University of Manchester

Geomechanics

for Petroleum Engineers

EART20422

Course Notes 2013-2014

School of Earth, Atmospheric and Environmental Sciences

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The Scope of Geomechanics

The theories of stress, strain and elasticity are the tools that we use to understand the mechanical behaviour of rocks. In the context of petroleum engineering, we are concerned about how rocks behave in reservoirs and around boreholes. This course is about these aspects, but it is important to put reservoir and borehole geomechanics into the wider context of the strength and mode of failure of rocks over a wide range of physical conditions.

Stress 2 types of force can act on a body, BODY forces and SURFACE forces. (a) BODY FORCES, e.g. gravitational, magnetic, inertial

No body contact required. Described as FORCE per UNIT VOLUME. (b) SURFACE FORCES, action by physical contact with other bodies (including fluids). Described using concept of STRESS or PRESSURE (= FORCE per UNIT AREA)

Resolution of forces on a surface

Sign Convention: Sign Convention: normal stress, compression +ve Shear stress, Right-handed +ve

Stress components in 3D

1st subscript is the NORMAL to the plane across which the stress component acts. 2nd subscript is the DIRECTION in which the component acts.

Using and , it is sufficient to write xx as x

Alternatively, we can use only to symbolize both normal and shear stress, in

which case BOTH subscripts are needed, but we can use the convenient shorthand

ij, where i, j = 1 to 3. In this case the 3 axes would be identified as x1 x2 x3.

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The array of terms can be written in matrix form ( 11 1213 )

(212223 )

( 31 32 33 )

Stress is a multicomponent quantity that is an example of a TENSOR quantity.

The terms are defined by reference to a reference frame, If the reference frame is

rotated the values of the terms change. It is the particular way that the values of the

terms change as the reference frame is rotated that defines stress as a tensor

quantity.

There are other examples of tensor quantities,

e.g. Strain (stress and strain are examples of FIELD tensors)

Magnetic susceptibility, permeability, piezoelectricity, elasticity (these are examples

of PHYSICAL PROPERTY tensors).

Symmetry of the stress tensor

Provided a body is in equilibrium with respect to ROTATION about any reference axis,

ij = -ji, i.e. the matrix representing

the tensor is symmetric about the leading diagonal

Homogeneous and Heterogeneous Stress

In a homogeneous stress field the values of the stress components do not change

from place to place. In a heterogenous stress field they do. For example the

distribution of stress in a hook supporting a load is immensely variable, and within

folded sedimentary layers this is also true.

Principal Stresses

For any state of stress, there is always an orientation of the reference axes for which

the shear stress components all become zero. The values of the normal stress

components in this reference frame are called PRINCIPAL STRESSES and are

labelled 1, 2 and3 (in order of decreasing magnitude). It is common practice to

represent graphically the shape of a stress field by drawing an orthogonal grid that

shows the variation of orientation of principal stresses from place to place. Where

the grid lines converge, the values of that stress component increase, and vice

versa. The orthogonal grid is called the STRESS TRAJECTORY pattern.

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This figure illustrates the stress trajectory pattern of a heterogeneously stretched

bar. To a certain extent the expected pattern of stress trajectories can be deduced

from commonsense, bearing in mind the form of the BOUNDARY CONDITIONS.

Some problems of stress distribution can be solved ANALYTICALLY; more general

problems have to be solved NUMERICALLY.

Homogeneous stress in two dimensions

In two dimensions we have only 4 stress components ( only 3 are independent). The two commonest tasks in dealing with homogeneous stress states are: (i) Calculating the values of Principal Stresses in terms of components referred to an arbitrarily inclined reference frame. (ii) Calculating the values of normal and shear stresses acting on an inclined plane in terms of the principal stresses. Using the concept of resolution of forces across areas it is a simple matter to derive equations to solve these problems. They are stated below without proof.

(i) Values of Principal Stresses in terms of components referred to an arbitrary reference frame. _______________________

Thus 1, 2 = ½(x + y) + ½ (x + y)2 - 4(xy -

2)

_____________

= ½(x + y) + ½ (x - y)2 + 42

Orientations of the Principal Stresses,

tan 2 = 2 / (x – y)

(ii) Stresses in a plane in terms of the principal stresses

Here, we are interested in the values of normal and shear stress resolved along any

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arbitrarily inclined plane, at o to the max principal stress. This is of particular importance in geoscience, because this is how we calculate the normal and shear stress acting on an inclined fault plane. It is key to understanding the stability (or otherwise) of faults

n = ½(1 + 2) + ½(1 - 2) cos2

= + ½ (1 - 2) sin2

The + signs refer to the fact that the equations

also give the values of n and with the plane

and its normal transposed. (recall ij = - ji)

Note that is maximum when sin 2 = 1,

i.e. = 45o

and zero parallel to the principal stresses

(= 0 or 90o)

Variation of and n with

e.g. setting 2 = -½ 1

The Mohr Circle construction

Note that these equations take the same form as the parametric equations to a circle:

n = ½(1 + 2) + ½(1 - 2) cos2

= + ½ (1 - 2) sin2

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(What, you don‘t know the parametric equations to a circle? ) (You do now!) The similarity in form means that points on the periphery of a circle can be used to represent all the different combinations of shear and normal stresses on variously inclined planes with respect to the principal stresses. The normal stress components where the circle cuts the abscissa (when shear stresses are zero) are the principal stresses (maximum and minimum values of the normal stress components).

Coordinates A and B give and n resolved on 2 planes 90o apart

Note : tan 2 = 2 /( x - y) , from Pythagoras.

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Elasticity

Elastic deformations are the small (<1%), strains that accompany stresses are too

low to cause permanent deformation. The strains are sufficiently small that a

particular variety of finite strain theory, called infinitesimal strain theory, can be

applied. Elastic strain is recoverable on removal of load and it is through elastic

strains that stresses are transmitted from place to place throughout a material.

Elastic strain is (to a good approximation) linearly proportional to applied stress.

This proportionality is called Hooke‘s Law (Ut tensio, sic vis - As is the extension,

so is the force). The constants of proportionality are called Elastic Moduli. For an

isotropic, linear elastic solid, we can usefully define 4 elastic moduli:

Young’s Modulus, E = i / i

Shear Modulus, G = ij / ij

Bulk Modulus, K = / v , where we define = (x + y + z)/3

Poisson’s Ratio, v = x / y, for load applied in y direction

The diagram (right) defines the concept of Poisson‘s ratio. As the ratio of two strains, this is just a number. The other elastic moduli all have the dimensions of stress, as they take the form stress/strain. Note that for isovolumetric deformation Poisson‘s ratio is 0.5 . For deformations of real crystalline solids it is less than 0.5 , normally about 0.3, implying that elastic deformation of a crystal involves elastic volume reduction.

Note that for small strains (< 3%), volumetric strain v = dV / Vo

can usefully be approximated as (x + y + z)/3 (but NOT for large strains) Longitudinal and shear strains are defined in the same way as for finite strains. Interdependence of elastic moduli

For an ISOTROPIC (make sure you understand what this word means) material, only 2 elastic constants are independent:

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These are the interrelationships:

K = E / 3(1- 2) ; G = E / 2(1 + v)

From which the other relations between groups of 3 can be derived.

Elastic Stress-Strain Equations

For a triaxial state of stress (1 2 3) 0 we can write:

1 = 1 / E – v2 /E – v3 /E = {1 – v(2 + 3)}/E

Similarly 2 = {2 – v(1 + 3)}/E

and 3 = {3 – v(1 + 2)}/E

Elastic Strain Energy

This is the potential energy stored in an elastic solid by virtue of the work done on it by the external forces to produce the strained state. Work done per unit volume = Force x displacement Volume

W = Force . l = . d Area lo

But = E , thus for a total strain ,

W =

= ½ (the area under the stress/strain curve)

Using principal stresses and strains in 3D

W = ½ )

Using the stress strain equations to eliminate strains:

W = 1 {

2E

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Introduction to Strain

Homogeneous and heterogeneous deformation. Finite and infinitesimal strain. Strain measures :

Changes in the length of lines: conventional strain e = (l1 - lo)/lo = l/lo , (1+e) = l1/lo

where l1 = current length and lo = original length. Quadratic elongation,

= (l1/lo)2 = (1+e)2

Natural or logarithmic strain, = ln (1+e) Changes in angles:

Shear strain, = a/b = tan Homogeneous Strain Straight lines remain straight, lengths of lines and angles change : The Mohr circle construction can be applied to strain just as it can be to stress, though we will not explore that avenue here. Homogeneous deformation of a circle to an ellipse The deformed circle illustrates graphically the amount of deformation

and is called the finite Strain Ellipse. In 3 dimensions a circle would be deformed into an ellipsoid.

The maximum and minimum values of strain are the Principal Strains,

and correspond with the principal axes of the ellipse.

Principal axes are also lines of no finite shear strain. Intersection of initial circle with strain ellipse are directions of no finite longitudinal strain (NFLS). These divide the ellipse into areas where lines have been shortened from areas where they have been stretched.

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Area change: In general strain involves volume change (or area change in 2D). Some strain can take place at const volume (or area).

Area change is given by (1 2 )0.5 – 1, x100 to express in percent.

Note area change can be either increase or decrease. Strain trajectories, These are orthogonal sets of lines (like stress trajectories) that are everywhere parallel to the principal axes of strain. They are useful for representing graphically variation of strain from place to place in a heterogeneously strained body. Note how strain along a particular trajectory increases as the trajectories converge. Ellipses of different eccentricities (do you understand this word? [= axial ratios]) can be used to represent variations in amount of distortional strain.

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Rock Deformation Mechanisms

Rocks are not continuous materials, like glasses, but are made of crystalline grains. This granular heterogeneity of structure, coupled with the possibility of intergranular porosity, profoundly affects how rocks can deform. The crystalline internal structure of grains also affects deformability. Rock deformation mechanisms can be classified into two main groups, and each mechanism is dominant over particular ranges of pressure/temperature conditions encountered within the Earth:

(a) cataclastic deformation, in which the rock distorts by brittle

fracturing at the grain scale or larger, and fragments are

displaced with respect to one-another.

(b) intracrystalline plasticity, in which the internal crystalline

structure of grains is distorted, effectively by the slippage of

atomic planes over one-another.

We will examine first the microstructural characteristics of these

different deformation mechanisms. Then we will consider the different

mechanical properties that are diagnostic of the different processes.

Cataclastic deformation

The rock distorts by brittle fracturing at the grain scale or larger, and fragments are displaced with respect to one-another.

There is no internal distortion of the crystal or lithic fragments. This is

how rocks deform in the upper crust, and the deformation is commonly

localized into fault zones.

The necessary accompanying volumetric expansion causes the

resistance to cataclasis to increase dramatically with pressure (depth of

burial), but the process is not very sensitive to temperature variations.

[note, this last sentence is profoundly significant, and one way or another

WILL figure in examination questions].

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Typical Structures and Microstructures:

Formation of oriented crack arrays, aligned preferentially parallel to the

greatest principal stress direction and originating at impingement points

between grains.

These crack arrays are called axial cracks. In ancient rocks they can

become cemented, so they are only seen as oriented fluid inclusion

arrays.

Axial crack arrays tend to coalesce,

or localize, after a critical density of them

has formed, to produce a planar fault,

typically at about 30° to 1 (above, max

stress is vertical)

In low porosity rocks such as limestones,

cracking may be mesoscale and pervasive

(see upper right).

Pervasively mesocracked rocks can form important hydrocarbon

reservoirs in which oil is held in the cracks, e.g. in Northern Italy.

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Strength and microstructural evolution by cataclasis depends strongly on

amount and nature of cement, and how much of the original

intergranular porosity remains.

Fault Rocks

The rocks that are produced in fault zones (ranging from >1 cm to 10‘s

of metres wide) are variously called fault gouge or cataclasites, and

are characterised by fragmentation and granulation of the protolith.

Fault gouges may be free of clays or other phyllosilicates, and be

made up of only fragmented quartz, feldspar of limestone (according to

the nature of the protolith), with various amounts of cement. However,

breakdown of feldspars or micas or growth of authigenic clay can result

in a clay-bearing fault gouge. These various characteristics determine

whether a fault zone permits or impedes the flow of fluids such as water,

oil and gas. Faults that are sufficiently impermeable can form reservoir

seals or can cause compartmentalization of reservoirs (the latter with

attendant production problems).

Clay-bearing, foliated fault gouge Cataclasite in fault zone in limestone

Intracrystalline Plasticity

Internal crystalline structure of grains

is distorted, effectively by the slippage

of atomic planes over one-another.

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More symmetric crystal structures can have several differently oriented,

crystallographically controlled slip planes.

The deformation is a

simple shear, hence is

constant volume, which

makes the process

insensitive to pressure.

However, it is strongly

sensitive to temperature.

From the mid-crust and

downwards, this is probably

the commonest deformation

mechanism in the Earth. At

shallow, upper crustal

depths (top 10 km) rock salt deposits are mobilized by this process and

this leads to the formation of important hydrocarbon traps and reservoir

cap rocks.

Typical Microstructural Features at the Grain Scale

These are:

(a) Flattened grains and

(b) Wavy extinction due to internal distortion of crystal structure.

(c) Subgrain formation

(d) Dynamic recrystallization, starting around the edges of grains.

This image below of deformed basic granulite shows all of these

features:

(image width about 4mm, crossed polars)

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Similar features (below) in experimentally deformed marble (700°C, original

grain size 200 micron).

(below) Dynamic recrystallization by grain boundary migration

(especially at higher temperatures). This progressively replaces the old,

strained microstructure by a new, strain free microstructure, often of a

different mean grain size (image right, experimentally deformed marble

800°C, original grain size 200 micron).

(c) and (d) are known as recovery processes, that progressively restore

the microstructure to an unstrained state.

Crystallographic Preferred Orientation (CPO)

This only develops to any significant degree with flow by Intracrystalline

Plasticity.

Recrystallized grain

size / stress

relationship

Recrystallized grain size

can be used as a

paleopiezometer.

Deformation at higher

stress produces smaller

recrystallized grains,

thus in naturally

deformed rocks we can

use recrystallized grain

size to measure the flow stress at the time of recrystallization: The graph above shows the experimentally-determined grain size vs stress relation for marble.

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Describing the strength of rocks

We can usefully consider this topic under three headings: (a) Brittle rocks of low porosity at low temperatures (<200oC) (b) Brittle rocks of high porosity at low temperatures (<200oC) (c) Rocks that deform by flow processes (at high temperatures, 300oC for most rocks, but only 80oC [or more] for rock salt) The strength and mode of failure of rocks is generally investigated in the laboratory using a triaxial testing machine. This consists of a thick-walled pressure vessel containing a cylindrical specimen between two loading pistons. Depth of burial is simulated by pressurizing a hydraulic fluid (liquid or gas confining pressure). The pressure is transmitted to the rock via a rubber or thin-walled metal jacket. The end-load is increased above hydrostatic by advancing the loading piston into the vessel. The amount of this excess load (differential load) at failure is a measure of strength. High temperature can be achieved either by heating the whole pressure vessel (external heating) or by means of a smaller furnace inside the pressure medium and placed around the specimen. Pore fluids under pressure up to the value of the confining pressure can be introduced via a hollow loading piston. We have 8 of these machines in the lab at Manchester.

Testing Machines

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(Above right: Iron-jacketed specimen assembly)

Specimens can be tested in compression (above left), extension (lower left) or torsion (right, photo David Olgaard)

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Modes of Rock Failure

This term refers to whether rocks fail (a) by faulting (localized deformation) or (b) by ductile flow (distributed deformation) The images show examples of faulting in the lab and in nature :

(left) single shear fault (photo John Ramsay) (right) conjugate faults (source unknown).

(above) in lab. (below) in nature

Stress - Strain characteristics

The results of mechanical tests (typically taken to about 15% strain) are usually plotted as stress/strain curves. They are very different for the cases of faulting and flow:

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Deformation in the upper continental crust (topmost 15 km) is usually by cataclastic faulting. That is, cracking that becomes localized into a brittle fault (brittleness pertains to cracking). Cataclasis is the name given to deformation processes involving cracking. Prior to permanent deformation, in the elastic regime the slope of the stress/strain plot is the Young‘s modulus. Deformation in the lower crust (lowermost 15 km) generally is by flow, leading to folding in layered rocks or pervasive deformation in isotropic rocks like granitoids. Deformation involving cracking usually results in a small amount of volume expansion (about 1 or 2%), as the fractured pieces no longer fit together. The volume expansion is resisted by confining pressure which increases as depth increases. Thus rocks deforming by cataclastic processes get stronger as depth increases. [remember, you were warned about the importance of understanding this point] On the other hand, increasing temperature with depth has little effect on resistance to cataclastic deformation. Deformation involving flow processes can be almost at constant volume, and hence are insensitive to variation in confining pressure. On the other hand, flow processes are very sensitive to temperature and to rate of deformation (higher temperatures or slower rates lead to lower strength. Thus rocks get weaker at greater depths [higher temperatures]). This is exactly opposite to cataclastic deformation. Thus the variation of rock strength with crustal depth looks like this (the details vary with rock type): The flow curve shifts to the left (lesser depth) as the strain rate is decreased and vice-versa. The faulting to flow transition usually is near mid-crustal depths. Because seismicity is associated with brittle deformation, the faulting to flow transition also corresponds to the depth limit of seismicity in the crust.

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How quickly do temperature and pressure increase with depth? The rate of temperature increase with depth is called the geothermal gradient. It is very variable according to the thermotectonic regime. In oceanic trenches it is low, about 12 °C/km; about 22 °C/km in stable continental crust; about 30 °C/km in orogenic (mountain building) regions and can be as high as 150 °C/km in volcanic areas.

The rate of pressure increase (the vertical component of normal stress,z) with depth is called the geobaric gradient. To a useful approximation it is given by :

z = m g z

in whichm is average crustal rock density, g is the gravitational acceleration

and z is depth. For typical crustal rocks dz / dz ~ 35 MPa / km, because the density of compact crustal rocks is about 2700 kg/m3. Thus the pressure at the continental Moho is about 1000 MPa (= 1 GPa). [1 MPa = 10 atmospheres = 145 psi] However, if rock density is much more variable with depth, perhaps at shallow crustal depths because progressive pore compaction and diagenetic mineral transformations are taking place with depth, a more precise calculation of pressure at a given depth is required. Assuming density varies with depth

according to (z), the pressure at depth z below the surface is given by z

z = g z) dz 0

(z) for sedimentary basins can be obtained from density measurements on core samples. The thermobaric regime of the crust has a profound effect on how rocks deform and on the conditions for the formation of oil and gas.

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Describing brittle/cataclastic deformation (i) - the Mohr-Coulomb failure criterion This describes well the brittle failure of rocks of less than about 10% porosity over the range of upper crustal P/T conditions. A linear relationship between principal stresses at failure is assumed:

Consider a rock subjected to an initial stress state such that 1 = 2 =3z

(wherez is the normal stress in the vertical direction), i.e. stresses are equal in all directions and principal stresses are horizontal and vertical. This is called a hydrostatic stress state. E. M. Anderson, who was famous for his quantitative work on stress and faulting in the first half of the 20th century, called this a standard state.

Normal Faults Reverse Faults Wrench Faults

Now imagine 1 gradually increased until the rock fails. Failure is usually in

shear, with the formation of a fault inclined at to 1. is usually about 30°.

(1 - 2) at any instant is known as the differential stress, and its particular value at failure is called the strength of the material. Anderson suggested that different types of fault formed in response to

different stress regimes in the Earth. Thus when the vertical stress is1 (i.e.

the greatest stress), normal faults form. When 1 is horizontal and the least

stress (3) is vertical, thrust faults form. When the intermediate stress (2) is vertical, wrench faults form. Potentially, 2 faults may form in a given stress state, forming a conjugate

pair that intersect in the 2 (intermediate stress) direction. Commonly, the first one to form relieves the load so the second fault fails to form.

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Extensional and Contractional faults A more general way of describing faults is to relate the stresses to the local layering, which make a marker for the sense of fault offset. Thus faults that try to thicken or duplicate the sequence are contractional (with respect to the layering) and those that excise (cut-out) part of the sequence are extensional in character. One can still talk about extensional or contractional tectonic regimes with respect to the orientation of the Earth‘s surface, though in the general case the principal stresses are not necessarily parallel and perpendicular to the Earth‘s surface as assumed by Anderson for his standard state.

Returning to our 2D state of stress (1 and 2 only, also known as plane stress), imagine increasing the value of the standard state stress (e.g.

increasing depth of burial). Now, at the greater depth, as we increase (1 - 2) to produce failure, we find that the differential stress is greater than before. This can be demonstrated in experiments, when the initial hydrostatic state is called the confining pressure. As the confining pressure is increased the rock becomes stronger. This is a characteristic of all brittle rocks. Experimental results can be represented in 2 ways:

1) A plot of 1 against 2 at

failure (or (1 - 2) against 2) (right) :

This is usually a straight line, in which case it can be represented by

1 = a + b 2, where a and b are material properties, and can be extracted from the intercept and slope of the line. 2) On a Mohr diagram (right), as a

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set of circles of increasing diameter. The circles are delimited by a straight line, called the Mohr envelope, that defines the strength of the material.

The Mohr envelope can be described by = o + n, where o and are material properties called the cohesive strength and coefficient of internal friction, respectively. Because the above are simply two alternative methods of reporting the same data, there exist linear transformations between the material parameters, given by

a = 2ob and b = (1 + sin )/( 1 - sin

= tan , where is the slope in degrees of the envelope. is called the angle of internal friction (commonly about 30°). These two equations are equivalent statements of the Mohr-Coulomb failure criterion. The angle of friction is related to the angle of the fault that forms at failure by

= 45° - /2 The diagram below shows the relationships between the Mohr circle world and the real world :

The Role of Pore Fluid Pressure - the Law of Effective Stress Pore fluid pressure contributes to the overall state of stress in a rock. Whereas the applied (far-field) stresses try to push the grains together, pore fluid pressure tries to push them apart. A liquid cannot support a shear stress, only a hydrostatic normal stress, thus pore pressure reduces the effective applied normal stress components by an amount equal to the hydrostatic pore pressure. This can be represented on a Mohr diagram. The pore pressure shifts all points on the circle to the left by an equal amount. The diameter of the circle (and hence shear stress components) remain unchanged, but the effective confining pressure is reduced.

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Because the Mohr-Coulomb criterion has a positive slope, the action of raising the pore pressure can result in failure. The effect on the Mohr-Coulomb criterion is given by:

(1 - 2) = a + (b - 1) (2 - p) =o + (n - p) where p is the pore fluid pressure. The following diagrams illustrate how raising the pore pressure effectively prises the two sides of a fault apart and reduces its ability to support frictional stress (figures by Steve Hickman). How does pore pressure vary in the Earth? If the pores of a rock at depth z are connected via continous channelways to

the free surface, the ratio of pore pressure p to overburden pressure z is equal to the ratio of density of the column of fluid to density of the column of rock:

= fluidgz / rockgz = fluid /rock The density of crustal rock of moderate porosity is about 2.5 times fluid

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(water) density, hence the ‘normal‘ value of is expected to be about 0.4.

Various processes can cause to increase above 0.4 : 1. Compaction and cementation - reduces pore volume hence raises pore

pressure

2. Erosion - removes overburden pressure, so rises (provided fluid cannot leak away).

3. Osmosis - Separation of aqueous fluid reservoirs of different salinities by a

semipermeable membrane (e.g. clay-rich) layer causes fluid diffusion from the less saline fluid to the more saline fluid.

4. Production of excess fluid through diagenetic or metamorphic reactions

(dehydration of clays, evaporites and other hydrous minerals). 5. Thermal expansion on heating.

> 0.4 in a reservoir causes blowout when the reservoir is pierced by drilling, unless it is prevented by appropriate engineering.

can rise to become equal to the least principal stress (= 1). Rocks are

weak in tension so any attempt for to exceed 1 results in hydraulic fracture of the rock (the excess fluid pressure blows the rock apart against the least principal stress). In the case of a reservoir this means failure of the seal and leakage of fluid into adjacent, non-overpressured rocks. Hydraulic fractures tend to propagate along the maximum principal stress trajectory. In mid-crustal rocks, hydraulic fractures are often preserved as veins filled with hydrothermal mineralization such as quartz. Tectonic brecciation is also thought to be produced in zones of high fluid pressure, because the reduction of effective confining pressure makes it easier to bring about the dilatation of the rock mass necessary to accommodate fractures.

Examples of hydrothermal quartz veins filling cracks (source unknown):

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(left) quartz veins later deformed Quartz solubility is high in

(a) high P/T (Pressure/Temperature) water and (b) when there is dissolved salt in the water. When a crack forms, the dilatation drops the pore pressure and reduces quartz

solubility, causing precipitation. In metamorphic complexes quartz is dissolved as water is evolved in dehydration reactions at high P and T at depth, and precipitates as the solute fluid works its way up the metamorphic pile and is depressurized. Early-formed veins are themselves later deformed as deformation progresses. Metamorphic rocks can lose a lot of volume in this way. A simple criterion for hydraulic fracture is:

p - 3 = T, where T is the tensile strength of the rock. i.e. cracking occurs

when pore pressure exceeds 3 by the amount of the tensile strength (on the order of 0.1 MPa) Shape of the Mohr-Coulomb criterion when a rock has tensile strength

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Byerlee‘s rule for rock-on-rock sliding friction Jim Byerlee, of USGS, proposed in 1977 (from laboratory measurements) that to a good approximation the friction coefficient of rocks is nearly independent of rock type. This has become known as Byerlee‘s rule, and is widely employed in estimating the behaviour of the brittle upper crust.

The open symbols in this graph are some of the data assembled by Byerlee (compared to in-situ stress measurements in the KTB borehole [a 9 km deep hole drilled in southern Germany during the 1990s]). At shallow depths the friction coefficient is about 0.85, decreasing to about 0.6 at greater depths.

Description of the Failure of Porous Rocks in the Upper Continental Crust - Critical State Mechanics

Rocks with porosities less than 10% behave according to the linear Mohr-Coulomb description over most of the upper continental crust (top 15 km). Mohr-Coulomb failure involves a small amount of dilatation (on the order of 1 or 2%) to allow for the formation of cracks, and this is why such rocks get stronger at higher pressures. More porous rocks, including many reservoir rocks, require a more complex description of failure because volume loss often occurs during burial and deformation. How does volume loss (compaction) occur? At shallow depths (down to about 3 or 4 km), compaction is Mechanical, involving initially rearrangement of grains and, at higher pressures, grain fracture and sliding between fragments. At greater depths, there is a transition to Chemical compaction and porosity loss, involving interpenetration of grains by solution at contact points

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and precipitation of cement in pore spaces. Primary porosity of loosely compacted sand is about 42%. Mechanical compaction typically will only reduce this by up to 10%. Chemical compaction can reduce porosity (eventually) to near zero. Generally, older rocks have progressively lower porosities, first because they have had more time for the processes to operate, and second because they have usually been more deeply buried. Elimination of porosity in (siliciclastic) sediments usually takes up to about 50 Ma (limestones are quicker). Mechanical compaction is largely time-independent - a given effective pressure produces a given amount of compaction. Chemical compaction is generally time-dependent. A rock compacts more at (a) a higher temperature and (b) over a longer time period. This is to be expected as chemical processes such as reaction and diffusion are temperature and time dependent according to the Arrhenius rate law:

rate exp(-H/RT) The above figures show data from an experimental study carried out in this dept a few years ago on the rate of chemical compaction of a synthetic silty mudstone. It illustrates the temperature and time-dependence of compaction rate. This type of study leads to a model from which the amount of compaction at any given depth and elapsed time can be predicted.

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Porosity loss by mechanical compaction

This is usually described by the normal compaction (or consolidation) curve: The hydrostatic pressure here is effective pressure, the total applied pressure less the pore fluid pressure. Thus if pore pressure equals applied pressure, there is nothing to drive the compaction. Rocks wholly or partially overpressured in this way remain underconsolidated (more porous than they would normally be for a given depth of burial). This is good for hydrocarbon reservoirs because porosity is not lost by compaction, but it does mean that once fluid starts to be extracted, compaction will occur and the rock column will subside (i.e. a production platform will subside - a very expensive mistake, e.g. Ekofisk platform in 1979), unless the fluid removed is replaced by pressurized water, for example. Erosion leads to an overconsolidated state. That is, the porosity was previously reduced by a greater amount than that which corresponds to the effective pressure the rock is now subjected to. Knowing the consolidation state of reservoir rocks is vital to the management of a reservoir. To a good approximation, the value of effective pressure necessary to start compaction x the mean grain size is independent of rock type. The graph (right) illustrates this. It is effectively the normal consolidation curve in log/log coordinates, and represents a very useful rule-of-thumb for porous sandstones in particular. The graph shown (right) is based on Zhang et al. J.Geophys. Res, 1990, 95, 341-352, but seems to apply to all sandstones.

Yield in Porous Materials

Porous materials, then, can yield even under effective hydrostatic stress, by the permanent collapse of pore spaces. Compaction can also be induced by combined hydrostatic and deviatoric stresses, until a balance between

30

pore compaction and dilatancy due to new crack formation occurs. This balance is called the critical state. It corresponds to deformation at constant porosity, at constant flow stress, at the peak on the yield surface. This figure (right) illustrates a pressure-sensitive yield surface at low mean pressures that closes to produce a yield point (P*) on the hydrostat. This is called a capped yield surface. (note: effective mean stress

= [(1 + 2 + 3)/3] - P ) To a good approximation, the critical state line is independent of rock type. Whilst the yield surface is expanding, deformation is dilatant (this is equivalent to Mohr-Coulomb behaviour). As it contracts it is compactive. The contracting region (of negative slope) is called the region of shear-enhanced compaction. This is because the rock compacts, even though the effective pressure is less than P*, because the non-hydrostatic part of the stress state can assist in bringing about compaction. Stress paths link the yield and critical state lines. As porosity progressively collapses in the post-yield region, the rock hardens until the critical state line is reached. The above example of Penrith sandstone (initial porosity 25%) shows how the capped yield surface diagram (below) relates to the corresponding

31

stress/strain data: The figure (right), shows data for 3 sandstones of different porosities (Penrith 25%; Darley Dale 13%; Tennessee 7%), showing the capped yield surfaces, the stress paths linking yield and critical state, and the critical state line (independent of rock type). From Cuss, Rutter & Holloway, 2003, Int. J. Rock Mech Min. Sci., 40, 847-862, also the figures below. The stress paths have a slope of 3, which is a property of the triaxial deformation method, because increasing the differential stress by x increases the mean stress by x/3). These stress paths stay on the same line as differential load is increased because these are drained tests. That is, pore pressure is kept constant by allowing fluid to flow into or out of the specimen according to whether deformation is dilatant or compactive. Undrained behaviour is considered later. Yield curves can be linked into a continuous yield surface in a plot of deviatoric stress (proportional to differential stress) vs effective mean stress vs porosity (x grain size), to show how the yield surface expands as porosity is reduced and how the critical state line separates regions of dilatant and compactive deformation. The yield surface (right) incorporates the individual yield curves for the 3 rocks above.

32

The effects of porosity differences between different rocks can be taken out by normalizing the plot of differential stress vs effective mean stress. We can divide mean pressure and differential stress by P* for that rock. This tends to collapse all yield surfaces onto a common curve: This means that one can compute the expected behaviour of a given sandstone from knowledge only of its porosity and grain size. This seems likely to prove a powerful unifying concept, and is presently a very active area of research. (Fig. above from Wong et al., J.Geophys Res. 1997)

This type of approach (called Critical State Theory) has proved particularly valuable in the description of the behaviour of soils and weak rocks (including partially molten rocks and rocks undergoing deformation with dehydration reactions). How do strain and porosity vary between yield and critical state? These data for porous (dehydrated) serpentinite at 450°C illustrate what happens. Porosity change (progressive compaction) and strain accumulation contours tend to track the yield curve.

33

Above data from: Rutter, Llana-Funez & Brodie, 2008. J. Struct. Geol, 165, 639-649. What about undrained behaviour? If fluid is prevented from leaving or entering the pores of the rock during deformation, it is forced to deform at constant volume. Thus if the pores try to dilate the pore pressure drops, or it rises if the pores try to compact. This causes stress paths to deviate from the simple linear paths characteristic of drained loading. In nature, for example, undrained loading during compaction causes pore pressure to rise, so the formation becomes overpressured. The figure (right) illustrates how such variations in pore pressure cause the stress path to deviate. It is vital for petroleum geologists and engineers to understand these concepts. They underpin modern approaches to reservoir evaluation and management.

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Describing high temperature plastic flow The main mechanism of high temperature flow (flow from mid-crustal depths and deeper) is by intracrystalline plasticity - involving internal distortion of grains by intracrystalline slip.

Intracrystalline plastic flow is a constant volume process and hence is

pressure-insensitive. On the other hand the flow stress is reduced markedly by increasing temperature or decreasing deformation rate, unlike cataclastic

deformation. Absolute temperature (T), flow stress () and deformation rate =

(d/dt ) are linked by the equation

d/dt = A f() exp (-H/RT) in which A is an empirical constant, H is the activation enthalpy for flow and R is the gas constant.

--------------------------------------------------------------------------------------------- A note about Strain rate, = d/dt : dimensions are sec-1.

-2 -4 -6 -8 -10 -12 -14 -16 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! --- lab. strain rates--- --- natural strain rates --- log10 strain rate (sec-1)

------------------------------------------------------------------------------------------------ f() = n at low stresses (or high temperatures) and

f() = exp (B) at high stresses (or low temperatures), where B is an empirical constant. Exponent n takes a value between 3 and 7, depending on the material. Taking logs to base e : at high temperature

ln d/dt = ln A + n ln - H/RT and at lower temperature :

ln d/dt = ln A + B - H/RT

Thus at constant T, eq.1 implies a linear relation between ln d/dt and ln ,

with slope n and eq.2 a linear relation between ln d/dt and , of slope B.

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At constant stress the relation between

ln d/dt and 1/T is linear. This is called an Arrhenius diagram, and the slope is -H/R. A collection of experimental data for flow stress might look like:

Typical n values:

Halite 5.5 Calcite 7.6 Quartz 3.4 Olivine 3.0

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Relation between cataclasis and intracrystalline plasticity

At progressively lower confining pressures, as T is increased (or d/dt increased), there is a mechanism transition from cataclasis to plasticity (sometimes erroneously called the brittle-ductile transition). Application to the Earth In the Earth, confining pressure increases with depth, temperature increases with depth, thus there is a mid-crustal transition from cataclasis to intracrystalline plasticity, and a mid-crustal strength maximum. Recall the diagram below-left.

The strength maximum may correspond to the depth limit of crustal earthquakes, which should be restricted to the brittle part of the crust.

To a first approximation, the upper

crust is made of quartz/granite, the lower crust of feldspar/mafic minerals and the upper mantle of olivine. Thus the lithosphere shows a coarse lithological stratification, which should be reflected in the strength profile with depth.

The resultant strength profile (right) is called Rheological Stratification of the Lithosphere : (Rheology is the study of flow) In order that Plate Tectonics can work, the stress distribution with depth at a plate boundary must meet the failure envelope everywhere (Whole Lithosphere Failure). Before this happens, strong intervals may behave as elastic struts that hold the lithosphere together. These constitutive flow laws can be used to extrapolate laboratory

37

mechanical data to geological conditions, so that mathematical models of geological deformation of the lithosphere can be made. In summary, such extrapolations indicate that : (a) at low temperatures and pressures (uppermost one-third of the continental crust), rocks fail by brittle faulting which can be seismogenic. Strength increases with depth because the process is temperature- and rate-insensitive. (b) with increasing depth there is a transition to ductile, plastic flow. Because it is pressure-insensitive, temperature dependence dominates and strength decreases with depth. If there are also variations in rock type with depth (e.g. across the Moho), corresponding fluctuations in rock strength also occur. This leads to the concept of rheological stratification of the lithosphere.

What gives rise to in-situ stresses in the Earth? Knowledge of in-situ stress state can be vital for geo-engineering. However, it is one of the most difficult things to measure. In-situ stress can arise from: (a) Forces arising from plate tectonics, e.g. ridge push and trench pull. These generate a maximum compressive stress linking constructive to destructive plate boundaries (e.g. world stress map showing orientation of maximum compressive stress - European part shown on right http://www.world-stress-

map.org). (b) Forces arising from lithospheric bending due to surface or buried loads. (c) Stresses arising from uplift, burial and compaction. (d) Local radial stresses due to features like diapiric uplift (e.g. salt) and pluton emplacement. (e) Stresses induced through volume changes associated with mineral transformations (e.g. in mantle transition zone, or serpentinization).

38

Storage of stresses in rocks (residual stresses) These arise from arrays of defects in crystals (e.g. dislocations), or from cementation of a stressed array of crystals, e.g. during burial and cementation or by crystallization of residual melt in a stressed matrix. The release of stored stresses by cracking during uplift and erosion is responsible for jointing in rocks. Item (c) above : Imagine burial with no lateral strain, because of the constraint of rocks to either

side. In this case z = gz, and a horizontal stress is induced through the Poisson

ratio effect, given by x = z / (1 - ). Such a stress path during burial is called a ko stress path, and is often invoked to describe (or infer) the stress state in a sedimentary basin because a basin can be regarded as a thin layer unable to slide on its contact with basement rocks underneath and prevented from lateral expansion by the wide expanse of neighbouring sediment on either side.

z (= 1) is given by gh, hence x is given by gh /(1 - ), and will be smaller.

The deviatoric (proportional to differential) stresses i‘ (where i = 1, 2 or 3) at

depth are obtained by subtracting the mean stress, () and pore pressure if present.

= (1 + 2 + 3)/3, hence 1‘ = 1 - = gh(1 +)/([3(1 - )]

and 2‘ = 2 - = - gh(1 - 2)/[3(1 - )] Thus the horizontal deviatoric stress is tensional, although the total stress is compressive. Thus compaction of sediments in nature is never likely to take place under hydrostatic stress. Only for very weak rocks might stress approach hydrostatic during normal burial. The ko path (right) is usually flatter than the critical state line, suggesting that compaction occurs in the shear-enhanced compaction regime. The ratio of stresses

x/z in the zero lateral strain path is called ko (or the coefficient of earth pressure at rest), and its value is commonly about 0.4 in sands and higher (about 0.7) in clay rich rocks (the critical state line has a slope near 1).

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Since the vertical stress is largest the stress state would be expected to give rise to normal faulting. Faulting as a mode of failure would only be possible by ‘trading‘ compaction for the lateral extension usually associated with normal faulting. Consider the ‘inverse‘ problem, ko uplift and erosion. A pluton crystallizing at depth under hydrostatic stress has the overburden removed under conditions of no lateral strain (see Turcotte & Schubert, Geodynamics (Cambridge) p.109). Thus erosion will induce stresses in the pluton as it approaches the surface. The

change in vertical stress dz is -g dz, progressively restoring it to zero. Thus the change in horizontal stress becomes :

dx = dz /(1 - ) from its original value gz. The horizontal stress at

the surface becomes gz - /(1 - ) = (1 - 2)gz/(1 - ). This can be a substantial horizontal compressive stress, that can cause surface-parallel cracking called sheeting, and

lead to a characteristic erosion pattern of granitic terrains (see images below).

How high are in-situ differential stresses?

The figure (left) shows how vertical

stressz (lithostatic gradient) and pore

with = 0.4) pressure (hydrostatic gradient are expected to vary with depth in a sedimentary basin. The shaded area shows how pore pressure might be enhanced by compaction below a seal horizon. Excess pore pressure is easily measured in a borehole,

gz, but the vertical stress is calculated from principal stresses in the horizontal plane are estimated from hydrofrac

40

measurements. (source: Engelder, Stress in the Lithosphere)

The graph left shows variations of horizontal principal stress Shmax and Shmin in the San Andreas fault pilot borehole, about 1.8 km from the fault zone (from Zoback, Reservoir Geomechanics). The pore pressure gradient Pp is also shown. The stresses are consistent with values expected in a strike-slip fault regime with a friction coefficient of 0.8 (equivalent to laboratory-determined values).

This graph compares stresses measured in the KTB (German continental deep borehole) with laboratory friction values. The in-situ stresses are closely comparable to the laboratory-determined values (coefficient of friction between 0.85 and 0.6), indicating that the upper crust can sustain stresses over long time periods just below those needed to initiate slip on faults. This map (right) shows max principal stress orientations on either side of the San Andreas fault (data from world stress map). The orientations are at a high angle to the fault zone, meaning that resolved shear stresses along the fault zone are very low, yet the fault still moves.

There is also a lack of a heat flow anomaly along the fault zone, consistent with a weak fault.

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Finding the explanation for this paradox of a weak San Andreas fault in a strong crust is the main reason behind the present deep drilling project. Are the fault rocks anomalously weak, or is there high fluid pressure in the fault zone?

Stress Transfer around Faults

A slip event on a fault causes a redistribution of stress around a fault, leading to a stress change that can transfer stress to nearby faults and possibly trigger an earthquake a short time afterwards. This is called stress triggering of earthquakes. It has been successfully applied in California and in Turkey. In the latter case, a succession of earthquakes in recent years have propagated from east to west along the North Anatolian fault. Stress transfer calculations place a likely short-term future earthquake in the Sea of Marmara just south of Istanbul.

The figure (right) shows calculated stress changes around a fault segment following an earthquake. These changes affect the stress on nearby faults, potentially destabilizing them. Note: the stress changes are very small. (below) Stress changes arising from earthquakes on faults around Landers (S. California) eventually led to the triggering of a large earthquake on the Landers fault segment. This triggering mechanism can result in earthquake storms.

(Figures source: Mechanics of

Faulting, by C. Scholz)

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Displacements following earthquakes can now be measured by satellite using Interferometric Synthetic Aperture Radar (InSAR) with a resolution on the order of 1 cm (above image). This allows stress transfer calculations and modelling of coseismic displacement fields to be made with great accuracy.

Halokinesis and Salt Tectonics Under most crustal conditions, halite (NaCl, rocksalt) is much weaker (less resistant to intracrystalline plastic flow) than any other

common rock type. Halokinesis refers to the movement of salt by flow. Halite is also much less dense than other rocks, even than porous siliciclastic sediments, at about 2200 kg/m3 compared to 2500 to 2700 kg/m3. Bedded salt is also very impermeable, thus it (and other evaporite rocks such as gypsum) can form very effective stratigraphic oil traps and seals.

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The weakness and low density of salt mean that it is possible for salt bodies to rise through the overburden under the driving force of gravity, punching its way through along fault planes or by upwarping and puncturing overlying sediments (diapirism). Mud can also be mobilized in the same way to produce mud volcanoes.

(above: hot oil globule rising through cooler immiscible oil in a ‘lava lamp’ http://www. fotosearch.com/photos-images/lava-lamp.html) (Note: Diaprism is a manifestation of gravity tectonics, but the term gravity tectonics also applies to the emplacement of extensional nappes in the late orogenic collapse of mountain belts). (above) NASA satellite image

of salt intrusions in SW Iran – look on Google Earth. There are good examples of salt tectonics in the oilfields of the Gulf of Mexico, the North sea and in N. Europe. In SW Iran the salt in many places has punched right through to the surface, forming salt glaciers that can flow downhill at ca 1 m/day when there is rainfall. Salt layers beneath the seafloor can also flow down slopes, mobilizing the overlying sediment.

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(Above: Salt domes in northern Germany (http://my.opera.com/nielsol/blog/salt-tectonics) and beloe, offshore Angola (http://en.wikipedia.org/wiki/Salt_tectonics). A great deal of salt tectonics imagery is available by searching salt tectonics on Google.

In the column of salt forming a diapir, the z (salt) = 2200 g z whereas in the

sediments on either size z (sed) = 2600 g z. The pressure difference is approximately 400 g z. Thus the surrounding sediments act like a heavy plunger pushing on the underlying bed of salt and pushing it into the lower pressure region which is the diapir root. Thus the higher the diapiric column, the greater is the resultant upward driving force. Salt tectonics is important (and leads to gas/petroleum traps in many parts of the world, e.g. Holland and N Germany, Iran, Gulf of Mexico. During the late Miocene (Messinian stage, ~ 8 Ma), the whole of the Mediterranean basin dried out when the Straits of Gibraltar were closed off, forming salt deposits on the ocean floor (the Messinian salinity crisis). These are now buried under a thin sediment veneer. Similar salt deposits now lying on the continental slope beneath the Red Sea are able to slide downslope, deforming the sedimentary cover above. As the salt rises higher it can reach a level of reduced or neutral buoyancy, as the less compacted overlying sediments become less dense. It can then spread out laterally with a range of intrusive geometries. Updoming of surrounding sediments due to diapirism, and formation of sealing structures in faults by penetrating salt, mean that evaporite structures can form a range of oil trap and seal geometries.

oooOOOooo

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Nature of Problems in Solid Mechanics

Problems may be classified as:

(a) Statically Determinate

Provided it can be usefully assumed that materials are wholly rigid, undeformable

elastically or plastically, so that the problem of stress can be solved entirely from the

equilibrium condition.

(b) Statically Indeterminate

Both stress AND strain components must be found, so solutions must satisfy the

equilibrium equations, the stress-strain equations and the strain compatibility

conditions.

We will consider each of these in turn.

Examples of Statically Determinate Problems

(a) A picture suspended by an inextensible

string.

The problem is solved by writing down the equilibrium

condition:

cosW

Wcos

46

Where T is the tensile force in the string.

(b) A thin walled pressure vessel under internal pressure,

P. We require the tensile stress T in the vessel wall. The thin

wall means we can ignore the variation of the stress across the

thickness of the shell:

Internal pressure x area = wall stress x circumference x

thickness

Pr rthence P rt

MORE ABOUT STRESS

Homogenous Stress in 3D and Stress Invariants

It can be shown that principal stress magnitudes are given as the 3 roots of a cubic equation (i.e eigenvalues of the characteristic equation of the matrix of terms in the stress tensor:

3 – I1

2 + I2 – I3 = 0

Where:

I1 = 1 + 2 + 3 = x + y + z

I2 = 12 + 2 + 13 = xy + yz + xz – xy2 – yz

2 – zx

2

I3 = 12 = xyz + 2xyyzzx - xxy

2 – yzy

2 – zxz

2

These are called the STRESS INVARIANTS. They are properties of the stress state that do not depend on the choice of reference frame. It is not surprizing the the values of the principal stresses can be obtained from invariants, because they do not depend on choice of reference frame. Strain energy too can be expressed in terms of invariants. Orientations of principal stresses are best obtained by the iterative method based on the radius-normal property of the representation quadric.

The Principle of Superposition states that 2 stress fields (referred to the

SAME coordinate FRAME) can be summed up adding their corresponding

47

components:

ij = Sij + ij

Thus a total stress tensor can be broken down into the sum of a MEAN STRESS :

(= 1/3 ij kk) (where ij is the Kroneker delta and summation is implied)

and a DEVIATOR stress, ij‘

(ij‘ = ij - 1/3 ij kk)

Deviator = total stress - mean stress

We can also separate the elastic strain energy into Wtotal = Wvolume change + Wdistortion

thus Wvolume = I12 (1-2)/[12G(1+)]

Wtotal can be expressed as 3Wvolume - 3 I2/ 6G

Hence Wdistortion = (I12 - 3I2)/ 6G = I2‘/ 2G

where I2‘ is the second invariant of the deviatoric stress tensor.

Infinitesimal Strain in 2 Dimensions

Distortion of a rectangle. Note: all changes at length and angle are SMALL (<1%), so that it can be assumed all products of derivatives and 2nd and higher derivatives can be neglected. u and v are displacements along x and y respectively.

48

Line AB changes to A1B1, which is approximately equal to A1B1’ because is small. Similarly A1D1 A1D1’

A1B1’ = x + u . x : A1D1’ = y + v . y

x y

B1B1 = v , : D1D1’ = u

x y

The infinitesimal strains are:

x = A1B1 - AB = x (1 + u

/x) - x = u

________ ______________ __

AB x x

Similarly yv

y

tan = = v and tan u

x y

Because angle DAB was originally 90°, = = +

vu

xy

Note: this is the TOTAL shear strain. To form a symmetric tensor it is usual to define

+ )

2

The components x y defined above are not independent, because they

arise from gradients of only 2 displacements, u and v.

x = u : x = 2u : 2x = 3u

x y xy y2

xy2 = P

y = v : y = 2v : 2y = 3v

y x xy x2

x2y = Q

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= v + u : = 2v + 2u : 2 = 3v + 3u

x y x x2 xy

xy x

2y y2x = R

Note that P + Q + R, which provides an equation that relates x, y and

2 = 2ex + 2y

xy y2 x2

This is called the STRAIN COMPATIBILITY EQUATION. There is only one in 2D, but six of them in 3D. These conditions must be satisfied for any stress distribution in an elastic solid to be valid.

Variations in Stress through a Body

Let us assume that the stress components vary with distance x, y, z through a body at the rates

xyzxyxyxy

x yz xyz

Considering only the forces in the x direction first, we can find the resultant force in

that direction. X is any body force along x (force/volume). Consider a block of

material of dimensions x, y, z:

Resultant force = stress x area - stress x area = Fx

Resultant force due to normal stresses:

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(x + x x) y z - x y z

x

stress area

Plus forces due to shear stresses and body forces:

+ (yx + yx . y) xz - yx xz

y

+ (zx+zx.z) xy - zxxy + X xyz = Fx

z

Force = mass x acceleration = a . xyz ( xyzmass )

= 2u xyz

t2

u = displacement in x direction. Divide throughout by volume (xyz) of body and collect terms :

x + yx + zx + X = 2u .

x y z t2

Similarly in y and z directions :

xy + y + zy + Y = 2v .

x y z t2

xz + yz + z + Z = 2w .

x y z t2

These are called the stress equations of motion.

If 2u = 2v = 2w = 0,

t2 t2

t2

the body is at rest or in steady motion, and equations are called the STRESS EQUILIBRIUM equations. They must be satisfied for all valid stress distributions. In 2D there are 2 equilibrium equations:

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x + yx = 0 : xy + y = 0

x yxy

Generalizing using summation convention in suffix notation, the equations can be written

j = 0

x

Stress trajectories: In 2D these can be used graphically to illustrate variation in

stress through a body. Stress trajectories are everywhere parallel to the principal stress axes. Consider stretching with necking of a ductile bar :

Trajectories are ORTHOGONAL Total force must be constant, therefore stress trajectories CONVERGE towards regions of increased stress and DIVERGE toward regions of lower stress.

Local orientation of 1 and 2 are given by tan 2 = 2

x - y

From the theory of stress functions, can often be obtained as a function of x & y,

from which the local orientations of principal stresses can be plotted on a grid of points, and the directon field sketched in. A quick graphical method is the Method of Isoclines:

Writing the slope of the stress orientation as dy/dx, we may have dy = f(x,y)

dx

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Solutions to this D.E. are y = f (x), a family of orthogonal curves which are the

stress trajectories. If not analytically soluble, the ISOCLINES may be determined:

e.g. if dy = x2, Set k = x

2 hence y = x

2

dx y y k

(this particular DE is This is a family of parabolas easily soluble!) for different k values

The stress trajectories intersect the isocline at a constant angle to the x axis (given

by k), thus the trajectory pattern can be sketched :

Plastic Yield Criteria

The Mohr-Coulomb criterion is reasonably satisfactory for rocks failing by brittle faulting at low temperatures, but for plastic yield (no mean pressure dependence) of rocks and metals at low temperatures, other criteria are better. The two simplest ones are the TRESCA and VON MISES criteria.

Tresca This states that

yield occurs when the max

shear stress (1 - 2)/2 reaches a critical value k. If

2 = 0, we can plot If we add the third dimension

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and all principal stresses are non-zero, the yield surface becomes a regular

hexagonal prism centred on the hydrostat1 = 2 = 3.

von Mises

He suggested yield occurred when the 2nd deviatoric stress invariant reached a critical level C2. The criterion can be written as:

6C2 = (1 - 2)

2 + (2 -3)

2 + (3 - 1)

2

The significance of C can be seen from a consideration of yield under simple

uniaxial loading, 1 = u ; 2 = 3 = 0

6C2 = 2 u

2, hence C = u / 3

The von Mises yield criterion effectively states that yield occurs at a critical value of

distortional strain energy, for Wdist = I2‘/2G

Octahedral Stresses Imagine a plane inclined equally to the 3 principal stresses. This is the Octahedral Plane.

The normal stress across it n oct = (1 + 2 + 3)/3 = I1/3 The max resolved shear stress across it

oct is

oct = (1/3) {(2 - 3)2 + (3

- 1)2 + (1 - 2)

2} = (2I2/3)

thus the von Mises criterion is equivalent to the statement that yield occurs when

oct reaches a critical value.

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Rocks yield in a pressure-sensitive way at low pressures, but flow is pressure-insensitive at higher pressures (right).

Relation between Tresca and von Mises criteria

In 2D the von Mises is an ellipse that circumscribes the Tresca hexagon. i.e. they are equivalent at 6 points. In 3D a von Mises circular cylinder circumscribes the Tresca hexagonal prism, thus deviations from the hydrostat in any direction produces yield. Experiments show that von Mises describes data better. Tresca is a more conservative criterion than von Mises. There are other variants on the Tresca and von Mises criteria that we will not consider further here.

Statically indeterminate problems Here we require to find all the stress and strain components (total 6 in 2D, therefore need 6 equations, these are the stress equilibrium, strain compatibility and elastic stress strain equations).

If we first consider PLANE STRESS (y, xy, zy = 0).

The solid is assumed ISOTROPIC and ELASTIC and HOMOGENEOUS. The stress/strain relations become:

x = (x - z)/E (1)

z = (z - x)/E

y = -(/E)(x + z)

(ie: plane stress does not give rise to plane strain unless x = - z)

zx = xz = xz / G : G = E

2(1 + ) The equilibrium equations without body forces are:

x + = 0 : + z = 0 (2a) (2b)

x z x z

55

The compatibility condition is:

2x + 2z = 2 (3)

z2 x2

xz

Substituting (1) into (3) we can eliminate strains:

2 (x - z) + 2(z - x) = 2(1 + ) 2

z2 x

2 xz (4)

Differentiation of (2a) wrt x and (2b) wrt z gives:

2x + 2 = 0 : 2 + 2z = 0

x2 zx xz z2

Adding:

2 2 = - 2x - 2z (5)

zx x2 z

2

Substitute (5) into (4) to eliminate shear stress terms:

2 (x - z) + 2 (z - x) + (1 + ) { 2x + 2z } = 0

z2 x2

x2 z2

rearranging causes the terms to cancel out:

2x + 2z - (2z + 2x) + 2x + 2z + (2z + 2x) = 0

x2 x2

z2 x2

z2 z2

z2 x2

2 (x + z) + 2 (x + z) = 0

x2 z2

OR ( 2 + 2 ) (x + z) = 0

x2 z2

OR 2 (x + z) = 0 (6)

2 is the Laplacian operator, and (6) is called the STRESS COMPATIBILITY

CONDITION. Stress states satisfying (6) also satisfy (1), (2) and (3).

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Note that: x + z = 1 + 2 = I1

Also = hydrostatic part of ij = I1/2 = P Therefore, for any hydrostatic pressure distribution

2 P = 0

This is Laplace’s equation. It also applies to ANY steady-state distribution of potential (e.g. electrical potential, ionic concentration). Solutions take the form

P = f (x, z), i.e a contour map of the potential in space.

Laplace’s equation is the first of a family of partial differential equations that collectively describe an enormous range of physical phenomena:

2 = k is Poisson’s equation

2 = k is the heatflow, or transport equation. Its solutions

t describe monotonic approaches of the potential towards equilibrium with time.

2 = k Has solutions periodic in time and is the wave equation

t

2 = ik Has solutions periodic in time but with quantized energy

t levels (amplitudes of the waves). This is Schroedinger‘s equation.

PLANE STRAIN

Here xy = zy = y = 0 (i.e. y = 0 or a constant)

y

Using y = 1/E (y – (x + z)) and setting y = 0 :

y = (x + z)

i.e. stress y must be applied to keep y = 0 The stress/strain equations become:

57

x = 1/E {(1 – 2) x – (1 + )z}

z = 1/E {(1 – 2) z – (1 + )x}

xz = xz = 2(1 + ) xz G E

The equilibrium and strain compatibility conditions are the same as for plane stress. Combining the 3 sets of equations leads to the same stress compatibility equation,

2 I1 = 0

AIRY STRESS FUNCTIONS

The Airy stress function, , is defined as:

x = 2 : z = 2 : xy = - 2

z2 x2

xz

diffn. w.r.t. x diffn. w.r.t. z

x = 3 = - 3

x z2x z xz2

Add

x + = 0

x z

i.e. is defined in such a way that it satisfies the stress equilibrium conditions.

Substitute for x and z in the stress compatibility equation, 2(x + z) = 0

( + ) ( 2 + 2 ) = 4 + 24 + 4 = 0

x2 z2

x2 z2

x4 z2x2

z4

OR (

2) = 0

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OR 4= 0

This is the BIHARMONIC EQUATION. Any function (x,z) satisfying this is

automatically a valid stress distribution in PLANE STRESS or PLANE STRAIN, i.e. a 2D stress distribution.

If 1 and 2 are solutions to 4 = 0, then1 +2 is also a solution. i.e. the

superposition principle applies. Examples of simple solutions : (A) Polynomial Functions. Any polynomial to 3rd degree automatically satisfies biharmonic equation, because the 4th order derivatives are zero. e.g. Let

= c + ax + bx2 + cy

2 + dxy + ex

3 + fy

3 + gx

2y + hxy

2

linear quadratic cubic terms terms terms

1 2 3

Linear terms = 1 When differentiated twice to get stresses, they are all zero.

Linear terms give rise to NO STRESSES.

Quadratic terms e.g. 2 = k1x2 + k2 y

2

Hence x = 2k2 : y = 2k1 : xy = 0

Therefore x and y are1 and 2

This represents:

x and y are constant through the material The stress-strain equations can then be applied to find the strains:

x = 1 (2k2 - 2k1) = 2 (k2 - k1) E E

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y = 2 (k1 - k2) = z = -2 (k1 + k2) E E

z = 0 only when k1 = -k2

To find displacements

x = u u = x dx

x

= 2(k2 - k1)x + f (y, z) + C E

Boundary conditions must be specified before displacements can be found.

The term in xy, set = dxy, x = 0, y = 0, = d i.e. the xy term gives rise to a homogeneous shear stress Overall, the quadratic terms describe a HOMOGENEOUS, GENERAL STRESS STATE The Cubic Terms

(a) = ex3

x = 0

y = 6 e x

= 0

y increases linearly with x alone

(b) = fy3, x = 0 : y = 6fy ; = 0

i.e. x increases linearly with y.

This is the type of stress distribution that occurs in a bending elastic beam: Stresses are proportional to distance from neutral surface.

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(c) Terms in x2y and y

2 x

Let = gx2y x = 0 ; y = 2gy ; = 2gx

Thus y = Linear function of y

Similarly, for = hyx2

x = 2hx ; y = 0 ;= 2hy

Thus the cubic terms give rise to a heterogeneous stress state where components vary linearly with x and y. For polynomial terms of higher degree the stress components vary non-linearly.

(B) Periodic Functions that satisfy the biharmonic

e.g. = ey cos x

x = 2 = ey cos x 4 = e

y cos x

y2 y4

y = 2 = - ey cos x 4 = e

y cos x

x2 x2

= 2 = + ey sin x 2 4 = - 2 e

y cos x

xy x2y2

add = 0 hence this function satisfies the biharmonic eq.

As y increases the stress components increase exponentially. In the region y=0 the components look like:

Plotting of stress trajectories :

Use tan 2 = 2 = -22xy

x - y (2 /x2

) - (2 / y2)

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Here, tan 2 = 2ey sin x / 2e

y cos x = tan x

2 = x, = x/2 Using isoclines method,

dy/dx = tan x = tan2

When x = 90°, = 45°

Isoclines are k = tan x

When k = 1, x = 45°, = 22.5°

hence x = tan-1

k, a family of

straight lines parallel to y axis. Stress trajectories are similar to those in a buckled viscous layer.

Delineation of probable failure zones

Mohr Coulomb or von Mises etc. criteria may be used to map out 1/2 conditions

that may lead to failure: e.g. for Mohr Coulomb _________________

1, 2 = ½(x + y) + ½ (x - y)2 + 42

for each point in the material, or may be expressed as

1, = f1 (x, y); 2, = f2 (x, y)

Thus 1 (x, y) = a + b 2 (x, y)

Geological Example of Cartesian stress functions

The best known ‘classical’ study is that of Hafner (1951) “Stress distributions and faulting” BGSA 62, 373-398. Hafner used Anderson’s “standard state” concept to simplify the problem:

1 = Anderson‹s Standard state (Hydrostatic stress)

2 = Deviator (Supplementary stress, causing deformation)

These can be superimposed by addition:1 + 2

Boundary conditions at Earth‘s surface, = 0, @ z = 0

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also z = 0 at z = 0. z = f(z) is assumed given by z = gz. If there is NO LATERAL STRAIN, the horizontal stress induced by gravity is

x = z / (1 - )

x = z only if = 0.5 (incompressible fluid). However, for the “standard state”, we assume

x = z = gz

and = 0

With pore fluid pressure contribution, z = gz - P(z)

Derivation of Supplementary stress states (2) If we assume the supplementary stress state consists only of a horizontal stress, the implications can be explored:

thus z = 0

i.e. 2 /x2 = 0 for all values of z.

This can be integrated twice to find the range of possible supplementary stress functions :

2 = z dxdx = af1(z) + b

hence 2 = axf1(z) + bx + cf2(z) + d This must satisfy the biharmonic equation, therefore the 2nd order derivatives of

f1(z) and f2(z) must be zero, constants or linear functions of z.

The stress components therefore become :

x = 22 / z2 = ax 2f1(z)/ z2

+ c 2f2(z)/ z2 : z = 22 / x

2 = 0

= -22 / xz = - (af1(z) + b) = -af1(z)/ z

By examining combinations of linear functions, constants or zero for f1(z) and f2(z) Hafner found that 3 groups of equations can be found that satisfy the surface boundary conditions,

z = 0 : = 0 : f1(z)/z = 0

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The 3 groups are as follows :

x z

cz + f 0 0 Group I ax 0 -az Group II axz 0 -az

2/2 Group III

Linear combinations of these solutions for supplementary stresses are also possible. Hafner explored all of them, and concluded that superposition of solutions

I and II were geologically most interesting (note g z has to be added to the normal stress components to get the total stresses) : Superposition of solutions I and II

x = ax + cz + f + gz ; z = gz ; = -az These can be illustrated graphically as follows :

Orientations of principal stresses obtained from: tan 2 = 2 = -2az

x- z (ax+cz+f)

= ½ tan-1

( -2az ) ax+cz+f _____________

Values of 1, 2 computed from ½(x + z) + ½ (x + z)2 + 42

________________

= ½(ax+cz+f+2gz) + ½ (ax+cz+f)2 -4a2z

2

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Boundaries of potential failure areas can be delineated by

1 (x, z) = A + B (2 (x, z)) Resultant Pattern : Geological ‘realization’ A foreland fold and thrust belt! Hafner carried out sandbox experiments to illustrate this geometry of upward-curving thrusts.

Stress Functions in Plane Polar Coordinates

Relation between Cartesian and polar coordinates

x = r cos

y = r sin

Stress components therefore become , r and

r

The body of theory is comparable to the Cartesian case so the derivations will not

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be repeated. The stress equilibrium equations become :

r/ r + /r + (r - )/r = 0 in the radial direction and

/r + /r + 2/r = 0 in the tangential direction. The infinitesimal strain-displacement equations are :

r = u /r ; = u/r + v/r) ; = u/r + v/r - v/r The strain compatibility equation is

2r / r = 2r / 2 + r 2r / r

2 - r r / r

The stress/strain equation have the same form as in Cartesians. The stress compatibility equation also looks the same :

2(r + ) = 0 and may be derived as before. Alternatively one may convert

the Laplacian operator into plane polars. The result is :

( 2 + 1 2 + 2 )

(r + ) = 0

r2 r

2 2 r

The Airy Stress Function. This is defined in a corresponding way to the Cartesian case, such that it automatically satisfies the equilibrium equations:

r = 1 12 : = 2

rr r2 2

r2

11 - 1 2

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r r r2 r r

Substitution of these definitions into the stress compatibility equation results in the biharmonic equation as before.

The circular hole in an infinite plate This is the most important result in the theory of stress functions in polar coordinates. Finding the solution to the stress distribution is an example of a Boundary Value Problem. One way of solving problems of stress distribution is to make intelligent guesses about the form of the stress function and then to find out what problems it solves. This can be done with this problem. The boundary conditions are:

At x = Sx = S1 (max principal stress in the far-field)

At y = Sy = S2

As xy the stress field is uniform and stress trajectories are horizontal and vertical.

(r)r = ½ (Sx + Sy) + ½(Sx - Sy) cos 2

r = + ½(Sx - Sy) sin 2

also, at r = a, r = = 0 The form of the appropriate stress function (part of the general series solution of the biharmonic eqn) is:

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= A log r + Br2 + (Cr2 + Dr4 + Er-2 + F) cos 2

The first two terms on the right hand side do not depend on , but the rightmost does. A, B, C, D, E, F are constants that must be determined using the boundary conditions. Differentiating to get the stress components:

r = 1 12 = A/r2 + 2B + (-2C - 6Er-4 - 4Fr-2) cos 2

__ __ + __ ___ ___

rr r2 2

= 2= -A/r2 + 2B + (2C +12 Dr2 - 6Er-4 ) cos2

r2

11 1 2 = (2C + Dr2 - 6Er-4 - 2Fr-2) sin 2

_ ( _ __ ) = _ __ - _ ___

r r r2 r r

These stress components can be set at the far-field boundary and around the periphery of the hole. Coupled with the condition that the solution must be true for

all and stresses must not become infinite in the far-field, 6 simultaneous conditions to be satisfied are obtained: D = 0: 2B = ½(Sx + Sy) : -2C = ½(Sx - Sy) : A/a2 + 2B = 0 -2C - 6Ea-4 - 4 Fa-2 = 0 : and 2C - 6Ea-4 - 2Fa-2 = 0 These are solved to give the non-zero constants: A = -a2 (Sx + Sy)/2 : B = (Sx + Sy)/4 : C = -(Sx - Sy)/4 E = -a4 (Sx - Sy)/4 : F = a2 (Sx - Sy)/2 And substituted in the general equations for the stress components to give the stress components :

r = ½(Sx + Sy)(1 - a2/r

2) + ½(Sx - Sy)(1+ 3a

4/r

4 - 4a

2/r

2)cos2

= ½(Sx + Sy)(1 + a2/r

2) - ½(Sx - Sy)(1+ 3a

4/r

4)cos2

= + ½(Sx - Sy)(1- 3a4/r

4 + 2a

2/r

2)sin2

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Add +P a2/r2 to normal stress terms if pore pressure P present - see below where a is the radius of the hole. This is called the Kirsch solution. Note that the

stress components consist of a part dependent on r only and a and r dependent part. For uniaxial stress (Sy = 0), the stress trajectory pattern looks like

Special Cases : When Sx = Sy = P these reduce to

r = = P (1 - a2/r

2) : = 0

= P (1 + a2/r

2)

This is the case of the infinitely thick-walled pressure vessel, I.D. = 2a, subject to all-round hydrostatic pressure. By subtracting P at all points, we obtain the case of the hole in an infinite plate subject to internal hydrostatic pressure, P, and zero load at r = infinity.

r = Pa2/r

2 : = -Pa

2/r

2 : = 0

Note: the stresses die away as 1/r

2

This corresponds to the volcanic centre under internal magma pressure, P, or a

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wellbore overpressured by the weight of drilling mud. Tangential stresses are tensile at r = a, so radial dykes (cracks) can develop.

Returning to the external pressure case, max occurs at the hole boundary, r = a,

hence

r = P(1 - 1) = 0 : = P(1 + a2/r

2) = 2P

thus is twice the applied pressure. This is the simplest case of stress intensification at a geometric imperfection.

Max shear stress is also at r = a, = (r)/2 = (2P - 0)/2 = P. This is in a plane oriented at 45° to the boundary. The principal stress directions are radial and tangential and are isoclines. The max shear stress trajectories are a possible slip line field, and define spiral surfaces. Slip on these tends to close the hole, and plastic borehole collapse (e.g. in salt) may occur in this way. Here is an example quotation of this behaviour from a drilling engineer: “ When drilling in the southern North Sea Gas Basin it was possible for a 5" drill hole to close up sufficiently to jam the bottom hole assembly (BHA) in the time it took to drill 31 feet in salt. This would occur with diesel based drilling mud sufficiently weighted to prevent hole collapse. > This would occur at depths of a few thousand feet (6-7000ft) at relatively low temperatures. Jarring would be required to free the BHA, although a few were left down hole after. > So salt can move laterally into a free space 25-30mm in 15 minutes at temperatures of approximately 50-70C ! “

Stabilizing effect of drilling mud Drilling mud is heavier than water and, provided it is sufficiently viscous that it does not penetrate into the wallrock pores, where it would add to pore pressure, exerts an internal radial pressure on the wellbore (viscosity increasing agents can be added to the drilling or hydrofrac fluid to compensate for the decrease of

viscosity with temperature at depth in the hole). This counteracts the (hoop stress) to a degree (makes it less compressive) and reduces the risk of failure that would tend to collapse the wellbore. If the mud weight is too great, it puts the hoop stress into tension and induces radial tensile cracks. Sometimes additives to the

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mud are used to reduce penetration of the mud into the wallrock. When one principal stress is zero (Sy = 0) :

Max tangential stress occurs when cos 2 = -1, i.e. at = /2 or 3/2 & r = a.

= 3Sx (compressive)

Min tangential stress occurs when cos 2 = 1, i.e. at = 0 or and r = a.

= - Sx, i.e. a TENSILE stress.

2) Non-hydrostatic loading - Borehole breakouts and drilling-induced tension crack formation When one principal stress is zero (Sy = 0) :

Max tangential stress occurs when cos 2 = -1, i.e. at = /2 or 3/2 & r = a.

= 3Sx (compressive)

Min tangential stress occurs when cos 2 = 1, i.e. at = 0 or and r = a.

= - Sx, i.e. a TENSILE stress. Thus the following possibilities for failure around the hole are as illustrated in figure right:

Failure of the borehole wall 90° to the compression direction leads to the formation of borehole breakouts. The failure can either involve tensile fracturing, causing fragments to spall into the hole, or shear failure, with the same result. In a porous sandstone, if failure spreads around the hole (e.g. if the differential stress is small) the hole diameter increases (forming a washout), this leads to sand

production. Extensive borehole wall failure can result in as much sand as oil being transported up the hole. A tension crack running into the wall parallel to Sx is stable, and does not penetrate far from the borehole wall. The orientation of the breakouts can be sensed with caliper gauges or borehole televiewer images (e.g. right, the horizontal axis is angular distance around

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the hole, the vertical axis is depth range (1 metre), the arrows point to the breakout features: source Zoback, Reservoir Geomechanics, Cambridge 2007, and leads to an estimate of the orientation of the maximum and minimum principal stresses in a plane normal to the borehole. This is the main source of stress orientation evidence used for the construction of the world stress map. The occurrence of breakouts and drilling-induced tensile cracking in the televiewer images indicates relatively high differential stress is present.

(Photo above) Optical thin section of an experimentally produced breakout in sandstone. Breakout features form parallel to the minimum stress. Cuss, Rutter & Holloway, 2003, Int. J. Rock Mech Min. Sci., 40, 847-862 Hydraulic fracture in borehole walls. A single tension crack running parallel to the maximum compressive stress can be induced by pumping up the borehole fluid pressure until the crack forms when the fluid pressure exceeds the least principal stress in the borehole wall by the amount of the tensile strength. As fluid continues to be pumped into the hole, the crack extends farther into the wallrock, and the injection pressure becomes equal to the least principal stress. This is the basis of the hydrofrac method, which has two uses (see Zoback, Reservoir Geomechanics). (Do not confuse hydrofrac with drilling-induced tension cracking). (a) for stimulating enhanced hydrocarbon recovery by increasing the effective surface area of the hole (a widely used technique in secondary oil recovery), (b) to measure the magnitude of in-situ stress. The tension crack forms when the fluid pressure exceeds the minimum stress by the amount of the tensile strength (which is always small). Together with other data, the form of the Kirsch solution can also be used to estimate the maximum principal stress.

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To carry out a hydrofracture test, the bottom of the hole is packed off with an inflatable packer (inflated with compressed air), through which the overpressured fluid (water) can be pumped. To hydrofrac a section of the hole above the bottom, a double packer (straddle packer) is used.

Hydrofrac assembly with straddle packer (above), and single packer details (left) (source of figures, Engelder, Stress in the Lithosphere).

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Graph (above) showing pressure cycles applied during a hydraulic fracture test (also known as a Leak-off test [LOT]). Formation of the fracture stops the pressure rise as fluid is forced into the expanding crack. Fracture occurs when the fluid pressure exceeds the minimum stress in the plane normal to the borehole by the amount of the tensile strength of the rock (usually about 1 MPa). In this way the minimum stress SHmin is obtained. Unwanted hydrofrac causes loss of circulation. Note: further transient stress state modification can be caused by the cooling effect of the drilling mud. This usually helps borehole stability. Borehole breakout width and the problem of finding SHmax Variation of stress components around (vertical) borehole wall: If the unconfined compressive strength is known, the breakout width can be predicted. Conversely, breakout width can help constrain SHmax. At higher SHmax, the breakout gets wider for a given rock strength (see figure below). Whether a tension crack is induced in the SHmax direction depends on the relative values of Shmin and SHmax. If Shmin is zero this is most favourable for a tension crack because a tensile stress = -SHmax is induced at the bore wall. A tension crack appears

on the televiewer image as a thin line halfway between the breakout orientations. If effective Shmin is compressive

the sinusoid is lifted above

the zero line and a drilling-induced tension crack cannot form. The mean value of Sv (= Sz) is given by the overburden pressure, but it varies sinusoidally around the borehole wall (right) owing to

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the Poisson ratio effect and the SHmax variation. SHmax from hydrofrac experiments For relatively shallow (<2km) boreholes a ‘classical‘ method (Haimson & Fairhurst 1970) was developed for estimating SHmax from hydrofrac experiments. Leaving aside the effects of mud weight and thermally induced stresses, knowing starting pore pressure Pp and tensile strength of the formation T, it is possible to show that SHmax = 3 Shmin - Pb - Pp + T in which Pb is called the breakdown pressure and is obtained from the hydrofrac test. After an initial hydrofrac has formed, Pb is the pressure that has to be re-applied to start the fracture running unstably. For deeper holes methods based on the observation of wall failure are more reliable. Figure (right) shows stress measurements in the vertical San Andreas fault pilot borehole (you have seen this figure before). SHmax and Shmin are shown as squares and circles. Sv is the vertical stress profile calculated from

Sv = g z. Note that Shmin is approximately equal to Sv. SHmax is the greatest stress, acting in the horizontal plane. The relative values of SHmax, Shmin and Sv indicated the type of faulting implied by the Anderson theory of faulting. In this case either strike- slip or thrust (reverse) faulting is equally likely.

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Pp shows pore fluid pressure profile expected for a value of 0.4 (normal hydrostatic gradient). The dashed line shows expected value of SHmax assuming friction coefficient of 0.8. The value of SHmax may in this case be limited by frictional sliding along favourably oriented small faults, and the data show consistency with respect to laboratory friction measurements.

Images of Experimental Hydraulic Fracturing

(a) Shale

(b) In transparent perspex (PMMA)

Here, pink wax has been used as the hydrofracture fluid. Because it is more viscous than water, the crack grows more slowly. Two hydrofracs have been produced from opposite sides of the central hole. The crack tip extends beyond the limit of fluid penetration. the separation is called the fluid lag.

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Final note: Obtaining stress magnitudes from borehole failures and hydrofrac tests is generally a complex business that requires careful interpretation of each situation according to borehole orientation and rock properties. This is even more true for deviated and horizontal boreholes. Refer to Zoback‘s book on Reservoir Geomechanics and original references therein for a more thorough treatment of these issues.

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Obtaining information about in-situ stresses from sonic logs

A sonic logging tool carries an array of ultrasonic transducers that can send P and S wave pulses along a section of a borehole. Measuring speed of sound in the borehole wall carries information about the elastic properties of the wallrock, and can for example be used to detect when a different rock type or a zone of pore fluid overpressure is entered. Microcracking generally results in a reduction of sonic velocity and a reduction of the Vp/Vs ratio. If the microcracks have a preferred orientation, the rock is rendered seismically anisotropic.

Microcracks produced during progressive loading are generally strongly oriented parallel to the

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direction of applied max stress (perpendicular to the min stress). Figure (above) shows volumetric expansion in Merrivale Granite resulting from formation of axially-oriented microcracks. Crack formation affects elastic properties hence acoustic velocities e.g. for Vp (data of Colin Sayers, Schlumberger).

Here, microcracking is axisymmetric along the maximum loading axis (33), so in the 22 and 11 directions waves propagate across the cracks, hence are slowed down. At a small stress, the pre-existing (intrinsic) rock anisotropy is reversed as a result of the formation of stress-induced microcracks.

Shear Wave Birefringence An array of planar-oriented microcracks causes shear wave anisotropy and shear wave splitting (birefringence). The fast wave is that vibrating parallel to the crack array, that is generally parallel to the max stress direction in the horizontal plane (or the plane perpendicular to the borehole). Thus a log of fast Vs azimuth indicates orientation of max local stress: (sources: Schlumberger)

Shear wave splitting is caused by the intrinsic

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rock properties (bedding fabric, crystallographic preferred orientation, pre-exisiting microcracks). Pre-existing cracks tend to close normal to max stress and open normal to min stress, so can carry information about the stressed state of the rock.

Relative slowness of the split shear waves is frequency dependent (dispersion, velocity is slower at high frequencies), but when the birefringence is wholly or partially stress-induced, there is a crossover effect produced, the magnitude of which is proportional to the differential stress.

This effect has in recent years been used to develop patented methods for inverting in-situ stresses from sonic logs.

The theoretical aspects are complex, and interpretation of the results

must be done with care and take into account all information available.

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An excellent description of anisotropic elasticity is given by J.F. Nye,

Physical properties of crystals: their representation by tensors and

matrices. Cambridge Univ. Press. 1957. Super bedtime reading.

Further developments from the circular hole in the infinite plate - the elliptical hole. It is convenient to change the coordinate system to elliptical cordinates, of which plane polars is a special case (when C = 0). The system is of a series of confocal ellipses and hyperbolae:

The transformation from Cartesian to elliptical coordinates is:

x = C sinh sin

y = C cosh cos

where 2C is the distance between the foci. labels a sequence of confocal

ellipses and labels a series of confocal hyperbolae.

If a particular ellipse o corresponds to the boundary of an elliptical hole, long axis

inclined to the y axis at angle , and subject to far-field principal stresses P1 parallel to y and P2 parallel to x, by the methods of conformal transformations it can be

shown that the variation of tangential stress t around the hole boundary are given by:

t = (P1 + P2) sinh 2o +(P1 - P2)(exp 2o cos2() -cos 2)/(cos 2o - cos 2)

o describes hole shape, and describes position around the boundary.

The relations between o and the major and minor semi-axes a and b of the ellipse are:

b/a = sinh o / cosho = tanh o

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Griffith (1924) supposed that the weakness of brittle solids was due to crack growth fro pre-existing flaws (Griffith cracks), such that when the most favourably oriented one became critically stressed, it would grow and cause tensile failure of the material. There are two ‘most critical‘ orientations: (a) with the crack normal to max tensile stress - crack grows in-plane. (b) a crack at 30% to max compression, crack grows out of plane towards max

compressive stress. This results in axial cracking.

A failure criterion can be derived, that predicts when the most unfavourably oriented flaw will start to grow:

(a) when the least principal stress is zero or tensile, 1 < -3T, where T is the tensile strength of the material, and is in principle measurable.

(b) When 1 > -3T, in which case the failure envelope becomes

(1 - 3)2 /(1 + 3) = - 8T

This is a parabolic failure criterion. Cohesive strength is predicted to be - 2T, and uniaxial compressive strength is - 8T. Remember, this criterion predicts the stress conditions at which cracks start to grow. In tension, one growing crack can cause

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failure. In compression, MANY cracks must grow and link to cause failure in the form of a shear fault surface, generally requiring higher stresses than for the first crack to grow.

This criterion transforms into 2 = 4 T (T - n) in the Mohr coordinate frame. The graphical relations look like this:

Modified Griffith Criterion (McLintock and Walsh). If cracks are closed (in the compressive stress field), so friction can act across them, a modified (linear) criterion can be derived:

1 = c +3 {(1+ 2) + }/{(1+ 2) - } For tensile stresses, the original parabolic Griffith criterion applies. Griffith‘s Energy Balance Criterion Griffith argued that a tensile crack will grow when the energy available around the stressed crack tip exceeds the surface energy required to make the additional crack surface area by rupturing atomic bonds. This led to a condition for crack growth:

L = ( 2 E/ C)0.5

where L is the applied tensile load, E is Young‘s modulus and is the specific surface energy for the solid.

The Flat Crack

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When the crack becomes very thin, and length 2C, the equations for the stress field can be simplified by shifting the origin from the centre of the ellipse to a focus, in practice to the crack tip.

Coordinates for the stress components at point r, can be given either using plane polars or Cartesians, General form:

r = K / 2r f1 (

= K / 2r f2 (

r K / 2r f3 (

Note the functions of r and are separated. Stress components die away as 1/ r, i.e. the stress field is long range. K is called the stress intensity factor, and is proportional to the applied load. The value of K (= Kc) at the point of crack growth is a measure of the brittle tensile strength and is called the fracture toughness. It is a true material property that does not depend on the loading method used to determine it. This approach to fracture is called Irwin‘s Fracture Mechanics, and underpins modern understanding of brittle failure.

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Crack Loading modes Mode I (opening mode) Mode II (Edge) Mode III (antiplane strain) These are the three end-member loading modes, real cracks can find themselves under combined mode loading. In the above equations, there is a K value for each mode KI , KII and KIII . KI is the easiest to determine experimentally.

Typical values: soda-lime glass 0.7 units are MPam ( = N m-3/2 ) marble 1.0 granite 1.5 basalt 2.5 crack-resistant, tough metal alloys 50.

The forms of f() in the stress component equations are different for different modes and also whether Cartesians or polar coordinates are being used. For Mode I in plane polars:

f1 ( = cos (/2) { 1 - sin (/2) sin (3/2)}

f2 ( = cos (/2) { 1 + sin (2) sin (3/2)}

f3 ( = cos (/2) {sin (/2) cos (/2) cos (3/2)}

Crack Extension Force, G

G = strain energy release rate per unit length of crack front. G = dW/dC.

At Griffith equilibrium, G = 2, and the crack grows if G > 2.

G is porportional to Kc . In uniaxial tension, KI = a (C); G = (a)2 C/E

Sub-Critical Crack Growth (stress-corrosion cracking) At sub-critical (K < Kc) cracks can grow slowly with the assistance of thermally activated corrosion reactions at the crack tip. The intensified stress at the crack tip stretches the atomic bonds and makes them more reactive.

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In minerals like quartz,all the bonds are Si - O (strong), and the reaction is a hydration to produce a silanol (double hydrogen) bond at the crack tip: 2( O - Si - O ) + H2O = O - Si - OH l l HO - Si - O This is weaker than a Si-O bond so breaks as soon as it forms, and the crack propagates one atomic step. Crack velocity v depends on temperature T (Arrhenius rule), K, and the concentration of environmental water CH2O v = A KI

n CH2O m exp(-H/RT) n is about 10 for quartz, but can range up to 50 or more. m is of order 1. Taking logs linearizes this equation.

K - v diagram

Regime 1: velocity is controlled by stress corrosion reaction. Regime II: velocity does not depend on K, but is limited by the rate at which water (the reactant) can diffuse or flow to the crack tip. Regime III: fast crack growth, independent of the corrosion reaction. Subcritical crack growth is responsible for brittle, time-dependent creep.

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