Gas Prediction in Tight Sandstone Reservoirs Based on a Seismic Dispersion Attribute Derived from Frequency-Dependent AVO Inversion

Abstract

Accurate gas prediction is crucial for identifying gas-bearing zones in tight sandstone reservoirs. Traditional seismic techniques, primarily grounded in elastic theory, often overlook inelastic dispersion effects inherent to such formations. To overcome this limitation, we introduce a gas prediction approach utilizing a dispersion attribute derived from frequency-dependent inversion based on an AVO equation parameterized by a gas indicator and related properties. Rock physics modeling, based on multi-scale fracture theory, reveals the frequency-dependent gas indicator is highly responsive to variations in porosity and gas saturation. Seismic AVO simulations exhibit distinguishable signatures corresponding to these variations, supporting the potential to estimate reservoir properties from pre-stack seismic data. Synthetic data tests confirm that the values of the proposed dispersion attribute increase with increasing porosity and gas saturation. Additionally, the calculated dispersion attribute exhibits a strong positive correlation with gas content, validating its effectiveness for gas evaluation. Field application results further demonstrate that the proposed dispersion attribute shows prominent anomalies in sandstone reservoirs with high gas content. Compared to the conventional P-wave dispersion attribute, the proposed dispersion attribute exhibits superior reliability in detecting gas-rich zones. These results demonstrate the utility of the method in predicting gas-bearing regions in tight sandstone reservoirs.

1. Introduction

With the continuous rise in global energy demand, tight sandstone gas reservoirs have emerged as a vital unconventional hydrocarbon resource, offering substantial exploration potential and favorable development prospects. The basis for high-quality tight sandstone reservoirs is significant gas enrichment, as sufficient gas accumulation is essential to sustain productivity. However, tight sandstone reservoirs are often characterized by low porosity, low permeability, intricate pore structures, and strong heterogeneity. Such geological complexities pose significant challenges to the accurate identification of sweet spots [1,2].

At present, the integration of rock physics methodologies provides an effective framework for quantifying fluid-related variations in elastic and seismic responses. Rock physics templates (RPTs) have been extensively utilized for the identification of porosity and fluid [3,4,5,6]. Additionally, numerous fluid indicators, derived from various combinations of elastic parameters, have demonstrated success in identifying reservoir fluids [7,8,9,10,11]. Concurrently, the direct estimation of reservoir properties from pre-stack seismic data has gained prominence due to its potential to minimize cumulative interpretation errors. As a result, AVO equations incorporating different fluid indicators have been developed to facilitate the direct inversion of reservoir parameters [12,13,14,15,16,17].

The aforementioned approaches primarily depend on elastic properties for fluid prediction. However, numerous studies have shown that gas saturation and porosity can induce observable seismic wave dispersion and attenuation. Laboratory experiments have validated the significant influence of fluids on P-wave velocity dispersion and attenuation behavior [18]. Concurrently, several theories have been proposed to describe frequency-dependent velocity variations resulting from fluid mobility within hydrocarbon-bearing formations [19,20,21,22,23]. Moreover, studies that integrate rock physics with seismic forward modeling have revealed that fluid-induced dispersion and attenuation exert notable effects on AVO responses [24,25]. These findings suggest that incorporating seismic wave dispersion into the development of seismic analysis methods offers a promising avenue for improving gas detection in tight sandstone reservoirs.

Contemporary seismic analysis techniques often utilize a frequency-dependent amplitude variation with offset (FD-AVO) methods to extract dispersion attributes from seismic data. The foundational FD-AVO framework was initially proposed by Wilson [26], and subsequent developments have introduced a range of dispersion attributes linked to P-wave reflection behavior [27,28,29,30,31,32,33,34,35,36] and fluid bulk modulus variations [37,38], primarily aimed at hydrocarbon detection. In recent years, machine learning methods have been incorporated into the framework of the FD-AVO inversion method [39,40]. Nonetheless, a key limitation of these existing approaches is their predominant focus on fluid effects while neglecting the influence of porosity in gas prediction. To achieve a more accurate evaluation of gas content, it is crucial to design dispersion attributes that account for the combined influence of fluid properties and porosity. The aforementioned approaches have been applied across different lithologies, including carbonates [3,4,5], sandstone [6,9,10,11,12,13,14,15,17,18,28,29,30,31,33,35,37,38,39,40], shale [27,34], and volcanic rocks [32,36]. Furthermore, seismic methods for fluid detection in various reservoirs have been thoroughly summarized and reviewed [41].

This study introduces a method for gas prediction in tight sandstone reservoirs using seismic dispersion attributes derived from a FD-AVO inversion method. Initially, an FD-AVO inversion framework is established by extending an AVO equation parameterized by a gas indicator and other elastic properties into the frequency domain to extract a dispersion attribute linked to gas content. The effectiveness of the proposed dispersion attribute for gas content identification is then assessed using synthetic seismic data. Subsequently, the method is applied to field data for gas prediction, with well log-derived gas content values serving to validate the reliability of the inversion results. Finally, a comparative analysis is conducted to demonstrate the advantages of the proposed dispersion attribute over conventional P-wave velocity dispersion.

2. Methods

2.1. FD-AVO Inversion of Dispersion Attribute D_F for Gas Prediction

Figure 1 illustrates the framework for extracting the dispersion attribute D_F using the FD-AVO inversion method. Based on the traditional P-wave reflection coefficient equation, a new AVO equation parameterized by indicators of gas content, porosity, and related elastic parameters has been proposed. Subsequently, the pre-stack seismic data are subjected to spectral decomposition. The resulting spectral components are used to compute the dispersion attribute D_F through the FD-AVO inversion method.

Jin et al. [17] introduced a gas indicator F for the prediction of gas content in tight sandstones:

$$F={\displaystyle \frac{1}{K_{\mathrm{f}}\mu }}$$

(1)

where K_f represents the fluid bulk modulus and μ denotes the shear modulus of the rock skeleton. Based on this, an AVO approximation was proposed to enable the direct inversion of the indicator F:

$$\begin{array}{ll}{R}_{\mathrm{PP}}\left(\theta \right)& =-{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left({\gamma }_{sat}^2-{\gamma }_{dry}^2\right)}{4{\gamma }_{sat}^2}}{\displaystyle \frac{\Delta F}{F}}-{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left({\gamma }_{sat}^2-2{\gamma }_{dry}^2\right)+8{\sin }^2\theta }{2{\gamma }_{sat}^2}}{\displaystyle \frac{\Delta \left(\varphi I_S\right)}{\varphi I_S}}\\ & +\left[{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left({\gamma }_{sat}^2-{\gamma }_{dry}^2\right)+4{\sin }^2\theta }{2{\gamma }_{sat}^2}}-{\displaystyle \frac{{\tan }^2\theta }{2}}\right]{\displaystyle \frac{\Delta \rho }{\rho }}\\ & +{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left(3{\gamma }_{sat}^2-5{\gamma }_{dry}^2\right)+16{\sin }^2\theta }{4{\gamma }_{sat}^2}}{\displaystyle \frac{\Delta \varphi }{\varphi }}\end{array}$$

(2)

where I_S is the S-wave impedance, ρ represents density, ϕ is porosity, and θ denotes incidence angle; γ_sat and γ_dry are the P- and S-wave velocity ratio of saturated rock and dry frame, respectively.

Based on Equation (2), the frequency-dependent AVO equation can be expressed as follows:

$$R_{\mathrm{PP}}\left(\theta ,\omega \right)\approx R_{\mathrm{PP}}\left(\theta ,{\omega }_0\right)+A\left(\theta \right)\left(\omega -{\omega }_0\right)D_{\mathrm{F}}+B\left(\theta \right)\left(\omega -{\omega }_0\right)D_{\varphi I_{\mathrm{S}}}$$

(3)

where

$$A\left(\theta \right)=-{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left({\gamma }_{\mathrm{sat}}^2-{\gamma }_{\mathrm{dry}}^2\right)}{4{\gamma }_{\mathrm{sat}}^2}}$$

(4)

$$B\left(\theta \right)=-{\displaystyle \frac{\left(1+{\tan }^2\theta \right)\left({\gamma }_{\mathrm{sat}}^2-2{\gamma }_{\mathrm{dry}}^2\right)+8{\sin }^2\theta }{2{\gamma }_{\mathrm{sat}}^2}}$$

(5)

$$D_{\mathrm{F}}={\displaystyle \frac{\partial }{\partial \omega }}\left({\displaystyle \frac{\Delta F}{F}}\right)$$

(6)

$$D_{\varphi {\mathrm{I}}_{\mathrm{S}}}={\displaystyle \frac{\partial }{\partial \omega }}\left[{\displaystyle \frac{\Delta \left(\varphi I_{\mathrm{S}}\right)}{\varphi I_{\mathrm{S}}}}\right]$$

(7)

Derivations from Equation (2) to (7) are provided in Appendix A.

2.2. Multi-Scale Fracture Theory

Within the multi-scale fracture theory proposed by Chapman [42,43], the effective stiffness matrix for an equivalent medium containing pores, microcracks, and fractures is expressed as follows:

$$C_{\mathrm{ijkl}}=C_{\mathrm{ijkl}}^0-{\varphi }_{\mathrm{p}}{C}_{\mathrm{ijkl}}^1-{\epsilon }_{\mathrm{c}}{C}_{i\mathrm{ijkl}}^2-{\epsilon }_{\mathrm{f}}{C}_{\mathrm{ijkl}}^3$$

(8)

where C⁰_ijkl corresponds to the isotropic background stiffness coefficient; C¹_ijkl, C²_ijkl, and C³_ijkl characterize the contributions of pore, microcrack, and fracture, respectively; ϕ_p represents porosity; ε_c quantifies microcrack density; and ε_f describes fracture density.

Scanning Electron Microscopy (SEM) analysis (Figure 2a) of the study area reveals a variety of pore types within the tight sandstones, including dissolution-enhanced intragranular pores, dissolved intergranular pores, and tectonic fractures. This pore system can be effectively modeled using a dual-porosity approach, where total porosity (ϕ) is partitioned into spherical pores (ϕ_p) and microcrack porosity (ϕ_c), as illustrated in Figure 2b. Accordingly, Equation (8) can be simplified by omitting the large-scale fracture parameter ε_f:

$$C_{\mathrm{ijkl}}=C_{\mathrm{ijkl}}^0-{\varphi }_{\mathrm{p}}{C}_{i\mathrm{ijk}}^1-{\epsilon }_{\mathrm{c}}{C}_{\mathrm{ijkl}}^2$$

(9)

The frequency-dependent P-wave velocity V_P(ω) and S-wave velocity V_S(ω) are obtained from the stiffness components C₃₃₃₃(ω) and C₄₄₄₄(ω) in Equation (9):

$$V_{\mathrm{P}}\left(\omega \right)=\sqrt{{\displaystyle \frac{C_{3333}\left(\omega \right)}{{\rho }_{\mathrm{s}}}}}$$

(10)

$$V_{\mathrm{S}}\left(\omega \right)=\sqrt{{\displaystyle \frac{C_{2323}\left(\omega \right)}{{\rho }_{\mathrm{s}}}}}$$

(11)

where ρ_s denotes the density of the fluid-saturated tight sandstone.

3. Results

3.1. Synthetic Data Tests

Based on the multi-scale fracture theory outlined in Section 2.2, we quantitatively analyze the frequency-dependent characteristics of the elastic wave velocity and the indicator F in tight sandstone reservoirs. Figure 3a shows the computed V_P dispersion curves at three different porosity (ϕ) values while maintaining a constant gas saturation (S_g). The corresponding frequency-dependent variations in the indicator F are presented in Figure 3b. The analysis reveals that the dispersion characteristics gradually intensify as porosity increases.

Figure 4 further explores the frequency-dependent V_P and F characteristics under varying S_g conditions, with a constant ϕ. Both V_P and F exhibit notable frequency-dependent variations as S_g increases. The observed frequency-dependent behaviors of V_P and F under different gas saturations and porosities provide valuable insights for gas prediction using seismic dispersion attributes.

Moreover, the efficacy of the proposed dispersion attributes derived from FD-AVO inversion is validated through synthetic data tests. A layered model of a tight sandstone layer encased in mudstone is depicted in Figure 5. The P-wave velocity, S-wave velocity, and density of the mudstone are 4200 m/s, 2000 m/s, and 2600 kg/m³, respectively. In contrast, the P-wave velocity, S-wave velocity, and density of the tight sandstone are 4500 m/s, 2200 m/s, and 2460 kg/m³. Synthetic seismic data are generated by integrating the propagator matrix method with rock physics modeling [44,45]. The tight sandstone reservoir is modeled with a thickness of 10 m. For seismic simulation, a Ricker wavelet with a central frequency of 40 Hz is employed, and the incidence angles range up to 30°.

Figure 6a–c show the simulated AVO responses for three reservoir models with different porosities (ϕ = 0.05, 0.1, and 0.15) at a fixed gas saturation (S_g = 0.7). The tight sandstone reflection signatures exhibit clear angle-dependent amplitude variations. Analysis reveals that reflection amplitudes attenuate with an increasing incidence angle across all porosity conditions. Additionally, models with higher porosity show enhanced amplitude attenuation at higher angles, indicating porosity-dependent AVO response characteristics. Figure 6d presents the variations in the dispersion attribute D_F for the three models with different porosities. The inversion process was carried out using the method outlined in Section 2.1, with a frequency bandwidth of 5–100 Hz. Quantitative analysis shows more pronounced D_F responses in models with higher porosity in the tight sandstone.

Figure 7a–c present the modeled AVO responses for distinct gas saturation values (S_g = 0.3, 0.5, and 0.7) at a fixed porosity (ϕ = 0.15). The analysis reveals that waveform polarity reversals occur at high incidence angles for the low gas saturation model (S_g = 0.3). Additionally, reflection amplitudes increase at higher angles for models with higher S_g values. Figure 7d shows the calculated D_F values, which progressively increase with higher S_g conditions.

Figure 8 further illustrates the root-mean-square (RMS) magnitudes of the dispersion attribute D_F under conditions of varying ϕ and S_g. The D_F response shows a clear enhancement with increasing values of both ϕ and S_g, indicating its dependence on these parameters. This behavior supports the conclusion that D_F serves as an effective indicator for estimating gas content (S_g × ϕ).

Furthermore, the dispersion attribute D_F, computed for varying S_g and ϕ conditions in Figure 8, is transformed into a cross-plot of D_F versus S_g × ϕ, as shown in Figure 9. The relationship demonstrates a nonlinear growth pattern, characterized by an initial steep increase followed by a more gradual rise. The red trendline represents the central tendency, and these results confirm a distinct positive correlation between D_F and S_g × ϕ.

3.2. Real Data Applications

The methodology developed in this study is applied to field data from a tight sandstone gas reservoir. Figure 10 shows the two-way travel time (TWT) contours of the target reservoir horizon. The TWT distribution reveals a relatively smooth structural relief across the study area, with three gas-producing wells (B, C, and D) and two no gas-producing wells (A and E) indicated.

Figure 11, Figure 12 and Figure 13 present comparative analyses of logging data, cross-well pre-stack seismic traces, predicted dispersion attribute D_F, D_Vp results, and the volumetric fraction of minerals for the three wells. The elastic properties of clay and quartz are shown in

Table 1. The properties of clay and quartz.

	Clay	Quartz
VP (km/s)	3.90	6.05
VS (km/s)	1.50	3.90
ρ (g/cm3)	2.58	2.65

[46]. The seismic inversion process utilizes a frequency bandwidth of 5–100 Hz. The frequency range used is optimized empirically, according to the bandwidth and characteristics of real seismic data. Notably, compared to D_Vp, the seismically derived D_F shows a strong agreement with logging data for gas content (S_g × ϕ). The intuitive comparison results confirm the superiority of D_F over D_Vp. The target intervals with high S_g × ϕ values can be consistently identified by high-value D_F anomalies. In comparison, Figure 14 presents a cross-plot of S_g × ϕ versus V_P/V_S and S_g × ϕ versus λρ, demonstrating that the conventional elastic parameters V_P/V_S and λρ show less sensitivity in effectively indicating gas content. These results emphasize the potential of D_F as a more effective indicator for identifying gas-bearing zones than both D_Vp and traditional elastic parameters.

Figure 15 presents the seismic profile across wells A, B, and C, with the bounding surfaces of the target reservoir delineated by two black horizons. Additionally, Figure 16 shows the corresponding time–frequency analysis results obtained through continuous wavelet transform (CWT), displaying spectral decompositions at six discrete frequencies: 10, 20, 30, 40, 50, and 60 Hz. These spectrally decomposed seismic attributes serve as the foundational data for FD-AVO inversion.

For comparison, Figure 17 and Figure 18 show the dispersion attributes D_F calculated using the method presented in this study and D_Vp derived from the conventional framework [34]. Figure 19 and Figure 20 display the V_P/V_S and λρ, both computed from the elastic inversion results. The results are superimposed with S_g × ϕ logging data. It is observed that gas-bearing tight sandstone formations within the target zones exhibit high D_F and D_Vp and low V_P/V_S and λρ responses. In comparison, D_F anomalies demonstrate an improved sensitivity to the zones with high gas content than D_Vp, V_P/V_S, and λρ, as evidenced by the close alignment between elevated D_F values and productive intervals with high S_g × ϕ in Wells B, C, and D. Meanwhile, the tight formations in dry wells A and E with very low S_g × ϕ show no D_F anomalies in the target interval.

4. Discussion

Accurate gas prediction is crucial for delineating gas-bearing zones in tight sandstone reservoirs. This study proposes a gas prediction methodology utilizing a dispersion attribute derived from frequency-dependent inversion. To verify the effectiveness of the proposed method with the theoretical test, this study adopts the multi-scale fracture theory [42,43] to calculate the frequency-dependent P-wave and S-wave velocities, which are then used to generate synthetic data. The multi-scale fracture theory primarily focuses on the dispersion mechanism caused by fluid squirt flow within complex pore structures. In future research, additional rock physics models can be incorporated to investigate other dispersion mechanisms. Using multi-scale fracture theory, we conducted rock physics modeling to analyze the frequency-dependent characteristics of indicator F under different porosity and gas saturation conditions [Figure 3b]. The modeling results reveal the significant sensitivity of indicator F to variations in both ϕ and S_g. However, conventional seismic inversion and interpretation methods have typically relied on elastic theories for fluid identification, overlooking inelastic dispersion effects in tight sandstone reservoirs. To address these challenges, this paper presented a frequency-dependent pre-stack inversion method to extract seismic dispersion attributes, enhancing gas prediction by utilizing the inelastic properties of tight sandstones. This method is based on an AVO equation specifically parameterized by the gas indicator F and associated properties.

As demonstrated in Figure 6a–c and Figure 7a–c, we simulated and analyzed the seismic AVO responses of tight sandstone reservoirs with varying ϕ and S_g configurations by integrating rock physics modeling with the propagator matrix method. In these simulations, the tight sandstone exhibits inelastic properties due to fluid flow within multi-scale fracture systems. The results revealed distinct AVO signatures associated with variations in ϕ and S_g, indicating the feasibility of identifying porosity and fluid using pre-stack seismic data. Furthermore, based on the method outlined in Section 2.1, we computed the seismic dispersion attribute D_F for gas content (S_g × ϕ) assessment in tight sandstones. Synthetic data tests validated the capability of the proposed D_F for reliably estimating variations in ϕ and S_g [Figure 6d and Figure 7d]. Quantitative analysis systematically demonstrated a significant increase in D_F values with higher ϕ and S_g values (Figure 8), confirming the sensitivity of D_F to both parameters. Notably, D_F exhibited a clear positive correlation with the product S_g × ϕ, as shown in Figure 9, further supporting its reliability for gas content assessment.

Finally, the proposed method and dispersion attribute D_F were applied to field data for gas prediction in the study area, as shown in Figure 10. As illustrated in Figure 11, Figure 12 and Figure 13, the results indicate that the seismically derived D_F demonstrates a strong correlation with the well log-derived S_g × ϕ. Target sandstones with high S_g × ϕ values were identified by high-value D_F anomalies in all three wells. These findings support the feasibility of using D_F as a reliable indicator for predicting gas-bearing tight sandstones in the investigated region. Comparative analysis further revealed that D_F anomalies (Figure 17) exhibit a significantly stronger correlation with gas-enriched intervals than the conventional P-wave dispersion attribute D_Vp (Figure 18), confirming the superior reliability of D_F for gas content prediction in tight sandstones.

Seismic elastic inversion and quantitative seismic interpretation methods have been employed to estimate porosity and gas saturation [4,6,17]. Different from the aforementioned conventional elastic inversion approaches, this study proposes a dispersion attribute obtained from frequency-dependent inversion. A key advantage of this method lies in its utilization of inelastic properties derived from seismic frequency information, offering a method for characterizing gas content (S_g × ϕ). However, the presented dispersion attribute provides a qualitative evaluation of gas content. Future research could focus on developing quantitative indicators to achieve a more accurate characterization of reservoir properties by using formal uncertainty quantification methods such as the Bayesian inversion or Monte Carlo simulations.

Meanwhile, rock physics modeling approaches that account for various factors such as mineralogical heterogeneity and elastic–inelastic coupling effects may be incorporated, offering deeper insights into the seismic dispersion characteristics associated with porosity and fluid. In addition, further research will explore the simultaneous inversion for multiple reservoir parameters (such as porosity, gas saturation, and fluid type) and combine seismic dispersion attributes with other seismic indicators to enhance predictive reliability and compensate the limitations of single-attribute approaches. Finally, the proposed method can be further applied to diverse geological settings, reservoir types, and well data in further study.

5. Conclusions

This study presents a gas prediction methodology based on a dispersion attribute derived from frequency-dependent inversion, utilizing an AVO equation specifically parameterized by the gas indicator F and associated properties. Rock physics modeling, grounded in multi-scale fracture theory, demonstrated the significant sensitivity of the frequency-dependent indicator F to variations in both porosity (ϕ) and gas saturation (S_g). Seismic AVO modeling results revealed distinct signatures linked to variations in ϕ and S_g, suggesting the feasibility of predicting porosity and fluid using pre-stack seismic data. Synthetic data tests confirmed the ability of the proposed seismic dispersion attribute D_F to reliably estimate variations in ϕ and S_g. Moreover, D_F showed a clear positive correlation with the product of S_g × ϕ, underscoring its effectiveness in gas content evaluation. Field data applications indicated that target sandstones with high well log-derived S_g × ϕ values were identified by elevated D_F anomalies. Comparative analysis further demonstrated that D_F anomalies exhibited a significantly stronger correlation with gas-enriched intervals than the conventional P-wave dispersion attribute D_Vp, validating the superior reliability of D_F for gas content prediction in tight sandstones. These findings suggest that the proposed seismic dispersion attribute provides an effective indicator for identifying gas-bearing areas in tight sandstones.