ISBN No 81-901729-6-7 Structural Geology for Petroleum Geoscientists Published by Association of petroleum Geologists, rd 3 Floor, Geology Division, S&T Building, KDMIPE, ONGC, 9-Kaulagarh Road, Dehradun 248001, India
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ii
Ass A ss oc i ati on of Petro Pet ro l eum Geol og i st s Spec i al Pu b l i cat i on 3
IIT Roorkee
iii
To
My Mother and Teachers
iv
Structural geology is obviously one of the more important subjects for geoscientists working in petroleum industry. Folds and faults f aults in deformed rocks make traps for hydrocarbon accumulation. Also, large-scale deformations, the so-called tectonics, control the architecture of petroliferous sedimentary basins. It is the primary job of a structural geologist to interpret geological map and field data, and infer geometry of large scale folds and faults. However, geoscientists with varied specializations and working with different kinds of data may also be called upon to make structural interpretations. For example, lineament maps prepared from air photographs or satellite based images are commonly interpreted in terms of crustal-scale deformation or seismologists working with seismic reflection profiles routinely interpret subsurface structural geometry. It is imperative that geoscientists with different specializations working in oil industries have basic working knowledge on structural geology. A number of excellent textbooks on structural geology are now available but it appears that many a petroleum geoscientists are reluctant to pick up any of these books. This is probably due to the fact that the scopes of these books are much wider than the requirements of petroleum geoscientists. In this publication, the focus is on topics that I think should be of common interest to most petroleum geologists and geophysicists. I hope this will be particularly useful to those who did not have a thorough grounding in structural geology during their college/university days. Also, students in a bachelor level structural geology course may find this book useful. Detailed discussion on all the topics covered in this publication can be found in any standard textbook on structural geology. A list of such textbooks is given in the reference section, copies of which are on my desk all the time. I am grateful to Dr. James Peters, Secretary, Association of Petroleum Geologists for his continuous encouragement. But for his persistent demand that t he manuscript be completed within a fixed time frame, I would have taken eternity to finalize the same. Dr. Premanand Mishra, Dr. R. Krishnamurti and Mrs. Mamata Gupta are thanked for reading the manuscript cover to cover and locating innumerable mistakes. However, I alone remain responsible for the mistakes that escaped scrutiny. I am also grateful to my friends and colleagues at IIT Roorkee (formerly University of Roorkee) who maintain a congenial academic and social environment where individuals can tread the path of their choice. But for the full support of my family, this publication would not see the light of the day.
Roorkee 05 September, 2006
Dilip K. Mukhopadhyay
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Preface 1.
Introduction
1-5
2.
Planes and lines
7-14
3.
Force and stress
15-23
4
Mohr circle
25-32
5.
Strain
33-41
6.
Stress-strain relation
43-48
7.
Brittle fracture criteria
49-59
8.
Faults: Morphology and classification
61-74
9.
Thrust faults
75-86
10.
Normal faults
87-97
11.
Strike-slip faults
99-105
12.
Folds: Geometry and classification
107-120
13.
Fault-related folding
121-135
14.
Balanced cross sections: Introduction
137-145
15.
Section Construction
147-157
16.
Section restoration
159-164
17.
References
165-167
vi
The rocks in many of the sedimentary basins with good prospect for hydrocarbon reserves are highly deformed. The deformation is manifested by the presence of largescale faults and folds, some of which act not only as traps for hydrocarbon accumulation but also control depositional systems of source, reservoir and seal facies. Structural geology is the subject that deals with deformed rocks. It is one of the key subjects in hydrocarbon exploration and research. A reliable interpretation of subsurface structural geometry is essential for locating drill wells. Structural geology is closely allied to engineering mechanics, fluid dynamics and material science. There is, however, an important difference between engineering mechanics and structural geology (Fig. 1.1). Most of the deformation processes in the crust are very slow. As a result geological structures develop over long periods of time, usually in millions of years. Further, most of the rocks are very highly heterogeneous materials. Consequently, structures we see in nature are end products of very slow deformation processes in highly heterogeneous materials. We try to interpret the deformation process and the initial condition from the end product. In engineering problems, one generally studies the effects of various deformation processes on undeformed and relatively homogeneous materials. Another serious problem in structural geology, indeed in most subjects in earth sciences, is that we have to deal with incomplete, sometimes conflicting, data set. Therefore, structural inferences are interpretative and non-unique, and require validation. But unfortunately, there is a general lack of enthusiasm for validation of structural inferences!
1
Initial
Final
Engineering
Structural Structural geology
? Figure 1.1. The difference between engineering mechanics and deformation in the crust. In engineering mechanics, the effects of known stress systems on initially homogeneous undeformed materials are usually studied. In structural geology we see the end product of deformation in rocks, which are mostly heterogeneous material. We try to infer the deformation process and initial condition from the end product.
1.1 1.1 Aspects Aspects of s tructu ral study
Three different aspects of deformed rocks are analyzed in structural geology, viz., geometric, kinematic and dynamic.
•
Geometric analysis analysis is the qualitative description of size, shape and orientation of a structure. Determination of orientation of fold axis from dip/strike data is an example of geometric analysis. Interpretation of large-scale folding from outcrop-scale structural data and map pattern is another example of geometric analysis. Stereographic projection is a powerful tool for geometric analyses of structures.
•
Kinematics Kinematics is a branch of mechanics that treats motion in an abstract framework, without any reference to force or mass. In structural geology, kinematic analysis is a mathematical description of movement of material points during deformation in a rock. The stress or the rheological properties of rocks are not taken into account during kinematic analyses.
•
Dynamic analysis analysis involves understanding applied forces that produce deformation in the rock. The palaeostress analysis is an example of dynamic analysis. Dynamics also include how rocks are strained in response to imposed stress. 2
Also, three different types of studies are undertaken by structural geologists, viz., observational, experimental and theoretical.
•
In observational studies studies deformation features in rocks are studied at different scales varying from submicroscopic through outcrop in the field to global scales. Typically such studies involve description of the geometry of different structures and their order of formation. Also, variations in orientation of different structures seen in outcrops are studied in order to interpret the geometry of map-scale structures.
•
The main aim of experimental structural studies studies is to reproduce in the laboratory the structures seen in nature in order to gain insight into different factors that control the geometry of different structures. One problem with experimental studies is that the strain rates of laboratory experiments are much faster than expected in natural deformations. Therefore, experiments are commonly carried out on analog materials.
•
In theoretical studies studies different types of structures are numerically modelled through the application of various physical laws and using analytical or numerical methods.
We look at deformed rocks at different scales. Following terms are used to denote approximate scale of observation of deformed rocks.
•
Global: Global: Structural features observed in the scale of the world. Mid-oceanic ridges, subduction zones and orogenic belts are observed in global scale.
•
Regional: Regional: Generally denotes a scale of the order of a physiographic province or basin, such as Dharwar craton, Himalayan fold-thrust-belt, Satpura basin etc.
scale: Structural features seen on a map, which correspond to • Map or Macroscopic scale: an area much bigger than an outcrop. The area covered by a map may vary from several tens of square meters, to several tens or hundreds of square km. Mesoscopic: Structural features observed in an outcrop or handspecimen. • Mesoscopic:
3
Microscopic: Deformation features seen in a thin section under an optical • Microscopic: microscope.
•
Submicroscopic: Submicroscopic: Deformation features that cannot be seen even at the maximum magnification of an optical microscope but can be seen with TEM, SEM etc.
Depending on how pervasive at a particular scale of observation, a structural feature may be penetrative or non-penetrative (Fig. 1.2).
•
Penetrative: Penetrative: We consider a structural element to be penetrative, or homogeneously distributed, if the spacing of the structure is small compared to the volume of the rock under observation.
•
Non-penetrative: Non-penetrative: If the spacing of a structure is large as compared to the volume of the rock under observation, then the structure is non-penetrative.
(a)
(b)
Figure 1.2. The faulting is not penetrative in larger scale (a) because it is not uniformly developed but in the scale of outcrop (b) the same faulting is penetrative.
1.2 Geometry vs. strain vs. stress Rocks accumulate permanent strain in response to an imposed stress during deformation. The geometric features are the manifestations of permanent strain. Thus in nature the sequence is stress
→ strain → geometry.
Although everything starts with
stress, we do not observe stress directly. This is because stress is an instantaneous quantity, i.e., it exists only at the moment it is applied. What we study in structural geology is strain but in most rocks strain cannot be measured because strain markers are uncommon. In deformed rocks, what we observe most of the time is geometry. We 4
can carry out geometric analysis in most of the deformed rocks but can carry out strain measurements if we are very lucky to have strain markers. It is only in very rare cases it is possible to do stress analysis.
5
6
The geometries of geological structures are described in terms of planar and linear structural elements. Planar structural elements include bedding planes, axial surfaces of folds, joint surfaces and fault surfaces. Hinge lines of folds, intersection lineation produced by intersection of planar elements, mineral lineation and slickenside lineation on fault surfaces are examples of linear structural elements. The following terms are used to describe the attitude of planes and lines: Attitude: It is a general term used to indicate orientation of a structural plane or • Attitude: line. The orientation is related to geographic coordinates, i.e., north-south and east-west, and the horizontal. The attitude is specified in terms of bearing and inclination.
•
Bearing: Bearing: It is the angle between a line and a specified geographic coordinate direction, measured in a horizontal surface. The geographic coordinate direction is usually north direction, but can also be east, south or west. Note that an inclined line has to be projected onto a horizontal surface before bearing can be measured.
•
Inclination: Inclination: It is the angle between a plane or a line and an imaginary horizontal line, measured downward in a vertical plane.
•
Strike: Strike: The line of intersection between a plane of interest and an imaginary horizontal plane is called strike line. Note that the strike line must be a horizontal line. The bearing of the strike line is the strike angle, or simply strike, of the plane of interest.
7
N (North)
Trend Plunge
D
Strike Strike line C Pitch Dip
A
B
L i n e
Plane G
H
E
Strike line
F
Figure 2.1. Block diagram showing different terms used to describe attitude of planes and lines.
•
Dip: Dip: Dip is the inclination of a plane of interest measured on a vertical surface oriented perpendicular to the strike line. Dip (or true dip) dip) is the maximum inclination of a plane.
• Apparent dip: dip: Inclination of a plane measured in any direction other than the perpendicular direction to the strike line. The apparent dip is always less than the dip (or true dip). dip). Apparent dip in the direction of strike is always zero.
•
Plunge: Plunge: It is the inclination of a line from an imaginary horizontal line measured on a vertical plane.
•
Trend: Trend: It is the bearing of a line. If the line is inclined, then it is necessary to vertically project the line onto an imaginary horizontal plane before trend can be measured.
•
Pitch: Pitch: We measure pitch (or rake) of a line that lies on a plane. Pitch is the angle between the line and an imaginary horizontal line, both lying on the same plane.
Let us consider a block ABCDEFGH in which ABCD and EFGH are horizontal planes and the other four planes are vertical (Fig. 2.1). In this block, CDEF is the plane of interest containing a line DF. The direction from A to D is the geographic north direction. The lines CD and EF are the strike lines for the plane CDEF. The line AB is the vertical projection of EF onto the horizontal plane ABCD. Note that the strike lines 8
CD and AB are parallel though the strike lines are at different elevations. The distance between AB and CD is a function of dip of the plane, higher the dip smaller the distance. The ∠NDC is strike of the plane CDEF. Note that a strike line has two bearings, 180° away from each other. The ∠BCF (or ∠ADE) is the dip (or true dip) of the plane CDEF. The dip direction is D to A or C to B. The ∠BDF is one possible apparent dip of the plane CDEF. The ∠BDF is the plunge of the line DF. Note that the lines DB and DF are contained in the same vertical plane and DB is the vertical projection of the line DF onto a horizontal plane. DB with bearing ∠NDB is the trend of the line DF. ∠CDF is the pitch of the line DF measured from C side of the strike line DC. It is important to remember that the terms strike and dip/dip direction describe the attitude of planes, and trend and plunge give attitude of lines. N
A 0
120
W
E
B S Figure 2.2. A line AB makes 120 °, measured clockwise, with the geographic north. The bearing of the line can be stated equivalently as N120°, N300°, E30°S, S60°E, N60°W or W30°N. Note that the suffix indicates the direction from which bearing is measured.
The bearing of a line is usually stated in two different conventions. For example the bearing of the line AB in Fig. 2.2 can be stated as N120 ° (i.e., 120 ° clockwise from N) or E30°S (i.e., 30° from E towards S). It may also be stated equivalently as N300 °, S60°E, N60°W or W30°N. Orientations of planes and lines observed in an outcrop can be measured directly using a clinometer, which is a magnetic compass with provision to measure inclination. The use of a clinometer is best demonstrated in an outcrop. The bearing of a line is 9
measured with respect to geographic geographic North Pole (or true north direction) but the magnetic needle in a clinometer obviously points towards magnetic magnetic North Pole. We all know that true and magnetic North Poles are not same. In 2003, the magnetic North Pole was located at about 78 °18’N latitude and 104 °W longitude near Ellef Ringes Island, northern Canada and about 700 km from the geographic North Pole. The angle between magnetic N-S line and geographic N-S line is called declination, which varies from place to place on the surface of the earth. The magnetic north direction must be corrected to get the direction of the geographic north. Most clinometers have provision to make such a correction. The declination in India is close to zero and, therefore, we can take magnetic north as determined with a clinometer as true north direction. Two aspects of attitude of planes need to be remembered. Firstly, many a cross sections are drawn on seismic reflection profiles. In some cases we have no choice with our first cross section, such as in offshore or areas covered with alluvium or in desert. In seismic reflection profiles, the horizontal axes are distance and the vertical scale is two-way-travel time (TWT). Obviously, the horizontal and vertical scales are different, and therefore, the “dip” of reflectors we see in the profiles are not true dips. Velocity models are required in order to convert TWT into depth. Further, artifacts, such as diffractions or velocity pull-ups/pull-downs may also give false dips of reflectors. Structural interpretations based exclusively on seismic reflection profiles without depth conversion and consideration for artifacts may give distorted picture. Secondly, we prefer to draw structural cross sections perpendicular to the dominant orientations of strikes of bedding planes/axial planes and/or trends of fold axes. However, more commonly we find that in some parts, the dip directions are oblique to the line of sections. In such cases, apparent dips should be used while constructing cross sections. In order to understand the spatial relations between angular components shown in Fig. 2.1 and defined above, three-dimensional visualizations of problems involving orientations of planes and lines are very important. Once the ability to visualize in 3D is developed, more efficient and quicker methods, including readily available softwares, may be applied to solve real life structural problems.
10
D'
B
D
A
K' K d
d
d
C
N
P
S
J Q
R Plane P
I F
E M L
H G
Figure 2.3. Visualization of a three-point problem in a block diagram. The problem here is to determine dip and strike of the shaded plane in the block. See text for details.
Consider the shaded plane P, i.e., the plane AIGS, in Fig. 2.3. The plane occurs at different “elevations” at the four edges of the block, i.e., at A, I, G and S at edges AE, BF, CG and DH, respectively. We know that a strike line for a plane can be drawn if we can find two points on the plane at the same elevation with respect to a datum plane (e.g., mean sea level). On a geological map, we look for two intersections between a rock contact and a topographic contour in order to draw a strike line. Points S and J are such points and a line through them gives us a strike line at the elevation given by the plane PQRS. The line KD is the projection of line JS onto the plane ABCD. Note that along the strike line KD, the plane P is always at a depth of d. The line L′D′, which is parallel to KD and passes through A, is also a strike line but at the elevation given by the plane ABCD. Plane ANLE is a vertical plane oriented perpendicular to strike lines. Line AM is the intersection between vertical plane ANLE and the plane P. Therefore,
∠NAM is the dip of the plane and AN (i.e., from A towards N) is the dip direction. Block diagrams are useful to visualize problems involving angular components of planes and lines, such as three-point problem, determination of strike/true dip from apparent dips or determination of apparent dip in any direction from true dip/strike. However, solving problems graphically with the help of 3-D diagrams is never easy. The solutions may be obtained by projecting everything onto one plane. In Fig. 2.4, the 11
lines AI, AM and AS (traces of the plane along three vertical surfaces) have been projected on the horizontal surface ABCD by rotating 90 ° along lines AB, AN, and AD, respectively. This type of graphical construction may be employed to find solutions to several types of problems involving dip and strikes of planes, for example: (1) determination of true dip from apparent dips in the directions, such as, AB and AD, (2) determination of apparent dips from true dip/dip direction, and (3) solution to three point problems wherein dip/strike of a plane can be determined from known depths of occurrence of the plane at different locations. North
K' A I
D'
J dip
d B
S d
K d N dip direction
M D
Figure 2.4. Construction of 2D projection diagram on a horizontal surface to the 3D problem shown in Fig. 2.3. See text for details.
The angular relationships between planes and lines can be readily determined using a stereonet. These days almost all earth scientists use easily available computer software for stereographic analyses of orientation data. However, it is extremely important to be able to visualize in 3D the orientations of lines and planes in a stereogram (Fig. 2.5). Consider that AB AB is the trace of an inclined plane P on a horizontal outcrop surface and the line contains a line OL OL (Fig. 2.5a). If a sphere is drawn centered at point O, then the plane P (and its extension above the surface) intersects the sphere as a great circle (Figs. 2.5b). This great circle is a spherical projection of the plane in 3D and must be projected onto 2D space before any orientation problem can be solved. The equatorial circle in Fig. 2.5b is the surface on which planes and lines are projected. In order to get the projection of a plane, lines are drawn connecting each point on the part of the great circle in the lower hemisphere to 12
zenith point Z. Five such lines joining points 1 though 5 to Z are shown in Fig. 2.5c. Each of these lines intersects the equatorial circle and the arc traced by the loci of such intersection points is the projection of the plane P. The arc 1-2’-3’-4’-5 1-2’-3’-4’-5 is the projection of plane P and is called a great circle. A view down from Z will look like Fig. 2.5d and is called a stereogram. The limiting circle is called primitive, which also represents a horizontal plane. The line OL OL is projected as a point L′. Note that both the plane and the line have lost one dimension after projection, as happens with any projection diagram. Z W
Z
N
A O
S
B
E N
A
O
W
E B
S
L
1 2'
W
N L'
2
E
3' s
4'
5
Plane P L
(a)
Plane P
L 3
(b)
4
(c) Z
N
N
1
W
2'
E L' 4'
S
O
P'
W
3'
(d)
Great circle
N E
W
Pole O' dip
dip
E
S
5
(e)
(f)
P
S
erivation of stereographic projection diagram. See text for details. Figure 2.5.D
If a large number of planes are plotted as great circles, the stereogram may become cluttered. In such a situation, planes are plotted as poles instead of as great circle. In Fig. 2.5e line OP OP is perpendicular to plane P and is projected at point P' P' on the equatorial plane. So, P' P' is pole to plane P. The primitive, primitive, i.e., the limiting circle represents the great circle of a horizontal plane and a straight line passing through center is a vertical plane. Therefore, great circles for planes with gentle dips will plot closer to primitive and steeper planes will have their great circles closer to the center (Fig. 2.5f). For poles it is just the reverse. Poles to planes with gentle dips will plot close to the center, whereas the planes with steep dips will have their poles closer to 13
primitive. Note that if planes are gently dipping, their poles will have steeper plunge and plunge of poles to steeply dipping planes will have gentler plunge. Similarly, lines with gentle plunges will plot closer to primitive and steeply plunging lines will plot closer the center. Using stereographic projections we can solve some of the three point problems quickly and efficiently. This technique is also useful for statistical analyses of orientation data.
14
This section gives elementary descriptions of force and stress. For a more detailed treatment, readers are referred to Ramsay (1967), Means (1976) or any recent textbook on structural geology.
3.1 Forc e Forces acting in rocks produce deformation structures that structural geologists study. Therefore, it is necessary for us to understand how forces are distributed in the earth and how they produce different types of structures. Since we usually study old deformation, we do not see or measure forces directly in rocks because forces are instantaneous. Rarely we can see or feel the effect of forces acting in the earth as at the time of an earthquake. Although forces acting on rocks cause deformations in them, structural geologists usually talk in terms of stress. Force and stress are not exactly the same though they are closely related, as we will see. High school physics textbook tell us that a force is an influence, which has an intention to set a body at rest in motion or to change the velocity and direction of a body in motion or to change the shape of a body. Note that a force may not be able to do any of these but it has to have the intention. This, in essence, is the Newton’s first law of motion. Forces may be balanced or unbalanced. The forces are balanced if the summation of all forces acting on a body is zero otherwise the forces are unbalanced. When balanced forces act on a body it does not change its position at rest (or of uniform motion) and it appears as if no force is acting on it. Unbalanced forces can move a stationary body or they can stop/slow down/accelerate a moving body. Force is a vector quantity having 15
magnitude as well as direction. A force is called a body force force if it can work from a distance and depend on the amount of material affected. The gravitational force is a body force. The forces that act across a surface of contact between two adjacent parts are called surface forces. forces . Tectonic forces, which drive lithospheric plates, are surface forces. These two types of forces are closely related in the earth. If a force F acting on a body of mass m produces acceleration a in the body then according to Newton’s second law of motion (* indicates multiplication): F=m*a The unit of force in S.I. system is newton (N), which is the force acting on a body of 1 kg mass produces an acceleration of 1 meter (m) per second (s) per second. Therefore, 1 N = 1 kg * 1 m s -2 If we consider a force that produces an acceleration of 1 cm s -2 on a body of mass 1 g, then the unit is dyne (dyn): 1 dyn = 1 g * 1 cm s -2, and since 1 kg = 10 3 gm and 1 m s -2 = 10 2 cm s -2 1 N = 105 dyn Most of the time we use the terms weight and mass interchangeably but they are not the same. When a body of mass m is allowed to fall freely, its acceleration is that of gravity g gravity g and the force acting on it is its weight W , so W=m*g The value of g of g varies from place to place, but for our purpose it can be taken to be a constant with a value of 9.8 m s-2. So, a free falling body of mass 1 kg will have a weight of W = 1 kg * 9.8 m s -2 = 9.8 N
16
Therefore, the weight of a body is the gravitational force exerted on it by the earth. Weight, being a force is a vector quantity. The direction of this vector is the direction of the gravitational force, i.e., towards the center of the earth. So when we say weight is 1 kg, what we actually mean is the gravitation force exerted on a body of mass 1 kg! Newton and dyne are wholly formal units of force but they are not familiar and difficult to relate to everyday experiences. One kg, on the other hand, is very familiar and is used frequently as somewhat informal unit of force. If we put a block of iron of mass 10 kg on a tabletop, the force exerted by the iron block on the table is 98 N or 98 * 105 dyn. Alternately, we may say that the iron block exerts a force of 10 kg on the table. Let us consider the uppermost cubic meter of a granite cube in an outcrop with the cube separated from the surrounding rock by open joints on its four vertical sides. We now wish to calculate the force acting on the basal surface of the granite cube. This force will be sum of the force exerted by the atmosphere acting on the top surface of the cube plus the force exerted by the cube. The mass of a column of atmosphere occurring on top of the cube is 9700 kg and the mass of a cubic meter of granite ( ρ = 2.7 g cm -3) is 2700 kg. The force acting on the base of the cube can be stated as 12400 kg or more formally as: F = (9700 + 2700) kg * 9.8 m s -2 = 121520 N = 121520 * 10 5 dyn Let us consider a plane half way down the above-mentioned cube of granite. The top half of the cube plus the atmosphere on the top will exert a force across the plane and this force (10.83 * 10 4 N) can be represented by a vertically downward pointing vector. According to Newton’s third law of motion, the lower half of the cube will exert an equal and oppositely directed force. In this case the two force vectors will point towards each other and particles on either side of the plane will be pushed closer together. Such forces are called compressive force. force . If the force vectors point away from each other, the particles on either side of the plane will be pulled away from each other and the force is tensile force. force . Compressive forces are given positive sign and tensile forces are given negative sign (Fig. 3.1b).
17
A force vector directed across a plane can have any orientation with respect to the plane. If the force vector is oriented perpendicular to the plane it is a normal force ( F ( F n ), ), if it is oriented parallel to the plane it is a shear force (( F F s). A force vector may not be oriented either normal or parallel to the plane. Such a force vector can be resolved into normal force and shear force, just like any other vector (Fig. 3.1a). The shear force can be further resolved into forces parallel to any convenient coordinate directions (Fig. 3.1a). Sign conventions for different kinds of forces are shown in Figs. 3.1b,c.
3.2 Stress If we place a 10 kg weight on top of a cubic meter of quartzite nothing happens. But if we put the same 10 kg weight on a grain of sand it may get pulverized. Intuitively we can say that the grain of sand “felt” a lot more force than the cube of quartzite although both of them were under the same force. In order to express this we need to define a new term called stress. stress.
z A
F y
Fn
Plane Pl ane P
(b)
(a)
F AB
B
compressive (+ve)
Fx Fs
B
A FBA Plane Pl ane P
Fy
x
(c)
FBA F AB
tensile (-ve)
Plane Pl ane P +ve
-v e
Figure 3.1. (a) Resolution of force vectors. The total vector F (bold indicates vector) acting on a plane P can be resolved into normal force ( F n ) and shear F s). The shear force can further be resolved into force vectors parallel to x force ( F F x) and y ( F y) co-ordinate axes. (b) Sign and notation conventions for surface ( F forces: compressive and tensile forces are considered +ve and –ve, respectively. F AB indicates force exerted by body B towards body A and A and F BA BA indicates force exerted by body A A towards body B (c) Sign conventions for shear forces: counter-clockwise and clockwise shear forces are +ve and –ve, respectively.
Let us again consider the uppermost cubic meter of granite cube in an outcrop with the cube separated from the surrounding rock by open joints on its four vertical sides. 18
The force on the 1 m 2 basal plane of the cube is 121520 N or 12400 kg. If we normalize force by the area we get 121520 N m -2 or 12400 kg m -2. Note that these values are same as 12.152 N cm-2 and 1.24 kg cm -2, respectively. In other words, we get the same normalized force no matter what the area of the basal plane we take so long we have 1 m high rock column and the column of atmosphere on top. This normalized force, i.e., force divided by area, is called stress ( σ): Stress (σ) = force (F) / area (A) Let us consider a situation where non-uniform forces are acting across a plane ( P). P). The stress across a small part of the plane ( ∆ A) A) is given by σ = ∆F
/ ∆A
If we take an infinitesimally small area we may consider it to be a point, p. p. The stress across the plane P at point p will be given by σ =
dF / dA
Stress on a plane is a vector quantity because it is the product of a vector ( ∆F) and a scalar (1/ ∆A). Stress has magnitude equal to the ratio of force to area and a direction parallel to the force across the plane. The formal unit of stress is pascal (Pa): stress = force / area = (kg m s -2) m-2 = N m -2 =Pa In the earth, most stresses are significantly larger than a pascal, so we frequently use megapascal (Mpa) 1 Mpa = 106 Pa = 10 bar = 9.8692 atm Like any other vector, a stress vector can be resolved into components to any convenient reference directions (Fig. 3.2). Obviously, stress vectors can be added vectorially so long as the stress vectors are related to a single plane. Stresses acting 19
perpendicular and parallel to the plane are called normal stress and shear stress. A stress vector acting across a plane P at point p has a stress vector of equal magnitude but oriented in opposite direction. If the two vectors point towards each other the stress is compressive (+ve), otherwise the stress is tensile (-ve). Note the similarities between Figs. 3.1 and 3.2.
3.3 3.3 Stress Stress ellipse elli pse There is another meaning of the word stress. As discussed above, there are two parallel but oppositely oriented stress vectors with the same magnitude at a point p p on a plane P. There may be infinite number of differently oriented planes, all of them passing through point p point p.. At point p point p and for each of the planes there are two parallel but oppositely oriented stress vectors. A collection of all the stress vectors associated with planes of all possible orientations but passing through the same point p p is also called stress - stress at a point. But this stress is not a vector but a tensor. The loci of heads (for tensile, Fig. 3.3a) or either tails (for compressive, Fig. 3.4b) of all the stress vectors acting at a point trace an ellipse in two dimension, called stress ellipse. It is obvious that there is no stress ellipse for a situation where some of the stress vectors are compressive, and others are tensile. One of the planes P with the associated stress vectors are also shown in Figs. 3.3a and 3.4a, note that the stress vectors are not perpendicular to the plane P. This P. This is generally true and a stress ellipse should not be thought of as collection of normal stresses. However, there are four stress vectors in an ellipse those act across perpendicularly oriented planes. These vectors are coincident with the major and minor axes of the stress ellipse. In other words, these are normal stresses and are called principal stress axes. The planes across which principal stress axes operate are called principal planes of stress (Figs. 3.3b and 3.4b). Equation of a stress ellipse has the same form as any other ellipse. If in a stress ellipse:
•
the centre of the ellipse is centered at the origin, i.e., at x = 0 and y = 0 in x-y coordinate system
•
major and minor axes are coincident with x and y axes, respectively 20
•
the major and minor radii of the ellipse are the magnitudes of σ1 and σ2, respectively
then, the equation of the stress ellipse is ( σx2 / σ12) + (σy2 / σ22) = 1
z
σ B A
A
σ n
σ
σ B A
Plane P Plane P
y
σ x σ s
A
σ A AB B
(b) B
B
compressive (+ve)
σ y x
(a)
(c)
σ A B
tensile (-ve)
Plane P +ve
-v e
Figure 3.2. Resolution, and notation and sign conventions of stress vectors. Note the similarity with Fig. 3.1. (a) The The total total stress vector (bold indicates vector) acting on a plane P can be resolved into normal force ( n ) and shear force ( s). The shear force can further be resolved into force vectors parallel to x ( x) and y ( y) co-ordinate axes. (b) Sign and notation conventions for surface forces: compressive and tensile forces are considered +ve and –ve, respectively. A and BA indicates force AB indicates force exerted by body B towards body A exerted by body A A towards body B (c) Sign conventions for shear forces: counter-clockwise and clockwise shear forces are +ve and –ve, respectively.
Principal planes of stress
σ AB
B A p
σ 2 P l la n a e P
σ 1
σ BA
(a)
σ 2
σ 1
(b)
Figure 3.3. (a) Stress ellipse given by loci of heads of all the stress vectors acting at point p point p on plane P. AB and BAa ssociated with plane P are also shown. Stress vectors in this case point away from point p and thus they are tensile. Note that AB and BA are not perpendicular to plane P. (b) Two pairs of stress vectors oriented perpendicular to the planes across which they operate. These stress vectors are called principal called principal stresses stresses and coincide with the major and minor axes of stress ellipse. The planes oriented perpendicular to the principal stress directions are called principal called principal planes of stress. stress.
21
Principal planes of of stress
σ AB
B A p
σ 2 P l la n a e P
σ 1
σ BA
(a)
σ 2
σ 1
(b)
Figure 3.4. Stress ellipse for a situation where all the stress vectors are compressive. See caption to Fig. 3.3.
3.4 Stress ellipsoid A stress ellipsoid represents collection of stress vectors in three dimensions (Fig. 3.5), all of them acting at a point. It is exactly analogous to stress ellipse in two dimensions except that there are three principal stress directions and three principal planes of stress.
σ 1 σ 2 σ 3
Figure 3.5. Stress ellipsoid, i.e., stress in 3D. Note that there are three principal stresses and three principal planes of stress.
The three principal stresses are σ1, σ2, and σ3 with magnitudes σ1 > σ2 > σ3. Three principal planes of stress define three stress ellipses containing σ1-σ2, σ2-σ3 and σ1-σ3 axes of the stress ellipsoid. One, two or three principal stresses may have non-zero values. In triaxial state of stress all the three principal stresses have non-zero values, in biaxial state of stress only two principal stresses have non-zero values and in uniaxial 22
state of stress only one principal stress has non-zero value. If σ1 ≠ σ2 ≠ σ3 it is called general general or polyaxial stress. stress . In axial stress two of the three stresses have same magnitude. If σ1 > σ2 = σ3 it is called axial compression compression and the ellipsoid is a prolate spheroid. In axial extension σ1 = σ2 > σ3 and the stress ellipsoid is an oblate spheroid. In hydrostatic stress magnitudes of all the three stresses are same, i.e., σ1 = σ2 = σ3. The ellipsoid is a sphere and the stress is called pressure, which is the only kind of stress that can exist in a fluid at rest. The equation of a stress ellipsoid is (σx2 / σ12) + (σy2 / σ22) + (σz2 / σ32) = 1
23
24
Normal (σn) and shear ( σs) components of stress on a real or imaginary plane are important for the understanding of theories of development of faults and joints. The equations of normal and shear stresses in terms of principal stresses ( σ1-σ2, σ1-σ3 or σ2-
σ3 space) are most useful in understanding the basic concept of stress. Finally, pictorial representation of how normal and shear stresses vary with the change in orientation of the plane with respect to principal stresses is very illustrative; this we do with the help of a diagram called Mohr circle. The equations for the normal and shear stresses can be derived from force-balance problems, which assumes that if a body is in equilibrium (i.e., it does not move or spin) then all the forces in any one direction sum to zero. Note that we should always balance forces and not stresses. We should first find the forces acting in any particular direction, determine stresses in terms of the forces, and then derive the expressions for normal and shear stresses. Consider a prismatic body in the earth with two of the bounding planes oriented parallel to the maximum and minimum force vectors (Fig.4.1). Another bounding plane P has an area A and whose orientation may be specified by the angle θ made by the F 1) force direction. Therefore, the areas of perpendicular to the plane with maximum (( F the left and right faces of the prism are Asin are Asinθ and Acosθ , respectively. Resolution of the magnitudes (F (F 1 and F 2) of forces acting on the three bounding planes is shown in Fig. 4.1. Resolved forces are parallel and perpendicular to the plane P. Forces trying to push the prismatic body upward must be equal to the forces trying to push the prismatic body downward, if the body has to remain in equilibrium. Therefore, from the balance of forces acting across the plane P (Fig. 4.1): F1n + F 3n = F 1 cos θ + F 3 sin θ 25
or, F n = F 1 cos θ + F 3 sin θ
(4.1)
And, from balance of forces acting parallel to the plane P we obtain: F1s + F 3 cos θ = F 3s + F 1 sin θ or, F1s – F 3s = F 1 sin θ - F 3 cos θ or, Fs = F 1 sin θ - F 3 cos θ
(4.2)
Fn F1n
F1
Plane P of area A
F1 F3 F1s
Fs
F3n
F3
F3s
θ
Perpendicular to plane P F3 cosθ
θ F3 sinθ
F3
A s i n
s θ c o A
θ
F1 sinθ
θ F1
F1 co cos sθ
θ
Figure 4.1. A prismatic body bound by planes parallel to F 1 and F 3 directions, and a plane P with area A A and whose normal is inclined at θ to the F 1d irection. As the body is at rest, fforces orces acting in any one direction must sum to zero. Force trying to move the prism from left to right is exactly balanced by the force acting from right to left and force trying to move the body from top to bottom is balanced by the force acting from bottom to top. Resolution of magnitudes of F 1 and F 3 forces on the three bounding surfaces are shown on the figure.
We now calculate the stresses in terms forces from the relation, stress = force / area:
σn = F n / A or, Fn = A σn
σs = F s / A F s = A σs
(4.3)
and
σ1 = F 1 / (A cos θ)
σ3 = F 3 / (A sin θ)
or F1 = σ1 A cos θ
F 3 = σ3 A sin θ
26
(4.4)
Replacing (eq. 4.3) and (eq. 4.4) in (eq. 4.1) and (eq. 4.2), we obtain
σn = σ1 cos 2θ + σ3 sin 2θ
(4.5)
σs = σ1 sin θ cos θ + σ3 sin θ cos θ
(4.6)
Using the identities cos 2θ = (1 + cos2 θ)/2, sin2θ = (1 - cos2 θ)/2 and sinθ cos θ = sin2 θ in eqs. 4.5 and 4.6, and collecting terms we obtain
σn = ½ ( σ1 + σ3) + ½ (σ1 - σ3) cos2θ
(4.7)
σs = ½ ( σ1 - σ3) sin 2θ
(4.8)
From the above two equations we can calculate σn and σs on any plane, given θ, σ1, and σ3. The above equations are derived for the σ1-σ3 principal plane. We can derive similar equations for σ1-σ2 and σ2-σ3 principal planes also. Equations (4.7) and (4.8) define a circle with σn and σs as x and y axes, respectively. The center of the circle (on the x-axis) and radius are given by ( σ1 + σ3)/2 and (σ1 -
σ3)/2, respectively. This circle is known as the Mohr circle for stress. σ1 325 bars 70
O
σs, bars 100
O
70
Q
P
P O
140
O
75 bars
σ3
20
Q
σ3 100
− 100
200
σn, bars
O
40
300
σ1
400
− 100
(a)
(b)
Figure 4.2. Construction of a Mohr circle. (a) A very small rectangular block showing state of stress and orientations of two planes on which normal and shear stresses are to be determined using Mohr circle. (b) Mohr circle for the state of stress as shown in (a). See text for details.
27
Let us consider a very small rectangular block in which there is a plane P inclined at 70° from the horizontal. σ1 is vertical and has a magnitude of 325 bars, and σ3 is horizontal with a magnitude of 75 bars. A line perpendicular to plane P makes 70° with σ1 direction, i.e., θ is 70 ° (Fig. 4.2a). We want to show pictorially the magnitudes of σn and σs on this plane; we can, of course, calculate the same using equations 4.7 and 4.8. A pair of orthogonal coordinate axes is drawn with σn and σs as horizontal and vertical axes, respectively (Fig. 4.2b). The two axes are graduated in units of bars with same scale on both the axes. On the positive side of σn, points are located at 325 and 75 bars and marked as σ1 and σ3, respectively. We locate a point half way between σ1 and σ3, i.e., at 200 bars, i.e., ( σ1 + σ3)/2. Taking this point as center we draw a circle that passes through both σ1 and σ3. The resultant circle is Mohr circle of stress for a state of stress with magnitudes of σ1 and σ3 at 325 and 75 bars, respectively. We find a point P on the Mohr circle such that a line drawn from P to the center makes an angle of 140 ° (i.e., 2 θ) from the positive end of the σn axis. The coordinates of point P are 80.3 and 104.2 bars, which are the σs and σn, respectively. The plane Q (Fig. 4.2a) inclined at 20 ° ( θ = 70 °) from horizontal has σs and σn values of 80.3 and 295.8 bars, respectively. So it seems that every point on Mohr circle represents a plane and the coordinates of the point equal normal and shear stresses associated with the plane. Therefore, the circle is the loci of infinite number of points, each of which represents stress on a plane whose orientation is specified by θ. Fig. 4.3 shows how the different terms in equations 4.7 and 4.8 are related to coordinates of point P on the Mohr circle. Figs. 4.2, 4.3 are drawn for the principal plane of stress containing σ1-σ3; this is two dimensional. We can also draw Mohr circles for σ1-σ2 and σ2-σ3 principal planes. The three Mohr circles in Fig. 4.4 represent three-dimensional state of stress. The largest, intermediate and smallest circles represent Mohr circles for σ1-σ3, σ1-σ2 and σ2-σ3 surfaces, respectively.
28
σn = [( [(σ σ 1 + σ 3)/ )/2 2 + (σ 1 − σ 3)/ )/2] 2] cos2θ
+σs
σn
P 1/2( σ1−σ 3) sin2 θ
1/2( σ1−σ 3)
−σn
σ3
2θ
1/2( σ1+σ 3)
σs
σ1
+σn
1/2( σ1−σ 3) cos2θ
−σs
Figure 4.3. Mohr circle showing significance different terms in equations 4.7 and 4.8.
σs
σ1-σ3 σ1-σ2 σ2-σ3
−σn
σ3
σ2
σ1
σn
−σs
Figure 4.4. Mohr circles for state of stress in three dimensions. The three circles represent three principal planes of stress.
Sign conventions for the Mohr diagram are as follows: compressive normal stresses and shear stresses represented by counterclockwise or sinistral pair of arrows are positive and tensile normal stresses and shear stresses represented by clockwise or dextral pair of arrows are negative (Figs. 4.2, 4.3); θ is positive for planes whose normal can be located towards counterclockwise direction from σ1, otherwise θ is negative (Fig. 4.5).
29
Plane P
Plane Q zP
+ θ
σ1
− θ
(a)
σ1
zP
(b)
he sign convention for angle θ. (a) +ve θ. (b) –ve θ. Figure 4.5.T
Two perpendicular planes (P ( P and Q, Fig. 4.6) with θ = ± 45 ° (or 2 θ = ± 90 °) have the largest absolute magnitude of shearing stress, which is equal to the radius of the Mohr circle [or, ( σ1 - σ3)/2 for σ1-σ3 principal plane]. Shear stresses for planes oriented perpendicular to any of the principal stress directions are zero. It may sound logical to presume that shear fractures (i.e., faults) should form at ± 45 ° to σ1. As we will see (section 7) shear fractures usually develop at angles less than this. σs P
σ3 P
Q
(σ1−σ3)/2
σ1
−σn
+ 90
σ3
σn
O
- 90
O
σ1
−(σ −( σ1−σ3)/ )/2 2
(a)
(b)
Q
−σs
Figure 4.6. Maximum shearing stress is possible on planes oriented at ± 45 ° (2 θ = ± 90°) to σ1 axis.
Different classes of two-dimensional state of stress at a point are shown in Fig. 4.7. In hydrostatic tension (lithostatic tension for rocks), stress across all planes is tensile and equal and the Mohr circle is a point on the negative side of the σn axis (Fig. 4.7a). If the stress across all planes is compressive and equal, the state of stress may be called hydrostatic compression compression and the Mohr circle is a point on the positive side of the σn axis (Fig. 4.7b). The term “hydrostatic” refers to the stress experienced by a fluid at rest. However, this term is also widely used to describe similar state of 30
stress in solid. Both the principal stresses can be either positive or negative and the states of stress are are general tension (Fig. 4.7c) or or general compression compression (Fig. 4.7d), respectively. In uniaxial tension tension only one principal stress is non-zero and it is tensile (Fig. 4.7e), whereas in uniaxial compression compression only one principal stress is non-zero and it is compressive (Fig. 4.7f). In many states of stress one principal stress is tensile and the other principal stress is compressive (Fig. 4.7g,h,i). Pure shear stress stress is a special class of state of stress where the two principal stresses have the same magnitude but opposite sign (Fig. 4.7i). In such cases, planes of maximum shear stress are also planes of pure shear, i.e., normal stresses across these planes are zero. Except for lithostatic tension (Fig. 4.7a), all other classes of stress are possible in the earth. σs
e g
c −σn
i
d
h f b
a
σn
−σs
ohr circles for various types of state of stress in two dimensions. Figure 4.7.M
Any non-hydrostatic state of stress, either in two-dimensions or in threedimensions, can be decomposed into two parts: a mean stress stress ( σm) and a stress deviator or deviatoric stress stress ( σd). The mean stress is the average of the principal stresses:
σm = ( σ1 + σ2)/2
in two dimensions, and
σm = ( σ1 + σ2 + σ3)/3
in three dimensions.
The deviatoric stress is defined as:
σd = σn - σm 31
and is a measure of how much the normal stress in any direction deviates from the mean or hydrostatic stress. Along three principal stress directions we have three principal deviatoric stresses whose magnitudes are given by:
σ1d = σ1 - σm σ2d = σ2 - σm σ3d = σ3 - σm Note that sum of the right hand terms in the above equations is zero. For example, a state of stress with the values of three principal stresses 750, 1050, and 1560 bars can be thought of as combining a mean stress of 1120 bars and three deviatoric stresses of –370, -70, and 440 bars. Mean stress in two-dimension is a state of hydrostatic tension or compression and locates the center of the Mohr circle; deviatoric stress is a pure shear stress. Given mean stress and deviatoric stress we can construct the Mohr circle. The deviatoric stress is responsible for distortion (or strain) in a body. The distortion may be elastic (i.e., reversible) or plastic (i.e., permanent). Most deformations are result of differential stress stress rather than the absolute magnitudes of principal stresses, except for dilation. Differential stress is the difference between the magnitudes of maximum and minimum principal stresses ( σ1 - σ3). The mean stress can be thought of as hydrostatic (or isotropic) part of the stress system and causes only volumetric changes (dilation) in the material. Mean stress also controls the strength of materials. For example, fracturing is inhibited with increasing mean stress.
32
5.1. 5.1. Deform Deformation ation When force is applied to a rock body, the particles within the rock body are displaced and the rock is said to be deformed. deformed. Two types of deformations are usually recognized (Fig. 5.1).
•
Homogeneous deformation: deformation : If particles arranged in straight lines in an undeformed rock remain so after deformation, then the deformation is called homogeneous. Another definition of homogeneous deformation is that all parallel lines of particles remain parallel after deformation.
•
Inhomogeneous deformation: deformation : In this type of deformation, straight lines of particles become curved after deformation and parallel lines of particles loose their parallelism after deformation.
(a)
(b)
(c)
Figure 5.1. Homogeneous and inhomogeneous deformations. (a) Original undeformed grid. (b) Homogeneous deformation wherein straight lines remain straight and parallel lines remain parallel after deformation. (c) If straight lines become curved and parallel lines loose their parallelism, the deformation is inhomogeneous.
Mother Nature does not draw a grid, as in Fig. 5.1a, before deforming a rock for our convenience! In some cases, however, Nature preserves features that can be used to determine type and amount of deformation. For example, the branches in plant fossil 33
Neuroteris Neuroteris are approximately straight and parallel before deformation (Fig. 5.2a). The straightness and parallelism may be preserved (Fig. 5.2b) or destroyed (Fig. 5.2b) after deformation indicating homogeneous or inhomogeneous nature of deformation, respectively.
(b) (a)
(c)
Neuroteris showing homogeneous (b) Figure 5.2. Undeformed (a) plant fossil Neuroteris and inhomogeneous (c) deformations
Four independent geometric processes contribute to the total displacement of particles during deformation:
•
Rigid-body translation translation is the movement of the entire body through space in such a way that the shape does not change. The movement vectors for all the particles in any external coordinate system have the same orientation and magnitude (Fig. 5.3a).
•
Rigid-body rotation rotation also involves movement without any change in shape. However, in this case the body rotates about a single point, which is fixed with respect to an external reference frame (Fig. 5.3b).
•
Distortion Distortion produces change in the shape of the body due to movement of particles with respect to each other (Fig. 5.3c).
•
Volume change, change, as the term implies, change in volume of the body without any change in shape. Volume change is also called dilation, dilation, although volume change can be either positive or negative (Fig. 5.3d).
34
( a)
(c)
( b)
(d)
(e)
Phillipsia Geometric processes leading to different types of Figure 5.3. Trilobite Phillipsia deformations, as shown by trilobite Phillipsia. Phillipsia . (a) Rigid-body translation. (b) Rigid-body rotation. (c) Distortion, i.e., change in shape. (d) Dilation, i.e., volume change. (e) A combination of all the four types of deformation.
Note that the descriptions of distortion and volume change do not require an external reference frame. The four processes are not mutually exclusive – Fig. 5.3e shows a deformation which includes all the four processes. Distortion and dilation together make up strain, strain , which involves movement of particles relative to each other. In a more quantitative sense, strain is a mathematical description that relates the size and shape of a body before and after deformation. A rock body may undergo rigid-body movement, either translation or rotation or both, but it is almost impossible to determine the exact amount of rigid-body movement. However, the strain in a rock can be precisely determined if the rock contains objects of known, original shape and/or size (e.g., Figs. 5.2, 5.3. Note that deformation and distortion (and strain) are not exactly the same although it is not uncommon to find them used interchangeably in the literature. Further it is important to remember that translation/rotation of a rock body may or may not be accompanied by internal strain, i.e., distortion/dilation. There are several ways strain can be measured but all of them involve measurement of some kind of change change from an initial undeformed to a final deformed state. The changes that are generally measured are changes in lines, lines, angles angles and volume. 35
5.2 5.2 Change Change in l ine lengt h Changes in line lengths, called longitudinal strain, strain , can be measured in different ways, viz., extension (e), stretch (S), quadratic elongation ( λ), and logarithmic or natural strain (ε). If Li is the initial undeformed length of a line, L d is the final deformed length of the same line, and ∆L is the change in the length of the line then (Fig. 5.4): e = (Ld – L i) / Li = ∆L / Li S = Ld / L i = (1 + e)
λ = (L d / L i)2 = (1 + e) 2 ε = log e (L d / L i) = loge (1 + e) Elongation can be either positive or negative depending on whether a line has extended or shortened. Stretch is always positive whether a line has extended or shortened. It has a value of 1.0 if there is no change in length of a line, S < 1.0 for shortening and S > 1.0 for extension. All the four parameters for longitudinal strain are dimensionless. They are not independent, if we know one we can calculate the others; which one to use for a particular problem depends entirely on convenience. However, logarithmic strain is realistic for several reasons. For example, if one line contracts to half of its original length and another line expands twice its original length, the elongations of the lines are 0.5 and 1.0, respectively. For the same deformed lines logarithmic strains are -log e2 and + loge2. For very large shortening of a line, elongation tends towards –1.0, but the logarithmic strain approaches - ∝. The stress-strain curves for isotropic materials are straight (i.e., linear) if logarithmic strain is used. In nature we almost always measure L d from deformed linear objects, such as boudinaged quartz vein (Fig. 5.4b). We can put the boudins back into their original position and determine Li, assuming no volume change. Note that in Fig. 5.4a the length of the line has decreased (shortened) but the line in Fig. 5.4b has increased (extended). In both the cases the change in the length of the line is 2.46 mm but parameters describing longitudinal strain are different.
36
Undeformed (length L ) i
Deformed Deform ed (length L d )
∆L
Deformed Deform ed (length L d )
∆L
2 cm
Undeformed (length L ) i
(a)
(b)
Figure 5.4. Longitudinal strain. (a) A line with initial length L i = 9.15 cm has shortened to 6.69 (= Ld). The elongation (e), stretch (S), quadratic elongation ( λ), and natural strain ( ε) are -0.27, 0.73, 0.53 and -0.31, respectively. (b) A more practical scenario wherein boudinaged quartz vein can be used to determine both Li and L d. The e, S, λ and εa re 0.37, 1.37, 1.87 and 0.31, respectively.
5.3 Change in angle Shear strain Shear strain is is a measure of change in angle between two originally perpendicular lines. Consider a rectangle ABCD, which after deformation becomes a parallelogram A’B’CD (Fig. 5.5). The angle between lines AD and CD has changed from 90 ° ( ∠ADC) before deformation to α ( ∠A’DC) after deformation. We can state the shear strain in two different ways (Fig. 5.5):
• Angular shear (ψ): This gives the change in angle, i.e., ψ = 90 ° - α. •
Shear strain (γ): This represents displacement (distance x) of a particle at a distance y from a particle that does not move. From Fig. 5.5:
γ= tanψ = x/y or, x = y tanψ = y γ if y is unit distance then x = γ = tan ψ Shear strain can be determined if appropriate markers are present in rocks. For example, we can determine shear strain from deformed trilobite Phillipsia Phillipsia because of inherent bilateral symmetry (Fig. 5.6). Similarly, well-preserved worm burrows or mud cracks and stratification surfaces can be used to determine shear strain. The original perpendicular line (e.g., line AD in Fig. 5.5) may move either in a clockwise direction or in an anticlockwise direction with respect to the original
37
orientation. Clockwise and anticlockwise shear strains are given negative and positive signs, respectively. x A
A’
y
B
B’
ψ
α D
C
Figure 5.5. Shear strain illustrated by a rectangle ABCD that has changed into a parallelogram A’B’CD after deformation. ∠ADA’ or ∠BCB’ is the angular shear (ψ). Shear strain, γ = tan ψ = x/y. If y is of unit length, γ = x.
α ψ
(a)
(b)
Phillipsia shows shear strain. Morphology of Phillipsia is Figure 5.6. Trilobite Phillipsia such that two mutually perpendicular imaginary lines can be drawn (a). These lines can be used to determine shear strain from a deformed fossil.
5.4 5.4 Change Change in v olume ol ume Volumetric strain or change in volume during deformation is called dilation ( ∆). If Vi is the initial volume and V d is the volume after deformation and ∆V is change in volume after deformation, then dilation is given by:
∆ = (V d - V i) / Vi = ∆V / Vi Although to dilate is to enlarge, dilation can have positive (i.e., enlarge) or negative (i.e., contract) values. In two dimensions, we can only determine change in area. 38
5.5 5.5 Strain Strain ellipse and ellip soid Strain ellipse (in 2D) and ellipsoid (in 3D) are elegant ways to depict homogeneous deformation of a body as a whole. If particles lying on the periphery of a circle are subjected to homogeneous deformation, the particles will trace an ellipse after deformation. This ellipse is called a strain ellipse. In three dimensions, a sphere in the undeformed state turns into an ellipsoid in the deformed state. Let us look at the strain ellipse from a different viewpoint (Fig. 5.7). A circle describes a collection of straight lines with equal length but of different orientations, all passing through one point, which is the centre of the circle. Each of the lines connects two particles on either side of the circle. During deformation particles will move with respect to each other and the length of the lines will change. Stress ellipse (or ellipsoid in 3D) describes a collection of straight lines in deformed state, all passing through the same point, which is the centre of the ellipse. Obviously, there is no stress ellipse for inhomogeneous deformation because straight lines do not remain straight. z z
λ3
1 1
λ1
x
x
(b) (a)
Figure 5.7. Strain ellipse. See text for discussion.
If the radius of the initial circle is taken to be of unit length, the major and minor axes of the ellipse can be represented by √γ1 (= 1+e 1) and √γ3 (= 1+e 3), where √γ1 and √γ3 are maximum and minimum elongations (Fig. 5.7). So, the equation of strain ellipse centered at origin is, x2/γ1 + z2/γ3 = 1
39
Similarly, the equation of strain ellipsoid is x2/γ1 + y2/γ2 + z2/γ3 = 1 where, γ1 > γ2 > γ3. The three elongations directions are usually taken parallel to x, y, and z co-ordinate axes, respectively. Plane strain is a type of deformation where γ2 = 1, i.e., along the intermediate axis of the strain ellipsoid there has not been any shortening or elongation.
5.6 5.6 Fini Fini te and and i nfin itesimal st rain When we look at a diastrophic structure in nature, such as a fold or a distorted fossil, we know that the rock has undergone some amount of strain. However, it is important to remember that the strain that we may observe and measure in rocks did not develop instantaneously but accumulated in small increments over a period of time. This is because, like most natural processes, deformations are also very slow. Therefore, we observe the end product of a series of deformed states and straining of rocks should be considered as progressive deformation. The final state of strain is called finite strain and small incremental strains are known as infinitesimal strains. It is possible that a line that shows finite extension may have undergone shortening at some stage progressive deformation. Shortening Lines of no finite elongatio n
Extension
Figure 5.8. Initial undeformed circle superimposed on strain ellipse. Two lines can be drawn by joining opposite points of intersection between the circle and ellipse. The lengths of these two lines have not changed during deformation. They are termed as lines of no finite longitudinal strain and separate area where w here all the lines have undergone finite extension (shaded area) from the area where all the lines have undergone finite shortening.
40
5.7 5.7 Pure Pure and s imp le shear During a progressive deformation, all successive incremental strain ellipses may have the same orientation, i.e., the axes of the strain ellipses remain parallel. Strain is considered non-rotational and this type of strain is called pure shear (Fig. 5.9a). If the strain ellipses change their orientations during progressive deformation, it is called simple shear.
λ1 λ1 λ3
λ3
(a)
(b)
Figure 5.9. (a) Non-rotational strain or pure shear. The major and minor axes of the successive strain ellipses are coincidental. (b) Rotational or simple shear. The strain ellipse undergoes rotation with respect to an external fixed reference frame during progressive deformation.
41
42
If a material is stressed, it gets strained. Rheology Rheology is the material response to applied stress. Much of the classical theoretical and experimental principals of stress-strain relations were developed in material science where materials are usually taken to be homogeneous and isotropic. The behaviour of natural earth materials, i.e., rocks and minerals, is often extremely complex because the rocks are neither homogeneous nor isotropic. Even if a rock is approximately homogeneous to start with, it develops fabric(s) during deformation and becomes heterogeneous. Nevertheless, stress-strain relationships for homogeneous and isotropic materials can be used as a first approximation for rocks. In the simplest form, materials respond to stress in two different ways. When the stress is withdrawn, the material may return to original shape and size, and strain is said to be recoverable. recoverable. Otherwise the strain is permanent is permanent.. We can give stress parallel to the axis of a cylindrical rock and measure strain. The results can be plotted on a stress-strain graph. At the initial stage stress-strain curves are usually straight with steep slopes (Fig. 6.1a). The straight line implies that there is a constant ratio between stress and strain and the steep slope means that small strain accumulates for large incremental increase in stress. The most important aspect of this part of the stress-strain curve is that the strain is completely recoverable, i.e., the material returns to original dimension (zero strain) when stress components all drop to zero. This kind of material behaviour is called elastic. elastic. The area under the straight line curve is a measure of stored elastic energy. In some elastic materials, there is no time lag between change in stress and corresponding change is stress. In other words, the linear relation between stress and strain is instantaneous. This material behaviour is known as Hookean Hookean behaviour. It is obvious that the Hookean behaviour and elasticity are not exactly same. The following parameters are used to describe the properties of elastic materials (see Fig. 6.1b): 43
•
Young’s Modulus: Modulus: In Fig. 6.1b the elongation parallel to the axis of the cylinder e is given by (ld – l i)/li. The stress is proportional to the strain for elastic material. The proportionality constant, E, is called Young’s Modulus. Modulus. E = σ / e This is for simple shortening and extension. The Young’s Modulus describes how difficult it is to give longitudinal strain in a material. Higher the absolute value of E, it will be more difficult to extend or shorten a material. Note that E has same dimensions as stress (e.g., stress). σ Initial
Deformed
li ld
Stress, σ
e L = (l (l d - li )/ )/li eT = (w (wd - wi )/ ) /wi
Area is a mea sure of s tored elastic energy
(a)
Strain, ε
(b)
wd wi
Figure 6.1. Stress-strain diagram for elastic deformation. The area under the curve represents stored elastic energy. (b) Longitudinal (e ( eL) and transverse (e (eT ) elongations.
•
Poisson’s Ratio: Ratio : If eL and eT are longitudinal and transverse elongations, respectively, then from Fig. 6.1b: eL = (l d – l i) / li
and
e T = (w d – w i) / wi
The Poisson’s Ratio ( ν) is simply the ratio between eT and e and eL: = eT / / eL ν =
44
So, ν is a dimensionless quantity. For small strains, the change in volume ( ∆) is given by:
∆ = ( σ / E) (1 - 2 ν) For constant volume deformation ν has a value of 0.5 no matter how high the stresses get. If volume decreases with compressive strain or increases with tensile strain, ν will have value less than 0.5. For most rocks ν varies between 0.25 and 0.33. Constant volume deformation indicates incompressible material. For porous rocks with or without fluids volume change may be significant during deformation.
•
Rigidity modulus: modulus: If a cube of elastic material is subjected to shear stress σ s it may undergo shear strain γ, which is directly proportional to σ s. The proportionality constant is called rigidity modulus, modulus of rigidity or shear modulus (G). Thus G = σs / γ Like Young’s Modulus, rigidity modulus also has the dimensions of stress. Shear strain induces change in shape of a body. Therefore, rigidity modulus describes resistance of an elastic body to change in shape.
•
Bulk Modulus: Modulus: This elastic property given the relationship between change in pressure [∆P, i.e., hydrostatic stress, σ mean )/3] and consequent σ1 + σ 2 + σ 3 )/3] mean = ( σ volume change or dilation (∆) of a block of elastic material. Thus the Bulk Modulas, K, is given by: K = σmean / ∆ The inverse of Bulk Modulus is known as compressibility. Obviously, the Bulk Modulus is a measure of ease or difficulty with which an elastic material can be compressed.
45
The above elastic moduli are constant throughout isotropic materials but they vary from place to place in anisotropic rocks geologists have to deal with. Further, the four elastic moduli are not independent of each other but are related to each other through some simple relations: G = E/[2 (1 + ν)] = [3 K (1 + 2 ν)]/[(2 (1 + ν)] If stress is continued to be increased, the rock may suddenly fracture in the elastic range (Fig. 6.2a) and the stored elastic energy is released in the forms such as elastic waves, sound and heat. This is known as brittle failure failure and the deformation is called brittle deformation. The value of the stress at which brittle failure takes place is called brittle strength, strength , rupture strength strength or fracture strength. strength. Under many conditions, including conditions of common laboratory experiments, the rocks undergo permanent strain instead of fracturing if elastic limit is exceeded. If stress is withdrawn strain is not completely recovered. The materials showing such this behaviour of permanent straining are plastic materials and the deformation is called plastic deformation or ductile deformation. The point on the stress-strain curve where the changeover from elastic to plastic deformation takes place is called yield point, which can be difficult to locate precisely during an experiment. The stress at the yield point is called yield stress, yield-point stress or yield strength. Yield stress is not a constant for any one rock type but varies with temperature, confining pressure, fluid pressure and strain rate. Two kinds of deformations can be envisaged above elastic limit:
•
Perfect plastic: plastic: The material may continuously accumulate permanent strain at the yield stress, i.e., the stress-strain curve will have zero slope (Fig. 6.2b). The material is then called perfectly called perfectly plastic. plastic.
•
Strain hardening: hardening : In most deformed rocks, the stress-strain curve above yield point has a positive slope (Fig. 6.2c). The slope of the curve is gentler than that of elastic region and the curve is markedly non-linear. Strain can increase in this situation only if stress is raised above initial yield stress; this process is called 46
strain hardening or work hardening hardening (Fig. 6.2c). This strain-hardened deformation is usually termed as plastic or ductile deformation. The rocks in this realm seem to flow somewhat like a fluid and a plethora of structures develop. However, it must be remembered that rocks are not fluid like water, which cannot sustain shear stress at rest. If stress is withdrawn after some amount of plastic deformation, such as at point X in Fig. 6.2c, the curve falls on the strain axis in an approximately linear fashion (XY) giving the total amount of permanent strain (OY, Fig. 6.2c) . If the stress is immediately reapplied, the curve goes back along the linear path (YX) to the original position on the plastic deformation curve. With increase in stress the curve continue along the original plastic deformation path. The strain hardening may be thought of as elastic part of the curve and yield strength moving continuously towards the right hand side of the stress-strain diagram. With continued increase in stress the rock finally ruptures at a point known as ultimate strength strength (Fig. 6.2c).
brittle strength
Fracture
Stress, σ
yield strength
perfect plastic
Stress, σ
ultimate strength
X
yield strength
rupture
Stress, σ elastic
elastic
elastic
plasticc plasti (strain hardening)
permanen t strain
(a)
(b) Strain, ε
(c) Strain, ε
Y Strain, ε
Figure 6.2. Stress-strain relationships of elastic, perfect plastic and plastic (strain hardened) behaviour of materials.
47
48
Fracturing is an important process of rock deformation, particularly in the upper part of the crust. Fractures are surfaces along which rocks loose cohesion, i.e., rocks break along fracture surfaces. There are two types of fractures depending on the relative motion between the two sides of a fracture. If there is no relative motion the fractures are called joints.
•
Extension fracture: fracture : In this type of fracture, relative motion between the two sides is perpendicular to the fracture surface. They signify overall extension perpendicular to the fracture surface. This type of fracture is also called Mode-I fracture (Fig. 7.1a).
•
Shear fractures: fractures : In this type of fracture, relative motion between the two sides is parallel to the fracture surface. This type of fracture implies shearing movement parallel to the fracture surface. The direction of relative motion may be perpendicular or parallel to the edge of the fracture surface, they are called Mode-II (Fig. 7.1b) and Mode-III (Fig. 7.1c) fracture, respectively. A mixed mode fracture has components of both extension perpendicular to fracture and shearing parallel to fracture.
(a)
(b)
( c)
Figure 7.1. Types of fractures depending on relative motion between two sides. (a) Mode-I extension fracture. (b) Mode-II shear fracture. (c) Mode-III shear fracture.
49
We also recognize two broad types of fractures in rocks in macrosopic scales depending on quasi-mechanical behavior of rocks at the time of fracturing. They are brittle and ductile fractures formed during brittle and ductile deformations, respectively (Fig. 7.2).
•
Brittle fracture: fracture : Brittle deformation occurs at the rupture strength in the elastic regime with the formation of brittle fractures, which are surfaces across which material loses cohesion. Rocks do not show any change in shape such that the broken pieces fits together to give the original shape and size of the rock body (Fig. 7.2b).
(a )
(b )
(c)
(d)
rittle and ductile deformations. (a) Undeformed cylinder. (b) Brittle Figure 7.2.B fracture. (d) Ductile deformation. (d) Brittle-ductile deformation.
•
Ductile fracture: fracture : In ductile deformation, the rocks show permanent strain that smoothly varies through the deforming material and no clear cut fracture develops in macroscopic scale (Fig. 7.2c). This term does not signify any specific deformation mechanism. Also, ductile deformation need not necessarily signify plastic deformation mechanism. It is a general term for macroscopic flow of rocks that can be accomplished by brittle deformation or plastic deformation or a combination of both. If the ductile deformation is accomplished by fracturing and rotation of individual grains or grain aggregates, the deformation is called cataclastic flow. flow. If the ductile deformation is dominated by flow of individual grains through dislocation glide and climb, and diffusion, the deformation is called plastic called plastic flow. flow . After some amount of ductile deformation, a deforming rock may fail and develop fractures. This type of fracture is called ductile fracture, fracture,
50
which may be considered as transitional behavior between brittle fracture and plastic flow (Fig. 7.2d). The type of fracture that may develop depends on the specific state of stress acting on a rock body (Fig. 7.3):
•
Hydrostatic compression: compression : Rocks can withstand unlimited hydrostatic compressive stress with volume change as the only manifestation of deformation (Fig. 7.3a). This type of state of stress is possible in the earth, especially at great depths.
•
Hydrostatic tension: tension: Rocks break apart into random pieces if they are subjected to high enough hydrostatic tensile stress when tensile stress exceeds cohesive stresses that hold the particles in a rock together (Fig. 7.3b). This is a very unlikely state of stress in the earth.
•
Uniaxial compression: compression : If σ2 and σ3 are zero or close to zero, as in uniaxial compression, longitudinal splitting takes place parallel to compressive σ1 (Fig. 7.3c). The fractures formed by longitudinal splitting tend to be more irregular in orientation and shape that other extensional fractures.
•
Uniaxial tension: tension : If σ1 and σ2 are zero or close to zero, as in uniaxial tension, tension fractures form perpendicular to tensile σ3 (Fig. 7.3d).
tension: Extension fractures can also form perpendicular to σ3 if σ1 = σ2 > • Axial tension:
σ3 > 0. Note that σ3 here is compressive (Fig. 7.3e). compression : Under this state of stress conjugate shear fractures • Axial or confined compression: oriented at angles less than 45 ° to σ1 form (Fig. 7.3f). The two fractures do not form simultaneously but form sequentially. In homogeneous material, it is not possible to predict which one of the two will form first. The line of intersection between two fractures can have any orientation on a plane perpendicular to σ1. The possible orientations of shear fractures are tangent to a cone whose axis is
σ1. Displacements are parallel to fracture surfaces. •
Triaxial stress: stress : Conjugate shear fractures form, as in axial compression. But in this case, the intersection between the shear fractures is parallel to intermediate 51
stress axis σ2. The diehedral angle between the conjugate shear fractures is less than 90°, which is bisected by σ1.
+σs
+σs
σ1 = σ 2 = σ 3 p (pressur e)
σ 1 = σ 2 =σ3
+σn
(a)
+σn
(b) +σs
σ2 = σ3 = 0
σ1
+σs
+σn
σ1
σ1 = σ2 = 0
σ3
(c)
σ3
+σn
(d)
σ3
+σs
+σs
σ1
σ 2 = σ3
σ1 σ3
σ1
+σn
σ1 = σ2
σ2 = σ3
σ1
+σn
(f)
(e)
+σs
σ1
σ3 σ3
σ2
σ1
+σn
σ2
(g)
Figure 7.3. Types of fractures depending state of stress in a rock body. In each diagram state of stress is shown by a Mohr diagram and orientations of fractures related to stress axes are shown. (a) Hydrostatic compression. (b) Hydrostatic tension. (c) Uniaxial compression. (d) Uniaxial tension. (e) Axial tension. (f) Axial or confined compression. (g) Triaxial stress.
Faulting is a consequence of shear stresses in the crust. Therefore, deviatoric stress components in a stress system must have significantly large values for faults to develop. It is important to understand the stress conditions for brittle failure of rock because this gives us an insight on how faults and fractures develop during orogenic 52
processes. Laboratory experiments on rock deformation using triaxial testing apparatus have been most illustrative in this regard. A schematic diagram showing the basic features of such an experimental apparatus is shown in Fig. 7.4a. In more sophisticated equipment pore fluid pressure and temperature can also be applied and controlled. The three principal stresses can be controlled but two of them are equal. The forces applied are perpendicular to the sample surfaces so that the principal stresses are either parallel or perpendicular to the axis of the cylindrical sample. Axial load oad
σa ( σ1 or σ 3) Steel housing hou sing
Piston Pisto n
Cylindical rock sample
σc
Confining pressure pressure (by a fluid)
σ14
4
Weak, impermeable rubber/cop per jacket r a b k , )
σ13
3
σ11
σc
σ
1
σ1
1
(b)
Fracture
1
(
(a)
Fracture
σ12
c
σ 2 − a
σc = σ 2 = σ3 σc = σ 1 = σ2
σs
0
σa ε0
σ14 4
Coulomb fracture fracture behaviou beh aviourr
p
+σs
Fracture
Unstable
+2θ
σ1
1
2
3
−2θ
p’
4
5 kbars kbars
σn
−σn
Transitional tensile tensile behaviou beh aviourr
+σn
0
+2θ
Τ0
−σ n
θ = (90 +φ)/2
p
σ3
Tensile fracture fracture behaviou beh aviourr
(e)
φ
+σ s
Mohr/fracture/ Mohr/fracture / failuree envelope failur envelope Stable
σc
σn
5 kbars kbars
-3
σs
(d)
σ13 3
(c)
-10
-5
Axial strain, ε x 10
σ12 2
−2θ
p’
−σs
(f)
−σ s
Figure 7.4. Derivation of fracture envelope. (a) A simplified sketch of a triaxial experimental apparatus. A cylindrical sample placed in an impervious rubber/copper jacket is given an axial load through a hydraulic ram and a radial confining pressure through a fluid. (b) Stress-strain diagram for a hypothetical experiment at 0.5 kbar confining pressure. (c) Mohr diagram for the same hypothetical experiment. Each Mohr circle represents a discrete state of stress during experiment. The largest Mohr circle represents state of stress at fracture. (d) Mohr circle at fracture showing two possible orientations of shear fractures ( p p and p’) p’) oriented at ± 2 θ to σn axis. (e) Mohr diagram showing Mohr circles at fractures at different confining pressures. Mohr/fracture/failure envelope is tangent to all the Mohr circles at two points except at very low confining pressure. The Mohr envelope divides the Mohr diagram into stable and unstable regions. (f) Different types fracture behaviour along the Mohr envelope. See text for discussion.
53
σ1
+σ n
The rock ro ck samples are cut into cylindrical shape, jacketed by weak and impermeable layer of copper or rubber and put in a piston cylinder. The sample is surrounded by a chamber filled with fluid through which confining pressure ( σc) can be varied and controlled. The rock sample is subjected to an axial load ( σa) through a piston driven by hydraulic ram. The axial load can be either σ1 or σ3 ( σa = σ1 or σa = σ3). The state of stress can, therefore, be either axial compression ( σa = σ1 > σ2 = σ3 = σc) or axial extension ( σa = σ3 < σ2 = σ1 = σc). The triaxial testing apparatus can be modified for experiments under uniaxial tension, but such experiments are rare. Data obtained from such experiments are usually graphed in Mohr diagrams, and stress-strain diagrams with stress and strain plotted on vertical and horizontal axes, respectively. Let us consider a series of experiments on a dry homogeneous rock in a triaxial rig and demonstrate the results of the experiments in a stress-strain diagram as well as in a Mohr diagram (Fig. 7.4, Suppe 1985). The experiments are conducted under axial compression such that σa = σ1 > σc. We first increase both σ1 and σc together until both of them reach a value of 0.5 kbar and the rock sample acquires an initial natural strain (ε0) parallel to the axis of the cylindrical sample, i.e., parallel to σ1. Note that the difference between axial load and confining pressure is zero at this stage (i.e., σ1 - σc = 0). We now keep increasing the axial stress keeping the confining pressure constant. The strains and Mohr circles at three discreet axial stresses ( σ11, σ12 and σ13) are shown in Figs. 7.4a and 7.4b, respectively. If at anytime during this stage of deformation the axial stress is returned to 0.5 kbar, the axial strain returns to ε0. Eventually, the rock sample fails at an axial stress of σ14. Let us assume that this stress is 4.5 kbar. We take the sample out of the rig and find that one shear fracture has developed oriented at 24 ° to the axial stress direction, i.e., the axis of the cylinder. If we repeat the experiment with another sample of the same rock we get a fracture oriented at the same angle but on the other side of the axis, i.e., together they form a conjugate pair. However, we can not predict which of the two possible fractures will yield during a particular experiment. We say that the fracture strength of the rock is 4.5 kbar at a confining pressure of 0.5 kbar. The largest Mohr circle in Fig. 7.4c represents the state of stress at fracture. We now keep only the Mohr circle at fracture and remove the all other Mohr 54
circles for convenience (Fig. 7.4d). The orientations of the two fracture planes ( p ( p and p' p')) are also shown in Fig. 7.4d. We now conduct similar fracture experiments at increasing confining pressures and plot Mohr circles at fracture at each confining pressure in the same diagram (Fig. 7.4.e). For each Mohr circle there will be a pair of fracture planes represented by p by p and p' p'.. Loci of these points give us a curve called Mohr, or fracture or failure envelope, which divides the Mohr diagram into stable and unstable parts (Fig. 7.4e). The state of stress in the stable part does not lead to any kind of fracturing. The state of stress in the unstable part leads to catastrophic fracturing of rock. The Mohr envelope shows three types of brittle fracture behaviour (Fig.7.4f):
•
Tensile fractures: fractures : Tensile fracture behaviour is shown at the point where the Mohr envelope crosses the negative side of the σn axis. The Mohr envelope is tangent to the Mohr circle only at this point so that, only one set of fractures oriented perpendicular to maximum tensile stress can develop. The tensile strength, T 0, is independent of other principal stresses and varies between 50 and 200 bars for most common rocks. Note that T 0, is the value at the point where Mohr envelope intersects σn axis.
•
Coulomb fracture behaviour : The fracture behaviour of rocks in the straight line segment of the Mohr envelope (continuous line in Fig.7.4f) is called the Coulomb fracture behaviour. Obviously, fracture strength increases linearly with confining pressure in this part of the Mohr envelope. This behaviour is common for many rocks at intermediate confining pressure with
σ1
approximately greater than |5T0|. In order to describe the Coulomb fracture behaviour, we can write a linear equation of the form:
|σs*| = C + σn tan φ
(7.1)
|σs*| = C + µ σn
(7.2)
where, σs* is the critical shear stress, i.e., the shear stress at fracture; c and µ are the intercept and slope, respectively; and φ is the slope angle of the line. The above equation is commonly called Coulomb fracture criterion. criterion . The constants C 55
and µ describes the failure properties of rocks. C is the cohesive shear strength, strength , which represents resistance to shear fracture on a plane with zero normal stress, and µ and φ are called coefficient of internal friction and angle of internal friction, respectively. Whenever, the state of stress on a plane of some orientation satisfies equations (7.1) or (7.2), a shear fracture may develop on that plane. For a triaxial stress, the criterion is satisfied on two planes where Mohr circle is tangent to the Mohr envelope. These two planes are symmetrically oriented about σ1 (or σ3) axis and intersect along σ2 axis. They form a set of conjugate shear planes. planes . Shear fracture may develop along any one of these two planes but the fracture criterion does not predict which of the two orientations will be preferred at failure. The θ and φ are related by the relation
θ = (90 ° + φ)/2
(7.3)
The Coulomb fracture criterion may also be expressed in terms of principal stresses, such as:
σ1 = S + K σ3
(7.4)
where, S and K are compressive strength strength and earth-pressure coefficient coefficient and given by S = 2 C [√(µ2 + 1) + µ]
(7.5)
K = [√(µ2 + 1) + µ]2
(7.6)
The fracture criterion in the form of equation (7.4) is plotted on a graph of maximum vs. minimum principal stresses although Mohr circle does not plot on such a graph.
•
Transitional tensile behaviour : This behaviour is shown by rocks with state of critical stress lying in between Coulomb fracture behaviour and tensile fracture behaviour on the Mohr envelope. This behaviour is shown above a critical value of tensile stress σ1, which is about three times T 0. In this segment, the fracture strength increases nonlinearly with increasing confining pressure and equation 56
(7.3) is invalid. Fractures are oriented in the range 0 to 30 ° to the maximum compression. Joints probably form in state of stress characteristic of tensile and transitional tensile segment. Although fracturing of rocks in nature is a complex process, the above discussion should be considered as a predictive tool for brittle fracturing. There are several factors that control fracture formation in rocks, such as confining pressure ( σc) and differential stress (σ1 - σc), as we have already seen. Among other factors, fluid pressure plays a major role in significant modification of Mohr circles. Fluid is an important agent in many geological processes including fracturing. We discuss the effect of fluid pressure (P f ) on Mohr circle in some detail (Fig. 7.5).
Mohr envelope
Unstable
+σ s
+σ s
Mohr envelope
Unstable
Stable
Stable
p
−σ n
T0 σ3e
State of effective stress
2θ
P f σ1e
σ3
σ1
+σ n
−σ n
σ3e
State of applied applied stress stre ss
(a)
σ3
σ1e
σ1
+σ n
2θ
p' −σ s
P f
−σ s
State of effective stress stress
State of applied applied stress stre ss
(b)
Figure 7.5. Effect of pore fluid pressure on Mohr diagram. (a) Low differential stress leads to tensile fractures. (b) Relatively large differential stress leads to shear fractures.
Water incorporated in the intragranular spaces during sedimentation form pore fluids. Dehydration reactions during diagenesis and metamorphism release H 2O-rich fluid into pore spaces. If all the pore spaces are interconnected all the way to the surface, we can calculate fluid pressure ( P f ) at a given depth using a an equation we learnt in high-school physics, viz., P = ρ g g h, where P, ρ , g and h are pressure, density, acceleration due to gravity and depth (i.e., height), respectively. Similarly, we can calculate lithostatic pressure or vertical normal stress, σ nL. Taking densities of water ( ρ w) and rock ( ρ r r) as 1.0 and 2.3 gm/cm 3, respectively, we get: 57
ω = P f /σnL = ( ρw g h)/( ρrg h) = 0.4
(7.7)
So it seems that the pore fluid pressure should be about 40% of lithostatic pressure. However, it does not always work that way. In many deep drill wells in sedimentary basins, the ratio ω approaches unity. In metamorphic petrology this ratio is one unless otherwise specified. It is unreasonable to assume constant density for easily compressible fluid although the same assumption may not be too bad for rather incompressible rocks occurring in the upper part of the crust. The value of ω should increase, as density of fluid increases with depth. It is for nothing that a metamorphic petrologist will argue that water and fluids are not exactly same things! Secondly, pore spaces may be not interconnected all the way to the surface. There may be impermeable barriers preventing fluids at depth communicating with fluids at surface and building up fluid pressure (formation overpressure). The effect of fluid pressure on fracturing of rocks can be best explained through the concept of effective stress. When a component of stress is lowered by an amount equal to the pore fluid pressure, it is called effective stress. Thus,
σne = σn - Pf σ1e = σ1 - Pf σ2e = σ2 - Pf
(7.8)
σ3e = σ3 - Pf σce = σc - Pf σemean = ( σ1 + σ2 + σ3 - 3P f)/3 where superscript e denotes effective stress. Shear stresses are not affected because fluid at rest cannot sustain shear stress. The size of the Mohr circle remains between state of effective stress and state of stress with zero fluid pressure. This is because differential stresses in both cases are same, i.e., (σ 1 - σ 3) = (σ 1e - σ 3e). The only effect is seen in translating of the Mohr circle towards lower compressive stress (i.e., towards left) in the Mohr diagram by an
58
amount equal to P f (Fig. 7.3). The equations describing fracture criterion also remain same except that σne replaces σn: |σs*| = C + σne tan φ = C + ( σn – Pf) tanφ
(7.9)
σ1e = S + K σ3e, or (σ1 - Pf) = S + K (σ3 - Pf)
(7.10)
Shifting of Mohr circle to lower compressive stress may lead to hydraulic fracturing via following processes:
•
If the differential stress [( σ 1 - σ 3) or (σ 1e - σ 3e)], which represents the size of the Mohr circle, is small and σ 3e = T 0, then extension fractures can form (Fig. 7.3a).
•
If the differential stress is relatively large, the P f may drive the Mohr circle far to the left to touch the fracture envelope and shear fracturing may occur (fig. 7.3b).
Putting it together we may state that given the initial state of stress in the rock is within the stable part of the Mohr diagram, the fluid pressure may be increased to a level that the Mohr circle for state of effective stress may touch the fracture envelope for fracturing to occur. In order to increase the permeability of hydrocarbon reservoir rocks, artificial hydraulic fracturing is carried out by pumping fluids through oil wells. Strength and ductility of rocks increase with mean stress. The pore fluid pressure reduces mean stress and makes rocks weaker and more brittle. Therefore, high fluid pressure may lead to brittle fracturing at depths normally associated with ductile deformation.
59
60
A fault is a brittle shear fracture fracture along which appreciable appreciable amount of differential displacement has taken place. This rather straightforward and generally accepted definition of faults has one serious problem, i.e., with the word appreciable. The displacements along faults may vary from microscopic to hundreds or thousands of km on very large faults, such as transcurrent faults. Therefore, what constitutes a displacement to be appreciable depends on the scale of observation. For example, a geologist working on regional faults pattern may dismiss a fault seen at an outcrop with centimeter/millimeter-scale displacement as minor shear fractures or microfaults. microfaults. A fault with few hundred meters of displacement may lead to the formation of a good trap for hydrocarbon accumulation but may not show up on a regional-scale seismic survey. A fault may occur as one one sharp plane of discontinuity (Fig. 8.1a) or as a zone of closely spaced planes of discontinuity (Fig. 8.1b). The discontinuity surfaces are the brittle shear fractures. Faults at depth may not have such sharp discontinuities; instead the deformation may be distributed in a diffused zone. Such zones of distributed deformation are called ductile shear zone (Fig. 8.1c). The maximum strain in a ductile shear zone is at the middle of the zone and gradually decreases to zero at the shear zone wall. In some cases a fault zone may have components of both brittle and ductile deformations (Fig. 8.1d).
(a )
(b )
(c )
(d )
Figure 8.1. Styles of faulting. (a) Brittle fault on a single surface. (b) Brittle fault zone consisting of several subsidiary fault surfaces. (c) Ductile shear zone. (d) Brittle-ductile shear zone.
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8.1 8.1 Fault Fault t erminolo gy The terminology associated with faults is complex and ambiguous. In this section and in the sections to follow, the terms used are most prevalent, have precise meaning and most useful in describing faults (see Fig. 8.2):
•
Blind/emergent: Blind/emergent: If tip line does not reach the ground surface the fault is blind. When the tip line is exposed to the ground surface it is an emergent fault. A blind fault may become emergent through erosion. emergent fault
blind fault
compressional decollement
tip fault trace
tip line
tip line
fault surface
extensional decollement hangingwall cut-off point
emergent fault
tip blind fault compressional detachment footwall cut-off point
listric fault
hangingwall hangingwall cut-offf line cut-of
footwall cut-offf line cut-of
hangingwall hangingwa ll block
footwall footwall block
extensional detachment
N D
V
N : Net slip slip D: Dip slip S : Strike-slip component H : Horizontal Horizontal component V : Vertical component
S
H
Splay faults isolated splay
diverging splay
b.p.
rejoining splay
connecting splay
b.l. b.p.
t.l. b.l.
b.l. b.p. - branch point b.l. - branch line t.l. - tip line
m a i n f a u l t
Figure 8.2. Diagrams illustrating different geometric terms associated with faults.
62
•
Branch line: line: The line of intersection between any two fault surfaces is known as the branch line. If the faults crop out at the surface, this line will appear as the branch point. In sections, they appear as branch point.
•
Cut-off line: line: The line along which a marker horizon is truncated by a fault. For the same marker horizon there are two cut-off lines on two sides of a fault (e.g. hangingwall cut-off line and footwall cut-off line for inclined faults). In sections, they appear as cut-off point.
•
Décollement: Décollement: A low-angle regional fault (compressional or extensional), which is usually parallel to a soft or incompetent strata.
•
Detachment fault: fault: A low-angle regional fault (compressional or extensional) that cuts through gently-dipping or horizontal strata.
•
Fault trace: trace: The line along which the fault plane cuts the ground surface is known as fault trace. Blind faults do not have fault trace!
•
Hangingwall/footwall: Hangingwall/footwall: For an inclined fault, the block above the fault surface is known as hangingwall and the block below the fault plane is known as footwall. In a tunnel parallel to the fault surface, the hangingwall literally hangs overhead and the footwall lie under the foot of an observer. For a fault with vertical dip, there is no hangingwall or footwall.
•
Horse: Horse: A mass of rock surrounded on all sides by fault.
•
Listric fault: fault: A fault with smoothly curving surface is called a listric fault.
•
Separation: Separation: The distance between points on cut-off lines on either side of fault. The separation distance varies depending on the direction in which it is measured. The horizontal separation separation is the separation in a horizontal plane and may or may not equal the strike-slip component.
•
Slip on a fault: fault : The distance between two points on the fault plane those were together before faulting, represents total displacement or net slip. slip. The net slip is a vector quantity. The net slip can be resolved into strike-slip strike-slip and dip-slip components. The dip-slip component can further be resolved into a horizontal component component and a vertical component. component .
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•
Splay: Splay: A secondary fault (i.e., smaller in size and displacement) that emerges from a main fault. There are different types of splays. A connecting splay connects two faults whereas a rejoining splay splay rejoins the same fault from which it branched off. When a splay fault branches off and diverges away from a fault, it is called diverging splay. splay. From an isolated splay splay no other fault branches off.
•
Tip line: line: Individual faults are always of limited extent and displacement die away to zero. The line along which fault displacement becomes zero is known as tip line (tip or tip point in two dimensions). Tip line may emerge on the erosion surface as tip point or it may remain blind.
8.2 8.2 Fault Fault classif ication Faults are classified on the basis of relative movement between the walls. This scheme of classification is adequate under most situations (Fig. 8.3).
•
Dip-slip faults: faults: These faults have dominant dip-slip component. The strike-slip component is small or negligible. They are of two types: (1) Dip-slip faults in which hangingwall moves down relative to the footwall are called normal fault. fault . The dip of normal faults is usually 50° or more. They are sometimes called extensional fault, fault, as they are more common in extensional tectonic set up, such as divergent plate margins. (2) Reverse faults faults are those in which hangingwall moves up relative to footwall block. A reverse fault in which dip of the fault surface is less than 45° are called thrust fault. fault. They are sometimes called compressional fault, as they are more common in compressional tectonic set up, such as convergent plate margins.
•
Strike-slip faults: faults: Strike-slip faults are those in which strike-slip component is dominant, i.e., predominantly horizontal displacement. They are steeply inclined, often vertical. They are characteristic of translational tectonic set up. Large-scale sub-vertical strike-slip faults are sometimes called transcurrent or wrench fault. They are of two types: sinistral sinistral (syn. left-lateral or left-handed)
64
when the relative movement is counterclockwise or dextral dextral (syn. right-lateral or right-handed) when the relative movement is clockwise.
•
Oblique-slip faults: faults : In these faults the strike-slip and dip-slip components are approximately same. They may be right normal, right reverse, left normal and right reverse depending on whether the opposite walls move clockwise or counterclockwise and whether hangingwall moves up or down relative to footwall. ( a) Dip-s lip fa fault (n (net slip = di p s lip)
Normal
( b) Strike- slip fault (net s lilip = strike s lilip)
Reverse Sinistral
(c) Oblique-slip fault (dip slip strike slip)
Ri gh t n or m al
Ri gh t r ev ers e
Dextral
(d) Rotational fault
Lef t no rmal
Lef t rev er s e
Figure 8.3. Classification of faults based on relative displacements between the walls.
The normal, reverse and strike-slip faults are the most common types of faults. They typically form in extensional, compressional and translation tectonic regimes, respectively. The terms strike fault, dip fault, transverse fault, wrench fault, tear fault, transcurrent fault, hade, throw, and heave should not be used because some of them are obsolete while others are imprecise and may create confusion.
8.3 8.3 Anderson's t heory of faulting There is no single comprehensive theory of mechanics of faulting. The theory of faulting as presented by E. M. Anderson (1951) is the simplest and the most enlightening (Fig. 8.4). There are two assumptions involved in the Anderson's theory of faulting: 65
•
Shallow level faults are Coulomb fractures . A consequence of this assumption is that there are two possible orientations of Coulomb fractures with respect to the principal stress directions. The line of intersection of two fractures is parallel to intermediate principal stress axis, compression,
σ1,
σ2.
The direction of maximum principal
bisects the acute angle between the fractures and the obtuse
bisector is parallel to the least principal compression direction, shortens parallel to
σ1 and
extends parallel to
σ3 as
σ3.
The material
a result of slip along the
fractures.
•
In shallow-level faulting one of the principal stress directions is vertical . This is due to the fact that the earth-air interface is a plane of no shear. Vertical
σ1, σ2 and σ3
leads to normal, strike-slip and reverse faults, respectively. These are the three most common types of faults, as noted earlier. σ1
σ3 σ2
σ1
σ2
(a)
σ2
σ1
σ3
D U
U D
σ1
σ2
σ3
σ3
σ3 (b)
σ2
σ1 σ2 (c)
σ1
σ3
Figure 8.4. Anderson’s theory of faulting. Vertical σ1, σ3 and σ2 lead to normal faulting (a), thrust faulting (b) and strike-slip faulting (c), respectively. D and U are downthrown and upthrown sides.
The Anderson's theory of faulting does not predict some of the commonly observed features associated with faulting, such as, high-angle reverse faults, very low-dipping and very high-dipping normal faults, listric faults and transform faults. Yet, it remains a simple and elegant way of describing how faults form in different tectonic settings.
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8.4 8.4 Recogniti Recogniti on of faults Under the present-day scenario of doing everything on a computer monitor it has become imperative to emphasize an obvious fact, viz., faults must be recognized on geological criteria. Several categories of geological features are used to recognize faults. They are not mutually exclusive; often more than one category of features are found associated with a single fault.
•
Geomorphic features: features : The most obvious geomorphic feature that may betray the presence of a fault is fault scarp scarp (Fig. 8.5). Fault scarps are continuous linear features characterized by sudden break in topographic slopes. They may indicate either active or inactive faults. However, the original scarp associated with an old and inactive fault may not survive erosion for very long. Regular fault scarps are more commonly associated with normal faults (Fig. 8.5a). Subsidiary smaller scarps are usually present parallel to the main scarp. The uplifted block is usually cut by V-shaped side valleys. The erosional debris derived from the uplifted block and brought along the side valleys form alluvial fans on the downthrown block. Successive movements along faults may leave perched alluvial terraces on the side valleys. The scarps formed due to thrust faulting tend to be irregular (Fig. 8.5b). An interesting feature associated with thrusting is that the fault often overrides the debris derived from the uplifted hangingwall and deposited in front of the scarp. In strike-slip faulting the scarp is usually small and of local importance (Fig. 8.5b). Deflected or offset geomorphic features, such as, river channels, hogbacks and ridges may indicate the presence of strike-slip faulting. The scarps associated with large faults may show up as lineament satellite images and air photographs. But it must be remembered that a majority of lineaments drawn on satellite images are not faults. Unfortunately, there is a rising but very unscientific tendency in some quarters to draw lineaments on satellite images, construct a rose diagram and deduce stress axes. Surface exposures of faults must be sought out during
67
fieldwork because much critical information can be gathered through field studies of faults.
Normal fault Main fault scarp scarp
Strike-slip fault
Thrust fault
Perched terrace terrace Subsidiary fault scarp
Irregular scarp Erosion debris
Alluvial fan
(a)
(b )
Local scarp
Deflected river channels channels Offset hogbacks
(c )
Main fault Subsidiary fault
eomorphic features associated with three main types of faulting. Figure 8.5.G
2
1
2 km
F1
N 1
F2
F3
2 3 F1
1
2
2
3
3
4
F2 7
3 2
8 1 2
(a)
9
(b)
Figure 8.6. Recognition of faults from map patterns. (a) Offsets of rock units trace faults. Symmetric repetition of rock units in the N-S direction around rocks 1 and 4 are due to folding. The fault surfaces trace lines of discontinuities along which three rock units meet. (b) Omission and repetition of rock units trace faults F1 and F2.
•
Geological map and stratigraphy: stratigraphy : Large faults are relatively easy to recognize in regions of moderate to excellent exposures of rocks through systematic mapping. One should be careful, however, because some of the geologic features are common to both faults and unconformities. Faults are recognized on the basis of truncation and offset offset of one or more rock units (Fig. 8.6a). 68
Truncation and offset occur along a line on a map that defines a discontinuity along which three rocks meet at a point. Such a map pattern may also indicate angular unconformity if rocks on two sides of the discontinuity are of different ages. In case of faults, same rock units should be present on both sides of the discontinuity. This relation may not be valid for large overthrusts, which may bring older rocks to lie over younger rocks or may even bring metamorphic rocks on top of sedimentary rocks. The map patterns due to truncation and offset vary considerably depending on the orientations of faults relative to the orientations of rock layers affected by faulting, and amount and direction of slip. The map patterns may become even more complex if the terrain had an earlier history of folding and faulting. A common effect of truncation and offset is an apparent horizontal shifting of the rock units. There may not be any truncation and offset if the strike of the fault is same as the strike of the rock units. In such cases, faults can be recognized on the basis of repetition and omission of omission of strata if the stratigraphy of the rock units is known (Fig. 8.6b). The line along which a packet of rock units are repeated (fault F1 in Fig. 8.6b) or some ofthe rock units are omitted (fault F2 in Fig. 8.6b) marks a fault plane. Symmetric repetition of rock units about a particular rock unit due to folding (see Fig. 8.6a) should not be confused with simple repetition due to faulting. In drill wells, missing or repetition of beds can be used to predict faults.
•
Fault rocks: rocks: Large-scale faulting with significant amount of displacement lead to development of characteristic textures and structures within the rocks present in the fault zone. Rocks with such characteristic textures and structures are collectively called fault rocks (Wise et al. al. 1984), which can be broadly divided into cataclasite cataclasite and mylonite. mylonite. Cataclasites or cataclastic rocks is a general term that refers to rocks fractured into clasts or ground into powder and signify brittle deformation. Individual clasts are sharp, angular and internally deformed. Cataclasites usually do not have any planar and linear fabrics. Fault gouge gouge and fault breccia breccia are results of cataclastic deformation within fault zones. Incohesive (i.e., friable) cataclasitic rocks are characteristic of faulting above 69
depths of 1-4 km; below this depth cataclastic rocks are cohesive. Frictional heating during brittle faulting can be sufficient high to melt a small portion of the rock. The melt may intrude into surrounding fractures and quenched to give veins of pseudotachylite. pseudotachylite. Mylonitic rocks form if faulting occurs at depths exceeding 10 to 15 km. These are fine-grained rocks with grain size reduction via dynamic recrystallization and neomineralization. Mortar texture with highly strained clasts in a matrix of fine-grained recrystallized grains is a typical texture in these rocks. Planar and linear fabrics are common in mylonites. This texture is a characteristic feature of ductile shear zones. The development of textures and structures in fault zones depend on several parameters including strain, strain rate, temperature, pressure and pore fluid pressure. Therefore, it is not possible to make simple correlation between depth of faulting and type of textures and structures in fault rocks.
8.5 8.5 Sepa Separatio ratio n and di spl acement acement Marker beds or quartz/calcite veins may get displaced during faulting and provide unambiguous evidence of faulting. Fault separation distance distance is the distance measured between hangingwall and footwall cut-off lines of the same horizon. The separation distance varies depending on the direction along which it is measured. Out of infinite number of separation distances, only the one measured in the direction of net slip will give the true displacement. displacement. Obviously, fault separation and displacement are separate parameters. Care must be taken to interpret offset marker layers because the geometry of offset depends on the orientation of the layers prior to faulting, in addition to the orientation of the displacement vector. For example, the marker beds will not show any offset if the displacement direction and cut-off lines are parallel. For this reason, horizontal beds affected by strike-slip faulting do not show any offset. In Fig. 8.7a, a set of inclined beds are affected by strike-slip faulting. The marker layers show offset on both horizontal and vertical surfaces (Fig. 8.7b). Horizontal and vertical offsets give horizontal separation and separation and vertical separations, separations , respectively. Note that they are not necessarily equivalent to strike-slip and dip-slip components. So, what 70
appears on a cross section (or on a seismic reflection profile) to be a normal fault displacing sub-horizontal beds may actually be strike-slip fault affecting inclined beds. Fig. 8.7c shows a set of inclined beds affected by normal faulting. If the fault scarp is completely eroded away to give a flat topography, the fault will appear like a strikeslip fault. Oblique-slip faults show offset on both map view and cross section, suggesting strike-slip faulting and dip-slip, respectively. Therefore, three dimensional reconstruction and presence of lineation may be required for unambiguous interpretation of offset markers on map as well as on section. strike-slip fault
(a )
(b )
normal fault
(c )
(d )
Figure 8.7. Fault separation and displacement. (a,b) Inclined beds affected by strike-slip faulting. Horizontal and vertical offsets suggest strike-slip and normal faulting, respectively. (c,d) Inclined beds affected by normal faulting. Offsets on horizontal surface indicate apparent strike-slip faulting.
8.6 8.6 Dete Determin rmin ation o f fault displ acement cement When we describe movement on a fault plane, we consider the movement of one block while keeping the other block fixed owing to the fact that it is not possible to have a reference frame independent of both the faulted blocks. For example, we define a normal fault as the fault in which the hangingwall goes down relative to the footwall. We could as well state that in a normal fault footwall goes up relative to the hangingwall. Alternately we could also say that in normal fault both the blocks go up, footwall goes up more than hangingwall. Finally, it is equally valid to state that both the blocks go down, hangingwall goes down more than the footwall. Simply put, we are unable to determine absolute movements on faults. The faulting that caused the 71
devastating 1964 Alaska earthquake near Anchorage is a rare example where absolute movement of both the fault blocks with respect to an external reference frame (sea level) could be determined (Platker 1965). Measurements at one place where normal faulting had occurred showed that both the fault blocks had moved upward – one block had moved more than the other. We can only hope to determine the orientation of the displacement vector and the sense of shear movement in most outcrops of fault or fault zone. The magnitude of the displacement can rarely be determined. Differential displacements often lead to finely polished fault surfaces, called slickenside slickenside surfaces. Ridge-in-groove Ridge-in-groove or striation striation lineations commonly form on slickenside surfaces in the direction of displacement. Growth of crystal fibers may result from fluid flowing through small spaces opened up along fault surfaces. The crystal fibers trace slickenfiber lineations, which also parallel displacement direction. A large number of shear sense indicators have been propose d for both brittle and ductile shear zones, some of which can be rather tenacious in most outcrops. A few “better” indicators are discussed below:
•
Drag fold: fold: If the walls of a fault undergo ductile deformation, then marker layers may be dragged into fold forms (Fig. 8.8). These folds are called drag folds whose hinge lines are parallel to the cut-off line and oriented at high angles to displacement direction. The sense of shear displacement is opposite to the curvature of the fold. Drag folds do not form if the cut-off lines are at low angle to the slip direction. Reverse drag drag associated with rollover anticlines have opposite sense of displacement to that of drag folds.
(a )
(b)
(c )
folds giving giving sense sense of displaceme displacement. nt. The beds beds are dra gged Figu Figure 8.8 8.8.. (a,b) Drag folds into the faults in the direction opposite to the sense of displacement. (c) Opposite sense of displacement in reverse drag in rollover anticline.
72
•
Steps associated with slickenfibers: slickenfibers : Exposed slickenside surfaces with slickenfiber lineation commonly have small steps (also called chatter marks) oriented approximately normal to the lineation (Fig. 8.9a). All the steps on the slickeside surface face in the same direction. The missing block on the outcrop (i.e., the block eroded away) is thought to have moved in the direction opposite to the stepped surface. The surface feels smooth to the hand if rubbed in the direction of the movement of the missing block.
slickenfiber slicken fiber
steps
(a)
(b )
s-surface s-sur face c-surface c-surf ace
(c)
(d )
median line
center line
(f)
(e)
Figure 8.9. Examples Examples of of small-sc small-scale ale struc structural tural features features as as shear shear sense sense indicator indicators. s. (a) Slickenfiber lineation and steps. (b) En-echelon gash veins and asymmetric folds. (c) S-C fabric. (d) Bookshelf gliding of pophyroclasts. (e) σ-structure. (f) δstructure.
•
EnEn-echelon gash veins and asymmetric folds folds:: These features develop in ductile shear zones. En-echelon gash veins are tension veins formed normal to maximum stretching direction (Fig. 8.9b). Acute angles formed by gash veins and shear zone walls typically point in the direction opposite to the direction of shear displacement. The axial planes of asymmetric folds are oriented normal to the 73
maximum shortening direction. The sense of shear is in the direction of the acute angle between axial plane and the shear zone wall.
•
S-C fabric: fabric: Schistosity is well-developed in many ductile s hear zones (Fig. 8.9c). Owing to heterogeneous strain the schistosity surfaces ( s-surfaces) are usually sigmoidally curved. Within shear zones, zones of high and low strains parallel the shear zone walls and give rise to a schistosity (c ( c-surfaces) oriented parallel to the shear zone walls. Overall the shear zone displays what is known as s-c fabric. The acute angle between s and c c surfaces point to the shear direction.
•
Bookshelf sliding: sliding : If fractured clasts or porphyroclastic minerals such as feldsp ar or micas are caught up in shear zones, the individual parts rotate in the direction of shear resulting in bookshelf structure, which may be visible to naked eyes. If the fractures initially make high angle to the shear plane, the shear sense in the fractured clasts is opposite to that of in the matrix (Fig. 8.9d). If the initial angle is low, the shear sense on the fractures is same as it is the matrix.
•
-structures: These structures are formed around porphyroclasts and are σ - and δ -structures: best seen in oriented thin sections under microscope (Passchier and Simpson 1986). The shape of the asymmetric pressure shadow trails containing dynamically crystallized grains defines these two types of structures. In σ structure structure (Fig. 8.9e), the median lines drawn through the pressure shadow do not cross the center line drawn through the center of the clast and oriented parallel to the average cleavage surface. In δ -structure -structure (Fig. 8.9f), the median lines cross center lines. The orientations of pressure shadows relative to sense of shear are shown in Figs. 8.9e, f.
All the shear sense indicators in Fig. 8.9 are right-handed or dextral. If we look at the same indicators from the backside of the page, they will all appear to left-handed or sinistral. Therefore, it is not sufficient to state that the displacement is dextral or sinistral. The movement of blocks with respect to geographic coordinates should also be known for definitive interpretation. 74
Low angle (dip < 45°) reverse faults are frequently termed as thrusts thrusts or thrust faults. faults. Thrust faults are much more common than steeply dipping reverse faults. The thrust faults are dip-slip faults with zero or negligible strike-slip component of displacement. They are abundant in the upper crustal levels of the external zones of compressional orogenic zones. Consequently, they are also referred to as contractional faults contractional faults by some workers. They bring older rocks to lie over younger rocks and often result in large-scale vertical duplication of regionally sub-horizontal strata. Thrusts cause elevation of hangingwall relative to footwall giving rise to irregular fault scarps. They range in scale from millimeters to meters, through tens to hundreds of kilometers in fold-thrust belts to thousands in kilometers in convergent plate margins. Over the years a plethora of thrust fault-related terminologies have been proposed in the literature, many of them with obscure meaning. T erminologies defined here are mostly after McClay (1992) who made a brave attempt to bring in a sense of sanity in an otherwise chaotic and often confusing set of terminologies littered through literature.
9.1 Ramp -flat thrust geometry Thrust surfaces usually have stair-case geometry geometry (Fig. 9.1). Though the thrust trajectories shown in Fig. 9.1 are idealized, many thrust have approximately stair-case trajectory. A thrust surface may be considered to have two sides, a hangingwall side and a footwall side. The following terms are used to describe flat-ramp geometry of thrust faults:
•
Flat: Flat: Sub-horizontal or gently-dipping (at the time of initiation) part of a thrust. Flats usually propagate through weaker layers.
75
Cut-off line
Fault Fau lt surface surface
Ramp Ra mp
Flat
Flat
R a m m p
Flat
(a)
Ramp
Flat
Flat
R a Flat m p
(b )
Cut-off p oin ointt Hangingwa Hangin gwall ll side of the thrust thrust
Tectonic transport direction directi on HWR H WR HWF
HWR H WR HWF
Lateral Lateral ramp
Oblique ramp
Frontal Frontal ramp
(c ) FWF
FWR
FWF FWR Footwal Fo otwalll side of the thrust thrust
FWF
(d )
Figure 9.1. (a,b) Flat-ramp-flat (stair-case) thrust trajectory in 2-D and 3-D. (c) Hangingwall and footwall blocks separated to illustrate ramps and flats in the two blocks. FWF: footwall flat, FWR: footwall ramp, HWF: hangingwall flat, HWR: hangingwall ramp. (d) Ramp in 3-D showing frontal, oblique and lateral ramps.
•
Ramp: Ramp: The moderately-dipping part (at the time of initiation) of a thrust. R amps usually climb up-section across stiffer layers.
•
Hangingwall flat (HWF): (HWF) : The portion of the thrust where the fault is parallel to the bedding surfaces on the hangingwall side.
•
Hangingwall ramp (HWR): (HWR) : The portion of the thrust where the fault is oblique to the bedding surfaces on the hangingwall side.
•
Footwall flat (FWF): (FWF) : The portion of the thrust w here the fault is parallel to the bedding surfaces on the footwall side.
•
Footwall ramp (FWR): (FWR): The portion of the thrust where the fault is oblique to the bedding surfaces on the footwall side.
•
Frontal ramp: ramp: The strike of the ramp i s perpendicular to the regional tectonic transport direction.
•
Lateral ramp: ramp : The strike of the ramp is p arallel to the regional tectonic transport direction.
•
Oblique ramp: ramp : The strike of the ramp is oblique to the regional tectonic transport direction.
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•
Thrust trajectory: trajectory : The trace of a thrust on a cross section. Thrust faults always cut up-section in undeformed rock but may cut down section in previously folded terrains.
Flats and ramps on thrust tr trajectory, an and ha hangingwall footwall fl flats and ra ramps are different aspects of flat-ramp geometry.
9.2 9.2 Thru Thru st v ergence Thrust vergence vergence (Fig. 9.2) refers to the direction towards which hangingwall moves relative to footwall. Thrust faults are most common in fold-thrust belts (FTBs) typical of contractional orogenic setting. The undeformed sedimentary basin in front of an FTB is called foreland foreland and the internal part of an FTB is known as hinterland, hinterland, which may contain metamorphic rocks involved in ductile deformation. (Fig. 9.2):
•
Hinterland-vergent thrust: thrust: The hangingwall of the thrust fault mo ves towards the hinterland.
•
Hinterland-dipping thrust: thrust: The dip of the thrust is towards the hinterland. This type of thrust is also called forethrust because the vergence is towards the foreland.
•
Foreland-vergent thrust: thrust: The hangingwall of the thrust fault moves towards the foreland.
Ramp anticline
Hinterland Foreland vergent
(a)
Ramp anticline Hinterland vergent
Hinterland dipping (forethrust) (forethrust)
Foreland dipping (backthrust)
(b)
Figure 9.2.( a) Thrust vergence. (b) Pop-up structure.
77
Foreland
•
F oreland-dipping thrust: thrust: The dip of the thrust is towards the foreland. This type of thrust is also called backthrust because the vergence is towards the hinterland.
•
Pop-up: Pop-up: The portion of hangingwall block that has been uplifted by a combination of forethrust and backthrust (Fig. 9.2b).
9.3 Thru st sheet sh eet (Fig (Fig.. 9.3) 9.3)
•
Thrust sheet: sheet: The areal extent of the hangingwall block of a regionally important low-angle thrust is com monly much greater than the thickness. Such a tabularshaped hangingwall block is called a thrust sheet. sheet. Thrust sheets are given the same name as the underlying thrusts, such as Jutogh thrust sheet and Jutogh thrust in Himachal Himalayas.
•
Thrust nappe: nappe: The French word nappe nappe means a sheet and, therefore, thrust sheet and thrust nappe should be sy nonymous. However, the term thrust nappe is reserved for thrust sheets with significant movement (>10 km) relative to footwall.
Allochthon: An adjective used to describe a thrust sheet that has moved large • Allochthon: distance from its original position. The rocks within an allochthon allochthon are thus geologically out of place and are called allochthonous. allochthonous. Obviously, allochthon and nappe are closely related terms.
•
Parautochthon: Parautochthon: An adjective used describe a thrust sheet that has smaller relative displacement as compared to allo chthon.
Autochthon: A large region of rock that has not moved from its original position • Autochthon: is called autochthon. autochthon. The rocks within an autochthon are called autochthonous. autochthonous. The basement rocks underlying a thrust are autochthonous.
•
Overthrust: Overthrust: A large thrust in which the hangingwall (i.e., the thrust) has actually moved relative to footwall.
•
Underthrust: Underthrust: A large thrust in which the footwall (i.e., the thrust) has actually moved relative to hangingwall.
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•
Klippe: Klippe: The erosional remnant of a once continuous thrust sheet above a horizontal or gently-dipping thrust is known as klippe. klippe.
•
Window: Window: If erosional processes have penetrated to deeper levels of a thrust sheet, then outcrops of footwall can be exposed. A clo sed outcrop of footwall rocks framed by overlying thrust sheet is called window.
•
Root zone: zone: The region in the hinterland direction where a thrust sheet passes into the subsurface is known as root zone. zone. No sense of “origin” or ‘genetic overtones” should be attached to this term.
Allochthonous ro cks
Klippe
Window Root zone
Autochthonous rocks ro cks
5 km
Figure 9.3. Cross section of thrust sheet showing allochthon and autochthon as well as window and klippe.
9.4 Ramp anticline In association with ramp-flat thrust trajectory, anticlines form in the hangingwall just above the ramps. Such anticlines are called ramp anticline or hangingwall anticline (Fig. 9.4). They form due to movement of the thrust sheet up and over the ramp. Different components of ramp anticlines are as follows:
•
Leading anticline/syncline pair : If the ramp anticli ne is flat crested, the anticline and syncline pair located towards the transport direction of the ramp anticline.
•
Trailing anticline/syncline pair : If the ramp anticline is flat crested, the anticline and syncline pair located in the direction opposite to the transport direction of the ramp anticline.
•
Forelimb: Forelimb: The fold limb of the ramp anticline located towards the transport direction is termed a s forelimb.
•
Backlimb: Backlimb: The fold limb of the ramp anticline located opposite to the forelimb is termed as backlimb. 79
•
Regional: Regional: It is the original elevation of a particular stratigraphic unit or datum surface, measured at a place where it is not involved in any thrust related structure.
•
Leading edge: edge: The edge of a thrust sheet towards the transport direction.
•
Trailing edge: The edge of a thrust sheet in the direction opposite to the transport direction.
Hinterland
Transport
Foreland
direction trailing anticline trailing syncline trailing edge
backlimb
axial surface
leading anticline ramp anticline
leading syncline forelimb
ramp anticline leading edge
Regional
Figure 9.4. Diagram illustrating different components of ramp anticlines.
9.5 9.5 Thrus Thrus t s ystems Thrust systems are made up of a group of link thrusts that are geometrically, kinematically and mechanically related. Majority of the thrust systems can be grouped into either duplexes or imbricate thrust systems. systems . A duplex is duplex is an array of thrusts linking a floor thrust (sole thrust) thrust) at the base to a roof thrust thrust at the top (Fig. 9.5a). A closely related array of thrusts all of which merge into a floor thrust is called imbricate thrust system such that thrust sheets overlap like roof tiles (also called imbricate fan or schuppen structure) (Fig. 9.5b). There is no roof thrust in this thrust system. It may be difficult to distinguish between an imbricate fan and a duplex whose roof thrust has been eroded (eroded duplex). duplex ). A third type of thrust system, called triangle zone, zone, is also fairly common in fold thrust belt.
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Duplex
Roof thrust
Imbricate fan
(a)
Link thrusts thrusts
Floor thrust
(b )
Figure 9.5. Two main types of thrust systems – duplex (a) and imbricate fan (b). Note that in imbricate fan there is no roof thrust.
Duplexes Geometry of a duplex depends on ramp angle, ramp height, initial thrust spacing and displacement on individual thrusts. By varying these parameters, a bewildering variety of duplex geometry can be obtained. A three-fold classification of duplexes is shown in Fig. 9.6 (McClay 1992, modified after Mitra 1986): (1) independent ramp anticlines and hinterland-dipping duplexes, (2) true duplexes including ramp anticline footwall, ramp anticline hangingwall and frontal zone of ramp anticline, and (3) overlapping ramp anticline including antiformal stack and foreland-dipping duplex. The different types of duplexes are as follows (Figs. 9.6, 9.7):
•
Independent ramp anticline anticline:: The final thrust spacing is much larger than displacement on individual thrusts leading to formation of widely spaced ramp anticlines. Ramp anticlines do not interfere with each other.
•
Overlapping ramp anticlines: anticlines : The horses stack up on top of each other in such a way that the overall geometry is like an antiformal arch.
stack : This is a variation of overlapping ramp anticlines in which the • Antiformal stack: overlapping horses have coincident trailing branch lines.
•
Breached duplex: duplex: A duplex in which out-of-sequence movement on link thrusts have breached or cut through the roof thrust.
•
Corrugated or bumpy roof thrust: thrust : A duplex in which the roof thrust is corrugated.
•
Hinterland-dipping duplex: duplex : Both link thrusts and bedding dip towards hinterland. With increased displacement, independent ramp anticlines grade into hinterland-dipping duplex. 81
•
Foreland-dipping duplex: duplex : At least part of link thrusts and bedding within the duplex dip towards foreland. With increased displacement, hinterland-dipping duplex may grade into foreland-dipping duplex.
•
Frontal zone of ramp anticline: anticline : The duplex is at the front of a ramp anticline.
•
Passive roof duplex: duplex : A duplex in which sequence above the roof thrust has not been displaced towards the foreland. Indepe ndent ramp a nticline
Hinter land-slo pping d uplexes
Increa sed displace ment
Ramp a nticl ine foot wall
True duplex
Ramp a nt icline h angingwa ll
Second orde r dupl exes
Frontall zone of ramp anticline Fronta anticline
Overlapping ramp anticlines leading to a ntifo ntiformal rmal stack
Fore land-dipp ing duplex
Increa sed disp lacement
Figure 9.6. Classification of duplexes (after Mitra 1986, modified by McClay 1992).
•
Planar roof duplex: duplex : A duplex in which top of the roof thrust is planar.
•
Ramp anticline footwall: footwall : Duplex is in the footwall of the ramp anticline.
•
Ramp anticline hangingwall: hangingwall : Duplex is in the hangingwall of the ramp anticline.
•
Truncated duplex: duplex: A duplex that is truncated (or beheaded!) by an out-ofsequence thrust. 82
•
True duplex: duplex : Parts of all the link thrusts and roof thrust are parallel to the frontal ramp.
Roof thrust thrust
Floor thrust thrust
(a) Planar roof duplex
Corrugated roof thrust thrust
(b) Antiformal stack
Breaching Breaching thrusts
(c) Corrugated or bumpy roof duplex (d) Breached duplex Passive roof roof thrust
Tip of buried thrust
(e) Truncated duplex (f) Passive roof duplex
ome examples of variation in duplex geometry. Figure 9.7.S
Imbricate thrust systems
•
Leading imbricate fan: fan : Maximum displacement is on the leading (i.e., lowermost) thrust.
•
Trailing imbricate fan: fan : Maximum displacement is on the trailing (i.e., highest) thrust.
•
Blind imbricate fan: fan : All the thrusts are buried below the erosion surface. Folding at higher structural level compensates the displacement along buried thrusts. Erosion Ero sion level level
(a) Leading imbricate fan
(b) Trailing imbricate fan
(c) Blind imbricate fan
Figure 9.8. Types of imbricate fans. (a) Leading imbricate fan with maximum slip on the frontal most thrust. (b) Trailing imbricate fan with maximum slip on
83
the trailing thrust. (c) A blind imbricate fan with folding as the surface manifestation.
Triangle zones A combination of two thrusts with the same basal detachment but with opposing vergence forms a triangular zone (Fig. 9.9a). A pop-up pop-up structure (Fig. 9.2b) is also type of triangular zone. An intercutaneous thrust wedge wedge (Fig. 9.9b) is bound by a floor thrust at the base and a passive roof thrust at the top.
(a ) Tri angl e zo ne
(b ) I nte rcut aneo us wed ge
Figure 9.9.T riangle zone (a) and intercutaneous wedge (b).
Evolution of hinterland-dipping hinterland-dipping duplex Kinematic modelling of duplexes and imbricate fans requires that the values of several parameters, such as ramp angle, ramp height, initial and final ramp spacing and slip on individual thrusts be assumed (Mitra 1986). By varying these parameters different geometric forms of thrust systems can be modelled. The progressive development of a simple hinterland-dipping duplex is illustrated in Fig. 9.10 (Boyer and Elliot 1982; Mitra 1986; Ramsay and Huber 1987). In the beginning, a major thrust sheet with flat-ramp-flat geometry and total slip Si develops. The lower and upper flats define two décollement surfaces, both of which are located at the same stratigraphic horizon. At the initial stage of duplex formation, a fracture propagates in the footwall from the lower flat and rejoins the upper flat, tracing a new ramp with initial ramp spacing as Ri.. Movement along the fracture generates a small horse, which moves forward by an amount S. The horse climbs up the new ramp and folds the overlying thrust sheet. The small horse then becomes inactive. Subsequently, a new fracture propagates from the footwall, the new horse thus formed climbs up the ramp carrying piggy-back style the earlier formed horse. A roof thrust and a floor thrust form with 84
horses occupying the area in between. The entire process can be repeated as many times as desired. In this model, the height and angle of the ramps, slip on individual ramps and spacing between ramps are same during sequential development of ramps. This leads to rather regular geometry of the duplex. By varying these parameters different types of duplexes can be generated. For example, by increasing ramp spacing and keeping all other parameters same we could we could have generated independent ramp anticlines instead of hinterland-dipping duplex shown in Fig. 9.10. Or, we could generate an overlapping ramp anticline, antiformal stack or foreland-dipping duplex by increasing displacements on the thrusts.
Majo r thr ust she et
S o
Incip ient fr acture
S1
Fore land
Lower an d upp er glide h ori zons
Horse
S 2
Roof thrust
Floor thrust
S 3
Figure 9.10. Series of diagrams illustrating progressive development of hinterland-dipping duplexes (after Mitra 1986).
9.6 9.6 Thru Thru st s equences This refers to the sequence of development of thrusts (Fig. 9.11). It is a very important parameter needed for proper interpretation of geometric and kinematic evolution of a thrust belt as well as for balancing and restoration of sections.
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•
Breaching: Breaching: An early-formed thrust cut by a later (i.e., younger) thrust formed in the hangingwall. The later thrust has the same vergence as the older thrust.
•
In-sequence thrusting: thrusting : A thrust sequence that has formed progressively and in order in one direction only. There are two types: forward-breaking forward-breaking sequence and break-back break-back sequence.
1
(a)
(b)
(c)
3
2
3
4
1
2
4
2
1
3
Figure 9.11. Thrust sequences. Numbers indicate the order in which thrusts developed. (a) Forward-breaking in-sequence thrusts. (b) Break-back insequence thrusts. (c) Out-of-sequence thrusts.
•
Out-of-sequence thrusting thrusting:: Opposite to in-sequence thrusting (Morley 1988). The out-of-sequence thrusts commonly cut through and displace pre-existing thrusts.
•
Forward breaking sequence: sequence : The sequence of thrusting in which new (younger) thrusts form in the footwalls of older thrusts and all the thrusts verge towards foreland.
•
Break-back sequence: sequence : The sequence of thrusting in which new (younger) thrusts form in the hangingwalls of older thrusts and all the thrusts verge towards hinterland.
86
•
Piggy-back thrust sequence: sequence : This sequence occurs when older thrusts are carried piggy-back style by younger thrusts; essentially same as forward-breaking thrust sequence.
•
Synchronous thrusting: thrusting : Two or more thrusts move together.
Normal faults are dip-slip faults in which hangingwall moves down relative to footwall (Fig. 10.1). Absolute movement on a normal fault can rarely be determined and four possible absolute movements produce the same relative movement:
•
hangingwall goes down but footwall remains fixed
•
footwall goes up but hangingwall remains fixed
•
both hangingwall and footwall blocks move downward but hangingwall moves more than the footwall
•
both hangingwall and footwall blocks move upward but footwall moves more than the footwall Well hangingwall hangin gwall block fault surface surface
net slip
footwall footwa ll block
(a)
(b)
Figure 10.1. Normal fault. (a) Hangingwall in a normal fault goes down relative to footwall. Net slip gives the orientation of the slip vector, which is oriented down the dip of the fault surface. Younger beds lie above older beds across the fault surface. (b) In a drill well, normal faults may be recognized from missing beds.
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The displacement vectors (i.e., the net slip) in normal faults are oriented almost down the dip of the fault plane such that they have either zero or negligible strike-slip component (Fig. 10.1). In previously undeformed sedimentary terrains, younger beds lie above older beds across fault surfaces where a part of the stratigraphic succession goes missing. Missing strata in drill wells commonly suggest normal faulting. It should be remembered that the nature of omission and offset of strata depends on the orientation of the fault surface relative to the orientation of beds involved in folding. Therefore, caution must be exercised while interpreting seismic reflection profiles owing to the fact that a normal fault with zero or negligible dip-slip component may look similar to a strike-slip component with small dip-slip component. The name “normal” does not indicate that this type of fault is more common than the other types. This term originated in British coal mines where such faults were common. If a coal seam was truncated by a fault, the “normal” practice was to continue the drive for some distance and sink a shaft to find the missing coal seam. The normal faults are commonly thought to be steeply inclined with a dip of about 60 °. More than three thousand fault dip measurements from 122 faults in British Coalfields show that 70 ° dip (standard deviation = 9 °) on normal faults is a better approximation than the commonly accepted value of 60 ° (Walsh and Watterson 1988). They are the dominant structural elements in areas where crustal rocks undergo subhorizontal stretching accompanied by subvertical shortening, such as oceanic and continental rift zones (Fig. 10.2). The steep dips of normal faults are in conformity with Adersonian model of faulting. However, it is now widely recognized that low-angle normal faults are also common in extensional tectonics. Increased interest in normal faults and extensional tectonics in the last two to three decades has resulted in a set of complex and often confusing terminology. For a glossary of normal fault and related terminology see Peacock et al. al. (2000) and references therein.
10.1 Horst-and-graben structure
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Normal faults with about 50 °-70° dip commonly occur in conjugate pairs and lead to the formation of horst-and-graben horst-and-graben structure (Fig. 10.2) (Cloos 1955, 1968). The bisector of the acute angle (about 40-60°) between a pair of conjugate normal faults is subvertical and is parallel to the maximum compressive stress direction. The minimum compressive stress direction is horizontal and parallel to the bisector of the obtuse angle between the conjugate faults. Therefore, moderately to steeply-dipping conjugate normal faults are in conformity with Andersonian model of faulting. Spectacular and active horst-and-graben structures can be found at mid-oceanic ridges and continental rift zones such east African Rift. Many inactive and old horst-and-graben structures, buried under growth or later sediments have been imaged through seismic reflection profiling (Fig. 10.3).
•
Horst: Horst: It is an elongated uplifted region bound on either side by sub-parallel normal faults dipping away from area of uplift. Horsts are commonly bound by grabens or half grabens (Fig. 10.2a).
Suez Rift-border fault Rift-border Intra-rift fault fault Syn-post-rift Pre-rift/basement Pre-rif t/basement Dip domain domain
Vertical shor ten ing
Graben
Hors t
N
Graben
Sinai Rift- borde r fau lt
(a)
Rift- borde r fau lt
Horizo ntal extension
Gulf of Suez Egyp t
Conjugate normal fau lts dis pla cing each other
(b)
Figure 10.2. (a) Schematic diagram of horst-and-graben structure in areas of horizontal extension showing principal structural elements of a rift system. (b) Map of northern part of Red Sea rift system (after Khalil and McClay 2002).
89
•
Graben: Graben : It is a long and narrow region of subsidence bound on either side by sub-parallel normal faults that dip towards each other. Grabens are commonly bound by horsts (Fig. 10.2a).
•
Half-graben: Half-graben: In some terrains, subsidence is only due to one controlling normal fault and a graben is flanked by only one horst. The structure on the downthrown side is then termed as half-graben (Fig. 10.4) . A half-graben typically contains a sedimentary wedge that thickens towards the fault.
•
Rift zones: zones: These are narrow and elongated zones of subsidence consequent upon crustal-scale extensions (Fig. 10.2). They are present both on continents (e.g., East African rift zone) and on ocean floor (e.g., mid-oceanic ridge). Horstand-graben structures commonly occur in rift zones. Owing to thinning of the crust, high heat flow and magmatic activity may be associated with rift zones. 0
Lr. Tertiar Tertiar y
) c 1 e s ( T W T 2
Up. Tertiary
Palaeo zoic
Palaeo zoic
3
Figure 10.3. Buried asymmetric graben structure imaged through seismic reflection profiling, Basin and Range province, Nevada, USA (after Effimol and Pinezich 1986).
•
Rift-border fault: fault : Rift zones are usually bound by a pair of normal faults dipping towards the middle of the zone. These faults represent the boundary of the rift zones and are called rift-border fault fault or basin-margin fault fault (Fig. 10.2).
•
IntraIntra-rift fault: fault: Faults of smaller magnitude than rift-border faults occur inside the rift zone and are called intra-rift faults (Fig. 10.2).
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Horst half-grabens half-gr abens
half-grabens half-gr abens
Listric fault
Detachment Detachment fault
Detachment Detachment fault
Figure 10.4. Main faults with concave upward listric trajectory gradually become detachment fault at depth. Subsidiary imbricate faults in the hangingwall may terminate (left hand side) or merge (right hand side) with the detachment fault. Half-grabens are bound by only one main fault.
•
Compatibility problem Compatibility problem:: Synchronous movement along a crossing pair of conjugate system of normal faults leads to the creation of opening between fault walls. Since rocks are weak materials, large openings can not be sustained in the crust except within few hundred meters from the surface. This is the so-called compatibility problem, which can be accounted for if the movement along a set of conjugate faults is sequential rather than synchronous (Fig. 10.5). Conjugate normal faults displace each other owing to the compatibility problem (Fig. 10.2a). This is one of the geometric features that can be used to establish if two sets of faults form a conjugate pair or not.
?
1
?
2 3
Figure 10.5. Compatibility problem associated with steeply-dipping conjugate normal faults. Synchronous movement along both the faults opens up unacceptable gaps. Alternate movements along two sets of faults avoid opening of gaps. Note that fault number 1 is displaced by fault number 2, which in turn is displaced by fault number 3. Faults 1 and 3 form a single set.
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•
Growth sediments: sediments : During normal faulting the downward displacement of hangingwall block creates low relief relative to the footwall block. Sediments derived through eroding footwall may fill this low relief and are called growth sediments (Fig. 10.3).
10.2 Low-angle normal faults The traditional view of extensional provinces is that their structural characters are manifestations of high-angle normal faults giving rise to horst-and-graben structure. This geometry is predicted by Andersonian model of faulting. However, detail surface mapping and seismic imaging of many extensional terrains show that low-angle normal faults are also quite common. An initially high-angle normal fault may acquire gentler dip due to block rotation. Alternately, a steeply-dipping basin boundary fault may become gradually gentler with depth. As the dip becomes gentler the high-angle brittle fault near the surface may grade though an intermediate brittle-ductile shear zone to low-angle ductile shear zones at depth (Ramsay 1980). Such faults with listric fault trajectory may become or merge with detachment or décollement fault at depth.
10.3 10.3 Books helf or domi no faulting Strongly rotated fault blocks may be a result of bookshelf or domino model faulting (Mandl 1984, 1987; Ramsay and Huber 1987). In this type of faulting, a system subparallel faults undergoes progressive rotation of beds and faults as extension continues. Progressive rotation takes place in such a way that steepening of beds is accompanied by decrease in dips of faults. Bookshelf faults generally pass into a detachment fault or into a zone of ductile deformation.
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Extension Extensi on fractures fractu res
(a) block rotati on
(b) block rotation r otation growth sediments s ediments new faults fa ults
old faults
(c)
(d)
Figure 10.6. “Bookshelf” or “domino” model of faulting. See text for discussion.
The initial fractures probably develop as vertical extension fractures (as opposed to Coulomb fractures) within retaining walls in response to horizontal stretching (Fig. 10. 6a). As the unfaulted walls move away, the faulted blocks collapse sideways and undergo rotation with normal sense of shearing displacement exactly the same way as books in a library bookshelf falls sideways if a book is removed (Fig. 10.6b). The angle of rotation faults is same as the dip of the beds if the rotation of fault block is rigidbody type and originally beds were horizontal,. The half-grabens formed above the rotating blocks may be filled up with syntectonic sediments called growth sediments. sediments. The growth sediments are cut by faults and sediments are thicker in the hangingwall near the fault than in the footwall. This suggests that faulting was active during sedimentation. Such faults are called growth called growth faults. faults. After a certain amount of rotation, it may be mechanically more efficient to develop new, steeply oriented fractures rather than continue to rotate on the old faults (Fig. 10.6c). The bedding planes and old faults continue to rotate as block rotation is transferred onto new faults (Fig. 10.6d). The old faults may become sub-horizontal or may eventually dip in the direction opposite to which they were initiated. They may apparently look like thrust faults but can still be 93
recognized as normal faults because younger rocks in the hangingwall will ride over older rocks in footwall. This idealized and rather naïve model does reproduce geometric relations in differently oriented fault blocks commonly found in extensional basins. However, very large horizontal stretching in extensional basins cannot be accounted by bookshelf faulting alone because even with high angle of rotation extension remains rather low.
10.4 Extensional detachment faulting Low-angle extensional detachment faults (Figs. 10.4, 10.7) are common in many extensional basins, such as the Basin and Range province of western US (e.g., Davis et al. al. 1980). Such faults are not predicted in Andeson’s theory of faulting. The total extensions in such terrains are typically very high and much of the horizontal extensions are accommodated along the detachment fault. The hangingwall undergoes high-angle normal faulting and the variation in dips of strata suggests that the faulted blocks underwent variable amount of block rotation. The extreme extension often strips off the unmetamorphosed cover rocks and leads to upwelling of the basement rocks (Fig. 10.7). Tectonic erosions at such places expose of basement rocks, called metamorphic core complexes. complexes . The detachment faults run close to the contact (but not necessarily along the contact) between the cover rocks and metamorphic crystalline basement complex. The basement rocks below the detachment are mylonitized and the cover rocks above the detachment show cataclastic deformation. Unmetamorphosed cover rocks
Mylonitized Mylon itized crystalline crystalline rocks
Upwelling Upwell ing of footwall due to stripping stripping of of hangingwall hangin gwall
Detachment Detac hment fault fault
Figure 10.7. Diagrammatic cross-section through metamorphic core complex showing stripping of fault blocks in the hangingwall exposing underlying metamorphosed basement rocks.
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10.5 10.5 List ric fault Normal faults often show smoothly curved trajectory and such faults are called listric normal faults faults (Fig. 10.4). Listric geometry allows smooth integration of movement on a near-surface and steeply dipping normal fault with movement on a detachment fault at depth. Displacement of hangingwall on listric faults also opens up large gap and sets up compatibility problem (Fig. 10.8). To solve for the compatibility problem, different kinds of structures develop in association with listric normal faults (Gibbs 1984):
•
Rollover anticline: anticline : The displacement of hangingwall on a concave upward listric fault opens up a large gap, which is not allowed in nature (Figs. 10.8a,b). Beds in the hangingwall may steepen to form an anticlinal fold and allow the hangingwall and footwall to remain in contact with each other (Fig. 10.8c). This fold is called rollover or rollover anticline. If cross sectional area and bed thickness remains constant during folding then another gap develops in the hangingwall. Thinning of folded layers and/or shearing in the hangingwall can close this gap.
fault: Antithetic faults (also called counter faults) are subsidiary to a • Antithetic fault: dominant fault or fault set and the antithetic faults dip in the direction opposite to the direction in which the dominant fault dips (Fig. 10.9). The sense of shear in antithetic faults is opposite to that of the dominant fault. Development of a set of antithetic faults in the hangingwall may solve to a large extent the compatibility problem associated with listric faulting (Fig. 10.8d). The antithetic faults may also have listric trajectory.
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gap
(a )
(b ) Fold
(c )
Faults
(d )
Figure 10.8. Compatibilty problem in listric fault (after Twiss and Moores 1992). (a) Incipient fracturing of with listric trajectory. (b) Movement of hangingwall opens up a large and geologically untenable gap. (d) Compatibility problem may be solved through distribution deformation in the hangingwall leading to the formation of rollover anticline. (d) Compatibility problem may also be solved through antithetic faulting in the hangingwall. Note that small gaps are present below the faulted blocks.
Graben
Main fault
Main fault Synthetic faults
Antithet ic Antithetic faults
Figure 10.9. Antithetic and synthetic faults associated listric main faults in a graben.
•
Synthetic faults: faults: These faults are minor or subsidiary faults with same sense of shear and similar orientations as the related major fault (Fig. 10.9). Antithetic and synthetic faults may form a conjugate pair.
•
Rider : Formation of a second, and then sequential subsidiary faults cutting back into the undeformed footwall give rise to wedge-shaped segments in crosssection between faults. These are called riders (Fig. 10.10a). The riders develop in-sequence towards the footwall (Fig. 10.11). Each rider is bound by a roof fault and a floor or sole fault. fault. The roof faults on riders passively rotate on the lower active fault. Riders can be both synthetic and antithetic. At high extensions, riders can get detached from each other may look like horst-and-graben structure in seismic sections (Figs. 10.10c,d).
96
riders only on footwa ll
riders only on hang ingwall 8
Footwal l
(a)
Hangingwall Hangi ngwall R i d R e r R o o S i o l f d e o e r f a u r F l t
1
l o o r f a u l t
7 2
3
4
5
6
(b)
Riders
Growth sediments
Detached riders at high ex tensions
(c)
(d)
Sole fault
Figure 10.10. Structures associated with listric faults.
•
Listric fan: fan: A set of synthetic riders form a listric fan, also called horsetail faults (Fig. 10.10a).
•
Reverse listric fan: fan : A set of riders antithetic to the main listric fault but rest on the hangingwall is called reverse listric fan (Fig. 10.10b).
•
Roof fault: fault: The topmost fault in a listric or reverse listric fan (Fig. 10.10a).
•
Floor fault: fault : The lowermost fault in a listric or reverse listric fan (Fig. 10.10a).
•
Extensional duplex: duplex : An extensional duplex is made up of stacked up horses and bound by a roof fault and floor fault (Fig. 10.11). An extensional duplex formed in association with listric normal faulting has geometry similar to a duplex formed during thrusting. Only sole fault in an extensional duplex is active and the roof fault consists of sequential faults. They commonly form where the main fault has ramp-flat trajectory.
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Central ridge 5 4
List ric fan fa n Counter fan
3 2
1
1
2
3
4
4
3
2
1
Extensiona Ext ensiona l duplex du plex
Figure 10.11. A composite diagram showing listric fan, counter fan and extensional duplex. The numbers indicate sequence of fault development.
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99
Strike-slip faults are commonly vertical faults with displacement predominantly in the horizontal direction (Fig. 11.1a). The dip-slip component is zero or small compared to strike-slip component. The sense of displacement of a strike-slip fault is either clockwise or counterclockwise and they are called dextral (syn. right-handed or rightlateral) or sinistral (syn. left-handed or left-lateral), respectively. If we stand on a fault and face one of the blocks, then if the block facing us has moved to right then it is righthanded (dextral) and it is left-handed (sinistral) if it has moved to the left (Fig 11.1b). Strike-slip faults are also known by several other names, such as lateral-, wrench-, tearand transcurrent fault. The descriptive meaning of the term "strike-slip" is definitive and simple and, therefore, should be preferred over all other names.
sinistral net slip
fault surfac e
(a)
(b)
dextral
Figure 11.1. (a) Strike slip fault showing displacement vector, which is essentially horizontal. (b) Sinistral (left-handed) and dextral (right-handed) sense of displacement.
11.1 11.1 Transfor m fault Transform fault fault is a special type of strike-slip fault, most spectacularly seen on an ocean-floor map cutting across mid-oceanic ridges. These faults link (or transform) other major tectonic features along plate boundaries, such as graben systems of mid100
oceanic ridges (Fig. 11.2). There are three unusual aspects about displacement on transform faults: (1) displacement remains constant throughout the fault and then ends abruptly, (2) the actual sense of displacement is opposed to what can be inferred from offsets of geologic features, and (3) two sub-parallel faults may have opposite sense of movements. The offset graben system in Fig. 11.2 suggests sinistral movement but the actual displacement is dextral.
transform fault
sense of diplacement opposite to ridge offset
mid-oceanic rift system
Figure 11.2. A strike-slip fault joining (or transforming) mid-oceanic rift systems. Such strike slip faults are called transform faults. Note that the actual sense of displacement is right-handed although offset rift suggests left-handed displacement. The right-handed displacement on the transform fault is in conformity with the extension at the rift.
11.2 Transpression and transtension zones A strike-slip fault or a set of parallel strike-slip faults affecting a set of subhorizontal beds do not generate much subsidiary structures. Consequently, such faults may appear to be rather uninteresting. However, strike-slip faults may occur in enechelon echelon sets or they may have bends (or jogs). Displacement in such situations results in the formation of strike-slip duplex, duplex , which is a set of horizontally stacked horses bound on both sides by parts of the main strike-slip faults. They are also called flower-, tulip- or palm-tree structure. structure. In an en-echelon set, a set of parallel strike-slip faults are oriented at the same angle to a reference line, called bearing line (Fig. 11.3). Adjacent faults in enechelon sets show right (Figs. 11.3a,b) or left (Figs. 11.3c,d) sense of strike shift. Similarly, curved strike-slip faults may be considered to have right-handed (Figs. 101
11.4a,b) and left-handed (Figs. 11.4c,d) jogs. These right- and left-handed senses of shift and sense of shear on fault surfaces lead to two situations:
line of bearing
(a )
(b )
(c )
(d )
Figure 11.3. En-echelon strike-slip fault system. All the faults in an enechelon system are inclined at a constant angle to a reference line, called line of bearing. The sense of shift of adjacent faults is either right-handed (a, b) or left-handed (c, d). The sense of shear displacement on the faults are either left-handed (a, c) or right-handed (b, d).
(a)
(b)
(c)
(d)
Figure 11.4. The strike-slip faults with bends (or jogs) in strike line. (a,b) Right-handed jog. (c,d) Left-handed jog. (a, c) Right-handed shear displacement. (b, d) Left-handed shear displacement.
•
The shear sense on fault and sense of en-echelon shift or jog are same (Figs. 11.5a,b). In this case, translation due to faulting is accompanied by extension set up in the zone of overlap between en-echelon faults or in the zone where fault trace is curved. Such combination of translation translation and extension extension is termed as transtension. transtension. Structures typical of extensional deformation may develop in transtension zones. The most common large-scale structure is pull-apart basin, basin, which is a rhomb-shaped graben (Fig. 11.6.. Normal faults in a pull-apart basin are oriented initially at about 45-50 ° to the strike-slip fault. This angle increases 102
as normal fault block rotates with increasing shear. These basins may get sediments derived through erosions and denudation of topographically higher regions bounding the graben. On a smaller scale, local depressions may lead to the formation of sag ponds, ponds, which may be site of temporary or permanent lake. The normal faults in the graben may have concave upward shape and merge with the main strike slip fault. The faults may have significant amount of both strike-slip and dip-slip components. Such a structure is called normal (or negative) negative) flower (or tulip) tulip) structure (Fig. 11.7a). zone of transtension
(a)
(b) zone of transpression
(c)
(d)
Figure 11.5. (a,b) Zones of transtension (translation + extension) developed due to same sense of strike-slip displacement and sense of en-echelon shift or jog. (c,d) Zones of transpression (translation + compression) developed due to opposite sense of strike-slip displacement and sense of en-echelon shift or jog. sedimen t deposi t
norma l faults
(a)
thrus t fault
fold axi al trace
(b)
Figure 11.6. Structures develop in zones of transtension (a) and transpression (b).
103
•
The shear sense on fault and sense of en-echelon shift or jog are opposed (Figs. 11.5c,d). A combination of translation translation and com pression pression follows and a zone of transpression transpression forms. The compression may to local vertical uplift of a rhombshaped region, called pressure ridges. ridges. On a larger-scale, the compression may lead to the formation of folds and thrusts. The axial traces and fault traces will be oriented initially at 45-50 ° to the strike-slip faults. With increasing shear along the main strike-slip fault, the axial and fault traces may rotate towards the main fault. The thrust faults with concave downward shape may also have significant strike-slip component. These thrust faults may have the geometry of a reverse (or positive (or positive)) flower structure (Fig. 11.7b).
(a)
(b)
Figure 11.7. (a) Normal or negative flower structure. (b) Reverse or positive flower structure.
11.3 11.3 Slip rate The rate at which fault blocks move past each other may not be uniform and frontal part of fault may move at a different velocity than the rear portion (Fig. 11.8). This leads to a compatibility problem, which are solved through the development of subsidiary structures. If the frontal part moves at a faster rate than the rear part then the fault block will be stretched and extensional structures such as normal fault may form. If, on the other hand, the frontal part moves at a slower rate than the rear part
104
then the fault block will be compressed and structures such as reverse faults or folds may form.
r e t s a F
r t e a s F
e r o w l S
r e w l o S
r e w o l S
r t e a s F
Figure 11.8. Diagram showing zones of extension and compression developed due to different slip rates at the front and rear of a fault block.
R 2 Ma in st rike Main slip fau fault lt P
R1
Figure 11.9. Subsidiary shear fractures developed in association with righthanded strike-slip fault.
11.4 Subsidiary shear fractures The strike-slip faults may be associated with wide range of subsidiary shear fractures. The most important of these are Riedel shears or R shears. shears. They were originally recognized in laboratory experiments in which a layer of clay was deformed overlying a sharply defined vertical strike-slip fault generated by two rigid blocks sliding past each other. Two sets of en-echelon shear fractures developed in the clay layer, one lying at 10-15° and the other at 75-80 ° to the underling fault surface (Fig. 11.8). They are designated as R1 and R2, respectively. The R1 has the same sense of movement as the 105
main strike-slip fault, but the R2 has opposite sense of movement. That is, if the main strike-slip fault has dextral sense of shear then the R1 and R2 shears have dextral and sinistral sense of displacement. In other words, the R1 and R2 are synthetic and antithetic respectively. Another type of subsidiary shear, called P shears, synthetic to main fault and symmetrical to R shears may also develop. Subsidiary shears on large scale can form a complex and anastomosing network of faults. The geometry and kinematics of such a complex network of faults may be difficult to interpret.
106
107
Folds are regular wavelike undulations traced by sideways deflections of layers or surfaces in rocks. They occur in scales ranging from microscopic, through outcrop or mountain sides and cliffs to tens of km in orogenic core zones. Folds are very common in metamorphic tectonites formed in response to ductile deformation in the deeper part of the crust. They also form in the shallow crustal depths in sedimentary or very lowgrade metamorphic rocks in the flanks of major orogenic zones. The terminology for the purpose of geometric description of folds has evolved over a long period of time. The terminology is rather extensive, not always consistent and some of them have genetic implications. However, we recognize that descriptive terminology should be devoid of any genetic implications. Folds are traced by layers, such as sedimentary beds, veins, dikes and metamorphic and igneous bands. Folds may also be considered to have been traced by single surfaces, such as bedding and cleavage surfaces. The geometry of single folded surfaces should be treated separately from the geometry of folded layers.
12.1 12.1 Curvatur e of a fold ed surf ace A curved line or surface can always be represented by a mathematical equation of the form z = f(x) f(x) or z= f(x, y). y). The first differential of the equation gives the tangent (line or plane), which is a measure of slope at a point. The second derivative gives the variation in the orientation of the tangent and represents curvature at a point. For example, a straight line has zero curvature and a circular arc has constant curvature. The following terms are used to describe the variation in curvature of a single folded surface (Fig. 12.1):
108
•
Hinge line: line: The hinge line is the line along which the folded surface has maximum (positive or negative) curvature. On a cross section, hinge line will obviously be represented by a point, called hinge point. point. Both hinge line and hinge point are simply called hinge. hinge. The hinge line can also be thought of as loci of points of maximum curvature. The hinge line of a fold need not be a straight line; it may be horizontal or plunging.
fold limb
hinge zone
crest poin t
n e i n l g e h i n
hinge point
fold domain
e i n l n i o x f l e i n
inflexion point trough point
n e i l e g n i h
hinge point
fold domain
hinge line depression
hinge line culminati on
kink fold
n e i n l i n g e h i n
hinge point
concentric con centric fold fold
hinge point hinge point
n e l ii n n g e h i n hinge point
Figure 12.1. Geometry of single folded surface.
•
Inflexion line: line : Inflexion lines mark zero curvature on a folded surface. The sign of curvature changes (i.e., the sign of the second derivative) across the inflexion lines. In other words, the sense of curvature of the folded surface changes, for example from convex up to convex down. On a section, inflexion line is represented by a point, called inflexion point. point . 109
•
Crest and trough: trough : Crest and trough (line or point) of a fold represent highest and lowest, respectively, topographic elevations on a folded surface. Crest and trough may, but not necessarily, coincide with hinge.
•
Hinge line culmination and depression: depression : If the plunge of hinge line varies, then there should be areas where hinge line will attain highest and lowest elevations, called hinge line culmination and depression, respectively.
•
Crest/trough line culmination and depression : Crest lines or trough line may also have culminations and depressions, as with hinge lines.
•
Fold axis: axis: It is an imaginary line on the folded surface, which when moved parallel to itself generates the folded surface. Such folds are also called cylindrical folds. folds. The non-cylindrical non-cylindrical folds surfaces do not have this property. Hinge lines are straight in cylindrical folds. The terms fold axis and hinge line are sometimes used synonymously, although they are not the same.
12.2 12.2 Fold Fold domain dom ain The folded surface between two successive inflexion lines is called a fold domain and constitutes one fold (Fig. 12.1). Hinge zone and limbs are parts of a fold domain.
•
Hinge zone: zone: The surface region around the hinge line where the curvature is high is called a hinge zone. Although a single hinge per fold domain is common in most folds, a single fold domain can have more than one hinge.
•
Fold limb: limb: The surface region between inflexion line and hinge zone is known as limb of a fold. Fold limbs can be straight or curved. A fold domain commonly has two limbs.
•
Concentric fold: fold: If the fold domain is part of a perfect circular arc, then the fold is called a concentric fold. Hinge points in such cases are taken at the midpoints of each of the fold arcs.
•
Kink fold: fold: Folds with straight limbs and sharp hinge are called kink or chevron folds. The curvature at the hinge is infinite and at limbs it is zero. Inflexion points are taken at midpoints on the straight line segments of the folds.
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Neutral antifom
synform Vertical Younger
Older
Older
Younger
synformal syncline
synformal anticline
Younger antiformal anticline
Older
antiformal syncline Older
(e) Monocline
saddle
Younger
inverted saddle
sheath fold
hinge line
hinge line
(d)
(k)
hinge line
non-cylindrical folds
eometric terms based on closure of fold domains. Figure 12.2.G
12.3 12.3 Fold cl osu re Following is a set of names given to folds depending on the closure of fold domains (Fig. 12.3):
• Antiform: Antiform: If the fold domain closes upward (-ve curvature), then the fold is called antiform. Anticline: In an anticline older rocks are located in the core of the fold. • Anticline:
•
Synform: Synform: In a synformal fold, the fold domain closes downward (+ve closure).
•
Syncline: Syncline: If younger rocks are located in the core of the fold, the fold is syncline.
• Monocline: Monocline: Regional step-like folds in which otherwise horizontal or shallowly dipping strata abruptly bend to steeper inclination within a very narrow zone.
•
Homocline: Homocline: A set of uniformly and gently dipping beds constitute a homocline.
•
Neutral fold: fold: When the fold domain closes sideways, the folds are called neutral fold. 111
•
Vertical fold: fold : A type of neutral folds in which both hinge lines and limbs are vertical.
•
Overturned fold: fold : If the two limbs of a fold dip in the same direction, it is called overturned fold. In this type of fold, one of the limbs rotates more than 90 ° from initial horizontal orientation.
•
Dome: Dome: A dome is an antiformal fold domain with hinge line culmination. Such a fold has an approximate shape of a dome.
•
Basin: Basin: A dome is a synformal fold domain with hinge line depression. Such a fold has an approximate shape of a basin.
•
Saddle: Saddle: A type of antiform with hinge line depression giving rise to shape of a saddle.
•
Inverted saddle: saddle: A type of synform with hinge line culmination.
•
Sheath fold: fold: If the hinge line curves more than 90 °, the fold domain may acquire the shape of sheath of a knife or sword. A fold with this shape is called sheath fold.
Combination of antiform/anticline and synform/syncline gives four possible geometry, viz., antiformal anticline, synformal syncline, antiformal syncline and synformal anticline. The geometries of antiformal syncline and synformal anticline require inverted stratigraphy. In deformed sedimentary terrains stratigraphy is rarely inverted and antiform/anticline and synform/syncline are, and can often be, used interchangeably.
12.4 Fold tightness The angle between tangents to the folded surface drawn through successive inflexion lines is called called interlimb angle angle (Fig. 12.3). The interlimb angle provides the degree of tightness tightness of a fold. Following names are given to folds on the basis of interlimb angle (Fleuty 1964):
• •
Gentle: Gentle: Interlimb angle 180° - 120° Open: Open: Interlimb angle 120° - 70° 112
• • • •
Close: Close: Interlimb angle 70° - 30° Tight: Tight: Interlimb angle 30° - 0° Isoclinal: Isoclinal: Interlimb angle 0° Elastica: Elastica: Interlimb angle negative
The above terms cannot be used in combination. For example, a fold cannot be tight isoclinal but the tightness of a set of folds can range from range from tight to to isoclinal. inter-limb angle
o
180
o
120 gentle
tangent at i2
tangent at i1
o
70 open
o
30
close inflexion point
i1
i2
tight
inflexion point
isoclinal
o
0
Figure 12.3.I nterlimb angle and tightness of fold (after Fleuty 1964).
12.5 12.5 Folded l ayers ayers Geometry of a folded multilayer is equivalent to the geometry of a set of stacked up folded surfaces. Several terms are used to describe the geometry of folded multilayers, the most import being the axial surface surface (Fig. 12.4): surface : Surface formed by joining successive hinge lines in a multilayered • Axial surface: sequence is called axial surface. surface. Axial surface need not be planar in a multilayered sequence but when this surface is planar, it is called axial plane. plane. It is important to remember that axial surface is not defined for a single folded surface and axial surface need not necessarily divide a fold into two symmetric halves.
• Axial trace: trace : Trace of the axial surface on the topographic surface. The term axial trace has nothing to do with fold axis. Axial trend (i.e., the trend of hinge line) and axial trace may or may not be parallel. 113
•
Inflexion surface: surface : Inflexion surfaces are formed by joining successive inflexion lines in a multilayered sequence. A fold domain is bound by two adjacent inflexion surfaces. When two inflexion surfaces bounding a fold join, the fold ceases to exist.
•
Conjugate fold: fold : A fold domain may have converging pair of axial surfaces and such folds are called conjugate folds. Obviously, conjugate folds have two hinges. The conjugate fold changes into a single-hinged fold where the two axial surfaces meet.
•
Polyclinal fold: fold: In this type of fold, there are more than two hinges in one fold domain. Such folds are however rather rare.
inflexion line surfaces
axial surfaces surfaces hinge line two axial surfaces surfaces merge to give one axial surface surface
hinge line fold ceases to exist at this point
axial plane
Figure 12.4. Axial surfaces of folds.
12.6 12.6 Fold symmetr sy mmetr y The following terms are associated with symmetry of folds (Fig. 12. 5):
•
Fold train: train: A fold train is a series of folds with alternating sense of curvature.
•
Wavelength: Wavelength: Wavelength of a continuous and regular waveform is defined as the distance between two successive maxima or between any two points in the same phase. However, a fold train rarely has the geometry of a regular waveform and wavelength cannot be measured directly. The distance between two successive inflexion points in a fold domain can be measured and this distance is the half wavelength of the fold.
114
Amplitude: Amplitude is the perpendicular distance between the median line • Amplitude: and the point on the folded surface that has the maximum deflection from the median surface.
Fold domain dom ain 1 A W/2 W/2
median surface
A A: Am Amplitude W: Wavelength
Fold domain dom ain 2
axial plane
symmetric symm etric fold
asymmetric asymm etric fold fold
parasitic paras itic fold fold harmonic harm onic folds folds
disharmonic disha rmonic folds
median medi an surface surface
anticlinoria-sy anticlin oria-synclinor nclinoria ia
old symmetry. Figure 12.5.F
surface: It is a surface formed by joining successive inflexion lines. • Median surface:
•
Symmetry: Symmetry: If the median surface and the axial surface are perpendicular to each other and the axial surface divides the fold domain into mirror symmetric quarter halves, the fold is symmetric. Otherwise the fold is asymmetric. asymmetric. The symmetric folds are m-shaped m-shaped and asymmetric folds are either s-shaped or zshaped. shaped.
•
Harmonic/disharmonic folds: folds : If the wavelength, amplitude and geometry of folds in a multilayered sequence are similar along the axial surface then the folds are harmonic, otherwise folds are disharmonic. disharmonic . The variation in the geometry of folds across axial surfaces has nothing to do with this geometric description.
•
Polyharmonic fold: fold: If a fold train has two or more orders of folds it is called polyharmonic fold. Each order of fold has its own characteristic wavelength and amplitude.
115
•
Parasitic folds: folds: In a polyharmonic fold, the folds with smallest wavelength and amplitude are called parasitic folds.
Anticlinorium: It is a very large anticlinal polyharmonic fold. • Anticlinorium:
•
Synclinorium: Synclinorium: It is a very large synclinal polyharmonic fold.
12.7 12.7 Fold att itude it ude The orientation or attitude of folds in three-dimensional space is important for proper description of folds. Fleuty (1964) proposed a complete set of descriptive name of folds based on the dip of axial plane and plunge of hinge line (Figs. 12.6, 12.7):
•
Upright: Upright: Folds with sub-vertical dip (>90 °) of axial planes are called upright. Depending on the plunge of the hinge line, the upright folds are pre-fixed with sub-horizontal, sub-horizontal, gently-plunging, gently-plunging, moderately-plunging moderately-plunging and steeply-plunging. steeply-plunging. Dip of axial a xial surface t h g i r p 0 0 90 U80
0
0
y d e l n p i l e e c t n S i
y l e t d a r e e i n d o l c n M i
0
60
30
0
d y e l n t i n c e l n G i
0
10
l a t n o z i r o h b u 0 S 0
Sub-horizontal 0
10
Recumbent Gently plunging
e 30 g n i h d l o Moderat Moderately ely f f plunging o e g n u 60 l P
0
t h g i r p U
e d n i i l n c I n
0
Steeply plunging 0
Reclined
80
Sub-vertical 0
90
Vertical
Figure 12.6. Names of folds based on the orientation of axial plane and hinge line (after Fleuty 1964).
116
•
Vertical: Vertical: Folds with sub-vertical (>90 °) dip of axial plane and plunge of hinge line are called vertical folds. As a consequence of this geometry, the folded surface is also sub-vertical everywhere.
•
Recumbent: Recumbent: Folds with less than 10 ° dip of axial plane as well as plunge of hinge line are called recumbent folds. The limbs of recumbent folds are also subhorizontal but dip at the hinge zone is vertical.
•
Reclined: Reclined: In a reclined fold, dip direction of the axial surface is the same as the trend of the hinge line. An equivalent statement is that the hinge plunges down the dip of the axial surface.
•
Inclined: Inclined: In this type of fold the dip of axial plane and plunge of hinge line vary between 10-80°.
(a) Upright
(b) Ve Vertical rtical
( c) Recumbent
( d) I nclined h or iz ontal
( e) Upr ight m ode ra rate ly inclined
( f) Moderately inclined m oderately plunging
(g) Reclined
Figure 12.7. Three-dimensional geometry and stereographic projection diagrams of different types of folds based on the attitude of folds as defined by Fleuty (1964).
117
12.8 12.8 Fold Fold classifi cation Folds can be very precisely classified on the basis of variation of thickness. Historically, the description of variation in layer thickness developed around two geometric models, viz., parallel viz., parallel fold fold and similar fold fold (Fig. 12.8) (van Hise 1896). In parallel fold the orthogonal thickness (i.e., thickness measured orthogonally across the layer) remains constant throughout the fold. These folds are usually found in competent layers surrounded by less competent matrix. Parallel folds do not continue for long distances along the axial surfaces but die along a décollement surface. In similar folds the thickness measured parallel to the axial plane remains constant throughout the fold. The orthogonal thickness in similar folds vary considerably with the limbs thinned and the hinges thickened. A special property of similar folds is that they can continue indefinitely along axial surfaces. Most of the natural folds show significant departure from these two ideal end member models.
T
t
Class 1A
t0 = T0
Class 1B Pa P arallel
1.0
Class 1C
t' t T
t'
- Orthogonal thickness at - Axial Planar thickness at =
t / t 0
dip
C l a s s 2 S i m i l a r
dip 0.5
Class 3
0.0 0
30
0
0
60
Figure 12. 9. t'-α classification of folds (after Ramsay 1967).
118
0
90
Ramsay (1967) gave a very precise scheme of classification of folds based on variation in layer thickness in quarter wavelength of folds (Fig. 12.9). Two parameters are defined, viz., limb dip ( α) with respect to a datum taken as tangent at the hinge point and thickness ( t ') as a ratio between thickness measured at the hinge and at α
α
limb dip. Three classes are recognized: class 1 with t α' > cos α, class 2 (similar fold) with tα' = cos α and class 3 with t α' < cos α. Class 1 folds are further divided into class 1A with tα' >1, class 1B (parallel fold) with t α' = 1.0 and class 1C with t α' <1 (but > cos α). This classification is also compatible with dip isogon isogon classification of folds (Fig. 12.10). Dip isogons are lines joining points of equal dip (measured with respect to the datum) measured on either sides of the fold. Fold classes 1, 2 and 3 have convergent, parallel and divergent dip isogons, respectively. In classes 1A and 1C, the isogons are strongly and weakly convergent, respectively. Dip isogons in class 1B folds are oriented perpendicular to layering. All measurements and constructions are carried out on profile profile section, i.e., sections perpendicular to hinge lines.
Class1, convergent isogons, inner curvature > outer curvature
1A strongly convergent convergent isogons
Class 2, Similar
1B , Pa ral le l isogons perpendicular to layering
1C weakly convergent isogons
Class 3
- dip
parallel isogons, parallel curvature curvat ure of two surfaces same
diverging isogons, diverging curvature of outer outer arc > inner arc
dip - dip
isogon
isogon isogon
Figure 12.10. Dip-isogon classification of folds (after Ramsay 1967).
119
12.9 12.9 Kink fold s Kink fold geometry is commonly used for the purpose of structural modelling in sedimentary terrains, such as fold-thrust belts although perfect kink geometry is rare in nature. However, many folds in these terrains have straight limbs and sharply curved hinges of small areal extent. No significant error is introduced while modelling if such folds are treated as kink folds. Advantages of kink fold geometry include dip panels can be easily constructed, axial surfaces can be unambiguously defined and easy mathematical description of fold. In kink kink and chevron chevron folds (Fig. 12.11), the limbs are straight and hinge zones are sharp (i.e., angular) with infinite curvature (because radius of curvature is zero). However, the hinge zones in natural kink/chevron folds may not be perfectly angular but may be slightly curved within a small areal extent. This type of fold occurs in strongly anisotropic rocks such as thinly-stratified sedimentary rocks or low- to medium-grade metamorphic rocks with slaty cleavage/schistosity. The terms kink fold and chevron fold are not synonymous but they are closely related. Chevron folds are roughly symmetrical with approximately equal limb lengths and have one set of sub-parallel axial surfaces. fold train is usually continuous across the axial surfaces for some distance. kink folds, on the other hand, are asymmetric with markedly different limb lengths. There is usually a discrete zone in the rock within which the layers have different orientation as compared to the orientation outside. In an ideal kink fold (Figs. 12.11; Fail 1969), dips of layers abruptly change across an imaginary planar boundary, called a kink plane, plane, which represents axial plane. The kink axis axis (or hinge line) is given by the intersection between kink plane and folded surface. A kink band band is the area between two adjacent kink planes whose kink axes are parallel. The line along which two kink planes join is called kink junction axis. axis . If the kink junction axis lies in the plane of layering then the fold is cylindrical, otherwise the fold is noncylindrical. If layers do not change thickness, the kink plane bisects interlimb angle (i.e.,
γ1 = γ2, in Fig. 12.11).
120
kink band
kink plane Kink ax axis axial axial planes pla nes
Kink ax axis
1
T1 t1
1
2
t2
2
T1
t1 t2
T2
T2
axial axial planes pla nes
α α
eometric characteristics of kink folds. Figure 12.11.G
Kink folds are peculiar in the sense that the folds are both similar and parallel. If the two limbs have different thickness, then γ1 ≠ γ2 but individual limbs still have the geometry of both parallel and similar folds. There is a region near the hinge where orthogonal thickness cannot be defined. There is only one set of sub-parallel kink planes in monoclinal kink bands. bands . However, kink bands may occur in two differently oriented sets producing conjugate kink bands. bands . In other words, two sets of kink bands may be inclined towards each other. When two kink planes meet they annihilate each other and only one kink plane extends from the kink junction axis (Fig. 12.11). This property can be used to construct cross section in areas where kinking is the preferred mode of deformation.
121
Rigid-body translation of faulted blocks along non-planar fault surfaces leads to creation of voids (Figs. 13.1a, b). As noted earlier, large voids cannot be sustained in the crust. In order for the two faulted blocks across a non-planar fault surface to remain in tight contact with each other, there must be distortions within at least one of the fault blocks. Layered rocks may develop folds when they slip past a bend on a faulted surface (Figs. 13.1c, d). The fold geometry is controlled by the shape of the fault. These folds are called fault-related folds because folding is a consequence of faulting. Such folds are common in deformed sedimentary basins in both compressional and extensional tectonic settings. The emphasis here is on thrust-related folds because they have been described more frequently from fold-thrust belts.
(a )
(c )
Fracture
gap
(b)
gap
(d)
Figure 13.1. Compatibility problem arising out of a rigid-body translation of hangingwall block along a fault with a sharp bend. (a) Incipient fracture with a sharp bend. (b) Gaps open up due to rigid-body translation of hangingwall block. (c, d) A kink band develops and fills up the gaps. The geometry of the kink band is controlled by the shape of the fault. Note that the gap above the footwall on the left side of (d) is yet to be filled up.
Since we do not understand mechanics in sufficient detail as yet, fault-related folds are usually modelled kinematically and the geometric consequences are evaluated 122
against naturally occurring folds and related structures. " Kinematics" Kinematics" is a branch of mechanics that treats motion in an abstract framework, without reference to force and mass. Since the publication of a classic paper by Suppe (1983), a number of kinematically valid fault-related folding models have become available. However, faultbend folding, folding, fault-propagation folding folding and detachment folding folding may be considered as endmember models and they will be described in detail in this section.
13.1 13.1.. Ass umpt ions For the purpose of kinematic modelling of fault-related folds several reasonable assumptions (or boundary conditions) must be made. This is a legitimate exercise for any kind of modelling.
•
Fault shape: shape: Faults are taken to have sharp bends leading to “ideal” stair-case trajectory although fault bends may not be as sharp as bends in ideal stair-case trajectories. Flat-ramp trajectories of thrust faults have approximately stair-case geometry. The fault bend controls the location and initiation of axial surfaces of fault-related folds.
•
Fold shape: shape: Folds are assumed to have kink fold geometry. Straight limbs and sharp hinges with infinite curvature of kink folds make the model calculations easier and simple. This is not a bad assumption because the folds in deformed sedimentary terrains usually have straighter limbs and hinges of small areal extent that approximate kink-fold geometry. The assumption of kink fold geometry does not introduce large error so long as axial surfaces and dip panels can be unambiguously determined.
•
Layer thickness: thickness: Layer thickness, bed length and cross sectional area are assumed to remain constant during folding. In some models thickness of limbs are allowed to change during folding.
•
Deformation: Deformation: Plane strain is assumed, i.e., the material points are not allowed to move in and out of section plane. Deformation is essentially accomplished by slip parallel to bedding with or without simple shear perpendicular to bedding. 123
Rocks do not undergo any internal deformation, i.e., deformation is constant volume.
13.2 13.2 Kinematics of kink fold s Kink folding with constant layer thickness is commonly accomplished by flexuralslip mechanism wherein layers slip past each other (Figs. 13.2, 13.3). The type of layerparallel slip in kink folds developed through layer-parallel compression (i.e., buckling; Fig. 13.2) is different from those developed due to faulting (Fig. 13.3). In buckling, the sense of slip on two limbs are opposite in the two limbs and amount of slip decreases from a high value near inflexion surface to zero at the axial surface Fig. (13.2). Material points cannot move past the axial surface.
dextrally sheared vein
sinistrally sheared vein
Axial surface
(a )
(b )
Figure 13.2. Flexural-slip due to layer-parallel compression. Sense of shear on the two limbs is opposite to each other. The amount of slip (i.e. shear strain) decreases to zero at the hinge. Consequently, the material points cannot pass through the axial surfaces.
Any pre-existing vein will be displaced on the limbs and shear strain will be positive and negative in adjacent limbs (Fig. 13.2b). Further, sense of shear can be used to locate the hinge of the fold. The location and orientation of axial surfaces in faultrelated folds are controlled by bends on the fault surface (Fig. 13.3). Sense of shear displacement will be same on both the limbs and material points can move past axial surfaces. Any pre-existing vein will show same sense shear displacement everywhere (Fig. 13.3c). Ramp portions will always have shear but the flat portions may or may not
124
have shear strain. In either case, if the layers across the axial surface maintain constant layer thickness, the axial plane must bisect the interlimb angle. post-deformation position pre-deformation position material point
ψ'
ψ ψ
(a )
ψ
(b ) sinistrally sheared vein vein
(c) Figure 13.3. Flexural-slip during fault-related folding. The material points can roll through axial surfaces (a, b). (a) Shear strain is set up on the ramp part although there is shear on the flat part. (b) Shear strain in the ramp part is different from shear strain on the flat part. (c) Sense of shear displacements on the two limbs of the kink fold is same, as given by sheared veins.
13.3 13.3 Fault Fault -bend fol din g In the model of fault-bend folding folding (Suppe 1983), a fracture with a staircase or flatramp-flat trajectory forms rapidly followed by movement of one or both the fault blocks. If the rocks are layered they may fold in response to riding over a bend in the fault; the folds thus formed are called fault-bend folds. If we move the hangingwall keeping the footwall fixed, a flat-crested anticline forms over the ramp and the fold is called a ramp anticline anticline (Figure 13.4). Names of different parts of ramp anticline and the angular parameters used for kinematic modelling of fault-bend folding are shown in Fig. 13.4. Progressive development of a fault-bend fold (FBF) caused by a simple step in décollement with folding confined to the hangingwall block is shown in Fig. 13.5 (Suppe 1983). At the time of initiation of folding two axial planes B and B’ form at the 125
lower bend Y and two axial planes A and A’ form at the upper bend X. With continued slip, the axial plane B’ climbs up the ramp and axial plane A’ moves along the upper flat. The material points roll through axial planes A and B. The axial planes A and B do not move. Since the footwall is fixed and only hangingwall moves, we can say that axial planes A and B are attached to the footwall and axial planes A’ and B’ are attached to the hangingwall. As the axial planes B’ climbs up the ramp, the fold amplitude increases but the width of the flat crest is reduced. The width of the two kink bands AA’ and BB’ also increases with increasing slip. When Y’ reaches X (upper bend), the axial plane B’ gets attached to the footwall and stops moving. At the same instant, the axial plane A is transferred to the hangingwall and starts moving along the upper flat. The ramp anticline stops growing in amplitude but width of the flat crest keeps increasing with continued slip. Transport direction
trailing anticline trailing syncline trailing edge
leading anticline
ramp anticline
backlimb
δ
forelimb
γ
γ
leading syncline leading edge
φ
β
θ
Figure 13.4. Different parts of and angular parameters in a ramp anticline. θ - initial cut-off angle, β - final cut-off angle, φ - change in the dip of fault (fault shape), γ - half of interlimb angle (Fold shape), δ change in dip across axial surface.
Suppe (1983) recognized several angular parameters, which can be used to describe the fault and fold geometry (Fig. 13.4): change in dip of fault ( φ), axial angle (i.e. halfinterlimb angle) of fold ( γ), initial cut-off angle ( θ), final cut-off angle ( β), and change in dip across axial surface ( δ = 180 ° - 2 γ). If the lower flat is parallel to bedding then θ is also the step-up angle. The angular parameters φ and γ represent fault and fold shapes respectively. For a simple step from one décollement to another (i.e., θ = φ), γ is related to θ by the following equation:
126
⎡ sin 2γ ⎤ φ = θ = tan −1 ⎢ 2 ⎥ ⎣1 + 2 cos γ ⎦
(13.1)
For a general case ( θ ≠ φ), θ, φ and γ are related by the following equations:
⎡ − sin( γ − θ) [sin(2 γ − θ) − sinθ] φ= ⎢ ⎣ cos( γ − θ)[sin(2 γ − θ) − sin θ] - sinγ
⎤ ⎥ ⎦
(13.2)
β = θ - φ + (180 ° - 2 γ) = θ - φ + δ , where δ = 180 ° - 2 γ A A'
(13.3) B'
B
X X'
a
Y'
Y
B'
A
B
A'
X
X'
Y'
b
Y B'
A
B
A'
X’
Y'
X
c
Y
Figure 13.5. Progressive development of fault-bend fold (after Suppe 1983)
Suppe (1983) provides a graph (Fig. 13.6), which is a pictorial representation of eqs. 13.1–13.3. The graph allows a quick analysis of possible range of solutions to a given 127
problem. For example, for θ = φ maximum cut-off angle cannot be greater than about 30°. This suggests that initial dip of most thrust faults should be 30 ° or less. Another interesting point to note is that for an "anticlinal" bend in the fault, γ is a double-valued function of θ and φ. Folds with larger and smaller values of γ are called first-mode (Mode-I) (Mode-I) (Fig. 13.7a) and second-mode (Mode-II) (Mode-II) (Fig. 13.7b) folds, respectively. If the angular parameters of a fault-related fold are not related to each other through eqs. 13.1–13.3, then it is either not an FBF or assumptions for kinematic modelling are invalid.
900
8 5 0 0
7 5 0
6 5 0 0
5 5 0 0
4 5 0 0
M o d - M o e - I d e - I I
1 2 0 0 0
0
3 5 0
2 5
0
5
β = − 5
5
0
0
15
1 5
0
− 2 5
0
0
30
− 3 5 0
0
− 1 5
− 6 − 5 5 0 0 5 0 0
− 4 5 0 0
5 0
1 5 0 0
0
1 0 0 0 0
γ 600
, e l g n a l a i x A
900
0
45
φ = 1 0 0
− 7 5 0
3 0 0
0
60
0
60
4 5 0
0
75
φ=90
β = 1 4 0 0
6 0 0 0
0
7 5 0 0 φ =
γ γ
0
30
φ
9 0 0 0 0
30
β γ γ
β Anticline
φ
θ
0
φ=θ
−5
θ
Syncline
00
0
0
900
600
300
00
- 600
Initial Initi al cut-off cut-off angle, angle, θ (ant (anticlin iclines) es)
- 300
- 900
Initial Initi al cut-off cut-off angle, angle, θ (s (synclin ynclines) es)
Figure 13.6. Graph showing relationships (equations 13.1-13.3) between different angular parameters of fault-bend folds (after Suppe 1983). The left and right sides of the graph are for anticlinal bend and synclinal bend, respectively, in the fault. The graph shows that initial cut-off angle for anticlinal bend cannot be more than about 30 ° for a simple step up in décollement (φ = θ). Also γ will have two values for every value of φ, for anticlinal bend in the fault; fold with larger and smaller γ values are called Mode-I and Mode-II folds, respectively.
a
b Mode- I fault -bend fo ld (larger axial angle)
Mode- II fau lt-bend fol d (sma ller axia l angle)
Figure 13.7. Mode-I and Mode-II fault bend folds.
128
13.4 Fault-propagation folding
A fault may not propagate rapidly through rock sequence as a clean fracture but may propagate gradually as slip accumulates. In such a case, at each instant during fault propagation, slip decreases upsection to zero at the fault tip and the shortening is transferred to a fold developing above the fault's tip. This kinematic process is called fault-propagation folding (Suppe and Medwedeff 1984, 1990; Suppe 1985; Mitra 1990). The folds formed above the tip of the propagating fault are called fault-propagation folds (Figs. 13.8). The primary geometric assumptions are the same as listed in section 13.1. The relations between the step-up or footwall cut-off angle ( θ), interlimb halfangles of faulted ( γ*) and unfaulted ( γ) units, and dip of the forelimb ( δ) are as follows (Fig. 13.8d; Suppe 1985, modified by Mitra 1990): cot θ + 2 tan ( θ/2) = 2 cot γ* - cot 2 γ*
(13.4)
γ = γ* + ( θ/2)
(13.5)
δ = 180 - (2 γ* + θ)
(13.56)
It follows that for a given θ, the geometry (i.e., interlimb angle and limb dip) of the nascent fold is maintained throughout its history. In foreland fold-thrust belts, θ usually varies between 15°-30°; the corresponding values for γ* and δ are 21.6 °-38.8° and 58.1° (overturned)-72.4 ° respectively. Therefore, in an ideal model, the faultpropagation folds are usually asymmetric and tight with steep to overturned forelimb. The ramp anticline is sharp-hinged with only one axial plane (AB', Fig. 13.8), up to the plane that locates the fault tip, i.e., the contact between faulted and unfaulted layers. Beyond this plane the axial surface bifurcates (axial planes A and B’) and the anticline is flat crested. The axial planes of the leading and trailing synclines terminate at the fault tip and at the fault bend, respectively. As the fault tip climbs up through the section with increasing slip, the axial plane AB’ grows in length as the width of the flat crest decreases.
129
A
A A'
B'
B' B
A'
B AB'
a
c
AB'
fault tip
slip
fault/fold fault/fol d initiation point point A
B'
A'
δ
B
γ
γ
γ∗ γ∗
AB'
faulted layer layer
d
b
unfaulted unfau lted l ayer
θ
Figure 13.8. Progressive development of fault-propagation fold (after Mitra 1990; Medwedeff and Suppe 1990). Angular parameters used for kinematic modelling are shown in (d).
13.5 13.5 Modified models o f f ault-bend ault-bend and fault-propagation f oldi ng The theories of fault-bend and fault-propagation folding are powerful end-member models, which can be modified to produce a variety of complex fold shapes. A few examples of such modifications are described below:
•
Breakthrough structures in fault-propagation folding: folding : In the fault-propagation folding model, the fault may propagate self similarly all the way to the surface or it can be halted at any instant depending on the rock properties. In the latter case, the folding ceases and the fault may break through in a fracture mode. The "breakthrough" thrust may propagate in several ways, e.g., along a décollement surface, synclinal axial surface, steep forelimb of the anticline and anticlinal axial surface (Mitra 1990; Suppe and Medwedeff 1990). Few examples of faultpropagation breakthrough structures are shown in Fig. 13.9.
slip
(a)
Decollement Decolleme nt breakthrough
slip
(b)
Anticlinal breakthrough breakthrough
slip
(c)
Synclinal breakthrough breakthrough
Figure 13.9. Examples of breakthrough structures associated with faultpropagation folding (after Suppe and Medwedeff 1990) 130
• Multi-bend fault-bend folding: If a thrust has sufficiently large slip, the beds may slip past more than one bend in the fault producing "multi-bend fault-bend" folds (Suppe 1983; Medwedeff and Suppe 1997). Examples of multi-bend faultbend folding with two bends in the ramp portion are shown in Fig. 13.10. Medwedeff and Suppe (1997) show that multiple fault bends give rise to complex fold shapes by a combination of two processes: a process of kink-band interference, and a set of processes associated with the generation of new dip panels and axial surfaces as hangingwall cut-offs are displaced past successive fault bends in the footwall. In theory, curved faults can be approximated by an arbitrary number of straight segments. In practice, a small number of straight segments generate a high degree of complexity and adequately models fold geometry. Consequently, a curved ramp can be modeled as a quasi-curved ramp. Also multi-segment ramps lead to proliferation of non-parallel axial surfaces that produce quasi-curved fold shapes.
Bend 3
Bend 3
a
Bend 2
Bend 2
b
Bend 1
Bend 1
Figure 13.10. Multi-bend fault-bend fold (after Medwedeff and Suppe 1997). (a) Synclinal bend. (b) Anticlinal bend.
•
Simple shear in the faulted layers: layers : In the ideal models of fault-related folding there is no layer-parallel simple shear within the hangingwall block. Consequently, the beds do not undergo layer-parallel shear until they enter the fault-bend fold and the fault surface is always the "active slip surface". If this condition is relaxed, i.e., if layer-parallel simple shear within the thrust sheet is allowed, then the fold shape can be modified in many different ways. The ideal theoretical shape of a fault-bend fold associated with a simple step in décollement is a flat-crested anticline (Fig. 13.5). Suppe (1983) shows that if the thrust sheet undergoes pervasive layer parallel simple shear, the two axial surfaces of the flat-crested anticline progressively annihilate each other forming 131
a new axial surface resulting in a sharp-crested fold (Fig. 13.11). The annihilation involves locking of the fault surface and migration of the active slip surface progressively to higher bedding surfaces, resulting in a simple shear in the hangingwall of the thrust sheet above the lower décollement. The active slip surface is always in the bed in contact with the branch in the axial surface. All the slip is absorbed in the annihilation and the thrust sheet is immobile along the lower décollement, beyond the anticlinal axial surface.
A c t i v e s l i p s u r f a c e
A c t i v e s l i p s u r f a c e
a
α
b
Figure 13.11. Development of sharp-crested anticlinal fault-bend fold through annihilation of flat-crest. Annihilation takes place as active slip surface migrates up section from the ramp (After Suppe 1983).
•
Combined fault-propagation and fault-bend folding : In the model of faultpropagation folding, the folding initiates as soon as the ramp begins to step up from the décollement (Fig. 13.12). The fold acquires its basic geometry at this stage and continues to grow self similarly with the propagation of the fault. Also all the beds in the hangingwall cut by the fault are folded through the anticlinal axial surface. Chester and Chester (1990) made an interesting modification to this model wherein they suggest the existence of a pre-existing ramp (or fracture) (Fig. 13.12). Folding is initiated when the pre-existing ramp is activated without any change in dip. In a way, it is similar to fault-bend folding where fracture forms first followed by folding. The difference is that in Chester and Chester's (1990) model there is no upper flat and the ramp continues to propagate without change in orientation. The fold above the fault tip is a faultpropagation fold whereas the fold at the ramp-flat intersection is a fault-bend fold. These two folds are separated by an unfolded region. Another important
132
geometric difference is that in this model some of the lower layers in the hangingwall cut by the fault are not folded through the anticlinal axial surface.
fault tip
a
b fault/fold initiation initiation point
Fault-propagation Fault-prop agation fold No deformation deformation Fault-bend fold
slip
c
d
Figure 13.12. Progressive development of a fault-related fold, which combines fault-propagation and fault-bend folding models (after Chester and Chester 1990).
•
Forelimb thinning/thickening: Jamison (1987) noted that interlimb angle and forelimb dip in many natural fault-bend or fault-propagation folds are different from values that are predicted from simple models with constant layer thickness. Forelimb thinning/thickening was suggested to be a solution to this problem (Fig. 13.13).
resid ual
a
b
Figure 13.13. Forelimb thinning and thickening model of Jamison (1987)
Forelimb thickening leads to larger interlimb angle and smaller forelimb dip than the predicted values. For a given value of θ, forelimb thinning leads to folds with smaller interlimb angle and larger forelimb dip but forelimb thickening leads to folds with larger interlimb angle and smaller forelimb dip. The thickening/thinning occurs only in the forelimb, the thickness remains constant in the remainder part of the beds. 133
Also, the basic geometry of the fold is acquired at the time of inception. However, the shape of a fold-propagation fold can be modified in the case of a décollement breakthrough; the part of the fold that retains the original geometry is called the residual. residual. Jamison (1987) also demonstrated that simple shear parallel to the thrust sheet thins the forelimb and reduces the interlimb angle and thus changes the fold shape. In their detailed modelling of fault-propagation folding, Suppe and Medwedeff (1990) developed a theory, called the "fixed front anticlinal axial surface" theory, in which forelimb thinning/thickening was considered as a possible variable.
13.6 13.6 Décol Décol lement f old Décollement folds (also called detachment folds) are the third end member of faultrelated folds (Fig. 13.14).
Lift-up f old
Box fo ld slip
slip
(a)
(b)
Footwa ll syncl ine
Footw all syncl ine + breakt hroug h slip
(c)
fault t ip
slip
(d)
Figure 13.14. Décollement (or detachment) folds (after Mitra and Namson 1989; McNaught and Mitra 1993).
These folds develop in response to shortening above a décollement surface or a thrust that is parallel to bedding. These folds are not associated directly with ramps. They require a ductile décollement layer (e.g., salt or shale) that can infill the space generated at the base of the fold. Décollement folds are rootless and commonly disharmonic. Box folds folds (Fig. 13.14a) and lift-off box folds (Fig. 13.14b) are two types of décollement folds (Mitra and Namson 1989). There can be breakthrough associated 134
with these folds that can leave a syncline, called footwall syncline syncline (McNaught and Mitra 1993) stranded in the footwall (Fig. 5.12a,b).
13.7 Rollover anticline Rollover or rollover anticline (Hamblin 1965; Gibbs 1984) is folding in the hangingwall block as consequence of slip on a listric normal fault. Such folds are widespread in extensional tectonic set up. Kinematic modelling of folds above a curved fault is difficult. However, we can model rollover on normal fault with one sharp bend. This can be followed by modelling with multiple bends that approximate a listric fault. Fig. 13.15 shows a model of rollover development for a single sharp bend on a normal fault (Xiao and Suppe 1992). In this model a fracture with a bend forms through the rock sequence (Fig.13.15a) and before the hangingwall starts to move. Therefore, this model is similar to the fault-bend fold described in section 13.3. As the hangingwall was pulled away, a void is created (Fig. 13.15b). The hangingwall collapses to fill the void and a kink band bound two axial planes form (Fig. 13.15c). It is assumed that the orientation of the collapse surface will be given by coulomb failure criteria (see section 7). The upper axial plane is called the active axial plane because it is the locus of active folding and the material points roll through this axial plane. The active axial plane remains fixed to the footwall at the bend. The inactive axial plane is anchored to hangingwall block and is located at the boundary between deformed and undeformed rocks. In the next step, the hangingwall moves and collapses to fill the void increasing the width of the kink band (Figs. 13.15d, e). In reality the movement and collapse are not stop-and-go type as in the model. The collapse and widening of the kink band is continuous with increase in slip. Normal faulting decreases elevation on the hangingwall side of the fault leading to the formation of basin. The basin may be filled up with syntectonic sediments, which are called growth strata. The geometry of rollover gets suitably modified by growth strata (Fig. 13.16). A detailed description of rollover development during active sedimentation is give by Xiao and Suppe (1992).
135
Active axial surface
(a)
Inactive axial Inactive axial surface
(c) Coulomb failure surface 700
(b)
Void
(d) Active axial surface
Inactive axial Inactive axial surface
(e)
Figure 13.15. Kinematic model of rollover development (after Xiao and Suppe 1992).
Growth wedge
(a)
h t t a w a o r r t G s
Zone of most recent deformation Growth axial surface
(c)
(b)
Active axial surface
h t w a o t r a r g - t s e r P
Inactive axial surface
Figure 13.16. Progressive development of rollover with growth sedimentation (after Xiao and Suppe 1992).
136
137
14.1 14.1 Basic ph ilos ophy The whole purpose of balancing structural cross sections is to limit the interpretation to that which can be considered to be structurally reasonable. reasonable. Two interpretations of a faulted anticline are shown in Fig. 14.1. The orientations of the faults suggest that they are conjugate faults. The structural interpretation in Fig. 14.1a is unreasonable because the two faults have normal and reverse senses of displacement but we know know that conjugate faults should have same sense of displacement. Further, the layer thickness varies in a way for which there is no obvious explanation. But in Fig. 14.1b both the faults have normal sense of displacement and the fold follows parallel (class 1B) geometry. Thus the interpretation in Fig. 14.1b is reasonable. Most of the published cross sections are schematic schematic and many of them are reasonable. A cross section across frontal fold-thrust belt of NW Himalayas is shown in Fig. 14.2. Although the section is schematic yet it is reasonable in the sense that the interpretation is in conformity with structures seen in other fold-thrust belts. However, we should avoid constructing schematic sections based on imagination or intuition that may lead to structurally unreasonable or untenable interpretation. It is desirable that geometrically geometrically and kinematically valid kinematically valid constraints should be applied during and after section construction so that our subsurface structural interpretation is reasonable in terms of present knowledge about how rocks deform. A balanced cross section is simply a better cross section as compared to a free-hand drawn schematic and unbalanced cross section. It is important to remember that a balanced cross section is constructed using limited data set and a few reasonable assumptions. Techniques of cross-section balancing are described in Dahlstrom (1969), Suppe (1985), Ramsay and Huber (1987), Marshak and
138
Woodward (1988), Woodward et al (1989), Mitra and Namson (1989), Mitra (1992) and others.
(a)
(b)
Figure 14.1. A simple example of “unreasonable” (a) and “reasonable” structural interpretation. Interpretation in (a) is “unreasonable” because the conjugate faults have opposite senses of movement and layer thickness variation is unexplainable. But in (b) both the faults have normal sense of movement and the fold has class 1B geometry.
T M F
SW
a n u s t t S o t h r u
r h i i p u m u k p o a t t a g u t t w a l u r a s s e u a D t h r J t h r u
r o o r u s t t P a t h r u
r p u m t t a l u s B T M P a t h r u
NE
20 km Siwa lik
Sub athu-Dhar msa la
Le sser Him aya Zo ne
Metamorphic basement basement
Figure 14.2. A schematic yet reasonable cross section across Himalayan foldthrust belt, Kangra area, Himachal Pradesh (Ranga Rao 1989, unpublished ONGC report, in Biswas 1994). Although the section is not balanced, the interpretation follows structural styles in fold-thrust belts.
14.2 14.2 Defini Defini tion of balanced balanced cro ss s ection Dahlstrom (1969) formally introduced the concept of balanced cross section in the literature and suggested two rules to be followed while constructing a balanced cross section:
•
Bed lengths must be preserved during deformation. In other words, deformed and undeformed bed lengths must match or “balance”.
•
There are only a limited suite of structures that can exist in a specific geological environment. The “ foothills family of structures” structures” in the Canadian foothills 139
comprise of concentric folds, décollement, thrusts (usually low angle and often folded), tear faults, and late normal faults. Elliot (1983) gave a more restrictive definition of a balanced cross section:
•
A balanced cross section must be both admissible and viable.
•
The structures drawn on a section should be those that can be seen in the field in outcrops, cliffs, mountain sides etc.
•
The use of these structures leads to an admissible admissible cross section.
•
If a section can be restored (i.e., retrodeformed) to an unstrained state, it is a viable viable cross section.
There are areas where exposures are scanty owing to heavy vegetations, such as Himalayan foreland belt. In such areas it may not be possible to decide admissible structures as suggested by Elliot (1983). Therefore, a preferred definition is: c ross section is a geometrically correct c orrect (i.e. admissible) deformed-state cross • A balanced cross section, which can be restored (i.e., viable) to an undeformed state or to an earlier less deformed state through kinematically valid steps.
14.3 14.3 Two aspects of balanced balanced cros s secti ons From the foregoing discussion, it is apparent that there are two aspects of balancing a structural cross section:
•
Construction of a valid structural cross section, which depicts the present day subsurface structural geometry. geometry . This is the so-called deformed-state cross section, which geologists have been constructing through ages. The key issue here is that the section we construct must contain the geometry of structures we actually see in the field or the geometry of structures present in other areas with similar tectonic setup. For example, if we are to construct a cross section across a rift basin, the section must dominantly contain normal faults and associated 140
structures. Or, if our section is across a fold-thrust belt, the main structural architecture must be controlled by thrusts and associated structures. This exercise results in admissible deformed-state cross section .
•
The deformeddeformed -state cross section should be restored to undeformed state (i.e., validated) . Cross sections are essentially interpretations based on incomplete data and require extrapolation and interpolation. So, how do we increase confidence in our interpretation? Simply put, we validate. If we cannot validate a cross section, we reject the interpretation. We then reinterpret the available data and construct a new section and try to validate. A balanced cross section is the one, which can be validated. We know know that sediments are commonly deposited with (sub-) horizontal layering. If we undeform (i.e., restore or retrodeform) an admissible cross section drawn across a deformed sedimentary basin and find that the layers become horizontal then we have validated our cross section.
We have not said anything about a “balanced cross section” being a correct interpretation. Balancing a cross section is akin to inverse modelling. Like all inverse modelling, the solution we arrive at is non-unique. It is possible to construct more than one balanced sections with the same data set. Therefore, it is important to remember that balancing is a necessary but not a sufficient criterion for a correct structural interpretation . Balancing even a mildly complex structural section is a time consuming, tedious and frustrating exercise without any guarantee of success within the fixed timeframe of a project. However, while non-balanced and schematic sections are almost always wrong, a balanced cross section is probably correct! However, practicing the techniques of cross-section balancing, particularly in deformed sedimentary basins, should lead to better and more reliable cross sections, even if a section cannot be rigorously balanced. 14.4 14.4 Exampl es
A few examples will illustrate the requirements for balanced cross section. The hypothetical deformed-state cross section in Fig. 14.3a seems to be quite reasonable with thrusts and related ramp anticlines. The steeply dipping parts of the thrusts may 141
be explained by rotation caused by younger faults in the footwall. However, if the section is restored (Fig. 14.3b) keeping the bed length and sectional area constant, the thrust trajectories become unreasonable. Therefore, the deformed-state cross section is not a balanced cross section and should be discarded. The hypothetical section in Fig. 14.4a is a balanced cross section because it can be retrodeformed into an admissible restored section. Fig. 14.5a shows a section across NW Himalayan foreland fold thrust belt, Dehra Dun (Mishra and Mukhopadhyay 2002). The Mohand anticline is a multibend fault-bend fold with about 12% foreland thinning. A multi-bend faultpropagation model with about 47% forelimb thinning is appropriate for the Santaurgarh anticline. The Santaurgarh anticline is related to a blind thrust with anticlinal breakthrough structure (Santaurgarh thrust). The admissible restored section of the deformed-state section is given in Fig. 14.5b. Therefore, the cross section in Fig. 14.5 is a balanced cross section. It is emphasized that both deformed and restored section should be given because unless an admissible restored section is given, balancing is not demonstrated.
Deformed Deform ed
(a) Restored Restor ed
(b)
Figure 14.3. The hypothetical cross section in (a) is unacceptable because the restored section is not admissible. Deformed Defo rmed
(a) Restored Rest ored
(b)
Figure 14.4. The hypothetical section in (a) is a balanced cross section because both the deformed-state section and restored sections are admissible.
142
Santaurgarh Thrust Santaurgarh anticline Main Frontal Thrust (MFT)
Mohand anticline
Main Boundary Thrust (MBT) Topography
Doon Gravel
P Alluvium
L
Doon Gravels/ Alluvium Up. Siwalik Fm
?
Mid. Siwalik Fm Lr. Siwalik Fm
Dharmsala Gr
(a) Detachment Detachme nt
Subathu Gr
L
P MFT
(b)
?
5 km
Figure 14.5. Balanced (a) and restored (b) cross sections across Himalayan foreland fold-thrust-belt, Dehra Dun area (after Mishra and Mukhopadhyay 2002).
14.2 14.2 Types Types of cros s sectio ns There are different types of cross sections and the subtle differences between them should be understood clearly:
•
Deformed-state cross section: section : Any structural cross section showing the presentday sub-surface structural geometry.
cross-section : Deformed-state cross section in which the depicted • Admissible cross-section: structures conform to the available data, and are also geometrically and kinematically admissible.
•
Restored cross section: section : A cross section that has been 'pulled apart' in the sense that displacements on faults have been removed and folded beds have been straightened.
• Admissible restored section : It is a restored section in which the restored beds are horizontal and the restored fault trajectories are admissible. In particular, ramp dips should not be more than 30-35 ° and thrust trajectories should not have zigzag shape unless they are out-of-sequence thrusts.
•
Viable cross section: section : A deformed-state cross section that can be restored to an admissible restored section.
143
•
Balanced cross section: section : A deformed-state cross section that is both admissible and viable. Therefore, a balanced cross section has been restored to undeformed state and has been tested for viability.
14.1 14.1 Fold-thrus Fold-th rus t belt s (FTB) (FTB) Fold-thrust belts (FTBs) form when sedimentary basins are deformed by compressional tectonics. Much of the concepts of balanced cross sections have emerged from fold-thrust belts. Consequently, section construction and restoration are described with reference to fold-thrust belts (sections 15 and 16). But these techniques are equally applicable for extensional basins. The characteristic features of fold-thrust belts include:
•
Narrow and elongated basins and occur at the margin of orogenic belts.
•
Composed mostly of sedimentary rocks.
•
Deformed in response to compressional tectonics.
•
Deformation is partitioned mostly in the cover rocks and basement remains largely unaffected – the so-called thin-skinned tectonics (Fig. 14.6). The cover sequence and basement are separated by a major thrust of décollement.
Hinterland
Foreland NE
SW
(a)
Basement
Regional detachme nt
Hinterland
20 km
Foreland SE
NW
mean sea level
20 km
t in e b a s e m e n n l i a l l a n c r y s t a i a P r e c a m b r i
(b)
Basal detachment
Figure 14.6. Cross sections across foothills of (a) Canadian Rockies (after Price 1981) and (b) NW Himalayas (after Mukhopadhyay and Mishra 1999) showing essential feature of thin-skinned tectonics where cover sequences are deformed but basement remain largely unaffected.
144
•
Presence of hinterland-foreland pair (Fig. 14.7). The undeformed sedimentary basin in front of the FTB is called foreland foreland and the metamorphic core with ductile deformation is called hinterland. The external zone zone contains the FTB. The terms external zone and foreland are sometimes used interchangeably.
•
Folding is commonly a consequence of thrusting.
•
Axial and fault traces are parallel to each other and parallel to the trend of the orogenic belt. High Him alaya sedi mentary Zon e High Him alaya Cr ystalline Zone Lesser H imalaya Zo ne Sub-Himalaya Zone
N T R A N S
ITSZ: Indu s Tsangpo Tsangpo Su ture Zone STDS: South Tibet detachment Sy sytem MCT: Main cent ral Thrust MBT: Main Boundar y Thrust MFT: Main Frontal Thr ust
H I N T E E R X T L A E R N N F A D O R L 0
30
E L A N D
Delhi
250 km
H I M AL AY A
Z O N E
T e t r a c t o n n s i c p o r t
ITSZ
M B T
0
80
M C T M F AL L L U U V T V I IA L
STDS
LAII N PLA
0
90
Figure 14.7. Geological map of the NW Himalayas (simplified after Gansser 1981) showing hinterland, foreland, external zone and tectonic transport direction.
The frontal part of the Himalayas shows all the characters of a fold-thrust belt. The Sub-Himalaya Zone consisting of Tertiary sedimentary rock sequence and the Precambrian sedimentary and very low-grade metamorphic rocks of the Lesser Himalaya Zone together constitute the Himalayan FTB (Fig. 14.7).
14.6 14.6 Appl icabili ty The concepts and techniques of balancing cross sections originated in foreland foldthrust belts, and in particular in the frontal fold-thrust belt of the eastern Canadian Rocky Mountains, which are characterized by folded and faulted, non-metamorphosed sedimentary sequences that lie above a gently hinterland-dipping detachment or 145
décollement. These concepts and techniques, sometimes referred to as Rocky Mountain Principles, Principles, have been successfully applied in many foreland fold-thrust belt in different parts of the world. These techniques can also be applied in extensional terrains. The most important application of this technique is found in hydrocarbon exploration and exploitation where reliable subsurface structural interpretation is of paramount importance for fiscal planning. A balanced cross section across Bolivian fold-thrust belt is shown in Fig. 14.8a (Baby et al. al. 1995). The balanced cross section was used to derive burial and uplift histories. This in turn helped in defining the gas and oil windows for detailed exploration (Fig. 14.8b). It is not a mere coincidence that structural geologists working for oil companies authored many of the early publications on balanced cross section. This technique is also important for academic research because a balanced cross section helps us to understand how an orogenic front develops and propagates in time and space, and to estimate orogenic shortening. In turn, these may help us to understand the geodynamic evolution of an orogenic belt. Therefore, in the techniques of cross-section balancing there is a convergence of interests of academicians and explorationists!
5
SW
NE Isiboro anticline
Late Paleogene-Neo Paleogene-Neogene gene Jurassic-Cretaceou JurassicCretaceouss Up. Carb.-Lr. perm. Carboniferous Dev.-Sil.-Ord. Dev.-Sil. -Ord. Cambrian-Precambrian
m K0 -5
(a)
10 km
5
m K0
Oil window Oil & Gas Zone Gas window
-5
(b)
Figure 14.8. An example of application of cross-section balancing in hydrocarbon exploration in Bolivian fold-thrust belt (Baby et al. al. 1995). (a) Balanced cross section. (b) Suggested oil-gas windows based on burial-uplift histories derived from balanced cross section.
146
147
Construction of cross sections is one of the most important tasks undertaken by explorationists and structural geologists. Sub-surface structural interpretations are communicated through cross sections. Exploration strategies and financial planning for explorations for hydrocarbon and other natural resources are based to a large extent on structural cross sections. Therefore, cross sections should be constructed with utmost care and, as far as possible, they should be quantitative rather than purely qualitative. The techniques of balanced cross section have been very useful for quantification of structural interpretations in sedimentary basins deformed due to in compressional tectonics.
15.1 Strategic Considerations
•
Line of section: section : Inherent in the methods of cross-section balancing is plane strain. For this reason the line of section must be chosen strictly parallel to the tectonic transport direction. However, a line of section within ± 10 ° of the tectonic transport direction does not result in significant error (Woodward et al. al. 1989). The tectonic transport direction can be determined using bow-and-arrow bow-and-arrow rule, as illustrated in Fig. 15.1a (Elliott 1976). This direction can also be determined from stretching lineations or from the orientations of the axes of small-scale folds. If the fault traces and axial traces are parallel on the map (Fig. 15.1b), then a section line oriented perpendicular to fault/axial traces is acceptable in most situations. In most FTBs, the tectonic transport direction is approximately known.
•
Ramp-flat geometry: geometry: The thrust faults in fold-thrust faults usually assumed to have ramp-flat trajectory with sharp bends for ease in section construction. 148
Slight curvature near the bend can be ignored without adding significant error. Curved faults can be considered to have multiple bends with straight line trajectories in between bends.
Thrust tip
Thrust
Axial traces Direction of tectonic transport transport
Direction of tectonic transport transport
(a)
(b)
Figure 15.1.( a) “Bow-and-arrow” rule to determine tectonic transport direction, which is given by the perpendicular to a line which connects the two exposures of a thrust tip line in map view and is in the direction from tip connector to the fault (Elliot 1976). (b) The direction perpendicular to the fault and axial traces also gives the tectonic transport direction.
•
Depth to Detachment: Detachment: When we construct deformed-state cross section in foldthrust belts, what we essentially try to do is fill up the space between the topographic surface and the basal detachment! So it is important to know the depth and dip of this basal detachment. It may be known from seismic reflection profiles or borehole litholog data. In the absence of such data, a commonly used method is depth-to-detachment calculation, which assumes area conservation during deformation. The method of depth-to-detachment calculation is illustrated in Fig. 15.2.
1
2'
2
3'
3
4
5
S = lo - l
Area A
1'
4'
5'
Area B d
Shortening, S = l o - l
Area Are a A = Are Area a B =dx S
d = ar ea A/S
Figure 15.2.D epth-to-detachment calculation.
149
•
Reference lines: lines : There are two types of reference lines, viz., pin line line and loose line (Fig. 15.3). A pin line is a line in the deformed-state cross section from which all restoration measurements are made. Pin lines can be regional or local. Regional pin lines are perpendicular to stratification and are located in the undeformed foreland. Local pin lines lines are located within the thrust belt. Loose lines are usually marked near the trailing edge of the cross section. A loose line can be considered to be chain of marker points in the layered sequence. It is particularly useful in tracking layer-parallel simple shear, which may not be obvious in deformed-state cross section.
Local pin lines
Foreland Regional pin line
Loose line
Figure 15.3. Different types of pine lines.
•
Template: Template: A restoration template shows the assumed pre-tectonic attitudes of the rocks shown in the deformed-state cross section. Restoration is usually done on this template. Template can be constructed if stratigraphy of the area is well known.
•
Sequence of thrusting: thrusting : It used to be assumed that thrusts in fold-thrust belts form in a forward-breaking sequence. But now we know that break-back sequence, out-of-sequence (Morley 1988) and synchronous thrusting could be important. Knowledge of sequence of thrusting is particularly useful for restoration. Faults should be restored in the reverse order than they formed.
150
•
Slip on a fault: fault : If the slip on a fault surface is assumed to be constant then matching hangingwall and footwall cut-offs is simple. However, for a blind thrust this is not true. The slip should become zero at the fault tip and the shortening is transferred to the overlying fold.
•
Balanced forward modelling: modelling : It should be an integral part of section construction. As many balanced forward models as possible should be constructed in the vicinity of each fault. Then, we can attempt to fit these models like a jig-saw puzzle. Sections that contain forward modelled structures have better chance of being restorable.
•
Previous sections: sections : Do not throw away previous sections, if available. Modify such sections after proper evaluation in terms of admissibility and viability. One can avoid repeating the same mistake made by previous workers. Of course if there is no previous section available one has to start from scratch with raw data.
•
Data: Data: All available data should be gathered and compiled, such as surface geological map, stratigraphy, regional geology and tectonic setting, dip data collected at the surface, dipmeter data from borehole logs, lithologs, seismic reflection profiles etc. Obviously more data we have more confidence we will have in our cross section. Subsurface data, such as, seismic reflection profiles, are not absolutely essential, essential , for balancing a cross section, which can be done from geological map and surface dip data alone. Of course, for an offshore project or if the area is covered by alluvium the first data is commonly seismic reflection profile. It should be remember that even with today's improved data acquisition and processing techniques, seismic data often leave much to be desired.
15.4 Parallel Folds In some geological environment, notably in sedimentary sequences, the folds are usually parallel, i.e., they belong to class 1B. In such folds, layers do not show any appreciable thinning of limbs or thickening of hinges, the orthogonal thickness measured perpendicular to layering remains constant at all points around the fold. In multi-layered sequence, the folds are conformable or harmonic. The folded surfaces are 151
smoothly curved or concentric in some parallel folds, whereas in others angular or sharp hinge zones separate domains in which layers are nearly straight. These folds are called concentric and kink folds, respectively. An interesting feature of kink folds is that they are parallel as well as similar (class 2). Parallel folds can be extrapolated data to depth using either Busk method (Fig. 15.4) or kink method (Fig. 15.5). The Busk method of section construction is appropriate for concentric parallel folds, whereas the kink method is appropriate for parallel folds with angular hinge zones and straight limbs. These methods produce reliable cross sections only if the assumption that the folds are parallel is valid. For non-parallel folding we should use dip-isogon method or one of the several orthographic projection methods to construct sections. The Busk and kink methods are described here because they, particularly kink method, are used most frequently in fold-thrust belt. Any kind of dip data can be extrapolated to depth using either Busk or kink method. The dip data can be in the forms of outcrop dips, formation tops or bottoms taken from outcrop or well log information, or dip meter data. However, it is important to be consistent.
Busking The Busk method (Fig. 15.4) allows us to construct the traces of bedding planes in a section plane from surface or subsurface measurements of the attitudes of the folded layers. The geometric basis of this method is the assumption that the folded layers are everywhere tangent to circular arcs. A consequence of this assumption is that the trace of each folded layer in a profile can be divided into a number of segments each of which is either a portion of a circular arc or straight line. Along each circular arc segment, the dip changes smoothly and continuously. Adjacent circular arcs are connected by inflexion points or by straight-line segments. Lines perpendicular to dips are drawn from the position of dip measurement data. The perpendicular lines from two adjacent dip data intersect at a point that represents a radius for a curvature of an arc, which is utilized to project the beds between the two data points. The method is described in Fig. 15.4. For example, in Fig. 15.4 two dip data 152
(1,2) are shown at two locations (A, B). We draw perpendicular (C, D) to dips 1 and 2, the perpendiculars intersect at O. We then draw to circular segments with OA and OB as radii. This gives us the fold segment between C and D. One of the problem of Busking is that, singularities singularities (points of infinite curvature) often appear in Busk construction of folds. Singularities are rarely observed in natural concentric parallel folds.
L I G B
G C
B
A
C
A F
D
F
D E
O
H
J
E
O
usk method of extrapolation of surface dip data to depth. Figure 15.4.B
Kinking Kink or constant dip-domain method method (Fig. 15.5) has proven to be extremely useful for extrapolating data to depth. Many fold-thrust belts contain folds, which display a kink or dip-domain geometry. Therefore this method has become very popular for constructing cross sections in such belts.
Angular
folds
produce
domainal
dip
patterns patterns on maps. A dip domain is an area in which strata have nearly constant dip or dip varies within a small range. Adjacent dip domains are separated by narrow zones, representing hinge zones, in which dips change rapidly. In kink methods we use two inherent geometric features of kink folds to extrapolate surface data to depth: if layers do not change thickness, the axial plane bisects interlimb angle and only one axial plane extends from the junction where two axial planes meet. The kink-method of section construction is rather straightforward and the technique is explained in Fig. 15.5. First we locate boundaries between dip domains (Fig. 15. a) and draw the axial planes with orientation that bisects the angle given by the dips in two adjacent dip domains (Fig. 15.5b). Where two axial surfaces meet, a new axial plane 153
starts bisecting the angle between two intersecting axial planes. Within the area between two axial planes, straight beds are drawn. Round-hinged folds can be approximated as closely-spaced small kinks (Fig. 15.6).
Ground surface
Domain Boundary
B
A
(a)
Dip
B
A
C
D
Domain Boundary (Axial Surface)
A
Branch point
B
Dip Domain X
( b)
Dip Domain Y
(c)
ink method of extrapolation of surface dip data to depth. Figure 15.5.K
round-hinged fold can be modelled using kink-fold geometry. Figure 15.6.A
15.5 15.5 Drawing a Cros Cros s Section Figure 15.7 illustrates how surface dip data can be extrapolated to depth and subsurface fold geometry can be deciphered. The dip data and stratigraphy are given on the left side of the diagram. At locations A locations A and B contact between sandstone and shale are exposed. The dip data shows five planar domains (domain 1-5) suggesting kink fold geometry. At domain boundaries attitude of axial planes are deduced from the bisectors of the adjacent domain dips. The axial surfaces are extrapolated to depth. 154
Wherever two axial surfaces meet, a new axial surface emerges whose orientation is given by the bisector of the two axial planes. The beds are then extended using the dip domain data and the fold is constructed.
Data 1 n i a m o D o
30
Solution Sol ution
2 n i a m o D o
0
3 n i a m o D
4 n i a m o D
o
o
40
60
5 n i a m o D o
0
o
70
o
75 Well
o
40
A B
A
60 o
B
Sandstone A
Sandstone A
Shale A
Shale A
Sandstone B
Sandstone B
Shale B
Shale B
Sandstone C
Sandstone C
Shale C
Shale C
75 o
Sandstone D
60 o
Sandstone D
xtrapolation of surface data to depth using kink method to deduce Figure 15.7.E the geometry of the large fold.
Maxi mum Maximum ramp height
Data Maximum dept h of fau lt
Solution 1 : Maximum depth o f fault
Solution 3 : Maximum ramp height
Figure 15.8. A hypothetical example of section construction showing range of possible solutions. See text for discussion.
155
Fig. 15.8 shows an exercise of how surface dip data and known stratigraphy are used to deduce buried thrust. The dip data show five dip domains, three horizontal dip domains are separated by two dip domains where dips are steeper. Overall the fold geometry is that of a flat-crested anticline, so we guess that it is a fault-bend fold. If this is the case there has to be a thrust at depth. Let us suppose we know that the tectonic transport direction is towards left. Note that the axial angle ( γ) is known and back limb dip can be used to infer ramp dip or cut-off angle ( θ). With available information we cannot deduce the exact location of the fault. However, we can find range of possible solutions. Solution 1 is based on maximum possible depth of the fault and solution 2 gives us maximum ramp height. We recognize that an exact solution is not possible in this example but the range of possible solution can be useful for planning further exploration strategies. The examples shown in Figs. 15.7 and 15.8 are hypothetical. A real-life example of section construction with limited data is shown in Fig. 15.9 (Suppe 1983). Fig. 15.9a shows the available data near the crest of the Hokou-Yangmei anticline, Taiwan FTB. The Well A encountered a double thickness of the distinctive Pliocene Chinsui Shale and normal thickness of formations below Chinsui Shale, suggesting that the small fold on which Well A sits does not extend below Chinsui Shale. Two guesses were made, as shown in Fig. 15.9b, both involving a simple step of a thrust fault from one décollement to another in the Chinsui Shale. In solution 1, a thrust steps up to the north and in solution 2 a thrust steps up to the south. The important angular observations are that the dip at the base of the Chinsui Shale is 5° whereas the minimum dip of the Chinsui Shale, between two wells is 32°. Therefore, we choose 32° - 5° = 27° as θ = φ in solution 1 and β in solution 2. Using Suppe's (1983) equations (or graph) we obtain
= 34° for
solution 1 with 34° - 5° = 29° as the predicted surface dip. This predicted dip is much greater than the observed surface dip of about 16°, so discard this solution was discarded. For solution 2 we obtain θ = φ = 22° and 22° - 5° = 17° as the predicted surface dip, in good agreement with the observation. Therefore solution 2 may be considered viable. We can now compute how the shallow fault in solution 2 will be folded by the deeper anticline ( γ = 58°). The cross-cutting fault block is in the footwall, 156
convex towards the fault; therefore it is a "syncline" and φ = 57° and β = 15°, which are in reasonable agreement with surface dips. The final interpretation of the structural geometry using solution 2 is shown in Fig. 15.9c. In this example, a hypothesis was invoked (fault-bending over a simple step up of décollement) and tested against the available data and a solution was found.
Well B North
South
0
dip = 32
Thickness doubled
-1
0
Chinsui shale dip = 5
Km
0
-2
-3
(a)
Data
0
16
160 β=? (34 (3 4 0)
θ = φ0 = 22 0
32 θ = φ0 0 = 32 32 −5 = 270
(b)
β = 270
φ = 0? (57 (5 7) β=? 0 (15 (1 5)
Solution1 (rejected)
γ γ = 58
(c)
0
Fina l interpret ation based on solution 2
Solution 2 (accepted)
Figure 15.9. Actual example of quantitative section construction, HokouYangmei anticline, Taiwan (Suppe 1983). With available data (a), two solutions are guessed (b). Solution 1 leads to conflict with surface dip data. Solution 2 is in conformity with surface dip data. (c) Final interpretation based on solution 2.
A different approach is illustrated in Fig. 15.10, where the trajectory of Main Frontal Thrust (MFT) and the geometry of the Mohand anticline, Dehra Dun re-entrant have been constrained (Mishra and Mukhopadhyay 2002). The available data are shown in Figs. 15.10a,b; the MFT trajectory and the basal detachment were approximately constrained from published ONGC seismic reflection profile and well data. Several forward models were made, three of which are shown in Figs. 15.10c-e. A model based on multi-bend fault-bend folding with 12% forelimb thinning and uniformly tapering
157
layers conforms to the surface dip data and interpreted litholog and "best" explains the geometry of the Mohand anticline (Fig. 10e). The above examples show that it is possible to construct quantifiable structural cross sections. N
Dun
Grav els
Alluvium
20
Mid. Siwalik Siwalik
Up. Siwalik Siwalik
Dun Gravels
Well MhA
15 22
Topography
30 11 30
MFT
33
MFT and B asal detachment Approximated from subsurface data
MhA
5 km
23
36
V I U M A L L U
MFT?
(a)
5 km
(c)
(b)
MhA : Mohand Anticline
5 km
Multi-bend fault -bend folding m odel with MFT emergent
(d)
5 km
Topography
12% forelimb thinning
sl s l i i p 4.0 0 k km
(e)
5 km
Doon Gravels
Up. Siwalik
Mid. Siwalik
Lr. Siwalik
Dharmsala
actual example of section construction, Mohand anticline, Dehra Figure 15.10.A Dun area Himalayan FTB (Mishra and Mukhopadhyay 2002). (a) and (b) Available data. (c) Solution assuming multi-bend folding model with MFT emergent. (d) Solution assuming multi-bend folding model with MFT blind. (e) Solution assuming multi-bend folding model with two synclinal bends on MFT, uniformly tapering layers, and 12% forelimb thinning. This section is in conformity with surface dip data.
158
159
Restoration (or Restoration (or check for viability) viability) of a cross section is the actual process of pullingback back cross sections in contractional terrains and pushing forward forward cross sections in extensional terrains. Restoration is done on the restoration template, which should incorporate changes in stratigraphic thickness, unconformities and any structures believed to have been present prior to the deformational event which will be restored. Sections are sometimes partially restored, restored, which means restoration to an earlier less deformed state. Regional sections should be fully restored. Except possibly for very simple structures, restored section should invariably accompany deformed-state section, if it is claimed that the section is balanced. There are several reasons for restoring a section: (1) It is the ultimate test of whether the deformed-state section is reasonable or reasonable or not. A cross section that does not restore is most likely a bad section. (2) The restored section gives us the original undeformed length and comparing it with the length in deformed-state cross section allows us to estimate total shortening (or extension). (3) The restoration process glaringly reveals commonly made mistakes and errors in section construction. Restoration is not merely measuring lengths (or areas) in deformed-state section and stretching them out, and somehow matching them. Sections must always be sequentially restored, i.e., one fault should be restored at a time . This requires that we must always take into account the sequence of faulting. The displacement on a younger fault should be removed first before the displacement on an older fault is removed. Also, out-of-sequence thrusts should be restored first followed by restoration of in sequence thrusts. In other word, restoration of faults should be carried out in an order reverse of the order in which they form. At every stage of restoration, all all the structures must essentially be admissible. If this not the case the section is not balanced even if the section can be restored to an undeformed state. Step-wise restoration has two 160
additional benefits: (1) it helps us understand how a fold-thrust belt evolves through space and relative time, and (2) burial and uplift history can be worked out that may be useful in understanding source-rock maturation and hydrocarbon migration and accumulation. The iterative process involved in restoring a section to its undeformed state and then changing the original section if it does not restore can be tedious, frustrating and very time consuming. Therefore, it is important to spend the maximum time and effort to correctly construct the first section using forward-modelled structural geometry. This is because even minor changes in one part of the section will invariably result in changing the remainder of the section. The ultimate objective is to construct a geometrically reasonable cross section within a limited amount of time. There are two methods of restoration: equal line-length restoration (also called sinuous bed method) and equal-area restoration. The methods are briefly discussed below.
16.1 Equal line-length restoration This is the most commonly applied restoration method. Line-length restoration is used when it is believed that there has been no stratigraphic thickness changes as a result of deformation, in any of the stratigraphic horizons included in the cross section. In other words, the line lengths (i.e., contacts between stratigraphic units in 2D) remain constant through the deformation. We can restore sections merely by stretching out the lines to return them to regional regional level and dip. If thicknesses do not change during deformation, areas will also remain constant. Consequently, we do not need to compare areas of beds or thrust sheets. If the folds are kink style, bed lengths can be measured with a divider or a good-quality ruler. If the folds are concentric, the measurements obviously become more involved. Before we start restoration a pin line and a loose line at either end of the section are to be established. The first step in the restoration is to match hangingwall cut-offs to respective footwall cut-offs, starting with the frontal thrust sheet if there is no out-ofsequence thrusting The measured bed lengths are then laid out as straight-line 161
segments. The ends of these straight-line segments then define the footwall trajectory of the second fault. This procedure is repeated successively for all the faults. Fig. 16.1 shows a fault-bend fold. For the purpose of restoration two reference lines are chosen. The pin line and the loose lines are located in the leading and trailing edges, respectively. Point A was at the upper bend (A') before deformation, so A is pulled back to this point. In so doing, we also pull back the entire rock package as well as the reference lines. Now keeping the pin line fixed, we straighten all the lines keeping the length of the lines constant. In Fig. 16.2, there are two faults in the deformed section; fault 1 is younger and fault 2 is older. We first restore fault 1 and then restore fault 2 following the same procedure as in Fig. 16.1. Note that the dip of fault 2 has changed in restored section. One surprise is that we find significant layerparallel simple shear in the restored section that is not obvious in the deformed-state section.
Deformed
Loose line line 3 1
Restored
Pin line line 1'
4
2
5
2'
3'
4'
5'
6'
6
A
A'
he line-length method of section restoration involving one fault. Figure 16.1.T Deformed
Restored Pin line
Loose line L2
L1
Shear
1
2 θ2
θ1
L1
L2 2
ψ
θ3
1 θ1
he line-length method of section restoration involving two faults. Figure 16.2.T
16.2 16.2 Equal Equal area restoratio resto ratio n Area restoration is used in regions where it is believed that changes in bedding thickness have occurred as result of deformation. The changes in bedding thickness 162
must be the result of plane strain within the cross sectional plane in order for the area balancing to be valid. The changes in bedding thickness must not be result of material moving in and out the plane of section during deformation. Under this condition, if there has been no overall volume change then the cross-sectional area of any rock unit shown in the deformed-state cross section has not changed during deformation. However, line-lengths (e.g., distances between given thrust faults as measured along specific bedding planes) may have changed. In such a situation, areas of rocks and not line lengths are measured for the purpose of restoration.
3 4
2 1
X
A = 3.2 sq. km
Y
X'
1'
2'
3'
4'
Y'
A = 2.0 sq. km
A = 3.2 sq. km
Figure 16.3. Equal-area method of restoration (adapted after Marshak and Woodward, 1988).
Equal-area restoration involves measurement of area (A) of the deformed thrust sheet and the original thickness (t) of the unit. Assuming plane strain, the restored average length (L) is given by A/t. Fig. 16.3 shows general methodology of area restoration. The shaded unit in the rather unusual-looking ramp anticline has undergone extreme thickening and has an area of 3.2 km 2. We take the thickness at the trailing edge as the original thickness. The line-length restoration of the shaded unit results in an area of 2.0 km 2, which is about 38% too small. In order to keep the area constant, the undeformed length of the unit must be increased as shown in the lower diagram.
163
16.3 The key-bed method Mitra and Namson (1989) pointed out that area restoration does not ensure that the units are balanced or the fault trajectories in the restored section are geologically reasonable. This is because only one average length is calculated for each unit and the top and bottom of the unit are assumed to be equal to the average length. This makes the undeformed sheet to be a parallelogram. Key bed X'
A2
A1
L2k
L1k
Y'
(a) W'
Deformed Def ormed section sec tion
Z' Y
X A2
A1 L2a
L1a
(b) W
Area restoration r estoration X
Z L1k
Y*
L2k
A1
A2 L2a
L1a
(c)
Ld L2
Area and a nd key-bed key- bed restoration restor ation W
Z*
L1
Figure 16.4. Combined equal-area and key-bed method of section restoration (after Mitra and Namson 1989).
This method is illustrated in Fig. 16.4. The deformed-state section is shown in Fig. 16.4a. Equal-area restoration (Fig. 16.4b) results in two parallelogram shaped restored thrust sheets. The orientation of the thrust YZ and the right-side reference line can be varied considerably maintaining a constant area, as shown by dashed lines. The equalarea restoration method can be improved by combining it with key-bed method of 164
restoring individual bed lengths. The key-bed method identifies a thin competent key unit that undergoes minimum penetrative deformation and area change so that it can be restored using line-length method. The top layer is assumed to be key bed whose length remained unchanged. Fig. 16.3c shows combined key-bed and equal-area restoration. Note the back shear in restored diagram which is not obvious in the deformed section.
16.4 EVAL UATING AND IMPROVING A CROSS SECTION Restoration of a cross section allows us to evaluate whether our interpretation, i.e., the deformed-state cross section, is reasonable reasonable or not. In the restored section, trajectories of all the faults must be admissible admissible in the sense that they should be gently to moderately dipping towards hinterland (except back thrusts). Also they should not have zigzag trajectories, except for out-of-sequence thrusts. The line length and/or area should be equal between deformed and restored; or if they are not then there should be proper explanation. We should restore one fault at a time, starting from the youngest fault (Mukhopadhyay and Mishra 1999, 2005). At each stage of restoration, the restored section must be admissible. If a section does not lead to an admissible restored section, then we would need to suitably change the deformed-state section until an acceptable restoration is obtained. The restoration process itself can potentially locate mistakes and problem areas that may need modification.
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Suggested Suggested reading reading The following textbooks have influenced my thinking on structural geology over the years. They always have pride of place on my desk. One should not be surprised if treatment of any of the topics here is similar to that in any of these books. Davis, G. H. (1984). Structural geology of rocks and regions. John Wiley & Sons, New York. Hobbs, B. E., Means, W. D. and Williams, P. F. (1976). An outline of structural geology. John Wiley & Sons, New York. Means, W. D. (1976). Stress and strain. Springer-Verlag, New York. Ramsay, J. G. (1967). Folding and fracturing of rocks. McGraw-Hill, New York. Ramsay, J. G. and Huber, M. I. (1983). The techniques of modern structural geology. Vol. 1: Strain analysis. Academic Press, London. Ramsay, J. G. and Huber, M. I. (1987). The techniques of modern structural geology. Vol. 2: Folds and fractures. Academic Press, London. Suppe, J. (1985). Principles of structural geology. Printice-Hall, New jersey. Twiss, R. J. and Moores, E. M. (1992). Structural geology. W. H. Freeman and Co., New York
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