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Scientific Reports volume 14, Article number: 25773 (2024 ) Cite this article diffraction of other shaped holes
Low-frequency shadows beneath gas reservoirs can be regarded as a time delay relative to the reflection from the reservoir zone, but they cannot be reasonably explained by the high-frequency attenuation or velocity dispersion observed in normal P-waves. According to the new asymptotic theory for porous media, seismic P-waves undergo multiple conversions between fast and slow modes during seismic waves passing through layered permeable reservoirs at low frequencies, and changes in the velocity and amplitude (i.e., energy) of slow P-waves can lead to low-frequency shadows. In this study, a forward analysis was performed on the dispersion and attenuation of fast and slow P-waves within the seismic frequency band based on the asymptotic theory for porous media; the results revealed that fast P-waves do not undergo frequency dispersion and attenuation within the seismic frequency band and that slow P-waves are the primary contributor of dispersion and attenuation. In addition, methods used to calculate the frequencies at which low-frequency shadows occur were analyzed and are discussed. Finally, S-transform time-frequency analysis method was used to calculate and analyze the low-frequency shadows of three-dimensional seismic data acquired from work area M in Sichuan. The low-frequency shadow anomalies determined by this method were found to be highly consistent with those identified based on the data acquired from wells in the target reservoir. These results indicate the good application performance of this method.
Low-frequency seismic amplitude anomalies have been successfully used to detect hydrocarbons in seismic exploration and development1,2,3. Seismic interpretation method based on the bottom-simulating reflectors (BSRs) will produce strong reflection and low-frequency with shadow seismic response characteristics4. However, the mechanism behind the occurrence of this type of low-frequency anomaly has yet to be reasonably explained with available theories. Other researchers have made huge contributions in this area of investigation5,6,7,8,9. Such as Ebrom attributed the cause of low frequency shadow to stack-related and other mechanisms. He proposed ten mechanisms that could lead to the occurrence of low frequency shadow, and these mechanisms may decrease the actual frequency by mis-stacking5. In fact, based on the review of related literature, there is no consensus on the causes of low frequency shadow. In addition, Barnes mentioned that only few examples of low-frequency shadows convincingly reveal the presence of gas6. Under normal circumstances, several different types of low-frequency anomalies appear beneath gas reservoirs. In most cases, low-frequency amplitude anomalies are present below hydrocarbon-rich reservoirs but are not markedly delayed in time relative to the reflection from the reservoir. The occurrence of this type of anomaly is usually ascribed to the relatively considerable attenuation of low-frequency seismic signals10,11. Attenuation is often explained with dissipative and viscous attenuation models or models describing seismic waves in nonhomogeneous fluid-saturated porous media10,11,12. Low-frequency anomalies can also result from the tuning response caused by gas or other fluids present in a reservoir2. One type of low-frequency shadow amplitude anomaly is relatively significantly delayed in time relative to the tuning response from thin layers and is delayed by 100 ms or longer relative to reservoir roofs. This phenomenon cannot be explained by the high-frequency attenuation or velocity dispersion observed in normal P-waves. Geophysicists have attempted to explain this phenomenon through high-frequency attenuation theories that include factors such as the heterogeneity of reservoir zones and the fluid wavelength–scale10,13. However, the wavelength-scale fluid pressure in a nonhomogeneous fluid-saturated porous medium is meaningful only at frequencies higher than the characteristic frequency and usually can only be obtained under experimental conditions; in addition, its attenuation within the seismic frequency band is often negligible13,14. Further, high attenuation can be obtained from highly nonhomogeneous reservoir models. However, the conditions defined for this type of representative model are strict. For example, this type of nonhomogeneous reservoir must consist of a layered structure composed of alternating saturated fluids with markedly different properties (e.g., a gas and water) and arranged in a patch-like distribution15. However, even this high-frequency attenuation mechanism fails to explain reflection delays of more than 100 ms. In contrast, the asymptotic analysis of Biot’s theory can provide a relatively reasonable explanation for the phenomena observed beneath reservoirs, including reflection delays and the formation of low-frequency energy. Silin and Goloshubin (2010) noted that this type of delayed low-frequency response is caused by fast-slow-fast P-wave conversions in layered, permeable fluid-saturated porous media16. In an attempt to determine the mechanism behind the occurrence of low-frequency shadows in gas reservoirs, Chabyshova and Goloshubin (2014) presented a relatively reasonable explanation for the low-frequency energy and time delays observed in a model of the propagation of P-waves in a typical layered sandstone structure by performing forward calculations using the asymptotic method associated with Biot’s model for elastic porous media17. Some scholars explored the use of low-frequency shadows to detect the presence of gas in reservoirs based on previous research18,19,20. In addition, some oil companies, such as China National Offshore Oil Corporation, have discovered anomalies in fan-shaped lakebed reservoirs by investigating low-frequency shadows. Low-frequency shadows have been successfully applied in the oil and gas exploration field; however, previous studies do not reasonably explain the occurrence of low-frequency shadows and time delays. In this study, these phenomena were reasonably explained with the asymptotic theory for porous media. Forward simulations of the velocity dispersion and attenuation of fast and slow P-waves based on the theory for porous media reveal that the attenuation of P-waves is caused primarily by slow P-waves. Moreover, the application of low-frequency shadows in the detection of the presence of gas in reservoirs was analyzed based on three-dimensional (3D) seismic data acquired from work area M. In practice, Silin and Goloshubin’s equation yields low response frequencies with relatively large errors. Therefore, the S-transform, known for its high time-frequency accuracy, was employed to extract the attributes of low-frequency shadows through frequency scanning. The final results are highly consistent with the predictions produced based on drilling data, confirming that this method can effectively detect oil and gas.
According to Biot’s theory, even the vertical incidence of a P-wave at a permeable interface can produce four types of waves21,22: fast and slow reflected P-waves (denoted by RFF and RFS, respectively) as well as fast and slow transmitted P-waves (denoted by TFF and TFS, respectively). Fast P-waves are generated due to the compressibility of rock during the propagation process, while slow P-waves are formed as a result of the motion of the porous fluid relative to the matrix. Slow P-waves rely heavily on fluid mobility. Compared to fast P-waves, slow P-waves undergo considerable frequency-variable attenuation and slow velocity dispersion.
In a porous medium, the laws of the conservation of mass and momentum reflect the displacement of the rock matrix and require continuity in the Darcy velocity, total pressure, and pressure of the fluid at reflecting interfaces. The asymptotic expressions of the reflection and transmission coefficients of a vertically incident seismic wave at a reflecting interface are given below:
where \(\:{R}_{0}^{FF}\) and \(\:{R}_{1}^{FF}\) are the zero- and first-order terms of the asymptotic expansion of the reflection coefficient, respectively, \(\:{T}_{0}^{FF}\) and \(\:{T}_{1}^{FF}\) are the zero- and first-order terms of the asymptotic expansion of the transmission coefficient, respectively, i is the imaginary unit, and \(\:\varepsilon\:\) is a comprehensive dimensionless parameter of the fluid, which is defined as follows:
where \(\:{\rho}_{f}\) is the density of the fluid, \(\:\eta\:\) is the coefficient of viscosity of the fluid, \(\:\kappa\:\) is a combination of permeabilities, and ω is the angular frequency.
In the asymptotic expansions, the zero-order reflection and transmission coefficients can be expressed as follows:
where \(\:{{Z}_{A}}^{FF}\) and \(\:{{Z}_{B}}^{FF}\) are the impedances of the fast P-waves in the upper and lower layers, respectively. The definition of \(\:Z\) is given below:
where \(\:M\) is the elastic parameter of the rock. According to Biot21,22, the parameters in Eq. (6) can be expressed as follows:
The first- and zero-order reflection and transmission coefficients can be expressed as follows:
In addition, \(\:{v}_{f}=\sqrt{\frac{M}{{\rho}_{f}}}\) , and \(\:{\gamma}_{\kappa\:}={\kappa}_{A}/{\kappa}_{B}\) is the ratio of the permeability of one phase of the two-phase medium to that of the other. In the above equations, \(\:{K}_{g}\) is the bulk modulus of the solid particles, \(\:K\) is the bulk modulus of the dry rock, \(\:\mu\:\) is the shear modulus of the dry rock, \(\:\varphi\:\) is the porosity, \(\:{\rho}_{g}\) is the density of the solid particles, \(\:{K}_{f}\) is the bulk modulus of the fluid, \(\:{{R}_{1}}^{FF}\) and \(\:{{T}_{1}}^{FF}\) are the first-order reflection and transmission coefficients of fast-to-fast converted P-waves, respectively, \(\:{{R}_{0}}^{SF}\) and \(\:{{T}_{0}}^{SF}\) are the zero-order reflection and transmission coefficients of fast-to-slow converted P-waves, respectively, \(\:{{R}_{1}}^{FS}\) and \(\:{{T}_{1}}^{FS}\) are the first-order reflection and transmission coefficients of fast-to-slow converted P-waves, respectively, and \(\:{R}_{SS}\) and \(\:{T}_{SS}\) are the reflection and transmission coefficients of slow-to-slow converted P-waves, respectively. See elsewhere16,17,23 for the detailed equations.
When physical measurements are unavailable, the bulk and shear moduli of dry sandstone can be calculated using the following equations:
where \(\:{K}_{sat}\) and \(\:{\mu}_{sat}\) are the bulk and shear moduli of water-saturated sandstone, respectively, and \(\:{V}_{p}\) and \(\:{{V}_{S}}^{\:}\) are the velocities of the P- waves and shear waves, respectively.
Dispersion and attenuation of fast and slow P-waves.
The asymptotic theory for porous media gives the following velocity dispersion and attenuation equations for fast and slow P-waves:
where \(\:{\zeta}_{0}^{F}={\gamma}_{\beta\:}+{{\gamma}_{K}}^{2}/{\gamma}_{\beta\:}{\gamma}_{\rho\:}\) , \(\:{\zeta}_{1}^{F}=1/{\gamma}_{\beta\:}{({\gamma}_{\beta\:}+{{\gamma}_{K}}^{2})\left(\right({\gamma}_{\beta\:}+{{\gamma}_{K}}^{2})/\gamma}_{\rho\:}{-{\gamma}_{K})}^{2}\) , \(\:{\gamma}_{\rho\:}\) is the ratio of the density of the medium matrix to that of the fluid, \(\:{V}^{F}\) and \(\:{V}^{S}\) are the velocity dispersion of fast and slow P-waves, respectively, \(\:{a}^{F}\) and \(\:{a}^{S}\) are the attenuation coefficients of fast and slow P-waves, respectively, and \(\:{v}_{b}\) and \(\:{v}_{F}\) are the velocities of the rock and the fluid, respectively.
According to the asymptotic theory for porous media, the reflection amplitude of a fast P-wave, \(\:{A}_{F}\) , in a thin permeable reservoir (Fig. 1) can be expressed as follows:
Conversion between fast and slow P-waves at the reflecting interface of thin layers.
In the above equation, \(\:{A}_{0}\) is the amplitude of the normally incident fast P-wave, VS is the velocity of the slow P-wave, \(\:{a}_{s}\) is the attenuation coefficient of the slow P-wave16, ε is a dimensionless parameter that relies on the imaginary unit i, ω is the angular frequency, and H is the thickness of the thin layer. The reflection amplitude of a fast P-wave in a highly permeable thin layer (with a thickness of 0.5 m or less) can reach the order of 10–4 and can be delayed by 100 ms or longer at low frequencies (5–10 Hz). Fast P-waves undergo almost no reflection in thin layers that differ considerably in their impedance23. The relatively fast attenuation of slow P-waves prevents them from being observed in seismic exploration. However, the absence of the separation of the energy of any wave at a reflecting interface during propagation is unrealistic. In fact, waves produce fast P-waves and slow P-waves conversions at the reflection interface. Due to their very low conversion energy, slow P-waves are often overlooked in seismic exploration. In the presence of multiple layers, a normally incident P-wave incurs a transmission loss at each interface, and its total transmission loss is the sum of its transmission losses in all the layers, which can be expressed in terms of the product of its transmission losses incurred in all the layers that it passes.
If a P-wave is reflected in the mth layer, its total transmission loss is the sum of the transmission losses incurred in all the layers above the mth layer, which can be expressed as follows:
where Td and Tu are the downward and upward transmission coefficients, respectively. Apart from incurring a transmission loss at the interface of two consecutive layers, a P-wave also undergoes absorption-induced attenuation in a homogeneous layered medium. The decay of the amplitude of a seismic wave in an absorptive medium is a function of the propagation distance. Its amplitude can be expressed as follows:
where A0 is the initial amplitude, H is the propagation distance of the seismic wave (i.e., the thickness of the thin layer), and \(\:{a}_{s}\left(\omega\:\right)\) is the attenuation coefficient of the slow P-wave. Then, the reflection of a seismic wave in the mth layer of a medium composed of interbedded permeable thin layers can be described as
where \(\:{R}_{0 m}\) is the reflection coefficient of fast-to-fast, fast-to-slow, or slow-to-fast converted waves in the mth layer.
In a medium composed of multiple thin layers (each with a thickness of 0.5 m or less), fast-slow-fast converted P-waves do not undergo complete attenuation due to reflection; instead, their amplitudes change significantly and are markedly delayed. This phenomenon is particularly more pronounced at low frequencies. Therefore, in a permeable thin-layered fluid-saturated porous medium, Biot slow waves or fast-slow-fast converted P-waves can generate a low-frequency amplitude response. P-wave conversion response in models describing reservoirs consisting of multiple interbedded thin sandstone layers can be used to find an asymptotic solution for seismic waves in fluid-saturated porous media. Asymptotic analysis is based on Biot’s theory, which describes the propagation of elastic waves in fluid-saturated porous media21,22. An asymptotic solution is an approximate solution that contains only the first term of the solution obtained from Biot’s theory and can therefore be applied within the low-frequency range of the seismic frequency band16. Asymptotic solutions are often used to estimate the reflection and transmission properties of plane waves propagating in layered saturated porous media. Through a forward simulation of a thinned (20-m-thick) sandstone reservoir, Chabyshova and Goloshubin17 confirmed the presence of low-frequency energy superposition and time delays.
According to the asymptotic theory for porous media16, the function of the reflection factor can be expressed in terms of the permeability \(\:{\upkappa\:}\) as follows:
where \(\:{\varrho}_{f}\) is the density of the fluid. This function has the following maximum value:
When \(\:\sqrt{{\left|\varepsilon\:\right|}_{max}}\:=\:\frac{1}{H}\frac{\kappa\:}{\eta\:}\sqrt{\frac{2 M{\varrho}_{f}}{{\gamma}_{\beta\:}+{{\gamma}_{K}}^{2}}}\) , the peak low frequency is
\(\:{f}_{MAX}\) ≈ 8 Hz when M = 10+ 10 Pa, \(\:{\gamma}_{\beta\:}+{{\gamma}_{K}}^{2}\) ≈ 2.5, \(\:{\upkappa\:}\) = 1 Darcy, \(\:{\upeta\:}\) = 10−3 Pa s, and H = 0.5 m (extracted from Silin and Goloshubin (2010)). However, \(\:{f}_{MAX}\) values calculated using Eq. (28) coupled with multiple models derived from real-world logging and rock physics data are much lower than 8 Hz, which can be mainly attributed to the following two factors. (1) The value of (\(\:{\gamma}_{\beta\:}+{{\gamma}_{K}}^{2}\) ) calculated based on real-world values of the rock physics parameters is much greater than 2.5. (2) \(\:{\upkappa\:}\) is much lower than 1 × 10–3 μm2 Darcy. Therefore, the obtained \(\:{f}_{MAX}\) value is very small, suggesting that Eq. (28) cannot be directly applied to real-world scenarios. However, low-frequency analysis can indeed reveal the presence of low-frequency shadows in some seismic data. Therefore, data acquired from known wells coupled with time-frequency analysis and low-frequency scanning should be employed in practice to determine the frequencies at which low-frequency shadows occur.
To further analyze the attenuation and dispersion characteristics of fast and slow P-waves, the parameters of the rock matrix and porous fluid required for forward modeling were extracted from the data acquired from well Y1 in western Sichuan (Table 1) and then used to perform forward calculations based on Eqs. (17)—(20).
Figure 2 shows the velocity dispersion and attenuation curves of a fast P-wave and a slow P-wave. As shown in Fig. 2a, the velocity of the fast P-wave undergoes almost no changes as the frequency increases. A similar phenomenon can be observed for the attenuation factor in Fig. 2b. These findings suggest that these equations are ineffective for determining the velocity dispersion and attenuation of fast P-waves within the seismic frequency band. In contrast, Fig. 2c reveals notable changes in the velocity and attenuation factor of the slow P-wave with frequency, suggesting that slow P-waves can undergo velocity dispersion and attenuation at any time within the seismic frequency band, which is the primary cause of low-frequency shadows.
Velocity dispersion and attenuation curves of a fast P-wave and a slow P-wave. (a) Velocity dispersion and (b) attenuation of a fast P-wave. (c) Velocity dispersion and (d) attenuation of a slow P-wave.
The low frequency shadow gas-bearing detection profile (a) gas well w1 (b)dry well w2 (c) dry well w3.
The map of favorable area of reservoir and low-frequency shadow gas-bearing detection. (a) Favorable area of reservoir (b) low-frequency shadow gas-bearing detection.
Planar distribution of seismic amplitude.
Planar distribution of instantaneous frequency.
The asymptotic theory for porous media provides a relatively reasonable explanation for the presence of low-frequency shadows and time delays beneath gas reservoirs. Forward modeling based on the asymptotic method for analyzing porous media demonstrates that under the asymptotic theory for porous media, slow P-waves indeed undergo velocity dispersion and attenuation within the seismic frequency band, which is the primary cause of low-frequency shadows beneath reservoirs.
Accurate methods for determining the frequencies associated with low-frequency shadows used to detect the presence of gas in reservoirs are currently lacking. These frequencies can be determined based on data acquired from known wells in conjunction with time-frequency analysis and frequency scanning.
Real-world experiments involving the use of low-frequency shadows to detect the presence of gas in reservoirs in multiple areas reveal that not all gas reservoirs are incidental to low-frequency shadows and time delays and that gas anomalies associated with most reservoirs appear in the surrounding areas.
The data that support the findings of this study are available from the first author, Shoucheng Xu, upon reasonable request.
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This work was supported by National Natural Science Foundation of China (Grant No. 42172175) , Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu University of Technology (PLC201710) and Department of Science and Technology of Guangxi Zhuang Autonomous Region, China (No. AB23026062).
College of Energy (College of Modern Shale Gas Industry), Chengdu University of Technology, Chengdu, 610059, Sichuan, China
Shoucheng Xu, Xiuquan Hu, Jintao Mao, Chao Zhang & Boqiang Wang
State Key Laboratory of Reservoir Geology and Development Engineering, Chengdu University of Technology, Chengdu, 610059, Sichuan, China
School of Geoscience and Technology, Southwest Petroleum University, Chengdu, 610500, Sichuan, China
Exploration and Development Research Institute, Sinopec Southwest Oil and Gas Branch Company, Chengdu, 610041, Sichuan, China
Sinopec Petroleum Engineering Geophysics Co., Ltd., South Branch, Chengdu, 610041, Sichuan, China
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Shoucheng Xu: Presenting ideas, Conceptualization, Formula analysis, Resources, Literature investigation, Writing - Original Draft. Xiuquan Hu: Conceptualization, Dispersion analysis, Literature investigation, Formula analysis, Writing - Original Draft. Duo Xu: Literature investigation, Formula analysis Writing -Original Draft. Yang Lei: Literature investigation, Formula analysis, Dispersion analysis, Data interpretation. Jintao Mao: Data collection, Illustrations - Review and Editing. Chao Zhang: Data collection, Illustrations - Review and Editing. Boqiang Wang: Data Curation - Review and Editing.
The authors declare no competing interests.
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Xu, S., Hu, X., Xu, D. et al. Analysis and application of low frequency shadows based on the asymptotic theory for porous media. Sci Rep 14, 25773 (2024). https://doi.org/10.1038/s41598-024-76870-4
DOI: https://doi.org/10.1038/s41598-024-76870-4
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