Introduction

Quantitative prediction of reservoir parameters is a crucial part of establishing reservoir geological model. Accurate prediction of reservoir parameters is of great significance for improving the accuracy of history fitting by reservoir numerical simulation, for studying the distribution of remaining hydrocarbon in the reservoir, and for formulating reservoir development adjustment plan. Therefore, improving the accuracy of quantitative reservoir description through logging and seismic data is important.

There are many ways to obtain the parameters of tight reservoirs from various logging curves, such as gamma ray and SP curves, which can be used to calculate clay volume, as well as compressional wave interval transit time and density curves used for porosity calculation. However, these logging curves can only provide the reservoir information around the wellbore, and it is required to perform seismic reservoir characterization to identify the characteristics of lateral distribution of reservoir properties. At first, seismic inversion has been carried out based on reconstructed curves. However, in tight sandstones, it is difficult to distinguish a reservoir from the surrounding rock using the P-impedance without any other additional information from post-stack seismic data. For this reason, geophysicists have reconstructed more reservoir-sensitive logging curves that can distinguish between reservoirs and non-reservoirs and have carried out relevant inversion based on combining these pseudo-curves with seismic data (Wan et al. 2015; Jiang and Qi 2016; Wang et al. 2017; Zhu 2017; Yu et al. 2019). However, due to the lack of physical basis and poor interpretation performance, this method has been abandoned largely with the increasing popularity of pre-stack seismic data.

Quantitative seismic interpretation is the ultimate goal of favorable zone evaluation and reservoir characterization. Currently, three methods are utilized widely. The first method is seismic multi-attribute inversion of petrophysical properties (Song et al. 2002; Wu et al. 2004; Yang et al. 2005; Wu et al. 2008; Wu et al. 2011a, b). The idea of this method is to determine quantitatively linear or nonlinear relationship between petrophysical properties and seismic attributes and then to calculate petrophysical properties by multiple linear regression or neural networks. The second method is geostatistical petrophysical property inversion (Sun and Peng 2007; Liu et al. 2009). This method performs petrophysical property inversion based on stochastic simulation in geostatistics. The deficiency of this method lies in that (a) its rock physics mechanism is unclear and (a) massive logging data are required to build the linear and nonlinear relationship. The third method is petrophysical property inversion based on rock physics (Mukerji et al. 2001). Originally, this method combines rock physics with pre-stack seismic inversion to estimate petrophysical properties of reservoirs. Bachrach (2006) further used a rock physics model for inversion of porosity and fluid saturation. Spikes et al. (2007) proposed a probabilistic seismic inversion method to estimate the petrophysical properties of reservoirs using well and seismic data. Larsen et al. (2006) presented a fluid inversion method based on the Markov chain. Grana and Rossa (2010) built upon the previous research achievements to perform inversion of reservoir properties by combining rock physics with Bayesian inversion, which produced convincing results. Zhang et al. (2020) proposed a linearized rock physics inversion for estimation of petrophysical parameters.

Owing to its solid rock physics basis, the method of quantitative seismic interpretation based on the rock physics template (RPT), which builds links between reservoir properties and seismic data, has become popular (O Degaard and Avseth 2003; Chi and Han 2009; Gupta et al. 2012; Hermana et al. 2016; Li et al. 2019; Yuan et al. 2019a, b). However, the accuracy of results obtained from this method usually depends on the reliability of the rock physics model used; therefore, for example, huge quantities of rock physics model needed to be developed for tight sandstone (Ba et al. 2016, 2017). In practice, this method is often difficult to operate for two reasons. Firstly, rock physics modeling is very complex. To deliver good results, several adjusting parameters, such as compressional velocity, shear velocity, density and pore aspect ratio for pure quartz and clay mineral, need to be optimized, but pore aspect ratio, as an important factor affecting elastic parameters, is often unavailable in the field. Secondly, due to the diversity of rock physics models, it is required to calibrate from massive logging data and laboratory testing data in order to determine the reliability of RPT and to select the appropriate model for specific geological setting.

To predict reservoirs, elastic impedance (EI) inversion has been used widely in the industry (Connolly 1999; Avseth et al. 2010) because EI can be obtained by performing a process similar to post-stack acoustic impedance inversion on partial stack seismic data given a certain angle of incidence. However, EI is problematic because its value changes drastically with incident angle, which greatly restricts its application in reservoir characterization. Whitcombe et al. (2002) fixed the range variability problem by normalization of EI values. The extended elastic impedance (EEI) method, which combines partial stack seismic data, compressional, shear velocity and density logging curves, has been used widely for qualitative fluid detection and reservoir characterization (Syed et al. 2015; Pranata et al. 2017). The application of EEI has also been recently extended to anisotropic media to derive anisotropic elastic impedance for fracture and fluid detection (Pan et al. 2019).

Geophysicists have also done a lot of work on fluid detection in tight sandstones. Yang et al. (2017) used the compression coefficients extracted from seismic data to perform qualitative fluid detection in tight sandstones. Jiang et al. (2017) analyzed the feasibility of using the Russell fluid factor for fluid detection in tight sandstones based on forward modeling and applied it to real data, delivering reliable detection. Yuan et al. (2019a, b) adopted the propagator matrix forward modeling and rock physics constraints to develop a pre-stack stochastic frequency-dependent velocity inversion method and derived associated hydrocarbon attributes to indicate tight sandstone gas reservoirs.

Tight sandstone has weak amplitude variation with offset (AVO) response due to its extremely low porosity, which makes it very difficult for conventional AVO analysis to reflect a reservoir’s porosity and fluid variations. Therefore, to capture weak AVO response, it is essential to develop a quantitative AVO characterization method to predict reservoir properties from pre-stack seismic data. In this paper, a relatively complete, convenient and practicable workflow was designed for quantitative prediction of reservoir parameters such as clay volume and porosity. This workflow combines statistical rock physics analysis with the EEI theory. It is suitable not only for quantitative characterization of tight reservoir parameters, but it is also worthy of extended application for characterization of other types of reservoir.

Geologic Setting

Regional Geology

The Sichuan Basin has the largest proven reserves and highest cumulative production of conventional natural gas (Dai et al. 2014) and tight gas (Zou et al. 2018) in China. Multiple tectonic movements have taken place in this basin, creating the five current structural systems, namely the Southwest Sichuan intermediate fold belt, the West Sichuan intermediate fold belt, the Middle Sichuan weak fold belt, the East Sichuan strong fold belt and the North Sichuan weak fold belt (Shen et al. 2007). The study area in AnYue Gas Field (Fig. 1), one of the most important tight gas fields with enormous geologic reserves in the Sichuan Basin, is located in the Middle Sichuan weak fold belt (Zhao et al. 2013). The area has a monoclinal structure with NE–SW inclination (Fig. 2). There is no obvious correlation between natural gas production and structure in this area. Several high-production wells are located in upper structures, and many such wells also exist in lower structures. For this reason, it is crucial to predict reservoir properties quantitatively to provide reliable reference for well deployment.

Figure 1
figure 1

Regional tectonic background and location of the study area in the ChuanZhong weak fold belt. The study area of AnYue gas field is a reputable gas field in China

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Figure 2
figure 2

Subsea depth map of the study area overlaid with gas production bar charts. Gas production increases with length of bars

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Sedimentary Geology

It has been revealed in previous studies that the Sichuan Basin is a composite and complicated basin (Zhao and Zhang 2003). During the Upper Triassic sedimentary period in the central Sichuan Basin, the structure is featured with relatively flat regional structural relief, and the shallow water continental lake basin was developed widely with coal-bearing strata, forming a “sandwich” structure of source rock and reservoir inter-bedding each other (Jiang et al. 2007; Xu et al. 2007). Such sandwich structure resulted in multi-reservoir–caprock association in the Triassic strata (Bian et al. 2009). The target layer of this study is in the Upper Triassic Xu2 section (Fig. 3) with total strata thickness of 70–100 m, but the thickness of effective reservoir is very thin, namely 3–10 m. The effective reservoir has developed mainly in the high-energy channel of delta plain sub-facies, and the rock types include mainly feldspathic lithic sandstone, lithic arkose and quartz lithic sandstone (Yang et al. 2007).

Figure 3
figure 3

Stratigraphy of Triassic and Jurassic units in the study area

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Seismic and Well Data

More than 100 wells with complete drilling and production data have been drilled in the study area. The logging curves for these wells have gamma ray, caliper, compensated neutron log, density, deep resistivity, shallow resistivity and P-sonic and S-sonic delta time curves. Part of the area covered by 3D seismic survey is about 800 km2. The main in-lines extend along NE–SW direction, the direction of cross-lines is approximately N–S, and the bin size is 25 × 25 m (Fig. 2). Primitive gathers have been undergone static, velocity spectrum analysis, normal move-out, dip move-out, pre-stack time migration to obtain AVO preserved pre-stack gather, and gather conditioning procedure such as mute and trim statics in the study area, satisfying the conditions for performing pre-stack seismic inversion.

Methodology and Workflow

When the compressional and shear wave velocities (Vp, Vs) and density (ρ) are known from well logs, the other parameters and EEI can be calculated as in Eq. (1). This formula includes two adjustable parameters: \(\chi\) and K. It is also important to optimize these two parameters, because they affect directly the ability of EEI to predict reservoir properties.

$${\text{EEI}}\left( \theta \right) = {\text{AI}}_{0} \left[ {\left( {\frac{\text{AI}}{{{\text{AI}}_{0} }}} \right)^{\cos \chi } \left( {\frac{\text{GI}}{{{\text{GI}}_{0} }}} \right)^{\sin \chi } } \right]$$
(1)

where \({\text{AI}} = \rho V_{\text{P}}\), \({\text{AI}}_{0} = \rho V_{{{\text{P}}0}}\), and \({\text{GI}} = {\text{AI}}_{0} \left[ {\left( {\frac{{V_{\text{P}} }}{{V_{{{\text{P}}0}} }}} \right)\left( {\frac{{V_{\text{S}} }}{{V_{{{\text{S}}0}} }}} \right)^{ - 8K} \left( {\frac{\rho }{{\rho_{0} }}} \right)^{ - 4K} } \right]\) where \(K = \left( {\frac{{V_{\text{s}} }}{{V_{\text{p}} }}} \right)^{2}\), the K value being set frequently to be a constant; \({\text{AI}}_{0}\), \(V_{{{\text{P}}0}}\), \(V_{{{\text{S}}0}}\), \(\rho_{0}\) are the average values of compressional impedance, compressional velocity, shear velocity and density in the target layer, respectively, which were estimated by statistical analysis of each parameter; and \(\tan \chi = \sin^{2} \theta\), where \(\theta\) is the incident angle.

Figure 4 shows an integrated workflow for the prediction of clay volume and porosity, as well as the optimization of \(\chi\) and K. The detailed steps of this workflow are the following.

Figure 4
figure 4

Workflow for quantitative reservoir parameters inversion

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  1. 1.

    Calculate values of EEI for varying angles \(\chi\) from the Vp, Vs, density curves of wells. Draw the cross-plots EEI for different angles and reservoir parameters (clay volume and porosity), and then obtain correlation coefficient (CC) and fitting formula through linear fitting.

  2. 2.

    Find automatically the maximum CC and the corresponding angle \(\chi_{i}\) by scanning values of CC derived from step 2.

  3. 3.

    Extract P and G attributes from pre-stack gather by fitting the AVO curves, and calculate EEI reflection cubed from the P and G attributes using the best angle \(\chi_{i}\) calculated in step 2.

  4. 4.

    Load the horizon, perform spike elimination and smoothing on the horizon, conduct high cut filtering of the existing well EEI(\(\chi_{i}\)) curve, and then build the initial model constrained by structure framework from horizons and well interpolation.

  5. 5.

    Perform model-based inversion to obtain EEI(\(\chi_{i}\)) cubes using the initial model built in step 4.

  6. 6.

    Convert EEI(\(\chi_{i}\)) cubes to reservoir parameters for reservoir characterization using the formula obtained in step 1.

EEI Analysis on Wells

Whitcombe et al. (2002) stated that different values of \(\chi\) could yield EEI expressions for S-wave impedance, bulk modulus, Lamé’s parameter, shear modulus and Vp/Vs ratio. Inappropriate values of K will lead to great errors between the EEI and Zoeppritz formulas (Abbasi et al. 2018), the ability of EEI to characterize reservoirs varies with \(\chi\), and thus, it is important to find the optimum value of χ and K before EEI inversion. Here, average values of AI, GI, Vp, Vs, Vs/Vp ratio and density (Fig. 5, Table 1) were obtained through statistical rock physics analysis of well data, and then, the value of K was deduced from the average value of Vs/Vp ratio. Thus, 0.6063 was used here as the value of K, which is distinct from the value given by Connolly (1999). The optimum values of \(\chi_{i}\) for clay volume and porosity prediction were found, respectively, by analyzing values of CC among the reservoir parameters and EEI.

Figure 5
figure 5

Histograms of different elastic parameters from logging data

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Table 1 Rock physics parameters statistics
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After the value of K was obtained, with each known variable on the right-hand side of Eq. (1), the values of EEI corresponding to different angles, \(\chi\), were calculated (Fig. 6). On this basis, values of EEI for different \(\chi\) angles and reservoir parameters were cross-plotted, and the least squares method was used for linear fitting of these two variables. The CC and regression formula for different angles were obtained, and then, the optimum angle \(\chi_{i}\) corresponding to the maximum correlation coefficient was found for the purpose of generating EEI reflection data volume.

Figure 6
figure 6

Logging curve sections

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The optimum \(\chi\) angle reflecting lithology and porosity was found by cross-plotting CC vs. \(\chi\)(Fig. 7). Values of EEI for different angles were cross-plotted with reservoir parameters as well, and CCs were obtained through linear fitting (Fig. 8). The regression equations for reservoir parameters and EEI at optimum \(\chi\) angle were obtained by linear fitting. Thus, the EEI can be converted into clay volume and porosity as follows:

$$V_{\text{clay}} = 4.6365 \times 10^{ - 5} \times {\text{EEI}}\left( {\chi_{i = 68} } \right) - 0.14794$$
(2)
$${\text{Por}} = - 3.104 \times 10^{ - 5} \times {\text{EEI}}\left( {\chi_{i = 12} } \right) + 0.43579$$
(3)

where Vclay is the clay volume and \({\text{EEI}}\left( {\chi_{i = 68} } \right)\) is the EEI when \(\chi\) is equal to 68; Por is the porosity and \({\text{EEI}}\left( {\chi_{i = 12} } \right)\) is the EEI when \(\chi\) is equal to 12.

Figure 7
figure 7

Search of the optimum value of angle \(\chi\) (CC stands for correlation coefficient). (a) Plot of EEI-Vclay CC vs. \(\chi\). (b) Plot of EEI-porosity CC vs. \(\chi\). The red point in (a) shows that the optimum χ angle based on the clay volume curves is 68° and the corresponding CC is 0.63047. The red point in (b) shows that the optimum χ angle based on the porosity curves is 12° and the corresponding CC is 0.84468

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Figure 8
figure 8

Cross-plots of EEI vs. reservoir parameters (CC = correlation coefficient). (a) EEI (x-axis) vs. clay volume (y-axis). (b) EEI (x-axis) vs. porosity (y-axis). The red line in each cross-plot is the fitting line

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EEI Inversion

Generation of EEI Reflection Coefficient Data Cube from Pre-stack Gather

AVO intercept and gradient were obtained by linear fit of AVO curve from PP pre-stack gather (Fig. 9) based on the Shuey formula (Shuey 1985), the P attribute is the intercept of the fitted line, and the G attribute is the gradient of the same fitted line. The intercept and gradient attributes from AVO analysis can also be combined to calculate additional AVO attributes for reservoir characterization, such as pseudo-S-wave, fluid factor and Poisson’s ratio contrast (Smith and Gidlow 1987; Ross 2002; Stovas et al. 2006; Veeken and Rauch-Davies 2006; Wu et al. 2011a, b). Based on the calculation of P and G, the reflection coefficient data cube for the most sensitive angle \(\chi\) was calculated according to Eq. (4) and used for subsequent sensitive EEI inversion.

$$R\left( {\chi_{i} } \right) = P*\cos \left( {\chi_{i} } \right) + G*\sin \left( {\chi_{i} } \right)$$
(4)
Figure 9
figure 9

Diagram of intercept and gradient attributes fitted from pre-stack gather

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Model-Based Seismic Inversion

Based on the input well EEI curve, horizon, seismic wavelet and EEI reflection coefficient cube, constrained model-based EEI inversion was performed (Russell and Hampson 1991). The detailed process of such inversion is as follows. Based on the dedicated well to seismic tie, depth domain well data were transformed to that in the time domain, and the reliable wavelets were extracted from well and seismic data. Under the constraint of interpretation horizon, the well curve was filtered and interpolated for a low-frequency model, which was used as an initial guess model. The model was perturbed taking into account the rock physics limit, and the corresponding synthetic seismic was generated after each model perturbation and compared with real seismic to find the error. When such error was greater than the given value, model perturbation was continued until the error became smaller than the given value, at which point the iteration is stopped, and the final EEI model was output as the inversion result.

As shown in Figure 10, the black line represents the initial model, the red line represents the inversion result, the blue line is the EEI curve from the well, and the synthetic (red wiggle) record from the inversion result is very close to the real seismic data (black wiggle). As shown in Figure 10, the EEI inversion result is close to the EEI calculated from well logs and these two parameters fall on the perfect fitting line in close proximity to each other (Fig. 11).

Figure 10
figure 10

Quality control on EEI inversion

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Figure 11
figure 11

Cross-plot from EEI inversion and well logs

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Reservoir Parameters Conversion from EEI Cube

Clay volume and porosity cubes (Figs. 12 and 13) were calculated from the EEI inversion results using Eqs. (3) and (4). Comparisons of clay volume and porosity derived from EEI inversion with the well data are shown in Figure 14. As shown in the figure, predicted values of clay volume and porosity of wells are positively correlated with the corresponding values obtained from well logs, indicating that the calculation results are reliable, and therefore, reservoir parameters such as clay volume and porosity can be determined effectively through EEI inversion.

Figure 12
figure 12

Clay volume section derived from EEI inversion

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Figure 13
figure 13

Porosity section derived from EEI inversion

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Figure 14
figure 14

Comparison of reservoir parameters derived from EEI inversion (predicted clay volume, predicted porosity) and those obtained from well logs: (a) clay volume from well logs (Vclay) vs. predicted clay volume; (b) porosity from well logs (Por) vs. predicted porosity

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Figure 15 shows thickness planar distribution maps for sandstones and porous sandstones, and the overlaid bar charts represent gas production. These two maps indicate that thicker and more porous sandstones (porosity > 5%) are distributed in the northeastern parts of the study area, and the predicted results are consistent with production data from the drilled wells. High-production wells have been drilled to thicker and more porous sandstones.

Figure 15
figure 15

Thickness planar distribution maps for (a) sandstones and (b) porous sandstones. Overlaid on the maps are gas production bar charts; gas production increases with length of bars (see also Fig. 2)

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Conclusions

EEI-based inversion provides a new idea for quantitative prediction of reservoir parameters. The relationships between reservoir parameters and EEI are established from statistical analysis of well data and are extended to EEI data volume obtained from seismic inversion. This provides a convenient and quick way to predict reservoir parameters quantitatively. This method is applicable not only for exploration areas with few wells, but also for development areas with abundant wells.

The values of \(\chi\) and K obtained from well analysis are the keys to accurate prediction of reservoir parameters. The EEI at a certain large \(\chi\) angle mainly reflects lithologic information, while the EEI at a certain small \(\chi\) angle is better correlated with porosity. Because \(\chi\) is proportional to the incident angle, it is essential to collect seismic data with long offset for the sake of predicting the reservoir properties. However, the ability of the proposed method to identify thin layers is still relatively finite due to the limited resolution of deterministic inversion. In the future, EEI geostatistical inversion will be trialed for improvement to identify thin reservoirs.