M. Levent Kavvas

Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fields 1. Development of models

Unsaturated flows within subsurface regions control many large-scale hydrological and environmental processes. This study addresses the issue of spatial averaging of unsaturated flow equations at field scales. Two models for horizontally averaged unsaturated flow have been developed from two different approaches in this study. The spatially horizontally averaged Richards equation (SHARE) model is represented by a system of two coupled partial differential equations for the mean water saturation in each horizontal soil layer and the cross-covariance of the saturated hydraulic conductivity and the water saturation in each horizontal soil layer in a heterogeneous field. As an alternative to the spatial averaging/perturbation approach which was used for SHARE, a spatially horizontally averaged form of Green-Ampt model is obtained by field scale spatial horizontal averaging of the local soil water dynamics which are represented approximately by rectangular profiles. This strategy leads to analytical solutions for average water content and the results can be upscaled to large spatial areas. The computational effort required to evaluate such analytic expressions is trivial in comparison with that of the numerical solution of Richards equation. The averaged Green-Ampt model, though approximate, yields good results when large variations exist in the soil properties in the horizontal directions.

Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fields 2. Numerical simulations

Two models for horizontally averaged unsaturated flow have been developed from two different approaches in the first (Chen et al., this issue) of these companion papers. In this paper the results from both the spatially horizontally averaged Richards equation (SHARE) model and the averaged Green-Ampt model are compared with the results from a three-dimensional finite difference model of unsaturated flow which is perceived as the reference solution. The results of the averaged Green-Ampt model show very good agreement with the averaged results from the three-dimensional model, while SHARE model results are applicable only when fluctuations in soil parameters are small with respect to their mean values. It is also shown that methods of simple parameter averaging (arithmetic or geometric averages) with the local Richards equation does not yield meaningful results in heterogeneous soils. This study suggests that spatially horizontally averaged simplified models (such as the averaged Green-Ampt model) are attractive alternatives to perturbation models (such as the SHARE model) in heterogeneous fields. Due to their simplicity in formulation, accuracy in predicting average behaviors, and minimal requirement of computer effort, the spatially horizontally averaged simplified models can be easily implemented in large-scale models, such as atmospheric mesoscale models. boundary and initial conditions. The average saturation at each depth was obtained by areal integration of the local scale water saturation over the specified area under the assumption of independent vertical soil columns at field scales. Relative occurrence frequency curves of the spatially varying parameters were required for the solution of the average quantities. The second model developed was the spatially horizontally averaged Richards equation (SHARE) model. This model was developed by using spatial averaging and regular perturbation techniques under the assumption of no source or sink in the study area. The SHARE model was expressed as a system of two coupled one-dimensional partial differential equations in terms of the mean saturation and the cross-covariance of saturation and saturated hydrau-

Physical basis for a time series model of soil water content

A first-order autoregressive Markovian model AR(1) is formulated on the basis of the hydrologic budget and soil water transport equation. The model predictions compared well with neutron probe measurements of soil moisture content, and the statistical moments were conserved. The applied water events were white noise in structure, and the random shocks generated from the flow dynamics simplifications have a statistical mean of zero and were uncorrelated for all time lags. The derived AR(1) model parameter is used to compute the mean diffusivity of the soil, which is in agreement with reported lab measurements and field estimates obtained from cumulative evaporation measurements made with two large lysimeters.

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.

On the coarse-graining of hydrologic processes with increasing scales

The transformation of hydrologic processes from nonstationarity to stationarity with increasing scales is investigated. The fundamental mechanism of this transformation is the coarse-graining of hydrologic processes by aliasing and averaging operations. These operations are investigated by means of spectral relationships and hydrologic conservation equations. It is shown that hydrologic conservation equations at large scales may still be parsimonious due to the coarse-graining of hydrologic processes with increasing scales. When a larger scale process is formed by averaging a smaller scale process, the high frequency components of the smaller scale process are eliminated by the averaging operation, rendering the average hydrologic conservation equations to be quite simple in form. It is also shown that for the ensemble average form of a hydrologic conservation equation to be equivalent to its volume-average form, the parameter functions of that conservation equation at the immediately smaller scale must be ergodic.

Areally-averagei overland flow equations at hillslope scale

Abstract Microscale-averaged inter-rill area sheet flow and rill flow equations (Tayfur & Kavvas, 1994) are averaged along the inter-rill area length and rill length to obtain local areally-averaged inter-rill area sheet flow and rill low equations (local-scale areai averaging). In this averaging, the local areally-averaged flow depths are related to the microscale-averaged flow depths at the outlet sections (downstream ends) of a rill and an inter-rill area by the assumption that the flow in these sections has the profile of a sine function. The resulting local areally-averaged flow equations become time dependent only. To minimize computational efforts and economize on the number of model parameters, local areally-averaged flow equations are then averaged over a whole hillslope section (hillslope-scale areal averaging). The expectations of the terms containing more than one variable are obtained by the method of regular perturbation. Comparison of model results with observed data is satisfactory. The co...

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.

Stochastic solute transport under unsteady flow conditions: Comparison of theory, Monte Carlo Simulations, and field data

In this, the second of two papers concerning the stochastic description of solute transport under unsteady flow conditions, we show how the ensemble-averaged solute transport equation derived in the companion paper [Wood and Kavvas, this issue] can be solved. A two-dimensional analysis is conducted under conditions that are representative of the Borden aquifer, and a solution to the ensemble-averaged solute transport equation is found numerically. The analytical model suggested by Dagan et al. [1996] is adopted for the velocity field mean and Lagrangian covariance functions. The numerical solution provides the ensemble-averaged concentration field under transient flow conditions; from this concentration field the first- and second-order moments of the ensemble-averaged solute plume are calculated. The ensemble-averaged plume moments compare favorably with the moments calculated using the approach of Dagan et al. [1996], with the plume moments from a Monte Carlo analysis, and with plume moments measured in the field. In our approach the Darcy-scale dispersion is not neglected, and it is shown that this dispersion term has a small but significant influence on the resulting solutions.

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.