Chap. 10] 



ELECTRICAL METHODS 



711 



profile at right angles to the power electrode base. If ri = a is the dis- 

 tance to the first potential electrode and Ri = 6 the distance to the 

 second one, and R2 = r2 = <» , 



P = 



2xa5 V 

 (b-a)' r 



(10-3U) 



7. The single probe method corresponds to ar^'angement 4, with one 

 power electrode in infinity. Since n = 0, r2 = 00 , Ri = o, and R2 = <» , 



p = 2x0 



(V. - V) 



(10-310 



Arrangement 1 is most frequently employed, for both vertical electrical 

 drilling and resistivity mapping. Next in order is probably 2, then fol- 

 lows 4. Arrangements 5 and 6 are con- 

 venient for vertical electrical drilling and 

 are used in the potential-drop-ratio proce- 

 dures with one additional potential elec- 

 trode. Arrangement 5 is also applied in 

 electrical logging. 



C. Potential Functions for Layered 

 Media 



1. GeneraV^ If a difference in resistivity 

 exists on a formation boundary (see Fig. 

 10-49) its effect may be represented by 

 placing a plate with definite transmission 

 and reflection characteristics in the boun- 

 dary. Assume a source at P above the 



plate. Considering the phenomenon for the moment as one of light 

 transmission, an observer at A facing the plate would see the point P 

 by looking at its image /. The light at A would be that received di- 

 rectly from P plus the amount reflected by the plate and appearing to 

 come from the image /. If the dimming of the apparent source at /, 

 due to reflection, be indicated by a factor k, the light and by analogy 

 the potential at A is equal to its amount at the source diminished by 

 the geometric effect of distance (1/r) plus the amount reflected, so that 



Fig. 10-49. Conditions on 

 boundary between two media 

 of different resistivity. 



47r \ri n/ 



(10-32a) 



'* For a more rigorous discussion of the theory of images than the one given 

 here to illustrate merely its elements, see J. H. Jeans, Mathematical Theory of Elec- 

 tricity and Magnetism, 4th ed., Cambridge U. Press, 1923, pp. 200-201. 



