Research on Rock Physics Modeling Methods for Fractured Shale Reservoirs

Abstract

The Sichuan Basin is a significant region for exploration and development of shale gas in China, and it is essential to clarify the impact of deep shale gas reservoir parameters on cost-effective development at scale to ensure national energy security. Rock physics modeling is a significant means of communicating the physical and elastic parameters of rocks. A rock physics modeling method applicable to fractured shale gas reservoirs is proposed for the current situation of complex fluid relationships in shale gas reservoirs and unclear characteristics of gas identification seismic response. In this paper, based on the Self Consistent Approximation (SCA) model and the differential effective medium (DEM) model, the anisotropic source of shale is used as a starting point to add bound water, kerogen, clay, and brittle minerals, the Schoenberg linear slip theory is used to add fracture disturbance effects, and then the Brown–Korringa model is used to perform fluid replacement under anisotropic conditions. Finally a rock physics model applicable to fractured shale gas reservoirs is obtained, and the established rock physics model is used for analysis of elastic parameters, Thomsen parameters, and fracture weakness parameters. Rock physics tests were performed on shale in southern Sichuan as an example. The experimental results show that the model established by the process can accurately invert the longitudinal and transverse wave velocities of the shale, which can provide a conceptual basis for the study of fractured shale gas reservoirs.

1. Introduction

The shale gas resources in the Sichuan basin are extremely rich; moreover, as a new clean energy source, the exploration and development of shale gas play a very significant role in optimizing the energy structure of China. Shale gas reservoirs belong to an unconventional reservoir, in which the petrophysical modeling technology for unconventional reservoirs is being improved on conventional reservoirs, because it is not strictly applicable to unconventional reservoirs and is currently seeking to improve the technology [1]. In the last decade, innovative developments in unconventional energy sources have made rock physics increasingly popular and a very active area of research. Because shale has a complex variation in composition content and numerous parameters associated with it, carving the elastic properties of the rock necessitates the use of rock physics modeling theory. A rock physics model can establish the relationship between physical and elastic parameters, indicating the crucial role played by rock physics in the interpretation of geophysical data, and rock physics modeling has become an integral part of oil and gas volume exploration, field development, and production monitoring.

For physics modeling of rocks, Vernik et al. [2] and others showed that the main source of anisotropy in shale is the directional arrangement of mineral fractions, based on previous scholars’ studies on the source of anisotropy in rocks. Carcione et al. [3] quantitatively calculated the effect of an organic matter mixture composed of kerogen and fluid on the elastic anisotropy of shale using the equivalent medium theory. Guo et al. [4] pointed out that the directional arrangement of clay and cheese roots is the main cause of strong anisotropy in shale, which is discussed in rock physics modeling. The effect of different compaction indices on shale anisotropy was discussed by rock physics modeling. Dong et al. [5] developed a petrophysical model of constituent mud shale applying SCA and DEM models and discussed the effects of variations in mud content and different pore morphology on modeling results. Deng et al. [6] declared a modeling approach based on SCA and DEM models based on the microscopic results and characteristics of the shale and validated the Longmaxi Formation shale by logging data. Nie et al. [7] used a digital core modeling approach to analyze the relationship between different reservoir parameters and rock physics properties. In considering the influence of fractures in rock physics modeling, the Hudson model and the Schoenberg linear slip model are the most commonly used equivalent models [8]. Sevostianov et al. [9] proposed the anisotropic equivalent field theory, characterizing the interaction between heterogeneous media as the effect of an equivalent stress field, which is used to add inclusions into anisotropic media. Chapman et al. [10] analyzed the impression of the actual morphology of fractures and investigated the consequences of different scales of fractures on rock anisotropy. Zhang et al. [11] established a link between fracture rock elasticity parameters and fracture parameters by constructing a petrophysics model to characterize fractures in shale reservoirs. Tang et al. [12] studied the effects of fracture density and fluid saturation of vertically fractured (HTI) media on P and S wave velocities using anisotropic fluid replacement theory. Chen et al. [13] proposed an inversion method for fracture parameters based on azimuthal anisotropic elastic impedance using the fracture medium rock physics theory. Guo et al. [14] introduced the Backus average theory and the Chapman multi-scale fracture model to quantitatively describe clay directional arrangement and fracture development in a shale reservoir during petrophysical modeling. Chen et al. [15] proposed using fracture flexibility inversion and a linear slip model to find natural cracks in inspected seismic data. Tan et al. [16] demonstrated the generality of rock physics templates for brittleness prediction by adding the effects due to vertical fractures using rock physics. Wei et al. [17] used the petrophysical equivalent model to convert the equivalent of a small-scale inhomogeneous fracture medium to a homogeneous anisotropic medium to establish the relationship between fracture characteristic parameters and seismic attributes. Hu et al. [18] used a SCA model to make up diverse components to compose a petrophysics model of shale with complex pore structure; the effect of fractures on anisotropy was found to be greater than that of clays, which are the primary dominating element for shale anisotropy.

Regarding the current situation of complex fluid relationships in shale gas reservoirs, unclear characteristics of gas identification seismic response, and large differences in different fluid identification factors, based on previous studies, it is found that the reason is an incomplete consideration of factors in the process of shale rock physics model construction. The conventional rock physical modeling process generally uses the Wood model formula to calculate the bulk modulus and shear modulus of the equivalent fluid, then uses Voigt-Reuss-Hill to estimate the bulk modulus and shear modulus of the rock matrix, calculates the equivalent density of the fluid and the equivalent saturated rock density, and uses the Krief empirical formula to obtain the equivalent bulk modulus and shear modulus of the rock skeleton. Finally, the equivalent bulk modulus and shear modulus of saturated rock are calculated using Gassmann’s equation. such as modeling without considering clay, kerogen, brittle minerals, fractures, voids, fluids, and anisotropic sources simultaneously. Therefore, in this paper, based on the orthogonal anisotropy of the actual shale reservoir, the K-T model is used to add bound water to the clay, and then the SCA and DEM models are jointly used to model the clay, kerogen, brittle minerals, and voids. The Schoenberg linear slip theory is then used to add fracture perturbations, and the log interpretation results are used to determine the fluid proportions and calculate the fluid modulus. The Brown–Korringa model is then used to perform fluid replacement under anisotropic conditions, and finally a more accurate shale rock physics equivalent model is established for each anisotropic cause. Additionally, based on this model, elastic parameters, Thomsen parameters, and fracture weakness parameters are analyzed.

2. Rock Physics Modeling Theory

2.1. Rock Physical Boundary Theory

Today, the more widely used theoretical models of rock physics boundaries are the Voigt-Reuss-Hill (VRH) bound and the Hashin–Shtrikman (HS) bound [19]. For a general rock physics model, if we want to calculate its equivalent elastic modulus, we need to know the volume content of its individual components and the corresponding elastic modulus, as well as the geometric particulars of how these different constituents are made up with each other. Generally speaking, hard materials have larger equivalent modulus values, and flexible materials have smaller equivalent modulus values. When we need to estimate the modulus, we can use the VRH average, which is the count average of the upper Voigt limit and the lower Reuss limit. The HS boundary is the same as the VRH boundary and is also a way to estimate the equivalent modulus of a mixture within a certain range, but the difference is that the range is a little narrower compared to the VRH, and the accuracy of the HS equivalent also depends mainly on the consideration of the geometric details, but when the geometric details are not known at all, the equivalent modulus calculated by HS is said to be the narrowest allowable upper and lower limits. The formula for the HS boundary is as follows:

$$K^{H{S}_{-}^{+}}=K_1+\frac{f_2}{{(K_2-K_1)}^{-1}+f_1{(K_1+\frac{4}{3}{\mu }_1)}^{-1}}$$

(1)

$${\mu }^{H{S}_{-}^{+}}={\mu }_1+\frac{f_2}{{({\mu }_2-{\mu }_1)}^{-1}+\frac{2f_1(K_1+2{\mu }_1)}{5{\mu }_1(K_1+\frac{4}{3}{\mu }_1)}}$$

(2)

where $K_1$ and $K_2$ are expressed as the bulk modulus of the first and second constituents. ${\mu }_1$ and ${\mu }_2$ are the shear moduli of the corresponding constituents. $f_1$ and $f_2$ are the capacity contents of the two constituents, and $K^{H{S}_{-}^{+}}$ and ${\mu }^{H{S}_{-}^{+}}$ are the equivalent results for HS. At the same time, there is a certain difference in the order of the phases. Generally speaking, when the hard material is wrapped around the softer material, the equivalent result obtained at this time is the upper limit, and the opposite is the lower limit.

2.2. Inclusion Theory

The Kuster–Toksoz (KT) model is considered the earliest inclusion theory, and the KT model, under the assumption of an isotropic background medium, sequentially adds minerals and fluids to the background media to finally acquire the equivalent elastic modulus of the mixture [20]. Additionally, both the DEM theory and the SCA theory are the equivalent medium theories of the inclusions model applied in this paper. We first introduce the DEM theory, which is divided into isotropic DEM and anisotropic DEM, and we focus on the anisotropic DEM theory. On the basis of isotropic theory, Hornby [21] derived a differential equivalent medium model for anisotropic conditions, and the isotropic parameters shear modulus $\mu$ and bulk modulus $K$ in the isotropic DEM theory formulation were replaced by the tensor matrix $C$. The shape factors $P$ and $Q$ that control the shape of the inclusions are replaced by the tensor matrix $G$ by the following equations:

$$\frac{d}{d_v}({\underset{~}{C}}^{DEM}(v))=\frac{1}{(1-v)}({\underset{~}{C}}^i-{\underset{~}{C}}^{DEM}(v)){\underset{~}{K}}^i(v){\underset{~}{C}}^{DEM}(v).$$

(3)

The expression of $K$ is:

$$\underset{~}{K}={[\underset{~}{C}(\underset{~}{I}+\widehat{\underset{~}{G}}(\underset{~}{C^{\cdot }}-\underset{~}{C}))]}^{-1}$$

(4)

where ${\underset{~}{C}}^{DEM}$ and ${\underset{~}{C}}^i$ are the stiffness matrices of the background medium and the inclusions, respectively, ${\underset{~}{C}}^n$ is the stiffness matrix of the nth module material added, $\underset{~}{I}$ is the unit tensor, $v$ is the volume fraction of the material, and $G$ is the Eshelby tensor, the shape factor controlling the shape of the inclusions.

For the SCA theory, the same anisotropic SCA theory is applied. As with the anisotropic DEM model, the isotropic parameters shear modulus $G$ and bulk modulus $K$ in the original model formulation are replaced by tensor matrices $C$ [22]. The shape factors $P$ and $Q$, which control the appearance of the inclusions, are replaced by the tensor matrix $G$.

2.3. Fluid Replacement Theory

Fluid substitution is an essential part of constructing a rock physics model, and for different model assumptions (e.g., high or low frequency), different fluid substitution formulations can be chosen. In this paper, we mainly apply the Brown–Korringa model because it is based on anisotropic fluid replacement theory. Brown and Korringa [23] derived a theoretical equation for the relationship among the gelling elastic modulus of an anisotropic rock skeleton and the gelling modulus of that skeleton when it is filled with fluid. It is the form of Gassmann’s formula for the anisotropic condition:

$$S_{ijkl}^{dry}-S_{ijkl}^{sat}=\frac{(S_{ij\alpha \alpha }^{dry}-S_{ij\alpha \alpha }^o)(S_{kl\alpha \alpha }^{dry}-S_{kl\alpha \alpha }^o)}{(S_{\alpha \alpha \beta \beta }^{dry}-S_{\alpha \alpha \beta \beta }^o)+({\beta }_{fl}+{\beta }_o)\theta }$$

(5)

where $S_{ijkl}^{dry}$ is the equivalent elastic flexibility tensor of the rock matrix, $S_{ijkl}^{sat}$ is the equivalent elastic flexibility tensor of the saturated fluid rock, $S_{ij\alpha \alpha }^o$ is the equivalent elastic flexibility tensor of the composition minerals, ${\beta }_{fl}$ and ${\beta }_o$ are the compressibilities of the minerals and pore fluid, respectively, and $\theta$ is the void ratio for a shale matrix satisfying $S_{\alpha \alpha \beta \beta }^{\min eral}={\beta }_{\min eral}$. The recurring subscripts in Equation (5) satisfy the Einstein summation convention, which can be used for an anisotropic rock fluid replacement problem.

2.4. Fracture Equivalent Theory

Fracture rock physics modeling can establish a link between the fracture parameters of the rock and the elastic characteristics of the fractures, including influencing parameters such as fracture density, fracture porosity, fracture pore aspect ratio, fracture pore filler, and fluid type. Some fracture parameters are not available but are critical for fractured reservoirs, so it is necessary to rely on fracture equivalence theory to establish ties to predict fracture parameters. This paper applies Schoenberg’s linear slip theory. Schoenberg (1980) [10] came up with the linear slip interface theory (also called linear slip theory) for interfaces with discontinuous displacements by studying the effects of layered media and interfaces on wave propagation and reflection. The linear slip model proposes the concept of flexibility and gives an equation for the equivalent flexibility matrix of the crack media model.

$$S=S_b+S_f$$

(6)

where $S_f$ is the equivalent flexibility and $S_b$ is the equivalent flexibility of the background medium caused by a fracture in the medium.

Schoenberg and Helbig (1997) [24] obtained the weakness $\delta$ and flexibility $Z$ of the fracture system by normalizing the density of physical quantities.${\delta }_N$, ${\delta }_T$, and ${\delta }_H$ are dimensionless physical quantities describing the weakness of the fracture and can be defined as:

$$\left\{\begin{array}{l}0\le {\delta }_N=\frac{Z_N\rho c_{{11}_b}}{1+Z_N\rho c_{{11}_b}}<1\\ 0\le {\delta }_T=\frac{Z_T\rho c_{{44}_b}}{1+Z_T\rho c_{{44}_b}}<1\\ 0\le {\delta }_H=\frac{Z_N\rho c_{{66}_b}}{1+Z_H\rho c_{{66}_b}}<1\end{array}\right.$$

(7)

where $Z_N$ is the normal flexibility, $Z_T$ is the vertical tangential flexibility, and $Z_H$ is the horizontal tangential flexibility. Using the fracture density $e$ to estimate ${\delta }_N$, ${\delta }_H$, and ${\delta }_T$ [25], it can be written as:

$${\delta }_N=\frac{4e}{3g(1-g)[1+\frac{1}{\pi g(1-g)}(\frac{k^{\ast }+4/3{\mu }^{\ast }}{\mu })(\frac{1}{\alpha })]}$$

(8)

$${\delta }_T=\frac{16e}{3(3-2g)[1+\frac{1}{\pi (3-2g)}(\frac{{\mu }^{\ast }}{\mu })(\frac{1}{\alpha })]}$$

(9)

$${\delta }_H=\frac{\mu {\delta }_V}{c_{{66}_b}}$$

(10)

where $g={(V_P/V_s)}^2$, $V_P$, and $V_s$ are the longitudinal and transverse wave velocities of the setting equivalent medium, respectively, $\alpha$ is the aspect ratio of the fracture, $\alpha =3\varphi /4\pi e$, $\varphi$ is the volume fraction of the fracture, $k$ is the bulk modulus of the background equivalent medium, $\mu$ is the shear modulus of the background equivalent medium, and $k^{\ast }$ and ${\mu }^{\ast }$ denotes the bulk modulus and shear modulus of the fluid component contained in the fracture, respectively.

Zhang (2018) [24] uses the anisotropy representation method for weakly anisotropic media, which includes the Thomsen anisotropy parameters:$\mathsf{\epsilon }, \mathsf{\delta }, \mathrm{and} \mathsf{\gamma }$.The anisotropy parameter is obtained from the stiffness matrix.

$$\epsilon =\frac{c_{11}-c_{33}}{2c_{33}}, \gamma =\frac{c_{66}-c_{44}}{2c_{44}}, \delta =\frac{{(c_{13}+c_{44})}^2-{(c_{33}-c_{44})}^2}{2c_{33}(c_{33}-c_{44})}$$

(11)

3. Fractured Shale Rock Physics Modeling Method

The main objective of this paper is to construct an accurate and sensible rock physics model of shale by analyzing the anisotropic causes of shale and simulating the equivalent effect by adopting the appropriate modeling theory. For rock physics modeling of shale, the first application of the established sandstone model was found not to achieve good results because the anisotropy of sandstone is very weak. Therefore, the anisotropic medium theory and more complex and realistic rock physics models were developed to meet the actual generation requirements.

For shale rock physical modeling, the influencing factors can be divided into the influence of the adopted rock mineral components and the influence of the adopted theoretical method. For rock physical modeling, the first thing is to get more accurate rock mineral component parameters, such as relative content, corresponding bulk modulus, shear modulus, density, etc. There are different parameters in different regions, and there will be certain errors when using the parameters measured by previous people. Next, for the inclusions theory used in the modeling method, the choice of matrix and inclusions, shape of inclusions, porosity, etc. will also have an impact on the modeling results. For the shale rock physics model, the mixture of clay and kerogen is usually shown to be oriented, and at the same time, there are a great quantity of pores and fractures distributed in the shale. In this research, the shale is equated to a mixture of kerogen and clay, brittle minerals, fractures, pores, and fluids based on anisotropic sources in modeling. The previous geological data shows that the brittle minerals in the target area mainly include quartz, calcite, and pyrite. Pore space can be separated into normal pore space and fracture pore space, and there is a very small part of invalid void because the content of a small proportion is generally treated as matrix. The fracture porosity is controlled by the fracture density and the fracture aspect ratio, and the pore geometry is expressed by the pore aspect ratio. The fracture parameters can be used to classify the pore types in the rock. The organic matter in the work zone is mainly kerogen, which is considered anisotropic along with clay, while the mixture of gas and water is a fluid. For the directional arrangement and distribution of the main anisotropic sources, namely clay and kerogen, previous authors have also considered them as fluids, but there are significant differences from the actual situation. Therefore, by using the SCA model for brittle mineral mixtures to avoid significant differences in the mechanical properties of brittle minerals, a combined application of the SCA and DEM models under anisotropy is used to simulate the mixture of kerogen and clay.

In this paper, the SCA and DEM models under anisotropy are combined to simulate the equivalent elastic characteristics of the mixture of clay and kerogen, which can ensure the interconnection of kerogen and clay and avoid the asymmetry of the elastic modulus of the mixture due to the different order of kerogen and clay addition. Then use the anisotropic DEM model to add pores to the shale matrix, and then vertical fracture perturbation is added using the Schoenberg linear slip model, and finally the Brown–Korringa model is used for fluid replacement under anisotropy. The modeling flow includes the following steps: (1) due to the presence of bound water in the well logging data, bound water was added to the clay using the K-T model for comprehensive consideration, (2) the combined use of SCA and DEM models under anisotropy to simulate clay and kerogen, both to ensure the connectivity of biphasic mixtures and to avoid the influence of DEM on the choice of background media, (3) then for brittle materials, to solve the problems caused by the significant differences in mechanical properties of brittle minerals, an isotropic SCA model is used for the simulation, (4) the brittle minerals mixture was added as inclusions to the background medium of the anisotropic DEM model to obtain a shale matrix model, (5) the anisotropic DEM model was used to add the empty pore space to obtain the dry shale model, (6) a model of dry shale containing fractures is obtained by adding the effect of fracture perturbation using Schoenberg’s linear slip theory, and (7) the well logging interpretation results were used to determine the fluid proportions and calculate the fluid modulus, and then the Brown–Korringa model was used to perform fluid substitution under anisotropic conditions to finally obtain a saturated shale rock physics model. The modeling flow is shown in Figure 1.

4. Shale Rock Physics Modeling, Validation, and Analysis

4.1. Practical Applications

The shale rock physics model established above was used to conduct theoretical experiments and to test its accuracy by predicting the $V_P$ and $V_s$. Figure 2 shows the well curve of an A well in the shale reservoir of the Wufeng Formation in the Sichuan Basin. The POR black curve represents the porosity curve; the DTC blue curve represents the P-wave slowness curve; the DTS red curve represents the S-wave slowness curve; the DEN green curve represents the density curve; the VOL_CALCITE curve represents the calcite content in the lithology profile as the red lithology display; the VOL_ILLTE curve represents the clay content as the gray lithology display; the VOL_PYTITE curve represents pyrite content as yellow lithology; and the VOL_QUARTZ represents quartz content as brown lithology.

It can be seen that the well logging data in the depth interval of 3780–3810m in the corresponding Wufeng Formation (within the red solid line box) of the target formation have changed significantly, for the longitudinal and transverse wave time differences have increased significantly, indicating that the $V_P$ and $V_s$ have decreased; the porosity has increased significantly and the density has decreased significantly. The combination of the above points indicates that the pore storage space of the reservoir in the destination section has become larger and the pore space is mainly filled with fluid. For the lithology profile, the entire section is led by quartz and clay as the main minerals, but in the destination section, calcite content increases sharply, quartz content and clay content decrease, especially quartz content, which decreases extremely noticeably, and pyrite content is extremely small in the entire section but increases relatively in the destination layer.

The actual well logging data was used for model trial calculations, and the component parameters used for the counts are shown in

Table 1. Details of the target reservoir components.

Components	Bulk Modulus $\left(\mathbf{GPa}\right)$	Shear Modulus $\left(\mathbf{GPa}\right)$	Density $\left({\mathbf{g}/\mathbf{cm}}^3\right)$
Pyrite	147.4	132.5	4.93
Calcite	76.8	32	2.71
Quartz	37	44	2.65
Clay	20.1	7	2.6
Kerogen	2.9	2.7	1.3
Water	2.25	0	1
Gas	0.01	0	0.1

. The volume content of each component used is: 31% clay, 9% calcite, 44% quartz, 3.5% casein, 0.9% pyrite, 6.6% bound water, 5% total porosity, and an assumed fracture density of 0.1. The actual oil content in the workings is very small, so the reservoir is supposed to contain only gas and water, and the above data are taken within the allowable range in the actual workings. The target reservoir components are composed of quartz, calcite, clay, pyrite, and organic matter, and the elastic modulus and density parameters used in the model are shown below.

In the use of anisotropic SCA and DEM models, in addition to the relative content of each mineral component, fluid type, saturation, porosity, etc., which are routinely utilized, there are four crucial parameters: inclusion aspect ratio, pore aspect ratio, each mineral particle aspect ratio, and fracture aspect ratio when adding the effect of fracture perturbation using linear slip theory, which are usually not directly available. In this paper, these four parameters are usually not directly available, so they are inferred by different methods under approximate conditions.

For the brittle mineral particle aspect ratio reference Gui et al. [26] used digital image processing to extract and count the results of microscopic feature parameters taken as 0.8. In the formation, the pore fractures are susceptible to the influence of diagenesis, formation pressure, and compaction at different depths, resulting in different degrees of flattening of the pore shape. In general, the pore aspect ratio and fracture aspect ratio decrease as the depth deepens. Therefore, the inversion of the pore aspect ratio and the fracture aspect ratio is carried out by the process shown in Figure 3. The fracture density, fracture aspect ratio, and fracture porosity satisfy the equation $\alpha =3\varphi /4\pi e$( $\alpha$ is the fracture aspect ratio, $\varphi$ is the volume fraction of the fracture, $e$ the fracture density).

As shown in Figure 3, the two aspect ratio parameters that have been determined at the beginning, i.e., the aspect ratio of brittle mineral particles and the aspect ratio of particles of cheese root and clay mixture, are input randomly for the pore aspect ratio and fracture aspect ratio, then brought into the established shale rock physics modeling process to calculate the longitudinal wave velocity under equivalent conditions, then to calculate the difference with the real value, and then repeatedly replace the different pore aspect ratios. The pore ratio with the smallest difference from the true longitudinal velocity at this time is the pore aspect ratio we want.

4.2. Shale Rock Physics Model Validation

To compare the superiority of our rock physics model, we first simulate a conventional rock physics model, which first adds non-clay minerals to the rock background using VRH averaging, then simulates the clay and pore space according to the SCA model, and finally obtains the rock physics model and performs $V_P$ and $V_s$ prediction. The conventional rock physics modeling process does not integrate the SCA+DEM model with integrated anisotropy for simulation compared to the method proposed in this paper, nor does it integrate various factors such as bound water, brittle minerals, fracture disturbance, pore space, and fluid replacement. In general, conventional methods are not as comprehensive as the methods proposed in this paper. What is shown next is a comparison of the $V_P$ and $V_s$ obtained by rock physics modeling applicable to fractured shale gas reservoirs proposed in this study and the results obtained by conventional rock physics modeling, as shown in the following figure.

Figure 4a shows the longitudinal velocity comparison, where the green curve $\mathrm{vp}$ is the $V_P$ measured by the actual well logging data, the blue curve $\mathrm{vpf}$ is the $V_P$ obtained by the rock physics modeling method proposed in this paper, and the red curve $\mathrm{vpf}1$ is the $V_P$ obtained by the conventional rock physics modeling. Figure 4b shows the comparison of the $V_s$, where the green curve $\mathrm{vs}$ is the $V_s$ measured by the actual well logging data, the blue curve $\mathrm{vsf}$ is the $V_s$ obtained by the rock physics modeling method proposed in this research, and the red curve $\mathrm{vsf}1$ is the $V_s$ obtained by the rock physics modeling. Figure 4c shows the absolute error of $V_P$ prediction, where the blue curve $\mathrm{vpf}$ is the error obtained by the rock physics modeling method proposed in this paper and the actual well logging data, and the red curve $\mathrm{vpf}1$ is the error obtained by the conventional rock physics modeling method and the actual well logging data. Figure 4d shows the absolute error of $V_s$ prediction, the blue curve $\mathrm{vsf}$ is the error obtained by the proposed rock physics modeling method and the actual well logging data, and the red curve $\mathrm{vsf}1$ is the error obtained by the conventional rock physics modeling method and the actual well logging data. From the comparison between the predicted speed and the actual logging speed, it can be seen that the conventional rock physics model has a large error in predicting the $V_P$ and $V_s$, which does not meet the application requirements. By applying the rock physics modeling method for fractured shale reservoirs proposed in this paper, the calculated $V_P$ and $V_s$ from the equivalent model are in good agreement with the well logging data, and the error between the actual well logging data is small, indicating that the application requirements are met by using the rock physics modeling method proposed in this paper. This indicates that the rock physics modeling method put forward in this study meets the application requirements.

4.3. Analysis of Anisotropy Parameters of the Shale Rock Physics Model

In order to avoid the influence of parameters selected from different well area data, the elastic and fracture parameters of fractured shale were inverted using logging data as well as petrophysical models, and then analyzed. Firstly, we invert the elastic parameters of shale rocks using the established rock physics model and analyze them. As shown in Figure 5, according to the inverse of the elastic parameters, Figure 5a shows that the Young’s modulus value range varies between 22 and 27 $\mathrm{GPa}$. Figure 5c shows that the shear modulus varies between 8 and 10$\mathrm{GPa}$, both of which have similar trends. Figure 5d shows that the Poisson’s ratio varies between 0.31 and 0.45. Figure 5b shows that the bulk modulus varies between 20 and 45$\mathrm{GPa}$, where the $V_P$ and $V_s$ increase and the bulk modulus follows.. Figure 5e shows that the pore aspect ratio ranges from 0.6 to 0.9. It can be seen that the bulk modulus and Poisson’s ratio transformations are larger in the destination layer segment.

Next, the Thomsen parameters of shales obtained using the shale petrophysical modeling method proposed in this paper are analyzed, as shown in Figure 6a–c, where the Thomsen parameters of rocks $\epsilon$, $\gamma$, and $\delta$ are calculated using the established petrophysical model. The Thomsen parameters vary drastically above and below the fractured shale reservoir, indicating that the Thomsen parameters of the inverse performance are consistent with the actual logging data.

Finally, the rock physics model is used to invert the fracture weakness parameters. From Figure 6b,c and Figure 7a, the subsurface fracture locations can be identified from the logging data according to the ${\delta }_N$, ${\delta }_H$, and ${\delta }_T$ variation characteristics combined with the subsurface logging interpretation results. It can be seen that the longitudinal and transverse wave velocities of the rocks show a decreasing trend when there are fractures in the subsurface, and the absolute values of the Schoenberg linear sliding model fracture rock weakness parameters ${\delta }_N$, ${\delta }_H$, and ${\delta }_T$ and the Thomsen anisotropy parameters $\epsilon$, $\gamma$, and $\delta$ increase. It indicates that the rock physics modeling method proposed in this paper is better applied to the inversion of anisotropic parameters.

5. Conclusions

In this research, a relatively comprehensive rock physics modeling process for fracture-bearing shale is established from anisotropic genesis to address the relatively simple problem of current shale rock physics models. A typical rock physical model of fracture-bearing rocks is established based on the actual work area well logging data and the predicted $V_P$ and $V_s$ match well by comparing the actual well logging data. The following conclusions and insights are drawn from the study of this paper. In the anisotropic modeling of shale, the SCA+DEM model under anisotropy is used comprehensively to ensure the connectivity of the kerogen and clay while avoiding the problem of large errors caused by the different order of adding fillers in the DEM model. The SCA model was used to add brittle mineral mixtures to avoid the bias induced by the significant diversity in their mechanical properties. For the aspect ratios of kerogen and clay, brittle mineral aspect ratios, and pore aspect ratios involved in the modeling process, the reservoir segment inversion can improve the accuracy and verify the correctness of the model to a certain extent, among which the pore aspect ratios and fracture pore aspect ratios of the formation are influenced by geological factors. The longitudinal and transverse wave velocities of the shale show a decreasing trend when fractures are present in the ground, and the absolute values of the fracture rock physics parameters ${\delta }_N$,${\delta }_H$, and ${\delta }_T$, and the Thomsen anisotropy parameters $\epsilon$,$\gamma$, and $\delta$, increase according to the Schoenberg linear sliding model.