A method for computing the development of scour depth with time at the abutments in flood was elaborated. The method was confirmed by experimental data. A theoretical analysis of this method allowed us to determine the influence of the basic river flow and bed parameters on the relative depth of scour at the abutments. It was found that the relative depth of scour depended on the flow contraction, the relative grain size of the river bed and its distribution in depth, kinetic parameter of the flow, relative depth and velocity of the flow, Froude number with respect to the slope, unsteadiness of the flow, time of scour, and duration of the flood. According to the earlier investigations, the scour at bridge abutments also depended on the shape of abutments and the angle of flow crossing. Graphical dependences of the relative scour depth on different parameters were presented.