Quantitative analysis of debris torrent hazards for design of remedial measres (Hungr and Morgan...

Quantitative analysis of debris torrent hazards for design of remedial measures

Thurber Consultants Ltd., Suite 100, 1281 West Georgia St. , Vancouver, B.C., Canada V6E 3J7

AND

Kellerhals Engineering Ltd., P.O. Box 250, Heriot Bay, B.C., Canada VOP I HO Received March 20, 1984 Accepted June 28, 1984

Debris torrents, which are rapid flows of soil and organic debris down steep mountain channels, are a major natural hazard in many parts of British Columbia.

A series of recent occurrences along the Squamish Highway, north of Vancouver, led the provincial government to initiate a systematic study of debris torrents in this area. This 2 year study involved hazard risk assessment and resulted in the design of a comprehensive system of remedial measures comprising debris retention barriers and basins, channel improvements and diversions, and the reconstruction of bridges. It was necessary in the course of this effort to formulate new design procedures covering the dynamic behaviour of debris torrents, as there was no accepted practical methodology suitable for Western Canadian conditions.

This paper provides an approach to determining magnitude (volume of debris material involved), frequency, peak discharge, velocity, conditions for deposition, runout distance, behaviour in bends and run-up against barriers, and dynamic thrust and impact loadings. The procedures are based on some of the more practical concepts available in the specialized literature, supplemented by simple original theories and calibrated against several recent debris torrent events from British Columbia for which sufficiently detailed observational data exists. To be generally applicable, the procedures require a wider and more thorough calibration. They are presented here as working hypotheses that can be used as a guide to the assembly of additional data and provide a rational design tool to supplement the application of experience and judgement.

Key words: debris torrents, debris flows, slope hazards, landslide dynamics, remedial measures, engineering design

Des torrents de dCbris, Ccoulements rapides de dCbris de sols et de matikres organiques dans les ravins montagneux abrupts, reprksentent un risque nature1 important dans de nombreuses regions de la Colombie Britannique.

Une sCrie rCcente de tels phCnomknes le long de la route de Squamish, au nord de Vancouver, a amen6 le gouvernement provincial i lancer une Ctude systCmatique des torrents de debris dans cette rCgion. Cette Ctude, d'une durCe de 2 ans, comportait 1'Cvaluation de la probabilitC de risque et a conduit i 1'Ctablissement d'un vaste systkme de mesures correctives comprenant des barrikres et des bassins de retention des dCbris, des amCliorations et des dCtournements de lits et la reconstruction de ponts. Dans le cours de cette Ctude, il a Ct6 nCcessaire de formuler de nouvelles procCdures d'analyse concernant le comportement dynamique des torrents de debris, puisqu'il n'existait aucune mCthode pratique acceptCe applicable aux conditions de l'Ouest du Canada.

Cet article prCsente une approche permettant de determiner l'importance (volume de matCriau de debris implique), la frkquence, le debit de pointe, la vitesse, les conditions de diposition, la distance de propagation, le comportement en virage et la remontCe le long d'obstacles, la poussCe dynamique et les forces d'impact. Les procCdures sont fondCes sur certains des concepts les plus pratiques prCsentCs dans la 1ittCrature spCcialisCe, appuyCes par des thCories originales simples et calibrCes sur plusieurs Cvknements recents en Colombie Britannique pour lesquels des observations suffisamment dCtaillCes Ctaient disponibles. Pour devenir d'application gCnCrale, les procCdures prCsentCes demandent une calibration plus large et plus complkte. Elles sont prCsenttes ici comme des hypothkses de travail qui peuvent &tre utilisCes comme guide lors de la collecte de donnCes additionelles et comme outil de design rationnel pour supporter le jugement et l'expkrience.

Mots cle's: torrents de dCbris, coulCes de dCbris, risques de talus, dynamique des glissements, mesures correctives, ingCniCrie.

Can. Geotech. J. 21, 663-677 (1984) [Traduit par la revue]

Introduction and interior regions of British Columbia. The area along A debris torrent can be described as the rapid the Squamish Highway north of Vancouver (British

movement of saturated soil, rocks, and other debris Columbia Highway 99) contains 26 mountain stream down a steep mountain channel (Swanston and Swanson basins each with over 1000 m of relief, which have 1976). ' Debris torrents commonly occur in the coastal produced at least 14 debris torrent events over the last 25

years. During the past 3 years alone, five major events '"~ebris flow" is a better-established international term to have occurred (Lister 1984), resulting in the loss of

describe the same phenomenon (Vames 1978). The word "mudflow" (Cuny 1966) has also been used in other regions. life of twelve persons and the destruction of or damage ~h~ term -debris is highly descriptive and well suited to nine bridges and six houses. The existence of debris to the particular character of the coarse-grained events of the torrent hazards along the highway ~-~cessitates operating Pacific Coast Mountains and has found strong regional expenditures for bridge maintenance, debris removal, acceptance in British Columbia and the Northwest United regular patrols, traffic delays, and preventative evacua- States. tion of residents. With continuing urbanization of many

CAN. GEOTECH. J. VOL. 21, 1984

P o i n t o f i n i t i a t i o n ( A l t i t u d e 1100 m , a . s. 1.)

S t a r t o f d e p o s i t i o n

B r i d g e demol i shed

End o f r u n o u t ( i n o c e a n )

FIG. 1. A coastal debris torrent basin after an event of a 20 000 m3 magnitude, which cost nine lives (M-Creek, Squamish Highway, October 1981).

of the debris fans, the future toll could undoubtedly be much higher.

In the fall of 1982, the British Columbia Department of Transportation and Highways authorized a study of the debris torrent hazards along the highway (Thurber Consultants Ltd., 1983). Subsequently, a range of remedial measures have been designed, including debris retention basins, channel improvements and diversions, torrent-proof bridge reconstructions, and deflecting dykes. Most are presently under construction. This substantial design effort required suitable design methods to obtain parameters for the safe and optimal sizing of the various structures. As there was no established methodology available in Canada to deal with the quantitative aspects of debris torrent behaviour, it was necessary to develop new techniques. The methods developed were based on a selection of the more practical and generally applicable concepts available in the specialized litera-

ture, especially that from Japan. These were calibrated against several debris torrent events from the vicinity of the project area.

The methods are semiempirical and have so far been applied only within southern coastal British Columbia (except as noted below). They are presented here as working concepts which, if suitably calibrated on a regional basis, could be used in any alpine region. A particular exception should be made with regard to debris flows with significant proportion of colloidal particles (clay or volcanic ash).

The nature of debris torrents in Western Canada Debris torrents begin in the steep upper reaches of

small drainage basins during periods of high runoff, collecting large quantities of loose material from the entire length of the channel and depositing on the surface of the debris cone (Figs. 1 and 2). Two or more branches

HUNGR ET AL. 665

FIG. 2. Damage due to the debris torrent shown in Fig. 1. The house was pushed off its foundation but not entirely destroyed, being close to the end of deposition where velocities were low.

of a given stream may be involved in a single event. Many debris torrent events occur in two or more surges, spaced over several hours. Individual surges have short durations measured only in minutes, and are commonly associated with abundant water flooding.

A typical surge through the lower reaches of a mountain creek begins by the rapid passage of a steep bouldery front, followed by the main body of the torrent. This consists of unsorted coarse particles ranging from gravel to boulders and logs, floating in a sluny of liquefied sand and finer material. Both the proportion of fines and the water content increase in the later stages of the surge, forming a liquid "after flow," which gradually merges into normal flood flow. Upon reaching flatter gradients or a less confining channel, the surge tends to decelerate and deposition can occur. The liquid after flow may break through the coarse deposits and continue further down the cone.

There appears to be a continuous spectrum of phenomena ranging from debris torrents to bed load. True debris torrent surges, as discussed herein, can be distinguished from mass bed load movements by the substantial rise of the debris wave above the flood flow water level at peak discharge. The mechanics of the latter would be influenced by water overflow and its associated tractive forces.

Owing to their relatively common occurrence, and potential for long reach and high velocities, debris torrents present a major hazard to developments and structures built in the vicinity of mountain creeks and on alluvial fans (Miles and Kellerhals 198 1). The danger may not be obvious, since the periodicity of debris torrents is irregular on any individual creek and long periods of dormancy often permit full reestablishment of forest cover over affected areas. Correct identification of streams subject to debris torrents is a major problem outside the scope of this presentation (see Thurber Consultants 1983).

The concept of a design event is analogous to that of a design flood in hydrology, although presently available data do not permit the evaluation of return periods (see below). We define the design event as the largest and most mobile debris torrent that could reasonably occur during the life of the structure under consideration. Those parameters of the design torrent required for the purposes of mitigation design are discussed below together with examples and requirements for further research.

Event magnitude or debris voIume Event magnitude is the total volume of coarse and fine

debris material transported to the fan in the course of a single event, irrespective of the number of surges. It serves not only to determine needed storage space in retention reservoirs but also as an overall index of the severity of the event, which can be used to scale other parameters. Reported magnitudes of debris flows range from several hundred m3 to 500 000 m3 (Hampel 1977). The largest reported Canadian case was 175 000 m3 (Jackson 1979).

Magnitude obviously depends on the size of the source area and the vulnerability of the source materials to be mobilized or rapidly eroded under flood flow conditions.

Some authors (e.g. Hampel 1977) base magnitude predictions on the volume of storm runoff. However, given the fact that most debris torrents are accompanied by profuse flooding and derive from highly saturated surficial soils, it would seem more logical to base the prediction on a balance of debris availability, rather than that of water.

In the first approximation, magnitude could be correlated with the size of the drainage basin. The ratio between the design torrent magnitude and the drainage area of a given creek, referred to as the "area debris yield rate" (m3/km2) should be a constant for drainage basins of different sizes but of similar topography, geology, and climatic and hydrologic conditions. Table 1 shows that, even within a narrow region such as the coastal British Columbia, the yield rate varies within a factor of six, while much greater variability applies in general. The yield rate approach would therefore appear to be suitable only for preliminary estimates. It may, however, be the best approach towards estimating the potential of certain basins that have been burnt by forest fire (L. Jackson, personal communication).

A different approach is to recognize that the direct source of the debris is in most cases the stream channel and its immediate vicinity. Magnitude should therefore be proportioned to the length of channel (including any major tributaries) upstream of the depositional area, to the point (points) of origin. The proportionality constant, referred to as the "channel debris yield rate" (m3/m) varies by about a factor of three, as shown in

666 CAN. GEOTECH. J. VOL. 21. 1984

TABLE 1. Area debris yield rate and channel debris yield rate for several past debris torrents of the British Columbia coast ranges

Drainage Mean gradient Area debris Channel debris Magnitude area of the channel* yield rate yield rate

Event (m3) Surges (km2) (deg) (m3/km2) (m3/m)

M-Creek, Squamish Highway, 1981 20 000 ?

Charles Creek, Squamish Highway, 1981 20 000 1

Alberta Creek, Squamish Highway, 1983 10 000 6

Wahleach A, Fraser Valley, 1983 55 000 ?

Port Alice, Vancouver Island, 1975 22 000 1 (Nasmith and Mercer 1979)

"From the point of initiation to the apex of the alluvial fan. tBased on combined length of two major branches of the debris torrent (3000

Table 1. It should be somewhat dependent on the drainage area, as discussed below.

The variation in the channel debris yield rate within a given region obviously results from differences in the topographic and geological character of various channels, in other words, channel erodibility. Ideally, one would wish to establish a classification of stream channels taking into account the gradient, type of bed material, height and steepness of side slopes if made of erodible material, and the general stability condition of the channel as reflected by vegetation. A few basic classes, characteristic of the coastal British Columbia streams, are suggested in Table 2, together with tentative estimates of the channel debris yield rate as back-calculated from events shown in Table 1.

To estimate the design debris torrent from parameters such as those given in Table 2, one needs to sum the combined lengths of channel in each category downstream of an estimated point of initiation, multiplied by the respective yield rate. Point sources of debris, such as individual landslides in creek banks, may add 10% or more to the total magnitude. Their potential may, however, be difficult to identify, in which case they should be accounted for by a conservatively selected yield rate.

The point of initiation would, in the case of the B.C. coast, be taken as the highest point of any suspect branch with a sufficiently steep gradient. The suggested gradient limits are 30" in the case of shallow channels with stable banks and 15" in the case of channels incised in unconsolidated materials where a debris slide from the bank could cause damming of the stream. All major branches should be assumed to be active simul- taneously.

It would be desirable to refine Table 2 by adding more categories, although this would require much more data

m) and without allowance for point sources.

concerning past events than is presently available. An alternative approach is to estimate the channel debris yield rate directly in the course of a detailed inspection of a stream channel (VanDine 1984). The volume of loose erodible material is inventoried by an engineering geologist, who can measure the surface area likely to be affected by debris torrent erosion but must guess at the likely depth of movement. This is perhaps the most reliable method available for channels with a bedrock base, although it is somewhat subjective. The estimate of erosion depth is very difficult in channels with loose base and banks.

A further refinement of the channel debris yield rate method is to account for the width of the channel. The debris volume produced from a length L of a channel with uniform erodibility and width B can be calculated:

[ I ] M = L B e l

where M is the magnitude in m3 and e' is the mean erosion depth in m. The width B can either be measured directly, or related to the square root of the drainage area Ai tributary to the stream sector under consideration (e.g. Kellerhals 1970). Dividing the stream into n sectors, we have

where Li is the sector length in m, ei is a "channel erodibility coefficient7' with the dimensions of m3/(m.km2), and Ai is measured in kmz.

Geometrical considerations indicate that the relation- ship between the channel debris yield rate and the erodibility coefficient for channels of uniform descrip- tion depends upon drainage area size and shape. It can be shown, for example, that the two constants would be

HUNGR ET AL. 667

TABLE 2. Tentative estimated channel debris yield rates for typical stream channels of the southern coast ranges

Channel debris Erodibility Channel Gradient yield ratet coefficient

tY Pe (deg) Bed material Side slopes Stability condition* (m3/m) (m3/(m.km))

A 20-35 Bedrock Nonerodible

B 10-20 Thin debris Nonerodible or loose soil (bedrock) over bedrock

C 10-20 Deep talus or Less than moraine 5 m high

D 10-20 Deep talus or Talus, over moraine 5 m high

E 10-20 Deep talus or Talus, over moraine 20 m high

*Prior to the expected debris torrent event. +For drainage areas of 1-3 km'. For other drainage areas use [2].

approximately equal in the case of an elongated rectangular drainage area of about 2 km2. Based on this, the tentative erodibility coefficients shown in Table 2 equal the corresponding yield rates. The introduction of channel width is important only where it is necessary to interpolate between greatly different drainage areas with similar erodibility characteristics.

It must be emphasized that all constants referred to above are applicable only on a narrow regional basis, as they incorporate climatic, geological, and even biologi- cal factors.

Frequency of occurrence Debris torrents in coastal B .C. are triggered primarily

by concentrated "cells" of high precipitation, occurring within relatively common storm events with apparent return periods of only 2-5 years, and combined with snowmelt (Miles and Kellerhals 198 1 ; Schaefer 1980; Evans and Lister 1984). Their frequencies therefore do not coincide directly with known precipitation patterns. On the other hand, insufficient historical data exists in Canada to permit direct statistical treatment. Debris torrent frequencies therefore cannot at present be predicted quantitatively.

Qualitative estimates can be made based on visible and geologic evidence (including dendrochronology) and recollections of residents. These should, however, be assessed carefully since dormancy periods of up to 200 years have been recorded in Europe (Hampel 1977). In this regard, there is an important distinction between streams with bedrock bed and those underlain by loose unconsolidated deposits. The former require gradual accumulation of detritus from one event to another, and are therefore temporarily stable after an event. The

Stable, practically 0-5 0-5 bare of soil cover

Stable 5-10 5-10

Stable 10-15 10-15

Side slopes at repose 15-30 15-30

Side slopes potentially Up to 200 Not applicable unstable (landslide (consider as area) point source)

latter, on the other hand, might be destabilized by an event and this can lead to a period of frequent activity, followed by a return to the dormant state. Charles Creek on the Squamish Highway, which flows through exten- sive talus deposits, for example, produced five major events in 15 years. M-Creek, in the same area, on the other hand, has a recurrence period in excess of 50 years (Skermer 1981).

Debris discharge The flow of a debris torrent has an unsteady, pulsating

character. Many authors (e.g. Morton and Campbell 1974; Pierson 1980; Takahashi 1981) who have observed debris torrents in motion describe each surge as consisting of a steep bouldery front, followed by a swell of finer debris, which gradually diminishes and becomes diluted and turbulent. The maximum discharge occurs shortly behind the front and may be of a relatively short duration. For example, eyewitnesses of the second surge of the November 1983 Charles Creek event described an initial wave of debris 4.5 m high, momentarily reaching an estimated discharge of 290 m3/s for approximately 5 s. A film of the subsequent flow shows a discharge of 120m3/s, continuing for 35 s. The full duration of the surge was estimated to be 2-3 min.

Another example is a film of a debris surge in the Kamikamihori Valley of Japan, obtained by Professor H. Suwa of Kyoto University. The film shows a front moving at a velocity of approximately 4 m/s through a check dam. The peak discharge of approximately 85 m3/s was achieved only 2 s after the front. Then, over the next 1 min, the discharge gradually decayed to about 10 m3/s in turbulent flow.

It is not known what determines the peak depth of a

668 CAN. GEOTECH. J. VOL. 21, 1984

@ @ E V E N T - see Table 3 @ @) @@ except :

9 - A l b e r t a C r . , 1983 0 1 1 111 10- Port A l i c e , 1973 1

0 I I I I

0 I

10 2 0 3 0 40 50

M A G N I T U D E (1000 m 3

M E T H O D OF E S T I M A T I N G DISCHARGE:

e y e w i t n e s s

superelevation ([13])

+ weir f o r m u l a

model test

0 veloci ty equation ([5])

flood discharge ( [ 3 ] )

FIG. 3. Correlation between debris torrent event magnitude and maximum discharge.

surge or its duration. It is likely that random factors such as the uarticular mechanism of initiation and accidental tempo;ary self-damming of the flow in the headwaters play an important role. The randomness of peak discharge is shown by a comparison between the 198 1 and 1983 events on Charles Creek, both of which were of similar magnitude. The first comprised numerous pulses with peak depths of less than 2 m and an estimated maximum discharge of 90m3/s. In contrast, the 1983 event consisted of two major and approximately four minor surges, with an observed maximum discharge of 290 m3/s.

The peak discharge of seven of the major events recorded in the Lower Mainland over the last 3 years has been determined and correlated in Fig. 3 with the event magnitude. The discharges were calculated from flow cross sections based on observed mud lines, combined with velocity estimates given by eyewitnesses, by superelevation analysis, and by the uniform laminar flow equation (see below). One direct discharge estimate is based on the broad-crested weir formula a ~ ~ l i e d

1 1

at a waterfall and another is a release discharge of a model sluny in a physical model test (Northwest Hydraulic Consultants Ltd. 1976).

An upper bound to this data is taken as an empirical correlation between the maximum discharge and event magnitude, even though it is acknowledged that a considerable scatter must exist below the line.

Another method of estimating discharge, suggested indirectly by both Hampel (1977) and Takahashi (1978), uses the criterion of water balance. A simple application can be based on the following assumptions: (a) The debris torrent surge forms by the dilution of itz situ saturated debris by mixing with flood water. (b) The surge peak is long, so that longitudinal transfer of water between the surge and the channel in front and behind can be neglected. Under these assumptions, the in situ porosity of the debris material, no, is increased by an amount equal to the ratio of water flood discharge to debris discharge, Qf/Q, to obtain the surge porosity, n. From this:

where Qf is the discharge of the associated flood. The results of this calculation are shown for three

selected creeks in Fig. 3, plotted against the estimated magnitude. The 50 year flood discharge and a porosity decrease of 15% were selected for these examples. This method strongly underestimates peak discharge. This is probably because the short debris surges, travelling at a higher velocity than the flood, incorporate significant water in the longitudinal direction, contrary to the above assumption (b).

A third method, which could be used in cases where there is little historical data, is to assume that the peak discharge results from a concentration of liquid debris behind a temporary dam somewhere in the channel, which subsequently bursts, producing a wave of debris. The discharge can be conservatively estimated using the "dam break" formula neglecting viscosity and assuming that the surge travels downstream without significant dissipation:

where g is the gravity acceleration and h and b are the height and width of the temporary reservoir. For example, the peak discharge of the 1983 Wahleach A2 debris torrent, estimated in Fig. 3 as 566 m3/s, would correspond to a dam 30 m wide and 7.6 m high. This is not unreasonable, considering that the torrent travelled in a deeply incised valley whose talus walls were subject to landsliding.

Flow velocity The best documented peak velocity observations from

the recent debris torrent events of coastal B.C. are summarized in Table 3. They are derived from eyewitness reports and superelevation data on banking in bends (see below). A plot of this data on a depth versus discharge per unit width diagram (Fig. 4) indicates a strong dependence of velocity on depth, which is typical for laminar flow. A comparison shown in Fig. 4

2 ~ l s o known as Ted Creek.

TABLE 3. Summary of velocity estimates

Flow cross section Suoerelevation Bed

Mean Mean slope Banking Curve Mean* depth width Area angle height radius velocity Discharge?

Event Location (m) (m) (m2) Shape (deg) (m) (m) (m/s) (m3/s) Comment

M-Creek, 198 1, end of gorge

M-Creek, 198 1, beneath hwy. bridge

Charles, 1981, end of gorge

Charles, 1983, upstream of bridge

Charles, 1983, below 1st fall

Charles, 1983, end of gorge

Wahleach A, 1983, end of gorge

Wahleach B, 1983 end of gorge

u

Rect.

Rect.

Trap.

Rect .

Rect .

Rect .

Rect.

Near base of fall, non- uniform flow

Discharge derived from 1

Eyewitnesses report low s velocity, no specifics c z

0 Minimum velocity from io

n four eyewitness -I

accounts > r

Discharge derived from 5

Assume channel erosion complete

Volcanic rock

*Velocity derived from the superelevation formula (eq. [13]), unless noted otherwise. ?Discharge calculated as mean velocity times maximum flow cross-sectional area.


0 0 0 0

6

5

slope angle = 8.2"

4 - E - I + a W 0

3 5 3 LL

I 3 I - X 4 2

2

1

t I I I I I I I I 1 I 1 I 1 2 3 4 5 10 2 0 3 0 50 100 200

U N I T D I S C H A R G E ( m 3 / ( s . m))

FIG. 4. Velocity /depth relationships applicable to the peak of a debris torrent surge. (Note: observation 4 adjusted by a shape factor to account for confined flow.)

indicates that turbulent flow, such as that of water, would have a strongly different trend. Velocity formulas such as Sribny's (Gol'din and Lubashevski 1966), which are derived from the Manning equation, would therefore appear inapplicable to the phenomenon under consideration.

There is a general consensus in the literature that the flow regime near the peak of a debris flow surge is, in fact, laminar (e.g. Pierson 1980; Takahashi 1981). Circumstantial evidence pointing to this conclusion includes the relative calmness of flow surfaces, the existence of boulder concentrations at the front, and the formation of levees, all of which imply a strong velocity gradient. Experimental criteria based on Reynolds and Hedstrom numbers also point in the same direction (Jeyapalan 1980).

Given the assumption of laminar flow, the mean velocity, v, further depends on flow resistance. Several

flow models have been suggested for the uniform flow of debris, all of which can be summarized by two equa- tions, the first being

where y is the unit weight of the debris, S is the slope, h is the flow depth, k is a cross-section shape ~oeff ic ient ,~ and v is the apparent dynamic viscosity of the debris.

Equation [5] relates to the Newtonian viscous flow model (Sharp and Nobles 1953; Cuny 1966), but can be extended to Bingham and pseudoplastic flows (Yano and Daido 1965; Johnson 1970) by replacing the dynamic viscosity with an apparent viscosity dependent on two material constants.

3k is equal to 3 for a wide rectangular channel, approximately 5 for a trapezoidal one, and 8 for a semicircular channel.

HUNGR ET AL. 67 1

The second equation describes a dilatant model based on the empirical work of Bagnold (1954) with nonplastic grain-fluid dispersions. For a wide channel:

where E, is a dimensional coefficient inversely propor- tional to the volume concentration of solid particles in the debris. This model was suggested for debris torrents by Bagnold (1 968), Breitfuss and Scheidegger (1 974), and Takahashi (1978, 1980).

The two criteria are rheologically quite similar, i.e., they do not represent very different velocity distribu- tions (Takahashi 1978). It is also possible that some flows move in the transition regime between the two models.

The maximum depth - velocity trends of both Canadian and Japanese4 (Okuda et al. 1980) observations plotted in Fig. 4 correspond well to [6] with a E, equal to 3.25 (m-'I2 s-I). Equation [5] with a viscosity of 3.0kPa-s provides a similarly good fit at flow depths greater than 2 m, although it appears to underestimate the velocities at smaller depths. Either equation can be used as a reasonably accurate basis for velocity predictions. The design of stream channels and debris barrier spillways in the Squamish Highway project was based on [5].

The empirical indication of Fig. 4, showing that data from different events and locations can be fitted by a flow equation with a unique viscosity value, is surpris- ing. One would expect viscosity to depend on the concentration of solid particles within the flow. The coefficient E, is indeed known to do so from Bagnold's (1954) work. The following section shows that, if the analysis is confined to that region of the moving surge which is near its point of maximum discharge (peak), the concept of a unique concentration and viscosity can be plausible. Fortunately, it is the peak velocity and flow depth that are critical for design purposes and are reflected in the mud-line observations of debris torrent channels.

Debris torrent surge profile Debris torrent material is characteristically well

graded (Pierson 1980; Hungr 198 1). Nevertheless, it could be regarded as a two-phase dispersion of solid grains "floating" in a heavy slurry of finer particles suspended by water turbulence. The coastal British Columbia torrents typically contain less than 5% of silt and clay-sized particles. An eyewitness of the November 1983 torrent on Charles Creek compared the appearance of this interstitial fluid to sandy mine tailings.

?he Japanese data represent 3 years of velocity observations at one station in the Kamikamihori Valley (at Dam 6a). The average bed slope is given as 8.2' at this location.

Bagnold (1954) used the principle of effective stress to show that the friction slope Sf of such a dispersion depends primarily on the volume concentration C of solid particles (those that are not suspended by turbulence):

(@ - P) c [71 Sf = ( u - p)C + p

tan a

where a is the dynamic friction angle of the particles, equal to approximately 30" for well-graded cohesionless material (Hungr 198 l ) , o is the rock density, and p is the bulk density of the slurry of fines and water.

Both the content of coarse grains and consequently Sf are known to be the largest in the bouldery surge front and to decrease with distance towards the tail. One could assume the velocity of the entire surge as uniform in accordance with the theory of uniformly progressive flow (Chow 1959). Then, placing the origin of a moving coordinate system at the front and measuring x in the upstream direction, the surface profile of the surge is given by the roll wave equation:

where h is flow depth and S bed slope. At the surge peak, where h is maximum, Sf must always equal S. There- fore, from [7], C at the surge peak is a function of bed slope only. Further, because the parameter E, of [6] depends mostly on C, the flow resistance (viscosity) of the peak must be uniquely related to bed slope. This is not to say that C changes in response to bed slope. Rather, the surge peak moves its position relative to the front, so as to find the location where the appropriate concentration exists.

Since velocity estimates are usually made in the vicinity of the fan apex, where only a limited range of slope angles occur, the viscosity of the peak flow can reasonably be assumed as unique and transferable from one event to another and also among different streams. An exception would perhaps arise in materials with substantial content of plastic fines.

Ahead of the peak, the friction slope exceeds the bed slope and, according to [6], the surge front must rise upstream. Behind the peak, the friction slope falls below the bed slope by virtue of the lower concentration, and a gradual decrease in flow height is observed. The relative lengths of the front, peak, and tail regions are determined by the overall gradation of the debris and the maturity of the longitudinal sorting. Nevertheless, the apparent viscosity of the surge peak is nearly constant and equal to approximately 3 x lo6 times that of water for the cases introduced so far.

Figure 5 shows a typical continuous record of a debris torrent surge obtained in Japan, which confirms the concept of constant velocity and varied discharge as described above. The presented viscosity values would,


FIG. 5. Surface velocity and discharge of a debris torrent surge in the Kamikamihori Valley, Japan, recorded by a spatial filter speedometer (Okuda et al. 1980).

in this case, apply only within the period between 8 and 15 s. The peak region would be approximately 35 m long. The concave tail profile of the surge may be due to the gradual onset of turbulence, which causes a renewed increase in the friction slope.

m m m - f low conf ined i n a channel

L f l ow depos i t ing on an 0 O open runout surface V) W 1

a 4 E V E N T - see Table 3

EO O m @ 0 + I I I I I

a SUGGESTED R A N G E 0 - FOR DESIGN ul

.- z u

Angle of deposition U N I T D I S C H A R G E (m3/ (s .m))

Field observations indicate that there is a certain limiting slope angle, below which a debris torrent surge loses its ability to travel at uniform velocity, decelerates, and begins to deposit the bulk of its material.

The conditions for the beginning of deposition would be given by [7], if one could assume uniform concentration. However, owing to the longitudinal decrease in concentration, [7] and [8] merely indicate that the surge peak gradually swells and moves back with falling bed slope angles.

The friction slope of the frontal boulder accumulation, which is relatively free of fines, appears to be around 16", since partial deposition of the frontal mass in the form of levees and boulder pockets usually begins to occur below this angle. The same angle is obtained from [7] on the assumption that the material is poorly graded and that the carrying fluid is water. However, provided that the flow is well confined within the channel, the more liquid material flowing behind can build up a sufficient depth gradient to continue pushing the frcnt forward. Whenever the confinement is inter- rupted, perhaps by a short wider channel section, a pocket of debris is deposited (Fig. 6). When the flow finally reaches a sufficiently wide channel, the surge peak collapses and the bulk of the coarse frontal material is rapidly discarded to the sides and full-scale deposition begins (Broscoe and Thomson 1969).

Figure 7 is a summary of the bed slope angles at the start of deposition for several events, correlated with the estimated discharge per unit width. All of the full-scale discharge events observed so far in the south coastal region began deposition near 10- 12", including torrents derived from both crystalline and volcanic rocks. In

FIG. 6. Bed slope at the onset of deposition, as a function of discharge.

FIG. 7. Lower channel of the November 1983 torrent on Charles Creek, Squamish Highway. The pocket of boulders in the right forefront was deposited in response to a momentary widening of channel at a subdivision road. The abutments of a destroyed railway bridge and tom rails can be observed in middle distance.

every case, the point of deposition also coincided with the exit from a confined channel. The angle of 10" appears significant to the start of deposition and, as will be shown later, also approximates in many cases the friction slope during subsequent decelerating movement. The same angle has independently been desig- nated as the deposition angle by an empirical study in Japan (Ikeya 198 1).

HUNGR ET AL. 67 3

Two low-discharge events on Charles Creek deposited on slopes of approximately 14". In one of these, the deposition occurred in a slightly widened channel at the foot of a waterfall, and in the second the front initially stopped at a low timber bridge. This raises the question whether channel confinement is more decisive or less decisive as a factor than the bed slope angle. If it is more, one could conceive high discharge events either passing through considerably flatter angles than 10" while confined, or depositing at steeper angles when allowed to expand. Such examples have not yet been documented in this area and the question has still to be resolved.

For the interim, it is suggested to assume a range of deposition angles between 10 and 14" for unconfined flows and 8 and 12" for channelled ones. For the purposes of design, one should select the conservative end of the range, i.e., upper when it is needed to keep the flow contained and moving within the channel, and lower when estimating runout and velocity.

The degree of confinement should, of course, be considered in relation to the discharge. The suggested criterion for sufficient confinement is a depth-to-width ratio of not less than 0.2. Thus, within a given channel cross section, low discharge surges will deposit at steeper angles than large ones. Trapezoidal channel cross sections with narrow bases should therefore be used where it is necessary to convey a wide range of debris discharges without deposition.

Runout distance Once deposition begins, gradual deceleration of the

surge front follows within a certain runout distance. Snow avalanche runout is calculated from energy or

momentum conservation principles, assuming that the friction slope of the avalanche during runout remains a constant, or a function of velocity only (Voellmy 1955; Salm 1966). A similar approach can be applied to debris torrents.

Takahashi and Yoshida (1979) formulated the momentum conservation equation for a steadily moving debris front travelling over a surface with a constant low inclination, after leaving a steeply angled channel. When simplified and combined with the concept of a constant friction slope, Sf (Appendix), the equation reduces to

where X, is the runout distance,

gh, cos 0, [lo] v = v,c0s(0,- 0) I [ l 11 G = g(Sf cos 0 - sin 0)

and 0 is the runout slope angle, 0, is the entry channel

slope angle, and v, and h, are the entry velocity and depth respectively.

Assuming that Sf equals tan 10" and that v, and h, can be obtained from the design discharge by means of [5] or [6], the runout distance can now be calculated. Figure 8 shows very good agreement with field obsewa- tions. This is important, because the application of the runout equation incorporates all of the techniques developed above: discharge, velocity, as well as deposition angle.

The velocity of the decelerating front can also be predicted from the same theory (Appendix):

where x is the distance from the entry. The theory describes the deposition of a single debris train, at peak discharge. Experimental (Takahashi and Yoshida 1979) and field observations both show that successive debris trains deposit beside each other in a fanlike manner.

Superelevation and run-up Superelevation of debris torrent flow in bends of the

channel is important for two reasons. Firstly, it needs to be determined for design, to provide sufficient freeboard on the outside of bends and to design deflector dykes. Secondly, differential height of mud lines in curving channels can be used to estimate the velocity of past events (Fig. 9).

In analogy with snow avalanches (Mears 1981), superelevation of debris torrents can be calculated by the forced vortex equation:

where Ah is the elevation difference between the two sides of the flow, b is the surface width of the flow, v the mean velocity, r the mean curvature radius, and k a correction coefficient defined below.

The forced vortex equation is more appropriate to debris torrents than the free vortex equation used to calculate superelevation for water, owing to the high viscosity and vertical sorting, which restrict lateral displacement within the flow.

Mizuyama and Uehara (1981) and Ikeya and Uehara (1982) showed by experiments and detailed field observations that the actual superelevation may be 2.5-5 times larger than that predicted by [13], possibly because of the same causes as cited in the previous paragraph. To achieve conservative predictions, it is recommended to use k = 5 to calculate superelevation for design and k = 2.5 to calculate velocity from superelevation data. The latter method was used to calculate some of the velocities shown in Table 3.

Run-up against barriers placed in the path of a debris torrent can be estimated from [9], with the angle 0 now

674 CAN. GEOTECH. J. VOL. 21. 1984

Port A l i ce . 1973 (Nasmi th and Mercer 19791

i 1

( incomple te runout) ,,/

/" Refer t o Table 3

for event numbers

0 200 400 600

OBSERVED R U N O U T ( m l

FIG. 8. Comparison of runout distances calculated from [9] and observed.

FIG. 9. Superelevation in a double channel bend of the Wahleach B torrent, near Hope, B.C. The flow reached up to the eye level of the person standing on the left bank, but was approximately 3 m lower on the opposite side. The estimated velocity and discharge based on these observations are 7.9 m/s and 284 m3/s respectively.

being negative and equal to the run-up slope angle. When different values are tried, the angle 15" always gives the maximum vertical run-up distance. For the purposes of design, one should therefore assume that a debris surge could deposit a wedge of material in front of the barrier, upon which the subsequent flow could climb to the maximum height. Run-up design could, under such conditions, use Fig. 10, which is based on [9].

For purposes of comparison, Fig. 10a also indicates the corresponding velocity head v,2/2g. In most cases,

based on the c r i t i c a l run-up angle of 15'

o 1 I I I

1 0° 1 5 O 20° APPROACH SLOPE ANGLE ( 0 )

c r ~ t ~ c a l run-up surface

FIG. 10. Maximum run-up prediction by means of [9]

the predicted run-up is substantially greater than the velocity head, because of the thrust of material travelling behind the front.

Dynamic thrust and impact In the case of barriers that are nearly perpendicular to

the flow direction, the most significant thrust is that of the debris front in which there is little lateral velocity variation. The front stops at the barrier after the initial impact and forms an inert mass, which will, to some extent, shield the structure from subsequent loading. Dynamic thrust loading should therefore be calculated using the momentum equation, taking the entire peak of the surge as a single flow prism travelling with a uniform velocity equal to the mean velocity.

The momentum equation is

[I41 F = p ~ v ~ s i n p

where F is the total thrust force, A is the flow cross- section area, p the debris density, v the mean velocity, and p the least angle between the face of the barrier and the flow direction.

It is recommended that the above load should be distributed over an area as wide as the expected debris

HUNGR

FIG. 11. Concrete bridge beam demolished by point impact during the November 1983 debris torrent on Charles Creek, Squamish Highway. Notice torn prestressing cables suspended from mid-span.

train, but about 1.5 times greater in height, to account for the forming of a stagnant wedge in front of the toe of the bamer.

Point impact loads due to boulders carried in the flow may be more important for the design of certain structural elements. For example, Fig. 11 shows pre- stressed bridge beams destroyed by the November 1983 torrent on Charles Creek. The same surge tore a con- tinuously welded standard rail stretched freely across the creek after the railway bridge deck was washed out by a previous surge.

The magnitude of point impact depends on the momentum of the largest particle involved on contact deformation and on the deflection of the impacted structure.

With regard to boulder size, there is practically no limit, given the presence of glacial deposits in most regions of Canada. The fast-moving boulder size is therefore limited only by the capacity of the design torrent to carry it. It is suggested that the design boulder be assumed a sphere with a diameter equal to the flow depth.

Contact deformation is important only in the case of inflexible structures such as abutments and heavy bridge decks. The impact force can be estimated using the Hertz equation (Timoshenko and Goodier 1969). How- ever, the elastic contact forces thus estimated are extremely conservative when the momentum of the boulder exceeds a few hundred kg.m/s, and should be divided by a factor of about 10 to account for contact crushing.

For flexible elements such as beams, structural deflection is more important than contact deformation. Equating the kinetic energy of the boulder with work expanded in bending deflection, one obtains

where P is the impact force in kN, M is the boulder mass in Mg, v is the velocity in m/s, and K is a stiffness factor.

For a simple beam:

where EI is the bending stiffness and L the length. This indicates that flexible structural members are more efficient in resisting impact than stiff ones. For example, a steel beam is more efficient than a concrete beam with the same bending capacity.

Application The quantitative approaches described in this paper

have been used extensively by the authors to facilitate solution of many design problems related to debris torrents. Some applications are provided below to illu- strate procedures.

To determine the area subject to direct overrun by debris, it is necessary to estimate the design torrent magnitude by [I] or [2], discharge from Fig. 3 or [4], and runout from [9]. The rules for recognition of the start of deposition are summarized in Fig. 6. The end of deposition is defined by extending the runout distance past the point where the slope angle equals 10". Areas beyond the end of deposition are likely to be affected only by flooding. Equation [12] can be used to outline the high velocity hazard region on the depositional area.

To design a bridge clearance, the magnitude and discharge must be determined as above. Velocity and flow depth can be obtained from Fig. 4, making adjustments for channel shape if necessary. It is recommended that a generous clearance above the estimated " flow surface be provided to allow for rapidly moving projecting particles, such as logs. A safety clearance of 3 m is being used on the Squamish Highway. The bridge entry should have a prismatic, straight channel as far as possible, with a gradient exceeding the estimated deposition angle for a range of discharge values.

To design a deflection wall or embankment, the magnitude, discharge, depth, and velocity of the design torrent is required as above. The banking height can be estimated by [13] and the dynamic thrust by [15].

Conclusions This paper details a first attempt at quantitative

prediction of debris torrent behaviour in Canada. Design of debris torrent remedial measures in other countries, especially in Europe, relies heavily on site-specific experience of long duration, sometimes of several centuries (Eisbacher 1982). Such experience is not available in Canada. The authors have therefore at-


tempted to combine theoretical understanding of the phenomenon with detailed observations, to obtain a semiempirical design method.

It is emphasized that these procedures require more thorough calibration to be generally applicable. Their use in design may supplement, but should by no means replace, the use of experience and informed judgement! The importance of collecting quantitative data such as debris volumes, flow cross sections, and velocities from as many debris torrent events a s possible must be stressed. With these provisions, it is believed that the methodology represents a rational approach to design and can constitute a useful tool in the development of facilities to cope with debris torrent hazards.

Acknowledgements The authors are indebted to the British Columbia

Ministry of Transportation and Highways and in particular those personnel who created a stimulating and productive contract atmosphere under which much of this work was carried out. Thanks are also due Mr. D. F. VanDine of Thurber Consultants, who provided the authors with recent research articles obtained during his recent visit to Japan, and who reviewed the manuscript.

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Appendix. The runout equation of Takahashi and Yoshida (1979)

+ {(a - p)C + p)h ,v~cos (0, - 0)

+ .tgh; cos 0, cos (0, - 0) { (a - np)Ck + np)

- & ( a - n p ) g C h cos 0 tan ci

where h is the mean depth of the surge front; x is the distance travelled by the front from its entry to the runout surface; a , p, and C are the grain density, interstitial fluid density, and grain concentration of the debris (assumed constant); n is a coefficient to account for the gradual settling of fines in the fluid, which can be assumed equal to 1.0 for well-graded soils (Takahashi and Yoshida 1979); k is a coefficient of effective lateral earth pressure within the debris; and ci is the dynamic friction angle of the solid grains.

The first term represents the rate of change of momentum in the front. The second is the gravity driving force, the third is the momentum flux from the upstream channel, the fourth is the static lateral thrust force transferred from the fluid still in the channel to the front, and the fifth is the frictional resisting force at the base.

T o this must be added the equation of continuity, which equates the volume of the runout debris train with the volume discharged from the channel:

where t is time. Substituting from [A21 to [ A l l , assuming k = 1.0 as

appropriate for a fluid-like dispersion, rearranging and substituting from [7] for the terms involving density and concentration, one obtains

d gh, cos 0, [A31 - (vt) = V, cos (0, - 0)

d t

- t(Sfg cos 0 - g sin 0) = V - G t

This can be solved with boundary conditions at the entry and end of runout to obtain [9]-[I I].

If a similar solution was made neglecting the third and fourth terms of [A!.], one would recover the runout equation of Voellmy (1955) with zero turbulence:

Assumptions v,' 2 - vu After leaving from a steep channel with bed angle 0,, [A41 XL = - -

g(Sf cos 0 - sin 0) G velocity v, and flow depth h,, the flow front travels at a uniformly decreasing velocity, v, over the runout This simplified equation tends to underestimate run- surface, inclined at 0. Both v, and h, are conservatively out by a factor of approximately two, but would be assumed constant with time. Momentum conservation appropriate for surges of very short duration, where the gives momentum influx is terminated early.

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