# A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network
## 1 Introduction  
In the past decade, oil production from unconventional subsurface resources such as ultra-tight organic-rich shale has increased significantly [1]. According to data released by the IEA, unconventional oil will account for more than 30% of total oil production by 2030 [2]. In unconventional oil production, the stored oil flows through the natural and hydraulic fractures formed in the rock matrix to reach the production wells [3]. Therefore, understanding the physical process of oil migration in fractures is a critical objective. However, because the mechanical behavior of a fractured rock matrix shows obvious heterogeneity and anisotropy [4, 5], it is difficult to scale the structure of the fractured rock matrix that the oil migrates through. Moreover, the complexity of the fracture network is a huge challenge for predicting the interaction between the liquid flow and the surrounding rock. Depending on their geological origin, natural fractures can consist of structural fractures formed under tectonic stress, non-structural fractures caused by factors such as dehydration and weathering, and organic-matter-related fractures formed due to abnormal hydrocarbon generation and the thermal evolution of organic matter [6]. [PERSON] et al. identified 2235 fractures in shale core samples, of which structural fractures accounted for 77.5% [7]. [PERSON] et al. analyzed the fractures and faults of the Jurassic Entrada Formation in the northernmost part of Paradox Basin located at the northwestern edge of the Colorado Plateau in central Utah. And identified structural fracture corridors propagating in an approximately vertical direction from the bottom to the top [8], as shown in Figure 0(a). [PERSON] et al. further demonstrated the vertical propagating behavior of a structural fracture based on the characteristics of the Kongdong fault section in the Wangguantum area [9]. Using seismograms, [PERSON] et al. predicted that structural fractures may also be the main fracture type in sedimentary basins [10; 11]. As shown in Figure 1b, structural fractures disrupt the sealing sequence of a rock matrix and allow vertical or sub-vertical fluid migration in the rock matrix. In recent years, three main types of numerical models have been developed to describe the hydrocarbon migration process in a fracture network: equivalent continuum models, discrete fracture models, and hybrid models. The equivalent continuum model (ECM) employs the continuum method to capture the rock mass seepage based on the permeability coefficient tensor, assuming that the fractures have a sufficient number of random occurrences and are connected to each other [12]. [PERSON] et al. [13] and [PERSON] et al. [14] simulated a fracture network using logarithmic mesh encryption technology combined with the double porosity model to describe the changes of pressure and saturation near the fracture better. Therefore, the flow pattern in the complex fracture network of a shale oil reservoir can be characterized more accurately. The discrete fracture model (DFM) determines the evolution of the migrating fluid in each fracture by solving the mass conservation equation associated with the suitable boundary condition [15]. [PERSON] and [PERSON] [16] established an embedded discrete fracture model (EDFM) to capture the interaction between the fluid and the surrounding rock matrix efficiently. Both the DFM and EDFM require a clear understanding of the spatial distribution of each fracture. They have higher accuracy than the ECM and can accurately describe the occurrence and distribution of fractures. However, their complexity leads to long simulation times and meshing difficulties in poor-quality meshing. The hybrid model is a combination of the ECM and DFM and is the latest technology. It can simulate continuously distributed micro-fractures in the ECM part while capturing explicit fractures using the DFM. It is expected to satisfy calculation accuracy requirements while saving computing resources [17]. For example, [PERSON] et al.