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Gravity-driven size segregation is important in mountain streams where a wide range of grain sizes are transported as bedload. More particularly, vertical size segregation is a multi-scale process that originates in interactions at the scale of particles with important morphological consequences on the reach scale. To address this issue, a volume-averaged multi-phase flow model for immersed bidisperse granular flows was developed based on an interparticle segregation force (Guillard et al. 2016) and a granular Stokesian drag force (Tripathi and Khakhar 2013). An advection-diffusion model was derived from this model yielding parametrisations for the advection and diffusion coefficients based on the interparticle interactions. This approach makes it possible to bridge the gap between grain-scale physics and continuum modelling. Both models were successfully tested against existing Discrete Element Model (DEM) simulations of size segregation in bedload transport (Chassagne et al. 2020). Through a detailed investigation of the granular forces, it is demonstrated that the observed scaling of the advection and diffusion coefficients with the inertial number can be explained by the granular drag force dependency on the viscosity. The drag coefficient was shown to be linearly dependent on the small particle concentration. The scaling relationship of the segregation force with the friction coefficient is confirmed and additional non-trivial dependencies including the inertial number and small particle concentration are identified. Lastly, adding a size ratio dependency in the segregation force perfectly reproduces the DEM results for a large range of small particle concentrations and size-ratios.