A NUMERICAL MODEL OF WATERSHED EROSION AND SEDIMENT YIELD by Vicente Lucio Lopes A Dissertation Submitted to the Faculty of the SCHOOL OF RENEWABLE NATURAL RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN WATERSHED MANAGEMENT In the Graduate College THE UNIVERSITY OF ARIZONA 1987 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Vicente Lucio Lopes entitled A numerical model of watershed erosion and sediment yield and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy ci 7S-/-7 ) Date (f) 1 )/frx,e-1; /./Y7 Dat I 5-/ 1147 Date Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. / _6??7 et /k'Dissertation Di ector - j-,f - Date STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: TO Guida, Angelo, Helena, and Junior iii ACKNOWLEDGMENTS The author expresses his sincere gratitude to his dissertation director, Dr. Leonard J. Lane, for his constant guidance and assistance. His enthusiasm toward this projéct was always encouraging. The author also wants to express his gratitude to his academic advisor, Dr. Martin M. Fogel, for his guidance, encouragement and assistance during this project. The author is also indebted to Dr. David A. Woolhiser for providing substantial guidance and encouragement toward the completion of this project. His support is sincerely appreciated. The author is grateful to Dr. Gordon S. Lehman and Dr. Simon Ince, members of the author's committee. Their advice and consideration are greatly appreciated. The assistance of Carl Unkrich is gratefully acknowledged. Without his expertise in FORTRAN programming and assistance in writing the source code of the computer model, this project would have suffered enormous delay. The author is grateful to the Aridland Watershed Research Unit of the U.S. Department of Agriculture, Agriculture Research Service (USDA-ARS) in Tucson, Arizona, for providing all the data and computer facilities necessary to the development of this project. His gratitude is also with the personnel from the USDA-ARS Aridland Watershed Research Unit for their assistance, encouragement, and friendship notably, Dr. Kenneth G. Renard, Dr. Edward D. Shirley, Roger Simanton, Jeffrey Stone, Fatima Lopez, Bob Wilson, Sue Anderson, and Tom Econopouly. Their support is greatly appreciated. The author is grateful to Bob Wilson and Sue Anderson for their professionalism in handling the drawings (Bob) and typing (Sue) of this manuscript. iv V The author is grateful to Escola Superior de Agricultura de Mossoro (ESAM) and Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) for the financial support during the four years he spent studying in the United States. Finally, the author is deeply grateful to his wife Margarida G. Lopes (Guida) for her constant encouragement, concern, and understanding during completion of his studies. This project might never have been completed without her love. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS viii LIST OF TABLES xi ABSTRACT xii 1. INTRODUCTION 1 Problem Statement Objectives Approach Benefits 3 4 5 2. LITERATURE REVIEW 6 Rainfall Excess Modeling Infiltration and Rainfall Excess Unsteady Surface Runoff Modeling Kinematic-Wave Modeling Erosion and Deposition Modeling Early Soil Loss Equations The USLE Approach Process-based Hillslope Erosion Modeling Watershed Erosion Models USLE-based Watershed Erosion Models Physically-based Watershed Erosion Models 6 7 11 12 15 15 18 20 24 24 27 32 3. MATHEMATICAL MODEL Infiltration Component Equations for Infiltration Surface Runoff Component Equations for Overland Flow The Kinematic Approximation Numerical Solution Equations for Channel Flow The Kinematic Approximation Numerical Solution Erosion-Deposition Component Equations for Hillslope Erosion and Deposition Sediment Continuity Equation Sediment Entrainment Rate by Shear Stress Sediment Deposition Rate Sediment Entrainment Rate by Rainfall Numerical Solution Equations for Channel Erosion and Deposition vi 32 33 35 35 38 39 40 42 43 43 46 46 49 51 52 53 55 TABLE OF CONTENTS--Continued Page Sediment Continuity Equation Sediment Entrainment by Channel Flow Sediment Deposition Rate Numerical Solution 56 57 58 58 4. DESCRIPTION OF WESP SYSTEM 60 General Description Watershed Segmentation Input File Generator WESP System Computational Sequence 60 62 65 65 68 5. PARAMETER ESTIMATION AND MODEL TESTING Input Data Parameter Estimation Rainfall Simulator Studies Estimation of Infiltration Parameters Estimation of Overland Flow Resistance Parameter Estimation of Erosion Parameters Small Watershed Studies Geometric Representation of Watersheds Infiltration Parameters Surface Flow Resistance Parameters Estimation of Erosion Parameters Storm Characteristic Data Test Results Simulation Results on Rainfall Simulator Plots Simulation Results on Small Experimental Watersheds Discussion of Test Results 6. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 71 71 73 73 76 76 78 80 82 90 90 94 94 94 96 111 111 126 126 127 129 Summary Conclusions Recommendations APPENDIX A: WESP VARIABLE NAME LIST 131 APPENDIX B: WESP OUTPUT SAMPLE 138 REFERENCES 140 vii LIST OF ILLUSTRATIONS Page Figure 2.1. Infiltration rate as a function of time under a constant 9 flux i o 3.1. Definition sketch of overland flow system 37 3.2. Definition sketch of finite difference scheme 41 3.3 Trapezoidal channel geometry (looking downstream) 45 3.4 Definition sketch of hillslope erosion/deposition system 47 4.1. Cascade of n planes receiving lateral inflow and discharging 61 into the j-th channel element 4.2. Delineation of overland flow planes and channel elements on WS 63.011 63 4.3. Representation of a typical plane for WS 63.011 64 4.4 Schematic representation of WS 63.011 for WESP 66 4.5. Information flow in program WESP 69 5.1 Location of Walnut Gulch Experimental Watershed 72 5.2 Schematic diagram of rainfall simulator plot 75 5.3 Walnut Gulch Experimental Watershed 63.105 83 5.4 Walnut Gulch Experimental Watershed 63.103 84 5.5 Schematic representation of WS 63.105 for WESP 85 5.6. Schematic representation of WS 63.103 for WESP 86 5.7. Location of cross sections between flume 103 and weir 101 87 and selected cross sections 5.8. Dry run on bare plot: (a) hydrograph (b) sedigraph 97 5.9. Wet run on bare plot: (a) hydrograph (b) sedigraph 98 vi ii ix LIST OF ILLUSTRATIONS--(Continued) Page 5.10. Very wet run on bare plot: (a) hydrograph (b) sedigraph 99 5.11. Dry run on clipped plot: (a) hydrograph (b) sedigraph 100 5.12. Wet run on clipped plot: (a) hydrograph (b) sedigraph 101 5.13. Very wet run on clipped plot: (a) hydrograph (b) sedigraph 102 5.14. Dry run on natural plot: (a) hydrograph (b) sedigraph 103 5.15. Wet run on natural plot: (a) hydrograph (b) sedigraph 104 5.16. Very wet run on natural plot: (a) hydrograph (b) sedigraph 105 5.17. Entrainment and deposition rates for dry run on bare plot 106 5.18. Entrainment and deposition rates for wet run on bare plot 106 5.19. Entrainment and deposition rates for very wet run on bare plot 107 5.20. Entrainment and deposition rates for dry run on clipped plot 107 5.21. Entrainment and deposition rates for wet run on clipped plot 108 5.22. Entrainment and deposition rates for very wet run on clipped plot 108 5.23. Entrainment and deposition rates for dry run on natural plot 109 5.24. Entrainment and deposition rates for wet run on natural plot 109 5.25. Entrainment and deposition rates for very wet run on natural plot 110 5.26. Storm event of 750705 on WS 63.105: (a) hydrograph (b) sedigraph 112 LIST OF ILLUSTRATIONS--(Continued) Page 5.27. Storm event of 750717 on WS 63.105: (a) hydrograph (b) sedigraph 113 5.28. Storm event of 750913 on WS 63.105: (a) hydrograph (b) sedigraph 114 5.29. Storm event of 750712 on WS 63.103: (a) hydrograph (b) sedigraph 115 5.30. Storm event of 750907 on WS 63.103: (a) hydrograph (b) sedigraph 116 5.31. Storm event of 750913 on WS 63.103: (a) hydrograph (b) sedigraph 117 5.32. Storm event of 760906 on WS 63.103: (a) hydrograph (b) sedigraph 118 5.33. Storm event of 760910 on WS 63.103: (a) hydrograph (b) sedigraph 119 5.34. Storm event of 770901 on WS 63.103: (a) hydrograph (b) sedigraph 120 5.35. Storm event of 780725 on WS 63.103: (a) hydrograph (b) sedigraph 121 5.36. Comparison of sediment yields for seven events on Watershed WS 63.103 as estimated from measured runoff and sediment concentration and as simulated using the mean values of the erosion parameters K r KR , and a 123 LIST OF TABLES Table Page 3.1. Elements of a trapezoidal channel 44 5.1. Infiltration and surface roughness parameters for rainfall simulator plots 77 5.2. Erosion parameters for rainfall simulator plots 79 5.3. Mean and range values of optimized erosion parameters for for rainfall simulator plots 81 5.4. Geometry of Watershed 63.105 88 5.5. Geometry of Watershed 63.103 89 5.6. Computational sequence for Watershed 63.105 91 5.7. Computational sequence for Watershed 63.103 92 5.8. Infiltration and surface roughness parameters for small watersheds 93 5.9. Erosion parameters for small watersheds 95 5.10. Simulated sediment yield on small watersheds using mean values of erosion parameters 122 xi ABSTRACT A physically based, distributed parameter, event oriented, non- linear, numerical model of watershed response is developed to accommodate the spatial changes in topography, surface roughness, soil properties, concentrated flow patterns and geometry, and land use conditions. The Green and Ampt equation with the ponding time calculation for an unsteady rain is used to compute rainfall excess rates. The kinematic wave equations are used to describe the unsteady one-dimensional overland and channel flow. The unsteady and spatially varying erosion/deposition process on hillslopes and channel systems is described dynamically using simultaneous rates of sediment entrainment and deposition rather than the conventional approach using steady state sediment transport functions. To apply the model the watershed is represented by a simplified geometry consisting of discrete overland flow planes and channel elements. Each plane or channel is characterized by a length, width, and a roughness parameter. For channel elements, a cross-section geometry is also needed. A modular computer program called WESP (Watershed Erosion Simulation Program) is developed to provide the vehicle for performing the computer simulations. Rainfall simulator plots are used to estimate infiltration parameters, hydraulic roughness, and soil erodibility parameters for raindrop impact and overland flow. The ability of the model to simulate watershed response (hydro- graph and sedigraph) to a variety of rainfall inputs and antecedent soil moisture conditions is verified using data collected on two small watersheds. xii xiii The good agreement between the simulated watershed response and the observed watershed response indicates that the governing equations, initial and upper boundary conditions, and structural framework of the model can describe satisfactorily the physical processes controlling watershed response. CHAPTER 1 INTRODUCTION Problem Statement Soil erosion is the removal of soil particles from the land surface by the action of water and wind, and refers to the dynamic processes of entrainment (detachment) and transport of sediments in watersheds. The erosion process is controlled by hydrodynamic, gravitational, and inter-particle electrochemical forces. The hydrodynamic forces are the principal external agents of soil water erosion and are essentially due to rainfall impact and shear by running water. The electrochemical forces act in opposition to the erosive forces and characterize the soil's resistance to the erosive action of the external agents. The erosion process begins on hillslopes with detachment of soil particles through raindrop impact and subsequent entrainment (detachment) by overland flow as soon as the excess water exceeds the static surface storage. As overland flow concentrates in gullies and stream channels, entrainment, transport, and deposition by concentrated flow occur. Entrainment and transport of noncohesive sediment particles by running water is controlled by slope length, slope steepness, particle size and weight distribution, and the forces exerted on the sediment particles by the flow. Entrainment and transport of fine, cohesive sediment particles will also be controlled by electrochemical inter-particle forces which make the erosion process even more complicated. When the external forces are diminished the settling process predominates and net deposition occurs. 1 2 The harmful effects of erosion and sediment on the physical environment are well documented in the literature. Soil erosion degrades the soil's resources by loss of soil water storage capacity, decreased infiltration rate (surface sealing) and thus increased opportunity for runoff and evaporation, loss of soil nutrients, and to a lesser extent, by increased weed production, reduced seed germination, and decreased root development. Sediment has great capacity for absorption of plant nutrients, pesticides, and other chemicals and therefore is a major carrier of pollutants. Soil and water conservation planning requires improved understanding of the physical and chemical properties of sediment in respect to specific erosion sites. The ability to predict soil erosion and sediment yield under current and alternative land and water management schemes is an important step in identifying the underlying causes of sediment damages, determining the sources of sediment causing such damages, and formulating feasible control measures. There is a current perspective in soil erosion science that process-based mathematical models provide an approach to improved understanding of the fundamental erosion and deposition processes and to improved erosion assessment and control technology (ARS,1983). However, any attempt to model the erosion and deposition processes as they occur in watersheds is seriously constrained by the complexity of an open system with component processes and state variables that may change rapidly In space and time. Therefore, a simplified representation must be used to model the complex erosion and deposition processes. For simplicity, the erosion process has been modeled under steady state assumptions (Meyer and Wischmeier, 1969; Foster and Meyer, 1972; Komura, 1976; Meyer et al., 1983; and Rose, 1985) even though the erosion process is clearly unsteady 3 because of the variation of rainfall intensity during a storm. Unsteady systems have been modeled using the kinematic wave equations with analytical solutions to the governing equations (Hjelmfelt et al.,1975, Shirley and Lane, 1978; Lane and Shirley, 1982; Singh and Prasad, 1982; Croley, 1982; Singh, 1983; Croley and Foster, 1984). However, the results with these models have been limited because of the simplifying assumptions required to solve the equations. Their application has been restricted to • modeling uniform overland flow over plane surfaces. Watershed erosion and sediment yield models with the kinematic approximation of the unsteady, spatially varied overland and channel flow equations and with erosion functions for detachment, transport and deposition of sediment have been developed (Bennett, 1974; Simons et al.,1975, Smith, 1976; Li,1979; Borah et al., 1981; Smith, 1981; and Alonso and DeCoursey, 1983). However, most of these models use empirical functions of sediment transport capacity developed in recirculating flumes for steady state and equilibrium conditions of sediment transport which can be a poor representation of the unsteady, nonuniform erosion and deposition processes occurring on hillslope and channel systems in small watersheds. Objectives The goal of this study is to accomplish the following three objectives: 1. Develop a numerical model for watershed erosion and sediment yield based on infiltration processes, unsteady surface runoff hydraulics, and fundamental erosion and deposition mechanics; 2. Develop a computer program (Watershed Erosion Simulation Program - WESP) and methodology for applying the model. 3. Describe parameter estimation techniques and model testing procedures using data from runoff-erosion plots and small watersheds. 4 Approach To address the first objective a numerical model is presented which accommodates the spatial changes in topography, surface roughness, soil properties, concentrated flow patterns, concentrated flow geometry, and land use conditions. The model consists of an infiltration component based on the Green-Ampt equation as modified by Mein and Larson (1973) with the ponding time calculation presented by Chu (1978), a runoff component based on the kinematic approximation to the dynamic equations of unsteady, nonuniform free-surface flow (Woolhiser and Liggett, 1967; Kibler and Woolhiser, 1970; and Rovey et al., 1977) and an uncoupled physically-based erosion-deposition component for simulating the dynamic processes of entrainment (detachment), transport, and deposition of sediment on hillslopes and channel systems (Meyer and Wischmeier, 1969; Bennett, 1974; Croley, 1982; and Rose, 1985). The model is capable of simulating the watershed erosion and deposition processes under current and modified land use conditions. To apply the model, the watershed is segmented into a sequence (cascade) of discrete overland flow planes and channel segments in which the fluid flow is assumed one-dimensional, and the kinematic equations are used to describe the unsteady flow. A four-point implicit finite difference scheme (Rovey et al., 1977) is used to solve the flow equations. A four-point implicit finite difference scheme was developed to solve the continuity equations for advective sediment transport (dispersion neglected) on 'hillslopes and stream channels. The computer program WESP (Watershed Erosion Simulation Program) in objective two is described. Methods used to apply the model are illustrated with examples. 5 Parameter estimation techniques and model testing procedures, as stated in objective three, are described using data from runoff-erosion plot studies and two small watersheds located on the U.S. Department of Agriculture, Agricultural Research Service Walnut Gulch Experimental Watershed. Benefits This dissertation and the research leading to it will be helpful in contributing directly to the development of the mathematical models proposed as part of the USDA Water Erosion Prediction Project (WEPP). Specifically, this research will contribute to the conceptual structure, mathematical formulation, computer algorithm, initial evaluation, and related applications of the proposed WEPP models. A major benefit of the research reported herein will be a modular and documented computer program describing watershed erosion and sediment yield. The modular structure is designed to facilitate substitution of different components and subroutines as research improves understanding of the processes controlling erosion and sediment yield on small watersheds. Moreover, the computer program WESP will serve as a "benchmark" or "standard" for evaluation of alternative and simplified models representing erosion and sediment yield on small watersheds. CHAPTER 2 LITERATURE REVIEW Estimation of soil erosion and sediment yield from watersheds is necessary for developing plans to control erosion and sediment damages. The relationships among rainfall, runoff, soil properties, land use and management practices, erosion, deposition, and sediment yield have been studied by many researchers for many years. This review of literature presents briefly the information most relevant to this dissertation on erosion modeling; from the early soil loss equations to more recent process-based hillslope erosion modeling, on infiltration and rainfall excess modeling, kinematic wave modeling, and on USLE-based and physically-based watershed erosion models. Rainfall Excess Modeling The first major component needed in constructing an event-based soil erosion and deposition model is rainfall excess or direct surface runoff. Rainfall excess is computed by subtracting the hydrologic abstractions or losses from input rainfall. The losses to be abstracted are: 1) interception losses, 2) evapotranspiration losses, 3) depression storage, and 4) infiltration losses. There are three basic approaches to modeling rainfall excess: 1) each of the four losses described above are modeled separately and linked together to generate rainfall excess, 2) the hydrologic abstractions are lumped in a single model, and 3) interception losses, evapotranspiration losses, and depression storage 6 7 are ignored and only infiltration losses are used to compute rainfall excess. In this study the last approach is used. Infiltration and Rainfall Excess Rainfall excess has been represented as the positive difference between instantaneous rates of rainfall and infiltration, or zero if infiltration capacity rate exceeds rainfall rate. Physically-based watershed erosion simulation models use the infiltration theory and equations to compute rainfall excess. Infiltration is the term commonly applied to the physical process of water entry into the soil through the surface or through a shallow hole or pit dug in the soil. Excellent reviews of the infiltration process have been presented by Philip (1969), Morel-Seytoux (1973), and Hillel (1980). Much progress has been made in soil physics and porous media flow for theoretically describing unsaturated soil water movement. Some of this recent progress has been associated with rapid developments in digital computers which has facilitated numerical solutions of the partial differential equations describing the infiltration process. If the soil surface is sprinkled at a steadily increasing rate i(t) sooner or later the supply rate will exceed the soil limited rate of absorption which has been defined by Hillel (1971) as the infiltrability. This corresponds to the infiltration flux or infiltration capacity f(t) resulting when water at atmospheric pressure is freely available at the soil surface. The infiltration process can be described as follows: i) As long as i(t) < f(t), water infiltrates as fast as it arrives and the actual rate of infiltration f o (t) is equal to i(t). The process is called supply-or flux-controlled. In case of constant supply rate i(t) i , a uniform moisture profile develops, and is termed nono ponding infiltration by Rubin (1966). 8 ii) When i(t) exceeds f(t), the soil-infiltrability determines the actual infiltration rate f o (t) f(t) and thus the process becomes profile-controlled. The soil surface becomes saturated and the excess of water i(t) - f(t) will accrue over the surface or run off. Rubin (1966) divided this into two phases: pre-ponding and post-ponding infiltration and called the instant at which the surface is saturated (under steady supply flux) the ponding time. Figure 2.1 summarizes the different features of the infiltration process under a steady supply rate 1. 0 . Soil infiltrability (or infiltration capacity) and its variations with time depend on: 1) time from the beginning of rain, 2) the initial moisture content (or pressure head): the wetter the soil is initially the lower the infiltrability will be and the faster the steady-state infiltrability will be reached, 3) hydraulic conductivity: the higher the saturated hydraulic conductivity of the soil will be, the higher its infiltrability will be, 4) surface conditions, and 5) the presence of impeding layers inside the soil profile. Land use and management practices can affect all but the first of these factors. Empirical formulas were historically developed to analyze the results of infiltrometer tests, and the resulting equations were applied to describe a rate of infiltration decreasing from an initial maximum to a final minimum rate (for instance, Kostiakov, 1932; Horton, 1940). These empirical formulas present some important limitations: 1) the parameters have little or no physical meaning and they cannot be determined or estimated from knowledge of the soil, 2) they are selected to provide the correct qualitative shape of the infiltration curve, 3) they cannot account for changes in initial and boundary conditions, and 4) most of them cannot accurately predict the ponding time and thus are poor models for rainfall excess pattern. 9 t init i olly ponded infiltration 1 1 pre- pond ing post - pond ing infiltrotion infiltration ponding - point -7 - 7— n RUNOFF • ollam Mom K. 411m O T nn•n• n•n• nnn• PTIME Figure 2.1. Infiltration rate as a function of time under a constant flux i o (after Rubin, 1966). 10 Physically-based infiltration models seem to constitute a more realistic representation of infiltration and much research on the flow phenomenon has been done in the last two decades. There are numerous infiltration models which have either been derived from soil physics consideration for certain simplifying assumptions such as the Richard's equation, or have been conceptually derived such as the Green and Ampt (Green and Ampt, 1911) and Philip (Philip, 1969) equations. These models may be more appropriate for rainfall excess modeling than the empirical models. However, they have been derived for particular cases and must be modified to account for more general situations. The Green and Ampt model has been the subject of considerable developments in the literature because of its simplicity and its satisfactory performance for a great variety of hydrologic problems. It has been extended to soils of nonuniform initial moisture content (Van Duin, 1955; Bouwer, 1969), to layered soils (Childs and Bybordi, 1969), and to crust-topped soils (Hillel and Gardner, 1970; Ahuja, 1974). It has been applied to infiltration into homogeneous soils from constant rainfall (Mein and Larson, 1973; Swartzendruber, 1974), as well as from unsteady rainfall (James and Larson, 1976; Chu, 1978). Also lumped parameter forms of the Green and Ampt model have been used to represent the infiltration component of the rainfall-runoff process on watershed modeling (Dawdy et al., 1972; Brakensiek and Ostad, 1977). Finally, recent research has focused on relating Green-Ampt equation parameters to measurable soil properties (Brakensiek, 1977; Brakensiek et al., 1981; Brakensiek and Rawls, 1982; Rawls et al., 1982; Rawls et al., 1983; and Brakensiek and Rawls, 1983). All the infiltration models presently available have been developed either from data of limited area, runoff plots or from physical and 11 computer models where some uniformity of soil properties was implicitly assumed. The difficulties with the application of these models to watershed erosion modeling arise from spatial variability of model parameters or soil properties and initial and boundary conditions at the watershed level. In addition, the problem of computing rainfall excess becomes even more complex due to temporal and spatial variability of rainfall. Therefore, more research is needed to consider spatial variability of soil properties, initial conditions and temporal and spatial variability of rainfall input to model rainfall excess. Unsteady Surface Runoff Modeling Whenever and wherever the rate of rainfall exceeds the soil infiltrability at the soil surface, the excess water begins to accumulate in static surface storage. The capacity of this storage is governed by the extent to which geometrical surface irregularities and surface tension can develop forces to balance the increasing gravitational forces. When the local static storage capacity is exceeded and surface tension forces are overcome, the excess water flows downhill under the influence of gravity, and surface runoff begins as a thin sheet flow from very small areas having little topographic relief. The next flow type is rill flow which is found in microchannels which gather the sheet flow from adjacent interrill contributing areas in a continuous fashion along their path. The rills merge to form larger concentrated flow channels which in turn merge to form even larger concentrated flow paths called stream channels. This complex spectrum of flow geometries and flow types is commonly called surface runoff. Surface runoff from a watershed is a nonlinear process which has been modeled through two general approaches: 1) systems approach (for 12 example, Amorocho and Orlob, 1961), and 2) hydrodynamic approach (for example, Woolhiser and Liggett, 1967; Chow and Ben-Zvi, 1973; Singh and Woolhiser, 1976). The systems approach develops input-output relationships without making any explicit assumptions regarding the internal structure of the system. The hydrodynamic approach requires the assump- tion that certain general laws of physics hold and further requires a geometrical representation of the actual watershed (Singh and Woolhiser, 1976). The physical laws required in the hydrodynamic approach are the equations of continuity of mass and momentum (Chow, 1959). Woolhiser and Liggett (1967) have shown that the simplified hydrodynamic approach based on the kinematic wave theory is applicable to many overland flow situations. Several researchers have applied the kinematic wave theory to model surface runoff from natural watersheds (for example see Brakensiek, 1967a,b; Kibler and Woolhiser, 1970; Eagleson, 1972; Li et al., 1975; Borah et al., 1980; Alonso and DeCoursey, 1983). Several alternate geometric representations (Wooding, 1965; Brakensiek, 1967a,b; Harley et al., 1970; Kibler and Woolhiser, 1970; Li et al.,1975; Alonso and DeCoursey, 1983) have been hypothesized and in- corporate varying degrees of geometric abstraction. In general, one geometric representation that has been made by several researchers is that a watershed may be represented by a network of overland flow planes and channel segments (Kibler and Woolhiser, 1970). Kinematic-Wave Modeling Kinematic wave modeling is one of a number of approximations of dynamic wave modeling. The dynamic wave model describes one-dimensional shallow-water waves (unsteady, gradually varied free-surface flow) and 13 consists of the continuity equation and the equation of motion with appropriately prescribed initial and upper boundary conditions. In the kinematic wave approximation, a number of terms in the equation of motion are neglected, and the equation of motion simply states that the friction slope is equal to the bed slope. Thus, the kinematic wave model is described by the continuity equation, a uniform flow equation, such as the Chezy or Manning equation, and the usually imposed initial and upper boundary conditions. The development of kinematic wave theory occurred late in the development of the theory of free-surface flow in open channels. It is based on the early developments in the study of steady varied surface flow and the late developments in the study of unsteady surface flow. The equation of motion and the water continuity equation were presented by Barre de Saint Venant in 1871. These equations are known as De Saint Venant partial differential equations of unsteady flow. Saint Venant attempted to integrate the continuity equation and the equation of motion by setting the channel slope equal to the energy friction line. This approach is similar to kinematic wave theory. The principal theoretical work on kinematic waves was done by Lighthill and Whitham (1955). They named the theory "kinematic wave" and investigated the general properties of waves and shock waves based on the theory. A large amount of work has been done on kinematic wave since the work by Lighthill and Whitham (1955). Work has been done on applying the kinematic wave theory to channel and overland flow, determining when the theory is applicable, and describing the properties of waves and solution techniques to the continuity equation. 14 Application of kinematic wave theory to channel routing has been described by Henderson (1963), Brakensiek (1967a,b), and Weinmann and Laurenson (1979). Application of kinematic wave theory to overland flow routing has been reported through a large number of papers presented in professional journals, technical reports, and symposia. Some relevant examples are those presented by Henderson and Wooding (1964), Woolhiser and Liggett (1967), Kibler and Woolhiser (1970), Schaake (1970), Li et. al (1975), and Borah et. al (1980). Application of kinematic wave theory to watershed modeling has been attempted with success by many researchers. The most relevant ex- amples are those by Wooding (1965), Bennett (1974), Simons et. al (1975), Rovey et. al (1977), and Alonso and DeCoursey (1983). Analytical solutions of the kinematic wave model can be obtained for many simple cases and considerable insight into the phenomenon can be obtained from them. However, in many cases numerical techniques such as finite difference methods or finite element methods are employed. The details involved in applying these techniques are described by Liggett and Gunge (1975). Numerical methods using the method of characteristics may be used when the lateral inflow rate changes rapidly, when the rating function changes with distance in a manner that renders analytical solutions difficult or impossible or when shocks are present. Both implicit and explicit techniques have been used for numerical solutions of the kinematic wave model. Kibler and Woolhiser (1970) used a very accurate (second order) Lax-Wendroff scheme for overland flow problems and compared its accuracy with solutions by characteristics and 15 an implicit scheme. Several implicit methods have appeared in the literature, such as those by Brakensiek (1967a,b), Cunge (1969), Li et al. (1975), and Rovey et al. (1977). Erosion and Deposition Modeling The Universal Soil Loss Equation (Wischmeier and Smith, 1960, 1965, and 1978) has been the most widely applied erosion model since the late 1960's, particularly in the USA. Despite its widespread use and the breadth of experience which it incorporates, the equation suffers from the conceptual defect that rainfall and soil factors (among others) cannot simply be multiplied together because of effects such as the subtractive effect of infiltration on overland flow from a given rainfall. A fuller understanding of the soil erosion process must therefore be based on more fundamental principles. Recent research on hillslope erosion modeling has concentrated on three main aspects: 1) the rates of sediment detachment and transport by raindrop impact, 2) the rates of transport and detachment by overland flow, either under ruled or unrilled conditions, and 3) the interaction between transport versus detachment in determining actual transport and erosion rates. All recent erosion modeling efforts implicitly or explicitly work within the constraints of a mass balance framework, which forms a common link between short and long term erosion models. Early Soil Loss Equations The basis for mathematical relationships to describe the soil erosion process began with efforts such as those by Cook (1936) to identify the major variables involved. Cook listed three major soil erosion factors: 1) the susceptibility of soil to erosion (soil erodibility), 2) the potential erosivity of rainfall and runoff including the influence of 16 degree and length of slope, and 3) the protective action of vegetative cover. The use of equations to estimate field soil loss began when Zingg (1940) published the results of his comprehensive study on the effect of degree of slope (S) and slope length (L) on soil loss. Using data from other researchers and his own experiments, Zingg recommended the relationship: A — CS 137 0.60 L (2.1) where A is the average soil loss per unit area from a land slope of unit width, and C is a constant. Smith (1941) evaluated the effects of mechanical conservation practices for four combinations of crop rotation and soil treatment on one soil and added crop (C) and supporting practice (P) factors to the Zingg equation: A — CS 7/5 L 315 P (2.2) A graphical method was developed for selecting the necessary conservation practices on soils in the Midwest. The C-factor included the effects of weather and soil as well as cropping system. Smith also introduced the concept of a specific annual soil loss limit for Midwestern soils. Ellison (1947) showed experimentally the effect of raindrop kinetic energy on soil particle detachment which he described by the following equation: E — KV 433 107065 D (2.3) where, E is the soil in t ercepted in splash samplers, during a 30-min. period (grams), K is a constant, V is the velocity of drops (fps), D is 17 the diameter of drops (mm), I is the rainfall intensity (in/hr.). Ellison defined the detaching force, for a given rainfall, as the product of the kinetic energy and the rainfall duration. Musgrave (1947) reported the results of analysis of soil loss measurements for several stations in the United States. Although not stated explicitly, the equation proposed by Musgrave was: E = IRS 135 035 L P 30 1 75 ' (2.4) where E is the soil loss in acre-in., I is the inherent erodibility of the soil (in), R is a cover factor, S is the degree of slope (percent), L is the length of slope (ft), and P 30 is the maximum 30-min. amount of rainfall, 2-year frequency,(in). The Musgrave equation was widely used for estimating gross erosion from watersheds, primarily because its highly generalized factor values were more easily assigned than were factors based on more specific conditions. Smith and Whitt (1948) proposed a "rational" equation for es- timating soil loss (A, Equation 2.5) for the claypan soils of Missouri using the effects of slope steepness, length of slope, crop rotations, conservation practices, and soil groups. The equation presented was: A = CSLKP (2.5) The C-factor was the average annual soil loss from claypan soils for a specific rotation on a 3% slope, 90 feet long, and farmed up and down slope. The other factors for slope (S), length (L), soil group (K), and supporting practice (P), were dimensionless multipliers to adjust the value of C to other conditions. The need for adding a rainfall factor to satisfactorily apply this equation over several states was acknowledged. Lloyd and Eley (1952) presented a graphical alternative to the 18 Musgrave (1947) equation for use "on a specific set of conditions". They tabulated values for major conditions found in the Northeastern United States. They stressed the need for practical methods of applying research findings to field conditions. Van Doren and Bartelli (1956) evaluated the faptors affecting soil loss for Illinois soils and cropping conditions. They proposed the following relationship: A= f(T,S,L,P,K,I,E,R,M) (2.6) where, A is the annual estimated soil loss, T is the measured soil loss, S is the steepness of slope, L is the length of slope, P is the practice effectiveness, K is the soil erodibility, I is the intensity and frequency of 30-min. rainfall, E is the previous erosion, R is the rotation effectiveness, and M is the management. The reference value for T was 3.5 tons per acre from Flanagan silt loam on 2% slope of 180-ft. length cropped continuously to corn. Estimates for other conditions were made using s 1.50 , L 0.38 (L<200 ft.) and 0 .6° ( 1 >200 ft.). Other factor values were given in tables and graphs. The USLE Approach The "Universal Soil Loss Equation" (USLE) (Wischmeier and Smith, 1960) was introduced in its present form at a series of Regional Soil Loss Prediction Workshops in 1959-62. The complete presentation of the USLE was in Agriculture Handbook 282 (Wischmeier and Smith, 1965), which has been revised (Wischmeier and Smith, 1978). The Universal Soil Loss Equation is: A = RKLSCP (2.7) 19 where, A is the computed mean annual soil loss per unit of area (metric ton/hectare), R is the rainfall erosivity factor for a specific location (megajoule-cm/hectare-hour), K is the soil erodibility factor for a specific soil horizon (metric ton-hour/megajoule-cm), L is the dimensionless slope-length factor, S is the dimensionless slope-steepness factor, C is the dimensionless cover and management factor, and P is the dimensionless supporting erosion-control practice factor. The USLE includes the six major factors that affect upland soil erosion by water: rainfall erosiveness, soil erodibility, slope length, slope steepness, cropping and management techniques, and supporting conservation practices. It is the result of methodical statistical analyses of erosion studies conducted at many locations in the United States dur- ing a half century of research. The USLE was developed as a method to predict long-term average annual soil loss from interrill and rill field areas. It was designed to meet the need for a convenient working tool for conservationists, technicians, and planners. The primary need was a relatively simple technique for predicting soil loss rates for specific situations. To overcome the situations where the USLE is not applicable, several modifications have been proposed. All the modifications presented so far, however are preliminary and regional because of limited data. Therefore, caution is recommended when trying to apply the modified equations to conditions different from which they were developed. Renard et al. (1974) modified the USLE to estimate sediment yield from semiarid rangelands of the Southwest United States. They introduced a channel erosion factor into the original USLE equation. The channel erosion factor is similar to the sediment delivery ratio used to predict sediment yield at an outlet. 20 Williams (1975) modified the USLE by replacing the rainfall ero- sivity factor with a runoff factor (volume of runoff x peak runoff rate) for an individual storm. The new equation, commonly referred to as the Modified USLE or MUSLE, eliminates the need for a sediment delivery ratio and allows estimation of sediment yields for individual storm events. Onstad and Foster (1975) modified the USLE by changing the rainfall erosivity factor to account for rainfall and runoff separately. This modification allowed estimation of sediment yield for single storm events. They also presented a method for estimating the relative proportions of interrill and rill sediment in the total sediment yield. Process-based Hillslope Erosion modeling In spite of the complexity of the soil erosion and deposition processes, process-based mathematical modeling promises to be the most viable way to estimate the time-dependent and spatially varying erosion response to various land use and management programs. A process-based model, as defined here, is a symbolic mathemati- cal representation of an idealized situation that has the important physical properties of the real system. The real system is the process as it actually is. In the present study the real system is the erosion-deposition process and all its facets. Foster (1982) listed some advantages of the process-based modeling approach over regression analysis when estimating time-dependent erosion/deposition rates and sediment yields: 1) it can be extrapolated more accurately to different land use conditions, 2) it represents the erosion/deposition process more accurately, 3) it can be applied to more complex conditions (spatial variability of surface characteristics and 21 soil properties can be included), 4) it is more accurate for estimating erosion/deposition and sediment yield on a single storm event basis. Research on process-based hillslope erosion and deposition modeling has taken two basic approaches. The first assumes for simplicity steady state erosion even though the erosion/deposition process is unsteady. Major models using this type of approach are those developed by Meyer and Wischmeier (1969), Foster and Meyer (1972), Komura (1976), Meyer et al. (1983), and Rose (1985). Meyer and Wischmeier (1969) presented relationships for the major erosion subprocesses which formed the conceptual basis of most subsequent erosion models. Foster and Meyer (1972) published a paper on a closed-form solution to the equation for steady-state overland soil erosion, which demonstrated the ability of models in this class to provide insight into the spatial variability of the erosion process on hillslopes and into the separable interrill and rill erosion process. Komura (1976) used the Kalinske equation of motion for sediment transport and the dynamic equation for spatially varied flow with lateral inflow to derive general equations for the estimation of soil erosion rates on uniform slopes by overland flow. Meyer et al. (1983) presented a quasi-steady one-dimensional mathematical model to simulate soil losses and sediment size distributions from cropped flatland fields. The authors used the kinematic-wave approximation for water movement and sediment transport. The resulting equations were solved by the method of characteristics to yield steadystate relationships. Rose (1985) described the processes of sediment detachment by rainfall impact, sediment entrainment (detachment) by overland flow, and 22 sediment deposition as simultaneously occurring at different rates during a storm event. The resultant sediment concentration is determined by the relative magnitude of these different rates. Rose presented a first-order partial differential equation expressing mass conservation of sediment in an overland-flow, which he reduced to an ordinary differential equation by assuming steady-state conditions, obtaining an analytic model of the erosion/deposition process on a plane surface. The second type of approach to modeling soil erosion and deposition on hillslopes has its focus in attempting to mathematically describe the major significant features of the erosion/deposition process without steady state assumptions. The kinematic-wave approximation to the dynamic flow equations has been largely used to drive the hydraulics of the erosion/deposition process, which has been modeled using the continuity equation for advective sediment transport plus some empirical relationships for detachment by rainfall impact and shear stress and a steady state sediment transport capacity function. In general the flow and (coupled) sediment equations have been solved analytically using the method of characteristics or numerically by finite difference methods. Analytical kinematic-wave runoff models have been restricted in application because of the simplifying assumptions required to solve the equations (constant rainfall intensity and constant infiltration rate, for instance). Major contributions to the development of analytical kinematic-wave models are those presented by Hjelmfelt et al. (1975), Shirley and Lane (1978), Lane and Shirley, (1982), Singh and Prasad (1982), Croley (1982), Singh (1983). Hjelmfelt et al. (1975) presented a mathematical model of the hillslope erosion/deposition process based on the sediment continuity equation and relationships for interrill erosion and rill erosion, using 23 the kinematic wave equations to describe overland flow. They solved the coupled partial differential equations for overland flow with interrill and rill erosion with constant and uniform rainfall excess for the rising and steady state portions of the flow hydrograph. Shirley and Lane (1978) and Lane and Shirley (1982) used the same approach presented by Hjelmfelt et al. (1975) for modeling hillslope erosion process but solved the coupled overland flow-erosion equations over the entire flow hydrograph using the method of characteristics, and then integrated the equations to produce a sediment yield equation for the entire runoff event. Singh and Prasad (1982) modified the modeling approach used by Hjelmfelt et al. (1975), and Shirley and Lane (1978) by formulating the partial differential equations for overland flow and erosion on an infiltrating plane. They used the method of characteristics to solve the special case of constant and uniform rainfall and infiltration (or constant and uniform rainfall excess) on a sloping plane. Croley (1982) modeled sediment flux to fluid flow in a rill system by assuming directly simultaneous sediment entrainment (detachment), deposition (settling), and lateral inflow, instead of representing their difference (net erosion or net deposition). Croley used the kinematic flow equations and the continuity equation for sediment to derive erosion equations for a rilled surface by assuming prismatic rill development in which the rills are identical and have a triangular shape. The method of characteristics was used to solve these equations for the unsteady flow case with uniform rainfall excess. Sediment concentration was given by steady-state sediment transport capacity equations. Singh (1983) derived analytical solutions to kinematic equations for erosion from a sloping plane subject to rainfall of finite duration. 24 Singh presented complete solutions for both equilibrium and partial equilibrium cases, and discussed briefly the properties of these solutions. In spite of the great effort in attempting to present analytical solutions to coupled unsteady runoff-erosion models, the simplifying as- sumptions necessary to analytically solve the kinematic wave equations for overland flow restrict the formulation and application of the erosion model. In general, the erosion equations are presented with simplified formulations to describe the complex erosion process. Therefore, aside from the problem of establishing the appropriate upper boundary condition, initial condition, and stability criterion, numerical solutions seem to be less restrictive in allowing more sophisticated formulations of the erosion model. Li (1979) described several numerical methodologies for water and sediment routing on hillslopes and stream channels. Watershed Erosion Models Many watershed erosion and deposition models have been developed to date. They can be loosely divided into two types: 1) models based on the USLE that utilize the USLE, extended USLE (such as MUSLE) or its parameters; and 2) models that attempt to represent the physical system through physically-based mathematical relationships. They can be divided into continuous simulation models or event-based oriented models. They also can be divided into distributed parameter or lumped parameter models. USLE-based Watershed Erosion Models Crawford and Donigian (1973) developed the Pesticide Transport and Runoff (PTR) computer model to estimate runoff, erosion, and pesticide losses from field-sized areas. The hydrologic component of the PTR model is the Stanford Watershed model (Crawford and Linsley, 1962). The 25 erosion component of PTR was developed by Negev (1967) and consists of relationships for sheet and rill erosion which include the detachment and transport of soil particles by overland flow. The Stanford Watershed Model was one of the first computer-based models of hydrologic simulation developed for basin-size areas. Donigian and Crawford (1976) incorporated - a plant nutrient component with the basic PTR computer model to develop the Agricultural Runoff Model (ARM). Frere et al. (1975) developed the Agricultural Chemical Transport model (ACTMO) to estimate runoff, sediment yield, and plant nutrients from field- and basin-sized areas. The hydrology component is based on the USDAHL computer model (Holtan and Lopez, 1971). The erosion/sediment transport component of ACTMO is a modification of the USLE to reflect both rainfall and runoff detachment and transport processes (Foster et.al, 1977). The erosion component estimates the contribution of rill and interrill sources to total sediment load. ACTMO includes a chemical component and was developed for small watershed areas. Bruce et al., (1975) developed a storm event-based parametric computer model for water, sediment, and chemicals, called WASCH. The hydrologic component of WASCH consists of a retention function, a characteristic function, and a variable state function. Two-stage convolution is used to produce nonlinear watershed response. The sediment component of WASCH uses the rill-interrill erosion model developed by Foster and Meyer (1975). Sediment transport capacity in the WASCH computer model is a function of overland flow discharge rather than velocity. Beasley et al. (1980) developed a distributed deterministic computer model referred to as ANSWERS which was designed to simulate runoff and erosion from large watersheds having agriculture as their primary land use. The hydrologic component of ANSWERS is the model developed by 26 Huggins and Monke (1966). The erosion component of ANSWERS consists of a modification of the USLE. Two soil detachment processes were included: 1) rainfall detachment, described by Meyer and Wischmeier (1969), and 2) overland flow detachment, described by Foster (1976). Sediment transport of both overland and channel flow is based on the sediment transport capacity. Channel erosion is assumed to be negligible, and only deposition is allowed in channel flow. In order to use the ANSWERS computer model, the watershed is divided into square uniform elements. The hydrologic response of each element is computed by an explicit backwater solution of a storage form of the continuity equation. The CREAMS computer model (Knisel, 1980) consists of three major components: hydrology, erosion, and chemistry. The hydrology component estimates storm runoff when only daily rainfall data is available by using the SCS curve numbers, or estimates runoff by the Green and Ampt equation when infiltration parameters are available. The erosion component considers the basic processes of soil detachment, transport, and deposition. Interrill detachment is described by a modification of the USLE for a single storm event. The sediment transport capacity of the overland and channel flow is derived from Yalin's equation (Yalin, 1963). Khanbilvardi et al.(1983) used the USLE parameters and factors to compute soil loss from interrill areas in an erosion model with separated rill and interrill components. Rill erosion was assumed to be the result of balancing the rill flow detachment and rill flow sediment transport capacity. Williams and Nicks (1983) described a computer model called SWRRB (Simulator for Water Resources in Rural Basins) which was developed for simulating water and sediment yields from large ungaged rural watersheds throughout the United States. The computer model includes three 27 general components: 1) a hydrology component for predicting surface runoff for daily rainfall (using the SCS curve number), peak runoff rates (a modification of the Rational Formula), percolation, return flow, evapotranspiration (Ritchie, 1972), and water balance for ponds and reservoirs. The computer model uses a short-cut flood routing method based on travel time at the peak flow rate and at a low flow rate, 2) a weather component for generating precipitation (Nicks, 1974), maximum and minimum air temperature, and solar radiation, and 3) a sediment yield component (Williams and Berndt, 1977) for computing sediment yield and routing sediment through ponds, reservoirs, flood plains, and stream channels. Williams (1983) presented an overview of the EPIC computer model (Erosion-Productivity Impact Calculator). The EPIC model is a comprehensive computer model, including several components: 1) hydrology, 2) weather generator, 3) erosion and sediment yield, 4) nutrients, 5) soil temperature, 6) crop growth, 7) tillage, and 8) plant environment control. The hydrology, weather generator, and erosion and sediment yield components of the EPIC computer model are very similar to those in the SWRRB model (Williams and Nicks, 1983) and CREAMS model (Knisel, 1980). The erosion component includes a wind erosion equation (Woodruff and Siddoway, 1965). Physically-based Watershed Erosion Models Physically-based mathematical models are formal models of real systems in which the governing physical laws are well-known and can be described by ordinary or partial differential equations. Watershed erosion and deposition can be simulated by using the equations of continuity and momentum for unsteady free surface flow, and the continuity equation 28 for advective sediment transport (dispersion neglected). The kinematic wave theory has provided the basis for the development of several physically-based watershed erosion-deposition models with potential for assessing erosion and sediment problems from disturbed watersheds. Some significant contributions to physically-based watershed erosion-deposition modeling are those described by Bennett (1974), Simons et al. (1975), Smith (1976), Li (1979), Borah et al., (1981), Smith (1981), and Alonso and Decoursey (1983). There are no analytical solutions for the flow and erosion equations as applied to watersheds and therefore numerical techniques are required. Li (1979) described several of these and discussed the efforts at Colorado State University in the computer-based numerical modeling of watershed response. Bennett (1974) divided the watershed erosion process into upland erosion and lowland or channel erosion, and used the concepts of water continuity, momentum, and sediment continuity to formulate an erosion and sediment yield model. Simons et al.(1975) developed a computer model to simulate the processes of interception, evaporation, infiltration, detachment by raindrop impact, and erosion by overland flow and channel flow. The effect of particle size distribution on flow detachment rate and flow transporting capacity, and the processes of degradation and aggradation in the channel system were also considered in the model. The governing equations were the water continuity, the momentum, and the sediment continuity equations. The kinematic wave approximation with numerical solution was used to solve the overland flow and concentrated flow equations. The MeyerPeter and Muller's equation for bed load and the Einstein equation for suspended load were used in computing the sediment transport capacity. 29 Simons et al. (1977) presented a simplified version of their early watershed erosion simulation model to be used on a single plane and which they called "a physical process model". In the simplified version, instead of routing flow over time and space using a finite difference scheme, they averaged the physical processes over both time and space to obtain an approximation of the more complex model. Smith (1976) described an erosion simulation model in which the differential equation for continuity of suspended sediment was incorporated into a numerical kinematic model for hydraulic response of a watershed surface. Smith used several examples from hypothetical watersheds to demonstrate the model sensitivities relative to choices of empirical functions for soil detachment rates from rainfall and flowing water, choice of sediment transport functions, and accuracy of numerical hydraulic simulation. Borah et al.,(1981) described a physically-based, distributed parameter computer model called SEDLAB for simulation of runoff response to precipitation. The model uses the continuity equation, the momentum equation, and equations for flow resistance to describe water routing for channel flow and overland flow planes. The kinematic approximation is used to solve the general equations of motion for flow routing. Van Liew and Saxton (1984) revised the SEDLAB simulation model to include overland and channel flow resistance, new infiltration methods, and improved sediment transport equations. KINEROS (KINematic EROsion Simulation) (Smith, 1981) is a computer-based watershed erosion model which incorporates erosion, sediment transport and pondage components into a previous computer model called KINGEN, described by Rovey et al.,(1977). KINEROS is a physically- based, nonlinear, distributed-parameter model that may be used in designing, 30 analyzing and managing small urban and agricultural watersheds (Smith, 1981). Watersheds up to several square kilometers in size are simulated with KINEROS by geometric simplification of the topography into a network of rectangular planes, channels and storage elements. In the distortion required to produce a rectangular area, the main features preserved are slope and the mean length of overland flow path. Rovey et al., (1977) give a detailed description of the geometric representation used in KINEROS. The major hydraulic processes, infiltration, and unsteady surface water flow, are simulated by solving numerically the differential equations representing those processes. KINEROS predicts rainfall excess using the Smith and Parlange infiltration equation (Smith and Parlange, 1978). The unsteady, free surface flow resulting from the rainfall excess pattern is simulated by using the kinematic approximation to the free surface flow equations (Woolhiser and Liggett, 1967). The erosion component of KINEROS uses the sediment continuity equation presented by Bennett (1974) for advective sediment transport (dispersion neglected). The model computes detachment by rainfall impact using an equation presented by Meyer and Wischmeier (1969). Channel flow erosion rate is estimated to be proportional to transport capacity deficit and deposition rate is approximately equal to excess. One of six steady flow sediment transport capacity functions may be selected by the user of KINEROS, including a tractive force relation, a function by Bagnold (Kilinc and Richardson, 1973), the "unit stream power" function of Yang (1973), and the sediment transport functions of Yalin (1963), Ackers and White (1973), and Engelund and Hansen (1967). All the sediment transport functions use a representative particle size and particle density. The hydraulic component of program KINEROS contributed directly to the development of the runoff component of program WESP (Watershed 31 Erosion Simulation Program) by providing the conceptual structure and mathematical formulation. Alonso and DeCoursey (1983) described a computer model called SWAM (Small Watershed Model) which was developed to simulate the effect of changes in land use and management on the hydrologic, sediment, and chemical response of agricultural areas not greater than 10 Km2 in size. The model simulates a watershed by subdividing the prototype into interconnected segments, each segment characterized by a uniform distribution of physical properties and model parameters. Four distinct types of segments are identified in SWAM: 1) source areas, 2) channels, 3) reservoirs, and 4) groundwater. The source area field simulates both surface and subsurface processes and is described in detail by Smith and Knisel (1983). Although the scientific study of watershed runoff, erosion and deposition processes has a relatively short history, significant advances have been made in developing physically based watershed erosion models. The continuity and momentum equations may be simplified to the kinematic equations for most overland flow cases and many open channel flow situations. To apply the kinematic equations to practical situations, one must first decide on the method of spatial representation of a watershed and level of geometric detail to be preserved. Then an appropriate infiltration model must be selected and linked to the overland flow model. Erosion equations with process based components for detachment by raindrop impact, detachment by shear, sediment transport and deposition can be formulated and driven by hydraulics of unsteady, spatially varied surface flow with the kinematic approximation. CHAPTER 3 MATHEMATICAL MODEL This Chapter presents the governing equations and the appropri- ate initial and upper boundary conditions used to describe the runofferosion-deposition processes in this study. Infiltration Component The first major component needed in building an event-based watershed erosion and deposition model is rainfall excess, which is sometimes called direct runoff. In general, abstractions or losses are subtracted from input rainfall resulting in rainfall excess which is routed to the watershed outlet. Hydrologic abstractions from rainfall are: 1) interception losses, 2) evapotranspiration, 3) depression storage, and 4) infiltration losses. In this model development only infiltration losses will be considered as abstractions from rainfall input. There are two distinct stages of infiltration during a rainfall event: 1) a stage in which the rainfall intensity is heavy and ground surface is ponded with water, and 2) a stage in which the rainfall in- tensity is light and there is no surface ponding. Under a ponded surface the infiltration process is independent of the effect of the time distribution of rainfall and the infiltration occurs at the infiltration capacity rate. Under unponded surface conditions, the infiltration occurs at the rainfall rate and all of the rainfall infiltrates into the soil. These two distinct stages of infiltration are well defined for a steady rain in which infiltration starts with an unponded surface and 32 33 later changes to a stage with surface ponding, which lasts until the end of the rainfall event. There is at most one ponding time in a steady rain. However, for an unsteady rainfall event, there may be several periods when the rainfall intensity exceeds the infiltration rate and the infiltration process may change from one stage to another and shift back to the original stage in a recurrent fashion. Equations for Infiltration A conceptual infiltration model utilizing Darcy's law was proposed by Green and Ampt (1911). The Green and Ampt equation is a simplified representation of the infiltration process and assumes that: 1) there exists a distinct and precisely definable wetting front, 2) suction or soil water potential at this wetting front remains constant regardless of time and position, 3) the soil profile is homogeneous, and 4) the distribution of antecedent soil moisture is uniform in the soil profile. It can be written as (Mein and Larson, 1973): f(t) = dF/dt K s (1 + N s /F(t)) (3.1) where, f(t) is the infiltration rate (m/s), F(t) is the cumulative depth of infiltrated water (m), t is the time variable, in seconds, K s is the effective soil hydraulic conductivity (m/s), and N s is the soil moisturetension parameter (m). The suction term, N s , in equation (3.1) can be computed as: N s (1 - S e )pS (3.2) where where, S e is the relative effective saturation = 8./0s (0 S :51) e ' O.1 is the initial soil moisture content, s is the soil moisture content 34 at saturation, p is the effective porosity (0 -sp- 1), and S is the average suction at the wetting front (m). Mein and Larson (1973) used the Green and Ampt equation to model infiltration during a steady rain. Chu (1978) developed two time parameters to modify the Green and Ampt equation to describe infiltration during an unsteady rain. Chu reported good agreement between the calculated result and the measured data. The infiltration process during an unsteady rain was modeled by Chu as follows: 1) Without surface ponding for the period from t' to t: R(t) R(t') F(t) P(t) - R(t') i(t) 0 f(t) r(t) (3.3) (3.4) (3.5) (3.6) where R(t) is the cumulative rainfall excess (m), P(t) is the cumulative rainfall (m) and is a continuous function of time when recorded by a weighting-type raingage, i(t) is the rainfall intensity (m/s), r(t) is the rainfall excess rate (m/s), t is the time in seconds measured from the beginning of rainfall, and t' is the time prior to the time t without surface ponding in seconds. 2) With surface ponding for the period from t' to t: R(t) P(t) - Fp(t) - D R(t,) F(t) F (t) f(t) K s (1 + N s /Fp ) r(t) Ii(t) - f(t) fo for for P - F - D > R(t) P - F - D R(t') (3.7) (3.8) (3.9) (3.10) for for G — D and i > f G < D or i f (3.11) (3.12) 35 where G is the depth of surface pondage (m), D is the depth of water retained on the surface without causing runoff (m), and Fp(t) is the cumulative infiltration because the surface is ponded (m) and which can be computed by the implicit function (Chu, 1978): F(t)/N s - ln(1 + F(t)/N) K s (t - t p + t s )/N s(3.13) where t is the ponding time in seconds and can be computed by the implicit function (Chu, 1978): P(t ) - R(t') - K N /(i(t ) - K ) = 0 s s (3.14) where t s is a shift of the time scale due to the effect of cumulative infiltration at the ponding time (referred to as the pseudotime, by Chu (1978)) in seconds. When t has been computed by equation (3.14) t s is solved by (Chu, 1978): K s t s /N s(P(t p ) - R(t'))/N s - ln(1 + (P(t p ) - R(t'))/N s ) (3.15) Surface Runoff Component The second basic component in process-based watershed erosion and deposition modeling is the runoff component. The kinematic wave equations have been used as a simplified one-dimensional flow approximation to the full equations of motion under almost all conditions of overland flow and for many conditions associated with stormwater in open channels. Equations for Overland Flow The equations of spatially-varied, unsteady and one-dimensional flow over a plane were described by Woolhiser and Liggett (1967). The definition of the overland flow system under consideration is shown in 36 Figure 3.1. A plane of unit width, length L o and slope S o is receiving rainfall at a rate i(x,t). The one-dimensional continuity equation with lateral inflow on a plane is: ah/at + a(uh)/ax (3.16) r and the momentum equation is: au/at + uau/a x + gah/ax g(S 0 - S f ) - ru/h (3.17) where, h(x,t) is the local depth of flow (m), u(x,t) is the local mean flow velocity (m/s), t is the time in seconds, x is the distance in the direction of flow (m), r(x,t) is the lateral inflow rate per unit area (m/s), g is the acceleration of gravity (m/s 2 ), S is the slope of the o plane, and S f is the friction slope. Equation (3.17) assumes that over pressure introduced by rainfall is negligible, that the velocity component of the rainfall in the x direction is zero, that the sine of the slope angle, 0, is approximately equal to the slope, and that the velocity distribution coefficient fi is equal to one. Modeling overland flow with one-dimensional equations represents significant abstraction and simplification. Actual overland flow occurs in complex mixes of sheet flow and small concentrated flow areas. The routes of concentrated flow are often determined by irregular microtopographic features which vary in the downstream direction (x) and in the lateral direction (y). The lateral inflow, r(x,t), in equations (3.16) and (3.17), is often represented as the positive difference between instantaneous rates of rainfall and infiltration, or as zero if infiltration capacity rate exceeds rainfall rate. This positive difference is called rainfall 37 RAINFALL i(x,t) H H IH1HHH L c) Figure 3.1. Definition sketch of overland flow system (after Woolhiser and Liggett, 1967). 38 excess. In solving equations (3.16) and (3.17), a typical assumption is that a block of rainfall can be partitioned into infiltration and rainfall excess. Rainfall excess is then routed as if the surface were impervious, which is a significant simplification (Smith and Woolhiser, 1971). Moreover, infiltration is usually assumed to be uniform over the overland flow surface, while in reality, infiltration rates vary significantly. The assumption of spatially uniform infiltration, and thus rainfall excess is a serious limitation in most current modeling approaches. The kinematic approximation The assumption of the kinematic approximation is that the friction slope is equal to the plane slope. That is, the gradients due to gravity and friction components dominate the other terms of the momentum equation and the water surface slope is assumed to be equal to the plane slope (Lighthill and Whitham, 1955; Henderson, 1963; Woolhiser and Liggett, 1967). Then the simplified momentum equation is: S o S f (3.18) Equation (3.18) can be used to write a parametric equation for the local velocity as: m-1 u —ah(3.19) where a and m are parameters related to surface roughness and geometry. Equation (3.19) can be substituted into equation (3.16) to yield: m-1 811/8x r ah/at + amh (3.20) The relevant upper boundary and initial conditions are of the form: 39 h(0,t) = 0 for t 0 (3.21) h(x,0) — 0 for x 0 (3.22) Manning's turbulent flow formula is: u (1/n)RH 2/3 Sf 2 (3.23) where RH (x,t) is the hydraulic radius (m) and n is the Manning friction factor of flow resistance. For planes and wide channels RH — h. This approximation and the substitution of equation (3.18) into equation (3.23) 1/2 and m 5/3. results in equation (3.19) with a — (1/n)S o Woolhiser and Liggett (1967) showed that solutions to the kinematic wave equations are good approximation to the solutions to the shallow water equations, provided the kinematic flow number is larger than about 20. However, the kinematic flow number refers to the accuracy with which the kinematic wave solutions approximate solutions to the shallow water equations for sheet flow on a plane. The kinematic flow number is not a measure of how well the shallow water equations, with one-dimensional flow and spatially uniform parameters, approximate overland flow on natural surfaces. Numerical Solution Equation (3.20) can be solved analytically for many initial and boundary conditions, if shocks are not present. However, in most of the cases it is necessary to use numerical solutions. Kibler and Woolhiser (1970) investigated several different methods of numerical solutions of equation (3.20). In this study, the numerical procedure proposed for solving equation (3.20) subject to (3.21) and (3.22) uses a second order nonlinear finite-difference scheme (Rovey et al., 1977). The Taylor series expansion was applied to linearize the nonlinear equations obtained 40 by the discretization of the flow continuity equation 'based on the four point implicit finite-difference scheme. The scheme is unconditionally stable with respect to choice of values for the time step At and the distance step Ax. However, values for these variables must be carefully chosen to ensure satisfactory accuracy. Using a four-point implicit scheme, equation (3.20) can be expressed as: O rh i+1_ h i At L j+1 + \ j ] (1151 ' )11 Ax (1_0) r At L j \In _ r(hi j+1, (ma Ax jj j r(hi+i)m L j+1 wri+1 + - (3.24) (1-w)r i in which 0 is the weighting factor for distance, and w is the weighting factor for time. For a definition sketch of the notation in equation (3.24), see Figure 3.2. Rovey et al. (1977) described the following approximate stability criterion for this finite difference scheme: amh m-1 At/Ax < 1 (3.25) so that for a fixed length increment, Ax, and the maximum depth h max occurring at time t, (3.26) At < Ax/(amhm-1 ) insures that stability exists at all points on the surface. Equations for channel flow Equations (3.16) and (3.17) written for a stream channel are: 3A/3t + a(uA)/ax qA(3.27) and au/at + uau/ax + gah/ax g(S 0 - S f ) - qAu/A (3.28) 41 -8) (8) i+I (w t Om o i (j+ 1) J X — 11m- X • UNKNOWN 0 KNOWN - Figure 3.2. Definition sketch of finite difference scheme. ) 42 where all variables are the same as those for equations (3.16) and (3.17), A(x,t) is the cross-sectional area of flow (m 2 ), and q A is the lateral inflow per unit length of channel. Notice that upstream inflow will be described as a upper boundary condition. The kinematic approximation: Free surface concentrated flow in stream channels can be computed using the kinematic approximation to the equations of unsteady, gradually varied flow (Henderson, 1963; Brakensiek, 1967a,b; Rovey, et al., 1977; Weinmann and Laurenson, 1979). The difference between routing runoff over planes and through channels is that upstream inflow to a plane is given in discharge per meter of width of the plane, while upstream inflow to a channel is the total discharge from the upstream channel. For watershed computations, a channel is assumed to have negligible width and, therefore, rainfall does not contribute directly to the channel. The lateral inflow to a channel is the discharge per meter of width received from an adjacent plane. With the kinematic approximation, equation (3.28) can be written for a stream channel as: Q aARHm 1 (3.29) - Assuming that Q(x,t) can be expressed as a function of A(x,t), equation (3.27) can be rewritten as: (3.30) m/at + dQ/dA8A/8x qA subject to the upper boundary and initial conditions: Q(0,t) Q0(t) for t 0 (3.31) 43 and, Q(x,0) = 0 (3.32) x > 0 for Substituting RH (x,t) = A(x,t)/WP(x,t), where WP(x,t) is the wetted perimeter, Q(x,t) can be related to A(x,t) by equation (3.29) as: m/wPm-1(3.33) Q = aA The functional relationships of dQ/dA terms in equation (3.30) for trapezoidal channels can be found from the set of geometrical relationships in Table 3.1 presented by Rovey et al. (1977), from geometry of Figure 3.3. Numerical Solution The finite-difference form of equation (3.30) subject to (3.31) and (3.32) was presented by Rovey et al. (1977): i +1 j+1 0 Ai ] j+1 [16, At + J Ax 1udAj (-1-Q1 1-L40j .+1 w i+1 Aj At Aj dA i+1 Aj+1 (3.34) A.j 1i - w-4Ai+1 + (1- )q where (12 dA a [(A)m 1 dA + - - dA Ad 1] H (3.35) dA where 0 and w are weighting factors for space and time respectively. The i+1 value of the unknown, Aj+1 , must be solved by an iterative technique and the terms dQ/dA must be evaluated before equation (3.34) can be solved. Erosion-Deposition Component The third component in a process-based erosion and deposition model is the erosion-deposition component itself. Soil erosion is the 44 Table 3.1. Elements of a trapezoidal channel (after Rovey et al., 1977). Geometric or Hydraulic Element Wetted perimeter at depth H Variable Name Relationship WP(H) B.001 + CO2 CO1 1/ZL + 1/ZR CO2 .11 + 1/ZL Discharge at depth H Q(H) a(H.001.(B+H/2)) /(B.001+CO2) Area at depth H A(H) H.001.(B+H/2) H(H) (H.001.(B+H/2)) DQH(H) a H(H).[C01.(B+H).N-(B.001 dQ/dH 2 • J1 + 1/ZR 2 N N-1 N+1 /(B.001)+H.0O2) N + H•CO2)-(N-1).0O2.(H.001. (B+H/2)] B 2 + (2.A/C01) Depth at area A H(A)-B+ dQ/dA DQA(H) DQH(H)/(C01.(B+H)) Top width at area A TW(A) 1.1112 + 2A/C01 ) • CO1 45 Figure 3.3. Trapezoidal channel geometry (looking downstream). 46 removal of soil particles (sediment) from their environment. Within the context of this study, it refers to the processes of entrainment (detachment) and transport of sediment by water. Due to the extremely complex nature of the erosion and deposition processes any attempt to model erosion and deposition must first break down the overall erosion and deposition process into several subprocesses that can be studied individually. In the present modeling approach the erosion and deposition process is conceptually divided into two phases: 1) hillslope erosion and deposition, and 2) channel erosion and deposition. The development of erosion and deposition equations for hillslope and channel systems is based upon the development of flow equations for unsteady and spatially varying overland flow and channel flow. Equations for Hillslope Erosion and Deposition To develop a mathematical model of erosion and deposition on hillslopes, expressions are needed to describe the rate at which each separate source and sink contributes to sediment concentration in the overland flow. Despite imperfect knowledge concerning the phenomenon and rate of ruling, this study assumes that sediment flux can be represented approximately without explicit description of rill features when those are present. The definition of the hillslope erosion/deposition system under consideration is shown in Figure 3.4. A control volume of the overland flow system is receiving sediment inflow at rates e i (x,t) and eR (x,t) and losing sediment at a rate d(x,t). Sediment Continuity Equation The continuity equation for sediment transport normally used for one-dimensional flow on hillslopes is (Bennett, 1974; Foster 1982): 47 FLUX IN 41M.Mn , Sediment Moss c(x,t):: Volume FLUX OUT 4=11n• Figure 3.4. Definition sketch of hillslope erosion/deposition system. 48 a(ch)/at + a(cq)/ax — 0 (3.36) Dispersion terms have been neglected in equation (3.36). The term a(cq)/ax is the buildup or loss of the sediment load with distance; a(ch)/at is the storage rate of sediment within the flow depth; c(x,t) is 3 the sediment concentration (Kg/m ); and 0(x,t) is the sediment flux to 2 the flow (Kg/m /s). The other variables are as defined earlier. The hillslope erosion/deposition process, can be represented by two different processes during a storm: 1) sediment entrainment (detachment) and 2) sediment deposition (settlement). Sediment entrainment on hillslopes is accomplished by two different subprocesses. These are: 1) sediment entrainment (detachment) by rainfall, and 2) sediment entrainment (detachment) by shear stress. Entrainment and deposition may occur simultaneously at different rates. The resultant sediment concentration is determined by the relative magnitude of these processes. Entrainment increases sediment concentration by upward flux of sediment from the soil surface into the overland flow. Deposition (settling) reduces sediment concentration by downward flux of sediment from the overland flow. It will be shown below that the rate of operation of each of these processes depends on different factors, and thus they should be separately represented in a mathematical model. This dynamic representation of the erosional system allows the system behavior to emerge automatically from the quantitative representation of the processes. The sediment flux to the flow, 0(x,t), for a single overland-flow plane might be written as: 0 e R - d + e I (3.37) in which e R (x ' t) is the rate of sediment entrainment (detachment) by 2 shear stress (Kg/m /s), d(x,t) is the rate of sediment deposition 49 2 (settling) (Kg/m /s), and e I (x,t) is the rate of sediment entrainment by 2 rainfall impact (Kg/m /s). Equation (3.36) describes flow, erosion and deposition which are uniformly distributed across the slope. Therefore, variables such as h(x,t) are averages and q(x,t) and (x,t) are expressed on a per width or per area basis, although the processes may actually occur on a limited area (Foster, 1982). Equation (3.36) also reflects an important concept, that entrainment is not the same thing as net erosion, and settling is not the same thing as net deposition. Entrainment (upward flux) is the actual rate at which detached sediment particles lying on the soil surface are picked up and carried into the flow region. Settling (downward flux) is the actual rate at which suspended sediment particles fall to the soil surface. In many turbulent flows that carry sediment, there is a continuous exchange of particles between the flow and the soil surface at any given location (Croley, 1982; Meyer, et al., 1983; Rose, 1985). Consequently, entrainment and settling can both take place simultaneously at a location in those cases when net deposition or net detachment do not occur along the soil surface. However, if settling exceeds entrainment, then net deposition occurs, and, conversely, when entrainment exceeds settling, net erosion takes place. If settling is equal to entrainment, then the transport process is in equilibrium at that point. Sediment Entrainment rate by Shear Stress Entrainment and transport of sediment occurs when the forces tending to entrain and transport sediment exceed those tending to resist removal. Water flowing over the soil surface exerts forces on the soil particles that tend to move or entrain them. On bare soil and stream beds, the forces that resist the entraining action of the flowing water 50 differ according to the particle size and particle size distribution of the sediment. For coarse sediment, the forces resisting entrainment are caused mainly by the weight of the particles. Finer sediments that contain appreciable fractions of silt or clay, or both, tend to be cohesive and resist entrainment mainly by cohesion rather than by the weight of the individual particles. Also, in fine sediments groups of particles (aggregates) are entrained as units whereas coarse noncohesive sediments are moved as individual particles. Sediment entrained (detached) by shear stress can be represented by a relationship expressing the entrainment rate as proportional to a power of the average shear stress acting on the soil surface (Croley, 1982; Foster, 1982): eR KR (T)15 (3.38) 1.5 . ․ ), soil detachability factor for shear stress (Kg.m/N R is a and r(x,t) is the average "effective" shear stress assuming broad shallow 2 flow (N/m ). where, K In most hillslope erosion modeling developments, the entrainment function has a critical shear stress component which represents the minimum requirement for initiation of motion of sediment. Although test results conducted in a single rill support the hypothesis of a threshold for initiation of particle motion in rill erosion (Foster, 1982), it is difficult to evaluate critical shear stress on a field scale or for small watershed applications. Also values suggested for critical shear stress from studies conducted in single rills are not applicable to the broad sheet-flow approach in overland flow erosion modeling (Foster, 1982). Furthermore, there are always fine particles of sediment detached by the action of wind or other elements between storm events, which will be 51 available to be transported by sheet flow as soon as rainfall exceeds infiltrability on the soil surface, without any resistance to removal. Therefore, it seems that Equation (3.38) may be a physically based and adequate function to represent entrainment rate by shear stress on hillslope erosion and deposition processes. Equation (3.38) says that the amount of sediment entrained by shear stress is related to the magnitude of an "effective" shear stress only and not to a "critical" shear stress. Sediment deposition rate The mass rate of sediment deposition (downward flux) can be expressed (Einstein, 1968) as: d — EV S C a where (3.39) is a coefficient depending on the soil and fluid properties (di- e mensionless), V s is the particle fall velocity (m/s), and C a (x,t) is the 3 actual near-to-the-bottom sediment concentration (Kg/m ). After multiplying and dividing equation (3.39) by C(x,t), the sediment concentration in transport, the deposition rate is given by: d — EV S RC C (3.40) C (x t)/C(x,t) is the concentration profile ratio. Assuming a ' that the sediment concentration distribution is uniform with flow depth where R above the soil surface, or that R — 1.0 (no distinction between near-tothe-bottom load and suspended load), the mass of sediment deposition (downward flux) on an overland-flow plane can be expressed as: d — EV C (3.41) The particle fall velocity in Equation (3.41) can be computed using Rubey's equation: 52 (7 s - 7) VS F oi 7 gds (3.42) and, 36 2 Fo — v 3 + gd s3 ( lis 12 7 - 1) 36v 2 7 gd3(& s 7 1) (3.43) where 7 and 7 s are the specific weight of water and sediment respectively 2 ty of water (m /s), and d S is a repre(N/m 3 ), v is the kinematic viscosity sentative particle size (m). Sediment entrainment rate by rainfall Sediment entrainment rate by rainfall, e i (x,t), is a function of the rate of detachment by raindrop impact and the rate of transport of sediment particles by shallow flow (Foster and Meyer, 1972). On steep slopes, the rate of detachment by raindrop impact limits entrainment by rainfall, whereas the rate of transport of sediment particles by shallow flow limits the rate of sediment entrainment by rainfall on flat slopes (Foster et al., 1977). A simple functional form of detachment by raindrop impact incor- porates rainfall intensity, i(x,t), as a measure of the erosivity of raindrop impact (Foster, 1982). If the rainfall intensity is uniform over the region of interest then i(x,t) i(t) and then: eI ai 2 (3.44) where a is a coefficient to be determined experimentally. Lane and Shirley (1985) included rainfall excess in equation (3.44) to reflect rate of sediment transport by shallow flow on hillslopes. They assumed a 53 simple equation for sediment entrainment rate on hillslopes: e .2 . I = K I 1 (r/i) where K I (3.45) K ir is a coefficient to measure soil detachability by rainfall im- 4 pact (Kg-s/m ). The ratio of rainfall excess rate, r(x,t), to rainfall intensity, i(x,t), can be interpreted as a measure of normalized runoff intensity for sediment transport by shallow flow. Notice that when r(x,t) 0 (pre-rainfall excess phase), or when i(x,t) 0 (post-rainfall phase) there is no sediment entrainment by rainfall, and when r(x,t) = i(x,t), sediment entrainment rate by rainfall is not limited by rainfall excess. Numerical Solution Analytical solutions have been proposed to solve equation (3.36). However, the results have been limited by the simplifying assumptions required to solve the overland flow equations. In general, in attempting to describe the erosion/deposition processes in the field, equation (3.36) should be less restrictive with respect to the formulations of the functions in equation (3.37). In general, for most of the formulations of the functions in equation (3.37) , there is no analytical solution for equation (3.36). Therefore, numerical techniques are required. The relevant upper boundary and initial conditions of equation (3.36) are: C(0,t) K i i(t)r(t)/(eV s + r(t)) for t ^ t p (3.46) and, C(x,t p ) K I i(t p )r(t p )/(a s + r(t p )) for x where t is the time of ponding. 0 (3.47) 54 The numerical procedure proposed for solving equation (3.36) subject to the above upper boundary and initial conditions uses a fourpoint implicit finite difference scheme. The strategy of solving equation (3.36) is to find the sediment concentration at the advanced time and i+1 in terms of known values. The finite difference formudistance step C lation of equation (3.36) is as follows: O[(ch) j+1 1-4-1 - (ch)1.- 1 + (1-6) [(ch) .11.-4-1 - (ch)1 + At j+lj At J (3.48) _ (co il = ( ) i+11 + (1-)1- (c ) i w F„i+1 Ax L g j+ 1JJ - cg _I Ax L cg 'j -1-1j 1+1 + (1 0)e i+1 ] + (1 0[0e 1 + (1-0)e- i ] w [0e Rj+1 R K. R. j+1 - - J J 40c11 1:14-0.-0c1 1]-(1-w)[0d1 +1 + (1-0)dl + J J we i+1 + (1-w)e 1 where 0 is the weighting factor for distance, and w is the weighting factor for time. This scheme can be either explicit or implicit, depending on the values of the weighting factors 0 and w. If 0 = 1 and w - 0, the numerical scheme becomes an explicit scheme, and is subject to the Courant condition to maintain stability (Kibler and Woolhiser, 1970). If 0 = 0.5 and w > 0.5, the scheme is unconditionally stable; however, for accuracy, it has been recommended to maintain the Courant condition. For a definition sketch of the notation in equation (3.48), see Figure 3.2. i+1 Rearranging equation (3.48), the unknown Cj i can be computed explicitly as: 55 Cj+1 +1 — w[0e R i. +1 + (1-0) (e R 1: 4-1 - d1 4-1 ) + ell + 3+1 J (3.49) (1-0[0(eRj4.1 i - d 1 ) + (1-0) (e R i. - dl) + efl + j+1 J [8(ch)1 4.1 - (1-0) ((ch)1 1-1 - (ch)1)]/At + / Pcq)1 1-1 - (1-w) ( (cq)1 41. - (cq)1)]/Ax / l ei+1 wi+1 i At hj+1 + Ax clj+1 4- why s) Equations for Channel Erosion and Deposition Erosion and deposition in channel systems are very difficult to model due to complex interactions between independent variables. The complex interrelationships involve not only the available sediment supply from hillslopes but also flow velocities, flow depths, slope and energy gradient, density, temperature, chemical composition of the water-sediment mixture, particle size distribution, shape, density of bed load and bed material, bank roughness and bed forms of the deformable stream bottoms, and transmission losses into the stream beds. Most factors affecting channel erosion and deposition change not only with time and with distance along a channel segment but also with flow depth and with lateral distance at an individual cross-section. These complex interrelationships have not been well understood and the literature is crowded with empirical relationships, most of them based on experimental observations conducted in recirculating flumes under steady state conditions. The total sediment load of a stream is usually divided into two parts. One part is the bed load, which consists of the sediment that 56 moves by skipping, sliding, or rolling and that always remains very close (generally within a few grain diameters for uniform sediment) to the stream bed. The other part is the suspended sediment load, which may be anywhere within the turbulent flow and is maintained in the flow by the upward components of the turbulent flow or by colloidal suspension if the sediment particles are very small. In the present model development no distinction is made regarding the mode of sediment transportation. The channel flow erosion-deposition equations estimate the total sediment concentration in transport during a flow event. The sediment flux to fluid flow in a channel is represented with physically-based mathematical functions for simultaneous sediment entrainment (detachment), deposition (settlement), and sediment lateral inflow from hillslopes. For a detailed discussion of the mechanics of erosion and deposition processes in stream channels refer to Graf (1971), Shen (1971), Bogardi (1978), ASCE (1977), and Vanoni (1984). For a more recent reference on the state-of-art of erosion and deposition modeling in alluvial streams refer to Dawdy and Vanoni (1986). Sediment Continuity Equation The continuity equation for sediment transport in one-dimensional flow in a single channel element is (Bennett, 1974): aAc/at + 8CQ/8x e r - d + q s (3.50) where e r (x,t) is the rate of sediment entrainment (detachment) by channel flow (Kg/m/s), d(x,t) is the rate of sediment deposition (settling) (Kg/m/s), and q s is the lateral sediment inflow from adjacent overland flow planes (Kg/m/s). The other variables are the same as defined earlier. Equation (3.50) is subject to the following upper boundary and 57 initial conditions: C(0,t) Co(t) for t 0 (3.51) for x 0 (3.52) and, C(x,0) — 0 where Co(t) is the incoming sediment concentration from above the channel. In the channel erosion/deposition phase, the bed profiles of the lowland stream channels are assumed to be relatively stable as opposed to the longitudinal profiles of gullies on upland areas, which can change rapidly in short time periods. The bed material in lowland streams is assumed to be predominantly coarser than silt size (0.062 mm) and deposited by the stream in the recent geologic past. Sediment entrainment by channel flow A general equation, initially developed for bed-load transport capacity, has been used to model entrainment (e ) by channel flow r (Croley, 1982; Foster, 1982): e ta(r - r ) c r 0 n for r for r < re r (3.53) in which, 7RH S f(3.54) T and, r c 6(7 s - 7)d (3.55) 58 2 1.5 where a is a coefficient for sediment entrainment (Kg-m /N -s), r(x,t) is the average shear stress (N/m 2 ), r is the average critical shear c 2 stress for the representative particl size (N/m ), n is an exponent, and is a coefficient depending on the sediment and fluid properties (dimensionless). The other variables are as defined earlier. When the average shear stress, acting on a particle or aggregate of sediment has reached a value that, if increased even slightly, will put the particle or aggrega- te into motion, critical or threshold conditions are said to have been reached. The problem of determining critical conditions for entrainment of sediment is one of the many complicated erosion/deposition problems faced when applying watershed erosion/deposition models. Sediment Deposition Rate The rate of sediment deposition (downward flux) is proportional to the sediment concentration and an effective particle fall velocity (Mehta, 1983). The deposition rate in kilograms per second per meter of channel width can be computed as: d = eTWV C s (3.56) in which TW(x,t) is the flow top width (m) and the other variables are as defined before. Numerical Solution There is no analytical solution to solve equation (3.50) and therefore numerical solution must be used. The numerical solution proposed to solve equation (3.50) subject to (3.51) and (3.52) uses the same scheme and procedure used for solving equation (3.36). Again, the finite difference scheme is formulated implicitly with C i+1 as the only unknown at the advanced time and distance step. Equation (3.50) can be 59 written in terms of finite differences as: At [ (CA) i+1 + j1 (CA) 1•4_ (1-0) j+1 j At i+1 - (CQ) i+1 . ] + (±)— Ax [ (CQ )j+1 j 4 4 wq Oe pAi.--1-1 _ )j (1: 0 A. Ax (CA) ] ..._ (3.57) J . 1 Q - )] (C i. — [(CQ) jj +1 i+1 + (1-0)e i+1] + (1-0 [0e i + (1-0)e i r. ] r. r. rj+1 j+1 J 3 0(11:1 + ( i+1 1 0)dll - - (1-0[0d1 +1 + (1-0)431] + + (1-w)q where 0 is the weighting factor for distance step, and w is the weighting factor for time step. Equation (3.57) can be rearranged to give: c i+1 = wi p e j+1 i+1 rj+1 (1 0[0(e - 4. (1-8) le i+1 - 6.7s(TW .i+1) + .4 i+1] ) j+ s r. (3.58) J - rj+1 i eV 5 (TWC)1 14 ) + (1 0)(e rii - - a 5 (TWC)1) + q si] + [0(CA)1 +1 - (1-0) [(CA)1 4-1 - (CA)1)]/At + [w(CQ)1 4-1 - (1- ) [(CQ1 +1 - (CQ)1)]/Axl/ 0 i+1 + A (-A7E j+1 w i+1 + Z7c Qj+1 w007 07 1+1 ) sTW. j+ 1 ) CHAPTER 4 DESCRIPTION OF WESP SYSTEM This Chapter describes the computer program WESP (Watershed Erosion Simulation Program) and the methodology for applying the model. General Description WESP (Watershed Erosion Simulation Program) is a computer-based watershed model developed for use in soil erosion research. The source code was written in standard FORTRAN 77 for portability and consists of a main program and eleven subroutines. WESP is a physically-based, event-oriented, numerical model developed to simulate the dynamic erosional and depositional behavior of small watersheds. Watershed geometry is represented by a simplified configuration consisting of sequences Of discrete overland flow planes and channel segments. Figure 4.1 shows a schematic representation of an nplane cascade receiving lateral inflow r and discharging into a channel segment. The kinematic wave equations are used to describe the unsteady overland and channel flow. The continuity equations for overland and channel flow are solved numerically using a four-point implicit finite difference scheme. The continuity equations for sediment on the overland flow planes (hillslope erosion component) and channel elements (channel erosion component) are solved numerically using a four-point implicit finite difference scheme. WESP was developed with the general purpose of being used in soil erosion research on small watersheds. The two specific purposes of the 60 61 0:1) -0 a.) 7 a (r) 0 .i..) 0 .,-i 60 t .,-1 to $.4 crl 4 U CI) -,-, -0 -0 cO e—I C.) ,-4 st 4) 14 0 60 • ,-.1 4+ a) 62 model are: 1) to understand and simulate the dynamic erosion/deposition system on small watersheds, and 2) to provide a "benchmark" program to verify and test the accuracy of subsequent model simplifications for application purposes. To run program WESP the user has first to complete the following steps: 1) set up a segmented representation of the watershed geometry, and 2) generate an input file. The following is a brief description of these steps. Watershed Segmentation To use program WESP the watershed has to be first segmented into a cascade of elements represented by rectangular planes and channel elements. Each element is characterized by a uniform distribution of physical properties and model parameters. The delineation of plane and channel elements is conducted using topographic maps. The degree of geometric distortion introduced by the simplified geometric representation is a function of the size and number of elements used in the representation (Lane and Woolhiser, 1977). Once the watershed boundaries have been defined the user must proceed with the delineation of the overland flow planes. The definition of planes should be based on soil, slopes and surface cover characteristics. The plane boundaries should be either streamlines or contour lines. Figure 4.2 shows the U.S. Department of Agriculture, Agricultural Research Service Walnut Gulch Experimental Watershed 63.011 with the delineation of overland flow planes and channel elements. During the discretization process an attempt is made to minimize geometric distortion by preserving the areas and lengths of flow paths for each plane element. Figure 4.3 shows a plane representation for a portion of Watershed 63.011. 63 64 I. -J 0. os to 0 00w Olommr Ci ci PI 6 6 6 o ••••• —•c—'—oc 65 Channel elements are selected such that they may be treated as prismatic elements having uniform hydraulic properties. Each channel element is assigned a trapezoidal or triangular cross-section. Outflows from other elements can enter a channel in two ways: 1) as uniformly distributed lateral inflow from adjacent overland flow planes, and 2) as point inflow from upstream channels or plane elements (see Figure 4.1). Figure 4.4 shows the schematic representation of Watershed 63.011 for WESP. Input File Generator This is the entry point for every new watershed simulation. Program INPUT was developed to generate input files containing all data other than rainfall input needed to run a given watershed simulation. Rainfall data are stored in separate files. The watershed data are entered interactively. The user is led step by step through the input file building process by messages generated on the monitor's display. The input consists of data subsets comprising the computational sequence and model parameters. Program INPUT augments the input data by internally computing all secondary parameters and initial values that can be derived from primary input data. Once the input file is completed, it can be used to perform repeated simulation runs under different rainfall inputs. The user can use another program called WESPAR to make changes in any data entry contained in an already existing WESP input file. WESP System The program WESP has three major components: 1) hydrology component to process rainfall input (histograms) and to compute rainfall excess rates using the Green and Ampt equation, 2) surface runoff 66 •nn•n•• I • =11•nn Og 1, 11 Cud 6, 1 o 67 component to route rainfall excess through overland flow planes and channel elements using the kinematic wave equations with numerical solution, and 3) erosion and deposition component to compute entrainment rates by rainfall impact and by runoff, and sediment deposition rates on overland flow planes and in channel elements. Sediment is routed through plane and channel elements using the sediment continuity equation with numerical solution. Surface runoff is considered in the "Hortonian" overland flow sense. Dispersion is neglected in the sediment transport equations. The name of each subroutine and its principal function in the program is indicated below: PROGRAM WESP routes water and sediment on small watersheds. It calls SUBROUTINES READER, RAIN, INFIL, CLERK, PLNFLOW, CHNFLOW, and WRITER. SUBROUTINE READER reads data describing plane and channel characteristics. It is called from PROGRAM WESP several times according to the execution order of each element. SUBROUTINE RAIN reads rainfall data and places them in arrays TPRECP and PRECP as time-intensity breakpoint pairs. This subroutine is called from PROGRAM WESP. SUBROUTINE INFIL computes infiltration rates on the planes using the Green and Ampt equation and generates rainfall excess patterns. It is called from PROGRAM WESP. SUBROUTINE CLERK performs the bookkeeping in the temporary storage location. It is called from WESP. SUBROUTINE ITER solves general nonlinear implicit equations using the Newton-Rapson iteration scheme. It is called from SUBROUTINES INFIL, PLNFLOW, and CHNFLOW. 68 SUBROUTINE IMPLCT computes residuals for ITER through EQN1 (for overland flow equation), EQN2 (for channel flow equation), EQN3 (for discharge equations), and EQN4 (for infiltration equation). It is called from ITER. SUBROUTINE PLNFLOW routes overland flow through the planes using a four-point implicit finite difference scheme. It is called from PROGRAM WESP. SUBROUTINE PLNSED computes entrainment rates by rainfall impact and shear stress from overland flow and sediment deposition rates on the planes and routes sediment through the planes using a four-point implicit finite difference scheme. It is called from SUBROUTINE PLNFLOW. SUBROUTINE CHNFLOW routes concentrated flow through channels with trapezoidal and triangular cross sections using a four-point implicit finite difference scheme. It is called from PROGRAM WESP. SUBROUTINE CHNSED computes entrainment rates by shear stress due to concentrated flow and sediment deposition rates in the channels and routes sediment through the channel network using a four-point implicit finite difference scheme. It is called from SUBROUTINE CHNFLOW. SUBROUTINE WRITER writes out the output file. It is called from PROGRAM WESP. FUNCTION ABSTRACT (to be developed) supplies abstraction rates from channel beds (transmission losses) to EQN2 in SUBROUTINE IMPLCT. Figure 4.5 shows the information flow in program WESP. Computational Sequence The computational order of WESP is such that all inflows required by any element at any stage of the simulation come from elements previously processed. This sequence is determined by the user during the 69 ( START ) OBTAIN CONTROL PARAMETERS Y LOAD RAINFALL DATA PRINT OUT HYDROGRAPH / SEDIGRAPH ( READ INFORMATION END ) ON ELEMENT NO YES — PROCESS CHANNEL– I * COMPUTE RAINFALL EXCESS RATES r –PROCESS PLANE —1 ADD INFLOW AND SEDIMENT DISCHARGE FROM CONTRIBUTING ELEMENTS PLNFLOW 1 1 1 1 CHNFLOW 1 Y CHNSED 1 Figure 4.5. Information flow in program WESP. PLNSED 1 70 watershed segmentation process by following the flow path through the cascade of elements. The order in which the elements appear in the flow path defines the computational sequence. As a general rule to establish the computational order, the number "one" element is assigned to one of the most upstream elements in the cascade and then the flow path is followed down to assign the computational order of other elements. If the next element is not on a tributary branch, it is assigned the next consecutive execution number; if the next element is on a tributary branch, the next consecutive element number is assigned to the most upstream contributing element on the tributary. This process is repeated until the watershed outlet is reached and all elements have been assigned an execution order number. During a simulation run the outflows from some elements are retained in temporary storage location, while other elements are processed until the junction between converging channels is reached. At this point the outflows in temporary storage location are combined and the storage location is released to be used in subsequent calculations. CHAPTER 5 PARAMETER ESTIMATION AND MODEL TESTING This Chapter describes parameter estimation techniques and model testing procedures used to verify the proposed watershed runoff-erosion model. Two sets of data were used: 1) data from rainfall simulator plots, and 2) data from two small experimental watersheds. The rainfall simulator plots and the two small watersheds are located on the U.S. Department of Agriculture, Agriculture Research Service Walnut Gulch Experimental Watershed near Tombstone, in Southeastern Arizona (Figure 5.1) which is operated by the Aridland Watershed Management Research Unit of the USDAARS in Tucson, Arizona. The details on input data, parameter estimation procedures, and model testing results are presented below. Input Data Three types of data are required to run program WESP. They are: 1) watershed characteristics, 2) erosion and deposition parameters, and 3) storm characteristics. The watershed characteristics include information on: 1) watershed geometry (surface area, elevations, and surface roughness), 2) channel network (slopes, cross-sectional areas, and hydraulic roughness), 3) soils (saturated hydraulic conductivity, antecedent soil moisture condition, and effective porosity), and 4) land uses. The erosion and deposition parameters are: 1) soil detachability parameters for raindrop impact and shear stress 71 72 1( • CO GNISE COUNTY WALNUT GULCH EXPERIMENTAL WATERSHED PERIPHERAL AREA WATERSHED AREA DETAILED ............ Ç! 3 4 SCALE IN MILES WATERSHED BOUNDARY SUGWATERSHED BOUNDARIES DRAINAGE (MAJOR) • RUNOFF MEASURING FLUMES Figure 5.1. Location of Walnut Gulch Experimental Watershed. 73 (overland and concentrated flow), 2) particle size (sediment) characteristics: distribution, shape, and density, and 3) parameters for critical shear stress (in stream channels) and sediment deposition (on hillslopes and stream beds). The storm characteristics include: 1) rainfall intensities (histogram), and 2) areal distribution. The storm characteristic data and the antecedent soil moisture conditions change from storm to storm. The watershed characteristic data and the erosion and deposition parameters are assumed to be time-invariant unless land use conditions change on the watershed. Most of the input data are measured directly from topographic maps, for example, surface areas, flow lengths, and slopes. Hydraulic roughness and most of the erosion/deposition parameters must be estimated experimentally or by optimization when data on storm hydrographs and sedigraphs are available. Parameter Estimation The model parameters were estimated in two stages. In the first stage rainfall simulator plot data were used to estimate: 1) the saturated hydraulic conductivity and the moisture-tension parameter of the Green and Ampt infiltration equation, 2) hydraulic roughness, and 3) the soil erodibility parameters for raindrop impact and runoff. In the second stage, data from two small experimental watersheds were used to: 1) verify the applicability of the erosion parameters estimated from rainfall simulator plots to overland flow on a watershed scale, and 2) estimate channel erodibility parameters for concentrated flow. Rainfall simulator Studies The rainfall simulator plots and the rainfall simulator operated on the Walnut Gulch Experimental Watershed were described in detail by 74 Simanton and Renard (1982). The rainfall simulator plots are 3.1 x 10.7 m, and have slopes of 9-12%. A diagram of the experimental setup is shown in Figure 5.2. Simulator runs were begun in the Spring of 1981 and have been repeated during the Spring and Fall seasons of each year. Three rainfall simulator plots were used in this study. The plot data were selected from the simulator runs made in the Fall of 1982. The three selected plots had the same type of soil (Bernardino series) but were treated differently: 1) with natural vegetation (natural), 2) with vegetation removed (clipped), and 3) with erosion pavement and vegetation removed (bare). Each plot was subjected to an initial 60-min rainfall simulation (dry run), followed 24 hours latter by a 30-min application (wet run), which was then followed 30 min later by another 30-min application (very wet run). A constant rainfall rate was applied in all simulations. Rainfall amount and intensity were measured with a recording gage placed between each plot pair (see Figure 5.2). Runoff was collected at the lower end of the plots in flumes with water-level recorders for measuring runoff rates. Sediment samples were collected manually in liter sample bottles at the flume exit during the simulation period. Sampling intervals were dependent on changes in the runoff rate, with frequent sampling when runoff discharge was changing rapidly. The time when the sediment sample was collected was recorded for later relation to the runoff hydrograph and development of the sedigraph and calculation of sediment yield. Sediment samples were analyzed for total sediment concentration. Sediment discharge and sediment yield were calculated using sediment concentration values and the runoff hydrograph. 75 ROTATING BOOM RAINFALL SIMULATOR SPRINKLER NOZZLE PATH ICM z !METER Figure 5.2. Schematic diagram of rainfall simulator plot (after Simanton and Renard, 1982). 76 Although the flumes used to measure runoff at the plot exit were designed to minimize sediment deposition, a significant amount of sediment was trapped in the flumes. This experimental error was most evident on the clipped and natural plots. Estimation of Infiltration Parameters The saturated hydraulic conductivity, Ksat, and the moisturetension parameter, Ns, in the Green and Ampt infiltration equation were estimated from plot data using the following procedure (L. J. Lane, personal communication): The saturated hydraulic conductivity, Ksat (Ks as shown in Column 5 in Table 5.1) was estimated from the very wet run plot data as the final infiltration rate. Using Ksat, the soil moisture-tension parameter (Ns as shown in Column 6 in Table 5.1) was estimated by optimization using the dry run plot data and a program called INFPAR which calculates rainfall excess using the Green and Ampt infiltration equation. The Ns value was optimized by trial and error to fit the measured runoff volume. Using the soil moisture data, the average suction at the wetting front, S, was estimated using the equation below: Ns — (1 - Se)pS (5.1) where Se is the relative effective saturation, and p is the effective porosity. The wet run was used to verify estimates of Ksat and S comparing the calculated rainfall excess and measured runoff volumes. Estimation of Overland Flow Resistance Parameter Once the infiltration parameters were estimated, the overland flow resistance parameter, a, was optimized to fit the measured 77 ri Ln 04 cr, c.4 cr, N. -.1- 0-00 4 . 0 0 0 0 r--- ri —1. 0 Hi ..2. —I 0 CYN en 0 0 01 en 0 0 % 0 m —1. 0 cd ZO I 00 l's• • Cf) 0 Lr% r••• Ln Ln• NI ri Ln Cn) Cn en Lin • 0 a) •HE 1.4 Q.. C.) r- en en 00 en CV Ce) %D tf) en II o r- en 0 œ c0 un II 00 N. 44-.1 1/4.0 c0 -a- tr 00 Lf1 100 f"-- 1.1"1 L0 00 'D NI on Lt4 Cfl Cf) en 4- ,10 00 4- en Ce) en en 43 Ln en en r1,1 cn en ‘43 a% co q) en NI NI .--1 • 0 Ni NI --1 0 NI Ni H 0 CO 0 ri 0 c0 0 rl 0 00 0 ri• 0 LM .-1 H 0 in .-I ri 0 en ,--I H• 0 a) 40 P:1 a) $.4 CO PI a) a) Pa a. • r-I •-1 r...) a. • H c.) a. • H HA C.) 4-1 0 4-) 0 4.4 CO A ....... 1"... 0 r- NI —1 3 ..... r.... 0 c0 NJ —I cl ..... 01 ...1. 0 NI en .-I ......, ....• 4.....? 0 0 %D r- ce) cn —1 ,--4 .r.. ...... ..-.. I=1 ..... in 0 %0 C.4 ..-( ......• in 0 0 cn —+ ......• Ill 0 H en ,-i .....• r0 ON 04 —I • o•—••. J•J (1) a) a) 4 )..) 0 0 O) HI CO *14 $4 J../ I•i 0 4.) 0 ri E •,-4 co C•1 op Cln H 1-1 '-4 0 4-1 )1 0 H 1.44 to 4-1 0 1-4 a. • "0 "0 Ci) a),.= .L.) ce (41.4 0) 0) —44-I a) cc co 3 ,—I 3 ... a) 3 Il •• ›N $4 ..0 ii A .....• CO ai a ri C.) CI) $.1 0 4.4 CO •H 0 El ..--4 •,-1 0 CO 4.) 0 CD ^0 a) c.) a) 3.) A CV . 0 .1-1 4-1 CO 1-1 V. • •..I001 CO 00 0 •1-1 1.4 0 -0 '0 a) -I-1 H ra. (1., CO cl) 4.) CO H ,-1 'Hi 0 4-1 0 • 14 CO $.1 1.1 0 ea 4.J ai 0 o c.) en . 4.4 0 a) • v-i C.) .1-1 i4-I 4a) 0 C) co co a) 0 4 Ce 0 0 )-4 CO b0 0 • 1-1 0 0 Co Z 4* 78 hydrograph. The optimization program SIMPLEX developed by Nelder and Mead (1965) was used with a least square objective function. The Manning's roughness parameter (Column 7 in Table 5.1) was found using: n (5.2) jSo/a where a is the overland flow resistance parameter and So is the slope of the overland flow plane. Estimation of Erosion Parameters The hillslope erosion and deposition component of WESP has three unknown parameters: a soil erodibility parameter for overland flow, K R , a soil erodibility parameter for rainfall impact, K r and a parameter for sediment settling, e . There are no theoretical or experimental ref- erences on KR and K 1 from the literature at the present time. Information exists on e values for overland flow (Davis, 1978) and channel flow (Einstein, 1968). Although the reported values of flow and e — e — 0.5 for overland 1.0 for channel flow were obtained for steady state condi- tions, these values were used in this study. The values of K 1 and KR in Table 5.2 were optimized by trial and error to fit measured sediment yields. The starting values of the erosion parameters were chosen as follows: At an equilibrium condition the rate of sediment entrainment (upward flux) is equal to the rate of sediment deposition (downward flux) such that: 0 e d + e — 0 R I (5.3) At the time the rainfall simulator is turned off e I — 0 because i(t) 0. At that instant, it can be assumed that e R is approximately 79 a (p C .4.-4 CV CV ri I".. .-4 CV • + • 0 0 0 0 CV 1-.1 N. CO 4 N. v-I ON V) ON N. ON N. ON .0 In r•-4 • r•-I • Cn • Cn 0 0 1-4 o 0 .o Cn • 0 a) v-1 ON VI . 1n1 CV a ...t Cn rnI 1-4 0 0 ..t Ce) Cr) CO CV I-1 • • 0 0 in 0 s•-.1 -. r•-• CO CO h. ON st N as cc) CO N-r-i vnI Cn Cn ....I N .4. .-4 m 0 0 .-1 Cy) • 0 0 Cr) CT ON Q in - 4 • C0 Ctl 0% o•i VD 'Q••••4 • 0 0 N. Lin 0 e."... 0 N.. ..1 CV r-4 Ill CV CV 0 tf) CV C4o CV ...I tn N 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 CV . 0 0 0 0 . 0 co 0 0 0 . 0 0 0 ....^. ..1. El •••-n .....r... 1...4 4 CO c) I .-4 b0 04 ,yd ..-.n ....., 0 0 0 0 0 0 0 0 0 0 o 0 in in o 1/40 -4 0 tr) 0 r-- a) —I ON tfl Crl Cel • . ,..o in —I .-4 v-1 In1 ..„1- 0 c4 ce) ....t 0 %.0 re) ,-.4 r-i 4 0 cz) —1 • • 40 CV in in in 0 0 0 N.'.00 ...I ce) CV CI ..-4 .—.1 .-4 Crl ,-4 .-I • 1..4 N. N. 0 0 N. CV CO CV ON CV .--i .—c .-4 N. 0 80 equal to d. From Equation (5.3): e R - d = 0 (5.4) or, that KR (r) 1.5 — eVsCe (5.6) with r iheSo, and he — (qe/a) 0. where he is the overland flow depth at the plot exit corresponding to qe, which is the water discharge at equilibrium, and Ce is the sediment concentration at equilibrium. The solution of Equation (5.6) provided a starting estimate for KR . Starting with K 1 — 0 and KR estimated as above, the values of K I and KR were adjusted by trial and error to obtain the measured sediment yield. Values of K1 and KR in Table 5.2 are the final estimates from the optimization. Table 5.3 shows the range of variability of erosion parameters estimated from rainfall simulator plots and mean values of K1 and K R for bare, clipped, and natural plots. Small Watershed Studies The studies on small watersheds had two objectives: 1) to verify the applicability of the erosion parameters estimated from rainfall simulator plots when applied to a watershed scale, and 2) to estimate channel erodibility parameters. Two small watersheds were selected on Walnut Gulch Experimental Watershed for this study: Watershed 63.105 (2344.7 square meters) and Watershed 63.103 (34792.2 square meters). These subwatersheds have similar physical characteristics, and are typical of the many thousands of hectares of semiarid rangeland with mixed grass and brush cover in 81 r•4 r.-I 0 Li..1 C • ri 03 $.4 cnI a% (-4 %.0 —1 0 0 . 0 on C) 0 . 0 0 0 0 i..1 0 (1-1 01 IQ tv J.J G) E $4 al 0. 0 0 ..4 co o 1.1 G) 9:1 G) ta •,-1 o 0 .--i . El ..-1 J.) O. -...t 0 in o CV 01 0 . 4/N . ,-. .-4 I I ..-1 t4.4 0 co G) 0 ,--i cd › • co 41) 4../ b0 0 O .--1 O O. $4 1.4 •0 0 O 4.) O 0 .--1 O 0 cd 0 G) • ,-I Z co 0 0 0 0 ON en 0 .-I . . O) I.. .-1 0 $.4 D.0 .6) .1-I O r-4 PZ C.) 0 Z 82 Southeastern Arizona, Southwestern New Mexico, and Northern Sonora, in Mexico. The soils are a well-drained, gravelly loam formed in calcareous old alluvium. The soil surface has minimal vegetative basal cover, with up to 60% gravelly erosion pavement. Grazing has been eliminated on the studied watersheds since 1963. Precipitation and runoff characteristics on these watersheds were studied in detail by Simanton and Osborn (1983). Figures 5.3 and 5.4 show watersheds 63.105 and 63.103 respectively. Three storm events in Watershed 63.105 (WS 105) and seven storm events in Watershed 63.103 (WS 103) were used in this study. Most of the sediment samples on WS 105 and WS 103 were collected by a modified Chickasha pump sampler with an intake system having slots on an arm suspended from the flume floor to a float at the water surface. Openings in the arm were constructed at five vertical positions to obtain an aliquot of the water-sediment mixture from several flow depths. The residual sediment trapped in the flume was added to the measured sediment load to give an estimate of the total sediment load. In late 1977, the Chickasha pump sampler on Watershed 63.103 was replaced by an automatic total-load sediment sampler. The total load sampling apparatus was described in detail by Renard et al. (1986). Geometric Representation of Watersheds Watershed 63.105 was represented by 10 overland flow planes and 2 channel elements. Watershed 63.103 was represented by 16 overland flow planes and 6 channel segments. Schematic representations of watersheds 63.105 and 63.103 for WESP are shown in Figures 5.5 and 5.6 respectively. Typical channel cross-sections for WS 103 are shown in Figure 5.7. Tables 5.4 and 5.5 provide summaries of the geometric representation of WS 105 and WS 103 respectively. The parameters ZL and 83 ,, , — i / / — / 1 s SI ' 1 Z I I on 6 N I .0' a/ •MININ n•••... ..w. n 84 85 2 1 3 1 4 i 5 1 , 6 --n 7 --... 8 --op. i ONINInlew 1•11n11. 12 1 1 •-n•••• 1 .11- 10 MII•nn Figure 5.5. Schematic representation of WS 63.105 for WESP. 86 I 1 2 Jr Figure 5.6. Schematic representation of WS 63.103 for WESP. 87 Figure 5.7. Location of cross sections between flume 103 and weir 101 and selected cross sections (after Osborn and Simanton, 1985). 88 Table 5.4 Geometry of Watershed 63.105. ZR ELEM Area Length Bottom Width and ZL (m) (m) (m) Slope (m) 1 180.2 15.5 11.6 0.142 * * 2 220.0 18.6 11.8 0.035 * * 3 222.0 13.7 16.2 0.081 * * 4 284.3 16.5 17.3 0.191 * * 5 205.3 22.9 9.0 0.089 * * 6 295.4 22.5 13.1 0.065 * * 7 267.6 16.5 16.3 0.203 * * 8 207.2 22.9 9.1 0.100 * * 9 * 4.3 * 0.038 0.000 0.250 10 157.9 11.3 14.0 0.158 * * 11 304.7 14.0 21.7 0.099 * * 12 * 18.0 * 0.038 0.000 0.250 89 Table 5.5. Geometry of Watershed 63.103. ELEM Area 2 Length Width Bottom ZR and ZL (m) Slope (m) 51.8 40.6 0.034 * * 8195.3 101.2 81.0 0.054 * * 3 1969.5 40.5 48.6 0.053 * * 4 2561.2 39.9 64.2 0.089 * * 5 1973.2 78.0 25.3 0.061 * * 1.20 10.00 # (m ) 1 2101.8 2 (m) 6 * 80.1 * 0.036 7 2569.9 48.5 53.0 0.056 * * 8 1905.1 44.5 42.8 0.069 * * 58.8 * 0.043 0.60 9 * 1.00 10 2703.3 57.0 47.4 0.053 * * 11 1131.5 44.2 25.6 0.043 * * 12 946.2 42.1 22.5 0.048 * * 13 1050.8 29.0 36.2 0.104 * * 14 * 25.6 * 0.038 0.60 15 1883.9 33.8 55.7 0.048 * * 16 2773.4 56.7 48.9 0.082 * * 17 * 52.1 * 0.032 1.80 5.00 1.00 18 867.5 35.1 24.7 0.052 * * 19 1385.5 51.8 26.7 0.055 * * 20 774.1 34.4 22.5 0.093 * * 21 * 22.9 * 0.036 0.60 0.80 22 * 3.05 * 0.032 1.80 1.00 90 ZR given in Tables 5.4 and 5.5 describe the channel cross-sections defined as shown in Figure 3.3. Tables 5.6 and 5.7 give the computation sequence for WS 105 and WS 103 respectively. The computation order of program WESP is such that all inflows required by any element at any stage of the simulation come from elements previously processed. The computational sequence shown in Tables 5.6 and 5.7 is the order for the computation of flow and sediment routing of the elements shown in Figures 5.5 and 5.6. The numbers in Columns 2 through 6 in Tables 5.6 and 5.7 indicate the linkage between elements in the computational sequence (see Appendix A for WESP variable name list). The symbol "0" is used to indicate there are no upstream inflow elements or lateral inflow elements. Input to the model consisted of measured quantities and estimated parameters. Areas and lengths of elements were measured directly from maps. Slopes were estimated by inspecting profiles drawn from topographic maps. Infiltration parameters Lumped infiltration parameters were used during the simulations. The saturated hydraulic conductivity of the soils (gravelly loams) of Watersheds 63.105 and 63.103 was assumed to be 2.0 mm/h. The moisture- pressure term, Ns, was optimized by trial and error for each individual event to fit the measured runoff volume. Values of Ns in Table 5.8 were the results of these optimizations. Surface Flow Resistance Parameters. Lumped surface roughness parameters for overland and channel flow were used in the simulations. The average Manning roughness coefficient for a plane surface with assumed turbulent flow was 0.06. The average Manning roughness coefficient for stream channels was assumed to be 0.03. 91 Table 5.6. Computational Sequence for Watershed 63.105 ELEM Contributing Channel Contributing Plane NTOP NLEFT NRIGHT NCHN1 NCHN2 1 0 0 0 0 0 2 0 0 0 0 0 3 2 0 0 0 0 4 3 0 0 0 0 5 4 0 0 0 0 6 0 0 0 0 0 7 6 0 0 0 0 8 7 0 0 0 0 9 5 1 8 0 0 10 0 0 0 0. 0 11 0 0 0 0 0 12 0 10 11 9 0 92 Table 5.7. Computational Sequence for Watershed 63.103. NTOP 1NLEFT 1 Contributing Channel Contributing Plane ELEM NRIGHT NCHN1 NCHN2 1 0 0 0 0 0 2 1 0 0 0 0 3 0 0 0 0 0 4 3 0 0 0 0 5 0 0 0 0 0 6 2 5 4 0 0 7 0 0 0 0 0 8 7 0 0 0 0 9 0 0 8 6 0 10 0 0 0 0 0 11 10 0 0 0 0 12 0 0 0 0 0 13 12 0 0 0 0 14 11 13 0 0 0 15 0 0 0 0 0 16 15 0 0 0 0 17 0 16 0 9 14 18 0 0 0 0 0 19 18 0 0 0 0 20 0 0 0 0 0 21 19 0 20 0 0 22 0 0 0 17 21 See WESP variable name list in Appendix A for explanation of variable names. r") 93 Co) 0 Cr) 0 C•1 0 Cel 0 en 0 en 0 0 0 0 0 0 VD VD 0 0 nID 0 VD 0 VD 0 ..0 0 0 . 0 . - . 0 0 0 r-- r-- 0 en 0 • r 4 0 • 4 -..1- . %.0 in . ON C11 VD v nI gnI in 0 Cl) tr Ln N- rS• •n•I un 0 .% 0 N- 0 0 ....t op In 0 • .--/- cn en tin Lin 0 0 0 0 0 0 4:, 0 %.0 0 VD 0 0 0 0 0 4 0 . cq 0 • eV 0 • en r-.• ...1- c0 -..1* en q::. I-- Ce) un . If) . VD (NI .-I inI 1n1 un vn 0 • 0 0 .-4 in 0 . 0 . Len r4 c0 0 c-1 cnI .0 Cr. 1/40 .-4 . 0 01 . 0 —1 ,./J %JD en 0 ,nI Cl)CA CA Cl)CA h. on esi r-• ce) n•••I vnI ,nI 0 ..1 I-- 0 Ch 0 as In tfl r-- 1.-- 0 In 1--- un r-—I 0 r4 01 .--I Csl ,•4 1n4 n••4 r•-• VD 0 -.4 CO en 0 Ln 0 . ...t ON en rnI ON 0 WI r•-• 0 . rn 0 r-- 0 Ce) 0 . 1n1 ••••4 0 Co) 0 0 Crl 0 . 0 en en 0 0 . 4rnI en 0 en 0 r.4 11 CA CA CA Cl) , ..o 0 0 .—I .--i 0 Ch 0 r-- In eV r-0 CO an 0 .0 r-- ON 0 %.0 1**-- 1`..N- 94 Estimation of Erosion Parameters The channel erodibility parameter, a, was optimized by trial and error to fit the estimated total sediment yield for individual events, given estimates of K I and K R obtained from optimization studies on runoff-erosion plots (see Table 5.3). A starting estimate for a (a coefficient for sediment entrainment by channel flow) was assumed to be the same as K R The final estimates of a are shown in Table 5.9. The coefficient for critical shear stress, 6, and the characteristic particle size, . ds, were assumed to be 0.047 and 0.120 mm, respectively, for all simulation runs. Storm Characteristic Data The rainfall intensities for the 10 storm events used in this study were obtained from weighing recording raingages located on the subwatersheds. Because the watersheds are very small and each watershed had only one recording raingage, each storm event used in this study was assumed to be uniformly distributed over the entire watershed area. Antecedent soil moisture conditions were not available. Test Results In the numerical computations, a time increment At of 5 seconds was chosen for the rainfall simulator plot simulations. For the small watersheds a 30-sec. time increment was used. These time steps were chosen using a stability criterion described by Rovey et al. (1977) (see equations 3.25 and 3.26). The weighting factors 0 — 0.5 and w — 0.6 were used for the rainfall simulator plots and overland flow planes. The values 0 — 0.5 and w — 0.8 were used for the channels. Test results on parameter estimation for rainfall simulator plots and small watersheds are presented in the following sections. 95 a) (NJ Ce) en .4 Lñ V) en 0 CV C4 c rr- U) 0 ." •D in 04 0 Le•N - -4 cl) a) "0 E "cl W cr) U) .1-) Do El .M c0 Cn O 0 .0 col 0 CV 1‘) CA c",1 0 --I' ce -4- .1- Crs m P.. 0 04 if) ...1* Ce) • 0. co CU 0 n••-4 (1.) 0 ) 0 en e c» 0 -.4 CUo 5 6 çtli 114 bo 00 04 ci1 .0 .-A -4 0 0 . 0 -4 0 0 . 0 0 0 0 v-1 0 0 . 0 0 0 . 0 0 00 04 n0 c0 0 0 -I 0 0 0 0 in -4--+ 0 . 0 .0 .0 0 0 . 0 00 .0 0 0 0 •,-4 co co 0 $4 14 W 4.1 rzl el 4- ) I cal) W tin cc) 0 o0 0 in eq 0 0 . 0 0 0 0 0 0 0 0 0 . 0 cNi r. un on un un v4 0 0 0 0 CV 0 0 0 0 0 CV 0 0 . 0 1/4r1/4 r- CV 0 0 1/1 0 0 a) 0 0 co CU .ra s_., WE 05-i ..., co co CU ... o) 04 .--4 ,--1 = 1 Z 04 0 0 co It) . -I 0 CU .-4- E's *---..r-HœQ I100 X ,M .-.• 0 .1- - m . .-I 0 0 en 1.1 0 0 en . r.-I 0 - en . 1-.4 0 %.10 un C4 0 --1- en 4-4 0 - en . 1.n1 - 4' 0 ...1- 0 .0 un 0 en en 1.-.1 4..4 C4 CU 0 s-, 14 0 C.) 4-1 W0 54 0 4-1 C.) .1-1• ,4 0 W o) "0 4.1 • r4 D 4-1 A-) a) 3 8 a) 3 >1/4 I-I 4.1 cl) 3 >1/4 U) a) › › .4_ 1 a) 3 >1/4 14 A .14 W 3 4.) W 3 4-) W 3 a) 3 14 A Q) CU 0 ir $.4 0 • al 3 01 it1 0 4-4 • .0 0 01 0 4-.4 1-4 m .0 Cn co 0 01 c 1/40 cs .C) 01 0 01 0 O 4-4 • m .0 0 1-4 • Cn .0 •,4 .0 0 ••-t 0 4-1 g a) •1/4 0)4-J• >1/4 4.1 0 14 clj trf r-- o rs- ,--4 r0 In I-- re) ---1 a\ CD tf) r-- C•I Is, VI •0 r--0 0 cr, CD -4 cr, CD In fs.- V) 1.-- lll r-- n.0 1--- -I cg) Cr1 0 0 .-4 ON CD .0 1"-- .-I .-I 0 C:" CD vr-- .-I in GO esi h. CD co r-- CTI 96 Simulation Results on Rainfall Simulator Plots The comparisons of the simulated and the measured hydrographs are shown in Figures 5.8(a) through 5.16(a) and sedigraphs for all rainfall simulations on the rainfall simulator plots used in this study are shown in Figures 5.8(b) through 5.16(b). As mentioned before, an experimental error resulted in sediment trapped in the flumes located at the end of the plots. The residual sediment trapped in the flumes was added to the sediment measured through the exit of the flume to give the total sediment yield. The effect of this experimental error was most evident on the clipped and natural plots as shown in Figures 5.11(b), 5.12(b), 5.13(b), and 5.16(b). The agreement between the measured hydrographs and sedigraphs was satisfactory. The variability of the entrainment (source) and deposition (sink) terms with time for different cover conditions is shown in Figures 5.17 through 5.25. The fact that the entrainment by rainfall term (e I ) has a similar behavior as the entrainment by shear stress term (e R ) makes it evident that unique parameter identification may not be possible with the rainfall simulator data set used in this study. Table 5.3 shows estimated mean values of parameters for bare, clipped, and natural surface treatments, respectively. It is alsmost certain that the parameter values obtained by trial and error until the computed sediment yield equals the measured sediment yield are not unique. An appropriate objective function incorporating possibly the sum of square of deviations of observed and computed sediment concentrations and squared deviations of total sediment yield should be formulated and minimized. Furthermore, it should be recognized that the goodness of fit of the model to the observed data is also a function of the structure of the mathematical model, the accuracy of the data used, and the method of fitting the model to the data. 97 00.0 70.8 60.8 58.0 38.8 20.0 18.8 AMM 0 0 0 0 0 0 6 6 TIMECMIN) (a) 40.8 35.8 m 30.8 S 25.0 • 20.8 = u 15.8 w 10.8 5.88 .880 aa a; TIMECHIM) (b) Figure 5.8. Dry run on bare plot: (a) hydrograph; (b) sedigraph 98 80.0 78.0 50.0 48.8 30.8 20.8 10.8 . 080 Q tri ea Q ii in cn Cu w4 TIMEMIN) (a) 40.8 35.0 30.0 25.0 28.0 15.8 10.0 5.00 .088 co co ui6 ,•1 N Timunim Q Q cn (b) Figure 5.9. Wet run on bare plot: (a) hydrograph; (b) sedigraph ▪ 99 88.8 78.0 PLOT ID: 13784 *---40EISEPVED SIMULATED 68.0 - • ........ 20.8 10.8 . 80 0 es) Cu (a) 48.0 35.8 • 30.8 m S 25.8 . 28.8 0 1..1 15.8 w 10.8 5.80 . 888 Figure 5.10. Very wet run on bare plot: (a) hydrograph; (b) sedigraph. 100 70.8 68.8 58.8 48.0 38.0 28.0 10.0 .088 m a; m m cio v. TIMECHIN) (a) 5.80 4.58 4.00 3.58 3.00 2.50 2.08 1.50 1.88 .580 .880 Figure 5.11. Dry run on clipped plot: (a) hydrograph; (b) sedigraph. 101 70.6 50.8 40.8 30.0 20.0 18.0 (a) 1 3.00 2.50 2.00 1.50 I I t t_ 1 1 1 PLOT ID: 13005 co---loMEAS.THRU FLUME SIMULATED TOTAL - - - - .. - ........... .s. ......,....- .................... - - - me I- I m co cn . co u; cn (b) Figure 5.12. Wet run on clipped plot: (a) hydrograph; (b) sedigraph. 102 78.8 68.8 50.0 40.8 38.0 28.8 10.8 .eee m m m CD 111 ru cr, m m TIMEOIM) (a) e.ee 2.58 2.88 1.58 1.08 .588 .088 Figure 5.13. Very wet run on clipped plot: (a) hydrograph; (b) sedigraph. 103 4e.e 35.8 38.8 25.8 2e.e 15.8 10.8 5.80 .eee m eo m v THEMIN) (a) 2.ee 1.88 1.60 1.40 1.28 1.08 .888 .488 .288 .888 co Cu to to ei cn ai Nt In T111 E 01111) (b) Figure 5.14. Dry run on natural plot: (a) hydrograph; (b) sedigraph. 104 58.8 45.8 48.8 35.8 38.0 25.8 28.8 15.8 10.8 5.80 .880 1.4 .200 .888 • U) m 4.4 ir; et; MI 44 co. rn TIME(MIN) (b) Figure 5.15. Wet run on natural plot: (a) hydrograph; (b) sedigraph. 105 58.8 45.0 48.8 35.8 30.0 u. 25.6 u. or as.e • 15.8 18.0 5.88 .080 m m co m m• . in m co 4.1 m ail Cu m I); .4 Cu TIHEMIN) (a) 2.80 1.80 1.68 1.40 1.28 Lee .800 .688 .488 .208 .008 W W e4 W ir; ..1 W fi; ni co. TInE(MIN) (b) Figure 5.16. Very wet run on natural plot: (a) hydrograph; (b) sedigraph. 106 . 288 PLOT ID: 13204 . 158 . 188 nnI .858 . 00 —.85 —.10 —.15 —.28 co co. CD. Figure 5.17. Entrainment and deposition rates for dry run on bare plot. .n8 PLOT ID: 13604 .150 , ........... . 180 —.05 —.18 eR ex .858 . 88 \ • WW1 d ------------------------ —.15 -.28 1 Figure 5.18. Entrainment and deposition rates for wet run on bare plot. _ 1 PLOT ID: 1 3704 t_ eg • .050 cr .88 .r4 -n N.11 0 -.65 cr -.le d co In /I ru v4 Figure 5.19. Entrainment and deposition rates for very wet run on bare plot. .180 PLOT ID: 12605 EhR . .060 .... ..... • •••••••••••• ..... .840 .828 .888 -.82 -.94 osi et) to ru co es; TIMEMIN) Figure 5.20. Entrainment and deposition rates for dry run on clipped plot. 107 ▪ 108 1 1 i 1 I i PLOT ID: 13005 .888 • ea, .868 1._ ........... „ .................................... .048 ez .820 .088 Q. E -.82 • -.04 g •••11 -.86 .. 0 -.88 d .. ... ..... -.10 • CD 60 CO CD 63 63 61 co 6; TIMECIIIM) Figure 5.21. Entrainment and deposition rates for wet run on clipped plot. .100 _1 PLOT ID: 13105 .880 .868 ............ • ................ .040 .820 .808 -.82 -.84 .111•11 -.06 -.08 -.10 1 1I CI Cf3 61•61 11; CD I 1I 1- 6; TIME(MIN) Figure 5.22. Entrainment and deposition rates for very wet run on clipped plot. 109 . 688 . 408 .288 .ee -.20 -.48 -.68 to to In TIMECIIIN) Figure 5.23. Entrainment and deposition rates for dry run on natural plot. .600 1 FLOT ID: 12807 .488 .200 .................... ................. ex .08 -.20 - •=1. d -.48 -.60 Figure 5.24. Entrainment and deposition rates for wet run on natural plot. 110 . 400 ▪200 . 00 -.20 -.40 -. 6 0 W W W W wl W.; tr; wi CU ero cu TIME(flIN) Figure 5.25. Entrainment and deposition rates for very wet run on natural plot. 111 Simulation Results on Small Experimental Watersheds The verification of the proposed watershed runoff-erosion model is demonstrated by comparing simulated and measured hydrographs and sedigraphs for individual rainfall events on Watersheds 63.105 and 63.103. The comparisons of the simulated and the measured hydrograph and sedigraph for all storm events used in the analysis are given in Figures 5.26 through 5.35. The agreement between the measured hydrographs and the simulated hydrographs was satisfactory. For the sedigraphs, the simulations were made to fit total estimated sediment yield rather then the sedigraphs which were obtained from sampled sediment by the Chickasha pump sampler. As mentioned in earlier sections, the Chickasha pump sampler does not measure total sediment load. The agreement between the observed pattern of sediment concentration and the simulated pattern of sediment concentration per event was surprising given that parameter optimization was designed to fit estimated total sediment loads and not the measured sediment concentrations from the Chickasha pump sampler. The mean values of K I' K R' and a for the three events used in this study on Watershed 63.105, and for the seven events used in this study on Watershed 63.103, with the respective estimates of sediment yield are given in Table 5.10. The error in Column 9 of Table 5.10 indicates the percentage of error on the estimates of sediment yield when using the mean values instead of the optimized parameter values per event. Figure 5.36 shows comparisons between estimated and measured sediment yields for WS 63.103 using the mean values of erosion parameters from Table 5.10. Discussion of Test Results The applicability of the proposed watershed runoff-erosion model to simulate storm event hydrographs and sedigraphs in small runoff- 112 88.0 78.0 68.0 50.8 48.0 38.0 20.0 .000 6) CII ID CD 6) • 60. CD 6) IS tr, TIMECHIM) (a) 20.0 t 18.0 14.0 0 12.8 lit I t 1_ WS 105 750705 e---0MEAS.THRU FLUME SIMULATED TOTAL r, 16.8 1 II . 18.8 o 8.00 6.80 Ui 4.80 2.88 ......... .800 cu TIMOIN) (b) Figure 5.26. Storm event of 750705 on WS 63.105: (a) hydrograph; (b) sedigraph. 113 280. 188. 160. 148. 120. 180. 88.0 68.0 40.8 28.0 .888 co. so. CU ,r4 (a) 28.8 18.8 16.0 14.8 12.8 18.8 8.88 6.88 4.00 2.80 .080 so a.) ca . ca. ea" ea TIME(MIM) (b) Figure 5.27. Storm event of 750717 on WS 63.105 (a) hydrograph; (b) sedigraph. 114 58.0 45.8 48.8 35.0 = 38.0 = 25.0 U. 2 28.0 15.8 18.8 5.88 .808 8.88 7.00 6.80 5.00 4.00 3.00 2.80 1.08 .008 a) ea cu a) la ea ea ea ea co . TIMEMIM) (b) Figure 5.28. Storm event of 750913 on WS 63.105: (a) hydrograph; (b) sedigraph. 115 tee. 90.6 80.0 78.0 68.0 50.0 48.0 30.8 20.0 10.8 .880 CD CD CD . TIME CIlIti ) (a) 58.8 45.0 48.8 35.8 38.8 25.8 28.8 /5.0 10.0 5.80 .008 (b) Figure 5.29. Storm event of 750712 on WS 63.103: (a) hydrograph; (b) sedigraph. 116 28.0 18.0 16.0 14.0 12.0 me 8.08 6.00 4.00 2.00 co Cu cu 40.8 35.0 WS 103 750907 e MEAS. THRU FLUME SIMJLATED TOTAL o 38.0 — 25.0 20.0 15.0 10.0 5.00 ..... ...... .000 .. 6 6 TIMOIN) (b) Figure 5.30. Storm event on 750907 on WS 63.103: (a) hydrograph; (b) sedigraph. 117 3e.e 25.e 2e.e 15.0 le.e 5.ee .eee T T T 513.0 45.0 40.0 35.e 30.0 25.e 2e.e 15.0 10.0 5.00 .000 T T co co 16 Cu fD 61 CD TIME(MIN) (b) Figure 5.31. Storm event of 750913 on WS 63.103: (a) hydrograph; (b) sedigraph 118 58.0 45.8 40.0 35.8 38.8 25.0 29.8 15.8 18.8 5.00 CU 50.8 45.8 40.8 35.8 38.8 25.0 28.8 15.0 10.8 5.88 .888 Figure 5.32. Storm event of 760906 on WS 63.103: (a) hydrograph; (b) sedigraph 119 14.0 12.0 is.e 8.08 6.80 4.80 2.00 .080 cu eto TIMEMIN) (a) 1 50.0 45.0 WS 103 760910 0---OMEAS.THRU FLUME 40.0 SIMULATED TOTAL 35.0 30.0 25.0 20.0 15.0 10.0 5.80 . .. ...... ... . . ..... • a) cu TIME(MIN) (b) Figure 5.33. Storm event of 760910 on WS 63.103: (a) hydrograph; (b) sedigraph 120 20.8 18.0 16. 0 14.8 12.e 18.0 8.80 6.08 4.88 2.88 .808 CD azi ru mum) (a) 30.0 25.8 28.8 15.8 18.8 5.88 .888 ea es:1 er; Cu TIMECMIN) (b) Figure 5.34. Storm event of 770901 on WS 63.103: (a) hydrograph; (b) sedigraph 121 40.0 35.8 38.8 25.0 20.8 15.0 18.8 5.00 .000 CD co CD CD. CD .4 CD. CI.1 .-4 (a) 50.0 45.0 48.8 35.0 30.8 25.8 20.8 15.0 18.0 5.80 .000 co co CD co CD CD eo 6 6 Tr CD. CD 6 co ( .1 CD. CD W CV .4 .4 TIME(MIN) (b) Figure 5.35. Storm event of 780725 on WS 63.103: (a) hydrograph; (b) sedigraph 122 ...,t . N. + OD cr) N. H e9 .-I + I./1 .-I 4" eV c0 eV ON r-. . as c-4 0 Coe 0 Ln ...I .-1 111 ...1" 1.-1 I I + I + en 0 a. N4 NJ .-1 en c0 c0 %D C en• • -A rn- 1-4 LO sn r-. 0 o qt, 0 0 en CV in O N1 0 . en en 0 e".• 01 o en ("I + • fn- co c-4r c•.1 LS en 0 0 0 0 0 0 0 0 0 0 0 C ep 0 0 en en en en en Ts 1n• 0 0 0 0 0 0 0 0 00 0 0 0 0 0 as o CU o G) . r-- -t 0 1-- -t 0 Ts 1n•• en h. Ts en --t 0 -40 ...t 0 0 0 0 -.I* 0 . 0 -t 0 -40 . 0 -t --t CC . 0 0 I,. . 0 0 0 Tn ..-1 "0 a) 3-1 co CU 0) -I -I' 0 0 0 0 0 0 0 0 . 0 --t .--I ,-.1 1-1 -.1* -I- 0 0 0 0 0 0 . 0 0 0 -t en en en -.I- -.1- 0 0 --t en 0 0 0 0 0 0 -.1en en 0 0 en 0 0 . a) ^0 • 14 (1) 00 00 CO . CO %D . CO %.0 00 n0 00 00 ‘.0 ••••1 v -4 1-4 CO CO H %D CO CO %0 . • 1-1 CU M M 0 0 0 0 C ••••4 • ••••I • en en en en r., %0 %D %.0 %0 %D N. en NJ I-- cn n10 0 0 .-.4 v--4 1.-I 0 ..--1 0 ...-1 CA as LIN 0 0 r-- 1-- 0 40 1-- N-sn 0 In N. Cr% r-- 0 0 0 N. r-- N. In in in In in • H in cr+ 0 n1:) Tn C.) en %LS H . -I til 0 N. e-1 N. 0 c0 1,.. 1n.. CA 0 o 0 .14 CO a) 123 [03 ESTIMATED FROM PUMP SAMPLER El ESTIMATED FROM TRAVERSING SLOT SAMPLER D 20000 SIMULATED 15000 10000 7 o 5000 0 z 2 3 4 5 7 EVENT NUMBER Figure 5.36. Comparison of sediment yields for seven events on Watershed WS 63.103 as estimated from measured runoff and sediment concentration and as simulated using the mean values of the erosion parameters K r KR , and a. 124 erosion plots and small watersheds was demonstrated by comparing simulated and measured storm event hydrographs and sedigraphs. Satisfactory results were obtained for different size storms and different surface treatment conditions. This verifies that the model can be used for runoff-erosion research as well as for synthesizing missing data, and predicting the response of watersheds to various types of watershed management practices. The data shown in Table 5.10 suggest the order of simulation error introduced by using mean values of the erosion parameters (K 1 , KR , a) rather than individual, and optimal values for each event. Sediment yield data from Table 5.10 are shown for seven events on Watershed 63.103 in Figure 5.36. Notice that except for the first event (Figure 5.29 and row 4 in Tables 5.9 and 5.10) the simulated yields follow the estimated yields quite well. Moreover, for the last two events where sediment concentration data on total load was measured, the simulated and measured sediment yields are in close agreement. The data shown in Table 5.9 illustrate the magnitude of variability in erosion parameter estimates required to match the estimated sediment yield. The soil erodibility parameter for rainfall impact, K 1 , varies by about a factor of 2. The soil erodibility parameter for shear stress, K R , varies by about a factor of 7, and the channel erosion parameter, a, varies by about a factor of 16. Again, notice that the erosion parameter values for the last two events (last row in Table 5.9) are quite comparable. The corresponding parameter estimates from the rainfall simulator plots (see Table 5.3) suggest that K I varies by about a factor of about 1.3 to 2 within the treatments and by about a factor of as much as 3 between the treatments. The soil erodibility parameter for shear stress varies by about a factor of 2 to 7 within treatments and by 125 about a factor of as much as 66 between treatments. Therefore the parameter variability shown in Table 5.10 for the small watersheds is consistent with parameter variability found for rainfall simulator plots. Finally, notice that K1 and KR for the bare plot (Table 5.3) are about an order of magnitude larger than those estimated for the clipped and natural plots. While the sample sizes used to represent erosion parameter estimates and their variability are too small for statistical interpretations, they do suggest the following: 1) The variation in K1 and KR is consistent from the natural and clipped plots to the small watersheds, 2) of magnitude larger on the bare plots K I and R are an order than on the clipped and natural plots and than on the two small water- values of K sheds, 3) the entrainment ratio (e i /(e i-FeR )) was 0.21, 0.17, and 0.36 on bare, clipped, and natural plots, respectively, showing that entrainment by rainfall represented 21%, 17%, and 36% of the total entrainment on bare, clipped, and natural plots, respectively, 4) It is almost certain that the erosion parameter values obtained by trial and error until the computed sediment yield equals the measured sediment yield are not unique, 5) sensitivity analysis will be necessary to look at model sen- sitivity to parameter estimation, 6) a large number of events will probably be required to quantify the mean values and the variability of the erosion parameters. In the absence of these large sample sizes, the treatments imposed on experimental plots and small watersheds will have to be severe and not subtle to reflect statistically significant differences in erosion parameters. CHAPTER 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary This study was concerned with the development and testing of a physically based mathematical model for simulating watershed response (surface runoff, erosion, and sediment yield) from field-sized areas and small watersheds. A distributed parameter, event-based, numerical model of watershed response was developed to accommodate the spatial changes in topography, surface roughness, soil properties, concentrated flow patterns, concentrated flow geometry, and land use conditions. The Green and Ampt equation with the ponding time calculation for an unsteady rain was used to compute rainfall excess rates. The kinematic wave equations were used to describe the unsteady overland and channel flow on small watersheds. This approach further requires a geometrical representation of the actual watershed which was represented in the model by a simplified configuration consisting of sequences (cascades) of discrete overland flow planes discharging into channel elements. The unsteady and spatially varying erosion/deposition process on hillslopes and channel systems was described dynamically using simultaneous rates of sediment entrainment and deposition rather than the conventional approach using steady state sediment transport functions. A modular computer program called WESP (Watershed Erosion Simulation Program) was written in FORTRAN 77 to provide the vehicle for performing the computer simulations. Parameter estimation was first conducted on rainfall simulator plots to estimate infiltration parameters for the Green and Ampt 126 127 equation, hydraulic roughness, and soil erodibility parameters for raindrop impact and runoff. The applicability of the erosion parameters estimated from rain- fall simulator plots when applied to a watershed scale under a variety of rainfall inputs and antecedent soil moisture conditions was verified using data collected on two small experimental watersheds. The small watersheds used in this study are the only scale at which data sets from real watersheds are available to conduct tests on distributed, physically based watershed response models. Finally, the modular structure of the computer program developed during this study will facilitate substitution of different components and subroutines in the future as research improves understanding of the processes controlling hydrologic response on field-sized areas and small watersheds. The following sections present the conclusions and recommenda- tions for further research based on the model development and model testing procedures conducted during this study. Conclusions The following conclusions can be drawn based on the model development, parameter estimation, and model testing results: 1. The form of the source (entrainment) and sink terms for the equations describing conservation of sediment mass on hillslopes and channel systems are mathematically consistent and incorporate appropriate initial and upstream boundary sediment concentrations. 2. The watershed response model developed during this study and described in Chapter 3 simulated the hydrologic response (hydrograph and sedigraph) of rainfall simulator plots and small upland watersheds very 128 well. The good agreement between the simulated response and the observed response was due to parameter fitting. However, the shape of the simulated and observed responses were not optimized and their agreement indicates that the governing equations and structural framework of the model can satisfactorily describe the processes of surface runoff, erosion and deposition on hillslopes and small watersheds. 3. The unsteady and spatially varying erosion and deposition processes can be described as a system with a continuous exchange of particles between the flow and the loose boundary (soil surface on hillslopes and stream beds in stream channels). Simultaneous sediment entrainment (detachment) rates by shear stress on hillslopes can be described by a relationship expressing the entrainment rate as proportional to a power of the average shear stress acting on the soil surface. Simultaneous sediment entrainment rate by rainfall impact can be described by a relationship including rainfall intensity as a measure of the erosivity of raindrop impact, and rainfall excess to reflect the sediment transport rate by shallow flow on hillslopes. Simultaneous sediment deposition rate (settlement) can be represented by a relationship including particle fall velocity and the sediment concentration in transport. 4. Although test results conducted in a single rill support the hypothesis of a threshold for initiation of particle entrainment by rill flow, all suggested values are not applicable to hillslope erosion model- ing when using the broad sheet-flow approach. In this study it was shown that the threshold for initiation of particle entrainment by sheet-flow can be neglected when modeling erosion on hillslopes. 5. Water and sediment routing was performed using numerical procedures. A four-point implicit finite difference scheme was used for 129 water and sediment routing. The implicit finite difference schemes used are unconditionally stable with respect to choice of values for time step and distance step. However, values for these variables must be carefully chosen to ensure satisfactory accuracy. The approach described by Rovey et al. (1977) can be used as an approximate stability criterion for selecting the time step given the distance step (see equations 3.25 and 3.26). 6. The watershed model produced satisfactory results but the lack of information on erosion and deposition parameters (they need to be determined by experiment and optimization) and the current form of the model limit the application of the model as a design tool. However, the model does have application in further research and also as a comparative tool or a "benchmark" for eavlauation of alternative and simplified models of watershed response. Recommendations Recommendations for future research can be made based upon the above conclusions and upon experience gained during the model development and simulation runs. More research into parameter estimation and model validation is needed if physically based mathematical models such as the one developed in this study are to be used to simulate hydrologic response from fieldsized areas or small watersheds. Information is needed to obtain accurate estimates of parameter values for the infiltration equation using soil texture, soil profile characteristics, ground cover, and management factors. The influence of spatial variability on infiltration parameters is yet to be determined. The infiltration component of the model should be extensively tested on field data for a variety of soil and cover conditions. Initial soil moisture content should be estimated using a water 130 balance model. Development of a technique for disaggregation of daily rainfall into rainfall intensity patterns would extend the applicability of this model to a larger number of watersheds for which there are extensive records of daily rainfall but no data on rainfall intensities. Further information is needed on the hydraulic roughness coefficient in overland flow routing as related to ground cover and management effects such as grazing. More research is needed into the estimation of erosion and deposition parameters on hillslopes and channel systems. It was obvious from the simulation runs that accurate values for entrainment and deposition parameters are crucial to erosion/deposition simulations. Therefore, it is recommended that a field data collection program be developed to collect basic erosion/deposition and hydraulic roughness data from rainfall simulator plots and small experimental watersheds. A major research need on field experiments is on how to lump the processes from plot to hillslope and then to watershed scales. Extension of the model to particle size distribution rather than a single characteristic particle size is recommended. This extension would allow separation of parameter values into the appropriate particle types rather than using average values. Finally, more research is necessary to continue the development of modeling techniques to describe more accurately the hydrologic response of field-sized areas and small watersheds. APPENDIX A WESP VARIABLE NAME LIST 131 132 PARAMETERS in WESP: PARAMETER (MAXSTOR-5000, MAXSTEP-500, MAXSET-5) MAXSTOR : The size of dynamic storage arrays QSTORE and QSSTORE, MAXSTEP : The maximum number of time steps in the simulation, MAXSET : The maximum number of rainfall intensity patterns per water shed during an event VARIABLES in WESP: Explanation of symbols on "Usage": I — Input variable (does not change in subroutines), 0 — Output variable (changed in subroutines), I/O — Computed inside the program given input (changed in subroutines), INT — INTernal (used for internal control and calculations) VARIABLES IN COMMON /CONTROL/: Definition Type Usage NSIM I*4 INT Number of time steps (Time driven loop), IFULL I*4 INT Maximum number of storage blocks available Variable in arrays QSTORE and QSSTORE, ITOP I*4 INT Index to outflow from element specified by NTOP(stored in arrays QSTORE and QSSTORE), ILAT I*4 INT Index for combined lateral inflow (stored in arrays QSTORE ans QSSTORE), IOUT I*4 INT Index to storage location for computed outflow in arrays QSTORE and QSSTORE, DT R*4 I Time step, DX R*4 I/O Step size, 133 NK I*4 I/O Number of step sizes, ET R*4 I/O Elapsed time for transmission losses, I,J I*4 INT Loop counters VARABLES IN COMMON /PARAM/: Variable Type Usage DURAT R*4 I Definition Duration (in minutes) of the event simulation GLEN R*4 I Characteristic length (longest cascade of planes), XLENGTH R*4 I Length of a plane or channel element WIDTH R*4 I Width of a plane (WIDTH-0 indicates a channel), Longitudinal slope of a plane or channel, SLOPE R*4 I ALPHA R*4 I/O Slope-resistance coefficient, POWER R*4 I/O Exponent in discharge equation, ABASIN R*4 I THETA R*4 I/O Area of watershed, Spatial weighting factor in the numerical equations, OMEGA R*4 I/O Temporal weighting factor in the numerical equations, NTOP I*4 I ID number of the plane contributing to the top of the current plane or channel, NLEFT I*4 I ID number of the plane contributing to the left bank of the current channel, NRIGHT I*4 I ID number of the plane contributing to the right bank of the current channel, 134 NCHN1 I*4 I ID number of the contributing upstream channel segment, NCHN2 I*4 I ID number of a second upstream channel converging with the first, ZL R*4 I Slope of left side of channel (COTAN of angle to horizontal), ZR R*4 I Slope of right side of channel (COTAN of angle to horizontal), BOTTOM R*4 I Bottom width of channel. VARIABLES IN COMMON /STORE/: Variable Type Definition Usage ILIST I*4 INT QSTORE R*4 0 Array for element ID's Large array partined into blocks for dynamic storage of all computed outflows, QSSTORE R*4 0 Large array partioned into blocks for dynamic storage of all computed sediment outflows. VARIABLES IN COMMON /PLN/: Variable Type Usage NPRECP I*4 I/O NSET I*4 I Definition Number of rainfall breakpoints, Number of different rainfall patterns occurring on the watersherd during an event, M I*4 I TPRECP R*4 I/O Index for rainfall pattern, Time corresponding to a given rainfall intensity, 135 PRECP R*4 I/O RE R*4 0 Rainfall excess rate, KS R*4 I Saturated hydraulic conductivity, NS R*4 I Moisture-tension parameter, FF R*4 0 Infiltration rate, H1 R*4 I/O Rainfall intensity, Flow depth on the plane at previous time step, H2 R*4 I/O Flow depth on the plane at current time step, Ql R*4 I/O Flow rate on the plane or channel at previous time step, Q2 R*4 I/O Flow rate on the plane or channel at current time step. VARIABLES IN COMMON /CHN/: Definition Variable Type Usage QLAT1 R*4 I/O Lateral inflow at previous time step, QLAT2 R*4 I/O Lateral inflow at current time step, QT0P1 R*4 I/O Inflow from top of channel at previous time step, QT0P2 R*4 I/O Inflow from top of channel at current time step, Al R*4 I/O Flow cross-sectional area at previous time step, A2 R*4 I/O Flow cross-sectional area at current time step, CO1 R*4 I/O Geometric parameter for channel crosssectional area, 136 CO2 R*4 I/O Geometric parameter for channel crosssectional area, QSLAT1 R*4 I/O Sediment lateral inflow at previous time step, QSLAT2 R*4 I/O Sediment lateral inflow at current time step, QSTOP R*4 I/O Sediment inflow from the top of element at current time step, HYRAD1 R*4 I/O Hydraulic radius at previous time step, HYRAD2 R*4 I/O Hydraulic radius at current time step, TOPWD1 R*4 I/O Top width at previous time step, TOPWD2 R*4 I/O Top width at current time step. VARIABLES IN COMMON /SED/: Variable Definition Type Usage VISC R*4 I Kinematic viscosity of water, GRAV R*4 I Acceleration of gravity, GAMWAT R*4 I Specific weight of water, GAMSED R*4 I Specific weight of sediment, SEDSIZE R*4 I Representative particle size, CLAMBDA R*4 I Parameter for critical shear stress, RKI R*4 I Paramerter for sediment entrainment by raindrop impact, RKR R*4 I Parameters for sediment entrainment by shear stress, Cl R*4 I/O Sediment concentration at previous time step, C2 R*4 I/O Sediment concentration at current time step, 137 ER1 R*4 I/O Sediment entrainment by overland or channel flow at previous time step, ER2 R*4 I/O Sediment entrainment by overland or channel flow at current time step, SHEAR R*4 I/O Shear stress, SHEARC R*4 I/O Critical shear stress, FALLVEL R*4 I/O Particle fall velocity (Computed with Rubey's equation), El R*4 I/O Sediment entrainment by rainfall impact. APPENDIX B WESP OUTPUT SAMPLE 138 139 STORM EVENT OF 750717 IN WS 63.105 TIME (MIN) INFIL (MM/H) EXCESS RUNOFF SEDIMENT (MM/H) (MM/H) (Kg/m**3) 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00 26.8428 7.8291 6.0332 5.2172 4.7276 4.3930 4.1464 3.9551 3.8014 3.6745 3.5675 2.5400 0.5410 0.5410 0.5410 0.5410 0.5410 0.5410 0.0000 0.0000 0.0000 0.0000 1.2700 1.2700 0.0000 11.2572 159.8109 108.2668 73.7828 117.1924 117.5269 56.8136 79.8649 11.4386 3.9455 0.2425 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 65.7969 143.0770 80.6426 103.2103 106.3364 85.2722 63.8718 35.0123 14.3902 5.9207 2.4769 1.2433 0.7040 0.4347 0.2873 0.2003 0.1458 0.1097 0.0850 0.0673 0.0544 0.0446 0.0372 0.0314 SEDIMENT YIELD (KG) - 1500.6393 RUNOFF VOLUME (MM) - 59.6531 RUNOFF PEAK (MM/H) - 143.8512 0.0000 12.6406 12.0977 9.9383 11.2996 11.3239 9.8412 9.6330 8.0941 6.7484 5.5368 4.3143 3.3702 2.6757 2.1753 1.8148 1.5506 1.3523 1.1998 1.0799 0.9836 0.9047 0.8392 0.7838 0.7365 REFERENCES CITED Ackers, P., and White, W. R., 1973. Sediment transport: New approach and analysis. Journal of the Hydraulics Division, Proc. ASCE, 99(HY11): 2041-2060. Agricultural Research Service, 1983. Agricultural Research Service Program Plan, USDA, ARS, 32p. Ahuja, L. R., 1974. Applicability of the Green-Ampt approach to water infiltration through surface crust. Soil Sci., 118(5):283-288. Alonso, C. V., and DeCoursey, D. G., 1983. Small watershed model. Proc. of the Natural Resources Modeling Symposium, U.S. Department of Agriculture, Agricultural Research Service, Pingree Park, CO, pp.4046. American Society of Civil Engineers (ASCE), 1977. Sedimentation Engineering. ASCE Manuals and Reports on Engineering Practice, No. 54, Vanoni, V. A. (Editor), New York, NY, 745 pp. Amorocho, J., and Orlob, G. T., 1961. Nonlinear analysis of hydrologic systems. University of California, Water Resources Center, Contrib. No. 40, 147 pp. Beasley, D. B., Huggins, L. F., and Monke, E. J., 1980. ANSWERS: a model for watershed planning. Trans. Am. Soc. Agr. Eng., 23(4):938-944. Bennett, J. P., 1974. Concepts of mathematical modeling of sediment yield. Water Resources Research, 10(3):485-492. Bogardi, J., 1978. Sediment transport in alluvial streams. Akademiai Kiado, Budapest, 826 pp. Borah, D. K., Prasad, S. N., and Alonso, C. V.,1980. Kinematic wave routing incorporating shock fitting. Water Resources Research, 16(3): 529-541. Borah, D. K., Alonso, C. V., and Prasad, S. N., 1981. Stream Channel Stability, USDA Sedimentation Laboratory, Oxford, Miss., Appendix I. Bouwer, H., 1969. Infiltration of water into non uniform soil. Journal of the Irrigation and Drainage Division., Proc. ASCE, 95(IR4):451-462. Brakensiek, D. L., 1967a. Kinematic flood routing. Trans. Am. Soc.Agr. Eng., 10(3):310-343. Brakensiek, D. L., 1967b. A simulated watershed flow system for hydrograph prediction: a kinematic application. Proc. of the Int'l. Hydrology Symposium, Fort Collins, CO. 140 141 Brakensiek, D. L., 1977. Estimating the effective capillary pressure in the Green and Ampt infiltration equation. Water Resources Research, 13(3):680-682. Brakensiek, D. L., and Onstad, C. A., 1977. Parameter estimation of the Green and Ampt infiltration equation. Water Resources Research, 13(6):1009-1012. Brakensiek, D. L., and Rawls, W. J.,1982. An infiltration based runoff model for a standardized 24-hour rainfall. Trans. Am. Soc. Agr. Eng., 25(6):1607-1611. Brakensiek, D. L., and Rawls, W. J., 1983. Agricultural management effects on soil water processes. Part II: Green and Ampt parameters for crusting soils. Trans. Am. Soc. Agr. Eng., 26(6):1753-1757. Brakensiek, D. L., Engleman, R. L., and Rawls, W. J., 1981. Variation within texture classes of soil water parameters. Trans. AM. Soc. Agr. Eng.,24(2):335-339. Bruce, R. R., Harper, L. A., Leonard, R. A., Snyder, W. M., and Thomas, A. W., 1975. A model for runoff of pesticides from small upland watersheds. Journal of Environmental Quality, 4(4):541-548. Childs, E. C., and Bybordi, M., 1969. The vertical movement of water in a stratified porous material. 1- Infiltration. Water Resources Research, 5(2):446-459. Chow, V. T., 1959. Open-channel hydraulics. Mc-Graw Hill Book Co., New York, NY, 680 pp. Chow, V. T., and Ben-Zvi, A., 1973. Hydrodynamic modeling of two-dimensional watershed flow. Journal of the Hydraulic Division. Proc. ASCE, 99(HY11):2023-2040. Chu, S. T., 1978. Infiltration during an unsteady rain. Water Resources Research, 14(3):461-466. Cook, H. L., 1936. The nature and controlling variables of the water erosion process. Soil Sci. Soc. of America, Proc., 1:487-494. Crawford, N. H., and Donigian, A. S.,Jr.,1973. Pesticide transport and runoff model for agricultural lands. Report No. EPA 660/274-013, Office of Research and Development, U.S. Environmental Protection Agency, Washington, DC. Crawford, N. H., and Linsley, R. K.,1962. The synthesis of continuous streamflow hydrographs on a digital computer. Technical Report No. 12, Stanford University, Dept. of Civil Eng., Stanford, California. Croley, T. E., II, 1982. Unsteady overland sedimentation. Journal of Hydrology, Elsevier Scientific Publishing Company, 56(1982):325-346. Croley ,T. E.,II, and Foster, G. R., 1984. Unsteady sedimentation in nonuniform rills. Journal of Hydrology, Elsevier Scientific Publishing Company, 70(1984):101-122. 142 Cunge, J. A., 1969. On the subject of a flood propagation computation method. (Muskingum Method). Journal of Hydraulic Research, 7(2):205230. Davis, S. S., 1978. Deposition of nonuniform sediment by overland flow on concave slopes. M. S. Thesis. Purdue University, West Lafayette, IN, 137 pp. Dawdy, D. R., and Vanoni, V. A., 1986. Modeling alluvial channels. Water Resources Research, 22(9):71S-81S. Dawdy, D. R., Lichty, R. W., and Bergmann, J. M., 1972. A rainfall-runoff simulation model for estimation of flood peaks for small drainage basins. USGS Prof. Paper No. 506-B. Donigian, A. S.,Jr., and Crawford, N. H., 1976. Modeling pesticides and nutrients on agricultural lands. Report No. EPA 600/2-76-043, Environmental Protection Technology Series, Office of Research and Development, U.S. Environmental Protection Agency, Washington, DC, 317p. Eagleson, P. S., 1972. Dynamics of flood frequency. Water Resources Research, 8(4):878-898. Einstein, H. A., 1968. Deposition of suspended particles in a gravel bed. Journal of the Hydraulics Division, Proc. ASCE, 94(HY5):1197-1205. Ellison,W.D., 1947. Some effects of raindrops and surface flow on soil erosion and infiltration. Trans. Am. Geophys. Union 26(3):415-429. Engelund, F., and Hansen, E., 1967. A monograph on sediment transport in alluvial streams. Teknish Vorlag, Copenhagen, 62 pp. Foster, G. R., 1976. Sedimentation, general. Proc. of the National Symp. on Urban Hydrology, Hydraulics, and Sediment Control. Univ. of Kentucky, Lexington, Ken. Foster, G. R., 1982. Modeling the erosion process. In: C. T. Haan, H. P. Johnson, and D. C. Brakensiek (editors), Hydrologic modeling of small watersheds, Am. Soc. Agr. Eng.,monograph No. 5, 259-380. Foster, G. R. and Meyer, L. D., 1972. A closed-form soil erosion equation for upland areas. In: H.W.Shen (Editor), Sedimentation: Symposium to Honor Professor H.A.Einstein, Fort Collins, CO, Chap. 12:12.1-12.19. Foster, G. R., and Meyer, L. D., 1975. Mathematical simulation of upland erosion using fundamental erosion mechanics. In: Present and prospective technology for predicting sediment yields and sources. USDAARS, ARS-S-40, pp. 190-207. Foster, G. R., Meyer, L. D., and Onstad, C. A., 1977. An erosion equation derived from basic erosion principles. Trans. Am. Soc. Agr. Eng., 20(4):683-687. Frere, M. H., Onstad, C. A., and Holtan, H. N., 1975. ACTMO, an agricultural chemical transport model. ARS-H-3, U.S. Department of 143 Agriculture, Agricultural Research Service, U.S. Government Printing Office, Washington, DC. Graf, W. H., 1971. Hydraulics of sediment transport. McGraw Hill Book Co., New York, NY, 544 pp. Green, W. H., and Ampt, G. A., 1911. Studies on soil physics, I. The flow of air and water through soils. Journal Agr. Sci., 4(1):1-24. Harley, B. M., Perkins, F. E., and Eagleson, P. S., 1970. A modular distributed model catchment dynamics. MIT , R. M. Parsons Laboratory for Water Resources and Hydrodynamics, Cambridge, Mass., Report No. 133. Henderson, F. M., 1963. Flood waves in prismatic channels. Journal of the Hydraulic Division, Proc. ASCE, 89(HY4):39-69. Henderson, F. M., and Wooding, R. A., 1964. Overland flow and groundwater flow from a steady rainfall of finite duration. Journal of Geophys. Research, 69(8):1531-1540. Hillel, D., 1971. Soil and water: physical principles and processes. Academic Press, New York, 288 pp. Hillel, D., 1980. Fundamentals of soil physics. Academic Press, New York, 438 pp. Hillel, D., and Gardner, W. R., 1970. Transient infiltration into crusttopped profiles. Soil Sci., 109:69-76. Hjelmfelt Jr., A. T., Piest, R. F. ,and Saxton, K. E., 1975. Mathematical modeling of erosion on upland areas. Proc.of the 16th Congress, International Assoc. for Hydraulic Research (IAHR),Sao Paulo, Brasil, 2:40-47. Holtan, H., N., and Lopez, N. C., 1971. USDAHL-70 model of watershed hydrology. Technical Bulletin No. 1435, U.S. Department of Agriculture. Horton, R. E., 1940. Approach toward a physical interpretation of infiltration capacity. Soil Sci. Soc. of America, 5:399-417. Huggins, L. F., and Monke, E. J., 1966. The mathematical simulation of the hydrology of small watersheds. Purdue University, Water Resources Research Center, Technical Report No. 1. 130 pp. James, L. G., and Larson, C. L., 1976. Modeling infiltration and redistribution of soil water during intermittent application. Trans. Am. Soc. Agr. Eng., 19(3):482-488. Khanbilvardi, R. M., Rogowski, A. S., and Miller, A. C., 1983. Modeling upland erosion. Water Resources Bull., 19(1):29-35. Kibler, D. F., and Woolhiser, D. A., 1970. The kinematic cascade as a hydrologic model. Hydrologic Papers No. 39, Colorado State University, Fort Collins, CO, 27pp. 144 Kilinc, M., and Richardson, E. V., 1973. Mechanics of soil erosion from overland flow generated by simulated rainfall. Hydrology Paper No. 63, Colorado State Univ., Fort Collins, CO, 54 pp. Knisel, W. G.,Jr,(editor), 1980. CREAMS: a field scale model for chemicals, runoff, and erosion from agricultural management systems. Conservation Research Report No. 26, U.S. Dept. Agr., 643pp. Komura, S., 1976. Hydraulics of slope erosion by overland flow. Journal of the Hydraulic Division, Proc. ASCE, 102(HY10):1573-1586. Kostiakov, A. N., 1932. On the dynamics of the coefficient of water-percolation in soils and on the necessity for studying it from a dynamic point of view for purposes of amelioration. Trans. 6th Corn. Int. Soc. Soil Sci., Moscow, Part A:17-21. Lane, L. J., and Shirley, E. D., 1982. Modeling erosion in overland flow. Proc. Workshop on Estimating Erosion and Sediment Yield on Rangelands, USDA-ARS, Agricultural Reviews and Manuals, ARM-W-26, Tucson, AZ, pp. 120-128. Lane, L. J., and Shirley, E. D.,1985. Erosion and sediment yield equations: solutions for overland flow. Paper presented at the Workshop on USLE Replacement, Nat. Soil Erosion Lab., West Lafayette, IN, 22pp. Lane, L. J., and Woolhiser, D. A., 1977. Simplifications of watershed watershed geometry affecting simulation of surface runoff. Journal of Hydrology, Elsevier Scientific Pub., 35(1977):173-190. Li, R. M., 1979. Water and sediment routing from watersheds. In: H. W. Shen (editor), Modeling of Rivers. Wiley-Interscience, New York, pp. 9-1-9-88. Li, R. M., Simons, D. B., and Stevens, M. A., 1975. Nonlinear kinematic wave approximation for water routing. Water Resources Research, 11(2):245-252. Liggett, J. A., and Cunge. J. A., 1975. Numerical methods of solution of the unsteady flow equations. In: Mahmood, K., and Yevjevich, V.,(Editors), Unsteady flow in open channels. Water Resources Publications, 89-180. Lighthill, M. H., and Whitham, G. B., 1955. On kinematic waves, I. flood movement in long rivers. Proc. Royal Soc. of London, Series. A., 229:281-316. Lloyd, C. H.,and Eley, G. W., 1952. Graphical solution of probable soil loss formula for northeastern region. Journal of Soil and Water Cons. 7:189-191. Mehta, A. J., 1983. Characterization tests for cohesive sediments. In: H.T. Shen (Editor), Proc. of the Conference on Frontiers in Hydraulic Engineering. ASCE/MIT, Cambridge, Mass., 79-84. 145 Mein, R. G. and Larson, C. L., 1973. Modeling infiltration during a steady rain. Water Resources Research, 9(2):384-394. Meyer, L. D., and Wischmeier, W. H., 1969. Mathematical simulation of the process of soil erosion by water. Trans. American Soc. of Agr. Eng., 12(6):754-758,762. Meyer, L. D., Alonso, C. V., and Harmon, W. C., 1983. Modeling soil losses from nearly flat fields. American Soc. of Agr. Eng. paper No. 83-2091, 15pp. Morel-Seytoux, H. J., 1973. Two phase flows in porous media. Advances in Hydrosciences, 9:119-202. Musgrave, G. W., 1947. The quantitative evaluation of factors in water erosion : a first approximation. Journal of Soil and Water Cons. 2(3):133-138. Negev, M. A., 1967. Sediment model on a digital computer. Technical Report No. 76, Stanford University, Dept. of Civil Eng., Stanford, California. Nelder, J. A., and Mead, R., 1965. A simplex method for function minimization. The Computer Journal, 7(4):308-313. Nicks, A. D., 1974. Stochastic generation of the occurrence, pattern, and location of maximum amount of daily rainfall. Proc. of of the Symposium on Statistical Hydrology, Tucson, Arizona, pp.154-171. Onstad, C. A., and Foster, G. R., 1975. Erosion modeling on a watershed. Trans. of Am. Soc. Agr. Eng., 18(2):288-292. Osborn, H. B., Unkrich, C. L. and Frykman, L., 1985. Problems of simplification in hydrologic modeling. In: Hydrology and Water Resources in Arizona and the Southwest. Proc. of the 1985 meeting of the Arizona Section of the American Water Resources Ass , and the Hydrology Section of the Arizona-Nevada Academy of Science, Las Vegas, Nevada, 15:7-20. Philip, J. R., 1969. Theory of infiltration. Advances in Hydroscience, 5:215-296. Rawls, W. J., Brakensiek, D. L., and Saxton, K. E., 1982. Estimation of soil water properties. Trans. Am. Soc. Agr. Eng., 25(5):1316-1320, 1328. Rawls, W. J., Brakensiek, D. L., and Soni, B., 1983. Agricultural management effects on soil water processes. Part I: Soil water retention and Green and Ampt infiltration parameters. Trans. Am. Soc. Agr. Eng., 26(6):1747-1752. Renard, K. G., Simanton, J. R., and Osborn, H. B., 1974. Applicability of the universal soil loss equation to semiarid rangeland conditions in the Southwest. In: Hydrology and Water Resources in Arizona and the Southwest. Proc. of the 1974 meeting of the Arizona Section of the 146 American Water Resources Ass , and the Hydrology Section of the Arizona Academy of Science, Flagstaff, Arizona, 4:18-31. Renard, K. G., Simanton, J. R., and Fancher, C. E., 1986. Small watershed automatic water quality sampler. Proc. of the 4th Fed. Interagency Sedimentation Conference. Las Vegas, NE, Vol.1, 1-51-1-58. Ritchie, J. T., 1972. A model for predicting evaporation from a row crop with incomplete cover. Water Resources Research, 8(5): 1204-1213. Rose, C. W., 1985. Developments in soil erosion and deposition models. In: Advances in Soil Science, Springer-Verlag, New York, 2:1-63. Rovey, E. W., Woolhiser, D. A., and Smith, R. E., 1977. A distributed kinematic model of upland watersheds. Hydrologic Papers No. 93, Colorado State University, Fort Collins, CO, 52pp. Rubin, J., 1966. Theory of rainfall uptake by soils initially drier than their field capacity and its applications . Water Resources Research, 2(4):739-749. Schaake, J. C., 1970. Modeling urban runoff as a deterministic process. Proc. of the Institute on Urban Water Systems, Fort Collins, CO, 343-383. Shen, H. W. (Editor), 1971. River Mechanics. Colorado State University, Fort Collins, CO, 2 vols. Shirley, E. D. and Lane, L. J., 1978. A sediment yield equation from an erosion simulation model. In: Hydrology and Water Resources in Arizona and the Southwest. Proc. of the 1978 meeting of the Arizona Section of the American Water Resources Ass. and the Hydrology Section of the Arizona Academy of Science, Flagstaff, Arizona, 8:90-96. Simanton, J. R., and Osborn, H. B., 1983. Runoff estimates for thunderstorm rainfall on small rangeland watersheds. Hydrology and Water Resources in Arizona and the Southwest, Office of Arid Land studies, University of Arizona, Tucson, AZ, 13:9-15. Simanton, J. R., and Renard, K. G., 1982. Seasonal change in infiltration and erosion from USLE plots in Southeastern Arizona. Hydrology and Water Resources in Arizona and the Southwest, Office of Arid Land Studies, University of Arizona, Tucson, AZ, 12:37-46. Simons, D. B., Li, R. M., and Stevens, M. A., 1975. Development of models for predicting water and sediment routing and yield from storms on small watersheds. USDA Forest Service, Rocky Mountain Forest and Range Experiment Station. 130p. Simons, D. B., Li, R. M., and Ward, T. J., 1977. A simple procedure for estimating on-site soil erosion. Proceedings of the International Symp. on Urban Hydrology, Hydraulics, and Sediment Control, University of Kentucky, Lexington, Kentucky, 95-102. 147 Singh, V. P., 1983. Analytic solutions of kinematic equations for erosion on a plane: II. Rainfall of finite duration. In: Advances in Water Resources, 6:88-95. Singh, V. P. and Prasad, S. N., 1982. Explicit solutions to kinematic equations for erosion on an infiltrating plane. In: V. P. Singh (Editor), Modeling Components of Hydrologic Cycle. Proc. of the International Symposium on Rainfall-Runoff Modeling, Mississippi State University. Water Resources Publication,Littleton, CO, 515538. Singh, V. P., and Woolhiser, D. A., 1976. A nonlinear kinematic wave model for watershed surface runoff. Journal of Hydrology, Elsevier Scientific Publ., 31(1976):221-243. Smith, D. D., 1941. Interpretation of soil conservation data for field use. Agr. Eng. 22:173-175. Smith, D. D. and Whitt, D. M., 1948. Evaluating soil losses from field areas. Agr. Eng. 29:394-396. Smith, R. E., and Woolhiser, D. A., 1971. Overland flow on an infiltrating surface. Water Resources Research, 7(4):899-913. Smith, R. E., 1976. Simulating erosion dynamics with a deterministic distributed watershed model. Proc. of the 3th Fed. Inter-Agency Sedimentation Conference, Denver, CO, 1.163-1.173. Smith, R. E., 1981. A kinematic model for surface mine sediment yield. Trans. of Am. Soc. Agr. Eng., 24(6):1508-1514. Smith, R. E., and Knisel, W. G., 1983. Summary of methodology in the CREAMS2 model. Proc. of the Natural Resources Modeling Symposium, U.S. Dept. of Agriculture, Agricultural Research Service, Pingree Park, CO, pp.33-36. Smith, R. E., and Parlange, J. Y., 1978. A parameter-efficient hydrologic infiltration model. Water Resources Research, 14(3): 533-538. Swartzendruber, D., 1974. Infiltration of constant-flux rainfall into soil as analyzed by the approach of Green and Ampt. Soil Sci., 117(5):272-281. Van Doren, C. A. ,and Bartelli, J. J., 1956. A method of forecasting soil losses. Agr. Eng. 37:335-341. Van Duin, R. H. A., 1955. Tillage in relation to rainfall intensity and infiltration capacity of soils. Netherlands Journal of Agr. Sci., 3:182-186. Van Liew, M. W., and Saxton, K. E., 1984. Dynamic flow resistance for hydrologic simulations. Journal of Hydraulic Engineering, Proc. ASCE, 110(12):1719-1732. Vanoni, V. A., 1984. Fifty years of sedimentation. Journal of Hydraulic Rngineering, Proc. ASCE, 110(8):1022-1057. 148 Weinmann, P. E., and Laurenson, E. M., 1979. Approximate flood routing methods: a review. Journal of the Hydraulic Division, Proc. ASCE, 105(HY12):1521-1536. Williams, J. R., 1983. The EPIC model - an overview. Proc. of the Natural Resources Modeling Symposium, U.S. Department of Agriculture, Agricultural Research Service, Pingree Park, CO, pp.23-30. Williams, J. R., 1975. Sediment-yield prediction with universal equation using runoff energy factor. In: Present and Prospective Technology for Predicting Sediment Yields and Sources, Proc. of Sediment-yield Workshop, U.S. Dept. Agr., Sedimentation Lab., Oxford, MI, 244-252. Williams, J. R., and Berndt, H. D., 1977. Sediment yield prediction based on watershed hydrology. Trans.of Am. Soc. Agr. Eng., 20(6): 11001104. Williams, J. R., and Nicks, A. D., 1983. SWRRB, a simulator for water resources in rural basins: an overview. Proc. of the Natural Resources Modeling Symposium, U.S. Department of Agriculture, Agricultural Research Service, Pingree Park, CO, pp.17-22. Wischmeier, W. H., and Smith, D. D., 1960. A universal soil-loss estimating equation to guide conservation farm planning. Trans. of the 7th Cong. Int. Soil Sci. Soc., 1:418-425. Wischmeier, W. H., and Smith, D. D., 1965. Predicting rainfall-erosion losses from cropland east of the Rocky Mountains. Agr. Handbook No. 282, U.S. Dept. Agr., Washington, D.C., 47pp. Wischmeier, W. H., and Smith, D. D., 1978. Predicting rainfall erosion losses - a guide to conservation planning. Agr. Handbook No. 537, U.S. Dept. Agr., Washington, D.C., 58pp. Wooding, R. A., 1965. A hydraulic model for the catchment-stream problem. I. Kinematic wave theory. Journal of Hydrology, 3(1965):254-267. Woodruff, N. P., and Siddoway, F. H., 1965. A wind erosion equation. Soil Sci. Soc. of America., 29(5):602-608. Woolhiser, D. A., and Liggett, J. A., 1967. Unsteady, one-dimensional flow over a plane - the rising hydrograph. Water Resources Research, 3(3):753-771. Yalin, M. S., 1963. An expression for bedload transportation. Journal of the Hydraulic Division, Proc. ASCE, 89(HY3):221-250. Yang, C. T., 1973. Incipient motion and sediment transport. Journal of the Hydraulic Division, Proc. ASCE, 99(HY10):1679-1704. Zingg, A. W., 1940. Degree and length of land slope as it affects soil loss in runoff. Agr. Eng. 21:59-64.