- Email: [email protected]
Modelling stability and surging in accumulation slides
Engineering Geology, 33 (1992) I-9 Elsevier Science Publishers B.V., Amsterdam
1
Modelling stability and surging in accumulation slides R.O. Davis Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand (Received February 4, 1992; revised version accepted June 3, 1991)
ABSTRACT Davis, R.O., 1992. Modelling stability and surging in accumulation slides. Eng. Geol., 33: 1-9. An idealized model for certain classes of debris flows and avalanches is developed. Slopes characterized by a steep feeder segment which loads a flatter accumulation segment will be focused on. Such slopes have been observed to exhibit episodic surging motions. The simplified model allows considerable insight into the behavior of such slides. An approximate stability criterion is derived explaining the surging motions. It is shown that, in general, one cannot predict the onset of surging due to the inability to precisely characterize the interaction between the feeder and accumulation slopes. Other results include an approximate analysis for the mean acceleration which may occur during surging.
Introduction This article considers the stability of a class of mudslides and other mass movements which share a particular characteristic longitudinal profile. The characteristic profile has a steep feeder slope lying immediately above a flatter accumulation slope. Slow movement of the feeder slope materials leads to increasing loading of the flatter accumulation slope materials. If sufficient loading occurs, the accumulation slope may fail rapidly, resulting in a surge, which may involve large velocities and displacements. Mudslides, avalanches, and other mass movements which may be identified by this characteristic shape and behaviour will be referred to here as accumulation slides. There are numerous occurrences of accumulation slides, many of which have been studied in the literature. Several examples are noted in articles by Hutchinson (1970), Hutchinson and Bhandari (1971), Hutchinson et al. (1974), and Prior and Stephens (1972). Instability and surging in the fiat portion of these slides is attributed to undrained loading of the flat segment caused by creeping Correspondence to: R.O. Davis, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. 0013-7952/92/$05.00
motions in the feeder slide. Surges have been observed (see Hutchinson et al., 1974) with velocities estimated to be on the order of 10 m/min involving volumes of hundreds of cubic metres. In this paper a simplified analysis of an accumulation slide will be presented. Both the feeder and accumulation slopes will be represented by infinite slope stability models. The effect of undrained loading of the flatter slope due to motions of the feeder slope is modelled by connecting the two with a simple spring-dashpot combination. Stability criteria for overall behaviour of the model are derived using an effective stress analysis. Motions of the feeder slope are then analysed, assuming the accumulation slope remains stationary. Finally, surging motions of the accumulation slope may occur if the combination of feeder slope loading and increasing piezometric level are sufficiently large. The model shows that these surges will be characterized by high velocities. The model also indicates that the onset of any particular surge is nearly impossible to predict. This agrees with the observations of Hutchinson et al. (1974).
Model formulation Figure 1 (top) illustrates the profile of a typical accumulation slide. Both the upper feeder slope
© 1992 - - Elsevier Science Publishers B.V. All rights reserved.
R.O. DAVIS
2
Fig. 1. (top) Typical accumulation slide profile;(bottom) idealized model for accumulation slide. and the lower accumulation slope rest on a continuous slip surface which is assumed to lie parallel to the upper surface. Subscript 1 refers to the feeder slope, 2 to the accumulation slope. The piezometric surface is assumed to lie parallel to the failure surface, a distance hw, above the failure surface. Horizontal dimensions of both the upper and lower slopes are assumed to be sufficiently great so that side and end effects may be ignored and infinite slope conditions apply. Figure 1 (bottom) shows an idealized model of the slope in Fig. 1 (top). Rigid blocks representing both the feeder and accumulation slides are resting on planar surfaces which are parallel to the slope profile in Fig. 1 (top). The block masses ml are equal to the masses of material contained within the two slides, and the blocks may move with velocities vi. Interaction between the feeder and accumulation slides in Fig. 1 (top) is idealized by a spring and dashpot connecting the two blocks in Fig. 1 (bottom). Clearly Fig. 1 (bottom) oversimplifies the physical problem shown in Fig. 1 (top), particularly with regard to the interaction between the two sliding masses. The principal effect of the feeder slope on the accumulation slope is thought to be undrained loading of the head of the accumulation slope due to creeping motion in the feeder slope (Hutchinson and Bhandari, 1971). This process
would be physically far more complex than the simple elastic and viscous forces connecting the two blocks in Fig. 1 (bottom). Nevertheless, the model in Fig. 1 (bottom) has the virtue of simplicity and clearly allows motions of the feeder slide to affect the stability of the accumulation slide. Moreover, as we will show below, the model in Fig. 1 (bottom) is sufficient to represent all the main response features commonly observed in accumulation slides. A second consequence of the model illustrated in Fig. 1 (bottom) is that the masses ml and m 2 are invariant. In reality the feeder slope mass m 1 is affected both by addition of material from upslope and by transfer of material to the accumulation slope. The accumulation slope mass m 2 is similarly changing. For reasons of simplicity, we will not attempt to model these mass flux problems here. Motions of the two-block system shown in Fig. 1 (bottom) are described by these three ordinary differential equations: ml I"1 ~--- W I sin/~ - $1 - F
(la)
sin
(lb)
m 2 I)2 = W 2 F=
f12 -- $ 2 ~- F
k(l~ l - v2) + c(131 -
i)2)
(lc)
where the superposed dot denotes differentiation with respect to time, and: Wi = ml; g = block weight; g = acceleration of gravity; Si=sliding resistance on failure surface; F = combined elastic and viscous forces; k =spring constant; c = d a s h p o t constant; and/~i = slope angle. The sliding resistance terms $1 and $2 must be provided by constitutive equations for the strength of the materials in the feeder and accumulation slopes. Presumably residual strength conditions will apply or will be quickly achieved following the initiation of motion, and the cohesion component of strength will be small or negligible when compared to frictional strength. A conventional effective stress model for frictional strength will be •used with: Si = Wi(1 - ~i) cos f l i f ( v i )
(2)
Here, ~i represents the effect of piezometric elevation on the effective stress which acts on the failure
MODELLING STABILITY AND SURGING IN ACCUMULATION SLIDES
use Eq. 2 to find:
surface:
7whw, ~i--
3
(3)
sin/3 i - (1 - ~l) cos/31f(0) =
7h i
where 7 and Yw are the unit weights of the slide material and the pore water, respectively, and hwi and hi are as shown in Fig. ! (top). The function f(vi) will be taken as: ( = t a n 4, f(vi) l ~
for vi > 0 forvi=0
(4)
where 4 represents the effective stress friction angle appropriate to the slide material. Here f represents the mobilized strength on the failure surface. If vi=0, only part of the strength may need to be mobilized to hold the block in static equilibrium. This condition is expressed by the less than or equal sign in the second part of Eq. 4. If the block is in motion, the strength is fully mobilized and f is then equal to tan 4. Note that the product Wi(1-~i) is equal to the buoyant weight of the slide material. Thus, Eq. 2 is exactly the usual strength term found in conventional infinite slope analyses, based on effective stresses. The principal difference between this analysis and a conventional infinite slope analysis is the presence here of two sliding masses. Also, it should be noted that, whereas Fig. 1 (bottom) would make it appear that the upper block may move onto the lower sliding surface, given that sufficient displacement occurs; this in fact will not happen in the context of Eqs. (la,b,c). The motion of the upper block is fully controlled by Eq. l a, and it is clear that the block will always be driven by the gravity force Wlsin/31. In reality, the displacement of the feeder slide will be small unless a surge occurs in the accumulation slide. The focus of the model is to examine the onset of the surge, not to attempt to accurately describe the slope geometry after the surge has taken place. Static stability There are well defined limits for static behaviour of the two-block model shown in Fig. 1 (bottom). Setting time derivatives equal to zero in Eqs. l a and l b, we can add the resulting equations and
_
m2 [sin/32 - (1 - ~2) cos/32f(0)]
ml
(5)
Here, f(0) is the mobilized strength on the failure surface. Since the velocities Vl and v2 are both zero, f(0) is not precisely defined, as shown in Eq. 4. Note that the force F does not enter Eq. 5. The limiting value off(0) for which static behaviour is possible is tan 4. If we set f(0) equal to tan 4, Eq. 5 becomes: sin/31 -
- - COS
m2
-
ml
fll tan
41 =
[sin/32 - cos/32 tan 42]
(6)
where the angles 41 and 42 are defined by: tan 4i
=
(1 - ~i) tan 4
(7)
Next, use the identity sin ( x - y) = sin x cos y - sin y cos x to rewrite Eq. 6 as: sin (/31 - 41) = ~o sin (42 - 132)
(8)
where: Lf -
m 2 cos 41
(9)
m 1 cos 42
Equation 8 represents limiting conditions for static equilibrium. It may be solved for any one of the parameters to give the critical value of that parameter in terms of the other parameters. For example, if we solve Eq. 8 for /31 we have: /3~c = 4~ + sin- ~ [Se sin (42 -/32)]
(10)
and this value of/31 represents the steepest feeder slope angle for which the slide may remain static, for given values of/32, 41, 42, and L~°. It should be noted that since 41 and 42 both depend upon water table depth, the values of these two parameters may be changing. Nevertheless, for given values of 4~ and 41, Eq. 10 gives the steepest possible value of/31 for which the feeder slope may remain static. The weakest possible condition for the slope occurs when the water table coincides with the upper surface. In that circumstance, ~1 = 0~2 ~ 0.50. The approximate equality
4
R.O. DAVIS
here follows from the observation that the pore fluid will usually be about half as dense as the slide material. Figure 2 shows graphs of Eq. 10 for a particular value of ~b1 = q~2, and several values of the mass ratio mz/ml. In the figure, /31c is graphed versus /32, and q~l and (~2 a r e both taken as I0 °. Equality of q~l and ~b2 implies cq=~2 which, from Eq. 3, implies the ratio of hwi to hi is the same in both feeder and accumulation slopes. Note that, if/3z approaches the value of ~bl = q~2, then/31 must do likewise, regardless of the value of mz/m 1. Note also the near linear nature of the graphs shown. This reflects the fact that the arguments of the sine function in Eq. 8 are never large, and sin x ~ x for small x. In fact, Eq. 8 may be well approximated by: /31 -- (~1 = ~'(~(~b2 -/32)
(l l)
so long as /31-q51 and 4)2-/32 are both relatively small. Finally, note how important the mass ratio m2/ m 1 is in determining the critical value of/31. For small values of mz/m a, ~ never rises far above the value of q~l =q~2 = 10°. Even in the extreme case where f12--*O°, the critical value of/31 is only 12° for m2/m~ =0.2. On the other hand, when m2/ m~ is large, /3a¢ takes values far greater than 10°. This result is physically intuitive. When the mass
of material in the accumulation slide is much larger than that in the feeder slide, the stability of the system is enhanced. The opposite is true in the reverse situation.
Analysis of motion In this section we will consider typical displacements of accumulation slides, such as those illustrated schematically in Fig. 3. The figure illustrates the displacement history of both feeder and accumulation slopes in response to raising piezometric levels. The upper part of the figure shows the increasing piezometric elevation, perhaps in response to rainfall. The lower part of the figure shows the displacement time histories of both the upper feeder slide (shown as the dashed line) and the lower accumulation slide (solid line). At some particular time, denoted to, the rising water level triggers instability in the feeder slide. It begins to move downslope at more or less constant velocity, while the accumulation slide remains stationary. Displacement of the feeder slide results in increased load on the accumulation slide and, at some later time, denoted ts, the combination of load from the feeder slide and rising piezometric level leads to slipping of the accumulation slide. Both slides now begin to move with rapidly increasing velocity and a surge results. We will model this behaviour, first considering the time period to to ts in which the feeder slide is
~1c2520 ~ ~
slFeeder ope~ //"~//.//I ] Accumulofion /~ i ~ /~slope
i~ Q'~
~S
100
I
".~o
15 -
5 10 ~2[degreesJ I
Fig. 2. Graph of Eq. 10 with O t = 0 2 = 10°.
fo
T~ine
fs
Fig. 3. Typical accumulation slide behaviour.
MODELLING STABILITYAND SURGING IN ACCUMULATIONSLIDES
creeping, and then times following ts where both slides are in motion.
Wi sin fli - Wi[ 1 - cq(to)] cos fli tan 4) - F o = 0
Lower block stationary
It is important to note here that to can be determined from Eq. 15 only if Fo is known. For times greater than to, so long as the lower block remains stationary, the motion of the upper block will be governed by Eq. 13, which has the solution:
First consider conditions in which the lower block in Fig. 1 (bottom) is stationary. Setting v2 = 0, Eqs. la,b,c become:
(15)
ml vl =
1
W1 sin f l l - Wl(l - ~1) cos flltan qS- F
(12a)
v 1 = W(t - to) + ~12g cos fll tan q5 :i I
F= kVl + cfl
(12b)
Here ud(t-to) is a transient term given by:
We can combine these equations to have:
~1 q - 2 2 1 O l l ) 1
q- O)2Vl = g c o s flltan ~b ~1
W(t - to) ----e - ' h ' ° l ( t - ' ° ) { C 1 (13)
where: e) 1 =
(14a) C
21 - 2 kx/-~l
(14b)
represent the natural frequency and fraction of critical damping associated with the upper block. Note the presence of &~ on the right-hand side of Eq. 13. This appears since the piezometric surface within the slope may be changing. For static water table conditions, the right-hand side of Eq. 13 would be zero, and the resulting equation would represent free vibration of a damped harmonic oscillator with natural frequency ~1. In fact, the block would not oscillate, since any upslope motion would induce a change of direction of the frictional force, immediately bringing the block to rest. Thus extended motions of the upper block can only occur in the presence of a rising water table. Now suppose the upper block begins to move at time to. At this instant, the right-hand side of Eq. 12a is exactly equal to zero. The value of F will be Fo, the force acting between the two blocks before motion commences. Prior to the first motion of the block, the rising piezometric level causes cq to increase. The right-hand side of Eq. 12a will be negative, but at time to, increasing c¢1 brings the value to zero. That is, at t = to:
(16)
sin c o l J l -- 22(t - to)+
C2cos col x/1 - 2~(t - to)}
(17)
Constants C1 and C2 are determined by requiring both vl and ~1 to be zero at t = to. The transient term in Eq. 16 is of much less interest than the second term, which represents the mean velocity of the feeder slide. Let Vlm denote the mean velocity: 1 ~21m = ( D ~ g c O s /~1
tan ~b ~1
(18)
We find that the mean velocity is directly proportional to the rate of increase of water level within the feeder slope. Equation 18 can be used to estimate the feeder slope velocity, but it is first necessary to estimate the value of the natural frequency, col. One way to estimate 0) 1 is as follows. Let a denote the average cross-sectional area of the feeder slide, and let L denote its length. Then the spring constant k may be represented by:
Ea
k = ~-
(19)
where E denotes Young's modulus for the feeder slide material. Using Eq. 19 in Eq. 14a, and noting that ml must equal paL where p denotes mass density, we find: C ~°1 - L
(20)
where C; is the so-called bar velocity x / ~ , the velocity of elastic waves in the feeder slope ma-
6
R.O. DAVIS
terial. Typically, for unconsolidated materials found in mudslides and avalanches, C; will be on the order of 100 m/s. If the length of the feeder slide is on the order of 100 m, then col will be roughly 1.0 s - 1. Returning now to Eq. 18, consider a feeder slope with //~=18 °, and suppose 05=15 ° . Then, with col= 1 s -1, we have: v l m = l x9.81xcos18xtan15xoi
Eq. 21a and 21b to have:
P "" l'l
ml
# ,
v'2=
+ -
(22)
m2
Then differentiating Eq. 21c, and making use of Eqs. 21a, 21b and 22, we find: F + 22-05F + 052F= B
(23)
Here:
1
B = kg{sin//1 - sin//2 - [cos//1(1 - cq)
= 2.5oi 1
-cos/32(1 - c~2)] tan 05} The maximum value of cq is roughly 0.5 (when the piezometric surface coincides with the upper surface), and the rate of change might typically be on the order of 0.1 to 1.0 day 1. Thus the range of feeder slope velocities would be 0.2 to 2 m/day. These values are in good agreement with measured values (Hutchinson et al., 1974).
05 =
Next suppose the combination of rising water level and upslope loading are sufficient to cause the lower block to begin to move. The time at which this occurs is denoted t s. All three of Eqs. 1 now apply. We will consider the special case in which both ~1 and ~2 remain constant for t > t s . That is, the piezometric levels in both slopes are assumed to abruptly become stationary at t =tS. The reason for making this assumption is to simplify the analysis. We will find this assumption has little effect since the motions following t~ will be extremely rapid and the time interval over which Eqs. 1 apply will be quite short. We begin by rewriting Eqs. la,b,c. m1171 = Wlsin//1 - Wlcos fll(l - ~ l ) t a n 05- F (21a) m 2172 =
W 2 sin//2
-
W2 cos//2(1
-
(24b) C
2- 2,,~
(24c)
and: 1
L o w e r block sliding
(24a)
-
1
ml
+
1
(24d)
m2
The solution to Eq. 23 consists of a transient and a constant part: 1
F = ~ ( t - ts) + ~ B
(25)
CO-
The transient term ~ is similar to ~ in Eq. 17 and has little interest here. The constant term shows that the force F will have a mean value equal to B/d) 2 for times greater than ts. We can now use Eq. 25 in 21a and 21b to find the acceleration of either block. After some manipulation we have: 1
171=A---~(t-ts), ml
1
172=A+--~(t-ts)
(26)
m2
where: A = g{ml[sin//1 - cos/31(1 - ~1) tan 05]+ m
~2) tan 05 + F m2[sin/32 - cos f12(1 - ~2) tan 05]}
(27)
(21b) F=k(vl
- v2) + c(~1 - v2)
(21c)
Taking ~l and c~2 to be constant, we differentiate
We see from Eq. 26 that both blocks have the same mean acceleration, which is given by Eq. 27. The transient accelerations have a similar form but are exactly out of phase.
7
MODELLING STABILITY AND S U R G I N G IN ACCUMULATION SLIDES
Note that the mean acceleration depends only upon the slope geometry, the friction angle qS, and the mass ratio, mz/m~. If we take the following typical values: /~1 =
18°,
q5=15 ° ,
fi2 = 6°,
m2/ml = 3
~=~2=0.5
Then we find A=0.024 g. This value is in the range expected for accumulation slides. For a mean acceleration of 0.02 g, the slide velocity one minute after ts would be about 12 m/s. The displacement after one minute would be about 350 m. If this displacement is larger than the length of the accumulation slide, as seems likely, we would expect the entire surge to begin and end in less than one minute. Discussion
We can use the two-block model proposed here to gain insight into the mechanics of accumulation slides. First, the stability criterion developed in Eq. 8 can now be viewed in a slightly different light. In deriving Eq. 8, we assumed the strength on both upper and lower failure surfaces was fully mobilized. In fact, it is now clear that the feeder slide will slip first, and hence its strength will be fully mobilized before that of the accumulation slide. Strength on the accumulation slide failure surface becomes fully mobilized at the instant t~ when surging commences. This might suggest that Eq. 8 is invalid since it was assumed in Eq. 5 that both slides were stationary. In fact, Eq. 8 is still a good approximate criterion for surging stability since the acceleration of the feeder slide is small during the time interval between to and ts. Thus Eq. 5 is approximately correct, and Eq. 8 can be used to assess whether or not surging is possible for any slide geometry. It is useful to note once again the importance of the mass ratio mz/m ~. It plays a central role in determining whether or not surging may occur, and also figures strongly in Eq. 27, where the mean acceleration of the surge is determined. Typically, in Eq. 27, the term multiplied by m2 will be negative since it represents the safety margin of the lower block without the force F. The term multiplied by ml will of course be positive. It
becomes clear then that greater values of m2/ml will result in smaller values for A, and hence less energetic surges. It is also interesting to consider possible histories of creeping motions in the feeder slide. Continuous downslope creep is possible only in the presence of increasing piezometric levels. If the piezometric level within the slope becomes stationary or begins to fall, the slide will quickly come to rest. One can easily envision circumstances in which the feeder slope may begin to creep, but, because of cessation of rain, will again come to rest. The result of such an excursion is an increase in the force F. There will subsequently be some relaxation as the viscous part of F dissipates, but there will remain a net increase. This is much like what one would expect in the actual accumulation slide. Undrained loading of the accumulation slide due to motion of the feeder slide may also dissipate somewhat as drainage and partial consolidation take place in the soft materials, but an overall increase in load will result. This of course makes a surge more likely at some future time. It is precisely this mechanism of racheting increases in F which results in almost total unpredictability of when a surge may occur. The difficulty of predicting surging was commented on by Hutchinson et al. (1974) in relation to a particular accumulation slide which occasionally endangered human life. In our model, it is impossible to predict the value of ts, even if piezometric levels in both slopes were continually monitored. The reason for this is that the initial force F o at the onset of motion in the feeder slide is, in general, unknown. To illustrate the unpredictability of ts, we can consider two simulations using the two-block model, each with identical properties but with different initial values of F o. Results are illustrated in Fig. 4, where time histories of block velocities are shown in dimensionless form. In the figure, velocity has been normalized by any convenient characteristic velocity VR, while time and force are made dimensionless as follows:
T= t
'
if-
F VR~
8
R.O. DAVIS 008 [ (b)
OO8
LJ c3
H u
007 006
[L
.005 0"3 .004 co LI 003 Z o 002
OO6 005 t /
o041 l ~J
o"3
z 001 LJ 0 F3 0
OO7
/y"-",,,.. . /. 2
003 f 002 [
" ,-',
"-... _...- --.. _--} c~
8 ]0 12 ]4 ]6 ]8 '20 '22' O I NU',4S I ONLESS - I ME (c)
;i
.
_
0
2
4
6 8 10 12 14 16 ~18 '20~ ' -~22 OlNENSIONLESS T[NF
$
~
,3:6 O; 4
/~ rrJ tJJ
3;2 J]C' 8O8
OOB /,~ 004 L • ~ 002 ! (1 o , , , o 2 U)
4
6
8
1o
1~
DIMENSiO"4LESS
14
1 6 ~ i ~ ' / ~
~
TINE
Fig. 4. Two simulations of accumulation slide response• (a) Velocity time history of feeder slide (dashed line) and accumulation slide (solid line) for ~ o = 0.008. (b) Similar to (a) but with .~-o = 0.004. (c) Time histories of . ~ for both conditions.
In Fig. 4(a), velocities of the feeder slide (dashed line) and the accumulation slide (solid line) are shown for an initial dimensionless force ~ o equal to 0.008. Figure 4(b) presents similar data but for the case where ~ o is equal to 0.004. Figure 4(c) shows the histories of ~ for both simulations, the ~ o = 0.008 case being the dashed line. The data in Fig. 4 were generated by numerically integrating Eqs. 21 after recasting them in dimensionless form. A linearly increasing piezometric level was used in both simulations. The details are not of interest here but are briefly discussed in the appendix below. The reason for showing these results is to illustrate how different initial conditions (different values for ~ o ) affect the time interval ts-to. In Fig. 4(a) the time interval between the onset of creep in the feeder slope and the start of the surge is approximately 13 time units. In Fig. 4(b), the interval is longer, roughly 17 time units. The only difference between the two simulations is the value of ~ o . As Fig. 4(c) shows, remains constant, equal to o~ o, until the feeder
block becomes unstable and begins to creep. Once creep begins, ~ follows essentially the same trajectory in both simulations, until the critical condition for the accumulation slide is reached and the surge begins. This happens in both simulations at the same time. The difficulty with prediction is that, in the field, the only measurable time is to, the onset of creep in the feeder slope, and the interval between to and ts depends upon ~ o , which is unknown. Thus for all practical purposes, even with the best possible measurements of piezometric levels, creep rates, and geometry, no prediction of ts can be made. Conclusion This paper presents an idealized model for accumulation slides. The model exhibits features which are commonly identified with accumulation slides, and permits insight into the mechanics of such slides. An approximate stability criterion for surging motions of accumulation slides is determined,
MODELLING STABILITYAND SURGING IN ACCUMULATIONSLIDES
depending upon slope geometry, strength of slope materials, and the ratio of masses of accumulation and feeder slides. This mass ratio plays an important role in determining stability, and, in circumstances where surging occurs, in determining the acceleration of the surge itself. A simple expression for the mean acceleration of the surge is derived. It is also shown that creeping motions of the feeder slide can occur only in the presence of rising piezometric levels, and that episodic creep leads to a racheting of the force exerted by the feeder slide on the accumulation slide. Finally, reasons behind the unpredictability of surging are discussed. Any slope with more or less concave geometry and mantled with unconsolidated materials may be considered as a candidate for accumulation slides. The stability model presented here may be useful in assessing whether or not such a slope may be prone to surges, and estimating the motions which may be involved.
Appendix Equations 21a,b,c were placed in dimensionless form as follows:
(28a)
dT
- F cos flz[tan f12 - (1 - ~2) tan ¢] + (28b)
{dV1 dr2) dr
- vl-
V2 q-
2~ d r
d-T]
where: I~i
F
v i - VR' T= t0)2, ~ 0)2 = ~ 2
'
/~2 -
c 2k x / ~ z '
VR~/m21c F-
g 0)2 VR
Here VR is any convenient reference velocity. The simulations shown in Fig. 4 were obtained by numerically integrating Eqs. 28a,b,c using a R u n g a - K u t t a - F e h l b e r g fourth-order scheme. The parameters ~1 and ~2 were taken to be equal, both linear functions of T, increasing from 0 to 0.5 over 20 time units. The mass ratio m z / m I was taken as 3, and the value of F set equal to 0.35. These values were chosen deliberately, not to represent a realistic slope, but to emphasise the mean velocity plus transient nature of the feeder slope velocity prior to the surge. If more realistic parameter values were used, with VR on the order of 10 m/s, the value of F would be larger .by a factor of 104 or more. The period of the transient motion would then be far shorter and the transients would be dissipated far more rapidly. By using these unrealistic parameter values, the role of the transient motion superposed on the mean velocity is much more visually apparent in Fig. 4.
References
dVl = F cos ill[tan 131 - (1 - ~1) tan ¢] - m ~ dT ml
d V2
9
(28c)
Hutchinson, J.N., 1970. A coastal mudflow on the London Clay cliffs at Belthinge, North Kent. Geotechnique, 20: 412-438. Hutchinson, J.N, and Bhandari, R.K., 1971. Undrained loading, a fundamental mechanism of mudflows and other mass movements. Geotechnique, 21: 353-358. Hutchinson, J.N, Prior, D.B. and Stephens, N., 1974. Potentially dangerous surges in an Antrim mudslide. Q. J. Eng. Geol., 7: 363-376. Prior, D.B. and Stephens, N., 1972. Some movement patterns of temperate mudflows: examples from north-east Ireland. Bull. Geol. Soc. Am., 83: 2533-2544.
1
Modelling stability and surging in accumulation slides R.O. Davis Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand (Received February 4, 1992; revised version accepted June 3, 1991)
ABSTRACT Davis, R.O., 1992. Modelling stability and surging in accumulation slides. Eng. Geol., 33: 1-9. An idealized model for certain classes of debris flows and avalanches is developed. Slopes characterized by a steep feeder segment which loads a flatter accumulation segment will be focused on. Such slopes have been observed to exhibit episodic surging motions. The simplified model allows considerable insight into the behavior of such slides. An approximate stability criterion is derived explaining the surging motions. It is shown that, in general, one cannot predict the onset of surging due to the inability to precisely characterize the interaction between the feeder and accumulation slopes. Other results include an approximate analysis for the mean acceleration which may occur during surging.
Introduction This article considers the stability of a class of mudslides and other mass movements which share a particular characteristic longitudinal profile. The characteristic profile has a steep feeder slope lying immediately above a flatter accumulation slope. Slow movement of the feeder slope materials leads to increasing loading of the flatter accumulation slope materials. If sufficient loading occurs, the accumulation slope may fail rapidly, resulting in a surge, which may involve large velocities and displacements. Mudslides, avalanches, and other mass movements which may be identified by this characteristic shape and behaviour will be referred to here as accumulation slides. There are numerous occurrences of accumulation slides, many of which have been studied in the literature. Several examples are noted in articles by Hutchinson (1970), Hutchinson and Bhandari (1971), Hutchinson et al. (1974), and Prior and Stephens (1972). Instability and surging in the fiat portion of these slides is attributed to undrained loading of the flat segment caused by creeping Correspondence to: R.O. Davis, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. 0013-7952/92/$05.00
motions in the feeder slide. Surges have been observed (see Hutchinson et al., 1974) with velocities estimated to be on the order of 10 m/min involving volumes of hundreds of cubic metres. In this paper a simplified analysis of an accumulation slide will be presented. Both the feeder and accumulation slopes will be represented by infinite slope stability models. The effect of undrained loading of the flatter slope due to motions of the feeder slope is modelled by connecting the two with a simple spring-dashpot combination. Stability criteria for overall behaviour of the model are derived using an effective stress analysis. Motions of the feeder slope are then analysed, assuming the accumulation slope remains stationary. Finally, surging motions of the accumulation slope may occur if the combination of feeder slope loading and increasing piezometric level are sufficiently large. The model shows that these surges will be characterized by high velocities. The model also indicates that the onset of any particular surge is nearly impossible to predict. This agrees with the observations of Hutchinson et al. (1974).
Model formulation Figure 1 (top) illustrates the profile of a typical accumulation slide. Both the upper feeder slope
© 1992 - - Elsevier Science Publishers B.V. All rights reserved.
R.O. DAVIS
2
Fig. 1. (top) Typical accumulation slide profile;(bottom) idealized model for accumulation slide. and the lower accumulation slope rest on a continuous slip surface which is assumed to lie parallel to the upper surface. Subscript 1 refers to the feeder slope, 2 to the accumulation slope. The piezometric surface is assumed to lie parallel to the failure surface, a distance hw, above the failure surface. Horizontal dimensions of both the upper and lower slopes are assumed to be sufficiently great so that side and end effects may be ignored and infinite slope conditions apply. Figure 1 (bottom) shows an idealized model of the slope in Fig. 1 (top). Rigid blocks representing both the feeder and accumulation slides are resting on planar surfaces which are parallel to the slope profile in Fig. 1 (top). The block masses ml are equal to the masses of material contained within the two slides, and the blocks may move with velocities vi. Interaction between the feeder and accumulation slides in Fig. 1 (top) is idealized by a spring and dashpot connecting the two blocks in Fig. 1 (bottom). Clearly Fig. 1 (bottom) oversimplifies the physical problem shown in Fig. 1 (top), particularly with regard to the interaction between the two sliding masses. The principal effect of the feeder slope on the accumulation slope is thought to be undrained loading of the head of the accumulation slope due to creeping motion in the feeder slope (Hutchinson and Bhandari, 1971). This process
would be physically far more complex than the simple elastic and viscous forces connecting the two blocks in Fig. 1 (bottom). Nevertheless, the model in Fig. 1 (bottom) has the virtue of simplicity and clearly allows motions of the feeder slide to affect the stability of the accumulation slide. Moreover, as we will show below, the model in Fig. 1 (bottom) is sufficient to represent all the main response features commonly observed in accumulation slides. A second consequence of the model illustrated in Fig. 1 (bottom) is that the masses ml and m 2 are invariant. In reality the feeder slope mass m 1 is affected both by addition of material from upslope and by transfer of material to the accumulation slope. The accumulation slope mass m 2 is similarly changing. For reasons of simplicity, we will not attempt to model these mass flux problems here. Motions of the two-block system shown in Fig. 1 (bottom) are described by these three ordinary differential equations: ml I"1 ~--- W I sin/~ - $1 - F
(la)
sin
(lb)
m 2 I)2 = W 2 F=
f12 -- $ 2 ~- F
k(l~ l - v2) + c(131 -
i)2)
(lc)
where the superposed dot denotes differentiation with respect to time, and: Wi = ml; g = block weight; g = acceleration of gravity; Si=sliding resistance on failure surface; F = combined elastic and viscous forces; k =spring constant; c = d a s h p o t constant; and/~i = slope angle. The sliding resistance terms $1 and $2 must be provided by constitutive equations for the strength of the materials in the feeder and accumulation slopes. Presumably residual strength conditions will apply or will be quickly achieved following the initiation of motion, and the cohesion component of strength will be small or negligible when compared to frictional strength. A conventional effective stress model for frictional strength will be •used with: Si = Wi(1 - ~i) cos f l i f ( v i )
(2)
Here, ~i represents the effect of piezometric elevation on the effective stress which acts on the failure
MODELLING STABILITY AND SURGING IN ACCUMULATION SLIDES
use Eq. 2 to find:
surface:
7whw, ~i--
3
(3)
sin/3 i - (1 - ~l) cos/31f(0) =
7h i
where 7 and Yw are the unit weights of the slide material and the pore water, respectively, and hwi and hi are as shown in Fig. ! (top). The function f(vi) will be taken as: ( = t a n 4, f(vi) l ~
for vi > 0 forvi=0
(4)
where 4 represents the effective stress friction angle appropriate to the slide material. Here f represents the mobilized strength on the failure surface. If vi=0, only part of the strength may need to be mobilized to hold the block in static equilibrium. This condition is expressed by the less than or equal sign in the second part of Eq. 4. If the block is in motion, the strength is fully mobilized and f is then equal to tan 4. Note that the product Wi(1-~i) is equal to the buoyant weight of the slide material. Thus, Eq. 2 is exactly the usual strength term found in conventional infinite slope analyses, based on effective stresses. The principal difference between this analysis and a conventional infinite slope analysis is the presence here of two sliding masses. Also, it should be noted that, whereas Fig. 1 (bottom) would make it appear that the upper block may move onto the lower sliding surface, given that sufficient displacement occurs; this in fact will not happen in the context of Eqs. (la,b,c). The motion of the upper block is fully controlled by Eq. l a, and it is clear that the block will always be driven by the gravity force Wlsin/31. In reality, the displacement of the feeder slide will be small unless a surge occurs in the accumulation slide. The focus of the model is to examine the onset of the surge, not to attempt to accurately describe the slope geometry after the surge has taken place. Static stability There are well defined limits for static behaviour of the two-block model shown in Fig. 1 (bottom). Setting time derivatives equal to zero in Eqs. l a and l b, we can add the resulting equations and
_
m2 [sin/32 - (1 - ~2) cos/32f(0)]
ml
(5)
Here, f(0) is the mobilized strength on the failure surface. Since the velocities Vl and v2 are both zero, f(0) is not precisely defined, as shown in Eq. 4. Note that the force F does not enter Eq. 5. The limiting value off(0) for which static behaviour is possible is tan 4. If we set f(0) equal to tan 4, Eq. 5 becomes: sin/31 -
- - COS
m2
-
ml
fll tan
41 =
[sin/32 - cos/32 tan 42]
(6)
where the angles 41 and 42 are defined by: tan 4i
=
(1 - ~i) tan 4
(7)
Next, use the identity sin ( x - y) = sin x cos y - sin y cos x to rewrite Eq. 6 as: sin (/31 - 41) = ~o sin (42 - 132)
(8)
where: Lf -
m 2 cos 41
(9)
m 1 cos 42
Equation 8 represents limiting conditions for static equilibrium. It may be solved for any one of the parameters to give the critical value of that parameter in terms of the other parameters. For example, if we solve Eq. 8 for /31 we have: /3~c = 4~ + sin- ~ [Se sin (42 -/32)]
(10)
and this value of/31 represents the steepest feeder slope angle for which the slide may remain static, for given values of/32, 41, 42, and L~°. It should be noted that since 41 and 42 both depend upon water table depth, the values of these two parameters may be changing. Nevertheless, for given values of 4~ and 41, Eq. 10 gives the steepest possible value of/31 for which the feeder slope may remain static. The weakest possible condition for the slope occurs when the water table coincides with the upper surface. In that circumstance, ~1 = 0~2 ~ 0.50. The approximate equality
4
R.O. DAVIS
here follows from the observation that the pore fluid will usually be about half as dense as the slide material. Figure 2 shows graphs of Eq. 10 for a particular value of ~b1 = q~2, and several values of the mass ratio mz/ml. In the figure, /31c is graphed versus /32, and q~l and (~2 a r e both taken as I0 °. Equality of q~l and ~b2 implies cq=~2 which, from Eq. 3, implies the ratio of hwi to hi is the same in both feeder and accumulation slopes. Note that, if/3z approaches the value of ~bl = q~2, then/31 must do likewise, regardless of the value of mz/m 1. Note also the near linear nature of the graphs shown. This reflects the fact that the arguments of the sine function in Eq. 8 are never large, and sin x ~ x for small x. In fact, Eq. 8 may be well approximated by: /31 -- (~1 = ~'(~(~b2 -/32)
(l l)
so long as /31-q51 and 4)2-/32 are both relatively small. Finally, note how important the mass ratio m2/ m 1 is in determining the critical value of/31. For small values of mz/m a, ~ never rises far above the value of q~l =q~2 = 10°. Even in the extreme case where f12--*O°, the critical value of/31 is only 12° for m2/m~ =0.2. On the other hand, when m2/ m~ is large, /3a¢ takes values far greater than 10°. This result is physically intuitive. When the mass
of material in the accumulation slide is much larger than that in the feeder slide, the stability of the system is enhanced. The opposite is true in the reverse situation.
Analysis of motion In this section we will consider typical displacements of accumulation slides, such as those illustrated schematically in Fig. 3. The figure illustrates the displacement history of both feeder and accumulation slopes in response to raising piezometric levels. The upper part of the figure shows the increasing piezometric elevation, perhaps in response to rainfall. The lower part of the figure shows the displacement time histories of both the upper feeder slide (shown as the dashed line) and the lower accumulation slide (solid line). At some particular time, denoted to, the rising water level triggers instability in the feeder slide. It begins to move downslope at more or less constant velocity, while the accumulation slide remains stationary. Displacement of the feeder slide results in increased load on the accumulation slide and, at some later time, denoted ts, the combination of load from the feeder slide and rising piezometric level leads to slipping of the accumulation slide. Both slides now begin to move with rapidly increasing velocity and a surge results. We will model this behaviour, first considering the time period to to ts in which the feeder slide is
~1c2520 ~ ~
slFeeder ope~ //"~//.//I ] Accumulofion /~ i ~ /~slope
i~ Q'~
~S
100
I
".~o
15 -
5 10 ~2[degreesJ I
Fig. 2. Graph of Eq. 10 with O t = 0 2 = 10°.
fo
T~ine
fs
Fig. 3. Typical accumulation slide behaviour.
MODELLING STABILITYAND SURGING IN ACCUMULATIONSLIDES
creeping, and then times following ts where both slides are in motion.
Wi sin fli - Wi[ 1 - cq(to)] cos fli tan 4) - F o = 0
Lower block stationary
It is important to note here that to can be determined from Eq. 15 only if Fo is known. For times greater than to, so long as the lower block remains stationary, the motion of the upper block will be governed by Eq. 13, which has the solution:
First consider conditions in which the lower block in Fig. 1 (bottom) is stationary. Setting v2 = 0, Eqs. la,b,c become:
(15)
ml vl =
1
W1 sin f l l - Wl(l - ~1) cos flltan qS- F
(12a)
v 1 = W(t - to) + ~12g cos fll tan q5 :i I
F= kVl + cfl
(12b)
Here ud(t-to) is a transient term given by:
We can combine these equations to have:
~1 q - 2 2 1 O l l ) 1
q- O)2Vl = g c o s flltan ~b ~1
W(t - to) ----e - ' h ' ° l ( t - ' ° ) { C 1 (13)
where: e) 1 =
(14a) C
21 - 2 kx/-~l
(14b)
represent the natural frequency and fraction of critical damping associated with the upper block. Note the presence of &~ on the right-hand side of Eq. 13. This appears since the piezometric surface within the slope may be changing. For static water table conditions, the right-hand side of Eq. 13 would be zero, and the resulting equation would represent free vibration of a damped harmonic oscillator with natural frequency ~1. In fact, the block would not oscillate, since any upslope motion would induce a change of direction of the frictional force, immediately bringing the block to rest. Thus extended motions of the upper block can only occur in the presence of a rising water table. Now suppose the upper block begins to move at time to. At this instant, the right-hand side of Eq. 12a is exactly equal to zero. The value of F will be Fo, the force acting between the two blocks before motion commences. Prior to the first motion of the block, the rising piezometric level causes cq to increase. The right-hand side of Eq. 12a will be negative, but at time to, increasing c¢1 brings the value to zero. That is, at t = to:
(16)
sin c o l J l -- 22(t - to)+
C2cos col x/1 - 2~(t - to)}
(17)
Constants C1 and C2 are determined by requiring both vl and ~1 to be zero at t = to. The transient term in Eq. 16 is of much less interest than the second term, which represents the mean velocity of the feeder slide. Let Vlm denote the mean velocity: 1 ~21m = ( D ~ g c O s /~1
tan ~b ~1
(18)
We find that the mean velocity is directly proportional to the rate of increase of water level within the feeder slope. Equation 18 can be used to estimate the feeder slope velocity, but it is first necessary to estimate the value of the natural frequency, col. One way to estimate 0) 1 is as follows. Let a denote the average cross-sectional area of the feeder slide, and let L denote its length. Then the spring constant k may be represented by:
Ea
k = ~-
(19)
where E denotes Young's modulus for the feeder slide material. Using Eq. 19 in Eq. 14a, and noting that ml must equal paL where p denotes mass density, we find: C ~°1 - L
(20)
where C; is the so-called bar velocity x / ~ , the velocity of elastic waves in the feeder slope ma-
6
R.O. DAVIS
terial. Typically, for unconsolidated materials found in mudslides and avalanches, C; will be on the order of 100 m/s. If the length of the feeder slide is on the order of 100 m, then col will be roughly 1.0 s - 1. Returning now to Eq. 18, consider a feeder slope with //~=18 °, and suppose 05=15 ° . Then, with col= 1 s -1, we have: v l m = l x9.81xcos18xtan15xoi
Eq. 21a and 21b to have:
P "" l'l
ml
# ,
v'2=
+ -
(22)
m2
Then differentiating Eq. 21c, and making use of Eqs. 21a, 21b and 22, we find: F + 22-05F + 052F= B
(23)
Here:
1
B = kg{sin//1 - sin//2 - [cos//1(1 - cq)
= 2.5oi 1
-cos/32(1 - c~2)] tan 05} The maximum value of cq is roughly 0.5 (when the piezometric surface coincides with the upper surface), and the rate of change might typically be on the order of 0.1 to 1.0 day 1. Thus the range of feeder slope velocities would be 0.2 to 2 m/day. These values are in good agreement with measured values (Hutchinson et al., 1974).
05 =
Next suppose the combination of rising water level and upslope loading are sufficient to cause the lower block to begin to move. The time at which this occurs is denoted t s. All three of Eqs. 1 now apply. We will consider the special case in which both ~1 and ~2 remain constant for t > t s . That is, the piezometric levels in both slopes are assumed to abruptly become stationary at t =tS. The reason for making this assumption is to simplify the analysis. We will find this assumption has little effect since the motions following t~ will be extremely rapid and the time interval over which Eqs. 1 apply will be quite short. We begin by rewriting Eqs. la,b,c. m1171 = Wlsin//1 - Wlcos fll(l - ~ l ) t a n 05- F (21a) m 2172 =
W 2 sin//2
-
W2 cos//2(1
-
(24b) C
2- 2,,~
(24c)
and: 1
L o w e r block sliding
(24a)
-
1
ml
+
1
(24d)
m2
The solution to Eq. 23 consists of a transient and a constant part: 1
F = ~ ( t - ts) + ~ B
(25)
CO-
The transient term ~ is similar to ~ in Eq. 17 and has little interest here. The constant term shows that the force F will have a mean value equal to B/d) 2 for times greater than ts. We can now use Eq. 25 in 21a and 21b to find the acceleration of either block. After some manipulation we have: 1
171=A---~(t-ts), ml
1
172=A+--~(t-ts)
(26)
m2
where: A = g{ml[sin//1 - cos/31(1 - ~1) tan 05]+ m
~2) tan 05 + F m2[sin/32 - cos f12(1 - ~2) tan 05]}
(27)
(21b) F=k(vl
- v2) + c(~1 - v2)
(21c)
Taking ~l and c~2 to be constant, we differentiate
We see from Eq. 26 that both blocks have the same mean acceleration, which is given by Eq. 27. The transient accelerations have a similar form but are exactly out of phase.
7
MODELLING STABILITY AND S U R G I N G IN ACCUMULATION SLIDES
Note that the mean acceleration depends only upon the slope geometry, the friction angle qS, and the mass ratio, mz/m~. If we take the following typical values: /~1 =
18°,
q5=15 ° ,
fi2 = 6°,
m2/ml = 3
~=~2=0.5
Then we find A=0.024 g. This value is in the range expected for accumulation slides. For a mean acceleration of 0.02 g, the slide velocity one minute after ts would be about 12 m/s. The displacement after one minute would be about 350 m. If this displacement is larger than the length of the accumulation slide, as seems likely, we would expect the entire surge to begin and end in less than one minute. Discussion
We can use the two-block model proposed here to gain insight into the mechanics of accumulation slides. First, the stability criterion developed in Eq. 8 can now be viewed in a slightly different light. In deriving Eq. 8, we assumed the strength on both upper and lower failure surfaces was fully mobilized. In fact, it is now clear that the feeder slide will slip first, and hence its strength will be fully mobilized before that of the accumulation slide. Strength on the accumulation slide failure surface becomes fully mobilized at the instant t~ when surging commences. This might suggest that Eq. 8 is invalid since it was assumed in Eq. 5 that both slides were stationary. In fact, Eq. 8 is still a good approximate criterion for surging stability since the acceleration of the feeder slide is small during the time interval between to and ts. Thus Eq. 5 is approximately correct, and Eq. 8 can be used to assess whether or not surging is possible for any slide geometry. It is useful to note once again the importance of the mass ratio mz/m ~. It plays a central role in determining whether or not surging may occur, and also figures strongly in Eq. 27, where the mean acceleration of the surge is determined. Typically, in Eq. 27, the term multiplied by m2 will be negative since it represents the safety margin of the lower block without the force F. The term multiplied by ml will of course be positive. It
becomes clear then that greater values of m2/ml will result in smaller values for A, and hence less energetic surges. It is also interesting to consider possible histories of creeping motions in the feeder slide. Continuous downslope creep is possible only in the presence of increasing piezometric levels. If the piezometric level within the slope becomes stationary or begins to fall, the slide will quickly come to rest. One can easily envision circumstances in which the feeder slope may begin to creep, but, because of cessation of rain, will again come to rest. The result of such an excursion is an increase in the force F. There will subsequently be some relaxation as the viscous part of F dissipates, but there will remain a net increase. This is much like what one would expect in the actual accumulation slide. Undrained loading of the accumulation slide due to motion of the feeder slide may also dissipate somewhat as drainage and partial consolidation take place in the soft materials, but an overall increase in load will result. This of course makes a surge more likely at some future time. It is precisely this mechanism of racheting increases in F which results in almost total unpredictability of when a surge may occur. The difficulty of predicting surging was commented on by Hutchinson et al. (1974) in relation to a particular accumulation slide which occasionally endangered human life. In our model, it is impossible to predict the value of ts, even if piezometric levels in both slopes were continually monitored. The reason for this is that the initial force F o at the onset of motion in the feeder slide is, in general, unknown. To illustrate the unpredictability of ts, we can consider two simulations using the two-block model, each with identical properties but with different initial values of F o. Results are illustrated in Fig. 4, where time histories of block velocities are shown in dimensionless form. In the figure, velocity has been normalized by any convenient characteristic velocity VR, while time and force are made dimensionless as follows:
T= t
'
if-
F VR~
8
R.O. DAVIS 008 [ (b)
OO8
LJ c3
H u
007 006
[L
.005 0"3 .004 co LI 003 Z o 002
OO6 005 t /
o041 l ~J
o"3
z 001 LJ 0 F3 0
OO7
/y"-",,,.. . /. 2
003 f 002 [
" ,-',
"-... _...- --.. _--} c~
8 ]0 12 ]4 ]6 ]8 '20 '22' O I NU',4S I ONLESS - I ME (c)
;i
.
_
0
2
4
6 8 10 12 14 16 ~18 '20~ ' -~22 OlNENSIONLESS T[NF
$
~
,3:6 O; 4
/~ rrJ tJJ
3;2 J]C' 8O8
OOB /,~ 004 L • ~ 002 ! (1 o , , , o 2 U)
4
6
8
1o
1~
DIMENSiO"4LESS
14
1 6 ~ i ~ ' / ~
~
TINE
Fig. 4. Two simulations of accumulation slide response• (a) Velocity time history of feeder slide (dashed line) and accumulation slide (solid line) for ~ o = 0.008. (b) Similar to (a) but with .~-o = 0.004. (c) Time histories of . ~ for both conditions.
In Fig. 4(a), velocities of the feeder slide (dashed line) and the accumulation slide (solid line) are shown for an initial dimensionless force ~ o equal to 0.008. Figure 4(b) presents similar data but for the case where ~ o is equal to 0.004. Figure 4(c) shows the histories of ~ for both simulations, the ~ o = 0.008 case being the dashed line. The data in Fig. 4 were generated by numerically integrating Eqs. 21 after recasting them in dimensionless form. A linearly increasing piezometric level was used in both simulations. The details are not of interest here but are briefly discussed in the appendix below. The reason for showing these results is to illustrate how different initial conditions (different values for ~ o ) affect the time interval ts-to. In Fig. 4(a) the time interval between the onset of creep in the feeder slope and the start of the surge is approximately 13 time units. In Fig. 4(b), the interval is longer, roughly 17 time units. The only difference between the two simulations is the value of ~ o . As Fig. 4(c) shows, remains constant, equal to o~ o, until the feeder
block becomes unstable and begins to creep. Once creep begins, ~ follows essentially the same trajectory in both simulations, until the critical condition for the accumulation slide is reached and the surge begins. This happens in both simulations at the same time. The difficulty with prediction is that, in the field, the only measurable time is to, the onset of creep in the feeder slope, and the interval between to and ts depends upon ~ o , which is unknown. Thus for all practical purposes, even with the best possible measurements of piezometric levels, creep rates, and geometry, no prediction of ts can be made. Conclusion This paper presents an idealized model for accumulation slides. The model exhibits features which are commonly identified with accumulation slides, and permits insight into the mechanics of such slides. An approximate stability criterion for surging motions of accumulation slides is determined,
MODELLING STABILITYAND SURGING IN ACCUMULATIONSLIDES
depending upon slope geometry, strength of slope materials, and the ratio of masses of accumulation and feeder slides. This mass ratio plays an important role in determining stability, and, in circumstances where surging occurs, in determining the acceleration of the surge itself. A simple expression for the mean acceleration of the surge is derived. It is also shown that creeping motions of the feeder slide can occur only in the presence of rising piezometric levels, and that episodic creep leads to a racheting of the force exerted by the feeder slide on the accumulation slide. Finally, reasons behind the unpredictability of surging are discussed. Any slope with more or less concave geometry and mantled with unconsolidated materials may be considered as a candidate for accumulation slides. The stability model presented here may be useful in assessing whether or not such a slope may be prone to surges, and estimating the motions which may be involved.
Appendix Equations 21a,b,c were placed in dimensionless form as follows:
(28a)
dT
- F cos flz[tan f12 - (1 - ~2) tan ¢] + (28b)
{dV1 dr2) dr
- vl-
V2 q-
2~ d r
d-T]
where: I~i
F
v i - VR' T= t0)2, ~ 0)2 = ~ 2
'
/~2 -
c 2k x / ~ z '
VR~/m21c F-
g 0)2 VR
Here VR is any convenient reference velocity. The simulations shown in Fig. 4 were obtained by numerically integrating Eqs. 28a,b,c using a R u n g a - K u t t a - F e h l b e r g fourth-order scheme. The parameters ~1 and ~2 were taken to be equal, both linear functions of T, increasing from 0 to 0.5 over 20 time units. The mass ratio m z / m I was taken as 3, and the value of F set equal to 0.35. These values were chosen deliberately, not to represent a realistic slope, but to emphasise the mean velocity plus transient nature of the feeder slope velocity prior to the surge. If more realistic parameter values were used, with VR on the order of 10 m/s, the value of F would be larger .by a factor of 104 or more. The period of the transient motion would then be far shorter and the transients would be dissipated far more rapidly. By using these unrealistic parameter values, the role of the transient motion superposed on the mean velocity is much more visually apparent in Fig. 4.
References
dVl = F cos ill[tan 131 - (1 - ~1) tan ¢] - m ~ dT ml
d V2
9
(28c)
Hutchinson, J.N., 1970. A coastal mudflow on the London Clay cliffs at Belthinge, North Kent. Geotechnique, 20: 412-438. Hutchinson, J.N, and Bhandari, R.K., 1971. Undrained loading, a fundamental mechanism of mudflows and other mass movements. Geotechnique, 21: 353-358. Hutchinson, J.N, Prior, D.B. and Stephens, N., 1974. Potentially dangerous surges in an Antrim mudslide. Q. J. Eng. Geol., 7: 363-376. Prior, D.B. and Stephens, N., 1972. Some movement patterns of temperate mudflows: examples from north-east Ireland. Bull. Geol. Soc. Am., 83: 2533-2544.