# Hydraulic Fracture Propagation near the Cavity in a Poroelastic Media
## 1 Introduction  
Hydraulic fracturing (HF) is used extensively to increase production in the oil and gas industry as well as for solving various problems associated with mining, underground construction, and the management of nature. The idea behind HF, which is relatively simple to implement, has led to its widespread adoption. It has been applied in projects related to the recovery of heat from geothermal sources [1; 2; 3; 4], the storage of carbon dioxide [5; 6; 7], underground bioleaching [8; 9], and the disposal of drill cuttings and waste materials [10; 11]. This emphasizes the importance of preparatory work on planning HF operations for the formation of fractures of a given direction, including the construction and analysis of adequate mathematical models. The main research areas of the mine modification of the method are associated with improving the safety of underground operations. When mining coal seems, HF is used to increase the efficiency of the production of methane from coal by increasing the gas recovery [12; 13; 14] and creating impervious protective screens in the rock mass near mine workings [15; 16]. In the case of the extraction of coal seams by longwall mining, hydraulic fracturing is applied for the controlled roof's collapse to prevent it from hanging [17; 18; 19]. This method has many variations for in situ stress measurement based on the analysis of breakdown, shut-in, and reopening pressures [20; 21; 22]. This helps to control the integrity of the mine's walls and in planning further activity. Calculations of the paths of cracks during their propagation are critical for the design of inseam hydraulic fracturing in the underground environment. Modeling errors that occur at this stage can lead to undesirable consequences, i.e., the emergence of cracks inthe sides of the working, intersection with adjacent wells, and uncontrolled destruction of the massif. There are several factors that have significant impacts on the development of fractures. These factors include the stress-strain state of the rock mass, the condition of the borehole walls (presence of casing and perforations) and their directions, the characteristics of the loading that is being applied, i.e., the design and other features of the downhole tool, the rate of injection of the fracturing fluid, and the properties of the fracturing fluid, as well as the location of the crack relative to mine workings. Modern directions of research are related to the development and verification of the analytical and numerical models of the propagation of hydraulic fractures. Various analytical models are used, including the Kristianovic, Geertsma, and de Klerk (KDG) Model [23, 24], the Perkins, Kern and Nordgren (PKN) Model [25, 26], and some variations of these models. Despite the fact that these models are based on the assumption of a non-varying and homogeneous stress field, they are suitable for quick analyses, and they allow improvements in numerical methods and approaches. In contrast, numerical models based on various techniques, such as the boundary element method (BEM) [27], the discrete element method (DEM) [28], the finite element method (FEM) [29], and damage mechanics-based approaches [30, 31, 32] can take into account, for example, the complex geometry of hydraulic fractures under excavation stress conditions. The propagation of cracks close to hollow inclusions has been considered in several studies, some of which were related to modeling the hydraulic fracture phenomenon near underground work areas. A two-dimensional model for the HF of the roof in the stope was established to investigate the crack-growth features [33]. The numerical simulation was based on the 8203 working face of the Datong Coal Mine in China as the prototype. The results that were obtained showed that the deflection angle of the hydraulic fracture increases as the mining width increases, but it decreases as the side pressure coefficient and the fractured horizon increase.