Research Article  Open Access
Jiong Liu, Junrui Ning, Xiwu Liu, Chunyuan Liu, Tiansheng Chen, "An Improved Scheme of FrequencyDependent AVO Inversion Method and Its Application for Tight Gas Reservoirs", Geofluids, vol. 2019, Article ID 3525818, 12 pages, 2019. https://doi.org/10.1155/2019/3525818
An Improved Scheme of FrequencyDependent AVO Inversion Method and Its Application for Tight Gas Reservoirs
Abstract
AVO inversion is a seismic exploration methodology used to predict the earth’s elastic parameters and thus rocks and fluid properties. It is built up on elastic theory and does not consider the seismic dispersion in real strata. Recent experiments and theory of rock physics have shown that in hydrocarbonbearing rocks, especially in gasbearing ones, the change of seismic velocity with frequency may be pretty remarkable for fluid flow in pore space. Some scholars proposed methods about seismic dispersion, such as frequencydependent AVO inversion, to forecast oil and gas reservoirs underground. In this paper, we demonstrate an improved scheme of frequencydependent AVO inversion, which is based on conventional SmithGidlow’s AVO equation, to extract seismic dispersion and predict the hydrocarbon underground. A simple model with gasbearing reservoir is devised to validate the inversion scheme, and further analysis indicates that our scheme is more accurate and reasonable than the previous scheme. Our new scheme applied to the tight gas reservoirs in Fenggu area of western Sichuan depression of China finds that regions with high dispersion gradients correlate well with regions with prolific gas. Analysis and case studies show that our scheme of frequencydependent AVO inversion is an efficient approach to predict gas reservoirs underground.
1. Introduction
The amplitude versus offset (AVO) is one of the most widely used geophysical techniques to predict oil and gas reservoirs [1]. Since Ostrander [2] developed the AVO technique to identify “bright spot” reservoirs, many scholars have studied AVO extensively and got many successful cases [3–9].
AVO approaches mentioned above are based on the assumption that subsurface rocks are elastic. However, recent developments in seismic rock physics theory and practical studies have revealed that underground strata are viscoelastic, and seismic wave always has noticeable dispersion and attenuation for fluid flow in pore space of rocks [10–16]. However, the dispersion characteristics of seismic wave in the actual medium are not considered in the conventional AVO techniques.
In rocks bearing oil, gas, or water, the seismic velocity is dependent on frequency. Especially in those bearing abundant gas, there may be remarkable seismic dispersion. The more fluid the rocks hold, the greater the seismic wave disperses at seismic frequency band. Therefore, seismic velocity dispersion may be used to identify the fluid in porous rocks underground. This has attracted some researchers to study and apply dispersion in hydrocarbon exploration. Chapman et al. [17] studied the dispersion effects on the variance of AVO based on a porous medium model and verified the potential of seismic dispersion for the detection of oil and gas reservoirs. Wilson et al. [18] proposed a frequencydependent method to extract dispersion information from prestack seismic data and predicted the gas reservoirs in North Sea. Wu et al. [19] presented a new technique of spectrum decomposition to improve the accuracy of Wilson’s method. Some successful cases have been reported by frequencydependent AVO analysis [20–23]. These researches show that seismic dispersion may be a tool to predict hydrocarbon reservoirs underground. However, there is an imperfection in Wilson’s scheme of frequencydependent inversion because it cannot give a reasonable explanation for the high Swave dispersion in its results.
In this paper, we develop an improved scheme of frequencydependent AVO inversion to predict tight gas reservoirs. Analysis and case studies show that the new frequencydependent AVO inversion scheme is more reasonable and accurate than the previous scheme.
2. FrequencyDependent AVO Inversion
In this part, Wilson’s scheme of frequencydependent AVO is briefly introduced first. Then, our improved scheme is derived in detail and analytical comparison between two schemes is made. Finally, a signal decomposition technique, which is used to obtain seismic records at different frequencies in frequencydependent AVO inversion, is illustrated.
2.1. Wilson’s Scheme
AVO inversion is a seismic exploration methodology used to predict the earth’s elastic parameters and thus rocks and fluid properties. Zoeppritz derived the formulations of reflectivity and transmissivity when a plane wave impinges on an interface of different strata underground and developed the theoretical work for AVO theory. Assuming the difference of elastic parameters across an interface is small, Aki and Richards [24] derived an approximation equation of reflectivity. Smith and Gidlow substituted the Gardner’s relation between density and Pwave’s velocity into AkiRichard’s equation and obtained the following AVO equation named SmithGidlow’s equation [25]: where is Pwave’s reflectivity, and are the average of P and Swave velocities of strata on both sides of the interface, is the incident angle, and and are, respectively, P and Swave velocity difference between the adjoining medium.
The SmithGidlow’s equation has been widely used in geophysics. But it is based on elasticity, and it does not consider the seismic dispersion in real rocks.
Considering the seismic velocity is dependent on frequency in rocks, Wilson et al. [18] extended the SmithGidlow’s equation into frequency domain and derived a frequencydependent AVO equation. The equation’s expression is where denotes frequency and is reference frequency. Coefficients and are
In equation (2), and are called P and Swave dispersion gradients and their expressions are
and denote the dispersion magnitude of the P and Swave at the reference frequency . In rocks bearing oil, gas, and water, the seismic velocity is dependent on frequency. Especially in rocks bearing abundant gas, there may be remarkable seismic dispersion. Therefore, and can be regarded as attributes to predict the fluids underground.
2.2. An Improved Scheme
Although some cases of success have been reported by Wilson’s frequencydependent AVO inversion, there are some problems in the equation. The item is assumed to be independent of frequency during the derivation of equation (2). The assumption contradicts the premise that seismic velocities are frequency dependent.
In this paper, we derive an improved scheme of frequencydependent AVO inversion without the unreasonable assumption in Wilson’s scheme. The following is the derivation process of the improved one.
First, the SmithGidlow’s equation (1) can be recast into where
Generally Pwave reflection coefficient , P, Swave velocities and the ratio are frequencydependent in real strata. So, equation (7) can be extended into frequency domain, whose expression is
Expanding equation (10) at a reference frequency by Taylor series, neglecting second and higher order items of , we then get the reflection coefficient of Pwave in the frequency domain where and are
Because there is a relationship as equation (11) can finally be simplified as
Equation (15) is our frequencydependent AVO inversion equation in this paper.
From the equation of (12), we can see that denotes the velocity derivative versus frequency of Pwave and we call it Pwave dispersion gradient. Since seismic dispersion in gas reservoirs always is strong, can be used as a tool to detect the “sweet spots” of gas reservoirs. On the other hand, represents the derivatives of mixing P and Swave versus frequency, which we call mixed dispersion gradient. It is more complicated than . Since the underlying physical mechanism of Swave’s dispersion in fluidbearing rocks is unclear presently, only Pwave’s dispersion gradient is used to predict the gas reservoirs and is ignored in the rest of the paper.
It is should be noted that there is reference frequency in the formulation (10). The choosing of always determines the results of frequencydependent AVO inversion. If is located in the domain where seismic dispersion is not remarkable, the inversion results may be suppressed by noise, and the results will deteriorate. We choose the dominant frequency of seismic wavelet as because the dispersion at the vicinity of dominant frequency usually is obvious in real data.
Our scheme of frequencydependent AVO inversion is inspired by Wilson’s scheme, but it is improved. First, there is no unreasonable assumption that is frequency independent in Wilson’s scheme. So our scheme is more accurate in theory. Second, is included in the unknown term in the new scheme and does not need estimating before inversion. While is included in the inversion matrix in Wilson’s scheme, and it needs estimating. So our scheme is more convenient than Wilson’s.
2.3. Smoothed Pseudo WignerVille Distribution
Obtaining seismic records at different frequencies is an important step of frequencydependent AVO inversion. Many methods can decompose seismic data into parts in different frequency bands, such as Fourier transform, shorttime Fourier transform (STFT) and wavelet transform etc. Here, a signal decomposition technique named smoothed pseudo WignerVille distribution (SPWVD) is introduced. The comparison between SPWVD and other methods, such as STFT and wavelet transform, is also made to test the accuracy of SPWVD.
WignerVille distribution (WVD) is one of the most effective approaches to decompose nonstationary signal on the timefrequency plane via an energy distribution function. The timefrequency decomposition of a signal by WVD is expressed as [26] where is the time delay and is the analytic signal corresponding to . Different from shorttime Fourier transform, WVD does not have a window function and avoids the contradiction that the time resolution and frequency resolution mutually constrain during STFT. However, as defined in equation (16), WVD is not linear, which means the WVD of two signals’ sum is not equal to the sum of each signal’s WVD because of a cross term in the WVD sum. These cross terms generate a false energy distribution which is one of most important issues when WVD is used for timefrequency analysis of multicomponent nonstationary signal. To mitigate the impact of cross terms, modified approaches called smoothed pseudo WignerVille distribution (SPWVD) have been proposed, and it is defined as [26]: where is time delay, and is frequency offset. is the time smoothing window while is the frequency smoothing window. If both and are real symmetric functions, . The selections of two smoothing windows are independent of each other. It is possible to attenuate the cross terms in WVD by the two smoothing windows. For example, the two smoothing windows and could both be Gaussian functions. in which is the smoothing function in timefrequency space and and are coefficients designed to adjust the bandwidth of the Gaussian functions.
For a synthetic or a seismic trace, the magnitudes of decomposed components at different frequencies are different for two factors. One is different reflection coefficients at different frequencies, which is caused by dispersion. The other is the different magnitude of wavelet components. Frequencydependent AVO inversion considers the reflection coefficients at different frequencies. So, the effects of the wavelet must be removed before frequencydependent AVO inversion. An efficient spectrum balance technique [27] is adopted in this paper.
In the following frequencydependent AVO inversion, the SPWVD technology is used to obtain seismic records at different frequencies.
2.4. Validation the Scheme of FrequencyDependent AVO Inversion
In this part, a simple model with horizontal layers is devised to test our scheme of frequencydependent AVO inversion. Comparison of results by Wilson’s scheme and ours is also made for the model.
The simple model is composed by three horizontal layers as shown in Figure 1. The top two layers are nonreservoir and are regarded as elastic medium without dispersion. The bottom layer is gasbearing reservoir with strong dispersion, and poroelastic medium is adopted to represent it. The parameters of each layer are shown in Table 1. Parameters and are P and Swave velocities, is the density, and is the porous volume; and are water and gas saturation. For layer 1 and layer 2, there are no or because they are elastic medium.

Reflection coefficients at the interface between layer 1 and layer 2 can be computed easily by AkiRichard’s equation or SmithGidlow’s equation. Then, corresponding synthetics can be obtained by convolution between the coefficients and wavelet. But the synthetics at the interface between layer 2 and layer 3 are more difficult to get because the seismic velocity is dependent on frequency. We first compute the seismic velocities at different frequencies on patchy model [28] and derive the corresponding reflection coefficients from seismic velocities. Then, the frequency components of the wavelet are computed by fast Fourier transform, and the components are multiplied with reflectivity at different frequencies. Finally, the results are transformed into time domain to get the synthetics by inverse fast Fourier transform. Because the patchy model of poroelasticity only considers the Pwave dispersion caused by porous fluid, the synthetics include the effects of Pwave dispersion and those of Swave are not included. This characteristic will be used to test the rationality of frequencydependent AVO inversion by Wilson’s scheme and ours later.
Figure 2 shows the prestack synthetics at thirteen incidence angles 2^{o}, 3^{o} … 13^{o}, which are computed by Ricker wavelet with 30 Hz dominant frequency. The peak at 0.1 s coincides with the interface across the top two nonreservoir layers and the trough at 0.2 s with the interface between the nonreservoir and gasreservoir layers.
SPWVD technique is used to decompose the synthetics. Figure 3 shows the original decomposition (Figure 3(a)) and balanced (Figure 3(b)) results of the synthetic at frequencies 26 Hz, 28 Hz, 30 Hz, 32 Hz, and 34 Hz with 6° incidence angle. We can see that the effects of seismic wavelet are eliminated by spectrum balance technique. The remaining energy differences with various frequencies at 0.2 s are caused by seismic dispersion.
(a)
(b)
Figure 4 is the inversion results based on the new inversion scheme for the layered model. To compare our scheme with the previous one, the inversion by Wilson’s scheme is made and also shown in Figure 4. From Figure 4(a), we can see the value of , the Pwave dispersion gradient of our results, is large at the interface between elastic and poroelastic medium, while the value at the elasticelastic interface is rather small. The inversion result is consistent with the model condition that dispersion effects exist on the elasticporoelastic interface and do not exist on the elasticelasticone. The model results also show that the Pwave dispersion gradient can be used to predict subsurface reservoirs and our frequencydependent inversion scheme is workable. Similarly, the peak of inversion result by Wilson’s scheme matches the interface between elastic and poroelastic medium too. But the value is less than ours at the peak. In Figure 4(b), the mixed dispersion gradient of our results also matches the interface of elastic and poroelastic medium. The Swave’s dispersion gradient of Wilson’s scheme has similar character. However, the Swave’s dispersion gradient should be zero because Swave is set to be independent of frequency in the model. Obviously, the result of is not reasonable. Through analysis, we think that is caused by unreasonable assumption that is frequency independent in equation (2). The inversion results show that Wilson’s scheme is not reasonable for Swave dispersion gradient. On the contrary, the nonzero value of at the elasticelastic interface in our results can be interpreted reasonably by the term of because Pwave is frequency dependent. Our scheme is more reasonable than Wilson’s.
(a)
(b)
Since the mixed dispersion gradient has similar characters with that of Pwave dispersion gradient, in following application, we only employ Pwave dispersion gradient to detect gas reservoirs underground.
3. The Application of FrequencyDependent AVO Inversion
In this part, we first introduce the geology background of Fenggu area briefly. Then, our scheme of frequencydependent AVO inversion is applied to predict the gas reservoirs in this area.
3.1. Geology Background
The Fenggu structure is located in eastern end of the XiaoquanHexingchangXinchangFenggu structural zone in western Sichuan depression, China (Figure 5). In late Triassic, a thick set of sediment was deposited in Xujiahe formation. T_{3}x^{2}, T_{3}x^{3}, T_{3}x^{4}, and T_{3}x^{5} members are positioned from bottom to top in Xujiahe formation (Figure 6), with T_{3}x^{1} member absent but developed in other parts of western Sichuan depression. Geology studies show that the gas source in Xujiahe formation of Fenggu area is mega.
In Upper Triassic Xujiahe formation, Fenggu area, T_{3}x^{4} member is considered highly prospective for gas. The favorable sedimentary include plain river, frontal mouth bar of fan deltas, and meandering stream deltas facies. Rock physics tests show that reservoir porosity is primarily composed of secondary corrosion pores, remaining intergranular pores and developed micro fracturing. Since the average porosity and permeability are low, the reservoirs in T_{3}x^{4} member of Xujiahe formation are tight gas reservoirs. In the following, we will detect gas reservoirs in T_{3}x^{4} member.
3.2. FrequencyDependent AVO Inversion in Fenggu Area
The scheme (15) of frequencydependent AVO inversion is applied to 3D seismic data of T_{3}x^{4} member, Xujiahe formation, Fenggu area. First prestack angle gathers are extracted from prestack seismic recordings (offset gathers) by seismic velocity. Subsequently, the SPWVD technique is used to derive angle gathers at different frequencies from prestack angle gathers. Next, the scheme (15) is applied on the angle gathers at different frequencies for inversion, and the dispersion gradient of Pwave is computed. Since the places with large Pwave dispersion gradient always correspond with hydrocarbon reservoirs, the gas reservoirs underground are predicted by the Pwave dispersion gradient.
Figure 7 shows the Pwave dispersion gradient cube of T_{3}x^{4} member in the Xujiahe formation, Fenggu area. The color bar represents the amplitude of Pwave dispersion gradient.
Figure 8 is the inversion profile of Pwave dispersion along a seismic line through well cf175. The two red curves denote the top and bottom of T_{3}x^{4} member. Colored blocks on wellpath indicate logging fluid interpretation results, in which yellow one represents highquality gasbearing reservoir while the thickness of the color blocks denotes the abundance level of reservoirs. Inversion results indicate that there is a region with high value of Pwave dispersion gradient at well cf175 (marked by the purple ellipse), which means it may be a reservoir with gas. The test of well cf175 proves it is a good conventional gas reservoir. Our inversion results match the well tests in this region.
Figure 9 is another inversion profile through well cf563. Red blocks on wellpath represent fractured gas reservoirs. Our inversion shows that the three domains marked by red ellipses may have abundant gas because of their high value of Pwave dispersion gradient. The well tests prove the regions are fractured or conventional gas reservoirs. Our inversions do match the well tests.
Figure 10 is the slice of P–wave’s dispersion gradient along the first set sand which is near the bottom of T_{3}x^{4} member. Inversion results show that the west and the southeast of the area is the favorable zone of gas reservoirs for the high value of Pwave dispersion gradient.
In the object area, five wells reaching the bottom of T_{3}x^{4} member have been drilled, and Table 2 shows the logging results of the first set sand of T_{3}x^{4} member.

Comparison of P–wave’s dispersion and the well tests in Table 1 reveals the following: the Pwave dispersion gradient value at cf563 is large and logging indicates the presence of a prolific gas reservoir; the Pwave dispersion gradient value is small at Fg22 and logging of this well shows that this area constitutes with mudstones; and inversion values at cf125 an cf21 are moderate and loggings indicate these regions are poorquality gas reservoirs. It should be noted that region at well Cf175 is near the domain with large Pwave dispersion gradient but the inversion value is not large. It can be seen more clearly from Figure 9 that the region near the bottom of T_{3}x^{4} at well cf175 has moderate Pwave dispersion gradient. Our predictions at well cf175 still match with the well tests. The comparison indicates that our predictions are right and the frequencydependent AVO inversion by our scheme can be used to detect tight gas reservoirs underground.
3.3. Comparison of FrequencyDependent AVO Inversions by Different Schemes
Frequencydependent AVO inversion by Wilson’s scheme is also done on seismic data of Fenggu area. Figure 11 is the inversion profile through well cf563. At location 2, the value of Pwave dispersion gradient is high and the prediction matches the well test. But at locations 1 and 3, the values of Pwave dispersion gradient are not high while well tests demonstrate the two places are good conventional and fractured gas reservoirs. The predictions by Wilson’s scheme mismatch with well tests at the two locations. However, the predictions by our scheme (Figure 9) match well tests. The comparison for this profile shows that our scheme is more accurate than Wilson’s.
Figure 12 is the inversion profile through well cf125 by the new scheme (Figure 12(a)) and Wilson’s (Figure 12(b)). P–wave’s dispersion gradients of the domain marked by red ellipse, which is near the bottom of T_{3}x^{4} member, are low, and well test demonstrates the place is a poor gas reservoir (white block). Two inversion results match well test. However, comparison shows that result by our scheme has better continuity and more contrast than that by Wilson’s.
(a)
(b)
4. Discussion and Conclusions
Dispersion is related to porous fluid when seismic wave propagates in reservoirs. In regions where porous fluid is abundant, the seismic dispersion is always great at seismic frequency band, especially in gas reservoirs. Frequencydependent AVO inversion is indeed a method to extract Pwave dispersion gradient from prestack angle gathers. Further prediction of oil and gas reservoirs can be made by inverted dispersion gradient.
In this paper, we derived a scheme of frequencydependent AVO inversion. It is motivated by Wilson’s scheme, but is improved. The comparison between inversion results for a layered model by two different schemes shows that our scheme is more reasonable. Further practical application in Fenggu area is made on our inversion scheme. The predictions of gas reservoirs by our scheme of frequencydependent AVO inversion match well tests. That verifies the effectiveness of our inversion scheme in prediction of gas reservoirs.
There is a point to be noted that the Gardner’s equation about density and Pwave velocity is employed in the derivation of our inversion scheme. Since it is just an empirical equation, it may need modification for special regions, and the scheme of frequencydependent AVO needs corresponding modification too. By this treatment, better inversion results can be expected. This task will be undertaken in our future work.
Data Availability
All model data during this study are listed in this manuscript; researchers can replicate the analysis. But the real seismic data in the application section, which is very large, are not available because they involve business secrets.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study is financially supported by National Science and Technology Major Project (Grant no. 2017ZX05005004002) and NSFC and Sinopec Joint Key Project (U1663207). We also acknowledge Miss Jie Li for her patient editing.
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