THE USE OF TWO-ELECTRODE AND SCHLUMBERGER FILTERS FOR COMPUTING RESISTIVITY AND EM SOUNDING CURVES*

Different sets of filter coefficients for the linear filter technique for the computations of resistivity and EM sounding curves are evaluated for several electrode and coil configurations. Instead of this procedure, the two-electrode filter can be used for computations of Wenner, Schlumberger, and dipole—dipole apparent resistivity model curves by defining convolutional expressions which contain the new input functions in terms of the resistivity transform function. Similarly, the Schlumberger filter performs the computations of dipole—dipole apparent resistivity model curves. The Wenner, Schlumberger, and dipole—dipole filter functions are defined in terms of the two-electrode filter using the new convolutional expressions. A relationship between the Schlumberger and dipole—dipole filter functions is given. The above arguments are adopted for the computations of EM sounding curves. It is shown that the EM filter for the horizontal coplanar loop system (which is identical to the two-electrode filter) performs the computations of the mutual coupling ratios for perpendicular, vertical coplanar, and vertical coaxial loop systems. In the same way, the Schlumberger filter can be used to compute vertical coaxial sounding curves. The corresponding input functions are defined in terms of the EM kernel for all convolutional expressions presented. After these considerations, integral expressions of the mutual coupling ratios involving zero-order Bessel function are derived. The mutual coupling ratio for the vertical coaxial loop system is given in the same form as the mutual coupling ratio for the vertical coplanar loop system.     

THEORY FOR THE BIPOLE-DIPOLE METHOD OF RESISTIVITY SOUNDING*

A theory for the bipole-dipole method of resistivity sounding is developed. Bipole-dipole apparent resistivities are related to Schlumberger apparent resistivities at two spacings. The theory can also be used to compute exact dipole-dipole apparent resistivity curves providing an improvement over the existing techniques which involve far field approximations. A comparison of bipole-dipole and dipole-dipole systems reveals the similarity between the two. However, the resolution of the bipole-dipole system depends on the azimuth angle. The flexibility of the theoretical expressions lead to a generalized field scheme independent of the bipolar or dipolar nature of the current source.     

DIRECT INTERPRETATION OF TWO-ELECTRODE RESISTIVITY SOUNDINGS *

This paper describes the procedure for interpreting the apparent resistivity data measured with the two-electrode array directly with the help of kernel function. The calculation of kernel function from the observed resistivity curve is done by the method of decomposition. In the method of decomposition the resistivity curve is approximated by a sum of certain functions, whose choice is only restricted by the requirement that the contribution to the kernel function corresponding to them should be easily computable. A few such functions are classified. These, and the standard curves for corresponding kernel functions obtained by utilising an integral expression for two-electrode array expressing the kernel explicitly in terms of the apparent resistivity functions, are plotted on log-log scale. The determination of layer parameters, that is, the layer resistivities and thicknesses from the kernel function can be carried out by a method proposed by Pekeris (1940).     

ON THE INTERPRETATION OF AIRBORNE ELECTROMAGNETIC DATA FOR THE TWO-LAYER CASE*

For helicopter-borne electromagnetic systems, the distance between the transmitting and the receiving coils is small compared with the altitude above ground. For this case, a major simplification can be made for the calculation of model curves. Some two-layer curves for the interpretation of frequency measurements are presented. A very simple procedure is demonstrated for the conversion of the relative secondary field into apparent resistivity and apparent distance for the mapping of airborne electromagnetic data. Furthermore, an approximation is described for the determination of the thickness and the resistivity of a layer lying on a perfectly conducting half-space.     

TRANSFORMATION OF DIPOLAR,WENNER AND TWO-ELECTRODE CURVES TO SCHLUMBERGER APPARENT RESISTIVITY SOUNDING CURVES*

A simple mathematical technique based on regressional analysis allows the transformation of dipolar, Wenner and two-electrode apparent resistivity sounding curves to Schlumberger ones. The algorithm is suitable for a programmable pocket calculator and the accuracy is very high. This has been demonstrated by comparing Schlumberger master curves with transformations of master curves for the other configurations for the same model.     

THE USE OF A SQUARE CONFIGURATION IN RESISTIVITY PROSPECTING *

Consideration is given to the use of a configuration of four electrodes set in a square array for resistivity measurements. It is found that, by passing current successively between different pairs of electrodes, an apparent resistivity can be determined which is both more sensitive to the position of the array centre and less dependent on orientation than the measures usually obtained with colinear arrays of electrodes. At the same time the observations made enable the degree of the departure of local conditions from conditions of lateral homogeneity to be assessed. Theoretical and practical examples of the use of this electrode system are given and the use of the system both as a tool in mapping and in depth investigations is considered. It is shown that provided electrode spacings are suitably arranged the results of a probe carried out using the square array can be interpreted by conventional methods. The system is shown to have particular advantages in the investigation of lateral resistivity variations and the reduced dependence on orientation makes possible the recasting of interpretation data in an orientationally invariant form with a consequent drastic reduction in the number of type curves required for a particular problem.     

TRANSFORMATION OF RESISTIVITY SOUNDING MEASUREMENTS OBTAINED IN ONE ELECTRODE CONFIGURATION TO ANOTHER CONFIGURATION BY MEANS OF DIGITAL LINEAR FILTERING*

The technique of digital linear filtering is used for transformation of apparent resistivity data from one electrode configuration into another. Usually filter spectra are determined via the discrete Fourier transforms of input and output functions: the filter characteristic is the quotient of the spectra of the output function and input function. In this paper, the transformation of the apparent resistivities is presented for four electrode configurations (Wenner, the two-electrode, Schlumberger, and dipole configurations). In our method, there is no need to use the discrete Fourier transform of the input and output functions in order to determine the filter spectrum for converting apparent resistivity in one electrode configuration to any other configuration. Sine responses for determination of the derivative of apparent resistivities are given in analytical form. If the filter spectrum for converting the apparent resistivity to the resistivity transform for one electrode configuration is known, the filter spectra for transforming the apparent resistivity to the resistivity transform for any electrode configurations can be calculated by using newly derived expressions.     

MATRIX METHOD FOR THE TRANSFORMATION OF RESISTIVITY SOUNDING DATA OF ONE ELECTRODE CONFIGURATION TO THAT OF ANOTHER CONFIGURATION1

Matrix equations are derived to transform the resistivity sounding data obtained in one type of a four-electrode array to the corresponding resistivity sounding data that would be obtained using a different four-electrode array. These expressions are based primarily on recent work in which we have established a linear relation between the apparent resistivity and the kernel function by using a powerful exponential approximation for the kernel function. It is shown that the resistivity sounding data of two different four-electrode arrays have a linear relation through an essentially non-singular matrix operator and, as such, one is derivable from the other for a one-dimensional model and it can also be extended to two-dimensions. Some numerical examples considering synthetic data are presented which demonstrates the efficiency of the method in such transformations. Two published field examples are also considered for transformation giving a reliable interpretation.     

NEW RESULTS IN RESISTIVITY WELL LOGGING*

Two independent theoretical analyses show that, spacing for spacing, the two-electrode normal device investigates deeper than the seven-electrode focused Laterolog 7. The first analysis, published earlier, compares the radii of investigation of the different sondes from the portion of the ground that contributes maximum to, or fifty percent of, the measured signal. The second relates to the nature of the departure curves for the two sondes for an infinitely thick resistive formation pierced by a bore hole. With increasing spacing, the apparent resistivity for the normal device rises much faster and asymptotically approaches the true resistivity much earlier than that for Laterolog 7. These two analyses also prove that, in a Laterolog 7, the radius of investigation increases as O₁O₂ decreases relatively to A₁A₂. Since any complex point electrode system is equivalent to a superposition of elementary dipoles or two-electrode devices, it follows that no resistivity sonde of discrete point electrodes can have a radius of investigation larger than that for the normal device of the same spacing. Laboratory measurements in a model tank confirm these theoretical results. For thin highly resistive formations we find from theoretical and model studies that the normal device is markedly superior to Laterolog 7 as it gives apparent resistivity values much closer to the true. It has other advantages as well. Thus, our results on the comparative performances of the normal and Laterolog 7 devices are at variance with those published so far since the introduction of Laterolog 7 in 1951. For reasons given in the text, the chief characteristic of Laterolog 7—namely, focusing the current from the central electrode in a thin horizontal sheet into the target formation—does not endow it with any special advantage. The main reason is that the measured potential in a Laterolog 7 is caused not only by the current focused into the target formation but also by the currents from the outer two electrodes which flow, due to the same focusing process, entirely through the adjacent formations and the mud column. Indeed, the contribution to the measured signal by the outer two power electrodes is considerably larger than that from the central.     

Behavior of electric potential fields over random conductivity,layered earth models

Solutions are derived for the potential distributions over one-layer and two-layer random conductivity earth models for the direct current resistivity method. Monopole and Wenner arrays are considered. It is shown that the random potential due to a spatially stationary Gaussian random conductivity is also Gaussian but not spatially stationary. The ensemble and sample statistics of the random kernel and apparent resistivity are examined. Representative curves of these functions are presented to show the effects of a random conductivity profile on them. It is shown that the effect of the random nature of the conductivity profile on an apparent resistivity sounding curve is greatest for small electrode spacings.     

EM AND IP RESPONSE OF A STEEL WELL CASING FOR A FOUR-ELECTRODE SURFACE ARRAY. PART II: NUMERICAL RESULTS*

Some numerical results for the apparent resistivity of a homogeneous half space containing a vertical steel well casing as measured by a dipole-dipole array are presented. The basic model includes the electromagnetic and induced polarization effects by allowing for frequency dependent eddy currents and an electrochemical boundary of the well casing. It is shown that the well casing produces an apparent complex resistivity response that could be mistaken for an induced polarization anomaly existing above a hydrocarbon deposit. This response of the well casing is strongly dependent on frequency.     

USE OF THE RESISTIVITY METHOD IN GEOLOGICAL MAPPING—CASE HISTORIES FROM RANIGANJ COALFIELD,INDIA*

The effectiveness of the electrical resistivity method has been studied using various configurations with different spacings over the Southern Boundary Fault in the northwestern part, and across a dolerite dyke named “Salma dyke” in the central part, of the Raniganj Coalfield, India. It has been observed that the delineation of the fault and the dyke was made possible under differential tropical weathering conditions. Geoelectric sections across the fault and the dyke have been prepared on the basis of Schlumberger sounding results. In profiling, Wenner, two-electrode, half-Schlumberger and part of Al-Chalabi's configurations were applied over the Southern Boundary Fault. Across the dyke, alpha-, beta-, and gamma-Wenner, Schlumberger, half-Schlumberger and two-electrode configurations were employed. Azi-muthal two-electrode sounding was also studied over the fault. The interpretation of the results of apparent resistivity profiles across the Southern Boundary Fault suggests that the Wenner and two-electrode configurations possess certain diagnostic features which help in mapping a single lithological contact, provided sufficient resistivity contrast exists. Although Schlumberger configuration seems to be quite suitable, other configurations may also be usefully employed over the dyke.     

用有限元法正演京津唐地区二维构造的大地电磁测深资料

本文阐述在大地电磁测深工作中,用有限单元法,模拟计算京津唐地区二维地质构造的大地电磁响应。 文章前部分,论述了用有限单元法正演二维大地电磁测深曲线的原理及其数学公式推导。文章后部分,根据两条测线实测的大地电磁测深结果并结合其它地球物理探测资料,给出了两个二维剖面模型,并计算了两种极化的视电阻率曲线。两条实测剖面分别是横切唐山地震带的马兰峪—柏各庄剖面及横过北京地区北北东构造走向的文化营—王辛庄剖面     

水平分层大地的交流视电阻率

本文给出了水平分层大地视电阻率的一种改进的定义,在这种改进了的视电阻率的远区曲线中,假极值效应有所压低,曲线的起伏度变得比原来的大,视电阻率的值也较接近于地层的真实电阻率.这些特性对于作出正确的判断都是有利的.     

ON THE COMPLETENESS OF DATA SETS WITH MULTIELECTRODE SYSTEMS FOR ELECTRICAL RESISTIVITY SURVEY

This paper describes how, using a surface linear array of equally spaced electrodes, potential data can be obtained for use in electrical resistivity imaging. The aim is to collect a complete data set which contains all linearly independent measurements of apparent resistivity on such an array using two-, three- or four-electrode configurations. From this primary data set, it is shown that any other value for apparent resistivity on the array can be synthesized through a process of superposition. Numerical tests show that such transformations are exact within the machine error for calculated data but that their use with real field data may lead to noise amplification.     

FINITE-DIFFERENCE EVALUATION OF APPARENT RESISTIVITY CURVES*

The problem of numerical evaluation of apparent resistivity curves is treated by finite difference modeling. The models proposed are set up in cylindrical coordinates and yield the potential field due to a point source located in a radially symmetric environment. The Schlumberger configuration, widely used for surface measurements, is emphasized. However, the treatment is equally applicable to other similar situations such as the computation of synthetic electric logs when the resistivity of the borehole fluid is different from that of the surrounding uniform or stratified medium. Moreover, the individual layers may not necessarily be isotropic. The medium under investigation is discretized by using a very coarse system of horizontal and vertical grid lines whose distance from the source increases logarithmically; consequently, the physical dimensions of the medium can be made “infinite” without affecting the numerical size of the model. Finer features such as a thin but anomalously resistive or conductive bed which would ordinarily be missed in coarse discretization are accurately taken into account, since the calculations are done in terms of the Dar Zarrouk parameters derived from the exact resistivity distribution of the model. This enables one to compute the potential field by inverting a small sparse matrix. When the medium comprises only a few layers, the efficiency of the finite-difference model is comparable to that of the known analytical methods; for more complicated structures, however, the finite-difference model becomes more efficient. The accuracy of finite-difference results is demonstrated by comparing them with the corresponding analytically obtained data.     

多层地电断面视电阻率的高精度计算

目前地电的方法在我国已应用到地震预报的研究中,为了了解与地下力学场变化相联系的地电变化,需要了解台址下勘探体积内的电阻率分层情况,从而正确地认识各种影响因素对地电阻率的作用。     

Theoretical study of slope effects in resistivity surveys and applications

When electric soundings are made over an irregular terrain, topographic effects can influence the values of apparent resistivity and lead to erroneous 1D interpretation. A 3D finite-element method has been applied to study the topographical effect of a slope on Schlumberger soundings parallel to the strike. When the resistivity survey is performed at the top of the slope, the apparent resistivity values can be two times higher than in the flat-earth case, depending on the angle (α) and height (H) of the slope, and on the distance (X) between the sounding and the slope top. The results are presented as nondimensional curves which can be used for evaluating topographic anomalies for any value of the parameters α, H and X. It is numerically shown that the topographic effects can be removed from measurements on horizontally layered structures with an irregular earth surface. Real measurements were performed in different geological conditions over an irregular terrain. The correction method based on the nondimensional curves has been applied to the data and has enabled the determination of the correct layered ground configuration using 1D interpretation.     

A NUMERICAL METHOD TO COMPUTE THE RESISTIVITY TRANSFORM FROM WENNER SOUNDING DATA*

A numerical technique to compute the resistivity transform directly from the observed Wenner sounding data has been developed. In principle, the procedure is based on a decomposition method and consists of two steps: the first step determines a function that approximates the apparent resistivity data and the second step transforms this function into the corresponding kernel by an analytical operation. The proposed method is tested on some theoretical master curves. A high degree of precision is achieved with very little computer time. The applicability is shown on two field examples.     

Combined straightforward inversion of resistivity and induced polarization (time-domain) sounding data

It is proposed that the Straightforward Inversion Scheme (SIS) developed by the authors for 1D inversion of resistivity sounding and magneto-telluric sounding data can also be used in similar fashion for time-domain induced polarization sounding data. The necessary formulations based on dynamic dipole theory are presented. It is shown that by using induced polarization potential, measured at the instant when steady state current is switched off, an equation can be developed for apparent ‘chargeability–resistivity’ which is similar to the one for apparent resistivity. The two data sets of apparent resistivity and apparent chargeability–resistivity can be inverted in a combined manner, using SIS for a common uniform thickness layer earth model to estimate the respective subsurface distributions of resistivity and chargeability–resistivity. The quotient of the two profiles will give the sought after chargeability profile. A brief outline of SIS is provided for completeness. Three theoretical models are included to confirm the efficacy of SIS software by inverting only the synthetic resistivity sounding data. Then one synthetic data set based on a geological model and three field data sets (combination of resistivity and IP soundings) from diverse geological and geographical regions are included as validation of the proposal. It is hoped that the proposed scheme would complement the resistivity interpretation with special reference to shaly sand formations.