geoch4app

C O N T E N T S

1. PURPOSE AND SCOPE

2. FUNDAMENTALS

3. STRESS

4. LABORATORY TESTING AND MATERIALDESCRIPTION

5. "ADVANCED" ROCK MECHANICS ANDROCK PHYSICS

6. EXAMPLE OF THE USE OF ROCKMECHANICS

7. A FEW PROBLEM AREAS

Appendix Rock Mechanics

12

LEARNING OBJECTIVES:

At the end of this appendix, students should be able to:

distinguish between external loads and internal reactions

discuss the yielding of a material in relation to its ability to resist loads

describe the characteristics of the major deformation mechanisms: fracture,dislocation-associated, solution-associated

explain what is meant by stress at a point, and especially to recognize that thisconcept applies to locations within a body, not on its surface

derive shear and normal tractions on an arbitrary plane, given a state of stress andorientation information

explain the Mohr-Coulomb yield criterion

explain the relationship between displacements and strains

describe laboratory evidence for precursive events leading up to macro-fracture

describe simple stress/strain relationships: elastic, viscous, plastic, visco-elastic,elastico-viscous

summarize the evidence, arising from laboratory work and other studies, concerningthe role of the major deformation parameters: lithology, confining pressure,temperature, rate, effective stress

Department of Petroleum Engineering, Heriot-Watt University 3


1. PURPOSE AND SCOPE

In the discipline of Petroleum Geoscience, the topics of Structural Geology (Chapter4) and Geophysics (Chapter 5) both require an understanding of Rock Mechanics. Anappreciation of Rock Mechanics is also needed in order to understand the behaviourof stress-sensitive reservoirs. Rock Mechanics is not an isolated subject, and anappreciation of the geological context is very important to enable a student to gain apractical understanding of the importance and applications, and implications andlimitations, of Rock Mechanics. In order to obtain these benefits, this Appendixshould be consulted in parallel with Chapters 4 and 5. To further encourage thislearning approach, the Appendix is treated like the other Chapters in this module, andlists a set of Learning Objectives.

The first section of the Appendix considers the fundamental concepts of deformation,and relates rock deformation to the continuum mechanics context. The next sectiondescribes the important generalizations that can be derived from laboratory testing ofrock behaviour, and also discusses how this understanding is cast into mathematicaldescriptions of the mechanical behaviour of rocks. The following section illustratessome general applications of rock mechanics concepts. The next section is concernedwith some concepts that are often mis-understood, and whose terms are frequentlymis-used, thereby leading to the potential for confusion and errors in communication.A final section provides exercises and suggestions for further learning.

2. FUNDAMENTALS

2.1 General Deformation ConceptsRocks are subjected to various forces in the crust:

Gravitational loading produced by the weight of the overburden

Tectonism, nominally related to Plate Tectonics

Those forces associated with variations in pressure and temperature

Such forces work together or in competition to create a state of stress in the rock.Stress can be loosely described as a normalisation of all the forces acting at a point.In general, stress states vary from point to point. Associated with stress is a distortion.Strain is a concept that allows the normalisation of such changes in shape that the rockundergoes in response to the applied loads.

The loading-induced stress magnitude may be sustainable by the rock (the rockbehaviour - strain - in this mode is usually considered to be elastic, and recoverable,once the loading is removed; see below). Such a mode of deformation is of littleimportance in the geological record (it does not produce any permanent deformation),but this mode IS of importance in reservoir management, and it is the mode that isrelevant to the passage of seismic waves (Chapter 5). On the other hand, loading CANproduce a large-enough stress magnitude that the rock may yield. Structural Geology(Chapter 4) is concerned with the study of such deformations, in particular, their

14

shapes and relationships to other geological aspects.

Rock Mechanics seeks to explain the processes by which deformation occurs, andhence to provide a genetic explanation for observations about the end result ofdeformation processes (i.e., folds, faults, fractures, passage of sonic waves, etc). Agrowing area of interest is the prediction of deformations that may happen as a resultof human or natural acts. The operation of a petroleum reservoir is one such situationthat is clearly of direct relevance.

2.2 What is deformation ?The notion of deformation concerns changes in geometry, and the causes of thesechanges. Prior to a deformation event, rocks exist in some form (used in the geometricsense), and following the deformation, they exist in another form. This change of formcan be segmented into three components (Fig. 1):

Initial Object(tree is reference)

Translation Rotation Distortion

Translation is a rigid-body movement (the plateau was uplifted).

Rotation is a rigid-body movement (the block was tilted).

Distortion is the change in shape of a non-rigid body (the fossil was squashed). Thisis the aspect of deformation that is usually referred to as strain (but the quantitativeformulation of strain also has a component of rigid-body rotation; see below).

These components can each be expressed as displacements. However, the concept ofdisplacement implies a reference frame. In Geology, we usually see the end productof deformation, so you might expect that we would typically choose to use a coordinatesystem linked to the end state. In fact, it is more usual to use a coordinate system linkedto the initial state (which is frequently not known with any certainty!). The reason forthis choice is because we think of deformation as a process, and a process movesforward, so the teaching of the subject is easier if we start at the beginning of aprocess and describe how things change with time. The fact that analysis of real

Figure 1Deformation = Translation+ Rotation + Distortion



deformation is made slightly more awkward by this choice is, however, an unfortunateconsequence. Some methods (not discussed herein) use a coordinate system thatmoves with the deforming body. Because of these alternative approaches (differentconventions), there is a non-negligible potential for confusion.

2.3 Deformation mechanismsWhen rocks distort, the fundamental processes that are operative occur at the atomicscale. There are three main classes of deformation mechanism (Fig. 2):

starting granular aggregate

cataclasis (grain cracking,grain-boundary sliding)

pressure solution (contactsclosed, grain shapesaltered); in this example,export of mass

dislocations (grain shapechanges by solid-stateprocesses)

after application of vertical load

These classes have contrasting characteristics:

Fracture/cataclasis (mass is conserved, but volume may change; breaking ofatomic bonds and sliding of grains/crystals; dependent on mean stress)

Pressure solution (mass may or may not be lost; effect related to normal tractionacross grain boundaries; reprecipitation of dissolved material in dilatant zones, ortransported away by moving fluids)

Dislocations (mass is conserved; various distortions and transformations of thecrystal lattice; temperature dependent; recovery and recrystallization processesoperate)

For different rock materials (types of rock), it is possible to determine the conditionsunder which one of these mechanisms dominates the resulting deformation (i.e. whichmechanism can produce the most distortion in a given time). This knowledge allowsspecialists to provide estimates of the crustal location under which a given deformationhas occurred (this is possible because pressures and temperatures vary with locationin the crust). Such knowledge also makes it possible to (in some cases) develop

Figure 2Illustration of the threeclasses of deformationmechanism

16

suitable mathematical models of deformation processes. We will not attempt here toattain this degree of competence. A readable article on these topics can be found inPhysics Today, Sept 1996, p. 24-29.

2.4 Strength and YieldingIt is an empirical observation that rocks become deformed. This means that they ceasebeing un-deformed. Presumably, this change happens as a consequence of appliedloads (external and internal). As loading increases, the rocks develop reactions tothese loads (see below). At some point, the rocks become unable to (continue to) resistthe load increases. In other words, there is a limit to the amount of reaction that theycan generate. The change from resisting to non-resisting (or to a lesser resistance) iscalled yielding. The notion of strength (see below) is one way of expressing themagnitude of rocks ability to resist loads.

2.5 DamageA useful concept is to treat rock deformation as damage. Damage consists ofpermanent (i.e. not elastic, or non-recoverable) distortion of rocks, and includes anyof the end-products of the three classes of deformation mechanism noted above. Suchdamage can be distributed throughout a rock mass. In contrast, deformation can beconcentrated, or localised. The transition from damage to localisation, or indeed,from localised deformation to a general state of damage, reveals a great deal about theevolution of physical conditions (e.g. pressure, temperature, etc), and about the initialand subsequent properties of the rocks.

2.6 StrainIn Structural Geology, we observe resultant geometries (i.e. the end-products ofchanges produced by loading and the consequent yielding). We often are able to inferthat the rocks involved in a deformation had some particular previous geometry(different from what we now observe). For example, we know that rocks are (usually)deposited as regular layers, so if we observe rocks whose layering is not flat, we candecide that deformation has taken place. In such a case, we are able to deduce that therehas been a CHANGE in geometry, and we can often quantify those changes. Strainanalysis is a quantitative way of expressing the distortions that are produced bydeformation.

Strain in 2-DA line is defined by its end-points, so any two points in a body can be joined torepresent a notional line. If these points experience relative movement during adeformation, the distance between them can be altered. A change in length is calleda longitudinal strain, so a longitudinal strain can be determined from the change inlength of a line drawn between any two such points.

We usually refer to a line by giving it a name composed of the symbols applied to itsend-points. If there are two points labeled A and B, there is a line defined by them,and the line is called AB (Fig. 3). By convention, we refer to points A and B AFTERdeformation by adding a prime: e.g. A, B. Thus there are two lines: AB and AB,with the first being the line before deformation, and the second the line afterdeformation.



AB

A'

B'

Before After

We use the symbol e to represent a longitudinal strain. Such a strain is defined as:

elength A B length AB

length AB=

-( ) ( )

( )

The value of e can range from -1 to +infinity. If the original length of the line was oneunit, then e is actually the change in length of the line.

Distortion may also alter the angular relationship between lines (Fig. 4). For each line(e.g. line OP), it is possible to draw a perpendicular line (e.g. line OQ). The angularshear strain, , in the direction of the line OP', is the angle (in radians) by which theperpendicular line OQ' has been altered away from the perpendicular:

90

0 P

Q

Before Deformation

0

P'

Q'

After Deformation

p'

p'

1 unit

By convention, clockwise rotations are taken as negative angles, and anticlockwiserotations are taken as positive.

The shear strain, , is defined as:

g y= tan( )

If the length of the perpendicular line is one unit, then = .

The state of strain is a concept that encompasses all of the distortions of the body: thechanges in line lengths of all lines, and all angular changes. It is perhaps easiest tounderstand this notion if we re-cast the above discussion in terms of spatial coordinates,x and y. If we retain the convention of using a prime to indicate a value following

Figure 3Convention for naminglines in a before andafter state

Figure 4Definition of angular shearstrain

Equation 1Definition of longitudinalstrain

Equation 2Definition of shear strain

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deformation, all points (x,y) in a body map onto a new set of points (x,y), and therelationship between the before and after states can be expressed mathematically,through a series of expressions known as coordinate transformation equations:

x f x yy f x y

( , ) ( , )=

=

1

2

Because strain is basically the distortion element of deformation, whole-body trans-lations and whole-body rotations are eliminated from these coordinate transforma-tions. In effect, this means that the coordinate system is attached to the original x,ypoint in the body, but that the origin moves and rotates with the whole body. Thefunctions f1 and f2 do not have any constant terms, and can be expressed in a generalfashion as:

x ax by

y cx dy

= +

= +

When the equations are expressed in matrix form, the two by two matrix is called thestrain matrix.

x

yx y

a b

c d

=[ ]

A useful visual display of the state of strain is the strain ellipse (Fig. 5). The strainellipse is simply the distorted shape produced by subjecting an original unit circle(one whose radius = 1) to the same deformation as indicated by Equation 5.

y y

x x1 unitb a

The major and minor semi-axes of the ellipse, a and b, respectively, are defined interms of the principal longtitudinal strains, e1 and e2, so that:

a e

b e

= +

= +

1

11

2

The term principal strain refers to a particular longitudinal strain in a direction suchthat, on the plane normal to that direction, there are no shear strains. In a general 2-Dstate of strain, there are two principal directions that are perpendicular to each other.e2 is usually negative (shortening), so b is usually less than 1. If the distortion is onethat conserves area, then the area of the distorted ellipse is equal to the area of theoriginal unit circle.

Figure 5The strain ellipse

Equation 6Major and minor semi-axesare related to pricipallongitudinal strains

Equation 3Coordinate transformationsare functions of spacevariables

Equation 4General form of coordinatetransformations for 2Dstrain

Equation 5The 2D strain matrix is aco-ordinate transformation



For a (realistic) state of strain, it is common to have to describe a set of sequentialcoordinate transformations that operate on the original body. In general, thesetransformations are history-dependent, and the strain path is very important (Fig. 6).

1 unit

1 + e11 + e2

0 X

Y

1 + e1

1 + e2

Starting Shape

SameShape,Different Path

Homogeneous Simple Shear 1

Homogeneous Pure Shear 2

Rigid BodyRotation (through angle ) 3

Elongations, Deformations 1 and 2(No strain in deformation 3)

(+)

(-)

0e

OY1

OX1

OX2

OY2Process

Strain in 3-DAlthough it is convenient to introduce concepts by assuming a two-dimensionalsimplification (as above), the real world is fully three-dimensional. We need to brieflylook at how strains are analysed in three dimensions. The strain ellipsoid (Fig. 7) is alogical extension of the 2-D strain ellipse.

Figure 6Two paths to yield the samefinal strain state, showingdifferent histories of strainstates

110

1 unit

Z

X

Y

Original UnitSphere and Strain Ellipsoid

By convention, the axes of the strain ellipsoid (which define the principal straindirections) are designated as X ( = 1 + e1), Y ( = 1 + e2), and Z ( = 1 + e3). X, Y andZ are NOT the same as the coordinate directions x, y, and z (this represents is anunfortunate choice of symbols!). The 3 x 3 matrix defining the coordinate transfor-mations is called the strain matrix:

x

y

z

a b c

d e f

g h i

x y z

= [ ]

One way of classifying different states of strain is to determine the principal strain ratios:

Re

eR

e

eR

e

e

R R R

XY XZ YZ

XZ XY YZ

=

+

+=

+

+=

+

+

=

1

1

1

1

1

11

2

1

3

2

3

, ,

These ratios can be plotted in a Flinn Diagram (Fig. 8) to identify flattening (one shortand two long axes) and constrictional (one long and two short axes) strain fields.

Figure 7The strain ellipsoid

Equation 7The 3D strain matrix

Equation 8Definitions of strain radius



Apparent Constriction1 k

Apparent Flattening

k = 1 k = RXY-1RYZ-1

RXY

RYZ1

1

1 < k < 0

If the volume of the strain ellipsoid equals that of the original unit sphere, there is novolumetric strain. Natural strains can, hovever, result in volume increases (dilatant)or volume decreases (compactant).

2.7 Pure shear v Simple shearA pure shear deformation is one in which the principal axes of strain do notexperience any rotation during the distortion (refer to Fig. 6). A simple sheardeformation is one in which displacements occur only parallel to the direction of theshear zone (in this case, the principal axes do rotate). Some Geoscientists seem tothink that it is useful to distinguish between these two types of strain (Fig. 9). Sucha distinction can only be made for the case of homogeneous strain (strain that does notvary with position), however. Many real deformations of interest affect large volumesof rock, and the resulting strains are heterogeneous (spatially variable). In suchsituations, there is little value in attempting to decide which word to apply. Neverthe-less, the concepts are a useful adjunct in the analysis of some common deformationsituations.

Figure 8Constrictional andflattening strains defined bystrain ratios

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1 + e11 + e2

1 + e11 + e2

Pure ShearPricipal axes of strain

do not rotate

Simple ShearPricipal axes of strainprogressively rotate

Examples of pure shear deformationsFlattening strains (which produce s-fabrics) can be associated with: boudinage,pressure solution, cleavage, schistosity. Fault/fracture networks can also produce thisstate of strain. During the compaction of basinal sediments (often a uni-axial strain,with large shortening in the vertical direction, and no strain in the lateral directions),large flattening strains are imposed onto the rocks.

Constrictional strains can also be produced by faulting, but this state of strain is moretypically seen in high-grade metamorphic rocks (re-crystallization is important here)where it produces l-fabrics.

There are, of course, many intermediate situations that produce elements of bothflattening and constriction, resulting in l-s fabrics.

Simple shear and Shear zonesA shear zone is (generally speaking) conceptually divided into: the wall rocks (i.e.,more-or-less intact rocks), and a finite-thickness of material within which thedeformation is localised (Fig. 10). According to the definition given in Chapter 4,shear zones are, therefore, faults. If the wall rocks of the fault move parallel to theboundaries of the shear zone, then the definition of simple shear applies inside the faultzone. Because this condition (motion parallel to the walls) is either true, orapproximately true, in a large number of examples, it is often taken that shear zonesand simple shear are synonymous, and that deformations observed in shear zones canbe directly explained through strain analysis of the simple shear model. Note that shearzones can be parrallel with the layers.

Figure 9Pure shear and simpleshear strain states



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7

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One set of ideas concerning deformations in shear zones derives from studies of faultpatterns that are observed on maps of active strike-slip fault zones. Another relatedset of concepts derives from the experimental deformation of rocks under confiningpressure in the laboratory. Both of these approaches identify a suite of commonly-observed fabrics that occur within shear zones (Fig. 11). Depending on where thedeformation occurs (i.e. at what pressures and temperatures), and on whether thestrain is dilatant or compactant, different specific expressions of deformation areobtained.

R

P

R

Y

Another style of shear zone is represented by the bookshelf model (Fig. 12). In thisconcept, an array of faults occurs above a basal detachment (a bigger fault). The early-formed faults experience slip, and they, and the rocks around them, rotate. Substantiallateral extension can be produced by this kinematic scenario. If extension becomesvery large, the rotated faults become unsuitable for slip, and new faults are formed.The process can continue, leading to very complicated zones - perhaps with faults thatstarted as normal faults, but that have been rotated so far that they are now in theconfiguration of reverse faults.

Figure 11Fracture fabrics typical ofshear zones

Figure 10A shear zone

114

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Block rotation

New sediments Old faults

New faultsNew sediments

New sediments

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Rotation ofnew fault blocks

Passive rotationof earlier faults

Complicated strain fields / FlowRecall that strain is the variation of displacement. If all points of a body move thesame, this is simply a translation, and the strain (variation of movement) is nil. Now,imagine something (say, some rock mass) that is flowing (in a non-turbulent fashion).Flow often has a velocity gradient (it might be useful to recall the velocity profile forflow in a pipe), so the displacements, and hence the strain, may well be spatially-variable (heterogeneous), as well as evolving with time. If some marker plane existsin the material (say, a rock layer), it can track the flow. If the non-uniform flow iscrossing the plane, then the plane will become distorted - or folded, as that term wasdefined in Chapter 4. Such flow-folds are more typical of metamorphic conditionsthan those found in petroleum basins. However, there are situations where flow-foldsmay need to be considered: in and associated with diapirs, and in a variety of soft-sediment types of deformation (e.g. deformations that occur before sediments arelithified into rocks).

3. STRESS3.1 ReactionsAlthough it seems obvious: an important idea is to distinguish between the insideof a body (internal), and things that are outside of it (external). This considerationleads to the important notion of the boundary of a body. The reason for emphasisingthis point is that we are all able to easily relate to loads (displacements, forces) thatoccur outside of bodies and that may act on their boundaries, but we have less directexperience with how such loads affect the interiors of bodies.

Figure 12Shear produced by rotationsof fault-bounded blocks(bookshelf model)



For example: consider some body of arbitrary shape. A potato serves as a goodexample (Fig. 13). Now, either in your imagination, or, in actuality, push on the potatowith your hands (or, perhaps, several people might push on the potato).

Each hand is exerting a force on the boundary of the potato (Fig. 14). If all of the handscooperate, the potato can remain stationary (static: not moving), although underload.

We know from the principles of mechanics that a static potato MUST be in forcebalance and moment balance (that is the definition of static). Somehow, the wholesystem of hands/potato/hands must be in balance. What is balancing the externalloads? The potato has reacted to the imposed loads, such that there are forces INSIDEthe potato that exactly balance the external loads. The forces generated within thepotato as a result of the external loads are called reactions. They disappear when theexternal loads are removed. Internal loads (for example, gravitational attraction) alsoproduce reactions.

3.2 TractionsA vector has both magnitude and direction. In mechanics, we use vectors to representforces. In the preceding potato example, the forces applied to the boundary of thepotato (by peoples hands) can be represented (e.g. Fig. 14) by a set of force vectorspointing towards the potato (if there are many hands, the picture would look a bit like

Figure 14A static, free body subjectedto boundary loads

Figure 13An object (potato) subjectedto a set of loads

116

a hedgehog). Equally, the reaction forces inside the potato could be graphicallydepicted by a set of vectors. Although this last statement is correct, it is not practicalto achieve such an image without some further analysis. This is because thearrangement of reaction forces within the potato is likely to be complex.

One of the ways to deal with this potential for variability in internal forces is toconsider only very small regions. If the region is small enough, there wont be anyvariation in either magnitude, or direction, of the force vectors across that region. Toillustrate this idea, let us continue with the imaginary (or real) potato. First, slice thepotato in half (Fig. 15). The principle here is that, if we can arrange the forces alongthe planar, cut surface to be EXACTLY the same as the forces that acted on thisimaginary (internal) plane before the cut was made, then the remaining half of thepotato will have EXACTLY the same internal arrangement of forces as before. If thisis the case, the remaining part of the potato will not "know" that there was a cut.

This process of cutting the potato can continue (Figs. 16, 17), until all that remains isa small cube (Fig. 18).

Figure 17A third cut through theobject

Figure 15Equivalent surface forcesacting on imaginary planeinside free body

Figure 16A second cut through theobject



If the surface (boundary) forces on each face of the cube are exactly the same as theforces that were on these planes before the potato was cut (or, as in classicalapproaches, if we only imagine that we are cutting the object), then the region insidethe cube will not know that it is an isolated body. If the cube is small enough to meetthe criterion of no variation in force across any face, then we now can term this ahomogeneous state.

However, because the cube is very small, the force magnitude on each face is minute.The forces may also differ depending on the exact size of the cube. These difficultiesare overcome by a process of normalization. In this approach, the forces are dividedby the area over which they operate, and the resulting quantity is a force per unitarea. This entity is called a traction. Tractions have magnitude and direction,because they are vectors.

Because they are vectors, tractions can be resolved into components that are perpen-dicular to the surface in question, and components that are parallel to it. These arecalled normal tractions and shear tractions, respectively. In a 3-D situation, theremay be two shear tractions (by convention, we consider two directions that areperpendicular to each other) lying in the plane of the surface. In a general state (Fig.19), each of the six faces of the cube has three tractions operating on it (a hedgehog,indeed!).

z

yx

zz

yy

zy

yz

zx

yx xy

xz

xx

The usual convention in Geoscience is to consider compressive tractions as positive,and tensions as negative. This is opposite to the convention adopted in much ofEngineering. The rationale for making this choice is because most of Geology isconcerned with the subsurface, where the forces are almost exclusively compressional.

Figure 19Resolved shear and normaltractions acting on a cube

Figure 18After three further cuts toleave a cube withequivalent surface forces

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3.3 State of stressAlthough the preceding thought experiment helps in comprehending how the spatialvariations of forces inside a body can be addressed (e.g. by imagining cubes with theirsurface tractions at a whole range of internal points), this is not a practical way toproceed. Moreover, a problem arises because the same force, acting at the same place,but on two planes that are oriented differently (say, one horizontal, and one vertical),will be expressed as different tractions on those separate faces. What is needed is away to express the relationship between the tractions that are resolved onto differentsurfaces.

The necessary concept is that of the state of stress. The stress state is a mathematicalconstruct (a black box) that calculates all tractions. In strict terms, the mathematicsare accomplished via a second-order tensor:

t

t

t

n n n

T T TT T TT T T

1

2

3

1 2 3

11 12 13

21 22 23

31 32 33

=[ ]

where: t is the traction vector, n is the unit normaal vector, andT is the stress matrix.

This tensor T relates one vector to another. The normal vector, n (components n1,n2, n3, or nx, ny, nz), defines the surface (Fig. 20) for which we wish to know the tractionvector, t (components t1, t2, t3). We can see that the components of T are the tractionsacting on the coordinate planes (e.g. use normal vectors (1,0,0), etc). For this reason,the tensor T is often called the stress tensor. The state of stress at a point is the notionembodied in this tensor. Although there is a single state of stress at any point, thetractions acting on different surfaces will (in a general case) be different.

T21

T22

T11

T12

T13

T23

T32

T31

T33

X3

X2

X1

n

Spatial variability of the state of stress is indicated by variations in the stress tensoras a function of location (e.g. x, y, z locations). Such variations in reaction might beproduced by the arrangement of boundary loads (perhaps all the hands are squeezingonly one end of the potato), or by variations in material properties (one part of auniformly-loaded potato might be rotten, and thus be unable to develop large reactionforces).

Equation 9The "Stress Matrix" as aco-ordinate transformation

Figure 20Tractions on an arbitraryplane



Unfortunately, many textbooks, and many Geologists, are not especially careful intheir use of terms. You may find books that define stress as a force per unit area. Thisis wrong, since this is the definition of a traction. Stress is defined as the relationshipof tractions at a point, so it is a higher-order concept.

If we adopt a sensible coordinate system (usually, a cartesian system with x, y, z), thestate of stress at any point can be resolved onto the planes that are aligned in thecoordinate directions. The tractions acting on the planes are often called the stresscomponents. The normal stress components (i.e. tractions operating perpendicular tothe surface) are usually designated with the Greek symbol sigma (), and the shearstress components (i.e. tractions operating parallel to the surface) are usually depictedby the Greek symbol tau (). Subscripts differentiate the coordinate directions: thenormal stress component (i.e. the traction) acting in the x-direction (i.e. on a surfacethat is perpendicular to the x-axis) is called

x (sometimes

xx), and the shear stress

component acting on that same surface, and oriented in the y-direction, is called xy

(sometimes xy).

It is extremely unfortunate that a degree of sloppiness has crept in to this topic. It israre to hear anyone use the term traction; instead, people usually use the wordstress for these quantities. If it was common that people said stress component,this would not be too bad, since this usage implies a traction resolved onto a plane. But,you will often hear stress components referred to without the component: i.e.someone will say the normal stress instead of the normal stress component. Thisprocess has continued to the point where some people will just say stress. Ifeveryone was an expert, this would be less of a problem, since the context often makesit clear what is really meant. But there is a danger that novices may get confused, andthink that stress components, and the state of stress, are the same thing.

3.4 Stress-Strain Relationships - Simple RheologiesA practical definition of rheology is the relationship between stress and strain. Amore precise person might insist on strain being replaced by strain rate, butGeologists are familiar with the notion that strain accumulates, and so they arecomfortable with the idea that any strain is the product of a strain history (with somerate attached). So, a working (im-precise) definition of rheology is either: therelationship between stress and strain, or the relationship between stress and strainrate. The term "constitutive relationship" also is used to refer to stress-strain links.

It might come as a surprise to learn that Geologists have not developed an acceptablerheological model for rock deformation. This statement is NOT the same as sayingthat we dont USE rheological models. We adopt simple models, and we use them toexplore the implications of deformation processes. However, the simple models allhave limitations. Problems arise if we fail to remember how these limitations affectthe applicability of any predictions that are made.

There are a few simple rheological types that should be remembered:

Elastic (strain is linearly-proportional to stress; no time consideration, so response

120

is instantaneous; strain is fully recoverable if load is removed)

Viscous (strain rate is function of stress; in Newtonian viscosity, rate is linearly-proportional to stress; no recovery of strain)

Plastic (permanent deformation is produced once stress reaches a critical level)

Elastico-viscous, Visco-elastic, Visco-plastic (combinations that attach recoveryto flow, etc)

Each of these rheologies is expressed via mathematical relations that have simplegraphical expressions (Fig. 21). In practice, rheologies are used to formulatemathematical representations of rock deformation which allow simulations to bedeveloped through computer models.

Elastic

Viscous

Plastic

Elastico -Viscous

Visco-elastic

SimplestRealistic Behaviour ofRock material

Name Mechanical ModelStress - Strain

Behaviour Characteristics

Time independent, totally recoverable, occurs and recovers instantaneously

Time dependent, permanent strain

Time dependent, and totally recoverable with infinite time

Some permanent strain, some instantly recoverable, some recoverable in time

Time independent, permanent strain

Some of the stain recoverable, some permanent(dashed line recoverable elastic strain)

= c

= k

yield = const.

32

1 3

2

4. LABORATORY TESTING AND MATERIAL DESCRIPTIONS

4.1 Laboratory testing of rocksIn the first half of the Twentieth Century, technological advances made it possible toundertake laboratory testing of rock materials under realistic conditions. Pressurevessels allowed the samples to be subjected to high confining pressures like thoseappropriate to the deep subsurface. Axial pistons, driven by gear trains or hydraulically,produced differential loading. Heating coils permitted samples to be subjected totemperatures like those typical of the subsurface.

Around the world, several labs began investigating Rock Mechanics. In the UK, this

Figure 21Generalised (one-dimensional) relationshipsbetween stress and strain(rate) for the simplerheological models



work was undertaken at Newcastle in support of mining problems. In the petroleumarea, the seminal work on this topic was undertaken by Shell Research in Houston,Texas. The Rock Mechanics laboratory there was directed by John Handin, who latertaught at Texas A&M University. When Handin moved to A&M in the early 70s, hetook with him many of the staff members of the former Shell lab, although some wentto other universities. Meanwhile, additional Rock Mechanics laboratories wereestablished in many places around the world. These groups of Geoscientists, and theirsubsequent students, provided the data and ideas (through publications) upon whichour present knowledge is based.

The following gives a very brief description of the basic laboratory methods used todetermine the properties of rocks under realistic conditions.

A sample of rock is cored from a larger block of rock. Usually, this is done with awater-lubricated/cooled core bit mounted in a special drill press. The ends of the coreare sawn perpendicular to the core axis (with a water-cooled/lubricated rock saw) togive a length approximately two times the diameter of the core, and the ends are thensmoothed via a grinding machine. The core is then jacketed with an imperviousmaterial (often, a type of heat-shrink plastic) to allow a radial confining pressure tobe applied via a fluid system (the jacket keeps the fluids out of the rock). Otherarrangements can include the jacketing within other components of the apparatus.

The jacketed rock is placed into a pressure vessel (Fig. 22). This is a large-ish steelcylinder that is designed to withstand very high internal pressures. Through the endsof the pressure vessel, pistons pass through seals and attach to the ends of the core.These pistons can be pushed together, providing an axial load that affects the rock.

O-Ring Seals

Polyolefin Jacket

Pressure Line

Confining FluidPressure Vessel

Packing Gland

PackingGland

UpperPiston

Specimen

LowerPiston

CompensatingPiston

SupportingRing

The fluid pressure serves to impose a confining pressure onto the rock. Pressure hasthe same units as a traction (mass length-2 time-2), so the radial pressure of the confiningfluid is believed to induce an identical radial stress component in the rock. The forcenecessary to move the pistons can be measured, and this force, divided by the area ofthe end of the core, is an axial traction (it is assumed to be uniform across the end ofthe specimen). Thus, we know the radial and axial tractions that affect the rocksample.

Figure 22Rock sample in a pressurevessel

122

Displacements can also be measured. The motion of the pistons is usually determinedoutside of the pressure vessel. Since the pistons are very stiff, it is assumed that thedisplacement of the ends of the core is the same as the motion of the pistons. Becausethe original length of the rock core is known, the axial strain can be determined.Motions inside the pressure vessel, such as distortions of the rock, can be determinedvia devices called strain gauges (also spelled as gages). These consist of small stripsof metal that become stretched or contracted, and whose resulting changed electricalresistance is used to calculate surface strain components. These components can thenbe used to establish the full state of strain.

During an experiment, the forces and displacements are recorded (usually whilekeeping a constant radial pressure). As noted above, these data can be transformedinto axial tractions and axial strains. Such data are used to produce a stress-strainplot (Fig. 23; as noted earlier in this Appendix, this usage of these terms is not correct).The shapes of the plots are typically interpreted in terms of the simple rheology modelsnoted above (e.g. linear portions are equated with elastic behaviour).

Peak Load

RuptureYield(Start of Inelastic Behaviour)

Axial Shortening

Axia

l Tra

ctio

n - R

adia

l tra

ctio

n("S

tress

Diffe

renc

e")

4.2 Failure (Fracturing/Faulting)Here, we are going to focus on fracture as an example of yielding. The reasons formaking this particular choice relate to the need to use the example as a basis fordiscussing some fundamental aspects of rock mechanics (fracture is a good way to dothis), and to the fact that fractured rocks have a profound role in applied RockMechanics.

A fracture is a material discontinuity. By this it is meant that the rock on either sideof a fracture is no longer connected (it was connected before the fracture happened).Fractures are usually planar features, but, in detail, their surfaces may be irregular.Joints in rocks are fractures. Faults in rocks might have been fractures (but there areother ways of developing faults), but, if so, once there has been significant motion, theterm fracture cannot be applied to these features.

Figure 23Stress-strain plot fromtypical rock experiment



We cannot, for obvious reasons, typically observe the formation of fractures in theirsubsurface environment. To partly overcome this limitation, Rock Mechanicists havelearned to produce fractures in rock, in the laboratory. The laboratory experimentsconsistently create conjugate sets of fractures (e.g. assemblages). These sets aresymmetric about the load axis, and the acute angle between the fractures is 50-60o.This angle is bisected by the load axis (Fig. 24).

Load Axis

Fracture Sets With Sense of Shear

Acute Angle

The notion is that, if we can explain the mechanics of fracture creation in thelaboratory, we can (perhaps) infer the same mechanics to be operative in nature. Inorder to do this, we need to develop further the mathematical representation of stressstates.

4.3 Tractions on an (Arbitrary) Internal PlaneAlthough the real world is decidedly three-dimensional, we will simplify ourmechanics to a two-dimensional approximation. This approach is fully justified inmany cases, and this simplification makes it much easier to draw images and to writeequations.

If we know the tractions that are acting on two perpendicular planes (here, let usassume that these are the planes normal to the x and y directions (Fig. 25), so the knowntractions are

x and y), we can calculate the tractions (normal and shear) acting on

any other (arbitrary) plane via simple formulae:

s s q t q q s q

t s s q

n x xy y

y x

= + +

= -

cos ( ) sin( ) cos( ) sin ( )

( )sin( ) cos

2 22

(( ) (cos ( ) sin ( ))q t q q+ -xy2 2

Figure 24Conjugate fracturesproduced in the laboratory

Equation 10Normal stress componentand shear stress componentas a function of tractions oncoordinate planes and theorientation of the arbitraryplane

124

x

nx

y

y

y

x

yx

yx

xy

xy

The normal and shear tractions vary in a smooth, systematic fashion as a function ofthe orientation of the plane (Fig. 26).

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

s

t

60 120 1800

4.4 Tractions (Normal and Shear) on Fracture PlanesWhen experiments are conducted in the laboratory, we know the external loads thatare imposed on the rocks to make them fracture. The axis of the cylinder is subjectedto the greatest load (usually), and the circumference of the cylinder is loaded by theconfining pressure. We can measure the angle(s) (relative to the axis of the cylinder)of the fractures that are created (with a protractor).

If we use the formulae given in Equation 10, we can calculate the normal stress(component) and the shear stress (component) acting on each fracture plane at thepoint of fracture (we know

axial and radial at the time when the fractures occurred, andthe angle of the fracture). If we take a series of experiments, each of which has adifferent pressure for the confining fluid, we note that each experiment also has adifferent axial load to cause fracture.

Now we can plot the - pairs representing the tractions needed to cause each fracture(Fig. 27). Such a graph usually produces a smooth curve, often approximating astraight line.

Figure 25Geometry for Equation 10

Figure 26Variation of normal andshear tractions as afunction of the angle of theplane



s

t tractions causing failure

The yielding (failure) of the rock is defined by the conditions of the curve that joinsthese points (Fig. 28). When these particular conditions (of and ) are obtained (asa consequence of reactions developed due to loading), the rock fractures. Such a curveis called a criterion (one of: a yield criterion, a fracture criterion, a failure criterion).

s

t

failure criterion

4.5 Principal stressAbove, we noted that the normal and shear tractions vary in magnitude depending onthe orientation of the plane on which they are acting (refer to Equation 10, Fig. 26).The normal tractions attain a maximum and a minimum on planes that are 90 apart.Note also that these maximum and minimum values of normal tractions occur onplanes where the shear tractions are zero. The shear tractions also attain maximum andminimum values. This occurs on planes that are oriented at 45 to the ones where thenormal tractions reach their maximum or minimum.

There is a special name given to the normal tractions that attain maximum andminimum values: they are called the principal stresses. (Note that the spelling is withthe -al ending, instead of the -le ending). Principal stresses act on planes of noshear stress (no shear traction). (Here is an example of the difficulty of theterminology. The definitions refer to stresses when they should say stresscomponents or tractions. In the case of principal stresses, they are NEVERreferenced in this (proper) way!)

We stated previously that the state of stress at a point can be fully described by givingthe normal and shear tractions that act on two mutually-perpendicular planes. It canbe shown that the two shear tractions acting on perpendicular planes are equal in value,but opposite in sign (this argument is based on static equilibrium, and gets tangled upwith sign conventions). Thus, there are only three independent values needed to fullydefine a (2-D) state of stress at a point: (

x, y, xy).

Figure 27Plot of shear and normaltractions necessary to causefracture of a given rock

Figure 28Fracture criterion

126

Because principal stresses are related to other tractions by means of an angle, it isequally possible to define the state of stress by the two principal stresses and theirorientation relative to the coordinate system (again, three values only are necessary).The different ways of defining the same thing (the state of stress) are simplyexpressions of a more general concept: each is merely a transformation of the other.

A special convention is used to refer to principal stresses. (Actually, there arevariations in use, and this makes it potentially confusing. We will stick to a singleconvention.) Because they are normal tractions, the symbol is still used, but insteadof the subscripts referring to coordinate directions, the subscripts are used to referencethe maximum (1) and minimum (2) values (compression is positive, so maximumis the most compressive). But do remember that the real world is three-dimensional!Because of this, some Geologists prefer to call the minimum principal stress 3, eventhough there are only two pricipal stresses in a 2-D analysis.

The formulae given in Equation 10 for normal and shear tractions can be re-writtenin terms of principal stresses. A clever substitution using a trigonometric identity(involving a double angle) leads to equations that have been memorized by countlessstudents:

n =+( )

+( )

( )

=

+( ) ( )

1 2 1 2

1 2

2 22

22

cos

sin

4.6 Mohr-CoulombEquations 11 describe a circle in - space. This circle, which fully describes the 2-D state of stress, is called Mohrs circle. If is the angle between the 1 direction andthe normal to the plane upon which one desires to know the tractions (refer to Fig. 25),these values are found by drawing a radius line with the double angle 2 (Fig. 29). Thepoint where the radius intersects the circle gives the coordinate pair (,) defining thetractions operating on that plane. A large number of additional techniques exist thatuse Mohrs circles (including one called the pole method), but we will not addressthem here.

s

t

s s3 1

2q

tractions "calculated"by Mohr circle method

Let us now return to our example, where we calculated tractions on the fracture planesobserved in laboratory tests. We can now plot the related Mohrs circles for each

Figure 29Simple Mohr circleoperations to determinetractions on a plane

Equation 11Normal and shear tractionsas functions of the principalstresses and the angle of theplane



experiment. We see that the fracture criterion line is tangent to the set of Mohrscircles (Fig. 30). This means that we did not need to know either the orientation of thefractures, or to calculate the tractions on the fracture planes. All we needed to do wasto plot a series of Mohrs circles (defined by 1 and 3, e.g. the axial and radialpressures) and draw a line tangent to them.

s

t

failure criterion

stress statesrepresented by circles

Now, we are in a position to make a prediction about fracturing. We can say thatfracture will occur if a Mohrs circle (representing a state of stress) is tangent to thefailure envelope (Fig. 31). The point of tangency gives the tractions that are actingon the fracture plane. A line drawn from the point of tangency to the centre of the circledefines the angle 2, and therefore, the orientation of the fracture relative to theprincipal stress direction.

s

t

s s3 1

2q

fracture angle (q) calculatedby Mohr circle method

How does a Mohrs circle describing the state of stress come to be tangent to the failureenvelope? In the laboratory, we can increase the axial load while holding theconfining pressure constant. In this situation, the differential stress ( = 1 - 3) isincreased. The Mohrs circle becomes larger (its diameter grows), but its leftintersection remains fixed (Fig. 32). This process can continue until the circlebecomes tangent to the failure envelope, at which point failure occurs. Thus, a Mohrscircle represents a state of stress at some time - it is not a fixed thing at all, but merelya graphical display that allows us to better visualize the mechanical state.

Figure 30The line tangent to a set ofMohrs circles is thefracture criterion line

Figure 31Determining the angle ofthe fracture plane

128

s

t failure when enlarging circlebecomes tangent

s3 s1 s1 s1 s1

The approximately-linear shape of the failure criterion line (Fig. 33) has beenrecognised for well over a century. A number of eminent individuals have beenassociated with analyses of these empirical relationships. The names Coulomb, Mohr,and Navier are applied to the criterion in various combinations (e.g. Mohr-Coulomb,Navier-Coulomb, etc). (It should be noted that Mohr was particularly concerned withthe non-linearity of the criterion, which we are not emphasizing here.)

s

t failure criterion

t0

f

t t m s t f s

f

m

= + = +

=

=

0 0 tan

"angle of internal friction"

"coeffiicient of internal friction"

4.7 Shear / Extension FracturesIn laboratory experiments, we often observe two conjugate sets of fractures (refer toFig. 24). These two sets are often symmetrical about the load axis (same angle, ) andwe can calculate that they had equal-in-magnitude, but opposite-in-sign (one clock-wise, one anti-clockwise), shear tractions operating on them at the time of fracture.The normal tractions on the two planes are identical compressions (figure 34).

3 1

2

2

same normal stress;equal, but opposite-signed, shear stresses

Figure 32Mohrs circles representinga changing state of stress

Figure 34Predicting two conjugatefractures

Figure 33The Mohr-Coulomb failure(fracture) criterion



The fractures comprising this assemblage are often referred to as shear fractures (eventhough the micro-mechanics of the fracture process is not fully known - see below).If we continue the experiment beyond the point where the fractures initiate, then thesefractures develop shear offsets. Although shearing is involved, the fractures do notoccur on the planes of maximum shear stress. Instead, they occur on planes where theratio of shear stress to normal stress is maximised (Fig. 35). The interpretation of thissituation is that the shear stress is trying to cause the fracture, and the normal stressis trying to prevent it. The most favourable location for failure is where the tendencyto fail (shear) is least resisted by the tendency to not fail.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 10 20 30 40 50 60 70 80 90

Angle, q (degrees)

Trac

tions

(MPa

)

Normal Stress Component

Shear Stress Component

Ratio: Shear StressNormal Stress

max

Sometimes in laboratory experiments, but more often in natural examples, we observethat the two sets of conjugate shear fractures are bisected by other fracture surfaces(Fig. 36). These extra fracture surfaces rarely exhibit shear displacements, and insteadseem to only have movements that produce the opening of a gap. In the next section,and later in this Appendix, we see that there is a theoretical basis for expecting anopening-mode fracture that is parallel to the maximum principal stress. For thisreason, it is common to include this bisecting fracture set within the notion of a fractureassemblage (e.g. Fig. 24).

Two sets of shear fracturesOne set of extension fractures

Equals the "normal" conjugate assemblage

Figure 35Ratio of shear stress tonormal stress on potentialfracture planes

Figure 36A complete fractureassemblage, consisting oftwo shear fractures and anextension fracture

130

4.8 Micro-mechanics of FracturingThe preceding discussion has been restricted to macroscopic issues. If we (briefly)turn our attention to the micro-mechanics of fracture, we find a considerable degreeof complexity. This complexity is relevant to a full appreciation of how fracturesimpact fluid flow, and particularly, why there is such a degree of variability ofpermeability near fracture surfaces. However, dont fear! This material is providedas background. You are not expected to fully grasp the issues noted in these nextparagraphs (and will not be examined on them). Indeed, many of these topics leadto heated arguments amongst so-called specialists.

The speciality of fracture mechanics identifies three modes of fracturing (Fig. 37):

Mode I: Extension (opening) motion perpendicular to the fracture surface

Mode II: Shear motion perpendicular to the fracture edge (motion is aligned alongthe propagation direction; this is the leading edge of the fracture)

Mode III: Shear motion parallel to fracture edge (motion is perpendicular topropagation direction; this is the side of the fracture)

Extension (Mode I)

Shear (Mode II)

Shear (Mode III)

How do these micro-mechanics concepts relate to macroscopic fracture? Based onlaboratory work, we know that microscopic extension fractures develop within grains/crystals as rocks are subjected to loading. The notion is that these microfracturesrepresent tensile failure of grains caused by point-to-point loading caused by grain/grain contacts. In other words, grains form chains that carry the loads, much likethe stones in an archway carry the loads. One idea is that the microfractures that arecreated by this loading link up to form macroscopic fractures.

An old idea that is related to this issue was proposed by Griffith. He suggested thatreal materials possess numerous microscopic flaws. Under loading, these developstress concentrations (similar to what Fracture Mechanics says), and these concentra-tions cause some of the flaws (those that have favourable orientations) to grow(become longer). Others have since taken the Griffith crack approach to developa macroscopic criterion that is functionally similar to the Mohr-Coulomb criteriondeveloped from empirical approaches. The only snag is that observations do not fullysupport such a development history for macrofractures created in the laboratory, orthose that exist in nature (petrographic examinations do not always find theseprecursor cracks associated with macrofractures).

Figure 37Three modes of fracturing



We can safely say that the jury is still out on the phenomenological basis for fracturein rock. The empirical approach, however, has proven robust in terms of predictionsand explanations. Clearly, there is room for further research to create a betterunderstanding of this important topic.

4.9 Flow of RocksA common conception is that rocks are very strong and essentially permanent.Mountains, and other rock outcroppings, seem to have always been there. Indeed,buildings and other works that have been constructed from stones are amongst themost long-lived of human creations.

Given this background, it is not surprising that non-Geologists commonly find itdifficult to believe that rocks can flow. Yet, the evidence for this behaviour isabundant. Salt glaciers creep across the land surface in the Middle East. Mud-rockdiapirs rise up and divert the mouth of the Mississippi River. Gigantic nappes areexposed in the sides of Alpine Valleys (Fig. 38). The solid mantle flows and drivesthe plates around the Earths surface.

yyyyyzzzzz{{{{{|||||yz{|yyyyyyzzzzzz{{{{{{||||||

Several 10's of km

A key concept that is necessary for understanding rock flowage is that of geologicaltime. The long times that are available mean that the strain rates can be very low. Asan example, consider a laboratory test on a rock cylinder. To make it simple, let usassume that the cylinder was 10 cm long before deformation, and that it was shortenedto only 9 cm length, producing a longitudinal strain of 0.1 (10%). (Strains do not haveunits, since they are ratios.)

At a typical laboratory shortening velocity rate the strain may be of 10-4/sec, so thisdeformation will take 103 sec, or about 20 minutes. If the shortening rate is 10-8/sec(about as slow as can be achieved in the laboratory), the experiment will take 107 sec,or about 116 days. It has been estimated that long-term geological strain rates are onthe order of 10-14/sec (based on the motions of the plates). Using this rate, theshortening of the core would take 1013 sec, or 317,000 years. With many millions ofyears available in natural situations, very large strains can be achieved.

At slow strain rates, rocks are less likely to yield by fracturing, and other deformationmechanisms can operate to achieve changes in shape. What has been found inlaboratory studies is that, at low strain rates, rocks strain at a rate that is determinedby the level of stress. In other words, if a larger load is applied, they yield (flow) ata faster rate, and if a smaller load is applied, they flow at a slower rate.

Figure 38Large nappe in the SwissAlps. Note overturning ofrocks on the bottom of thenappe.

132

Another type of rock flowage is cataclastic flow. Here, rocks deform by fracturing,but in this case, the fracturing becomes very intense (closely-spaced), producing anaggregate of small rock fragments bounded by discontinuities. In bulk, suchaggregates of pieces can flow (change shape) in a fashion very like the way that loosesand can flow. In the latter example, the sand grains are not distorting, but simplymoving relative to one another. Cataclastic flow is similar, with the rock fragmentssliding and rotating, except that further breakage of the pieces is common during theflow. There is no scale limit to cataclastic flow. The process occurs at all scales, withthe pieces ranging from sub-millimetre in size to the scale of tectonic plates.

So, bulk changes of shape are possible for deforming rocks. Sometimes this occursby mechanisms that do not produce discontinuities, and sometimes it happens viaprocesses that are dominated by the formation of and movement along discontinuities.

4.10 DuctilityThe usual way to depict the behaviour of rocks deformed in the laboratory is to preparea (so-called) stress-strain curve from the force and displacement records (refer to Fig.23). If we use the terms derived from ideal rheological models, most rocks exhibit aninitial elastic response (but see a later section for a caution concerning this).Yielding is usually taken to mean the end-point of the elastic portion of the curve. Insome experiments, yielding occurs by the development of one or two fractures thatresult in a complete loss of strength. In other experiments, post-yield loading producescontinued deformation without a complete loss of strength (Fig. 39).

500

400

300

200

100

0 2 4 6 8 10

326

165

84.5

23.50

3 (MPa)

Axial Strain (%)

(1

-

3) (M

Pa)

1

1

3 3

For a given rock, these contrasting behaviours are most-readily associated withvariations in the confining pressure. In the laboratory, the meaning of confiningpressure is quite clear. Confining pressure is the pressure of the fluid that surroundsthe rock specimen (and which we equate with the radial stress,

r). However, the

Figure 39A range of post-yieldbehaviour



meaning is less definite when referring to natural conditions. It is typical for confiningpressure to mean the least compressive principal stress (3). However, the term isoften used in a very general way (non-quantitative) where it simply is used to contrastrock behaviour at two different depths in the crust.

We can define a term ductility to mean the amount of post-elastic strain that aspecimen can attain before catastrophic failure. For a given rock, we find that ductilityincreases with increasing confining pressure (Fig. 40). In the samples illustrated, thedeformation mechanism continues to be cataclastic (fracture dominated), even thoughthe specimens can be described as more ductile.

4.11 Effective StressAnother parameter that exerts a major control on rock behaviour is pore pressure.Throughout most of the subsurface - and certainly in all areas of interest to PetroleumGeoscience - the pore spaces of rocks are filled with a fluid. Typically, this fluid isaqueous, although gas and petroleum phases occur (and our job is to find and producethem!). At all locations, the fluid has a pressure. In general, the fluid pressureincreases as a function of depth. For normal-salinity seawater, this gradient (thehydrostatic gradient) is approximately 0.45 psi/ft (10.1 kPa/m), although there aremany things which can complicate this simple relationship.

Pore pressure has a profound mechanical effect. The pore pressure counteracts thecompression arising from all of the normal traction components (including theprincipal stresses), but it does not affect the shear traction components. As aconsequence, the rocks experience an effective state of stress:

T T T

T T T

T T T

T P T T

T T P T

T T T P

where

T

P

eff

eff

eff

p

p

p

ijeff

p

11 12 13

21 22 23

31 32 33

11 12 13

21 22 23

31 32 33

pore pressure effective stress state

=

=

=

:Equation 12Definition of an effectivestate of stress

Figure 40Increasing bulk ductilitycaused by increasingconfining pressure

134

The principle of effective stress was first validated in the area of soil mechanics. It wassubsequently shown to apply in rock mechanics testing (Fig. 41); see below foradditional comments on effective stress). Rocks which have the same effective stressstates (which may have been created by different combinations of framework stressand pore pressure states) deform in a similar fashion. In practice, this means that rockdeformation can be controlled either by depth of burial (which largely controls theframework confining pressure), or by alterations of the pore fluid pressure. Increasedfluid pressure generally makes rocks behave as though they were at shallower depths.

Pc=100,Pp=50

Pc=200,Pp=150

Pceff =50,

Pceff =50,

Pceff =50,

300

400

200

00 4 8 12

0

Axial Shortening (%)

(M

Pa)

(M

Pa)

Pc=50,Pp=0 (dry)

Pceff = Effective Confining Pressure (MPa)

Pc = Confining (radial) pressure (MPa)

Pp = Pore pressure (MPa)

Pceff = 200

Pceff = 150

Pceff = 50

Pceff = 100

Pceff = 0

Pceff = 25

Same rock, Same effective confining pressure

Same rock, Different effective confining pressure

Figure 41Laboratory resultsillustrating the principle ofeffective stress



In a Mohrs Circle display, pore pressure moves the circle to the left or to the right(increase or decrease of pore pressure, respectively), but it does not change the sizeof the circle (Fig. 42). Referring to Equation 12, the pore pressure affects only thenormal tractions (left-right motion), and not the shear tractions (no vertical changes).

s

t failure can occur if circlemoves far enough left

increased pore pressuremoves circle to left,without changing diameter(normal tractions changed,shear tractions unaltered)

4.12 General Rules About Rock BehaviourRocks can exhibit a wide range of behaviours following yielding. There is also a largedegree of variability in the load needed to cause yield. The main parameters thatcontrol these behavioural differences are:

Lithology (composition)

Grain size

Porosity / cementation

Effective confining pressure (equivalent depth of burial)

Strain rate (loading rate)

Temperature (depth of burial)

For a given rock (e.g. a quartzose, well-sorted, medium-grained, moderately-cementedsandstone having a moderate porosity [10%]), a dominant control on the variations indeformation behaviour (as illustrated in Figure 40) is the effective confining pressure(Fig. 43). Increasing effective confining pressure alters the behaviour from low-ductility, catastrophic failure towards greater strength, but with a ductile response (butremember that this is a macroscopic expression, and may be accomplished byfracture-dominated mechanisms).

Figure 42Mohrs circle view ofeffective stress

136

Diff

eren

tial S

tress

(1

-

3)

Effe

ctive

Con

finin

gPr

essu

re In

crea

sing

Strain

Temperature and StrainRate are Constant

For the same rock, a strain rate decrease, or an increase in temperature, will result ingreater ductility, but lower strength (Fig. 44).

Diff

eren

tial S

tress

(1

-

3)

Tem

pera

ture

Incr

easin

g at

Co

nsta

nt S

train

Rat

e or

St

rain

Rat

e De

crea

sing

at

Cons

tant

Te

mpe

ratu

re

Strain

Effective Confining Pressureis Constant

Although these variations in behaviour are described in terms of a single rock type, thegeneral responses - e.g. changes in strength and ductility - are applicable to all rocktypes. The curves illustrated (in figures 43, 44) summarise the first order deformationalresponse of all common rocks.

Now that we see how changes in environmental conditions affect a single rock type,we need to determine how to extend these results to other lithologies. For this, we

Figure 44Strain rate and temperatureeffects

Figure 43Variations in behaviourassociated with changes ineffective confining pressure



compare the behaviours of different rock types under similar conditions (Fig. 45). Forthis purpose, we assume a reasonable increase in temperature and pressure with depth.The curves shown represent typical responses, but it should be remembered that theremay be a considerable variance in these values if the rocks concerned are differentfrom the examples used.

0

4km

8km

50 100Peak Strength (MPa)

Dep

thDuctility (%)

Rock Salt

Rock Salt

Shale

Shale

Limestone

Limestone

(Accounting for usual geothermal and pore -pressure gradients)

Sandstone

Sandstone

Dolostone

Dolostone

QuartziteQuartzite

Granite

Granite

Rocks that are otherwise similar, but that have dissimilar grain sizes, also havedifferent characteristics. Small grain sizes tend to be associated with greater strength,and lesser ductility, as contrasted with larger grain sizes. Mono-mineralic rocks (e.g.a sandstone with grains of quartz only) tend to be stronger than otherwise-equivalentpoly-mineralic rocks (e.g. a sandstone with grains of quartz, feldspar, mica).

4.13 Elastic Behaviour ?The initial, linear portion of stress-strain curves is often assumed to represent anelastic response of the rock. If it were truly elastic, then everything that occurs duringthis loading should be recoverable. We now know that this is not the case.

Technology allows us to attach devices to rocks inside a pressure vessel to listen to anynoise they make as they deform. During the initial loading, these devices record thesounds produced by cracks that form in the rock (these sounds are called acousticemissions, and they represent micro-earthquakes). These fractures do not go awaywhen the load is removed, and so they are not part of an elastic process. Theirpresence in the rock (as small voids - fractures have small, but finite, apertures, oropenings) represents a volume gain or dilatancy that is non-recoverable. Micro-cracking increases exponentially as yield is approached. If rocks are loaded part-wayto yield, but are un-loaded before they reach yield, nearly all of the volumetric strainis recovered, although the newly-formed cracks produce a small amount of permanentdilatancy. For many practical purposes, we can consider the initial portion of theloading curve as elastic, but we should be aware that there may be associated effects(e.g. on fluid flow, or on sonic properties) that should not be ignored.

Figure 45Representation of thebehaviours of differentlithologies with depth

138

4.14 Volumetric StrainsThe deformation-induced dilatancy caused by microcracks can be expressed in a moredefinite fashion as macrofracture occurs. Volumetric strains associated with fractur-ing can be as much as a few percent, with important effects on fluid flow. This modeof behaviour has long been recognised, and underlies the important topic of fracturedreservoirs.

However, there is another major deformation mode that causes compactional volu-metric strains. As porosity reduction is nominally associated with a decreasedpermeability, this deformation mode is of considerable importance to many aspectsof the Petroleum System. Compactive volumetric strain can occur in the deformationof lithified rocks (see a later section), but this mode is of primary importance in thedeformation that occurs as sediments compact in a basin, losing their porosity. Thislatter situation is often neglected in treatments of rock deformation. It is, of course,well known in the area of soil mechanics.

Many rocks exhibit a transition from the dilatant mode to the compactant mode as afunction of confining pressure (Fig. 46; here the plot shows mean stress versusvolumetric strain). When plotted on a plot, such experiments indicate a failurecriterion whose slope decreases with increasing confining pressure (Fig. 47).Interestingly, the ability to predict fracture angles is degraded in that portion of thecurve where the slope is reduced. It seems clear that the Mohr-Coulomb approach isnot appropriate for explaining these compactant volumetric strain changes.

P (M

Pa)

m

200

100

0.1DVVo

(%)

IsotropicCompression

Dilatant Volumetric Strain

CompactantVolumetric Strain

s

t

failure criterion

note change in slope

Figure 47Mohr-Coulomb failureenvelope for experimentsillustrated in Figure 46

Figure 46Changes in volumetricstrain behaviour in a set ofexperiments



Figure 48Poro-plastic yield surface

4.15 A New Material DescriptionA useful development in rock mechanics is to treat yielding as a plastic phenomenon,and to adopt the formalism of plasticity theory. The experimental responses depictedin Figure 44 (and in a large number of other examples) represent permanent porositychanges as a consequence of deformation. Because there is a permanent deformation,and because there is a porosity change, the appropriate term to apply is poro-plasticity.In plasticity, yielding is associated with a mathematical fomulation in which statevariables (e.g. temperature, pressure, strain, etc) are used to define a yield surface.When the actual state plots on the yield surface, yielding is predicted. This is verysimilar to the Mohr-Coulomb approach where, if a pair plots on the yield(fracture) criterion, then failure is predicted.

A characteristic poro-plastic yield surface (Fig. 48) depicts two contrasting types ofyielding. Dilatant yielding (porosity increase, typical of classical views of fracturing)occurs when the stress state (here it plots as a single point) contacts the yield surfaceon its nearly planar portion. Compactant yielding (porosity loss) occurs when thestress state contacts the yield surface on the curved portion. This approach showsconsiderable promise for attacking a number of important problems in Rock Mechanics.

Pm Yield Surface

Dilatant

Compactant"Stress Difference"

f

5. ADVANCED ROCK MECHANICS AND ROCK PHYSICS

In this section of the Appendix, we introduce some further topics in Rock Mechanicsand Rock Physics. These subjects build on the knowledge developed in the previoussections, and they serve as bridges to various applications of the subject that aPetroleum Engineer is likely to encounter in his/her employment. However, it shouldbe noted that each topic is still given only a very light treatment each represents justa brief overview of a considerable body of knowledge, hence the quotes aroundadvanced.

5.1 Dis-continuum rock mechanicsEarlier in this Appendix, we focused on fracturing as an important form of yielding.Certainly, fractures are almost ubiquitous in rocks, and fractures can have importantconsequences for fluid flow. Here, our purpose is to describe how the mechanicalbehaviour of a fractured rock mass can be better understood in terms of simplefrictional laws.

140

Friction is concerned with sliding that occurs (or does not occur) along an interfacebetween two bodies (Fig. 49). The idea is that there is a normal force, F

n, acting

perpendicular to the interface, and a shear force, Fs, acting parallel to the interface. The

body lying above the interface, and the body lying below it, are considered to becompletely rigid, and the forces are assumed to be constant throughout the bodies.Under these conditions, the forces can be converted to tractions by dividing each bythe area (A) of the surface contact between the upper body and the one below. Thus,we have a normal stress (component),

n (= F

n/A), and a shear stress (component), t

(= Fs/A).

F

F

n

s

A

Experimentally, we find that there is a linear relationship between the normal stressand shear stress (components) needed to make the upper block slide along theinterface:

t m s

t

s

c n

c

n

=

=

=

critical shear stress (on surface) normal strress (on surface) experimental constant of proportionalm = iity

The constant in this equation, , is taken to be a property of the interface. It is calledthe coefficient of friction.

The experiments that are performed to determine the value of m also reveal thatEquation 13 represents a criterion (Fig. 50). By this we mean that, at all conditionswhere the shear stress is LESS than that given by Equation 13 (for any value of normalstress), no sliding occurs, AND that shear stresses larger than those given by Equation13 cannot be achieved (for any particular normal stress). The magnitude of the shearstresses is limited because frictional sliding occurs when the magnitude of t equals thatspecified by the criterion, and larger values are therefore unattainable.

s

t sliding criterion

Figure 49Simple configuration fordefining frictionalprocesses

Equation 13Proportional relationshipbetween shear stress andnormal stress, determinedexperimentally

Figure 50The criterion for frictionalsliding



The form of the frictional sliding criterion (e.g. Fig. 50) is superficially similar to theMohr-Coulomb failure criterion (e.g. Fig. 51). This similarity seems to invite us toplot the two criteria together (Fig. 49). If we apply our knowledge concerning the useof Mohr Circles (see above), we might anticipate that it would be possible to answer(e.g. in a simple mechanical thought-experiment) the questions: will sliding occur, orwill failure (fracture) occur?

s

t

failure criterion

sliding criterion

This nave approach can, unfortunately, lead to a very wrong prediction (Fig. 52). Thereason for this problem is that the frictional sliding criterion applies to a particularsurface (i.e. one that has a particular orientation), and specifically, to the shear stressand normal stress acting upon it. In a given stress state, these components vary as theorientation of the (virtual, or, in this case, real) surface is varied (see Equation 10). Foran isotropic rock (our usual assumption in these arguments), the failure criterion doesnot concern itself with orientation. Thus, SOME pre-existing planes in the rock massmight be subject to sliding (in a given stress state), while others would not. Argumentsconcerning sliding versus fracture MUST account for the orientations of the (poten-tial) sliding surfaces.

s

t

failure criterion

sliding criterion

Planes within THESEorientations would haveslipped, but other planes would not.NO failure is predicted.

The preceding few paragraphs illustrate how we can understand the sliding versusfracture behaviour of a simple (idealised) situation. In rock masses that containnumerous discontinuities (potential sliding surfaces), with a variety of orientations,the difficulty of making a prediction is greatly increased. This is especially true if weconsider that the state of stress may vary from place to place in the rock mass. Suchvariations could be caused by the way that an applied load interacts with the intactblocks of rock and their bounding discontinuities. A variety of theoretical andobservational studies indicate that stress fields are distorted by the presence ofdiscontinuity planes. Predicting the mode of deformation (sliding or failure) could bevery difficult in practical situations since it may not be correct to assume a simple stateof stress that was the same everywhere (see also below for other comments on thisissue).

Figure 51Combined plot of failureand sliding criteria

Figure 52Predicting sliding requiresthat the orientations beconsidered

142

5.2 Poro-elasticityPoro-elasticity is an extension of the basic rock mechanics concepts presented earlier.This theory seeks to explicitly address the compressibility of porous rocks whose porespaces are filled with mechanically-active fluids. The theory is appropriately appliedto situations whose state of stress is below the yield (failure) criterion for thatmaterial.

The basic idea is that a fluid-filled porous rock can be described in terms of twocomponents: the rock framework (consisting, nominally, of grains, plus cement andinter-granular material), and the pore space (Fig. 53).

fluid-filled pores grains

Each of these two components is compressible. The fluid in the pore spaces is subjectto volumetric changes in response to pore-pressure changes, or, alternatively, the fluidpressure can be altered as a result of a change in the volume of the pore space (in aclosed system). The framework also responds to loading changes: increasing thecompressive loading of the framework produces a pore-space reduction. A pore-pressure increase causes a shrinking of the framework components, effectivelyincreasing the porosity (Fig. 54). The usual approach in poro-elasticity is to assumethat the stress state is isotropic (e.g.

x = y = z = mean), so allowing this state to be

called a pressure. Many treatments of this subject simply refer to the frameworkloading as the confining pressure (P

c).

note contraction of grainsand increase of pore volume

Figure 53A schematic two-dimensional representationof a fluid-filled porous rock

Figure 54Volumetric changes of thepore space and framework



There is an obvious potential for coupling between these two components (e.g. thefluid-filled pores, and the rock framework). A rock loaded in such a way as to produceincreased framework compression will tend to cause a pore-space reduction. Thefluid in the pores will either flow away (while maintaining the constant pore pressure;this is called a drained condition), or it will be forced to occupy less volume (in anundrained case), producing an increase in pore pressure. In the case of fluidwithdrawal (e.g. during production of a reservoir), the framework can expand inresponse to the lowered fluid pressure, and the elastic energy released in this way willcause even more fluid to flow because of the resulting reduction of pore space. Poro-elastic constitutive laws (not shown here) are expressed as equations relating stress,strain, and pore pressure, but there is always an ADDITIONAL equation describingfluid flow (usually in the form of Darcys Law).

Three types of compressibility can be distinguished: bulk compressibility (rock +pores, together), pore compressibility, and rock compressibility. We can experimen-tally determine these parameters for cases where either the rock-framework load isconstant (in a lab test, by keeping the confining pressure constant), or where the porepressure is constant. For the rock compressibility, the confining pressure and porepressure are varied together.

CV

VP

CV

VP

CV

VP

CV

VP

Cpore

bcb

b

c P

bb

b

c P

c

c P

c P

mn

P

c

P

c

= -

=

= -

=

1

1

1

1

f

f

f

f

ff

f

f

f

f

= Compessibility (m = b: bulk; m = : pore)V = Volume (s = b; s = )P = Pressure (t =

s

t

:

c:c: confining; t = p: pore)

The considerations described in the preceding paragraphs mean that we need to re-visit our discussion of the concept of effective stress. The revised idea is that porepressure changes have effects much like those previously indicated (e.g. Equation 12),but the effects may not be 100% of those specified previously. Accounting forcompressibility, effective stress is:

Equation 14Definitions ofcompressibilities

144

s s d a

d

ijeff

= -

=

=

ij ij p

ij

P

i ji j

10,

,

The term a is called the Biot coefficient. It is treated as a material parameter i.e. eachrock has its Biot coefficient. a ranges from 0 to 1, and values close to one indicate thatthe simple approach to effective stress (e.g. like that given earlier in this Appendix)is more nearly valid.

A final parameter is worth including here. The Skempton Ratio, B, is the change inpore pressure per change in confining pressure, in undrained conditions. B rangesfrom 0 to 1. Low values for B tend to be associated with very compressible fluids (suchas gas). Experimental determinations of B are still som

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