Modelling discontinuous well log signal to identify lithological boundaries via wavelet analysis: An example from KTB borehole data

Identification of sharp and discontinuous lithological boundaries from well log signal stemming from heterogeneous subsurface structures assumes a special significance in geo-exploration studies. Well log data acquired from various geological settings generally display nonstationary/nonlinear characteristics with varying wavelengths and frequencies. Modelling of such complex well-log signals using the conventional signal processing techniques either fails to catch-up abrupt boundaries or at the best, do not provide precise information on insidious lithological discontinuities. In this paper, we have proposed a new wavelet transform-based algorithm to model the abrupt discontinuous changes from well log data by taking care of nonstationary characteristics of the signal. Prior to applying the algorithm on the geophysical well data, we analyzed the distribution of wavelet coefficients using synthetic signal generated by the first order nonstationary auto-regressive model and then applied the method on actual well log dataset obtained from the KTB bore hole, Germany. Besides identifying the formation of layered boundaries, the underlying method also maps some additional formation boundaries, which were hitherto undetected at the KTB site. The results match well with known geological lithostratigraphy and will be useful for constraining the future model of KTB bore hole data.     

A heuristic search method for optimal zonation of well logs

Optimal zonation of well-log data, that is, determining an optimal number of major segments such as waveforms in a log, may be achieved by employing a criterion of minimum variance (within a segment) and a heuristic search of potential boundary (link) points of digitized log data. This new method is based on an algorithm originally devised by D. M. Hawkins and D. F. Merriam in 1973. Their method can be improved by introducing a heuristic search procedure, thereby decreasing computer time by 7- to 50-fold, depending on the number of data points and configuration of the logs. Time saving is proportional to the size of the data set. Three examples—one hypothetical and two real—are used to illustrate the modification of the Hawkins and Merriam algorithm.Presented at the Geological Society of America Annual Meeting, Orlando, Florida, 1985.     

Multiscale segmentation of well logs

Examination of well logs leads to the observation that they exhibit characteristic behavior over a wide range of scales, differing from a foot to hundreds of feet. This behavior is not accounted for in conventional segmentation methods. The segmentation method that is proposed here aims to solve this problem. It is based on a multiscale representation of the well log. This is a representation of the well log at different levels of smoothness. The behavior of edges, which are associated with extrema of the first derivative, across scales is of great importance in the segmentation. Analysis of this behavior leads to a multiscale segmentation of the well log, in which a fine-scale segment is a part of one coarse scale segment only. In this way the geologist is able to analyze the log at different scales simultaneously. An extension of this approach is not to preserve all fine-scale information but to zoom in only on parts of the log where it is considered to be of interest.     

小波分析在高分辨率层序地层划分中的应用

介绍了小波分析的基本原理和步骤,探讨了测井信号小波分析在高分辨率层序地层划分中的适用性及其意义。对测井信号进行小波多尺度分解后可获得不同级次的旋回周期,将钻井岩相标定的各个级别旋回层序界面与测井信号小波旋回周期对应,然后选取合适的尺度进行地层层序划分。划分结果证明小波变换非常适合中期及其以下级别旋回层序的划分。该方法对提高旋回层序的划分精度与储层评价、预测精度具有重要的意义。      

Spectral Analysis based on Wavelet Transform Maxima: Identification of Multi-order Stratigraphic Boundaries and Cycles

In sedimentology, stratigraphic sequences and cycles are ordered by time spans and physical scales, such as thickness, and bounded by discontinuities, including unconformities or flooding surfaces. Spectral analysis based on wavelet transform (WT) maxima is proposed and used as a quantitative tool to identify multi-order stratigraphic boundaries and cycles in well log data. The proposed spectral analysis is based on quantitative analysis on the center frequencies and resolutions of Gaussian wavelets in time and frequency, and uses a combination of the WT maxima based on both the first order Gaussian wavelet having a high time resolution and the seventh order Gaussian wavelet having a high frequency resolution. WT maxima spectra, which can characterize the evolution of WT maxima across scales and periods along WT maxima lines concerned with sequence boundaries, are used to detect dominant spectral peaks corresponding to the time-period domain WT maxima and to determine WT maxima spectral slopes. The WT maxima spectral slopes are helpful for discriminating sequence boundaries from intrasequence cyclic variations in well log data, and the time-period domain WT maxima are used to relate the detected boundaries to relevant cycles. The interval WT maxima spectra and the stationary index, related to the WT maxima spectra, are introduced as indicators that can be used for the hierarchical ordering of the boundaries and cycles. Application of the proposed method to well log data shows that the suggested method is efficient in identifying multi-order sequences that relate well to the Milankovitch cycles.

     

Ensemble Kalman filtering for non-linear likelihood models using kernel-shrinkage regression techniques

One of the major limitations of the classical ensemble Kalman filter (EnKF) is the assumption of a linear relationship between the state vector and the observed data. Thus, the classical EnKF algorithm can suffer from poor performance when considering highly non-linear and non-Gaussian likelihood models. In this paper, we have formulated the EnKF based on kernel-shrinkage regression techniques. This approach makes it possible to handle highly non-linear likelihood models efficiently. Moreover, a solution to the pre-image problem, essential in previously suggested EnKF schemes based on kernel methods, is not required. Testing the suggested procedure on a simple, illustrative problem with a non-linear likelihood model, we were able to obtain good results when the classical EnKF failed.     

Automated Segmentation of Resistivity Image Logs Using Wavelet Transform

We describe a wavelet-transform-based method for automated segmentation of resistivity image logs that takes into account the apparent dip in the data and addresses the problem of discriminating lithofacies boundaries from noise and intrafacies variations. Our method can be applied to borehole measurements in general, but might have an advantage when applied to resistivity image logs as it addresses explicitly the large variability in facies segments recorded with a high-resolution multiple-sensor tool. We have developed an algorithm based on this method that might outperform other existing segmentation methods in the cases of low to moderate dip. We made a detailed comparison of the segmentation from our method with the one done by a geologist to delineate different lithofacies blocks in a well drilled in a deepwater depositional environment. Our results show considerable success rates in reproducing the geologically defined lithofacies boundaries, and the generality of our procedure suggests it could also be applied to other depositional environments.     

An Initial Guess for the Levenberg–Marquardt Algorithm for Conditioning a Stochastic Channel to Pressure Data

A standard procedure for conditioning a stochastic channel to well-test pressure data requires the minimization of an objective function. The Levenberg–Marquardt algorithm is a natural choice for minimization, but may suffer from slow convergence or converge to a local minimum which gives an unacceptable match of observed pressure data if a poor initial guess is used. In this work, we present a procedure to generate a good initial guess when the Levenberg–Marquardt algorithm is used to condition a stochastic channel to pressure data and well observations of channel facies, channel thickness, and channel top depth. This technique yields improved computational efficiency when the Levenberg–Marquardt method is used as the optimization procedure for generating realizations of the model by the randomized maximum likelihood method.     

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.     

Empirical Maximum Likelihood Kriging: The General Case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.     

A new algorithm using image colour system transformation for rock grain segmentation

Summary The present paper is concerned with the development and application of an effective automatic algorithm of image analysis in order to detect grain boundaries on microscope images of Redziny dolostone and Wisniowka quartzite. The algorithm utilises sets of 6 colour images for each measurement field on thin sections, which are recorded using an optical polarizing microscope in different polarization set-ups. The proposed method is based on an image pre-processing procedure that is focused on colour system transformation, followed by rock grain boundaries segmentation using the image analysis methods for each colour image. For the image pre-processing procedure, several colour system transformations were selected and compared. By using the alternative colour systems that concentrate on colour information we are able to minimise the effects of internal micro-structures in the grain boundaries segmentation procedure. The grain boundary maps obtained confirm that the use of an approximately perceptually uniform colour system as an image pre-processing procedure can significantly improve the rock grain segmentation. This newly-developed method may facilitate petrographical and stereological studies of rock structures.     

局部时频变换域地震波吸收衰减补偿方法

地震信号在地下传播时会受到地层吸收衰减的影响,从而降低了地震资料的分辨率。因此地震波吸收衰减补偿是地震资料处理中的一项重要环节。本文研究的地层吸收衰减补偿方法主要基于局部时频变换(LTFT),该方法能够调节选取谱分解的频率范围和频率采样间隔,解决了短时傅里叶变换固定时窗和小波系数无法提供波形频率的精确估计值问题,适用于非平稳地震信号的时频分析。在求取地层Q值的方法中,频谱比值法具有高效简单的特点,有着广泛的应用范围。本文假设地下介质为层状变Q模型,使用局部时频变换将信号转换为时频域,通过频谱比值法求出各层的Q值,最后根据Kolsky衰减模型来补偿地震信号。理论模型测试和实际资料处理的结果表明,本文提出的方法能够有效恢复衰减信号,提高地震资料的分辨率。     

Automatic soil-boundary location from transect data

Spatially distributed soil data possess a short-range erratic variation, an irregular longer range pattern, and maybe multivariate. In order to reveal a pattern or meaning in them, they are usually classified by drawing boundaries. A method is described for determining boundaries automatically on transects. A small portion of a sampled transect is taken and divided about its midpoint, and Mahalanobis' generalized distance, Dor D²,between the two halves calculated from the sample data. The procedure is repeated for portions of the same length at positions one-sampling interval apart along the transect. High peaks on the resultant series of D² identify the boundaries. The length of portions is set equal to, or somewhat less than, the expected average distance between boundaries, and is determined by constructing correlograms of principal components. The lag distance over which fairly steady decay occurs is related closely to the distance between boundaries. The procedure is illustrated with data from a 6-km transect in Oxfordshire and shows good agreement with boundaries drawn by combined air-photo interpretation and field judgment. A means of extending the procedure to two dimensions is suggested.     

On estimating the parameters in mixtures of circular normal distributions

The method of maximum likelihood used in analyzing data which are bimodal in the range (0, )is lengthy and not always successful. Two alternative methods are proposed and the results compared with those obtained by using the method of maximum likelihood on orientation data from clasts and shells. It is shown that a procedure based on the Method of Moments gives comparable results and is much more practicable.     

砂泥岩薄互层测井资料解释方法

从分析薄互层的特征入手,采用反褶积方法对单条测井曲线做提高分辨率的处理,并调整电阻率测井曲线,使其与提高了分辨率的泥质含量及孔隙度测井曲线的纵向分辨率协调、在此基础上,按一定的解释模型计算出孔隙度和饱和度等参数。实际计算表明,本方法改善了储集参数的评价,降低了含水饱和度。     

A computational procedure for joining separate map sheets

Map sheets have been often used as a basic spatial unit for managing spatial data produced from paper maps. This often results in incompatibility between adjacent map sheets, because spatial objects do not cross the boundaries smoothly and even the boundaries themselves do not match their neighbors exactly. To solve the problem this paper proposes a computational procedure for joining separate map sheets to obtain seamless spatial data. Line objects digitized separately in different map sheets are considered, which are frequently used to represent road networks, gas pipelines, and boundaries of polygon objects. The procedure consists of three steps: (1) extraction of end nodes, (2) detection of matching nodes, and (3) transformation of the map sheet. Each step goes interactively so that unexpected errors can be avoided by human observation. To test the validity of the procedure, map sheets are combined containing the road network data of Tokyo 23-ku area, Japan. This revised version was published online in July 2006 with corrections to the Cover Date.     

A generalized image and boundaries of the Bazhenov Formation: constraints from well log data for Salym-type sections (West Siberia)

A technique akin to Galton’s composite portraiture is suggested for creating a generalized image of a stratigraphic unit based on well log data from a group of well sections. The procedure begins with superimposing stratigraphically equivalent section points according to pairwise correlation models followed by weighted summation of similar logs. The superimposition highlights persistent common features of the sections, while their minor individual details become suppressed. A generalized model of the Bazhenov Formation in the Salym area has been constructed from gamma-ray, apparent resistivity, and self-potential logs. Radioactivity and resistivity anomalies, which are often used as constraints on the Bazhenov Formation boundaries, show a significant spatial discrepancy. According to the correlation, the formation volume in each well depends on the depth interval corresponding to the hypostratotype. The results were used to map the Bazhenov Formation thickness and its standard deviation, as well as average values of different logs.     

一种提高储层裂缝识别准确度的方法

油田勘探开发中,储层裂缝的高效、准确识别一直是一个难题。常规测井裂缝识别方法方便但准确度低,电成像测井裂缝识别方法纵向分辨率高、识别准确但人工识别繁琐。为了解决常规和电成像测井裂缝识别方法各自缺点带来的问题,本文提出了一种既高效又准确的储层有效裂缝识别方法。该方法以裂缝在电成像测井上的响应特征为依据,选取裂缝层段为样本,构建常规测井裂缝综合识别因子Y₁和电成像测井裂缝识别因子Y₂,将两因子结合构建储层裂缝识别因子Y₃。利用Y₃识别裂缝,采样点间隔为0.002 54 m,远小于常规测井的0.125 m,比常规测井识别裂缝准确度高;自动拾取裂缝效率远高于繁琐的人工识别,比电成像测井识别裂缝省时省力。将该方法应用到王府地区火山岩储层裂缝识别中可快速准确地识别储层裂缝,准确率达到80%左右,对其他类型储层裂缝研究有一定的借鉴意义。     

Estimation of Geological Attributes from a Well Log: An Application of Hidden Markov Chains

Geophysical well logs used in petroleum exploration consist of measurements of physical properties (such as radioactivity, density, and acoustic velocity) that are digitally recorded at a fixed interval (typically half a foot) along the length of the exploratory well. The measurements are informative of the unobserved rock type alternations along the well, which is critical for the assessment of petroleum reservoirs. The well log data that are analyzed here are from a North Sea petroleum reservoir where two distinct strata have been identified from large scale seismic data. We apply a hidden Markov chain model to infer properties of the rock type alternations, separately for each stratum. The hidden Markov chain uses Dirichlet prior distributions for the Markov transition probabilities between rock types. The well log measurements, conditional on the unobserved rock types, are modeled using Gaussian distributions. Our analysis provides likelihood estimates of the parameters of the Dirichlet prior and the parameters of the measurement model. For fixed values of the parameter estimates we calculate the posterior distributions for the rock type transition probabilities, given the well log measurement data. We then propagate the model parameter uncertainty into the posterior distributions using resampling from the maximum likelihood model. The resulting distributions can be used to characterize the two reservoir strata and possible differences between them. We believe that our approach to modeling and analysis is novel and well suited to the problem. Our approach has elements in common with empirical Bayes methods in that unspecified parameters are estimated using marginal likelihoods. Additionally, we propagate the parameter uncertainty into the final posterior distributions.     

Estimating heat flow from an unsteady temperature log of submarine borehole BDP-98 (Lake Baikal)

We have applied the method of one temperature log suggested by Kutasov in 1987 to process an unsteady temperature log of the 674 m deep submarine borehole BDP-98-2 (Akademichesky Ridge, Lake Baikal) and found it suitable to reconstruct the primary steady thermal gradient not disturbed by drilling. The steady gradient we derived using a special formalism, with reference to drilling conditions and measured thermal properties of sediments, was 63 mK/m, more than two times the unsteady gradient at different depths. Heat flow calculated with this gradient and a mean thermal conductivity of 1.1 W/(m·K) was 70 mW/m², which is consistent with earlier geothermal data from the same area. Thus, the one-log procedure is a useful tool to predict the original undisturbed thermal gradients and estimate approximate heat flows if a single unsteady temperature log is available.