Abstract: The bed-boundary condition is an important problem in non-equilibrium sediment transport.In this study,this challenge is addressed by first transforming spatial variables into time variables in the context of unbalanced sediment transport differential equations.Building upon the Muskingum method,a Muskingum-type bed-boundary condition is proposed and the saturation recovery coefficient is analyzed.This innovative approach creates a basis for the establishment of non-equilibrium sediment transport models.The saturation recovery coefficient is found to be a product of two main factors:The ratio of bottom sediment carrying capacity to depth-averaged sediment carrying capacity,and a weighting coefficient on the bottom sediment concentration by the bottom special sediment concentration and bottom sediment carrying capacity.The latter is an essential parameter that interacts with the bottom sediment concentration,impacting the vertical sediment concentration distribution.The bottom special sediment concentration,representing a balanced vertical sediment distribution,further contributes to this complex equation.Applying the proposed model yields promising results.Simulations closely match the measured values of depth-averaged sediment concentration,bottom sediment concentration,and vertical sediment concentration distribution,preliminarily proving that the proposed method is feasible.Influence analysis on the weighting coefficient of the Muskingum-type bed-boundary condition indicates that it critically affects the saturation recovery coefficient and vertical sediment concentration distribution.This influential parameter warrants in depth exploration in further studies.