## 1 Introduction  
With remarkable economic growth, coal consumption in China accounted for over 50 percent of the country's total energy consumption in 2022 [1]. However, in coal mining, there can be a series of gas accidents, such as gas explosions and coal gas outbursts. These accidents are mainly due to inherent characteristics (such as large coal reservoir stress, low permeability, and high gas content) of coalbeds in China [2]. Moreover, coalbed methane is a highly valuable and clean energy source. Therefore, degassing coal seams could be a double-win strategy for coal mines' safety and fulfilling energy consumption [3]. The permeability of coal seams can be improved by fracturing coal, which can increase gas flow and enhance the effectiveness of gas drainage [4]. Currently, hydraulic fracturing [5], hydraulic slotting [6], and loose blasting [7] are the main technologies used to increase the permeability of coal seams. These measures can improve gas recovery from coal seams, but they all have limitations. In recent years, the physical field excitation method (such as heat injection, acoustic vibration, and electromagnetic field excitation) has the object of many studies. Formation microwave heating treatment (FMHT) has been proposed to artificially fracture coals and improve gas extraction efficiency. Compared to traditional treatment methods, the interaction between coal and microwaves has the advantages of high efficiency, low energy consumption, and time savings [8]. Microwave energy can selectively heat receptive components (such as pyrite and water) in the coal matrix, resulting in differential thermal stresses, inducing fractures within the coal matrix, and then improving coalbed permeability. A systemic understanding of microwave energy on the petrophysical characterization of coal is essential for FMHT's application in the field. [PERSON] et al. [9] found that coal samples dehydrate and shrink after microwave irradiation, making them more brittle and easier to fracture. [PERSON] et al. [10] found that low-rank sub-bituminous coal demonstrated more extensive fracturing after microwave radiation according to scanning electron microscopy. [PERSON] et al. [11] found that the crack propagation of coal cores after microwave irradiation depends on the receptive component distribution and original crack distribution, using camera images and the ultrasonic method. [PERSON] et al. [12; 13] found that the initial cracks in the coal extended and gave rise to secondary cracks after microwave radiation by scanning electron microscopy. In addition, the microwave fracturing effect on dried coal/pyrite mixtures was studied by camera image, leading to the finding that fracturing enhances as the pyrite proportion increases. [PERSON] et al. [14] discovered that moisture within the coal was rapidly heated and gasified by microwave energy. This resulted in volume expansion, which caused the extension of pre-existing fissures and the emergence of new fractures. [PERSON] et al. [15] validated the effect of isotropic stress on cleat frequency and the distribution of coal cores after microwave irradiation by micro-focused X-ray computed tomography. [PERSON] et al. [16] discovered that the comminution of coal samples increased with microwave power and finally tended towards an upper limit. The above research has confirmed that the pores and fractures of coal develop significantly under microwave radiation. However, the variation in the spatiotemporal electromagnetic field and crack propagation inside the coal cores during microwave radiation are difficult to accurately measure in laboratory experiments. Numerical simulations have been used to study the effect of microwave radiation on rock cores. Discrete element-based models can simulate the initiation and propagation of cracks, but they are not good at calculating the distribution of the electromagnetic field [17; 18; 19; 20; 21; 22; 23]. Therefore, the finite-element software COMSOL 6.0 has been used to study rock fracturing induced by microwave radiation. [PERSON] et al.