For rocks composed of non-conducting matrix minerals and saturated with water, an empirical relationship known as Archie's Law is useful in analysis of electrical properties. Archie's Law is commonly written
(2-1)
where 
FLUID equals the
electrical properties of the fluid in the pores, F is
the porosity (ratio of void volume/total volume), and A and m are constants
that depend on the geometry of the pores. For many rocks, A = about 1 and
m = about 2. See Keller, G.V. (1982) for a broader discussion. Papers
discussing various A and m values for specific rocks (shaly sands, clean sandstones, etc.)
have been published in the journal Geophysics.
Note: there are several versions of Archie's Law that attempt to include the effects of partial saturation (water-gas or water-oil) or mixed fluids in the pores, or, the air water mixes in the vadose zone. We will not attempt to use these formulas in this course but it is necessary to point out the fact that Equation 2-1 is not the only way in which Archie's Law is written.
Archie's Law is not valid for rocks containing a significant percentage of clay. Clay provides for conductive matrix, rendering a fundamental assumption invalid. Graphite, native metals, and minerals with metallic lusters are also electrical conductors, but these are far scarcer than clays.
The electrical resistivity of a fluid depends on the amount of ionic material in solution and on the temperature of the liquid. Figure 2-1 is a nomogram that allows you to compute relationships between temperature, TDS (equivalent NaCl salinity) and fluid resistivity - as well as the relationship between rock resistivity, fluid resistivity and "formation factor" (A *F-m). There are methods for calculating equivalent NaCl based on TDS involving a mixture of components but I do not have that at my fingertips (please inform me if you come across this information and I will incorporate it into my class notes). If you ever find such information is critical to your research, contact me and we should be able to find it.
NOTE: Based on Figure 2-1, Archie's Law can written where
F is porosity. This is the same as equation 2.1 with A = 1.
FIGURE 2.1:
Nomogram relating resistivity, formation factor, salinity and temperature. Any of these parameters may be determined if the other three are known. After Meidev (1970). This nomogram was developed for geothermal fields. Dr. Meidev is an international expert in the use of electrical resistivity in geothermal exploration.