URTeC: 2455478

VMD Based Sedimentary Cycle Division for Unconventional Facies Analysis Fangyu Li*, Tao Zhao, Yin Zhang, and Kurt J. Marfurt, University of Oklahoma.

Copyright 2016, Unconventional Resources Technology Conference (URTeC) DOI 10.15530/urtec-2016-2455478

This paper was prepared for presentation at the Unconventional Resources Technology Conference held in San Antonio, Texas, USA, 1-3 August 2016.

The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents of this paper have

not been reviewed by URTeC and URTeC does not warrant the accuracy, reliability, or timeliness of any information herein. All information is the responsibility of, and, is subject to

corrections by the author(s). Any person or entity that relies on any information obtained from this paper does so at their own risk. The information herein does not necessarily reflect

any position of URTeC. Any reproduction, distribution, or storage of any part of this paper without the written consent of U RTeC is prohibited.

Summary

Seismic facies analysis is an important tool for quantitative interpretation. It is usually based on machine learning

technique to integrate well logs and seismic attributes. For unconventional reservoirs, the target layer is usually quite

thin, which increase the difficulty of facies recognition. In order to improve the accuracy and robustness of

unconventional facies analysis, we purpose to utilize a new category of seismic attributes, sedimentary cycle

components, as the constraint of the optimization function. In the traditional seismic facies analysis, it is of great importance to use well logs as the supervised information, but the well logs are only at a few limited locations.

Seismic data has the best coverage, so we adopt the sedimentary information from seismic decomposition as a

supportive attribute in the machine learning process.

As a time-frequency analysis technique, Hilbert-Huang transformation (HHT) is introduced into seismic facies

analysis as the sequence stratigraphic constraints. But the data-driven HHT method has some drawbacks. Recently,

varitional mode decomposition (VMD) method is proposed to separate more robust and reasonable intrinsic modes

from the data. We use VMD to decompose seismic data into Intrinsic Mode Functions (IMF), and different IMFs have

different characteristics and indicate different sedimentary information at different geological time scales, which can

be used as the supervised information.

We analyze different kinds of typical models of sedimentary cycle and their IMFs to verify the reliability and precision of the proposed method. Then in the field applications, we apply the sedimentary components from VMD as

a geological time constraint in the self-organizing map (SOM) based seismic facies analysis. The field applications on

Barnett shale/ Marble falls limestone show good correspondence between facies classification results and the logging

data, with more reasonable unconventional layering features.

Introduction

Seismic data includes all kinds of elastic waveform expressions from underground geological features as well as

coherent and uncoherent noises. As an important tool to unveil certain subtle geological information, signal

decomposition shows people more hidden information in the data than superficially. As the most classic spectral

analysis tool, Fourier transform gives us the stationary frequency coefficients of different sigmoid functions. But, seismic signal spectral components change along the traces and depth, termed “non-stationary” and need to be

analyzed via a time variant approach. Time frequency analysis (TFA) methods develop from Fourier transform, for

instance, short-time Fourier transform (STFT) and continuous wavelet transform (CWT) are classical TFA tools

(Partyka et al., 1999; Sinha et al., 2005). But these methods are bound by the Heisenberg uncertainty principle with a

tradeoff between time and frequency resolutions (Tary et al., 2014). The highest vertical resolution is achieved by a

method based on a matching pursuit (MP) approach, whereby the waveforms in a mother wavelet library are matched

to a seismic trace in an iterative process according to the highest spectral energy (Wang, 2007; Wen et al., 2015).

Apparently, the performances of MP methods depend on the configuration of wavelet library and fitting methods,

while it also occasionally fails to consistently match wavelets to the relatively low energies at the low/ high

frequencies.

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Empirical mode decomposition (EMD) (Huang et al., 1998) was proposed to analyze non-stationary signal and has

been widely used (Klplan et al., 2009). EMD is a data-driven signal decomposition method, so it is not based on

predefined wavelet library and can show the intrinsic hidden properties of the interest signal. But instinct mode

function (IMF) from EMD is not based on bandlimited assumption, so EMD attacks all energy at high wave-number.

Regarding to this drawback, variational mode decomposition (VMD) (Dragomiretskiy and Zosso, 2014) was proposed

for decomposing a data into an ensemble of band-limited IMFs. Liu et al. (2015) utilized EMD to characterize the sedimentary cycle modes and got good results. But as the number of IMF outputs from EMD can’t be controlled, it is

hard to apply EMD on seismic interpretation, because it is not reasonable to imagine two adjacent traces have totally

different sedimentary patterns. Thus, we propose to use VMD as the seismic decomposition approach, as the number

of “mode” in VMD is predefined and the results are more robust. (In this study, IMF is used as the term of signal

decomposition results from both EMD and VMD.)

Self-organizing map (SOM) is an excellent seismic facies analysis/classification tool that captures the information

residing in multiple seismic attributes by reorganizing data samples based on their topological relationship. For

traditional SOM, the distance information in the input attribute space is lost when projected into the 2D latent space,

Zhao et al. (2016) adopted a distance-preserving approach to constrain facies to better reflect the degree of diversity in

input attribute space. However, till now all the SOM applications are temporally unaware, which means the patterns of

facies on an SOM map could only follow patterns presented in input attributes without geology sequence constraint. As the well logs and outcrops can’t cover all the interest areas, so the stratigraphy and lithology information should

come from other ways (Sun et al., 2014; Zhang et al., 2015). The sedimentary information from seismic based on

VMD can help define layers that are otherwise not well defined on seismic attributes.

In this study we first introduce EMD and VMD methods, then use a synthetic example to show their different

properties. After applying on sedimentary cycle models, we adopt the VMD method to decompose the seismic

amplitude signal into a user-defined number of modes, and select one of the modes as an indicator of the sedimentary

cycle by calibrating with well logs. In order to add such sedimentary cycle constraint to SOM facies analysis, we

modify the SOM objective function and workflow. In the field applications of Barnett shale and Marble falls

limestones, we identify some subtle layers invisible using traditional unconstrained SOM facies analysis.

EMD and VMD

EMD decomposes a data series into a finite set of IMFs, which represent different oscillations embedded in the data.

They are constructed to satisfy two conditions: (1) the number of extrema and the number of zero-crossing must be

equal to or differ at most by one; and (2) at any point the mean value of the envelope defined by the local maxima and

the envelope defined by the local minima must be zero. The IMFs are elementary amplitude/frequency modulated

harmonics that can model the nonstationarity and the nonlinearity of the data (Huang et al., 1998). It is easy to tell that

the number of EMD outputs can't be controlled, the algorithm would stop when the residue falls below the error

threshold, but very small unexpected interferences can change the final results.

VMD decomposes a real input signal into a number of modes that have specific sparsity properties while reproducing

the input. Meanwhile, the sparsity prior of each mode is chosen to be its bandwidth in spectral domain. Each IMF has

a localized frequency component, which represents certain hidden information. The IMFs are extracted concurrently

instead of recursively, leading to its high efficiency. VMD is achieved by solving the following optimization problem:

2

2

k

k k

j t

t k ku

k k

jt u t e s t u f

t

,

min . .

, (1)

where ku and k are modes and their center frequencies, respectively.

Artificial Signal Decomposition Example

In order to compare EMD and VMD, we design a mixed signal with a lower background frequency and a gapped

higher frequency in the middle section. The analyzed signal is 1 2s s s with

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1

2

sin 2 0.055 ( 1) 1 500

sin 2 0.69 ( 1) 101 350

0

s t if n

t if ns

else

, (2)

Figure 1 show IMFs from EMD and VMD. The two components can be clearly resolved by VMD and separated

signals are basically consistent with the original ones. Though EMD can decompose the signal, it generates seven

different IMFs, which is hard to be used in signal analysis. So we believe that VMD method can decompose seismic

signal without generating too many IMFs. (The threshold in EMD is very hard to determine.)

Figure 1: Artificial signal decomposition example. Mixed single and IMFs from (Left) EMD and (Right) VMD.

Synthetic Sedimentary Pattern Identification

Sedimentary cycle is the result of the periodic occurrence of any depositional event. It is mainly caused by the earth's crust periodic oscillation. To be simplified, we build four kinds of cycle model: a) Normal cycle model: the thickness

of sandstone and mudstone increases with depth, while the grain-size of sediment change from fine to coarse. b)

Inverse cycle model: regression and parameters are contrary to the normal cycle. c) Inverse-normal cycle model: the

thickness of the sand shale inter-bed increases at first, and then decreases with depth. The grainsize of sediment

change from fine to coarse then to fine. d) Normal-inverse cycle model: the parameters of the normal-inverse cycle

model are contrary to the inverse-normal cycle model, whose energy and frequency first increases then decreases.

Figure 2 shows the reflectivity series, seismic traces, and IMFs from VMD of the four different models above. From

the previous example in Figure 1, we can be confident that VMD is better than EMD in the current application situation, so only VMD is used to decompose the seismic signal in this example. Notice that the sedimentary changes

are clear on IMF3 compared to the reflectivity. (But, from the following applications, we can find the sedimentary

pattern from different data will show up on different IMFs. We should be careful about it.) The key for division of the

stratigraphic sequence is dividing the sedimentary cycle correctly, which can be done by calculating the logging data.

But, well logs cannot cover the whole survey, and sometimes we even don’t have the logging data. So, we want to

extract the sedimentary cycle from the seismic using a signal decomposition method, which should be data-driven and

robust. From the previous introduction and example, VMD becomes our choice.

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Figure 2: Reflectivity series, seismic traces and IMFs from VMD of (a) Normal, (b) Inverse, (c) Inverse-normal and (d) Norma-inverse models.

Field Application of VMD Sedimentary Cycle Identification

The field data set is from the Fort Worth Basin, Texas. The Barnett Shale reservoir falls between the Marble Falls and Viola Limestones which form the fracture barriers (Perez and Marfurt, 2014). A thin Forestburg Limestone which can

acts as an imperfect fracture barrier separates the reservoir into the Upper Barnett and the Lower Barnett sections.

Figure 3 shows the seismic data and inverted seismic impedance with the interpreted Marble falls, Upper Barnett

shale, Forestburg, and Lower Barnett Shale, with the IMFs from VMD.

As our objective is to analyze the sedimentary cycle, in this example we choose Marble falls limestone to be the target

area. The seismic trace is extracted from 10 ms above Marble falls horizon and 10 ms below Upper Barnett shale

horizon. It is hard to find the sedimentary cycle directly from seismic. Note the IMFs from VMD of the Marble falls

Limestone in Figure 3. From the IMF1, we can clearly see a Normal-inverse model pattern, shown in Figure 2(d).

From the gamma logs in Figure 3, we can also see a similar curve in the middle of Marble falls (denoted by dashed

red arrows). So, besides the synthetic example in Figure 2, this field application also persuades us that VMD like

signal decomposition methods can be used in sedimentary pattern characterization.

Figure 3: Field application. (Left) Seismic profile. (Middle) Inverted seismic impedance. (Right) IMFs from VMD of Marble falls Limestone.

Improved SOM Methodology

As motioned above, there is no geological time constraint in traditional SOM seismic facies analysis, which

sometimes leads to unreasonable classification results. Based on a term defined by the gradient of a mode calculated

using VMD, the distance now is modified as:

1

(1 )N

i PVi grad PVgrad

i

d A A V V

, (3)

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where 𝑑 is the weighted distance between a multiattribute sample vector and a prototype vector; 𝑁 is the number of attributes; 𝐴 and 𝐴𝑃𝑉 are the attribute values of a sample vector and a prototype vector, respectively; 𝑉𝑔𝑟𝑎𝑑 and

𝑉𝑃𝑉𝑔𝑟𝑎𝑑 are the gradient of a mode from VMD for a sample vector and a prototype vector, respectively; and 𝛼 is a weight between 0 and 1. The complete workflow of the modified SOM facies analysis is shown in Figure 4.

Figure 4. Workflow of the proposed VMD constrained SOM facies analysis.

We apply the proposed workflow on the Fort Worth Basin data set. In this unconventional resource play, interbedded

limestone and shale layers are presented, with the Upper and Lower Barnett Shales being the dominant shale

reservoirs. To constrain the SOM analysis using VMD derived mode, we first use VMD to decompose the seismic

signals into three independent modes, which represents long, medium, and short time sedimentary cycles.

Figure 5 shows the decomposed three modes of the seismic trace between Upper Barnett and Marble falls Limestone

in Figure 3. By comparing the gradient of the three modes with the Gamma Ray log, we find the gradient of VMD 3

matches the pattern of the Gamma Ray log the best, which means the gradient of VMD 3 is an appropriate

representation of the sedimentary cycle at this area. Therefore, we use the gradient of VMD 3 as the constraint in the

subsequent SOM analysis.

Figure 5. (a) Seismic amplitude from a trace between Upper Barnett and Marble falls from Figure 3. (b) IMFs from VMD. (c) The gradient of VMD

3. (d) Comparison between the Gamma Ray log and the gradient of VMD 3.

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Improved SOM Application

We use P-impedance, S-impedance, Mu/Lambda, and Poisson’s Ratio as input attributes to feed into the SOM

algorithm (Figure 6), which are selected to represent the variation in lithology caused by sedimentation.

Figure 6. P-impedance, S-impedance, Mu/Lambda, and Poisson’s Ratio as input attributes of the SOM.

SOM projects these four attributes into a 2D latent space with a 2D colorbar (Matos et al., 2009), which helps to

define facies that are not easily identified on these attributes. Figure 7 gives the SOM facies maps with and without

VMD constraint at t = 1.28 s. In this figure we observe the map with VMD constraint has more details. To better

compare the results, we take vertical sections at AA’ (Figure 8), BB’ (Figure 9) and CC’ (Figure 12).

Figure 7. SOM facies maps generated (a) without VMD constraint and (b) with VMD constraint at t = 1.28 s.

In Figure 8, we can clearly observe that different formations are delineated with more distinct colors when a VMD

constraint is added. More specifically, a thin layer between Marble Falls Limestone and Upper Barnett Limestone

presents in Figure 8b cannot be traced in Figure 8a (black ovals in Figure 6). This thin layer behaves high Gamma Ray

on the well log (Figure 3), while the Marble Falls Limestone and Upper Barnett Limestone are of low Gamma Ray.

Such response on Gamma Ray log has verified the extra layer identified on the SOM facies map with VMD

constraint.

Figure 9 gives vertical sections of SOM facies maps without and with VMD constraint at line BB’. Similar to Figure

6, we can also identify a higher contrast of color in Figure 9b comparing to Figure 9a, which means the facies are

more distinct. In the Upper Barnett Shale formation, a layer that is not precisely defined in Figure 7a is now presented

much more clearly (black ovals), and we observe lateral variation within this layer, represented by variation of colors.

Figure 10 gives a close look at trace X1 and X1’, with overlaying trace display. Comparing to trace X1, the difference between two adjacent formations is more dramatic in X1’, which results in a better separation between different

layers. To further understand the geologic meaning of the layer picked out by SOM with VMD constraint, we use the

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VP/VS ratio at line BB’ to look for evidence of this layer (Figure 11). There is obviously a layer of high VP/VS ratio

in the Upper Barnett Shale formation, which corresponds to the layer identified in Figure 7b, and the high VP/VS ratio

correlates with the blueish color facies. Such details are not presented in Figure 9a, where the VMD constraint is not

used.

Figure 8: Vertical sections along line AA’ (location shown in Figure 7)

through SOM facies maps generated (a) without VMD constraint and (b)

with VMD constraint. Note the layers are better separated in (b) at the

Marble Falls Limestone and Upper Barnett Limestone formations. Also

the facies map in (b) presents more contrast in color which leads to an

improved definition of different layers. The blue curves are Gamma Ray

logs at well A.

Figure 9: Vertical sections along line BB’ (location shown in Figure 7)

through SOM facies maps generated (a) without VMD constraint and

(b) with VMD constraint. Note the formation highlighted by the black

ovals is clearly presented in (b) but vague in (a). Trace X1 and X1’ are

discussed in Figure 10.

Figure 10: Trace display of the SOM facies at traces X1 and X1’.

Note the layer at t = 1.255 s is defined more clearly when a VMD

constraint is introduced in the SOM.

Figure 11: Vertical section along line BB’ (location shown in Figure

5) through VP/VS ratio. The black oval highlight a layer with high

VP/VS ratio within the Upper Barnett Shale. This layer corresponds

to the layer in blueish color identified in Figure 7b, which is also

marked with a black oval. In contrast, this layer is not well defined

in Figure 7a, in which the VMD constraint is not used.

Figure 12 shows vertical sections of SOM facies maps without and with VMD constraint at line CC’ in Figure 7.

Similar to the interpretation above, we can dig out more subtle geological information from Figure 12b. The black arrows highlight there is a thin layer missed without VMD supervision on Figure 12a. Figure 13 gives the supportive

information from Vp/Vs ratio attribute. It is common for shale reservoir to have layering sub-structures, but without

the sedimentary circle constraint, the original SOM will miss this thin layer, which is important for unconventional

plays characterization.

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Figure 12: Vertical sections along line CC’ (location shown in Figure 7)

through SOM facies maps generated (a) without VMD constraint and (b)

with VMD constraint. Note the layers are better separated in (b) at the

Upper Barnett shale formations.

Figure 13. Vertical section along line CC’ (location shown in Figure

7) through VP/VS ratio. The black arrows highlight a layer with high

VP/VS ratio within the Upper Barnett Shale. This layer corresponds

to the layer in blueish color identified in Figure 12b, which is also marked with black arrows.

Conclusions

For unconventional resource characterization, we explored the feasibility of constraining the SOM facies analysis

using sedimentary cycle information. The constrained SOM facies map provides more details, and shows layers that

are more likely being overlooked on unconstrained SOM facies map. The sedimentary cycle is estimated by

decomposing seismic amplitude signal into a finite number of modes using VMD. However, the geological meaning

of such modes is not well understood, and these modes need to be carefully calibrated with well logs. Furthermore,

different layers with the same VMD gradient are not distinguishable, and each seismic trace is decomposed

individually. Therefore, a spatial/temporal constraint may further improve the SOM performance.

Acknowledgement

We thank Devon Energy for providing the seismic and well log data. Financial support for this effort is provided by

the industry sponsors of the Attribute-Assisted Seismic Processing and Interpretation (AASPI) consortium at the University of Oklahoma. We thank colleagues for their valuable input and suggestions. The prestack inversion was

performed using licenses to Hampson and Russell, provided to the University of Oklahoma for research and

education courtesy of CGG.

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