Debris flow blocking river is a common mountain disaster chain, and however, there is a scarcity of quantitative approaches for assessing this particular disaster chain. To tackle this issue, we have developed a mathematical model using the framework of depth-averaged theory and its associated computational method. The model effectively captures the multistage process of debris flow blocking river at a catchment scale. It encompasses the dynamics of runoff, debris flow, and the river, ensuring the transfer of mass and momentum throughout the entire chain. To facilitate a more intuitive transition between the various secondary induced disasters associated with debris flow blocking river, two additional state variables are introduced. The presented computing method solves the model equations by integrating an HLLC Riemann solver into a second-order accurate finite volume method. To validate the effectiveness of this approach, two laboratory experiments and the 2020 Meilong debris flow blocking river event are simulated, and the obtained results are consistent with the available data. Moreover, this approach is employed to estimate the impact of scission on the chain process of debris flow blocking river. The simulated results showcase whether the transition between the various sub-disasters can successfully transpire under the influence of chain scission. This study can provide a basis for quantitatively assessing the chain process of debris flow blocking river as well as finding the optimization scheme to prevent this disaster chain.