Soil erosion by overland flow with rainfall


Material Information

Soil erosion by overland flow with rainfall
Physical Description:
xxix, 400 leaves : ill. ; 28 cm.
Zapata, Raúl Emilio, 1956-
Publication Date:


Subjects / Keywords:
Soil erosion -- Mathematical models   ( lcsh )
Rain and rainfall   ( lcsh )
Civil engineering thesis Ph. D
Dissertations, Academic -- Civil Engineering -- UF
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1987.
Includes bibliographical references.
Additional Physical Form:
Also available online.
Statement of Responsibility:
by Raul Emilio Zapata.
General Note:
General Note:

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 024127055
oclc - 19712592
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Full Text







Copyright 1987


Raul Emilio Zapata

To Carmencita,

Raul Enrique and Manri Luz


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.



ACKNOWLEDGMENTS . . . . . . . . . . .. iv

LIST OF TABLES . . . . . . . . . . . . x


LIST OF SYMBOLS . . . . . . . . . . .

ABSTRACT . . . . . . . . . . . . .


I. INTRODUCTION . . . . . . . . .



1.1 The Soil Erosion Problem . . . . . .
1.2 Purpose and Scope of This Study . . . .

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 . Simolified Solutions.



Kinematic Wave Method . The Law of Resistance . Boundary Shear Stress .
Kinematic Approach . .
Dynamic Approach . . . Entrainment Motion and
Critical Shear Stress . Flow Velocity . . . Turbulence . . . .
Overland Flow Erosion Studies . .


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 . . . . .


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 . . . . .


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 Time-Mean Bed Shear Stress Evaluation
4.2.7 Grain Reynolds Number Evaluation .
4.2.8 Dimensionless Sediment Transport
Parameter . . . . . .. Evaluation of . . . Evaluation of V and .
4.2.9 Sediment Transport Data in Diagrams .












* *

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


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 su-Values . . . . . . . .. .327
5.6 Final Remarks . . . . . . . .. .345


6.1 Conclusions . . . . . . . .. 352
6.2 Recommendations . . . . . . .. .355

APPENDICES . . . . . . . . . . . . .. 359





REFERENCES . . . . .


. . . . . . . . . 361

. . . . . . . . . 371

S. . . . . . . . 373

. . . . . . . . . 400


Table Page


TERMS OF Sf . . . . . . . . . . .. 48



RICHARDSON'S DATA . . . . . . . . . .. .195

TRANSPORT VARIABLES . . . . . . . . . .197

FOR FIGURES IN THIS STUDY . . . . . . . .211


RELATIVE ERROR . . . . . . . . . .. 258
FACTOR . . . . . . . . . . . . .. 262


WITH SAME BED SLOPE . . . . . . . . .. 279


5.9 ANALYSIS OF VARIANCE . . . . . . . . .. .286



ESTIMATED ERROR OF DATA . . . . . . . .. .300


PREDICTED MODEL SOLUTION . . . . . . . .. .306

GIVEN WATER DEPTH . . . . . . . . .. 318

5.16 PREDICTED su AND p-VALUES . . . . . . . .. .333

3.22) . . . . . . . . . . . . .. 365





Figure Page

2.1 Definition Sketch . . . . . . . . . .. 44

2.2 Erosion-Deposition 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 Grain-Size 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 su-Value Using C, = 2.2 x 107 ft-2 (Constant) 245

4.9 Predicted Su-Value Versus Measured Sediment
Concentration using C* = 2.2 x 107 ft-2 (Constant) . 246

4.10 Comparison of Observed and Predicted Dimensionless
Sediment Transport . . . . . . . . .. .252

4.11 Predicted and Required su-Values . . . . . .. .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 D-Values 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 4-Values . . . . .. 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


5.16 Required su-Value Versus Measured Sediment
Concentration, Cs . . . . . . . . . .. 341
5.17 Predicted su-Values 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 Su-Values . . . . . . . .. . 370


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 cross-sectional 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

C5 = Su-value 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

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 = Darcy-Weisback friction factor

c' = 8g SfOI/(1.481"8 0.2)

c" = C'NM1"8

D = Drag Coefficient

D5O = mean equivalent spherical raindrop-size 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 raindrop-size diameter

DF = soil detachment by runoff

DI = soil detachment by rainfall
D = *S 1
2 K du I
dzI z=h

D* = LoDF/Tco

d = ds = grain-size 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


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 10-4 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

F-test 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


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 30-minute
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 = K-value 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

ns = number of straight lines into which the grain-size 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 B-2)
1 x gh
S2 2Iq

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
SDI = soil properties effect constant for soil detachment by
Se = slope of energy grade line

SEE = standard error of estimate

Sf = friction slope

SGs = specific gravity of particle is fluid in water


SH = total head slope
STF = soil properties effect constant for soil transport by

STI = soil properties effect constant for soil transport by

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 t-value 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 time-mean longitudinal velocity

u' = local longitudinal velocity fluctuation

uh = lateral velocity of water moving away from waterdrop impact
Umax = maximum local time-mean velocity at distance ymax
from the bed surface

Ut = it + uj = instantaneous velocity near top of grains on the
Ut = time-mean velocity near top of grains on the bed


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

v = time-mean vertical velocity

v' = vertical velocity fluctuation at a given location

v* = shear velocity = 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 = time-mean 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


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 t-Student 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


= 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

= soil shear strength

To = instantaneous bed shear stress

To = time-mean bed shear stress

Tcr = time-mean bed shear stress when p = 10-3

Tcrs = time-mean bed shear stress for sloping beds when p = 10-3

Tf = time-mean shear stress due to form roughness

Tg = time-mean shear stress due to grain roughness

Ts = time-mean shear stress at the water surface

S= sediment transport intensity function

@ = angle of repose


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


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



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 time-mean bed shear stress is

obtained from the longitudinal momentum equation including additional


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.



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

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

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 quasi-steady 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


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
Soil erosion caused by the combined actions of rainfall and

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


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



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


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

stress-strain 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 sediment-laden 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-


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


Erosion 1301.75

Erosion S So1'35

Erosion XL 0


130 = maximum amount of rain in 30 minutes of rainfall

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 mass-terminal 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 10-6 < ReA < 0.01

20 pm < De < 1 mm with 0.01 < ReA < 300

1 mm < De < 7 mm with 300 < ReA < 4000


De = equivalent spherical drop diameter

Re = D De = drop Reynolds number

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 cumulative-volume 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 distribution-rainfall 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 mean-drop 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 size-rainfall 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


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 30-minute 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)

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

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


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

(kg *


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-


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 co-authored a series of articles

in which the individual characteristics of the waterdrop splash were


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 one-third 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

three-drop 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


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 three-drop 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 Navier-Stokes 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 Navier-Stokes

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 micro-relief.

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 cone-shaped 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 physically-based 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 grain-size 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 bell-shaped 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 grain-size 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

time-dependent 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 well-sorted 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, soil-water 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

Savat (1981) presented results of splash erosion in which net
discharge of sediment (downstream splash--upstream 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 grain-size 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 drop-liquid-solid domain. The drop-solid do-

main was described with analytical relationships from the conserva-

tion of momentum on a sloped bed. For the drop-liquid-solid domain

the water layer depth effects were related to the erosion rates by an

exponential form. The results showed that the drop-solid domain is

time dependent while the drop-liquid-solid 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 man-made 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 one-fiftieth (1/50) of that

from bare soil and sheet flow losses to one-third (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 open-channel 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 semi-empirical 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 one-dimensional 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 one-dimensional 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

a(AUm) + $I) -m I VD B sin(9 + f) OL UmqL
at ax
= gA sin e- g A Sf g a.[A(h cos 0 + h,)]


A = cross-sectional 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-

01 = momentum correction factor for the raindrop's velocity

OL = momentum correction factor for the lateral flow velocity

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

a(h~m) + D(Oh~m2
) (h ) =gh(S6 Sf) g a [h( h cos e + h.)]


+ 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 so-called 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
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.


VDcos(9 + a)


VD sin( 0 + )0

I t
I (
,' Surface

z-2 cos9 /

Z= h



Definition Sketch (based on Chen and Chow, 1968).


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 so-called 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.


0 0





0 0
0 .-I -I
'-4 0 0







Ln CM 0
00 o -4


a 1

a 4-
4' CMJ

0 E ^
a C 13

c 4- 2

r 0 t

LL. cn (n


U M El 2

LL.. -4 : 44-

1 0 1 1 1


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 -




V4-4 -4
00 0 0
-4v-4 -4 -4

00 0 0
0 0 0 0
4J> 4-) 4-) 4-)

CV .t LA -
I 1 0
00 0 0
0- 0-4 v-4
00C 00


0 0 .-4

0 4.

0 C
0 0


4-) 4J-
4- 4-







E 0

4) 4-
4) 4
*1- a

^3 44
40 0
3 0

IL II .r-
LL '4-
'- C
IIC *.-

*j -ft

K 4 ')


m MC

0 4-3 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
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





01b CMJ

*-1 C'M
o i~

o o

C'.' 4m
1-: CM
*O m
V-4 V-4

*c* CM'

o o

to co
%0 Ln

I- I

*~ *.

0 CI

























3 .0


r0 C-


3 0

S- 0
t0 0

o o

U) *r-

4.) (

o #,

0 C>

EU '-4
.0 -



03 4.)0

S-0 I 5-

i- Eu


0. 01

OflU 0

CM. 00
CM. 0
C'.' -4
-4 -4


CM. 0
o o

o U)
o 0 0

0 00


o o o

>- >- cc,


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)


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


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)


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


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. 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)
ak and bk are coefficients expressed by the following

Laminar Flow bk = 3 ak = g-V

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

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 non-wetting
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,

dx = q akbk h =bk U0 (2.26)
dt Dh

a~k bk-1
= = = 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 rainfall-runoff relationship. On the other side, the kinematic

wave model was able to better represent the rainfall-runoff 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. The Law of Resistance

The Darcy-Weisbach 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 cross-sectional 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
(Ref = -f) and the boundary roughness.

For laminar flows over smooth boundaries the relationship is

Cf = 24 (2.29)

For laminar flow over rough boundaries Cf can be represented by

Cf =K:- (2.30)

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,


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)

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)

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)

(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 Darcy-Weisbach equation and solved
for cf to obtain

8g NM2
cf = 8gN2 (2.36)
1.49 h1/6
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)

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 Darcy-Weisbach 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-


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 flow-mean 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 Darcy-Weisbach 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 Darcy-Weisbach equation (Equation

2.28) and the flow Reynolds number definition to obtain the expres-


1 = 5.75 tog(Ref(cf/8)1/2) + constant (2.39)

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

(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 Darcy-Weisbach 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. Boundary Shear Stress

Kinematic Approach. The time-mean 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), Abdel-Razaq 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 one-third 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 hot-film sensor agreed very well with

the computed T values from the one dimensional spatially varied flow
equation developed by Yen and Wenzel (1970), Equation 2.20a. This

showed the applicability of the one-dimensional dynamic equation of

spatially varied flow for practical purposes.

Yoon indicated that, for a constant flow Reynolds number, T
increased appreciably with increasing rainfall intensities. This

happened for Ref-values 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 K-value. So both

studies clearly showed that the flow Reynolds number and rainfall

intensity are the most important parameters affecting the time-mean

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
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-
pression for cf, Equation 2.33, but greater than the T0 calculated

assuming uniform flow, Equation 2.28. The T0o-values were later

used in that study to develop empirical equations for the sediment

discharge of that sloped area subjected to rainfall. 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 time-mean bed shear

stress (TO) will be considered as the time-mean 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


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 erosion-deposition criteria based on

the cross-sectional mean flow velocity (5m) required to move parti-

cles of certain size. Figure 2.2 shows this basic erosion-deposition

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.


00 ]f| ill I 1 11 I 1 1
~I I Hl l

500 0 0 111

30 000 I00m

Figure22 Erosion-Dpsto 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...i-i.,..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

Figure 2.2. Erosion-Deposition Criteria for Unifom Particles
(after Hjulstrom, 1935; Graf, 1984; reprint with pemission of Water
Resources Publications).

The second theory is based on the time-mean 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 well-known 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)


' 1


"4 )

A r-4
c ir>

kIB. -

-- Ch
Z' ,4J 11:
icvJ w- ff, T/3 C

C "0

c- c

. -- "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 -

v CS nnf E1
a. 0- M1 < cQ
~ ^*tu~N~LLL.

^ ~- 0
_______L a ------ -o
i. S
a *= ^*
c 'A>. '


vcr = time-mean critical shear velocity

SGs = specific gravity of particles if fluid is water

V*ds = particle Reynolds number

This entrainment function, Eh, and the particle Reynolds

number were used to construct the well-known 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 time-mean 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