Question

The erosion rate of a streambank is calculated by the excess shear stress equation: e=k(a-c) where e is the erosion rate in m/s, k is the erodibility coefficient in m^3/ N s, a is the applied shear stress (Pa) and c is the critical shear stress (Pa). The applied shear stress is calculated using the equation a=den*g*d*s where den is the density of water (1000 kg/m^3), g is the acceleration due to gravity (9.8 m/s^2), d is the depth of water (m) and s is the bed slope (m/m). The critical shear stress is a function of the silt and clay content of the streambank and is calculated using the following equation: c=0.1+0.1779(SC%)+0.0028(SC%)2-2.34E-5(SC%)3 where SC% is the percent of the streambank that is silt and clay. The erodibility coefficient is calculated using the equation k=0.20*c^-0.5. Develop a program in MATLAB that calculates the erosion rate of a streambank. The user should enter the following values: depth of water, bed slope and percent of silt and clay. The program should write out “The erosion rate is:”, the rate calculated and the units. Test the program using the following: d=1, s=.001, SC=35 and d=0.1, s=.004, SC=50. Document the erosion rate. Paste all input and output code into this Word document.

Answer 1

Streambank erosion is known to be a significant
source of sediment in many impaired streams (Si-
mon et al., 2000; Wilson et al., 2008; Fox and Wil-
son, 2010). Particle detachment models are often
employed to predict rates of streambank erosion due to
fluvial processes within a basin. Commonly, the erosion
rate of cohesive streambanks is simulated using the excess
shear stress equation (Partheniades, 1965; Hanson, 1990a,
1990b), which is defined as:
( )a
rd c ? = ??? k (1)
where ?r is the erosion rate (cm s-1), kd is the erodibility coef-
ficient (cm3
N-1 s-1), ? is the average hydraulic boundary
shear stress (Pa), ?c is the critical shear stress (Pa), and a is
an empirical exponent commonly assumed to be unity (Han-
son, 1990a, 1990b; Hanson and Cook, 2004). Using this
model, erosion initiates once ? exceeds ?c, and kd defines the
rate at which particles are detached after erosion is initiated.

In order to estimate kd as a function of ?c for cohesive
soils, Hanson and Simon (2001) suggested an inverse rela-
tionship between kd and ?c:
0.5 0.2 ? = ? d c k (2)

Hanson and Simon (2001) derived their relationship
based on 83 in situ JETs conducted on cohesive streambeds
in the Midwestern U.S. A wide data range was observed,
with ?c spanning six orders of magnitude and kd spanning
four orders of magnitude. A general inverse relationship
was observed between ?c and kd, suggesting that soils with
a low ?c have a high kd and vice versa. Their relationship
predicted the data with a coefficient of determination (R2
)
of 0.64 and was incorporated into streambank erosion and
stability models, such as the Bank Stability and Toe Ero-
sion Model (BSTEM), as a tool for estimating kd from ?c
(Midgley et al., 2012). This relationship was recently up-
dated by Simon et al. (2011) based on hundreds of JETs on
streambanks across the U.S.:
0.838 1.62 ? = ? d c k (3)

Analytical methods for the JET were first presented by
Hanson and Cook (1997, 2004), assuming that the rate of
variation in the depth of scour (dJ/dt) was the erosion rate
as a function of the maximum stress at the boundary, which
was determined by the diameter of the jet nozzle and the
distance from jet origin to the initial channel bed. There-
fore, the erosion rate equation for jet scour is written as
(Hanson and Cook, 1997):
?
?
?
?
?
?
?
?
? ?
? = c
o p
d
J
J
k
dt
dJ
2
2
for J >= Jp (4)
where J is the scour depth (cm), and Jp is the potential core
length from jet origin (cm). Accordingly, ?c was assumed to
occur when the rate of scour was equal to zero at the equi-
librium scour depth (Je):
2
?
?
?
?
?
?
?
?
? = ?
e
p
c o J
J
(5)
where ?o = Cf?wUo
2
is the maximum shear stress due to the
jet velocity at the nozzle (Pa), Cf = 0.00416 is the coeffi-
cient of friction, ?w is water density (kg m-3), Uo is the jet
velocity at the orifice (cm s-1), Jp = Cddo, do is the nozzle
diameter (cm), and Cd = 6.3 is the diffusion constant. Equa-
tions 4 and 5 can be incorporated in a dimensionless form
as the following equation:
( ) 2
2
1 * *
* *
J dJ
dT J
?
= (6)
where J
*
= J/Je and Jp
*
= Jp/Je. Stein and Nett (1997) pre-
sented the reference time (Tr) as follows:
d c
e r k
J T
? = (7)
and the dimensional time (T*
) was given as:
T*
= t / Tr (8)
where t is the time of a data reading or scour depth meas-
urement.
Equation 6 refers to the change in scour depth with time,
for time T*
. Integration of equation 6 gives the following
equation:
?
?
?
?
?
?
?
?
?
+
+ ? ?
?
?
?
?
?
?
?
?
+ ? = ? + *
*
*
*
* * * *
1
1
0.5ln
1
1 0.5ln
p
p
p p J
J
J
J
J T T J (9)
The Excel spreadsheet discussed by Hanson and Cook
(2004) used equations 4 through 9 to determine ?c and kd.
The critical stress (?c) was determined from equation 5
based on the equilibrium scour depth (Je). Blaisdell et al.
(1981) noted that it was difficult to determine the equilibri-
um scour depth due to the large time required to reach Je.
Therefore, the spreadsheet calculated the equilibrium scour
depth using the scour depth data versus time and a hyper-
bolic function for determining the equilibrium scour depth
developed by Blaisdell et al. (1981). The general form of
this equation is:
(f – fo)
2
– x
2
= A1
2
(10)
where A1 is the value for the semi-transfer and semi-
conjugate of the hyperbola, f = log(J/do) – x, x =
log[(Uot)/do], and fo = log(Je/do). From fitting the scour
depth data based on plotting f versus x, the coefficients A1
and fo can be determined using Microsoft Excel Solver, and
then Je can be determined (Je = do10fo). The spreadsheet
was then used to calculate kd by fitting the curve of meas-
ured data based on equation 9. The kd depends on the
measured scour depth, time, pre-estimated ?c, and the di-
mensional time function (Hanson et al., 2002).