Physics for Geologists, Second edition

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98 Stress and strain Figure 9.7 Mohr's circle. The stress to the left of the t-axis is tensional, that to the right, compressional. The diameter of the circle is the difference between the greatest and the least principal stresses. The line tangential to this circle is Coulomb's criterion, the slope of which is +. This angle is also the angle of internal friction of the material. that the shear stress is maximum when 2a = 190" and CL = +45" and tmax = :(al - a3). But the material may fail with the shear stress less than the maximum. We return to the Coulomb criterion for sliding (page 93), t = to + a, tan 4. If the properties of the material are known, this equation can be plotted on the Mohr diagram. (The Coulomb criterion is not necessarily linear, particularly at small stresses, but it is a good approximation for many materials.) The conditions at fracture are given when the Coulomb criterion is tangential to Mohr's circle. The slopes of the criterion are @ and -4, so the plane of fracture makes an angle of 45" f $12 with the axis of least principal stress, 03. It is because the internal angle of friction, 4, is usually close to 30" in rocks, and because the principal directions tend to be vertical and horizontal, that faults are commonly inclined at about 30" or 60" to the vertical for normal and thrust faults, respectively. Compaction, consolidation, lithification Compaction is the reduction of the bulk volume of a sediment or rock by the reduction of pore space in it. Consolidation is any process that changes loose sediment to a coherent rock (or magma or lava to a solid rock, but we are more concerned with sediment here). It embraces compaction of sediment but includes the addition of solid substances to what was pore space - cementation. Compaction is essentially gravitational: consolidation includes the deposition of cements. Lithification is a synonym for consolidation. It is reasonable to suppose that compaction, the loss of porosity due to compression of a particulate medium, is a function of porosity (f) or bulk density, and that the rate of loss of porosity with depth (2) is proportional to porosity, that is, Copyright 2002 by Richard E. Chapman
Stress and strain 99 from which we obtain by integration (remembering that the boundary conditions are that the sedimentary rock had an initial porosity, and that the porosity cannot be reduced below zero), The quantity b is called the scale length [L] (it is more convenient to divide by a large number than to multiply by a small one), and fo is the porosity at zero depth (which should be interpreted as the depth at which the sediment accumulated into the stratigraphic record). The value of b can be obtained from the data by solving Equation 9.17 for b. Equally, it is the depth at which the porosity of normally compacted mudrock reaches the value fo/e (or 0.368 fO). The value of fo is usually about 0.5, so b is usually about the depth at which about 18 per cent porosity is achieved in normal compaction. This equation is found to be a good summary of the compaction of mudrocks, with scale lengths varying from about 500 m to 4 km. When the scale length is large, the curve approaches a straight line, and it could be that sandstone compaction, which appears to be linear with depth, follows the same laws with a very large scale length. Bulk density (pb) is related to porosity by where pf and ps refer to the mean mass densities of the pore fluid and of the solids. Let us digress briefly to a practical application. In boreholes, it is much easier to measure electric or acoustic properties of rocks than it is to measure their porosity, but we are often more interested in the porosity. We are therefore concerned with the search for formulae relating one property with another. What then is the relationship between mudrock porosity and the sonic velocity, or its inverse, called the shale transit time (symbol Atsh) [L-IT, inverse velocity] in mudrock? The boundary conditions are that there will be some maximum value of porosity, fo, when the mudrock first accumulates into the stratigraphic record. This is usually close to 50 per cent. (Note that we are not concerned with the superficial porosities of 70 per cent and more in mud that has yet to accumulate into the stratigraphic record.) And the minimum porosity at depth will be close to zero. There will be a corresponding transit time, Ato, near the surface, and a limiting value of the transit time corresponding to zero porosity. This latter is called the matrix transit time, At,,,,;, or At,, . When seeking a relationship between two quantities of different dimensions, in this case L3/L3 and L-'T, it may be assumed that a dimensionless form will be required, and that the dimensionless porosity, f/fo, will be related to a dimensionless transit time involving Atsh, At0 and At,,. The boundary conditions are that when f = fo, Atsh = Ato; and when Copyright 2002 by Richard E. Chapman
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Physics for Geologists Second editi
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Copyright 2002 by Richard E. Chapma
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... vlii Contents Polarization 47 L
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Figures Copyright 2002 by Richard E
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Tables Copyright 2002 by Richard E.
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xiv Preface waves diminishes with d
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xvi Preface provides. This book is
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xviii Acknowledgements to K. Whaler
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xx Symbols mass refractive index bu
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2 Basic concepts Table 1.1 Dimensio
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4 Basic concepts Table 1.2 Basic SI
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6 Basic concepts pascal (Pa) and si
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8 Basic concepts all the relevant d
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10 Basic concepts relating them by
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2 Force The dimensions of force are
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14 Force The ratio of the Moon's ac
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16 Force Figure 2.1 The sum of the
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18 Force When a motorcycle takes a
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20 Force Figure 2.3 Torque is the p
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22 Force How does momentum differ f
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24 Force the weights - that is, the
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26 Force Figure 2.6 Wall-sticking i
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3 Gravity Universal gravity When di
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30 Gravity are significant to geolo
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32 Gravity Figure 3.2 The tractive,
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34 Gravity Mean sea level It might
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36 Gravity The benefits of space fl
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38 Gravity Figure 3.6 Profile of th
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40 Gravity Figure 3.8 The siphon at
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4 Optics Light is an electromagneti
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44 Optics Figure 4.2 Refraction. (a
Page 63 and 64: 46 Optics The coeficient of reflect
Page 65 and 66: 48 Optics only the light oscillatin
Page 67 and 68: 50 Optics called Iceland Spar. Two
Page 69 and 70: 52 Optics Diffraction If you shine
Page 71 and 72: 54 Optics Figure 4.7 Geometrical op
Page 73 and 74: 56 Optics Figure 4.9 The mirrors in
Page 75 and 76: 5 Atomic structure and age-dating P
Page 77 and 78: 60 Atomic structure and age-dating
Page 83 and 84: 66 Electromagnetic radiation radiat
Page 85 and 86: 68 Electromagnetic radiation Source
Page 87 and 88: Heat and heat flow The noun heat ha
Page 89 and 90: 72 Heat and heat flow Second Law: A
Page 91 and 92: 8 Electricity and magnetism Electri
Page 93 and 94: Electricity and magnetism 77 Figure
Page 95 and 96: Electricity and magnetism 79 rivett
Page 97 and 98: Electricity and magnetism 8 1 defin
Page 99 and 100: Electricity and magnetism 83 is kno
Page 101 and 102: 9 Stress and strain Pressure, p, is
Page 103 and 104: Stress and strain 87 than some natu
Page 105 and 106: These three elastic constants are r
Page 107 and 108: Stress and strain 91 Figure 9.1 New
Page 109 and 110: Stress and strain 93 Figure 9.2 Com
Page 111 and 112: Stress and strain 95 Figure 9.3 Ide
Page 113: Fracture Stress and strain 97 We fo
Page 117 and 118: 10 Sea waves The nature of water wa
Page 119 and 120: Sea waves 103 Figure 10.1 Wave moti
Page 121 and 122: Sea waves 105 refraction by islands
Page 123 and 124: Acoustics: sound and other waves 10
Page 125 and 126: Acoustics: sound and other waves 10
Page 127 and 128: 12 Fluids and fluid flow Introducti
Page 129 and 130: Fluids and fluid flow 1 13 Figure 1
Page 131 and 132: Fluids and fluid flow 115 Figure 22
Page 133 and 134: Fluids and fluid flow 117 and the f
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
Page 139 and 140: Fluids and fluid flow 123 ardentes,
Page 141 and 142: Fluids and fluid flow 125 Figure 12
Page 143 and 144: - 14 $ 12 E 10 K .- 3 8 r cll + 6 0
Page 145 and 146: Fluids and fluid flow 129 the liqui
Page 147 and 148: Some dangers of mathematical statis
Page 155 and 156: Appendix 139 What would happen if y
Page 157 and 158: Notes 141 and since x is very large
Page 159 and 160: References Allum, J. A. E. (1966) P
Page 161 and 162: References 145 -(1950) 'Mechanism o
Page 163: Further reading 147 Trigg, G. L. an
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Physics for Geologists, Second edition

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