chapter 8 geophysical quantities

7/27/2019 Chapter 8 Geophysical Quantities

1/17

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 andEnvironmental Engineering, 1998.

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 () AND RESISTIVITY ()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 ofresistance, the units of conductivity are sometimes given as mhos/meter or millimhos/meter. .

Resistivity is the inverse of conductivity ( = 1 / ). The units of resistivity are Ohm meters(m).

8.1.1. Factors Influencing Electrical Conductivity

Electrical conductivity of earth materials is influenced by the metal content (sulfides) in therock, porosity, clay content, permeability, and degree of pore saturation.

8.1.1.1 .Metal Content

All metal objects of interest in contaminated site assessments have a very large conductivitycontrast 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 (1982) show the resistivities of a large number of sulfide-bearingrocks (from 0.5 to 15 weight percent). A version of their figure is reproduced from Hearst

and Nelson (1985) in figure 225. As Hearst and Nelson point out, below 2% there is not

much correlation, whereas between 2% and 10%, there is a steady decrease in electricalresistivity 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.

8.1.1.2 Porosity

In the absence of metals, which conduct electronically, formation conductivity is related tothe 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 Archies Law is widely used and reasonably valid:

391


2/17

,m

f

w aF ==

(16)

where

.spaceporetheoftytortuaosiandvolumethetorelated"factor,formation"

space,porectedinterconneoffractiontheporosity,effective

sediments,atedunconsolidfor2typicallyconstant,empirical

sediments,atedunconsolidfor1typicallyconstant,empirical

whole,aasformationtheoftyconductivi

water,theoftyconductivi

f

=

=

=

=

=

=

F

m

a

w

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

,nFS=

(17)

where

constant.empiricalan

volume,poretheofsaturationfractional

=

=

n

S

Therefore, if the formation factor and groundwater conductivities of a saturated formationcan 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.

8.1.1.3 Clay Content

Clays and shales are hydrated minerals with high porosities and low permeabilities. Theminerals 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:

(18)),(1 +=F

where cations.leexchangeabtheoftyconductivi= can be estimated from its cation

exchange capacity. However, there are several problems in applying this apparently simple

equation.

392


3/17

Figure 225. Electrical resistivity of rocks with various wt % of sulfide. Samples each average overseveral cubic meters.

8.1.1.4 Permeability

Archies 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 thelaws of Darcy and Ohm, which predict electrical current and fluid flow, respectively.

Darcys Law:

,dh

dVkq = (19)

Ohms Law:

,dh

dVj = (20)

where

393


4/17

ly.respectivegradient,or voltagehead

density,current

density,fluid

=

=

=

dh

dV

j

q

Nonetheless, relationships between electrical conductivity and permeability are tricky andsite 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 226 and table 12.

Table 12. Comparison of electric and hydraulic properties.

Electrical Hydraulic

Transverse resistance: T = hii = Hl Transmissivity: Th = hiki= KlH

Longitudinal conductance: S= hi/i = H/l Leakance: Lh=ki/hi = Kt/H

Average aquifer resistivities: l,t Average hydraulic conductivities: Kl, Kt

Depending on the resistivity structure, surface resistivity soundings can often estimate eitherT or S for an aquifer sequence, but not H ort orl independently. If H = hi can beestimated from other data, then the average resistivities can be obtained.

Figure 226. A layered aquifer model.

394


5/17

Also, it is useful to be able to correct for known changes in the water quality between sites

since these will affect but not K. At the material level, both direct and inverse relationshipsbetween resistivity and hydraulic conductivity are quite possible.

Figure 227. Schematic relationship between hydraulic conductivity, porosity and resistivity

(Mazac et al., 1985).

For a clean aquifer, where Archies Law predicts an inverse relationship between resistivity

and (effective) porosity, effective porosity determines hydraulic conductivity, and an inverserelationship 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 and k. In situations where clay

content dominates the resistivity of a material, again a direct relationship between and kcan 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. and a direct vs. krelationship. Superimposed, for any given material, they show opposite trends. This is

shown schematically in figure 227.

8.1.1.5 Skin Depth

In electromagnetic methods, the electrical conductivity of the earth plays a pivotal role in thepenetration 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 morecharge in a given time. The depth at which a plane electromagnetic wave will be attenuated

toe

1(0.37) of its surface amplitude is called the skin depth, . The usefulness of the skin

depth concept is that it represents the maximum penetration of an EM method operating at

frequencyfin a medium of conductivity . he actual exploration depth may well be muchless than a skin depth owing to other factors, notably the geometry of the prospecting system.

Skin depth is related to conductivity as:

395


6/17

,)(4.503)( 21

21

== (21)where

ty.permeabilimagnetic

,

m

Seimensintyconductivi

Hz,infrequency

=

=

=

f

Figure 228. Skin depth as a function of resistivity and frequency.

The second formulation assumes a free space permeability of 4.10-7

. A chart of skin depthversus frequency is shown in figure 228.

8.1.2 Ground Penetrating Radar Attenuation

An issue related to conductivity is the attenuation () of radar signals by conductive soils andoverburden. This is usually quoted in decibels per meter and in the frequency range 100 to

1000 MHz is approximated as(Annan, 1991):

,m

69.12

1

=K

dB

(22)

where.

m

mSintyconductivi

constant,dielectricrelative

=

=

K

396


7/17

For example, over ground having a conductivity of 30 mS/m and K of 25, a typical GPR

signal would be attenuated at = 10.1 decibels/meter. A good radar system might have 100dB of sensitivity to use in ground transmission. In this environment, its penetration would be

limited to 5 meters (of two-way travel).

8.1.3 Induced Polarization (Ip) And Complex ResistivityThese properties are not widely used in environmental or engineering surveys but haveinteresting applications to the detection of clay and organics.

As previously defined, electrical resistivity, , is the constant of proportionality between theelectrical 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 is independentof 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 (f)='(f) + i"(f). In this case, the current is still linearlyproportional to the applied electrical field at each frequency, but it will have a phase shift inits response. In the diagram (figure 229), a complex has shifted the current sinusoid withrespect 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 theground 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 energyin 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 229. Schematic of the phase shift between an applied voltage and the resulting current

when is complex.

397


8/17

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 averageresistivity 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 sourcesare 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 230.

Figure 230. 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, insome circumstances, important IP and CR anomalies. In clays, ion exchange processes

effectively create different mobilities for cations and anions within the pores. This separatesor 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 thatoccur 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 anarbitrary choice. The simplest of complex resistivity units are ohm meters,but measured as a

real (') and imaginary (") part and as a function of frequency. Alternatively, it is

convenient to consider the resistivity as having a magnitude || = ('2+ "

2)

1/2[ohmmeters]

and a phase lag or lead = tan-1(''/') [radians or, more practically, milli-radians].

398


9/17

Figure 231. 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). Arelated 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. Chargeabilityis a measure of the area under the decay curve in figure 231.

Figure 232 showsthe 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 theoreticalresponse of Figure 231. 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 organicmolecules preferentially attach themselves to the clay surfaces, inhibiting cation exchange.

8.1.4 Dielectric Permittivity 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 thenegative charges moving in the opposite direction. Figure 233 shows a schematic result of

applying an electric field to a molecule.

399


10/17

Figure 232. Variation of resistivity (upper) and phase as function of frequency for some

montmorillonite clays (from Olhoeft, 1986).

Intrinsic polarization, p, has units of chargedistance per meter3, or coulombs/m2. The

dielectric permittivity relates polarization to the applied field:

,Ep = (23)

The permittivity is often expressed in terms of the permittivity of free space, 0, in terms ofthe dielectric constant .

,0 K= (24)

As the figure 235 shows, K varies from its free space value of 1 to a maximum of 80 forwater. 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 earthmaterials 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 23, has units of coulombs/(volt-meter) orfarads/m. From equation 23, K is dimensionless.

400


11/17

Figure 233. Schematic displacement of charge within a molecule by an electric field E.

8.1.4.1 Velocity of EM Radiation

The speed V of electromagnetic waves through a medium of permittivity and magneticpermeability (see next section) is:

,)( 21

= V (25)

8.1.4.2 Reflection Coefficient

The ratio R of the reflected to incident signal amplitude for an EM signal traveling frommedium 1 towards medium 2 is:

,)(

)(

21

22

1

1

21

22

1

1

KK

KKR

+

= (26)

8.1.4.3 Water Content

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. Onemodel, proposed by Calvert (1987), is:

,)1( 22 wmf KKK += (27)

This relationship is plotted in figure 234.

401


12/17

Figure 234. Dielectric constant of a water-saturated rock as a function of porosity (Kw = 81;

Km = 3).

8.2 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 levelsbelow which saturation and hysteresis are important). Magnetic susceptibility, k, is related

to magnetic permeability ()by:

),1(0 k+= (28)

where 0 is the magnetic permeability of free space, which is 4

.10-7. 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 235 shows the range of the dielectricconstant for different geologic materials including air (dielectric constant = 1) and water

(dielectric constant = 80).

Figure 235. Dielectric constant range for some common materials.

402


13/17

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 coilwith units of ampere-meter2.

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 theybecome magnetic. Magnetization, J,is the density of aligned dipoles per cubic meter, with

units (ampere/m).

The earths geomagnetic field, B,is the origin of most of the magnetization, J, found in

rocks, that is, the magnetization is induced by the present Earths magnetic field. The

relationship is:

,)1(

0

+=

kkBJ (29)

The exceptions are materials that have a permanent or remnant magnetization, acquired

elsewhere in a strong local magnetic field.

We detect magnetic objects in the subsurface by the way their magnetic fields distort the

earths geomagnetic field. These distortions are termed anomalies. It is generally safe to

assume that sediments are non-magnetic. Igneous or metamorphic rocks can haveappreciable and locally variable susceptibility.

The magnetic moment M of an object, assuming it is uniformly magnetized, can be estimated

as:

,JVM = (30)

where V is the volume of the object.

The unit of magnetic susceptibility, k, in the SI system is dimensionless. Magnetic

permeability () has units of Henrys per meter. The geomagnetic field (B) has units of forceper magnetic pole or Teslas. The practical unit of geomagnetism is thenano-Tesla, or

gamma.

8.2.1 Geomagnetic Field

The geomagnetic field of the earth is very similar to that of a large bar magnet placed at thecenter 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


14/17


15/17

8.4 POROSITY

Porosity, saturation, and density are related as a function of porosity as:

,1000)1( Smf += (32)

in which subscripts f and m refer to the formation and the matrix, respectively, and S is thefractional water saturation. A water density of 1,000 kg/m3 is assumed. Figure 238 showsthe density ranges of common materials.

Figure 237. Susceptibility range of common materials.

Figure 238. Density ranges in common materials.

8.5 SEISMIC VELOCITIES (VS, VP)

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


16/17

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 tothe 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 Vp and Vs are related to the

elastic moduli (Youngs modulus (E), Poissons ratio (), and Bulk modulus (k)) and the

density as follows.( ) smVssmVp /,/331.1

22 =+= (33)

Since liquids have no shear rigidity, shear waves cannot propagate through them. Velocities

have SI units ofmeters per second, sometimes also expressed as kilometers per second ormeters 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 0.35.

,)1( mwf SSS +=

(34)

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 239.

Figure 239. 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 (V) across aboundary ( is the density of the rock). In particular, for waves at normal incidence, the ratioR of reflected-to-incident amplitude is given by:

,)(

)(

1122

1122

VV

VVR

+

=

(35)

406


17/17

8.7 GEOMECHANICAL (ENGINEERING) PROPERTIES

Seismic velocities can be related to standard geotechnical properties. For example, Poisson's

ratio can be found from:

,)1(

)5.0(2

v

v

V

V

p

s

=

(36)

Figures 240 and 241 show the S and P wave velocities of seismic waves for a number of

different rock types.

Figure 240. S-wave velocity ranges for common materials.

Figure 241. P wave velocity ranges for common materials.

407

chapter 8 geophysical quantities

Documents