Hydraulic Fracture Propagation in Layered Formations

A. A. DANESHYMEMBER SPE-AIME

HALLIBURTON SERVICESDUNCAN, OKLA.

in which

. . . . . . . . . . .(2)

L [2(L2+h?)E(k)-L2K(k)]'

. . . . . . . . . . . . . . . . . .(3)

p-a=

hf fracture height

E(k) complete elliptic integral of the secondkind

L = fracture extent (length of a two-dimensionalfracture or radius of a penny-shapedfracture)

E Young's modulu s of material

(1 Poisson's ratio of material

y effective fracture surface energy of material

a least in-situ principal stress

A similar equation for a three-dimensional fractureis derived in Appendix A in the form of

K(k) complete elliptic integral of the first kind

k parameter of the elliptic integrals

Eqs. 1 through 3 show p to decrease with increasingL (Fig. 1). As the fracture becomes larger, it needsless pressure for propagation.

In deriving these equations, no allowance hasbeen made for fluid leak-off into the formation.Leak-off would have two effects: (1) seepage offluid into the rock requires energy and fluid inaddition to that necessary for the fracture volume

~77 E YP - a - (penny-shaped fracture),

- 2 L (l - (12)

p - a = _I 2 E Y 2 (two-dimensional fracture),l77L(l-(1 )

. . . . . . . . . . . . . . . . . . . . .( 1)

in which

fracture. The relationship between this pressureand material properties3 is

ABSTRACT

Most industrial hydraulic fractures are created inlayered formations. During propagation, thesefractures encounter various formations with differentphy sical and mechanical properties. This paperdiscusses the effect of those properties onpropagation of the fracture.

Most of the theoretical studies on fracturepropagation have been extensions of Griffith'swork. 1,2 Based on an energy criterion, Griffithdeveloped a relationship among fracture shape,material properties, and the external force neededfor fracture propagation. The energy source inhydraulic fracturing is the fluid pressure inside the

0037-9999/78/0002-6088$00.25© Society of Petroleum Engineers of AIME

INTRODUCTION

Original manuscript received in Society of Petroleum Engineersoffice Aug. 9, 1976. Paper accepted for publication June 14,1977. Revised manuscript received Aug. 31, 1977. Paper (SPE6088) was presented at the SPE-AIME 51st Annual FallTechnical Conference and Exhibition, held in New Orleans,Oct. 3-6, 1976.

This paper reports theoretical and experimentaldevelopments involving propagation of hydraulicfractures in layered formations.

Unobstructed fractures are shown experimentallyto propagate with a decreasing fracturing fluidpressure. This general trend is in agreement withtheoretical predictions. Restrictions in fracturepropagation result in an increase in fluid pressure.The relative fracturability of rocks can bedetermined by a direct experiment, the results ofwhich are clear, easy to interpret, and include allpertinent parameters, such as physical andmechanical properties of rocks, as well as thereactions between formation and fracturing fluid(for example, leak-off).

Fracturing experiments with layered samplesshow that with strong bonding between rocks it isdifficult to contain a fracture in a formation totally.

The strength of the interface between adjacentformations is shown theoretically to be an importantfactor in fracture containment. With a weak bonding,fracture containment is possible and is associatedwith slippage at the interface. The pattern ofpropagation then will depend on the relativemechanical properties of fractured formations.

FEBRUARY, 1978 33

EXP ERIMENTAL PROCEDURE

INJ ECTED VOLUME, CUIN, -10 2

0.5 1.0 1.5 2.0 2.5

TESTING PROCEDURE

All hydraulic fractures were induced with the aidof a closed-loop, servocontrolled hydraulicpressurizer. As described earlier, the radial dis­placement of the borehole, perpendicular tofracture direction, was used as a feedback signal.This displacement was measured by a strain-gaugedcantilever designed for that purpose. Fig. 4 showsa sample during the test. The instrument attachedto the center of the sample was used to seal theborehole and allow its pressurization.

The borehole in all samples was 1 in. in diameterand located at the- center of the largest face of thesample (Fig. 5). Fractures were induced fromopen holes. All samples were loaded parallel totheir largest dimension. The magnitude of this loadwas about 200 psi of the cross-sectional area underload. Fractures thus created were at the center ofthe sample and parallel to its largest dimension. Inaccordance with industrial hydraulic fractures,height would be the fracture dimension parallel tothe borehole axis, length would be the distancebetween the advancing tip and borehole, and widthwould be the opening of the two faces. Physicaland mechanical properties of the six rock typesused in this research are listed in Table 1.

During all tests, plots were made of fluid pressurevariations vs the injected volume. Two featuresof this plot are significant: (1) the area under thecurve represents the energy consumed in fracturingand (2) since injected volume is an indirect measureof fracture length, a rough measure of fluid pressurevariations and fracture length is given.

variations of fluid pressure vs fracture length andinjected volume. Also, plots were made of variationsin fracture length with displacements at variouspoints in the sample. These results were used forselection of a feedback signal for the closed-loop,servocontrolled hydraulic pressurizing unit. Fromthese results, it was decided that the radialdisplacement of a borehole wall, perpendicular tofracturing direction, was increasing monotonicallyand suitable for use as feedback (Fig. 3).

E • 6. *106 PSI

II • 0.15

r '0.1 LB IN/IN 2

PENNY-SHAPEDFRACTURE

ELLIPTICAL FRACTURE(h1' 20 FEET)

~ 300

~ 250:::>CJ)

~. 200a:a. 150C:l~. 100:::>t; 50«a:u.. 20 40 60 80 100 120 140

FIG. 1 - THEORETICAL FRACTURING PRESSUREVARIATIONS WITH FRACTURE LENGTH (NO FLUID

LEAK-OFF).

The main feature of our experimental setup forfracturing in layered rocks was control of fracturepropagation speed. The induced hydraulic fractureswere visible during testing and were slow enoughfor observation of the fracture and the fluid insideit.

Experiments in this project were preceded by anumerical analysis of hydraulic fracture propagationwith the aid of a finite-element program specificallydeveloped for that purpose. Hydraulic fractureswere simulated numerically in samples withdifferent geometries. Based on these results, plots(such as those in Fig. 2) were made showing

and (2) fluid leak-off affects fracture width causedby rock swelling and possible chemical reactions.

Daneshy4 has analyzed the role of rock propertiesaffecting fracture propagation in the absence offluid leak-off. These properties were given as E, v,and y. For practical purposes, the term determiningfracturability of rocks was given as E . y. Thispaper describes experimental methods of determiningfracturability in the presence of leak-off. It alsopresents a theoretical and experimental study offracture extension in layered rocks.

FIG. 2 - FINITE-ELEMENT RESULTS OFFRACTURING SAMPLES (LABORATORY

SIZE).

FRACTURE ....... F;]12.0"L!J 8H

6.0"

E- 6!106PSI

1/- 0.15r ·0.2 LB.lN/IN 2

0.02 0.040.06 0.08 O. 1 2BOREHOLE DISPLACEMENT,8H' IN,*10-

FIG. 3 - FINITE-ELEMENT RESULTS SHOWINGBOREHOLE DISPLACEMENT VS FLUID

PRESSURE.

*_ 2.0en£l-Ui 1.5a:::>~ 1.0wa:£l- 0.5o5~u.

FRACTURE_!_f \12.0"6.0"

E = 6. it 106 PSI

JI = 0.15

Y = 0.2 LBIN/IN2

0.5 1.0 1.5 2.0 2.5 3.0 3.5FRACTURE LENGTH,IN

5

10

15

Q5...JlL

(j) 200..

Wa::::>C/)C/)wa:0..

,....•

No

34 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

FEBRUARY, 1978

FIG. 5 - GEOMETRY OF VARIOUS SAMPLES.

35

EXPERIMENTAL RESUL TS created fractures were essentially two-dimensionaland could be considered Griffith cracks. Fig. 6shows the results for two example rock types -­Indiana limestone and Kasota. Kasota is the lesspermeable rock. Comparison of the Kasota results(Fig. 6) with those obtained by finite-elementtechnique (Fig. 2) show the two curves are similarin shape. Minor differences between the two curvescome from two sources. Kasota is not totallyimpermeable, and fracturing fluid is compressible,especially at low pressures (it was assumedincompressible in the finite-element analysis).Fig. 7 depicts a fractured sample after theexperiment.

Comparison of the Kasota results with Indianalimestone shows the influence of rock penneability(Fig. 6). As fluid leaks into the rock, the curve forpressure vs injected volume is pushed to the right,indicating larger volume. Fracturing fluid pressuresof the two rocks demonstrate their basic differencein fracturability; Indiana limestone will fractureeasier than Kasota.

In all tests of this type, fluid pressure declinedwith fracture extension, similar to the mannerpredicted by theory and described earlier. Becauseof fluid leak-off, a direct numerical comparison oftheoretical with experimental results could not bemade. However, the effect of leak-off onexperimental results was as would be expected.

During testing of these samples, it was observedthat the advancing fracture tip was always ahead offracturing fluid. The fluid occupied only 60 to 70percent of fracture length. This phenomenon, firstpostulated by Christianovich and Zheltov,S isthe basis for several theoretical studies of fracturepropagation. The tests reported here seem to be thefirst experimental verifications of this postulate.

CiSa.. 1500Wa:::>en 1000enwa:a..z 5000t=0w...,z 0.1 0.2 0.3 0.4 0.5

INJECTED VOLUME, CU IN.

FREE FRACTURE GROWTH

The first set of experiments was to check thevalidity of theoretical analyses and finite-elementresults. Samples of 3- x 12- x I-in. rock were coatedwith a flexible transparent material about 1/8 in.thick (Sample B in Fig. 5). The coating servedthree purposes: (1) it allowed fracturing of thesample without fluid squeezing out; (2) itstransparency allowed visual observation of thefracture and the fluid inside during the test; and(3) its flexibilIty allowed fracture opening withoutconsiderable restrain t at outside faces. The

FIG. 6 - EXPERIMENTAL RESULTS SHOWING FLUIDPRESSURE VARIATIONS FOR UNRESTRICTED FRAC­

TURE GROWTH.

RESTRAINED FRACTURE GROWTH

In the next phase of research, tests wereconducted to study fluid pressure variations in afracture that was restrained in its growth. Eachsample was coated with a stiff transparent material.Once the created hydraulic fracture reached thiscoating, it could not penetrate or deform the'coating. Thus, the fracture was free to move awayfrom but restrained along the borehole. Sample C inFig. 5 is one such specimen. Fig. 8 shows theresulting variations of fluid pressure with inj ectedvolume. Fluid pressure increases as the fracturepropagates away from the borehole, which increasesinjected volume as well. The fluid pressureincreased because the stiff coating on the specimen

TABLE 1 - MECHANICAL AND PHYSICAL PROPERTIES OFROCKS USED IN THIS RESEARCH

FIG. 7 - UNRESTRICTED FRACTURE GROWTH INCARTHAGE LIMESTONE.

Rock Type

Carthage limestoneIndiana limestoneKasotaPecos sandstoneBedford limestoneLuders limestone

E X 10-6 psi

10.06.06.71.54.74.0

L0.280.320.300.250.360.30

Porosity(percent)

0.918.410.318.011.717.4

Permeability(air, md)

0.007207.0

0.060.30.940.9

36 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

acted as if it :vere pinching the fracture, thusreducing its width. Width reduction made it difficultfor the fracture to propagate, and higher fluidpressure was needed. Fig. 9 shows a fracturedsample.

RELATIVE FRACTURABILITY

The relative fracturability of two rocks isdetermined by comparing the magnitude of the fluidpressure needed to extend identical fracture lengths

in the rocks. Curves simiiar to those in Fig. 6provide a way to study relative fracturabilities.

A simpler and more definitive way of determiningrelative fracturability of two rocks is to observefracture propagation in a sample composed of bothrocks. A composite sample was made by cementingslabs of the two rocks under study (Samples' D andE in Fig. 5). In Sample D, a 3- x 4- x I-in. slab ofCarthage limestone was cemented to a 3- x 6- x I-in.slab of Pecos sandstone. The borehole wascontained entirely in Pecos sandstone. Contactsurfaces between the two rocks were surface-groundand cemented together with a thin layer of adhesive.To account for the contribution of the adhesive,the 3- x 6- x I-in. slab of Pecos sandstone wasmade of a 3- x 4- x I-in. slab cemented to a 3- x2- x I-in. one. Thus, the induced hydraulic fracturewould encounter a thin layer of adhesive on bothsides of the borehole.

The composite sample so prepared was coatedwith a thin layer (about 1/8 in.) of the transparentflexible coating. This allowed an unrestrainedfracture growth.

The composite samples were prepared in pairs,so that two rocks in one sample were arrangedopposite two in another sample (D and E in Fig.5). The results obtained from one test could bereaffirmed by the other.

Fig. 10 presents results of the relativefracturability tests of Carthage limestone and Pecossandstone. (The lighter rock is Carthage and thedarker one is Pecos.) In the sample on the right,the fracture induced in Pecos did not penetrateinto Carthage and preferred extension in Pecos.

The photograph on the right shows the fractureextending through the cemented interface in Pecosall the way to the top. The bottom wing of thesame fracture has bent at the interface with Carthageand continued to stay in Pecos. This indicated thatthe fracture preferred propagating in Pecossandstone. In the photograph to the left, thehydraulic fracture begins in Carthage and extendsthrough both rocks, indicating that Pecos sandstonecannot stop a fracture begun in Carthage limestone.In fact, examination of this sample showed thatthe hydraulic fracture was extending in Pecossandstone faster than in Carthage limestone.

Composite samples present a quick and definitiveway of detennining the relative fracturability of tworocks.

FRACTURE EXTENSION IN LAYERED SAMPLES

The purpose of these experiments was to observevertical fracture growth in layered samples bondedtogether strongly.

Fracture extension in layered rocks wasinvestigated by tests with composite samples madeof two different rock types. A 6- x 12- x 2-in. slabof rock was sandwiched between two slabs ofanother 6- x 12- x I-in. rock to yield a 6- x 12- x4-in. sample. All faces were surface-ground beforebeing cemented to each other. The outside faces ofthe sample were surface-ground again. The boreholeextended through all three layers and was 4 in.long. It was cased at top and bottom with a I-in.open-hole section at its center to assure fracture

Ci5~1500a::=>(f)(f) 1000wa::0..

z 500o~uw...,z

BEDFORD

O. 1 0.2 0.3 0.4 0.5INJECTION VOLUME, CU IN

FIG. 8 - EXPERIMENTAL RESULTS SHOWING THEPROPAGATION OF VERTICALLY RESTRICTED

FRACTURES.

FEBRUARY, 1978

FIG. 9 - RESTRICTED FRACTURE GROWTH INCARTHAGE LIMESTONE.

37

FIG. 10 - EXPERIMENTAL RESULTS SHOWING THE RELATIVE FRACTURABILITY OF TWO ROCKS.

0.1 0.2 0.3 0.4 0.5INJECTION VOLUME,CU IN.

FIG. 11 - FLUID PRESSURE VARIATIONS DURINGFRACTURING OF LAYERED SAMPLES.

pressure on further extension. A more severe caseof temporary fracture stoppage is shown by CurveC, which corresponds to Carthage limestone betweentwo layers of Indiana limestone. After breakdownin Carthage limestone, the fracture continued itsfree propagation until it reached the interface. Atthis point, the fracture stopped at the interfacebecause its continued propagation into Carthagelimestone required additional fluid pressure. Oncethe fracture crossed the interfaces, fluid pressuredeclined as expected.

Figs. 12 and 13 depict cross-sections of layeredsamples after fracturing. The parts of the samplein front of each cross-section are removed for aclear demonstration of the created fractures.

initiation in the middle rock. The sample faces tobe fractured were coated with the transparentflexible material discussed earlier. Sample A inFig. 5 shows the final sample shape. Care wastaken to cement the rocks with a very thin layerof adhesive.

All fractures induced in layered samples crossedthe interface and extended to the outside layers.The only case of total fracture containment was ina sample of Pecos sandstone between two layersof Plexiglass. In this sample, the bond at theinterface broke as a result of fracturing andprevented fracture extension into the Plexiglass.

Fig. 11 shows three typical pressure behaviorsduring fracturing in layered rocks. Curve Acorresponds to Pecos sandstone between two layersof Luders limestone. The fluid pressure had ageneral decrease with increasing fracture extent,although the wavy shape of the curve indicates thediscontinuous nature of fracture propagation. Also,although the fluid pressure is decreasing, thegeneral shape of the curve is different from thosepresented in Fig. 6. Curve B corresponds to Pecos

sandstone inside two layers of Kasota. This curveindicates that when the fracture reached theinterface, it did not propagate immediately intoKasota. The pressure :emained constant for a shorttime during which the fracture continued to propagateinside Pecos sandstone. This extension and theincreased fluid pressure forced fracture propagationinto Kasota with a resulting decrease in fluid

~ 2000

Wa:$ 1500(/)UJa:a.. 1000

~i=l>UJ...,~

LEUDERS LIMESTONEA: PECOS SANDSTONE

LEUDERS LIMESTONE

PECOS SANDSTONEB: KASOTA

PECOS SANDSTONE

INDIANA LIMESTONEC: CARTHAGE LIMESTONE

INDIANA LIMESTONE

38 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

The rocks used in layered samples had differentphysical and mechanical properties, as well asfracturabilities. However, experimental resultsshow that these differences were insufficient tostop fracture growth at the interfaces, whichsuggests that restricted fracture heights may occurmore as a result of a weak interface than of therock properties. If so, restricted heights are morelikely to exist at shallower depths, where bondingbetween different rocks is expected to be weaker,than at greater depths. For strong interfaces,barriers may need to be defined as formations thatreduce vertical fracture growth rather than preventit altogether. Such formations can be identified bytheir relative fracturability.

THEORETICAL ANALYSIS

Consider a brittle formation, (a), sandwichedbetween two layers of another formation, (b). Thetwo interfaces between the rocks are designated(ab). Let E a , va', and Ya denote the mechanicalproperties of (a), and E b' vb' and Yb those of (b).The shear strength of the interface is denoted byr abo Let (a) contain a propagating hydraulicfracture.

The fluid pressure needed for fracture propagation

in (a) is governed only by (a) properties, as longas the fracture is contained totally. This pressureis necessary to overcome the resistance of (a) tofracturing. The magnitude of pressure is governedby an equation similar to Eqs. 1 through 3. Thefracture will continue to propagate in (a) as longas fluid pressure inside is sufficiently large. Themagnitude of this pressure depends on fractureextent and becomes smaller as the fracture isextended (Fig. 1).

Fracture behavior will be governed by theproperties of (a), (b), and (ab) at the interfaces. Ingeneral, the hydraulic fracture may encounter oneof the following situation.

1. Shear strength of the interface, Tab' is so lowthat the fracture-induced shear stress along (ab) islarger than rabo In such a case, the hydraulicfracture will stop at the in terface but will can tinueto propagate in (a). Shear failure will occur along(ab), and the fracture will have a shape similar tothat in Fig. 14. The properties of (b) will have noeffect on fracture propagation. When the fracturereaches the interface, the magnitude of fluidpressure in the fracture depends on its size - thelarger the extension, the less fluid pressure. For afracture that has just reached the interface, thelarger extensions normally will be associated withthicker formations. Since the value of fracture­induced shear stress at (ab) is directly proportionalto fluid pressure, the occurrence of a shear failureat (ab) depends on formation thickness. The thinnerthe formation is, the better will be the chances forfracture slippage at the boundary.

A special case is when rab = O. In this instance,the fracture always stops at the boundary. Such afracture has a constant width along its height. Itsgeometry can be predicted accurately by severalmethods. 6 ,7

2. Shear strength of the interface, rab' is large.In such cases, one of two situations may beencountered when the fracture reaches (ab). First,fluid pressure in the fracture is sufficient to extendit into (b). Fracture will propagate into (b) withoutdifficulty and will grow mostly in (b) whilecontinuing to extend in (a). The ratio betweenfracture extensions in (a) and (b) will depend ontheir mechanical and phy sical properties. Themathematical form of this dependence is not known

(a b)(b)

(a)FRACTURE

FIG. 13 - HYDRAULIC FRACTURE INDUCED IN ALAYERED SAMPLE.

FEBRUARY, 1978

FIG. 14 - VERTICAL FRACTURE PROPAGATION INTHE PRESENCE OF A WEAK INFLUENCE,

39

at present, although the relative fracturabilities of(a) and (b) can be determined experimentally. Notethat once the fracture grows into (b), Eqs. 1 through3 are no longer valid for computing the fluidpressure needed for fracture extension. In fact,fracture growth in (b) will force its extension in(a) with less fluid pressure than would otherwisebe necessary. This is primarily a geometry effect;a fracture opening in (b) will induce tensilestresses at its tip in (a), causing fracture extension,and vice versa.

Second, when fluid pressure in the fracture is notenough to extend it into (b), the hydraulic fracture

stops at the interface. Further extension of thefracture in (a) will require an increase in the fluidpressure because of the pinching effect of (b). Thefracture will grow into (b) whenever fluid pressurebecomes sufficient. At this point, the fracture willbreak into (b), and fluid pressure will begin onceagain to decrease. The rate of fracture extensionin (b) will be smaller than that in (a).

If (b) is less fracturable than (a), it is possiblefor fluid pressure to cause shear failure at (ab)before it extends into (b). The fracture will beginto slide at the interface as in the first situation.

If Tab is large and (b) is less fracturable than(a), the fracture again will be contained totally in(a). The hydraulic fracturing treatment may have tobe terminated because of pressure limitations. Theexact pattern of thi s type of fracture propagation isunknown now.

DISCUSSION

The theory and research presented is part of acontinuing effort involving fracture propagation inlayered rocks. Research so far supports the generalresults obtained from theory. First, it is observedthat unrestricted fracture propagation usually isaccompanied by a decrease in fracturing fluidpressure. Also, it is seen that obstacles to fracturegrowth result in pressure increase.

The most significant aspect of the describedexperimental procedure is its similarity to fieldfracturing. By using actual cores and fracturingfluids, such factors as penneability and porosityare incorporated in the experiments. These are theparameters whose influence on fracturing had notbeen investigated yet, either theoretically orexperimentally.

Experiments on the propagation of restrictedfractures has supported the validity of theoreticaldevelopments, even though these ignore rockpermeability and fluid leak-off. Facing a restriction,the fracturing fluid pressure increases to remove theobstacle. The primary factor in fracturing of layeredformations is the strength of the interface. Weakinterfaces are likely to stop fracture propagation,regardless of the relative properties of formations.Strongly bonded interfaces eventually will allowfracture extension through them.

More theory and research are needed before thequestion of fracture extension in layered rocks is

40

understood thoroughly. In particular, efforts shouldbe directed toward investigating the case of a rockbetween two dissimilar formations. Another con­sideration is the angle between the interface andthe fracture planes. Also, it will be useful if theinfluence of rock permeability on fracturepropagation could be isolated. The final goal shouldbe to determine the three-dimensional fracturegeometry from measurement of basic rock propertiesmade with either cores or logs.

NOMENCLATURE

A f created fracture surface area

E Young's modulus of rock

E(k) complete elliptic integral of the second kind

hi height of open-hole section

hf fracture height

K(k) complete elliptic integral of the first kind

k parameter of the elliptic integrals

L fracture lenpth

p fracturing fluid pressure

a least in-situ principal stress

y effective fracture surface energy

f-1 Poisson's ratio of rock

ACKNOWLEDGMENT

The author would like to thank David Meadowsfor assistance with the experiments and the manage­ment of Halliburton Services for permission topublish this paper.

REFERENCES

1. Griffith, A. A.: "The Phenomena of Rupture and Flowin Solids," Phil. Trans., Royal Society, London (1921)Vol. A221, 163-198.

2. Griffith, A. A.: "The Theory of Rupture," Proc., Int.Congo Appl. Mech. (1924).

3. Sneddon, 1. N.: "The Distribution of Stress in theNeighborhood of a Crack in an Infinite Solid," Proc .,Royal Society, London, Series A (Oct.-Dec. 1946)187.

4. Daneshy, A. A.: "Rock Properties ControllingHydraulic Fracture Propagation," paper SPE 5752presented at the SPE European Spring Meeting,Amsterdam, April 8-9, 1976.

5. Christianovich, S. A. and Zheltov, Yu, P.: ttFormationof Vertical Fractures by Means of Highly ViscousLiquid," Proc. Fourth World Pet. Congo (1955) Vol.2, 579-586.

6. Geertsma, J. and de Klerk, F.: "A Rapid Method ofPredicting Width and Extent of Hydraulically InducedFractures," ]. Pet. Tech. (Dec. 1969) 1571-1581;Trans., AIME, Vol. 246.

7. Daneshy, A. A.: "On the Design of Vertical HydraulicFractures," ]. Pet. Tech. (Jan. 1973) 83-93; Trans.,AIME, Vol. 255.

8. Daneshy, A. A.: "Three-Dimensional Propagation ofHydraulic Fractures Extending From Open Holes,"Applications of Rock Mechanics, Proceedings of ASCE15th Symposium on Rock Mechanics, E. R. Haskins,Jr. (ed.).

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

9. Key, P. L.: teA Relation Between Crack Surface Dis­placements and the Strain Energy Release Rate," Int.]. Frac. Mech. (Dec. 1969) Vol. 5, No.4, 287-296.

aE (k) = aE (k) . a (k 2)

aL a (k 2) aL

Or

will simplify Eq. A-2 to read

aE (k) = E (k) - K (k)a (k 2 ) 2k2

2(1-fl2) . L[2(hI2 +L2)E(k)-L2K(k)]

Y = (p -a)2 .3E [L2+hI2][E(k)]2

. . . . . . . . . . . . . . . . . . .(A-4)

a (k 2) = -2 L k2

aL hl2

(p -a) =

Consider an isotropic, homogeneous, linearlyelastic material with Young's modulus, E, Poisson'sratio, fl, and effective fracture energy, y. LethI denote ·the height of the pressurized open-holesection, or the interval between the highest andlowest perforations accepting fluid in a cased hole.

Daneshy8 has shown experimentally that a hydraulicfracture induced under such conditions will beelliptical in shape, with its major axis, hI' alongthe borehole, and the minor axis, L, perpendicularto it. The two parameters hi and L will be relatedto each od1 er by

FRACTURE EXTENSION CRITERIA FORA THREE-DIMENSIONAL HYDRAULIC

FRACTURE

APPENDIX A

hl=L2+h I2 . . . (A-I) . . . . . . .( A- 5)

For an elliptical crack with axes hi and L, Key9

has given the fracture extension criteria as For the special case of ~1 0, corresponding toeither a penny-shaped fracture (that is, horizontalfracture), or an elliptical fracture with a largevalue of L, one will obtain

L = hi'

h 2k2 = _1_ = 0

hl 2 '

IT2'

K(O)E(O)

fluid pressure in the fracture

least principal stress

complete elliptic integral of the secondkind.

p

a

E(k)

in which

AI denotes fracture surface area, which 10 thiscase would be p - a =

ITEy

Substituting A I from Eq. A-3 in A-2, using Eq.A-I and the following relationships,

A I = 21Th fL . . . .

a a aLaAf aL aA I

,

aL hi-- ,aA

I2IT (h1

2 + L2)

. . . . . . (A-3) Eq. A-5 is identical to the criterion given bySneddon 3 for a penny-shaped fracture.

It is important to note that under identicalmaterial conditions a penny-shaped fracture (Eq. 2)will need the highest pressure for fracture extension,and a Griffith fracture (Eq. 1) will require thelowest. An elliptical fracture (Eq. 3) will startwith pressures near Griffith values and as it growsl,arger will tend toward a penny-shaped fracture.

***

FEBRUARY, 1978 41