In this paper, a model is presented for modeling saturated granular-liquid free-surface flows, in which the volume-averaged mixture bulk velocity is employed to derive the balance equations for the mass and momentum of mixture flow. Additionally, an evolution equation of the slip velocity between granular-and liquid constituents is derived to describe the separation between these constituents. The frictional-collisional constitutive relation for granular-constituent is employed to determine the stress due to particles interaction. The governing equations for mixture flows are numerically solved by a finite difference two-step projection method. The volume of fluid (VOF) method is employed to track the free surface of the mixture flow in the present numerical model. Good agreements between numerical results and experimental data are observed by modeling the dam-break process of granular-liquid mixture flow, dam-break waves over the saturated erodible beds and surge waves induced by submarine landslides along an inclined plane. Furthermore, the difference between the volume-averaged mixture bulk velocity and mass-averaged mixture bulk velocity is found to vary as the instinct density ratio of granular-constituent and liquid-constituent and the volumetric concentration ns of the granular-constituent, and the evolution in the slip velocity during the process of the settlement of sediments is numerically analyzed. In this study, we developed a mathematical model for saturated granular-liquid free-surface flows, in which the volume-averaged mixture bulk velocity is employed to describe the balance equations for the mass and momentum of the mixture flow of granular-liquid flows and the evolution equation of the slip velocity between granular-constituent and liquid-constituent is derived to describe the separation between the granular-constituent and liquid-constituent. The stress due to the interaction of particles is determined based on the frictional-collisional constitutive relations. The finite difference two-step projection method is employed to numerically solve the governing equations and the volume of fluid (VOF) method is employed to track the free surface of the mixture flow in the present numerical model. Good agreements between numerical results and experimental data are observed. Finally, the role of slip velocity on the dynamics of granular-liquid flows is analyzed. A novel mathematical model for saturated granular-liquid free-surface flows is presented Good agreements between the numerical results and experimental data are observed The evolution of slip velocity plays a pivotal role in the dynamics of granular-liquid flows