1020 



EXPLORATION GEOPHYSICS 



Equipotential 

 Spheres 



drilling mud filling the drill hole and the effective resistance of 

 the earth material near the electrode. The contact resistance is 

 relatively constant between electrode and drilling mud through- 

 out the entire length of the drill hole, while the effective resist- 

 ance of the earth materials near the moving electrode varies. 



Since the resistances R^ and R2 remain relatively constant throughout 

 the measurements, the measured total resistance R is equal to a constant 

 resistance plus a varying resistance. The changes in the value of the resist- 

 ance in the immediate vicinity of the moving electrode A may therefore be 



plotted as the moving electrode traverses 

 the bore-hole. 



The theoretical potential drop 

 around a single electrode may be de- 

 rived. Assume the electrode to have 

 small dimensions so it can be treated as 

 a point electrode, and that it is sur- 

 rounded by a homogeneous and iso- 

 tropic medium of infinite extent, with 

 the bore hole of such small diameter 

 that its effect can be neglected. Refer- 

 ring to Figure 633, in the vicinity of A 

 the current will tend to flow radially outward equally in all directions, and the 

 equipotential surfaces will be spheres centered on A. 



The difference of potential dE between points located on an equipoten- 

 tial sphere of radius r and of potential E, and an equipotential sphere of 

 radius r + dr and of potential E + dE is expressed by a form of Ohm's 

 Law: 



Fig. 633. 



Lines of 

 Current Flow 



-Equipotential surfaces around a 

 point electrode. 



dE = I 



- T.P: 



dr 



47rr^ 



(2) 



where p is the resistivity of the medium. 



By integration, taking the absolute potential to be zero at great dis- 

 tances from A, the potential £ at a distance r from source A is obtained : 



Airr 



(3) 



Equation 3 still applies to the space surrounding the electrode if r is the 

 radial distance measured from the center of the electrode. This equation 

 may be rewritten : 



E 

 I 



P 



(4) 



The ratio E/I, which is seen to be proportional to the resistivity p of the 

 surrounding formation, may be interpreted as the resistance of a mono- 

 electrode of radius r, and it is in fact the resistance which would be meas- 



