Abstract
Heterogeneity analysis of conventional data, such as geophysical log data, has been still limited to the application of near-wellbore zone, which makes it difficult to optimize the hydraulic fracturing design and may render suboptimal performance. However, the fluctuation of multi-stage pumping data, manifesting nonlinear behavior of physical properties with shale reservoir during hydraulic fractures propagation stage, is usually ignored. In this study, the empirical mode decomposition technique (EMDT) was introduced and applied to the multi-stage pumping data to determine the respective Intrinsic Mode Functions (IMF). By using a relationship between the IMF number and its mean wavenumber, the heterogeneity index associated with far-wellbore shale reservoir was determined. The results indicate that the heterogeneity index from multi-stage pumping data is good coincided with the effective stimulation reservoir volume (ESRV) obtained from micro-seismic events. Not only that, but it also reveals that there is a strong correlation of heterogeneity index, IMF number, ESRV, and degree of heterogeneity within shale reservoir. This work has demonstrated that heterogeneity index analysis combined with EMDT has been significantly important and essential to quantify the degree of heterogeneity within far-wellbore shale reservoir from multi-stage pumping data, which contributes to optimizing the hydraulic fracturing design and improving good optimal performance.
1 Introduction
The hydraulic fracturing technique is widely used to improve economic production from shale gas resources. Generally, many strategies of hydraulic fracturing are planned prior to any hydraulic fracturing operation to ensure its effectiveness. However, by comparison, hydraulic fracturing requires much more analysis for many optimal designs in heterogeneous shale reservoir [1, 2, 3, 4]. It is the fact that because the deposition, diagenesis and tectonic effects of forming the shale reservoir varies, heterogeneity can be considered as the variation of rock properties as a function of their spatial locations [5]. They may vary dramatically to affect the hydraulic fracturing performance in heterogeneous reservoir [6]. In order to overcome technique and operation challenges related to hydraulic fracturing, many approaches are used to provide deep insight into the hydraulic fracturing performance.
Initially, many single parameters were respectively used to characterize the reservoir heterogeneity, such as the Coefficient of Variation (CV) and the permeability ratio, from rock core scale or pore scale in the laboratory [7, 8, 9, 10, 11, 12]. However, there are some problems that these parameters are only described heterogeneity from one side, and it is difficult to be the general application of reservoir scale. Meanwhile, some statistical hypotheses of rock properties, such as Weibull distribution [13, 14] and Normal distribution [15], can also be used to represent the reservoir heterogeneity in the numerical simulation. There is an undeniable fact that these methods contribute to improving deep insight into quantifying the reservoir heterogeneity. Nevertheless, it is a difficulty that these statistical hypotheses may be applied to address the engineering problems. Subsequently, many methods were introduced to quantify the reservoir heterogeneity based on the geophysical log data, such as the Lorentz coefficient [16], fractal or multifractal [17, 18, 19, 20, 21], the Empirical mode decomposition technique (EMDT) [22] and the Hilbert spectral analysis (HAS) [23, 24, 25]. Moreover, some comprehensive characterization methods considering multiple factors were also proposed to quantify the reservoir heterogeneity [12]. Nonetheless, limited by the detection range (different scale) of the geo physical instrument, there are still some bad applications, especially for far-wellbore shale reservoir away from detection range of conventional methods. Not only that, hydraulic fracturing curve linked with hydraulic fractures propagation range can exhibit subsurface information in the reservoir scale, which can help engineers to real-time master some formation properties of far-wellbore shale reservoir during hydraulic fracturing. The detection range (different scale) of different measuring methods are shown in Figure 1.
Schematic diagram of the detection range (different scale) within different measuring methods. (a) Schematic diagram of 3D; (b) Schematic diagram of 2D
Experimental analysis and geophysical log data analysis are self-evident for optimizing hydraulic fracturing strategies, whereas some other data should also be focused on heterogeneity analysis of shale reservoir after hydraulic fracturing, which is extremely significant for the development of adjacent shale reservoir. In recent years, production data have been used to describe the reservoir heterogeneity [1, 6]. What’s more, the pressure pulses, named as the water hammer signatures, frequently generated by pump shutoff or valve closure after a hydraulic fracture treatment have also been used to diagnose hydraulic fractures networks [26, 27]. Notably, when hydraulic fractures are propagating to the extent far-wellbore reservoir, the pressure changes of multi-stage pumping data within hydraulic fracturing can be observed and recorded [28]. In essence, the pressure changes of hydraulic fracture propagation stage can manifest the nonlinear behavior of physical properties within the shale reservoir, as shown in Figure 2. However, these abundant field data is usually ignored, which can be considered as the valuable wealth for evaluating reservoir heterogeneity due to its available at no additional cost.
Schematic diagram of hydraulic Fracturing Curve.
The EMDT, a suitable data-adaptive technique, was first established by N. E. Huang et al. [29] to analyze nonlinear and non-stationary data [23] at the end of the 20th century. And then, it was also combined with the Hilbert spectrum analysis method to constitute Hilbert-Huang transform (HHT), named by the National Aeronautics and Space Administration (NASA). The major advantage of EMDT, as the key part of HHT, is that the decomposition of any complicated signal can easily be operated, which is fully data adaptive. Gaci and Zaourar [22] proposed a method to perform heterogeneity analysis from geophysical logging data. And then, Gairola and Chandrasekhar [24] improved the heterogeneous method of geophysical logging by using EMDT combined with Hilbert spectrum analysis. The several applications of EMDT have found in a variety of specializations in geophysics, such as sequence stratigraphy, seismics, atmospheric sciences [25]. Not only but, it has been expanded into fault detection of mechanical devices, financial data and medical science [23]. However, application of the EMDT is rarely reported in hydraulic fracturing field. To this end, the EMDT was introduced and applied to multi-stage pumping data to evaluate heterogeneity degree within the shale reservoir by using the superiority of EMDT.
The main aim of the present study is the demonstration of the heterogeneity analysis of shale reservoir based on multi-pumping data. Firstly, the multi-stage pumping data manifesting nonlinear behavior of physical properties with shale reservoir were selected and considered as the original signal. And then, the empirical mode decomposition technique (EMDT) was introduced and applied to the multi-stage pumping data to determine the respective Intrinsic Mode Functions (IMF). At last, by using a fitting relationship between the IMF number and its mean wavenumber, the heterogeneity index associated with far-wellbore shale reservoir was determined. The organization of this work is as follows. Section 2 briefly introduces the basic theory and procedures of EMDT and heterogeneity index analysis. Section 3 describes the application of EMDT and heterogeneity index analysis to multi-stage pumping data considered for the present study. Section 4 provides the results and discussion, which can provide insights in heterogeneity analysis of shale reservoir and Section 5 summarizes the main conclusions of the present study.
2 Methods
To get the utmost out of these abundant field data, the pressure changes of fracture propagation were selected to quantify the degree of heterogeneity within shale reservoir, which can also be considered as the original signal of far-wellbore shale reservoir. The complete method and procedure can be summarized, as shown in Figure 3.
Flowchart representing the procedure of Empirical mode decomposition technique and Heterogeneity index analysis.
2.1 Empirical Mode Decomposition Technique (EMDT)
The EMDT is used to decomposition of different frequency components of the original signal (from highest to lowest), namely Intrinsic Mode Functions (IMF) without the need of a priori basis as Fourier and wavelet-based methods do [29,30], until the signal becomes monotonic, which is named residue and represents the overall trend of the original signal.
The operating procedure of EMDT is as follows [24]:
Step 1: Determine the local maxima and minima of the original signal (say, x(s)), where t represents the hydraulic fracturing time.
Step 2: Use the cubic spline to process the local maxima and minima to obtain the upper envelope and lower envelope and calculate its mean, m1.
Step 3: Calculate the first proto-mode, named as x1, given by the Equation (1).
Step 4: If the new extrema are contained in the first pro-mode (x1), it should be considered as the signal and steps 1-3 should be repeated to obtain the new first proto-mode, named as x11, calculated by using the Equation (2).
This procedure is repeated up to ‘k’ iterations, after which the first ‘k’ proto-mode (x1k) is given by the Equation (3).
In the present study, it is considered as the stop criterion for these iterations that the normalized squared difference between two successive sifting operations is given by the Equation (4).
Where N represents the number of data points, theoretically, the stopping criterion, SDk means that the sifting process stops only if (a) the number of local extrema and the number of zero-crossings must be equal or differ at most by one in the entire signal and (b) at any time point, the mean value of two envelopes (upper envelope and lower envelope) is zero for the entire signal. In this study, SD k ≤ 0.1 was selected as the stopping criterion to determine the first IMF (IMF1(s)). This implies that after ‘k’ iterations, once the stopping criterion is satisfied, then x1k is properly considered as the first IMF, which represents the highest frequency.
Step 5: To identify the second IMF (IMF2(s)), calculate the residue (r1(s)) by subtracting the first IMF (IMF1(s)) from the original signal (x(s)) and consider the residue (r1(s)) as a new signal and repeat preceding steps (steps 1-4). It is important to note that the frequency of the second IMF (IMF2(s)) will be lower than that of the first IMF (IMF1(s)).
Step 6: As the described steps 1-5, the entire decomposition process should be over only if (a) all IMFs are determined and (b) the residue (rn(s)) becomes the monotonic function, where no more IMF can be decomposed from the data. What’s more, it is noticed that all the estimated IMF can be synthesized to reconstitute the original signal, which is calculated by the Equation (5).
2.2 Heterogeneity Index Analysis
The previous studies have shown that the IMF number generated from a larger heterogeneous system are more than those from the smaller heterogeneous system [24]. However, heterogeneity index analysis can quantify the qualitative observations made through IMF number by using EMDT acted as a filter bank and provides a quantitative estimate of the degree of heterogeneity in the shale reservoir. The three-step procedure to determine the heterogeneity index is as follows.
Step 1: For each IMF, the IMF can be characterized by the mean wavenumber (km), which is calculated by the Equation (6) as an energy weighted mean wavenumber [22].
Where Sm(k) is the Fourier spectrum of mth IMF.
Step 2: For a given IMF, the plot can be drawn by the fitting relationship between the IMF number, mand logarithm km. The mean wavenumber km is inversely proportional to the IMF number m, as is depicted by the Equations (7) and (8) [22].
Where log k0 and log ρ are the intercept and slope of the straight line fitting the log(km) vs. m graph, respectively. So, the ρ value can be calculated from the antilogarithm of the slope of the straight fitting line.
Step 3: Indeed, the ρ value is connected with the number of scales needed to decompose the original signal using EMDT. Meanwhile, there also exists an inverse relationship between the heterogeneity degree and the estimated ρ value, namely smaller ρ values mean higher heterogeneity degree of shale reservoir and vice-versa [22, 24].
Interestingly, it can also be noticed that the expected ρ value is hovering around 2 by using the EMDT acted as a dyadic filter bank in the wavenumber domain, as shown by applications on stochastic simulations of fractional Gaussian noise and white noise [25]. Hence, the estimated ρ value can be used as a global measure of the complexity of the original signal.
3 Application of EMDT and Heterogeneity Index Analysis to Multi-Stage Pumping Data
3.1 EMD of multi-stage pumping data
The major shale reservoirs in the study area are buried in the Longmaxi Formation of the Weiyuan block, Sichuan Basin, China. The hydraulic fracturing was operated in the depth of 1,705 ~ 2,750m of a shale gas well, Wei XXX-H1, whose horizontal segment length is of about 1,045m. The multi-stage plug-and-perf technology planned by 11 stages was operated for forming the large effective stimulated reservoir volume (ESRV) with complex fracture networks. The multi-stage pumping data were selected from the hydraulic fracturing curve, as shown in Figure 4.
Multi-Stage Pumping Data of Hydraulic Fracturing.
The EMDT was applied to the multi-stage pumping data for evaluating the shale reservoir heterogeneity. To explicate the evaluation procedure, the Stage 5 and Stage 7 were taken as examples, respectively. The EMDT produced 11 IMFs and a residue for Stage 5 (Figure 5) and 9 IMFs and a residue for Stage 7 (Figure 6).
Intrinsic mode functions (IMF) obtained by empirical mode decomposition of pumping data within Stage 5.
Intrinsic mode functions (IMF) obtained by empirical mode decomposition of pumping data within Stage 7.
The reconstructed signals, given by synthesizing all the IMF and the residue of pumping data decomposed by the original signals with the extremely small error of the order of about 10−14 between the original and synthesized signals, as shown in Figure 7.
Reconstructed signal from the IMF for pumping data in (a) Stage 5 and (b) Stage 7. Note that the least error of the order of about 10-14 between the original and reconstructed signal, implying the chosen stopping criterion of to be properly justified (see Section 2).
3.2 Heterogeneity index analysis for shale reservoir using IMFs
The IMFs decomposed from the EMD of original pumping data were performed to calculate the mean wavenumber (km) by the Fast Fourier Transform. As explained above, the heterogeneity index analysis provides a quantitative estimated with the degree of heterogeneity in the shale reservoir. Figure 8 depicts the graph drawn between the logarithm of mean wavenumber of each IMF and the IMF number, plotted for Stage 5 (Figure 8(a)) and Stage 7 (Figure 8(b)). Compared with the Equation (8), logarithm of the slope of the best fit line can be determined. Finally, the heterogeneity index of Stage 5 (ρ = 1.7533 ± 0.0877) and Stage 7 (ρ = 2.2653±0.1133) can be respectively obtained by the antilogarithm of the slope of the best fit line.
Linear regression between the logarithm of the mean number of each IMF and the respective IMF number corresponding to (a) Stage 5 and (b) Stage 7, clearly revealing the higher degree of heterogeneity (low value) in Stage 5 compared to the lower degree of heterogeneity (high value) in Stage 7.
4 Results
4.1 Heterogeneity index of multi-stage pumping data
As the above procedures (see Section 3), the heterogeneity index of multi-stage pumping data were obtained and listed in Table 1, where ‘±’ symbol represents the 95% confidence intervals.
Heterogeneity index of the multi-stage pumping data.
| Number of Stages | Heterogeneity Index |
|---|---|
| 1 | 1.5473±0.0774 |
| 2 | 2.1704±0.1085 |
| 3 | 1.8946±0.0947 |
| 4 | 1.4785±0.0739 |
| 5 | 1.7533±0.0877 |
| 6 | 1.6715±0.0836 |
| 7 | 2.2653±0.1133 |
| 8 | 2.4895±0.1245 |
| 9 | 2.3018±0.1151 |
| 10 | 2.2794±0.1140 |
| 11 | 2.2689±0.1134 |
4.2 Results verification
Based on the above heterogeneity index analysis, the results are verified by checking the micro-seismic events of different fracturing stages. Firstly, it can be observed that complex networks distribution of multi-stages is extremely different, which looks like a ‘scorpion’ as shown in Figure 9(a). Meanwhile, Figure 9(b) indicates that the micro-seismic events of ‘heel’ (Stage 11) of the horizontal well is more than those of ‘toe’ (Stage 1) of the horizontal well.
Diagram of micro-seismic events mapping after hydraulic fracturing. (a) Top view; (b) Front view.
To further analyze the difference of multi-stage, the fracture dimensions of hydraulic fractures are quantified and characterized by a statistical method. Table 2 summarized the statistical data of multi-stage hydraulic fracturing from micro-seismic events. It can also be noticed that there are differences in the effective stimulated reservoir volume (ESRV) of multi-stage hydraulic fracturing, where the ESRV of Stage 8 is the largest and the Stage 1 is the smallest. Therefore, it clearly illustrates that there exists the heterogeneity of far-wellbore shale reservoir.
Statistic results of micro-seismic monitoring data.
| Number of Stages | Length/m | Width/m | Height/m | Number of Micro-seismic Events | Effective Stimulated Reservoir Volume (ESRV)/×104m3 |
|---|---|---|---|---|---|
| 1 | 302 | 149 | 64 | 118 | 348 |
| 2 | 436 | 231 | 52 | 673 | 664.8 |
| 3 | 465 | 298 | 58 | 594 | 882.4 |
| 4 | 374 | 364 | 59 | 364 | 593.6 |
| 5 | 304 | 258 | 72 | 801 | 649.6 |
| 6 | 404 | 240 | 87 | 694 | 605.6 |
| 7 | 330 | 329 | 76 | 695 | 756 |
| 8 | 493 | 429 | 88 | 1481 | 1417.6 |
| 9 | 497 | 408 | 76 | 1069 | 1135.2 |
| 10 | 555 | 484 | 83 | 808 | 1102.4 |
| 11 | 558 | 535 | 82 | 630 | 829.6 |
To verify the validity of heterogeneity index as an indicator of the heterogeneity degree, the CV is also used to estimate heterogeneity degree, named as the ratio of the standard deviation to the mean, which is usually a statistic to measure the degree of variation in the data. Generally, the smaller CV contributes to obtaining more complex networks. Meanwhile, it is noted that the probability of forming complex networks is the lower in the more degree of heterogeneity within the shale reservoir. In other words, the CV is negative with ESRV with complex networks, whereas the heterogeneity index (ρ) is a positive correlation with ESRV with complex networks.
As shown in Figure 10, the matching rate of the multistage sort is only 63.64% between CV and ESRV, where there exists abnormal change in Stage 1, Stage6, Stage 9 and Stage 11. However, the matching rate reaches 81.82% between ESRV and heterogeneity index, where there exists abnormal change in Stage 4 and Stage 3. It can be seen that the matching ratio between the heterogeneity index and ESRV improves about 20% compared with that between CV and ESRV. In addition, the relative trend of multi-stage with heterogeneity index is highly similar to that of ESRV in Figure 10.
Analysis Results of effective stimulated reservoir volume, CV and heterogeneity index of multi-stage, where black color indicates abnormal fracturing stage. Top figure: ESRV results of multi-stage; Middle figure: Coefficient of Variation of multi-stage (CV); Bottom figure: heterogeneity index of multi-stage.
Theoretically, the heterogeneity index of Stage 4 (ρ = 1.4785 ± 0.0739) is smaller than that of Stage 1 (ρ = 1.5473 ± 0.0774), the degree of heterogeneity within the fourth stage should be more than that of Stage 1. However, it can be observed that there exists a sudden drop at the close to 90 minutes of the original signal within Stage 4, as described in Figure 4. In fact, there are a small number of natural fractures acted as the channel where fracturing fluid flow. In essence, it can be the reason why the ESRV of Stage 4 indicates larger than that of Stage 1.
What’s more, it can be observed that the distribution of micro-seismic events within Stage 2 is relatively dispersed and overlap with that of Stage 2, which clearly shows the impact of inter-stage stress interference or stress shadow effect from Figure 9. In other words, hydraulic treatment of Stage 2 has significant effects on the stress field of Stage 3, which can promote the inter-connection with hydraulic fractures of Stage 3. Hence, the reason lies behind the ESRV of the third stage is larger than that of Stage 11 is inter-stage stress interference or stress shadow effect.
5 Discussion
To reduce the regional limitations of heterogeneity index as an indicator of heterogeneity degree within shale reservoir, another shale gas well, Ning HX-X located about 250 km away from Wei XXX-HX, is also selected to further generalize heterogeneity index. some studies indicate that the bigger Lorentz coefficient can be regarded as a proxy of the more heterogeneous reservoir. Therefore, the Lorentz coefficient is also used to evaluate heterogeneity degree of shale reservoir.
As shown in Figure 11, the relative order of 12 stages estimated by heterogeneity index is the same as ESRV with complex networks, whereas normal fracturing stages estimated by CV and Lorentz coefficient have 9 stages and 7 stages, respectively. The abnormal fracturing stages are also marked by black color in Figure 11. Compared with CV with 60% matching ratio and Lorentz coefficient with 46.67% matching ratio, the matching ratio of heterogeneity index reaches 80%, which highlights the higher superiority. Meanwhile, the relative trend of heterogeneity index is highly similar to that of ESRV, while the other two methods roughly exhibit reverse trend with ESRV in Figure 11.
Geographic location and analysis results of Ning HX-X. (a) Relative location graph of Wei XXX-HX and Ning HX-X; (b) Analysis Results of effective stimulated reservoir volume, heterogeneity index, Coefficient of Variation (CV) and Lorentz coefficient of multi-stage in Ning HX-X, where black color indicates abnormal fracturing stage.
In addition, some explanations can also help to further reveal deep insight of heterogeneity index as a proxy of heterogeneity degree within shale reservoir from multistage pumping data.
On the one hand, Figure 4 depicts that the fracture propagation pressure of first six stages is more than that of the last five stages, which is in good agreement with the order of heterogeneity index shown in Figure 10. Meanwhile, it can also be observed that the fracture propagation pressure of Stage 5 is more than that of Stage 7, which is consistent with Figure 8. There are differences in the fracture propagation pressure within multi-stage hydraulic fracturing, which implies there can be differences in the heterogeneity of shale reservoir. Not only that, the overall heterogeneity index of multi-stage is also statistically analyzed. As shown in Figure 12, the distribution of overall heterogeneity index ranges from 1.2 to 2.7, which is mainly distributed in 2.1~2.4, with counts of 5 and a frequency of 45.45%. Meanwhile, the average of the Weibull distribution is much closer to the central position of the main distribution ranges compared with the Normal distribution. Therefore, the overall distribution of heterogeneity index within multi-stage is more consistent with the Weibull distribution, which can contribute to assuming the variation of some parameters follow the Normal distribution or Weibull distribution in the numerical simulation [31, 32, 33, 34, 35].
Statistical analysis graph of heterogeneity index, where the purple number represents the counts of statistical results and black percentage number indicates the percentage of heterogeneity index of any stage. The Normal distribution and Weibull distribution are described by blue and pink lines, respectively.
On the other hand, it is a general understanding that the IMF number generated from a larger heterogeneous system are more than those from the smaller heterogeneous system [24]. An intercomparison of such information between Stage 5 and Stage 7 facilitates to correctly assess their heterogeneity index. Figure 10 illustrates that the heterogeneity index of Stage 5 is less than that of Stage 7 and the frequency of Figure 5 is much higher than that of Figure 6, such as IMF 1. Moreover, the EMDT as the filtering bank can obtain the fluctuation of high frequency, which can also reflect the degree of heterogeneity [36, 37]. By contrast, the fluctuation of high frequency in Figure 5 is more frequently than that of high frequency in Figure 6. This implies the degree of heterogeneity in Stage 5 is more than that in Stage 7, which is in good agreement with the relation that the IMF number of Figure 5 is more than that of Figure 6 [24]. Simultaneously, it can also be observed that the heterogeneity index (ρ) of the shale reservoir is a negative correlation with the IMF number, as shown in Figure 13 [22, 24, 37].
Relationship between IMF Number and Heterogeneity Index.
Figure 14 systematically summarizes the relationship of these parameters to help further understanding. There are some correlations between the three parameters (IMF number, Heterogeneity index and ESRV) and heterogeneity degree. At the same time, the three parameters also affect each other. There is a positive correlation between the IMF number and heterogeneity degree, while the other two parameters suggest the reverse relation with heterogeneity degree. Not only that, both the higher heterogeneity index and the lower IMF number contribute to obtaining the higher ESRV.
Relationship diagram among IMF number, Heterogeneity Degree, Heterogeneity Index, and ESRV. Note that the blue plus ‘+’ represents a positive correlation, the black minus ‘-’ stands for the negative correlation.
Conversely, due to the reverse relation between heterogeneity index and heterogeneity degree, there is also the negative correlation between the heterogeneity index and IMF number decomposed by the EMDT. In essence, the heterogeneity degree can be regarded as a bridge that both IMF number and heterogeneity are linked with ESRV.
However, this does not mean that both IMF number and ESRV can be used to evaluate the degree of heterogeneity within the shale reservoir. If the heterogeneity index is ignored, when equal IMF number are generated by EMDT, as the present Stage 5 and Stage 6 of Figure 13, it becomes difficult to determine, which Stage is more heterogeneous than the other [24, 37]. Similarly, the inter-stage stress interference can also disturb the results of ESRV. Therefore, the heterogeneity index can be a relative reliable evaluated indicator for determining the degree of heterogeneity within the shale reservoir. On the whole, it is quite worthwhile that the application of EMDT and HIA to multi-stage pumping data to improve the heterogeneity analysis of far-wellbore shale reservoir.
In addition, heterogeneity index as an indicator of heterogeneity degree within shale reservoir from multi-stage pumping data has some advantages compared with conventional evaluated methods based on core experiment and geophysical logging data: (1) abundant data source or database are based on hydraulic fracturing and (2) no additional measurement tools can reduce completion cost and (3) the evaluated method is easy and quick to operate for field engineers.
6 Conclusions
In this paper, use of the empirical mode decomposition technique reveals the intrinsic property of shale reservoir from multi-stage pumping data and heterogeneity index analysis are responsible for quantifying the degree of heterogeneity within shale reservoir, especially for far-wellbore shale reservoir. The heterogeneity analysis was studied according to multi-stage pumping data of a shale gas well, Wei XXX-H1, using the aforementioned methods. From the results of this study, there are some conclusions drawn that:
The multi-stage pumping data provides its worthiness over the other conventional geophysical log data and rock core analysis in unraveling the degree of heterogeneity within the far-wellbore shale reservoir.
Heterogeneity analysis results of multi-stage pumping data are in good agreement with ESRV derived from micro-seismic events and confirm that there is a positive correlation between the heterogeneity index and ESRV.
There is a negative correlation between the heterogeneity index and IMF number decomposed by the EMDT, whereas it is a positive correlation between the degree of heterogeneity within the shale reservoir and IMF number.
The combined EMDT and the heterogeneity index analysis have been significantly important and essential to quantify the degree of heterogeneity within far-wellbore shale reservoir.
Indeed, the universality of our methods can be affected and degraded by some uncertainties of shale gas development. However, it can still provide some engineering values for evaluating heterogeneity of shale reservoir. Therefore, heterogeneity index combined with abundant geological characteristics can contribute to further improving deep understanding of heterogeneity within shale reservoir in the future study.
Acknowledgement
This work was jointly supported by the National Natural Science Foundation of China (No.51674272), the National Science and Technology Major Project of China (No.2017ZX05009-003), the key project of jointly funded by the National Natural Science Foundation of China (No. U1762211).
References
[1] Parvizi, H., Rezaei-Gomari, S., Nabhani, F., Turner, A. Evaluation of heterogeneity impact on hydraulic fracturing performance. J. Petrol. Sci. Eng., 2017, 154, 344-353. doi:10.1016/j.petrol.2017.05.001.10.1016/j.petrol.2017.05.001Search in Google Scholar
[2] Ouchi, H., Foster, J.T., Sharma, M.M. Effect of reservoir heterogeneity on the vertical migration of hydraulic fractures. J. Petrol. Sci. Eng. 2017, 151, 384-408. doi:10.1016/j.petro1.2016.12.034.10.1016/j.petro1.2016.12.034Search in Google Scholar
[3] Ouchi, H., Agrawal, S., Foster, J.T., Sharma, M.M. Effect of Small Scale Heterogeneity on the Growth of Hydraulic Fractures. In Proceedings of SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, Texas, USA, 24-26, January 2017. doi:10.2118/184873-MS.10.2118/184873-MSSearch in Google Scholar
[4] Zhou, J., Huang, H., Deo, M. Numerical Study of Critical Role of Rock Heterogeneity in Hydraulic Fracture Propagation. In Proceedings of 50th U.S. Rock Mechanics/Geomechanics Symposium, Houston, Texas, 26-29, June 2016.Search in Google Scholar
[5] Tang, H.Y., Li, S.B., Zhang, D.X. The effect of heterogeneity on hydraulic fracturing in shale. J. Petrol. Sci. Eng., 2018, 162, 292-308. doi:10.1016/j.petrol.2017.12.020.10.1016/j.petrol.2017.12.020Search in Google Scholar
[6] Parvizi, H., Gomari, S.R., Nabhani, F., Monfared, A.D. Modeling the Risk of Commercial Failure for Hydraulic Fracturing Projects Due to Reservoir Heterogeneity. Energies, 2018, 11, 218. doi:10.3390/en11010218.10.3390/en11010218Search in Google Scholar
[7] Yang, X., Meng, Y.F., Shi, X.C., Li, G. Influence of porosity and permeability heterogeneity on liquid invasion in tight gas reservoirs. J. Nat. Gas Sci. Eng., 2017, 37, 169-177. doi:10.1016/j.jngse.2016.11.046.10.1016/j.jngse.2016.11.046Search in Google Scholar
[8] Wang, Y., Wang, L.H., Wang, J.Q., Jiang, Z., Wang, C.C., Fu, Y.N., Song, Y.F., Wang, Y.F., Liu, D.Z., Jin, C. Multiscale characterization of three-dimensional pore structures in a shale gas reservoir: A case study of the Longmaxi shale in Sichuan basin, China. J. Nat. Gas Sci. Eng., 2019, 66, 207-216. doi:10.1016/j.jngse.2019.04.009.10.1016/j.jngse.2019.04.009Search in Google Scholar
[9] Chen, Y., Mastalerz, M., Schimmelmann, A. Heterogeneity of shale documented by micro-FTIR and image analysis. J. Microsc., 2014, 256, 177-189. doi:10.1111/jmi.12169.10.1111/jmi.12169Search in Google Scholar PubMed
[10] Liu, P., Ju, Y., Ranjith, P.G., Zheng, Z.M., Chen, J.L. Experimental investigation of the effects of heterogeneity and geostress difference on the 3D growth and distribution of hydrofracturing cracks in unconventional reservoir rocks. J. Nat. Gas Sci. Eng., 2016, 35, 541-554. doi:10.1016/j.jngse.2016.08.071.10.1016/j.jngse.2016.08.071Search in Google Scholar
[11] Fan B.J., Shi L., Li Y.T., Zhang T.J., Lv L., Tong S.K. Lithologic heterogeneity of lacustrine shale and its geological significance for shale hydrocarbon-a case study of Zhangjiatan Shale. Open Geosci., 2019, 11(1), 101-112. doi:10.1515/geo-2019-0009.10.1515/geo-2019-0009Search in Google Scholar
[12] Wang, X.X., Hou, J.G., Liu, Y.M., Ji, L., Sun, J. Studying reservoir heterogeneity by Analytic Hierarchy Process and Fuzzy Logic, case study of Es1x formation of the Wang guan tun oilfield, China. J. Petrol. Sci. Eng., 2017, 156, 858-867. doi:10.1016/j.petrol.2017.06.066.10.1016/j.petrol.2017.06.066Search in Google Scholar
[13] Tang, C.A., Tham, L.G., Lee, P.K.K., Yang, T.H., Li, L.C. Coupled analysis of flow, stress and damage (FSD) in rock failure. Int. J. Rock Mech. Min., 2002, 39(4), 477-489. doi:10.1016/S1365-1609(02)00023-0.10.1016/S1365-1609(02)00023-0Search in Google Scholar
[14] Yang, T.H., Tham, L.G., Tang, C.A., Liang, Z.Z., Tsui, Y. Influence of heterogeneity of mechanical properties on hydraulic fracturing in permeable rocks. Rock Mech. Rock Eng., 2004, 37(4), 251-275. doi:10.1007/s00603-003-0022-z.10.1007/s00603-003-0022-zSearch in Google Scholar
[15] Gao, Q., Ghassemi, A. Pore Pressure and Stress Distributions Around a Hydraulic Fracture in Heterogeneous Rock. Rock Mech. Rock Eng., 2017, 50(12), 1-17. doi:10.1007/s00603-017-1280-5.10.1007/s00603-017-1280-5Search in Google Scholar
[16] Yue, C., Yang, X., Zhong, X., Pan, B., Wang, F. Evaluation of Formation Heterogeneity Using Lorentz Coefficient of Logging Curves. J. Jilin Univ. Earth Sci. Edit., 2015, 45(5), 1539-1546. (in Chinese with English abstract). doi:10.13278/j.cnki.jjuese.201505303.10.13278/j.cnki.jjuese.201505303Search in Google Scholar
[17] Liu, Y.F., Liu, Y.T., Sun, L., Liu, J. Multiscale Fractal Characterization of Hierarchical Heterogeneity in Sandstone Reservoirs. Fractals-Complex Geom. Patterns Scaling Nat. Soc., 2016, 24(3), 1650032. doi:10.1142/S0218348X16500328.10.1142/S0218348X16500328Search in Google Scholar
[18] Al-Zainaldin, S., Glover, P.W.J., Lorinczi, P. Synthetic Fractal Modelling of Heterogeneous and Anisotropic Reservoirs for Use in Simulation Studies: Implications on Their Hydrocarbon Recovery Prediction. Transp. Porous Media, 2017, 116(1), 181-212. doi:10.1007/s11242-016-0770-3.10.1007/s11242-016-0770-3Search in Google Scholar PubMed PubMed Central
[19] Li, Z., Tang, L., Jiang, Z.X., Liang, Z.K., Yu, H.L., Yang, Y.D., Xiao, L. Fractal characteristics of nanopores in lacustrine shales of the Triassic Yanchang Formation, Ordos Basin, NW China. Open Geosci., 2019, 11, 196-207, doi:10.1515/geo-2019-0016.10.1515/geo-2019-0016Search in Google Scholar
[20] Subhakar, D., Chandrasekhar, E. Reservoir characterization using multifractal detrended fluctuation analysis of geophysical well-log data. Physica A, 2016, 445, 57-65. doi:10.1016/j.physa.2015.10.103.10.1016/j.physa.2015.10.103Search in Google Scholar
[21] Liu Z.D., Zhao J.Z., Zhang P., Sun J.X. Evaluating the CBM reservoirs using NMR logging data. Open Geosci., 2018, 10(1), 544-553. doi:10.1515/geo-2018-0043.10.1515/geo-2018-0043Search in Google Scholar
[22] Gaci, S., Zaourar, N. On exploring heterogeneities from well logs using the Empirical Mode Decomposition. General Assembly of the EGU Division on Energy, Resources and the Environment (ERE), Vienna, Austria, 27 April - 2 May 2014. doi:10.1016/j.egypro.2014.10.347.10.1016/j.egypro.2014.10.347Search in Google Scholar
[23] Huang, N.E., Wu, Z.H. A review on Hilbert-Huang transform: method and its applications to geophysical studies. Rev. Geo-phys., 2008, 46(2), RG2006. doi:10.1029/2007rg000228.10.1029/2007rg000228Search in Google Scholar
[24] Gairola, G.S., Chandrasekhar, E. Heterogeneity analysis of geophysical well-log data using Hilbert Huang transform. Physica A, 2017, 478, 131-142. doi:10.1016/j.physa.2017.02.029.10.1016/j.physa.2017.02.029Search in Google Scholar
[25] Huang, N.E., Shen S. S. Hilbert-Huang Transform and Its Applications; World Science Publishing Co. Pte. Ltd, Singapore, 2005, 149-169.10.1142/5862Search in Google Scholar
[26] Hwang, J., Szabian, M.J., Sharma, M. Hydraulic Fracture Diagnostics and Stress Interference Analysis by Water Hammer Signatures in Multi-Stage Pumping Data. In Proceedings of SPE/AAPG/SEG Unconventional Resources Technology Conference, Austin, Texas, USA, 24-26, July 2017. doi:10.15530/URTEC-2017-2687423.10.15530/URTEC-2017-2687423Search in Google Scholar
[27] Iriarte, J., Merritt, J., Kreyche, B. Using Water Hammer Characteristics as a Fracture Treatment Diagnostic. In Proceedings of SPE Oklahoma City Oil and Gas Symposium, Oklahoma City, Oklahoma, USA, 27-31, March 2017. doi: 10.2118/185087-MS.10.2118/185087-MSSearch in Google Scholar
[28] Panjaitan, M.L., Moriyama, A., McMillan, D., Dunaeva, A., Rutledge, L., Xu, J., Parkhonyuk, S., Kabannik, A., Korkin, R.,Warren, M., et al. Qualifying Diversion in Multi Clusters Horizontal Well Hydraulic Fracturing in Haynesville Shale Using Water Hammer Analysis, Step-Down Test and Microseismic Data. In Proceedings of SPE Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, Texas, USA, 23-25, January 2018. doi: 10.2118/189850-MS.10.2118/189850-MSSearch in Google Scholar
[29] Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Snin, H.H., Zheng, Q., Yen, N.C., Tung, C.C., Liu, H.H. The Empirical Mode Decomposition and the Hubert Spectrum for Nonlinear and Non-Stationary Time Series Analysis. Proc. R. Soc. A, 1998, 454, 903–995. doi:10.1098/rspa.1998.0193.10.1098/rspa.1998.0193Search in Google Scholar
[30] Flandrin, P., GonÇAlvÈS, P. Empirical Mode Decompositions as Data-Driven Wavelet-Like Expansions. Int. J. of Wavelets, Multiresolution and Information Processing, 2004, 2, 477-496. doi:10.1142/s0219691304000561.10.1142/s0219691304000561Search in Google Scholar
[31] Yang, T.H., Tham, L.G., Tang, C.A., Liang, Z.Z. and Tsui, Y. Influence of heterogeneity of mechanical properties on hydraulic fracturing in permeable rocks. Rock Mech. Rock Eng., 2004, 37(4), 251-275. doi:10.1007/s00603-003-0022-z.10.1007/s00603-003-0022-zSearch in Google Scholar
[32] Wangen M. Finite element modeling of hydraulic fracturing in 3D. Comput. Geosci., 2013, 17(4), 647-659. doi:10.1007/s10596-013-9346-2.10.1007/s10596-013-9346-2Search in Google Scholar
[33] Wang, D.B., Ge, H. K., Yu, B., Wen, D. S., Zhou, J., Han, D. X., Liu, L. Study of the influence of elastic modulus heterogeneity on in-situ stress and its damage in gas shale reservoirs. Nat. Gas Geosci., 2018, 29(5), 632-643. doi:10.11764/j.issn.1672-1926.2018.04.001.10.11764/j.issn.1672-1926.2018.04.001Search in Google Scholar
[34] Nagaso, M., Mikada, H. and Takekawa, J. The role of rock strength heterogeneities in complex hydraulic fracture formation - Numerical simulation approach for the comparison to the effects of brittleness. J. Petrol. Sci. Eng., 2019, 172, 572-587. doi:10.1016/j.petrol.2018.09.046.10.1016/j.petrol.2018.09.046Search in Google Scholar
[35] Su X.T., Yang Z.J., Liu G.H. Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials: A 3D study. Int. J. Solids Struct., 2010, 47(17), 2336-2345. doi:10.1016/j.ijsolstr.2010.04.031.10.1016/j.ijsolstr.2010.04.031Search in Google Scholar
[36] Moghtaderi A., Flandrin P., Borgnat P. Trend filtering via empirical mode decompositions. Comput. Stat. Data Anal., 2013, 58, 114—126. doi: 10.1016/j.csda.2011.05.015.10.1016/j.csda.2011.05.015Search in Google Scholar
[37] Xiao, Z.K., Ding, W. L., Hao, S. Y., Taleghani, A. D., Wang, X. Y., Zhou, X. H., Sun, Y. X., Liu, J. S., Gu, Y. Quantitative analysis of tight sandstone reservoir heterogeneity based on rescaled range analysis and empirical mode decomposition: A case study of the Chang 7 reservoir in the Dingbian oilfield. J. Petrol. Sci. Eng., 2019, 182, 106326. doi:10.1016/j.petrol.2019.106326.10.1016/j.petrol.2019.106326Search in Google Scholar
© 2019 W. Zhai et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.
