Physics for Geologists, Second edition

More documents

Recommendations

Info

1 18 Fluids and fluid flow that the total weight is less than the sum of the weights put into the container, unless the container was full of water to the brim before we put in the sand. In that case, the weight of water lost is equal to the weight of water displaced and this has to be taken into account. If we divide Equation 12.2a by the area of the base, we get This suggests that the pressure exerted by the solids is only acting on the proportion of area occupied by solids, (1 - f), and that the water is only acting on the proportion f. Karl Terzaghi (as we saw in Chapter 9) examined the question of buoyancy in concrete dams. He postulated that buoyancy only acts on the solid surfaces exposed to water in the pore spaces, but he found that the buoyant force was very close to yv(1- f), and that therefore the force of buoyancy acts on the total area (1 - f), irrespective of the areas of contact between grains. He later found the same results in clays. Consequently, he stated that the total stress in porous solids is divided between the effective stress, a, transmitted through solids and the pore-fluid pressure (he called it the neutral stress). To be a bit more precise, where S is the vertical component of total stress. This is known as Terzaghi's relationship. It is now known not to be exact, but it is sufficiently close for most geological purposes. (One cannot avoid a clash of symbols: this a is not the surface tension.) Reverting now to Equation 12.2a we see that that is but one possible partition. Substituting into Equation 12.3 the expressions for S and p, we can write The effective stress is due to the partial density of the solids less the water they displace, and it is this partition of total stress that agrees with experimental results (see Hubbert and Rubey 1959: 139-142). The result is important because it is the effective stress that leads to mechanical compaction of sediments and sedimentary rocks under gravity. Terzaghi called the water pressure the neutral stress because it has no direct r6le in the deformation of the grains. Another important result is that the depth of water over the top of the sand makes no difference to the effective stress. Imagine that there is a depth 1 of Copyright 2002 by Richard E. Chapman
water above a vertical thickness h of water-saturated sand: Fluids and fluid flow 1 19 The depth of water does, of course, contribute to the total stress, S. Reynolds numbers Osborne Reynolds (1842-1912) carried out some beautiful experiments in the 1880s on the flow of water in pipes, and was interested in the conditions that separated the stable from the unstable, the laminar from the turbulent. He found that it was a function of the internal diameter of the pipe and the velocity and viscosity of the water in it, but argued that since units are arbi- trary, there must be a dimensionless relationship. Noting that the dimensions of kinematic viscosity (v = qlp) are those of a length multiplied by a velocity, he took as his dimensionless number dV/v, where d is the diameter of the pipe and V is the velocity of the liquid (volume per unit area normal to flow, L~T-' LP2 = LT-l). This is now called a Reynolds number. It con- sists of a characteristic length multiplied by a characteristic velocity divided by the kinematic viscosity of the fluid. It has the indefinite article because there are different ways of defining the components. (Reynolds actually took the inverse of this but it was changed to the form given, presumably to give a large number rather than a small fraction, and that is how it is now defined.) In pipe and channel flow, the characteristic length is the hydraulic radius (which we shall come to shortly). What should it be in porous rocks? It is common practice to take the mean grain size, but this is a very indirect quan- tity because it is precisely the part of the rock that fluids do not flow through. We shall return to this problem when we consider water flow through porous rocks. Hydraulic radius With pipe flow, there is no difficulty in measuring the radius. But what if the pipe were square or elliptical? or if the pipe were not quite full? It has been found that the unifying parameter is obtained by taking the volume of water and dividing it by its wetted surface (if the dimensions remain constant, you can take the cross-sectional area divided by the wetted perimeter). This is called the hydraulic radius, and it has the dimension of length. The hydraulic radius of a pipe of circular cross-section and constant internal diameter d is If we never used anything but pipes that were full, this would not be neces- sary. But pipes are not always full, and water can flow in channels as well as pipes. Copyright 2002 by Richard E. Chapman
Page 1 and 2:
Physics for Geologists Second editi
Page 3 and 4:
Copyright 2002 by Richard E. Chapma
Page 5 and 6:
... vlii Contents Polarization 47 L
Page 7 and 8:
Figures Copyright 2002 by Richard E
Page 9 and 10:
Tables Copyright 2002 by Richard E.
Page 11 and 12:
xiv Preface waves diminishes with d
Page 13 and 14:
xvi Preface provides. This book is
Page 15 and 16:
xviii Acknowledgements to K. Whaler
Page 17 and 18:
xx Symbols mass refractive index bu
Page 19 and 20:
2 Basic concepts Table 1.1 Dimensio
Page 21 and 22:
4 Basic concepts Table 1.2 Basic SI
Page 23 and 24:
6 Basic concepts pascal (Pa) and si
Page 25 and 26:
8 Basic concepts all the relevant d
Page 27 and 28:
10 Basic concepts relating them by
Page 29 and 30:
2 Force The dimensions of force are
Page 31 and 32:
14 Force The ratio of the Moon's ac
Page 33 and 34:
16 Force Figure 2.1 The sum of the
Page 35 and 36:
18 Force When a motorcycle takes a
Page 37 and 38:
20 Force Figure 2.3 Torque is the p
Page 39 and 40:
22 Force How does momentum differ f
Page 41 and 42:
24 Force the weights - that is, the
Page 43 and 44:
26 Force Figure 2.6 Wall-sticking i
Page 45 and 46:
3 Gravity Universal gravity When di
Page 47 and 48:
30 Gravity are significant to geolo
Page 49 and 50:
32 Gravity Figure 3.2 The tractive,
Page 51 and 52:
34 Gravity Mean sea level It might
Page 53 and 54:
36 Gravity The benefits of space fl
Page 55 and 56:
38 Gravity Figure 3.6 Profile of th
Page 57 and 58:
40 Gravity Figure 3.8 The siphon at
Page 59 and 60:
4 Optics Light is an electromagneti
Page 61 and 62:
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 79 and 80:
Page 81 and 82:
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 and 114: Fracture Stress and strain 97 We fo
Page 115 and 116: Stress and strain 99 from which we
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: Fluids and fluid flow 117 and the f
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
show all

Physics for Geologists, Second edition

Create successful ePaper yourself

Delete template?

Save as template?