## 1 Introduction  
As the main seepage channels of fluids, fracture spaces of rock masses are found widely in rock mass engineering, such as dam foundations, slopes, underground caverns, etc. In the process of underground engineering, many rock hydraulics problems are related to the rock fracture pathway. Under the interaction of artificial engineering disturbance stress, in situ stress, and groundwater seepage pressure, the seepage behavior of groundwater in rock mass fractures affects the stability of fractured rock masses in engineering activities [1,2]. The safety of human engineering activities and the adverse effects of the geological environment on engineering activities are also often affected by the seepage behavior of fluids in rock fractures [3]. Therefore, evaluating and predicting the impact of fluid seepage behavior in rock fractures on the stability and safety of human engineering activities is a crucial new topic in hydrogeology and engineering geology at present and will be in the future. In the seepage of a fractured rock mass, the famous cubic law has been widely used in theoretical research and engineering practice. When the fracture surface is parallel and smooth, the cubic law describes the cubic relationship between the seepage flow and the fracture aperture. However, in practical engineering, natural rock fractures are rough, making it difficult for the fluid seepage to correspond to the assumptions of the cubic law. Therefore, many scholars have conducted seepage tests on artificial rough fractures and found that the parallel plate model hypothesis is no longer applicable in describing the seepage behavior of fluids due to the rough and uneven surface of fractures [4]. In the literature about fluid seepage behavior, the roughness of fractures represents the rough and uneven surface of fractures. Many experimental and numerical studies have confirmed the influence of roughness on fluid seepage behavior [5; 6]. [PERSON] et al. proposed a seepage model considering the roughness of fractures and found that the roughness of fractures is a vital cause of fluid seepage flow reduction and nonlinear seepage [7]. [PERSON] et al. revised the cubic law by introducing relative roughness to accurately determine the seepage behavior of a fluid in rough fractures. Based on the modified cubic law, many experimental studies were carried out in fractured channels. The results showed that the seepage behavior of fluids in fracture channels with different roughness differs from that described by the cubic law because of the influence of fracture roughness [8]. [PERSON] et al. found that the roughness of the fracture causes the weak inertia effect, causing the fluid seepage flow to decrease with the increase in fracture roughness [9]. It can be observed in the above mentioned studies that the roughness of the fracture is one of the most significant factors affecting the fluid seepage behavior and seepage flow reduction in rough fractures. In rough fractures, the fluid seepage path is usually tortuous, which easily affects the seepage behavior of the fluid [10]. The tortuous fluid seepage path is generally expressed by the tortuosity of the fracture [11; 12; 13]. The tortuosity of the fracture can cause additional seepage flow loss, which causes the fluid seepage behavior to deviate from the linear Darcy law [14]. [PERSON] et al. studied the effect of fracture tortuosity on the seepage flow in the seepage channel. They believed that the actual seepage flow corresponds to the ratio of the seepage flow calculated by means of the linear Darcy law to the tortuosity [15]. [PERSON] and [PERSON] suggested that the increase in fracture tortuosity is the reason for the decrease in fluid seepage flow. The results showed that the fluid flowing into the fracture is divided into several seepage channels according to the differing tortuosity of the fracture [16]. These seepage channels dispersed the fluid flowing into the fracture and resulted in a decrease in seepage flow.