Geology 228/278 Applied and Environmental Geophysics Lecture 3. Physical properties of earth materials in near-surface environment

Geology 228/278 Applied and Environmental Geophysics Lecture 3 Physical properties of earth materials in near-surface environment

Outline 1. Introduction 2. Mechanical properties 3. electrical properties: electric conductivity 4. Magnetic properties: permeability and susceptibility 5. Dielectric polarization: dielectric permittivity 6. Mix model: analytic model and empirical model Analytic mix model Empirical mix model Archie's law and Waxman-Smits relationship CRIM 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.

Include the density and the elastic properties of the earth materials These material properties are described by elastic modulii.

Young s modulus E Young s modulus is the stress needed to compress the solid to shorten in a unit strain. Poisson s ration ν E = F / A x / x Poisson s measures the relativity of the expansion in the lateral directions and compression in the direction in which the uni-axial compression applies. ν = y / x / y x

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. K = F / A v / v

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 modulii of zero. From the equations given above, notice that this implies that fluids and gases do not allow the propagation of S waves. µ = F / A y / x

Young s modulus E Young s modulus is the stress needed to compress the solid to shorten in a unit strain. E Poisson s ration ν 1 = σ z / Poisson s measures the relativity of the expansion in the lateral directions and compression in the direction in which the uni-axial compression applies. z ν = r z / / r z

Shear Modulus µ (cont.) µ = F / A x / y

Seismic Velocities related to material properties Vp- P-wave (compressive wave) velocity Vs- S-wave (shear wave) velocity So, seismic velocities are determined by the mechanic properties of the materials in which the seismic waves propagate through.

Seismic velocity vs material s mechanic properties 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 airfilled 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.

From: Sheriff and Geldart, Exploration Seismology, p69.

Property Units Iron Unsaturated Sand Saturated Sand P-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 1.0 Resistivity ohm-m 9E-08 1E+04 1E+02 Bulk Modulus GPa 100.2 37 Shear Modulus GPa 95.2 44 Poisson's Ratio (σ) 0.14 0.08 Young's Modulus N/m 2 6.74 Density g/cm 3 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

Seismic Refraction Results Profile Parallel to the Tennis Courts

The electric conductivity of earth materials The electric property of materials is described by electric conductivity or electric resistivity. Conductor: σ > 10 5 S/m; Semi-conductor: 10-8 < σ < 10 5 S/m; Insulator: σ < 10-8 S/m;

Electric Resistivity Ohm s Law: V = RI 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. R = L ρ A

Ohm s Law (discovered in 1827) V = IR Georg Simon Ohm (1787-1854)

It's Resistivity, NOT Resistance R = ρ = ρ L A RA L So the unit for resistivity is ohm-meter

Resistivity of Earth Materials Although 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.

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 10^6 Gabbro 1 x 10^3-1 x 10^6 Basalt 10-1 x 10^7 Limestones 50-1 x 10^7 Sandstones 1-1 x 10^8 Shales 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

Electric Conductivity Electric conductivity σ is the reciprocity of the electric resistivity ρ: σ =1/ ρ

Magnetic Permeability The magnetic constitutive relation: B = r µ H = µ µ H = µ 0(1 + κ ) H 0 where B - magnetic flux density H Magnetic field µ - Magnetic Permeability µ0 magnetic permeability in vacuum µr relative magnetic permeability κ magnetic susceptibility

B = µ 0 H + µ 0M = µ 0H + µ 0χH = µ 0 ( 1+ χ) H = µ 0µ rh

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.

The dielectric properties of a material are defined by an electrical permittivity, ε. The permittivity is dependent upon a materials ability to neutralize part of an static electrical field. For this, a dielectric material must contain localized charge that can be displaced by 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 complex permittivity.

Dielectric Permittivity The dielectric constitutive relation: D = εe = ε 0 ε r E where D electric displacement density E electric field ε0 electric permittivity in vacuum εr relative electric permittivity ε electric permittivity

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

Index of refraction (n) and dielectric constant ε r ε = ε / ε = n, or n = r 0 2 ε r

Value of the complex dielectric constant ε = ε ' + iε" is the parameter responsible for the observed phenomena in dielectric polarization Loss tangent tanδ = ε / ε

There are two more microsopic effects that cause ground to be chargeable 1)Membrane polarization 2)Electrode polarization

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.

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.

Domestic microwave Oven f = 2.45 GHz GPR f < 1.5 GHz Variation of ε' and ε" with frequency for water

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 his 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: ε ε = ε = ( ε s + ( ε 2 (1 + ω τ ε 2 (1 + ω τ ε ) ωτ where ε and ε s are the dielectric constants under high frequency and static fields respectively. s 2 ) 2 ) )

Since infra-red frequencies are often regarded as infinite for most purposes, ε results from atomic and electronic polarizations, whilst s results from the sum of all the polarization mechanisms described in a later section. The relaxation time, τ, was derived by Debye from Stoke's theorem: τ = 4πηr 3 kt where r is the molecular radius, η the viscosity, k Boltzman's constant, and T the temperature. If the Debye equations are plotted against wt with arbitrary values for ε and ε s 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].

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) 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 17.2 25 21.3 10 0.072 0.060 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 Average Soil 30-16 20 0.075 0.94 1.1 0.97 0.38 1.40 4.71 Skin depth (m) 1.0 2.6 0.7 0.2 Liu and Li: J. Appl. Geophys., 2001.

Table 1. Electromagnetic properties of some earth and engineered materials Material conductivity σ (milis/m) dielectric constant ε r dielectric permittivit y ε (picof/m) electromagnetic wave velocity v (m/µs) skin depth δ (m) transition frequency ω 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 7x10 6 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 5.8x10 9 0.27x10-6 Brewster & Annan (1994)

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

Parallel Plate Capacitors E Dielectric Plates E Dielectric plates arranged a) parallel and b) perpendicular to the electrodes. The analytical mix model are: 1 θ θ = + ε 1 2 ε = θ ε 1ε 1 + θ2ε 2 1 ε 2 parallel model series model

There are other theoretical models appears work quite well for sediments filled with water, one popular one is the complex refraction index model (CRIM): = = = + + = = + + = n i i i n i i i or n n n n 1 2 2 1 1 1 2 2 1 1...... ε θ ε θ ε θ ε θ θ θ

The 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 ε 0 =1, ε ice = 3.6, ε wat = 81, ε asph = 2.6-2.8, ε aggreg = 5.5-6.5 ε b = ( 1 φ) ε + φs ε + φ(1 S) ε g w a )

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.

Archie s law ρ = a φ m S n ρ w ρ effective formation resistivity; ρw pore water resistivity; φ porosity; S saturation; a 0.5-2.5; m 1.3-2.5; n ~2.

Maxwell-Smits relationship (empirical for shaly sand) σ 1 = ( σ F + w BQ v ) σ effective formation conductivity; σw pore water conductivity; Β constant coefficient; F Formation factor; Qv Cation exchange capacity;

1. Electrical conductivity and hydraulic conductivity From Ohm s law I = V R = σa dv dl From Darcy s law Q = ka dh dl

The effective medium theory (wavelength >> size of heterogeneity) ρ E v EM = 2 2 1 1 1 1 1 E d d E d d E + = 2 2 1 1 ρ ρ ρ d d d d + = The ray theory (wavelength ~ size of heterogeneity) 2 2 1 1 1 1 1 v d d v d d v RT + =

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 other means 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.

TABLE 1. Material Properties Material Density (kg/m 3 ) Dynamic Modulus (Pa) P-velocity (m/sec) Steel 7.9 2.4 x 10 11 5512 Concrete 2.4 3.5 x 10 10 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/(mmm), the shear modulus = 10^10 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?