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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?