage boundary effect-for bothclosed and constant pressure boundary casesprovided xe/xf 1 1; that is, if the fracture does

not extend to the outer boundary.

Comparison of the results for xe/xf = 1

for the closed and constant pressure casesindicates one important difference. The pressuredrops for the uniform-flux and infinite conduc­tivity cases for the closed reservoir are identi­cal, whereas for the constant pressure outerboundary case this is not so. This result is dueto the influence of the outer boundary. Forxe/xf = 1 in a closed reservoir, no gradientsparallel to the fracture exist; in the constantpressure reservoir this is not so.

From the viewpoint of field applications, thepressure behavior for vertically fractured wellslocated in other drainage shapes is also needed.These may be found in Ref. 65.

Some Practical Considerations

As mentioned earlier one of the problems inanalyzing pressure data by the semi-log approachis that it is difficult to locate the beginningof the pseudo-radial flow period. Inspection ofthe theoretical solutions, however, indicatesthat if the one-half slope line can be identifiedthen the correct semi-log line should startapprox­imately two cycles from the time of the end of theone-half slope line for an infinite-conductivityfracture. For a uniform-flux fracture the timefor start of the correct straight line is onecycle from the end of the one-half slope line. Ingeneral, data over a one-half cycle time periodwould be required to form a well-defined semi-logline. Only thus can the proper straight line beidentified and if early time data are analyzed at

-123-

the test site then the total time of testing canbe readily determined.

A second rule, which is probably more usefulthan the one stated above, is the "double-Llprule." In examining vertically fractured gas wells,Wattenbarger66 noticed that the dimensionlesspressure drop at the start of the semi-logstraight line is twice that of the dimensionlesspressure at the top of the one-half slope line.This result, strictly true only for the uniform­flux case, is the "double Llp rule." For the infinite­conductivity vertical fracture the pressure changebetween the end of the one-half slope line and thebeginning of the semi-log straight line is approx­imately 8. In any event it is clear that theratio of the pressure change must be at least 2.This rule is particularly useful in those caseswhere pressure behavior at an unfractured welldominated by wellbore storage is wrongly identi­fied as a fractured well (see Ramey67 for furtherdiscussion).

Determination of the Fracture Length from the

Effective Wellbore Radius

Gringarten, et al. 55,have shown that theeffective wellbor;-radius can be used to calculatefracture length if Xc/xf is known or if it islarge. This procedure is simple if one notes thatfor large penetration ratios values of xf/r~ areessentially constant. From Fig. 10 we see thatfor xe/xf > 2, xf/r~ ~ 2 for an infinite­conductivity fracture, and xf/r~ ~ e = 2.71828 fora uniform-flux fracture. The first step is toestimate the skin factor, s. The second stePinvolves determining xf/r~ from Fig. 10. Asr~ = rw exp (-s) an estimate of xf can now beobtained. For example consider a uniform-fluxfracture where xe/xf > 2. Noting that xf/r~ z

e = 2.71828 the fracture half-length is given by

The Effective Wellbore Radius Concept

As mentioned earlier, Prats46 has shown thatan infinite-conductivity vertical fracture,producing an incompressible fluid from a closedcircular reservoir, was equivalent to an unfrac­tured well with an effective radius equal to aquarter of the total fracture length for ratiosof the reservoir radius to the fracture half­length greater than 2. The same is true for awell producing a slightly compressible fluid underpseudo-steady state conditions. 52 We can see thatthese results also apply to a vertically fracturedwell in an infinite reservoir during the pseudo­radial period, because Eq. (4) can be written as

[0.000264kt 1 +0.80907}~CtJ,l xf 2

(2)

(17)

(18)

Approximate Determination of Formation

Permeability and Fracture Half-Length

There are a number of instances, particularlyin tight reservoirs, in which the linear flowperiod lasts for several hundred hours. Underthese conditions neither the type curve nor theconventional approach may be applicable. However,the last point on the half slope line may be usedto estimate an upper limit of the permeability­thickness product. Using the resultant value ofpermeability, a corresponding fracture length mayalso be calculated. The appropriate expressionsto be used are:

where Llp, and t are the pressure change and timecorresponding to the last available point on thehalf slope line. Equations (19) and (20) areapplicable for penetration ratios xe/xf » 1.Equations (19) and (20) may also be used if databeyond the half slope line are available but arenot sufficient to perform a type curve match or touse the semi-log graph. If natural fractures areto be analyzed in this fashi~n, then the right­hand sides of Eqs. (19) and (20) should bereplaced by 0.76 and 0.16, respectively.

(20)

(19)0.215

0.016

kh141. 2qBJ,l Llp

and

The effective well radius for an infinite­conductivity vertical fracture in an infinitereservoir is thus exactly one fourth of the totalfracture length. This, of course, is only validfor the pressure drop on the fracture during thepseudo-radial period, aii"d---mllst not be used forother conditions.

The effective wellbore radii for the resultsdiscussed in Figs. 8 and 9 are shown in Fig. 10.Here xf/Xc rather than xe/xf is used for conven­ience. The symbol r~ represents the effectivewellbore radius of a vertically fractured well andequals the product rw exp (-s) .12 The results areapplicable for pseudo-steady state flow (closed)and steady state flow (full recharge) conditions.Examination of the data in Fig. 10 indicates thatthe effective wellbore radius for the uniform­flux case is smaller than that for the infinite­conductivity case. Furthermore, the value of theeffective wellbore radius is essentially constantfor fracture penetration ratios, xe/xf > 2. It isalso evident that the outer boundary conditionsmust be considered if this concept is ·to be used todescribe pressure behavior at the well.

-124-

A Finite-Capacity Vertically-Fractured Well

in an Infinite Reservoir

Application of the Gringarten, et al., typecurves to hydraulically fractured wells-rn manyinstances produces results that are compatible withreservoir performance and design calculationsprior to treatment. But in some instances theresults are not compatible with design calculationsor production performance even though field datamatched the type curve very well. In manyinstances when data following large volume fracturetreatments (injection of several thousand gallonsof fluid and several hundred thousand pounds ofproppant) were analyzed, then computed effectivefracture lengths were small--of the order of a fewfeet. One of the potential reasons for thisanomaly appears to be the finite flow capacity ofthe vertical fracture. To date three groups havepublished results on the effect of finite fracturecapacity on pressure behavior. 5l ,68,69 A summaryof the work of the three groups follows. Letthe reader be cautioned that most of the resultspresented here represent only the beginning ofthe work which needs to be done to understand thepressure behavior of finite-capacity fractures.

Before proceeding to document the results inRefs. 51, 68, ~gd 69, let us refer back to thestudy of Prats which was published almostfifteen years ago. It appears that the results ofthis paper have been virtually ignored. (Sur­prisingly this paper appeared in the Society ofPetroleum Engineers Journal and not in the Journalof Petroleum Technology - only a small fraction ofthe SPE membership subscribes to this journal.)The effect of finite fracture capacity defined as"the product of fracture permeability and fracturewidth" was demonstrated in this paper. Pratsshowed that three parameters controlled thepressure distribution around a fractured well.They are,(i) the ratio of the fracture length tothe well radius, (ii) the ratio of the reservoirdrainage radius to the well radius, and (iii)the dimensionless fracture flow capacity, F~D

defined by:

in Fig. 11. It shows that the pressure distribu­tion for F~D = a is given by confocal ellipses.

For F~D = 00 the pEessure distribution is given bycircles concentric with the well exis. Pratsnoted that for F~D = 100 the pressure distributionwas essentially the same as that for an unfrac­tured well (radial flow). For intermediate valuesof F~D the pressure distribution lies in betweenthe two extremes. As the pressure draw-down orbuild-up curve is essentially the reflection ofpressure distribution in the reservoir, it isclear that finite fracture capacity can drasticallyinfluence the pressure draw-down or build-up trace.

Figure 12 presents the effect of fracturecapacity on the effective wellbore radius. Threeobservations are evident: (i) If F' > 28 thenany increase in the fracture lengtfiDwould be tot­ally ineffective, (ii) for F~D between 1 and 28

increases in productivity may be significantif the fracture length is increased and, (iii)since the formation flow capacity is fixed, thefracture capacity would have to be increased ifproduction increases are to be significant.

The most important message of Prats' paper,however, is the following: The dimensionless frac­ture flow capacity, F'D' determines well perfor­mance and productivit? increases--not fracturelength; that is, for a long fracture to be aseffective as a short one,the fracture flow capa­city would have to be much higher for the longerone. Fracture capacity dictates optimum fracturelength. Unfortunately this has not been recog­nized by reservoir or production engineers.

Let us now return to the discussion of theeffect of dimensionless fracture flow capacity onpressure transient behavior. Before proceedingfurther it should be noted that the variousresearch groups mentioned earlier have defineddimensionless fracture capacity somewhat differ­ently from Prats' definition. The Agarwal,et al.6~definition is ----

F'cD

(21)

11

2F'cD

68Cinco, ~ al. ,define dimensionless fracturecapacity as

(22)

The first two parameters describe the geometry ofthe system and the third is the measure of theability of the formation to carry fluids into thefracture relative to the ability of the fractureto carry fluids into the well. For a very effec­tive f-acture,that is one which has a great abilityto carry fluids, F~D _0 small and approaches the

limiting value of zero for infinite fracturepermeability. Likewise, for a very ineffectivefracture, F~D would be large and approaches

infinity for the limiting case of an unfracturedwell.

The effect of fracture capacity on the pres­sure distribution around a fractured well is shown

-125-

Ramey, et al. 5l , define dimensionless fracturecapacitya-;;:-

1

(kfw)D

(23)

(24)

Figure 14 presents the dataon a semi-log graph (PwD vs. log

Clearly some consistency in the nomenclature iscalled for. Personally, I prefer the definitionof Agarwal, et a1.

Figure 13 demonstrates the effect of dimen­sionless fracture capacity, FcD ' on the pressurebehavior at the well, when all other parametersare constant. The dimensionless fracture capacityranges from 10-1 to 500. Agarwal, et al., reportthat for practical purposes the infinit;­conductivity solution obtained by Gringarten, eta1. ,55 can be used if FcD >,. 500. -

The most important point to note in Fig. 13is that for small values of time the shape of thecurves for various FcD do not possess distinctivecharacteristics. Furthermore, there is a wideseparation between the various curves at smalltimes. Thus fracture capacity strongly influ­ences pressure behavior at early times. However,this separation diminishes as tDx increases.

f

Figure 13 may be used for type curve matchingto estimate k, xf and FcD ' The pressure match

should provide an estimate of the formation per­meability, k, the time match for the value of thefracture half-length, XI' and the appropriate FcDcurve for the value of fracture flow capacity,kfw. At the present time no method is availableto estimate kf and w separately. From a practicalviewpoint, however, the shapes of the curves areso similar that the probability of matching datawith an erroneous value of FcD is high. If typecurve matching is attempted, care and diligenceare needed. If the formation flow capacity isknown, the matching procedure is simplified andmore importantly becomes more reliable sincevalues of p D may be computed prior to matching.In this eve~t the tracing paper needs to be movedin only the horizontal direction during the match­ing process. Matching, even along these lines,can be difficult. Agarwal, et al. 69 , stronglyrecommend that pre-fracture pressure data bemeasured whenever possible. In extreme cases anumerical model may be needed to match field dataadequately. The need of the hour is to be able todevise a procedure for analyzing field dataconveniently and correctly.

The results in Fig. 13 also agree with thespeculation by some that fracture capacity can beone potential reason leading to apparent shortvertical fracture lengths that are calculatedfrom well tests when the solutions of Gringarten,et al. 55, are used. For example, data obtained forFcD = 2 can be matched with the similar para­meter value FcD = 500 by moving to the right onthe time scale. If this is done, an erroneousvalue of t D would be obtained,which in turnx

fwould result in a low estimate of xf'

shown in Fig. 13t Dx )' The

fstraight line shown in Fig. 14 corresponds to theslope of the straight line that would be obtainedfor plane radial flow. All of the curves in Fig.14 show a much shallower slope than 1.151 per logcycle. Since the time range of 10-5 ~ t Dx ~ 1

f

covers most of the times for which testing wouldbe carried out in low permeability reservoirs, itis doubtful that the radial flow response will beseen. If data were graphed on semi-log graphpaper to compute permeability from an apparentstraight line an optimistic estimate of formationpermeability would result. The error in theestimate would depend on the producing time.

Actually, for data beyond tDxf ~ 1, a

semi-log straight line with the proper slopeeventually results (see Fig. l5~ This straight linemay be used to determine formation permeability.For FcD = 10-1 the semi-log straight line beginsat tDx ~ 1. The time for onset of pseudo-radial

fflow is dependent on FcD and increases as FcDincreases. From the earlier discussion thisresult should be expected. Cinco, et al. 68 , andRamey, et al. 5l,point out that for times tDx ~ 5

-- f

pseudo-radial flow prevails for all values of FcDof interest. (Note that this assumes no boundaryeffects. )

Figure 15 also demonstrates the behavior ofthe uniform-flux fracture with respect to FcD 'For small times (tDx ~ 0.16) the dimensionless

I

fracture capacity of the uniform-flux fracture is500; for times greater than the time for the onsetof pseudo-radial flow (tDx >,. 2) it follows the

f

curve corresponding to FcD Z 4.4. For inter­mediate times, the uniform-flux solution changesfrom FcD = 500 to FcD = 4.4. Thus the uniform-

flux solution is essentially a variable fracturecapacity solution.

Agarwal et al. 69 , suggest that a graph ofp D vs. ~D is also useful in analyzing dataw xfwhen fracture capacity is important. Figure 16presents a replot of the data shown in Fig. 13along these lines. On a graph such as Fig. 16early time data for the infinite-conductivity orinfinite-capacity fracture will fallon a straightline-passing through the origin with a slope equalto 111 [see Eq. (5)]. As FcD decreases straightlines with the same slope can be seen, howeverthey do not pass through the origin. Agarwal,et al. 69 , have empirically correlated the PwDintercept as a function of FcD (see Fig. 9 of Ref.69). This correlation may be used to determineFcD ' But it should be noted that as FcD

decreases then the length of the straight linesegment decreases--and disappears for FeD = 1.

For practical applications the difficultiesinvolved in using this graph are essentially thesame as those for the log-log or semi-log graphs;that is, that the shape of the curves are not dis­tinct enough to permit any identification of thecorrect fracture solution or the appropriatestraight line.

Although it will not be considered here indetail, some information is available on theeffsItof the closed outer boundary. Ramey, etal. ,report that for values of FcD >, 300

-126-

solutions obtained were very close to those ofGringarten, et al. 55 , for all values of xe/xf.Note that the Ramey,et al., criterion for specify­ing a finite-capacitY-fracture to be an infinite­conductivity fracture is 'somewhat different fromthe Agarwal, et al. 69,value of 500. This differ­ence is mainlY-due to the precision used incomparing the solutions and should not be con­strued as an error on the part of either of thetwo research groups.

Finally it should be noted that there are anumber of other factors which can give an appear­ance of a small fracture capacity. These includethe effect of producing time on build-up data,non-Darcy flow 70 ,71.72, confining pressure69 , anddamage, to name only a few. Much work remainsto be done, particularly in improving the abilityto analyze field data. Let us now return to theconsideration of other aspects of fractured wellbehavior.

no mention of the exact nature of the skin exceptthat it is an infinitesimally thin steady stateresistance to flow. The impediment may existwithin the fracture, on the fracture surface orextend some distance into the formation. It isagain emphasized that Eq. (25) is not intended todescribe pressure behavior of wells intersectingfinite flow capacity fractures.

Wellbore Storage in Fractured Wells

As fractured wells normally have high produc­tive capacities wellbore storage should not beimportant. However, Ramey35 has shown that well­bore storage effects can be important in somecases. Theoretical studies of the wellborestorage effect in fractured wells have been pre-

'sented by Wattenbarger and Ramey74, and Ramey andGringarten75 (for the infinite-conductivityvertically fractured well), and by Raghavan73 (forthe uniform-flux case).

The Skin Effect in Fractured Wells (Uniform-

Flux or Infinite-Conductivity)

The basis for the above discussion is evidentif the skin effect is included in the solutionsfor fractured well behavior. Analogous to the pro­cedure for radial flow, the solution for the pro­ducing pressure at small times for a fractured wellwith a skin effect may be written as:

(26)

In practical applications, the most importantpoint to be noted about Figure 18 is that a trans­ition region exists between the unit slope and thehalf slope lines. In some cases when field dataare plotted on log-log coordinates no transitionis evident. In such cases it is probable that thevalue of L'lp may be in error. A detailed

where c is the unit storage factor and is identi­cal to that defined in Ref. 34. The CDx = 0

fcurve corresponds to a fractured well with nowellbore storage (uniform flux) in an infinitereservoir. 65 For large values of CDXf a line of

unit slope s~milar to that for unfractured systemsis obtained. However for small values of CDxfno unit slope line is evident for times of inter­est. (Actually a unit slope line does exist fordimensionless times smaller than that consideredhere.) As time increases, all curves becomeasymptotic to the C = 0 line. Figure 18 alsodemonstrates that i~Xtellbore storage is largethen the presence of the fracture would beobscured. Then the fracture would have to bedetected by comparing storage volume calculationswith wellbore completion data. 35 All of the aboveobservations are also applicable to the infinite­conductivity case. 74 ,75

Figure 18 is a log-log graph depicting thepressure behavior of a well producing via auniform-flux fracture which is controlled at earlytimes by wellbore storage. The well is assumedto be located in an infinite reservoir. Theparameter of interest in Fig. 18 is the wellborestorage constant defined by the relation:

(25)p D(tDx ) = IrrtDx + sw f f

where s is the skin factor. Equation (25) indi­cates that for small times the first term would besmall and thus the one-half slope line would beobscured. Therefore, a graph of (Pi - Pwf) vs. ton log-log coordinate paper would be flat. How­ever, a graph of Pwf vs. It would be a straightline on Cartesian graph paper.

In many instances, particularly in injectionwells, there is skin damage associated with thefractured systems. Interpretation of data fromthese wells can be difficult. A typical pressuretrace on log-log graph paper is shown as Curve A,in Figure 17. If pressure data, plotted on log­log graph paper, approach the half slope line fromabove one may be reasonably certain that skindamage exists. If the skin effect is fairlylarge, then no one-half slope line may be evident(Curve B). In that event it would be difficult toidentify a vertical fracture using log-log graphpaper. This flat data on log-log paper, however,will graph as a straight line on Cartesian graphpaper (L'lp or Pwf is plotted versus It ). It shouldalso be noted that it is possible to mistake theskin effect for a finite-capacity fracture.

The use of log-log and Cartesian graphs toidentify a resistance to flow in a fractured wellhas been discussed only recently73, though Ramey35makes passing reference to this possibility.

This visualization of the skin effect makes

*A unit slope line on log-log graph impliesthat wellbore storage is dominant. Data duringthe unit slope period cannot be used to estimateformation properties.

-127-

discussion of this aspect may be found in Refs. 67and 73.

Preliminary results on the effect of fracturecapacity on wellbore storage have been reportedby Cinco, et al. 68 They pointed out that forsmall timeS; tDxf ~ 10-5 the porosity and compres-

sibility of the fracture system also influence thepressure behavior. They demonstrated that forsmall times two dimensionless groups control well­bore pressures. These dimensionless groups aredefined as follows:

at the center of the formation with impermeableupper and lower boundaries in an infinite reser­voir was presented in 1973. The main objectiveof this work was to determine if the early timepressure behavior of a horizontally fracturedwell is distinctly different from that of eithera vertical fracture, or plane radial flow. Theresults obtained in Ref. 76 were used to preparethe curves shown in Fig. 21 where the dimension­less wellbore pressure drop per unit of dimension­less reservoir thickness is graphed as a functionof dimensionless time. The dimensionless thick­ness is the parameter of interest. For purposesof this discussion the dimensionless time anddimensionless thickness groups are defined,respectively, as follows:

w<P f cftA =---

1Txf<PCt

kf<P Ctand B =---

k<PfCft

(27)

(28)

0.000264kt<pct].lr~

(29)

In Eqs. (29) and (30) r f is the fracture radius,k is the horizontal permeability, and kz is theis the vertical permeability (see inset Fig. 21).

The curves corresponding to hDrf > 3 in

Fig. 22, and hD < 1 in Fig. 21 have shapesrf

which are different from those of the verticalfracture cases (see Fig. 3 and Fig. 5). Further­more if hDrf < 0.7 then there is an increase in

slope from one-half towards unity that has nocounterpart in the vertical fracture case. Thusit may be possible to distinguish between the twotypes of fracture from a well test. If 1 ~ hDrf~ 3, however, there is a possibility that hori­zontal and vertical fracture behavior will beconfused: The line for a uniform-flux verticalfracture in an infinite reservoir was found tomatch the horizontal fracture case of hDrf of

Fig. 22 is a semi-log graph of the same datapresented in Fig. 21 in terms of PwD vs. tDr •

fAs shown in Fig. 22, at long times the dimension-less pressure drop is a linear function of thelogarithm of time with a characteristic slope of1.15l/logv. Thus a semi-log graph of ~p or PWf

vs. t may be used to estimate horizontal permea­bility if the test is run for a long enoughperiod. Fig. 21 is ''easy to use for type-curvematching purposes because all curves have incommon an initial one-half slope straight line,corresponding to early time vertical linear flow(instead of horizontal linear flow, as for t~vertical fracture case). Also, a single curve isobtained for hDr ~ 100. For practical purposesthis curve repre~ents the situation in whichfluid is withdrawn via a single plane horizontalfracture in a reservoir of infinite extent in alldirections.

(30)if:z

A Horizontally-Fractured Well in an

Wellbore Storage and Skin Effect in a

Vertically-Fractured Well (Uniform-Flux)

where <Pf and Cft are the porosity and effective

compressibility of the fracture, respectively.Note that the product AB is equal to FcD /1T andA = CDXf since c = (2<PfCfthxfw). Figure 19 is a

graph of p D vs t D for a finite-capacityw xf

fracture with A = 10-1 and B = 107• Also shownare the results obtained by Ramey and Gringarten75for CDxf = 10-1• The results are in good agree-

ment. More work needs to be done in this area ofpressure analysis.

Figure 20 demonstrates several interestingand instructive features. For example, if thewellbore storage period is followed by a halfslope period and data do not approach the halfslope line from above, then damage is negligible.The solutions shown here also indicate that ifwellbore storage and skin are negligible, thenthe half slope line will be observed. Raghavan73has discussed application of the theoreticalresults shown in Fig. 20 to field data.

Figure 20 is a log-log graph of PwD vs. t Dxffor a vertically fractured well (uniform-flux)including the wellbore storage and skin effects.All lines start out with a line of unit slope andfor each value of CD are independent of s for

xfsmall times. Thus, at early times fractured wellbehavior is similar to that of an unfracturedwell (see Agarwal, et al. 34).

An analytical solution76 for a well with asingle horizontal, uniform-flux fractuEe located

Infinite Reservoir

-128-

"(··-i r} ;j "i 9 ~ ;t.l ~.1' tJ "J l

about 2.4 in Fig. 21 reasonably well. However,the dimensionless pressure scales are basicallydifferent in nature and in magnitude, and it islikely that results would appear questionable,should the wrong fracture type be selected.

infinite and the fracture extends over the entirethickness of the formation. The symbol h

DXfrepresents the dimensionless thickness of theformation and is defined by the expression:

A few expressions useful in well test analy­sis are summarized from Ref. 76. The initial ver­tical linear flow period (one-half slope on log-logcoordinates) is represented by: ~~

z

(34)

an Infinite Reservoir

which may be rearranged to the dimensional form:

The pseudo-skin factor is a function of 6w

and

hDx ' Figure 25 presents the pseudo-skin factorf

as a function of dimensionless thickness, h--nxf'and angle of inclination, 6w' If 6w and hDx are

fincreases thesmall, then s is small. As hDx

fpseudo-skin factor, s, increases. This impliesthat well productivity is affected considerablyby the angle of inclination, 6w' when the fracture

half-length, xf' is much smaller than the thick­ness, h.

For large values of dimensionless time, t Dx 'f

the dimensionless wellbore pressure drop, p ,iswD

a linear function of the logarithm of dimension~

less time with a characteristic slope equal to1.151. Thus long time data can be analyzed usingconventional semi-logarithmic techniques. Thetime for the start of the pseudo-radial flowperiod is a function of 6 and hD • For the case

w xf

shown here pseudo-radial flow prevails for dimen·sionless times, t Dx q 8. Cinco, et al. 78 , have

falso mentioned that at early times linear flow(perpendicular to the fracture surface) prevails.Thus the transient flow behavior of a well inter­cepting an inclined fracture in an infinitereservoir includes a linear flow period, a trans­ition region,and a pseudo-radial flow period.Qualitatively the above description also holds forother values of hDx • The duration of the various

fflow regimes depends on hDx and 6. It can also

f w

be seen that the dimensionless wellbore pressuredrop for an inclined fracture is always less thanthat for a vertical fracture. As the inclinationof the fracture, 6w' increases, the dimensionlesspressure drops are smaller. At long times(pseudo-radial flow) the difference in the pres­sure drop between the inclined fracture case andthe vertical fracture case becomes constant. Thisdifference can be handled as a pseudo-skin factor,s, defined by 78

(32)

(31)

(33)

+ 1. 80907

( )

1/22 0.000264

1T~IlCt

0.000264kt

~Ilctr~

PDr2 __f

1T

70.6qBIlkh

(kZ)1/2r~ (Pi-Pwf)

l41.2qBIl

The dimensionless wellbore pressure drop foran inclined fracture is shown in Fig. 23 forseveral values of the inclination of the fracture,6 ,78 (see Fig. 24 for a description of the geo-w

metry of the system). The results shown here areprobably most useful in analyzing pressure behav­ior at fractured wells in steeply dipping reser­voirs. The fracture conductivity is assumed to be

The proper time limits for the application ofeither Eqs. (32) or (33) depend upon hDr • The

fpseudo-skin factor which is the quantitativemeasure of the pseudo-skin effect during thepseudo-radial flow period may be obtained fromEq. (33) by subtracting the pressure drop due toan unfractured well. Further details are given inRef. 76. Application of the results presentedhere may be found in Ref. 77.

Wells Intercepting an Inclined Fracture in

At long times, the flow is the same as thatcreated by an unfractured well, with an addi­tional pressure drop which is referred to inthe petroleum engineering literature as thepseudo-skin effect. A long time approximationfor hDr < 1 can be written as:

f

-129-

A Limited Entry Vertical Fracture in an

Infinite Reservoir

In this section we shall briefly discuss theeffect of limited entry on the pressure behaviorof vertically-fractured wells. For brevity onlythe uniform-flux case will be considered. Here weshall use the term "limited entry" to describesituations where the vertical fracture height, hf,is less than the formation thickness, h.

Figure 26 is a graph of PwD vs. t Dxf for a

uniform-flux fracture located at the center of theformation. 79 Fig. 27 presents the details regard­ing the geometry of the system. The dimensionlessthickness, hD ,is 5. The term b, which is the

xfratio of the fracture height, hf , to the thicknessh, is the parameter of interest. In this paper thisratio will be described as the entry ratio. Thecase b = 1 corresponds to the complete entry (orhDx = 00) case--the Gringarten,et al.,solution. 55

f --

As can be seen from Fig. 26, all straight linesstart out with a slope of one-half which corres­ponds to the linear flow period. (For b = 0.1this period occurs earlier and is not shown.) Theduration of the linear flow period is a functionof the entry ratio and increases as the entryratio increases. This is to be expected'sincelarger values of b correspond to a greater frac­ture area. Following the linear flow period thereis a transition region and finally there is apseudo-radial flow period. It can be shown thatpressure data in this region will graph as astraight line with a slope of 1.151 per log cycleon semi-log paper. 79 As shown in Fig. 26 thestart of the semi-log straight line for b < 1occurs much later than that for b = 1. Thus ifconventional semi-log methods are used to analyzepressure data this observation indicates thattests should be run for a much longer period thanfor the complete entry case. In some instancesall of the data obtained during a test may corres­pond only to times prior to the onset of pseudo­radial flow.

The displacement of the curves shown in Fig.26 is a result of the additional pressure dropcaused by the convergence of flow into the openinterval. The magnitude of this additional pres­sure drop changes with time until the pseudo­radial flow period. During pseudo-radial flowthe magnitude of this additional pressure drop isconstant. This stabilized additional pressuredrop,which is a result of the fracture heightbeing less than the formation thickness,can bequantitatively described by the pseudo-skinfactor. It is a function of hD and b. Pseudo-

xf

skin factors for systems of interest are presentedin Ref. 79.

Figure 28 is a log-log graph of the same datashown in Fig. 26. However, in this graph theordinate is the product of the dimensionlesspressure drop and the entry ratio. Plotting theresults in this manner results in the curves forentry ratios, b < l,merging into the b = 1 curveat early times. Therefore all curves start out

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from the one-half slope line corresponding tob = 1. As the vertical component of flow beginsto affect pressure behavior the limited entrycurves leave the curve for b = 1. Ultimatelypseudo-radial flow develops and the pressure dropis a linear function of the logarithm of flowtime.

From a practical point of view, however,graphs such as Fig. 29 are more useful than thoseconsidered so far. Fig. 29 is similar to Fig. 28except that in this case the dimensionless frac­ture height, h fD , is the parameter of interest.This dimensionless fracture height is defined as

(36)

The advantage of this procedure is that it givesmore order to the graph. For example, in thisinstance all curves merge at early times into thecomplete entry curve just as for the case shownin Fig. 28. But Fig. 29 also permits display ofdata for several values of band h on the same

fDgraph without expanding the scale. This may bemore clearly seen in Fig. 30 where the dimension­less pressure drop for h fD = 5 and two values of b

are presented. The first deviation from the b = 1curve is independent of b and depends only on hfD •

After a period of transition, the effect of b canbe seen. Finally there is a pseudo-radial flowperiod corresponding to the semi-log straight line.The beginning of the pseudo-radial flow period is afunction of h

Dxf(or hfD and b). Figure 29 may be

used to obtain system parameters. If the test ,isrun for a long enough period then the permeability­thickness product, kh, the fracture half-length,xf' the vertical permeability, kz ' and the entryratio, b, can be determined. by type curve matchin~

This, of course, assumes that xe/xf is large.Further ,details may be found in Ref. 79. Typecurves for other cases such as the pressurebehavior for a limited entry infinite-conductivityvertical fracture are also presented in Ref. 79.

From the above discussion it can be concludedthat as hDxf or hfD increases, that is, as strati-

fication becomes more severe, the pressureresponse for the limited entry fracture is delayedin time. Furthermore, for any dimensionless timebeyond the linear flow period the dimensionlesspressure drop is higher for larger values of hfDor hD• Thus, in terms of real variables it canbe concluded that as the vertical permeabilitydecreases the pressure drops are larger and thepressure response is slower.

Raghavan,et al. 79 , also examined the effectof fracture location within the producing inter­val. After examining various fracture locationswithin the producing inte~~al, Raghavan,et al.concluded that the productivity for a given-Se~ ofconditions decreases as the fracture positiondeparts from the center of the producing interval.

Raghavan, et al. 79 , also delineated conditionsunder whic~i~would be possible to recognize thatthe fracture height is less than the formationthickness. They concluded that it would be diffi­cult to identify this condition from pressureversus time data if hfD > 5. They-also found that

the position of the fracture within the producinginterval was unimportant if b > 0.7.

Vertically Fractured Wells Producing at

Constant Wellbore Pressure

As mentioned earlier, the constant ternunalpressure case has not attracted attention--mainlybecause it is not readily apparent how this con­dition might affect pressure build-up after thewell is shut in at the sand face. Nevertheless,results for this wellbore condition are useful.If the formation permeability is low, then it maynot be possible to hold the rate constant for longperiods of time.

If the well is produced at s constant ratethen the wellbore pressure changes with time.However, if the pressure is held constant then therate would vsry as a function of time. Locke andSawyer80 have examined the change in rate versustime for an infinite-conductivity vertical fracturein an infinite reservoir and in a closed squaredrainage region. The results are shown in Fig.31. Here the reciprocal dimensionless rate, l/qD'

has been graphed vs. dimensionless time, t D ,andxf

xe/xf is the parameter of interest. The reciprocaldimensionless rate for the constant terminal pres­sure case is analogous to the dimensionless well­bore pressure drop, PwD' for the constant terminal

rate case and is defined as

The effect of fracture capacity on the dimen­sionless rate has also been investigated. Figure32 presents l/qD vs. t Dxf for a finite capacity

vertical fracture in an infinite reservoir. 69 Theparameter is FCD ' As in the constant rate case,

the curves for various values of FCD do not pos­sess distinctive characteristics. At early timesthere is a wide separation between different FcDcurves. As t Dx increases, the separation betweenthe curves decrtases. Agarwal, et al. 69 , have alsoshown that if FcD ~ 500, then the finite capacity

curves merge into the infinite-conductivity curvesof Locke and Sawyer.

Figure 32 may be used to estimate formationpermeability, k, fracture half-length, xf' andfracture capacity, kfw, by type curve matching.If the fracture capacity can be considered to beinfinite then the curves of Locke and Sawyer maybe used. The procedure is essentially the same asthat for the constant rate curves. The only dif­ference is that in the present instance theordinate is l/q rather than 6p.

It is obvious that care and diligence shouldbe exercised in analyzing data by the type curvemethod (or any other approach) if the fracturecapacity is important. The curves shown in Fig.32 have no distinct characteristics and the proba­bility of obtaining a match with the wrong value ofFeD is high. If an estimate of the formation flow

capacity is available then the type curve matchingprocedure is simplified and would be more reliable.Since various aspects of analyzing data for finite­capacity vertically fractured wells have beendiscussed already at length, further discussion isnot warranted here.

At small values of time it can be shown thatthe following relationship holds

Thus if we graph l/q vs. t on log-log paper oneshould obtain a straight line with a slope equalto 0.5. This observation can be used to identifya fractured well p~oducing at a constant wellborepressure. Figure 31 can be examined along thesame lines as Fig. 3; however, since the character­istics of the l/qD vs. t Dx curves are similar to

fthose of PwD vs. t Dxf for the constant rate case,

we will not examine these results in detail.

(39)PwD

Virtually all of the results discussed so farare strictly applicable to fluids of constant com­pressibility and viscosity. The theoreticaljustification for the application of these solu­tions to the analysis of gas well test data isbased on the work of Aronofsky and Jenkins8l andAl~Hussainy, Ramey and Crawford. 82 In applyingthese results to the flow of steam *gas) only, thedefinitions of the dimensionless pressure drop anddimensionless time need to be modified. Forapplication to steam (gas) wells the right handside of the definition of dimensionless pressuredrop, PwD' is modified as follows:

19.87 x 10-6 kh Tsc

Pressure Transient Analysis for 'Steam Wells

where q is the flow rate, measured in thousands ofcubic feet per day, and T is the reservoir temper­ature in OR. The subscript, sc, refers to stand­ard conditions of pressure and temperature, andm(p) is the pseudo pressure function defined

(38)

(37)kMpl41.2qB).l

Here 6p = Pi - Pwf = a constant, and all other

symbols have the same meaning as before.

-131-

by:82

L dp'\lZ

(40)*

durations of producing time and shut-in time werefound to have significant influence on pressurebehavior and the authors state that care shouldbe taken to insure that data are selected properlyto estimate formation properties and fracturelength.

Here Pb refers to a base pressure and Z is the

compressibility factor. The viscosity and com­pressibility terms in the definition of dimension­less time should be evaluated at the initial pres­sure, Pi' Thus the equation for the definition of

dimensionless time, t D ,isxf

0.000264kt

~Cti\liX~(41)

Recently, the type curve matching techniquehas been proposed to analyze pressure data infractured wells. 55 This method involves plottingthe pressure change versus shut-in time [(pws -

PWf,s) vs. ~tl on log-log graph paper. Here pwsis the shut-in pressure, p f is the pressure atw ,sthe time of shut-in,and ~t is the shut-in time.The principal advantage of the type curve approachis that the trial and error procedure inherent inthe semi-log methods can be avoided. The log-logmethod is also useful to insure that properstraight lines are chosen when data are analyzedby semi-log techniques.

Other expressions for dimensionless time should bemodified appropriately.

Here both the type curve and the conventionalmethods will be presented. The advantages anddisadvantages of both of these methods will bediscussed.

The Type Curve Approach for the Analysis of

P D(tD ,x !xf)-P D[(t+~t)D ,x !xflw xf e w xf e

Build-up Data

Recent papers by Gringarten, ~ al. 55 ,77,have demonstrated the usefulness of the type curvemethod to interpret pressure data obtained atfractured wells. The basis for the type curveapproach for analyzing build-up data is identicalto that for draw-down.

(2)

(42)

0.000264kt

~Ct\lX~

[p (tHt)-p f(t) 1ws wkh

141, 2qB\l

Pressure Build-up Equations for Type CurveAnalysis. Shut-in pressures for a fractured wellproducing at a constant rate, q, for a time, t,can be determined by superimposing an injectionwell starting at time, t, with the injection ratebeing equal to the production rate prior to shut­in. This results in a zero rate for times t + ~t.

Using the draw-down equation and applying theabove principle the basic equation for the analy­sis of build-up data by the type curve method isgiven by:34,86

where

Shut-in Pressure Behavior of

As mentioned earlier, Russell and Truitt54

were the first to present detailed information onthe transient pressure behavior of a verticallyfractured well. They also analyzed a limitednumber of pressure build-up cases and found thatthe straight-line slope on a Horner16 build-upgraph required significant correction as the frac­ture length increased. They also recommended theMuskat19 semi-log graph for estimation of staticformation pressure.

Vertically Fractured Wells

In 1968, Clark60 suggested a method for cal­culating fracture length using the results of theRussell and Truitt study. In 1972, Raghavan, Cadyand Ramey86 further extended the Russell and Truittstudy by examining an extreme variety of semi-logbuild-up methods (Horner,16 Miller-Dyes-Hutchinson17

and Muskat19 ). Raghavan, et al. 86 ,and Raghavanand Hadinot063 have pointe~out that the determi­nation of the permeability-thickness product bysemi-log build-up methods is a trial-and-errorprocess, as the slope of the build-up curve isinfluenced by both the fracture penetration ratio(ratio of drainage length to fracture length) andthe formation permeability. Also in Ref 86, the

*The pseudo pressure function is essentially atransformation which accounts for the variation influid properties. This transformation is known asthe Kirchoff Transformation83 in the heat conduc­tion literature, as the Leibenzon Transformation inthe Russian literature, and as the Matrix FluxPotential in the soil mechanics literature. Forhydrocarbon gases at low pressures «3000 psi) ithas been observed that the product (\lZ) is essen­tially constant. Thus in this region m(p)ocp2. Inthe high pressure region (p > 3000 psi) it can beshown that (p/\lZ) is reasonably constant. Thus inthe high pressure range m(p)ocp.

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In Eqs. (42) and (43), t is the producing time and6t is the shut-in time. If now (t + 6t) ~ t, wethen have:

and

i'~ ;:"il,~.) ",.".

(43)

'J ,;:; ,/ '/

The Semi-Log Approach: An Infinite-Conductivity

Vertically Fractured Well in a Closed Square

In the following we shall take the approachsuggested by Raghavan, et al. 86, and explore thecharacteristics of common build-up methods ofanalysis along the lines of the pressure build-uptheory suggested by Ramey and Cobb. 18 Our atten­tion will be restricted to the Miller, Dyes, andHutchinson and Horner methods.

kh [p (t+6t) - p f(t)]141. 2qB)l Ws w

(44)

Taking logarithms of both sides of Eqs. (44) and(43), respectively, we obtain:

Pressure Build-up Equations for Semi-LogAnalysis. Shut-in pressures for a fractured wellproducing at a constant rate, q, for a time, t,can be determined by superimposing an injectionwell starting at time t; the injection rate beingequal to the production rate before shut-in. Thisthen results in a zero production rate after timet, and thus at the well location we have: 18

(45)

khl41.2qB)l

log kh + log[p (t+6t)141. 2qB)l ws

(47)

Equation (47) serves as the basis for the Horneranalysis.

andlog O.000264k + log 6t

~Ct)lX~(46) For a well located in a closed reservoir the

Miller-Dyes-Hutchinson graph requires the pressuredifference <p - pws)' This difference can bedetermined from the following:

The Miller-Dyes-Hutchinson Method. Thismethod requires that build-up pressures be plottedas a function of the logarithm of shut-in time.Perrine87 first presented a dimensionless form ofthe Miller-Dyes-Hutchinson build-up curve in whichthe pressure difference was (p - pws)' Forunfractured wells in closed drainage systems, this

The volumetric average pressure, p, is of interestfor two reasons. The average pressure in thereservoir is a direct reflection of the quantityof fluids in place and is necessary to performmaterial balance calculations. Also, in a closed,bounded system the average pressure, p, is thelimit of the shut-in pressure, Pws' as build-uptime approaches infinity.

If actual build-up data are plotted as thelogarithm of the absolute difference between flow­ing pressure at the start of build-up and pressureafter the change vs. the logarithm of shut-in timethen the actual field data should be similar to alog-log graph on which PwD vs. t Dx have been

plotted. The difference between t£e two graphs isonly a linear translation of both coordinates,represented by the first terms on the right-handsides of Eqs. (45) and (46). If a pCDper match isobtained, then the formation permeability andfracture length can be estimated from the typecurve match. Both uniform-flux and infinite­conductivity fractures can be analyzed by thisapproach. Though the above derivation has speci­fically assumed a vertically fractured well in asquare drainage region it is applicable to any ofthe systems considered here.

The importance of Eq. (44) deserves emphasis.Equation (44) states that if t»6t, that is, theduration of the shut-in period is much smaller thanthe producing period prior to shut-in, then thepressure changes which form the build-up traceafter the well is shut in are identical to thedraw-down trace. This then implies that all of thecharacteristics discussed in the section on pres­sure draw-down behavior for the various systemsexamined here are applicable to the respectivebuild-up case.

PDskh (-p _ )

l4l.2qB)l Pws

(48)

-133-

graph offers a direct and simple extrapolation fromshut-iEo pressures, Pws' to the fully static pres­sure, p.

Figure 33 presents a Miller-Dyes-Hutchinsongraph for a vertically fractured well, for thefracture penetration ratio, xe/xf = 10.

The producing time prior to shut-in is the para­meter of interest. As in the unfractured wellcase, a single curve results for pseudo-steadystate production,prior to shut-in.

In the conventional Miller-Dyes-Hutchinsongraph for an unfractured well, a linear portionis evident for early shut-in times. This linearportion possesses a slope of 1.151 per log cycleand is inversely proportional to the permeability­thickness product. But Fig. 33 exhibits some sur­prising differences from unfractured well behavior.The maximum slope to be found on any of the curveson Fig. 33 is 0.90--much less than the expectedvalue of 1.151. Furthermore, there are no well­defined straight lines evident, and the maximumslope for each producing time decreases as pro­ducing time decreases.

Figure 34 presents build-up curves (pseudo­steady production) for all fracture penetrationvalues xe/xf' discussed in Ref. 86. The build-upbehavior for an unfractured well is also shown forpurposes of comparison. The maximum slope of thebuild-up curves decreases as fracture penetrationratio decreases. It is also evident that themaximum slope is significantly less than that forthe unfractured case for all fracture penetrationratios. --

Russell and Truitt54 described a similareffect for the Horner16 graph for vertically­fractured well data. This will be discussed in afollowing section. But Russell and Truitt didpoint out clearly that the reduced slope for aHorner graph could lead an analyst to compute apermeability-thickness value which could be toolarge. They pointed out that this could explainthe apparent opening of "new sand" after frac­turing.

At this stage is should be emphasized that theslope of a pressure build-up graph is not neces­sarily related to the slope of a draw-down graph.A straight line with the correct slope may appearin a draw-down test, but not on any of the con­ventional build-up graphs.

As pointed out by Raghavan, et al. 86 ,Figs. 33and 34 raise serious questions co;Cerning appro­priate interpretation measures for use withMiller-Dyes-Hutchinson graphs of vertically­fractured well data. In order to apply the Miller­Dyes-Hutchinson method to fractured well build-updata Raghavan, ~ a1., followed the suggestionRussell and Truitt had proposed for the Hornerbuild-up graph. Russell and Truitt had suggestedthat the maximum slope be read for the fracturedwell build-up data, and then the permeabilitycorrected to the true value. Figure 35 presentsthe permeability-thickness correction factors forthe Miller-Dyes-Hutchinson form of plotting for aninfinite-conductivity vertical fracture in aclosed square reservoir as a family of dashed

-134-

lines (xf/xe rather than xe/xf is used here forconvenience). The correction factor was obtainedfrom graphs similar to Fig. 33 by dividing theactual maximum slope by 1.151. The solid linerepresents a similar correction factor for aHorner-type build-up graph, and will be discussedlater. All of the Miller-Dyes-Hutchinson graphcorrection factors are considerably smaller thanthose for the Horner graph. This means that theapparent permeability-thickness found via theMiller-Dyes-Hutchinson graph could contain a muchgreater error than that from a Horner graph.

One appealing feature of the Miller-Dyes­Hutchinson graph is that knowledge of the produc­ing time is not required to prepare the graph.l~87

But it should be clear that this advantage is moreapparent than real. It is necessary to know theproducing time to be able to complete a Miller­Dyes-Hutchinson analysis properly. Productiontime would be required to enable selection of theproper line on Fig. 35 for permeability correction.This operation would require trial-and-error andthe following procedure is recommended:

1. From Fig. 3~ determine permeability usingthe pseudo-steady state line.

2. Calculate dimensionless producing time tocheck on the permeability correctionfactor.

3. Repeat the above procedure until theproper value of permeability is deter­mined.

If producing times were long enough that pseudo­steady production could be assumed safely, theabove procedure would be simplified.

The Horner Method. The Horner methodrequires a graph of the shut-in pressures versusthe logarithm of (t + bt)/bt; where t representsthe producing time prior to shut-in and bt repre­sents the shut-in time. The dimensionless Hornergraph can be prepared by means of Eq. (47).

Figure 36 presents a Horner-type build-upgraph for a vertically-fractured well with afracture-penetration ratio of 10. Producing timeprior to shut-in is shown as a parameter. As inthe case of the Miller-Dyes-Hutchinson graph, noextensive linear portion is evident in the build­up for any of the curves of Fig. 36. But allcurves do appear to approach a common value ofmaximum slope at long build-up times. The maximumslope is indicated by the dashed line in Fig. 36.Thus the duration of the production period doesnot appear to affect the maximum slope over therange of producing times considered. Inspectionof graphs similar to Fig. 36, but for otherfracture-penetration ratios, indicated that themaximum slope was affected by the fracturepenetration ratio, but not by the producing' time.

As mentioned earlier permeability-thicknesscorrection factors have been prepared by Raghavan,~ al., for the Horner graph. The results forall fracture-penetration ratios are shown as theheavy line on Fig. 35. Again, the correction

factors for a Horner-type graph are not functionsof the duration of the production period. Thusa single line is shown on Fig. 35.

The Semi-Log Approach: An Infinite-Conductivity

Vertically Fractured Well in a Constant Pressure

As the constant pressure square case is ofinterest in geothermal reservoir engineering weshall briefly examine the characteristics ofthe MDH and Horner methods for this case.

The Miller-Dyes-Hutchinson Method. Figure 37presents a typical Miller-Dyes-Hutchinson graphfor a vertically-fractured well in a constantpressure square. The producing time is theparameter of interest, and the fracture pene­tration ratio, x IX f = 15. Equation (47) was usedfor preparing th~ results displayed in Fig. 37.The rationale for using Eq. (47) is discussed inRef. 88. As in the case of the closed squaredrainage region the maximum slope for any of thecurves in Fig. 37 is O.965--much less than theexpected value of 1.151. This maximum slopedecreases as the producing time decreases and asxe/xf decreases.

An important difference between the resultsshown here and that shown in Fig. 33 must benoted. Unlike the closed case, the shut-in well­bore pressure eventually reaches Pi for all pro­ducing times. This is due to' fluid rechargeacross the constant pressure boundary. Followingthe procedure suggested by Raghavill\ et al. 86 ,correction factors can be prepared for this casealso. These are shown in Fig. 38. They are afunction of producing time for times prior tosteady state and the fracture penetration ratio,xe/xf'

The Horner Method. Figure 39 presents atypical Horner graph for a vertically fracturedwell in a constant pressure square (Xe/xf = 15).Unlike the closed square, the shut-in wellborepressure reaches Pi for all producing times due tofluid recharge. Again as in the case of theMiller-Dyer-Hutchinson graph, no extensive linearportion is evident. Correction factors necessaryto use the Horner method to estimate thepermeability-thickness product are presented inFig. 38. For this case the correction factorsare a function of producing time for xe/xf > 1.5.

A comparison of the shape of the build-upcurves shown in Fig. 39 with that for an unfrac­tured well in a constant pressure square shows animportant difference. Kumar and Ramey88 showedthat as producing time increases, the curves moveto the right,and suggested that a system underrecharge could be identified by this property. Inthe present instance, however, the curves move tothe left for small producing times before movingback to the right. Thus, the suggestion of Kumarand Ramey to identify a constant pressure boundarysystem from pressure data is not applicable tovertically fractured wells unless producing timesare very large.

~135-

The Uniform-Flux Fracture

Because of the obvious difficulties involvedin graphical differentiation of the Horner and MDHgraphs and associated problems involved in deter­mining correct slopes, the uniform-flux case hasnot been examined in detail. Furthermore, thetype curve approach is more advantageous to deter­mine permeability-thickness and fracture length.

Pressure Build-up Analysis for Finite-

Capacity Fractures

The basis for the type curve approach forfinite-capacity fractures is identical to thatdiscussed earlier. The draw-down type curves areapplicable if the producing time, t Dxf ' is much

greater than the largest build-up time. This is acritical assumption and has not been exploredfully in the literature. The effect of smallproducing time on build-up data is to give anappearance of a small fracture capacity. Thispoint is demonstrated in Fig. 40 where the effectof producing time on build-up data for aninfinite-conductivity vertical fracture is dis­played. 89 The shape of the curves shown here issimilar to those shown in Fig. 13. For example,the curve for tDxf = 10-1 may be matched with manyof the FcD curves shown in Fig. 13. This, inaddition to the difficulties mentioned earlier,indicates that analysis of build-up data forfractured wells of finite-capacity is a formidablechallenge.

To my knowledge the applicability of semi-logmethods to the finite fracture capacity system hasnot been investigated in any detail. However!considering the results that have been obtainedso far, work along these lines may not be fruitfuL

Determination of Static or Average Reservoir

Pressure

As mentioned in the section titled "Uses ofPressure Transient Data" one of the objectives ofa pressure test is to determine average reservoirpressure for material balance calculations. Forthe case of an unfractured well this may be esti­mated by extrapolating the proper straight line ona Horner or MDH graph to an appropriate shut-intime. However, for the case of a verticallyfractured well no simple method of extrapolationexists since a linear portion is not evident onthe semi-log graphs. A thorough discussion ofthis aspect is beyond the scope of this paper.Pertinent information on this subject may befound in Refs. 54, 63 and 86.

Application to Injection Wells

The preceding discussi0n also forms thebasis for the analysis of the shut-in pressurebehavior in injection wells. For type curve analy­sis the ordinate of the log-log graph should be(p f - P ) rather than (p ~ Pwf s). If thew ,s ws ws,

to investigate each parameter that affects pres­sure behavior individually and obtain the neces­sary solutions. This catalogue would be useful inidentifying the potential characteristics for eachspecific circumstance. It would also be useful inreducing the range of variables that need to beconsidered in analyzing field data. The 3econdavenue is to develop techniques whereby field datacan be analyzed conveniently and at the same timeinsure that the description of the fracture andreservoir are realistic and compatible with pro­duction performance.

semi-log approach is used than the resultspresented in the section titled "The Semi-logApproach: An Infinite-Conductivity VerticallyFractured Well in a Constant Pressure Square"should be used.

Shut-in Pressure Behavior for other Fractured

Systems

The shut-in pressure behavior for wellsintercepting horizontal or inclined fractures hasnot been examined in the petroleum engineeringliterature. The principal reason for thisappears to be the limited application of thesesolutions. It should be noted that the basis foranalyzing data by the type curve approach isidentical to that for a vertically-fractured well.Applicability of the semi-log techniques can alsobe investigated along the lines presented here.

Discussion and Summary

The main object of this survey is to documentmost of the recent work that has been conductedand which is available in the open literature(until Oct. 8, 1977). In doing so I have laboredunder one important restriction. I am aware thatseveral research groups (universities and indus­trial laboratories) are actively working in thisarea. Thus, it is possible that some of theproblems I have outlined here have been solved.Hopefully the results of any such investigationswill be presented shortly.

Judging from the work that has been presentedin the past few years, it is probable that newsolutions which include the effects of non-Darcyflow (within the fracture, on the fracture sur­face, or both), wellbore storage and skindamage, and confining pressure will be discussedin the open literature shortly. Furthermore, itis clear that the effect of fracture height onthe pressure behavior of finite flow capacityfractures will be available in the near future.Another problem which needs consideration is theeffect of the variation in fracture capacity withdistance on pressure response and deliverability.Undoubtedly solutions to most of these problemswill be obtained via the digital computer.

The availability of solutions for specificcases, however, does not necessarily imply thatit would be possible to analyze field data con­veniently. In some instances consideration ofone of the effects mentioned above would provideanswers which are compatible with productionperformance. In other instances a combination offactors would have to be taken into account. Insuch an event simple graphical techniques wouldbe inadequate and one would have to resort toparameter estimation techniques90- 94 (automatichistory matching, inverse problem solving).

In summary, it appears that two avenues areavailable for us to increase our understanding offractured well pressure behavior. The first is

-136-

A =A =bBBc =

cftc

t

CDXf

cr

hh f

h Drf

h =DX fh

fD=

hw

hwD

kk fk z

m(p)II [m(p) J=

m

m'

Pi

Psc

PwD =

Pwf

Pwf ,s

Nomenclature

drainage area, sq. ft.constant defined by Eq. (27)entry ratioformation volume factor, RB/STBconstant defined by Eq. (28)unit storage factor, RB/psifracture compressibility, psi- l

system compressibility, psi- l

dimensionless storage constant based onfracture half-length

relative fracture capacity defined byCinco, et al., dimensionless

dimensionles;-fracture capacity definedby Agarwal, et al.

dimensionless fracture capacity defined byPrats

formation thickness, feetfracture height, feet

dimensionless thickness based on fractureradius

dimensionless thickness based on fracturehalf-length

dimensionless fracture height

edge length of an inclined fracture

dimensionless edge length of an inclinedfracture

horizontal permeability, mdfracture permeability, md

vertical permeability, md

real gas pseudo pressure, psi2/cpdifference in real gas pseudo pressures,

psi2/cpslope of semi-log straight line, psi/log~,

psi2/log~, or psi2/cp/log~

slope in psi/~ or, psi2 /lhour,

psi/cp/lhour

fluid pressure, psidimensionless pressure drop

dimensionless shut-in wellbore pressuredrop

initial pressure in the system, psi

standard pressure, psi

dimensionless wellbore pressure drop

wellbore flowing pressure, psi

wellbore pressure at instant of shut-in,psi

wellbore shut-in pressure, psi

o u / Bp average reservoir pressure, psi

PDs MDH dimensionless wellbore pressure drop

q surface flow rate, STB/D, Mcf/DqD dimensionless flowrate

r = radius, feetr f horizontal fracture radius, feetrw

radius of well, feet

r' = effective wellbore radius, feetws skin factor, dimensionless

dimensionless time based on horizontalfracture radius

dimensionless time based on fracturehalf-length

shut-in time, hourstemperature, oRstandard temperature, oR

fracture width, feetdrainage length, feet

fracture half-length, feetcompressibility factorporosityviscosity, cpangle of inclination of the fracture from

the vertical

Subscripts

CD dimensionless capacityD dimensionlesse external boundary

DXf dimerisionless variable based on fracturehalf-length

f fracturei initialw = wellbore

ACKNOWLEDGEMENTS

The financial support of the Department ofPetroleum Engineering at the University of Tulsais acknowledged with deep gratitude. I am alsoindebted to Dr. Ram G. Agarwal of Amoco ProductionCompany, Tulsa, Oklahoma, for the stimulating dis­cussions we have had on the subject of finite flowcapacity fractures during 1972-1975. I am thank­ful to the Geothermal Program at the LawrenceBerkley Laboratory, University of California, forthe invitation to present this paper.

References

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TINSLEY. ~ Q.!.

VAN POOLLEN.et Q.!.PRATS--.....

~~~::::==-'_--m=:===~r=--cRAFT.!e91~

12

rw =9t inches0 re = 1050 feet~ 10:I:.... Xf/re = 0.5

J

~U«

8J

>:'uzwu 6lL.lL.W

;:4g

lL.

RELATIVE CONDUCTIVITY. Wkf Ik,ft

Fig. 1. Flow efficiency vs. relative conductivity for a vertically fractured well.

e------2 Xe----

FRACTUREPLANE

WELLBOREAXIS

t-lf.,j~ BOUNDARYSURFACES

------> DRAINAGEBOUNDARIESCLOSED ORCONSTANTPRESSURE

-----+--FRACTURE

~--- 2Xe ----lIl....

Fig. 2. Schematic diagram for a vertically fractured system.

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W0::om c-I ~-10-W cL 10~O(/)0::(/)0Ww-10::Z::>Q(/)(/)(/)ZWwo::~a..

o

DIMENSIONLESS TIME, tOXf

Fig. 3. Dimensionless wellbore pressure drop vs. dimensionless time for an infinite­conductivity vertically fractured well in a closed square drainage region.

APPROXIMATESTART OF SEMI-lOGSTRAIGHT LINE(m =1.151 IlOG"')

FRACTUREPENETRATIONRATIO, Xe/Xf

I

INFI NI TE-CONDUCT IVITY

END OF LINEARFLOW PERIODFOR Xe/Xf >1

END OF LINEARFLOW PERIODFORXeIXf=1

2

4

6

8

c3: 10 r---r--r-r'T'"l""TTTr----r---r-r.,..,...I"'TTl--r--r""T""T'1I'""1"T'i"1'"-r--n-,.-..,....,-rrTTl

C-

a.:­oa::oWa::::>(/)(/)wa::a..w0::oCD-I-IW~(/)(/)W-IZoU5zw~

o

DIMENSIONLESS TIME, tOXf

Fig. 4. Dimensionless wellbore pressure drop vs. dimensionless time for an infinite­conductivity vertically fractured well in a closed square drainage region.

-142-

UNI FORM-FLUX

APPROXIMATEEND OF LINEAR FLOW

FRACTUREPENETRATIONRATIO, Xe/Xt

I

tSTART OF SEM'-LOGSTRAIGHT LINE

00

DIMENSIONLESS TIME, t oxf

Fig. 5. Dimensionless wellbore pressure drop vs. dimensionless time for an uniform-fluxvertically fractured well in a closed square drainage region.

0~ 10

0-

ll:' UNIFORM-FLUX0a::0w 8a::::>(/)(/)wa:: 6a..

FRACTUREwa:: PENETRATION0 RATIO, Xe /Xtm-.J 4-.JW~(/)(/)w

2-.JZ0u;zw~ 015 10-2 I 10 102

DIMENSIONLESS TIME, fOX,

Fig. 6. Dimensionless wellbore pressure drop vs. dimensionless time for an uniform-fluxvertically fractured well in a closed square drainage region.

-143-

DIMENSIONLESS TIME, t I)A

Fig. 7. Dimensionless wellbore pressure drop vs. dimensionless time for an infinite­conductivity vertically fractured well in a closed square drainage region.

I 'CONSTANTPRESSUREBOUNDARY

~INFINITE­CONDUCTIVITY

1-2xf-l2x CLOSED

e BOUNDARY

wa::::>(f)(f)wa::0-

wa:: 10°0CD;:tjQ.wa:­::0~lSw...J

~ START OF SEMILOG(f) STRAIGHT LINE~ AOiOo----END OF LINEAR FLOW~ FOR Xe /Xf >1o I0-I '-:::-1-.llQ...l..lll.l.l.--;:-l-~J..J.J..L.U.l~J....-JL...J..J.J.IJ.I.l..-~...l...l...uJJ.l.I----l....l....L-I.l.u.Ll-::-..I.-L..L.I...u.w.

10-3 10-1 I 10 102 103

DIMENSIONLESS TIME, tOXf

Fig. 8. Dimensionless wellbore pressure drop vs. dimensionless time for an infinite­conductivity vertically fractured well in a square drainage region.

-144-

0 U J. .) 'J '9 a ) C)-:,,~? ". -'J c.,J c}

CONSTANTPRESSUREBOUNDARY

START OF SEMI-LOGSTRAIGHT LINE

~ UNIFORM-FLUX

~ CLOSEDl--2X

e--l BOUNDARY

2fi 10 .....--......,....,...,..,.,.,m--T"""""T'""T..".."..,..,...-....,......,....,.....,.,.,.,.,.....-T"""""T'""T..,..".,.".,-r....,......,...,...,.""m0­

Il:'oa:owa:=>~ 10wa:tl.Wa:om..J..JW~(f)

VIW..JZoiiizw

~ 101L..::--l.-L...I..J..J..ULI.l...,...-.l...-.I-1..LJ.J.J.LL--L-L....L...I.J..Uw-_.l...-J....J.-W.UJ.J,,=--L-'-..l...L..I..ULU

~ ~ ~

DIMENSIONLESS TIME, 'ox,XBL 7710-6713

Fig. 9. Dimensionless wellbore pressure drop vs. dimensionless time for an uniform-fluxvertically fractured well in a square drainage region.

4,..--.---.--r--......,...-...,.....-,..--.--...,.....-r----.

CONSTANT PRESSURE BOUNDARYCLOSED BOUNDARY

UNIFORM-FLUX

IN FINITE-CONDUCTI VITY

3

2 I---..,.-='=_~-- - - - - - - .... ~---...

-,!-..........xVl::lCi<l:a:L1Ja:o1Il...J...JL1J:tL1J>i=l>L1JLLLLL1J...J<l:l>oa:0..

@a:

IO!::--'----;:~-.L.~-!7---l.~;!-;;---l.-_=_=_-...l-~

RECIPROCAL FRACTURE PENETRATION RATIO, Xf IXe

XBL 7710-6723

Fig. 10. Effective wellbore radius for a vertically fractured well in a square drainageregion.

-145-

2

Fto • 10Feo = 0

o

DIMENSIONLESS DISTANCE, X/Xt- FRACTURE

..-X......X

WUzi=!(f)

ol2 2w-IZQ(f)zw~

o

XBL 7710-6720

Fig. 11. Effect of fracture capacity on the pressure distribution around a fracturedwell.

-x.....-~...<Ii:::>is<!Q:

WQ:olD..J..JW~W>twI.L.I.L.W

DIMENSIONLESS FRACTURE CAPACITY, F~O

XBL 7710-6710

Fig. 12. Effective wellbore radius vs. dimensionless fracture capacity.

-146-

u -~ -~.,

>:) c -J, ")I.) ;) ,,"_;..1 "t.,:! 'l",~1 0~

FRACTURED WELL IN ANINFINITE RESERVOIR,

Xe/Xf =00

00

o 10~e-n:­oa:owa::JVlVlWa:a..wa:oID-l-lW~VlVlW-lZoU5zw~

o

DIMENSIONLESS TIME, tDXt

XBL 7710-6714

Fig. 13. Dimensionless wellbore pressure drop vs. dimensionless time for a finite-capa­city vertically-fractured well in an infinite reservoir.

RADIAL FLOWSLOPE =1.151/10g-

2

3

o 4 r-~~"""'''''''''-'''-'''''''''''''.......-.....,........rrrnr--,....,-r-MTnr---'''''''''''''''T"TTTII~

e-n:­oa:owa::JVlVl

~wa:g-l-lW~Vl

~-lZoU5zw~

o

DIMENSIONLESS TIME, tOXt

XBL 771 0-6704

Fig. 14. Dimensionless wellbore pressure drop vs. dimensionless time for a finite-capa­city vertically-fractured well in an infinite reservoir.

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DIMENSIONLESSFRACTURE CAPACITY. FCD

0.633.14

6.2831.4500

c~5r--~T'TTTT1rrr---r""""nTT""TT-r-T"'TTTTT1"--~T"T"TTT1....-.-....,...n-nr-r--"7r-r"'TTrTm

0-n:-l?oUJ 4et::;:)

l2~0. 3UJet::g.JiLl 2::(/)(/)UJ.JZoiiizUJ;§OI-c:t:::I:W:il.ll..~...J..J.J.J..IJ.LL.....l-~.u..uI--JL.LLLUJJL--L....I.....L..u..Ll.:--~~.wJ

o ~ ~ dDIMENSIONLESS TIME, tDxf

XBL 7710-673B

Fig. IS. Late time draw-down data for a finite fracture capacity vertically fracturedwell in an infinite reservoir.

DIMENSIONLESS FRACTURE CAPACITY, FCD10'1

• (PWD) INTERCEPT

X TIME TO GET ONSTRAIGHT LINE

3

2

O~_.L-_.L-_.L-_.L-_.L-_..L-._..L-._..L-._..L-_.J

o 0.2 0.4 0.6 0.8

SQUARE ROOT DIMENSIONLESS TIME;YtOXf

4,..---r--....,...--,----,--,.----r--....,...--.,.---,r-=....-.cln:­Oet::oUJa::;:)(/)(/)UJa::0-UJa::oIII-.J-.JUJ::(/)(/)UJ-.JZoiiizUJ:::i:o

XBL 7710-6719

Fig. 16. Dimensionless wellbore pressure drop vs. square root dimensionless time fora finite-capacity vertical fracture.

-148-

o

'"C-

D:<IWC>Z<l:J:U

Wa:::)(f)(f)wa::0..

Fig. 17.

HALF SLOPE LINE

PRODUCING TIME, t

XBL 7710-6739

Effect of skin on fractured well behavior.

10

DIMENSIONLESS TIME, fOXt XBL 771 0-10226

Fig. 18. Dimensionless wellbore pressure drop vs. dimensionless time for a uniform-fluxvertical fracture with wellbore storage.

-149-

10

XBL 7710-10115

FINITE CAPACITY VERTICALFRACTURE IN AN INFINITE RESERVOIR, Xe/X, =00

• Q Q CINCO,!!.!!!:

RAMEY AND GRINGARTEN

COX, - 10-1

10-1

DIMENSIONLESS TIME, t OXf

A- <p, C"W _ 10-17T<p C, X,

B-~ _10-7k <p, CII

10 ~--,~,-r""'r'TTn--'T""'-rT'"T"'T"T'TT"-..,...-r-Y-T"T"I"T"T1r---'-r-T""'"'T'n"Ja~

0.

CLoa::owa::;:)(f)(f)wa::a..wa::olD-l-lW:::(f)

ffl-lZoVizw:::i:o

Fig. 19. Dimensionless we11bore pressure drop vs. dimensionless time for a finite-capa­city vertical fracture with we11bore storage.

10

"

SKIN fACTOR, 920

OIMENSIONLESS WELLBORESTORAGE CONSTANT, Cox,

UNifORM-fLUXWELL IN AN INfiNITE RESERVOIR, X./X, • 00

104 ':7-J...J...LU.!J."::;-'...u........W:;,-'--L.UJWJ:.-.........LJ.LWL.-'....u.ww10-4

o~

'" 10 I --------::>6;....--z~-__;p-51~ F-

awa:illUlwa:<l.wa:om--'--'~UlUl

':':JzoVizw::;:o

DIMENSIONLESS TIME, lox,

XBl7710·6725

Fig. 20. Dimensionless pressure drop vs. dimensionless time for a vertically fracturedwell with skin and we11bore storage.

-150-

0 u ~.,) Lj ') U ~) " ~J~Ji

DIMENSIONLESS TIME, tor fXBL 771 0-1 0224

Fig. 21. Dimensionless wel1bore pressure drop vs. dimensionless time for a uniform-fluxhorizontal fracture in an infinite reservoir.

5

2

3

4

6

ol 7r-"-rT1"TTl1rr--rT""TTTTrrr-r-rrTTTTTT-r-T"TTTlTTr-""17""-1""1'TTT!

a:oa:owa:i7lf3a:0-wa::o(II....I....IW::enenw....IZoUizw:!:is

DIMENSIONLESS TIME, torf

XBL 7710-6703

Fig. 22. Dimensionless wellbore pressure drop vs. dimensionless time for a uniform-fluxhorizontal fracture in an infinite reservoir.

-151-

10

DIMENSIONLESSTHICKNESS.hOXf =10

INFINITE-CONDUCTIVITY

WELL IN AN INFINITE RESERVOIR,Xe/Xf =00

DIMENSIONLESS TIME, tDXf

XBL 7710-6721

Fig. 23. Dimensionless wellbore pressure drop vs. dimensionless time for an infinite-con­ductivity inclined fracture in an infinite reservoir.

IMPERMEABLEBOUNDARIES

hW.j 2(J k . 2hWD =Xf COS Wi<; +Sin (JW

hw =EDGE LENGTHOF FRACTURE

YO

XBL 7710-671 B

,)

Fig. 24. Schematic diagram for an inclined fracture.

-152-

o u

CIl

cr~uItz

"UlI

oo::>w~

-210 L._"'-I--JL...-............J..u..LLl-_..L--....L....L...L..L.L.W

~ I ~

DIMENSIONLESS THICKNESS, hOXf

X8l 1710-6711

Fig. 25. Pseudo-skin factor vs. dimensionless thickness for an inclined fracture.

0.3 O.

ENTRY RATIO, b

0.1

APPROXIMATESTART OF PSEUDO­RADIAL FLOW,b<l~

ocf 16 r----r--r-r-'T""M,-rn---r-...,.........,...,....,....--.,--..,.....,...,.........-r...--~~ ............,..,.,~....,......,.. ....... ......0:1foW0:::::J(f)

~ 100:::a..

~

°~-.J

~ APPROXIMATE UNIFORM-FLUX~ START OF PSEUDO-~ RADIAL FLOW. b= I FRACTURE TO CENTERz APPROXIMATE OF FORMATION

0(f)- END OF LINEAR DIMENSIONLESS THICKNESS.FLOWr5 _I hOXf = 5

::"?: 10 L._::::"2--I--I->-.w..L.J.J.l

IO-_'""'->-->-...l-J..,LLUL--l--l-LJ-l..U.ll-_l.-..L...I..J..J...L.llJ...,2,........L.....l..J..J..l.J.l.U,03

£5 10 10 10

DIMENSIONLESS TIME, t DXf

XBL 7710-6700

Fig. 26. Dimensionless wellbore pressure drop vs. dimensionless time for a limited entryuniform-flux vertical fracture located at the center of the formation.

-153-

z'///Z =h//////.hJ///////////

~WELLBORE

I t:kFRACTURE

-+,.t'-M-LA,L~:,Lr~4LI--.,-~ X

FRONT VIEW

Y

----+i'.~"'b9--~X-Xf I ~+Xf

INFINITE-II CONDUCTIVITY,OR

UNIFORM-FLUX

PLAN VIEW

XBL 7710-6708

Fig. 27. Schematic diagram: a limited entry vertical fracture.

W0::aro-.J-.JW5:Vl 0Vl ~we.-.J.Clz~00..-0VlO::ZOW~w-0::O:J~Vl

XVl~w

00::-0..t(0::>­0::~ZW

o

0.1

~APPROXIMATE STARTOF PSEUDO-RADIALFLOW,b<1

UNIFORM-FLUX

FRACTURE AT CENTER OF FORMATION

DIMENSIONLESS THICKNESS, hox( 5

-210

10

l._o2--l.--l.....L...l.L.J..1.J.L.,,....-JL-J.-1..l.U.llL_.l.-..L..L.LLWL_....L-.L.I..1..L.ulJ'..--1..-'-L..U..I.JJJ

103

I 10 10

DIMENSIONLESS TIME, tDXf

XBL 7710-6712

Fig. 28. Draw-down data for a limited entry uniform-flux vertical fracture in an infinitereservoir.

-154-

o u

5

_-I0.75 0.5

DIMENSIONLESS FRACTUREHEIGHT, hfD

APPROXIMATE STARTOF PSEUDO-RADIALFLOW, b< I

UNIFORM-FLUX

FRACTURE AT CENTER OF FORMATIONENTRY RATIO, b =0.1

W~

oCO...J...JW:i:(f) ~(f)e.~..o2~00..cn@r50~~e~ 101X(f)~W

O~-a..~~

>­~~2W

DIMENSIONLESS TIME, tDXt

XBL 7710-6735

Fig. 29. Draw-down data for a limited entry uniform-flux vertical fracture in an infinitereservoir.

W~

oCO...J...JW:i:(f) ~(f)e.W..o...J ~2~

Q15(f)~20W

~~0:)~(f)

X(f)

O~i=o..<{~

>-~~2W

UNIFORM-FLUX

FRACTURE AT CENTER OF FORMATION

DIMENSIONLESS FRACTURE HEIGHT,hfD =5

DIMENSIONLESS TIME. tDXt

XBL 7710-6733

Fig. 30. Draw-down data for a limited entry uniform-flux vertical fracture.

-155-

w~a::(/)(/)W..JZ 10Q(/)

zw::Eo..J<tUoa::ll.uwa::

ocr......

DIMENSIONLESS TIME, tOXf

XBL 7710-6707

Fig. 31. Reciprocal dimensionless rate vs. dimensionless time for an infinite-conductivityvertical fracture in a closed square reservoir.

ocr......

w~a::(/)(/)W..JZQ(/)

zw::Eo..J<tUoa::ll.UWa::

FRACTURED WELL IN ANINFINITE RESERVOIR,

Xe/Xf =00

-310 -5

10

DIMENSIONLESS TIME, tOXt

XBL 7710-6701

.""Fig. 32. Reciprocal dimensionless rate vs. dimensionless time for a finite-capacity verti­

cal fracture producing at a constant terminal pressure.

-156-

U ;J

FRACTURE PENETRATIONRATIO, Xe/Xj • 10

INFINITE-CONDUCTIVITY

DIMENSIONLESS SHUT-IN TIME, At OA

XBL 7710·6734

Fig. 33. Miller-Dyes-Hutchinson build-up graph for an infinite-conductivity verticallyfractured well in a closed square.

WELL PRODUCED TO PSEUDO­STEADY STATE PRIOR TO SHUT-IN

INFINITE-CONDUCTIVITY

0.51.------';-:;-;;--

Vl 0.--.,..-.,.."T"'I".,.,..n---......,c....-T"T"l"TTT"-..,-..,....,......,..,."..---.-..,.."T'"..".,_-.-.."""T""...........C

Ie;a:­oCl::owCl::::>enenwCl::n.zoen~ 1.0:r:~::>:r:enw6Q: 1.5w 1-_--..J..J~

DIMENSIONLESS SHUT-IN T1ME;AtOA

XBL 7710·6705

Fig. 34. Miller-Dyes-Hutchinson build-up graph for a vertically fractured well in aclosed square--effect of fracture penetration ratio.

-157-

o 0.2 0.4 0.6 0.8 I

RECIPROCAL FRACTURE PENETRATION RATIO,Xf/Xe

XBL 7710-10223

Fig. 35. Permeability-thickness correction for a vertically fractured well at the centerof a closed square.

O..--r-r"TTTn-n--r--t-rn-r"""-"'-""""'TTT1-..,--rT"T"n'TTI

1.5 X10-2

PRODUCING TIME, fDA

1.5X 10-3

10

HORNER TIME RATIO, (HlH)/M

LINE OF COMMONMAXIMUM SLOPE ----1,..,.-,

FRACTURE PENETRATION RATIO,Xe IXf =10

INFINITE- CONDUCTIVITY

InC

D-

a:­o0::oI.LI0::=>(f)(f)I.LI 20::C-

o::I.LIZ0::o:I:

XBL 7710-6702

Fig. 36. Horner build-up graph for a vertically fractured well in a closed square.

-158-

i ~ U \;;.:~ \) J - -, ,I1.,,) \) -- , l.-P l

0

Vl0

Q.

n:00::ClW0::::>(J)(J)

W0::11.

Z0(J)

~I0I-::>

,- :r(J)w>-Cl

I0::W..J..J

~

10-4

I '-REGION STRAIGHTENEDBY MUSKAT GRAPH

FRACTURE PENETRATION RATIO, Xe/Xt' 15

INFINITE - CONDUCTIVITY

10-2

SHUT-IN TlME,AI DA

XBL 7710-6716

Fig. 37.

Fig. 38.

Miller-Dyes-Hutchinson build-up graph for a vertically-fractured well in aconstant pressure square.

MILLER-DYES-HUTCHINSONGRAPH FOR PRODUCING

TIME 'DAot

RESULTS SHOWN HERE ARE STRICTLYAPPLICABLE FOR THE INFINITE­CONDUCTIVITY CASE

XBL 7710·6728

Permeability-thickness correction for a vertically fractured well at the centerof a constant pressure square.

-159-

Fig. 39.

<Ilo

Q.

0:oa:oLUa:::>VlVlLUg: 2

a:LUza:oJ:

3

10 102

HORNER TIME RATIO, (1+111) lilt

XBl 7710-6706

Horner build-up graph for a vertically fractured well in a constant pressuresquare.

10

VI

?.pWVla:wa::~VlVlwa::Q.

~W..JZQ

~w~0

Icl10-4

INFINITE-CONDUCTIVITYWELL IN AN INFINITE RESERVOIR, Xe/Xf =00··

X START OF SEMI-LOGSTRAIGHT LINE

10-2 10-1

DIMENSIONLESS SHUT-IN T1ME,LHOXf

10

XBL 771 0-1 0222

Fig. 40. Effect of producing time on build-up data.

-160-