A 1D numerical model of the downslope flow and deposition of muddy subaerial and subaqueous debris flows is presented. The model incorporates the Herschel-Bulkley and bilinear rheologies of viscoplastic fluid. The more familiar Bingham model is integrated into the Herschel-Bulkley rheological model. The conservation equations of mass and momentum of single-phase laminar debris flow are layer-integrated using the slender flow approximation. They are then expressed in a Lagrangian framework and solved numerically using an explicit finite difference scheme. Starting from a given initial shape, a debris flow is allowed to collapse and propagate over a specified topography. Comparison between the model predictions and laboratory experiments shows reasonable agreement. The model is used to study the effect of the ambient fluid density, initial shape of the failed mass, and rheological model on the simulated propagation of the front and runout characteristics of muddy debris flows. It is found that initial failure shape influence the front velocity but has little bearing on the final deposit shape. In the Bingham model, the excess of shear stress above the yield strength is proportional to the strain rate to the first power. This exponent is free to vary in the Herschel-Bulkley model. When it is set at a value lower than unity, the resulting final deposits are thicker and shorter than in the case of the Bingham rheology. The final deposit resulting from the bilinear model is longer and thinner than that from the Bingham model due to the fact that the debris flow is allowed to act as a Newtonian fluid at low shear rate in the bilinear model.