geology 228 applied geophysics lecture 3lanbo/g228lect0603.pdf · geology 228 applied geophysics...

Geology 228Applied Geophysics

Lecture 3

Physical properties of earth materials in near-surface environment

Outline

1. Introduction2. Mechanical properties3. electrical properties: electric conductivity4. Magnetic properties: permeability and susceptibility 5. Dielectric polarization: dielectric permittivity6. Mix model: analytic model and empirical model

Analytic mix modelEmpirical mix modelArchie's law and Waxman-Smits relationshipCRIM model

7. Note on effective materials

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).

Petrophysics is the study of the physical and chemical properties that describe the occurrence and behavior of rocks, soils and fluids.

This course concerns the PHYSICAL properties.

Outline




The mechanical properties Include

density (M/V); elastic modulii;viscosity;plasticity;

of the earth materials. We only concentrate on the elastic properties (elastic modulii) for this course.

Young’s modulus E

Young’s modulus is the stress needed to compress the solid to shorten in a unit strain.

Poisson’s ratio ν

Poisson’s measures the relativity of the expansion in the lateral directions and compression in the direction in which the uni-axial compression applies.

zzE

/1

∆=

σ

zzrr

//

∆∆

−=ν

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 material in 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 it doesn'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.

vvAFK

//

∆=

Shear Modulus µ (cont.)

yxAF

//

∆=µ

Shear Modulus µ

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 edges of 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 large shear 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 moduliiof zero. From the equations given above, notice that this implies that fluids and gases do not allow the propagation of S waves.

xyAF

//

∆=µ

Seismic Velocities related to material propertiesVp- P-wave (compressive wave) velocityVs- S-wave (shear wave) velocity

So, seismic velocities are determined by the mechanic propertiesof the materials in which the seismic waves propagate through.

Any change in rock or soil property that causes ρ, µ, or K to change will cause seismic wave speed to change. For example, going from an 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 factors causing 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.

Seismic velocity vs material’s mechanic properties

From: Sheriff and Geldart, Exploration Seismology, p69.

Property Units Iron Unsaturated Sand Saturated Sand

Bulk Modulus GPa 100.2 37Shear Modulus GPa 95.2 44Poisson's Ratio (σ) 0.14 0.08Young's Modulus N/m2 6.74Density g/cm3 22.564 2.65 3.01P-wave velocity km/s 5.92 2.73 4.18S-wave velocity km/s 3.23 1.37 3.42Vp/Vs 1.83 1.22 1.99Porosity - 0.36 0.36Dielectric Permittivity 221 6.25 25Magnetic Permeability 17.834 1.0 1.0Resistivity ohm-m 9E-08 1E+04 1E+02

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 petrophysicsCalculated from field data at Otis MMR, Cape Cod, Massachusetts

P-wave velocity tomogram for theProfile Parallel to the Tennis Courts

Question: assume Poisson’s ratio of 0.25, specific gravity 2.6 for both sediment and bedrock, what are K, the bulk modulus for the sediment and bedrock respectively (assume vs=500 m/s, vb=3000 m/s)?

Outline




The electric conductivity of earth materials

The electric property of materials is described by electric conductivity (σ) or electric resistivity(ρ=1/σ).

Conductor: σ > 105 S/m;Semi-conductor: 10-8 < σ < 105 S/m;Insulator: σ < 10-8 S/m;

Electric Resistivity

• Ohm’s Law:

RIV =

where V-voltage, I-current, and R-resistance. The Resistance is proportional 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, ρ, is the resistivity, a material property to describe the capability to resist the electric current flow.

ALR ρ=

Georg Simon Ohm (1787-1854)

IRV =

Ohm’s Law (discovered in 1827)

It's Resistivity, NOT Resistance

LRA

ALR

=

=

ρ

ρ

So the unit for resistivity is ohm-meter

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 in pore fluids. Although water itself is not a good conductor of electricity, ground water generally contains dissolved compounds that greatly enhance its ability to conduct electricity. Hence, porosity and fluid saturation tend to dominate electrical resistivity measurements. In addition to pores, fractures within crystalline rock can lead to low resistivities if they are filled with fluids.

Material Resistivity (Ohm-meter)Air ∞Pyrite 3 x 10^-1Galena 2 x 10^-3Quartz 4 x 10^10 - 2 x 10^14Calcite 1 x 10^12 - 1 x 10^13Rock Salt 30 - 1 x 10^13Mica 9 x 10^12 - 1 x 10^14Granite 100 - 1 x 10^6Gabbro 1 x 10^3 - 1 x 10^6Basalt 10 - 1 x 10^7Limestones 50 - 1 x 10^7Sandstones 1 - 1 x 10^8Shales 20 - 2 x 10^3Dolomite 100 - 10,000Sand 1 - 1,000Clay 1 - 100Ground Water 0.5 - 300Sea Water 0.2

The resistivities of various earth materials are shown below.

Electric Conductivity• Electric conductivity σ is the reciprocity of

the electric resistivity ρ:

ρσ /1=

Outline

1. Introduction2. Mechanical properties3. electrical properties: electric conductivity4. Magnetic properties: permeability and

susceptibility5. Dielectric polarization: dielectric permittivity6. Mix model: analytic model and empirical model



Magnetic Permeability

• The magnetic constitutive relation:

HHHB )1(00 κµµµµ +=== rwhere B - magnetic flux densityH – Magnetic fieldµ - Magnetic Permeabilityµ0 – magnetic permeability in vacuumµr – relative magnetic permeabilityκ – magnetic susceptibility

HHHHMHB rµµχµχµµµµ 000000 )1( =+=+=+=

Magnetic Susceptibility 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 κ);• Iron, steel have the values of 10 -100.

Outline




The dielectric properties of a material are defined by an electrical permittivity, ε. The permittivity is dependent upon a material’s ability 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 displacement is time dependent in most materials so that a complex permittivity is required to adequately describe the system, ε* = ε’+ iε”. Since the polarization mechanisms that occur in these materials depend on frequency, temperature, and composition so will this complexpermittivity.

Dielectric Permittivity

• The dielectric constitutive relation:

EED rεεε 0==where D – electric displacement densityE – electric fieldε0 – electric permittivity in vacuumεr – relative electric permittivityε – electric permittivity

Mechanisms involved in Dielectric Polarization include:Electron polarization;Atomic polarization;Molecular polarization;

rr norn εεεε === ,/ 20

Index of refraction (n) and dielectric constant εr

"' εεε i+=∗Value of the complex dielectric constant

is the parameter responsible for the observed phenomena in dielectric polarization

εεδ ′′′= /tan

Loss tangent

There are two more microscopic effects that cause ground to be chargeable

1)Membrane polarization

2)Electrode 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.

Membrane 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.

Electrode polarization

Variation of ε' and ε" with frequency for water

Domestic microwave Oven f = 2.45 GHz

GPR f < 1.5 GHz

There is a clear maximum in the dielectric loss for water at a frequency of approximately 20GHz, the same point at which the dielectric constant ε' 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 this theoretical expressions for ε' and ε" in terms of other material properties, formed the basis for our current understanding of dielectrics. The dielectric constants, ε' and ε" are dependent on both frequency and temperature, the first of which is expressed explicitly in the Debye equations whilst temperature is introduced indirectly through other variables:

)1()(

)1()(

22

22

τωωτεε

ε

τωεε

εε

+−

=′′

+−

+=′

∞

∞∞

s

s

where ε∞ and εs are the dielectric constants under high frequency and static fields respectively.

Since infra-red frequencies are often regarded as infinite for most purposes, ε∞ results from atomic and electronic polarizations. The relaxation time, τ, was derived by Debye from Stoke's theorem:

kTr 34πητ =

where r is the molecular radius, η the viscosity, kBoltzman's constant, and T the temperature. If the Debye equations are plotted against wt with arbitrary values for ε∞ and εs as shown in the last Figure, then the similarity of these expressions to the experimental values shown in the next Figure is clear.

Debye expressions for ε' and ε" calculated as a function of [ωτ].

Outline




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 47.7Dry Sand 30 0 4 0.1 0.150 0.009 106Wet Sand 30 100 17.2

2521.310

0.0720.060

0.970.38

1.02.6

Dry Clay 30 0 4 10 0.150 0.94 1.1Wet Clay 30 100 17.7

1631.3100

0.0710.075

1.404.71

0.70.2

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

Material conductivity

σ(miliS/m)

dielectricconstant

εr

dielectricpermittivit

y ε(picoF/m)

electromagneticwave velocity

v(m/µs)

skin depth

δ(m)

transition frequency

ωt(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)

Schematic representation of soil matrix indicating relationship between air (A), soil particles (B) and water (C).

E E

Parallel Plate Capacitors

DielectricPlates

Dielectric plates arranged a) parallel and b) perpendicular to the electrodes. The analytical mix model are:

2

2

1

11εθ

εθ

ε ′+

′=

′ 2211 εθεθε ′+′=′

parallel model serial model

1 2

There are other theoretical models appears work quite well for sediments filled with water, one popular one is the complex refraction index model (CRIM), like serial model but sum on the square root of the dielectric constant:

∑

∑

=

=

′=+′+′=′

=++=

n

iii

n

iii ornnnn

12211

12211

...

...

εθεθεθε

θθθ

Complex Refraction Index Model (CRIM)

• The wavelength of the signal is much larger 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• ε0=1, εice = 3.6, εwat = 81,

εasph = 2.6-2.8, εaggreg = 5.5-6.5

))1()1( awgb SS εφεφεφε −++−=

Archie’s Law (for formation without or little 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 include considerations 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 the pores, 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.

wnmSa ρφρ −−=

ρ−effective formation resistivity;ρw−pore water resistivity;φ – porosity;S – saturation;a – 0.5-2.5;m – 1.3-2.5;n ~2.

Archie’s law

σ−effective formation conductivity;σw−pore water conductivity;Β – constant coefficient;F – Formation factor;Qv – Cation exchange capacity;

Maxwell-Smits relationship (empirical for shaly sand)

)(1vw BQ

F+= σσ

1. Electrical conductivity and hydraulic conductivity

From Ohm’s law

dLdVA

RVI σ==

From Darcy’s law

dLdHkAQ =

For example, in exploration and development of petroleum or water resources (or environmental cleanups), the properties of interest are the porosity, saturation, chemistry and mobility. These are in pursuit of the questions:

Is there any place in the rocks for fluids to exist? (porosity)

How much of the porosity is fluid filled? (saturation)

What kind of fluids are there? (chemistry) Can the fluids be moved? (mobility)

No matter which formula you use, as long as you have obtained the observation on the effective resistivity (conductivity), you have had some constraints on either the porosity and/or the saturation.

The pore fluid is a major player to determine the formation electric property!

whereεr−effective dielectric constant;θv−water content in terms of volume;

Topp model for dielectric constant of solid-water mix (empirical for strong water-dependence of shallow depth sediments)

362422 103.4105.51092.2103.5 rrrv εεεθ −−−− ×+×−×+×−=

32 70.7600.14630.903.3 vvvr θθθε −++=

or

Electromagnetic wave velocity, dielectric constant, and water content

2

2, rr

c cv orv

εε

= =

And by the Topp Model we have

Outline




Property Units Iron Unsaturated Sand Saturated Sand

Bulk Modulus GPa 100.2 37Shear Modulus GPa 95.2 44Poisson's Ratio (σ) 0.14 0.08Young's Modulus N/m2 6.74Density g/cm3 22.564 2.65 3.01P-wave velocity km/s 5.92 2.73 4.18S-wave velocity km/s 3.23 1.37 3.42Vp/Vs 1.83 1.22 1.99Porosity - 0.36 0.36Dielectric Permittivity 221 6.25 25Magnetic Permeability 17.834 1.0 1.0Resistivity ohm-m 9E-08 1E+04 1E+02

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 petrophysicsCalculated from field data at Otis MMR, Cape Cod, Massachusetts

The effective medium theory (wavelength >> size of heterogeneity)

ρEvEM =

2

2

1

1 111Ed

dEd

dE

+= 22

11 ρρρ

dd

dd

+=

The ray theory (wavelength ~ size of heterogeneity)

2

2

1

1 111vd

dvd

dvRT

+=

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 other means to detect the changes in elastic properties, we can use the effective medium theory to get the overall mixed, orbulk, 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.

Elastic property and seismic velocity of porous media –effective medium theory

TABLE 1. Material Properties

MaterialDensity(kg/m3)

Dynamic Modulus

(Pa)P-velocity(m/sec)

Steel 7.9 2.4 x 1011 5512Concrete 2.4 3.5 x 1010 3819

References

Mavko, G, T. Mukerji, and J. Dvorkin, The Rock Physics Handbook, 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/(m3), the shear modulus = 1010 Pa.2, if the Poisson’s ratio is 0.25 (this is known as the Poisson condition which can be a nominal value for the Poisson’s 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 its conductivity?

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