634 ELECTRICAL METHODS [Chap. 10 



In alternating electrical fields the current is not in phase with the e.m.f . 

 Current conduction is controlled by the resultant modulus of resistance 

 and reactance, which is known as impedance. Impedances are customarily 

 represented by conjugate numbers with the resistive component on the 

 real axis, the reactance on the imaginary axis, and the phase shift as the 

 argument of the complex quantity. 



Although most rock-forming minerals are insulators, it does not follow 

 that most rocks are poor conductors. This is true only if they are solid 

 throughout (such as some igneous rocks and chemical sediments). The 

 majority of rocks and formations are porous, are filled with more or less 

 conductive moisture, and act as electrolytes. Their conductivity depends 

 on four factors: (1) pore volume; (2) disposition of pores (grain packing); 

 (3) portion of pores filled with water; and (4) conductivity of the water, 

 which is composed of (a) a primary conductivity (that is, conductivity of 

 the water as it enters the pores), and (6) a secondary conductivity (acquired 

 by solution of mineral matter and therefore dependent on duration of con- 

 tact [stagnation]). These relations may be expressed by the following 

 formula : 



Pi = -Pi or (Tx = —-(Ti, (10-6) 



Vi c 



where px is the resistivity and <rx the conductivity of the rock, c a constant 

 depending on the arrangement of the pores, vi the pore volume, pi the 

 resistivity and <ti the conductivity of the water or other medium filling 

 the pores 



For specific arrangements of mineral grains of regular geometric shape, 

 it is possible to calculate not only the pore volume and thus the resistivity 

 as a function of the resistivity of the water filling the pores, but also the 

 relative effect of the conductivity of the grains compared with the effect 

 of the medium in which they are imbedded. Maxwell has derived a 

 general relation for the conductivitj^ ax of an aggregate consisting of a 

 medium with the conductivity <ri , in which spherical grains of the con- 

 ductivity 0-2 are imbedded in regular arrangement and in such a manner 

 that their distance is large compared with their radius.^" If the total volume 

 of the aggregate is Vx and the total volume of the grains Vi , and if the 

 ratio of these two volumes V2/vx — r, then the following relation obtains: 



2ai -h 0-2 — 2r(ai — a) /'1n^'7„^ 



(Tx = -K i j — 7 -^-o-i. (10-7aj 



^<Tl + 0-2 + r(<ri — (Ti) 



Hence, in terms of the (pore) volume Vi (if the pores are filled completely 

 with the medium of the conductivity <ri), 



i«See also J. N. Hummel, Beitr. angew. Geophys., 6(1), 32-132 (1935). 



