## 1 Introduction  
In order to address the challenges and limitations associated with conventional oil and natural gas resources, it is essential to actively promote the exploration and extraction of shale gas and enhance the availability of unconventional natural gas reserves. These initiatives play a critical role in optimizing the overall energy landscape and driving reforms in energy production and consumption [1]. Shale gas, being an unconventional form of natural gas primarily found in shale reservoirs, has garnered significant attention in recent years [2]. Hydraulic fracturing has emerged as a successful technique for improving reservoir permeability and enhancing the recovery of shale gas, making it a pivotal technology for shale gas development [3]. During the hydraulic fracturing process, proppants are introduced into the fractures and become embedded within the shale formation due to their interaction with the fracture surfaces under closure pressures; this embedding mechanism results in a reduction in fracture aperture and conductivity (Figure 1) [4]. As a result, the vital role played by proppant embedment in facilitating the stimulation effect of hydraulic fracturing becomes evident. Therefore, it is crucial to thoroughly investigate the influencing factors that affect both proppant embedment and fracture conductivity through a combination of experimental research and theoretical calculations. The main factors influencing proppant embedment depth and fracture conductivity include closure pressure, the size, arrangement, Young's modulus, and Poisson's ratio of the proppant, as well as Young's modulus and Poisson's ratio of the shale [5; 6; 7; 8; 9]. Among these factors, as production time increases, the fluid pressure within the fracture gradually decreases, and the closure pressure tends to stabilize. Proppants such as quartz and ceramics, due to their stable physical and chemical properties, typically do not react with hydraulic fracturing fluid; hence, the mechanical parameters related to proppants can be considered essentially unchanged over time, so the changes in fracture aperture caused by the proppant itself primarily result from deformation under closure pressure. However, shale is an intricate, porous material with multiple phases and scales, comprised of diverse inorganic solid minerals and organic components. When it comes into contact with hydraulic fracturing fluid, it undergoes changes in microstructural properties and degradation in mechanical characteristics such as elasticity, hardness, and strength. This process ultimately softens the shale, leading to reduced fracture conductivity and diminished shale gas production [10; 11]. Several previous studies have investigated the shale softening behaviors induced by fracturing fluid. [PERSON] et al. (2014) established a linear correlation between swelling strain and water content by examining the shale's wettability and water adsorption capacity [12]. [PERSON] et al. (2016) linked the decrease in fracture conductivity to the presence of clay content [13]. [PERSON] et al. (2020) examined the fracture mechanical properties of Longmaxi shale samples under different environmental conditions, concluding that clay mineral hydration and expansion were the primary factors contributing to fluid weakening effects. Moreover, the impact of interactions between solid minerals and fracturing fluids on shale microstructure (such as pores) and permeability has been discussed [14]. [PERSON] et al. (2018) noted expansion in the electrical double layer (EDL) content and composition, emphasizing that clay swelling within the hydraulic fracture network is a significant factor contributing to production decline [15]. [PERSON] and [PERSON] (2017) suggested that acid fracturing fluids underwent chemical reactions with carbonate-rich shale, resulting in alterations to the micropore structure within the shale [16].