**3. Results**

#### *3.1. Stress Balance along the Lamina*

A finite element simulation of lamina-induced fractures is conducted in the statics module of the ANSYS workbench based on two geo-bodies, i.e., the lamina and surrounding rock. The contact relationship was settled to the frictional model characterized by a friction coefficient representing the difference between the two separated geo-bodies. Here, this study defined the coefficient as the lamina friction coefficient (μ*lamina*) in the laminated rock, where μ*lamina* could be used to indicate different lamina lithofacies and modify the T–K failure criterion in the finite element simulation. The laminated rock model is shown in Figure 1, which was constructed with the laminated body and the surrounding rock body. Two parameters were introduced into the geo-model to characterize the laminated rock, i.e., the lamina angle (θ) and μ*lamina*. It should be emphasized that μ*lamina* of the laminated rock differs from the internal friction coefficient of intact rocks. The μ*lamina* is not only a physical factor representing the physical sliding behavior in simulation but also a geological factor indicating different lamina lithofacies. Therefore, the range of μ*lamina* may vary from the traditional internal friction coefficient. Its detailed values for different laminated rock types were tested using the triaxial compression technologies in Section 2.2.

**Figure 1.** Laminated rock model composed of the lamina and the surrounding rock. Red arrows indicate the decomposition of stress along the lamina surface, where θ indicates the lamina angle, σ1 indicates the axial stress, σ3 indicates the confining pressure, *Clamina* indicates the lamina cohesion, and f indicates the friction force between the lamina surface and surrounding rock.

Another critical factor for the failure criterion is the lamina angle (θ), indicating the intersection angle between the axial stress direction and the lamina dip direction, which is the complementary angle to the β angle (the acute angle between the direction of maximum principal stress and the discontinuity) in previous works [6,47]. A stress equilibrium equation was built based on the decomposition of stress along the lamina surface (Equation (1), Figure 1). In this equation, θ indicates the lamina angle, σ1 indicates the axial stress, σ3 indicates the confining pressure, and *Clamina* indicates the lamina cohesion, which is different from the cohesion of homogeneous rock.

$$
\sigma\_1 \cos \theta - \sigma\_3 \sin \theta = \mu\_{\text{laminar}} (\sigma\_3 \cos \theta + \sigma\_1 \sin \theta) + C\_{\text{laminar}} \tag{1}
$$

The Mohr–Coulomb rock failure criterion (Equation (2)) is usually used to explain the propagation condition for shear fractures. Here, an assumption was made that the propagation condition for bedding fractures meets the Mohr–Coulomb rock failure criterion. In other words, the lamina-induced fractures could be regarded as specially compressed fractures along the lamina surface in the intact rock. Then, Equation (3) was achieved when we solved the simultaneous Equations (1) and (2). It describes the relationship among the internal friction angle (ϕ) of intact rock, the lamina friction coefficient (μ*lamina*), and the lamina angle (θ) of laminated rock. Equation (4) describes the relationship among the cohesion (*S*0), the internal friction angle (ϕ) of intact rock, and the lamina angle (θ) of laminated rock. The final relationship (Equation (5)) was built among the cohesion (*S*0), the internal friction angle (ϕ) of intact rock, the lamina angle (θ), and the lamina friction coefficient (μ*lamina*) of laminated rock, when Equation (3) was put into Equation (4). Thus, an important function (Equation (5)) was used to modify the T–K criterion in the following section. Furthermore, the detailed derivation and physical interpretations are shown in Appendix A.

$$
\sigma\_3 = 2S\_0 \frac{\cos \varphi}{1 - \sin \varphi} + \sigma\_3 \frac{1 + \sin \varphi}{1 - \sin \varphi} \tag{2}
$$

$$\varphi = \arcsin \frac{\mu\_{\text{laminar}} \cos \theta + \sin \theta}{(\mu\_{\text{laminar}} + 1)\cos \theta + (1 - \mu\_{\text{laminar}})\sin \theta} \tag{3}$$

$$\text{So} = \frac{1}{2} \cdot \frac{1 - \sin \varphi}{\cos \varphi} \cdot \frac{\mathbb{C}\_{\text{laminar}}}{\cos \theta - \mu\_{\text{laminar}} \sin \theta} \tag{4}$$

$$S\_0 = \frac{1}{2} \cdot \frac{\frac{\cos \theta - \mu\_{\text{lamin}} \sin \theta}{(\mu\_{\text{lamin}} + 1)\cos \theta + (1 - \mu\_{\text{lamin}})\sin \theta}}{\cos \arcsin \frac{\mu\_{\text{lamin}} \cos \theta + \sin \theta}{(\mu\_{\text{lamin}} + 1)\cos \theta + (1 - \mu\_{\text{lamin}})\sin \theta}} \cdot \frac{\mathcal{C}\_{\text{lamin}}}{\cos \theta - \mu\_{\text{lamin}} \sin \theta} \tag{5}$$

#### *3.2. The Failure Criterion of Bedding Fractures*

Tien and Kuo (2001) proposed a common failure criterion (Equation (6)) for the intact bedded rocks, which demonstrates the characteristics of strength anisotropy revealed by the laboratory experiments, where k and n indicate the elastic constants of laminated rocks [6]. In addition, the T–K criterion is based on the nonlinear Hoek–Brown criterion for the homogeneous rock with no lamina.

$$\frac{\sigma\_{1(\theta)} - \sigma\_3}{\sigma\_{1(\theta=90)} - \sigma\_3} = \frac{k}{\cos^4 \theta + k \sin^4 \theta + 2nk \sin^2 \theta \cos^2 \theta} \tag{6}$$

Zhou et al. (2017) proposed a more efficient modified T–K criterion (Equation (7)) to replace the nonlinear Hoek–Brown criterion with the linear Mohr–Coulomb criterion, where ϕ indicates the cohesion of surrounding rock beyond the lamina and *S*0 indicates the cohesion of laminated rocks [47]. The main problem for Zhou et al.'s modified criterion is that the cohesion of laminated rocks (*S*0) was a constant in Equation (7), but it actually varied with the lamina angle (θ) and the lamina friction coefficient (μ*lamina*) when the compressed fractures primarily propagated along the lamina surface in the laminated rocks. Thus, θ and μ*lamina* could be used together, as two modified factors of *S*0 in the

T–K criterion, to predict the simulated failure behaviors in cm-scale laminated cores to the km-scale bedding formation.

$$\sigma\_1 = \sigma\_3 + \left(\frac{2\sin\varphi}{1-\sin\varphi} - 2S\_0\sqrt{\frac{1+\sin\varphi}{1-\sin\varphi}}\right)\frac{k}{\sin^4\theta + k\cos^4\theta + 2nk\cos^2\theta\sin^2\theta} \tag{7}$$

Here, Equation (5) was put into Equation (7) to modify the T–K criterion. Thus, a modified T–K criterion is proposed by Equation (8), where ϕ indicates the surrounding rock properties and the other parameters indicate the lamina properties. The modified criterion is composed of four critical variables, i.e., the maximum principal stress (σ1), minimum principal stress (σ3), lamina angle (θ), and lamina friction coefficient (μ*lamina*), which are advantageous and could also be used in the ANSYS finite element simulation.

$$\begin{array}{l} \sigma\_{1} - \sigma\_{3} = \quad \left( \frac{2 \sin \phi}{1 - \sin \phi} - \frac{\frac{\cos \theta - \mu\_{\text{lim}} \sin \theta}{(\mu\_{\text{lim}} + 1) \cos \theta + (1 - \mu\_{\text{lim}}) \sin \theta}}{\cos \arcsin \frac{\mu\_{\text{lim}} \cos \theta + \sin \theta}{(\mu\_{\text{lim}} + 1) \cos \theta + (1 - \mu\_{\text{lim}}) \sin \theta}} \times \frac{C\_{\text{laminar}}}{\cos \theta - \mu\_{\text{laminar}} \sin \theta} \sqrt{\frac{1 + \sin \phi}{1 - \sin \phi}} \right) \\ \quad \times \frac{k}{\sin^{4} \theta + k \cos^{4} \theta + 2nk \cos^{2} \theta \sin^{2} \theta} \end{array} \tag{8}$$

#### *3.3. Index of Bedding Fractures*

Li et al. (2018) simulated the distribution of the structural fracture of the Upper Triassic Yanchang Formation in the Ordos Basin by the ANSYS software with a simplified fracture index set. The simulation was proposed based on the relationship between the maximum principal stress and the failure strength, and represents the total fracture possibility based on the distance between the envelope line and the stress state point (Equation (9)) [14]. In the bedding fracture simulation, σ 1 (the maximum normal stress satisfied the critical rupture condition) is replaced by the failure criterion for the bedding fractures (i.e., Equation (8)) to indicate the index of bedding fractures, as shown in Equation (10).

$$\begin{cases} \begin{aligned} f &= \frac{\sigma\_1 - \sigma\_1'}{\sigma\_1}, \sigma\_3 > -T\_0 \\ \text{where} \quad \sigma\_1' &= \sigma\_3 + \frac{4}{\sqrt{1 + \mu^2 - \mu}} T\_0 + \mu \sigma\_3' \\ f\_{LF} &= 1, \sigma\_3 < -T\_0 \end{aligned} \tag{9} \end{cases} \tag{9}$$

$$\begin{cases} \begin{aligned} f\_{LF} &= \frac{\sigma\_{1} - \sigma\_{1}'}{\sigma\_{1}}, \sigma\_{3} > -T\_{0-\text{laminar}} \\ \text{where} \quad \sigma\_{1}' &= \sigma\_{3} + \begin{pmatrix} \frac{2\sin\eta}{(\mu\_{\text{min}}+1)\cos\theta} - \frac{\tan\eta - \mu\_{\text{min}}\sin\theta}{(\mu\_{\text{min}}+1)\cos\theta + (1-\mu\_{\text{min}})\sin\theta} \\ \frac{2\sin\eta}{1-\sin\eta} - \frac{(\mu\_{\text{min}}-1)\cos\theta + (1+\mu\_{\text{lamair}})\sin\theta}{(\mu\_{\text{min}}+1)\cos\theta + (1-\mu\_{\text{min}})\sin\theta} \end{pmatrix} \end{cases} \end{cases} \end{cases} \begin{cases} f\_{LF} &= \sigma\_{3} + \frac{\sigma\_{3}}{\sigma\_{1}} + \frac{\sigma\_{3}}{\sigma\_{1}} + \frac{\sigma\_{3}}{\sigma\_{1}} = \sigma\_{4} + \frac{\sigma\_{4}}{\sigma\_{1}} + \frac{\sigma\_{4}}{\sigma\_{1}} \end{aligned} $$

where *T0* is the tensile strength, *T0-lamina* is the tensile strength of laminated sandstone; μ is the coefficient of internal friction; *fLF* is the fracture index of lamina induced fractures.
