Summary

Interpretation of induction-logging data is difficult because of its diffusive nature and its large volume of investigation. To overcome these difficulties, various techniques have been applied in the past. These techniques include deconvolution filters, least-squares inversion based on the maximization of entropy and on neural-network techniques. Among them, the maximum-entropy method is the most robust. Unfortunately, it is computationally intensive and can only be carried out in computing centers. To reduce this intensity and accelerate the inversion process, a new method has been developed for induction-log inversion using quasi-Newton updates of the Jacobian matrix. Using the new method, 200 ft of log data can be inverted in 80 seconds on an 800-MHz notebook computer.

Introduction

Induction-logging instruments play a crucial role in formation evaluation for the exploration of hydrocarbons. It is primarily used to delineate between hydrocarbon-bearing zones and nonhydrocarbon zones. Formation resistivity derived from induction tools is used to estimate oil in place in the hydrocarbon-bearing zones, while the invasion profile generated by induction tools with several depths of investigation is used to gauge the permeability of the zone. Unfortunately, the interpretation of induction-logging data is a difficult task because of the limitation of the basic induction-tool physics. Specifically, (1) induction tools investigate large volumes surrounding the borehole, (2) induction-tool response depends nonlinearly on formation conductivity, and (3) modern array-induction tools such as Halliburton's High Resolution Array Induction (HRAI*; see Fig. 1) tool are highly complex, with an assortment of transmitters and receivers with different characteristics. As a result, the apparent resistivity profile generated by an induction tool is far from the true formation resistivity profile. This can be seen clearly in Fig. 2, where the raw responses of six arrays of the HRAI tool in the benchmark Oklahoma formation are plotted. The diffusive nature of an induction tool and the shoulder-bed effect clearly distort its response.

Since the introduction of the first induction tool by Doll,1 many attempts have been made to correct its adverse environmental effects (e.g., shoulder-bed effect, skin effect, etc.) with the goal of enhancing its resolution. An approach that is frequently used by log analysts to interpret induction-logging data is the familiar trialand- error method, whereby a log analyst starts with an assumed formation resistivity profile to generate a model response using predicative forward modeling, adjusts the input resistivity, and iterates until acceptable results are obtained. The obvious drawbacks of this approach are the amount of time consumed and the possibility of introducing bias by even the most experienced interpreter.

Traditionally, inverse filters were used to deconvolve induction logs. This is based on the recognition that induction logs can be approximated as a convolution of formation conductivity and a canonical tool-response function. Inverse filters have been used to deconvolve the raw logging data.2 This operation is normally carried out at the wellsite on the logging truck. Although this technique has been used widely for years, it is known to introduce spurious effects.3

An alternative approach to enhance induction-tool resolution is to carry out a full inversion. Least-squares inversion based on the iterative minimization of the sum of squared errors between log data and forward predictive model results were reported by Lin et al.4 and Gianzero et al.5 in the past. However, least-squares solutions are often questionable because of their high sensitivity to noise. To enhance the stability of the least-squares inversion method against noise, following the practice in image processing, Dyos6 introduced an entropy term to the objective function through a Lagrange multiplier. This was followed by the work of Freedman et al.7 Zhang et al.8 introduced three algorithms based on the maximum flatness, maximum oil, and minimum oil. Barber et al.9 applied the method of Freedman et al.7 to the processing of array-induction data. Without exception, all these gradient-based methods require the calculation of the Jacobian matrix, namely the apparent resistivity partial derivatives that represent the sensitivity of the apparent resistivity to various formation parameters such as bed boundaries and conductivity value in each bed. The calculation of the Jacobian matrix is time consuming for the following reason: for each iteration, if there is a total of M logging points, the number of forward modeling calculations necessary is proportional to M, but the number of calculations required by the Jacobian matrix is proportional to M2. Thus, with any realistic logging-data set of a few hundred feet, the computational cost associated with updating the Jacobian matrix at each iteration becomes prohibitively expensive, even on fast workstations. As a result, these algorithms are all slow. For example, the quoted processing speed in Ref. 9 is 200 ft/2 hrs on an UltraSparc* 2 workstation. Clearly, the infrequent application of full inversion in the interpretation of induction-logging data in the petroleum industry stems largely from the fact that it takes too much computing power and too much time.

Recently, artificial neural networks have been applied to the inversion of induction-logging data.10 The most advantageous feature of this method is its speed because it works as an inverse filter after the neural network has been properly trained. However, it does have its shortcomings. First, each subarray in an array-induction tool needs to be properly and judiciously trained with appropriate training data. This means that the developmental phase of this technique is much longer than any gradient-based inversion techniques, for which only the physical configurations of the transmitter and receiver arrays are needed. Second, it is possible that dubious results will occur if the formation resistivity profile in question lies outside the limits of the training set.

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