Testing various constitutive equations for debris flow modelling

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1 Hydrology, Water Resources and Ecology in Headwaters (Proceedings of the HeadWater'98 Conference held at Meran/Merano, Italy, April 1998). IAHS Publ. no. 48, Testing various constitutive equations for debris flow modelling THILO KOCH Swiss Federal Institute of Snow and Landscape Research, CH-8903 Birmensdorf, Switzerland Abstract On the basis of a curvilinear coordinate system and incorporation of a number of flow resistance laws, a one-dimensional model has been developed to simulate debris flows in mountain torrent channels. The model simulations are compared to debris flow events in the Kamikamihori Valley in Japan. The results suggest that the models with a Newtonian turbulent and a Voellmy fluid flow resistance, as well as a model with a new empirical flow resistance approach, are better at reproducing the observed velocities and flow-heights than the models with a Bingham, Newtonian laminar or dilatant grain-shearing flow resistance. The material constant for the dynamic part in the flow resistance law, the channel width and the total volume were found to be the most sensitive parameters for the flow simulation. INTRODUCTION Debris flows form one type of natural hazard often occurring in mountain regions. A debris flow is a rapidly flowing mixture of grains of various sizes and water initiated on a steep slope, usually travelling along torrents and coming to rest at a debris flow fan. A debris flow implies fluid-to-fluid, fluid-to-solid and solid-to-solid interactions, which are generally not well understood and not considered in most of the existing approaches to model debris flows. Most of the models developed have been restricted to the one-dimensional case, with mainly three types of constitutive equations: Bingham flow, Newtonian turbulent flow, or dilatant inertial. A more recent overview of constitutive laws to describe the rheology of debris flows is given by Jan & Shen (1993). The objective of this study is to evaluate a number of different model approaches by applying them to documented events and comparing simulated and observed flow behaviour. SIMPLE DEBRIS FLOW MODELS The debris flow mixture is described as a one-dimensional incompressible continuum. The shallow water approximation, averaging different constitutive equations and the mass and momentum balance laws, has been used. The pressuredependent deformation is assumed to comply with a Mohr-Coulomb plastic yield criterion. The model is formulated for a curvilinear coordinate system, following the torrent topography, where s is the downward coordinate. The depth-averaged

2 50 Thilo Koch equations of mass and momentum are described in a dimensionless form for a Coulomb-friction flow in Savage & Hutter (1991). A formulation in a dimensional form is presented by Hungr (1995) for several constitutive equations. I have followed Hungr (1995) to a larger extent. The numerical procedure used here is a 1-D version of a Lagrangian central finite difference scheme, completely described in Hutter & Koch (1991). The mass is directly connected with n cells (Fig. 1). The index i corresponds to the cell boundary points and j to the cell centres; k counts for the time. The calculations start with an initial geometry and velocity distribution for the mass using the geometric data (bed slope and width) of the torrent profile. The boundaries with heights Hf' x (at time k - 1) are connected with the depth-averaged velocities v*~ l/ (at time k- Vz, second-order time accuracy). As a the first step, we obtain the new coordinates s* from the previous coordinates sf~ ] and velocity v*" 1/ **=**-'+v,*- At where At is the time step. Second, the continuity equation is solved according to (D H B"+> V () where H) is an average cell height, ds* =sf +l -sf the cell length, B'. the local width of the cell boundary and F = V\ = constant, the volume of cell j. The resulting force, F.*, connected to boundary point i is composed of the tangential components of weight G*, the resisting force Tf and the pressure-distribution dependent component P t k F," = G: + p; with rpk G* =y,dsf//*,* since* (3) (4) Fig. 1 Definition of mesh cell notation for the numerical scheme used in the 1-D model.

3 Testing various constitutive equations for debrisflowmodelling 51 where y, = bulk unit weight, d sf = length, constructed from the neighbouring cells with length ds*, and a* = bed slope angle. The pressure term P k s acllpa: «sdh, -Y,M ds i-dsïh'b* sin a* (5) is different from Hungr (1995) and is in this implementation calculated by a Mohr- Coulomb plastic yield criterion (Hutter & Koch, 1991). The earth pressure coefficient with. \ act I pass IK. dv, K acl for ->0 ds-, K" ass îox -<0 ds, (6) act I pass K ss = 1 + ^/1-cos <j>/cos 5 cos -1 (V) Here, S = bed friction angle and (j> = internal angle of friction. This law has been successfully tested for dry granular flows, e.g. Koch et al. (1994). I expect it to be useful in debris flows when the granular particles stay in contact with each other. The resisting forces 7]* can be mainly written in the general, depth-averaged form ^=y,bfd,f[ Kf(vff + x, + (x c ); (8) with (t c )* =Hf cosaf tan ; (9) where = material constant, i y = cohesion and x c = coulomb yield strength. (3 and A, are exponents characterizing the material laws together with the existence of % y and In the case of a Bingham fluid the following cubic equation has to be solved in rpk. T} k Hf "m 3u,5>; It + {T?) 3 (10) where x B = Bingham yield stress and u. B = Bingham viscosity. The following equation is an explicit and, without f, an often-used formulation of v; v = CH K S B +f(x v,t c ) (11) Equation (11) is equivalent to equation (8) under the assumption that we have a

4 5 Thilo Koch stationary flow and neglecting P (usually P < < G,T ); it is also helpful in understanding how v is influenced by flow height H and slope 5. As a product of 0 (in equation (8)), we obtain the material constant C ; K and & are exponents and products of P and A. respectively, while f can contain terms of x }, and/or x c. Equation (11) is written in a non-discrete form because it is not used in the simulation routine. Table 1 lists five of the tested constitutive equations by presenting the values of P, X, K, 9 in equation (8) or (11) and marking the existence of x y and x c with a cross. The equivalent material constants are listed in 0. As previously mentioned, a Bingham laminar flow cannot be described with equation (8). Under the assumption that the Bingham yield stress is negligible (here x c -> 0), a flow resistance law for a Newtonian laminar fluid (Table 1) is obtained. The new velocity v* +l/ is obtained from the old velocity v*~" with gravity g, by solving the equation of motion: * + l/ = *-l/ + F k At r - ^ ( 1 ) The steps are repeated until the debris flow comes to rest or reaches the end of the path. Table 1 Comparison of flow resistance laws described in equations (8) or (11). Equations (8) (11) -(8), (11)- Approaches P X C K & T t c Newtonian laminar Dilatant inertial Newtonian turbulent Voellmy fluid KI f-i n C 1 K n- 1 % n l C KSB The approaches are described by using the values of the exponents p and X, K and S, respectively and, indicated with a cross, whether x y and/or x c are included. The material constants, equivalent to (equation (8)) are listed: \i= viscosity, n Manning's roughness coefficient, 4= lumped coefficient accounting for grain and concentration properties in granular flow, C = Chézy roughness coefficient, and K = coefficient from empirical investigations. SIMULATIONS AND FIELD OBSERVATIONS Comprehensive comparisons between model simulations and observations are made for debris flow events in the Kamikamihori Valley in Japan (Rickenmann & Koch, 1997). The longitudinal and bed-slope-angle profile of the Kamikamihori Valley used in the calculations is shown in Fig. (check dams were smoothed and the step-height sum reduced by 19 m). In the calculations, the model channel has a constant width of 10 m down to a fan apex at X = 1.9 km. From there, it widens to 15 m at X =.0 km; further downstream there is a widening of 10 m per 100 m longitudinal distance. This is a rough approximation of the known cross-sections.

5 Testing various constitutive equations for debris flow modelling 53 - Kamikamihori valley Slope angle [ ] 30 Middle-reach observation site CO 0- (D 0) =3 CQ_ 10 CD Zi i r Fig. Longitudinal and bed-slope profile of the Kamikamihori Valley (Japan). The arrow indicates where flow hydrograph had been measured. The model parameters have been adjusted to best reproduce the measured data on flow velocity along the flow path. For calculating the earth pressure coefficient, I set 5 = 4 and <b = 10, where 8 is found in equations with a XQ term (Table 1) to produce a stopping position of the wave front close to the measured value in the field and reasonable -values; <j> is assumed as there is only little influence of <j> on the results for long and fiat waves. The events of 3 August 1976 (Okuda et al., 1980), case A, and of 1 September 1979 (Suwa & Okuda, 1981), case B, are used here. Average debris wave-front velocities were obtained for segments along the flow path measured between signal wires tightened perpendicular to the torrent channel. Because local velocity variations are not known, I assumed an uncertainty margin of ±0% with regard to the measured values. For case A, the initial input hydrograph at X 0 = 0 km has a maximum flow depth H Q of.5 m, a length, L 0 of 70 m and a constant inflow velocity, v 0 of 6.5 m s" 1. The total volume, M 0 (sediment and water), corresponds to 6500 m 3. For each of the five approaches listed in Table 1, one simulation is shown in Fig. 3. The simulated velocities are reasonably close to the observed ones for the turbulent, the Voellmy and the KI models. For the Bingham and the dilatant models no good agreement could be obtained for the velocity development along the whole flow path. Simulations with the Newtonian laminar model show results similar to the Bingham model. While using values in a reasonable range, the first term in the brackets of equation (8) dominates the flow resistance for higher velocities, while the yield stress terms (x y,x c ) dominate the stopping process. In the case of the Voellmy and KI models, terms x y,x c produce a stopping wave in the range of the observed runout distance for the chosen parameters. For case B, the conditions of the input hydrograph are: X Q = 0.8 km, H 0 = 4.5 m, L 0 = 640 m, v 0 = 5 m s" 1, and M Q = m 3. Figure 4 shows the results of the same approaches, but with different material parameters. Again, the simulated velocities of the turbulent, the Voellmy and the KI models show an acceptable agreement with the measured ones, using almost the same flow resistance parameters as for case A. With the Bingham model, the velocity development can be reproduced somewhat better than for case A; however, the simulations started at a smaller bed

6 54 Thilo Koch turbulent: n = 0.15 Voellmy: 5 = 3.5, C? = 10 ms- - Bingham: T B= 1000 Pa, u^= 800 Pa s - dilatant: 5 = 31 rrrf- 5 s-i V observed (+/- 0%), 3 Aug 1976 y Fig. 3 Simulated and observed velocities for event A in the Kamikamihori Valley. The arrows indicate the range of observed runout distances. slope (Fig. ) than for case A. This and further experience with other events seem to confirm that the Bingham and the dilatant models cannot reproduce the flow behaviour for strongly varying slope angles. SIMULATION SENSITIVITY TO BOUNDARY AND INITIAL CONDITIONS The initial conditions and the exact cross-sections of the observed Kamikamihori events are poorly known. Therefore additional calculations have been made considering variations in the initial and boundary conditions. Figure 5 compares front velocities (case A event) for a turbulent model (reference case I: n = 0.15; turbulent model, see equation (11), with K = 0.67 (Table 1)) under the following conditions (original value in brackets): Case II: Input hydrograph: Max. depth H Q = 5.5 [.5] m, length L 0 = 135 [70] Case III: Case IV: Case V: Case VI: m. Volume: V 0 = 9735 [6500] m 3, H 0 unchanged, L 0 = 400 [70] m. Channel width: 8 [10] m, depth H 0 = 3 [.5] m, V 0 unchanged. More detailed channel slope profile, different channel widening on fan: at X = 160 m, widening to 0 m at X = 170 and at X = 1860 m additional widening of 6 m per 100 m longitudinal distance. Using a hydraulic radius calculated for a rectangular channel profile. Inflow velocity: v 0 = 5 [6.5] m s' 1. Case VII Figure 5(a) shows that using a different inflow depth but a constant volume (case II) strongly affects the front velocity during the initial phase. For X > 0.9 km, the front velocities of case II and the reference simulation are nearly identical. Using a larger volume (case III) or a smaller channel width (case IV) produces higher velocities along the whole flow path. Both cases result in higher flow depths, which yield the

7 Testing various constitutive equations for debrisflowmodelling I 8- f 6- > e 4- u_ - / i-. 11 i turbulent: n = 0.17 Vœllmy: ô=4.4, C = 10 m s- Bingham: T B= 800 Pa, 1^= 400 Pa s dilatant,=4 rrr 05 s- 1 Kl: K = m-33 S3 T = 400 Pa, 5 = 5 y \ ' "". _ V observed (+/-0'/o), 1 Sept 1979 ^M U "J 1 I I A A Fig. 4 Simulated and observed velocities for event B in the Kamikamihori Valley. The arrows indicate the range of observed runout distances. higher velocities. Figure 5(b) shows that using a hydraulic radius (case VI) or a smaller inflow velocity (case VII) obviously affects only the initial phase. Using a more detailed slope angle profile and a different channel widening on the fan (case V) produces local velocity variations which might be an improvement. A special effect is observed in the Bingham model: when there is a higher initial flow depth the wave becomes elongated, resulting in smaller flow heights downstream and thus a shorter travel distance. DISCUSSION Simulations for case A, case B and events not presented here (Saas Valley in Switzerland, lahars from Pine Creek and Muddy River at Mount St Helens, USA) lead to complementary results. Considering the front velocity along the flow path at steep variable slopes, there is better agreement between simulations and field observations for the Newtonian turbulent, the Voellmy and the KI models than for the dilatant, the Bingham and the Newtonian laminar models. Referring to Table 1, it can be seen that the better agreement is associated with flow resistance laws having smaller exponents K and &<0.5 in equation (11). Further evaluations, including models not presented here, confirm the preference of resistance laws close to turbulent clear water flow. For this reasons, models with smaller exponents seem to be more appropriate for simulations in torrents on steep and variable slopes. For less steep and less variable slopes, the differences in the results of the tested models is less significant. The Bingham model as well as the dilatant model may possibly also be suited to simulations of the flow behaviour on fans, where the bed slopes do not vary over too big a range. Simulations are generally more sensitive to input conditions, in particular to initial flow depth, for those flow resistance laws having

8 56 Thilo Koch a)? 1? fc 10.rT> n elo > c o M Cases I - IV: turbulent: n=0.15 Case I (reference simulation) Case II (new hydrograph) Case III (bigger volume) Case IV (smaller channel width) V observed (+/- 0%), 3 Aug 1976 T T (b) >>1U Cases I, V - VII: turbulent: n=0.15 Case I (reference simulation) Case V (detailed slope angle) Case VI (hydraulic radius) Case VII (smaller inflow velocity) W-S, V observed (+/- 0%), 3 Aug 1976 i 1 1 r Fig. 5 Simulated and observed front velocities for case A (debris flow events in the Kamikamihori Valley). All simulations are done with the turbulent model but for different initial and boundary conditions, (a) cases I through IV, (b) cases V through VII. higher exponents K, i.e. the Bingham, the laminar and the dilatant models. On the other hand, for the turbulent, the Voellmy and the KI models, variations in some initial conditions are damped relatively rapidly with increasing flow length. From a practical point of view, when performing simulations of debris flow events in nature, knowledge on initial and boundary conditions is only rough. Sensitivity examinations show relatively little influence of initial inflow velocity and height on the downstream flow behaviour for the preferred models. This also applies when considering side-wall effects (using a hydraulic radius) or a very detailed slope profile. Beside selecting an appropriate material constant, 0, the most accuracy is required for estimating the total volume and the channel width. These are the dominant factors controlling the dynamic behaviour of the tested models. The selection of the bed friction angle, S, and the internal angle of friction,, (equation (7)), also have only little influence on the flow behaviour. On the other hand, S in the Coulomb yield strength (equation (9)) is essential for the runout distance, which is very sensitive to values tan 5, close to tan a, in the deposition zone. 8 appears in the Voellmy, the KI and the Bingham Coulomb viscous flow. The Newtonian, dilatant and turbulent models do not simulate the stoppage. Connecting the turbulent model with a Coulomb yield strength solves this problem and produces a flow behaviour close to the Voellmy model. Acknowledgement I thank my colleague Dieter Rickenmann for the helpful discussions.

9 Testing various constitutive equations for debrisflowmodelling 57 REFERENCES Hungr, O. (1995) A model for the runout analysis of rapid flow slides, debris flows, and avalanches. Can. Geotech. J. 3, Hutter, K. & Koch, T. (1991) Motion of a granular avalanche in an exponentially curved chute: experiments and theoretical predictions. Phil. Trans. Roy. Soc. Lond. A334, Jan, C. D. & Shen, H. W. (1993) A review of debris flow analysis. In: Proc. XXVIAHR Congress, Technical Session B, vol. Ill, 5-3. Koch, T., Grève, R. & Hutter, K. (1994) Unconfined flow of granular avalanches along a partly curved surface, Part II: Experiments and numerical computations. Proc. Roy. Soc. Lond. A44S, Okuda, S., Suwa, EL, Okunishi, K., Yokoyama, K. & Nakano, M. (1980) Observations on the motion of a debris flow and its geomorphological effects. Z. Geomorph. N.F., Suppl. Bd 35, Rickenmann, D. & Koch, T. (1997) Comparison of debris flow modelling approaches. In: Debris-Flow Hazards Mitigation (Proc. Int. ASCE Conf., San Francisco), Savage, S. B & Hutter, K. (1991) The dynamics of avalanches of granular material from initiation to runout, Part I: Analysis. Acta Mechanica 86, Suwa, H. & Okuda, S. (1981) Topographical change caused by debris flow in Karnikamihori valley, Northern Japan Alps. Trans. Jap. Geomorph. Un. -,