Investigation and development of analytical and numerical solutions to the various components of physically-based distributed catchment models

The basic equations describing the movement of water in open channels are formulated. Different levels of approximations to the complete unsteady flow equations are discussed, and a brief review of previous work done in this area is presented. The properties of different numerical schemes for channel routing are evaluated analytically based on the linear form of the kinematic wave model. The validity of the numerical models is tested using two real sets of data. The first application is the Blue Nile river basin in Sudan, and the second application is the Pangani river basin in Tanzania. The equations governing the movement of water in the overland flow planes are formulated in different dimensionless forms. A variety of finite difference models are proposed to solve the overland flow problem. The validity of these numerical schemes is tested based on a set of numerical experiments. Properties of each scheme are investigated in terms of accuracy, stability and robustness. Sensitivity analysis of different numerical models under consideration is carried based on different combinations of channel characteristic parameters. The accuracy of the models is tested via computation at higher levels of approximation and with the aid of a simple mass balance check. A model is suggested to route the movement of water and sediment along the overland flow planes. The model utilizes the kinematic wave theory and provides simple numerical solutions for the two flow processes with the aid of the modified Lax-Wendroff and the MacCormack schemes. The former is used for solving the equations governing the movement of water while the later is for the movement of sediment. The validity of the water/sediment model was tested based on hypothetical data sets and the results are compared with approximate analytical solutions. An approximate quasi-analytical solution to the diffusion wave approximation of Saint-Venant equations was developed. The proposed model assumes that part of the momentum can be lumped and concentrated at the moving centre of gravity. The results from the developed model are compared with the four-point implicit numerical solutions of the full Saint-Venant equation. A criterion that can improve the numerical solutions of the Richard's equation governing the flow in the unsaturated zone is suggested. The proposed criterion assumes that there is a relationship between the mesh ratio (~z/Ot) and soil parameters of similar nature to the Cournat condition used in open channels. The imposing condition guarantees smooth, non-oscillatory infiltration profiles, and perfect mass balance conservation, while requiring lesser computational cost. A procedure is suggested to utilize the proposed runoff/sediment scheme and the proposed diffusion algorithm in a physically based distributed, model. The models incorporate two stages of routing; one representing the overland flow and the other channel flow. They are linked with one-dimensional numerical solution of Richard's equation. The performance of the overall models is evaluated using a set of rainfall runoff events from a small upland catchment in the United States of America known as the R-5 catchment. The results obtained are satisfactory.