**1. Introduction**

Multi-scale bedding fractures, including km-scale regional bedding fractures and cm-scale lamina-induced fractures, are caused by lamina dissolution or induced by regional tectonic stress [1–10]. The formation of sedimentary lamina and bedding underground exerts a strong control on the following fracture propagation [11–13], so the term lamina-induced fractures is used to indicate the cm-scale fractures forming along the core lamina under the influence of external forces and internal rock mechanical properties [14]. Moreover, the lamina-induced fractures also correspond to the km-scale regional bedding fractures, which are induced by the formation beddings. These fractures have been the focus of conventional and unconventional oil and gas reservoir characterization including shale, tight oil [14–23], and Carboniferous rocks [24]. The fractures have also played an important role in methane gas emissions from coal seams [25–27]. It has been found that the open state of bedding fractures in complicated tectonic zones could play an essential role in oil and gas di ffusion, emission, migration, and accumulation [14,17,28–33]. In addition, the lamina or bedding in the tight reservoirs coupled with the hydraulic fractures could induce a more complicated in-situ fracture network [34]. However, the quantitative simulation work of bedding fractures, despite the qualitative description, as mentioned above, is still lacking due to the complicated stress distribution caused by the lamina heterogeneity. It is challenging to distinguish the lamina or bedding from surrounding rocks and predict the failure behaviors under the e ffects of heterogeneity.

Multiple methods and softwares have been used to conduct numerical simulations and make predictions on the energy resources, e.g., artificial neural networks, ant tracking algorithms, petrophysical logging, and microseisms [35–38]. The ant tracking algorithm can be applied to detect small faults, but it is di fficult to extract detailed formation of the fractures due to the extremely low resolution and low coherence of data [39]. Even though the microseism is often used for artificial fracture detection and modeling in the hydraulic fracturing process of the horizontal wells, its roles are confined to the oil-gas exploration and development period [40]. In addition, the accuracy of logging interpretation strongly depends on the data amount, and the resolution of seismic data is too low to conduct a precise fracture simulation. Therefore, it is necessary to find out an e fficient method to make an accurate prediction for the multi-scale fractures without the need for a grea<sup>t</sup> amount of data.

In this study, a finite element simulation of bedding fractures was conducted based on the dynamic propagation condition for the tectonic bedding fractures. The finite element simulation is widely used to predict the present geological stress and fracture index, because it can provide a platform for researchers to focus on the geological or mechanical model, with the adjustability of the rock failure criterion and boundary stress conditions [41–43]. In addition, this study focused on the modified failure criterion of tectonic bedding fractures in the dynamic simulation environment.

The most challenging work for bedding fracture simulation is to build rational failure criteria in the finite element simulation environment considering four factors, i.e., maximum stress, minimum stress, lamina angle (or the km-scale regional stratigraphic dip), and di fferences between the lamina and surrounding rock in di fferent lamina lithofacies. Although some failure criteria have been proposed for the bedding fractures based on the mechanical tests of laminated rock, as shown in Table 1 [6,44–47], a coe fficient should be additionally proposed to indicate the di fferent lamina lithofacies and the di fference between the lamina and the surrounding rock in the laminated cores or the rock formations with bedding fractures.


**Table 1.** Failure criteria for the bedding fractures proposed by di fferent researchers.

In summary, the innovations of the multi-scale fracture (km-scale regional bedding fractures and cm-scale lamina-induced fractures in cores) simulations conducted in this work are reflected in the following aspects: (1) A modified Tien–Kuo (T–K) bedding failure criterion was proposed based on the stress equilibrium equation along the lamina, where two critical parameters (the lamina angle (θ) and lamina friction coe fficient (μ*lamina*)) were proposed to precisely characterize the laminated rock. The μ*lamina* values were tested based on the triaxial compression tests of di fferent lamina lithofacies. (2) A finite element simulation was conducted based on the modified T–K criterion to study the stress distribution in the laminated rock model composed of the independent lamina and the surrounding rock bodies. (3) The regional bedding fractures distribution of the Upper Triassic Yanchang Formation in the Ordos Basin was clarified based on the regional stratigraphic dip distribution, as well as the regional laminated rock lithofacies distribution proposed in this work, and the regional stress field, which was simulated in our prior simulation work [14].

#### **2. Materials and Methods**
