Sri Niwas

Estimation of hydraulic parameters of shaly sandstone aquifers from geoelectrical measurements

Electrical and hydraulic conductivities of shaly sandstones are described using a capillary approach for a granular, clay bearing material. The clays are assumed to occur as shale shells uniformly coating the insulated sand grains. The real and imaginary components of the complex electrical conductivity for this material model are written in terms of a unique solid matrix conductivity, treated as a volumetric property. Above a critical salt concentration the conductive contribution of the shells is independent of the electrolyte salinity. Under this condition the equation for the bulk conductivity of the sandstone can be expressed in a simplified form. However, below the critical concentration the matrix conductivity is dependent on the conductivity of the water saturating the shaly component and can be expressed only by the complete equation. The hydraulic conductivity for this model is expressed by a modified Kozeny-Carman equation. A new semi-empirical equation relates the hydraulic conductivity for such rocks, to their porosity, formation resistivity factor and the electrical conductivity of its solid matrix. These combined properties are described as the lithoporosity factor. In this new formulation the petrophysical parameters involved are easily determined from the electrical geophysical measurements. The performance of this equation is firmly tested with experimental laboratory data available in the literature. Its application is then extended to estimate the hydraulic parameters of a shaly sandstone aquifer in Bahia-Brazil, using either the borehole or the surface geoelectrical data. Examples are given to emphasize the combined use of electrical resistivity and induced polarization measurements in computing hydraulic properties.

Computation of apparent resistivities using an exponential approximation of kernel functions

We present a fresh approach to the mathematical computation of apparent resistivities in electrical prospecting. The method is based on an exponential approximation of the kernel function which reduces the integral equation for the potential over a layered earth to a simple algebraic equation. The coefficients in the approximation are obtained using a least-squares inversion technique. A single, unified matrix equation allows computation of apparent resistivity values for arbitrary four-electrode arrays over a layered earth. The unified G function automatically reduces to that of a symmetrical four-electrode array and dipole array function with the proper interelectrode separation. Computations for some two-, three-, and four-layer earth models (Schlumberger configuration), along with a few Wenner and radial dipole apparent resistivity values, demonstrate the versatility of this unified equation.

Straightforward inversion of vertical electrical sounding data

A straightforward inversion scheme (SIS) has been developed to interpret vertical electrical sounding data. This scheme does not require quasi-linearization of the inverse resistivity problem and thereby dispenses with the iterative process and the necessity of guessing the number of layers and their resistivities and thicknesses. The entire solution domain is divided into uniform thickness layers, whose scale must be judiciously selected for the desired resolution. The apparent resistivity formula can now be posed as an underdetermined matrix equation whose minimum norm solution is downward continued to obtain the reflection coefficients which, in turn, yield the vertical resistivity distribution. A recurrence relation has been developed especially for this purpose. In general, when data are expected to be noisy, a regressed minimum norm solution is used. Exhaustive tests of the algorithm have established its numerical efficiency. Results of six typical synthetic models, representing diverse geological conditions, as well as results of two field examples are included to demonstrate this claim.

Fast computation of Hankel Transform using orthonormal exponential approximation of complex kernel function

The computation of electromagnetic (EM) fields, for 1-D layered earth model, requires evaluation of Hankel Transform (HT) of the EM kernel function. The digital filtering is the most widely used technique to evaluate HT integrals. However, it has some obvious shortcomings. We present an alternative scheme, based on an orthonormal exponential approximation of the kernel function, for evaluating HT integrals. This approximation of the kernel function was chosen because the analytical solution of HT of an exponential function is readily available in literature. This expansion reduces the integral to a simple algebraic sum. The implementation of such a scheme requires that the weights and the exponents of the exponential function be estimated. The exponents were estimated through a guided search algorithm while the weights were obtained using Marquardt matrix inversion method. The algorithm was tested on analytical HT pairs available in literature. The results are compared with those obtained using the digital filtering technique with Anderson filters. The field curves for four types (A-, K-, H-and Q-type) of 3-layer earth models are generated using the present scheme and compared with the corresponding curves obtained using the Anderson sc heme. It is concluded that the present scheme is more accurate than the Anderson scheme

A simple method of interpreting dipole resistivity soundings

A computational method has been developed based on a linear relationship between apparent resistivity and the kernel function through a matrix operator. Forward modeling of dipole apparent resistivity values over a one-dimensional layered earth model as well as the inversion of electrical sounding data can be easily accomplished using this matrix operator. For a few two-layer, three-layer, and four-layer earth models, the apparent resistivity values can be calculated quickly and accurately. It is shown that the technique is adaptable to automatic interpretation in both the resistivity and the kernel domains.

Straightforward inversion of MT data using a normalized impedance function

We present a new algorithm for 1D magnetotelluric (MT) data inversion. It inverts a normalized impedance response function derived from the classical Cagniard impedance function. The scheme transforms the nonlinear problem of estimating layer resistivities and thicknesses into a linear problem of estimating the coefficients of power series of the new response function. This is achieved by working with a model where each layer has a thickness of constant penetration. The first coefficient of the series provides top-layer resistivity, which, in conjunction with the constant penetration parameter, then provides the layer thickness. The scheme employs a recurrence relation developed between the coefficients of the power series of two successive layers. This relation is used to continue downward and estimate the remaining layer resistivities and thicknesses. The scheme has been tested on a synthetic model and on three well-studied data sets relating to deep, intermediate, and shallow exploration.

Normalized impedance function and the straightforward inversion scheme for magnetotelluric data

This paper investigates the performance of normalized response function obtained by normalizing the Cagniard impedance function by a suitable factor and then rotating the phase by 45‡ to make it purely real for homogeneous half-space and equal to the square root of the half-space resistivity. Two apparent resistivity functions based on respectively the real and imaginary parts of this response function are proposed. The apparent resistivity function using the real part contains almost the same information as that yielded by the Cagniard expression while the one using the imaginary part qualitatively works as an indicator of the number of interfaces in the earth model. The linear straightforward inversion scheme (SIS), developed by the authors employing the concept of equal penetration layers, has been used to validate the proposed apparent resistivity functions. For this purpose, several synthetic and field models have been examined. Five synthetic models are studied to establish the veracity of the new functions and two well-studied published field data sets are inverted through SIS for comparison. We noticed that the new function and SIS compliment each other and lead to better understanding of the data information and model resolution.

Straightforward inversion scheme (SIS) for one-dimensional magnetotelluric data

This paper presents a Straightforward Inversion Scheme (SIS) for interpreting one-dimensional magnetotelluric sounding data. The basic steps of SIS are (i) parameterization of the layered model such that the layer thickness, expressed in units of its skin depth, is a constant (α); (ii) expansion of the reflection function at each interface as a power series in parameter u = exp(-2(1 +j)α√f);(iii) development of a recurrence relation between the coefficients of the same powers ofu in the power series of reflection functions of any two successive layers; (iv) estimation of the impedance power series coefficients using regressed minimum norm estimator; and (v) evaluation of layer resistivities and thicknesses using the inverse recurrence relation. The power of SIS is established by inverting four synthetic data sets and two field data sets. The effect of noise is extensively studied on a synthetic data set, deliberately corrupted with increasing levels of Gaussian random noise up to 25%. It is found that the scheme can retrieve broad features of the true model even with noise levels as high as 25%. On the basis of findings of different experiments conducted on SIS, it is concluded that SIS is an efficient, robust algorithm with high resolving power. Further, being linear, it is non-iterative and it dispenses with the requirement of having to choose an initial guess model.

EM2INV — A finite difference based algorithm for two-dimensional inversion of geoelectromagnetic data

AbstractThe paper presents an efficient finite difference based 2D-inversion algorithm, EM2INV, for geoelectromagnetic data. The special features of the algorithm are• optimal grid generation based on grid design thumb rules,• finite domain boundary conditions,• interpolation matrix that permits generation of response at observation points different from grid points,• Gaussian elimination forward matrix solver, that enables reuse of already decomposed coefficient matrix,• super-block notion that reduces the number of blocks with unknown resistivities and, in turn, the size of Jacobian matrix and• bi-conjugate gradient matrix solver for inverse problem which circumvents the need of explicit Jacobian matrix computation. The algorithm is tested rigorously by setting up exercises of diverse nature and of practical significance. The stability of the algorithm is established by inverting the synthetic response corrupted with Gaussian noise. The inversion experiments are aimed at studying• relative performance of response functions,• inversion quality of E- and B-polarization data,• efficacy of single and multi-frequency data inversion,• minimum number of frequencies and observation points needed for successful data inversion. It has been observed that the Magneto-telluric data deciphers better the vertical position of the target and Geomagnetic Depth Sounding data deciphers the horizontal variations in a better way. The conductive and resistive bodies are better resolved by inversion of E- and B-polarization data respectively. The results of multi-frequency inversion imply that the increase in the number of frequencies does not necessarily enhance the inversion quality especially when the spread of observation points is sufficiently large to sense the target. The study of a minimum number of observation points highlights the importance of single point inversion that furnishes useful information about the inhomogeneity.

Fast automatic solution of the inverse resistivity problem

The paper presents a fast automatic approach to solve the inverse resistivity problem, assisted by optimization, which is a non-linear model-fitting technique. The selected inverse problems are ill-posed and the inverse solution is defined by ‘best fit’ in the sense of least-squares. Formulations are presented in a systematic manner for Newton’s method, least squares method and Marquardt’s modification (ridge regression) method based on local linearization of non-linear problem. The convergence of least-squares method and Marquardt’s method, to provide a robust solution, are first tested on a theoretical model and effectiveness of Marquardt’s method is demonstrated, and then two-field apparent resistivity curves from Banda district, India are interpreted and discussed.