1972 Mechanics of Hydr Frac

8/12/2019 1972 Mechanics of Hydr Frac

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240 M. King Hubbert and David G. Willis

L E A S T P R I N C I P A L< S T R E S S

k.

FIG. 1Stresselement an d preferred plane of fracture.

During the last 2 years, the writers have beenengaged in a crit ical reexamination of thisproblem, and, because the results obtained havesustained the conjecture offered earlier by Hubbert (1953), the principal content of this paperis an elaboration of that view.STATE O F STRESS UNDERGROUND

The approach commonly made to thethe problem of unde rgrou nd stresses is to assume that the stress field is hydrostatic ornearly hydrostatic, the three principal stressesbe ing approximate ly equa l to one another andto the pressure of the overburden. That this assumption cannot generally be true is apparentfrom the fact that, over long periods of geologic time, the earth has exhibited a high degreeof mobility whereby the rocks have been repeatedly deformed to the limit of failure byfaulting and folding. In order for such deformation to occur, substantial differences between the principal stresses are required.

The general stress condition underground istherefore one in which the three mutually perpendicular principal stresses are unequal. Iffluid pressure were applied locally within rocksin this condition, and if the pressure were increased until rupture or parting of the rocks resulted, that plane along which fracture or parting would first be possible would be the oneperpe ndicu lar to the least principal stress. It ishere postulated that this plane is also the onealong which parting is most likely to occur(Fig . 1) .In order, therefore, to have a mechanical basis for anticipating the fracture behavior of therocks in various localit ies, i t is necessary toknow something concerning the stress statesthat can be expected. T he best available evidence bearing upon these stress conditions is

the failure of the rocks themselves, either byfaulting or by folding.The manner in which the approximate stateof stress accompanying various types of geologic deformation may be deduced was shownin a paper by Hubbert (1951); the rest of thissection is a para phr ase of that paper .Figures 2 and 3 show a box having a glassfront and containing ordinary sand. In the middle, there is a partit ion which can be movedfrom left to right by turning a handscrew. Thewhite l ines are markers of powdered plaster ofparis which ha ve no mech anical significance.As the partit ion is moved to the right, a normalfault with a dip of about 60 develops in thele f t-hand compar tment (Fig . 2) . With fur thermovement, a series of thrust faults with dips of

about 30 deve lops in the r ight-hand compart me n t (F ig . 3 ) .The general nature of the stresses which accompany the failure of the sand can be seen inFigure 4 . Adopt ing the usua l convent ion ofdesignating the greatest, intermediate, and leastprincipal stresses by


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Mecha nics of Hydraulic Fracturing 241the right, and o^ will be the vertical stress,which is equal to the pressure of the overlyingmater ia l . In the r ight-hand compartment , however, < Jwill be horiz ontal, increasing as thepartit ion is moved, and o3 will be vertical andequal to the pressure of the overlying material.A third type of failure, known as transcurrentfaulting, is not demonstrated in the sandboxexperiment. This type of failure occurs whenthe greatest and least principal stresses are bothhorizontal, and failure occurs by horizontalmotion along a vertical plane. In all three kindsof faults, failure occurs at some critical relationbe tween u1 and o-3.To determine this crit ical relation, i t is firstnecessary to obtain an expression for the valuesof the nor ma l stress


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FIG . 7Shear box for measuring ratio T/V at which slippage occurs.

loose sand, is approximate ly 30 . These c r i t i cal stress values may be plotted on a Mohr diagram, as shown in Figure 8. The two diagonallines comprise the Mohr envelopes of the material , and the area between them represents stable combinations of shear stress and normalstress, whereas the area exterior to the envelopes represents unstable conditions. Figure 8thus indicates the stabili ty region within whichthe perm issible values of In the case of normal faulting,the horizontal principal stress is progressivelyreduced, thereby increasing the radius of thestress circle until i t becomes tangent to theMohr envelopes. At this point, unstable conditions of shear an d norm al stress are reached,and faul t ing occurs on a p lane making an angleof 45 + /2 with the least stress. For sandhaving an angle of internal friction of 30, thenormal fault would have a dip of 60, whichagrees wi th the previous experiments . For thecase of thrust faulting, the least principal stresswould be vertical and would remain equal tothe overburden pressure , whereas the horizontal stress would increase progressively until unstable conditions occurred and faulting tookplace on a plane making an angle of 45 -f < p / 2with the least principal stress, or 45


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Mechanics of Hydraulic Fracturing 243

FI G 9 Mohr d iagram show ing possible range of horizonta l stress for a given vertical stress as. Horizontalstress can have any value ranging from approximately one third vertical stress, corresponding to normal faulting, to approximately three times vertical stress, corresponding to reverse faulting.cally and the Mohr envelopes become approximately parallel with the


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244 M. King Hubbert and David G. WillisOne additional modification in the theoretical analysis is needed before it is directly appli

cable to geologic phenomena. Sedimentaryrocks are both porous and permeable, and the :rpore spaces are almost invariably occupied byfluids, usually water, at some pressure, p. It isnecessary to know the effect which is producedby the fluid pressure upon the mechanical properties of the rock.This question was specifically investigated byMcHenry (1948) , who ran a la rge se r ies oftests on duplicate specimens w ith and w ithoutenclosure by impermeable jackets, using nitrogen gas to produce the pressure (p). He foundfor the unjacketed specimens that, when the axial compressive stress, S, was corrected for theopposing fluid pressure, p, the value of the re

sidual effective stress, o- = S p, at which failure occurred was to a close approximation constant and independent of the pressure (p) ofthe permeating fluid.This result is directly applicable to the behavior of rocks undergro und. Porous sedimentary rocks are normally saturated with fluid under pressure and constitute a mixed solid-fluidstress system. The stress field existing in thissystem may be divided into two partial stresses:(1) the hydrosta t ic pressure , p, which pervadesboth the fluid and solid constituents of the system, an d ( 2) an additiona l stress in the solidconstituent only. The total stress is the sum ofthese two.If, across a plane of arbitra ry or ientation, Sand T are the normal and tangent ia l components, respectively, of the total stress, and o-and T the corresponding com ponents of thesolid stress, then, by superposition,

S =


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Mechanics of Hydraulic Fracturing 245l imits. However, in most regions a certain typeof deformation is usually repetit ive over longgeologic periods of t ime, indicating tha t thestresses of a given type are psrsistent and notfar from the breaking point most of the time.The orientation of the trajectories of theprincipal stresses in space is largely determinedby the condition which they must satisfy at thesurface of the earth. Th is is a surface alongwhich no shear stresses can exist . For unequalstresses, the only planes on which the shearstresses are zero are those perpendicular to theprincipal stresses; therefore, one of the threetrajectories of principal stress must terminateperpendicular to the surface of the ground, andthe other two must be parallel with this surface.Thus, in regions of gentle topography and simple geologic structures, the principal stressesshould be, respectively, nearly horizontal andvertical, with the vertical stress equal to thepressure of the overlying material.Therefore, in geologic regions where normalfaulting is taking place, the greatest stress, altshould be approximately vertical and equal tothe effective pressure of the overburden,whereas the least stress, cr3, should be horizontal and most probably between one half andone third the effective pressure of the overburden.

However, in regions which are being shortened, either by folding or thrust faulting, theleast stress should be vertical and equal to theeffective pressure of the overburden, whereasthe greatest stress should be horizontal andprobably between two and three times theeffective overburden pressure.

In regions of transcurrent faulting, both thegreatest and least stresses should be horizontal,and the intermediate stress,


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2 4 6 M. King Hubbert and David G. Willis

BLAINE ANHYDRITE(DR Y,24 C, 0-2 00 0 ATM PRESSURE)2 2 - 4 3

4000 5000K G / C M

FIG. 12Mohr envelopes fo r Blaine Anhydri te measurements by John Handin).A large part of the region of West Texas andthe Mid-Continent is a region of tectonic relaxation characterized by older normal faults. Thesi tua t ion here i s somewhat more ambiguousthan that of the Gulf Coast because faulting inthese regions is not now active. However, be

cause evidence of horizontal compression islacking, i t is sti l l reasonable to assume that arelaxed stress state in these areas is the moreprobable one at present.In contrast, California is in a region whereactive tectonic deformation is occurring atpresent, as indicated by the recurrence of earthquakes, by extensive folding and faulting of therocks during Holocene time, and by slippagesalong faults and measurable movements of elevation bench marks during the last few decades. All three of the types of stress patterndescribed probably occur in different parts ofthis region; but, in areas sti l l undergoing activecompression, the greatest stress must be essentially horizon tal, wh ereas the least stress wouldbe the effective weight of the overburden.It should be understood that the foregoinganalysis of faulting is used only as a means ofestimating the state of stress underground, andthat the shearing mechanism of faulting is quitedist inc t f rom the mechanism of producing hydraulic fractures, which are essentially tensionphenomena . However , an understanding of theregional subsurface stresses makes it possible to

analyze the stress conditions around the borehole and to determine the actual conditions under which hydraulic tension fractures will beformed.STRESS DISTORTIONS CAUSED BYB OR EHOLE

The presence of a well bore distorts thepreexisting stress field in the rock. An approximate calculation of this distortion may be madeby assuming that the rock is elastic , the borehole smooth and cylindrical, and the boreholeaxis vertical and parallel with one of the preexisting regional principal stresses. In general,none of these assumptions is precisely correct,but they will provide a close approximation tothe actual stresses. The stresses to be calculatedshould all be viewed as the effective stressescarried by the rock in addition to a hydrostaticfluid pressure, p, which exists within the wellbore as well as in the rock. The calculation ismade from the solution in elastic theory for thestresses in an infinite plate containing a circularhole with its axis perpendicular to the plate.This solution was first obtained by Kirsch(1898) and la te r was g iven by Timoshenko(1934) and by Miles and Topping (1949) .Expressed in polar coordinates with the center of the hole as the origin, the plane-stresscomponents a t a poin t 6,r, exterior to the holein a plate with an otherwise uniform uniaxialstress, dA, are given by


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M e c ha nic s o f Hy dra u l ic Fra c tur ing 247


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1 Ap

CT -Ap

FI G . 15- -Stresses caused by a pressure Apwithin well bore.EfFect of Pressure Applied in Borehole

The application within the borehols of afluid pr ess ure in excess of the or igina l fluidpressure produces additional stresses. For anonpenetrating fluid, these stresses may be derived from the Lame solution for the stresses ina thick-walled elastic cylinder, which was givenby Timoshenko (1934) . I f the oute r radius ofthe cylinder is allowed to become very largeand the external pressure is set equal to zero,the solution becom es applicab le to the well-bore problem, and the radial, circumferential,and vertical stresses become

a, = + Ap\p is the increase in fluid pressure inthe well bore over the original pressure, a is thehole radius, and r is the distance from the centerof the hole.The circumferential stresses due to a pres

sure Ap in the well bore are shown in Figure15 . The stresses given are those caused by Apalone, and to obtain the complete stress field itis necessary to superpose these stresses uponthose caused by the preexisting regional stresseswhich have been calculated. This method isil lustrated in Figure 16, in which a pressureequal to 1.6 uA is applied to the well bore forthe case in which < JBI


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Mechanics of Hydrau lic Fracturing 24 9turing by pressure applied in well bores areeffectively zero, and that the pressure requiredto produce a parting in the rocks is only thatrequired to reduce the compressive stressesacross some plane in the walls of the hole tozero.As the pressure is increased, the plane alongwhich a fracture will commence will be thatacross which the compressive stress is first reduced to zero. In the case of a smooth cylindrical well bore, this plane must be vertical andperpendicular to the least principal regionalstress. For the cases il lustrated in Figure 14,the least compressive stress across a verticalplane at the walls of the hole ranges from twiceaA to zero . Therefore , the down-the-hole pressure required to start a vertical fracture witha nonpenetrating fluid may vary from a value oftwice the least horizontal regional stress to zero,depending upon the a/aA ra t io .It can be seen from equation 7 that pressureinside a cylindrical hole in an infinite solid canproduce no axial tension; thus, i t is suggestedthat i t is impossible to initiate horizontal fractures. However, under actual conditions in wellbores,end effects should o ccur at well botto ms orin packed-off intervals in which axial forcesequal to the pressure times the area of the crosssection of the hole would be exerted upon theends of the interval. Furthermore, irregularit iesexist in the walls of the borehole which shouldpermit internal pressures to produce tension.

In particular, as has been suggested by Bug-bee (1943), the initial fractures may be jointswhich have sep arated sufficiently to allow theentrance of the fluid, in which case it is onlynecessary to apply sufficient pressure to holdopen and extend the fracture.Injection pressuresOnce a fracture hasbeen started, the fluid penetrates the parting ofthe rocks and pressure is applied to the wallsof the fracture, thereby reducing the stressconcentration that previously existed in thevicinity of the well bore; the pressure, Ap , re quired to hold the fracture open in the case ofa nonpenetrating fluid is then equal to thecomponent of the undistorted stress field normalto the plane of the fracture. A pressure onlyslightly greater than this will extend the fractureindefinitely, provided it can be transmitted tothe leading edge, as can be seen from an analysis of an ideally elastic solid (Fig. 17). Thenormal stresses across the plane of a fracturenear i ts leading edge are shown for the casein which the applied pressure, Ap , is slightlygreater than the original undistorted stress

FIG. 17Stresses in vicinity of a crack in a stressedelastic material when pressure acting on walls ofcrack is slightly greater than stress within the material.

field, < JA.This s olution is derived directly fromthe solution for the stresses in a semi-infinitesolid produced by a distributed load, whichwas presented by Timoshenko (1934) .The tensile stress near the edge of the fracture approaches an infinite magnitude for aperfectly elastic material. For actual materials,this stress will still be so large that a pressure Aponly slightly greater than aA will extend thefracture indefinitely. The minimum down-the-hole injection pressure required to hold openand extend a fracture is therefore slightly inexcess of the original undistorted regionalstress normal to the plane of the fracture. Theactual injection pressure will , in general, behigher than this minimum because of frictionlosses along the fracture.Pressure behavior during treatment A c omparison of the breakdown and injection pressures required for nonpenetrating fluids and forvarious values of aB/a A shows that there are,in general, two types of possible down-the-holepressure behavior during a f rac tur ing t rea tment(Fig . 18) . The pressures, Ap , are increasesmeasured with respect to the original fluidpressure in the rocks. In one case the breakdown pressure might be substantially higherthan the injection pressure, a situation whichwould probably correspond to a horizonta lfracture from a relatively smooth well boreor to a vertical fracture under conditions inwhich the two horizontal principal stresses,< TAa nd uB, were nearly equal. In the secondcase, there is no distinct pressure breakdownduring the t rea tment , indica t ing tha t the pressure required to start the fracture is less than,or equal to, the injection pressure. Such a situation would correspond to a horizontal or vertical fracture starting from a preexisting open-


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250 M. King Hubbert and David G. WillisF L U I D P R E S S U R E D I F F E R E N C E B E T W E E N

' TH E F O R M A T I O N A N D T H E W E L L B O R EF L U I D P R E S S U R E I N T H E W E L L B O R E

F L U I D P R E S S U R E I N T H E F O R M A T I O N

( 0 )

F L U I D P R E S S U R E D I F F E R E N C E B E T W E E N/ T H E F O R M A T I O N A N D T H E W E L L B O R E

F L U I D P R E S S U R E I N T H E W E L L B O R E

F L U I D P R E S S U R E I N T H E F O R M A T I O N

( t > )FIG. 18Idealized diagram of two possible types of pressure behavior during fracture treatmentdepending upon various underground conditions.

ing or to a vertical fracture in a situation wherethe ratio < rB/a A of the horizontal principalstresses was greater than 2.0.Effec t of Pene tra t ing Flu ids

Wh en a penetr ating fluid is used in a fractu ring operation, a more complicated mechanicalsituation exists. As noted previously, the totalnormal s t ress S across any plane may be resolved i nto the sum of a residu al solid stress o-and the fluid pressure p, o r S a + p.Fu rthe rm ore , with a non pene trating fluid, anincrease in pressure (Ap) equa l to


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Mechanics of Hydraulic Fracturing 251all the forces exerted upon the rock containedwithin a column of unit area of cross sectionnormal to the fracture, orF r d(Av) r T = - -^-dn = - \cl(Ap) = Apo. (11).4 Jo dn J Apo

In order for the fracture to be held open andextended, this outward-directed force per unitarea m ust be equal to


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252 M. King Hubbert and David G. Willisof depth in an area of incipient normal faulting.

The value of Sz/z is approx imately equal to1.00 psi per foot of depth for normal sedimenta ry rocks in most a reas. Under normal hydrostatic fluid-pressure conditions, p/z is abo ut0.46 psi per foot of depth. Substituting thesevalues into equation 18 givesP/ z ^ 0.64 psi/ft

as the approximate minimum va lue tha t shouldbe expected in the Gulf Coast.Let us consider the values of P which wouldoccur under conditions in which the originalfluid pressure was other than hydrostatic. Inthose cases of an original fluid pressure lessthan hydrostatic, i t can be seen from equation17 that P would be correspondingly reduced.However, where abnormally high original fluidpressures prevail , P would become higher unt i l ,in the limit, when the original pressure p a p proaches the to ta l overburden pressure Sz, Palso approaches the total overburden pressureand fracturing will occur at pressures onlyslightly greater than the original fluid pressure.Walker (1949) has descr ibed an in te rest ingexample of lost circulation which might be explained on the basis of the foregoing analysis.In a Gulf Coast well dril l ing below 10,000 ft(3,0 50 m ) , the specific weight of the drill ingmud , which was a li t t le over 18 lb/g al, ha d tobe kept constant to within 0.3 lb/gal, or about2 percent, to prevent either lost circulationwhen the density was too high or kickin g bythe formation fluids when the density was toolow.

F I E L D E V I D E N C EPresent field data derived from experiencewith hydraulic fracturing, squeeze cementing,and lost circulation are fully consistent with theforegoing conclusions. In the Gulf Coast area,recent normal faulting indicates that verticalfractures should be formed with injection pressures less than the total overburden pressure. Inthe Mid-Continent and West Texas regions, oldnormal faul t ing , a l though represent ing moreambiguous evidence, also favors vertical

fracturing.Howard and Fast (1950) have summarizedthe pressure data from 161 squeeze-cementingand acidizing jobs performed in the Gulf Coastand West Texas-New Mexico a reas. Also , publ ished da ta by Harr ison et al. (1954) and Scot tet al. (1953) describe injection pressures forlarge samples from hydraulic-fracturing opera

tions in the Gulf Coast, Mid-Continent, andWest Texas regions. With but few exceptions,the injection pressures have been substantiallyless than the total overburden pressure, thusimplying that vertical fractures are actually being formed.In addition to the preceding da ta, the occurrence of lost circulation throughout the GulfCoast area at pressure substantially less thanthat due to the weight of the overburden supports the conclusion that the least stress shouldbe horizontal in this area.In much of California, however, tectoniccompression is taking place, and in these areashorizontal fractures should occur with injectionpressures greater than the total overburdenpressure . Al though compara t ive ly few frac turing operations have been performed in California, extremely high pressures are required withinjection pressures commonly greater than theoverburden pressure (W. E. Hassebroek, persona l c ommun . ) .A phenomenon very similar to artificial formation fracturing, but on a much larger scale,is that of dike emp lacem ent. It has beenpointed out by Anders on (1 951) tha t igneousdikes should be injected along planes perpendicular to the axis of least principal stress. Thissituation is entirely analogous to that for artificial formation fracturing. A remarkable fieldexample of the effect of a regional stress pattern upon the orientation of igneous dikes is theSpanish Peaks igneous complex in Colorado.A map of this area is shown in Figure 19,and a photograph of West Spanish Peak fromthe northwest, showing dikes cutting flat-lyingEoce ne strata, is given in F igure 20. Ode(1957) has made a mathemat ica l so lu t ion ofthe regional stress field which would most l ikelyresult from the presence of the structural features in the area. A comparison of the radial-dike system with the mathematical solutionshows the dikes to be almost exactly perpendicular to the trajectories of the least principalstress.E X P E R I M E N T A L F R A C T U R I N G D E M O N S T R A T I O N

In order to verify the inferences obtainedtheoretically, a series of simple laboratory exper iments has been performed. The genera lprocedure was to produce fractures on a smallscale by injecting a frac turin g fluid into aweak elastic solid which previously had beenst ressed . Ordinary ge la t in (12-percent so lu t ion)was use d for the solid, bec ause it was sufficiently weak to fracture easily, was readily


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Mechan ics of Hydrau lic Fracturing 253

FIG. 19Dike pattern of Spanish Peaks area, Colorado.

Warn.FIG. 20Photograph of West Spanish Peak from northwest, showing dikes cutting flat-lying Eocene strata(G. W. Stose, U.S. Geological Survey).


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FIG. 21Experimental arrangement for producingleast stress in a horizontal direction.molded with a simulated well bore, and was almost perfectly elastic under short-time application of stresses. A plaster-of-paris slurry wasused as a fracturing fluid because it could bemade thin enough to flow easily and could alsobe allowed to set, thus providin g a pe rma nentrecord of the fractures produced.

In a model experiment conducted in thisway, the stress distributions are entirely independ ent of scale. Provid ed the m aterial is elastic, similitude will exist no matter on whatlength scale the experiment is conducted.The experimenta l a rrangement consisted of a2-gal polyethylene bottle with its top cut off,used as a container in which was placed a glass-tubing assembly consisting of an inner moldand concentric outer casings. The containerwas sufficiently flexible to transmit externallyapplied stresses to the gelatin. The procedurewas to place the glass-tubing assembly in theliquid gelatin and, after solidification, to withdraw the inner mold leaving a wel l bor ecased above and below an open-hole section.Stresses were then applied to the gelatin in twoways. Th e first meth od (Fig . 21) w as tosqueeze the polyethylene container laterally,thereby forcing it into an elliptical cross sectionand producing a compression in one horizontaldirection and an extension at right angles in theother. The least principal stress was thereforehorizon tal, and vertical fractures would be expected in a vertical plane, as shown in Figure21.

In other experiments the container was

wrapped with rubber tubing stretched in tension (Fig . 22) , thus producing radia l compression and a vertical extension. In this case, theleast principal stress was vertical, and horizontal fractures would be expected.The plaster slurry was injected from an aspirator bottle to which air pressure was appliedby means of a squeeze bulb.Four experiments were performed undereach of the two stress conditions, and in everycase the fractures were formed p erpen dicula rto the least principal stress. A vertical fractureis shown in Figure 23 and a horizontal fracturein Figure 24 .The saucer shape of the horizontal fractureis a result of the method of applying thestresses. As the gelatin is compressed on allsides, it tends to be displaced vertically but isrestrained by the walls of the container. Thus ashear stress is produced, causing the least principal stress to intersect the container at an angle from above. Therefore, when the fracturesare formed normal to the least principal stress,they turn upward near the walls of the container, producing the saucer shape shown inFigure 24 .

A further variation in the experiment consisted of stratifying the gelatin by pou ring andsolidifying alternate strong and weak solutions.One experiment was performed in th is way un-dsr each stress condition. The vertical fractureis i l lustrated in Figu re 25 , in which th e we akgelatin appeared to fracture slightly more read-

FIG. 22Experimental arrangement for producingleast stress in a vertical direction.


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Mechanics of Hydraulic Fracturing 255i ly than the strong gelatin. Figure 26 shows ahorizontal fracture in stratified gelatin. In thiscase, the fracture is not saucer sha ped but appears to have followed a plane of weakness created by bubbles between two gelatin layers.SI G N I FI C A N C E O F V E R T I C A L FR A C T U R I N GI N R E SE R V O I R E N G I N E E R I N G

In view of the foregoing evidence, i t now appears fairly definite that most of the fracturingproduced hydraulically is vertical rather thanhorizontal, so the significance of this fact in reservoir engineering should be mentioned. In geologically simple and tectonically relaxed areas,the regional stresses should be fairly uniformover extensive areas so that the horizon talstress trajectories in local areas should benearly rectil inear. Consequently, when numerous wells in a single oil field are fractu red, thefractures should be collimated by the stressfield to almost the same strike.There are serious implications, as Crawfordand Collins (1954) have pointed out, with respect to the direction of dr ive and the sweepefficiency in secondary-recovery operations. Ifthe direction of drive should be parallel withthe strike of the fractures, then the flow would

FIG. 23Vertical fracture produced under stress con-ditions illustrated in Figure 21.

FIG. 24Horizontal fracture produced under stressconditions illustrated in Figure 22.be effectively short-circuited and the sweep efficiency would be very low. However, if thedrive were normal to the strike of the fractures,the flow pattern would approximate that between parallel l ine sources and sinks and thesweep efficiency would approach unity.This circumstance emphasizes the need,which is becoming increasingly more urgent,for the development of reliable downhole instruments by means of which not only the ver-

1'IG. 25Vertical fracture in stratified gelatin.


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FIG. 26Horizontal fracture in stratified gelatin.

t ical extent, but also the azimuth of the fractures, can be determined.Since the foregoing paragraphs were written,these theoretical inferences have been strikinglyconfirmed by the fracturing experience duringwaterflood operations of the North Burbankfield, Oklaho ma. A ccord ing to Z. Z. Hu nte r(1956), the initial pilot flood was based on theconventional five-spot pattern of injection andproducing wells, but the results were anomalous. The injection wells were broken down atvery low pressures (as low as one fourth of theoverburden pressure) , and producing wel ls eastand west of injection wells were commonly bypassed by the flood. Finally, a sudden influx ofwater occu rred in the isolated StanleyStr inger sandstone ( in the Burb ank sandstone) 1 mi (1 .6 km) east of the flood area.Cumulative experiences of this kind, supplemented by fracture observations in orientedcores, led to the conclusion that the fractureswere essentially vertical and oriented east andwest. This realization led to a change of procedure wherein line drives were instituted from

east-west rows of fractured injection wells to alternate rows of fractured producing wells.Greatly increased oil production without a corresponding increase in the water-oil ratio resulted.Another question to be considered concernsthe vertical migration of fluids. It is obviousthat vertical fractures will facilitate the verticalmigration of fluids where the fractures intersectpermeabi l i ty barr ie rs . They may in th is manner

interconnect several separate reservoirs in lenticular sandstones imbedded in shales, and mayin fact tap some such reservoirs not otherwisein communication with the fractured well.There is a danger, however, where a reservoiris overlain by a thin permeability barrier and awater-bear ing sand (or sandstone) , tha t a ver t i cal fracture may also permit the escape of theoil and gas into the barren s ands (or sandstones) above.A related question is that of the effect on water production of a vertical fracture which extends across the oil-water interface. In order toobtain an approximate idea of what this effectmay be, consider a reserv oir comp osed of athick sand which is homogeneous and isotropicwith respect to permeability. If productionprior to fracturing is from an interval wellabove the water table, the water will form a radially symmetrical cone, with a slope whosesine at any point is given by

- g ra d *9 I (19)where p0 and pw are the densities of oil andwater, respectively, g is the acceleration ofgravity, and $ n is the potential of oil (Hubber t , 1940, 1953b) .The oil potential at a given point is definedby

P*o = gz + (20)w he re z is the elevation of the point with respect to sea level and p is the gauge pressure.Then, by Darcy's law, the volume of fluidcrossing a unit area in unit t ime will be

kp0 A *q = g r a d *o , (21)where k is the p erm eability of the sa nd a nd //, isthe fluid viscosity. Substitution into equation 19gives, for the tilt of the oil-water interface,

(22)Pw po gkpoHence, the sine of the angle of t i l t is proportional to the rate of flow, q, of the oil along theinterface.We hav e now on ly to consider the flow p atterns about the well without and with verticalfracturing. Without fracturing the flow converges radially toward the well with a rapidlyincreasing flow rate and a corresponding steepening of the cone. With fracturing, for thesame rate of oil production from the well, theflow pattern approximates that of l inear flowtoward a ver t ica l -p lane sink . The maximum


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Mec han ics of Hydra ul ic Frac tur ing 2 5 7values of the flow velocity, q, for this case willbe very much less than for the radial-flow case.Hence, for a given rate of oil production, a vertical fracture across the oil-water interface in auniform sandstone, instead of causing an inc rease of water product ion , ac tua l ly shouldserve to reduce markedly the water coning and,consequently, to decrease the production of watera result in accord with reports of field experience wherein fracturing near the water table has not resulted in increased water production.C O N C L U SI O N S

From the foregoing analysis of the problemof hydraulic fracturin g of wells, the followinggenera l conc lusions appear to be warranted .1. The state of stress underground is not, ingeneral, hydrostatic, but depends upon tectonicconditions. In tectonically relaxed areas characterized by normal faulting, the least stress willbe approximately horizontal; whereas, in areasof tectonic compression characterized by folding and thrust faulting, the least stress will beapproximately vertical and, provided the deformat ion is not too great, approxim ately equ al tothe overburden pressure .2. Hydraulically induced fractures should beformed approximate ly perpendicula r to theleast principal stress. Therefore, in tectonicallyrelaxed areas, they should be vertical, whereas,in tectonically compressed areas, they should

be horizontal.3. Rupture or breakdown pressures are affected by the values of the preexisting regionalstresses, by the hole geometry including anypreexisting fissures, and by the penetratingquality of the fluid.4. M inim um injection p ressures dependsolely upo n the magn itude of the least prin cipalregional stress and are not affected by the holegeometry or the penetrating quality of the fluid.In tectonically relaxed areas, the fracturesshould be vertical and should be formed withinjection pressures less than the total overburden pressure. In tectonically compressed areas,provided the deformation is not too great, the

fractures should be horizontal and should require injection pressures equal to, or greaterthan, the to ta l overburden pressures.5 . It does not appea r to be mech anicallypossible for horizontal fractures to be producedin re la tive ly undeformed rocks by means of total injection pressures which are less than thetotal pressure of the overburden.6. In geologically simple and tectonically relaxed areas, not only should the fractures in a

single field be vertical, but ihey also shouldhave roughly the same direction of strike.7. Vertical fractures intersecting horizo ntalpermeability barriers will facili tate the verticalflow of fluids. However, in the absence of suchbarriers, vertical fractures across the oil-wateror gas-oil interface will tend to reduce the coning of water or gas into the oil section for agiven rate of oil production.

R E F E R E N E SAnde rson, E. M., 1951, The dynamics of faulting anddyke forma tion with applications to Britain : E dinburgh and London, Oliver and Boyd, 2d ed.Bugbee, J. M., 1943, Reservoir analysis and geologicstructure: AIM E Trans., v. 151, p. 99.Clark, J. B., 1949, A hydraulic process for increasingthe productivity of wells: AIME Trans., v. 186, p.1-8.Crawford, P. B and R. E. Collins, 1954, Estimatedeffect of vertical fractures on secondary recove ry:AI ME Trans., v. 201, p. 192.Dickey, P. A., and K. H. Andresen, 1945, The behavior of water-input wells: Drilling and Prod. Practices, v. 34.Harrison, Eugene, W. F. Kieschnick, Jr., and W. J.McG uire, 1954, The mechanics of fracture inductionand extension: AIM E Trans., v. 201, p. 252.How ard, G . C , and C. R. Fast, 1950, Squeeze cemen ting operations: AIME Trans., v. 189, p. 53.Hubber t, M. K., 1940, The theory of ground-water motion: Jour. Geology, v. 48, p. 785-944.1951, Mec hanical basis for certain familiar geologic structure s: Geol. Soc. America Bull., v. 62, no.4, p. 355-372.1953a, Discussion of paper by Scott, Bearden,and How ard, Roc k rupture as affected by fluidproperties : AIM E T rans., v. 198, p. 122.

1953b, Entrapm ent of petroleum un der hydrody-namic conditions: A m. Assoc. Petroleum GeologistsBull., v. 37, no. 8, p. 1954-2026.Hunter, Z. Z., 1956, 8V 2 Million e xtra barr els in 6years: Oil and Gas Jour., August 27, p. 86.Kirsch, G., 1898, Die Theorie der Elastizitat und dieBediirfnisse der Festigke itslehre: Zeitschr. Ver.Deutsch. Ingenieure, v. 42, p. 797.Mc Henr y, D ouglas, 1948, The effect of uplift pressu reon the shearing strength of concrete: Troisieme Con-gres des Grand s Barrages, Stockholm, Sweden.Miles, A. J., and A. D. Topping, 1949, Stresses arou nda deep well: AIME Trans., v. 179, p. 186Ode, H., 1957, Mechanical analysis of the dike patternof the Spanish Peaks area, Colorado: Geol. Soc.America Bull., v. 68, no. 5, p. 567-575.Reynolds, J. J., P. E. Bocquet, and R. C. Clark, Jr.,1954, A method of creating vertical hydraulic fractures: Drill ing and Prod. Practice, p. 206.Scott , P. P., Jr., W. G. Bearden, and G. C. Howard ,1953, Roc k ruptu re as affected by fluid prope rties :AIME Trans., v. 198, p. 111.Terzaghi, Karl, 1943, Theoretical soil mechanics: NewYork, John Wiley and Sons.Tim oshen ko, S., 1934, Theo ry of elasticity: New Y orkand Londo n, M cGraw-H ill .Walker, A. W., 1946, Discussion of paper by A. J. Tep-litz and W. E. Hasse broek, An investigation of oil-well cem enting : D rilling and Pro d. Practice, p. 102.1949, Squeeze cementing: World Oil, September, p. 87.

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1972 Mechanics of Hydr Frac