River, Coastal and Estuarine Morphodynamics: RCEM2011
© 2011
Floodplain heterogeneity and meander migration
MOTTA Davide
Department of Civil and Environmental Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
E-mail: dmotta2@illinois.edu
ABAD Jorge D.
Department of Civil and Environmental Engineering
University of Pittsburgh, Pittsburgh, Pennsylvania, USA
E-mail: jabad@pitt.edu
LANGENDOEN Eddy J.
US Department of Agriculture, Agricultural Research Service
National Sedimentation Laboratory, Oxford, Mississippi, USA
E-mail: eddy.langendoen@ars.usda.gov
GARCIA Marcelo H.
Department of Civil and Environmental Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
E-mail: mhgarcia@illinois.edu
ABSTRACT: The impact of horizontal heterogeneity of floodplain soils on rates and patterns of
meander migration is analyzed with a Ikeda et al. (1981)-type model for hydrodynamics and bed
morphodynamics, coupled with a physically-based bank erosion model according to the approach
developed by Motta et al. (2011). We assume that rates of migration are determined by the resistance to
hydraulic erosion of the soils, which is described by an excess shear stress relation. This relation uses two
parameters characterizing the resistance to erosion: critical shear stress and erodibility coefficient. The
spatial distribution of critical shear stress in the floodplain is generated on a regular grid with varying
degree of randomness to mimic natural settings and the corresponding erodibility coefficient is computed
with a relation derived from field-measured pairs of critical shear stress and erodibility. Centerline
migration and associated statistics for randomly-disturbed distribution based on the distance from the
valley axis are compared for sine-generated centerline using the Monte Carlo method. Two relevant
parameters are identified: (i) the standard deviation of the critical shear stress distribution, which is an
indicator of the local soil heterogeneity, determines centerline skewness and its variability, and can lead
to development of downstream skewness and complex planform features, while not significantly affecting
lateral migration; (ii) the cross-valley increase in soil resistance, which mainly constrains rates of lateral
migration also affecting skewness. The Monte Carlo approach, applied this time to the case of a natural
river alignment and purely random floodplain-soil distribution, shows that migrated centerlines present
larger variability the coarser is the scale of the floodplain heterogeneity (third key parameter for
describing the effect of floodplain heterogeneity on migration) and the increase in variability with the
heterogeneity scale is less then linear. Finally, when the approach for meander migration based on
hydraulic erosion is compared to the “classic” approach based on excess velocity at the outer bank, the
former approach produces more variability and shape complexity for equal stochastic variability of the
corresponding governing parameters, because of the presence of a threshold for bank erosion missing in
the case of classic approach for migration.
1 INTRODUCTION
A great quantity of two-dimensional (2D) depth-averaged analytical models have focused on the impact
of hydrodynamics and bed morphodynamics on the migration of meandering rivers (Ikeda et al., 1981;
Blondeaux and Seminara, 1985; Johannesson and Parker, 1989; Sun et al., 2001a; Zolezzi and Seminara,
2001) and on the degree of planform-shape complexity that can be obtained (Seminara et al., 2001;
Frascati and Lanzoni, 2009). Less attention has been given to the complexity resulting from the
heterogeneity of the floodplain soils, the quantification of its importance, and the identification of the
important parameters. Past studies have been all based on the concept of rate of bank erosion linearly
proportional, through a migration coefficient, to near-bank excess velocity (Howard, 1992, 1996; Sun et
al., 1996, 2001b; Posner and Duan, 2011), according to the method independently introduced by
Hasegawa (1977) and Ikeda et al. (1981). Howard (1992), Howard (1996), and Sun et al. (1996) studied
the development of heterogeneities in the floodplain caused by river migration and their influence on the
future development of the channel. In particular, Sun et al. (1996) focused on the change in erodibility
associated to different sedimentary environments such as point bar deposits, floodplain deposits, and
oxbow lake deposits, and were able to reproduce the formation of meander belts for long-term
simulations. Sun et al. (2001b) went a step further and simulated the development of size distribution as
channel migrates. Posner and Duan (2011) compared a deterministic model, adopting a constant
migration coefficient, with a stochastic model, where the instantaneous migration coefficient at each bank
location is a random variable satisfying either uniform or normal distribution, and compared the two
approaches against experimental data, finding that the stochastic approach yields more realistic
predictions of meandering-planform evolution.
In terms of planform shapes, Sun et al. (1996) claimed that the typical meander wavelength is
determined mainly by hydraulic factors (flow in the channel and inclination of the underlying floodplain)
and is independent of the difference in the erodibilities of sedimentary deposits. Perucca et al. (2007),
who focused on the role played by vegetation on meandering river morphodynamics and the feedback
between riparian vegetation dynamics and meandering dynamics, showed the possible occurrence of
peculiar meander shapes that do not show the usual marked upstream skewness, when the bank erodibility
is linked to the biomass density.
In this paper we discuss how soil floodplain distribution modulates planform shapes and rates of
migration. The model by Motta et al. (2011) is used, which allows for focusing on the horizontal
heterogeneity of the physical parameters for hydraulic erosion, which are erodibility and critical shear
stress. Motta et al. (2011)'s model quantifies the migration of meandering streams through a physicallyand process-based method that relates channel migration to the streambank erosion processes of hydraulic
erosion and mass failure. Hence, channel migration depends on measurable soil properties, natural bank
geometry, and both vertical and horizontal heterogeneity of floodplain soils. In general, the processes
responsible for bank retreat are classified as follows: hydraulic (fluvial) erosion, and cantilever, planar,
rotational, and piping streambank failure (Pizzuto and ASCE Task Committee on Hydraulics, Bank
Mechanics, and Modeling of River Adjustment, 2008). As shown by Constantine et al. (2009), using field
measurements, and by Motta et al. (2011), through modeling, mass failure mechanisms like planar
failures, although important, can be indirectly represented by modifying the parameters for hydraulic
erosion to quantify rates of migration, at least for the case of vertical homogeneity of the bank material.
Hydraulic erosion requires that the local boundary shear stress exceeds the critical value to initiate motion
of the soil particles. From a modeling perspective, the rate of lateral hydraulic erosion E* (with
dimensions of length over time) for each bank-material layer is modeled using an excess shear stress
relation, typically used for fine-grained materials (but also applicable to non-cohesive materials)
E* = M * (τ * τ c* −1) = k * (τ * −τ c* )
(1)
where M* is the erosion-rate coefficient (with dimensions of length over time), τ*c is the critical shear
stress, k* is the erodibility, and τ* is the shear stress acting on the bank. M*, τ*c, and k* are all site-specific.
The critical erosional strength denotes the cohesion strength provided by interparticle forces of attraction
or repulsion acting at the microscopic level, including electrostatic forces, Van der Waals forces,
2
hydration forces, and biological forces (Papanicolaou et al., 2007). Although Eq. 1 appears simple, in
practice it is necessary to define the erodibility parameters and the boundary shear stress τ*. These are all
highly variable, which explains why observed rates of fluvial erosion range over several orders of
magnitude (Darby et al., 2007).
Besides the complexity related to the experimental quantification of the parameters τ*c and k* (or M*),
there is that of their spatial distribution in the floodplain, which depends on the soil distribution that is the
result of the development of the fluvial system in time, through processes of channel deposition because
of lateral migration and overbank deposition formed by vertical accretion. Anomalies in the modern
floodplain can be the remnant of the evolution over geologic time scales. Therefore, depending on when
the computer modeling of meander migration begins, the spatial distribution of soil erodibility may vary.
However, a general characterization of a typical modern floodplain can be made, with a general decrease
of sediment size and consequent decrease in erodibility away from the channel. It is also recognized that
cutoff deposits generated by a meandering river tend to confine it to a strip in the flood basin, known as
meander belt (Allen, 1965; Sun et al., 1996).
As regards the shear stress acting on the river bank, it is the sum of the stress resulting from drag on
bedforms and the stress acting on the actual boundary (Constantine et al., 2009). Therefore, in Eq. 1, τ*
has to be interpreted as effective shear stress.
The present paper, after briefly illustrating the meander-migration model and the numerical techniques
for the representation of floodplain-soil distribution and the generation of the erodibility pattern, evaluates
the impact of horizontal heterogeneity of floodplain soils on meander-migration rates and patterns, using
a stochastic approach for comparing configurations characterized by different degrees of randomness and
spatial scale of variation.
2 MODEL OF MEANDER MIGRATION
The RVR Meander platform (Motta et al., 2011) merges and extends the functionalities of the first
version of RVR Meander (Abad and Garcia, 2006) and CONCEPTS (Langendoen and Alonso, 2008;
Langendoen and Simon, 2008; Langendoen et al., 2009), and is composed by a hydrodynamics/bed
morphodynamics module, a bank erosion module, and a migration module. Hydrodynamics and bed
morphodynamics are computed according to the linear model illustrated by Garcia et al. (1994), which is
a slightly modified version of Ikeda et al. (1981)’s and Johannesson and Parker (1985)’s models. It
provides an analytical solution of the 2D St. Venant Equation, written in streamwise and transverse
coordinates
1
∂U *
∂U *
C*
g
∂H * τ s*
U * * +V * * +
U *V * = −
−
* *
* *
* *
1+ n C
∂s
∂n 1 + n C
1 + n C ∂s* ρ D*
1
∂V *
∂V *
C*
∂H * τ *
U * * +V * * −
U *2 = − g * − n *
* *
* *
1+ n C
∂s
∂n 1 + n C
∂n
ρD
∂ (U * D* ) ∂ (V * D* )
1
C*
+
+
V * D* = 0
1 + n*C *
∂s*
∂n*
1 + n*C *
(2)
(3)
(4)
Superscript star indicates values with dimensions, s* and n* are streamwise and transverse coordinates, U*
and V* are streamwise and transverse depth-averaged velocities, g is the acceleration of gravity, H* is
water stage, τ*s and τ*n are streamwise and transverse bed shear stresses, ρ is water density, D* is water
depth, and C* is the local curvature. The bed morphology is described assuming that the transversal bed
slope is proportional to the local curvature through a constant of proportionality A named scour factor
(Ikeda et al. (1981)’s assumption)
η1*
Dch*
= − AC *n*
(5)
where η*1 is the bed elevation deviation from the bed elevation of the channel centerline at a particular
cross section and D*ch is the reach-averaged water depth for uniform flow. An analytical solution of the
linearized Eqs. 2-4, for low curvature and constant discharge Q and channel width 2B*, is obtained for
3
streamwise and transverse velocity and water depth, expressed as sum of their uniform-flow
reach-averaged value (subscript “ch”) plus a perturbation (subscript “1”), which varies in both streamwise
and transverse directions. The solution depends on upstream and local curvature, sinuosity (which affects
the slope of the channel), half-width to depth ratio β = B*/D*ch, reach-averaged Froude number Fch and
friction coefficient Cf,ch, and scour factor A. We report here just the solution for streamwise
depth-averaged flow velocity
(6)
U ( s, n ) = 1 + U1 ( s, n )
U1 ( s, n ) = a '1 ( n ) e
− a '2 s
(
+ n a '3 C ( s ) + a '4 e
− a '2 s
s
∫ C ( s )e
a '2 s
0
ds
)
(7)
and streamwise bed shear stress
(8)
τ s* = ρC f ,chU * U *2 + V *2
where the normalizations U=U*/U*ch, s=s*/B*, n=n*/B*, and C= B*C* are made, the parameters a’1, a’2, a’3,
and a’4 are function of β, F2ch, Cf,ch, A, and upstream boundary conditions on velocity and curvature (see
Garcia et al. (1994) and Motta et al. (2011) for details), and the friction coefficient Cf,ch is calculated,
following Engelund and Hansen (1967), as
⎛
⎛ D* ⎞ ⎞
C f ,ch = ⎜⎜ 6.0 + 2.5ln ⎜ ch * ⎟ ⎟⎟
⎝ 2.5d s ⎠ ⎠
⎝
−2
(9)
where d*s is a measure of the bed grain size. Here the friction coefficient is assumed to be constant
everywhere in the channel for a given channel planform configuration (i.e., a given D*ch).
Two alternative methods are implemented in the RVR Meander platform for the computation of bank
erosion and centerline migration: with the first method (migration-coefficient method, named MC method
hereafter) the outer bank displacement rate, equal to the centerline displacement rate (so that the channel
width keeps constant), is proportional to the excess velocity at the outer bank through a dimensionless
migration coefficient (Hasegawa, 1977; Ikeda et al., 1981):
R* = E0 (Ub* − U A* )
(10)
R* is the rate of migration (with dimensions of length over time), U*A is the cross-sectionally-averaged
velocity, and U*b is the depth-averaged near-bank velocity. The dimensionless migration coefficient E0 is
usually obtained via calibration against historic channel centerlines. With the second method
(physically-based method, named PB method hereafter), the centerline displacement rate is based on the
modeling of the physical processes responsible for bank retreat (Motta et al., 2011). As mentioned, just
the hydraulic erosion is here considered, which is expressed by Eq. 1. All applications shown in this paper
adopt the PB method, except for a case when MC and PB methods are compared for heterogeneous
floodplain.
3 NUMERICAL REPRESENTATION OF THE FLOODPLAIN AND GENERATION OF THE
PATTERN OF ERODIBILITY
3.1 Numerical representation of the floodplain
Floodplain is represented as a rectangular grid, characterized by constant grid spacing Δx* and Δy* in the
x and y* direction respectively. At each grid node, the parameters M* (or k*) and τ*c (for PB method) or
E0 (for MC method) are assigned. The value V* of those parameters at an arbitrary location (x*,y*) is
computed by inverse-distance weighting interpolation (Shepard, 1968):
*
4
4
V
*
(x , y ) =
*
*
∑V ( x , y ) w ( x , y )
*
*
i
i
*
i
*
*
i
(11)
i =1
4
∑w (x , y )
*
*
i
i =1
with
wi ( x* , y* ) =
1
( d ( x , y , x , y ))
*
*
*
i
*
i
p
=
1
⎛
⎜
⎝
(x
*
* 2
i
−x
) +(y
*
2 ⎞
− yi* ) ⎟
⎠
p
(12)
where (x*i,y*i) are the coordinates of one of the four vertices of the grid cell containing the arbitrary point
(x*,y*), d is the distance of the arbitrary point from one of the four vertices, and p is a positive real number.
p is here assumed equal to one. As the river migrates in the floodplain, the parameters τ*c, k*, M*, and E0
at right and left bank are computed with Eq. 11 according to their position in the floodplain. The values
are recomputed at the beginning of each migration time step. The extension of the floodplain grid has to
be set large enough to contain the possible migration of the centerline, and an algorithm verifies that the
boundary of the grid is not crossed by the channel (in that case the simulation stops, because no
extrapolation is carried out).
3.2 Generation of the pattern of erodibility
The horizontal distribution of critical shear stress τ*c (or E0, in case the MC method is adopted), defined
at the floodplain-grid nodes, can be generated in two different ways. The first corresponds to purely
random distribution, referred to as PR distribution hereafter; the second is a randomly-disturbed
distribution based on the distance from the valley axis, or RD distribution.
In the PR case, τ*c values are generated assuming a normal distribution (µ,σ), where µ and σ are mean
and standard deviation respectively, using the Box-Muller transform (Box and Muller, 1958) in the polar
form (Bell, 1968; Knopp, 1966). Because of physical consistency, whenever a negative normal number is
generated, it is re-generated until it is positive.
In the RD case, once defined the valley axis as a straight line in the form
(13)
y* = Mx* + Q
the local mean critical shear stress increases exponentially with the distance l from the valley axis
(τ [ Pa])
*
c
mean
= ae
bl [m]
(14)
where a is the mean critical shear stress at the valley axis, and b is a coefficient that governs the variation
of the mean critical shear stress in the cross-valley direction. The choice of the exponential form, besides
being among the simplest, is somewhat analogous to what derived analytically by Pizzuto (1987) to
model cross-valley grain size trends associated to overbank flows and to what adopted by Sun et al. (1996)
for the increase in time of the erodibility in oxbow lakes because of silt infilling (observe that time and
distance from valley axis can be thought as correlated). At each floodplain-grid node, the actual value of
critical shear stress is then generated using a Gaussian distribution characterized by mean given by Eq. 14
and a certain assigned standard deviation σ.
Regardless the way a value τ*c is assigned at a floodplain-grid node (PR or RD approach), the
corresponding value of erodibility k* is found using the widely used expression developed by Hanson and
Simon (2001), obtained for channels in the loess areas of the midwestern USA (Iowa, Nebraska, and
Mississippi) and based on in situ jet-testing measurements
k * ⎡⎣cm3 ( Ns )⎤⎦ = 0.2 (τ c* [ Pa ])
−0.5
(15)
that is equivalent to
5
M * [ m s ] = 0.2 ⋅10−6 (τ c* [ Pa ])
0.5
(16)
In both PR and RD cases, values of τ*c and k* (or M*) outside the floodplain-grid nodes are computed by
interpolation using Eq. 11.
4 TEST CASES
We evaluated the impact of horizontal heterogeneity of floodplain soils on rates and patterns of meander
migration for sine-generated and natural initial channel centerlines. Table 1 reports the run parameters,
including, for convenience, the value of Manning's roughness coefficient corresponding to the value of
the friction coefficient. Because reach-averaged Froude number, friction coefficient, and half-width to
depth ratio change over time as the reach-averaged depth varies with changing sinuosity, Table 1 reports
the values corresponding to a straight channel characterized by same width and inclined at the valley
slope.
A stochastic approach was followed, based on the Monte Carlo method, that rely on repeated generation
of the distribution of the erodibility properties in the floodplain, according to a certain PR or RD
distribution, and computation of the corresponding migrated centerline. In this way, a set of migrated
centerlines is obtained for a particular PR or RD floodplain-soil distribution.
Observe that, because only hydraulic erosion is considered here, vertical banks were assumed, because
bank shape has no effect on rates of migration. On the contrary, bank shape would have an impact on
rates of migration in case mass failure mechanisms, such as planar failure, were taken into consideration.
However, as observed by Motta et al. (2011), the effect can be indirectly accounted for by modifying the
resistance-to-erosion parameters τ*c and k*, at least for the case of vertically-homogeneous soil.
For both sine-generated and natural cases, we considered relatively short time scales (few decades), that
is periods before cutoff occurrence. In this case, all planform complexity is caused by floodplain
heterogeneity, which is the focus here, whereas cutoff would introduce additional complexity, here not
considered.
Table 1 Test case run parameters. (Q is discharge; 2B* is channel width; S0 is valley slope; F02 is squared Froude
number for straight channel characterized by valley slope; Cf0 is friction coefficient for straight channel characterized
by valley slope; β0 is half-width to depth ratio for straight channel characterized by valley slope; A is the transverse
bed slope parameter (scour factor); and n0 is the Manning's roughness coefficient.)
Q
2B*
S0
F02
Cf0
A
n0
β0
Case
Shape
3
[m /s]
[m]
[m/m]
[-]
[-]
[-]
[-]
[-]
1
Sine-generated
20.0
30.0
0.0005
0.026
0.0194
12.44
5.0
0.046
Mackinaw
2
46.2
38.0
0.0009
0.109
0.0082
17.06
5.0
0.030
River
4.1 Sine-generated channel
For the sine-generated case, focus is on meander migration in RD configurations, the quantification of
lateral migration and bend skewness, and the identification of the governing parameters in heterogeneous
floodplains.
4.1.1 Sine-generated channel: description
The centerline of a sine-generated channel is expressed as
⎛ 2π s ⎞
⎟
⎝ Λ ⎠
(17)
θ = θ o sin ⎜
where θ is the angle between centerline and valley axis, θ is its value at the crossover point, and Λ is
the length of the channel centerline over one meander wavelength.
We considered here a 2,040-meter long (2,000-meter long sinuous section with 20-meter long straight
0
6
entrance and exit sections) and 30-meter wide channel with Λ = 250 m and θ = 55°, which corresponds to
a relatively low sinuosity of 1.27. 200 equally-spaced nodes describe the initial centerline, yielding a node
spacing of about 10.2 m. The channel contains 17 bends. We simulated the centerline evolution for a
20-year period employing a time step of 0.2 years. Table 1 lists the various simulation parameters.
0
4.1.2 Sine-generated channel: statistics for migrated centerline
Following the approach described by Posner and Duan (2011), the valley axis allows for identifying
bends (lobes) based on the intersections of the centerline with the valley axis. For each lobe, statistics can
be computed using the (x*,y*) coordinates of the centerline nodes. While Posner and Duan (2011) used
such statistics to compare simulated and measured migrating bends, here the statistics allow describing
and comparing the migrated-centerline variability associated to different floodplain-soil configurations.
In particular, we considered the mean value Meany* of the coordinate y*, which quantifies lateral
migration, and the skewness Skewx* of the coordinate x*, where a positive (negative) value of Skewx*
represents upstream (downstream) skewness.
Figure 1 shows an example of evaluation of Meany* and Skewx* for all migrated-centerline lobes, except
the most upstream (Lobe 1) and downstream (Lobe 17) excluded hereafter in the analysis to focus on the
central bends, obtained for PR distribution of critical shear stress with µ = 6 Pa and σ = 1 Pa. The valley
axis is in this case characterized by y* = 34.4 m.
100
0.3
80
0.2
60
0.1
20
Skewx*
Meany* [m]
40
0
0
-20
-0.1
-40
-60
-0.2
-80
-100
-0.3
2
3
4
5
6
7
8
9
Lobe
10
11
12
13
14
15
16
2
3
4
5
6
7
8
9
Lobe
(c)
10
11
12
13
14
15
16
(d)
Figure 1 Sine-generated curve: example of migrated-centerline lobes (Figure b) for a certain PR floodplain-soil
distribution (Figure a) and corresponding values of Meany* (Figure c) and Skewx* (Figure d). Flow is from left to right.
In Figure b, the initial centerline is in red, the migrated centerline in black.
4.1.3 Sine-generated channel: meander migration in RD configurations
Table 2 reports the parameters of four Monte Carlo simulations for RD configuration, characterized by
different values of b (indicator of the cross-valley increase of soil resistance) and σ (indicator of the local
randomness of soil property). We selected a grid-cell size Δx* = Δy* of 60 m, which is equal to two times
the channel width, almost two times the amplitude of the initial centerline, and about three times the
wavelength of the initial centerline measured along the valley axis; moreover, the value of the grid-cell
size is large enough to produce very appreciable variability of the migrated centerlines (see Section 4.2.2
for in-depth analysis of the effect of the grid-cell size). One of the rows of the floodplain grid coincides
with the axis of the initial centerline at y* = 34.4 m.
Because of the shape of the initial centerline, several simulations were discarded for floodplain-soil
configurations producing migrated centerlines characterized by non-physical loops at the downstream end.
7
Therefore, 1000 floodplain-soil generations were used for each of the four Monte Carlo simulations for
sine-generated curve, in order to obtain a sufficiently large number of migrated centerlines for computing
meaningful metrics. Note also that a Monte Carlo approach will never completely represent an entire
statistical population; on the other hand, a sufficient number of runs can reasonably represent it.
Figure 2 shows, for each Monte Carlo simulation, an example of realization of floodplain-soil spatial
distribution and all the migrated centerlines. We generally observe, for higher value of σ, a higher
variability of migrated centerlines (compare Figures 2a,b (configurations RD1 and RD2) Vs. Figures 2c,d
(RD3 and RD4)) and, for higher value of b, less lateral migration and a more pronounced tendency
towards upstream skewness (Figures 2a,c (RD1 and RD3) Vs. Figures 2b,d (RD2 and RD4)).
Configuration
RD1
RD2
RD3
RD4
Table 2 Sine-generated curve: parameters for RD Monte Carlo simulations.
Grid size
Maxis [-]
Qaxis [m]
a [Pa]
b [1/m]
[m]
60.0
0.0
34.4
6.0
0.001
60.0
0.0
34.4
6.0
0.003
60.0
0.0
34.4
6.0
0.001
60.0
0.0
34.4
6.0
0.003
σ [-]
1.0
1.0
0.5
0.5
Figure 2 Sine-generated curve: example of realization of floodplain-soil spatial distribution and all 20-year migrated
centerlines for each of the four RD Monte Carlo simulations, characterized by different values of parameters b and σ.
(a) Configuration RD1, (b) RD2, (c) RD3, and (d) RD4 (see Table 2). Flow is from left to right. The initial centerline
is in red, migrated centerlines in black.
8
A box plot can be used to graphically visualize minimum, 25th percentile, median, 75th percentile, and
maximum of both Meany* and Skewx* values, as shown in Figure 3. Note that, for convenience, the
absolute values of Meany* are actually plotted in Figures 3a,c,e,g and 4a,b. In Figure 4 we compare, for
the four RD configurations and for each lobe, the median values of Meany* (Figure 4a), the difference
between 75th and 25th percentiles of Meany* (Figure 4b), the median value of Skewx* (Figure 4c), and the
difference between 75th and 25th percentiles of Skewx* (Figure 4d).
100
0.8
RD1 (low b , high σ )
90
80
0.4
60
Skewx*
Meany* [m]
70
50
40
30
-0.8
0
2
3
4
5
6
7
8
100
2
9 10 11 12 13 14 15 16
Lobe
(a)
3
4
5
6
7
8
0.8
RD2 (high b , high σ )
90
9 10 11 12 13 14 15 16
Lobe
(b)
RD2 (high b , high σ )
0.6
80
0.4
70
60
Skewx*
Meany* [m]
0.0
-0.2
-0.6
10
50
40
30
0.2
0.0
-0.2
-0.4
20
-0.6
10
-0.8
0
2
3
4
5
6
7
8
100
2
9 10 11 12 13 14 15 16
Lobe
(c)
3
4
5
6
7
8
0.8
RD3 (low b , low σ )
90
9 10 11 12 13 14 15 16
Lobe
(d)
RD3 (low b , low σ )
0.6
80
0.4
70
60
Skewx*
Meany* [m]
0.2
-0.4
20
50
40
30
0.2
0.0
-0.2
-0.4
20
-0.6
10
-0.8
0
2
3
4
5
6
7
8
100
2
9 10 11 12 13 14 15 16
Lobe
(e)
3
4
5
6
7
8
0.8
RD4 (high b , low σ )
90
9 10 11 12 13 14 15 16
Lobe
(f)
RD4 (high b , low σ )
0.6
80
0.4
70
60
Skewx*
Meany* [m]
RD1 (low b , high σ )
0.6
50
40
30
0.2
0.0
-0.2
-0.4
20
-0.6
10
-0.8
0
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Lobe
(g)
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Lobe
(h)
Figure 3 Sine-generated curve: box plots of Meany* and Skewx* for the four RD Monte Carlo simulations,
characterized by different values of parameters b and σ (see Table 2).
9
75th - 25th percentile of Meany* [m]
Median value of Meany* [m]
55
50
45
40
35
30
RD1 (low b, high sigma)
RD2 (high b, high sigma)
RD3 (low b, low sigma)
RD4 (high b, low sigma)
25
20
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Lobe
(a)
RD1 (low b, high sigma)
RD2 (high b, high sigma)
RD3 (low b, low sigma)
RD4 (high b, low sigma)
18
16
14
12
10
8
6
4
2
0
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Lobe
(b)
0.16
75th - 25th percentile of Skewx*
0.40
0.35
Median value of Skew x*
20
0.30
0.25
0.20
0.15
0.10
RD1 (low b, high sigma)
RD2 (high b, high sigma)
RD3 (low b, low sigma)
RD4 (high b, low sigma)
0.05
0.00
2
3
4
5
6
7
8
0.14
0.12
0.10
0.08
0.06
0.04
RD1 (low b, high sigma)
RD2 (high b, high sigma)
RD3 (low b, low sigma)
RD4 (high b, low sigma)
0.02
0.00
9 10 11 12 13 14 15 16
Lobe
(c)
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Lobe
(d)
Figure 4 Sine-generated curve: median value and difference between 75th and 25th percentiles for Meany* and Skewx*
corresponding to the four RD Monte Carlo simulations characterized by different values of parameters b and σ (see
Table 2).
From Figures 3 and 4, if b (indicator of cross-valley increase of soil resistance) increases, the value of
Meany* (indicator of lateral migration) generally decreases: this can be observed by comparing Figure 3a
(RD1) with Figure 3c (RD2), or Figure 3e (RD3) with Figure 3g (RD4), or from Figure 4a, where the
median values of Meany* are lower for higher b. As regards Skewx* (indicator of planform shape), high b
values produce a more enhanced tendency toward upstream skewness, which is visible especially from
Figure 4c: this can be explained as the result of the constrained lateral migration.
If σ (indicator of the local randomness of soil property) increases, Meany* values are not significantly
affected on average (Figure 4a); on the other hand, the possibility of getting downstream
(negatively)-skewed lobes increases (compare Figures 3b (RD1) and 3d (RD2) with Figures 3f (RD3) and
3h (RD4)) and both Meany* and Skewx* present a wider spectrum (Figures 4b and 4d).
4.2 Natural channel
For the natural channel case, focus is on the effect of the scale of heterogeneity, conditions for
development of downstream-skewed bends and generally complex planform shapes, and the comparison
between MC and PB stochastic approaches.
4.2.1 Natural channel: description
We tested different floodplain configurations for a reach on the Mackinaw River in Illinois, USA. The
reach is located in Tazewell County about 15 kilometers upstream of the junction of the Mackinaw River
with the Illinois River between the towns of South Pekin and Green Valley. The 1951 historic centerline
was taken as starting centerline.
An equidistant set of 350 centerline nodes with a spacing of 11.4 m describes the initial channel
centerline. Channel width is 38 m and valley slope is 0.0009 m/m (Motta et al., 2011). The mean
sinuosity of the study reach is 1.34. As assumed by Motta et al. (2011), we used a model discharge of
46.2 m3/s, that is, from an analysis of the discharge record between 1922 and 1956 at the USGS station
05568000 near Green Valley, between the average value (20.9 m3/s) and the maximum value (61.6 m3/s)
of the mean annual streamflow over the period. The upstream boundary for modeling was set in a straight
10
reach. Because the velocity distribution is not known there, a uniform profile in the transverse direction
was assumed for simplicity. Further, we used a 2 nd-order Savitzky-Golay filter with an averaging window
of 5 nodes applied every 10 iterations.
4.2.2 Natural channel: effect of heterogeneity scale
Monte Carlo simulations were run for different values of grid-cell size (Δx*, Δy*) with PR configuration
characterized by µ(τ*c) = 9.0 Pa and σ(τ*c)= 1.0 Pa (Table 3). The mean value is the same adopted for the
Mackinaw River by Motta et al. (2011) for simulations in homogeneous floodplain and recently
confirmed by jet erosion tests in situ. Because floodplain-soil values at grid nodes are random while
values in between nodes are obtained through interpolation (therefore they are deterministic), the grid-cell
size is an indicator of the spatial scale of randomness of the floodplain-soil properties. Figure 5 shows the
results for grid-cell size values respectively equal to 1, 2, and 3 times the channel width. 500
floodplain-soil generations were considered for each of the three cases, which is a number or realizations
that was visually judged as sufficient to describe the variability of the migrated centerlines.
Table 3 Natural channel: PR Monte Carlo simulations characterized by different scale of heterogeneity.
Configuration
Δx* = Δy* [m]
Δx*/2B* = Δy*/2B* [-]
2W*/2B* [-]
PR1
38.0
1.0
2.12
PR2
76.0
2.0
3.10
PR3
114.0
3.0
3.59
From Figure 5 we see that lower grid-cell size is associated to lower migrated-centerline variability. In
quantitative terms, the ratio was computed between the area occupied by all migrated centerlines (whose
perimeter was manually digitized) and the length of the migrated centerline obtained for homogeneous
floodplain characterized by mean floodplain-property values (τ*c = 9 Pa, k* = 6.7 · 10-8 m3/(Ns)). This
ratio with dimensions of length, indicated as 2W*, was then divided by the channel width 2B* to give a
dimensionless indicator of the migrated-centerline variability. From Table 3 we observe that 2W*/2B*
increases for increasing grid-cell size less than linearly. This implies, from a practical perspective, that a
finer-scale campaign of characterization of the floodplain soil is, as one could reasonably expect,
associated to less uncertainty on the predicted migrated centerline. On the other hand, the increase of
2W*/2B* with the scale of heterogeneity is less than linear and presents, at least in this case, a tendency
toward a plateau: our stochastic methodology may therefore provide a tool for determining the suitable
spatial density for field characterization of floodplains.
Figure 5 Continues in the next page
11
Figure 5 Natural channel: example of realization of floodplain-soil spatial distribution and all 40-year migrated
centerlines for each of the four PR Monte Carlo simulations, characterized by different values of grid-cell size Δx* =
Δy*. (a) Configuration PR1, (b) PR2, and (c) PR3 (see Table 3). Flow is from right to left. The initial centerline is in
red, migrated centerlines in black.
4.2.3 Natural channel: complexity of planform shapes
While the belts occupied by all migrated centerlines in Figure 5 show upstream skewness, typically
developed by meanders, a few cases are present where more complex shapes are obtained. Considering
for instance the set of PR2 simulations (Table 3 and Figure 5b), we could identify downstream-skewed
bends (Figure 6a,b), symmetric compound loops (Figure 6c), and compound bends (Figure 6d). In general,
large scales of heterogeneity are more likely to produce downstream-skewed and, in general, complex
planform features, given the higher migrated-centerline variability that can be obtained (see Section
4.2.2).
Figure 6 Natural channel: examples of downstream-skewed bends (L1 and L2), compound loops (L3 and L4), and
compound bend (L5). Flow is from right to left. The initial centerline is in red, migrated centerlines in black.
4.2.4 Natural channel: comparison of MC and PB approaches for heterogeneous floodplain
MC and PB stochastic migration approaches were compared by considering PR distributions of the
parameters E0 (for the MC approach) and τ*c or k* (for the PB approach) characterized by the same ratio
between their standard deviation σ and mean value µ (Table 4). In this way it was possible to compare the
effect of the spatial variability of those parameters on migrated centerlines. For the MC approach, the
mean value 5 · 10-7 for E0 was taken from Motta et al. (2011). In the case of the PB approach, when τ*c
values were randomly generated at the floodplain-grid nodes, the corresponding values of k* were
computed with Eq. 15, whereas, when k* values were randomly generated, the corresponding values of τ*c
were computed with the inverse of Eq. 15. A grid size Δx* = Δy* = 2B* = 38 m was used and a period of
20 years was simulated.
From Figure 7 we observe that the spatial variability of parameters τ*c or k*, which govern PB migration,
reflect in greater variability of migrated centerlines than in the case of E0, which governs the MC
12
migration. In other words, the PB approach intrinsically produces more centerline variability as result of
floodplain-resistance variability than the MC approach. Quantitatively, using the indicator 2W*/2B*
introduced in Section 4.2.2, the variables τ*c and k* respectively produce about 340 and 595% of the
migrated-centerline variability associated to the variable E0 (Table 4). This can be explained by observing
that, while the MC approach always predicts a “smooth” migration pattern because it does not account for
any threshold for erosion (Eq. 10), the PB approach using hydraulic erosion does (Eq. 1, where erosion
occurs for shear stress at the bank greater than the critical shear stress τ*c) and depends on two parameters
instead of one, τ*c and k*, even if correlated by Eq. 15.
Table 4 Natural channel: PR Monte Carlo simulations for comparison of effects of the spatial variability of the
parameters E0, τ*c, and k*.
Parameter
µ
σ
σ /µ [−]
2W*/2B [-]
-7
-8
E0 [-]
5 · 10
5.6 · 10
1/9
0.43
9.0
1.0
1/9
1.47
τ*c [Pa]
6.7 · 10-8
7.4 · 10-9
1/9
2.56
k* [m3/(Ns)]
Figure 7 Natural channel: example of realization of floodplain-soil spatial distribution and all 20-year migrated
centerlines for Monte Carlo simulations characterized by (a) PR distribution of E0 (MC migration method), (b) PR
distribution of τ*c (PB migration method); (c) PR distribution of k* (PB migration method). Flow is from right to left.
The initial centerline is in red, migrated centerlines in black.
5 DISCUSSION
We developed a methodology for assigning numerically on a regular grid the floodplain-soil properties
in terms of migration coefficient E0 (when using the classic MC approach for migration) or parameters for
hydraulic erosion τ*c and k* (for PB approach, of main interest here), combining random number
generation at grid nodes and interpolation in between nodes. Two different spatial patterns were analyzed,
purely random (PR) and randomly disturbed (RD), where, in the latter case, a general pattern with
increasing resistance to erosion was assumed away from the valley centerline.
We considered two application cases, corresponding to sine-generated and natural channel, for which we
conducted several numerical tests (Monte Carlo simulations) based on repeated generations of floodplain
13
configuration and computation of the corresponding migrated centerlines and associated metrics.
In the case of sine-generated channel, focus was put on RD configurations, and two metrics, Meany* and
Skewx*, were used to respectively quantify lateral migration and planform orientation of bends. Two
governing parameters were found, which indicate the local randomness of soil resistance to erosion (σ)
and the cross-valley increase of soil resistance (b). The parameter σ mainly governs the bend shape, and
larger values increase the possibility of occurrence of downstream-skewed bends and in general allow for
a wider spectrum of bend skewness. Roughly speaking, b controls lateral migration; however, deeper
analysis revealed that bend skewness is affected too, being higher b values associated to more pronounced
upstream skewness of bends.
For a natural channel, migrated-centerline variability as function of heterogeneity scale was evaluated by
defining a dimensionless metric (2W*/2B*) that quantifies the “spread” of the migrated centerlines
corresponding to a Monte Carlo simulation for PR floodplain configuration characterized by a certain
grid-cell size. The analysis showed that lower grid-cell size, i.e. lower scale of heterogeneity, corresponds
to lower migrated-centerline variability and that the increase of migrated-centerline variability with
heterogeneity scale is less than linear and seems to tend toward a plateau.
In presence of heterogeneous floodplain, even using a simple model for hydrodynamics and bed
morphodynamics like that adopted here, downstream-skewed lobes and complex planform shapes such as
symmetric compound loops and compound bends can be obtained, especially for large scale of
heterogeneity and natural-channel alignment.
Finally, comparison of stochastic MC and PB simulations showed that the PB approach for migration is
intrinsically associated to greater migrated-centerline variability as result of floodplain-resistance spatial
variability.
6 CONCLUSIONS
In this paper we performed an analysis of the impact of floodplain-soil heterogeneity on rates and
patterns of migration, considering horizontal heterogeneity and the mechanism of hydraulic erosion. We
showed, using a simple Ikeda et al. (1981)-type model for hydrodynamics and bed morphodynamics, that
floodplain-soil complexity can greatly contribute to planform complexity of meandering rivers and
produce downstream-skewed features, which an Ikeda et al. (1981)-type model cannot produce in
homogeneous floodplain. The simulations performed allowed for highlighting three key parameters that
characterize the effect of floodplain heterogeneity on channel planform complexity, respectively
quantifying the local randomness of soil resistance, the cross-valley increase of soil resistance, and the
spatial scale of heterogeneity.
Future work will include the analysis of mass failure mechanisms in heterogeneous floodplains, with the
goal of addressing the impact of vertical soil heterogeneity and bank-profile shape on rates and patterns of
migration.
Moreover, it will be interesting to couple the described procedure for channel migration in
heterogeneous floodplain with a model for hydrodynamics and bed morphodynamics that considers the
free response of the bed sediments, by fully coupling flow, sediment transport, and bed morphodynamics,
such as the second-order model by Johannesson and Parker (1989) or the fourth-order model by Zolezzi
and Seminara (2001). The analysis would allow for assessing how the planform complexity due to the
hydrodynamics compares to that caused by bank processes and floodplain heterogeneity. As shown by
Camporeale et al. (2007) both second and fourth-order models can give rise, or wash away, multilobed
planimetries in homogeneous floodplain, and Seminara et al. (2001) already showed that, without
invoking cutoff processes, spatial heterogeneity of bank erodibility or human constraints, in
super-resonant regime meander shape displays alternating upstream- and downstream-skewed loops, as
well as multiple loops.
Finally, moving to long-term simulations will allow for analyzing the coupled effect of cutoff processes
and floodplain heterogeneity on meander patterns.
7 ACKNOWLEDGMENTS
This research was supported by an agreement from the U.S. Department of Agriculture, Forest Service,
Pacific Southwest Research Station and using funds provided by the Bureau of Land Management
14
through the sale of public lands as authorized by the Southern Nevada Public Land Management Act.
This work was performed under Specific Cooperative Agreement No. 58-6408-8-265 between the
Department of Civil and Environmental Engineering at the University of Illinois at Urbana-Champaign
and the U.S. Department of Agriculture, Agricultural Research Service, National Sedimentation
Laboratory. Eric Waschle Prokocki is also acknowkledged for the interesting discussions on
floodplain-soil patterns.
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