A Rock Physics Modeling Method for Metamorphic Rock Reservoirs in Buried Hill

Abstract

The buried hills of the Archean metamorphic rocks in the Bozhong Depression of the Bohai Bay Basin are the main gas-bearing strata, with burial depths ranging from 4000 m to 5500 m. However, metamorphic rocks have internal structural characteristics, such as diverse mineral components, oriented arrangement of mineral particles, complex pore connectivity, variable crystal structures, orthogonal development of multiple sets of fractures, and uneven fluid filling. Compared with conventional reservoirs, they have obvious heterogeneity and anisotropy characteristics. Traditional rock physics modeling methods are no longer suitable for predicting the elastic and anisotropic parameters of metamorphic reservoirs. Therefore, we introduced a vector mixed random medium model to calculate the effect of the oriented arrangement of metamorphic rock minerals on the modulus of the rock matrix and introduced a metamorphic factor to describe the impact of metamorphic recrystallization and alteration metasomatism on the elastic modulus of the rock matrix. Practical applications have shown that the new, improved rock physics modeling method can better estimate the S-wave velocity and anisotropy parameters in wells compared to traditional rock physics modeling methods, providing a reliable basis for predicting fractured reservoirs in metamorphic rock at buried hills.

1. Introduction

China possesses extensive deep oil and gas reservoirs with significant resource potential. The country’s strategic focus on deep oil and gas reservoirs has intensified due to increasing fossil fuel demand and ongoing advancements in deep gas and oil technologies [1]. In recent years, Chinese oil and gas companies have expanded their exploration efforts towards fractured buried reservoirs, particularly deep metamorphic rock buried reservoirs. The Bohai Bay Basin alone currently accounts for over 25% of China’s oil resources in terms of confirmed reserves, annual output, and undeveloped assets [2]. Discovering high-quality buried-hill reservoirs remains a primary objective for the Chinese oil and gas industry [3].

Metamorphic rocks themselves are not inherently favorable reservoirs and require the formation of cracks and secondary pores through multiple geological processes such as tectonic movements, weathering, erosion, alteration, and metasomatism to become viable reservoirs [4]. Archean buried-hill reservoirs are predominantly characterized by secondary dissolution pores and structural fractures [5,6]. The development of structural cracks within the rock intersects different stages of fractures, forming a network. Concurrently, physical weathering and dissolution generate expansion and dissolution pores along fractures, enhancing fracture aperture and establishing a well-connected reservoir space [7]. These reservoirs exhibit dual storage characteristics dominated by cracks, secondary dissolution pores, and macroscopic structural fractures. Therefore, employing appropriate methodologies for predicting reservoirs of fractured metamorphic rock is crucial.

Due to the different rock structure characteristics of various reservoirs, many scholars have been using the idea of equivalent media to establish rock physical models for different types of reservoirs. Xu and White (1995) introduced the Xu White model as a framework for characterizing the elastic properties of sandstone, incorporating considerations of mud content, porosity size, and pore shape in the determination of rock velocity within mudstone–sandstone formations [8]. Ruiz et al. (2010) used the Berryman self-consistent model to study the physical modeling of tight sandstone rocks and predict the elastic modulus of tight sandstone [9]. Yin Xingyao et al. (2016) introduced an anisotropic rock physics model for tight sandstone using Hudson theory and the anisotropic Gassman fluid replacement equation. They also forecasted the S-wave velocity and anisotropic characteristics specific to tight sandstone [10]. Xu and Payne (2009) divided the pores of carbonate rocks into four types: clay-containing pores, intergranular pores, microcracks, and hard pores when constructing models for carbonate rocks. They extended the Xu White model and applied it to carbonate reservoirs [11]. Zhang Guangzhi et al. (2013) developed an anisotropic rock physics model for carbonate reservoirs with high-angle fractures by integrating inclusion models and linear sliding models to add pores and fractures of different shapes [12]. Hornby et al. (1994) combined the self-consistent model with the differential equivalent model to construct a rock physics model for shale and characterize the seismic elastic properties of Kimmeridge shale [13]. Dvorkin et al. (2007) proposed a new method for modeling shale rock, which divides the pores in the rock into effective pores and muddy pores. They believe that the muddy pores are not connected, and only fluid replacement is performed in the effective pores, improving the rock physics model of shale rock [14]. Guo Zhiqi et al. (2016) introduced the compaction index of clay minerals into the Backus average theory, considering the anisotropy caused by the directional arrangement of clay, and using Chapman’s multi-scale theory to consider the anisotropy caused by fracture systems, constructed an anisotropic rock physics model for shale [15].

Research on seismic rock physics models has predominantly focused on reservoirs such as clastic rocks, carbonate rocks, tight sandstones, and shale, with limited attention given to rock physics modeling techniques for buried-hill metamorphic reservoirs. As exploration of deeply buried-hill gas and oil reservoirs intensifies, there is an urgent need to develop accurate seismic rock physics models for fractured reservoirs within metamorphic buried hills. This paper aims to address this gap by constructing a rock physics model tailored to metamorphic rock reservoirs in buried hills. By examining the mineral composition, mineral particle arrangement, pore structure, crack development, and metamorphic processes of these rocks, this study seeks to accurately predict the elastic and physical parameters of metamorphic rock reservoirs. This will provide precise prior information essential for the prediction of oil and gas reservoirs within metamorphic rocks.

2. Materials and Methods

The study area is located in the southwestern part of the 19-6 structural belt within the Bozhong Depression of the Bohai Bay Basin. To the west, it is adjacent to the southwestern sub-sag of the Bozhong Depression and the Chengbei Low Uplift. To the north, it borders the Shaleitian Uplift and the main depression of the Bozhong Depression. To the southeast, it is situated near the southern sub-sag of the Bozhong Depression and the Binnan Low Uplift. The southern boundary connects with the Huanghekou Depression. Overall, the area is characterized by a complex, faulted anticline structure with multiple periods of faulting in various directions. It features a unique structural pattern of intra-sag uplifts and distinctive oil and gas accumulation characteristics. The advantageous geographical location of the study area is illustrated in Figure 1.

1.

Reservoir lithology characteristics

The rock types within the buried hills of the Bohai Bay Basin exhibit diversity. Through analysis of drilling cores and rock thin sections, we identified that buried-hill reservoirs predominantly consist of Archean metamorphic rocks. The main rock types are metamorphic granodiorite, granodiorite gneiss, plagioclase granulite, fractured granodiorite gneiss, fractured plagioclase granulite, felsic fragmented porphyry, and granodiorite primary mylonite [17]. The overall degree of metamorphism is not high.

Buried-hill reservoirs contain a variety of rock-forming minerals, typically consisting of pale minerals like quartz and feldspar, dark-colored minerals (biotite, hornblende, etc.), and other minerals [18]. Under the same structural stress, light-colored minerals have strong brittleness, relatively small compressive and shear abilities, and are prone to cracking, while dark-colored minerals have weak brittleness, strong plasticity, high compressive and shear abilities, and are prone to plastic deformation, less prone to cracking, and poor storage performance [19]. Figure 2 shows the composition of the buried-hill metamorphic rock reservoir in the research area, with an average of 49.03% feldspar, 35.13% quartz, and 13.79% biotite. With the increase in dark minerals in the rock, the performance of the reservoir decreases.

2.

Reservoir porosity and permeability characteristics

The physical characteristics of buried-hill reservoirs from the Archean period indicate low porosity and permeability. Logging data reveal porosity levels ranging from 2.4% to 12.5%, averaging 5.3%. Well testing reveals permeability values ranging from 0.01 to 11.81 mD, and the average is 0.733 mD. After examining the permeability and porosity of 36 core samples from the reservoir section (it can be seen in Figure 3), it was determined that the porosity levels varied between 1.7% and 10.8%, with an average of 4.1%. Out of the total samples, 24 were below 5%, making up 67% of the total. The permeability ranges from 0.04 to 13.32 mD, and the average is 0.456 mD. When cracks develop, the permeability of the rock samples significantly increases, and the permeability of the samples in the crack development section can reach 13.32 mD.

3.

Characteristics of reservoir space

The storage capacity of the Archean buried-hill reservoir in the study area primarily consists of secondary dissolution pores and structural fractures, supplemented by fractured intergranular pores, microporous fractures, and other types, with minimal presence of primary pores [20]. The fractured intergranular pores are caused by physical weathering and mainly exist in the weathering zone of the buried mountain reservoir, with relatively small vertical and horizontal pores. The dissolution pores formed by acidic solution leaching are mainly secondary pores formed by the dissolution of particles such as feldspar, often distributed in a bead-like manner with blurred edges and irregular shapes. Due to tectonic movements at different stages, the metamorphic rock reservoirs in the work area have developed a large number of fractures, which are orthogonal to each other and form a fracture network.

At the same time, under the influence of physical weathering and dissolution, the expansion and dissolution pores generated by infiltration along the fractures not only increase the opening degree of the fractures but also form a well-connected reservoir space with dual storage space characteristics dominated by fractures and secondary dissolution pores coexisting with macroscopic structural fractures [21]. The basic composition of the buried-hill metamorphic rock reservoir in the study area is shown in Figure 4.

4.

Reservoir development characteristics

The Bohai Archean metamorphic rocks reservoir has undergone numerous tectonic events since the Indosinian era, resulting in the development of numerous expansion fractures. Among them, horizontal fractures are affected by the pressure of the overlying strata, resulting in a small degree of opening, which limits their impact on production capacity. In contrast, high-angle fractures exhibit a relatively larger opening, providing favorable storage and permeation pathways for oil and gas [22].

Description of core observation

Fracture development characteristics are evident from core observations of the buried-hill metamorphic rock reservoir, as illustrated in Figure 5. Rocks in the upper weathering zone exhibit extensive fracturing, forming a network-like structure. As depth increases within the weathering zone, rock fragmentation decreases, yet multiple stages of intersecting fractures remain visible. The lower section of the weathering zone shows reduced weathering, primarily featuring structural fractures, some of which are filled with clay minerals. In the interior zone of the buried hill, fractures predominantly exhibit high-angle structural types, lacking a distinct network structure. The rock matrix is relatively dense, with significant fragmentation observed only near high-angle faults where fractures are well developed.

Surface outcrops and subsurface fractures were analyzed in the research area, with the statistical findings presented in Figure 6. Fracture inclination angles in the study region ranged from low to high, with low-angle fractures making up just 7% of all fractures, medium-angle fractures accounting for 25%, and high-angle fractures making up the remaining 68%. In the northwest direction, medium-to-low-angle fractures primarily form, but under compressive stress, these low-angle fractures are mostly closed and ineffective. Conversely, in the northeast direction, medium-to-high angle fractures predominate, and under tensile stress, these are mostly open and effective.

The field outcrops reveal a high degree of rock fragmentation in the weathered zone of the reservoir, with most cracks exhibiting strong resistance to compaction and displaying a prominent high-angle network development characteristic. With increasing depth of the formation, the weathering and leaching impact on the outcrop diminishes gradually, leading to a deterioration in crack development, a decrease in the number of cracks, and an increase in rock density [25].

Development mode of reservoir fractures

The buried-hill metamorphic rock reservoir has undergone multiple phases of structural stress superposition and reformation, primarily controlled by the early Indosinian and Yanshanian faults. The relationships between fractures and their filling patterns are complex. Based on the distribution of fractures and the cutting relationships between fracture sets observed in the outcrop area, it is inferred that the NNW- and NNE-trending fractures developed earlier and are conjugate. The NE- and NW-trending fractures are also conjugate shear fractures, with their extension limited and displaced by the NNW- and NNE-trending fractures, which formed later, as illustrated in Figure 7a.

From the overall distribution characteristics of the fractures, the NNW- and NNE-trending fractures formed as conjugate fractures under N-S horizontal compressive shear stress, while the NE- and NW-trending fractures formed under E-W horizontal compressive stress. After experiencing strike–slip compressive reformation during the Yanshanian period and extensional reactivation during the Himalayan period, most of these fractures are now either open or partially filled, resulting in the present fracture network system, characterized by two sets of orthogonally developed high-angle extensional fractures, as shown in Figure 7b.

5.

Vector mixed random medium model

In cases where the contact relationship and geometric structure type of rock particles are unclear, Voigt and Reuss boundary models assume that the rock matrix minerals are subjected to the same strain or stress. Analyzing the mineral composition data acquired from logging allows for the identification of the upper and lower bounds of the rock matrix’s elastic modulus. These bounds can then be averaged using the Hill method to derive the final elastic modulus of the rock matrix [27,28].

The Archean metamorphic rocks in the Bohai Sea have experienced significant regional metamorphism, displaying a distinct gneiss-like structure with visible directional arrangements of light and dark minerals on the core. The overall rock particles exhibit directional distribution along a certain direction, and using a mixed-type autocorrelation function alone from the horizontal and vertical directions is obviously unable to accurately describe this special-oriented gneiss-like structure of metamorphic rocks.

Therefore, on the basis of the mixed autocorrelation function, an angle factor is added to obtain a vector mixed random medium model for characterizing the gneiss-like structure of rocks. The expression of the autocorrelation function is [29,30]:

$$\phi (x,z)=\exp \left\{-{\left[{\displaystyle \frac{{(x\cos \theta +z\sin \theta )}^2}{a^2}}+{\displaystyle \frac{{(x\sin \theta -z\cos \theta )}^2}{b^2}}\right]}^n\right\}$$

(1)

In the formula, a and b are the autocorrelation lengths in the x and z directions, respectively, which can control the average particle size of the rock matrix particles in the horizontal and vertical directions. The values can be determined based on the particle size structure of the rock slices; n is the roughness factor that can control the degree of rounding of rock particle boundaries, with a value range of 0–1. The angle factor $\theta$ describes the oriented arrangement direction of rock matrix particles, with a value range of 0–2 π. When $\theta =0$, the vector mixed random medium model can degenerate into a mixed random medium model. The rock matrix is simulated based on a vector mixed random medium model, as shown in Figure 8.

Therefore, the elastic modulus of the homogeneous background medium in the rock matrix can be determined first using the VRH model, and then the random disturbance caused by non-uniform particle structure can be calculated using the random medium model. The two are superimposed to obtain the final modulus of the mixed minerals in the rock matrix. This method considers the influence of rock particle geometry and oriented arrangement on the elastic properties of rocks and can provide reliable data support for rock physics modeling of metamorphic rock reservoirs.

By adjusting the VRH model to compute the mixed minerals in the metamorphic rock matrix using vector mixed random medium theory, the double pore model is then applied to determine the elastic modulus of the rock matrix. This calculation includes the addition of unconnected pores containing bound water, as demonstrated in Equation (2) [31].

$$\begin{array}{l}{K}_m={\displaystyle \frac{\left(1-{\varphi }_{mat}\right)K_0+{\varphi }_{mat}{\displaystyle \sum _{i=1}^n{v}_i{K}_w{P}_i}}{\left(1-{\varphi }_{mat}\right)+{\varphi }_{mat}{\displaystyle \sum _{i=1}^n{v}_i{P}_i}}}\\ {\mu }_m={\displaystyle \frac{\left(1-{\varphi }_{mat}\right){\mu }_0}{\left(1-{\varphi }_{mat}\right)+{\varphi }_{mat}{\displaystyle \sum _{i=1}^n{v}_i{Q}_i}}}\end{array}$$

(2)

In the formula, $P_i$ and $Q_i$ are the geometric factors of the different isolated pore, respectively; $K_w$ is the bulk modulus of bound water; $K_m,{\mu }_m$ is the bulk modulus and shear modulus of the rock matrix.

Using an improved rock physics modeling method suitable for Bohai buried-hill metamorphic rock fractured reservoirs, the S-wave velocity and fracture parameters of the reservoir were predicted. Compared with the results of seismic rock physics models for other lithological reservoirs, the prediction error of the rock physics model for buried-hill metamorphic rock reservoirs is within 10%, and the model has good prediction accuracy and applicability.

6.

Construction process of seismic rock physical model for buried-hill metamorphic rock reservoirs

The Bohai buried-hill metamorphic rock reservoir is influenced by various geological processes, including tectonic activity, weathering and erosion, and metamorphic recrystallization. It exhibits characteristics such as diverse mineral components, directional arrangement of mineral particles, complex pore connectivity, variable metamorphic crystal structure, orthogonal development of multiple fracture sets, and uneven fluid filling within the rock’s internal structure.

We refer to the process of establishing classic seismic rock physics models and study the method of constructing equivalent rock physics models for fractured metamorphic rock reservoirs. The modeling diagram is shown in Figure 9.

The specific construction process is as follows:

The initial step involves arranging various mineral particles directionally to create a rock matrix, followed by calculating the equivalent elastic modulus of the mixed minerals within the matrix.

Based on the composition content of minerals such as quartz, feldspar, and biotite provided by logging data, the VRH model is used to select Formula (3) to first calculate the bulk modulus $K_{VRH}$ and shear modulus ${\mu }_{VRH}$ of the homogeneous background medium in the rock matrix. Reusing the vector mixed random medium model to calculate the random disturbance $\sigma$ caused by the shape of mineral particles in the rock matrix and the oriented arrangement structure in the form of gneiss, the volume modulus $K_o$ and shear modulus ${\mu }_o$ of the mixed minerals in the rock matrix are obtained by superimposing the two:

$$\begin{array}{l}{K}_o=K_{VRH}(1+\sigma ),\\ {\mu }_o={\mu }_{VRH}(1+\sigma ).\end{array}$$

(3)

The subsequent step involves introducing unconnected pores containing bound water into the rock matrix and then calculating the resulting composite elastic modulus.

The pores within the study area predominantly consist of primary intergranular pores, secondary dissolution pores, and microcracks. Analysis of rock thin sections reveals a strong correlation between secondary dissolution pores and microcracks, while primary intergranular pores are sparsely distributed and few in number. It is assumed that the connected pores of the rock are composed of secondary dissolution pores and microcracks, and isolated pores are composed of primary intergranular pores. So, the various pores of rocks can be represented as:

$$\begin{array}{l}{\varphi }_p+{\varphi }_s+{\varphi }_f={\varphi }_t,\\ {\varphi }_s+{\varphi }_f={\varphi }_{con},{\varphi }_p={\varphi }_{iso}\end{array}$$

(4)

In the formula, ${\varphi }_p,{\varphi }_s,{\varphi }_f$ represent the porosity of primary intergranular pores, secondary dissolution pores, and microcracks, respectively; ${\varphi }_t,{\varphi }_{con},{\varphi }_{iso}$ represent total porosity, connected porosity, and isolated porosity.

The proportion factor $\nu$ of various pores relative to the total pores can be expressed as:

$$v_i={\displaystyle \frac{{\varphi }_i}{{\varphi }_t}}$$

(5)

In the formula, ${\varphi }_i$ represents the porosity of the different type of pore.

Because unconnected pores containing bound water are challenging to flow through and displace, they are considered in the fluid displacement equation by assuming their equivalence to a portion of the rock matrix. Therefore, the volume ratio ${\varphi }_{mat}$ of unconnected pores relative to the rock matrix can be expressed as:

$${\varphi }_{mat}={\displaystyle \frac{{\varphi }_t-{\varphi }_{con}}{1-{\varphi }_{con}}}$$

(6)

Hence, the rock matrix is considered to consist of mixed minerals and isolated pores containing bound water. The dual pore model can then be employed to assess the impact of adding these isolated pores on the elastic modulus of the rock matrix, as demonstrated in Equation (2).

The third step involves incorporating dry, connected pores into the rock matrix and then calculating the equivalent elastic modulus of the resulting isotropic rock skeleton.

Applying the dual pore model to assess the impact of incorporating dry connected pores on the elastic modulus of the rock skeleton [31,32]:

$$\begin{array}{l}{K}_{dry}={\displaystyle \frac{\left(1-{\varphi }_t\right)K_0+\left({\varphi }_t-{\varphi }_{con}\right){\displaystyle \sum _{i=1}^n{\upsilon }_i{K}_w{P}_i}}{\left(1-{\varphi }_t\right)+\left({\varphi }_t-{\varphi }_{con}\right){\displaystyle \sum _{i=1}^n{\upsilon }_i{P}_i+{\varphi }_{con}{\displaystyle \sum _{i=1}^n{\upsilon }_i{\widehat{P}}_i}}}},\\ {\mu }_{dry}={\displaystyle \frac{\left(1-{\varphi }_{mat}\right){\mu }_0}{\left(1-{\varphi }_t\right)+\left({\varphi }_t-{\varphi }_{con}\right){\displaystyle \sum _{i=1}^n{\upsilon }_i{Q}_i+{\varphi }_{con}{\displaystyle \sum _{i=1}^n{\upsilon }_i{\widehat{Q}}_i}}}}\end{array}$$

(7)

In the formula, ${\widehat{P}}_i$ and ${\widehat{Q}}_i$ are the geometric factors of the different connected pore; $K_{dry}$ and ${\mu }_{dry}$ are the bulk modulus and shear modulus of the dry rock skeleton.

The fourth step involves evaluating how metamorphism, recrystallization, and alteration impact the elastic modulus of the dry rock skeleton.

On the one hand, metamorphism and recrystallization can cause the original matrix mineral crystals to regrow and increase in particle size, resulting in closer contact between matrix mineral particles and a harder rock skeleton. On the other hand, the light-colored minerals in the rock matrix are easily dissolved and hydrolyzed by acidic fluids due to alteration and metasomatism, forming dissolution pores and making the rock skeleton appear relatively soft. Therefore, drawing on the modeling approach of the Pride model, the influence of metamorphism on the elastic modulus of the dry rock skeleton is characterized by the introduction of metamorphic factors $\gamma$ and porosity ${\varphi }_t$ [33,34]:

$$\begin{array}{l}K={\displaystyle \frac{K_{dry}}{1+\gamma {\varphi }_t}},\\ \mu ={\displaystyle \frac{{\mu }_{dry}}{1+1.5\gamma {\varphi }_t}}\end{array}$$

(8)

In the formula, $K,\mu$ are the corrected bulk modulus and shear modulus of the dry rock skeleton, respectively; $\gamma$ is a metamorphic factor that reveals the influence of metamorphism on the contact relationship between rock matrix particles, with a value range of $2<\gamma <20$. When $\gamma$ is small, it indicates strong metamorphism and recrystallization, and the rock is relatively dense. Conversely, when $\gamma$ is big, it indicates strong alteration and metasomatism, and the rock is relatively loose.

The metamorphic factor is a parameter unique to the rock physics model of buried-hill metamorphic rock reservoirs, which cannot currently be obtained from logging data and rock thin sections. Only by constructing the objective functional of the measured and predicted longitudinal wave velocities and continuously adjusting the size of the metamorphic factor based on the simulated annealing method can the relative error between the predicted and measured longitudinal wave velocities of the rock physics model be minimized, thereby obtaining the metamorphic factor in the well.

The fifth step considers the influence of crack development and calculates the equivalent stiffness matrix of an anisotropic dry rock skeleton.

Analysis of core and imaging logging data reveals the predominant development of two orthogonal sets of high-angle fractures in the study area. Based on this observation, the reservoir in the study area can be approximated as an OA medium comprising two sets of high-angle fractures within an isotropic background rock matrix. Utilizing the Schoenberg linear sliding model allows for the determination of the equivalent stiffness matrix of the fractured dry rock skeleton [35]:

$$C^{dry}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{22}& C_{23}& 0& 0& 0\\ C_{13}& C_{23}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]=\left(\begin{array}{cc}{C}_1& 0\\ 0& C_2\end{array}\right)$$

(9)

In the formula, $C_1,C_2$ are as follows:

$$\begin{array}{l}{c}_1={\displaystyle \frac{1}{d}}\left(\begin{array}{ccc}M\left(1-{\delta }_{N1}\right)\left(1-r^2{\delta }_{N2}\right)& \lambda \left(1-{\delta }_{N1}\right)\left(1-{\delta }_{N2}\right)& \lambda \left(1-{\delta }_{N1}\right)\left(1-r{\delta }_{N2}\right)\\ \lambda \left(1-{\delta }_{N1}\right)\left(1-{\delta }_{N2}\right)& M\left(1-r^2{\delta }_{N1}\right)\left(1-{\delta }_{N2}\right)& \lambda \left(1-r{\delta }_{N1}\right)\left(1-{\delta }_{N2}\right)\\ \lambda \left(1-{\delta }_{N1}\right)\left(1-r{\delta }_{N2}\right)& \lambda \left(1-r{\delta }_{N1}\right)\left(1-{\delta }_{N2}\right)& M\left[\left(1-r^2{\delta }_{N1}\right)\left(1-r^2{\delta }_{N2}\right)-4r^2{g}^2{\delta }_{N1}{\delta }_{N2}\right]\end{array}\right),\\ c_2={\displaystyle \frac{1}{d}}\left(\begin{array}{ccc}\mu \left(1-{\delta }_{T2}\right)& 0& 0\\ 0& \mu \left(1-{\delta }_{T1}\right)& 0\\ 0& 0& \mu {\displaystyle \frac{\left(1-{\delta }_{T1}\right)\left(1-{\delta }_{T2}\right)}{\left(1-{\delta }_{T1}{\delta }_{T2}\right)}}\end{array}\right)\end{array}$$

(10)

In the formula, $d=1-r^2{\delta }_{N1}{\delta }_{N2}$, $g=\mu /\left(\lambda +2\mu \right)$, $r=\lambda /\left(\lambda +2\mu \right)$; $M,\mu ,\lambda$ are the longitudinal wave modulus, shear modulus, and Lame parameter of isotropic background rocks, ${\delta }_{T1},{\delta }_{N1}$ are the tangential and normal weakness of the first group of cracks, ${\delta }_{T2},{\delta }_{N2}$ are the tangential and normal weakness of the second group of cracks.

The sixth step adds the fluid to the rock and calculates the equivalent stiffness matrix of saturated fractured rock.

Due to the fact that the Bohai buried-hill metamorphic rock reservoir belongs to a gas bearing low porosity, low permeability, and tight reservoir, assuming that the rock pores are filled with unevenly distributed natural gas and water, the Brie model is chosen to calculate the bulk modulus $K_f$ of the mixed fluid. The equivalent stiffness matrix of saturated fractured rock can be obtained by substituting the stiffness matrix of the fractured dry rock skeleton into the Gassmann fluid replacement equation derived by Huang et al., which includes the parameters of crack weakness, as shown in Equation (11) [36].

$$C^{sat}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{22}& C_{23}& 0& 0& 0\\ C_{13}& C_{23}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]$$

(11)

In the formula,

$$\begin{array}{l}{C}_{11}^{sat}=M\left(1-{\delta }_{N1}-r^2{\delta }_{N2}\right)+{\displaystyle \frac{{\left(K_m-K\left(1-{\delta }_{N1}-r{\delta }_{N2}\right)\right)}^2}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{22}^{sat}=M\left(1-r^2{\delta }_{N1}-{\delta }_{N2}\right)+{\displaystyle \frac{{\left(K_m-K\left(1-r{\delta }_{N1}-{\delta }_{N2}\right)\right)}^2}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{33}^{sat}=M\left(1-r^2{\delta }_{N1}-r^2{\delta }_{N2}\right)+{\displaystyle \frac{{\left(K_m-K\left(1-r{\delta }_{N1}-r{\delta }_{N2}\right)\right)}^2}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{12}^{sat}=\lambda \left(1-{\delta }_{N1}-{\delta }_{N2}\right)+{\displaystyle \frac{\left(K_m-K\left(1-{\delta }_{N1}-r{\delta }_{N2}\right)\right)\left(K_m-K\left(1-r{\delta }_{N1}-{\delta }_{N2}\right)\right)}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{13}^{sat}=\lambda \left(1-{\delta }_{N1}-r{\delta }_{N2}\right)+{\displaystyle \frac{\left(K_m-K\left(1-{\delta }_{N1}-r{\delta }_{N2}\right)\right)\left(K_m-K\left(1-r{\delta }_{N1}-r{\delta }_{N2}\right)\right)}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{23}^{\mathrm{sat}}=\lambda \left(1-r{\delta }_{N1}-{\delta }_{N2}\right)+{\displaystyle \frac{\left(K_m-K\left(1-r{\delta }_{N1}-{\delta }_{N2}\right)\right)\left(K_m-K\left(1-r{\delta }_{N1}-r{\delta }_{N2}\right)\right)}{\left(K_m/K_f\right)\varphi \left(K_m-K_f\right)+K_m-K+K^2\left({\delta }_{N1}+{\delta }_{N2}\right)/M}},\\ C_{44}^{sat}=\mu \left(1-{\delta }_{T2}\right),\\ C_{55}^{sat}=\mu \left(1-{\delta }_{T1}\right),\\ C_{66}^{sat}=\mu {\displaystyle \frac{\left(1-{\delta }_{T1}\right)\left(1-{\delta }_{T2}\right)}{\left(1-{\delta }_{T1}{\delta }_{T2}\right)}}.\end{array}$$

(12)

3. Results

Based on the relationship between stiffness coefficients and elastic parameters, the longitudinal and transverse wave velocities of saturated rocks can be derived using the following formulas:

$$\begin{array}{l}C{V}_p=\sqrt{{\displaystyle \frac{C_{33}^{sat}}{\rho }}},\\ V_s=\sqrt{{\displaystyle \frac{C_{55}^{sat}}{\rho }}}.\end{array}$$

(13)

In the formula, $\rho$ is the density of the rock.

According to the relationship between the stiffness coefficient and the Tsvankin anisotropy parameter, the anisotropy parameters of longitudinal and transverse waves of saturated rocks can be obtained. The calculation formula is as follows:

$$\begin{array}{l}{\epsilon }^{(1)}={\displaystyle \frac{C_{22}^{sat}-C_{33}^{sat}}{2C_{33}^{sat}}},{\gamma }^{(1)}={\displaystyle \frac{C_{66}^{sat}-C_{55}^{sat}}{2C_{55}^{sat}}},\\ {\epsilon }^{(2)}={\displaystyle \frac{C_{11}^{sat}-C_{33}^{sat}}{2C_{33}^{sat}}},{\gamma }^{(2)}={\displaystyle \frac{C_{66}^{sat}-C_{44}^{sat}}{2C_{44}^{sat}}}.\end{array}$$

(14)

Based on the relationship between anisotropic parameters and crack strength, the corresponding strength parameters for two sets of cracks can be obtained, and the calculation formula is as follows:

$$\begin{array}{l}{\delta }_{N1}={\displaystyle \frac{{\epsilon }^{\left(2\right)}}{-2g\left(1-g\right)}},{\delta }_{T1}=-2{\gamma }^{\left(2\right)},\\ {\delta }_{N2}={\displaystyle \frac{{\epsilon }^{\left(1\right)}}{-2g\left(1-g\right)}},{\delta }_{T2}=-2{\gamma }^{\left(1\right)}.\end{array}$$

(15)

Due to the fact that the reservoir in the study area is mainly gas bearing, based on the assumption that the fractures are filled with gas, and using the relationship between fracture density and fracture weakness, the density size of two sets of fractures can be calculated. The calculation formula is as follows:

$$\begin{array}{l}{e}_1={\displaystyle \frac{3\left(3-2g\right)}{16}}{\delta }_{T1},\\ e_2={\displaystyle \frac{3\left(3-2g\right)}{16}}{\delta }_{T2}.\end{array}$$

(16)

Using the seismic rock physics model developed for the metamorphic rock of buried-hill reservoir, we can investigate the correlation between rock physical parameters and mineral composition, as illustrated in Figure 10.

Figure 8 shows that under constant reservoir parameters, an increase in dark minerals like biotite correlates with increased volumetric and shear moduli of the rock. The volumetric modulus serves as a critical indicator of storage capacity in metamorphic rock buried-hill reservoirs, with higher values suggesting reduced rock matrix porosity. Conversely, an increase in light-colored minerals, such as quartz and feldspar, is associated with higher rock crack density and metamorphic alteration, indicative of increased rock fragmentation and altered permeability characteristics.

The parameters required for constructing a metamorphic rock buried-hill reservoir model mainly include the following: (1) the bulk modulus, shear modulus, and density of the main rock-forming minerals (quartz, feldspar, and biotite) in the rock and the fluids in the pores; (2) the volume content of quartz, feldspar, and biotite, the porosity of various pores, and the water saturation of rocks; and (3) the pore aspect ratio. This information is mainly obtained through logging data and core data.

Logging interpretation data can provide information on the volume content of quartz, feldspar, and biotite, as well as the porosity and water saturation of various pores (intergranular pores, dissolution pores, microcracks). By analyzing the pore structure characteristics of the sample rock, we have statistically obtained the distribution range of pore aspect ratios for various types of pores. Among them, the pore aspect ratio of intergranular pores is low, with an average value of 0.17, while that of dissolution pores is high, with an average value of 0.35. The pore aspect ratio of microcracks is extremely low in the shape of a coin, with an average value of 0.05. This provides prior information for seismic rock physics modeling of buried-hills metamorphic rock reservoirs.

Based on these data, a seismic rock physics model was developed for the buried-hill metamorphic rock reservoir to forecast the S-wave velocity of well A within the study area. Comparative analysis against models derived from different rock types was conducted, and the findings are depicted in Figure 11.

Figure 11 illustrates that, in comparison to alternative models, the seismic rock physics model tailored for the buried-hill metamorphic rock reservoir most closely aligns with the measured S-wave velocity, exhibiting a prediction error of less than 10%. These results affirm the model’s robust applicability.

The crack density predictions based on the rock physics model of buried-hill metamorphic rock reservoirs are presented in Figure 12. The figure shows that within the weathering zone (4550–4600 m) of the buried hill, two sets of fractures exhibit significant development, attributed to factors such as tectonic activity, weathering processes, and ancient topography. These fractures appear networked in imaging logging data. With increasing depth, the dissolution effect diminishes, influenced by lithological barriers, and fractures in the inner part of the buried hill are primarily controlled by multiple tectonic events. These fractures alternate within single groups, characterized by high-angle distribution in imaging logging data. The model’s predictions regarding the density of these two fracture sets align closely with imaging data interpretations, consistent with the geological principles governing the study area.

Overlaying two sets of cracks to determine the total crack density, the comparison with crack densities derived from the bilateral resistivity empirical formula is depicted in Figure 13. The left panel displays results calculated via the empirical formula, the middle panel shows model predictions, and the right panel illustrates lithofacies productivity interpretations. High-yield lithofacies are marked in red, medium-yield in yellow, and low-yield in green.

Figure 13 reveals that the model’s predicted crack density closely aligns with lithofacies productivity interpretations, demonstrating superior agreement compared to results obtained from empirical formulas. Furthermore, the predicted crack density curve trend correlates more accurately with actual subsurface conditions.

Using the S-wave velocity and fracture parameters predicted by rock physics modeling as constraints and utilizing the derived anisotropic longitudinal wave reflection coefficient and Fourier coefficient equations containing fracture parameters, the Bayesian Fourier coefficient iterative inversion method under Cauchy sparsity constraint and low-frequency model constraint is used to achieve the prediction of fractured reservoirs of metamorphic rocks. The prediction results, as shown in Figure 14, show that the development of cracks in the interior of the buried hill exhibits an inclined strip distribution feature, which is consistent with geological understanding.

4. Discussion

Considering the structural characteristics of mineral particle-oriented arrangement, variable crystal structures, complex pore connectivity, and orthogonal development of two sets of fractures in buried-hill metamorphic rock reservoirs, the VRH model and vector mixed random medium model are used to calculate the random disturbance caused by the directional arrangement of mineral flakes in the rock matrix. The metamorphic factor is introduced to describe the impact of metamorphic recrystallization and alteration metasomatism on the elastic modulus of the rock matrix. The dual pore model is used to add isolated and connected pores of different shapes. The anisotropy caused by two sets of orthogonal development of high-angle fractures is considered through a linear sliding model, and a seismic rock physics model that conforms to the structural characteristics of buried-hill metamorphic rock reservoirs is constructed. If we use the rock physics modeling method proposed by Xu and White (1995) to predict the elastic and fracture parameters of metamorphic reservoirs [8], the prediction error will be much greater than 10%. This is because this method does not consider the directional characteristics of mineral particle arrangement in metamorphic rocks and the metamorphic process of metamorphic rocks. Similarly, other rock physics modeling methods proposed by others are not suitable for predicting the elastic and fracture parameters of metamorphic reservoirs.

5. Conclusions

A seismic rock physics model modeling scheme suitable for fractured reservoirs in buried-hills metamorphic rock is proposed based on the internal structural characteristics of diverse mineral components, directional arrangement of mineral particles, complex pore connectivity, variable crystal structures, orthogonal development of multiple sets of fractures, and uneven fluid filling in Bohai buried-hills metamorphic rock reservoirs. Referring to the construction process of classical seismic rock physics, combined with equivalent medium theory and random medium model, a seismic rock physics modeling scheme is proposed.

Based on the constructed seismic rock physics model, not only can the S-wave velocity of fractured reservoirs in buried-hills metamorphic rock be estimated to compensate for the missing S-wave velocity information in some well sections, but the variation characteristics of the estimated fracture density curve can also be analyzed. Combined with the interpretation results of imaging logging, it can guide the evaluation of effective reservoirs. Moreover, the predicted S-wave velocity and crack density results can provide reliable data support for subsequent 3D underground crack density inversion using seismic data.