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SOIL EROSION BY OVERLAND FLOW WITH RAINFALL BY RAUL EMILIO ZAPATA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987 Copyright 1987 by Raul Emilio Zapata To Carmencita, Raul Enrique and Manri Luz ACKNOWLEDGMENTS I want to express my most sincere appreciation to Dr. Bent A. Christensen, committee chairman, for the direction, advice and assis tance which he has given to me throughout my graduate studies at the University of Florida. His knowledge, moral support and patient guidance helped me to complete this study. Thanks are due to Dr. E. R. Lindgren for his teaching lessons in fluid mechanics and for serving on the supervisory committee. Thanks are also extended to Dr. L. H. Motz for serving on the super visory committee. Special thanks are extended to the University of Puerto Rico, Mayaguez Campus, for providing me the opportunity to improve my know ledge, securing me a leave of absence and financial support through out my studies. Thanks are also extended to the Center for Instruc tional Research Computing Activities at the University of Florida for the use of their facilities. Thanks are extended to Gail Luparello and Irma Smith for the quality typing, and also to Katarzyna Piercey and her husband, Dr. R. Piercey, for their beautiful drawings. To nmy wife, Maria del Carmen, whose loving support and encour agement has always inspired me and allowed me to have beautiful experiences with our children, Raul Enrique and Mari Luz, I give nmy deepest love, appreciation, and respect. I am also very grateful to my parents and family for their patience and understanding during this period of our lives. I wish to express nmy deep appreciation to Dr. L. Martin, Dr. F. Fagundo and their families for the friendship, guidance and help which they have provided to me and my family during our stay in this natural and beautiful city of Gainesville, Florida. I thank nmy fel low graduate students and neighbors for their friendship and encour age them to continue working hard to reach their goals. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . .. iv LIST OF TABLES . . . . . . . . . . . . x LIST OF FIGURES . . . . LIST OF SYMBOLS . . . . . . . . . . . ABSTRACT . . . . . . . . . . . . . CHAPTERS I. INTRODUCTION . . . . . . . . . SXV .xxviii 1.1 The Soil Erosion Problem . . . . . . 1.2 Purpose and Scope of This Study . . . . II. SOIL EROSION PROCESS AND REVIEW OF RELATED STUDIES . . . . . . . . . 2.1 The Soil Erosion Process . . . . 2.2 Initial Studies . . . . . 2.3 Raindrop and Rainfall Characteristics 2.3.1 Raindrop Characteristics . . 2.3.2 Rainfall Characteristics . . 2.4 Splash Erosion . . . . . . 2.4.1 Waterdrop Splash . . . . 2.4.2 Splash Erosion Studies . . 2.5 Overland Flow Erosion . . . . 2.5.1 Hydraulics of Overland Flow . 2.5.1.1 Simolified Solutions. 2.5.2 the Kinematic Wave Method . 2.5.1.2 The Law of Resistance . 2.5.1.3 Boundary Shear Stress . Kinematic Approach . . Dynamic Approach . . . 2.5.1.4 Entrainment Motion and Critical Shear Stress . 2.5.1.5 Flow Velocity . . . 2.5.1.6 Turbulence . . . . Overland Flow Erosion Studies . . e e 2.6 Rill and Gully Erosion . . . . . . 2.6.1 Rill Erosion . . . . . . . 2.6.2 Gully Erosion . . . . . . . 2.7 Soil Erosion Estimates and Prediction . . 2.7.1 Use of Existing Stream Sediment Trans port Equations in Overland Flow . . 2.7.2 The Universal Soil Loss Equation . . 2.7.3 Soil Erosion Models . . . . . 2.8 Soil Characteristics and Slope Effects in Soil Erosion. . . . . . . .. 2.8.1 Soil Characteristics . . . . . 2.8.2 Slope Gradient Effects . . . . 2.9 Recent Books on Soil Erosion . . . . . III. DEVELOPMENT OF THE SOIL EROSION EQUATION . . . 3.1 General Purpose and Considerations . . . 3.1.1 Basic Considerations and Assumptions . 3.1.2 Major Considerations and Assumptions . 3.2 Equilibrium Transport Condition . . . . 3.2.1 Evaluation of Ne . . . . . . 3.2.2 Evaluation of Nd . . . . . . 3.2.3 Evaluation of 1, .... . 3.2.4 Evaluation of Average Saltation Length, . . . . . . . . 3.2.5 General Equilibrium Transport Equation. 3.3 Relationships between Probability of Erosion and Bed Shear Stress . . . . .... 3.3.1 Criterion for Erosion ... ....... 3.3.2 Probability of Erosion . . . . 3.4 Sediment Transport Equation . . . . . IV. DATA, PROCEDURES, AND COEFFICIENTS EVALUATION . . 4.1 Introduction . . . . . . . A 0 n +% .7 C. uaiQ . . . . . . . . . . 4.2.1 Soil Properties . . . . . 4.2.2 Effective Grain Size Evaluation . . 4.2.3 Drag Coefficient Evaluation . . . 4.2.4 Angle of Repose Evaluation .. 4.2.5 Shields Entrainment Function and Critical Shear Stress Evaluations . 4.2.6 TimeMean Bed Shear Stress Evaluation 4.2.7 Grain Reynolds Number Evaluation . 4.2.8 Dimensionless Sediment Transport Parameter . . . . . .. 4.2.8.1 Evaluation of . . . 4.2.8.2 Evaluation of V and . 4.2.9 Sediment Transport Data in Diagrams . 100 100 102 103 104 114 123 144 144 148 151 153 153 153 154 156 157 159 159 165 166 169 169 187 189 191 191 191 192 199 200 202 203 204 204 204 204 205 206 * * 4.3 Evaluation of C* and su . . . . .. 214 4.3.1 Additional Considerations on the C*Value . . . .. . . . . 214 4.3.2 su Considerations . . .. .. .. 239 4.3.3 Procedure Used to Evaluate the Coefficients . . . . . .. 247 4.3.4 The Values of the Coefficients C2, C4, C5, C6, and m . . . . . .. 250 V. DISCUSSIONS AND MODEL VERIFICATION . . . .. .255 5.1 Introduction . . . . . . . .. 255 5.2 Error Analysis of the Data . . . . .. .255 5.2.1 Estimated Relative Error of q . . 257 5.2.2 Estimated Relative Error of I . . 259 5.2.3 Estimated Relative Error of .' . 259 5.2.4 Estimated Relative Error of the Slope Correction Factor of T . . . .. 259 5.2.5 Estimated Relative Error of .... 260 5.2.6 Discussion of the Error of the Longitudinal Slope Correction Factor 261 5.2.7 Discussion of the Estimated Relative Errors of D and . ..... . ..... .. 268 5.2.8 Other Possible Errors . . . . 281 5.2.9 Use of the Estimated Data Errors in Evaluation of the Coefficients . . 283 5.3 Error Analysis of the Model and the Predicted Values . . . . . . .. .284 5.3.1 General Statistics of the Model . . 285 5.3.2 Statistical Analysis of the Estimated Coefficient Values . . . . . 288 5.3.3 Discussion of Errors of the Model Predicted Values. . .. . . 296 5.3.4 Justification of the Least Squares Approximation Method . . . .. 307 5.4 The Saltation Length Process and the C Values . . . . . . . . .. 313 5.5 The suValues . . . . . . . .. .327 5.6 Final Remarks . . . . . . . .. .345 VI. CONCLUSIONS AND RECOMMENDATIONS . . . . . 352 6.1 Conclusions . . . . . . . .. 352 6.2 Recommendations . . . . . . .. .355 APPENDICES . . . . . . . . . . . . .. 359 A GENERAL NOTES IN THE EVALUATION OF THE ABSOLUTE ERROR AND RELATIVE ERROR OF VARIABLES . . . .. 359 viii B INFLUENCE OF CHANGE ON PROPOSED MODEL . C CONVERSION FACTORS REFERENCES . . . . . BIOGRAPHICAL SKETCH . . OF DIRECTION OF BUOYANT FORCE . . . . . . . . . 361 . . . . . . . . . 371 S. . . . . . . . 373 . . . . . . . . . 400 LIST OF TABLES Table Page 2.1 RANGE OF VALUES OF VARIABLES AND SOME DIMENSIONLESS PARAMETERS IN OVERLAND FLOW . . . . . . . 46 2.2 RELATIVE MAGNITUDES OF THE TERMS So, S1, S2 and S3 in TERMS OF Sf . . . . . . . . . . .. 48 2.3 SOME STREAM BEDLOAD TRANSPORT FORMULAS . . . . .. .105 4.1 RELEVANT DATA OBTAINED FROM KILINC AND RICHARDSON'S STUDY. 193 4.2 DIMENSIONLESS PARAMETERS CALCULATED FROM KILINC AND RICHARDSON'S DATA . . . . . . . . . .. .195 4.3 CRITICAL SHEAR STRESS AND DIMENSIONLESS SEDIMENT TRANSPORT VARIABLES . . . . . . . . . .197 4.4 IDENTIFICATION OF DATA POINTS AND GENERAL LEGEND FOR FIGURES IN THIS STUDY . . . . . . . .211 5.1 ESTIMATED RELATIVE ERROR OF THE DATA . . . . .. .256 5.2 SEDIMENT CONCENTRATION VALUES AND THEIR CALCULATED RELATIVE ERROR . . . . . . . . . .. 258 5.3 ESTIMATED ERROR OF THE LONGITUDINAL SLOPE CORRECTION FACTOR . . . . . . . . . . . . .. 262 5.4 ESTIMATED RELATIVE ERROR OF 1, T;, AND RELATED VARIABLES 264 5.5 DATA POINTS WITH POTENTIALLY LARGE ERRORS OF T; . . 270 0 5.6 AVERAGE RELATIVE ERROR OF 1 AND Cs FOR EACH DATA SET WITH SAME RAINFALL INTENSITY . . . . . . .. .279 5.7 AVERAGE RELATIVE ERROR OF D AND Cs FOR EACH DATA SET WITH SAME BED SLOPE . . . . . . . . .. 279 5.8 DATA POINTS WITH POSSIBLE LARGE ERRORS OF P . . .. 282 5.9 ANALYSIS OF VARIANCE . . . . . . . . .. .286 5.10 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 289 5.11 REQUIRED MINIMUM WATER DEPTH TO USE THE PROPOSED MODEL . 293 5.12 DATA POINTS WITH PREDICTED ERRORS LARGER THAN THE ESTIMATED ERROR OF DATA . . . . . . . .. .300 5.13 PREDICTED @VALUES AND THEIR ERRORS . . . . .. 305 5.14 SUMMARY OF DATA POINTS WITH LARGEST ERROR ON THE PREDICTED MODEL SOLUTION . . . . . . . .. .306 5.15 RAINFALL INTENSITY EFFECTS IN f(h/de) FUNCTION FOR GIVEN WATER DEPTH . . . . . . . . .. 318 5.16 PREDICTED su AND pVALUES . . . . . . . .. .333 B.1 ANALYSIS OF VARIANCE USING IMPROVED METHOD (EQUATION 3.22) . . . . . . . . . . . . .. 365 B.2 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 366 B.3 PREDICTED DVALUES AND THEIR ERRORS . . . . .. 367 B.4 PREDICTED su AND pVALUES OF THE IMPROVED METHOD ..... 368 LIST OF FIGURES Figure Page 2.1 Definition Sketch . . . . . . . . . .. 44 2.2 ErosionDeposition Criteria for Uniform Particles . .. 68 2.3 Shields' Diagram for Incipient Motion Including Mantz Extended Curve for Fine Cohesionless Grains . . .. 70 2.4 Typical Velocity Profile and Shear Stress Distributions for Flow With and Without Rainfall . . . . .. 82 2.5 Surfaces of Detachment, Transport and Maximum Erosion Rate . . . . . . . . . . .. 126 3.1 Particle Travel Distance . . . . . . . .. 160 3.2 Schematic Saltation Length Approach for Overland Flow . 163 3.3 Effect of Longitudinal Slope on the Saltation Length of a Grain . . . . . . . . . . .. 167 3.4 Forces Acting on a Grain About to Move for the Horizontal Bed and Sloping Bed Conditions . . . . . . 171 3.5 Incipient Motion for Uniform Flow Condition Following Ulrich's (1987) Approach . . . . . . . 174 3.6 Correction Factor on the Bed Shear Stress due to Longitudinal Slope for Ulrich's Coarse Material ..... 176 3.7 Evaluation of the Probability of Erosion . . . .. 188 4.1 GrainSize Distribution of the Sandy Soil Used by Kilinc and Richardson (1973) . . . . . . .. .201 4.2 o Versus 4': The Data Points . . . . . . . 207 4.3 $ Versus T' for Given Rainfall Intensity . . . .. 208 4.4 4 Versus T for Given Rainfall Intensity . . . .. 209 4.5 1 Versus T' vfor Given Bed Slope . . . . . .. .210 4.6 Initial Depth Required to Move Grain Under Very Shallow Water Depth Conditions . . . . . .. .220 4.7 Slope Effects on the Normal Component of the Rainfall Momentum Flux . . . . . . . .. .230 4.8 Predicted suValue Using C, = 2.2 x 107 ft2 (Constant) 245 4.9 Predicted SuValue Versus Measured Sediment Concentration using C* = 2.2 x 107 ft2 (Constant) . 246 4.10 Comparison of Observed and Predicted Dimensionless Sediment Transport . . . . . . . . .. .252 4.11 Predicted and Required suValues . . . . . .. .253 5.1 Bed Shear Stress Ratio . . . . . . . .. .267 5.2 T Values as Calculated by Kilinc and Richardson (1973) 272 5.3 ''Values for Given Bed Slope . . . . . . .. 273 5.4 'Values for Given Bed Slope . . . . . . .. 274 5.5 Observed DValues for Given Bed Slope . . . . .. .277 5.6 Predicted D and Estimated Error Ranges in Data for Rainfall Intensities of 2.25 and 4.60 in./hr . . .. 297 5.7 Predicted $ and Estimated Error Ranges in Data for Rainfall Intensities of 1.25 and 3.65 in./hr . . .. 298 5.8 Predicted 0 Versus Observed 4Values . . . . .. 303 5.9 Residual Values Versus the Natural Logarithm of Observed $Values . . . . . . . . .. 310 5.10 Normal Probability Plot of the Standardized Residual . 311 5.11 Saltation Length Depth Function, f(h/de) . . . .. 316 5.12 Slope Correction Factor for the Average Saltation Length. 320 5.13 Saltation Length Ratio, 2.2 x 107/C . . . . . 324 5.14 Predicted Probability of Erosion . . . . . .. .331 5.15 Changes in p/(1 p) Due to Errors in p Evaluation . 336 xiii 5.16 Required suValue Versus Measured Sediment Concentration, Cs . . . . . . . . . .. 341 5.17 Predicted suValues Versus Measured Sediment Concentration, Cs . . . . . . . . . .. 342 5.18 Relationship Between Cs and v*de/v . . . . .. .344 B.1 Correction Factor for Bed Shear Stress due to Longi tudinal Slope for Kilinc and Richardson's Silty Sand Material . . . . . . . . . . .. 363 B.2 Required SuValues . . . . . . . .. . 370 LIST OF SYMBOLS A = cross sectional area of water flow Ao = surface area exposed to falling raindrops A1 = constant of particle area A2 = constant of particle volume A3 = Ai (B1/(2A2))1/2 = constant Aj = dimensionless constant AI = increment of surface area a =2.5 ao = coefficient between 0 and 1 used by Onstad et al. (1976) aI = constant a2 = constant ab = thickness of the bedload transport layer, assume twice the size of sediment particles ad = I = constant in DuBoys formula af = coefficient relating detachment capacity to transport capacity of flow aI = coefficient which depends in soil characteristics ak = coefficient used in discharge per unit width equation of the kinematic wave method am = empirical coefficient ap = constant in velocity profile equation ar = coefficient to relate rainfall intensity to the roughness coefficient, K ay = 2.45 x(y/Ys)0"4 B = width of the crosssectional area of the flow Bo = buoyant force of a particle in a static fluid (horizontal water surface) b = 7.0 bk = coefficient used in discharge per unit width equation of the kinematic wave method bm = 2.1 Clf bp = constant in velocity profile equation br = coefficient to relate rainfall intensity to the roughness coefficient, K C = Chezy's coefficient CO = constant determined by Chiu for deep water flow conditions C1 = dimensional function for the saltation length (length 2) C2 = constant representing initial dimensionless water depth required to have incipient grain motion on a horizontal bed C3 = dimensionless constant C4 = dimensionless constant related to rainfall intensity influence in water depth function of the saltation length definition C5 = Suvalue when v*de/v = 1 C6 = (1/2.3)'(slope of the su versus t&n(v*de/v) curve) Ca = sediment concentration near the top of the bed layer Cc = canopy density cover factor CD = drag coefficient Cg = ground density cover factor Clf = clay fraction percent Cm = cropping management factor in USLE Cmi = cropping management factor for interril area xvi Cmr = cropping management factor for rill area Cs = Ct = total sediment concentration in the water flow Cte = temperature correction factor in energy equation. Park et al. (1983) Ctm = temperature correction factor in momentum equation, Park et al. (1983) C = dimensionless friction coefficient C* = AA3/A2C1 cf = DarcyWeisback friction factor c' = 8g SfOI/(1.481"8 0.2) c" = C'NM1"8 D = Drag Coefficient D5O = mean equivalent spherical raindropsize diameter for given rainfall intensity Dc = detachment capacity of flow Dco = detachment capacity of flow at the toe of the sloping bed De = equivalent spherical raindropsize diameter DF = soil detachment by runoff DI = soil detachment by rainfall v D = *S 1 2 K du I dzI z=h D* = LoDF/Tco d = ds = grainsize diameter dio = grain size with 10% of finer material d31 = grain size with 31% of finer material d35 = grain size with 35% of finer material d50 = grain size with 50% of finer material xvii d54 = grain size with 54% of finer material d57 = grain size with 57% of finer material dE = de value used in this study = 145 pm = 4.76 x 104 ft de = diameter of effective grain size E = rate of soil loss from USLE Eh = critical value of Shields' entrainment function EI = soil erosion on interrill areas EV = estimated coefficient value e = base of natural logarithm en = void ratio of the soil F = fraction of weight of the sediment that is finer than grain size d FD = resultant detachment force FDH = resultant detachment force for horizontal bed FDS = resultant detachment force for sloping bed FR = resultant force at incipient motion Fr = Um/(gh)1/2 = Froude Number Ftest value = statistical value used to test hypothesis f = infiltration rate f(en) = 0.685/(1 + en)0.415 f(h/de) = function to represent the water depth influence in the saltation length f(I) = function to represent the rainfall properties in the water depth function, f(h/de) f(IVt) = function to represent the rainfall parameter effects in the salvation length xviii f(N) = function to represent the longitudinal slope influence in the saltation length G = weight of grain in air at one atmosphere of pressure G* = sediment load relative to flow transport capacity at the toe of the sloping bed g = acceleration of gravity gs = sediment load (weight per unit time per unit width) gse = total soil loss mass per unit width in a storm event h = water depth, measured normal to bed surface hH = required water depth to have incipient grain motion on horizontal beds hI = initial depth required to reach incipient motion hm = average water flow depth hs = hH cos e hw = water depth plus loose soil depth ho = local water depth at distance x' from the bank h = overpressure head induced by the raindrop impacts over the hydrostatic pressure head. I = rainfall intensity 130 = rainfall intensity during the maximum measured 30minute rainfall intensity during the rainstorm event I = I f = rainfall excess K = roughness coefficient associated to cf and Ref KEA = rainfall kinetic energy per unit area KED = waterdrop kinetic energy KEt = rainfall kinetic energy per unit area per unit time Kf = soil erosivity factor in USLE Kfc = soil erosivity factor Kfe = soil erosivity factor for channel erosion Ko = Kvalue with no rainfall conditions k = roughness size of the bed surface L = lift force Le = overland flow length Lf = slope length factor in USLE Lo = total length of the sloping bed I = average saltation length I. = average total distance traveled by a particle before it is finally at rest SH = average saltation length on horizontal bed In = natural logarithm tog = logarithm to base 10 IS = average saltation length on sloping bed MA = rainfall momentum per unit area ML = exponential coefficient based on the bed slope and used in USLE Mt = rainfall momentum per unit area per unit time MUSLE = Modified Universal Soil Loss Equation by Williams (1975) m = dimensionless exponential coefficient used in f(e ) N = number of particles in motion ND = number of data points Nd = number of particles deposited per unit time and unit bed area Ndrop = number of raindrops collected on a given area per unit time Ne = number of particles eroded per unit time and unit bed area NM = Manning's roughness coefficient NMb = Manning's roughness coefficient for bare soil NMc = Manning's roughness coefficient for rough, mulch or vegetative covered soil n = normalized velocity fluctuation no = 3.09 = value of n corresponding to = 0 crs n = limit of integration to obtain probability of erosion from + Area2 n = limit of integration to obtain probability of erosion from Area1 ns = number of straight lines into which the grainsize distribution curve is divided OMF = overland momentum flux P = pressure P. = overpressure due to raindrop impacts Pf = erosion control practice in USLE Pfi = erosion control practice for interrill areas Pfr = erosion control practice for rill areas p = absolute probability that a particle is eroded Q = water flow discharge q = water discharge per unit width qL = lateral flow discharge per longitudinal unit length qp = storm runoff peak qs = volume of particles with size de transported per unit time and unit bed width R = REI = rainfall erosivity factor in USLE R' = hydraulic radius R = resistance radius ReA = VDDe/vA Rede = v*de/v Ref = Umn/h/v Res = vsde/v RMF = rainfall momentum flux RMFn = rainfall momentum flux normal to bed surface r = correlation coefficient r2 = coefficient of determination S = slope gradient factor in USLE So = tan e So = sin e Sl : ah (1 B2) 1 x gh S2 2Iq gh2 I VD cosa S3 gh SC( < ) = longitudinal slope correction factor SCU = 1 Y s tan e Ys Y tan P SDF = soil properties effect constant for soil detachment by runoff SDI = soil properties effect constant for soil detachment by rainfall Se = slope of energy grade line SEE = standard error of estimate Sf = friction slope SGs = specific gravity of particle is fluid in water xxii SH = total head slope STF = soil properties effect constant for soil transport by runoff STI = soil properties effect constant for soil transport by rainfall su =/ /U t Sy = (x Xcr)/Xcr Tc = transport capacity of flow Tco = transport capacity of flow at the toe of the sloping bed Th = total head t = time ti = time consumed for exchange of a particle at the bed tj = time period of the specific storm increment t(19,o0.75) = Student's tvalue for 19 degrees of freedom and 5% level of significance U+ =i/v Um = cross section mean velocity of overland flow USLE = Universal Soil Loss Equation u = u + u' = local instantaneous longitudinal velocity u = local timemean longitudinal velocity u' = local longitudinal velocity fluctuation uh = lateral velocity of water moving away from waterdrop impact area Umax = maximum local timemean velocity at distance ymax from the bed surface Ut = it + uj = instantaneous velocity near top of grains on the bed Ut = timemean velocity near top of grains on the bed xxiii uj = velocity fluctuation near top of grains on the bed VD = waterdrop velocity Vj = terminal velocity of raindrop with equivalent spherical raindrop size Dj VR = storm runoff volume Vt = mean terminal velocity of the raindrops v = v + v' = instantaneous vertical (normal) velocity at a given location v = timemean vertical velocity v' = vertical velocity fluctuation at a given location v* = shear velocity V.cr = critical shear velocity v s = (Ts/[p(h Ya)]1/ Vs = particle fall velocity Vs35 = fall velocity of particle with size d35 W = buoyant weight of grain We = p h m2/ r WS = weight of splashed soil by single waterdrop impact w = w + w' = instantaneous lateral velocity at a given location w = timemean lateral velocity w' = lateral velocity fluctuation XL = slope length x = longitudinal distance x' = distance measured across the flow from its bank x* = x/L0 Xcr = critical length to initiate erosion xxiv + Y = yv./v y = distance from the bed surface to a location in the water Ymax = distancefrom the bed surface to the location with maximum u ZH = depression storage elevation on a horizontal bed z = vertical distance from the bottom surface a = dimensionless energy correction factor for the velocity distribution of the flow al = level of significance for tStudent test a = LoDco/Tco B = dimensionless momentum flux correction factor for the velocity distribution of the flow B1 = constant of particle area 02 = (1 + nosu)/(1 + Su2)1/2 BI = dimensionless momentum flux correction factor for the distribution of the raindrop terminal velocity OL = dimensionless momentum flux correction factor for the lateral flow velocity distribution r = surface tension of water Y = specific weight of water Yd = specific dry weight of soil material including pore volume Ys = specific weight of soil grains Ax = longitudinal length increment 6 = thickness of the viscous sublayer = very small number compared to unity ' = very small value of SCU e = longitudinal bed surface inclination with respect to the horizontal XXV = Lo DIo/Tco K = von Karman constant X = instantaneous lift per unit area X1 = constant for saltation length = dynamic viscosity of water S= kinematic viscosity of water VA = kinematic viscosity of air S= water surface angle with respect to the horizontal p = mass density of water Oerr= standard deviation of the estimated error oy = uV u = soil shear strength To = instantaneous bed shear stress To = timemean bed shear stress Tcr = timemean bed shear stress when p = 103 Tcrs = timemean bed shear stress for sloping beds when p = 103 Tf = timemean shear stress due to form roughness Tg = timemean shear stress due to grain roughness Ts = timemean shear stress at the water surface S= sediment transport intensity function @ = angle of repose xxvi X = O/((Ys Y)ds) = 1/'" Xcr = cr/((Ys Y )ds) ' I= flow intensity function !Q = flow intensity function for sloping bed surfaces 0 = angle of the path of the falling raindrops with respect to the vertical axis W = angle of detachment of the resultant force with respect to the bed surface wH = detachment angle for horizontal bed wS = detachment angle for sloping bed xxvii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SOIL EROSION BY OVERLAND FLOW WITH RAINFALL By Raul Emilio Zapata December 1987 Chairman: Dr. Bent A. Christensen Major Department: Civil Engineering The objective of this study is to develop a soil erosion model for overland flow with rainfall based on physical concepts and obser vations. The basic sediment transport equation used by the proposed model is based on the modified Einstein equation for total load transport of noncohesive materials in open channels as presented by Chiu in 1972. The proposed model was developed as generally as pos sible in order to be valid for the case of deep water flows (i.e., rivers and open channels) as well as for very shallow flows (overland sheet flow) with or without rainfall. The proposed model provides a stochastic point of view of this random process, usually modeled using deterministic approaches. Rainfall effects on the erosion process are mostly represented in the changes of the boundary shear stress and in the local velocity fluc tuation at the top of the grains. The timemean bed shear stress is obtained from the longitudinal momentum equation including additional xxviii momentum terms due to the incoming rainfall flux. The local velocity fluctuations are assumed to be distributed according to the Gaussian 1 aw. Chiu's saltation process assumed the saltation length to be inversely proportional to the particle size and independent of the flow conditions. In this study the effect of the longitudinal bed slope, water depth, and rainfall parameters on the saltation length is included. However, distance traveled in air by particles splashed by raindrops and the number of particles which travel airborne cannot be fully accounted for by the proposed definition. An error analysis was conducted to the proposed model and the assumptions made during its development. Errors on the predicted values were found to be similar to data errors. The maximum relative error on the predicted values was 33%, which may be considered rea sonable for the complex process of soil erosion by overland flow with rainfall. The proposed model may be used in the future as an initial step toward an improved erosion model based on the physics of this complex process. xxix CHAPTER I INTRODUCTION 1.1 The Soil Erosion Problem Soil erosion due to precipitation and water flow is a very complex process. Due to its importance to the agricultural economy of most societies, it has been studied during the last two or three centuries, but more seriously so during the last fifty years. It has been recognized that soil erosion is a natural geological process shaping the topography of our planet's surface. In addition to caus ing a general loss of soil to the oceans, erosion reduces the fertil ity of the soil by carrying away nutrients and minerals that plants need for a healthy lifecycle; degrades the water quality of natural or manmade water courses, lakes and oceans; and creates problems to irrigation, navigation, and water supply systems. Severe soil ero sion may also result in severe structural problems, including com plete failure of manmade structures supported by the soil. The two major soil erosion agents are water and wind (Ellison, 1947). They can erode the soil surface by acting together or indi vidually. Wind erosion is very important in arid areas and where un protected soil surfaces (e.g., surfaces not protected by vegetation or any manmade material) are exposed to the wind. Soil erosion is also caused by rainfall and water flowing over the soil. Unprotected soil surface areas are eroded by the rainfall and the subsequent overland flow carries away the eroded soil particles. The present 1 study is only concerned with the soil erosion due to rainfall impact and flowing water. Most of the observations presented here are based on water. Nevertheless, wind erosion is recognized as an eroding agent which has a significant effect in the shaping of the earth's surface. The soil erosion problem has been widely studied by many scien tists and engineers, but it is still not well understood. The main reason is that soil erosion is a very complex process and involves so many variables that it is practically impossible to measure the influence of all of them in one study. In addition, the scale of this process is so small that it makes it practically impossible to measure the variables in an accur ate manner even with today's sophisticated equipment. Researchers cannot produce a model with dimensions longer than those of the pro totype because not all of the dimensions or variables can be modeled to the same scale (e.g., waterdrops larger than 7 mm are unstable (Blanchard, 1950) and the terminal velocity of the waterdrops is influenced by surface tension while in a larger model surface tension may be relatively negligible). Using the prototype size requires equipment with almost microscopic dimensions in order to measure flow parameters at different locations rather than average values with larger instruments. Therefore, scaling and instrumentation are problems that soil erosion researchers have to deal with. Soil erosion is also an unsteady and stochastic process. Des cribing it requires knowledge of how the variables change with time. In most of the studies, the investigators have selected some significant variables or parameters and studied their behavior under different conditions. Unfortunately, most of these studies were con ducted under simulated steady or quasisteady conditions. This is considering that the surface area under study is significantly larger than the area affected by a single raindrop impact and that the time period is substantially larger than the time increment between rain drop impacts. Under such conditions, the soil erosion process can be mathematically described as a simpler process, but the capability to predict changes in time and sometimes spatial changes is lost. There are other studies in which a specific part of the soil erosion process is considered and measured. Then a mathematical model is developed and used in describing that part of the erosion process. Some of the general topics related to soil erosion studied in the past are Relation of rainfall to runoff and soil erosion. Detachment and/or transportation of soil particles by raindrop impact. Detachment and/or transportation of soil particles by runoff. Soil erosion caused by the combined actions of rainfall and runoff. The erosion process related to hydraulic parameters. The influence of soil properties on the erosion process. The influence of longitudinal slope, slope length and/or slope shape on erosion. Measurement and prediction of soil loss from a given area. Eroded soil characteristics and properties. Nutrient and pollutant migration, etc. Most of the studies found in the literature seem to describe empirical approaches where, from observed data, equations are devel oped using some kind of a regression analysis. One disadvantage of this approach is that the developed equations are only valid for the specific conditions that existed during the observations. Extrapola tion, of course, cannot be recommended, and even interpolation may have its problems. The other disadvantage of using empirical ap proaches is that the resulting empirical relationships can only pre dict the mean values of the observations for a certain condition and any information relating to fluctuations is lost. Some of the varia bles upon which many of these studies are based are the kinetic ener gy or momentum of raindrops, the rainfall intensity, some of the soil properties, vegetative cover, slope and length of the ground surface and the conservation practice used in order to prevent soil loss. Attempts have been made to explain soil erosion by the basic physical laws. However, the results are usually limited to very nar row parts of the whole process. Continued research is definitely needed. Attempts to physically describe the effect of raindrops and sheet flow at the same time have been only partially successful and the literature on this topic is quite limited. Attempts of using existing sediment transport equations origin ally developed for water courses have been made, but the results have not been satisfactory because the boundary conditions are different. Modification of such equations in order to consider the effect of very thin surface flows and rainfall may however improve prediction. However, caution is advised in order to satisfy all boundary condi tions at any time and location. Real physical models describing the soil erosion process in overland flow are very rare. This study attempts to present a new vision of this erosion process which might help others in the future to better understand this very complex process. 1.2 Purpose and Scope of This Study The goal of the present investigation is to describe the soil erosion process using a stochastic approach similar to that original ly presented by Chiu (1972) for transport of cohesionless sediments by water and air. That approach is based on the saltation hypothesis and may be considered as a modification of the original stochastic theory presented by Einstein (1950). To reach that goal only physical considerations will be used. The proposed approach is intended to be as simple and general as pos sible. So it will allow use of the model at almost any flow condi tion if the required information is available. However, before use, it will be necessary to test the proposed approach at these not pre viously tested flow conditions, even though the required basic physi cal considerations are expected to be included into the proposed model of this investigation. The use of the stochastic approach requires some modifications in order to include the rainfall effect on the thin sheet flow. Those modifications will be presented in Chapter III and IV where the proposed approach is presented and developed using existing data from the literature. In addition, an error analysis of the used data and the predicted results is presented and discussed in Chapter V. The possible advantages and limitations of the proposed method are discussed. Other discussions and conclusions are also presented. Throughout this work the unit system used is the English Sys tem. However, in the review of other studies related to soil erosion the unit systems used in those studies are used. The proper unit conversion is presented in such cases. See also Appendix C for the conversion factors for units between the English System and the SI System. CHAPTER II SOIL EROSION PROCESS AND REVIEW OF RELATED STUDIES 2.1 The Soil Erosion Process The initial cause of soil erosion due to water is rainfall. When the raindrops impact a ground surface not covered by water, their kinetic energy will generate a splash of water in which thou sands of droplets will disperse in all directions (Mutchler and Lar son, 1971; and Mutchler, 1971). Some of these droplets will carry soil particles out of the area of impact. The amount of soil de tached and the distance traveled by the individual soil particles will be a function of the ground surface soil properties and the rainfall characteristics. If there is a water layer covering the ground surface, the ef fect of the splash on the soil surface will be mostly a function of the water layer thickness, the drop diameter and the soil properties (Mutchler, 1967; Mutchler and Young, 1975). These authors found that raindrop impacts are more erosive when the water depth is about one fifth of the drop diameter and that the impacts are practically non erosive when the soil is covered by water at a depth of about three drop diameters or more. Palmer (1963, 1965) also studied the effects of the impact of waterdrops with the shallow water layer. He studied the stress strain relationship on a surface covered by different water layer thicknesses that were impacted by different drop sizes. His maximum 7 reported strain values occurred when the water depth was about one drop diameter. Also, when the water depth was 20 mm (0.787 in.), the stressstrain relationship was found to be about the same as without the water layer. For a deeper water layer, the waterdrop impact effects became negligible. He also reported that when the depth of the water layer was three times the drop diameter or more, the soil loss was very small. If the water layer is moving, then there is an additional ac tive force creating soil erosion. Such overland flows usually have very small depths and their mean velocities are not high enough to produce erosive bed shear stresses. But the turbulence due to the raindrop impacts does make the increased detachment and transport of particles possible. This is because the raindrop impacts increase the energy and the momentum transfer in overland flows. In any rainfall event, the overland flow will be present after the topsoil is saturated and the rainfall intensity exceeds the infiltration capacity of the soil. Usually the depth and the veloci ty of the sheet flow increase as the water moves downslope because more rainfall is accumulated as the contributing area increases. This flow tends to move towards microchannels in which the accelera ting water tends to increase the scouring action forming rills or small channels that usually grow in dimensions in the downstream di rection. The rills may carry the water into bigger channels called gullies where the now sedimentladen flow continues its erosive ac tion. Finally, the water will reach a continuously flowing stream such as a river. Here the soil will be carried until its final depo sition in a reservoir, lake, delta, or ultimately, the ocean. Usually a particle which is detached at the highest point of a drainage area does not reach the river in the same rainfall event since the erosion process usually is relatively a slow moving process with respect to time. This is because the particle depends on an external force (i.e., raindrop impacts, flowing water or wind) to be detached and move downslope. If there is not any force capable of moving the particle, it will remain in the same location. Since the wind is excluded in this study, only during each rainfall event will the particle have a downslope displacement. So the total distance traveled by the soil particle will depend on the number of rain storms, the specific rainfall characteristics and the soil character istics. The presence of rills and gullies in any area will depend on the soil surface's properties, the steepness of the slope, and the presence of vegetation. Therefore, one may find areas with highly erosive soils in which gullies and maybe rills are absent due to the lack of slope. On the other hand, it is possible to find gullies and rills lacking even on very steep slopes if the soil is highly resis tant to erosion (Ellison, 1947). The raindrop impact effects in rills and gullies are usually considered negligible compared to the flow discharge effects. This is because the water depth in rills and gullies can be enough to sup press the detachment capacity of the raindrop impact. In addition, the flow discharge has enough velocity in itself to produce erosive bed shear stresses. However, the raindrop impacts and the corres ponding splashes are very significant in any area before the rills are generated and the areas between the rills. These two areas are usually referred to as the interrill areas. Now, it is necessary to know the extent of the rainfall effects on the soil erosion process. Young and Wiersma (1973) studied the relative importance of raindrop impact and flowing water to the ero sion process. This was accomplished by determining the source and mode of sediment transport on a laboratory plot under conditions of normal rainfall energy and greatly reduced rainfall energy. They found that decreasing the rainfall impact energy by 89% without reducing rainfall intensity, the soil losses decreased by 90% or more. It was thus demonstrated that the impact energy of raindrops is the major agent in soil detachment. For all three soils studied 80% to 85% of the soil loss originating in the interrill area was transported to a rill before leaving the plot. Thus, it was indica ted that the transport of detached particles from the plot was accom plished mainly by flow in the rills. From that study, Mutchler and Young (1975) found that the soil carried along by the splash energy was only 10% to 17% of the soil loss from interrill areas to rills. The remainder of the loss to rills was carried in the thin surface flow which without raindrop impacts carried little, if any, soil. Therefore, the conclusion was that the raindrop impact was the driving force in transporting soil in thin surface flows (sheet flows) to rills. There are many considerations about the soil erosion process by rainfall which have not been indicated in this section. It is better to review them individually in order to understand this process from single contributions of the factors and then joining them into a gen eral soil erosion process description. 2.2 Initial Studies Soil erosion has been studied extensively during the last half century mostly due to its importance to agriculture and food produc tion. Before the 1930s, the soil erosion problem was recognized but not considered as a major problem. Therefore, there was not much written about it and most of the literature available came from Euro pean studies which did not apply directly to many of the conditions found in the United States. During the 1930s, there was an increased need for studies re lated to soil erosion. It was realized that some of the most produc tive lands were removed from agricultural production because the water from rainfalls and the wind was carrying away the fertile top soils and nutrients which the plants needed. Since there was not much knowledge about the erosion process, the initial studies were basically concentrated on collecting data which could help to estab lish the magnitude of the problem and in studies to find some alter natives or conservation practices to control erosion. In addition, there were few studies in which the mechanism of soil erosion and their effects were considered too. Laws (1940) presented one of the first studies in which the relation of raindrop size to erosion and infiltration rates were considered. He also mentioned previous studies done by European and American scientists around the turn of the century, and referred to studies carried out at that time by himself and other researchers. Ellison (1944, 1945, 1947, 1950) contributed a series of papers in which he described the soil erosion process. It was the first time this process was described and studied in such detail. Ellison (1944) initially presented the current knowledge about the soil ero sion process and the factors which might affect the process. He developed an empirical equation for raindrop erosion (splash erosion) based on the rainfall intensity, the diameter of the waterdrop and the velocity of the drop. In 1945, he presented his experimental results of the effects of raindrop impact and flow in the infiltra tion capacity and the soil erosion. He divided the study in raindrop effects alone, runoff effects alone, and the combined effects. Like previous studies, many of his experiments were exploratory in nature and the data had only qualitative significance. Then Ellison (1947) proceeded to describe his approach to the soil erosion problem step by step. He postulated that the soil ero sion process was "a process of detachment and transportation of soil materials by soil agents." This definition described the process as composed of two principal and sequential events. In the first one, the soil particles are torn loose, detached from the ground surface and made available for transport, which is the second event. There fore, the erosive capacity of any agent was comprised of two indepen dent variables of detaching capacity and transporting capacity. The raindrop impacts and the surface flow runoff were the erosive agents he considered in his study. Wind was also recognized as an individu al erosive agent, but not included in Ellison's research. Ellison's approach was based on four different conditions (i.e., detachment and transportation of particles due to raindrop im pacts or surface runoff) to describe the soil erosion process. The detachment of soil particles by the erosive agents was related to the soil properties and conservation practices available to the area under study. Meanwhile, the transport of soil particles by the ero sive agent was considered to be a function of the transportability of the soil, the intensity of the transporting agent, and the quantity of soil already detached. The effect of slope and wind were mentioned as sources of splash transportation in Ellison's studies. The kinetic energy of the runoff, the slope, the surface roughness, the thickness of the water layer, and the turbulence generated by the raindrop impacts were mentioned as parameters for surface flow transportation. How ever, Ellison did not develop expressions to define each of these parameters. More work and knowledge were necessary before the fun damental relationships could be obtained. Musgrave (1947, 1954) presented a review of the knowledge on sheet erosion and the estimation of land erosion. Using data from the available literature and from his experiments, he indicated that the erosion was related to many variables expressed in the following proportionalities. Erosion 1301.75 Erosion S So1'35 Erosion XL 0 Area where 130 = maximum amount of rain in 30 minutes of rainfall (inches) So = slope gradient (percent) XL = slope length (feet) He also presented the relative amount of erosion for different vegetal covers. Adjustments between studied soils being exposed to different rainfall, slope and slope length conditions were made in order to present results of rate of erosion under a common basis. An example of this procedure was presented in his 1947 study. Ekern (1953) presented a good summary of the previous knowledge and information needed about the rainfall properties that affect raindrop erosion. Then he presented his approach to raindrop erosion based on the kinetic energy of the natural rainfall and discussed the rainfall parameters and soil factors needed to represent the erosion process. He recognized the use of simulated rainfall as a tool for obtaining a better understanding of the erosion process. However, he emphasized the need for the control of the rainfall parameters (i.e., rainfall intensity, and drop size, pattern, shape and velocity) in order to have the best representation of a natural rainfall while the soil erosion data is collected. Like Ekern, other authors have also discussed the use of simu lated rainfall for soil erosion research. Among them, Meyer (1965) and Bubenzer (1979) have presented detailed information about simula ted rainfall conditions. The general consensus of all these studies is that the drop size distribution, the drop velocity at impact and the rainfall intensity are the basic parameters which need to be con trolled and duplicated to the best possible accuracy. In the next sections of this chapter, a review of the erosive agents presented by Ellison (i.e., raindrop and surface runoff) are presented in more detail. 2.3 Raindrop and Rainfall Characteristics 2.3.1 Raindrop Characteristics It was mentioned before that the raindrop impacts are the ini tial cause for detachment of soil particles from the bed surface; they also provide the necessary turbulence to keep the particles in motion in the shallow overland flows. Not all raindrops which Impact the soil surface during certain periods of time are identical. So, it is necessary to study the raindrop characteristics in order to understand the erosion process due to rainfall. Raindrop character istics important in soil erosion are the drop mass, size, shape, and their terminal velocity. Falling raindrops in air are not completely spherical, but researchers have referred to an equivalent spherical diameter De based on the actual mass of the raindrop to discuss the variation in size between waterdrops. Laws (1941) presented velocity measurements of waterdrops with sizes ranging from 1 mm (0.039 in.) to 6 mm (0.236 in.) in diameter falling through still air from heights of 0.5 m (1.64 ft) to 20 m (65.6 ft). He also reported a few measurements of raindrop veloci ties in order to compare with earlier observations. Laws' measuring techniques consisted of a high speed photographic system, used to measure the drop velocity and the flour pellet method to determine the drop size. Laws' results showed that the waterdrops attained a terminal velocity after falling a certain height. The height re quired to reach terminal velocity increased as the drop size in creased for drop sizes of about 4 mm (0.157 in.) or less. Beyond that drop size the required height gradually decreased as the drop size increased. The variations in the drop shape and the consequent change in the friction resistance through the drop falling stage were related to that reduction of the required height to reach terminal velocity. Nevertheless, the terminal velocity always increased as the drop size (i.e., drop mass) increased. Later, Gunn and Kinzer (1949) presented what appears to be the most accurate fall velocity measurements available. Using electronic techniques to measure the fall velocity they were able to work from drop sizes so small (about 0.75 mm = 0.029 in.) that the Stokes Law was obeyed to up to (and including) drops large enough to be mechan ically unstable (about 6.1 mm = 0.24 in.). This work was done under controlled conditions in stagnant air at 760 mm Hg pressure, a tem perature of 20C (680F) and 50% relative humidity. The new observa tions resulted in generally larger values than those found by other researchers but approached more to the values obtained by Laws (1941). The new values were measurably smaller than Laws' values. The overall accuracy of the drop massterminal velocity measurements of Gunn and Kinzer's study was better than 0.7%. There are other studies dealing with the behavior of the fall ing raindrop. For instance, Blanchard (1950) studied the growth of larger waterdrops caused by collision with small drops, the breakdown of larger waterdrops and the deformation of the waterdrop with time. Jones (1959) considered the shape of the raindrops during rainstorm events and concluded that there was basically a mean shape which varied consistently with the mass of the raindrop. However, he also observed that the shape was the result of oscillations about a mean and that the tilt observed in the raindrop's major axis was associa ted with the wind speed and its prevailing direction in the atmos phere at the moment the measurements were taken. Likewise, Jameson and Beard (1982) studied the oscillating forms of the freely falling raindrops. Epema and Riezebos' (1984) study indicated that the oscillations are gradually damped and at terminal velocity their drop shape observations showed that the drops attain equilibrium and have an oblate shape. Comparison of the drop shape showed that the drops obtained in the laboratory (still air conditions) were more oblate than the drops of equivalent drop diameters in natural rain observed by Jones (1959). Some researchers have developed analytical approaches to des cribe the raindrop size, shape, and falling speed. Spilhaus (1948) assumed that a falling raindrop has an ellipsoidal shape. The sur face tension effect was combined with the aerodynamic deformation of the drop in order to maintain the steady shape and falling velocity. His theoretical values partially agreed with Laws (1941) data, but his approach was not able to describe the complex behavior of the falling raindrop in air. McDonald (1954) presented a better analy tical approach in which he concluded that under most conditions the surface tension, the hydrostatic pressure and the external aerody namic pressure were the three factors which had important roles in producing the characteristic deformation of large raindrops. Wenzel and Wang (1970) used a balance of forces approach to study freely falling drops. That is, neglecting minor forces, they considered the balance between the drag force, the buoyant force and the gravitational force. Solving for the drag coefficient, CD, and using data from Laws (1941) and Gunn and Kinzer (1949), they produced diagrams for the drag coefficient of falling waterdrops in air. A relationship between fall velocity and fall height was also obtained using the balance of forces equation in an integral form. Beard (1976) studied the waterdrop behavior in the atmosphere dividing the analysis in three physically distinct flow regimes 1 pm < De < 20 pm with 106 < ReA < 0.01 20 pm < De < 1 mm with 0.01 < ReA < 300 1 mm < De < 7 mm with 300 < ReA < 4000 where De = equivalent spherical drop diameter Re = D De = drop Reynolds number A VA VD = drop velocity VA = kinematic viscosity of air For each regime he developed equations, using the drop size and the physical properties of the drop and atmosphere, in order to estimate the drop axis ratio, the projected horizontal drop diameter and the terminal velocity. 2.3.2 Rainfall Characteristics To evaluate soil erosion by rain it is necessary to know about the rainfall intensity, the duration of the event, the size distribution of the raindrops at a given intensity, and the kinetic energy or momentum of the raindrops at a given intensity. Laws and Parsons (1943) presented the drop size distribution against rainfall intensity relationship. They used the mean raindrop size, D5o, as the value to represent the particle distribution for a given rainfall intensity. The mean drop diameter was defined as the abcissa of the point in the cumulativevolume curve having an ordinate of 50%. Their empirical equation was presented as 50 = 2.2310.182 (2.1) where D50 is in millimeters and the rainfall intensity, I, in inches per hour. They recognized that the raindrop size distribution at any rainfall rate they presented was only an approximation. A variabil ity of the drop size distribution from time to time for the same rainfall intensity was also recognized and the possibility that a similar raindrop size distributionrainfall intensity relationship could be found elsewhere was mentioned too. Chapman (1948) studied the effect of forest on the raindrop size distribution and on the striking force at the soil surface. He found that the volume of water striking the soil per unit area per unit time in a pine plantation and in an open area were approximately equal. The raindrop size distribution in the forest field showed a more flattened shape instead of the bell shaped frequency reported by Laws and Parson (1941) for open areas. This indicates that the forested area had a more uniform distribution of the water volume throughout the range of drop sizes. In addition, he observed that the meandrop size in the open field increased with increasing rate of rainfall, but for the pine area the mean drop was apparently unrelated to the rainfall intensity (at least within the range of rainfall rates measured). He also indicated that the raindrops could reach again near terminal velocities in the forested area be cause the soil did not have any other vegetation than trees. The trees provided with 8.5 m (27.9 ft) of free fall distance between the base of the canopy and the soil. Other researchers (e.g., Mihara, 1951; Hudson, 1963; Carter et al., 1974; McGregor and Mutchler, 1977; and Park et al., 1983) have presented raindrop sizerainfall intensity relationships different from the one proposed by Laws and Parson. Their basic differences are considered to be due to the geographic location, climatologic conditions, kind of rainstorm measured, the time at which the samples were taken during the rainstorm events, and the method used to measure the raindrop size. The difference in the proposed empirical relationship can be seen by presenting some of the proposed equations: Carter et al. (1974) D5O = 1.63 + 1.331 0.3312 + 0.02I3 (2.2) McGregor and Mutchler (1976) D5 = 2.76 + 11.40 exp(1.401) 13.16 exp(1.171) (2.3) Park et al. (1983) D50 = 0.33 I0.12 (2.4) where D5O has units of millimeters and I has units of inches per hour except Park et al. who used I in millimeters per hour. It should also be mentioned that Horton (1948) proposed a sta tistical distribution of drop sizes at different spatial locations of a storm in order to describe part of a thunderstorm model. Some fre quency distribution curves to corroborate the applicability of the model were also suggested. However, due to the complexity of the rainfall process many researchers have not followed Horton's approach. Instead they have looked for the important parameters which might affect the soil ero sion process and have concentrated their efforts on them. In terms of rainfall effects, researchers have studied the energy and momentum rainfall can provide to erode the soil surface. The kinetic energy of the rainfall can be obtained from the raindrop size distribution for the given rainfall intensity and the terminal velocity for each raindrop size. Based on that approach empirical equations have been proposed. Mihara (1951) proposed the relationship KEt = 21,400 11.22 (2.5) where KEt is presented as kinetic energy per unit area and time [erg/(cm2 min)] and I has units of mmn/lO min. Wischmeier and Smith (1958) proposed the relationship KEA = 916 + 331 log(130) (2.6) where KEA is the kinetic energy per unit area [(ft ton)/(acre in.)] and 130 is the rainfall intensity corresponding to the maximum measured 30minute rainfall intensity during the rainstorm event with units of in./hr. This equation is used in the Universal Soil Loss Equation which has been widely used to predict soil loss from laboratory and field areas during the last thirty years. Elwell and Stocking (1973) used the expression originally developed in Hudson's masters thesis (1965) for their Rhodesia, Africa, region KEA = 29.82 127.51 (2.7) I where KEA has units of (J/m2) and I has units of mm/hr. Carter et al. (1974) proposed the expression KEA = 429.2 + 534.0 130 122.5 1302 + 7.8 1303 (2.8) where KEA has units of (ft tons)/(acre in.) while 130 follows Wischmeier and Smith's (1958) definition and has units of in./hr. McGregor and Mutchler (1977) presented their expression as KEA = 1035 + 822 exp(1.22 130) 1564 exp(1.83 130) (2.9) where KEA and 130 have the same units as Wischmeier and Smith (1958). Kneale (1982) obtained for small rainfall intensities (0.1 mm/hr < I < 7 mm/hr) the expression tog KEt = 0.90 + 1.25 I (2.10) where KEt has units of J/(m2 hr). Park et al. (1983) have proposed the expression KEt = 211070 Cte 11.16 (2.11) where KEt has units of J/(ha hr), I is in mm/hr and Cte is a tem perature correction factor. Rogers et al. (1967) discussed some of the sources of error in calculating the kinetic energy of rainfall. They indicated that the sources of errors are variations in the raindrop size distribution even at different periods of the rainstorm with the same rainfall intensity and the measuring technique used to measure rainfall inten sity and wind effects. Recently, Mualem and Assouline (1986) proposed an analytical function to represent the raindrop size distribution which was cali brated for Rhodesia (Hudson, 1965) and Washington, D. C. (Laws and Parson, 1943) data. From it, the rainfall kinetic energy per unit mass and the rainfall kinetic energy per unit time expressions were presented as a function of rainfall intensity. The curves for rain fall kinetic energy per unit mass differed significantly from known empirical expressions obtained by other authors which used the same data. Their rainfall kinetic energy per unit time curve was found to have an insignificant deviation between both data places at low rain fall intensities, but became noticeable at higher values of the rain fall intensity. Similarly, there are some relationships giving the momentum of rainfall applied to a given surface and the rainfall intensity men tioned in the literature. Elwell and Stocking (1973) used the expression originally developed in Hudson's masters thesis (1965) MA = 75.3 155.2 I MA is rainfall momentum per m)/(s in2) and I has units Park et al. (1983) proposed unit area and has units of of mn/hr. the use of the expression Mt = 64230 Ctm I1 09 (2.13) where Mt is the rainfall momentum per unit area per unit time (kg m/s)/(ha hr), I has units of mm/hr and Ctm is the tem perature correction factor. Finally, the relationship between the total number of drops collected in a unit area per unit time, Ndrop [drops/(m2 s)], and the rainfall intensity I (mn/hr), Ndrop = 154 I0.5 (2.14) presented by Park et al. (1983) may also help in the future to im prove relationships for the soil erosion process. All of these equations presented here have certain conditions in order to be used correctly. The reader is referred to the origin al studies for more information. 2.4 Splash Erosion 2.4.1 Waterdrop Splash The study of the waterdrop impact and the consequent waterdrop splash was improved with the introduction of the high speed cameras. With this equipment, the different conditions and the time sequence of this process were studied in detail. Ellison (1950) originally presented sequences of photographs about the waterdrop splash on soil where (kg * (2.12) surfaces with different water layer thicknesses over the bed surface. The variation in splash characteristics with respect to changes in the water layer thickness was visually explained in those photo graphs. But it was not until the late 1960s that the interest on the waterdrop splash process and splash sequences were really studied thoroughly. Mutchler authored and coauthored a series of articles in which the individual characteristics of the waterdrop splash were presented. Mutchler (1967) studied the waterdrop splash at terminal velo city over different types of surfaces with and without a water layer covering it. He studied the effects of the drop diameter, the water depth, the roughness and the softness or hardness of the solid sur faces on the splash characteristics. A set of parameters were estab lished to describe the geometry of splash. For this he used the width of the crater of the splash, the height of the splash sheet wall, the radius of curvature of the splash sheet wall, the angle at which the sheet wall goes with respect to the water surface, and the angle at which the splash droplets are ejected from the splash sheet wall. Since these parameters changed their values with respect to time he used the characteristic shape occurring at the time of maxi mum sheet wall height to show the effect of the water layer depth on the splash. He concluded that the water depth had its greatest ef fect on the waterdrop splash at depths of about onethird of a drop diameter and that the splash geometry changed very little at water depth greater than one drop diameter. Mutchler and Hansen (1970) used data from Mutchler (1967) to develop empirical dimensionless equations to represent the waterdrop splash. They used the water layer depth to drop diameter ratio (h/De) as the only parameter needed to obtain the other dependent parameters already presented by Mutchler (1967). Mutchler (1971) also presented relationships for the splash droplet production by waterdrop impacts on a glass surface with a water layer over it. Using the h/De ratio, he presented relation ships for the number of droplets produced by one waterdrop impact, the mean droplet diameter size of the droplet size distribution and the standard deviation of that distribution. Discussions of how these parameters changed with the water layer depth and the waterdrop size or weight were also present. Mutchler and Larson (1971) studied the amount of splash that a waterdrop at terminal velocity could produce by impacting a water layer over smooth glass at various water depths. They presented empirical equations to predict the weight of water splashed which indicated a maximum splash weight at h/De of 0.14 and 0.20 for De equal to 5.6 mm and about 3 mm, respectively. The influence of greater depth became relatively insignificant at a depth of about threedrop diameters. They stated that without splash there cannot be splash transport. At shallow water depths splash weights greater than two times the waterdrop weight were observed. But as the water depth increased to three waterdrop diameters or more, the splash amount decreased and became relatively insignificant. Mutchler and Young (1975) presented a relationship for the rate of change of width of the crater with time. From this, they obtained an expression of the lateral (horizontal) velocity, uh, of the wa ter moving away from the impact site along the surface. They also obtained a rough estimate of the viscous bed shear stress To by us ing the equation To= du= uh (2.15) dz h/2 where du = rate of change of the horizontal velocity dz = increment of vertical distance in the water S= dynamic viscosity of water Based on these conditions an estimate of the minimum velocity required to detach soil particles from the surface and how long those shear stresses could last before they become smaller than the criti cal shear stress, Tcr was presented. From this approach Mutchler and Young were able to show that the erosive action of a waterdrop impact was effective very early after impact and thus in the vicinity of the center of impact. They also showed that for water layer depths equivalent to threedrop di ameters, the soil is essentially protected from raindrop impacts. Finally, it was also indicated that most of the water splashed from the area of impact came from the water layer and not from the water drop itself. Contemporary to Mutchler's works, Hobbs presented another ser ies of articles about waterdrop splash characteristics. Hobbs and Kezweeny (1967) measured the number of droplets produced by the im pact of a waterdrop on a water surface and the electric charge of these droplets. The number of droplets produced was a function of the fall distance of the waterdrop. A fall distance of 10 cm (3.94 in.) or less did not produce any splash, and for fall distances up to 200 cm (78.7 in.) the number of droplets was found to increase linearly with the fall distance. Mutchler's results cannot be com pared with the results obtained by Hobbs and Kezweeny because the latter study did not test fall distances higher than 200 cm (78.7 in.). Consequently, no terminal velocity of the waterdrop was reached in this latter study. It should also be mentioned that the latter study reported that nearly all of the spray droplets carried a negative charge and for the range of fall distances used it appeared that the fall distance had little effect on the charges carried by the spray droplets. Hobbs and Osheroff (1967) and Macklin and Hobbs (1969) also studied the effect of the water layer depth on the waterdrop splash but their major interest was the study of the Rayleigh jet produced by the returning (converging) fluid filling the crater created by the waterdrop impact. The waterdrop splash has also been studied analytically using the NavierStokes equations. Each study has established its assump tions and boundaries to the problem resulting in simplified equations which are solved by numerical analysis and computers. Harlow and Shannon (1967a, 1967b) solved the NavierStokes equations for the waterdrop impacting a water layer phenomenon by neglecting the viscosity. Also, surface tension effects at the water surface were not considered. Solutions were presented for waterdrop impacts onto a flat plate, into a shallow pool and into a deep pool. The changes in splash configuration with respect to time were pre sented for each case. Information about the pressures, velocities, droplet rupture and effects on compressibility were also presented for each water depth studied. Wenzel and Wang (1970) used a different numerical approach than the one used by Harlow and Shannon (1967a, 1967b) and included the surface tension. Their results only consider the initial stages of the waterdrop impact into stagnant water due to limitations in the time of execution of the program and economic restrictions. These initial stages included the period of time in which the waterdrop impacts the water layer and the water moves radially outward. The inward direction motion of the water was not included in the study. Their results included a maximum impact pressure model and a quanti tative discussion of pressure distribution, boundary shear, surface tension effect, free surface configuration and various forms of ener gy and their transformation during the impact process. Theoretical results from the computer solution were successfully verified with their experimental data of the impact pressure at the bottom surface of a pan at various water layer depths. They used waterdrops of various sizes falling at different impact velocities. Wenzel and Wang also showed that surface tension cannot be neglected in this kind of study. Huang et al. (1982) neglected the body (gravity) force, the viscosity forces and the surface tension in order to examine the raindrop impact on a smooth rigid surface. The maximum pressure was reported to occur at the contact circumference and that the lateral jet velocity at the rigid surface was considered to provide the cru cial mechanism in the raindrop soil detachment process. From this they implied that the three critical factors important in defining the soil resistance against the raindrop impact were the soil defor mation characteristics, the soil shearing strength, and the soil surface microrelief. Then, Huang et al. (1983) presented the deformation pattern of a solid material under a raindrop impact by numerical analysis of the assumed linear elasticity material. The deformation due to imposed impact loadings of: (a) a steady uniform load, and (b) a simulated raindrop impact load were compared and found to be completely differ ent. As an example, for the material with a low modulus of elasti city, a uniform depression was found under steady, uniform load, while a coneshaped depression was shown under the simulated raindrop impact. As the authors indicated, this study only presented the shape of surface deformation, but the interaction between the lateral jet stream and the irregularities of the soil surface were believed to be the ones which determined the amount of splashed soil. Recently, Wright (1986) presented a physicallybased model of the dispersion of splash droplets from a waterdrop impact on a slop ing surface. He considered the forces and momentum transfer acting at the moment of impact in order to obtain the velocity vectors of the droplets. The absorption of some of the waterdrop's momentum by the soil particles was considered as well as the air resistance ex erted on the droplets while they travel in the air. The effects of slope, wind, raindrop size and some soil properties on the droplet distribution were also included. The probability of a particular droplet size being transported was obtained from splash droplet size distribution obtained from Mutchler studies. Although the proposed model considered the soil absorption of the waterdrop momentum the model does not consider the detachment of soil particles which would be the next stage toward a model of soil erosion by rainsplash. 2.4.2 Splash Erosion Studies The literature shows many studies dedicated to the splash ero sion. There are studies about: techniques used to measure the splash erosion, soil and rainfall properties which are important in this process, mechanics of the process, rate of soil detachment with respect to time or to rainfall intensity, empirical relationships to represent the erosion rate of this process, etc. Not all of the pub lished studies can be presented here but at least a brief description of the current stage of this erosion process is presented. The most popular method used to measure the splash erosion con sists of exposing a small amount of soil in a cup to the direct impact of falling waterdrops with a known rainfall intensity. The amount of soil material that has been removed from the cup after a certain period of time is considered to be the soil loss due to splash on that soil material. Ellison (1944) and other studies have considered that the soil detachment at a given rainfall intensity decreases as time increases, while Bisal (1950) and others indicate that it is constant. The latter authors consider that the decrease in detachment rate is due to the obstruction created by the cup's wall as the soil surface progressively decreases with time and have proposed correction factors for the use of the splash cup technique (Bisal, 1950; Kinnell, 1974). Farrell et al. (1974) has also pro posed a correction factor for the geometric parameters (i.e., size and shape) of the soil containers used to measure splash erosion. The splash erosion has been related to soil characteristics (i.e., particle size distribution, presence of aggregates, organic content, and others), the bed slope, and the rainfall characteristics (i.e., rainfall kinetic energy (or rainfall momentum) per unit area and time, drop size, shape, and impact velocity). From it, each study has presented empirical equations to predict the amount of soil splash, detached or transported from a given surface area (Ellison, 1944; Ekern and Muckenhirn, 1947; Ekern, 1950; Bisal, 1960; Bubenzer and Jones, 1971; Quansah, 1981, and Gilley and Finkner, 1985). The use of the rainfall kinetic energy or the rainfall momentum in those empirical equations appears to be a preference of the au thors. However, some of these studies have presented their experimental data or statistical foundations to support the use of their rainfall parameter in their equation. Rose (1960) justified the use of rainfall momentum per unit area and time instead of using the kinetic energy per unit area and time. Meanwhile, Gilley and Finkner (1985) presented statistical analysis which indicates that the kinetic energy times the drop circumference is better. Apparent ly the literature shows that there is a majority of studies prefer ring the rainfall's kinetic energy more than the rainfall's momentum for the development of their splash erosion equations, but the use of any of these two rainfall parameter must be physically justified in each case. Bubenzer and Jones (1971) also studied the effects of drop size and impact velocity on the splash detachment. They found that small er drops produced less splash than the larger ones even though the kinetic energy, the total rainfall mass and impact velocity were almost constant. Therefore, more parameters are needed to describe the splash erosion. The effect of the bed slope is also very important in the splash erosion (e.g., Ekern and Muckenhirn, 1947; Ekern, 1950; Free, 1952; DePloey and Savat, 1968; Savat, 1981, and others) because the soil downslope transport increases as the bed slope increases. Free (1952) also indicated that the effect of the slope in relation to the direction of the storm was important in determining the amount of soil removed from the soil pans. Losses from pans facing the direc tion of the storm were found to be three times those from pans facing away from the direction of the storm. They indicated that this is due to the fact that the normal component of the raindrop increases if the bed slope is facing the direction of the storm. Mazurak and Mosher (1968. 1970) and Farmer (1973) have reported that for any soil grainsize there is a linear relationship between the soil detached by raindrop impacts and the rainfall intensity. Mazurak and Mosher studies were conducted by separating the soil par ticles or aggregates in ranges of sizes and testing each of them in dividually, while Farmer's study was for the mixture of sizes. In these studies the curves of soil detachability against particle size had a bellshaped form with a peak around the 200 pm size. Ekern (1950) also found that fine sand (175 pm to 250 urnm) gave the largest amount of soil transported. Farmer's results showed curves skewed toward the smaller sizes with the tendency to be a more skewed curve as the content of smaller sizes increased in the original soil. In addition, Farmer's study included some overland flow effects which changed the susceptibility to detach soil particles by raindrop im pact. Without overland flow the soil particle sizes in the range of 110 pm to 1450 pm were most susceptible to detachment by raindrop im pact, with the peak range from 238 pm to 1041 pm. Meanwhile, with overland flow the most susceptible size range was 219 pm to 2034 pm, and the peak ranged from 440 pm to 1336 pmurn. A previous study by Rose (1960, 1961) showed that soil detach ment by raindrop impacts and the rainfall intensity was not linear. This departure of linear characteristic was associated with the resulting breakdown of the structure of the aggregates in the soil by the raindrop impacts. DePloey and Savat (1968) used autoradiographies of radioactive sand to study the splash mechanism. Their results showed the impor tance in the splash phenomenon of the grainsize distribution of sands, the slope gradient, the angle of ejection, the distribution of grains around the point of impact of the raindrops, the characteris tics of the rain, and the physical properties of the soil. Using their data and physical considerations in developing a mass balance of the soil particles, they were able to describe the splash mechan ism for horizontal surfaces, sloped surfaces, and for segments of a convex slope. Morgan (1978) indicated that his results of rainsplash erosion from field studies of sandy soils confirmed the relationships between splash erosion, rainfall energy, and bed slope obtained in laboratory experiments by other researchers. He also reported that only 0.06% of the rainfall energy contributed to splash erosion and that the major role of the splash process is the detachment of soil particles prior to their removal by overland flow. Poesen (1981) studied the erodibility of loose sediments as a timedependent phenomenon. He indicated that the variations in the detachability of soil particles during the rainfall event could be explained by changes in water content (including the liquifaction and the development of a water layer on the surface), cohesion and granu lometric composition of the top layer. In his case the presence of a water film (less than one raindrop diameter size thick) decreased the detachment. So he concluded that findings by other researchers about the increase in detachment when the thin water layer was pre sent was not a universal phenomenon but that it might be limited to the materials and procedures used in each experiment. The relation ship of amount of soil detached by splash to the mean grain size had very similar shape to the relationships obtained in previous studies by Ekern (1950) and Mazurak and Mosher (1968). This relationship was reported to be very similar to the relation between grain size and the susceptibility to runoff and wind erosion established by other researchers. For this study the highest detachability was found to be for the fine very wellsorted sand with a mean grain size of 96 uI. Yariv (1976) had also considered the presence of water as a very important factor when he presented the concepts for a theoreti cal model to describe the mechanism of detachment of soil particles by rainfall in three stages: dry soil conditions, soilwater mixture conditions and soil with overland flow conditions. The model was proposed as a single general equation with changing coefficient val ues in order to describe the three stages of the mechanism. A sto chastic point of view was suggested by Yariv for the solution of the model. Savat (1981) presented results of splash erosion in which net discharge of sediment (downstream splashupstream splash) was found to increase proportionally to (sin )0.9 with respect to the bed's inclination with horizontal. This sediment discharge was also associated with the susceptibility to splash erosion of the soil which, like other previous researchers have indicated, is a complex function of the moisture content of the sand and its grainsize dis tribution. A technique was proposed in order to obtain the mean pro jected splash distance along the sloped plane surface. Park et al. (1982) used dimensional analysis to model the splash erosion of the two possible domains; the direct impact or drop solid domain, and the dropliquidsolid domain. The dropsolid do main was described with analytical relationships from the conserva tion of momentum on a sloped bed. For the dropliquidsolid domain the water layer depth effects were related to the erosion rates by an exponential form. The results showed that the dropsolid domain is time dependent while the dropliquidsolid domain is independent of time. Bed slope effects were also considered in both domains. Recently, Riezebos and Epema (1985) presented the importance of drop shape on the splash erosion. They found that for all test com binations together, the introduction of the observed drop shape in erosivity parameters only produced minor improvements in the relation between erosivity and detachment (or transport) by splash. However, when they used small fall heights and low fall velocities, as in many rainfall simulators and drop tests, the prolate drops produced a splash detachment which was two to three times higher than the one produced by oblate drops at impact. This was partly associated with the high splash erosion in areas below the vegetation. The presence of vegetative cover or any manmade cover over the soil will reduce the splash erosion because this cover will absorb most of the raindrop energy (Mihara, 1951; Free, 1952; Young and Wiersma, 1973, and others). The cover prevents surface sealing; con sequently, the infiltration is not drastically reduced and the rate of runoff is diminished. Free (1952) reported that the presence of straw mulch reduced splash loss to about onefiftieth (1/50) of that from bare soil and sheet flow losses to onethird (1/3). Osborn (1954) indicated that, in addition to the already men tioned soil and rainfall characteristics, the land use management and conservation practices also affect the splash erosion. Other soil characteristics not mentioned before were also related to this ero sion process. 2.5 Overland Flow Erosion 2.5.1 Hydraulics of Overland Flow Knowledge of the hydraulics of the surface water runoff is needed for the detailed understanding of the general soil erosion process. Many studies have been completely dedicated to this complex overland flow hydraulics. Surface runoff is the most dynamic part of the response of a watershed to rainfall. The runoff from a watershed can be subdivided in sheet flow; rills and gullies flow; and openchannel flow. Overland flow deals basically with the first two kinds of flows and it is the one which supplies water and sediment to the open channels. The equations used to describe the open channel hydraulics can be used to describe the overland flow hydraulics if additional terms are included in or der to account for the rainfall effects, the slope effects, and the very shallow flow conditions of overland flow. The most frequently used hydraulic parameters associated with overland flow with rainfall are water depth (h), mean flow velocity (Om), or discharge flow rate per unit width (q). These parameters have been related to the detachment and transport of soil particles in many empirical or semiempirical approaches. Usually, the studies are based on the correlation between the total flow discharge at the lower end of the area under study and the total soil loss of the area. Other studies have also considered the rate of change of water and soil loss with respect to time. Additional parameters which have been used in overland flow erosion studies are bottom shear stress, pressure at the bottom of the shallow water flow, roughness of the bed surface (with and without vegetation), longitudinal slope of the bed surface and its longitudinal length. Rainfall provides input of water over the area. Depending on infiltation, this may allow the flow discharge to change as the water flows downslope. For this reason this flow is usually called spa tially varied flow. The theory used to describe this flow is based on the principles of mass and momentum conservation. Keulegan (1944) first derived the onedimensional dynamic equation for two dimension al spatially varied flow considering the rainfall as lateral flow. Other derivations for this flow condition can also be found in Chow (1959, 1969), Grace and Eagleson (1965, 1966), Chen and Hansen (1966), Chen and Chow (1968), Yen and Wenzel (1970), Morgali (1970), Kilinc and Richardson (1973), and others. The presentation of terms in the equation may change depending on the assumptions and boundary conditions used in each study. In general form, the onedimensional conservation of mass equa tion for spatially varied flow for sloped beds was presented by Chen and Chow (1968) as A + a(AUm) = (I f) B cos0 e + q (2.16) at ax L The corresponding dynamic equation was presented as 2 a(AUm) + $I) m I VD B sin(9 + f) OL UmqL at ax (2.17) = gA sin e g A Sf g a.[A(h cos 0 + h,)] ax where A = crosssectional area f = infiltration rate t = time x = longitudinal distance B = width of the channel section 0 = angle of the path of falling raindrops with respect to the vertical axis 8 = momentum correction factor for the flow velocity distribu tion 01 = momentum correction factor for the raindrop's velocity distribution OL = momentum correction factor for the lateral flow velocity distribution qL = lateral flow discharge per longitudinal unit length g = gravitational acceleration h. = overpressure head induced by the raindrop impacts over the hydrostatic pressure head. Sf = friction slope The overland flow equations which are a special case of the channel flow, can be readily obtained by considering the discharge per unit length, q = AUm/B. Other considerations are that the lateral flow vanishes, B = 1, and the area becomes A = h(1) = h. Hence, q = Umh and the equations are expressed as follows: Conservation of mass for overland flow h+ mh) = (I f) cos (2.18) at ax Momentum equation for overland flow 2 a(h~m) + D(Oh~m2 ) (h ) =gh(S6 Sf) g a [h( h cos e + h.)] (2.19) + 0I I VD sin(e + a) which for nearly horizontal beds gives SO = sin e= tan e= S Figure 2.1 shows a sketch defining these overland flow parame ters. The right hand side terms of the continuity equation are the sources of water. For the sheet flow case, the assumption of water flowing in parallel streamlines toward the rills makes the lateral flow term (qL) equal to zero, but for rills or any open channel flows the lateral flow must be considered. The infiltration term is zero if the bed surface is zero. For overland flow studies researchers have also used the term rainfall excess, 1I = rainfall intensity minus infiltration rate. Overland flow will not exist if rainfall excess shows a negative or zero value. This term was proposed by Eisenlohr (1944) in a discussion of the one dimensional dynamic equa tion derived by Keulegan (1944) for overland flow. The terms in the momentum equation, or the socalled dynamic equation (Equation 2.19) in order of sequence from left to right have the following significance: (a) the unsteady term or local accel eration term; (b) the convection acceleration term; (c) the force due to the water weight and the friction loss term or boundary shear force term; (d) the pressure gradient term which includes the rain fall overpressure term; and (e) the momentum influx due to the fall ing raindrop's component in the longitudinal slope direction. Some studies have neglected the overpressure term but have included the q L term 0(I f + L)Um to account for the retarding effect of rainfall B excess and lateral inflow due to the mixing of the additional mass. This term is obtained when the continuity equation is multiplied by BUm and introduced in the momentum equation as a substitution of the convection acceleration term. 44 VDcos(9 + a) VD VD sin( 0 + )0 ^ I t I ( I I I II #"Control ,' Surface z2 cos9 / Z= h FL/OW ^7^^^n Horizontal Definition Sketch (based on Chen and Chow, 1968). g Figure 2.1. The literature presents studies in which the continuity equa tion and momentum equation are used for overland flow descriptions based on different assumptions and boundary conditions. There are studies for cases of steady or unsteady state conditions; flows over porous or Impervious surfaces; with socalled physically smooth or rough boundaries; under laminar or turbulent conditions; with fixed or loose boundaries, and with or without wind effects. In most stud ies the momentum correction factor 0 was assumed equal to unity due to the difficulties in obtaining the velocity distribution of the shallow overland flow. The use of 0 = 1 assumes uniform velocity distribution in the cross section. Usually the momentum influx due to rainfall (last term in Equation 2.19) has been neglected. This term may be important in cases of steeper slopes or under windy con ditions (Rogers et al., 1967, and Yoon, 1970). Consequently, the qualitative judgment of the results of each study must be based on the assumptions and methodology used by the authors. The possible general application of the results should also be restricted by the same considerations. The study of the relative importance of each term in Equations 2.18 and 2.19 may help to simplify these equations and allow the de velopment of simple hydraulic models based on these physical princi ples. Table 2.1 presents the range of values of variables and dimen sionless parameters in overland flow as reported by Grace and Eagle son (1965). These values were obtained from an extensive literature search in order to establish a similarity criterion for the modeling of overland flow. 00 Cl CD 0 0 Cl 0D 0 4 0 0 .4 0? 0 0 0 .I I '4 0 0 4' (Y) + 0 o C 0 0 4.) r4 0 *4 0 Ln CM 0 *.0o 00 o 4 I I I I I I I 4)LL _j .0 a 1 2 a 4 4' CMJ 0 E ^ a C 13 c 4 2 r 0 t LL. cn (n 4.) C a, 4 4 4, 0 u C 0 41) U M El 2 LL.. 4 : 44 LO 1 0 1 1 1 I 0 CVI 1 I m I 0)0 0 00C 'c  0 0 0 0 0 4() 4) 4J 4) 4) I 1 1 0 I o0 0 0 0 l 4 4 4  CM4 0 0 V44 4 I I I 00 0 0 4v4 4 4 00 0 0 0 0 0 0 4J> 4) 4) 4) CV .t LA  I 1 0 00 0 0 0 04 v4 00C 00 CMi 0 0 .4 LC 0 0 *4 0 0 4. 4) LA) 0 C 0 0 U 4) 4J 4 4 4) 4 0 >.A (n 43 0 u 4J 4.) 0 ai *4 <0 E 0 4) 4 4) 4 *1 a ^3 44 40 0 3 0 IL II .r LL '4 ' C IIC *. 0 *j ft K 4 ') 4 m MC 0 43 S. L (. ) a, > im C0 Robertson et al. (1966) and Yoon (1970) also presented the momentum equation for the case of steady spatially varied flow over an impervious surface with mild slope and discussed the significance of each term of their equation. Both studies used almost the same assumptions and presented the momentum equation in the form 2 ah (1 Bq ) = So Sf 2BIq + I VD cos (2.20a) 7x gh gh2 gh 1 = S Sf S2 + S3 (2.20b) in which B = 1 for Robertson et al. study, and S, S2 and S3 repre sents the simplified form of each term in Equation 2.20a. Table 2.2 shows the relative magnitude of the terms So, S1, S2 and S3 with respect to Sf. These values indicate that the most significant terms of the momentum equation are SO = sine and Sf and the remaining terms are at least two orders of magnitude smaller than SL or Sf. The contribution of these less significant terms (S1. S2 and S3) showed fluctuations which were due to the different testing conditions at the time the measurements were collected (i.e., rain fall intensity and bed slope). There are studies of overland flow with rainfall (e.g., Grace and Eagleson (1965, 1966) and Chen and Chow (1968)) which have indi cated that the pressure distribution is not hydrostatic. They have used an overpressure term in the momentum equation in order to EU x 4 x m 01b CMJ 4 *1 C'M o i~ o o C'.' 4m 1: CM *O m V4 V4 *c* CM' o o to co %0 Ln I I *~ *. 0 CI 4 V/) 6 L&J (0 M COj tV) z I LA (/) U. CD Vd) LLJ U5 i) LLJ c I 0 00 LJ O tJ L&J QC,. EU I *1 u I" 4J 4) CO L off. .9 0 0) 4. 0.. CL 4,l 3 .0 t.0 r0 C CJ 0 3 0 (A ON 4.) S 0 t0 0 LzJ o o 4) 0) C U) *r 4.) ( 4 o #, 0 C> c.0 CO0 0 EU '4 .0  i.0o r9 03 4.)0 S0 I 5 01 0) i Eu TOUII 3 0. 01 OflU 0 5 CM. 00 CM. 0 C'.' 4 4 4 0 0 4 CM. 0 o o U,, o U) o 0 0 0 00 i(A o o o > > cc, 9> EU w 4.) C C 0) 0 0 .0 0 0 0 ). , c; account for the increase in pressure due to the raindrop impacts and the vertical momentum influx of the raindrops. This overpressure term was presented by Chen and Chow (1968) as P* = BI PI VD cose cos (e + ) = p g h* (2.21) where P, = overpressure due to raindrop impact p = fluid density They considered that this overpressure was uniformly distribu ted over the cross section except at the free surface where P be comes zero in order to have atmospheric pressure at the free surface. This approach is based on the assumption that the total head, Th, over a vertical cross section is constant or Th = y cose + + a u = constant (2.22) Y 2g where P = y(h y) cose + h, a = energy correction factor y = distance from the bed surface to a location in the water This assumption created a discontinuity in pressure at an infinites imal small distance, Ay, from the free surface. Grace and Eagleson (1965) have considered that the overpressure distribution was linearly distributed from zero at the free surface to a maximum value of 2P, at the bed surface. They presented expres sions for the overpressure term based on vertical momentum equation and the order of magnitudes of each term. For the horizontal bed with no infiltration and vertical falling raindrops the overpressure term becomes P, = 0.5 p I VD (2.23) Other researchers have used the overpressure term when the momentum equation was presented in their studies (e.g., Kisisel, 1971; Kilinc and Richardson, 1973; and Shahabian, 1977). Kisisel and Shahabian studies also included the rainfall turbulence effect in h and following Grace and Eagleson's approach of linear overpressure distribution the h, expression was presented as 1 1 h = [ BI I VD cose cos (e + n) + v 2(h)] (2.24) where v'2(h) = variance of vertical velocity fluctuations at the free surface. However, Shahabian's results show that the overpressure term induced by the momentum influx of the raindrops seems to be a constant addi tion to the hydrostatic pressure except at the free surface where both the hydrostatic and overpressure terms are zero. This was based on measurements at locations between 0.05 to 0.70 the water depth. The magnitude of this overpressure with respect to other terms in Equation 2.19 is sometimes small and the overpressure term is usu ally neglected. The other reason to neglect this term is the collec tion of data for the evaluation of h, in special values of B0j, VD and Q. It should also be mentioned that the calculated P, value is an average pressure magnitude which is uniformly distributed over a large surface area (with respect to the raindrop impact area) and time, while the falling raindrops reached the overland flow randomly in space and time. Therefore, the maximum overpressure due to the rainfall will be larger than P, and very localized in space and time. Palmer (1963, 1965), and Wenzel and Wang's (1970) data present some of the rainfall effects on the pressure at the bed surface which is protected by a water layer. Unfortunately, the data were collect ed from stagnant water and overland flow effects were absent. No pressure data which might include rainfall effects and overland flow effects were found in the literature review of the present work. 2.5.1.1 Simplified Solutions, the Kinematic Wave Method Due to the complexity of the solution of the longitudinal mo mentum equation (Equation 2.19) with all of its terms, the research ers have used some assumptions and simplifications in order to obtain the magnitudes of the hydraulic parameters needed to describe the overland flow with rainfall. One of the simplest and most frequently used approach is the kinematic wave method. The kinematic wave method has been applied to overland flow over a sloping plane in many studies with good success as an approxi mation of these flow conditions (e.g., Lighthill and Whitman, 1955; Henderson and Wooding, 1964; Wooding, 1965a, 1965b, 1966; Woolhiser, 1969; Eagleson, 1970; Morgali, 1970; Muzik, 1974; Li, 1979; Lane and Shirley, 1982; Crowley II, 1982; Rose et al., 1983a, and others). This approach uses the continuity equation for unsteady spatially varied flow (Equation 2.18) and a simplified momentum equation in which all terms, except bed slope (So = sine) and friction slope (Sf) are neglected. This is based on the low numerical significance of these terms in comparison to the magnitude of SO and Sf. From this it is obtained that S'o = Sf. A relationship between flow discharge per unit width (q) and the flow depth is established by q = akhbk (2.25) where ak and bk are coefficients expressed by the following Laminar Flow bk = 3 ak = gV bk = 5/3 ak = 1.49 So0.5 (using Mann 0M ing's equa tion in Turbulent Flow English units) 8gS' 0.5 bk = 3/2 ak = CS0"5 = () (using f Chezy's equation) where v = kinematic viscosity NM = Manning's roughness coefficient C = (8g/cf)0"5 = Chezy's coefficient cf = Darcy Weisbach's friction factor based on pipe diameter Eagleson (1970) reported that experimental data from Horton (1938) showed that the bk value was about 2.0 for natural surfaces, and that further studies had supported that value for different kinds of surfaces (e.g., vegetated surfaces, clipped grass, and tar and gravel). The fluctuations of the bk exponent had been associated with roughness effects. Usually an increase in roughness is associa ted with the increase of the water depth which means a decrease of the exponent's value. Muzik's (1974) results showed that bk was exactly 1.66 = 5/3 for a galvanized sheet metal surface treated with a diluted solution of hydrocloric acid to change the nonwetting metal surface into a wetting surface. The value of ak is obtained based on known values of NM or C. Woolhiser (1975), Lane and Shirley (1975), Podmore and Huggins (1980), Engman (1986) and others have presented tables of typical values for Manning's NM and Chezy's C coefficients which can be used in overland flow studies. The method of characteristics is frequently used to solve the kinematic wave equations because it only has a single characteristic relation to solve, namely, bk1 dx = q akbk h =bk U0 (2.26) dt Dh since a~k bk1 = = = akh (2.27) h h Using this method, Henderson and Wooding (1964) proposed a series of relationships which allowed calculation of the surface run off from a sloped bed surface at any location along the bed surface and at any time. The method can also be used to produce the hydro graph at any point along the sloped plane. When the kinematic method is used for watershed modeling, the watershed is divided in segments with constant slope and the water flow is routed along the watershed segments (Woolhiser, 1975). Wool hiser (1969) also used the kinematic approach to model the overland flow on a converging surface on which the water moved toward a center point in a radial motion. Morgali (1970) presented computer solutions to this method and studied the behavior of the equations for both cases laminar and tur bulent flows. The variation of the flow regime along the sloped bed was also considered if rainfall and bed surface conditions were favorable and enough time for the test was allowed. His hydrograph results agreed very well with the observations. The only discrepan cies were observed on the rising segment of the hydrograph after the inflection point of the rising limb and before the equilibrium flow was reached at the downstream end of the bed surface. The reason for this is that the kinematic approach does not predict that inflection point in the rising limb. Muzik (1974) tested the kinematic wave method against the in stantaneous unit hydrograph method under laboratory controlled over land flow due to rainfall conditions. He concluded that runoff from an impervious surface generated by rainfall is a highly nonlinear process and any linear analysis of the process does not strictly ap ply. Linear models could only be used as a linear approximation of the rainfallrunoff relationship. On the other side, the kinematic wave model was able to better represent the rainfallrunoff rela tionship and predicted values which agreed very well with the ob served values. The model responded very well to changes in rainfall intensity and slope of the runoff plane, but as observed by other researchers, the kinematic wave model can sometimes overestimate the discharge because of the predicted lack of the point of inflection on the rising limb of the hydrograph. 2.5.1.2 The Law of Resistance The DarcyWeisbach friction factor cf is frequently used in overland flow studies. This is expressed as 8g R'Sf 2 To cf = 82 R 2 (2.28) m2 2 m P Um where R' is the hydraulic radius of the crosssectional area (cross sectional area of the flow divided by its wetted perimeter and usually assumed equal to the flow depth, h, of the overland flow). The friction factor is a function of the flow Reynolds number Uyi (Ref = f) and the boundary roughness. For laminar flows over smooth boundaries the relationship is Cf = 24 (2.29) Ref For laminar flow over rough boundaries Cf can be represented by Cf =K: (2.30) Ref where K is a parameter related to the characteristics of the bed surface and can be as large as 40,000 for dense turf (Woolhiser, 1975). For overland flow with rainfall, the raindrop impacts increase the K factor and it has been represented by the expression K = Ko + ar Ibr (2.31) where Ko is the K value without rainfall and ar and br are empirical coefficients. Tables with typical values for Ko, ar, and br are pre sented in Woolhiser's (1975) study. Woolhiser also indicated that for smooth boundaries (Ko = 24) the raindrop impact effect is important, but it becomes insignificant for vegetated surfaces (Ko > 3000). Izzard (1944) was among the first researchers to use this approach in his study of runoff over rough paved plots. His results suggested the following equation Cf = 27(0.21 I4/3 + 1) (2.32) Ref Shen and Li (1973), using data from various studies of overland flow with rainfall over smooth boundaries, proposed the following equation if Ref < 900. Cf = 24 + 27.162 I0407 (2.33) Ref The transition to an apparent turbulent regime has been report ed at flow Reynolds numbers from 100 to 1000. The higher values usu ally corresponded to the smooth boundaries. Shen and Li (1973) used Ref = 900 as the maximum Reynolds number in the laminar flow regime over smooth surface while Yoon (1970) established this maximum Ref in the range of 1000. Savat (1977) has presented a summary of other maximum laminar Reynolds numbers reported in the literature. He considered that a turbulent flow was believed to prevail when Ref 1 1000, the transi tional flow occurred when Ref 500 and a laminar flow when Ref < 250. Savat also indicated that the maximum laminar Reynolds number changed with changes on the bed slope as seen in other investigations. For the turbulent flow regime, there are many proposed rela tionships to use. The Blasius equation Cf = 0.233 (2.34) Ref0.25 (Woolhiser, 1975) can be used for smooth boundary flows without rain fall and a Reynolds number less than about 30,000. Robertson et al. (1966) used the same type of equation to express the friction factors for three different rough boundaries under rainfall conditions. Un fortunately, for the flow Reynolds number range tested in the study (400 < Ref < 4500), the coefficients of their equations changed for each rough surface studied. Another equation which is frequently used for the turbulent flow regime is Manning's equation in English units (assuming R' = h) 0 1.49 S1/2 h2/3 (2.35) m NM f Robertson et al. introduced the DarcyWeisbach equation and solved for cf to obtain 8g NM2 cf = 8gN2 (2.36) 1.49 h1/6 or 8g Sf1/10 NM9/5 NM 9/5 cf =  = c =__" (2.37) ( vRef)l/5 1.499/5 Refl/5 Refl/5 In this form the equation has a similar form to Blasius equation (Equation 2.34). Robertson et al. (1966) reported that in one of their three rough surfaces studied the value of the flow Reynolds number exponent was equal to 0.20 = 1/5 with correlation coefficient of 0.74 for the Ref range between 550 and 4500. For larger Ref values the effects of viscosity and rainfall are diminished and the friction factor is usually considered constant for that bed surface. Consequently, the Chezy's equation can be used since cf no longer depends on Ref. cf = 8 (2.38) C2 Yen et al. (1972) developed equations for the friction slope (Sf), the total head slope (SH) and the dissipated energy gradi ent (Se) from the general equations presented in Yen and Wenzel (1970) and in Yen (1972) for overland flow with or without rainfall over smooth boundary case. Then each loss gradient term was written in DarcyWeisbach resistance coefficient form (Equation 2.28) and showed that each slope term was numerically different to each other. The difference in magnitude among the coefficient depended on the flow conditions (i.e., with or without rainfall or lateral rainfall). Based on their results, they suggested that for steady spatially var ied flow computations the momentum equation was preferred to the energy equation or total head equation, particularly if the Darcy Weisbach's cf, Manning's NM, or Chezy's C coefficients are used as the resistance factors. Shen and Li (1973) also developed equations for the friction factor and other parameters for rainfall conditions over "smooth" surfaces based on the ratios of each parameter value (i.e., water depth, water discharge, mean velocity, boundary shear stress, Froude number and friction factor) under rainfall conditions and the equiva lent parameter without rainfall with the same flow discharge rate. This is like using the flow Reynolds number (Ref) as the scaling number. They also obtained Equation 2.33 to calculate the friction factor in the laminar range (Re < 900). For the very turbulent regime (Re > 2000), the friction factor was obtained from Blasius' equation but with a different coefficient value for rainfall condi tions. For the intermediate flow regime (900 < Ref < 2000) a lin early interpolated equation was proposed. These previous equations were obtained by regression analysis of their data and from the lit erature. Their statistical analysis indicated that the uncertainty in the selection of the friction factor for the computation of flow depth and boundary shear stress was not too sensitive and that the error in using incorrect friction factors was not cumulative with each step of their numerical model. Their equations were recommended under the conditions of being used only for 126 < Ref < 12,600, 0.5 in./hr (12.5 mm/hr) < I < 17.5 in./hr (445 mm/hr), 0.005 < So < 0.0108 and over a physically "smooth" boundary. Savat (1977) presented a good summary of the hydraulics of sheet flow on physically smooth surfaces. He also discussed some roughness conditions and presented equations for flowmean velocity, friction factor and Manning's NM. The variation of the exponents of the water depth and the bed slope terms in the equations due to the flow regime (i.e., laminar, transition or turbulent) were also discussed. His comparison with available literature suggested that sheet flow could be either laminar or purely turbulent, but that mixed flows prevailed on low slopes (under 5% slopes) combined with greater depths. He also indicated that in most cases sheet flows were supercritical, specially on steep slopes. Savat's equations and experiments indicated that the effect of raindrop impacts on the DarcyWeisbach friction factor, cf, did not exceed 20% in the case of laminar flow on gentle slopes. He also indicated that the rainfall influence diminished when the discharge or the Reynolds number increased as well as when the bed slope angle increased. Savat also used an equation for cf in hydraulically smooth turbulent flows, originally presented by Keulegan (1938), in which Savat rearranged by using the DarcyWeisbach equation (Equation 2.28) and the flow Reynolds number definition to obtain the expres sion 1 = 5.75 tog(Ref(cf/8)1/2) + constant (2.39) Cf1/ Julien and Simons (1985) also suggested the use of the equation originally proposed by Keulegan (1938), but they used Blasius' equation for this kind of flow. Their definition for hydraulically smooth flow was that the viscous sublayer, 6 = 11.6v(p/F ) 05, was greater than three times the size of the sediment particles, ds. When the thickness of the viscous sublayer is small compared to sediment size the flow is considered hydraulically rough and the logarithmic equation also given by Keulegan (1938) was considered to apply. This equation was presented by Julien and Simons as i/2 (89) = C = aI Xog(a2 h) (2.40) Cf s where al and a2 are constants. However, they used approximated power relationships such as Manning's equation to express the friction factor. Thornes (1980) also presented a similar expression to Equation 2.40 to obtain the friction factor which was originally used by Wol man (1955). Savat (1980) considered the resistance to flow in rough super critical sheet flow which is present on steep slope flows. However, the study only considered overland flow with no rainfall. Expres sions to obtain the DarcyWeisbach friction factor were obtained for both laminar and turbulent flow regimes based on the classical cf equations and compared with his laboratory results. The observed values were found to be higher than the ones obtained from classical equations. The discrepancies were associated with the great varia tion on the relative depth of standing and travelling waves usually found on steep slope flows, and due to the turbulence and wake forma tion around the bottom grains. 2.5.1.3 Boundary Shear Stress Kinematic Approach. The timemean boundary shear stress To is related to the friction slope (Sf) by the equation to =YR'Sf (2.41) For the case of overland flow the hydraulic radius, R', is equal to the water depth, h. Some researchers have used the water depth for their T0 calculations. Another frequently used approxi mation is the assumption of bed slope, So = sine, being equal to the friction slope based on the relative magnitude of the terms on the momentum equation (the kinematic wave method). Another reason for this assumption is the problem of estimating the friction slope especially under field conditions. Vegetation and cover material over the soil make it practically impossible to directly measure the parameters in order to calculate the friction slope. These assumptions lead to the equation o = y S (2.42) which some researchers have used as the real value for To while others have used it correctly as a first approximation only. Dynamic Approach. Another form to obtain To is by solving the dynamic equation (Equation 2.19) for Sf and obtaining T0 from Equation 2.41. Based on the assumptions made by the authors of each study, the representation of the dynamic equation may be slightly different. As mentioned before, Keulegan (1944) was the first to express that equation for the case of spatially varied flow like the case of overland flow with rainfall. Other articles which have pre sented their derivations for this equation or at least have presented possible methods to solve it are Woo and Brater (1962), Morgali and Lindsey (1965), Grace and Eagleson (1965, 1966) Ligget and Woolhiser (1967), AbdelRazaq et al. (1967), Chen and Chow (1968), Chow (1969), Morgali (1970), Eagleson (1970), Yen and Wenzel (1970), Yen (1972), Yen et al. (1972), Kilinc and Richardson (1973), and others. The dynamic equation can only be solved by numerical techniques due to the complexity of the equation. Keulegan (1944) recommended that before any approximate solu tion is attempted the dependence of the friction factor on the flow Reynolds number is required to be well known. Izzard (1944) was among the first to present that relationship from curve fitting of data collected from rougher paved plots. Izzard also obtained that the water depth was proportional to the cubic root of the longitudinal distance from the upper end of the slope. This rela tionship had a certain limit which was associated with the change in the flow regime from laminar to turbulent flow. The onethird power was also associated with Equation 2.25 used in the kinematic wave method given that the flow discharge per unit width is expressed as q = Ix. Yoon (1970) presented in his doctoral dissertation very signi ficant information about T over physically smooth surfaces when sim ulated rainfall was applied to overland flow. His measured T0 values obtained from a flat surface hotfilm sensor agreed very well with the computed T values from the one dimensional spatially varied flow 0 equation developed by Yen and Wenzel (1970), Equation 2.20a. This showed the applicability of the onedimensional dynamic equation of spatially varied flow for practical purposes. Yoon indicated that, for a constant flow Reynolds number, T 0 increased appreciably with increasing rainfall intensities. This happened for Refvalues of up to approximately 1000. The rainfall intensity effect became negligible as Ref further increased. He also showed that the relationship expressed by Equation 2.30 was val id for overland flow with rainfall and Ref < 1000 with the constant K Increasing with an Increasing rainfall intensity and with a small dependence of the bed slope. Later, Shen and Li (1973) indicated that the slope effect was not significant on the Kvalue. So both studies clearly showed that the flow Reynolds number and rainfall intensity are the most important parameters affecting the timemean boundary shear stress. Kisisel's (1971) study showed the same trend as Yoon's results. The increase in T and cf with the increase in the rainfall intensity 0 was equally observed over both physically smooth and rough surfaces and particularly for laminar and transitional flow Reynolds numbers. In this study the Cf values also increased when the bed slope increased. For the physically rough surface case the increase in Cf was slightly larger than that observed for flows with rainfall over the smooth surface. He indicated that for both surfaces, the main factor affecting the friction factor values was found to be the rainfall input. The studies of Yoon (1970), Kisisel (1971), and Shen and Li (1973) were conducted at nearly horizontal uniform slopes (So < 3%) with a fixed bed. These are ideal conditions in which the dynamic equation was found to be a useful tool to evaluate To and cf. Kilinc and Richardson (1973) also used the dynamic equation for that purpose too, but their study was conducted at steeper bed slopes (5.7% < So < 40%) and with a movable bed (silty sand). Kilinc and Richardson obtained fo introducing Equations 2.18 and 2.24 into Equation 2.19. Their study considered steady state conditions with B = 1, BI = 1, q = (I f)x, assumed that the in filtration rate (f) was constant along the bed slope and used h = q/um to express the water depth. The solution for T0 at the downstream end of the plot was obtained numerically using their ex perimental data. These To values which included rainfall effects were found to be less than the T calculated from Shen and Li's ex 0 pression for cf, Equation 2.33, but greater than the T0 calculated assuming uniform flow, Equation 2.28. The T0ovalues were later used in that study to develop empirical equations for the sediment discharge of that sloped area subjected to rainfall. 2.5.1.4 Entrainment Motion and Critical Shear Stress The discrete soil particles of the bed in any stream are subjected to tractive forces (e.g., shear stress) and lift which try to move the particles. They are, of course, also subjected to resist ing forces (e.g., buoyant weight and frictional forces) which will prevent the particle motion. When the tractive forces are equal to the resisting forces, the particle will be in an entrainment condi tion. Under this condition the magnitude of the timemean bed shear stress (TO) will be considered as the timemean critical shear stress value (Tcr) of the instantaneous tractive force to have been applied to the bed surface area. This critical value is basically a function of the particle density, size, shape and roughness, and the arrangement of the individual particles in the bed surface. If the acting force is larger than a critical value, the parti cle will be set in motion. The resulting modes of transport (i.e., rolling, saltation or suspension) depend on how much greater the act ing force is with respect to critical force. If the acting force is greater but nearly equal to the critical force of the particle roll ing or sliding will be the predominant mode of transportation. A greater acting force can make the particle start saltation motion and when the acting force is high enough, the particle will be suspended in the stream. So, for a given acting force, higher than the criti cal one, the larger size particles will usually move by rolling or saltation and the smaller size particles will predominantly move by suspension. Usually particles transported in suspension are referred to as the suspended load. The particles which move by rolling or saltation are referred to in the literature as bedload. There is also the so called washload which is made up of grain sizes finer than the bulk of the bed particles and thus is rarely found in the bed of the stream. These particles are usually washed through the section of the stream. Some authors have considered washload and suspended load as the same load in their sediment transport definition. The total sediment transport load is referred to as the sum of bedload and suspended load and/or washload depending on the definition used by the authors. In this study, the total load will be considered as the summation of bedload and suspended load with washload being included into the suspended load. The critical force required to begin the motion of particles has been associated basically with two theories. First, Hjulstrom (1935; Graf, 1984) presented an erosiondeposition criteria based on the crosssectional mean flow velocity (5m) required to move parti cles of certain size. Figure 2.2 shows this basic erosiondeposition criterion for uniform particles. It shows the limiting zone at which incipient motion starts and the line of demarcation between the sedi ment transport and sedimentation. The diagram also indicates that loose fine sand is the easiest to erode and that the greater resis tance to erosion in the smaller particle range must depend on the cohesion forces. 68 00 ]f ill I 1 11 I 1 1 ~I I Hl l 500 0 0 111 30 000 I00m Figure22 ErosionDpsto Crtei fo Unior Particle (afte Hi st Tronsporotion rpi priso o at R.5e s Pi ?^ ^ ;;: ^ ^ ;; ^ =; 1I =;:l^ EE;;E1 ~ 0 3 i! 1 l!11  I I '.., _:: :: : I : : I : :  0 .3 ^ i...ii.,..i . ... . [~l . ^ . _ . __ . I ll l l: ;:... ....... 0 .2 7 0 l ll l I I 1 0.1 _j __z: s Cf S C0 ( on ^ Cy fe in y inr in 0000 0 85 88 0 o 6 d6 0 (vC 8 Y 0 0 0 0 d.mm Figure 2.2. ErosionDeposition Criteria for Unifom Particles (after Hjulstrom, 1935; Graf, 1984; reprint with pemission of Water Resources Publications). The second theory is based on the timemean critical shear stress, .cr. DuBuat (Graf, 1984) used this approach during the late eighteenth century, but it did not become popular until the beginning of this century when Schoklitsch published his results (Graf, 1984). Since then, other researchers have used this approach too. In 1936, Shields (Graf, 1984) used the shear velocity, v*, which represents a measure of the intensity of turbulent fluctuations near the bottom boundary. This is related to the bed shear stress by the expression v* = (To/p)1/2 (2.43) Shield used this term in order to describe his wellknown en trainment motion approach which he presented in Figure 2.3. This diagram is a graphical representation of the threshold movement of particles. It was developed from a dimensional analysis for longi tudinal flows without the influence of raindrops. Shields considered the disturbing force to be the shear force and assumed that the resistance of the particle to motion should depend only on the form of the bed and the buoyant weight of the particle. He studied these forces for different flow conditions and showed that the threshold movement of particles could be represented by a single parameter called the entrainment function, Eh, defined as Iv 2 v d Eh = cr = *cr = Function (*S) (2.44) (Ys Y)ds gds(SGs 1) 70 ' 1 d0 "4 ) 4&J SO~ CN x C N A r4 c ir> kIB.   Ch Z' ,4J 11: icvJ w ff, T/3 C C "0 c c '00 .  "I, 0 a ."  E 2 00 I ci i #I on S. MP. No w !!^ *ERZ s 0 I E~ 0 50~( winJJJ ac 2 0 U  C.3I v CS nnf E1 a. 0 M1 < cQ ~ ^*tu~N~LLL. ^ ~ 0 _______L a i. S a *= ^* c 'A>. ' where vcr = timemean critical shear velocity SGs = specific gravity of particles if fluid is water V*ds = particle Reynolds number V This entrainment function, Eh, and the particle Reynolds number were used to construct the wellknown Shields' diagram of Figure 2.3. Using an analytical approach to the threshold movement of a single particle, White (1940) confirmed the general form used by Shields in spite of White's unfortunate omission of the hydrodynamic lift on the individual grains. In addition, Yalin (1963) derived a similar relationship of this diagram from a model for lift forces rather than for shear forces. This adds more confidence to the Shields' diagram. More recently, Mantz (1977) presented an extension of the Shields' diagram for the case of fine cohesionless grains (i.e., particle diameter in the range of 0.01 mm to 0.150 mm). The dashed line in Figure 2.3 shows the regression line proposed by Mantz based on his data and from the literature. The use of Shields' diagram in overland flow with and without rainfall was proposed by Foster and Meyer (1972a) as the method to obtain the timemean critical shear stress of the soil particles ex posed to overland flow. Also Kisisel et al. (1971) suggested using Shields' diagram to obtain ?cr which was required for use of the DuBoys' equation in overland flow 