In monsoon Asia, mud-type debris flows are of great concern since they are triggered by seasonal heavy rainfall characteristic of local climates. To capture the essential dynamics of such mud-type rainfall-induced debris flows, two-dimensional depth-averaged models are often considered effective alternatives to fully three-dimensional analyses. For this purpose, shallow water equation (SWE) based simulation is a common strategy due to its computational efficiency. While traditional SWE assume a negligible bottom slope and Newtonian fluid (pure water), debris flows 1) occur in mountainous regions where topography is steep and irregular, and 2) consist of a mixture of water and soil, exhibiting transitional behaviour from a static to a fluid state when yield strength is exceeded. Therefore, debris flow simulations using SWE must address 1) gravitational effects on steep slopes and 2) non-Newtonian, strain rate dependent behaviour. These challenges require an appropriate formulation of the momentum equations. In this study, we combine Steep Slope SWE (SSSWE) [1] with a strain-rate-dependent resistance model to represent mud-type debris flows in mountainous regions. The employed SSSWE defines the vertical depth as the vertical distance from the bottom to the flow surface, while ensuring the conservation of momentum and mass in slope-aligned coordinates. For the modified momentum equations, a pseudoplastic-type shear resistance model is used, where a strain-rate-dependent softening expression for the internal friction angle is introduced for the transitions from the peak stress value to a residual value. To assess the potential of the proposed model, various verification analyses are carried out. For examining its practical applicability, the qualitative comparison with the experiments is also conducted in terms of its flow paths and impact forces. The analysis demonstrated the potential of the present framework for predicting flow paths and impact forces, which could be useful for estimating damage in future mudslide disasters.