EM COUPLING IN MULTIFREQUENCY IP AND A GENERALIZATION OF THE COLE-COLE IMPEDANCE MODEL*

The Cole-Cole relaxation model has been found to provide good fits to multifrequency IP data and is derivable mathematically from a reasonable, albeit greatly simplified, physical model of conduction in porous rocks. However, the Cole-Cole model is used to represent the mutual impedance due to inductive or electromagnetic coupling on an empirical basis: this use has not been similarly justified by derivation from any simple physical representation of, say, a half-space, layered or uniform. A uniform conductive half-space can be represented as a simple subsurface loop with particular resistive and inductive properties. Based upon this, a mathematical expression for the mutual impedance between the two pairs of electrodes of a dipole-dipole array is derived and designated “model I”. It is seen that a degenerate case of model I is the Cole-Cole model with frequency exponent c= 1. Model I is thus more general than the Cole-Cole expression and must provide at least as good a fit to a set of field data. Provision for variation of c from unity could be made in model I equally well as for the Cole-Cole model although, at present, this would be a purely empirical alteration. Model I contains four parameters, one of which is, in effect, the resistivity of the half-space. Therefore only three parameters are involved in the model I expressions for normalized amplitude and for phase of the EM-coupling mutual impedance. Model I is compared with previously published “standard” values for two different dipole separations. Under particular constraints, model I is shown to provide better fits than the Cole-Cole model (with c= 1) over particular frequency ranges, specifically at very low frequencies and at moderately high frequencies where the model I phase curve follows the standard phase curve across the axis to positive values (negative coupling).