Erosion and Sedimentation
in the Pacific Rim (Proceedings of
the C o r v a l l i s Symposium, August, 1987).
IAHS P u b l . no. 165.
Estimation of debris flow hydrograph on varied slope bed
T. TAKAHASHI, H. NAKAGAWA S S. KUANG
Disaster Prevention Research Institute, Kyoto University,
Uji, Kyoto 611, Japan
ABSTRACT
A system of equations with which to predict
the hydrograph of a debris flow at an arbitrary position
in a channel whose bed varies in slope, width and
thickness and is composed of a graded mixture of particles is presented. This system not only can be used to
estimate the hydrograph
of a debris flow under an
arbitrary water supply but to determine the change in
concentration of particles in the flow by separating
them into coarse and fine components. This change in the
concentrations of the coarse and fine fractions produced
by erosion and deposition may account for the formation
of mud flows which contain little coarse sediment on
well graded sediment beds having an abundance of coarse
materials. Our application of this system of equations
to laboratory experiments and to the huge mud flow
generated by the eruption of the Nevado del Ruiz Volcano
in Colombia proved that it is a useful prediction method.
NOTATION
a
depth measured from the surface of the bed
a
depth in which T = T T /
B
the width of the stream channel
C
roughness coefficient
c
volume concentration of the fine fraction in muddy fluid
c
volume concentration of the coarse fraction in the total volume
c
critical c for an immature debris flow
Le
L
c
volume concentration of solids in the flow
c^ volume concentration of solids in the static bed
c. volume concentration of the fine fraction in the static bed
*F
c + volume concentration of the coarse fraction in the static bed
c\ ,. volume concentration of the coarse fraction in the static bed
DL
after deposition of debris flow
cT theoretical maximum concentration of coarse sediment in the flow
Loo
c
D
F
i
n
Q
r
t
u+
V
VL
T
equilibrium solid concentration in the flow
thickness of the bed, d
mean diameter of the coarse sediment
Froude number,
g
acceleration due to gravity
erosion(>0) or deposition(<0) velocity,
K
coefficient
Manning's roughness coefficient,
p
numerical constant
discharge,
q
unit width discharge,
R
hydraulic radius
inflow rate per unit length, s degree of saturation in the bed
time,
u
cross-sectional mean velocity
shear velocity assigned to the interstitial fluid
volume of fine sediment in the pillar-shaped space in the flow
having a height of h and a bottom area of unity
volume of coarse sediment in the space V„
F is defined
167
168
T.Takahashi et al.
x
length of the channel
(X, 0 numerical coefficients,
0
channel slope
0
original bed slope,
0
critical slope for a debris flow
0
critical slope for a land slide,
K
coefficient
p
density of water, n apparent density of muddy fluid in the void
pL, apparent density of the debris flow {=c (<j-p) + p}
p*m apparent density of the static bed {-c^ gKl-c^) ps}
0
density of the solid particles,
x overall tangential stress
X, turbulent Reynolds stress in the interstitial fluid
T,- critical tractive stress of the interstitial fluid
fc
X„ dispersive stress,
xT shearing resistance stress
X ^ nondimensional j ,
x*f
nondimensional xf
(j) internal friction angle in the bed
INTRODUCTION
Prediction of the hydrograph of debris flows that are debouched from
mountain torrents is indispensable for determining what is a
hazardous area, the design
of structural countermeasures, or
both. Although theories as to the mechanics and estimation of
discharges of debris flow exist, none is appropriate for predicting
the entire hydrograph generated in a real basin because a simple
case, in which constant rate of water is supplied from upstream in
an infinitely long prismatic channel of uniform thickness and with
uniform material composition of its bed always is assumed.
To bridge the gap between actual engineering needs and state-ofthe-art theoretical investigations, a new procedure that makes it
possible to estimate a debris flow hydrograph under conditions of
complex space and a complex water supply is needed. We here present
a new method with which to predict the debris flow generated by a
supply of water in a varied slope channel.
THEORETICAL ESTIMATION
Development and onset of a debris flow on a bed of varying slope
Consider a channel bed with a continually
changing slope (Fig.l), in which 0 is the
critical slope for the onset of a debris
flow and 0
the
critical slope for a
landslide to occur owing to an increment in
the underflow stage, defined (Takahashi,
1977) as
tan e 2 = . - J * i ^ _
tan *
(1)
3 | 2i
"
W
Kb)
em
] 1(a)
' e*te
Fig.l Varying slope bed
(2) a n d water seepage.
tan 6 = .— l T*^ g ~ p L- tan d>
2
c^(a-p)+p
As long as the surfaces of the seepage and overland flows are like
those in Figure 1, the entire bed should be stable, only a small
amount of bed load transport possibly taking place in regions 2 and
3.
When a surface water flow abruptly enters the channel upstream,
the bed will be eroded from the upper to the lower layer. The eroded
Debris flow hydrograph
169
sediment will be mixed with the water then run down, increasing the
discharge and solid concentration downstream. In the ranges 1(a) and
K b ) , the bed is essentially stable ( i.e., no sliding ) because the
seepage water surface does not reach the bed surface. Soon the flow
reaches range 2, in which the bed is saturated by seepage water.
Because the flow on the bed surface and the seepage water are in
contact in this region, the hydro-static pressure and tangential
stress of the flow are transmitted directly to the bed. The upper
part of the bed layer becomes unstable due to the imbalance in the
applied tangential and internal resisting stresses. If there were no
hindrance
downstream, this layer would yield immediately, but the
still stable bed just downstream from the flow front hinders
simultaneous movement, only some parts of the unstable layer
beginning to move and mingle with the flow. The debris flow that
develops progressively in this way proceeds to range 3 in which, if
the sediment concentration in the flow is too great, a part of the
sediment will be left and the rest continue to run down.
Note that the erosion in range 1(b) will enlongate range 2 in the
upstream direction and erosion in range 1(a) may induce sliding
at the surface of the bed rock. Such factors would affect the
characteristics of the hydrograph of a flow and might be one cause
of the intermittency observed in actual debris flows. Here, however,
we have neglected these phenomena. The case in which a slope change
between ranges 2 and 3 is abrupt and the conditions for stopping the
forefront of a debris flow is satisfied also is beyond the scope of
this study.
Fundamental equations
One-dimensional, unsteady flow in a sloping, open channel in which
there is erosion or deposition of sediment is described by momentum
(3) and continuity(4) equations
1 3q
2q 3q
.
q 2 , 3h
q2
— u "ii + - T 2 " , ' = sin6 - (cos 6
3 ) - - -v2n!p
gh 9t
gh 3x
gh 3x
C h R
--S-silc, + (1-C^)SH(1+K)- Ç *-T_I } _ _ A r ( 2 H . _i)
gh
*
*
pT
gh 2
pT
and 3h
3q
at
+
¥x
= l{c
(3)
J)
^
*+d-c*)s} + r
(4)
in which, K contributes to the increase or decrease in the momentum
of the flow and is approximated as 1 for erosion and 0 for deposition. When the Chezy or Manning resistance laws apply, p=l/2 and
p=3/2 for the Bagnoldian dilatant fluid. Other notations are the
same as presented in the previous section.
The constant C, which defines the resistance term, also differs
with the characteristics of the flow (Takahasi, 1984); in a
muddy debris flow in which the relative depth h/d is about 30 or
more the usual Manning type resistance formula is applicable, so
that,
C = R1/Vn
r
But in a stony debris flow in which h/d is
dilatant type resistance formula gives
less
than
30,
,.(5)
the
170
T.Takahashi et al.
In the upstream end of the channel, even if the relative depth is
less than 30, the quantity of sediment would not be large enough to
be dispersed as a whole flowing layer; consequently, an immature
debris flow would appear, in which
C = 0.7g l/2 h/(d L R)
The critical sediment concentration dividing the stony debris flow
and the immature debris flow is about 0.40^ .
In this study, we divided the solid component into two fractions; a coarse fraction whose particles are sustained in the flow
by the effect of collisions and a fine fraction whose particles are
suspended by turbulence in the interstitial fluid. The particle
diameters for the two fractions may change with variation in the
hydraulic condition of the flow, but, here we have assumed that they
are fixed values. The continuity equations for each fraction are
3V|
3t
+
3(qcL) =
3x "
{
ic^L ; i>,0
ic^L ; i < 0
(8)
and
3VF
7F
3{q(l-cL)cF}
+
'
,iciF
{
s^T--= i(i-c,DL)cF
; l ^0
;i<0
(9)
Change in the thickness of the bed layer is written
3D + X.
.
„
3D
„
=
Tï
J ï + 1 =
°°
and the bed slope
, ,3D>
6 = 8 0 - tan" (—)
(10)
(11)
Given the value of i, q, h, c , c , D and Q can be obtained, at
least numerically, at an arbitrary position in the channel under the
appropriate boundary conditions by solving the fundamental equations
which appeared earlier simultaneously. The next step is to give the
functional relationship of i.
The tractive force of the surface flow on a steep, unsaturated
sediment bed would erode the bed surface as in individual particle
transportation in a channel of lesser slope. Analogous to the given
bed load transportation formulae,
-J-
= K(T, f - T # f c )
(12)
can be assumed. Here, the overall shear resistance is the sum of the
shear stress elements produced by encounters between particles and
by the turbulence of the intergranular fluid; i.e.,
xG +
Tf
= {cL(a-pm)+pm}gh.sine
(i3)
in which the density of the intergranular fluid, D , is denser than
the pure fluid because of the loading of the fine sediment fraction.
It can be written
pm = cF(a-p)+p
(14)
Because T is approximated (Takahashi, 1977) by
T G = cL (a-pm)gh-cos6 -tan<t>
(15)
Debris flow hydrographs
(tan*
n
L ( tane _ 1 ) }
171
(16)
In equation (16), T becomes smaller as c
increases, finally
reaching a critical value that is less than that at which the flow
no longer erodes the bed. This critical T value is written T .
For dilute, individual particle transportation, T
is a function
of the diameter of the bed material, but for very dense, massive
particle transportation, even if T were much larger than the
critical tractive force, the bed particles would not be entrained
into the flow if ample void space were not present in the flow. This
means that Twould be determined by the maximum equilibrium
concentration of the particles rather than by particle diameter. In
this case, the equilibrium concentration is defined as
C
T»
=
ptan6
(a-p)(tan<t>-tane)
(I 7 )
According to an earlier study (Takahashi, 1977), c
is the asymptotic concentration when all the particles are sustained in motion
by the effect of dispersive pressure due to collisions, and the
interstitial fluid at this concentration is somewhat turbulent. If
the bed material contains fine particles suspended by this turbulence, as in the case presented, the maximum equilibrium concentration would be more dense. Thus, we can assume a maximum equilibrium
concentration for nonsuspended larger particles that is equal to
c ; i.e., c which corresponds to T
is equal to c .
Substitution of this assumed term xn equation (127 gives
i
,
„ 3*/ 1 /? r a -P„ ,tan<t> 1_ . , 1,', , tan$ „ , ,
,h
^ = K (I s i n
Ml
(18)
L e)
lJ - - ^ c L ( ^ 7 r e - l ) } - ( . : - _ - l ) ( c TTœ - c L ) - /?H ~ ^ ° - " '
- p
L^tane
'tane " » L'a
n
L
Note that on a bed steeper than 8„ calculated c„,„ value exceeds c.
2.
•poo
*
and even the maximum possible compaction value. Because no flow is
possible at such a high concentration, c would be replaced by the
maximum possible flow concentration; consequently, i=0.
The other asymptotic concentration obtainable by inserting T f = 0
in equation (16) has a physical meaning defined by
Pmtan9
c
Loo " "(a-p^Ktàn^-tane)"
<19)
If a highly concentrated flow enters a flatter region whose c
is
greater than c
and less than the c
of that region, it may pass
through with neither erosion nor deposition. Therefore,
for
cT >c T âc , i=0 and for c T Sc . iSO.
Loo L Too
L L°°
Next, consider a debris flow moving on a bed saturated with water
whose slope angle is between 0 and 8 . The applied shear stress in
the bed at depth a is
T = g sin6{(cTh+c#a)(a-p)+(h+a)p}
(20)
The resisting stress at the same ooint is
T = g cos6{c h(a-p )+c^a(a-p) }tan<(>
(21)
Above the depth a , T is larger than T ; hence, this bed layer
become unstable, a is obtained from equations (20) and (21);
a =.^I-co„_ {1..CL p ™ t a n < ^ c T / c T j ( c L a / c L H p / P m ) t a n e
! l
L
c -c T
c, p
tan$-tane
~
(22)
172
T.Takahashi et al.
As stated earlier, a whole bed layer with a thickness of a does
not flow a s soon as the flow front arrives; there is a delay before
( d L / u ) / a , the
the completion of erosion. Writing this delay
erosive speed obtained from equation (22) is
= a —T-m.c^-c
u-
£jD
tan<f>-(c T /c Too ) ( c L a 3 / c L ) ( p / p j t a n e q
tan<t>-tan8
-Lc
d
(23)
The value of c which gives i=0 in equation (23) is c^ . Therefore,
at that
if the run out debris flow has a c value larger than c
position, it will deposit some coarse particles but continue to run
down, thereby diluting the concentration. The amount of excess
coarse particles is h(c -c ) per unit area. Describing the time
necessary to deposit that amount as similar to the time for erosion,
(d /u)/g, the depositing speed is
(24)
'#DL
Kinematic wave approximation
Equation (3) can be simplified considerably by neglecting all the
terms except friction loss and the inclination of the bed. By this
simplification equation (3) is replaceable with
Q = ChRp(sin6)l/2
(25)
This simplification is valid when the Froude number is small, the
free surface of the flow is nearly parallel to the bed, and the
magnitudes of i and r are less than the velocity of the flow. The
validity of this simplification is examined later with equation (25)
instead of equation (3).
EXPERIMENTS
A
transparent
360 cm
plexiglass flume
,.
200cm
7cm wide and 4m
long
was used.
The bottom slope
of this flume was
30 °
at
the
upstream end and
15°
at
the
^\.*
downstream
end
^-s:
(Fig.2).
Boards
^
^
were
attached
Flew Controller
along
the side
walls
of
the
channel to control the lateral inflow.
Fig.2 Experimental flume.
Bed material A, whose size distribution is given in Figure 3,
was layered on the bottom of the channel at a thickness of 10cm.
Before beginning the experiment, seepage flow was produced on the
bed. The free surface of this flow appeared 150cm from the downstream end; therefore, the part of the bed upstream from this point
was unsaturated.
J^^Vsi
^^^^p£\
r
^0^
Debris flow hydrograph
173
A predetermined discharge of water
100
- I I lll({
I i i i
i i
**L
was given abruptly from the
upstream f(d)
(
*
)
end or laterally from both walls,
y / B
after which a debris flow began on the
50
bed and developed downstream. This
development
was
followed by TV
i 1 1 i f jL—•'l
Mil"
video-recorders through
the transdCmm)
parent side wall then analyzed. The
experimental data are given in Table 1. Fig.3 Particle size distribution in the material.
Table 1 Experimental d a t a •
1
HIM
7
Kun
Length of
Bed (cm)
Thickness of
Bed (cm)
/l
iIJ.
i
i
11
Water Supply
Discharge
(cc/s)
Position
Duration (s)
1
270
10
i Upstream End
200
40
2
270
40
270
1
!
350
3
10
10
200
40
'/
Side Walls
COMPARISON OF EXPERIMENTAL RESULTS WITH CALCULATED ONES
•/ / .
.
A
leap-frog
t(sec)
explicit
finite
difference
scheme
with
D
Ax=1.0cm
and fern)
A1» * V o
At=0.002s was
L
o \iT&-used
in our
Fig.4 Bed erosion(Runl).
Fig.6 Bed erosion(Run2),
calculations.
The
boundary
h
between
the
h
fcm)
(cm)
fine
and
:•»
coarse
fractions was
o a ^ o ' è t e o V V fix»
H-OC^.
assumed to be
o
1"
as
0.3mm;
•e * 1
o o
—»$-•—S-w
therefore, the
size distri*>t(sec) 3 0
t(sec) 3 0
bution of the
Fig.7 Depth-time (Run2).
coarse material Fig.5 Depth-time (Runl).
is given as line B in Figure 3, d equating 1.8mm. The degree of
saturation, s, is 1.0 in the reach of the saturated bed; s=0.8 is
assumed for the upstream unsaturated area.
Figures 4-6 and 5-7 compare the experimental results of Runs 1
and 2 with our calculated values. After trial and error computations
during our calculations, we found that the most appropriate values
were K=0.06 and a=0.0007. The tendency for bed erosion shown in
Figures 4 and 6 suggests that both equation (18) for the unsaturated
region and equation (23) for the saturated region predict the
erosion speed of the bed material well.
Time variations in the flow depth shown in Figures 5 and 7 are
evidence that our calculations are comparable to the experimental
values found for the upstream region, but predict a somewhat larger
Cofcutarton Experiment F\»itkxi
—
O
20cm
O
60cm
O
100cm
•™._—
0
160 cm
•
220cm
|Q9<
£>
Cctaulaiion
.
Expirlmtnt
O
0
0
Û
PoilHon
20cm
SOcm
KOcm
220cm
*^K«=
174 T.Takahashi et al.
depth in the downstream region. Taking into account that the experimental results were obtained from violently turbulent video images
of the flow surface that were hazy at the boundary between the fixed
and moving layers, the error in the experimental measurements would
be comparatively large. Therefore we believe that these figures
prove the validity of our resistance law and the representative
diameter of the bed as well.
The flow depth at each point in the figures is asymptotic to a
constant value. This is because of run out of all the materials and
the resulting exposure of the rigid channel bottom. The constant
depth therefore represents steady flow depth on the rigid bed. If
there is any discrepancy between the experimental and calculated
depths, it would be attributable to the inappropriateness of the
roughness coefficient for the rigid bed that was used in our calculations.
Results of our calculations of depth variation clearly show how
the debris flow hydrograph develops downstream. The peak discharge
increases and the shape of the hydrograph becomes triangular as the
flow proceeds downstream. This latter tendency may be why in many
real debris flows the peak discharge appears immediately after the
forefront and decreases with time. The abrupt increase in depth in
our calculations, e.g. at the 20cm point at approximately 13 seconds
(Fig.5) corresponds to the end of the bed erosion and this shock
wave is transmitted downstream. In the experiment, however, this was
not clear and may not have existed.
Figures 8 and 9 show the results
10
20
30
4C
n
of Run 3. The erosion rate at 120cm,
at which point the bed was unsatu^ » S^c>oo"»--._._
rated, compares fairly well with the
5
experimental value. But, at 220cm
(50cm above the downstream end of the
D(cm)
channel) our calculations fit the
10
experimental results fairly well up
Fig.8 Bed erosion in Run3.
to 24 seconds, after which time the
difference between the two results
increases. A similar tendency for
very small
erosion or deposition
shown in the experimental values at
(O
a O
oo O O O
220cm can be seen in Figure 4 as
\
well. This is attributable to the
existence of a fixed bed girdle at
t(sec)
the downstream end of the channel
which we have neglected in the computations. Fig.9 Depth-time(Run3),
0
r
i3
O
K 0 M
O
220MI
Q
APPLICATION TO THE ARMERO MUD FLOW
Outline of the Armero mud flow
On 13 November 1985, the Nevado del Ruiz volcano in Colombia erupted. Although the eruption was not an exceedingly large scale one, it
was accompanied by a small scale pyroclastic flow. The rapid melting
of the surface of the mountain's ice cap by this pyroclastic flow
triggered several disastrous mud flows. Of these, the one that
struck Armero City was the biggest, claiming 21,000 human lives.
Debris flow hydrograph
175
The peak discharge of
the mud
flow, estimated
from the superelevation of
the flood
mark at the
river's
bend immediately
upstream from Armero, was
Mt.Ruiz
about 30,000m / s . Mud and
debris were spread over an
area of about 30km . If the
thickness of this deposition is assumed to average
Fig.10 Armero City and the surroundlm, the total volume of the
ing river systems.
run off
sediment would be
X1000
about 30x10 m . The composition of the deposit in the
Armero area seems to be mainly
gravel with a mean diameter of
10cm; whereas, downstream from
Armero fine mud predominates.
The total volume of the runoff
sediment of the mud fraction
was much more than that of the
gravel fraction, so the flow
appeared as a mud flow.
The origins of this huge
amount of sediment is posited
as the beds of the Rivers
Azufrado and Lagunillas. The
size compositions of both river
beds should have contained many
coarse particles before the
mud flow, but the run out mud Qrt
flow
contained
few coarse
particles. The mechanism of
the development of this mud
flow as well as its hydrograph
are explained.
t(hour)
Fig.11 Calculated hydrographs
the mud flow along the River.
of
0.5 r
Q.
Fig.10 shows the positions
of the rivers and Armero City.
Calculation of the hydrograph
2 t(hour)
Fig.12 Calculated solid concentThe original bed slope, 9 , was
rations along the River.
obtained from a topographic map (scale, 1/100,000). The Azufrado
River has three tributaries; the Hedionda(R ) , Azufrado(R ) and
Planuela(R ) Rivers; whereas the Lagunillas River has only one
source (R.7. The widths of all four tributaries are assumed to equal
50m, and the widths of the main streams, the Azufrado and Lagunillas, also are assumed to be 50m. The bed thickness for the entire
river reach is assumed to be 20m, beyond which thickness no erosion
will take place. The properties of the bed materials are assumed to
be
=0.64
^ = 0 . 2 5 6 , c S F = 0 . 3 8 4 , C * D L = ° - 5 ' tarty =0.75, d =10cm,
0=2.65g/cm , a n d p = 1 . 0 g / c m
The
relative
depth of the flow, h/d.
L'
greatly
L
exceeded
30;
176
T.Takahashi et al.
consequently, a Manning type resistance law is applicable, and
n =0.04 is assumed. This n value gives the maximum discharge of the
mud flow in the straight channel reach just upstream from Armero
City which is almost the same with the value obtained at the channel
bend a little upstream. Because the channel slope read from the map
does not exceed 0 and before the mud flow the bed should have been
saturated by water, equation (18) was not used. From our experimental results, 10
was the value for a and (3.
The quantities of water suplied to the four tributaries were
determined from the estimated volume of the melted ice in each
basin. The duration of the melting ice flood was assumed to be 15
min; therefore, the discharges to the rivers were Q=831m /s for R
and R , Q=397m /s for R and Q=l,500m /s for R .
A difference scheme of Ax=200m and At=1.0sec was used in our
calculations, Figure 11 showing the results. Positions L , A , and
L
are given in Figure 10. The peak discharge of the flow at L
is
28,600m /s which agrees with the in situ estimation. This coincidence was brought by trial and error calculations in which different
properties were used for the bed material; but, it should be emphasized that the magnitude of the variables adopted are reasonable and
that our results prove the proposed method is a useful one.
Figure 12 shows the time variations for calculations of the fine
and coarse fractions. The peak for the solid concentration precedes
the peak for depth. The magnitude of c greatly exceeds that of c
which shows why a mud rather than a debris flow formed.
The total run out solid volume was estimated as 20.4x10 m . If
0^=0.65 is assumed for deposition, the total solid volume actually
deposited in the Armero area is about 19.5x10 m .
Separation of the flow's composition into coarse and fine
fractions was extremely important in order to explain the distribution of thickness and particle size in the actual deposition. The
method used to simulate the deposition process and its results can
be found elsewhere(Takahashi & Nakagawa, 1986).
CONCLUSIONS
Theoretical and experimental values were applied to an actual debris
flow to estimate its hydrograph and the sediment concentration on a
bed of varying slope. We found:
1) A system of equations that predicts the hydrograph and the
sediment concentration of the debris flow generated by a supply of
water on a varying slope bed.
2) Kinematic wave approximation usually gives good results.
3) Numerical coefficients in the formulae for erosion and deposition
speed( K, a and 3) can be determined by comparing the experimenal
results with calculated values.
4) The calculated results compare fairly well with the experimental
erosion rate and time variation in the flow depth.
5) The application of our calculation method to an actual, huge mud
flow proved that it predicts not only the hydrograph but the sediment concentration separation into coarse and fine fractions as
well.
Debris flow hydrograph
177
REFERENCES
Takahashi, T. (1977) A mechanism of occurrence of mud-debris flows
and their characteristics in motion. Annuals, Disaster Prevention Research Inst., Kyoto Univ., NO.20B-2, 405-435(in Japanese).
Takahashi, T. (1984) Dynamics of debris flow. Jour. Japan Soc. of
Fluid mech. , Vol.3, No.4, 307-317(in Japanese).
Takahashi, T. S Nakagawa, H. (1986) Estimation by computer simulation of a hazardous area produced by a mud flow. Proc. of Symp.
on Natural Disaster Science, Japan, No.5, 159-160(in Japanese).