122 A new concept of karst development based on hydrogeology and geophysics potential (M and N). The greater the spacing between the quadruplet of electrodes (A and B, M and N), the thicker the slice of ground surveyed around the borehole. Using this quadrupole configuration for surface measurements, the apparent resistivity ρa (in Ohm·m) of the ground through which the current flows is calculated using the following formula: ρa = K ΔV / I With I the current intensity, ΔV the potential difference between electrodes M and N, K a geometric factor depending on the electrode arrangement: K = 2π × (1/AM – 1/AN – 1/BM + 1/BN)–1 The factor 2π is used for surface measurements, as it represents the half-sphere of current displacement around the injection electrodes. This half-sphere is the result of the electrode positioning at the interface between two media: • the ground, through which the current propagates. • the infinitely resistant air, where the current does not propagate. In a vertical borehole, the acquisition device is placed inside the well. The contact between the electrodes and the formation is indirect, occurring through the borehole fluid, which is typically either a water-based drilling mud or fresh water. The resistivity of fresh water generally ranges from 10 to 1000 ohm·m, depending on factors such as ionic concentration, temperature, and the presence of dissolved solids. For water-based drilling muds, resistivity typically ranges from a few ohm·m to around ten ohm·m, depending on salinity and temperature. The apparent resistivity equation for a borehole configuration is similar to that used in surface measurements but must take into account the cylindrical geometry of the medium. In certain idealized cases—such as a narrow borehole within a homogeneous formation—apparent resistivity can be approximated using a formula analogous to that of surface configurations, with an adapted geometric factor K. This factor depends on electrode spacing, the relative position of the electrodes in the well, borehole diameter, borehole fluid properties, and the geological model. The borehole resistivity profile is then obtained by successively combining different quadrupoles along the measurement cable, with each electrode alternately playing the role of current injection or measurement (Fig. 3). Figure 3 Electrical resistivity measuring devices based on a Wenner-Schlumberger scheme, with a = MN (m) separation and n = acquisition data level (after D. Chapellier, 2001a).
RkJQdWJsaXNoZXIy NjA3NzQ=