RS_03

3-1. IntroductionThe National Academy of Sciences defines rockmechanics as the theoretical and applied science of the mechanical behavior of rock; it is that branchof mechanics concerned with the response of the rockto the force fields of its physical environment. Fromthis definition, the importance of rock mechanics inseveral aspects of the oil and gas industry can easilybe understood. The fragmentation of rock governs itsdrillability, whereas its mechanical behavior influencesall aspects of completion, stimulation and production.However, not until recently has this particular aspectof earth sciences started to play a predominant role inenergy extraction. The impetus was to explain, quali-tatively and quantitatively, the orientation of fractures(Hubbert and Willis, 1957), some unexpected reser-voir responses or catastrophic failures (e.g., less pro-duction after stimulation and pressure decline in wellssurrounding an injection well; Murphy, 1982), casingshear failure (Nester et al., 1956; Cheatham andMcEver, 1964), sand production (Bratli and Risnes,1981; Perkins and Weingarten, 1988; Morita et al.,1987; Veeken et al., 1991; Kooijman et al., 1992;Cook et al., 1994; Moricca et al., 1994; Geilikman et al., 1994; Ramos et al., 1994), rock matrix collapseduring production (Risnes et al., 1982; Pattillo andSmith, 1985; Smits et al., 1988; Abdulraheem et al.,1992) and borehole stability problems (Gnirk, 1972;Bradley, 1979; Guenot, 1989; Santarelli et al., 1992;Ong and Roegiers, 1993; Maury, 1994; Last et al., 1995).

The significant contribution as far as the orientationof fractures is concerned was provided by the workof Hubbert and Willis (1957; see Sidebar 3A), whichindicates ever-increasing differences between verticaland horizontal stresses within the earths crust. Untilthen, all design considerations were based on theassumption that an isostatic state of stress prevailedeverywhere.

As deeper completions were attempted, boreholecollapses and instabilities became more common andoften led to expensive remedial measures. The pri-mary cause of these problems is instabilities causedby large tectonic forces. The concepts developed bymining engineers that rocks are far from being inertwere found applicable (Cook, 1967; Hodgson andJoughin, 1967). Rocks are quite receptive to distur-bances, provided that some energy limits are notexceeded. If critical energies are exceeded, dynamicfailure is likely to occur, such as rock burst or casingcollapse. In addition, the importance of inherentdiscontinuities (e.g., faults, fissures) was realized(Goodman, 1976), especially as highly conductiveconduits. From this broader understanding of the roleof rock deformation, research focused on definition of the pertinent parameters required to properly char-acterize the targeted formations. Cores were taken notonly for the determination of permeability, porosityand lithology, but also to run mechanical tests undersimulated downhole conditions. Downhole tools weredeveloped to better characterize the formation in situ.

There will always remain some uncertainties on therelevance of laboratory-determined parameters to thefield situation, either because of the disturbance a coresample suffers during the coring and handling processor because of scale effects. There are also limitationson the use of simple constitutive laws to predict rockbehavior for heterogeneous, discontinuous, time-dependent and/or weak formations. Studies are cur-rently being conducted on these issues, and our under-standing and predictive capability of downhole rockbehavior can be expected to continue to progress.

This chapter briefly summarizes some of the mostimportant aspects of rock mechanics to characterizethe mechanical behavior of reservoirs and adjacentlayers, as applied to the stimulation process.

Reservoir Stimulation 3-1

Formation Characterization:

Rock MechanicsM. C. Thiercelin, Schlumberger DowellJ.-C. Roegiers, University of Oklahoma

3-2 Formation Characterization: Rock Mechanics

3A. Mechanics of hydraulic fracturing

Hubbert and Willis (1957) introduced several key conceptsthat explain the state of stress underground and its influenceon the orientation of hydraulic fractures. Reviewed here arethe fundamental experiments that Hubbert and Willis per-formed to validate these concepts.

State of stress undergroundThe general state of stress underground is that in which thethree principal stresses are unequal. For tectonically relaxedareas characterized by normal faulting, the minimum stressshould be horizontal; the hydraulic fractures produced shouldbe vertical with the injection pressure less than that of theoverburden. In areas of active tectonic compression andthrust faulting, the minimum stress should be vertical andequal to the pressure of the overburden (Fig. 3-20). Thehydraulic fractures should be horizontal with injection pres-sures equal to or greater than the pressure of the overburden.

To demonstrate these faulting conditions, Hubbert andWillis performed a sandbox experiment that reproduces boththe normal fault regime and the thrust fault regime. Figures3A-1 and 3A-2 show the box with its glass front and contain-ing ordinary sand. The partition in the middle can be movedfrom left to right by turning a hand screw. The white lines areplaster of paris markers that have no mechanical significance.As the partition is moved to the right, a normal fault with a dipof about 60 develops in the left-hand compartment, asshown in Fig. 3A-1. With further movement, a series of thrustfaults with dips of about 30 develops in the right-hand com-partment, as shown in Fig. 3A-2.

The general nature of the stresses that accompany thefailure of the sand is shown in Fig. 3A-3. The usual conven-

tion is adopted of designating the maximum, intermediate andminimum principal effective stresses by 1 , 2 and 3 ,respectively (here taken as compressive). In the left-handcompartment, 3 is the horizontal effective stress, which isreduced as the partition is moved to the right, and 1 is thevertical effective stress, which is equal to the pressure of theoverlying material minus the pore pressure. In the right-handcompartment, however, 1 is horizontal, increasing as thepartition is moved, and 3 is vertical and equal to the pres-sure of the overlying material minus the pore pressure. Thethird type of failure, strike-slip faulting, is not demonstrated in the sandbox experiment.

Next, the combination of shear and normal stresses thatinduce failure must be determined. These critical effectivestress values can be plotted on a Mohr diagram, as shown in Fig. 3A-4. The two diagonal lines form the Mohr envelopesof the material, and the area between them represents stablecombinations of shear stress and normal effective stress,whereas the area exterior to the envelopes represents unsta-ble conditions. Figure 3A-4 thus indicates the stability regionwithin which the permissible values of n and are clearlydefined. The stress circles can then be plotted in conjunctionwith the Mohr envelopes to determine the conditions of fault-ing. This is illustrated in Fig. 3A-4 for both normal and thrustfaulting. In both cases, one of the principal effective stresses

Figure 3A-1. Sandbox experiment showing a normal fault.

Figure 3A-2. Sandbox experiment showing a thrust fault.

Figure 3A-3. Approximate stress conditions in the sand-box experiment.

1 3

3 1

Sand

Figure 3A-4. Mohr diagram of the possible range of hori-zontal stress for a given vertical stress v . The horizontalstress can have any value ranging from approximatelyone-third of the normal stress, corresponding to normalfaulting, to approximately 3 times the vertical stress, cor-responding to reverse faulting.

1

3

v

Mohrenvelope

0

2a 2a

Points of fracture


3A. Mechanics of hydraulic fracturing (continued)is equal to the overburden effective stress v . In the case of normalfaulting, the horizontal principal stress is progressively reduced, there-by increasing the radius of the stress circle until the circle touches theMohr envelopes. At this point, unstable conditions of shear and nor-mal effective stress are reached and faulting occurs on a plane mak-ing an angle of 45 + /2 with the minimum stress. For sand with anangle of internal friction of 30, the normal fault would have a dip of60, which agrees with the previous experiments. The minimum princi-pal effective stress would reach a value at about one-third of the valueof the overburden effective stress (Eq. 3-58). For the case of thrustfaulting, the minimum principal stress would be vertical and remainequal to the overburden pressure while the horizontal stress is pro-gressively increased until unstable conditions occur and faulting takesplace on a plane making an angle of 45 + /2 with the minimum prin-cipal stress or 45 /2 with the horizontal. For sand, this would be adip of about 30, which again agrees with the experiment. Failureoccurs when the maximum horizontal effective principal stressreaches a value that is about 3 times the value of the overburdeneffective stress (Eq. 3-59). The intermediate stress, which is the mini-mum horizontal stress, is not defined by this process.

From these limiting cases and for a fixed effective vertical stressv, the effective horizontal stress may have any value between theextreme limits of 13 and 3 times v.

Orientation of hydraulic fracturesThe second important contribution of Hubbert and Willis work con-cerns the orientation of hydraulic fractures. When their paper was pre-sented, technical debate was occurring on the orientation of hydraulicfractures. A theoretical examination of the mechanisms of hydraulicfracturing of rocks led them to the conclusion that, regardless ofwhether the fracturing fluid was penetrating, the fractures producedshould be approximately perpendicular to the axis of minimum princi-pal stress.

To verify the inferences obtained theoretically, a series of simplelaboratory experiments was performed. The general procedure was toproduce fractures on a small scale by injecting a fracturing fluid intoa weak elastic solid that had previously been stressed. Ordinary gel-atin (12% solution) was used for the solid, as it is sufficiently weak tofracture easily, molds readily in a simulated wellbore and is almostperfectly elastic under a short-time application of stresses. A plaster ofparis slurry was used as the fracturing fluid because it could be madethin enough to flow easily and once set provided a permanent recordof the fractures produced. The experimental arrangement consisted ofa 2-gal polyethylene bottle, with its top cut off, used to contain a glasstubing assembly consisting of an inner mold and concentric outer cas-ings. The container was sufficiently flexible to transmit externallyapplied stresses to the gelatin. The procedure was to place the glasstubing assembly in the liquid gelatin and after solidification to withdrawthe inner mold leaving a wellbore cased above and below an open-hole section.

Stresses were then applied to the gelatin in two ways. The firstway (Fig. 3A-5) was to squeeze the polyethylene container laterally,thereby forcing it into an elliptical cross section and producing a com-pression in one horizontal direction and an extension at right angles inthe other. The minimum principal stress was therefore horizontal, andvertical fractures should be expected, as observed in Fig. 3A-6. Inother experiments, the container was wrapped with rubber tubingstretched in tension, thus producing radial compression and verticalextension. In this case, the minimum principal stress was vertical, anda horizontal fracture was obtained.

From these analyses and experiments, Hubbert and Willis con-cluded that

the state of stress, and hence the fracture orientations, is gov-erned by incipient failure (i.e., faulting) of the rock mass

in areas subject to active normal faulting, fractures should beapproximately vertical

in areas subject to active thrust faulting, fractures should beapproximately horizontal.

Figure 3A-5. Experimental arrangement for pro-ducing the least stress in a horizontal direction.

Favoredfracturedirection

Leastprincipalstress

Figure 3A-6. Vertical fracture produced understress conditions illustrated in Fig. 3A-5.

3-2. Basic concepts3-2.1. StressesIn considering a randomly oriented plane of area Acentered on a point P within a body across which aresultant force F acts (Fig. 3-1), the stress vector at that point is defined as

(3-1)

Therefore, this quantity is expressed as a force perunit area. In geomechanics, by convention, compres-sion is taken to be positive because the forces prevail-ing in the earth are usually compressive in nature.This resultant stress can be decomposed into a nor-mal component n and a shear component . Theshear component tends to shear the material in theplane A. It should be realized that an infinite amountof planes can be drawn through a given point varying,by the same token, the values of n and . The stresscondition, therefore, depends on the inclination. Con-sequently, a complete description of a stress mustspecify not only its magnitude, direction and sense,but also the direction of the surface upon which itacts. Quantities described by two directions, such as stresses, are known as second-order tensors.

In a two-dimensional (2D) situation, if x, y andxy are known (Fig. 3-2), the stress state on any planewith normal orientation at an angle from Ox can bederived using the following expressions:

(3-2)

(3-3)

These expressions are obtained by writing equilib-rium equations of the forces along the n and direc-tions, respectively. The moment equilibrium impliesthat xy is equal to yx. There always exist two perpen-dicular orientations of A for which the shear stresscomponents vanish; these are referred to as the princi-pal planes. The normal stresses associated with theseplanes are referred to as the principal stresses. In twodimensions, expressions for these principal stressescan be found by setting = 0 in Eq. 3-3 or, becausethey are the minimum and maximum values of thenormal stresses, by taking the derivative of Eq. 3-2with respect to the angle and setting it equal to zero.Either case obtains the following expression for thevalue of for which the shear stress vanishes:

(3-4)

and the two principal stress components 1 and 2 are

(3-5)

(3-6)


= lim

A

FA0

.

n x xy y= + +cos sin cos sin

2 22

=

12

2arctan xy

x y

12 2

1 212

14

= +( ) + + ( ) x y xy x y/

22 2

1 212

14

= +( ) + ( ) x y xy x y/

.

= ( ) +12 2y x xysin2 cos .

Figure 3-1. Force on a point P.

PA

F

Figure 3-2. Two-dimensional decomposition of normal andshear stresses.

y

B

n

y

x

xy

yxA xO

If this concept is generalized to three dimensions, itcan be shown that six independent components of thestress (three normal and three shear components) areneeded to define the stress unambiguously. The stressvector for any direction of A can generally be foundby writing equilibrium of force equations in variousdirections. Three principal planes for which the shearstress components vanishand, therefore, the threeprincipal stressesexist.

It is convenient to represent the state of stress at agiven point using graphical methods. The most widelyused method is the Mohr representation described inSidebar 3B. Other useful quantities are stress invari-ants (i.e., quantities that do not depend on the choiceof axes). For example, the mean stress m:

(3-7)

and the octahedral shear stress oct:

(3-8)

are two stress invariants typically used in failure criteria.

3-2.2. StrainsWhen a body is subjected to a stress field, the relativeposition of points within it is altered; the body deforms.If these new positions of the points are such that theirinitial and final locations cannot be made to corre-spond by a translation and/or rotation (i.e., by rigidbody motion), the body is strained. Straining along anarbitrary direction can be decomposed into two com-ponents, as shown in Fig. 3-3:

elongation, defined as

(3-9)

shear strain, defined as(3-10)

where is the change of angle between two direc-tions that were perpendicular prior to straining.

Consequently, strain (which is either a ratio oflengths or a change of angle) is dimensionless.Because stresses are taken as positive in compression,a positive longitudinal strain corresponds to a decreasein length, and a positive shear strain reflects an increase in the angle between two directions that were

perpendicular initially. Just as in the case of stresses,principal strains can be defined as longitudinal straincomponents acting on planes where the shear strainshave vanished. It should be pointed out that the analogybetween stress and strain analyses is not completelyvalid and that equilibrium equations and compatibilityequations have to be satisfied respectively for thestresses and for the strains. These relations put somerestrictions on the local variation of stress and strain inthe neighborhood of a point. For example, compatibili-ty equations ensure that the strained body remains con-tinuous and that no cracks or material overlaps willoccur. For further details on stresses and strains, thereader is referred to the classic works by Love (1927),


m x y z= + +( ) = + +( )13 13 1 2 3

oct = ( ) + ( ) + ( )[ ]13 1 2 2 1 3 2 2 3 2 1 2/

=

liml

l ll0

*

= ( )tan ,

3B. Mohr circle

Equations 3-2 and 3-3 can be used to derive n and as afunction of 1 and 2 (effective stresses are considered):

(3B-1)

(3B-2)

The angle is the angle at which the normal to the planeof interest is inclined to 1. These expressions provide theequation of a circle in a (n, ) plane, with its center locatedon the axis at 12(1 + 2) and of diameter (1 2) (Fig. 3B-1).This circle is known as the Mohr circle and contains all theinformation necessary to determine the two-dimensionalstress state at any orientation in the sample. The intersectionof this circle with the horizontal axis determines the maximumand minimum values of the normal stresses at a point in thematerial. The apex represents the maximum value of theshear stress. For a three-dimensional state of stress, similarcircles can be constructed for any two orthogonal directions.

n = +( ) + ( )1212

21 2 1 2 cos

= ( )12 21 2 sin .

Figure 3B-1. The coordinates of point M on the Mohrcircle are the values of normal stress and shear stressacross a plane with the normal oriented at to thedirection of maximum principal stress.

2 n n1

2

M

Timoshenko and Goodier (1970) and Muskhelishvili(1953).

3-3. Rock behaviorWhen a rock specimen or an element of the earth is submitted to load, it deforms; the higher the stresslevel, the more strain the rock experiences. It is animportant aspect of rock mechanics, and solidmechanics in general, to determine the relationshipbetween stress and strain (i.e., the constitutive equa-tions of the material under consideration). Various the-ories have been developed to describe, in a simplifiedway, this relationship. The simplest one is the theoryof elasticity, which assumes that there is a one-to-onecorrespondence between stress and strain (and, conse-quently, that the behavior is reversible). Because thisis usually the assumed case in hydraulic fracturing,most of the simulation models use the theory of elas-ticity. Other theories have been developed to bettertake into account the complex behavior of rock, espe-cially in compression. For example, the theory of plas-

ticity is particularly useful for predicting the stressconcentration around a wellbore or the behavior ofsoft materials during reservoir depletion.

3-3.1. Linear elasticityTo introduce the theory of linear elasticity, let us con-sider a cylindrical sample of initial length l and diam-eter d. The sample shortens along the loading direc-tion when a force F is applied to its ends (Fig. 3-4).According to the definitions in the previous section,the axial stress applied to the sample is

(3-11)

and the axial strain is

(3-12)

where l* is the resultant length.Linear elasticity assumes a linear and unique rela-

tionship between stress and strain. The consequenceof uniqueness is that all strain recovers when the


Figure 3-3. Normal and shear strain components.

Original

Deformed

P

O

l

P Q

/2

l

O

Q

O P

O P

1 2

4=

Fd

1 =l ll

*,

material is unloaded. In the case of a uniaxial com-pression test, this means that

(3-13)The coefficient of proportionality E is Youngs mod-ulus.

When a rock specimen is compressed in one direc-tion, not only does it shorten along the loading direc-tion, but it also expands in the lateral directions. Thiseffect is quantified by the introduction of an additionalconstant (Poissons ratio ), defined as the ratio of lat-eral expansion to longitudinal contraction:

(3-14)

where

(3-15)

where d* is the new diameter.The negative sign is included in Eq. 3-14 because,

by convention, expansion is considered negative and

Poissons ratio, by definition, is a positive quantity.These stress-strain relations can be generalized to fullthree-dimensional (3D) space by

(3-16)

where the shear modulus G is

(3-17)

Another coefficient that is commonly used is thebulk modulus K, which is the coefficient of propor-tionality between the mean stress m and volumetricstrain V during a hydrostatic test. In such a test, allthree normal stresses are equal and, consequently, alldirections are principal. For this case:

(3-18)

where V is the rock volume and V is its variation.In isotropic linear elasticity, only two elastic con-

stants are independent. For example, and as discussedpreviously, the shear modulus G and the bulk modulusK can be written as functions of E and . The mostcommonly used constants in reservoir applications aredefined in Sidebar 3C.

Elasticity theory can be extended to nonlinear andanisotropic materials. A nonlinear elastic materialdoes not have a linear relationship between stress andstrain, but recovers all strain when unloaded. Ananisotropic material has properties that differ in differ-ent directions. A common type is transverse anisot-ropy, which applies to materials that have a plane andan axis of symmetry (the axis of symmetry is the nor-mal to the plane of symmetry). This is particularlysuited for bedded formations where the bedding planeis the plane of symmetry. These materials, whichexhibit the simplest type of anisotropy, are character-ized by five elastic constants.


Figure 3-4. Sample deformation under uniaxial loading.

d

d

l

l

F

1 1= E .

=

2

1

,

2 =d dd

*,

x

x

y z

yy

x z

zz

x y

xy xy yz yz xz xz

E E

E E

E E

G G G

= +( )= +( )= +( )= = =

1 1 1; ; ,

G E=+( )2 1 .

m V V

K VV

K E= = =( ); ; ,

3 1 2

3-3.2. Influence of pore pressurePore fluids in the reservoir rock play an important rolebecause they support a portion of the total appliedstress. Hence, only a portion of the total stress, namely,the effective stress component, is carried by the rockmatrix (Fig. 3-5). Obviously, this effective stresschanges over the life of the reservoir. In addition, themechanical behavior of the porous rock modifies thefluid response. Two basic mechanisms highlight thiscoupled behavior (e.g., Detournay and Cheng, 1993):

An increase of pore pressure induces rock dilation. Compression of the rock produces a pore pressure

increase if the fluid is prevented from escapingfrom the porous network.

When the fluid is free to move, pore pressure diffu-sion introduces a time-dependent character to themechanical response of a rock: the rock reacts differ-ently, depending on whether the rate of loading isslow or fast compared with a characteristic time thatgoverns the transient pore pressure in the reservoir(itself governed by the rock deformation).

Hence, two limiting behaviors must be introduced:drained and undrained responses. One limiting case isrealized when a load is instantaneously applied to aporous rock. In that case the excess fluid pressure hasno time to diffuse and the medium reacts as if it wereundrained and behaves in a stiff manner. On theother extreme, if the pressurization rate is sufficientlyslow and excess pressure areas have ample time todrain by diffusion, the rock is softer. The stiffeningeffect is more important if the pores are filled with arelatively incompressible liquid rather than a relativelycompressible gas.

In 1923, Terzaghi first introduced the effectivestress concept for one-dimensional consolidation andproposed the following relationship:

(3-19)where is the total applied stress, is the effectivestress governing consolidation of the material, and pis the pore pressure. However, Biot (1941, 1956a) pro-posed a consistent theory to account for the coupled


3C. Elastic constants

Two independent constants characterize isotropic linear elas-tic materials. Several different conditions can be consideredand specific equations can be derived from the three-dimen-sional elasticity relations (Eq. 3-16):

Unconfined axial loadingSpecified: x or x, with y = z = 0

(3C-1)

(3C-2)

(3C-3)

where E is Youngs modulus, is Poissons ratio, and G isthe shear modulus.

Hydrostatic (isotropic) loadingSpecified: x = y = z and x = y = zThe volumetric strain V is equal to x + y + z.

(3C-4)

where K is the bulk modulus.

Plane strain loading (all x-y planes remain parallel)Specified: z = 0, with the added constraint that y = 0

(3C-5)

(3C-6)

where E is the plane strain modulus used in fracture widthmodels.

Uniaxial (laterally constrained) strainSpecified: y = z = 0

(3C-7)

(3C-8)

where C is called the constrained modulus and is used forearth stresses and plane compressive seismic waves.

E x x=

y z x= =

E G= +2 1( ) ,

K Ex V= = ( ) 3 1 2 ,

=E x x

= ( ) = ( )E E G1 2 12 ,

C E K Gx x= = ( ) +( ) ( )[ ] = + 1 1 1 2 43 z y x= = ( )[ ]/ ,1

Figure 3-5. Load sharing by pore pressure. Total stress =pore pressure + effective stress carried by the grains.

Force

Force

Pores

Grains

= p ,

diffusion/deformation processes that are observed inelastic materials. Such a strong coupling is due to thefact that any change in pore pressure is accompaniedby variation in the pore volume; hence, it affects theoverall mechanical response of the rock. This poro-elastic material behavior is similar to that of an elasticsolid when the stresses in Eq. 3-16 are replaced by thefollowing effective stresses:

(3-20)This relation rigorously governs the deformation

of a porous medium, whereas failure is controlled byTerzaghis (1923) effective stresses in Eq. 3-19 (Rice,1977; Rudnicki, 1985). The poroelastic constant varies between 0 and 1 as it describes the efficiency of the fluid pressure in counteracting the total appliedstress. Its value depends on the pore geometry and thephysical properties of the constituents of the solid sys-tem and, hence, on the applied load. It is related to theundrained Poissons ratio u, drained Poissons ratio and Skempton (1960) pore pressure coefficient B,defined as

(3-21)

where p represents the variation in pore pressureresulting from a change in the confining stress under undrained conditions. From these variables:

(3-22)

Only in the ideal case, where no porosity changeoccurs under equal variation of pore and confiningpressure, can the preceding expression be simplified to

(3-23)

where K is the bulk modulus of the material and Ks isthe bulk modulus of the solid constituents. Typically,for petroleum reservoirs, is about 0.7, but its valuechanges over the life of the reservoir. The poroelasticconstant is a scalar only for isotropic materials. It isa tensor in anisotropic rocks (Thompson and Willis,1991). Another important poroelastic parameter is theporoelastic stress coefficient , defined as

(3-24)

which describes the in-situ stress change caused byinjection and/or production. This is addressed later in the chapter.

3-3.3. Fracture mechanicsFracture mechanics studies the stability of preexistingdefects that are assumed to pervade a continuum.These inclusions induce high stress concentrations intheir vicinity and become the nucleus for crack initia-tion and/or propagation. Historically, Griffith (1921,1924) established the foundation of fracture mechan-ics; he studied propagation by considering the energyused in various parts of the fracturing process.

Griffiths original treatment expressed the conditionthat the total energy is unchanged by small variationsin the crack length. The different approach presentedhere states that the energy that is consumed by thecreation of new surfaces should be balanced by thechange in the potential energy of the system:

(3-25)where dWelas represents the change in elastic energystored in the solid, dWext is the change in potentialenergy of exterior forces, dWs is the energy dissipatedduring the propagation of a crack, and dWkin is thechange in kinetic energy. Energy dissipated as heat is neglected. To proceed further, it is assumed that theenergy dWs required to create the new elementaryfracture surfaces 2dA is proportional to the area created:

(3-26)where F is the fracture surface energy of the solid,which is the energy per unit area required to createnew fracture surfaces (similar to the surface tension of a fluid). The factor 2 arises from the considerationthat two new surfaces are created during the separa-tion process. The propagation is unstable if the kineticenergy increases; thus, dWkin > 0 gives

(3-27)where the strain energy release rate Ge is defined as

(3-28)

The onset of crack propagation, which is referred toas the Griffith criterion, is

(3-29)


= p.

B p=

,

dW dW dW dWelas ext s kin+ + + = 0,

dW dAs F= 2 ,

Ge F> 2 ,

Gd W W

dAeelas ext

=

+( ).

Ge F= 2 .

=

( )( ) +( )3

1 2 1u

uB

.

= 1 KK

s

,

=

( )( )

1 22 1

,

Another approach was developed by Irwin (1957).He demonstrated that, for a linear elastic material, themagnitude of the stresses in the vicinity of a stress-free crack follows an r1/2 relationship (Fig. 3-6):

(3-30)

where KI is referred to as the stress intensity factor forthe opening mode of deformation of the fracture, fij()represents a bounded function depending only on theangle referenced to the plane of the crack, and r isthe distance from the point of interest to the tip of thefracture. The negative sign is included because, byconvention, tensile stresses are negative.

The width w near the tip of a stress-free crack isalso a function of the stress intensity factor:

(3-31)

In Eq. 3-31, plane strain is assumed.The stress intensity factor is a function of the load-

ing parameters and of the geometry of the body.Hence, length is included in the unit to express KI. A fracture propagates when KI reaches a critical value,known as the critical stress intensity factor KIc or frac-ture toughness. For a perfectly elastic material, KIc is a material property. It must be evaluated experimen-tally. Experimental results show that for short cracklengths, KIc increases with crack length. When thisscale effect is observed, KIc cannot be considered amaterial property. This behavior is discussed in moredetail in Section 3-4.6.

The unit for KIc is pressure times the square root oflength. Fracture toughness is a measure of the resis-

tance of the rock to crack propagation. It must not be confused with the tensile strength of the rock To,although these two properties can be related by thefollowing formula:

(3-32)

where ac is a length scale (e.g., flaws or grain size)characteristic of the rock under consideration.

Irwins (1957) approach is similar to Griffiths(1921, 1924). It can be demonstrated that, for anisotropic and linear elastic material, the stress intensityfactor is related to the strain energy release rate by

(3-33)

As an example of this application to hydraulic frac-turing, the stress intensity factor for a uniformly pres-surized crack subjected to a far-field minimum stress3 is

(3-34)where pf is the pressure in the crack, L is the half-lengthof the crack, and plane strain is assumed. During prop-agation, the net pressure (pf 3) is therefore

(3-35)

Using Sneddons (1946) solution, the width at thewellbore ww is

(3-36)

A propagation criterion based on the stress intensityfactor is easily implemented in fracture propagationcodes. However, the concept of fracture surface energydoes not imply linear elasticity and can be used forfracture propagation in nonlinear materials where thestrain energy release rate is replaced with the J-inte-gral (Rice, 1968).

Stress intensity factors are not limited to openingmodes. Other modes exist (Irwin, 1957) to analyze 3D fracture propagation in complex stress fields (e.g., propagation from inclined wellbores) where thefracture changes direction during propagation. Finally,fracture mechanics has also been used to explain brit-tle rock fracture in compression (Germanovich et al.,1994).


ij I ijK

rf= ( ) +

2... , T

Ka

o

Ic

c

=

,

GE

Ke I=

1 2 2.

K p LI f= ( ) 3 ,

p KLf

Ic( ) =

3 .

wK

LL

EwIc

=

( )

4 1 2.

wE

K rI=( )8 1

2

2

.

Figure 3-6. Stress concentration near the tip of a crack.

y

Circ

umfe

rent

ialst

ress

,

Distance from the tip, r

For further details on fracture mechanics, the readeris referred to Cherapanov (1979), Kanninen andPopelar (1985) and Atkinson (1987).

3-3.4. Nonelastic deformationAs discussed in the next section, most rocks exhibitnonreversible deformations after unloading, or at leasta nonunique relationship between stress and strain.This means that rocks are not perfectly elastic materi-als, and a number of theories have been developed to model such behaviors. They include the theory ofplasticity, damage mechanics and time-dependentanalysis (creep). As an example, the theory of elasto-plasticity is briefly described.

Figure 3-7 shows the stress-strain relationship of acylindrical ideal sample. From O to point A, the rela-tion between stress and strain is linear, and the slopeof the curve is Youngs modulus E. The stress-strainrelation does not change if the sample is unloaded inthis region. This is the region where the theory ofelasticity applies. Beyond point A, the slope of thecurve decreases. Moreover, if the sample is unloadedin this region, say at point B, the unloading portiondoes not follow the same path as the loading portion

but is perfectly linear with a slope E. At zero stress,part of the deformation has not been recovered. Thisrepresents the plastic strain component in the theoryof elasto-plasticity. Point A is actually the initial yieldstress of the rock. During reloading, the samplebehaves as a perfectly elastic solid up to point B,which is the new yield stress. The increase of yieldstress with an increase of plastic strain is called strainhardening, and the decrease of yield stress with anincrease of plastic strain is called strain softening. Aperfectly plastic material is a material with no strainhardening or softening. As shown in this example, theyield stress is a function of the loading history of therock if the rock hardens or softens.

In elasto-plasticity, part of the strain is predicted bythe theory of elasticity; i.e., any strain increment asso-ciated with a stress increment is the sum of an elasticcomponent and a nonelastic component:

(3-37)where d is the total strain increment, de is the elasticstrain increment, and dp is the plastic strain incre-ment. Contrary to the elastic strain component, theplastic strain component cannot be recovered duringunloading. Predicting the plastic strain incrementrequires a yield criterion that indicates whether plasticdeformation occurs, a flow rule that describes how theplastic strain develops and a hardening law.

The yield criterion is a relationship between stressesthat is used to define conditions under which plasticdeformation occurs. In three dimensions, this is repre-sented by a yield function that is a function of thestate of stress and a hardening parameter:

(3-38)The hardening parameter h determines the evolution

of the yield curve with the amount of plastic deforma-tion of the material. Elasto-plastic deformation withhardening is important in the study of the stability offormations prone to sanding. Weak sandstones usuallyshow hardening behavior, which can be close to linearhardening. For further details on elasto-plasticity, thereader is referred to Hill (1951) and Chen and Han(1988).

3-3.5. FailureA failure criterion is usually a relationship betweenthe principal effective stresses, representing a limit


Figure 3-7. Stress-strain relationship for an elasto-plasticmaterial with strain hardening. OA = elastic, AB = plastic.

A

O

B

E E

d d de p = + ,

f h 1 2 3 0, , , .( ) =

beyond which instability or failure occurs. TheTerzaghi effective stress is used in failure criteria.Several types of criteria have been proposed in the lit-erature and have been used for various applications.The more popular criteria include the following: Maximum tensile stress criterion maintains that fail-

ure initiates as soon as the minimum effective prin-cipal stress component reaches the tensile strengthTo of the material:

(3-39)

Tresca criterion expresses that failure occurs whenthe shear stress (1 3)/2 reaches the characteristiccohesion value Co:

(3-40) Mohr-Coulomb criterion expresses that the shear

stress tending to cause failure is restricted by thecohesion of the material and by a constant analo-gous to the coefficient of friction times the effectivenormal stress acting across the failure plane:

(3-41)where is the angle of internal friction and Co isthe cohesion. The Mohr-Coulomb failure criterioncan be rewritten in terms of the principal stresses togive 1 at failure in terms of 3:

(3-42)where the coefficient of passive stress N is

(3-43)

The uniaxial compressive strength then becomes

(3-44)In a ((n p), ) plane, this criterion is a straight

line of slope tan and intercept Co. A rock fails assoon as the state of stress is such that the criterion is met along one plane, which is also the failureplane. Using the Mohr circle graphical representa-tion described in Sidebar 3B, this means that thestate of stress at failure is represented by a Mohrcircle that touches the failure envelope. The point of intersection can be used to determine the angle between the normal to the failure plane and thedirection of 1, as shown in Fig. 3-8. It can beshown that

(3-45)

Mohr failure envelope is a generalization of the lin-ear Mohr-Coulomb criterion. An example of a moregeneral model is the following nonlinear model:

(3-46)where A and n are obtained experimentally. Thefailure envelope can also be constructed graphically(see Section 3-4.5).As shown here, the Tresca and Mohr-Coulomb cri-

teria do not include the influence of the intermediatestress 2. Experimental evidence shows they are, inmany cases, good approximations. However, there areother criteria that include the effect of 2.

3-4. Rock mechanical property measurement

3-4.1. Importance of rock properties in stimulation

Most of the hydraulic fracture propagation modelsassume linear elasticity. The most important rockparameter for these models is the plane strain modulusE , which controls the fracture width and the value ofthe net pressure. In multilayered formations, E mustbe determined in each layer, as the variation of elasticproperties influences the fracture geometry. Elasticand failure parameters are also used in stress modelsto obtain a stress profile as a function of depth androck properties. These profiles are important for esti-


3 = p To .

1 3 2 = Co . = +4 2

.

1 3 = + ( )p A pc n , = + ( ) ( )C po ntan ,

1 3 = + ( )p N pc ,

N

= + tan2 4 2 .

c oC N= 2 .

Figure 3-8. Graphical representation of a state of stress atfailure.

M

Co

3 1 n

2

Mohr-Co

ulomb cri

terion

mating the stress variation between layers and, conse-quently, the geometry of hydraulically induced frac-tures. The parameters involved are Youngs modulus,Poissons ratio, the poroelastic coefficient and the fric-tion angle. The poroelastic stress coefficient controlsthe value of stress changes induced by pore pressurechanges that result from depletion, injection or frac-ture fluid loss.

The role of fracture toughness in hydraulic fractur-ing has been the subject of investigation in recent years(Shlyapobersky, 1985; Thiercelin, 1989; Cleary et al.,1991; Johnson and Cleary, 1991; Advani et al., 1992;SCR Geomechanics Group, 1993; Valk and Econ-omides, 1994). Laboratory measurements give valuesof KIc of the order of 1000 psi/in.1/2 (at least in theabsence of confining pressure), whereas fracture propa-gation models indicate that KIc must be at least 1 orderof magnitude larger to influence fracture geometry.These results are, however, a function of fracturegeometry and pumping parameters. Shlyapobersky(1985) suggested that in-situ fracture toughness, oftenreferred to as apparent fracture toughness, can be muchgreater than laboratory values because of scale effects.These effects include the influence of heterogeneities,discontinuities (Thiercelin, 1989), large-scale plasticity

(Papanastasiou and Thiercelin, 1993) and rock damage(Valk and Economides, 1994). In-situ determinationis, however, difficult to achieve.

Finally, rock failure must be considered in evaluat-ing the long-term stability of the rock around the frac-ture or at the wellbore. In particular, in weak forma-tions (chalk or weak sandstones) part of the rock col-lapses if the drawdown pressure is too high.

3-4.2. Laboratory testingUniaxial and triaxial tests are considered the most use-ful tests in the study of mechanical rock properties.The difference between them resides in the presenceor absence of confining pressure applied to the speci-men. A typical triaxial testing system is shown sche-matically in Fig. 3-9. It subjects a circular cylinder of rock to an axisymmetric confining pressure and alongitudinal or axial load. Generally, these loads aresimilar to the in-situ state of stress. Relationshipsbetween the mechanical properties of the rock and thedegree of confinement are obtained by performing aseries of tests using different stress and pore pressureconditions. Also, if the rock is anisotropic, an addi-tional series of tests should be performed using differ-


Figure 3-9. Triaxial testing configuration.

Loading ram

Pore pressureinletJacketingmaterialSample withstrain gaugeaffixedPore pressureoutletSphericalseat

Electrical feed-throughs

Data acquisition

Triaxial cell

Strain

Stre

ss

Loading frameTemperaturecontroller

Confiningpressuresystem

Porepressuresystem

ent orientations of the cylinder axis with respect to theplane of anisotropy. During the course of the test, theprimary information recorded is the deformation ver-sus load relationships (Fig. 3-10), from which bothYoungs modulus and Poissons ratio can be found.Because these primary elastic constants depend onconfining stress, temperature, pore saturation andpressure, it is extremely important that the laboratoryenvironment encompass the representative field situa-tion to obtain representative data.

The importance of good specimen preparation can-not be overemphasized, and the International Societyof Rock Mechanics (ISRM) recommended proceduresthat must be followed (Rock Characterization Testingand Monitoring; Brown, 1981). The end faces mustbe parallel; otherwise, extraneous bending momentsare introduced, making correct interpretation of theresults more difficult. In addition, because of the mis-match between the rock properties and those of thetesting platens, shear stresses that develop at the rock/platen interfaces create an additional confinementimmediately adjacent to the specimen ends. This dic-tates the use of specimens with a length:diameter ratioof at least 2 or the use of appropriate rock inserts oradaptive lubricant. The loading rate should also bemaintained between 70 and 140 psi/s to avoiddynamic effects. Finally, some rock types (such asshales) are sensitive to the dehydration of natural porefluids; care must be taken to preserve their integrity byavoiding drying cycles and contact with air duringspecimen recovery, storage and test preparation.

3-4.3. Stress-strain curveFigure 3-10 presents a typical stress-strain relationshipfor rocks. The test is conducted under constant con-fining pressure pc and constant axial strain rate. Mea-surements include the values of axial stress, axial strainand radial strain. When confining pressure is applied to the sample, the origin of the stress-strain plot isusually translated to remove the influence of hydro-static loading on the stress and strain (i.e., the axialstress is actually the differential a pc).

During the initial stages of loading, from O to pointA, the rock stiffens. This nonlinear regime is probablydue to the closing of preexisting microcracks pervad-ing the specimen. This particular region of the stress-strain curve is a signature of the stress history under-gone by the rock specimen during past geologic time,including the coring process. This characteristic is dis-cussed later as applied to in-situ stress determinations.

As the load increases further, the stress-strain curvebecomes linear (from point A to point B); this is theportion of the stress-strain curve where the rockbehavior is nearly elastic. If unloading occurs in thisregion, the strain returns almost to zero, usually alonga different path. This effect is called hysteresis andindicates that some energy dissipates during a cycle of loading and unloading.

When the rock specimen is loaded beyond point B,irreversible damage sets in. It is shown by a decreaseof the slope of the stress versus radial strain curve. Atthis stage, the damage is not seen on the axial strain.At point C, the axial strain also becomes nonlinear,and large deformations eventually occur. If the rock is unloaded in this region, permanent strains at zerostress are observed. Point D is the maximum load thatthe rock can sustain under a given confining pressure.Rock failure (i.e., when the sample loses its integrity)occurs at about this point. Some rocks, especiallythose with high porosity, may not exhibit a maximumpeak stress but continue to carry increasing stress (i.e.,continue to harden).

Another interesting rock characteristic is revealedby the volumetric strain, defined as the change in vol-ume with respect to the original specimen volume.For a triaxial test, the volumetric strain of the cylindri-cal specimen is a + 2r , where a is the axial strainand r is the radial strain. As seen on Fig. 3-11, thevolumetric strain versus axial stress can reverse itstrend upon reaching point E; i.e., the rock specimen


Figure 3-10. Stress-strain curves.

B

A

C

D

O

c

r a

starts to increase in volume under additional compres-sive load. This is referred to as dilatancy.

Dilatancy is responsible for the nonlinearity that isobserved in the radial strain and consequently in thevariation of volume. It is due either to the creation oftensile cracks that propagate in a direction parallel tothe axis of loading (in that case, point B shown in Fig. 3-10 is distinct from point C) or to frictionalsliding along rough surfaces and grains (in that case,point B is close to point C). Soft rocks under confin-ing pressure could show a decrease in volume insteadof an increase, because of compaction. This is typicalof chalk and weak sandstones. Compaction in cohe-sive rocks requires the destruction of cohesion, whichcould create a sanding problem during production.Finally, if the framework of elasto-plasticity is used,point B is the initial yield point. If the nonelastic com-ponent of the variation of volume is negative, the rockis dilatant; otherwise, the rock is compactant.

Brittle rocks and ductile rocks must also be differ-entiated. Brittle rocks are characterized by failureprior to large nonelastic deformation. Low-porositysandstones and hard limestones are typical brittlerocks. Ductile rocks are characterized by the absenceof macroscopic failure (i.e., theoretically, the rock willyield indefinitely). Salt, young shales and very highpermeability sandstones are typical ductile rocks.These behaviors, however, are functions not only

of rock type but also confining pressure, loading rateand temperature, with a general transition from brittleto ductile behavior with an increase in confining pres-sure, increase in temperature and decrease in loadingrate. Moreover, in porous rocks (e.g., sandstones,shales), a transition from dilatant to compactantbehavior with confining pressure is also observed.

3-4.4. Elastic parametersAs discussed previously, rocks are not perfectly elas-tic. Especially in soft rocks, it could well be difficultto find a portion of the stress-strain curves that exhib-its nearly elastic behavior. On the other hand, theknowledge of elastic parameters is of great impor-tance for engineering applications, and assuming, as a first approximation, that the rock behaves as an elas-tic material has significant advantages.

There are two main approaches to elastic parameterdetermination. The first one is to find elastic param-eters that can be used to predict as close as possiblethe behavior of the rock along an expected loadingpath. These parameters do not measure the real elasticcomponent of the rock but approximate the rockbehavior. This is the approach used in engineeringdesign, although the assumption underlying the mea-surement must be kept in mind. The other approach is to develop a test procedure that measures, as closeas possible, the elastic component of the strain. Thisapproach is useful if a correlation is sought betweendownhole measurements made using sonic tools andcore measurements. Because of the variety ofapproaches that can be used, it is essential to alwaysmention how elastic properties have been measured.

Elastic property measurement can be made understatic conditions or under dynamic conditions. Theratio of the dynamic to static moduli may vary from0.8 to about 3 and is a function of rock type and con-fining stress. In most cases, this ratio is higher than 1(e.g., Simmons and Brace, 1965; King, 1983; Chengand Johnston, 1981; Yale et al., 1995). Possible expla-nations for these differences are discussed in the fol-lowing. Static elastic properties, as measured in thelaboratory during sample loading (see the followingsection), are generally assumed more appropriate thandynamic ones for estimating the width of hydraulicfractures. Knowledge of dynamic elastic properties is,however, required to establish a calibration procedureto estimate static downhole properties from downhole


Figure 3-11. Axial stress versus volumetric strain.

E

O v

measurements, which are obtained essentially fromsonic tools (see Chapter 4). Static elastic properties

Static elastic properties are usually measured usingthe equipment described in Section 3-4.2. For clas-sification purposes, the ISRM proposed the follow-ing recommended procedures that use, for the mea-surement of Youngs modulus, the axial stressaxialstrain curve measured during the loading of thesample (Brown, 1981) (Fig. 3-12): tangent Youngs modulus Etthe slope at a

stress level that is usually some fixed percentageof the ultimate strength (generally 50%)

average Youngs modulus Eavdetermined fromthe average slopes of the generally straight-lineportion of the curve

secant Youngs modulus Esusually the slopefrom zero stress to some fixed percentage of theultimate strength (generally 50%).

Poissons ratio is determined using similar meth-ods and the axial strainradial strain curve.

These elastic constants must be adjusted to theproper reservoir conditions for design purposes.Moreover, for stimulation purposes an approachthat requires failure of the sample is not necessary.The best method is either to use the average modu-lus or to measure a tangent modulus at a state ofstress near the expected downhole effective state of stress. The average value simulates the effect ofthe width, causing stresses that are maximum at thefracture face and decay to zero away from the face.Ideally, the sample must be tested in a direction nor-mal to the expected hydraulic fracture plane (i.e., inthe horizontal direction if the fracture is expected to be vertical). The best way to reproduce downholeconditions is probably to apply a confining pressureequal to the effective downhole mean pressure (h + v + H)/3 p, where h, v and H are theminimum horizontal stress, vertical stress and maxi-mum horizontal stress, respectively. The tangentproperties are then measured using an incrementalincrease of the axial load. Terzaghis effective stressis used here rather than the Biot effective stress con-cept because the tangent properties are essentiallycontrolled by this effective stress (Zimmerman etal., 1986).

The second approach utilizes small unloading-loading cycles that are conducted during the mainloading phase. If the cycle is small enough, theslope of the unloading stress-strain curve is close to that of the reloading stress-strain curve (Fig. 3-13,Hilbert et al., 1994; Plona and Cook, 1995). Thisleads to the measurement of elastic properties thatare close to the actual ones and also close to thevalue determined using ultrasonic techniques. It isalso important to perform these measurements atthe relevant confining pressure and axial stress.

Elastic properties determined using sonicmeasurementsSonic measurements are conveniently used to deter-mine the elastic properties under dynamic conditionsin the laboratory. These properties are also calleddynamic elastic properties. To obtain them, a mech-anical pulse is imparted to the rock specimen, andthe time required for the pulse to traverse the lengthof the specimen is determined. Then, the velocity of the wave can be easily calculated. Again, thesemeasurements should be performed under simulated


Figure 3-12. ISRM-recommended methods to measureYoungs modulus: derivative of the stress-strain curve at point A is Et,

measured at 50% of the ultimate strength slope of the straight line BC is Eav slope of the straight line OA is Es.

A

B

C

O

failure

a

0.5failure

downhole conditions and can be conducted duringtriaxial compression tests (Fig. 3-14).

As also discussed in Chapter 4, two types of elas-tic body waves can be generated: compressional(also called P-waves) and shear (S-waves). Elasticwave theory shows that the velocities of P- and S-waves (uP and uS, respectively) are related to theelastic constants through the following relationships(in dry rocks):

(3-47)

(3-48)

where refers to the mass density of the rock speci-men and the relationship between the various elasticmoduli is as in Sidebar 3C. The subscript dyn refersto dynamic, as the values of the elastic constantsobtained by dynamic techniques are in general higherthan those obtained by static methods. This differ-ence is now believed to be due mainly to the ampli-tude of the strain, with the very low amplitudedynamic measurements representing the actual

elastic component of the rock (Hilbert et al., 1994;Plona and Cook, 1995). Because of poroelasticeffects and rock heterogeneity, the acoustic velocityis also a function of wave frequency. But in dryrocks, the influence of the frequency appears to be of second order compared with that of the strainamplitude (Winkler and Murphy, 1995). Conse-quently, when the dynamic and static small-ampli-tude loading/unloading measurements are compared,their values agree quite well (Fig. 3-15; Plona andCook, 1995).

Correlations can be established between static anddynamic moduli (Coon, 1968; van Heerden, 1987;Jizba and Nur, 1990). Coon demonstrated that thecoefficient of correlation can be improved if consid-eration of the lithology is included. These correla-tions allow an estimation of large-amplitude staticin-situ values from log data where core data are notavailable (see Chapter 4). Figure 3-15 suggestsanother procedure in which a corrective factor isfound by the ratio of the loading to unloading tan-gent moduli for low-amplitude static tests.

Scale effects in elastic propertiesThe elastic properties of rock are scale dependent,as are any rock properties. This means that thevalue of an elastic parameter that is determined on a laboratory sample may be quite different of that of a rock mass, mainly because of the presence of


Figure 3-13. Youngs modulus measured using small cycles(Hilbert et al., 1994). Youngs modulus at B is the slope ofline AB.

A

B

O

B

a

Figure 3-14. Ultrasonic pulse measurement.

Applied load

Applied load

Receiver transducer

Transmitter transducer

Environmentalchamber

Specimen

Platen

Platen

uC K G

E

Pdyn

dyn dyn

dyn dyn

dyn dyn

=

=

+

=

( )+( ) ( )

1 2

1 2

1 2

43

11 1 2

/

/

/

uG E

Sdyn dyn

dyn

=

= +( )

1 2 1 2

2 1

/ /

,

discontinuities in the rock mass. Various approachesare being developed to take this phenomenon intoconsideration (Schatz et al., 1993). An alternative isto determine the properties downhole, as describedin the next section. However, downhole measure-ments are usually limited to a scale on the order of3 ft, whereas a large fracture involves a scale on theorder of 100 ft. Rock imperfection on this scale canbe mapped by a combination of wellbore seismicand sonic measurements.

Elastic properties determined using downholemeasurements

Downhole measurements are made to estimate theelastic properties. Dynamic log measurements aredescribed in detail in Chapter 4. Other techniquesinclude direct downhole static measurements andinversion of the pressure response obtained during a micro-hydraulic fracturing test. A direct downholestatic measurement requires measuring the deforma-tion of a small portion of the wellbore during pres-surization. This can be done by using downholeextensiometers (Kulhman et al., 1993). Usually thistechnique yields only the shear modulus G. Pressureinversion techniques (Piggott et al., 1992) require

a fracture propagation model to invert the pressureresponse in terms of elastic properties. The geometryand mechanical assumptions of the fracture propa-gation model must be as close as possible to theactual situation. If the fracture propagates radially,this technique can extract an estimate of the planestrain modulus E (Desroches and Thiercelin, 1993).

Poroelastic properties

For isotropic rocks, it is generally recommended toconduct tests that measure the volumetric responseof the sample, as poroelastic effects are volumetricones. Three tests are usually made to measure thefive properties that characterize an isotropic poro-elastic material. All three tests involve hydrostaticloading but differ on the boundary conditionsapplied to the pore fluid. For the drained test, thefluid in the rock is maintained at constant pressure;for the undrained test, the fluid is prevented fromescaping the sample; and for the unjacketed test, thepore pressure is maintained equal to the confiningpressure. The reader is referred to Detournay andCheng (1993) for further information. Presentedhere is the determination of , which, with knowl-edge of the drained Poissons ratio, allows determi-nation of the poroelastic stress coefficient , whichis probably the most important poroelastic parame-ter for hydraulic fracturing applications. This mea-surement is conducted using the drained test, inwhich the volume change of the sample V and thevolume change of the pore fluid Vf are measuredas a function of an incremental increase of the con-fining pressure. The value of is then given by thefollowing relation:

(3-49)

As for the elastic properties, the test must be con-ducted with a confining pressure close to the down-hole mean stress. These properties must be tangentproperties and, for practical purposes, are a functionof the Terzaghi effective stress. Mathematical con-sideration and experimental results confirm thatporoelastic properties are controlled by the Terzaghieffective stress (Zimmerman et al., 1986; Boutcaet al., 1994).


Figure 3-15. Dynamic versus static Youngs modulus mea-surements (after Plona and Cook, 1995).

Dynamic ESmall-amplitude static ELarge-amplitude static E

20

0 2 4 6 8 10 12

1

2

3

4

5

6

7

8

9

10

11

Stress (MPa)

Youn

g's m

odulu

s (GP

a)

=VV

f.

3-4.5. Rock strength, yield criterion and failure envelope

The strength of a rock is the stress at which the rockfails (i.e., the rock loses its integrity). This strengthobtained with a uniaxial test is called the uniaxialcompressive strength c (UCS). The overall strengthof rocks is a relationship between the principal effec-tive stress components (in the sense of Terzaghi, seeSection 3-3.5). This relationship is called the failurecriterion, and its graphical representation is called thefailure envelope.

To obtain the failure envelope of a particular rocktype, a series of triaxial tests should be performedunder different confining pressures until failure of thespecimen occurs for each condition. There are variousways to represent the failure envelope. The classicapproach in rock mechanics is to plot the effectivestresses at failure for each test using a Mohr circle rep-resentation (see Sidebar 3B) of diameter (failure 3),where failure represents the ultimate strength of thespecimen measured under confinement 3 (Fig. 3-16).The envelope of these circles is a locus separating sta-ble from unstable conditions. It should be emphasizedthat the failure of rocks occurs when the matrixstresses reach a critical level; hence, the failure enve-lope represents a relationship between the effectivestress levels. Therefore, the knowledge of such a char-acteristic can also be used to put some limits on theallowable variation of the reservoir pore pressure dur-ing production. Indeed, a change in pore pressure cor-responds to a translation of the pertinent Mohr circlealong the normal stress axis.

A specific case is the study of pore collapse. Porecollapse is usually not associated with a sudden lossof integrity and therefore has to be detected from theinitial yield envelope rather than the failure envelope.In some instances, yield can be initiated under iso-tropic loading (Fig. 3-17). The portion of the yieldcurve that shows a decrease of the shear stress at yieldas a function of the confining pressure is characteristicof compactant materials. This usually occurs withpoorly consolidated rocks. In cohesive materialscompaction is associated with pore collapse andconsequently with cohesion loss. This is a potentialfailure mechanism of the matrix that could lead to the production of formation particles (e.g., sanding).

3-4.6. Fracture toughnessDetermining the value of fracture toughness requiresusing a sample that contains a crack of known length.The stress intensity factor, which is a function of theload and sample geometry, including the length of thepreexisting crack, is then determined. Testing mea-sures the critical load and, therefore, the critical stressintensity factor KIc at which the preexisting crack isreinitiated. Another approach is to measure the frac-ture surface energy and use Eq. 3-33. An exampleusing a simple geometry is discussed in Sidebar 3D.Testing the sample under downhole conditions is alsorequired because fracture toughness increases witheffective confining pressure and is affected by temper-ature. Various sample geometries have been proposed,but the most practical ones from an engineering point


Figure 3-16. Failure envelope. 1 = Mohr circle correspond-ing to uniaxial tensile test; 2 = Mohr circle corresponding touniaxial compressive test; and 3 = Mohr circle correspond-ing to triaxial test with effective confining stress c and fail-ure stress (i.e., ultimate strength) failure.

1

2

3

Failure envelope

To

To

= Tensile strengthc

= Uniaxial compressive strength

0 n3 c failure

Figure 3-17. Failure and initial yield envelopes for poorlyconsolidated sandstones.

n

Failure line

Yield envelope

Zone of pore collapse


3D. Fracture toughness testing

To illustrate the measurement of fracture toughness and theinfluence of a crack on material behavior, a bar of unit thick-ness containing a central crack of length 2L is considered(Fig. 3D-1). Although this is a simple geometry, in practicesuch an extension test is difficult to conduct. The crack lengthis supposed to be small compared with the bar width and thewidth small compared with the bar length. The stress intensityfactor for this geometry is given by

(3D-1)

where is the stress applied to the sample (i.e., F/2b).Figure 3D-1 also shows a plot of the load versus displace-

ment curve. The load increases to the point where the crackstarts to propagate. During stable crack propagation, the loaddecreases. If the sample is unloaded at this stage, the load-displacement curve exhibits a slope different from the oneobtained during initial loading. However, the displacement isrecovered upon complete unloading. This behavior is funda-mentally different from that of elasto-plasticity, and a perfectlybrittle behavior is exhibited. The change of slope is not amaterial property but is due to the increased length of thecrack. It can, therefore, be used to estimate the crack length.

The critical stress intensity factor is the value of KIc whenthe crack starts to propagate:

(3D-2)

where Fc is the critical load.It can also be demonstrated that the area OAB in Fig. 3D-1

corresponds to the energy dWs that was dissipated to propa-gate the crack from 2L to (2L + 2L). The strain energyrelease rate is, therefore, the dissipated energy divided by thecreated surface area 2L:

(3D-3)

A similar approach can be used if the crack is propagatedto the sample end; in that case:

(3D-4)

where dWs corresponds to the area under the load-displace-ment curve and the initial crack length is assumed to be smallcompared with the sample width (i.e., L b). Using thisapproach, there is no need to measure crack length. For lin-ear elastic behavior:

(3D-5)

The load-displacement curve shown in Fig. 3D-1 can also beused to determine the process zone behavior (Labuz et al.,1985).

K L Lb

LbI

12

11 2/

,

K Fb

L Lb

LbIc

c

21

21

1 2

/

,

G dWLes

=

2.

G dWbe

s=

2,

GE

Ke I=1 2 2

.

Figure 3D-1. Fracture toughness measurement. The shaded area on the left of the plot represents the energy requiredto propagate the crack from 2L to (2L + 2L).

2b

2L

O

F

F

F

Fc

2L + 2L

B

A

of view are those based on core geometries (Ouchter-lony, 1982; Thiercelin and Roegiers, 1986; Zhao andRoegiers, 1990; ISRM Commission on TestingMethods, 1988, 1995).

However, the existence of very large stress valuesnear the tip of the crack makes it difficult to develop a rigorous test configuration because a cloud of micro-cracks is created ahead of the crack tip. This is com-monly referred to as the process zone (Swanson andSpetzler, 1984; Labuz et al., 1985). The extent of thisnonlinear region must be limited so that it does notreach the edge of the laboratory sample. Also, thisprocess zone must be relatively small compared withthe size of the crack if linear elastic calculations are to be valid (Schmidt, 1976; Schmidt and Lutz, 1979;Boone et al., 1986).

The development of the process zone is one of thecauses of the scale effects that are observed in fracturetoughness testing; i.e., the determined value of frac-ture toughness increases with sample size. Modelingof process zone behavior can be conducted using theinformation obtained during tensile failure of a speci-men (see Sidebar 3D). Modeling can also give someinsight on the tip behavior of large-scale hydraulicfractures (Papanastasiou and Thiercelin, 1993).

3-5. State of stress in the earthThe propagation and geometry of hydraulic fracturesare strongly controlled by the downhole state of stress.In particular, it is generally accepted that the degree of fracture containment is determined primarily by thein-situ stress differences existing between layers. In theabsence of a meaningful stress contrast, other mecha-nisms such as slip on bedding planes (Warpinski et al.,1993) and fracture toughness contrast (Thiercelin et al.,1989) can have a role. Moreover, hydraulic fracturespropagate, in most cases, normal to the minimum stressdirection. Consequently, knowledge of the minimumstress direction allows prediction of the expected direc-tion of the hydraulic fracture away from the wellbore.

Stresses in the earth are functions of various para-meters that include depth, lithology, pore pressure,structure and tectonic setting. A typical example fromthe Piceance basin in Colorado (Warpinski and Teufel,1989) is shown in Fig. 3-18. The stress regime in agiven environment depends, therefore, on regionalconsiderations (such as tectonics) and local considera-tions (such as lithology). Understanding the interac-

tion between regional and local considerations isimportant as it controls the stress variation betweenlayers. In some stress regimes the adjacent layers areunder higher stress than the pay zone, enhancing frac-ture height containment; in others, the adjacent layersare under lower stress than the pay zone, and fracturepropagation out of the zone is likely, limiting lateralfracture penetration. Key regional stress regimes and


Figure 3-18. Stress profile for Well MWX-3 (Warpinski andTeufel, 1989).

50007600

6000 7000 8000

7400

7200

7000

6800

6600

6400

6200

Stress (psi)

Dept

h (ft)

Mudstone

Sandstone

the consequences of these regimes on the local state of stress in a reservoir are reviewed in the following.These regimes lead to the introduction of simple stressmodels that allow making rough estimates of thestress profile as a function of depth and rock properties.These models can also be used to obtain a calibratedstress profile from log and stress measurement infor-mation, as shown in Chapter 4. The influence of thevariation of temperature and pore pressure on the stateof stress is also analyzed. Finally, the influence ofindustrial intervention on the state of stress is pre-sented. Intervention includes drilling a hole anddepleting or cooling a formation.

3-5.1. Rock at restOne stress regime is when the rock is under uniaxialstrain conditions (i.e., there is no horizontal strain any-where). To estimate the state of stress that is generatedunder this regime, it is assumed that the rock is asemi-infinite isotropic medium subjected to gravita-tional loading and no horizontal strain.

Under these conditions, the vertical stress is gener-ated by the weight of the overburden and is the maxi-mum principal stress. Its magnitude, at a specificdepth H, is estimated by

(3-50)

where is the density of the overlying rock massesand g is the acceleration of gravity. The value of thisstress component is obtained from the integration of a density log. The overburden gradient varies fromabout 0.8 psi/ft in young, shallow formations (e.g.,Gulf Coast) to about 1.25 psi/ft in high-density formations. Assuming that quartz has a density of165 lbm/ft3, the overburden gradient ranges betweenthe well-known values of 1.0 and 1.1 psi/ft for brine-saturated sandstone with porosity ranging between20% and 7%, respectively.

With uniaxial strain assumed, the other two princi-pal stresses are equal and lie in the horizontal plane. If they are written in terms of effective stress, they area function of only the overburden:

(3-51)where Ko is the coefficient of earth pressure at rest and h is the minimum effective horizontal stress.Assumptions about rock behavior can be used to esti-

mate values of Ko. However, stress predictions usingthese assumptions must be used with great cautionand may not be applicable in lenticular formations(Warpinski and Teufel, 1989). Nevertheless, they areuseful for understanding the state of stress in the earthand can be used as a reference state (Engelder, 1993).

With the assumption of elasticity and for the bound-ary conditions outlined previously, Ko is

(3-52)

and the relationship between the total minimum hori-zontal stress h and the overburden v is, after re-arranging and using the Biot effective stress for ,

(3-53)

The dependence of horizontal stress on rock lithol-ogy results from the dependence of Poissons ratio on rock lithology. In most cases, the model predictsthat sandstones are under lower stress than shales asKo in sandstones and shales is about equal to 13 and 12,respectively. The use of Eq. 3-53 to obtain stress pro-files in relaxed basins is presented in Section 4-5.2.More complex elastic models that are associated withthis stress regime have been developed to considerrock anisotropy (Amadi et al., 1988) and topography(Savage et al., 1985).

For purely frictional materials, Ko can be approxi-mated by (1 sin) (Wroth, 1975), which gives thefollowing relationship for the total stresses:

(3-54)where is the angle of internal friction of the rock(Eq. 3-41), of the order of 20 for shales and 30 forsandstones. In this expression, the Terzaghi effectivestress concept prevails because this case involves fric-tional behavior.

This equation implies that rocks with a high valueof friction angle are under lower stress than rocks withlow value of friction angle; i.e., in general, sandstonesare under lower stress than shales. The observationthat models based on elasticity and models based onfrictional behavior give the same trend of stress contrastalways occurs, although the fundamental assumptionsfor these models have nothing in common.

For purely viscous materials (salt), Ko is simplyequal to 1 and the state of stress is lithostatic (Talobre,1957, 1958):


vH

H gdH= ( )0

,

Ko=

1,

h v p=

+1

2 .

h v p ( ) + ( )1 sin sin ,

= h o vK ,

(3-55)(a lithostatic state of stress as such does not requirethe uniaxial strain condition, and therefore, it defines a stress regime by itself).

Over geologic time, rock experiences, in variouscombinations and degrees, diverse mechanical behav-iors and various events. Behaviors include elastic,frictional and viscous behaviors, and events includethe occurrence of tectonic strain, variation of porepressure and temperature, erosion and uplift. Asreviewed by Prats (1981), these mechanisms lead todeviations from these simple reference states, some of which are briefly reviewed here.

3-5.2. Tectonic strainsTectonic stresses and strains arise from tectonic platemovement. In this section, the notion of tectonic strainis introduced, which is a quantity added to or sub-tracted from the horizontal strain components. Ifincremental tectonic strains are applied to rock forma-tions, these strains add a stress component in an elas-tic rock as follows:

(3-56)

(3-57)

where dH and dh are the (tectonic) strains with dH >dh. The resulting stress increments are not equal, withdH > dh, where dH is the stress increment gener-ated in the dH direction and dh is the stress incrementgenerated in the dh direction. These relations areobtained by assuming no variation of the overburdenweight and provide a dependence of stress on Youngsmodulus E. This means that the greater the Youngsmodulus, the lower the horizontal stress if the strainsare extensive and the higher the horizontal stress if thestrains are compressive. To understand this mecha-nism, the different layers can be compared to a seriesof parallel springs, the stiffness of which is propor-tional to Youngs modulus as depicted in Fig. 3-19.This model is actually a good qualitative description of the state of stress measured in areas in which com-pressive tectonic stresses occur. The model canaccount for situations where sandstones are underhigher horizontal stress than adjacent shales (Plumb et al., 1991; see also Chapter 4). The overburden stress

is a principal stress but not necessarily the maximum.The state of stress described in this section cannot be considered to define a particular stress regime(although one could speak of compressional stressregime) as it does not define a reference state. Only ifthe strains are high enough for the rock to fail are ref-erence states obtained, as discussed in the next section.

3-5.3. Rock at failureIf the strains are high enough, the rock fails either in shear or in tension. Three stress regimes can bedefined if the rock fails in shear. These stress regimesare associated with the three classic fault regimes(Anderson, 1951): normal, thrust and strike-slip faultregimes (Fig. 3-20). Stresses can be estimated by theadapted shear failure model. The simplest shear fail-ure model that applies to rocks is the Mohr-Coulombfailure criterion. A stress model based on this criterionassumes that the maximum in-situ shear stress is gov-erned by the shear strength of the formation (Fenner,1938). Hubbert and Willis (1957) used this criterionand sandbox experiments in their classic paper onrock stresses and fracture orientation (see Sidebar3A). As presented in Eq. 3-42, the Mohr-Coulombfailure criterion can be written to give 1 at failure in terms of 3. In sandstones and shales, N is aboutequal to 3 and 2, respectively.

If failure is controlled by slip along preexisting sur-faces, the compressive strength c can be assumednegligible. However, a residual strength may stillexist. The angle of internal friction is usually mea-sured by using ultimate strength data as a function ofthe confining pressure obtained during triaxial testing.This angle can also be measured by using residual


h v

d E d E dh h H

+1 12 2

d E d E dH H h

+1 12 2

,

Figure 3-19. By analogy, the stiffer the spring, the moreload it will carry.

Formation A

Formation B

Formation C

Stiff plateStiff

fixed plate

Constant displacement

strength data as a function of the confining pressureobtained during triaxial testing once the sample hasfailed. Using the residual angle of friction rather thanthe angle of internal friction in a failure stress modelshould be more consistent with the assumption thatthe minimum stress is controlled by friction along pre-existing planes. Generally, the residual angle of fric-tion is smaller than or equal to the internal angle offriction.

If the formation is in extension (i.e., normal faultregime, Fig. 3-20), the vertical stress is the maximumprincipal stress. The minimum principal stress is inthe horizontal plane and is therefore h. Equation 3-42becomes

(3-58)


Figure 3-20. The three fault regimes (Anderson, 1951).

Normal fault regime

Thrust fault regime

Strike-slip fault regime

v = 1

v = 3

v = 2

H = 2 h = 3

h = 2

h = 3

H = 1

H = 1

h vp Np ( )1 ,

in which the effect of strength is neglected. An equa-tion similar to Eq. 3-51 can be retrieved. However, ifthe rock is at failure, the coefficient of proportionalitycannot be considered as a coefficient of earth stress atrest. The most surprising and confusing result is that,in practice, Eqs. 3-53 and 3-58 give similar predic-tions, especially if, in the elastic model, is assumedequal to 1. The coefficient of proportionality in sand-stones and shales is, whether elasticity or failure isassumed, about equal to 13 and 12, respectively. Thissimilarity has been demonstrated in more detail forone area of East Texas by Thiercelin and Plumb(1994b).

If the formation fails under compressive tectonicstrain, the maximum principal stress is in the horizon-tal plane and is therefore H. In the thrust faultregime, the minimum principal stress is the verticalstress (Fig. 3-20):

(3-59)In this case, h is the intermediate principal stress

and is equal to or greater than the vertical stress.Horizontal hydraulic fractures could be achieved.

Thus, the principal stresses can be estimated andordered by looking at the fault regime. In practice,these considerations must be checked with downholemeasurements, as the state of stress may deviate fromthe expected ordering of stresses because of stress his-tory. These models assume that the fault plane wascreated under the current tectonic setting; i.e., the nor-mal to the fault plane makes an angle (/4 + /2) withthe direction of the maximum principal stress. Pre-existing faults can be reactivated under a state of stressthat differs from the one that created them. A Mohr-Coulomb stability criterion can still be applied, but Eq. 3-42 must be modified to take into considerationthat the fault plane orientation was not induced by thecurrent state of stress.

Another stress regime is associated with tensile fail-ure. Tensile failure is sometimes observed downhole,although it appears to contradict the general compres-sional regime of the earth. This mode of failure sim-ply states that 3 p = 0 (by neglecting the tensilestrength of the rock) and may be suspected if it isobserved from downhole images that the normal tothe plane of the preexisting fractures is the directionof minimum stress. This condition can occur in exten-sional regions with overpressured zones (where thepore pressure tends to be the value of the minimum

stress component) or when the in-situ stress ratio istoo large. As the rock is close to a uniaxial state ofstress, this regime can occur only in rocks with a com-pressive strength high enough to avoid normal fault-ing (as a rule of thumb, the uniaxial compressivestress must be equal to or greater than the effectiveoverburden stress). This condition is achieved for tightgas sandstones in some areas of the Western UnitedStates and East Texas.

Failure models also have an important role in provid-ing bounds for the in-situ stress. They represent a limitstate above which the rock is unstable in the long term.In the extension regime in particular, it is unlikely that a minimum stress value below the value predicted bythe failure model can be obtained.

3-5.4. Influence of pore pressureIt is of interest to understand what happens whendepleting or injecting into a reservoir. Elastic modelswith uniaxial strain conditions can be applied withsome confidence, as the variation of stresses occursover a short period of geologic time, although it isalways necessary to double check the assumptionsbecause failure models could well be the real physicalmechanism, as shown in the following.

If the material behaves elastically, and assuminguniaxial strain conditions, Eq. 3-53 gives

(3-60)The range of 2 is approximately between 0.5 and

0.7. Geertsma (1985) demonstrated the applicabilityof this model to stress decrease during depletion.

A failure model can also be applied. For example,Eq. 3-58 gives

(3-61)

If the coefficient of friction is 30, the coefficient of proportionality is 0.67. As previously, a strong sim-ilarity exists between the predictions from the elasticand failure models. To use a failure model, however,requires checking that the effective state of stress sat-isfies the failure criterion prior to and during the varia-tion of pore pressure. The effective stresses increaseduring depletion, although the total minimum stressh decreases.

Field data generally support the predictions of thesemodels and show that variation in the minimum stress


H vp N p ( ) .

d dph = 2 .

dN

Ndph

=

1.

ranges from 46% to 80% of the change in pore pres-sure (Salz, 1977; Breckels and van Eekelen, 1982;Teufel and Rhett, 1991).

3-5.5. Influence of temperatureTemperature variation also changes the state of stress(Prats, 1981). Cooling happens during uplift or theinjection of a cool fluid. This induces an additionalstress component in the horizontal plane, which usingthe uniaxial strain assumption again is

(3-62)

where dT is the temperature variation and T is thelinear thermal expansion coefficient. In this case, aninfluence of Youngs modulus on the state of stress isalso obtained. Cooling the formation reduces the nor-mal stress; hence, cool-water injection could lead totensile fracturing of the formation in the long term.

3-5.6. Principal stress directionFigure 3-20 indicates the expected direction of theminimum stress as a function of the fault regime(Anderson, 1951). In practice, it is observed that atshallow depths the minimum principal stress is thevertical stress; i.e., a hydraulic fracture is most likelyto occur in a horizontal plane. The transition betweena vertical minimum principal stress and a horizontalminimum principal stress depends on the regional sit-uation. In an extension regime, however, the mini-mum stress direction can be expected to be always inthe horizontal plane, even at shallow depths. This isusually not observed, probably because of the exis-tence of residual stresses and because vertical stress is usually the minimum principal stress at shallowdepths. In normally pressured sedimentary basins, theminimum stress is most probably in the horizontalplane at depths greater than 3300 ft (Plumb, 1994b).

Stress rotation may also occur because of topology.However, at great depths, rotation is induced mainlyby fault movement. In some situations, overpressur-ization has been observed to generate a change in theordering of stress, with the value of the minimum hor-izontal stress higher than that of the vertical stress.Finally, changes in structural or stratigraphic positioncan locally affect the stress direction dictated by the

far-field stress and the stress value. An example is thestress field at the top of the Ekofisk formation, wherethe maximum principal horizontal stress is orientedperpen

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