RESISTIVITY SOUNDING ON A THREE-LAYER TRANSITIONAL MODEL*

The expression for the potential due to a point source of current, placed on the surface of a horizontal three-layer earth, has been derived when conductivity in the second layer varies linearly with depth and changes abruptly at the boundaries. Master curves for Wenner and Schlumberger configurations have been presented for an insulating basement and for one value of conductivity gradient in the second layer.     

RESISTIVITY SOUNDING ON A MULTILAYERED EARTH CONTAINING TRANSITION LAYERS*

For curves over a horizontally stratified earth where any of the layers has a conductivity variation proportional to (1 +β·z)^N, where β and N are arbitrary constants and z is the depth to the layer, expressions for apparent resistivity for Wenner and Schlumberger sounding are derived. No assumption has been made about the continuity of conductivity at the interface. It is shown that most of the previous investigations in this connection can be regarded as particular cases of the present study.     

RESISTIVITY SOUNDING ON AN EARTH MODEL CONTAINING TRANSITION LAYERS WITH LINEAR CHANGE OF RESISTIVITY WITH DEPTH*

Mallick and Roy solved the problem of determining the apparent resistivity function for a three-layer stratification in which the central layer is a transition layer with linear change of the conductivity with depth. In the present paper the problem is solved for a transition layer with linear change of the resistivity with depth, a type of change that seems to be more common in nature than the type considered by Mallick and Roy. The solution is extended to layer stratifications involving an arbitrary number of transition layers and of homogeneous layers. The solution is given in the form of a modification of the recurrence relation that was derived by Pekeris for homogeneous layers.     

MATHEMATICAL ANALYSIS OF D.C. RESISTIVITY SOUNDING OVER A PARABOLOID*

A complete mathematical analysis is proposed for direct current resistivity prospecting over the surface of a layered paraboloid. The analysis evaluates the Green's function in parabolic coordinates for a current source at the vertex. The general solution is obtained as a Fourier-Bessel integral involving those curvilinear coordinates that have a kernel function which is similar to that of a half-plane containing inhomogeneous layers. This similarity permits the computation of a class of sounding curves over such an oval surface providing a way to analyse field data over hilly terrain.     

THEORETICAL ANALYSIS OF APPARENT RESISTIVITY OVER A DYKE OF ARBITRARY SHAPE*

This paper deals with the apparent resistivity as observed on the surface due to a dyke of arbitrary shape. In order to give a closed analytical solution it has been necessary to assume that the dyke is either perfectly conductive or resistive relative to the enclosing medium. Furthermore we have considered an infinite line source instead of a conventional point current source; however a simple integral transform is given to transform the point-source-data into the line-source-data. So the present study is equally useful where line sources are inconvenient to handle. Besides considering the conventional method of electrical surveying (bipole method) where the source and sink are separated by a finite distance, we have considered a new variation (unipole method) where the source and sink are separated by an infinite distance, and the source is split into two separate sources each of half strength. A series of apparent resistivity curves for both methods are presented for different parameters of the dyke. The usefulness of these curves lies mainly in the fact that they may provide the necessary guide-lines for semi-quantitative interpretation of the observed data.     

INDUCTIVE SOUNDING OF A STRATIFIED EARTH WITH TRANSITION LAYER RESTING ON DIPPING ANISOTROPIC BEDS *

The generalised three layer boundary value problem with a transition layer sand-witched between an isotropic overburden and dipping anisotropic substratum is discussed assuming that plane electro-magnetic waves are incident normally over the air-earth inter-face. The tangential electric (E_y) and magnetic (H_x) fields and the expression for surface-impedance (E_y/H_x) have been evaluated at the earth's surface. Through numerical analysis it is shown that changes in the values of the parameters m (coefficient of anisotropy), h (thickness of the transition layer), α (angle of inclination of the dipping beds), and b (conductivity ratio between substratum and upper layer) modify the amplitude and phase-variation curves (with skindepth) significantly.     

A STUDY ON THE DIRECT INTERPRETATION OF DIPOLE SOUNDING RESISTIVITY MEASUREMENTS OVER LAYERED EARTH *

Dipole sounding resistivity measurements over layered earth can be interpreted directly by adapting the procedure given by Koefoed (1968) for Schlumberger system. To carry out the first step of the interpretation leading to the determination of the raised kernel function, partial resistivity functions for the dipole method are derived and given in the form of standard curves. The second step involving the derivation of layering parameters from the kernel being independent of the electrode configurations remains unaltered. The applicability and limitations of the method are discussed.     

RESISTIVITY SOUNDING ON A LAYERED TRANSITIONAL EARTH*

Master curves are presented for three-layer earth-sections, the electrical resistivity in the intermediate layer of which has a linear variation with depth. A new approach is proposed to interpret the sounding data with the help of RMS difference in apparent resistivity values.     

TRANSFORMATION OF DIPOLE RESISTIVITY SOUNDING MEASUREMENTS OVER LAYERED EARTH BY LINEAR DIGITAL FILTERING *

The technique of linear digital filtering as developed for the direct interpretation of Schlumberger and Wenner soundings (Ghosh 1971) has been applied here for the derivation of the resistivity transform function from the field dipole measurements as the first step in directly interpreting dipole data. Filter coefficients for this transformation have been worked out for the radial-polar, perpendicular and parallel (30°) arrays of dipole sounding. The procedure combines speed with accuracy.     

RESISTIVITY SOUNDING ON A HORIZONTALLY STRATIFIED MULTI-LAYERED EARTH*

The electrical potential due to a point source of current placed on the ground surface is studied for a multi-layered earth consisting of homogeneous overburden of constant conductivity over a stack of transition layers where conductivity varies with depth according to power and exponential laws in even and odd layers, respectively. The general recursion relations are derived and are used to obtain expressions for the apparent resistivities for Schlumberger and Wenner electrode arrays. Their asymptotic behavior has been studied. The solutions for some particular cases are given: (i) odd layers with conductivities exponentially varying with depth while all even layers (and the first) have constant conductivities; (ii) even layer conductivities varying as a power law while odd number layers are of constant conductivity; and (iii) any two successive layers as transition layers and all others having constant conductivities. Further it is shown that Patella's theory is a particular case of the present study. It is concluded, therefore, that the present treatment is more general as all earlier models consisting of trasition layers can be derived from this study.     

UNCERTAIN RESISTIVITY SOUNDING AND EQUIVALENT MODELS*

Problems of the resistivity sounding method may be reparametrized by means of Dar Zarrouk—or generalized D.Z.—transformations. A complete set of equivalent models is subdivided into three classes: initial models, intermediate models, and all other models. Equivalent reparametrized models are subdivided into three analogous classes. There is no one-to-one mapping of intermediate primary and intermediate reparametrized models. Non-trivial equivalent models are generated from initial models via intermediate reparametrized models. A method is developed to find equivalent models relative to the uncertainty of resistivity soundings. There are different forms of approximate equivalence laws, apart from those commonly accepted. The equivalence depends considerably on the “flatness” of the sounding curve. For mean flatness, the electric models exhibit the (hp^±2)-equivalence rather than the (hp^±1)-equivalence.     

RESISTIVITY SOUNDING ON A MULTI-LAYERED EARTH WITH TRANSITIONAL LAYERS. PART II: THEORETICAL AND FIELD EXAMPLES *

This final part of our study of the resistivity sounding method in the presence of transitional layers deals with the direct application of the theoretical developments, previously reported, to both theoretical and field examples. The former are used to explain in more detail all steps of the direct procedure of interpreting resistivity measurements, when transitional layers with different dispositions are present in the assumed earth section. An alternate sequence of uniform and transitional layers is at first considered, then the case of a random sequence of these layers, and finally a sequence of only transitional layers. Then we give some logical considerations about the best fields of application of the new interpretation theory, with particular reference to hydrogeological, geological- structural, and mining exploration problems. Representative field examples are finally presented and discussed in detail for each of the exploration fields above listed. The comparison of the geophysical models obtained with the old interpretative scheme and the new system shows the greater resolving power of the new theory, provided that preliminary geological considerations, and where existing–borehole evidence justify the use of the new approach.     

I.P. AND RESISTIVITY MODEL STUDIES OVER THIN DYKES*

Induced polarization and resistivity model studies over thin dykes for varying resistivity contrasts, depth of burial, and dip angles show striking parallelism of the surface apparent resistivity contours with the boundary of the body. This effect may be utilized for the estimation of strike length of the body. Results show that intermediate values of the electrode spacing is satisfactory for detection under widely varying conditions of resistivity contrast and depth. The percentage frequency effect (P.F.E.) and the metal factor (M.F.) responses are found to be more sensitive to the variation in the depth of burial than the resistivity responses. Pseudosections for P.F.E. and M.F. are concentrated much closer to the body than the resistivity pseudo-sections.     

LABORATORY RESULTS IN RESISTIVITY LOGGING*

This paper is an experimental extension of the theoretical investigations by Roy (1975) on the relative performances of the Laterolog 7, normal and some other sondes in logging of resistive formations. Only infinitely resistive formations have been simulated and placed in a tank containing tap water (true resistivity 27 Ωm) as electrolyte—representing both the mud column and the adjacent formations. Two sets of laboratory results (Doll 1951, NN 1958, 1969), have been repeated and we find that, for both these sets, the performance of the normal device is by far the superior of the two. In addition, we have studied the effect of varying the spacings A₁A₂, O₁O₂ and AM of Laterolog 7, normal, and two new sondes—Laterolog 4 and modified unipole—for two bore hole diameters in each case. For formation thicknesses less than A₁A₂ or AM, the Laterolog 7 is unsuitable because its response is flat and close to the base-line value. The normal device is more diagnostic, although, in such a case, it registers a trough or a resistivity low even against a resistive formation. For bed thicknesses clearly greater than A₁A₂ or AM, the normal sonde is decidedly superior to Laterolog 7, since its anomalies are sharper and larger. When the formation thickness is equal to or only slightly larger than A₁A₂ or AM, Laterolog 7 is somewhat better as it records a readable positive deflection while the normal does not. However, one must remember that a single run of the conventional resistivity log includes two normals and a lateral at different spacings. Laterolog 4 and modified unipole can in many instances produce better logs than normal, other considerations apart. The results are consistent with our own theoretical predictions and experience in surface resistivity profiling. They do not, however, agree with the prevalent concepts on Laterolog 7 vis-a-vis normal sonde.     

RESISTIVITY SOUNDING ON A TWO-LAYER EARTH WITH TRANSITIONAL BOUNDARY*

A theoretical solution is obtained for the problem of a two-layer earth with transitional boundary. In practice, the transition layer can stand for the weathered zone in hard rock areas where the degree of weathering diminishes with depth. Master curves and tables of data are presented for the case when the lower half-space is infinitely resistive.     

THEORETICAL BROADSIDE RESISTIVITY PROFILES OVER AN OUTCROPPING DYKE

Theoretical broadside resistivity profiles over an outcropping dyke of infinite depth extent with three Wenner configurations (namely Alpha, Beta and Gamma), Schlumberger, and two-electrode configurations of various electrode spacings and for various reflection factors are presented. The broadside profiles qualitatively indicate that the shape of the anomaly is invariant with the electrode configurations. The various electrode configurations can be arranged in decreasing order of magnitude of anomalies as Beta Wenner, Alpha Wenner, Schlumberger, Gamma Wenner, and two-electrode. The broadside Wenner profiles also show larger anomalies compared to those in inline profiles.     

AN APPROXIMATE METHOD OF RESISTIVITY SOUNDING INTERPRETATION*

An approximate method of interpretation of resistivity sounding is presented, which may be described as a very crude manner of application of the exact direct interpretation method. The accuracy of this method is fairly low, the errors being in the order of 25%. The method is very fast in application and well suited for application to multilayer cases. The main advantage of the method is that it is in close and clear relation to the exact theory.     

AN ANALYSIS OF EQUIVALENCE IN RESISTIVITY SOUNDING*

A mathematical analysis is given of the phenomenon of equivalence in resistivity sounding, which is based upon the properties of the raised kernel function. Analysis of this function instead of the apparent resistivity function is justified because, as has been shown in a previous publication, variations in the apparent resistivity function lead to variations in the raised kernel function with relative values of the same order of magnitude The expression for the raised kernel function is expanded into a Mac Laurin series. Equivalence can occur only if the second order term of this series is negligible. The coefficient of the first order term depends on the resistivity and the thickness of the layer under consideration. There is an infinite set of combinations of values for these two quantities, for which the coefficient of the first order term has the same value. All these combinations represent equivalent layer distributions.     

BIPOLE-DIPOLE FIELD DIFFERENCE—A NEW RESISTIVITY SOUNDING TECHNIQUE*

The method of field differences using two parallel bipole sources is described. The system possesses much greater resolving power and investigating depth than the conventional Schlumberger configuration. Of the two variations, the longitudinal field difference has about two and half times greater depth of investigation than the transverse one. Practical advantage of the present technique over the Schlumberger system in reducing the ambiguity in interpretation due to equivalent layered structures is demonstrated on a theoretical example.