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Geology 228/278
Applied and Environmental
Geophysics
Lecture 3
Physical properties of earth materials
in near-surface environment
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Outline
1. Introduction
2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability and susceptibility5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical model
Analytic mix modelEmpirical mix model
Archie's law and Waxman-Smits relationship
CRIM model
7. Note on effective materials
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Introduction
People live on the surface of the earth, standing on rock and soil,inside a bubble of gas, growing food in and from the fluid and solid
constituents, and exploiting natural resources like minerals, water
and petroleum. How well the occurrence and behavior of the
physical and chemical properties and processes in rocks, soils and
fluids are understood determines how well
buildings and dams are supported by their foundations (civil
engineering);
food is grown (agriculture);
resources are developed (petroleum, mining and
hydrogeological engineering);
the environment is protected (waste management and
environmental remediation); and
energy or data are transmitted (power, electrical engineering
and telecommunications).
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Petrophysics is the study of the
physical and chemical propertiesthat describe the occurrence and
behavior of rocks, soils and fluids.
This course concerns the
PHYSICAL properties.
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Outline
1. Introduction
2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability and susceptibility5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical model
Analytic mix modelEmpirical mix model
Archie's law and Waxman-Smits relationship
CRIM model
7. Note on effective materials
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Include the density and
the elastic properties of the earth materials
These material properties are described by elastic modulii.
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Youngs modulus E
Youngs modulus is the stress needed to compress the solid
to shorten in a unit strain.
Poissons ration
Poissons measures the relativity of the expansion in the
lateral directions and compression in the direction in whichthe uni-axial compression applies.
xx
AFE
/
/
=
xx
yy
/
/
=
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Bulk Modulus K
Imagine you have a small cube of the material making up
the medium and that you subject this cube to pressure by
squeezing it on all sides. If the material is not very stiff, you
can image that it would be possible to squeeze the materialin this cube into a smaller cube. The bulk modulus
describes the ratio of the pressure applied to the cube to the
amount of volume change that the cube undergoes. If k is
very large, then the material is very stiff, meaning that itdoesn't compress very much even under large pressures. If
K is small, then a small pressure can compress the material
by large amounts. For example, gases have very small Bulk
Modulus . Solids and liquids have large Bulk Modulus.
vv
AFK
/
/
=
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Shear Modulus
The shear modulus describes how difficult it is to deform a
cube of the material under an applied shearing force. For
example, imagine you have a cube of material firmly
cemented to a table top. Now, push on one of the top edgesof the material parallel to the table top. If the material has a
small shear modulus, you will be able to deform the cube in
the direction you are pushing it so that the cube will take on
the shape of a parallelogram. If the material has a largeshear modulus, it will take a large force applied in this
direction to deform the cube. Gases and fluids can not
support shear forces. That is, they have shear modulii of
zero. From the equations given above, notice that thisimplies that fluids and gases do not allow the propagation of
S waves.
xy
AF
/
/
=
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Youngs modulus E
Youngs modulus is the stress needed to
compress the solid to shorten in a unit
strain.
Poissons ration
Poissons measures the relativity of the
expansion in the lateral directions and
compression in the direction in which the
uni-axial compression applies.
zzE
/
1
=
zzrr
//
=
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Shear Modulus (cont.)
yx
AF
/
/
=
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Seismic Velocities related to material properties
Vp- P-wave (compressive wave) velocityVs- S-wave (shear wave) velocity
So, seismic velocities are determined by the mechanic properties of the
materials in which the seismic waves propagate through.
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Seismic velocity vs materials mechanic properties
Any change in rock or soil property that causes , , or K to change
will cause seismic wave speed to change. For example, going froman unsaturated soil to a saturated soil will cause both the density and
the bulk modulus to change. The bulk modulus changes because air-
filled pores become filled with water. Water is much more difficult to
compress than air. In fact, bulk modulus changes dominate this
example. Thus, the P wave velocity changes a lot across water table
while S wave velocities change very little.
Although this is a single example of how seismic velocities can
change in the subsurface, you can imagine many other factorscausing changes in velocity (such as changes in lithology, changes in
cementation, changes in fluid content, changes in compaction, etc.).
Thus, variations in seismic velocities offer the potential of being able
to map many different subsurface features.
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From: Sheriff and Geldart, Exploration Seismology, p69.
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Property Units Iron Unsaturated Sand Saturated SandP-wave velocity km/s 5.92 4.18 2.73
S-wave velocity km/s 3.23 3.42 1.37
Vp/Vs 1.83 1.22 1.99
Porosity - 0.36 0.36
Dielectric Permittivity 221 6.25 25
Magnetic Permeability 17.834 1.0 1.0Resistivity ohm-m 9E-08 1E+04 1E+02
Bulk Modulus GPa 100.2 37
Shear Modulus GPa 95.2 44
Poisson's Ratio () 0.14 0.08
Young's Modulus N/m
2
6.74Density g/cm3 22.564 2.65 3.01
Values From:
Carmichael, Robert S.. 1989. Practical handbook of physical properties of rocks and minerals.
Mavko, G., and others. 1998. The rock physics handbook: tools for seismic analysis in porous
media.
Schon, J.H.. 1996. Physical properties of rocks: fundamentals and principles of petrophysics
Calculated from field data at Otis MMR, Cape Cod, Massachusetts
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Seismic Refraction Results
Profile Parallel to the Tennis Courts
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Outline
1. Introduction2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability and susceptibility5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical model
Analytic mix modelEmpirical mix model
Archie's law and Waxman-Smits relationship
CRIM model
7. Note on effective materials
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The electric conductivity of earth materials
The electric property of materials is described by
electric conductivity or electric resistivity.
Conductor: > 105 S/m;
Semi-conductor: 10-8 < < 105 S/m;
Insulator: < 10-8 S/m;
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Electric Resistivity
Ohms Law:
RIV=
where V-voltage, I-current, and R-resistance. The Resistance isproportional to the length of 2 points, and inversely proportional to the area
of the cross-section on which the current flow through. The proportional
coefficient, , is the resistivity, a material property to describe the capabilityto resist the electric current flow.
A
LR =
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Ohms Law (discovered in 1827)
IRV= Georg Simon Ohm(1787-1854)
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It's Resistivity, NOT Resistance
L
RAA
LR
=
=
So the unit for resistivity is ohm-meter
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Resistivity of Earth MaterialsAlthough some native metals and graphite conduct
electricity, most rock-forming minerals are electrical
insulators. Measured resistivities in Earth materials are
primarily controlled by the movement of charged ions inpore fluids. Although water itself is not a good conductor
of electricity, ground water generally contains dissolved
compounds that greatly enhance its ability to conductelectricity. Hence, porosity and fluid saturation tend to
dominate electrical resistivity measurements. In addition
to pores, fractures within crystalline rock can lead to lowresistivities if they are filled with fluids.
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The resistivities of various earth materials are shown below.
Material Resistivity (Ohm-meter)
Air
Pyrite 3 x 10^-1
Galena 2 x 10^-3
Quartz 4 x 10^10 - 2 x 10^14
Calcite 1 x 10^12 - 1 x 10^13
Rock Salt 30 - 1 x 10^13
Mica 9 x 10^12 - 1 x 10^14
Granite 100 - 1 x 10^6Gabbro 1 x 10^3 - 1 x 10^6
Basalt 10 - 1 x 10^7
Limestones 50 - 1 x 10^7
Sandstones 1 - 1 x 10^8Shales 20 - 2 x 10^3
Dolomite 100 - 10,000
Sand 1 - 1,000
Clay 1 - 100
Ground Water 0.5 - 300
Sea Water 0.2
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Electric Conductivity
Electric conductivity is the reciprocity of
the electric resistivity :
/1=
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Outline
1. Introduction2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability andsusceptibility
5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical modelAnalytic mix modelEmpirical mix model
Archie's law and Waxman-Smits relationship
CRIM model7. Note on effective materials
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Magnetic Permeability
The magnetic constitutive relation:
HHHB )1(00 +=== rwhereB - magnetic flux density
H Magnetic field
- Magnetic Permeability
0 magnetic permeability in vacuum
r relative magnetic permeability magnetic susceptibility
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HHHHMHB r 000000 )1( =+=+=+=
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Magnetic Susceptibili ty of rocks, minerals and iron steel
more rocks have a wide range: 1 ppm to 0.001; Magnetite ore can be as high as 150;
Some minerals are diamagnetic (negative ;
Iron, steel have the values of 10 -100.
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The dielectric properties of a material are defined by an
electrical permittivity, . The permittivity is dependentupon a materials abil ity to neutralize part of an static
electrical field. For this, a dielectric material must
contain localized charge that can be displacedby the application of a electric field (and in doing store
part of the applied field). This charge displacement is
referred to as polarization. Such a charge displacementis time dependent in most materials so that a complex
permittivity is required to adequately describe the
system, * = + i . Since the polarization mechanisms
that occur in these materials depend on frequency,temperature, and composition so will this complex
permittivity.
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Dielectric Permittivity
The dielectric constitutive relation:
EED r 0==
where
D electric displacement density
E electric field
0 electric permittivity in vacuum
r relative electric permittivity
electric permittivity
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Mechanisms involved in Dielectric Polarization include:
Electron polarization;Atomic polarization;
Molecular polarization;
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Index of refraction (n) and dielectric constant r
rr norn === ,/ 2
0
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Value of the complex dielectric constant
"' i+=
is the parameter responsible for the observed
phenomena in dielectric polarization
Loss tangent
= /tan
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There are two more microsopiceffects that cause ground to be
chargeable
1)Membrane polarization
2)Electrode polarization
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Membrane polarization
Membrane polarization occurs when
pore space narrows to within several
boundary layer thicknesses.
Charges accumulate when an electric
field is applied.
Result is a net charge dipole which
adds to any voltage measured at the
surface.
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Electrode polarization
Electrode polarization occurs when pore
space is blocked by metallic particles.Again charges accumulate when an
electric field is applied.
The result is two electrical double layers
which add to the voltage measured at
the surface.
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There is a clear maximum in the dielectric loss for water ata frequency of approximately 20GHz, the same point at
which the dielectric constant ' goes through a point of
inflexion as it decreases with increasing frequency.The 2.45GHz operating frequency of domestic ovens is
selected to be some way from this maximum in order to
limit the efficiency of the absorption.Too efficient absorption by the outer layers would
inevitably lead to poor heating of the internal volume in
large samples.
In his theoretical expressions for ' and " in terms of other
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In his theoretical expressions for and in terms of othermaterial properties, formed the basis for our current
understanding of dielectrics. The dielectric constants, ' and" are dependent on both frequency and temperature, thefirst of which is expressed explicitly in the Debye equations
whilst temperature is introduced indirectly through othervariables:
)1(
)(
)1(
)(
22
22
+
=
+
+=
s
s
where
and s are the dielectric constants under
high frequency and static fields respectively.
Since infra red frequencies are often regarded as infinite for
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Since infra-red frequencies are often regarded as infinite for
most purposes, results from atomic and electronicpolarizations, whilst s results from the sum of all the polarization
mechanisms described in a later section. The relaxation time, ,was derived by Debye from Stoke's theorem:
kT
r34
=
where r is the molecular radius, the viscosity, kBoltzman's constant, and T the temperature. If the Debye
equations are plotted against wt with arbitrary values for
and s as shown in the last Figure, then the similarityof these expressions to the experimental values shown in
the next Figure is clear.
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Debye expressions for ' and " calculated as a function of [].
Outline
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1. Introduction2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability and susceptibility5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical modelAnalytic mix model
Empirical mix modelArchie's law and Waxman-Smits relationship
CRIM model
7. Note on effective materials
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Table 1. Representative physical properties of basic constituents and composites of soil
Material Porosity
(%)
Water
Saturation
(%)
Dielectric
Constant
Electrical
Conductivity
(mS/m)
EM
Velocity
(m/ns)
Attenuation
(Np/m)
Skin
depth
(m)Air - - 1 0 0.300 0
Water - - 81 1 0.033 0.021 47.7
Dry Sand 30 0 4 0.1 0.150 0.009 106
Wet Sand 30 100 17.225 21.310 0.0720.060 0.970.38 1.02.6
Dry Clay 30 0 4 10 0.150 0.94 1.1
Wet Clay 30 100 17.7
16
31.3
100
0.071
0.075
1.40
4.71
0.7
0.2
Average Soil 30 - 16 20 0.075 0.94 1.1
Liu and Li: J. Appl. Geophys., 2001.
Table 1. Electromagnetic properties of some earth and engineered materials
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Material conductivity
(miliS/m)
dielectric
constant
r
dielectric
permittivit
y (picoF/m)
electromagnetic
wave velocity
v
(m/s)
skin
depth
(m)
transition
frequency
t(MHz)
reference
fresh water 12-50 81 735 33.3 95.1-22.8 16-68 Brewster & Annan (1994)
salt water 150 81 716 33.3 7.6 209 Daily, et al (1995)
freshwater ice 3.17 168.5 Arcone (1984)
air 2.5x10-14 1.0 8.85 300.0 - 0.28x10-11 Balanis (1989)
clay (dry) 1-10 10 88.5 94.9 141-14.1 11-113 Telford et al (1990)
clay (saturated) 100-1,000 7 62.0 113.4 0.98-0.1 161-1614 Ulrikesen (1982)
sand (dry) 0.001 4.5 39.8 141.4 63,412 0.25x10-1 Patel (1993)
sand (saturated) 0.1 30 266 54.8 4,227 0.38 Ulrikesen (1982)
dry concrete 5.6 49.6 126.8 Matthews et al (1998)
dry soil 4 3.9 34.5 151.9 13.7 116 Wakita et al (1996)
wet soil (20%) 13 14.4 127.4 79.0 15.6 102 Wakita et al (1996)
granite (dry) 1 x10-5 5 44.2 134.2 7x106 0.23x10-3 Ulrikesen (1982)
granite (wet) 1 x10-1 7 62 113.4 7,045 1.6 Ulrikesen (1982)
Texas aggregates 0.0012 5.1 45.1 132.8 59,889 0.27x10-1 Saarenketo at al (1996)
asphalt 6.8 60.2 115.0 Hugenschmidt et al (1996)
PCE 5.6x10-9 2.3 20.4 197.8 5.8x109 0.27x10-6 Brewster & Annan (1994)
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Schematic representation of soil matrix indicating
relationship between air (A), soil particles (B) and water (C).
Parallel Plate
Capacitors
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E E
DielectricPlates
Dielectric plates arranged a) parallel and b) perpendicular to theelectrodes. The analytical mix model are:
2
2
1
11
+
=
2211 +=
parallel model series model
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There are other theoretical models appears workquite well for sediments filled with water, one
popular one is the complex refraction index model
(CRIM):
=
=
=++=
=++=
n
iii
n
i
ii ornnnn
12211
1
2211
...
...
The Complex Refraction Index Model CRIM)
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The wavelength of the signal is muchlarger than the typical size of the
heterogeneity (pore size)
Contains two of a few pore materials(air, ice, water, and possible others),
and the solid matrix
0=1, ice = 3.6, wat = 81,
asph = 2.6-2.8, aggreg = 5.5-6.5
))1()1( awgb SS ++=
Archies Law (for formation
without or l ittle clay content)
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without or l ittle clay content)
Archie's Law (Archie, 1942) describes the relationship
between electrical resistivity and porosity, fluid saturation,
and fluid type in a rock. The injection of current and
measurement of voltage can result in determination of
porosity, saturation and fluid type. However, the
geometric factor and parameters in Archie's Law have
many of built in assumptions. These includeconsiderations of the rugosity of the borehole wall,
properties of the drilling mud, invasion of the mud into the
formation, morphology of the porosity, connectivity of thepores, wettability of the rock, presence or absence of clay
minerals, and more. Depending upon the choices made
about these assumptions, different interpretations result
for porosity, saturation and fluid type.
Archies law
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Archie s law
w
nm
Sa
=effective formation resistivity;
wpore water resistivity; porosity;S saturation;
a 0.5-2.5;m 1.3-2.5;
n ~2.
Maxwell-Smits relationship (empirical for shaly sand)
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p ( p y )
)(
1
vw BQF += effective formation conductivity;
wpore water conductivity; constant coefficient;F Formation factor;
Qv Cation exchange capacity;
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1. Electrical conductivity and hydraulic conductivityFrom Ohms law
dLdVA
RVI ==
From Darcys law
dLdHkAQ=
Outline
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1. Introduction2. Mechanical properties
3. electrical properties: electric conductivity
4. Magnetic properties: permeability and susceptibility5. Dielectric polarization: dielectric permittivity
6. Mix model: analytic model and empirical modelAnalytic mix model
Empirical mix modelArchie's law and Waxman-Smits relationship
CRIM model
7. Note on effective materials
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Property Units Iron Unsaturated Sand Saturated Sand
P-wave velocity km/s 5.92 4.18 2.73S-wave velocity km/s 3.23 3.42 1.37
Vp/Vs 1.83 1.22 1.99
Porosity - 0.36 0.36
Dielectric Permittivity 221 6.25 25
Magnetic Permeability 17.834 1.0 1.0Resistivity ohm-m 9E-08 1E+04 1E+02
Bulk Modulus GPa 100.2 37
Shear Modulus GPa 95.2 44
Poisson's Ratio () 0.14 0.08Young's Modulus N/m2 6.74
Density g/cm3 22.564 2.65 3.01
Values From:
Carmichael, Robert S.. 1989. Practical handbook of physical properties of rocks and minerals.
Mavko, G., and others. 1998. The rock physics handbook: tools for seismic analysis in porousmedia.
Schon, J.H.. 1996. Physical properties of rocks: fundamentals and principles of petrophysics
Calculated from field data at Otis MMR, Cape Cod, Massachusetts
The effective medium theory
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The effective medium theory
(wavelength >> size of heterogeneity)
EvEM=
2
2
1
1
111
Ed
d
Ed
d
E += 2211 d
d
d
d+=
The ray theory (wavelength ~ size of heterogeneity)
2
2
1
1 111
vd
d
vd
d
vRT +=
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Elastic property and seismic velocity of porous media effective medium theory
As long as the sizes of the pores, or the grains, or any other
significant heterogeneities associated with the pores, are much
smaller than the wave length of the seismic waves, or any othermeans to detect the changes in elastic properties, we can use the
effective medium theory to get the overall mixed, or bulk, property of
the porous media consisting of solid matrix and pore fluids.
If the means to measure the material property has a resolution close
to the size of the heterogeneity, we need to adapt the corresponding
assumption. In using the seismic wave methods again, it is the ray
theory. The following compares the differences.
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TABLE 1. Material Properties
Material
Density
(kg/m3)
Dynamic
Modulus
(Pa)
P-velocity
(m/sec)
Steel 7.9 2.4 x 1011 5512
Concrete 2.4 3.5 x 1010 3819
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References
Mavko, G, T. Mukerji, and J. Dvorkin, The Rock Physics Handbook,
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Cambridge University Press, 1998.
Knight, Ann. Rev. Earth Planet. Sci., 29:229-255, 2001.
Topp, Davis, and Annan, Water Resource Res. 16(3):574-582, 1980.
Debye. P. Phys. Zs. 36, 100, 1935.
Homework:
1, what is the seismic S-wave velocity in the near surface earth given:Density = 2500 kg/(mmm), the shear modulus = 10^10 Pa.
2, if the Poissons ratio is 0.25 (this is known as the Poisson condition
which can be a nominal value for the Poissons ratio of earth materials),
what is the P-wave velocity in the same material as in Question 1 (check
the relations of elastic parameters in the table).3, for water the relative dielectric constant is 81, what is the velocity of
radar wave in water? How many time of this value is slower than that in
the air?
4, for a soil sample the resistivity is 100 ohm-meter, what is itsconductivity?