Erosion and Sedimentation in the Pacific Rim (Proceedings of the Corva l l i s Symposium, August, 1987). IAHS Pub 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=TT/ 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 equilibrium solid concentration in the flow D thickness of the bed, d mean diameter of the coarse sediment F Froude number, g acceleration due to gravity i erosion(>0) or deposition(<0) velocity, K coefficient n Manning's roughness coefficient, p numerical constant Q discharge, q unit width discharge, R hydraulic radius r inflow rate per unit length, s degree of saturation in the bed t time, u cross-sectional mean velocity u+ shear velocity assigned to the interstitial fluid V volume of fine sediment in the pillar-shaped space in the flow

having a height of h and a bottom area of unity VT volume of coarse sediment in the space V„ is defined L F 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 f c 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-of-the-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 3 | 2i Kb) ] 1(a) 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 e2 = . - J * i ^ _ tan * ( 1 ) " W em ' e*te

Fig.l Varying slope bed tan 6 = .—lT*^g~pL- tan d> (2) a n d water seepage.

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 . q2 , 3h q2

— u "ii +-T2", ' = sin6 - (cos 6 3 ) - - -v 2 n !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) ( 3 )

gh * * pT gh2 pT ^J)

and 3h 3q at + ¥x = l{c*+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 = R 1 / V n ,.-r (5)

But in a stony debris flow in which h/d is less than 30, the dilatant type resistance formula gives

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 = 0.7gl/2h/(dLR)

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| + 3(qcL) = { ic^L ; i>,0

and

3t 3x " ic^L ; i < 0 (8)

^0 3VF 3{q(l-cL)cF} ,iciF ; l

7 F + ' s^T--= {i(i-c,DL)cF ;i<0 (9)

Change in the thickness of the bed layer is written 3D . „ Tï+ X = °

and the bed slope

3D . „

J ï + 1 = ° (10)

, ,3D> 6 = 80 - tan" (—) ( 1 1 )

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#fc) (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 ( i 3 )

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 171

(tan* n

L(tane_1)} (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 T- would be determined by the maximum equilibrium concentration of the particles rather than by particle diameter. In this case, the equilibrium concentration is defined as

ptan6 CT» = (a-p)(tan<t>-tane) (I7)

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 , „ * 1 /? r a-P„ ,tan<t> _ . , 1,', , tan$ „ , , ,h

^=K(sine) 3 /Ml--^c L(^ 7 r e-l)}1-(. :-_-l)(c T œ-c L)--

/?H ~ I^°- L" ' lJ- p L^tane 'tane " T» L'a n L

(18)

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 Tf=0

in equation (16) has a physical meaning defined by Pmtan9

cLoo " "(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 >cTâc , i=0 and for cTSc . 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™tan<^cT/cTj(cLa/cLHp/Pm)tane

L c -cT c, p tan$-tane ~ ! l (22)

172 T.Takahashi et al.

As stated earlier, a whole bed layer with a thickness of a does s soon as the flow front arrives; there is a delay before not flow a

the completion of erosion. Writing this delay erosive speed obtained from equation (22) is

( d L / u ) / a ,

= a —T-m.-c ^ - c u- £jD

-Lc

tan<f>-(cT/cToo) (cL a 3 /cL ) ( p / p j t a n e q tan<t>-tan8 d

the

(23)

Therefore, at that

run

The value of c which gives i=0 in equation (23) is c^ . if the run out debris flow has a c value larger than c position, it will deposit some coarse particles but continue to 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 terms except friction loss and the inclination of the bed. By simplification equation (3) is replaceable with

Q = ChRp(sin6)l/2

the this

This simplification is valid when the Froude number is small, free surface of the flow is nearly parallel to the bed, and magnitudes of i and r are less than the velocity of the flow. validity of this simplification is examined later with equation (25) instead of equation (3).

(25) the the The

EXPERIMENTS

360 cm A transparent plexiglass flume 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 (Fig.2). Boards were attached 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.

^-s:

,. 200cm

J^^Vsi ^^^^p£\

^0^ ^ ^

^\.* r

Flew Controller

Debris flow hydrograph 173

A predetermined discharge of water was given abruptly from the upstream end or laterally from both walls, after which a debris flow began on the bed and developed downstream. This development was followed by TV video-recorders through the trans­parent side wall then analyzed. The experimental data are given in Table 1.

100

f(d)

(*)

50

dCmm)

Fig.3 Particle size distri­bution in the material.

1

-

- I I l l l ( { I

i 1 1 i f jL—•' l

i i i H I M

y

7 / l

i i i I J .

/ B

i i 1 1

**L

-

M i l "

Kun

1

2

3

Table

Length of Bed (cm)

270

270

270

1 Experimental da ta

Thickness of Bed (cm) Position

10 i Upstream End

10 1 '/

10 ! Side Walls

•

Water Supply

Discharge (cc/s)

200

350

200

Duration (s)

40

40

40

COMPARISON OF EXPERIMENTAL RESULTS WITH CALCULATED ONES

A leap-frog explicit finite difference scheme with Ax=1.0cm and At=0.002s was used in our calculations. The boundary between the fine and coarse fractions was assumed to be as 0.3mm; therefore, the size distri­bution of the

D fern)

A1» *Vo L o \ i T - &-- •

/ /

. .

t(sec)

Fig.4 Bed erosion(Runl). Fig.6 Bed erosion(Run2),

h (cm)

|Q9<

Cofcutarton Experiment F\»itkxi — O 20cm

O 60cm O 100cm

• ™ . _ — 0 160 cm • 220cm

h fcm)

o a^o 'è teoVV fix»

H - O C ^ .

o o

o 1"

•e * 1

£> : • »

Cctaulaiion Expirlmtnt PoilHon O 20cm

. 0 SOcm 0 KOcm Û 220cm

* ^ K « = —»$-•—S-w

t(sec) 3 0 *>t(sec) 3 0

d equating 1.8mm. The is 1.0 in the reach of the saturated bed;

Fig.7 Depth-time (Run2). degree of s=0.8 is

coarse material Fig.5 Depth-time (Runl). is given as line B in Figure 3, saturation, s, 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

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 of Run 3. The erosion rate at 120cm, at which point the bed was unsatu­rated, compares fairly well with the experimental value. But, at 220cm (50cm above the downstream end of the channel) our calculations fit the experimental results fairly well up 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 220cm can be seen in Figure 4 as well. This is attributable to the existence of a fixed bed girdle at the downstream end of the channel

n

5

D(cm)

10

0 10 20 30 4C

^ r» i 3S^c>oo"»--._._

O K 0 M Q

O 220MI

Fig.8 Bed erosion in Run3.

(O a O oo OOO

\

t(sec)

which we have neglected in the computations. Fig.9 Depth-time(Run3),

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 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 lm, the total volume of the run off sediment would be 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 Q. flow contained few coarse particles. The mechanism of the development of this mud flow as well as its hydrograph are explained.

Fig.10 shows the positions of the rivers and Armero City.

Mt.Ruiz

Fig.10 Armero City and the surround­ing river systems.

X1000

t(hour)

Fig.11 Calculated hydrographs the mud flow along the River.

0.5 r

of

Calculation of the hydrograph

The original bed slope, 9 , was

2 t(hour) Fig.12 Calculated solid concent­rations 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 =0.384, C *DL = °- 5 ' 0=2.65g/cm , andp=1.0g/cm

The relative depth of the flow, h/d.

d =10cm, L

L '

tarty =0.75,

g r e a t l y 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).