**4. Discussion**

#### *4.1. Comparisons of Di*ff*erent Failure Criteria For Bedding Fractures*

To verify the modified T–K criterion, the testing data were compared to the calculated data using three criteria, i.e., the modified T–K criterion in this study, Zhou et al.'s modified T–K criterion, and the Jaeger criterion (Table 1). Figure 2 shows the triaxial compression test curves in the confining pressures of 10 and 20 MPa, where the samples are the artificial laminated rock in Zhang et al. (2017) (Figure 2a,b) and the naturally laminated rock in Zhou et al. (2017) (Figure 2c,d) [47,59]. The detailed criterion functions are listed in Table 1, and the detailed values of each criterion's parameters are listed in Table 2. Detailed peak strength values are listed in Table 3, where samples a-0 to b-90 are achieved

from Zhang et al. (2017) (Figure 2a,b), samples of c-0 to d-90 are referenced from Zhou et al. (2017) (Figure 2c,d), and samples of '1-1' to '6-3' are tested in this work, as shown in Figure 3. The lamina angle in this work was the complementary angle of the β angle in Zhou et al. (2017) [47,59].

**Figure 2.** Stress–strain curves of laminated rock tested in previous works. (**a**) Curves tested in Zhang et al. (2017) [59]; samples are artificial laminated cores tested at 10 MPa confining pressure. (**b**) Curves tested in Zhang et al. (2017); samples are artificial laminated cores tested at 20 MPa confining pressure. (**c**) Curves tested in Zhou et al. (2017) [47]. The peak strength values of these curves are also used to verify the modified Tien–Kuo (T–K) criterion, and the samples are the naturally laminated cores tested at 10 MPa confining pressure. (**d**) Curves tested in Zhou et al. (2017). The peak strength values of these curves are also used to verify the modified T–K criterion, and samples are the naturally laminated cores tested at 20 MPa confining pressure.


**Table 2.** Parameters used to fit the function curves of the failure criteria.

\*: The lamina cohesion in the Jaeger criterion; \*\*: The internal friction angle in the Jaeger criterion.

**Table 3.** Peak strength values of the stress–strain curves in Figures 2 and 4.



**Table 3.** *Cont.*

As shown in Figure 2, there was a much lower peak strength for those samples with the lamina angle of 30◦. Notably, black square points in Figure 3 indicate the compression peak strength values tested with the 20 MPa confining pressure in Zhou et al. (2017) (Figure 2d) [47]. Black round points in Figure 3 indicate the compressive peak strength values tested with the 10 MPa confining pressure in Zhou et al. (2017) (Figure 2c) [47]. In Figure 3, function curve 1 indicates the peak strength values fitted in Zhou et al. (2017)'s modified T–K criterion with a 10 MPa confining pressure [47]. Function curve 2 indicates the peak strength values fitted in Zhou et al. (2017)'s modified T–K criterion with a 20 MPa confining pressure [47]. Function curves 3 and 4 indicate the peak strength values fitted in the Jaeger failure criterion with 10 and 20 MPa confining pressures, respectively [44]. In addition, function curves 5 and 6 indicate the peak strength values fitted in our modified T–K criterion with 10 and 20 MPa confining pressures, respectively.

Generally, the modified T–K failure criterion has almost the same trend as the T–K criterion and even agrees better with the testing data in Zhang et al. (2017), as shown in Figure 3, especially for the medium-low-angle range (nearly 20◦–50◦) [59]. Moreover, the four variables in this criterion are in accordance with the ANSYS parameter environment and could provide us with a better failure criterion in the finite element simulation.

#### *4.2. Lamina Friction Coe*ffi*cients of Di*ff*erent Lamina Lithofacies*

It is necessary to use the lamina friction coefficient (μ*lamina*) to indicate the regional heterogeneity of lamina lithofacies in the geo-simulation work. In order to test the lamina friction coefficient (μ*lamina*) of the different lamina rock types, we tested 18 column samples with different lamina angles from six different laminated cores in triaxial compression experiments. Triaxial compression stress–strain curves are shown in Figure 4. Peak strength values obviously varied between the samples with the different lamina angles, except Core-1, which was similar to the intact core without lamina. The peak strength values were compared to the calculated values using the modified T–K criterion (Figure 5). Black points with the different shapes indicate the peak strength values of the six cores shown in Figures 4 and 5. Colored dotted lines indicate the function curve of the modified T–K criterion, where the different colors indicate the calculated data in different lamina friction coefficients (μ*lamina*). The values of the lamina friction coefficient (μ*lamina*) are 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1. In addition, the value of ϕ is 58◦, σ3 is 20 MPa, *n* is 2.20, *k* is 0.9, and *Clamina* is 20 MPa.

**Figure 4.** Stress–strain curves of the six laminated core samples with different lamina angles at a 20 MPa confining pressure. (**a**) Core-1; (**b**) Core-2; (**c**) Core-3; (**d**) Core-4; (**e**) Core-5; (**f**) Core-6. Blue lines indicate those samples in a high lamina angle range, cyan lines indicate those samples in a low lamina angle range, and red lines indicate those samples in a medium lamina angle range.

As indicated in Figure 5, the following results could be drawn. (1) Different lamina friction coefficients (μ*lamina*) indicate different laminated core types. The cores more strongly characterized by the obvious lamina, such as Core-4, Core-5, and Core-6, are typically correlated with a lower μ*lamina*. (2) The fitting data agreed well with the testing data, except Core-1, which is characterized by the inconspicuous and micro-cross-bedding lamina. Here, the strength values of Core-1 are nearly located in the same range due to the weak influence of these inconspicuous and micro-cross-bedding lamina within Core-1. In other words, the smaller the difference between the lamina and surrounding rock, the weaker the influence of the lamina in different angle ranges. Furthermore, the values of μ*lamina* of Core-4 and Core-5 are around 0.5, which are the laminated siltstone deposited by traction currents in the delta front sedimentary environment. The value of μ*lamina* of Core-6 is around 0.3, which is the laminated siltstone sedimented in the lacustrine turbidite environment. The value of μ*lamina* of Core-3 is around 0.9, which is the laminated mudstone sedimented in a still water environment. From the perspective of the regional finite element simulation, the μ*lamina*, as a critical variable, should be assigned differently according to the laminated rock types deposited in the different sedimentary environments during Yanchang Formation in the Ordos Basin.

**Figure 5.** Function curves calculated with the modified T–K criterion using different lamina coefficients (μ*lamina*). Black points in different shapes indicate the different laminated rocks. Notably, peak strength values of Core-1 (with inconspicuous lamina) in the red points do not agree with the calculated function curves.

#### *4.3. Stress Distribution in the Laminated Rock*

The laminated core model (cm-scale) in the finite element simulation is composed of two bodies, as shown in Figure 6, where the contact part between two bodies is settled as frictional with a sliding friction coefficient in accordance with the lamina friction coefficient in the failure criteria of this study. The failure behavior of the laminated rock in the special lamina angle range is determined by the stress distribution constrained by the constant value of μ*lamina*. Therefore, it is necessary to study the stress distribution of the laminated rock using the finite element simulation technology, especially for the different lamina angle ranges, i.e., the low, medium, and high lamina angle ranges.

Researchers have tried to simulate the compression behavior of tight intact rock and the laminated rock using finite element simulation technology [60,61]. To better understand the stress distribution during the compression process, the failure distribution around the compression samples was simulated by using the modified criterion in the ANSYS environment. The simulated samples were divided into two parts: The surrounding rock and the lamina body, with the lamina surface acting as the sliding surface. Additionally, the mechanical parameters of the surrounding rock used in the simulation were different from those of the laminated rock. Here, the surrounding rock was set as siltstone, while the laminated rock was claystone in the ANSYS software. In addition, the contact relationship, i.e., the lamina sliding surface, was set as frictional in the ANSYS software with the fractional coefficient value of 0.2. Another critical parameter, the lamina angle, was designed in the computer model, as shown in Figure 6a,c,e.

Accordingly, the failure criterion was defined according to the modified T–K criterion in the ANSYS software. Figure 6a,c,e show the simulated results, in contrast to the CT (computerized tomography) results of fractured samples (Figure 6b,d,f) with the low, medium, and high lamina angle ranges, respectively.

By contrast, for the high lamina angle (Figure 6a,b) outside the effective lamina angle range, stress was prone to concentrate on the surrounding rock body compared to the samples simulated in the medium and low lamina angle ranges (Figure 6c,e). Therefore, the typical X-conjugate fractures were prone to propagate without associated bedding fractures. Notably, the effective lamina angle indicates that those compressed fractures were generally induced by the lamina body on which the stress was primarily concentrated.

**Figure 6.** Finite element simulation results of the bedding fractures (i.e., lamina-induced fractures) and the computerized tomography (CT) photos of the fractured laminated samples. The color code for the finite element simulation results of (**<sup>a</sup>**,**c**,**<sup>e</sup>**) indicates the fractured index based on the modified T–K criterion. (**b**,**d**,**f**) show the CT photos of fractured samples, and the darker color indicates a bigger fracture aperture.

For the medium lamina angle in the effective lamina angle range, a shear fracture propagated between the two bedding fractures, as shown in the CT photo (Figure 6c,d), and was accompanied by some shear fractures between the laminated bodies. The fracture aperture was smaller than that in the core with a small lamina angle and was characterized by the shear sliding distance, as shown in Figure 6d. In addition, stress was prone to concentrate around the rock body rather than the sample simulated at a low lamina angle, which could be indicated by the CT photos (Figure 6a,c,d).

For the low lamina angle (Figure 6e,f) in the effective lamina angle range, the CT photo showed a bigger fracture aperture in the darker color. In addition, the simulated fracture index of the lamina was much higher than that of the surrounding rock, indicating a stronger propagation tendency for the bedding fractures caused by the stress concentration on the lamina.

Overall, stress in these compression samples was prone to concentrate on the lamina when the lamina angle was in the effective lamina angle (low- and medium-angle range). The low-angle lamina always induces fractures in an extensively open state with a bigger failure aperture, and the medium-angle lamina always induces fractures in a shear sliding trend. On the contrary, stress was prone to concentrate on the surrounding rock body when the lamina angle was outside of the effective range, and contributed less to the compressed bedding fractures in the laminated rock.

#### *4.4. Distribution of the Regional Bedding Fractures*

In order to simulate the regional bedding fractures generated in a particular tectonic period using the modified criteria, previous simulation studies of the conventional structural fractures at the regional scale are used as the tectonic stress field [14,19,20,62]. Thus, the bedding fractures of the Yanchang Formation in the Ordos Basin generated during the latest orogenic episode (i.e., Himalayan episode) have been simulated by ameliorating Li et al. (2018) [14]. Specifically, as a result, the bedding fractures were calculated (Figure 7d) based on the improved rock failure criteria for bedding fractures (Equation (10)), the stress field simulated in Li et al. (2018) (Figure 7a,b) [14], the stratigraphic dip distribution map (Figure 7c), and the lamina fraction coe fficient in di fferent lithofacies (Figure 7d). Here, the stratigraphic dip at the regional scale (i.e., the intersection angle between the horizontal direction and the dip direction at the regional scale) is equal to the lamina angle at the cm-scale, because the maximum principal stress is in the horizontal direction during the orogenic periods, where the stratigraphic dips of the Yanchang Formation are valued in the relatively e ffective range for the bedding fracture failure criteria, as mentioned before (Figure 7c). In addition, μ*lamina* is assigned di fferent values according to the di fferent fitting results in Figure 5. The laminated siltstone sedimented in a delta front environment (Figures 5 and 7d) is assigned 0.5, the laminated turbidite sandstone sedimented in a lacustrine turbidite environment (Figures 5 and 7d) is assigned 0.3, and the mud sedimented in a still water environment (Figures 5 and 7d) is assigned 0.9.

The finite element simulation environment, including the boundary condition, Himalayan tensile stress field, geological model, and petrophysical model, remained consistent with those in Li et al. (2018) [14]. The bedding fracture index in the Yanchang Formation was calculated based on four variables, i.e., the maximum principal stress (σ1), minimum principal stress (σ3), lamina angle (θ), and lamina friction coe fficient (μ*lamina*). The calculated bedding fracture distribution, as shown in Figure 8, is broadly consistent with the structural fractured regions simulated by Li et al. (2018) [14], such as the typical oilfield regions of "Dingbian," "Xin'anbian," and "Zhiluo" (Figure 7c). The di fference is that the simulated bedding fractures spread over an even larger area, which reveals that bedding fractures are more easily induced than the conventional structural fractures (i.e., the tensile, shear, and hybrid fractures) under the same stress condition. For the Himalayan tensile stress field of the Yanchang Formation in the Ordos Basin, it is obvious that the bedding fractures tend to propagate under the conditions of lower maximum principal stress, higher minimum principal stress, and higher stratigraphic dip values (Figure 7a–d). Generally, the bedding fractures distribution simulated in this work (Figure 7d) would be a supplement for the structural fractures simulated by Li et al. (2018) [14], and both of the two research works contribute to the further study of the natural fracture network underground (including the bedding fractures and the conventional structural fractures) for the Yanchang Formation of the Ordos Basin.

**Figure 7.** Calculation parameters used in the bedding fractures simulation. (**<sup>a</sup>**,**b**) Himalayan stress field during the Yanchang Formation simulated in Li et al. (2018) [14]; (**c**) stratigraphic dip map of tight reservoirs of Yanchang Formation; (**d**) lamina lithofacies distribution of tight reservoirs of Yanchang Formation.

**Figure 8.** Bedding fractures distribution of tight sandstone reservoirs of Yanchang Formation in Ordos Basin during the Himalayan episode.
