CHAPTER 8 GEOPHYSICAL QUANTITIES

Transcription

1 CHAPTER 8 GEOPHYSICAL QUANTITIES This chapter is substantially based on a report produced by the Environmental and Engineering Geophysical Society titled Applications of Geophysics in Geotechnical and Environmental Engineering, Many physical properties and quantities are involved in an understanding of geophysics and geophysical methods. In this section, the units of measurement are defined and some relationships established such as saturated porosity vs. dielectric constant. In addition, some of the physical properties of earth materials are presented. 8.1 ELECTRICAL CONDUCTIVITY (σ) AND RESISTIVITY (ρ) Electrical conductivity is the proportionality factor relating the current that flows in a medium to the electric force field that is applied. It is a measure of the ability of electrical charge to move through that material. Resistivity is the reciprocal of conductivity. The units of conductivity are Siemens per meter (S/m). The practical unit is millisiemens per meter (ms/m). Because Siemen, the unit of conductance, is the reciprocal of the Ohm, the unit of resistance, the units of conductivity are sometimes given as mhos/meter or millimhos/meter.. Resistivity is the inverse of conductivity ( ρ = 1 / σ ). The units of resistivity are Ohm meters (Ωm) Factors Influencing Electrical Conductivity Electrical conductivity of earth materials is influenced by the metal content (sulfides) in the rock, porosity, clay content, permeability, and degree of pore saturation Metal Content All metal objects of interest in contaminated site assessments have a very large conductivity contrast with their surroundings and can usually be readily detected with electrical and electromagnetic methods. Quantitative estimates of the metal content are not easily obtained. Nelson and Van Voorhis (198) show the resistivities of a large number of sulfide-bearing rocks (from 0.5 to 15 weight percent). A version of their figure is reproduced from Hearst and Nelson (1985) in figure 5. As Hearst and Nelson point out, below % there is not much correlation, whereas between % and 10%, there is a steady decrease in electrical resistivity The slope of resistivity vs. percent sulfides decreases ( i.e. conductivity increases ) noticeably beyond ten percent sulfides. This decrease quickens beyond 10%, suggesting that small veins are forming exceptionally conductive pathways. Note that IP effects are much more pronounced than resistivity anomalies at the low metal content Porosity In the absence of metals, which conduct electronically, formation conductivity is related to the volume and conductivity of the water in earth materials. The groundwater conducts through its ions, and its conductivity, therefore, depends strongly on the total dissolved solids. Within a porous, clay-free medium whose matrix is non-conducting, a relationship known as Archie s Law is widely used and reasonably valid: 391

2 where σ σ w f = F = aφ m, (16) σ = conductivity of the water, w σ = conductivity of the formation as a whole, f a = m = empirical constant, typically1for unconsolidated sediments, empirical constant, typically for unconsolidated sediments, φ = effective porosity, the fraction of interconnected pore space, F = "formation factor," related to the volume and tortuaosity of the pore space. As Hearst and Nelson point out, it is amazing that the conductivity of so many geological formations is well represented by this simple function of porosity. It holds true even to the very low porosities found in crystalline rocks. For a simple three-component system of air, water, and matrix, the relation where σ σ ϖ φ n = FS, (17) S = fractional saturation of n = an empirical constant. the pore volume, Therefore, if the formation factor and groundwater conductivities of a saturated formation can be measured (say by geophysics and sampling, respectively), the porosity can be approximately estimated. If F is known, then S can be estimated in a partially saturated medium Clay Content Clays and shales are hydrated minerals with high porosities and low permeabilities. The minerals themselves may not be very conductive, but their surface charge causes an excess of cations in the pore fluid immediately adjacent to the clay surfaces. The result is high conductivity near the clay surfaces, which can dominate the overall conductance if the pore water conductivity is low. A commonly quoted relationship (Waxman and Smits, 1968) is: 1 σ = F ( σ + σ ), φ ϖ χ (18) whereσ χ = conductivity of the exchangeable cations. σχ can be estimated from its cation exchange capacity. However, there are several problems in applying this apparently simple equation. 39

3 Figure 5. Electrical resistivity of rocks with various wt % of sulfide. Samples each average over several cubic meters Permeability Archie s relationship notwithstanding, a rock with a non-conducting matrix must be permeable as well as porous to conduct electricity. There is a clear symmetry between the laws of Darcy and Ohm, which predict electrical current and fluid flow, respectively. Darcy s Law: Ohm s Law: where dv q = k, (19) dh dv j = σ, (0) dh 393

4 q = fluid density, j = current density, dv dh = head or voltage gradient, respectively. Nonetheless, relationships between electrical conductivity and permeability are tricky and site specific. These relationships are sought at both the material (sample) level and on aquifer scales. An excellent summary is given by Mazac, et al. (1985). At the aquifer level, the resistivity, transverse resistance, and horizontal conductance, as measured by surface resistivity and EM soundings or well logs, are compared to average hydraulic conductivity, transmissivity, and leakance. These parameters are illustrated in figure 6 and table 1. Table 1. Comparison of electric and hydraulic properties. Electrical Transverse resistance: T = Σ h i ρ i = Hρ l Hydraulic Transmissivity: T h = Σh i k i = K l H Longitudinal conductance: S= Σh i /ρ i = H/ρ l Leakance: L h =Σk i /h i = K t /H Average aquifer resistivities: ρ l, ρ t Average hydraulic conductivities: K l, K t Depending on the resistivity structure, surface resistivity soundings can often estimate either T or S for an aquifer sequence, but not H or ρt or ρl independently. If H = Σhi can be estimated from other data, then the average resistivities can be obtained. Figure 6. A layered aquifer model. 394

5 Also, it is useful to be able to correct for known changes in the water quality between sites since these will affect ρ but not K. At the material level, both direct and inverse relationships between resistivity and hydraulic conductivity are quite possible. Figure 7. Schematic relationship between hydraulic conductivity, porosity and resistivity (Mazac et al., 1985). For a clean aquifer, where Archie s Law predicts an inverse relationship between resistivity and (effective) porosity, effective porosity determines hydraulic conductivity, and an inverse relationship with hydraulic conductivity can be expected. For different materials, however, hydraulic conductivity increases and porosity decreases with grain size, leading to a direct relationship between ρ and k. In situations where clay content dominates the resistivity of a material, again a direct relationship between ρ and k can be expected (as in the example above). Mazac, et al., (1985) shows a generic trend between different materials (clay to gravel) with an inverse k vs. φ and a direct ρ vs. k relationship. Superimposed, for any given material, they show opposite trends. This is shown schematically in figure Skin Depth In electromagnetic methods, the electrical conductivity of the earth plays a pivotal role in the penetration that can be obtained. Conductivity removes (attenuates) energy from the EM wave through the work done by moving charge. Higher frequency EM waves lose energy more quickly than low frequency waves because, conceptually at least, they move more charge in a given time. The depth at which a plane electromagnetic wave will be attenuated to e 1 (0.37) of its surface amplitude is called the skin depth, δ. The usefulness of the skin depth concept is that it represents the maximum penetration of an EM method operating at frequency f in a medium of conductivity σ. Τhe actual exploration depth may well be much less than a skin depth owing to other factors, notably the geometry of the prospecting system. Skin depth is related to conductivity as: 395

6 where 1 1 δ = ( πφµσ ) = 503.4( σφ), (1) f = frequency in Hz, Seimens σ = conductivity in, m µ = magnetic permeability. Figure 8. Skin depth as a function of resistivity and frequency. The second formulation assumes a free space permeability of 4π A chart of skin depth versus frequency is shown in figure Ground Penetrating Radar Attenuation An issue related to conductivity is the attenuation (α) of radar signals by conductive soils and overburden. This is usually quoted in decibels per meter and in the frequency range 100 to 1000 MHz is approximated as (Annan, 1991): σ db α = 1.69, 1 K m () where K = relative dielectric constant, ms σ = conductivity in. m 396

7 For example, over ground having a conductivity of 30 ms/m and K of 5, a typical GPR signal would be attenuated at α = 10.1 decibels/meter. A good radar system might have 100 db of sensitivity to use in ground transmission. In this environment, its penetration would be limited to 5 meters (of two-way travel) Induced Polarization (Ip) And Complex Resistivity These properties are not widely used in environmental or engineering surveys but have interesting applications to the detection of clay and organics. As previously defined, electrical resistivity, ρ, is the constant of proportionality between the electrical current and the applied electric field (Ohm's Law). If the applied field varies in time, the current behaves similarly. For example, if the applied field is a sinusoid with a zero crossing at time t1, then the current will be similar. The implication is that ρ is independent of the frequency or time behavior of the applied field. In fact, the resistivity (or conductivity) will almost always be frequency dependent to some degree, and is also complex. The resistivity then has the form ρ(f) =ρ'(f) + iρ"(f). In this case, the current is still linearly proportional to the applied electrical field at each frequency, but it will have a phase shift in its response. In the diagram (figure 9), a complex ρ has shifted the current sinusoid with respect to the applied field. An equivalent representation of the effects of a complex resistivity is the result of a sharp turn-off of the applied voltage. Normally, the current in the ground would also cease immediately. When the resistivity is complex, the current will continue to flow for a period of microseconds to as much as several seconds in some cases. Methods known as induced polarization (I P), spectral I P, and complex resistivity (CR) exploit these more general properties of the resistivity (conductivity) parameter. The process of shifting implies delay between cause and effect, and this, in turn, requires that the energy in the applied field be stored for an instant before being converted into current flow. In an electrical circuit, this storage could be depicted as a capacitor. Figure 9. Schematic of the phase shift between an applied voltage and the resulting current when ρ is complex. 397

8 There are several mechanisms in the earth that enhance this complex, frequency- dependent behavior of resistivity. Disseminated metal ores are by far the most (commercially) important target of I P surveys. The mechanism for storage/delay is at least conceptually related to the blocking of pores by metallic grains. Pore water ions build up on either side of the grain, giving the effect of a capacitor. A similar effect can be observed in some garbage dumps containing a lot of scrap metal. The disseminated metal may not increase the average resistivity substantially but are charged by an applied electric field. Their presence can be recognized by the slow voltage discharge after the applied field is turned off. These effects are most pronounced at frequencies below 1,000 Hz. IP effects arising from metallic sources are generally most pronounced at fairly low frequencies (below 100 Hz). However, in conductive terrain, maximum anomalies can occur at higher frequencies. Conductivity ranges of some materials are shown in figure 30. Figure 30. Conductivity ranges of some materials. Note this is a very variable parameter, and the ranges are approximate. Olhoeft (1986) describes three other active chemical processes that produce smaller but, in some circumstances, important IP and CR anomalies. In clays, ion exchange processes effectively create different mobilities for cations and anions within the pores. This separates or polarizes charge and is usually known as membrane polarization. This effect is much smaller than is observed for disseminated metals, but can be exploited in some cases to distinguish clay and contaminated aquifers, both of which will have a low direct current (DC) resistivity. Olhoeft (1986) also describes some poorly understood polarization effects that occur when organics react with clay minerals. The most promising aspect of this, from the standpoint of detecting organic wastes, is that measurable reduction of the IP response of clays following organic contamination has been recorded. The effect, however, is subtle. It is usual to talk of complex resistivity instead of complex conductivity, although this is an arbitrary choice. The simplest of complex resistivity units are ohm meters, but measured as a real (ρ ' ) and imaginary (ρ " ) part and as a function of frequency. Alternatively, it is convenient to consider the resistivity as having a magnitude ρ = (ρ ' + ρ " ) 1/ [ohm meters] and a phase lag or lead φ = tan-1(ρ''/ρ') [radians or, more practically, milli-radians]. 398

9 Figure 31. Schematic of decay time associated with complex resistivity, IP. In IP surveys, the percentage frequency effect PFE is defined as the normalized difference between the resistivity measured at two different frequencies (typically 0.1Hz and 10Hz). A related parameter, the chargeability(m), is measured by systems that record the time delay in current flow following an abrupt interruption or onset of the applied voltage. Chargeability is a measure of the area under the decay curve in figure 31. Figure 3 shows the CR response of a montmorillonite (clay) soil from the EPA Pittman site near Henderson, Nevada, taken from Olhoeft (1986). One of the two samples (triangles) is contaminated by waste that includes organics. Note the decrease in resistivity with frequency and the presence of a phase peak for both samples, consistent with the theoretical response of Figure 31. The contamination has lowered the resistivity of the second sample - clearly inorganic contaminants dominate - but the shift of the phase peak to lower frequencies is stated by the author to represent peptization, a process whereby organic molecules preferentially attach themselves to the clay surfaces, inhibiting cation exchange Dielectric Permittivity ε Permittivity is conceptually similar to electrical conductivity. It relates charge separation, rather than current, to the applied electric field. Materials that have no free charge carriers such as ions or electrons may still appear to pass current when a voltage is applied. That is, energy will be drawn from the voltage source to move charge. The charge that moves is bound to the molecules of the material, for example the positive charge on the nucleus and the negative charges of the electron shells. An applied field polarizes the charge distribution with the positive charges moving in one direction and the negative charges moving in the opposite direction. Figure 33 shows a schematic result of applying an electric field to a molecule. 399

10 Figure 3. Variation of resistivity (upper) and phase as function of frequency for some montmorillonite clays (from Olhoeft, 1986). Intrinsic polarization, p, has units of charge distance per meter3, or coulombs/m. The dielectric permittivity relates polarization to the applied field: p = εe, (3 ) The permittivity is often expressed in terms of the permittivity of free space, ε 0, in terms of the dielectric constant Κ. ε = Kε 0, (4) As the figure 35 shows, K varies from its free space value of 1 to a maximum of 80 for water. K is strongly frequency dependent in parts of the frequency spectrum, and should more properly be portrayed as complex. For our purposes, these aspects can be ignored. For this report, K is considered only at ground penetrating radar (GPR) frequencies, in the range 100 to 1,000 MHz. Permittivity is the primary factor influencing the speed of electromagnetic radiation in earth materials at GPR frequencies. Contrasts in velocity, in turn, produce reflections of electromagnetic energy within the Earth. Thus, K is the major influence on ground penetrating radar measurements. From equation 3, ε has units of coulombs/(volt-meter) or farads/m. From equation 3, K is dimensionless. 400

11 Figure 33. Schematic displacement of charge within a molecule by an electric field E Velocity of EM Radiation The speed V of electromagnetic waves through a medium of permittivity ε and magnetic permeability (see next section) µ is: V 1 = ( µε), (5) Reflection Coefficient The ratio R of the reflected to incident signal amplitude for an EM signal traveling from medium 1 towards medium is: Water Content 1 ( K1 K ) R =, (6) 1 1 ( K + K ) 1 1 With a dielectric constant of 80, water dominates the permittivity of rock water mixtures. There does not appear to be one widely accepted model for water-saturated rocks. One model, proposed by Calvert (1987), is: K f = (1 φ ) K + φ K, (7) m w This relationship is plotted in figure

12 Figure 34. Dielectric constant of a water-saturated rock as a function of porosity (K w = 81; K m = 3). 8. MAGNETIC SUSCEPTIBILITY (K) Magnetic susceptibility is a measure of the ability of a material to be magnetized. The proportional constant links magnetization to the applied magnetic field intensity (at levels below which saturation and hysteresis are important). Magnetic susceptibility, k, is related to magnetic permeability (µ) by: µ = µ 0 (1 + k), (8) where µ 0 is the magnetic permeability of free space, which is 4π The most magnetically susceptible materials are called ferromagnetic materials which contain iron, nickel, cobalt and many alloys of these materials. Of the several ferromagnetic minerals, magnetite predominates in the applications addressed here. In waste sites, iron and steel are the major sources of magnetic anomalies. Figure 35 shows the range of the dielectric constant for different geologic materials including air (dielectric constant = 1) and water (dielectric constant = 80). Figure 35. Dielectric constant range for some common materials. 40

13 There are several other magnetic quantities, and their relationship is sometimes confusing. The basic magnetic object is the magnetic pole, equivalent to the north end of a magnet. Magnetic poles (a) do not exist as discrete particles, and (b) always appear to come in pairs, and are, therefore, a dipole. A dipole is the result of electric charge in motion; one is made by passing a current I (amperes) around a circuit of area A (meters). Hence, the dipole strength or magnetic dipole moment, M is a vector perpendicular to the plane of that coil with units of ampere-meter. All matter consists of charge in motion, but for most materials, the resulting dipoles are randomly aligned and cancel. For certain materials, this cancellation is incomplete, and they become magnetic. Magnetization, J, is the density of aligned dipoles per cubic meter, with units (ampere/m). The earth s geomagnetic field, B, is the origin of most of the magnetization, J, found in rocks, that is, the magnetization is induced by the present Earth s magnetic field. The relationship is: kµ 0 J = B, (1 + k) The exceptions are materials that have a permanent or remnant magnetization, acquired elsewhere in a strong local magnetic field. (9) We detect magnetic objects in the subsurface by the way their magnetic fields distort the earth s geomagnetic field. These distortions are termed anomalies. It is generally safe to assume that sediments are non-magnetic. Igneous or metamorphic rocks can have appreciable and locally variable susceptibility. The magnetic moment M of an object, assuming it is uniformly magnetized, can be estimated as: M = JV, where V is the volume of the object. (30) The unit of magnetic susceptibility, k, in the SI system is dimensionless. Magnetic permeability (µ) has units of Henrys per meter. The geomagnetic field (B) has units of force per magnetic pole or Teslas. The practical unit of geomagnetism is the nano-tesla, or gamma Geomagnetic Field The geomagnetic field of the earth is very similar to that of a large bar magnet placed at the center of the Earth, with its south end oriented toward the north magnetic pole. The field is dipolar, vertically downward at the north magnetic pole, vertically upward at the south magnetic pole, and horizontal at the (magnetic) equator. It has a strength of roughly 30,000 gammas at the equator, 70,000 gammas at the poles. In the United States, it is acceptable for 403

14 the purposes of simple modeling to assume that a field declination of about 60 degrees has a strength of 55,000 gammas Susceptibility and Magnetite The susceptibility of most rocks can be related to magnetite content reasonably well as follows: k = 1. f ' 1., (31) Figure 36. Susceptibility as a function of magnetite content. where f is the volume percent of magnetite (Grant and West, 1965). A plot of this empirical relationship is shown in figure Magnetization of Soils Breiner (1973) points out that soils reflect their parentage, and that some, as a result, will be considerably more magnetic than others. Because magnetite tends to concentrate in sediments where streams carrying heavy metals lose velocity, concentrations can be expected in distributing streams, in alluvial fans, in glaciated terrain (particularly eskers, outwash, and beach deposits). In highly organic soils, maghemite, a relatively magnetic form of hematite, can be produced. As a result, the magnetic susceptibility of soils can be laterally variable and, because they are close to the magnetometer, give rise to very localized anomalies of significant magnitude. Figure 37 presents the susceptibility range of common materials. 8.3 DENISTY Density, σ, is the intrinsic unit mass of a material, defined as the mass of one cubic meter with units of kilograms per cubic meter (kg/m3). However, the old cgs units of grams-percubic-centimeter are still widely used. One gm/cc is equivalent to1,000 kg/m3. 404

15 8.4 POROSITY Porosity, saturation, and density are related as a function of porosity Φ as: σ = σ ( 1 φ) 1000Sφ, (3) f m + in which subscripts f and m refer to the formation and the matrix, respectively, and S is the fractional water saturation. A water density of 1,000 kg/m3 is assumed. Figure 38 shows the density ranges of common materials. Figure 37. Susceptibility range of common materials. 8.5 SEISMIC VELOCITIES (VS, VP) Figure 38. Density ranges in common materials. If the ground is stressed by an explosion or a hammer blow, it generates three fundamental types of elastic waves: P (primary, push-pull) waves; S (secondary, shear) waves, and surface waves. The P and S waves propagate through the body of the earth; the surface waves can exist only close to the free surface. 405

16 Only P and S are discussed in this section. P waves are characterized by having a particle motion in the direction of propagation, whereas S waves have particle motion transverse to the direction of propagation. P waves are the faster of the two, with velocities typically 50% higher than those for associated S waves. The wave velocities V p and V s are related to the elastic moduli (Young s modulus (E), Poisson s ratio (ν), and Bulk modulus (k)) and the density ρ as follows. ( m) / s, Vs = m s V p = / (33) Since liquids have no shear rigidity, shear waves cannot propagate through them. Velocities have SI units of meters per second, sometimes also expressed as kilometers per second or meters per millisecond. The P-wave velocity of a water/competent rock mixture obeys the following relationship (Wyllie's Equation) reasonably well up to porosities of S f = φs w + ( 1 φ) S m, where S is the slowness (1/V), and subscripts f,w, and m stand for formation, water, and matrix, respectively. Assuming a matrix Vp of 5,950 m/sec for sandstone, and a water velocity of 1,500 m/sec we get the porosity dependence shown in figure 39. (34) Figure 39. P-wave velocities as a function of porosity. Valid for competent rock only. Overestimates velocity for soft sediments. 8.6 REFLECTIVITY The cause of seismic reflections is contrasts in the seismic impedance (ρv) across a boundary (ρ is the density of the rock). In particular, for waves at normal incidence, the ratio R of reflected-to-incident amplitude is given by: ( ρ V R = ( ρ V ρ1v1 ), + ρ V ) 1 1 (35) 406

17 8.7 GEOMECHANICAL (ENGINEERING) PROPERTIES Seismic velocities can be related to standard geotechnical properties. For example, Poisson's ratio ν can be found from: V V s p (0.5 v) =, (1 v) (36) Figures 40 and 41 show the S and P wave velocities of seismic waves for a number of different rock types. Figure 40. S-wave velocity ranges for common materials. Figure 41. P wave velocity ranges for common materials. 407