# 1.8 Electromagnetic Rock Properties

In their seminal paper on the electromagnetic behavior of rocks,
Fuller and Ward (1970) discussed the expectation that the circuit
representation of rocks should include non-ideal elements such as
frequency dependent values of R and C. Consequently, the constitutive
relations (equations 1.6.1 to 1.6.3) would be, in general, functions
of frequency with σ, ε,
and μ frequency dependent and
complex. This leads to the concept of *effective EM properties* which are those actually measured by experiments. The *effective
conductivity*, σ_{e} is the parameter that relates to the component of total current
density that is in-phase with the electric field and the *effective
dielectric permittivity*, ε_{e} relates the out-of-phase current density to the electric field.
The consequences of this are that the effective parameters are not
the true quantities as defined in Section 1.7 but they are combinations
of them. Appealing to Maxwell’s equations in the frequency
domain, Fuller and Ward (1970) show the effective quantities to
be

σ_{e}(ω)
= σ'(ω)
+ ωε''(ω), |
(1.8.1) |

ε_{e}(ω)
= - σ''(ω)/ω
+ ε'(ω), |
(1.8.2) |

and the effective dielectric constant is

K_{e}(ω) =
- σ''(ω)/(ωe_{0})
+ K'(ω). |
(1.8.3) |

Examples of laboratory measurements of the effective dielectric constant and conductivity for various rocks as functions of rock type, water content, and frequency, are presented in Figures 1.10 and 1.11(Ward et al., 1968).

Figure 1.10. Effective dielectric constant K_{e} versus
frequency for selected earth materials (modified from Ward et al., 1968). |

Figure 1.11. Effective conductivity σ_{e}
versus frequency for selected earthmaterials (modified from Ward et al., 1986). |

### Low Frequency Resistivity of Rocks

The low frequency (i.e., approximately zero frequency or *direct
current*, *dc*) resistivity range found in natural materials
spans over 20 orders of magnitude, from highly conductive ore minerals
to highly insulating minerals such as micas. Figure 1.12 is a repeat of Figure 0.3 (Palacky, 1988).

Figure 1.12. Typical ranges of electrical resistivity (ohm-m) or conductivity (mS/m) for selected Earth materials (Palacky 1988).. |

Water is the most important variable in hydrologic resistivity studies because the solid rock matrix or soil grains are comparatively very good insulators (Figure 1.12). Consequently, the ions dissolved in water are the electric charge carriers responsible for the bulk material’s ability to conduct electricity. This is true for both "clean" (clay-free) and "shaly" (clay-rich) formations as will be explained below.

Factors affecting the resistivity of surficial earth materials (e.g., McNeill, 1990) are: 1) porosity, 2) texture which influences pore space interconnectivity, 3) water resistivity, 4) degree of water saturation, 5) temperature, and 6) clay content.

The first three variables are included in the
classical form of Archie’s empirical “law” (Archie,
1942) which assumes complete water-saturation
and no clay. Here, the bulk (total, t) formation resistivity, ρ_{t} is given by

ρ_{t} = ρ_{w} a φ^{-m} . |
(1.8.4) |

where a is called the coefficient of saturation, ρ_{w} is the resistivity of the water (or electrolyte), φ is the fractional porosity, and m is usually
called the cementation factor. The dimensionless
constants in Archie’s equation vary with
a = 0.6 to 1.0 (a = 1 is often implicitly assumed)
and m = 1.4 to 2.2 (Ward, 1990). The ratio ρ_{t} /ρ_{w}, referred to as the formation factor, equals φ^{-m} when a = 1 and is __~__ 4 for common values
(McNeill, 1990) such as m = 1.6 and φ = 0.4 (40
% saturated porosity).

Instead of the resistivity, its reciprocal the conductivity, is often reported instead, hence the alternate form of Archie’s law:

σ_{t} =
1/a σ_{w}φ^{m}. |
(1.8.5) |

Archie’s law (equation1.8.6 or 1.8.5) is
obviously a nonlinear expression since the functional
relationship, e.g., between σ_{t} and σ_{w} above,
is a power function of φ. This relationship is
a straight line when the log of σ_{t} versus the
log of σ_{w} is plotted as is commonly done.

Archie’s law (1.8.4) has been modified to
allow for any degree of water saturation, S_{w}
(the volume ratio of the pore water to the total pore volume, factor 4 above) resulting in

ρ_{t} =
(a ρ_{w}φ^{-m})/
S_{w}^{n} |
(1.8.6) |

The saturation exponent n is often assumed to
be 2. As pointed out by Ward (1990), this
equation is important in groundwater studies
since the fraction of water saturation, S_{w} can
be calculated if a, m, and n can be estimated
(e.g., a = 1, m = 1.6, and n = 2) and ρ_{w}, ρ_{t},
and φ can be independently measured. The water
resistivity, ρ_{w} can be measured in the field
or in the lab; ρ_{t} can be measured by borehole
electric logs or by field measurements; and φ can be measured by borehole porosity logs, in
the laboratory, or simply estimated. Use of Archie’s
saturation equation (1.8.6) or Archie’s
law (1.8.4) makes one very big assumption and
that is that the formation is clay-free. The
effect of clay content is discussed below.

The influence of factor 5 above, namely temperature,
results in a decrease in resistivity as temperature
rises (up to a few hundred ^{o}C) since the ion
mobility in water increases. Usually this is
not significant in very near-surface studies since
the temperature coefficient of resistivity is
about 2.5% per ^{o}C (Ward, 1990). However,
it can be a substantial effect depending on the
geothermal gradient and water salinity. This
is the basis for using resistivity methods in
prospecting for water-dominated geothermal systems.
The dependency of water resistivity on salinity
and temperature is illustrated in Figure 1.13
from Nesbitt (1993).

Figure 1.13. Electrical
resistivity versus temperature for varying concentrations of KCl and pressures (Nesbitt, 1993). |

The effect of factor 6 above, the clay content,
is the subject of numerous publications on the
resistivity of clay-bearing formations. Most
mathematical models use the fraction of clay
present or the cation-exchange capacity of the
specific clay mineral to empirically estimate
the bulk formation resistivity (or conductivity).
Recent nonlinear formulations of this problem
use a separate clay conductivity, σ_{s} (s for shale),
embedded in a host fluid of conductivity σ_{w}. This
serves to emphasize that the unusually high conductivity
of clay occurs only when clay particles are surrounded
by water resulting in an additional form of ionic
conduction. Conversely, this also emphasizes
that dry clay has high resistivity.

The abnormally low resistivity of clay minerals
is caused by mobile cations in the fluid that
are required to balance the negative surface
charge on most clay particles. Negative surface
charge is caused by broken bonds on grain edges
and crystal lattice substitutions on cleavage
surfaces. The ion exchange capacity of a mineral
is a measure of the number of cations required
to neutralize this negative surface charge. Every
mineral has an ion exchange capacity; clay minerals,
especially montmorilllonite and vermiculite,
have large values. But, because the cation exchange
phenomenon is surface dependent, it is also greater
for fine grain compared to coarse grain formations
with identical volumes of the same clay mineral. Since
both the mineralogy and grain size are important,
both clay minerals and clay-size particles exhibit
enhanced ion exchange capacity. In the presence
of a fluid the adsorbed cations form a double
layer attached to clay surfaces (Ward, 1990).
One layer is immobile and fixed to the clay particles.
The other layer, referred to as the diffuse layer,
varies in concentration in the fluid as a decreasing
function of distance from the clay surfaces.
The diffuse cations are free to move under the
influence of a voltage-causing electric field.
These loosely bound cations effectively increase
the charge density available for ionic conduction
in the electrolyte. Such an additional conductivity
contribution is referred to as *surface conductivity* or the *double layer effect*. It can exceed the
normal water conductivity by many times and is
thus of utmost importance in clay-rich zones.
The effect is generally more important when the
water conductivity, therefore, the intrinsic
ion content, or the porosity is low. It is important
to realize that when water is extracted from
a formation, the conductivity measured of the
sample is that of the intrinsic water conductivity, σ_{w}. It does not include any contribution from
the exchange cations which remain attached to
the mineral grains. Therefore, measured water
conductivity values should be considered as under-estimates
of the effective *ionic conductivity*, σ_{e}, as measured by EM geophysics when clay
is present.

Mathematical expressions that include all of the
six factors above have too many unknowns to permit
their general, practical application. However,
in some cases the local aquifer conditions can
be characterized and known or estimated parameters
permit application of expressions such as Archie’s
relations. It is expected that the effect of
clay becomes negligible in cases where the water
conductivity (therefore, salinity) is large resulting in σ_{w} >> σ_{s}. Surprisingly, when the water conductivity
is low, σ_{w} << σ_{s}, a form of Archie’s
law (1.8.5) is appropriate. However, estimates of conductivity
when σ_{w} __~__ σ_{s} easily can be wrong by orders
of magnitude when simply applying Archie’s
relations in clay-rich environments.

A free software program called SIGMELTS (Pommier and Le Trong, 2010) is available online to calculate the bulk conductivity of two-phase Earth materials using various models including Archie's law. The program also allows discrimination of variables such as temperature, pressure, and water content.

### High Frequency EM Properties of Rocks

Table 1.1 lists electromagnetic properties of various Earth materials at a frequency of 100 MHz (Davis and Annan, 1989).

Table 1.1. Dielectric constant, conductivity,
velocity and attenuation at 100 MHz for various geologic materials. |

**References**

Archie, G.E., 1942, The electrical resistivity log as an aid on determining some reservoir characteristics: Trans. Am. Inst. Min. Metal. Am. Petr. Eng., v.146, 54-62.

Davis, J.L. and A.P. Annan, 1989, Ground-penetrating radar for high-resolution mapping of soil and rock stratigraphy: Geophys. Prosp., 37, 531-551.

Feynman, R. P., Leighton, R. B., and Sands, M., 1963, The Feynman Lectures on Physics, vol. 1, 22.10, Addison-Wesley Pub. Co.

Fuller, B.D and Ward, S.H., 1970, Linear system description of the electrical parameters of rocks: IEEE Trans. Geosci. Elect., GE-8, 1, 7-18.

McNeill, J.D., 1990, Use of electromagnetic methods for groundwater studies, in Ward, S.H., Ed., Geotechnical and environmental geophysics, 01, Soc. Expl. Geophys., 191-218.

Nesbitt, B.E., 1993, Electric resistivities of crustal fluids: J. Geophys. Res., 98, 4301-4310.

Palacky, G. J., 1988, Resistivity characteristics of geologic targets, *in* Investigations in Geophysics vol. 3: Electromagnetic methods in applied geophysics-theory, vol. 1, edited by M. N. Nabighian, Soc. Expl. Geophys., 53–129.

Pommier, A., and Le Trong, E., 2010, “SIGMELTS: A web portal for electrical conductivity calculations in geosciences, Computers and Geosciences, vol. 37, 1450-1459.

Ward, S.H., 1990, Resistivity and induced polarization methods: in S. H. Ward, Ed., Geotechnical and environmental geophysics, vol. 1 – Review and tutorial, Soc. Explor. Geophys., 147-189.

Ward, S.H, Jiracek, G.R., and Linlor, W.I., 1968, Electromagnetic reflection from a plane-layered lunar model: J. Geophys. Res., 73, 1355-1372.