Heterogeneities analysis using the generalized fractal dimensions and the continuous wavelet transform. Application to t
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ORIGINAL PAPER
Heterogeneities analysis using the generalized fractal dimensions and the continuous wavelet transform. Application to the KTB boreholes Leila Aliouane & Sid-Ali Ouadfeul
Received: 16 April 2013 / Accepted: 30 June 2013 # Saudi Society for Geosciences 2013
Abstract The main goal of this paper is to identify heterogeneities from well logs data using the wavelet-based multifractal analysis. Firstly, the wavelet transform modulus maxima lines method is applied with a moving window of 128 samples to the raw well logs data. After that, the generalized fractal dimensions that correspond to the three first moments of the function of partition are estimated. Application to synthetic and real well logs data of the main and pilot Kontinentales Tiefbohrprogramm de Bundesreplik Deutschland wells shows that the information and the correlation dimensions can be used for heterogeneities analysis and lithofacies segmentation form well logs data. Keywords Multifractal . Wavelet transform modulus maxima lines . Generalized fractal dimension . Well logs data . KTB
Introduction The fractal analysis has been widely used in exploration geophysics. In gravity and magnetism, it is used for causative sources characterization (Maus and Dimri 1994, 1995, 1996; Fedi 2003; Fedi et al. 1997). In seismology, the fractal analysis is used for earthquake characterization (Ravi Prakash and Dimri 2000; Teotia and Kumar 2011; Dimri et al. 2005). In petrophysics, the fractal analysis is used for lithofacies classification and reservoir characterization. We cite for example the paper of Lozada-Zumaeta et al. (2012)
L. Aliouane LABOPHYT, Faculté des Hydrocarbures et de la Chimie, Université Mhamed Bougara, Boumerdès, Algeria S.0) is a translation in the space is the complex conjugate of ϕ
The analyzing wavelet must check the admissibility condition: ð þ∞ ϕðzÞdz ¼ 0 ð2Þ −∞
The second step consists to calculate the maxima of the modulus of the continuous wavelet transform; at this step, the first and the second derivatives of the modulus of the continuous wavelet transform are used. We call (a, b0) maxima of the modulus of the CWT at the point b0, if for all b→b0,|Tϕ(a,b0)|>|Tϕ(a,b)|. The function of partition Z(q, a) is a summation of the modulus of the CWT at the set of maxima L(b) with a q moment. The function of partition is given as (Arneodo et al. 1995): X T ϕ ða; bi Þq Z ðq; aÞ ¼ ð3Þ LðbÞ The spectrum of exponents is then derived from the function of partition, it is related to Z(q, a) by a power law for low scales: Z ðq; aÞ ¼ aτ ðqÞ if a→0
ð4Þ
The generalized fractal dimension D(q) can be obtained from the spectrum exponents by (Arneodo et al. 1988, 1995):
Arab J Geosci 1,12
Fig. 3 Generalized fractal dimensions obtained by WTMM analysis of a synthetic well log data
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1,08 1,06 1,04 1,02 1,00 0
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1,0 0,8 0,6 0,4 0,2 0,0 -0,2 -0,4 -0,6 0
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