Post on 02-Jun-2018
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Identification of StratigrFormation Interfaces
Wavelet and Fourier TranRaphael Ar
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Objective interpretation?
Formation boundary
criteria is subjective
Different interpretation for
each interpreter
well log is treated with
signal processing methods
Objective,apparently
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Modification to Fourier transform was implemented for
identification of lithologic boundaries
Short-time Fourier transform
(STFT):Segments signals using
windows
Combination of Wavelet transform
and Fourier transform was studied in
this paper to analyze SP and GR log toidentify stratigraphic interfaces
Wavelet transform (WT):
Uses wavelet functioncontaining both time and
frequency information
simultaneously
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Discrete Fourier Transform
=
/
x = /
Discrete Fourier
transform
Inverse discrete Fourier
transform
The depth in well log can be treated as time inprocessing the SP or GR log as a signal
Transformed data can be used to select frequency bandsto filter SP or GR log
In this paper, Fast Fourier Transform is used to efficiently
calculate discrete Fourier transform of SP or GR log
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Wavelet Transform
Provides varying time and frequency resolutions using
windows of different lengths
Consists of two kernel variables, phase (or location) andscale, by utilizing wavelet function
, 1
1
1 ,
is called wavelet coefficients
is conjugatecomplex of
a and b are scale and p
location variables)
Continuous wavelet tran
Shifting the wavelet func
phase (b) and stretching
wavelet function from th
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Wavelet Transform
Types of wavelet transform depends on wavelet
functions used
Haar function
Daubechies function
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Wavelet Transform
Discrete wavelet transform:
Signal x(t) is decomposed into several lower-resolution components,
approximation (cA) and detail wavelet coefficients (cD)
+
. ( )
()
()
1
2
1
2
1212
is a sorthog
functio
is eqphase
Reconstruction of the original signal
x is obtained by the sum of inverse
WT of approximation (cA) and its
detail (cD)
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Discrete Wavelet Transform
. Inverse WT of detail wavelet coefficient(cD) is used to reconstruct high-frequency
signal RcD
Approximation response (low-pass filtering) and
detail response (high-pass filtering) can be
calculated at the same time
Approximation (cA) represents low-frequency
component of signal x
Detail wavelet coefficient (cD) represents high-frequency
component of signal x
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Wavelet transform method
Well log data
(SP or GR)
Scale distributions
of waveletcoefficients Approximation
coefficients
(cA)
Detail
coefficients
(cD)
Stratigraphic
interfaces
Stratigraphic
interfaces
Identification of
interfaces
Continuous
wavelet transform
Discrete wavelet
transform
clo
c
h
H
a
tin
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Combination of wavelet
transform and Fourier
transform methods
Well log data(SP or GR)
Result of the discrete wavelet transform
Noisydata?
Detail coefficients (cD)
Reconstructed data(RcD)
Spectrum ofreconstructed data(RcDF)
Modified spectrum ofreconstructed data (MRcD
Modified reconstructeddata (MRcD)
Logarithmic distributiondata (LRcD)
Approximationcoefficients (cA)
Approximationcoefficients (cA)
Detailcoefficients (cD)
Approximationcoefficients (cA) Detailcoefficients (cD)
Stratigraphic interfaces
Level 1
Level 2
Level 3
NoYes
Inverse wav
Fourier trans
Filtering
Inverse Four
log transform
Interface identification
Decomposition
Decomposition
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Ideal well log data application of wavelet transform
method
Clean sand
formation
between 7420 and
7440 ft
Shale formation
above 7420 andbelow 7440 ft Ideal data has nonoise.Therefore the method
of combining WT and FT
is not necessary to be
applied
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Ideal well log data - application of wavelet
transform method
Applying continuous
wavelet transform
Wavelet coefficients
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Ideal well log data - application of wavelet
transform method
Applying Discrete
wavelet transform
Detail coefficients cD
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Field data
From co
were fo
units:
A-sand
B-sand
C-sandD-sand
SP log s
curve re
zone 1,
GR logfour for
of noise
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SP log data analysis
Applying continuous
wavelet transform
Four formations ofsand cannot be
clearly observed
Even the three
formation zones
observed in
conventional log
analysis cannot be
observed
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SP log data analysis
Applying discrete
wavelet transform
Formation
stratigraphy areshown from three
positive-negative
pairs of detailcoefficients cD
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SP log data analysis
Applying WT and FT
combination
(1)Data was less noisy, cD
from single level
decomposition were
calculated
(2)High-frequency
component is obtained
using inverse WT ofwavelet coefficient cD
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SP log data analysis
Applying WT and FT
combination
(3)RcD is transformed using
FFT to be analyzed
Dominant frequencies
are between 20-40 nHz
Chosen frequency
band is the center of the
spectrum, 30-31 nHz
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SP log data analysis
Applying WT and FT
combination
(4)Inverse FFT is applied to
chosen frequency
band, obtaining
modified reconstructed
data MRcDF
(5)Logarithmic transform
data is applied to
reconstructed data
The logarithmic distribution
shows four clear formation
stratigraphic zones
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GR log data analysis
Applying continuous
wavelet transform
To much noise causing
the wavelet coefficientsbecome sparse
Not possible to
determine the formation
stratigraphic from these
sparse scales
GR l d t l i
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GR log data analysis
Applying discrete
wavelet transform
The number of positive-
negative pairs of detailcoefficients is difficult to
be analyzed
If there is no core data, it
would be a painstaking
process to analyze this
noisy data
GR l d t l i
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GR log data analysis
Applying combination of
WT and FT
(1)Three level decomposition
is applied to filter out thenoise and obtain detail
coefficients cD
(2)Reconstructed signal RcD
is obtained using inverse
wavelet transform
GR log data analysis
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GR log data analysis
Applying combination of
WT and FT
(3) Dominant frequency is
chosen after FFT isapplied to RcD
Dominant frequencies
are between 0-20 and
30-45 nHz
(4)The strongest frequency
band is chosen, 8.5-10 nHz
GR log data analysis
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GR log data analysis
Applying combination of
WT and FT
(5)Inverse FT is applied to the
chosen frequency bands
to reconstruct GR signal
(6)Logarithmic transform is
then applied to the
reconstructed GR signal
The logarithmic distribution
shows four clear formationstratigraphic zones
Concluding remarks
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Concluding remarks
Well log can be treated with signal
processing methods to minimize subjective
results
Result of wavelet transform is similar to
conventional well log interpretation, which
causes difficulties in identifying formation
boundaries
Combination of wavelet transform and
Fourier transform provides objective
identification of boundaries