## 2 Materials and Methods
### Methods  
Numerous testing methods and apparatuses have been designed for determining the soil resistance of debris flow materials in the range from microscale to macroscale measuring systems (e.g., plate-plate, cone-plate, cylinder, vane, stirrer, slump, and large-scale devices). Figure 1 shows the rheometer used in this study. The commercial ball-measuring rheological system (BMS, RheolabQC, Anton Paar) consists of three balls and a container. The balls have three diameters, i.e., 10, 12, and 15 mm, whilst the container (cup) has a diameter of 115 mm with a sample volume of 500 cm\({}^{3}\). In this system, a homogenous soil sample, the mixture of soil and water, is placed into the container and the ball is immersed into the soil sample. The depth of ball penetration is the same for all tests. The distance between the ball and the bottom of the container is 20 mm. Torques are automatically recorded when rotating the ball. The shear rate control or shear stress control mode can be applied. The measurement of shear stress and the shear rate in a ball-measuring system can be performed for 1-5 mm diameter soil samples. Similar tests for large-particle fluids using a ball-measuring system have been performed by [19; 25; 26]. The edge and slip effects are assumed to be negligible. Potential slipping and gliding problems are negligible because of the ideal spherical shape used to minimize the edge effect during shearing. The system displays a wide gap between the sensor and soil sample container. In this context, it is applicable for both fine-grained and coarse-grained sediments in debris flow materials. However, there are some limitations in performing such experiments. The normal pressure effect is not taken into account. The temperature is fixed at 20 \({}^{\circ}\)C during shearing. However, it is expected that the variation in temperature is one of minor effects in the rheological behavior of sediments with large particle sizes. Two simple test methods were applied to satisfy the purpose of the study. Two types of rheological tests were performed: a flow curve test and a stress growth test. The first test was conducted to determine the dynamic yield stress (e.g., the Bingham yield stress and viscosity), and the second test was conducted to determine the static yield stress (e.g., the peak and residual shear stresses). As a result, the flow curves and stress growth methods could produce shear stress vs. shear rate and shear stress vs. shearing time relationships, respectively. To determine the yield strength of waste rock materials, the rheological models considered are as follows:  
\[\text{Bingham}:\ \tau=\tau_{c}+\mu_{h}\dot{\gamma}\dot{\ } \tag{1}\]  
\[\text{Bilinear}:\ \tau=\tau_{y}+\mu_{h}\dot{\gamma}^{n}+\left(\frac{\tau_{c} \dot{\gamma}}{\dot{\gamma}+\dot{\gamma}_{o}}\right) \tag{2}\]  
\[\text{Modified Bingham}:\ \tau=\left[\mu+\tau_{c}\frac{1-\exp(-m\dot{\gamma})}{ \dot{\gamma}}\right]\dot{\gamma} \tag{3}\]  
\[\text{Herschel-Bulkley}:\ \tau=\tau_{c}+K\dot{\gamma}^{n} \tag{4}\]  
\[\text{Power law}:\ \tau=\mu\dot{\gamma}^{n}, \tag{5}\]  
where \(\tau\) is the shear stress (Pa), \(\mu\) is the viscosity (Pa-s), and \(\dot{\gamma}\) is the shear rate (s\({}^{-1}\)). In Equation (1), \(\tau_{c}\) is the yield stress (Pa), which is called the Bingham yield stress, and \(\mu_{h}\) is the plastic viscosity (Pa-s). In Equation (2), \(\tau_{y}\) is the yield stress (Pa) obtained from the bilinear model, which is similar to the yield stress obtained from the Bingham fluid; the exponent \(n\) is the flow behavior index (dimensionless); and \(\dot{\gamma}_{o}\) is the shear rate related to the rheological transition from a Newtonian viscosity to an ideal plastic viscosity. The exponent \(n\) is also used in the Herschel-Bulkley and power law models to describe the flow behavior, e.g., if \(n<1\), pseudoplastic (shear thinning) behavior occurs, and if \(n>1\), dilatant (shear thickening) behavior occurs. Equation (3) is the modified Bingham model.