828 GEOPHYSICAL WELL TESTING [Chap. 11 



or camera drive is coupled to the cable winch, gives a direct graph of 

 apparent resistivity against depth. Resistivities and self-potentials are 

 usually recorded simultaneously as indicated schematically in Fig. 2-15. 

 In homogeneous ground, the relation between resistivity and potential 

 readings given at pages 710-711, apply. Since the electrodes are buried in 

 an infinite and not a semi-infinite medium, the spacing factors given in 

 these formulas change from 2ira to 47ra, so that, for the Schlumberger 

 arrangement, with the notation of Fig. 11-ld, 



P = -, Y ' (11-1; 



r — r L 



which follows directly from eq. (lO-31/i). Since r, r', and / are kept con- 

 stant, it is seen that the resistivities are directly proportional to the poten- 

 tial difference recorded. Formula (11-1) holds for homogeneous ground 

 only. In the presence of layers of different resistivity, an apparent instead 

 of a true resistivity is recorded. The relation of apparent and true resis- 

 tivities for given formation thicknesses and resistivities can be calculated 

 in the same manner as on page 719 since by rotating the geologic section 90° 

 the bedding planes are equivalent to the vertical formation contacts treated 

 in resistivity mappmg, assuming for the moment that the effect of the 

 drilling fluid is negligible. 



When a formation contact is passed with the four-electrode system 

 (Fig. 11-1/), an apparent resistivity variation as in Fig. 10-56 is obtained 

 with four breaks near the formation boundary instead of one. As the 

 number of electrodes is reduced, the number of peaks decreases, as illus- 

 trated in Fig. 11-2 for the Schlumberger electrode arrangement. The 

 two upper diagrams are for one formation boundary and the two lower 

 diagrams for two boundaries. The distance of the breaks is uneven and 

 corresponds to CF\ in one and P1P2 in the second case. The resistivity 

 of the second medium is approached in steps and not all at once. For 

 formations whose thickness is large compared with the electrode separa- 

 tion, the corresponding apparent resistivity curves may be derived by 

 joining curves such as those given in Fig. 11-2 (upper part). For thin 

 formations, the images of the current electrode, due to reflections on hoih 

 formation bovmdaries, must be considered, which means that resistivities 

 must be calculated as in a three-layer case. Results of such calculations 

 are illustrated in Fig. 11-2. It is seen that a formation whose thickness 

 is less than the distance of one potential electrode from the next current 

 electrode will not allow the apparent resistivity to come up to the value 

 of the true resistivity. This is possible only when the electrode separation 

 is smaller than the formation thickness. 



2 J. N. Hummel, Beitr. angew. Geophys., 6(1), 89-99 (1936). 



