fracturing stimulation book

5-1. IntroductionSince its introduction, hydraulic fracturing has been,and will remain, one of the primary engineeringtools for improving well productivity. This isachieved by

• placing a conductive channel through near-wellbore damage, bypassing this crucial zone

• extending the channel to a significant depth intothe reservoir to further increase productivity

• placing the channel such that fluid flow in thereservoir is altered.

In this last instance, the fracture becomes a toolfor true reservoir management including sanddeconsolidation management and long-termexploitation strategies. As first visualized (see theAppendix to this chapter), the concept of hydraulicfracturing was quite straightforward. This visual-ization is described in the following, and in general,for reasonably simple geology, the basic physics offracturing is straightforward and well established.Complexity arises from two directions: geologicreality and the inherent multidisciplinary nature ofthe fracturing process.

Historically, the control of fracturing has restedwith drilling and operations groups owing to thenature of field procedures using pumps, packers,pressure limits, etc. However, the final results (andthus design) are dominantly a production engineer-ing exercise, and fracturing cannot be removed fromintimate contact with reservoir engineering. At thesame time, designing a treatment to achieve thedesired results is also intimately connected with rockmechanics (which controls fracture geometry; seeChapters 3 and 4), fluid mechanics (which controlsfluid flow and proppant placement inside a fracture;see Chapter 6) and the chemistry that governs theperformance of the materials used to conduct the

† Deceased

treatment (see Chapters 7 and 8). However, thedesign must also be consistent with the physicallimits set by actual field and well environments.Also, treatments must be conducted as designed toachieve a desired result (i.e., full circle to the criticalrole of operations). Proper treatment design is thustied to several disciplines:

• production engineering

• rock mechanics

• fluid mechanics

• selection of optimum materials

• operations.

Because of this absolutely essential multidiscipli-nary approach, there is only one rule of thumb in frac-turing: that there are no rules of thumb in fracturing.

The multidisciplinary nature, along with the diffi-culty in firmly establishing many of the design vari-ables, lends an element of art to hydraulic fracturing.This is not to say that the process is a mystery nor is it to say that for most cases the basic physics con-trolling the process is not defined (see Chapter 6). Itsimply says that the multitude of variables involved,along with some uncertainty in the absolute valuesof these variables, makes sound engineering judg-ment important.

5-1.1. What is fracturing?If fluid is pumped into a well faster than the fluidcan escape into the formation, inevitably pressurerises, and at some point something breaks. Becauserock is generally weaker than steel, what breaks isusually the formation, resulting in the wellboresplitting along its axis as a result of tensile hoopstresses generated by the internal pressure. Themechanics of this process are described in Section3-5.7, and the simple idea of the wellbore splittinglike a pipe (shown as a cartoon in Fig. 5-1) becomesmore complex for cased and/or perforated wells and

Reservoir Stimulation 5-1

◆

Basics ofHydraulic Fracturing

M. B. Smith, NSI Technologies, Inc.J. W. Shlyapobersky,† Shell E&P Technology Co.

nonvertical wells. However, in general, the wellborebreaks—i.e., the rock fractures—owing to the actionof the hydraulic fluid pressure, and a “hydraulic”fracture is created. Because most wells are verticaland the smallest stress is the minimum horizontalstress, the initial splitting (or breakdown) results in a vertical, planar parting in the earth.

The breakdown and early fracture growth exposenew formation area to the injected fluid, and thus therate of fluid leaking off into the formation starts toincrease. However, if the pumping rate is maintainedat a rate higher than the fluid-loss rate, then thenewly created fracture must continue to propagateand grow (Fig. 5-2). This growth continues to openmore formation area. However, although thehydraulic fracture tremendously increases the forma-tion flow area while pumping, once pumping stopsand the injected fluids leak off, the fracture will closeand the new formation area will not be available forproduction. To prevent this, measures must be takento maintain the conductive channel. This normallyinvolves adding a propping agent to the hydraulicfluid to be transported into the fracture. When pump-ing stops and fluid flows back from the well, thepropping agent remains in place to keep the fracture

open and maintain a conductive flow path for theincreased formation flow area during production.The propping agent is generally sand or a high-strength, granular substitute for sand (see Section 7-7).Alternatively, for carbonate rocks, the hydraulic fluidmay consist of acid that dissolves some of the forma-tion, leaving behind acid-etched channels extendinginto the reservoir.

After the breakdown, the fracture propagation rateand fluid flow rate inside the fracture become impor-tant. They are dominated by fluid-loss behavior. Asintroduced by Carter (1957) and discussed in the fol-lowing (and in Chapters 6 and 9), the fluid-loss rateqL from a fracture can be expressed as

(5-1)

where CL is the fluid-loss coefficient, A is an elementof the fracture area (i.e., increased inflow area), t istime measured from the start of pumping, and τ isthe time when each small area element of a fractureis created or opened. As a direct consequence of thisrelation, the highest rate of fluid loss is always at thefracture tip. Newly created fracture area exists at that

5-2 Basics of Hydraulic Fracturing

Figure 5-2. Cross-sectional view of a propagating fracture.Figure 5-1. Internal pressure breaking a vertical wellbore.

qC A

tLL≈−

2τ

,

point (t – τ = 0 in the denominator), making qL in-stantly infinite.

Initially, fracture penetration is limited, and hencefluid loss is high near the wellbore. For that reason,the first part of a hydraulic fracture treatment con-sists of fluid only (no proppant); this is termed thepad. The purpose of a pad is to break down the well-bore and initiate the fracture. Also, the pad providesfluid to produce sufficient penetration and width toallow proppant-laden fluid stages to later enter thefracture and thus avoid high fluid loss near the frac-ture tip. After the pad, proppant-laden stages arepumped to transport propping agent into the fracture.This chapter describes the process for propped frac-ture treatments; acid fracture treatments are dis-cussed in Section 10-6.

However, because fluid loss to the formation is stilloccurring, even near the well, the first proppant isadded to the fluid at low concentrations. The prop-pant-laden slurry enters the fracture at the well andflows toward the fracture tip (Fig. 5-3). At this point,two phenomena begin. First, because of the higherfluid loss at the fracture tip, slurry flows through thefracture faster than the tip propagates, and the prop-pant-laden slurry eventually overtakes the fracture tip.Next, because of fluid loss, the proppant-laden slurrystages lose fluid (but not proppant) to the formation.

Thus, proppant concentration (i.e., volume fraction of solid proppant) increases as the slurry stages dehy-drate. The pump schedule, or proppant additionschedule, must be engineered much like handicappinghorse races, but with no single winner. Rather, allstages should finish at the right place, at the righttime, with the right final proppant concentration. Thepad should be completely lost to the formation, andthe first proppant stage should be right at the fracturetip (which should be at the design length).

As the proppant slurry stages move down the frac-ture, they dehydrate and concentrate. Slurry stagespumped later in the treatment are pumped at a higherconcentration. These stages are not in the fracture forlong prior to the treatment end (i.e., prior to shut-down) and are thus exposed to less fluid loss and lessdehydration. Ideally, the first proppant stage pumpedreaches the fracture tip just as the last of the padfluid is lost into the formation (a correctly handi-capped race), and this first stage has concentratedfrom its low concentration to some preselected,higher final design concentration. Meanwhile, theslurry concentration being pumped is steadilyincreased to the same final design concentration. At treatment end, the entire fracture is filled with thedesign concentration slurry. Design considerationsfor the final concentration are discussed later in thissection and in detail in Section 10-4.

The preceding description might be termed a “nor-mal” design, where the entire fracture is filled with auniform, preselected, design proppant concentrationjust as the treatment ends. If pumping continues pastthat point, there would be little additional fractureextension because the pad is 100% depleted. Con-tinued pumping forces the fracture to become wider(and forces the pressure to increase) because theincreased volume simply acts like blowing up a bal-loon. In some cases the additional propped widththat results may be desirable, and this procedure isused purposely. This is termed tip-screenout (TSO)fracturing.

At the conclusion of the treatment, the final flushstage is pumped. This segment of a treatment con-sists of one wellbore volume of fluid only and isintended to sweep the wellbore clean of proppant(Fig. 5-4). The well is generally then shut-in forsome period to allow fluid to leak off such that thefracture closes on and stresses the proppant pack.Shut-in also allows temperature (and chemicalbreakers added to the fluid while pumping) to reduce


Figure 5-3. Introducing proppant into the fracture.

��

the viscosity of the fracturing fluid (see Section 7-6.2).Ideally, this process leaves a proppant-filled fracturewith a productive fracture length (or half-length xf),propped fracture height and propped fracture width(which determines the fracture conductivity kfw).Here, xf is the productive fracture half-length, whichmay be less than the created half-length L or lessthan the propped length.

5-1.2. Why fracture?Hydraulic fracture operations may be performed on awell for one (or more) of three reasons:

• to bypass near-wellbore damage and return a wellto its “natural” productivity

• to extend a conductive path deep into a formationand thus increase productivity beyond the naturallevel

• to alter fluid flow in the formation.

In the third case, fracture design may affect and be affected by considerations for other wells (e.g.,where to place other wells and how many additionalwells to drill). The fracture becomes a tool for reser-voir management. Although these three motivations

are addressed separately in this section, they fre-quently overlap.

• Damage bypass

Near-wellbore damage reduces well productivity.This damage can occur from several sources,including drilling-induced damage resulting fromfines invasion into the formation while drillingand chemical incompatibility between drilling flu-ids and the formation. The damage can also bedue to natural reservoir processes such as satura-tion changes resulting from low reservoir pressurenear a well, formation fines movement or scaledeposition. Whatever the cause, the result is unde-sirable. Matrix treatments (discussed in Chapters13 through 20) are usually used to remove thedamage chemically, restoring a well to its naturalproductivity. In some instances, chemical proce-dures may not be effective or appropriate, andhydraulic fracture operations are used to bypassthe damage. This is achieved by producing a high-conductivity path through the damage region torestore wellbore contact with undamaged rock.

• Improved productivity

Unlike matrix stimulation procedures, hydraulicfracturing operations can extend a conductivechannel deep into the reservoir and actually stimu-late productivity beyond the natural level.

All reservoir exploitation practices are subjectto Darcy’s law:

(5-2)

where the all-important production rate q is relat-ed to formation permeability k, pay thickness h,reservoir fluid viscosity µ, pressure drop ∆p andformation flow area A. Reservoir exploitationrevolves around manipulating this equation. Forexample, pressure drop may be increased by usingartificial lift to reduce bottomhole flowing pres-sure, water injection to increase or maintain reser-voir pressure, or both. For other cases, in-situcombustion or steam injection is used to reducereservoir fluid viscosity and thus increase produc-tivity. For fracturing, as pictured in Fig. 5-5, oper-ations are on the formation area in the equation,with the increased formation flow area giving theincreased production rate and increased presentvalue for the reserves. (Strictly speaking, it is the


Figure 5-4. Flushing the wellbore to leave a propped frac-ture.

��

qkh p

x

A

h≈

µ

∆∆

,

flow shape that is altered, as discussed in detail inChapter 1.)

This is the classic use of fracturing, to increasethe producing rate by bypassing near-wellboreformation damage or by increasing exposure ofthe formation area and thus stimulating well per-formance beyond that for no damage. For a singlewell, treatment design concentrates on creatingthe required formation flow area to yield increasedproduction at minimal cost. More formally, thedesign should optimize economic return on thebasis of increased productivity and treatment cost.

• Reservoir management

Along with improving well productivity, fracturesalso provide a powerful tool for altering reservoirflow. In combination with the other parts of fielddevelopment, the fracture becomes a reservoirmanagement tool. For example, creating longfractures in tight rock (k < 0.1 md) enables fielddevelopment with fewer wells. However, evenfewer wells are required if the fracture azimuth is known and the wells are located appropriately(e.g., not on a regulatory-required square pattern).The actual philosophy shift for fracturing, from

accelerating production from a single well to reser-voir management, occurred with the application ofmassive stimulation treatments in tight gas forma-tions (see Appendix to this chapter). Although out-wardly a traditional application of fracturing topoorer quality reservoirs, these treatments repre-sented the first engineering attempts to alter reser-voir flow in the horizontal plane and the methodol-ogy for well placement (e.g., Smith, 1979).

Fracturing for vertical inflow conformance (i.e.,reservoir management) was successfully used inthe Gullfaks field (Bale et al., 1994), where selec-tive perforating and fracturing were used to opti-mize reserve recovery and control sand productionwhile maintaining (but not necessarily increasing)the required production rates. This is illustrated inFig. 5-6, where the bottom, low-permeabilityRannoch-1 zone was perforated to create apropped fracture that extends up and into thehigh-permeability (>1000-md) Rannoch-3 zone.Without fracturing, the entire zone can be perfo-rated, and a low drawdown allows a significantproduction rate on the order of 20,000 STB/D,sand free. However, sand production is triggeredby water breakthrough in the high-permeabilityzone (from downdip water injection). The result-ing wellbore enlargement caused by sand produc-tion acts to stimulate production from the high-permeability zone. To stop sand production, draw-


Figure 5-5. Increased flow area resulting from a fracture.

��

Figure 5-6. Fracturing for vertical inflow conformance.

200 40 60 80

Fracture penetration (m)

Rannoch-3

Rannoch-3

Rannoch-3

Rannoch-2

Rannoch-1

1820

1860

1880

1840

55004500

Stress (psi)

TVD(m belowsea level)

down must be reduced even more. The productionis then essentially 100% water coming from thestimulated high-permeability zone, and the wellmust be abandoned. This further diminishes pro-duction from the large reserves found in thedeeper zones with lower permeability.

Open- or cased hole gravel packing could beused to eliminate the sand production. However,such completions are less than satisfactory for tworeasons. First, the deeper, lower permeabilityzones can significantly benefit from stimulation.Second, significant scaling occurs with waterbreakthrough and quickly plugs the gravel pack.

The fracturing tool selected to manage theGullfaks field is termed an indirect vertical frac-ture completion (IVFC). The IVFC accomplishesseveral goals:

– Some (although choked) production is achievedfrom the main zone to enable the well to reachminimum productivity standards.

– Production from the lower, moderate-perme-ability zone is stimulated, maximizing reservesfrom this zone.

– Greater drawdown is allowed because the weakhigh-permeability rock is separated from theperforations, and greater drawdown increasesthe total rate and significantly increases recov-ery from the lower zones.

– If the upper high-permeability zone has sandproduction tendencies (as is typically the case),then producing this zone via the fracture totallyavoids the need for sand control.

– Any potential for water breakthrough in thehigh-permeability zone is retarded, and post-water-breakthrough oil production is signifi-cantly increased.

To achieve these goals, fracture conductivitymust be tailored by synergy between the reservoirand fracture models. Too much conductivityaccelerates production and the time to waterbreakthrough from the high-permeability mainzone. Also, too much conductivity, because ofsurface or tubular limits for the production rate,restricts drawdown on the lower zones, and thedesired, more uniform vertical production profileis not achieved. The fracture design goal is not tosimply accelerate the rate but to achieve maxi-mum reserves recovery with no sacrifice of rate

(as compared with a simple completion in whichthe entire zone is perforated).

Another example of reservoir management iswaterflood development utilizing fractures and a“line drive” flood pattern (i.e., one-dimensional[1D] or linear flow from injection fractures to pro-duction fractures). Knowledge of the fractureazimuth, combined with conductive fractures (orcorrectly controlled injection greater than the frac-ture pressure) results in improved sweep efficiencyand enables more efficient field development.

5-1.3. Design considerations and primary variables

This section introduces the primary variables for frac-ture design. Sidebar 5A summarizes how the designvariables originate from treatment design goals.

As mentioned previously, fracturing was con-trolled historically by operational considerations.This limited its application because fracturing isdominantly a reservoir process, and hence why areservoir is fractured and what type of fracture isrequired should be dominated by reservoir engineer-ing considerations. The permeability k becomes theprimary reservoir variable for fracturing and allreservoir considerations. Other, so-called normalreservoir parameters such as net pay and porositydominate the economics and control the ultimateviability of a project but do not directly impact howthe fracturing tool is employed. As discussed inChapter 12, postfracture productivity is also gov-erned by a combination of the fracture conductivitykfw and xf, where kf is the permeability of the prop-pant in the fracture, w is the propped fracture width,and xf is the fracture penetration or half-length.These variables are controlled by fracturing andtherefore identify the goals for treatment design.

The productive fracture half-length xf may be lessthan the created (or the created and propped) half-length L because of many factors (see Section 12-3).For example, the fracture width near the tip of afracture may be too narrow to allow adequatepropped width. As another example, vertical varia-tions in formation permeability, or layering, cancause the apparent productive length xf to be lessthan the actual propped length (Bennett et al., 1986).Similarly, this also makes the fracture height hf

important in several ways (Fig. 5-7):



5A. Design goals and variables

This discussion briefly summarizes the design goals of hydraulic fracturing that provide a road map for the major design variables.

Design goals

Design goals result from Darcy’s law (Eq. 5-2), in which the dimensionless term A/(∆xh) is defined by flow conditions and equalsln(re /rw ) for steady-state flow (as discussed in Chapter 1). For steady-state flow, Prats (1961) showed that a fracture affects produc-tivity through the equivalent wellbore radius rw and that rw is related to the fracture half-length or penetration xf by the dimension-less fracture conductivity (CfD = kfw/kxf). Cinco-Ley et al. (1978) extended these concepts for transient flow with the relation amongxf, rw and CfD shown in Fig. 5-11 for pseudoradial flow (where the pressure-depletion region >> xf but is not affected by externalboundaries). Thus, the primary design goals are fracture half-length or penetration and the fracture conductivity kfw, with their rela-tive values defined by CfD.

Design variables

Design variables result from material balance, rock mechanics and fluid mechanics considerations.The material balance is (Eqs. 5-10 through 5-12)

(5A-1)

where CL and Sp are fluid-loss parameters that can be determined by the results of a fluid-loss test (Fig. 5A-1) for which the filtratevolume divided by the exposed area VL/A = Sp + 2CL√t . Combining the relations in Eq. 5A-1 gives Eq. 5-13:

where fracture penetration L is related to pump rate, fluid loss, height, width, etc.Next is the elasticity equation (Eq. 5-14):

where pnet = pf – σc, and width is related to net pressure as a function of modulus and geometry and the pressure required to propa-gate the fracture (Eq. 5-21):

(5A-2)

where d is the characteristic fracture dimension and generally is the smaller dimension between hf and L.Third is the fluid flow equation (Eqs. 5-15 through 5-19), in which Eq. 5-15 (dpnet/dx = 12µq/hfw3) is combined with the width equation:

(5A-3)

where the pressure drop down the fracture is related to viscosity, pump rate, fracture length (and thus to fluid loss), etc. The netpressure distribution gives the fracture width distribution and thus the final propped fracture width (i.e., kfw). Hence the primarydesign variables are CL, hL, Sp, hf, E , KIc-apparent, qi, µ and σc .

Optimum design

The optimum design results from maximizing revenue $(rw´) minus the costs $(xf, kfw) by using the preferred economic criteria.

2Cw

Sp

Vol

ume

lost

/are

a, V

L/A

√time

Lq t

6C h t 4h S 2whi p

L L p L p f

≈+ +

,

w2p d

E'maxnet= ,

p (p ) at tip K 1 / dtip c Ic apparent= − ∝ −σ ,

pE'h

q L pnet

3

f4 i net tip

4

1/ 4

≈ { } +

κµ ,

V V V ; V 2Lh w, V q t and V 6C h L t 4Lh Sf i Lp f f i i p Lp L L p L p= − = = ≈ + ,

Figure 5A-1. Ideal laboratory fluid-loss data for spurtloss Sp and the wall-building or filter-cake fluid-losscoefficient Cw. If the total fluid loss is dominated by thefilter cake, then the total fluid-loss coefficient CL = Cw.


• In Fig. 5-7a, the fracture is initiated near the top of the interval, and hf is not large enough to con-tact the entire zone, which is clearly an importantreservoir concern.

• In Fig. 5-7b, the fracture grew out of the zone andcontacted mostly nonreservoir rock, diminishing xf

relative to the treatment volume pumped.

• In Fig. 5-7c, the fracture grew downward past theoil/water contact and if propped would possiblyresult in unacceptable water production.

In all these cases, as discussed in Section 5-4.2,fracture height growth is controlled by rock mechan-ics considerations such as in-situ stress, stress gradi-ents, stress magnitude differences between differentgeologic layers and differences in strength or frac-ture toughness between different layers. All these

rock mechanics considerations are related to the netpressure pnet:

(5-3)

where pf is the pressure inside the fracture and σc is theminimum in-situ stress (or fracture closure pressure).

For an ideal, homogeneous zone, closure pressureis synonymous with the minimum in-situ stress.However, such ideal conditions do not exist. Stress is a point value, and stress varies from point to point.For realistic in-situ conditions, closure pressurereflects the pressure where the fracture is grosslyclosed, although the pressure may still be greaterthan the minimum in-situ stress at some points. Forzones that are only slightly nonhomogeneous, theclosure pressure represents a zone-averaged stressover the fracture. However, other conditions may bemore complex. Consider the three-layer case of twolow-stress sandstone intervals with a thick interbed-ded shale. The correct closure pressure may be thezone-averaged stress over the two low-stress zones,without including the higher stress interbedded zone.

The fracture width is also of major importance forachieving the desired design goals. Typically, this isexpressed as the product of fracture permeabilitytimes fracture width; i.e., kfw is the dimensional con-ductivity of the fracture. Figure 5-8 is an ideal well-bore/fracture connection for a propped fracture thatis intended to bypass near-wellbore formation dam-age. To achieve the desired production goals, a nar-row fracture must, at a minimum, carry the flow that

p pnet f c= − σ ,

Figure 5-7. The importance of fracture height.

Oil

(a)

Oil

Water

(c)

Oil

(b)

Figure 5-8. An ideal wellbore/fracture connection for apropped fracture that is intended to bypass near-wellboreformation damage.

rw


would have been produced through the entire well-bore circumference (had there been no damage). Thefracture conductivity kfw must be greater than 2πrwk,where rw is the wellbore radius. For higher perme-ability formations that can deliver high rates withsufficient fracture permeability, fracture width andany variables that affect width become important. Asdiscussed in the following and in Section 6-2, widthis controlled by the fracture dimensions (hf and L),net pressure inside the fracture acting to open andpropagate the fracture, and another property, themodulus or stiffness of the rock.

As implied by the term hydraulic fracturing, fluidmechanics is an important element in fracturing. Thetwo dominant fluid mechanics variables, injection(pump) rate qi and fluid viscosity µ, affect net pres-sure inside the fracture (and thus width) and largelycontrol transport and the final placement of proppantin the fracture. These variables also have a role incontrolling the volume of fluid lost to the formationduring pumping. For example, high pump ratesreduce the total fluid loss because for a given volumepumped there is less time for fluid loss to occur.

Another key factor of a good design is selection of the fluid and proppant systems with performancecharacteristics (e.g., µ, CL, kf) that best meet therequirements for the fracture treatment (i.e., materialselection). In addition, the performance variables forthe materials must be properly characterized. Fluidsand proppants are addressed in Chapter 7, and theirperformance is discussed in Chapter 8.

Finally, all the design parameters must be moldedto be compatible with existing well conditions (i.e.,operational considerations). For example, it does lit-tle good to complain that the detailed design andanalysis done in planning a treatment for an existingwell call for a high pump rate of 60 bbl/min whenthe wellbore conditions limit the maximum allow-able pump rate to one-half that rate. Clearly, for newwells the operational considerations (detailed inChapter 11) should be an integral part of planningfor the drilling and completion process (e.g., welltrajectory for extended reach wells) (Martins et al.,1992c).

5-1.4. Variable interactionIt is clear that with major design considerations com-ing from multiple disciplines, the variables willreact, interact and interconnect in multiple ways and

that many of these interactions will be contradictoryor incompatible. This is discussed later, but an exam-ple is as follows. Consider a case where reservoirgoals require a long fracture. With deep penetrationinto the pay zone, getting good proppant transportdown a long fracture clearly requires high fluid vis-cosity. However, high viscosity increases the netpressure inside the fracture. This reacts with thestress difference between the pay and the overlyingand underlying shales and causes height growth,resulting in less penetration than desired, and thusless viscosity is required.

Inherent contradictions controlling fluid selectionabound:

• Good viscosity is required to provide good prop-pant transport, but minimal pipe friction is alsodesirable to reduce surface pump pressure.

• The fluid system is expected to control fluid loss,but without damage to the formation or fracturepermeability.

• Performance at high temperature, for long periodsof time, is required from a fluid system that doesnot cost much.

5-2. In-situ stressIn-situ stress, in particular the minimum in-situ stress(termed the fracture closure pressure for nonhomoge-neous zones, as discussed earlier) is the dominantparameter controlling fracture geometry. It is dis-cussed in detail in Chapter 3. For relaxed geologicenvironments, the minimum in-situ stress is gener-ally horizontal; thus a vertical fracture that formedwhen a vertical wellbore broke remains vertical andis perpendicular to this minimum stress. Hydraulicfractures are always perpendicular to the minimumstress, except in some complex cases, and even forthose cases any significant departure is only at thewell. This occurs simply because that is the leastresistant path. Opening a fracture in any other direc-tion requires higher pressure and more energy.

The minimum stress controls many aspects offracturing:

• At very shallow depths or under unusual condi-tions of tectonic stress and/or high reservoir pres-sure, the weight of the overburden may be theminimum stress and the orientation of thehydraulic fractures will be horizontal; for more


normal cases, the minimum stress is generally hor-izontal and the maximum horizontal stress direc-tion determines whether the vertical fracture willrun north–south, east–west, etc.

• Stress differences between different geologic lay-ers are the primary control over the importantparameter of height growth (Fig. 5-9).

• Through its magnitude, the stress has a large bear-ing on material requirements, pumping equipment,etc., required for a treatment. Because the bottom-hole pressure must exceed the in-situ stress forfracture propagation, stress controls the requiredpumping pressure that well tubulars must with-stand and also controls the hydraulic horsepower(hhp) required for the treatment. After fracturing,high stresses tend to crush the proppant and reducekf ; thus, the stress magnitude dominates the selec-tion of proppant type and largely controls postfrac-ture conductivity.

Therefore, the detailed design of hydraulic fracturetreatments requires detailed information on in-situstresses. An engineer must know the magnitude ofthe minimum in-situ stress for the pay zone andover- and underlying zones and in some cases mustknow the direction for the three principal stresses.For a simple, relaxed geology with normal pore pres-

sure, the closure stress is typically between 0.6 and0.7 psi/ft of depth (true vertical depth, TVD). Moregenerally, as discussed in Chapter 3, the minimumstress is related to depth and reservoir pressure by

(5-4)

where Ko is a proportionality constant related to therock properties of the formations (possibly to boththe elastic properties and the faulting or failure prop-erties), σv is the vertical stress from the weight of the overburden, pr is the reservoir pore pressure, andT accounts for any tectonic effects on the stress (fora relaxed, normal fault geology, T is typically small).Ko is typically about 1⁄3. For fracture design, bettervalues are required than can be provided by such asimple relation, and methods of measuring or infer-ring the in-situ stress are discussed in Chapters 3 and4. For preliminary design and evaluation, using Eq. 5-4 with Ko = 1⁄3 is usually sufficient.

5-3. Reservoir engineeringAs previously mentioned, because the ultimate goalof fracturing is to alter fluid flow in a reservoir,reservoir engineering must provide the goals for adesign. In addition, reservoir variables may impactthe fluid loss.

Figure 5-9. Fracture height growth. (a) Idealized fracture profile of the relation of fracture geometry to in-situ stresses. σh = minimum horizontal stress, σH = maximum horizontal stress. (b) Typical fracture vertical cross section illustrating therelation of the total fracture height hf to the “original” fracture height hfo. (c) Theoretical relation among hf/hfo, pnet and thein-situ stress difference ∆σ (Simonson et al., 1978).

1

23 4

Shale

σH

σvσh

∆σ

Shale

(a) (b) (c)

hf hfo p net

0.10.20.30.40.50.60.70.80.9

1 2 3

p net

/∆σ

hf/hfo

0

σ σc o v r rK p p T≅ −( ) + + ,

5-3.1. Design goalsHistorically, the emphasis in fracturing low-perme-ability reservoirs was on the productive fracturelength xf. For higher permeability reservoirs, the con-ductivity kfw is equally or more important, and thetwo are balanced by the formation permeability k.This critical balance was first discussed by Prats(1961), more than 10 years after the introduction offracturing, with the important concept of dimension-less fracture conductivity CfD:

(5-5)

This dimensionless conductivity is the ratio of theability of the fracture to carry flow divided by theability of the formation to feed the fracture. In gen-eral, these two production characteristics should bein balance. In fact, for a fixed volume of proppant,maximum production is achieved for a value of CfD

between 1 and 2, as discussed in Chapters 1 and 10,with an analogy to highway design in Sidebar 5B.

Prats also introduced another critical concept, theidea of the effective wellbore radius rw . As shown inFig. 5-10, a simple balancing of flow areas betweena wellbore and a fracture gives the equivalent valueof rw for a propped fracture (qualitative relationonly):

(5-6)

However, this simple flow area equivalenceignores the altered pore pressure field around a linear

fracture and also assumes infinite conductivity. Pratscorrectly accounted for the pressure distributionaround a fracture and provided a general relationbetween dimensionless conductivity and rw forsteady-state conditions (see Chapter 1). The relationshows that for infinite-conductivity fractures, theupper limit on rw is slightly less than that from theflow area balance in Eq. 5-6. For infinite kfw, Pratsfound

(5-7)

Cinco-Ley et al. (1978) later integrated this into afull description of reservoir response, including tran-


′ ≈r xw f

2π

.

′ =r xw f0 5. .

Ck w

k xfDf

f

= .

5B. Highway analogy for dimensionless fracture conductivity

A simplistic analogy for dimensionless fracture conductivityCfD is a highway system. The numerator of this dimension-less variable is kfw, which is the capacity of the highway orthe ability of the highway to carry traffic. The denominator iskxf; this is the ability of the feeder roads to supply traffic tothe highway.

The famous old U.S. highway known as Route 66 ran, formuch of its length, across sparsely populated areas wherefeeder roads were few, narrow and far between. The abilityof the feeder road network to supply traffic to the highwaywas limited (similar to the conditions existing when a prop-ped hydraulic fracture is placed in a formation with very lowpermeability). In this case, the width, or flow capacity, of thehighway is not an issue (kfw does not have to be large).What is needed (and was eventually built) is a long, narrow(low-conductivity) highway.

As a comparison, consider Loop 610, the “superhighway”surrounding the city of Houston. The feeder system is locat-ed in a densely populated area, and the feeder roads arenumerous and wide. Here, the width, or flow capacity, of thehighway is critical. Making this highway longer has no effecton traffic flow, and the only way to increase traffic flow is towiden (i.e., increase the conductivity of) the road. This isobviously analogous to placing a fracture in a higher perme-ability formation, with the postfracture production limited bythe fracture width (or, more accurately, limited by kfw).

If CfD is the ratio of the ability of a highway to carry trafficto the ability of the feeder system to supply that traffic to thehighway, clearly a highway should be engineered to approxi-mately balance these conditions. That is, a CfD value > 50 isseldom warranted, because a highway would not be con-structed to carry 50 times more traffic than the feeder systemcould supply. In the same way, a value of 0.1 makes littlesense. Why construct a highway that can only carry 10% ofthe available traffic? In general, an ideal value for CfD wouldbe expected to be about 1 to result in a balanced, well-engineered highway system.

A balance of about 1 is certainly attractive for steady-flowtraffic conditions that may exist through most of the day.However, during peak traffic periods the feeder system maysupply more traffic than normal, and if this rush hour or tran-sient traffic period is a major consideration, then a larger ratioof CfD may be desirable. Thus, a CfD of 10 may be desirablefor peak flow (transient) periods, as opposed to a CfD value ofapproximately 1 for steady-state traffic conditions.

Figure 5-10. Equivalent wellbore radius rw′.��

..

. ... . . .

... ..

. ... . .

. .... ... .

.. ..

. . . ..

..... . . ....

.... .. .. .

..

.. .

..

..

. . .... .

..

.

.

.... .

Flow area = 2πrwh

Flow area = 4xfh

2rw´ = xfπ

Flow area =2πrw´h


sient flow. For pseudoradial flow, Cinco-Ley et al. ex-pressed rw as a function of length and CfD (Fig. 5-11).

The chart in Fig. 5-11 (equivalent to Prats) can beused (when pseudoradial flow is appropriate) as apowerful reservoir engineering tool to assess possi-ble postfracture productivity benefits from proppedfracturing. For example, the folds of increase (FOI)for steady-state flow can be defined as the postfrac-ture increase in well productivity compared withprefracture productivity calculated from

(5-8)

where re is the well drainage or reservoir radius, rw

is the normal wellbore radius, and s is any prefrac-ture skin effect resulting from wellbore damage,scale buildup, etc. An equivalent skin effect sf result-ing from a fracture is

(5-9)

for use in reservoir models or other productivity cal-culations. Equation 5-8 provides the long-term FOI.Many wells, particularly in low-permeability reser-voirs, may exhibit much higher (but declining) early-time, transient FOI. The preceding relations are fortransient pseudoradial flow before any reservoirboundary effects; the case for boundary effects isdiscussed in Section 12-2.6.

5-3.2. Complicating factorsThese principal concepts give a straightforwardmethod for predicting postfracture production; how-ever, complications can reduce postfracture produc-tivity below the levels expected or give better pro-ductivity than that calculated. The major complica-tions include non-Darcy (or turbulent) flow, transientflow regimes, layered reservoirs and horizontal per-meability anisotropy (particularly any natural fissurepermeability).

For high-rate wells, non-Darcy or turbulent flowcan be an important factor that causes an increasedpressure drop along the fracture. This creates anapparent conductivity that is less than the equivalentlaminar flow conductivity. The apparent CfD is alsoreduced and productivity is less than that expected.Another complicating effect that can reduce produc-tivity from expected levels is formation layering,where a fracture is in multiple layers with signifi-cantly different values for porosity, permeability orboth. Unlike radial flow into a wellbore, average val-ues of permeability and porosity do not apply, andfor layered formations, postfracture performancefalls below simple calculations based on average per-meability (Bennett et al., 1986). These and othereffects are discussed in Section 12-3.

For lower permeability formations and for sometime period, postfracture performance is dominatedby transient flow (also called flush production) asdiscussed by Cinco-Ley et al. (1978). For transientconditions, reservoir flow has not developed intopseudoradial flow patterns, and the simple rw´ rela-tions are not applicable. In the example in Fig. 5-12,pseudoradial flow did not develop until about 48months. During the prior transient flow regimes,

Figure 5-12. Late development of pseudoradial flow.

Rat

e (M

cf/D

)

5000

2000

8000

1000

500

200

10012 24 36 48 60 72 84 96 108

Numericalmodel

Radial flow

Time (months)

Radial flow× FOI = 3.9

Gas wellk = 0.1 md, xf = 1000 fth = 50 ft, kfw = 200 md-ft

Figure 5-11. Equivalent wellbore radius as a function ofdimensionless fracture conductivity and fracture length.

Effe

ctiv

e w

ell r

adiu

sF

ract

ure

half-

leng

th

r w

x f,

0.1 0.2 0.5 1 2 5 10 20 500.01

0.05

0.1

0.2

0.5

Dimensionless fracture conductivity, CfD

CfD > 30, xf limitedrw´ = xf/2

CfD < 0.5, kfw limitedrw´ = 0.28 kfw/k0.02

s r rf w w= − ′( )ln /

FOIr r s

r re w

e w

= ( ) +′( )

ln

ln

/

/,


productivity was better than that predicted from thepseudoradial flow rw . The duration of the transientflow period is a function of permeability, CfD and xf

2

such that for moderate- to high-permeability wellsthe period is too short to have practical significancefor fracture design. However, it may be important forpostfracture well test analysis. For low-permeabilitywells with long fractures, transient flow may domi-nate most of the productive well life.

5-3.3. Reservoir effects on fluid lossReservoir properties such as permeability to reservoirfluid, relative permeability to the fracturing fluid fil-trate, total system compressibility, porosity, reservoirfluid viscosity and reservoir pressure all play a role influid loss while pumping (see Section 6-4). Thus, cer-tain reservoir information is required for treatmentdesign, as well as for specifying design goals.

5-4. Rock and fluid mechanicsRock and fluid mechanics (along with fluid loss)considerations control the created fracture dimen-sions and geometry (i.e., fracture height hf, length L and width w). These considerations all revolvearound the net pressure pnet given by Eq. 5-3.However, pnet, which controls hf and L, is itself afunction of hf and L, and the various physical behav-iors connecting height, net pressure, width, etc.,interact in many ways. This makes simple statementsabout the relative importance of variables difficult orimpossible. However, the basic physical phenomenacontrolling fracture growth are understood and arewell established.

5-4.1. Material balanceThe major equation for fracturing is material bal-ance. This simply says that during pumping a certainvolume is pumped into the earth, some part of that islost to the formation during pumping, and theremainder creates fracture volume (length, width andheight). It is the role of fracture models to predicthow the volume is divided among these three dimen-sions. The volume pumped is simply

(5-10)

where qi is the total injection rate and tp is the pump-ing time for a treatment. Equally simple, the fracturevolume created during a treatment can be idealizedas

(5-11)

where hf is an average, gross fracture height, w– is theaverage fracture width, L is the fracture half-lengthor penetration, and η is the fluid efficiency. Finally,as discussed by Harrington et al. (1973) and Nolte(1979), the volume lost while a hydraulic fracturetreatment is being pumped can be approximated by

(5-12)

where CL is the fluid-loss coefficient (typically from0.0005 to 0.05 ft/min1/2), hL is the permeable orfluid-loss height, and Sp is the spurt loss (typicallyfrom 0 to 50 gal/100 ft2). Because material balancemust be conserved, Vi must equal VLp plus Vf, andEqs. 5-10 through 5-12 can be rearranged to yield

(5-13)

showing a general relation between several impor-tant fracture variables and design goals.

Modeling of hydraulic fracture propagation inlow- to medium-permeability formations typicallyshows an average width of about 0.25 in. (±50%)over a fairly wide range of conditions (e.g., Abou-Sayed, 1984). Using this value, the effect of the pri-mary variables height hf and fluid-loss coefficient CL on fracture penetration L are investigated in Fig. 5-13. This is for a simple case of a constant0.25-in. fracture width. Figure 5-13a shows length as a strong, nearly linear function of hf; e.g., dou-bling hf cuts fracture penetration by 50%. For similarconditions, Fig. 5-13b shows that the fluid-loss coef-ficient is not as important; e.g., doubling CL reducesL by only about 20%. However, with fracturing, suchsimple relations are never fixed. As seen in Fig. 5-13c, for a higher loss case, doubling CL from 0.005to 0.01 reveals a nearly linear relation between CL

and L, just as for height in Fig. 5-13a. Basically, forFigs. 5-13a and 5-13b, the loss term (first term in thedenominator of Eq. 5-13) is small compared with thefracture volume term (third term in the denominator).Therefore, the fluid loss is relatively low and fractureV q ti i p= × ,

V h w L Vf f i= × × = ×2 η ,

V C h L t Lh SLp L L p L p≅ +6 4 ,

Lq t

C h t h S whi p

L L p L p f

≅+ +6 4 2

,


fluid efficiency (η, as defined in Eq. 5-11) is high. InFig. 5-13c, the loss term is large compared with thevolume term (high loss and low efficiency), and theloss coefficient becomes the dominant variable, withL less sensitive to variations in hf or equivalently w– ifit varies from the fixed value of 0.25 in.

5-4.2. Fracture heightEquation 5-13 demonstrates that fracture height hf

and fluid-loss height hL are important parameters forfracture design. Loss height is controlled by in-situvariations of porosity and permeability. Fractureheight is controlled by the in-situ stresses, in particu-lar by differences in the magnitude or level of stressbetween various geologic layers. More formally,height is controlled by the ratio of net pressure tostress differences ∆σ, as illustrated in Fig. 5-9, where∆σ is the difference between stress in the boundaryshales and stress in the pay zone. Ignoring any pres-sure drop caused by vertical fluid flow, the relationamong fracture height, initial fracture height, pnet

and ∆σ can be calculated as demonstrated bySimonson et al. (1978). This relation is included in Fig. 5-9c.

For cases when pnet is relatively small comparedwith the existing stress differences (e.g., less than50% of ∆σ), there is little vertical fracture growthand the hydraulic fracture is essentially perfectlyconfined. This gives a simple fracture geometry (Fig. 5-14a) and increasing net pressure (Fig. 5-14b).For cases when pnet is much larger than the existingstress differences, vertical fracture height growth isessentially unrestrained. Again, the geometry is afairly simple radial or circular fracture (Fig. 5-14c)and declining net pressure (Fig. 5-14b).

For more complex cases when pnet is about equalto ∆σ, fracture geometry becomes more difficult topredict, and significant increases in height can occurfor small changes in net pressure. Also, for this case,the viscous pressure drop from vertical flow retardsfracture height growth (see Weng, 1991), and theequilibrium height calculations in Fig. 5-9 are nolonger applicable.

Figure 5-13. Effect of hf and CL on L.

L (ft

)

1400

1600

1200

1000

800

600

400

200

0200 40 60 80 100

hf = hL = 100 ft

hf = hL = 200 ft

2:1

Time (min)

(a)

CL = 0.001 ft/min1/2

qi = 30 bbl/minw = 0.25 in.Sp = 0

L (ft

)

1400

1600

1200

1000

800

600

400

200

0200 40 60 80 100

Time (min)

CL = 0.0005

CL = 0.001

(b)

hf = hL = 200 ftqi = 30 bbl/minw = 0.25 in.Sp = 0

L (ft

)

500

600

400

300

200

100

0200 40 60 80 100

CL = 0.005

CL = 0.01

1.8:1

Time (min)

hf = hL = 100 ftqi = 30 bbl/minw = 0.25 in.Sp = 0

(c)


5-4.3. Fracture widthConsider a slit in an infinite elastic media (i.e., theearth). Also consider that the slit is held closed by a fracture closure stress but is being opened by aninternal pressure equal to the closure stress plus a netpressure pnet. Under these conditions (discussed in

detail in Chapter 6), the slit opens into an ellipticalshape, with a maximum width

(5-14)

where E ′ is the plane strain modulus (E ′ = E/(1 –ν2), ν is Poisson’s ratio and typically equals about0.2), and d is the least dimension of the fracture. For a confined-height fracture with a tip-to-tip lengthgreater than hf, d equals hf. This shows a direct rela-tion between net pressure and width and introducesan important material property, the plane strain mod-ulus. However, because typically ν2 < 0.1, the planestrain modulus seldom differs from Young’s modulusE by a significant amount.

5-4.4. Fluid mechanics and fluid flowThe major fluid flow parameters are the fluid viscos-ity (resistance to flow) µ and injection rate qi. Therate also effects the pump time and hence is impor-tant to fluid-loss and material-balance considerations,as discussed previously. Both parameters are criticalfor proppant transport, and both parameters alsoaffect net pressure and thus affect fracture height and width.

As an example, consider a Newtonian fluid flowinglaterally through a narrow, vertical slit (i.e., fracture) (Fig. 5-15). For laminar flow (the general case forflow inside hydraulic fractures), the pressure dropalong some length ∆x of the slit is

(5-15)

Assuming a simple case of a long, constant-heightand -width fracture with two wings and zero fluidloss (i.e., the flow rate in each wing is q = qi /2) andalso assuming zero net pressure at the fracture tip,

Figure 5-14. Relationship of pnet to stress differences.

3500 5000 100 200 300

5200

5100

5000

4900

Stress (psi) Fracture penetration (ft)

∆σ = 1500 psi

pnet < 1/3∆σ

(a) Depth(ft)

Net

pre

ssur

e (p

si)

2000

1000

500

200

100

50

20

101 2 5 10 20 50 100 200

Pump time (min)

Shut-in

∆σ = 1500 psi

∆σ = 50 psi

(b)

CL = 0.002qi = 20 bbl/minhfo = 100 ftE = 4E+6 psiµ = 200 cp

3200 3800 100 300 400 500200

4800

4900

5000

5100

5200

5300

Stress (psi) Fracture penetration (ft)

∆σ = 50 psi

pnet > 4∆σ

(c)Depth

(ft)

∆∆p

x

q

h wnet

f

= 123

µ.

Figure 5-15. Fluid flowing laterally through a narrow verti-cal fracture.

w

q = qi ⁄ 2q

v– = hfw

wp d

Enet

max ,=′

2


Eq. 5-15 is integrated from the fracture tip back tothe wellbore to give

(5-16)

For this long, confined-height fracture, hf is theminimum fracture dimension for Eq. 5-14, and thefracture width and net pressure are related by

(5-17)

Combining the two equations gives the proportionality

(5-18)

The exponent of 1⁄4 for this simple fracture geome-try and for Newtonian fluids implies that the fracturewidth is virtually constant; e.g., doubling the pumprate from 20 to 40 bbl/min increases the width onlyby about 20%. The same effect is found for all thevariables in Eq. 5-18. Generally, for non-Newtonianfluids, the exponent is approximately 1⁄3.

This relationship for fracture width can also beused with Eq. 5-17 to give net pressure expressed as

(5-19)

where κ is a constant (see Eq. 6-11) to provide anequality for this expression.

Thus, as a result of viscous forces alone, net pres-sure inside the fracture develops as a function of themodulus, height and (qµ)1/4. From the nature of thisrelation, however, it is clear that modulus and heightare much more important in controlling net pressurethan are pump rate and viscosity, the effect of whichis muted by the small exponent for the relation.

5-4.5. Fracture mechanics and fracture tip effects

The fluid mechanics relations show pnet related to modulus, height, fluid viscosity and pump rate.However, in some cases, field observations haveshown net pressure (and presumably fracture width)to be greater than predicted by Eq. 5-19 (Palmer andVeatch, 1987). In such cases the fluid viscosity has asmaller effect on fracture width than predicted byEq. 5-19. This is probably because the simple rela-

tion in Eq. 5-16 assumes no net pressure at the frac-ture tip; i.e., fracture tip effects or fracture propaga-tion effects are ignored. When tip effects are takeninto account, the fracture width is affected by bothfluid viscosity and tip effects (Shlyapobersky et al.,1988a, 1988b). As shown by Nolte (1991), tipeffects can be approximated by considering the netpressure within the tip region to equal ptip (asopposed to zero) in Eq. 5-16. For a positive tip pres-sure, the net pressure equation becomes

(5-20)

where ptip is the pressure required at the fracture tipto open new fracture area and keep the fracture prop-agating forward. This simple relationship serves toillustrate that there are always two components to netpressure: a viscous component and a fracture tip-effects component. The relative magnitude of thetwo effects varies from case to case, and because ofthe small exponent, the combined effects are muchless than the direct sum of the individual effects. Forexample, when the viscous component and the tipcomponent are equal, the net pressure is increased byonly 20% over that predicted when one of the com-ponents is ignored.

• Fracture toughness and elastic fracture mechanics

The fracture tip propagation pressure, or fracturetip effect, is generally assumed to follow thephysics of elastic fracture mechanics. In that case,the magnitude of the tip extension pressure ptip iscontrolled by the critical stress intensity factor KIc

(also called the fracture toughness). Fracturetoughness is a material parameter, and it may bedefined as the strength of a material in the pres-ence of a preexisting flaw. For example, glass hasa high tensile strength, but the presence of a tinyscratch or fracture greatly reduces the strength(i.e., high tensile strength but low fracture tough-ness). On the other hand, modeling clay has lowstrength, but the presence of a flaw or fracturedoes not significantly reduce the strength. Lab-oratory-measured values for the material propertyKIc show toughness ranging from about 1000 toabout 3500 psi/in.1/2, with a typical value of about2000 psi/in.1/2. These tests (after Schmidt andHuddle, 1977; Thiercelin, 1987) include a rangeof rock types from mudstones and sandstones to

pE

hq Lnet

f

i= ′ { }3 4

1 4/

/,κµ

pE

hq L pnet

f

i tip≈ ′ { } +

3

4

4

1 4

κµ/

,

pE w

hnet

f

= ′2

.

pq L

h wneti

f

= 63

µ.

wq L

Ei∝′

µ 1 4/

.


carbonates and consider confining pressures from0 to 5000 psi.

From elastic fracture mechanics, for a simpleradial or circular fracture geometry with a pene-tration of L, the fracture tip extension pressure is

(5-21)

and it decreases as the fracture extends. For evena small fracture penetration of 25 ft, this gives atip extension pressure of 29 psi, whereas viscouspressures (Eq. 5-19) are typically 10 or moretimes larger. Thus normal linear elastic fracturemechanics considerations indicate that fracturemechanics, or the tip extension pressure, generallyplays a negligible role for hydraulic fracturing.

• Apparent fracture toughness

Field data typically show fracture extension pres-sure to be greater than that given by Eq. 5-21,with 100 to 300 psi as typical values and evenhigher values possible. This difference is due toseveral behaviors not included in elastic fracturemechanics calculations. One important (and long-recognized) consideration is that the fracturingfluid never quite reaches the fracture tip; i.e., thereis a “fluid lag” region at the tip that increases theapparent toughness and tip pressure (Fig. 5-16). In other cases, tip pressure may be even greater.Other tip phenomena include nonelastic rockdeformation near the fracture tip and tip pluggingwith fines, with these mechanisms acting alone orin conjunction with the fluid flow and/or fluid lag

phenomena. Tip phenomena are discussed indetail in Chapters 3 and 6.

Measured values for tip extension pressure that are higher than predicted from laboratory-measured rock toughness KIc can be accounted for in hydraulic fracture calculations through theuse of the effective, or apparent, fracture tough-ness KIc-apparent (Shlyapobersky, 1985). In practice,because KIc-apparent is not a material constant, thetip effects should be defined or calibrated by frac-turing pressure data for a particular situation (seeSidebar 9B).

5-4.6. Fluid lossAs seen from the material balance (Eq. 5-13), fluidloss is a major fracture design variable characterizedby a fluid-loss coefficient CL and a spurt-loss coeffi-cient Sp. Spurt loss occurs only for wall-building flu-ids and only until the filter cake is developed. Formost hydraulic fracturing cases, the lateral (and ver-tical) extent of the fracture is much greater than theinvasion depth (perpendicular to the planar fracture)of fluid loss into the formation. In these cases, thebehavior of the fluid loss into the formation is linear(1D) flow, and the rate of fluid flow for linear flowbehavior is represented by Eq. 5-1.

This assumption of linear flow fluid loss giving theCL ⁄√ t relation has been successfully used for fractur-ing since its introduction by Carter (1957). The rela-tion indicates that at any point along the fracture, therate of fluid loss decreases with time, and anythingthat violates this assumption can cause severe prob-lems in treatment design. For example, fluid loss tonatural fissures can result in deep filtrate invasioninto the fissures, and the linear flow assumption mayno longer be valid. In fact, for the case of natural fis-sures if net pressure increases with time, the fluid-loss rate can increase, and treatment pumping behav-ior may be quite different from that predicted. Thetotal fluid loss from the fracture is controlled by thetotal fluid-loss coefficient CL, which Howard andFast (1957) decomposed into the three separatemechanisms illustrated in Fig. 5-17 and discussed in Section 6-4.

The first mechanism is the wall-building character-istics of the fracturing fluid, defined by the wall-building coefficient Cw. This is a fluid property thathelps control fluid loss in many cases. For most frac-turing fluid systems, in many formations as fluid loss

p KLtip Ic= π

48,

Figure 5-16. Unwetted fracture tip (fluid lag).

p1 = fracture pressurep2 ≤ reservoir pressure

Fluid lag

Closure stress

p1 p2


occurs into the formation, some of the additives andchemicals in the fluid system remain trapped on ornear the formation face, forming a physical filter-cake barrier that resists fluid loss.

Outside of the filter cake is the invaded zone,which is the small portion of the formation that hasbeen invaded by the fracturing fluid filtrate. Thismechanism is the filtrate effect, or invaded zoneeffect, and it is characterized by the viscosity or rela-tive permeability control coefficient Cv. As discussedin Chapter 6, Cv can be calculated, and this parame-ter is governed by the relative permeability of theformation to the fracturing fluid filtrate kfil, the pres-sure difference ∆p between the pressure inside thefracture (i.e., closure pressure + pnet) and the reser-voir pressure, and the viscosity of the fracturing fluidfiltrate µfil. This mechanism is usually most impor-tant in gas wells, where the invading fluid has muchhigher viscosity than the reservoir fluid being dis-placed, or where relative permeability effects pro-duce a filtrate permeability that is much less (<k/10)than the permeability to the reservoir fluid. Othercases are where a clean fluid is used such that no fil-ter cake develops or for fracturing high-permeabilitywells where no filter cake develops and high-viscos-ity crosslinked gel may be lost to the formation (i.e.,µfil is very high).

For fluid to leak off from the fracture, the reservoirfluid must be displaced. This sets up some resistanceto fluid loss, and this reservoir effect is characterizedby the compressibility coefficient Cc. As discussed inChapter 6, the parameter for this calculation is gov-erned by a pressure difference ∆p between the pres-sure inside the fracture (i.e., closure pressure + pnet)and the reservoir pressure, permeability to the mov-able formation fluid k, total system compressibilityfor the reservoir ct, and the viscosity of the reservoirfluid (gas or oil) µ. This parameter is usually moreimportant for a liquid-saturated reservoir (low com-pressibility and relatively high reservoir fluid viscos-ity) and when a filter cake does not develop.

Each of these three mechanisms provides someresistance to fluid loss, and all three act as resistorsin series (although the fluid-loss coefficient itself isdefined in terms of conductance, or the inverse ofresistance). The three mechanisms variously com-bine in different situations to form the total or com-bined fluid-loss coefficient CL, which is used forfracture design (see Chapter 6). This clearly complexsituation makes it desirable to measure fluid lossfrom field tests (just as permeability must be mea-sured from field flow, buildup tests or both) when-ever possible (see Chapter 9).

5-4.7. Variable sensitivities and interactions

The complexity of hydraulic fracture design comesfrom the interactions of the major design variables(hf, E, ∆σ, KIc and CL) and that different variablesaffect different aspects of fracturing in different ways.As discussed in Section 5-4.1 concerning the sensitiv-ity of fracture penetration to hf and CL, the impor-tance of various variables can change from case tocase. Several examples of this are discussed here.

• Net pressure

The magnitude of net pressure for a specific frac-ture treatment is a major concern, because theratio of net pressure to stress differences betweenthe pay zone and bounding zones controls fractureheight. Also, net pressure directly controls width.However, what controls net pressure varies signif-icantly from case to case.

In the case of hard-rock formations (i.e., forma-tions with values for Young’s modulus of 2 × 106

psi or greater) with height confinement and for

Figure 5-17. The three regions of fluid loss.

Pre

ssur

e

Invadedzone

Reservoircontrol

Closurepressure

+pnet

Distance into formation

Reservoir pressure

Filter cake


treatments pumping viscous fluids at normal frac-turing rates, the viscous term of the net pressureequation dominates any fracture tip effects.Toughness or tip effects become important forcases where fracture height is unconfined (e.g.,radial or circular fractures) or for very soft rocks(e.g., formations such as unconsolidated sandswith E ≤ 0.5 × 106 psi). For treatments using low-viscosity fluid or pumping at very low rates, theviscous term of the net pressure equation becomessmall, and fracture toughness becomes a dominantparameter. Although many cases fall into one ofthese extremes, neither effect should be over-looked for the prudent application of fracturing.

The magnitude of net pressure may also be con-trolled by in-situ stress differences between thepay and the bounding layers. Consider a casewhere barrier zones (e.g., formations with higherclosure stress) surround the pay zone (Fig. 5-9)and further assume that because of either viscousor toughness effects, pnet increases to the level ofthe stress differences. Massive height growth thenbegins, and only very small increases in the netpressure are possible. Net treating pressure is nowcontrolled directly by ∆σ and is essentially inde-pendent of both fluid viscosity and apparent frac-ture toughness effects. This case is illustrated inthe next section.

• Fracture height and net pressure

For a fracture with significant stress barriers andin a formation with a medium to high value forthe modulus, the viscous term in Eq. 5-20 controlsthe net treating pressure. In such a case, pnet

becomes a strong function of fracture height.However, as illustrated in Fig. 5-9, fracture heighthf is controlled by net pressure. To put it in anoth-er form, fracture height is a function of fractureheight. As discussed in Chapter 6, this is wherefracture models become important.

As an example, consider the case of a thin (h = 25 ft) sandstone pay zone in a hard-rock for-mation (E = 5 × 106 psi). Further assume that thiszone is surrounded by shales with an in-situ stress1000 psi greater than the stress in the sand, mak-ing them what would normally be consideredgood barriers to vertical fracture growth. As seenin Fig. 5-18a, even for pumping a moderate (50-cp)viscosity fluid at a moderate rate, net pressureimmediately jumps to a level slightly greater than

1000 psi (i.e., ∆σ is controlling pnet), and exten-sive height growth occurs. Because ∆σ is control-ling the allowable net pressure, increasing thefluid viscosity fourfold has essentially no effect onnet pressure after the first few minutes. The verti-cal fracture width profile plotted in Fig. 5-18bshows that for pnet about equal to ∆σ, fracturewidth in the bounding layers may be too small forproppant admittance. This is discussed in the sub-sequent section on proppant admittance.

Now consider the same case but with a 50-ftthick sandstone section. As seen in Fig. 5-19, pnet

stays below ∆σ for the 50-cp fluid case and little

Figure 5-18. Height growth example in a hard-rock forma-tion.

p net

(psi

)

2000

1000

500

200

100

50

20

00.20 0.5 1 2 5 10 20 50 100 200 500

Pump time (min)

µ = 200 cp

µ = 50 cp

∆σ = 1000 psiqi = 15 bbl/minhf = 25 ftE = 5 × 106 psi

Nolte-Smith plot

(a)

5100

5000

4900

3500 4000 4500 –0.2 0 0.2

In-situ stress (psi) Width (in.)

Depth(ft)

Perforatedinterval

(b)


height growth occurs. For a more viscous (200-cp)fluid, net pressure again approaches the stress dif-ference of 1000 psi, and again extensive heightgrowth occurs. These examples show that fractureheight is a function of fracture height.

Finally, consider the original (h = 25 ft) caseagain, but assume this is a soft-rock (unconsoli-dated sand with E < 0.5 × 106 psi) zone. Furtherassume that because of high permeability, fluidloss is much greater than for the previous twocases. Figure 5-20 shows pnet is much less than∆σ, with essentially no height growth. Also, theflat nature of the net pressure behavior in theNolte-Smith log-log plot of pnet versus time indi-cates that fracture tip effects are dominating netpressure behavior, as expected from Eq. 5-20.Chapter 9 discusses net pressure behavior and themeans to determine the controlling conditions.

• Fluid viscosity

Fluid viscosity provides an example of how vari-ables affect different parts of the fracturing pro-cess in different ways. Consider a case of radialfracture growth in a soft rock (E < 1 × 106 psi).Toughness dominates pnet and fracture width, andviscosity becomes unimportant in controlling frac-ture geometry. However, viscosity can remain acritical consideration for proppant transport if along fracture is desired and for fluid-loss control.

Further assume this case is a very high perme-ability formation, such that only a short fracture isrequired. Thus, high viscosity is not required forproppant transport. However, in this very high

permeability formation it is probable that the frac-turing fluid cannot build a filter cake to controlfluid loss, and the only fluid-loss control willcome from the viscosity (or invaded zone) effectCv (see Section 5-4.6). Viscosity is therefore amajor factor for fluid selection, despite having noeffect on geometry and not being critical for prop-pant transport.

5-5. Treatment pump schedulingThe fracture design process involves reservoir engi-neering to define the xf and kfw goals. It involvesrock mechanics to consider the possibility of obtain-ing a desired fracture geometry. It includes fluidmechanics considerations to confirm that therequired proppant transport is possible and rheologyto determine if the required fluid properties are pos-

Figure 5-19. Height growth example in a thicker hard-rockformation.

p net

(psi

)2000

1000

500

200

100

50

20

00.20 0.5 1 2 5 10 20 50 100200 500

Pump time (min)

µ = 200 cp

µ = 50 cp

∆σ = 1000 psi qi = 15 bbl/minhf = 50 ftE = 5 × 106 psi

Figure 5-20. Height growth example in a soft-rock formation.

∆σ = 1000 psiqi = 15 bbl/minhf = 25 ftE = 5 × 106 psi

µ = 50 cp

Nolte-Smith plot Confined heightTip-dominated behavior

p net

(psi

)

2000

1000

500

200

100

50

20

0.2 0.5 1 2 5 10 20 50 100 200 500

Pump time (min)

(a)

5100

5000

4900

3500 4500 100 200 300 400

In-situ stress (psi) Fracture penetration, L (ft)

Perforatedinterval

Depth(ft)

(b)


sible. It also includes material selection and on-siteoperational considerations as discussed in Section 5-6. The product of this process is a treatment pumpschedule. This includes the pad volume necessary to create the desired fracture penetration, along withacid or proppant scheduling to achieve the desiredpostfracture conductivity. For propped fracturing,pump scheduling includes fluid selection, proppantselection, pad volume, maximum proppant concen-tration to be used and a proppant addition schedule.After the design goals and variables are defined, theproppant addition schedule is usually obtained byusing a fracture simulator, although for many casesanalytical calculations based on fluid efficiency arealso easily implemented. Chapter 10 provides addi-tional detail for treatment design.

5-5.1. Fluid and proppant selectionFracturing materials are discussed in Chapter 7, andtheir performance characterization is discussed inChapter 8. The major considerations for fluid selec-tion are usually viscosity (for width, proppant trans-port or fluid-loss control) and cleanliness (after flow-back) to produce maximum postfracture conduct-ivity. Other considerations that may be major forparticular cases include

• compatibility with reservoir fluids and reservoir rock

• compatibility with reservoir pressure (e.g., foamsto aid flowback in low-pressure reservoirs)

• surface pump pressure or pipe friction considerations

• cost

• compatibility with other materials (e.g., resin-coated proppant)

• safety and environmental concerns (see Chapter 11).

Proppant selection must consider conductivity atin-situ stress conditions (i.e., the effect of stress onproppant permeability kf). Proppant size must also beconsidered. In general, bigger proppant yields betterconductivity, but size must be checked against prop-pant admittance criteria, both through the perforationsand inside the fracture (see Section 5-5.4). Finally,the maximum in-situ proppant concentration at shut-in must be selected, as it determines how much of thehydraulic width created by the fracture treatment willbe retained as propped width once the fracture closes.

5-5.2. Pad volumeFor a treatment using viscous fluid, the fluid carriesthe proppant out to the fracture tip. For these casesthe pad volume determines how much fracture pene-tration can be achieved before proppant reaches thetip and stops penetration in the pay zone. Once thepad is depleted, a fracture may continue to propagateinto impermeable layers until the proppant bridges inlow-width areas. Thus, pumping sufficient pad tocreate the selected length is critical. For treatmentsusing very low viscosity fluid (i.e., “banking”-typetreatments), proppant settles out of the fluid andessentially replenishes the pad. The pad volume mustonly be sufficient to open enough fracture width forproppant admittance, and the carrying capacity of thefluid, as opposed to the pad volume, determines thefinal propped length.

On the other hand, too much pad can in someinstances be even more harmful, particularly forcases requiring high fracture conductivity. The frac-ture tip continues to propagate after pumping stops,leaving a large, unpropped region near the fracturetip. Significant afterflow can then occur in the frac-ture, carrying proppant toward the tip and leaving a poor final proppant distribution. This afterflowoccurs because the widest section of the fracture isnear the wellbore at shut-in, and most of the prop-pant pumped is stored there. However, the highestfluid-loss rates are near the fracture tip. Thus, prop-pant-laden slurry continues to flow toward the tip ofthe fracture. Afterflow continues until either the frac-ture closes on the proppant, stopping proppant move-ment, or until proppant-laden slurry reaches the frac-ture tip. At that point the slurry dehydrates and stopsany additional fracture propagation. Ideally, ofcourse, it is better to have the proppant at the frac-ture tip at shut-in and thus minimize afterflow.

An ideal schedule for a normal treatment (asopposed to subsequently discussed TSO designs) isone where the pad depletes and proppant reaches thefracture tip just as the desired fracture penetration isachieved and also just as pumping stops. This is thesequence in Figs. 5-2, 5-3 and 5-4.

The critical parameter of the pad volume or padfraction fpad is related directly to the fluid efficiencyfor a treatment (Nolte, 1986b). This relation fromSidebar 6L gives the pad volume expressed as a fraction of the entire treatment volume:


(5-22)

That is, a treatment with an expected efficiency ηof 50% would require a pad fraction of about 1⁄3. Asdiscussed in Chapter 9, the efficiency for a specificformation and fluid system can be determined by acalibration treatment.

This discussion of pad volume has so far concen-trated on the fluid-loss aspects of the pad volume;i.e., the pad is pumped first to serve as a sacrificialstage of the treatment to enable the fracture to pene-trate into permeable formations. This important effectof the pad volume may be the critical aspect govern-ing the size of the pad for most applications. How-ever, hydraulic fracturing is complicated, in that mostthings are done for at least two reasons, whichapplies to pad volume specification. The second pur-pose of the pad volume is to create sufficient fracturewidth to allow proppant to enter the fracture (seeSection 5-5.4 on proppant admittance). Even for acase of very low fluid loss, some minimum pad vol-ume is required. Both of these aspects of the pad vol-ume must always be considered for treatment design.

• Propped width

A major design goal is fracture conductivity kfw,which consists of proppant pack permeability andpropped fracture width. Proppant permeability kf

is a function of the proppant selected, in-situstress and residual damage from fluid additives(see Chapter 8). Propped width is controlled bythe treatment design.

The effective propped width wp-eff is a functionof the average fracture width wf at shutdown (i.e.,hydraulic width at the end of pumping a treat-ment), proppant concentration C in the fracture atthat time (i.e., giving the ideal propped width wp)and the volume of proppant wlost that is lost on thefaces of the fracture to embedment, gel residue,etc. (usually expressed as lbm/ft2 “lost”). In termsof these parameters, the effective propped widthcan be expressed as

(5-23)

(5-24)

where F is the fill fraction (Fig. 5-21), the con-stant 8.33 converts the units to lbm/gal, γprop is the specific gravity (s.g.) of the proppant, C is thefinal in-situ proppant concentration at shut-inexpressed as pounds of proppant per fluid gallon(ppg), and φ is the porosity of the proppant pack,typically about 0.35.

Increasing the concentration from 8 (F ≈ 0.4) to 16 ppg (F ≈ 0.6) significantly increases thepropped fracture width (50% increase in the fillfraction). However, this large increase in proppedwidth is accomplished at the expense of additionalrisk to the job and to the well, because of eithersurface mechanical problems or an unexpectedtotal screenout somewhere in the fracture or in thenear-wellbore region between the well and the far-field fracture (see the discussion of tortuosity inSection 6-6). In practice, most treatments use amaximum concentration of about 8 to 14 ppg,although concentrations of 20 ppg have beenpumped.

Another manner of increasing propped width isto increase fracture width. Theoretical and numer-ical models generally show that the fracturewidth, while the fracture is growing, is relativelyinsensitive to the controllable job variables ofpump rate and fluid viscosity. For a simple frac-ture geometry, width is proportional to rate andviscosity raised to a small power. For Eq. 5-18with the exponent 1⁄4, doubling the pump rateincreases fracture width by only 18%, at theexpense of significant pipe friction and surface

w w w w F wp eff p lost f lost− = − = × −

FC

Cprop

=× +( ) × −( )8 33 1.

,γ φ

Figure 5-21. Fill fraction versus proppant concentration.

Fill

frac

tion,

F

0.8

1.0

0.6

0.4

0.2

0.040 8 12 16 20 24

Proppant concentration (lbm/gal)

% of fracture filled by proppant pack Pack porosity = 0.35

Proppant s.g. = 2.65(sand)

Proppant s.g. = 3.2(intermediate strength)

fpad ≈ −+

11

ηη

.


pressure increases. Viscosity is easily increased byan order of magnitude (e.g., 10 times increase in µincreases the width by 77%), but only at theexpense of using more fluid additives and withadditional conductivity damage potentially negat-ing the extra width.

Thus, the hydraulic fracture width is fairly fixed(±50%, at least in terms of the treatment’s control-lable parameters), and the proppant fill fraction hasa practical limit of about 0.5 (±0.1). Without TSOdesigns (discussed in the following) the final,effective propped width is almost fixed by nature.The goal for a normal fracture design is then toachieve a required kfw within these limits, withproppant concentration, proppant selection andfluid selection allowing a large range of values.

• Tip-screenout designs

As mentioned previously, as long as a fracture isfree to propagate, the hydraulic fracture width isrelatively insensitive to the controllable treatmentparameters of fluid viscosity and pump rate. Ifmore conductivity is required than can beachieved from a normal design, the only effectivemanner to increase the propped width is to stopthe fracture from propagating but to continue topump. This technique has come to be called TSOfracturing (Smith et al., 1984).

For a normal treatment, the pad volume isdesigned to deplete just as pumping stops. Whatwould happen if pumping simply continuedbeyond that time? If the pad is depleted, thenproppant-laden slurry will be located everywherearound the fracture periphery. If there is fluid loss,then this slurry will dehydrate and leave packedproppant around the periphery. Even with no fluidloss, the proppant may bridge in the narrow frac-ture width around the periphery, particularly inplaces where the width is extremely narrow as aresult of the fracture penetrating a boundary layer.In either case, any additional propagation isrestricted and further pumping causes an increaseof net pressure and thus an increase of fracturewidth. TSO designs are discussed in detail inChapter 10.

5-5.3. Proppant transportSeveral modes of proppant settling can occur whileproppant is being transported into a hydraulic frac-

ture (see Section 6-5). First is what may be termedsimple or single-particle settling. Behavior of thistype is governed by Stokes law, in which the veloc-ity of a single particle falling through a liquid medi-um is

(5-25)

where vfall is the settling rate in ft/s, dprop is the aver-age proppant particle diameter in in., µ is the fluidviscosity in cp, and γprop and γfluid are the specificgravity of the proppant and the fluid, respectively.The settling rate, and thus the efficiency with whichproppant can be transported into the fracture, isdirectly related to the fluid viscosity. This is usuallythe main consideration for how much viscosity isrequired for a fracture treatment. However, there areadditional considerations for calculating settling fol-lowing Stokes law. At low proppant concentrations(e.g., less than 1 or 2 ppg) particles may clump, pro-ducing an apparent diameter greater than the actualparticle diameter and accelerating settling. Higherparticle concentrations act to increase the slurry vis-cosity and retard settling (also known as hinderedsettling). The pump rate is also an important param-eter controlling proppant transport for simple settlingby Stokes law.

As shown in Fig. 5-22, for a Newtonian fluid thedistance D a proppant particle is transported into afracture, before that particle can fall from the top of

vd

fall

prop

prop fluid= × −( )1 15 1032

. ,µ

γ γ

Figure 5-22. Stokes law.

h

D

vfall

v1

D ⁄ h = v1/v2

v1 = fluid velocity

∝ qi/hw ∝ qi/h(µqi)1/4

∝ qi3/4/hµ1/4

vfall = fall rate ∝ 1/µD/h ∝ (qiµ)3/4/h

D is independent of h


the fracture to the bottom, is related to (qiµ)3/4. Thisdistance is independent of the fracture height and,more significantly, shows that for some given trans-port distance, less viscosity can be used at higherpump rates. This relation can be important for highertemperature applications, where fluid viscosity candegrade significantly with time. At higher rates (andhence shorter pump times), less viscosity is requiredfor proppant transport. Also, the fluid is exposed tothe high formation temperature for less time, so thefluid system maintains better viscosity. In general,considering how fluid viscosity degrades down afracture, including the effect of proppant concentra-tion increasing the effective slurry viscosity, andconsidering the non-Newtonian nature of mostfracturing fluids, if a fracturing fluid retains 50- to100-cp viscosity (at reservoir temperature and at ashear rate of 170 s–1) at the end of the fracture treat-ment, it will provide essentially perfect proppanttransport (Nolte, 1982).

The next mode of proppant settling is termed con-vection, and it was probably first included in fracturemodeling in the context of a fully three-dimensional(3D) planar model by Clifton and Wang (1988). Thistype of settling is controlled by density differences(i.e., buoyancy) between two fluids. For example, a proppant-laden fluid stage with an 8-ppg concen-tration has a slurry density of 11.9 lbm/gal (s.g. =1.44). If this slurry is placed directly next to a cleanfluid stage with a density of 8.5 lbm/gal (s.g. = 1.02),the heavier slurry will tend to sink and underride thelighter clean fluid, simply carrying the proppanttoward the bottom of the fracture. However, a treat-ment does not normally follow clean pad fluid with a heavy 8-ppg slurry. Rather, the treatment increasesproppant concentration slowly to account for fluid-loss effects and mitigate convection effects. Onlynear the end of pumping (when the need for trans-port decreases), when the initial proppant stageshave undergone significant dehydration, can a signif-icant density difference begin to develop. In general,rigorous numerical modeling of this phenomenashows convection is not a major factor during pump-ing (Smith et al., 1997). If excessive pad is used,such that a large unpropped region of the fractureexists after shut-in, convection can occur during theshut-in after flow, with potentially significant adverseeffects on the final proppant placement.

The third effect on proppant transport is termedmigration (see Chapter 6). In brief, a viscoelastic

fluid (which describes most fracturing fluid systems)flowing down a channel imparts a normal force toparticles entrained in the fluid such that the particlestend to migrate to and concentrate in the center ofthe channel. For low average concentrations, this canresult in a center core of high-proppant-concentrationslurry, with a region of essentially clean fluid oneither side. This heavier core of concentrated slurrytends to fall owing to its greater density, carrying theentrained proppant toward the bottom of the fractureat a faster rate than for a dispersed slurry (Nolte,1988b).

Finally, any calculations for proppant settling mustconsider geologic reality. Detailed examinations ofhydraulic fractures both at the wellbore using televi-sion cameras (Smith et al., 1982) or away from wellsin mineback tests (see Warpinski, 1985) show some-thing other than the smooth fracture walls assumedfor settling calculations. Small shifts and jogs of thefracture probably have no significant impact on fluidflow or on lateral proppant transport into the frac-ture. However, these small irregularities could signif-icantly impact settling. Calculations for proppant set-tling that ignore these effects will be a worst-casescenario.

5-5.4. Proppant admittanceProppant admittance is critical to hydraulic fracturingin two forms: entrance to the fracture through perfora-tions and entrance of proppant into the fracture itself.These effects were recognized early, and the originalfracture width models were used primarily for deter-mining a pad volume that would allow admittance bygenerating a fracture width greater than 2.5dprop,where dprop is the average proppant particle diameter.Before these models, operators were reluctant topump significant volumes of pad as it was consideredexpensive and potentially damaging.

The laboratory data in Fig. 5-23 (Gruesbeck andCollins, 1978) illustrate two important ideas:

• A minimum perforation diameter is required forproppant to flow through the perforations.

• Minimum perforation diameter is a function of theslurry concentration.

At low concentrations (e.g., 1 ppg), the perforationhole diameter must be only slightly greater than thatof the proppant particles. The required hole diameterincreases with concentration until at about 6 ppg


(solid volume fraction of about 0.20), the perforationhole diameter must be 6 times the average particlediameter.

This same trend applies for slurry flow down a nar-row fracture. An approximate proppant bridging orproppant admittance criteria can be derived by calcu-lating an equivalent hydraulic radius for a narrowslot, rhyd = w/2, where w is the average width of thefracture. For a round perforation hole, the hydraulicradius is d/4, where d is the perforation hole diameter.Equating the two hydraulic radius values shows that2w is equivalent to the diameter of a round hole.Using this along with two lines fitting the data of Gruesbeck and Collins leads to an approximateadmittance criteria for a hydraulic fracture:

• For a proppant solid volume fraction fv less than0.17, the average width must be greater than (1 + 2fv /0.17) × dprop.

• For fv greater than 0.17, the average width must begreater than 3dprop (i.e., a width greater than threeproppant grain diameters).

This approximate correlation also compares wellwith other experimental data from proppant-ladenslurry flowed through a narrow slot (van der Vlis etal., 1975), although the correlation may be optimisticfor low proppant concentrations. As shown in Table5-1, the behavior for bridging in a fracture is similarto bridging in perforation holes. At low proppantconcentrations, the average fracture width must beonly slightly greater than the average particle diame-ter. As the proppant concentration increases toward

6 to 8 ppg, the required average fracture widthincreases to 3dprop.

This critical width is important to the hydraulicfracturing process. Should proppant enter a part of thefracture where sufficient width does not exist, theproppant will bridge and no longer flow down thefracture. Additional slurry flowing in this directionwill cause proppant to pile up, dehydrate and blockthat part of the fracture. Should this occur near thewellbore, possibly as a result of some form of near-wellbore width restriction (see tortuosity discussion inSection 6-8), a total screenout can result with seriousconsequences for the success of the fracture treatment.

5-5.5. Fracture modelsClearly, developing a final treatment pump schedulemust consider many options. The interactive roles ofthe various major variables (hf, E, CL, KIc-apparent, µand qi) must be considered along with the variousroles of fluid viscosity for net pressure, width, prop-pant transport and fluid loss. In addition, the designmust consider the various roles of the pad volumeconcerning fluid loss and creating fracture width.Fracture simulators, or fracture placement models,provide the means to handle this complexity and toconsider the interaction of the multitude of variables.For this reason, a final schedule is generally devel-oped using a fracture geometry model. However, asdiscussed in Section 5-5.2, Sidebar 6L and Section10-4, in many instances an acceptable pump schedulecan be developed more simply for a treatment on thebasis of the expected fluid efficiency (as determinedfrom a calibration treatment). The use of a properlycalibrated fracture geometry model also enables theconsideration of multiple scenarios for designing the

Table 5-1. Proppant admittance criteria.

Proppant†

Concentration(lbm proppant/gal fluid) w— /dprop

Experimental CorrelationBridge Formation‡ Bridge

0.5 to 2 1.8 1.15 to 2.0

2 to 5 2.2 2.0 to 3.0

5 to 8 2.6 3.0

† Sand as proppant‡ Data from van der Vlis et al. (1975)

Figure 5-23. Proppant admittance through perforations(Gruesbeck and Collins, 1978).

Per

fora

tion

diam

eter

/av

erag

e pa

rticl

e di

amet

er

7

8

6

5

4

3

2

1

050 10 15 20 25 30 35

Sand concentration(lbm sand/gal fluid)

0.185 0.312 0.405Maximum particle concentration (vol/vol)

100-cp HEC solution

Tap water


optimum treatment for a specific application. Thisapproach is briefly discussed in Section 5-6.1.

5-6. Economics and operational considerations

The preceding discussion covers most of the techni-cal aspects of hydraulic fracturing (reservoir engi-neering, fluid mechanics, rock mechanics, etc.) andreviews the complex interactions that exist betweenthe various, often competing design variables.However, to complicate things further, hydraulicfracturing and treatment design are generally gov-erned by—or are at least sensitive to—two finalconsiderations: economics and field operations.

5-6.1. EconomicsAt the most basic level, hydraulic fracturing is abouttime and money: “economics.” Given reasonablegeologic continuity, a single well would, given suffi-cient time, drain an entire reservoir. However, theoperating costs of maintaining a well over thedecades required to accomplish this drainage wouldprobably make the entire operation unattractive froma commercial viewpoint. Alternatively, a single well

with a large hydraulic fracture may drain the reser-voir much faster, making the economics much moreattractive despite the additional cost of the treatment.Carrying this forward, 2, 10 or 100 or more wellscould be drilled and/or fractured. Between theseextremes is the optimum plan, which is the numberof wells, number of fractured wells or both that max-imize the economic value of the production com-pared with the development capital costs and theongoing operating costs.

As a simple example, the process (at least for asingle well) could proceed as pictured in Fig. 5-24(Veatch, 1986). First, reservoir engineering calcula-tions provide a production forecast for various com-binations of fracture half-length xf and conductivitykfw (including the case of no fracture at all). Basedon some future price forecast, this allows calculationof a present value, which is the future revenue fromthe production less future operating costs and dis-counted back to the present. Hydraulic fracturingcalculations based on fluid loss, fracture height, etc.,are used to determine the treatment volumes requiredto generate various combinations of fracture lengthand propped fracture width, and these calculationsare easily converted into estimated treatment costs.Some form of net revenue economic analysis is thenused to determine the best type of proppant, desired

Figure 5-24. Veatch (1986) economics diagrams.

xf = xf2kfw = kfw2 kfw = kfw2

kfw = kfw2

kfw = kfw2

kfw = kfw1xf = xf1

kfw = kfw1

kfw = kfw1

kfw = kfw1

No fracture

Trea

tmen

t vol

ume

Cum

ulat

ive

prod

uctio

n

Dis

coun

ted

reve

nue

($)

Cos

t ($)

Rev

enue

less

cos

t ($)Time xf

xf

xfxf


fracture length and other requirements for the opti-mum treatment.

There are, of course, many variations of this basicprocess. For example, full-cycle economics includesdrilling and other completion costs, along with frac-ture treatment costs, in determining the optimumfracture design. This type of analysis is usuallyappropriate in any case involving multiple wells(e.g., should a resource be developed using 10 wellswith huge fractures or 20 wells with smaller or nofracture treatments?). Point-forward analysis, on theother hand, considers only the fracture treatmentcosts (because drilling and other completion costsare already expended) and is most appropriate forworking over existing wells.

5-6.2. OperationsAs discussed in the preceding section, economicsprovides the final design consideration for hydraulicfracturing, whereas field conditions provide the prac-tical limits within which the design must fit. Evenbeyond defining these limiting conditions, however,any design is only as good as its execution; thus thetreatment must be pumped as designed. Field opera-tions and operational considerations impacthydraulic fracturing in two ways:

• prefracture condition of the wellbore, quality ofthe cement job, perforations, pressure limits, etc.,with these considerations defining practical limitsthat the design must meet

• quality assurance and quality control (QA/QC)before and during the actual treatment.

These operational considerations are discussed inChapters 7 and 11, with some of the major itemshighlighted in the following.

• Wellbore considerations

Some of the major wellbore considerations forhydraulic fracturing include

– size and condition of wellbore tubulars

– quality of the cement job for zonal isolation

– perforations

– wellbore deviation.

During a hydraulic fracture treatment, the pre-dicted surface pressure psurf and the hydraulichorsepower required for a treatment are related

to the hydrostatic head of the fluid in the wellborephead and the pipe friction ppipe friction:

(5-26)

(5-27)

Pipe friction is a major term, and thus the sizeof the well tubulars has a strong influence onallowable pump rates (because pipe friction is typ-ically related to ve, where v = qi/A is the flowvelocity down the tubing, and e is typically about1.1 to 1.7). Also, the strength and condition of thetubulars (along with the associated wellheadequipment) set an allowable surface pressure andthus indirectly set an allowable injection rate. Inaddition, the size, type and condition of the well-bore tubulars may limit (or prohibit) future work-over and recompletion opportunities.

A critical aspect of wellbore considerations is agood cement job around the casing or liner to pro-vide zonal isolation. In general, a fracture growswhere nature dictates, and the engineer has littlecontrol over fracture height growth. The only con-trol possible is the ability to specify where theperforations are placed and the fracture initiates. If that ability is compromised by a poor cementsheath around the casing that allows the perfora-tions to communicate directly with an undesiredinterval, then even this minimal level of control islost, and the hydraulic fracture treatment may beseriously compromised.

Another important consideration is the perfora-tions that allow the fluid to leave the wellbore andcreate the fracture. The number and size of theperforation holes are important for proppantadmittance, as discussed briefly in Section 5-5.4and in detail in Section 11-3.

• Quality assurance and quality control

Quality issues are critical for hydraulic fracturing.After proppant pumping starts, a treatment cannotbe stopped because of problems without signifi-cantly compromising the design goals. For thistime period, everything must work, including the wellbore equipment, pumping and blendingequipment and chemicals (i.e., the fluid system).To cite a simple example, if a treatment uses 10 tanks of batch-mixed fluid, and one of the

p p p psurf c net pipe friction head= + + −σ

hhp q pi surf∝ × .


tanks is bad, then the QA score is a relatively high90%. However, if the bad fluid is pumped justafter proppant addition starts, it may easily causetotal failure of the treatment, and if successfultreatment is critical to economic success of thewell, this causes total economic failure. Typically,this type of failure cannot be overcome withoutcompletely redrilling the well (refracturing opera-tions are usually a risky procedure), and thus 90%can be a failing grade for QA.

fracturing stimulation book

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