In this paper a numerical modeling formulation is presented for simulation of the development of the longitudinal profile and bed sediment distribution in sand-bed rivers. The objective of the model application, which is presented in the companion paper (Wright and Parker, 2005), is to study the development of two characteristics of large, low-slope, sand-bed rivers: (1) a downstream decrease in bed slope (i.e. concave upward longitudinal profile) and (2) a downstream decrease in characteristic bed sediment diameter (e.g. the median bed surface size D50). Three mechanisms that lead to an upward concave profile and downstream fining are included in the modeling formulation: (1) a delta prograding into standing water at the downstream boundary, (2) sea-level rise, and (3) tectonic subsidence. In the companion paper (Wright and Parker, 2005) the model is applied to simulate the development of the longitudinal profile and downstream fining in sand-bed rivers flowing into the ocean during the past 5000 years of relatively slow sea-level rise. ?? 2005 International Association of Hydraulic Engineering and Research.