# Spectral Decomposition and a Waveform Cluster to Characterize Strongly Heterogeneous Paleokarst Reservoirs in the Tarim Basin, China

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## Abstract

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## 1. Introduction

## 2. Geological Background

## 3. Materials and Methods

#### 3.1. Spectral Decomposition

#### 3.1.1. Continuous Wavelet Transform

#### 3.1.2. S-Transform

#### 3.1.3. Matching Pursuit

- (1)
- Set the time of the maximum envelope of the complex trace to be the time delay ${\mu}_{n}$, the instantaneous frequency to be the center frequency ${\omega}_{n}$, and the instantaneous phase to be the phase ${\varnothing}_{n}$.
- (2)
- Then, use Equation (7) to search for the optimal parameter ${\sigma}_{n}$ over a group of preselected, uniformly distributed $\mathsf{\sigma}$ values with fixed ${\mu}_{n},{\omega}_{n},{\varnothing}_{n}$ values.$${g}_{{\sigma}_{n}}\left(t\right)=arg\underset{{\sigma}_{n}\in \sigma}{\mathrm{max}}\frac{|\langle {R}^{\left(n\right)}f,{g}_{{\sigma}_{n}}\rangle |}{\Vert {g}_{{\sigma}_{n}}\Vert}$$
- (3)
- Update these four parameters for an optimal wavelet ${g}_{{\gamma}_{n}}$ by searching within a range D using Equation (8). The searching range around a parameter $\mathsf{\xi}$ is $\left[\mathsf{\xi}-\Delta \mathsf{\xi},\mathsf{\xi}+\Delta \mathsf{\xi}\right]$; for instance, $\Delta \mathsf{\mu}$ is the time-sampling interval.$${g}_{{\gamma}_{n}}\left(t\right)=arg\underset{{\gamma}_{n}\in D}{\mathrm{max}}\frac{|\langle {R}^{\left(n\right)}f,{g}_{{\gamma}_{n}}\rangle |}{\Vert {g}_{{\gamma}_{n}}\Vert}$$
- (4)
- After 3, the amplitude of the optimal wavelet ${g}_{{\gamma}_{n}}$ is$${a}_{n}=\frac{|\langle {R}^{\left(n\right)}f,{g}_{{\gamma}_{n}}\rangle |}{\Vert {g}_{{\gamma}_{n}}\Vert}.$$

#### 3.2. Waveform Cluster

- (1)
- Select the time window to extract the waveforms, ${x}_{j}\in {\mathcal{R}}^{d},j=1,\dots ,N$ where ${x}_{j}$ is the $j$th waveform in the time window, $d$ is the time sampling number, and $N$ is the number of waveforms;
- (2)
- Select appropriate values for $m$ and $c$ and a small positive number $\mathsf{\epsilon}$. Initialize the prototype matrix M randomly and set the step variable t = 0.
- (3)
- Calculate (at t = 0) or update (at t > 0) the membership matrix $\mathrm{U}$ by:$${\mu}_{ij}{}^{\left(t+1\right)}=1/({\displaystyle \sum}_{l=1}^{c}{\left({D}_{lj}/{D}_{ij}\right)}^{1/\left(1-m\right)}),fori=1,\dots ,candj=1,\dots ,N.$$
- (4)
- Update the prototype matrix M by:$${m}_{i}{}^{\left(t+1\right)}=({\displaystyle \sum}_{j=1}^{N}{\left({\mu}_{ij}{}^{\left(t+1\right)}\right)}^{m}{x}_{j})/{\displaystyle \sum}_{j=1}^{N}{\left({\mu}_{ij}{}^{\left(t+1\right)}\right)}^{m}),fori=1,\dots ,c.$$
- (5)
- Repeat steps 2–3 until ${M}^{\left(t+1\right)}-{M}^{\left(t\right)}<\epsilon $. The $j$th waveform is assigned to the $l$th cluster if ${\mu}_{l,j}$ is the maximum of all ${\mu}_{i,j},i=1,\dots ,c.$

## 4. Results and Discussions

#### 4.1. Choose the Reservoir-Sensitive Single-Frequency Data

- (1)
- The signal-to-noise ratio is significantly improved;
- (2)
- The recognition of small-scale caves and fractures is obviously improved, the energy is more concentrated, and the cave’s edge is much clearer (shown in the red box in Figure 5e);
- (3)
- Small fractures, such as the one marked by the blue arrow in Figure 5e, are clearer; and
- (4)
- The continuity of strata around the fracture-cavity reservoir is greatly increased.

#### 4.2. Verification of the Sensitive Single-Frequency Data

#### 4.3. Characterization of the Reservoir Distribution by a Waveform Cluster

- (1)
- The connectivity between wells is much better, such as the connection between Well 1, Well 2, and Well 3 in Block 1, and Well 4, Well 5, and Well 6 in Block 2. The river connectivity in Block 3 is significantly improved and the river width is widened. Previous studies on the karst development [2,3,14,46,88] and the actual drilling process show that Well 1 and Well 2 have a certain degree of connectivity (Figure 7d), which is not shown in the clustering results of the full-band data (Figure 7b). The near-shore karst platform and gentle karst slopes of the ancient channel were formed by strong hydrodynamic erosion, which can easily form a pipeline system “crossing the mountain”. The pipeline system is less damaged by the filling in the later stage, and the formed oil and gas reservoirs are larger. An increase in the connectivity of the paleo-channel may indicate a corresponding increase in reservoir connectivity.
- (2)
- The portrayed trend in channels is more obvious. Well 4, Well 5, and Well 7 in Block 2 of Figure 7b are randomly distributed, which suggests that there is no connection between them. The connectivity in Block 2 of Figure 7d is better and the trend in channels is more obvious, which indicate that a small channel branch may have formed. This provides a new perspective for understanding the crack caves in the Tahe Oilfield.

#### 4.4. Geological and Geophysical Interpretation of the Reservoir

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The location of the study area and an overview map of the Tarim Basin. (

**a**) Tectonic components of the Tarim Basin, including three uplifts and four depressions with east–west orientations (modified from [17]). The Tahe oilfield is located in the Northern Uplift of the Tarim Basin. (

**b**) The study area is in the center of the Tahe oilfield, which is located in the southwestern part of the Akekule Arch in the Northern Uplift of the Tarim Basin (modified from [18]). (

**c**) Cross Section A-A’, showing the paleokarst system of the Tahe Oilfield. The main reservoirs are located in the northern part of the Tahe Oilfield in the Yingshan Formation (O1–2 years), which is at depths exceeding 5500 m (modified from [2]). See Figure 1b for the cross-section location.

**Figure 3.**The comparison of time-frequency spectra of a synthetic seismic trace. From left to right, the method used is (

**a**) Continuous Wavelet Transform (CWT), (

**b**) S-Transform (ST), and (

**c**) Matching Pursuit (MP), respectively.

**Figure 5.**Seismic sections of different single-frequency data around Well 1. (

**a**)–(

**d**) represent 15 Hz, 20 Hz, 25 Hz, and 30 Hz, respectively. (

**e**) represents 35 Hz. (

**f**)–(

**h**) represent 40 Hz, 45 Hz, 50 Hz, and 55 Hz, respectively. (

**i**) represents the original full-band seismic section.

**Figure 6.**A comparison of synthetic traces (blue) with (

**a**) the full-band and (

**b**) a selected 35-Hz frequency profile.

**Figure 7.**A comparison of the reservoir distribution by the 35 Hz single-frequency data and the full-band data. (

**a**) and (

**c**) are the direct results of the waveform cluster, while (

**b**) and (

**d**) are the interpretation of the cluster results. (

**a**) and (

**b**) are the full-band data, while (

**c**) and (

**d**) are the 35 Hz single-frequency data.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Shan, X.; Tian, F.; Cheng, F.; Yang, C.; Xin, W.
Spectral Decomposition and a Waveform Cluster to Characterize Strongly Heterogeneous Paleokarst Reservoirs in the Tarim Basin, China. *Water* **2019**, *11*, 256.
https://doi.org/10.3390/w11020256

**AMA Style**

Shan X, Tian F, Cheng F, Yang C, Xin W.
Spectral Decomposition and a Waveform Cluster to Characterize Strongly Heterogeneous Paleokarst Reservoirs in the Tarim Basin, China. *Water*. 2019; 11(2):256.
https://doi.org/10.3390/w11020256

**Chicago/Turabian Style**

Shan, Xiaocai, Fei Tian, Fuqi Cheng, Changchun Yang, and Wei Xin.
2019. "Spectral Decomposition and a Waveform Cluster to Characterize Strongly Heterogeneous Paleokarst Reservoirs in the Tarim Basin, China" *Water* 11, no. 2: 256.
https://doi.org/10.3390/w11020256