ELECTRICAL METHODS 



479 



Case IV: Layers of Different Materials. — The discussions of cases 

 I, II, and III yielded expressions for the effective resistivity p of a homo- 

 geneous medium as a function of potential differences observed at the sur- 

 face, distances measured along the surface (e.g., electrode separations), 

 and the total current / passing between the energizing electrodes. In the 

 present case (IV), a new concept appears: namely, variation of effective 

 resistivity with depth. 



t.o 

 .9 

 .6 



.7 

 .6 



^. 



.4 

 .3 

 .2 



8 



10 II 



J 2 3 4 5 6 



Fig. 287. — Plot of the fraction of the total current penetrating below 

 a plane C located at a depth of d units versus the ratio of the depth d to 

 the electrode separation L. Ratio of conductivity of upper to lower layer 

 materials is 1:1. When the lower layer has a conductivity greater than 

 the upper layer, the ratio I^I becomes smaller. 



Suppose, for example, that for a given area explored by one of the 

 methods described under Case III, it is found that the calculated values 

 of p are the same for all electrode separations. Hence, if a plot of the 

 values of the resistivity as ordinate against the values of the electrode 

 separation as abscissa is a horizontal, straight line, one may infer that the 

 subsurface structure is uniform. Generally, however, a resistivity graph 

 of field data yields a complicated curve indicating that the average re- 

 sistivity varies with depth below the surface. In this case, the description 

 or deduction of subsurface conditions from the geoelectrical data may be 

 quite difficult. 



Physical considerations alone may yield useful information. Consider, 

 for example, a semi-infinite medium of resistivity p2 covered by a layer of 

 resistivity pi and of thickness A. The source and the sink are placed at 

 the boundary of the upper layer as shown in Figure 288. The following 



