## 2 Materials and Methods
### Experimental Method  
The relationship between stress and strain for debris flow is referred to as the rheological relationship. Hence, the parameters in the rheological relationship are known as the rheological parameters and mainly include the viscosity coefficient and the yield stress [22]. The viscosity coefficient is a fundamental physical index that characterizes the frictional properties of fluids, reflecting the ability of a fluid to resist deformation when it undergoes relative motion. Its generalized model is as follows: fluid layers are divided into many elemental layers which are parallel to the flow direction, and due to varying velocities between adjacent layers, there exists resistance to movement opposite to the flow direction, defined as viscosity. The value surpassed when an object undergoes deformation beyond its elastic limit is termed the yield value. In rheology, only when stress reaches a certain value does fluidity occur; this is known as the critical stress.  
As shown in Figure 3, the experimental equipment for this study was an Anton Paar MCR 52 intelligent high-speed rheometer (made in Austria), equipped with an automatic data acquisition system for real-time test data reading. The temperature range was controlled by Peltier elements from 5 \({}^{\circ}\)C to 70 \({}^{\circ}\)C. The air bearing contained an embedded normal force sensor, accurately determining the interface position. The rheometer allowed a maximum particle size of 10 mm for test samples and a shear rate measurement range of 0.1 s\({}^{-1}\) \(<\) \(\gamma\) \(<\) 100 s\({}^{-1}\). The cylindrical loading container was 115 mm in diameter, 48 mm in height, and 500 mL in volume. After loading the sample, the instrument was activated; pressure was outputted by an air compressor until the semiconductor refrigeration cycle controlled the temperature at a constant 20 \({}^{\circ}\)C. The rotor was controlled by software; it was gradually moved to the center of the debris flow slurry mixture, and its rotational shear function was rapidly activated to shear the debris flow slurry mixture, generating data for the rheological curve. This process allowed for obtaining the relationship curve between shear stress and shear rate of the Jiangjiagou debris flow slurry at different volume concentrations (\(C_{\rm v}\)). The shear rate gradually increased from 0 s\({}^{-1}\) to 30 s\({}^{-1}\), which typically corresponds to the real shear rate range for debris flow under natural conditions. The loading mode was set as uniform acceleration loading, i.e., linear loading. The total test time was 213.5 s, with a sampling interval of 3.5 s per data point, totaling 61 data points. In total, nine samples were tested, from a low \(C_{\rm v}\) value to a high \(C_{\rm v}\) value. This aimed to ensure the regularity of the sampled data points and indirectly monitor the acceleration stability of the rheometer based on the shear rate at different sampling points.  
Numerous studies have indicated that the majority of debris flow fluids can be described using the Bingham fluid, which combines the rheological equations of the Saint-Venant plastic body and the Newtonian viscous body, as shown in Equation (2)  
\[\tau=\tau_{\rm s}+\eta\gamma \tag{2}\]where \(\tau\) represents the shear stress (Pa), \(\tau_{\rm s}\) is the yield stress (Pa), \(\gamma\) stands for the shear rate gradient (s\({}^{-1}\)), \(\eta\) denotes the viscosity coefficient (Pa-s), and \(1/\eta\) is the flow coefficient, reflecting the strength of the flow. Thus, a larger viscosity coefficient indicates poorer fluidity, while a smaller coefficient signifies higher fluidity.