Rao S. Govindaraju

Modeling the erosion process over steep slopes : approximate analytical solutions

Abstract Analytical expressions are developed for the rainfall-runoff-erosion process on steep hillslopes subjected to time-varying rainfall events. The erosion equation is essentially represented as a first-order reaction with the reaction rate being represented by the soil erodibility. The analytical transient solutions are based upon the assumption that the flow and sediment discharge have a constant relationship as during steady-state conditions. The analytical solution for the sediment discharge performs well when compared with numerical and experimental results. The approximate analytical solution for the concentration profile is the asymptotic limit of the transient numerical solutions. An error analysis shows that the analytical solutions improve with increasing slope length and that the solution model presented here is applicable to a wide range of physical situations.

Dynamics of Moving Boundary Overland Flows Over Infiltrating Surfaces at Hillslopes

The three flow processes occurring on hillslopes (overland flows, streamflows, and subsurface saturated-unsaturated flows) are in dynamic equilibrium and interact continuously through their common boundaries. A physics-based, deterministic, distributed model incorporating internal coupling of the three components is developed to study the extent and location of saturated regions neighboring the streams. These saturated zones develop overland flow and are very responsive to rainfall and are therefore important contributors to the hillslope hydrograph. Using a wide channel which drains the water from the side hillslopes (resembling an open book) for a physical section, we study the response of these variable source areas (VSAs) to various hydrologic and topographic parameters. An infiltrating boundary condition, which allows for both the Horton and Dunne mechanisms of overland flow generation, is incorporated in the analysis. It is observed that the dynamic modeling of the VSAs is useful for determining the hillslope hydrograph properties.

Error analyses of simplified unsaturated flow models under large uncertainty in hydraulic properties

The functional relationships between soil hydraulic properties (K, θ and ψ) are needed for analyzing unsaturated water flow in soils and are generally of a nonlinear form. These relationships, estimated from field-measured data, usually show large variability because of measurement errors and natural spatial heterogeneity exhibited by many field soils. A probabilistic approach is often employed to account for this uncertainty in parameters whereby the coefficients in the resulting partial differential equation are treated as random quantities (usually characterized by their moments). The objective of this analysis is to evaluate the errors resulting when simplified solutions based on simple forms of the hydraulic properties are used rather than nonlinear and numerically complicated models. As an example of a parametric functional relationship for K-θ-ψ, we adopt the Brooks-Corey formulation in which a single parameter λ is considered as random to emulate the spread encountered in practice. For data that show a large enough spread and when one is interested in the mean behavior of K, θ or ψ, simple flow models such as the Green-Ampt model and the linear model are proposed as alternatives to the more accurate (but numerically demanding) Richards equation. To compare models, scaled error definitions are introduced and are expressed as a percentage of the saturated hydraulic conductivity of the soil. These time dependent errors are then computed under various boundary conditions. In general, the Green-Ampt solution is found to yield smaller errors than the linear model when compared with solutions from Richards equation. It is concluded that the use of simplified models is justified when field data exhibit sufficient spread, and a tight bound cannot be placed on the parameter estimates.

A simplified model for two-dimensional overland flows

Abstract Numerical models of two-dimensional overland flow equations are often prohibitively expensive due to the highly nonlinear nature of the flow equations and the dense mesh required for accurate solutions. These models are frequently under-utilized due to lack of sufficiently detailed data at the grid scale. Observed results of the outflow hydrographs show fluctuations due to variability in the surface topography and precision limitations in the measuring instruments. A new solution methodology is presented in this paper using an eigenfunction expansion which is then combined with the kinematic wave approximation. The computational effort required by this new method is negligible when compared to the usual numerical methods. The results from this method are compared with the more expensive numerical results and experimentally observed results. These comparisons suggested that the semi-analytical solution methodology is an attractive modeling tool for practical two-dimensional overland flow computations.

Characterization of the rill geometry over straight hillslopes through spatial scales

Abstract Experimental evidence is presented which shows that there exists a spatial scale over which ergodic assumptions are applicable for rilled hillslope surfaces. The property of expected spatial rill density (ESRD) is defined and this property is shown to stabilize after the averaging interval is more than 20 ft over the hillslope. The rilled hillslope geometry may be readily quantified in terms of this easily measurable quantity. The rill geometry has a significant influence on the surface distribution of overland flow and sediment transport. An analysis of the spatial scales suggests that a continuum representation may be utilized in modeling these hydrodynamic flow phenomena. For a complete characterization of the rilled hillslope geometry, the distributions of the rill widths and rill depths at various locations along the hillslope have been presented. These preliminary results show that the mean rill widths, the mean rill depths and the ESRD increase with increasing slope lengths. The distributions of the rill widths and rill depths suggest that these quantities may be modeled by truncated Gaussian or Gamma distributions at different spatial locations. The rill geometry properties may be related to hydrologic, and geomorphologic properties of the hillslope. It is suggested that the ESRD is a better descriptor of the rill geometry on the hillslope than the rill density because of the wide variations exhibited by the distributions of the rill widths.

Development of an approximate model for unsaturated flow with root water uptake under rectangular water content profiles assumption

Abstract The rectangular profiles assumption (as in the model of Green and Ampt) is utilized for developing an approximate model for one-dimensional, vertical, unsaturated flow with root water uptake. The simplified model reduces the non-linear partial differential equation (Richards equation) for water movement to non-linear algebraic equations whose solutions require less computer effort. The basic hydraulic principles (continuity and Darcys law) are retained in the approximate model. Soil surface boundary conditions of infiltration and redistribution are considered. Analytical solutions are available for the depth to the wetting front when the soil surface is held at saturation. Comparisons between the numerical solutions and approximate solutions are made for two distinct soil types. The approximate model predicts root water uptake quite accurately when compared with predictions from the numerical solution of Richards equation. It is concluded that such simplified models hold promise in a variety of practical applications.