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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
487
DOI: 10.4408/IJEGE.2011-03.B-054
NUMERICAL SIMULATION OF FLOOD AND DEBRIS FLOWS THROUGH
DRAINAGE CULVERT
J
oonGCHeol
PAIK & s
anG
-d
eoG
PARK
(*)
(*)
Gangneung-Wonju National University, Department of Civil Engineering, Korea
results show that the present model is a promising en-
gineering tool for practical simulations of such flows.
We elucidate the transient features of flood and debris
flows through culverts, based on numerical solutions
of such flows in the open channel with culvert at vari-
ous configurations of different bottom slopes between
the channel and the culvert.
K
ey
words
: debris flow, culvert, numerical simulation
INTRODUCTION
Over recent decades, there have been lots of ef-
forts to understand the propagation and deposition
behavior of the debris flows. Due to the huge diffi-
culty of the real-time field measurements, the major-
ity of debris flow researches have been carried out by
laboratory experiments and numerical simulations
with the parameters back-calculated or calibrated to
match previous field events. The physical modeling
has provided much of what we know about the rheol-
ogy of the debris flow, but their results suffer from
spatial scale effects and are approximately applica-
ble to large-scale events. The numerical simulation
has become an ideal approach for reproducing the
debris flow propagation.
The debris can include anything from the small-
est clay to boulders, trees and even parts of man-
made structures. Debris flow is generally considered
to contain more than 50% particles larger than sand
size (v
aRnes
, 1978) while mudflow is composed
ABSTRACT
In mountainous area, cross drainage culvert is
commonly used to allow water and debris to pass un-
derneath road or embankment. Flood and debris flows
typically undergo a sudden change of the flow depth
in the open channel with culvert due to the discontinu-
ity of the bottom slope and the cross-sectional area
near the culvert inlet, which can result in culvert fail-
ure due to blockage. In this study, we seek to improve
our understanding of the culvert flow and its transition
in such open channel. A second-order-accurate finite
volume method using a shock-capturing scheme with
TVD limiters has been developed for solving one-
dimensional shallow water equations with debris flow
resistance terms to predict the time-dependent behav-
ior of non-Newtonian debris flow through culverts. To
evaluate the numerical model, we first apply it to cal-
culate a saturated debris flow in a large scale experi-
mental channel. The comparison of the numerical re-
sults with experimental measurements shows that the
present model can reasonably well reproduce labora-
tory but fairly large scale debris flows. To investigate
the behaviour of the common flood flow, an immature
debris flow and debris flow in our laboratory flume
with a square culvert and an abrupt change of bed
slope, we conducted these flows using corresponding
flow resistance relations. The numerical results appear
to be in good agreement with our experimental meas-
urements which provide useful information on the
debris flow transient through culverts. The numerical
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J. PAIk & S.-D. PARk
488
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
low sediment concentration and debris flow at the same
flow rate condition. Recently, these flows had been also
investigated in our laboratory channel with rectangular
cross section and an abrupt slope change of the slope.
Finally, conclusions about behaviour of these flows
with same flow rate but different sediment concentra-
tions in the culvert-like experimental flume are drawn
NUMERICAL MODEL
GOVERNING EQUATIONS
The shallow water equations can be applied to both
dam break wave propagation and non-Newtonian mud/
debris flows with an appropriate flow resistance term
(J
in
& f
Read
, 1999; b
Rufau
et alii, 2000; n
aef
et alii,
2006). The continuity and momentum equations are
written in the conservative form as follows:
where
In above equation, t = time; x = the distance along
the longitudinal axis of the channel; h = the flow depth
normal to the local bed surface; u is the depth-aver-
aged velocity; q
i
= lateral inflow or outflow; g = the
gravity acceleration; S
0
= the bed slope given by the
bed inclination θ
where z(x,t) = the bed level respect to an arbitrary
horizontal reference. In this study, various flow re-
sistance relations for the flow resistance term S f
available for mud/debris flows are implemented in
the model.
If the solids concentration is less than about 0.02,
the flow contains bed load or suspended load depend-
ing on the turbulenceand viscosity.
predominantly of silt, with some clay and fine sand.
Distinct physical processes differentiate these type of
flows based on the rheology of the water-sediment
mixture. According to J
ulien
& l
eon
(2000), the
yield and viscous stresses are dominant in mudflows
and the dispersive stress is dominant in debris flows.
The sediment concentration also can be used to dis-
tinguish between stony debris flow, immature debris
flow (less than about 0.2), and turbulent flow (less
than about 0.02) (t
akaHasHi
, 1991). In numerical
simulations of such immature debris and debris flows,
practical problems are to determine representative
parameters, such as bulk viscosity and yield stress,
characterizing the solid-fluid mixture and to select
the appropriate flow resistance relations (n
aef
et alii,
2006). To investigate the distinct features of debris
flow compared to those of common flood flow, in this
study, we solved the governing equations with con-
sidering three resistance formula available for differ-
ent flow regimes: 1) the turbulent flow relation for the
bed load/suspended load flow; 2) the immature debris
flow model (t
akaHasHi
, 1991); and 3) the Voellmy
debris flow model.
In this work a 1D numerical model is developed by
employing a shock capturing method, which works in
the finite volume context with an approximate Riemann
solver, to simulate fluid mixture (mud/debris) flows.
The present model solves the time-dependent non-
linear one dimensional shallow water equations with
complex source terms by the Weighted Averaged Flux
(WAF) method using the HLL approximate Riemann
solver, with total variation diminishing (TVD) limiters
In the subsequent section, we first present the gov-
erning equations based on the shallow water equations
incorporated with various flow resistance relations to
determine the basal and/or internal friction slope and
the numerical methods. In P
aik
et alii (2010), the nu-
merical model was already evaluated by applying to a
dam break problem with analytical solutions account-
ing for a Coulomb-type behavior with constant fric-
tion angle on constant slope bottom and to a mudflow
which was experimentally investigated in a small scale
rectangular channel of a constant slope. In this study,
we apply the present model to a debris flow experimen-
tally reproduced in a large scale USGS flume to further
evaluate its computational performance. Subsequently,
we present and discuss numerical results of a flood
(water) flow, an immature debris flow with relatively
(1)
(2)
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NUMERICAL SIMULATION OF FLOOD AND DEBRIS FLOWS THROUGH DRAINAGE CULVERT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
489
where C
k
= S
k
Δtx Courant number related to the
wave speed S
k
; A
k
= second-order accurate WAF lim-
iter function which, in this study, is obtained using ei-
ther minmod or superbee limiter (t
oRo
, 2001); ΔF
k
i+1/2
= flux jump across wave k :
The flux limiter allows one to obtain a first-order-
accurate solution for the discontinuities and a second-
order-accurate solution elsewhere. To determine the
wave speeds and flux jumps, we employ the HLL
approximate Riemann solver which is based on the
estimates of the smallest and the largest wave speeds
arising in the Riemann solution:
where q
K
(K=L;R) , is given as follows:
The solution of the flow depth h in the star region
is obtained by the approximate Riemann solver (t
oRo
2001):
The HLL numerical flux is determined as
where n = pseudo Manning’s roughness coefficient
which accounts for both turbulent boundary friction
and internal collisional stresses.
where h
r
=A/P = the hydraulic radius, P = the wetted
perimeter, d
p
= the mean effective diameter of particles.
The Voellmy flow relation consists of a turbulent
Chézy coefficient accounting for velocity dependent
friction losses and a basal friction term to describe the
stopping mechanism, and presents the good behav-
iour regarding both the debris flow behavior and the
deposit characteristics (b
eRtolo
& w
ieCzoRek
, 2005;
n
aef
et alii, 2006; m
edina
et alii, 2008).
where θ is internal friction angle of the flowing mass
and C
Z
is the Chézy coefficient.
NUMERICAL METHODS
The method employed in this study works in the
finite volume context with an approximated Riemann
solver. Using the explicit conservative formulation of
the governing equations, the upwind scheme can be
applied in the form
where Δt = the time interval; Δx = the spatial step; n
= the time interval index; i = the spatial node index;
and F
i+1/2
= the numerical flux at the cell interface
x=x
i+1/2
. The weighted averaged flux (WAF) method
is a second-order extension of the Godunov upwind
method (t
oRo
, 2001). A total variation diminishing
(TVD) constrain is enforced on the scheme to avoid
spurious oscillations in the vicinity of discontinui-
ties. The resulting TVD version of the secondorder
WAF flux is
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
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J. PAIk & S.-D. PARk
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
RESULTS AND DISCUSSION
wATER-SATURATED DEBRIS FLOwS
In order to further evaluate the applicability of
the present numerical model, we first applied the nu-
merical methods for reproducing the debris flow in a
large scale USGS experimental flume which was ex-
perimentally and numerically investigated by d
enlin
-
GeR
& i
veRsion
(2001). The experiments with debris
flows of ~ 10 m
3
of water-saturated sand and gravel
(with 2% silt and clay by weight) were conducted at
the USGS debris flow flume which is a rectangular
concrete chute 95 m long and 2 m wide that slopes
31° throughout most of its length and flattens at its
base to adjoin a runout surface that slopes 2.5°. The
solid-water mixture was initially placed as a triangular
wedge against a vertical gate of 2m height, and sud-
denly released by the gate opening. In the experiment,
the flow was confined by concrete panels which effec-
tively extended the flow length 7.4m across the runout
surface. Details of the flume facility and experimental
methods have been reported in i
veRsion
(1997). Nu-
merical computation has been conducted using the
debris properties of ρ = 2000 kg /m
3
, C
Z
= 37m
1/2
s , θ
= 31°, and Φ = 7°. The computational domain is dis-
cretized using 841 grid points and the computational
time-step is set with a CLF number of 0.9.
Flow depths computed at locations 2 m, 33 m, and
67 m downstream of the vertical gate are compared
with experimental measurements and previously
simulated results of d
enlinGeR
& i
veRson
(2001) in
Fig. 1. As also observed in solutions of Denlinger and
Iverson, the most significant simulation errors occur
at the location 2m downstream of the gate. As already
pointed out by d
enlinGeR
& i
veRson
, it is because
the depth-averaged model does not account for reac-
tion forces exerted by the static bed in response to the
slope-normal acceleration, and it consequently pre-
dicts too much thinning just downstream of the gate.
Further downstream of the gate, however, the
computed results reveals that present numerical model
can reasonably well reproduce the flow.
COMPARISON OF FLOOD AND DEBRIS FLOwS
Understanding and modelling the flow behaviour
of flood flow and debris flow is a crucial prerequisite
for the development of the design criteria of culverts
in the debris flow potential regions. Laboratory experi-
ments are presented that model the debris flows with
The approximate Riemann solver offers a simple
way of dealing with dry bed conditions and the deter-
mination of the wet/dry frontvelocities (t
oRo
, 2001).
The wave speeds are determined as the exact dry front
speeds as follows:
The solution and integration of the system of
equations (1) using the fractional-step approach in-
volves a two-step solution procedure. In the first step,
only the homogeneous part of the system is solved
with the HLL approximate Riemann solver. In the
second step, the source term is taken into account by
solving the ordinary differential equation using the
first-order accurate, splitting method.
where U
adv
is the advected flow variable based on the
solution of only the homogeneous part of the system.
The friction term has been discretized by a full im-
plicit method (l
ianG
& m
aRCHe
, 2009)
where D =1-ΔΔn (∂S
f
/∂q
n
) is the coefficient derived
for a full implicit scheme.
The performance of the numerical model and the
grid sensitivity of the solutions had been evaluated by
comparing the numerical solutions computed on suc-
cessively refined grids with analytical solutions ac-
counting for a Coulomb-type friction law on the dry
bottom of the constant slope which is higher than the
internal friction slope and some experimental measure-
ments of debris flows. Numerical tests confirm that the
numerical model yields solutions in very good agree-
ment with the analytical solutions even near the discon-
tinuities at appropriately refined grid resolution. For a
detailed description of the numerical method and its
performance the reader is referred to P
aik
et alii (2010).
(13)
(14)
(15)
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NUMERICAL SIMULATION OF FLOOD AND DEBRIS FLOWS THROUGH DRAINAGE CULVERT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
491
diameter in the immature debris flow relataion is set to
be 0.8mm based on the particle size analysis. The total
simulation time is 120 s and the time step is specified
at every time with a CFL number of 0.6.
Depth profiles for three different flows computed
at the time of 120 s at the same flow rate are com-
pared in Fig. 2 where the flow depth is doubled for
visualization. As shown in the figure, the fresh water
flow, computed flow depth is appeared to increase
by 40% at the downstream channel. Note the overall
longitudinal depth profile of flood flow shown in Fig.
2(a) is in very good agreement with the experimental
measurement. Interestingly, the numerical result of
an immature debris flow reveals the emergence of the
debris flow surge as shown in Fig. 2(b). This kind of
debris flow surges were experimentally investigated
by d
avies
(1990). Consequently, the local depth of the
immature debris flow significantly increased, but the
deposition of sediments in the downstream channel
was not observed in the numerical results. In contrast,
various fluid-solid mixtures in a rectangular channel
which consists of two channel of different slope angles:
the upstream channel of 5 m length and 25° slope and
the downstream channel of the length of 3 m and a bed
slope of 6° including a culvert at various angles. The
measurement methods include six ultrasonic sensors
to measure flow depth and record stage hydrographs
and video cameras to assess the surface velocity and
interpret the flow processes of debris flows. The ex-
perimental measurements show that the flow depth of
debris flows dramatically increase as compared with
that of the clear water flow, and depends on the angle
of culvert slope as well as the sediment concentration.
We calculate a flood flow, a immature debris flow and
a debris flow in the experimental flume using flow re-
sistance relations of Eq. (3), Eq( 4) and Eq. (5), respec-
tively. Based on the calibration and the test of material
properties, the manning coefficient n in Eq. (3) is set
to be 0.01 s m
-1/3
while the Chezy coefficient in Eq.
(5) equals to 44.0 m
1/2 s
. The effective mean particle
Fig .2 - Computed depth profiles of (upper) flood flow,
(centre) immature debris flow, and (lower) debris
flow at t = 120s
Fig.1 - Comparison of [upper] measurement and predic-
tion of D
eNliNGer
& i
verSoN
(2001) with [lower]
present prediction
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J. PAIk & S.-D. PARk
492
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
the numerical results obtained by using the Voellmy
relations shows that the typical debris flow results in
the significant deposition of sediments in the down-
stream culvert of low bed slope. The computed debris
flow deposition is in good agreement with our flow
visualization shown in inset, as shown in Fig. 2(c).
We further compared the computed time history
of debris flow deposition at four different locations
with experimental measurements obtained by ultra-
sonic sensors in Fig. 3. As shown in the figure, the
patterns of sediment deposition in the computed flow
field are different from those of the measurements.
It is because of the assumption of a complete single
phase flow for the debris flow in the computation.
It should be note, however, that the final deposition
depth computed based on the single phase flow is
comparable to the experimental measurements. This
result provide useful information on the debris flow
transient through culverts.
CONCLUSIONS
A 1D debris flow model based on the shallow wa-
ter equations with complex source terms has been de-
veloped to simulate 1D non- Newtonian debris flows.
Three flow resistance relations for the flood flow, im-
mature debris flow and debris flows are incorporated
into the model to investigate the time-dependent be-
haviours of such flows in the culvert-like channel of
low bed angle. The governing equations are solved
by a second-order-accurate finite volume method em-
ploying Godunov-type schemes with spatially discre-
tized flux functions and TVD-limiters for high-resolu-
tion monotone solutions.
To evaluate the numerical model, we first apply
it to calculate a saturated debris flow in a large scale
experimental channel. The comparison of the numeri-
cal results with experimental measurements shows
that the present model can reasonably well reproduce
laboratory but fairly large scale debris flows. To in-
vestigate the behaviour of the common flood flow, an
immature debris flow and debris flow in our labora-
tory flume with a square culvert and an abrupt change
of bed slope, we conducted these flows using cor-
responding flow resistance relations. The numerical
results appear to be in good agreement with our ex-
perimental measurements which provide useful infor-
mation on the debris flow transient through culverts.
The numerical results show that the present model is
a promising engineering tool for practical simulations
of such flows. We elucidate the transient features of
flood and debris flows through culverts, based on nu-
merical solutions of such flows in the open channel
with culvert at various configurations of different bot-
tom slopes between the channel and the culvert
ACKNOWLEDGEMENTS
This research was financially supported by a grant
(Code#’08 RTIP B-01, RD-Flow Project) from Re-
gional Technology Innovation Program funded by the
Ministry of Land, Transport and Maritime Affairs of
Korean Government..
Fig. 3 - Experimental measurements (solid lines) and
numerical predictions (dashed lines) at four dif-
ferent sections (see Fig. 2 for section numbers,
S1- S6)
REFERENCES
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eRtolo
P. & w
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P., G
aRCía
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NUMERICAL SIMULATION OF FLOOD AND DEBRIS FLOWS THROUGH DRAINAGE CULVERT
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d.J. (1978) - Slope movement types and processes. In: s
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