Fujii, Y., Mufundirwa, A., Kodama, J. and Fukuda, D. (2014), Prediction ofTime and Volume of Rock Failure by Slope Method, Proc. The 3rd.
International Conference on Advances in Mining and Tunneling (ICAMT2014), pp. 285-289, Vung Tau, Vietnam, Oct. 22, 2014.

PREDICTION OF TIME AND VOLUME OF ROCKFAILURE BY SLOPE METHOD
Yoshiaki Fujii1*, Mufundirwa Azania2, Jun-ichi Kodama1 and Daisuke Fukuda1
1

Hokkaido University, Sapporo, Japan
MBADA Diamonds, Harare, Zimbabwe*
Corresponding author’s email: fujii6299@frontier.hokudai.ac.jp
2Abstract: The slope method by authors to estimate
failure time of rock is described. The method wascompared to the conventional inverse velocity method
and it was found that the slope method was slightly
better than the inverse velocity method. The method topredict volume of rock failure is also presented. Theabove methods are very useful since the both timing
and volume of rock failure can be predicted as longas either displacement or strain is observed andanalyzed in real time.
Key words: Prediction, Failure time, Failure volume,
Displacement rate
1.

INTRODUCTION
Failure time of rock is often said to beunpredictable. Failure time however can be predictedusing the slope method (SLO), which was developedby the authors [1], as well as the conventional inverse
velocity method (INV) [2] as long as either strain or
displacement is measured.
In this paper, the slope method is described firstwith a clear definition of the critical point at which
prediction begins. Asamushi landslide in Japan and arock slope failure at a Japanese limestone mine are
analyzed again with the newly defined critical point
and an improved smoothing. The results arerepresented by a slightly modified method and arecompared to the results using the conventional inverse
velocity method. Finally, methods to predict thefailure volume are proposed.

regression line since not only y data but also x datacontain errors.
Strain rate is calculated by the followingequations.
 i n   i
 d (3)  ti n  ti
 dt  t Tt t(4)T  i n i2Note that the strain rate is for t = T but time ofprediction is t = ti+n.
The critical point is defined as follows.
(1) Find the earliest point (CP1) whose strain rate isbigger than any point before CP1 and less than anypoint after CP1.
(2) A point just before CP1 is the critical point (CP).
Predictions are made using the data from thecritical point to the time of prediction. Displacement
can be used instead of strain.

t d/dt

tF
1OTHE SLOPE METHOD
Fukui & Okubo proposed the following equation
for strain of rocks under tertiary creep [3].
(1)   B log( tF  t )  C
where  is strain, tF is failure time, t is elapsed timeand, B and C are constants. Differentiating the bothsides with respect to t, we getdd(2)
t tF
Bdt
dt
where d/dt is strain rate. Eq. (2) means that failure
time can be evaluated as the slope of the regression
line taking strain rate on x-axis and, the product ofstrain rate and elapsed time on y-axis (Fig. 1). The
reduced major axis is used instead of a usuald/dt

2.

Fig. 1. Evaluation of failure time using the slope
method.

3.

dt / d600
Displacement (mm)THE INVERSE VELOCITY METHOD
Eq. (2) can be deformed as follows.
dt(5)t  tF  Bd
where dt/d is called inverse velocity and Eq. (5)means that failure time can be evaluated as the xintercept of the regression line taking the elapsed timeon x-axis and inverse velocity on y-axis.

500
400300200
1000050100150

Time (h)Fig. 3. Displacement vs. time for Asamushilandslide.

OtFFig. 2. Evaluation of failure time using the inversevelocity method.
4.

CASESTUDYONASAMUSHILANDSLIDE
The 100,000 m3 landslide near surface joints inliparitic tuff occurred at Asamushi, Japan in 1966,burying 80 m length of track and interrupting railroad
traffic for 26 days [4].
Measured displacement is shown in Fig. 3 anddisplacement rate was calculated for n = 1, 5 and 10(Fig. 4). The displacement rate for n = 1 is scatteredand gave poor prediction. The best predictions weregiven by n = 10. The plot for the slope method (SLOplot) and the inverse velocity method (INV plot) areshown in Figs. 5 and 6.
The results are shown in Fig, 7. In this figure, thepredicted life expectancy is calculated by subtracting
time of prediction from the predicted failure time. The
broken line shows the correct life expectancy.
Predictions exactly on this line are correct, predictions
above or below this line mean dangerous or safe
predictions, and negative life expectancy ismeaningless. The predictions using SLO and INVshow safe but large error. However, SLO prediction isslightly better since its error is smaller than that ofINV and it approaches the correct life expectancy atthe last hours while INV prediction goes intomeaningless region.

20Displacement rate (mm/h)

t100050100Time (h)150Fig. 4. Displacement rate for Asamushi landslide.
Red: n = 1, blue: n = 5 and green: n = 10.

Life expectancy (h)8006004002000

Displacement rate x time (mm)50Safe
SLO0
Meaningless60INV

80 100 120 140 160
Time (h)Fig. 7. Predictions by SLO and INV (n = 10) forAsamushi landslide.

(a) Whole plot

5.

CP2001001.522.5
Displacement rate (mm/h)(b) Magnified plot
Fig. 5. SLO plot for Asamushi landslide (n = 10).

0.6

Inverse velocity (h/mm)

Dangerous

2 3 4 5 6 7
Displacement rate (mm/h)

CASE STUDY ON ROCK SLOPE FAILURE[1]
A 500 m3 rock mass failure occurred in a
Japanese limestone mine in 2007. Displacementshows a significant increase from approx. a day
before failure (Fig. 8). The displacement rate for n = 1
shows large scatter (Fig. 9a) and that for n = 10 (Fig.
9b) is used for the analysis. The displacement rateincreased, decreased and increased again to the failure(Fig. 9b).
The predictions are in the meaningless region(Fig. 10). This is because the decrease in thedisplacement. However, using only the data after thedecrease (Fig. 11), the life expectancy was able to bepredicted (Fig. 12). The predictions contained large
dangerous error. However, the prediction by SLO isslightly better than that by INV since the error in SLO
prediction is smaller than that in INV prediction
although SLO prediction goes into the meaninglessregion at the last minutes.

0.50.4

180CP

0.3
0.2
0.1050100Time (h)150Displacement (mm)Displacement rate x time (mm)100170160Fig. 6. INV plot for Asamushi lanslide (n = 10).

05000

10000

Time (min)Fig. 8. Displacement vs. time for rock slope failure.

Life expectancy (min)Displacement rate (mm/min)

1
0.8
0.6
0.4
0.2
00

0.2
0.1
CP

Dangerous-50INVSLO
Meaningless
11700
11750
Time (min)Fig. 10. Predictions by SLO and INV (n = 10) forrock slope failure.

Displacement rate (mm/min)

Displacement rate (mm/min)

0.3Safe

05000
10000
Time (min)(a) Whole plot. Red: n = 1, blue: n = 5 and green: n= 10.

011650500.14
CP
0.12
0.1
0.08
0.06
11720 11730 11740
Time (min)

11700
11750
Time (min)Fig. 11. Displacement rate for rock slope failure.
(b) Moderately magnified plot for n = 10.

0.05

0CP
11660 11670 11680
Time (min)(c) Fully magnified plot for n = 10.
Fig. 9. Displacement rate for rock slope failure.

Life expectancy (min)Displacement rate (mm/min)

40
Dangerous30
INV
2010Safe

0Meaningless
SLO-1011740
11750
11760
Time (min)Fig. 12. Predictions by SLO and INV (n = 10) forrock slope failure.

6.

PREDICTION
OFTHEFAILURE
VOLUME
Prediction of failure volume is possible using thefollowing equations (Figs. 13 and 14) whichsummarize the data in Mufundirwa et al. [1] withcorrelation coefficients of 0.98 and 0.95, respectively.
(6)
V  3.02  10 11δ t 2.80
(7)
V  3.09  10 5 u C1.902
3where V (m ) is the failure volume,  t (s) is the timefrom the critical point to failure and uC (mm/min) isthe displacement rate at the critical point. Data byother researchers [5-8] are located around theregression line (Fig. 14) implying the validity of Eq.
(7).
Plotting the critical displacement and the timefrom the critical point to failure, we get the following
equation.
(8) t  3.71  10 5 u C0.623
The time from the critical point to failure can beroughly estimated only from the critical displacement
rate. Eq. (8) is very convenient to obtain the firstrough prediction of failure time.
Those data in the previous researches should be
examined again in future since the critical point was
not strictly defined although there will not be big
differences in the critical displacement rates.
7.

CONCLUDING REMARKSThe slope method to estimate failure time of rockwas described. The method was compared to theconventional inverse velocity method and the slope
method was slightly better than the inverse velocity
method.
The method to predict the volume of rock failure
was also presented. The timing and volume of rock
failure can be predicted by the above methods as longas displacement or strain is observed and analyzed inreal time.
However, necessary reinforcements are carriedout for dangerous slopes and it is rare that a slope

collapse occurs at a rock slope under observation. The
slope method can be used even for this kind of rockslopes to confirm the effectiveness of thereinforcements and that an imminent failure has gone.
REFERENCES[1] Mufundirwa, A., Fujii, Y. and Kodama, J., A
New Practical Method for Prediction ofGeomechanical Failure-time, Int. J. Rock Mec. Min.
Sci., Vol. 47, No. 7, pp. 1079-1090, 2010.
[2] Rose, N.D. and Hungr, O., Forecasting
Potential Rock Slope Failure in Open Pit Mines using
the Inverse-velocity Method. Int. J. Rock Mech. Min.
Sci., Vol. 44, pp. 308–320, 2007.
[3] Fukui, K. and Okubo, S., Life Expectancy andTertiary Creep for Rock, Proc. MMIJ Fall Meeting,pp. 91–94, 1997 in Japanese.
[4] Saito, M., Forecasting Time of Slope Failure
by Tertiary Creep. Proc. of 7th International
Conference on Soil Mechanics and Foundation
Engineering, Mexico City, 2, pp. 677–683, 1969.
[5] Wyllie, D.C., Munn, F.J., The Use ofMovement Monitoring to Minimize Production
Losses due to Pit Slope Failures. In: Proc. of the FirstInternational Symposium on Stability in Coal Mining.
Vancouver, BC; Miller Freeman San Francisco, pp.
75–94, 1979.
[6] Zavodni, Z.M., Broadbent, C.D., Slope
Failure Kinematics. Canadian Institute of Mining
Bull, Vol. 99, pp. 69–74, 1980.
[7] MacRae, A.M.R., Case Histories ofDeformation Measurements in Canadian Surface
Mines, Proc. 4th Canadian Symposium on Mining,
Surveying and Deformation Measurements. Canadian
Institute of Surveying, Ottawa, pp. 255–278, 1982.
[8]Cruden,
D.M.,
Masoumzadeh,S.,
Accelerating Creep of the Slopes of a Coal Mine.
Rock Mechanics and Rock Engineering, Vol. 20, pp.
123–135, 1987.

10
Vaiont landslide8

log volume (m3)6Asamushi landslide

4Rock mass failure

20|r| = 0.98

-2
Lab. uniaxial creep test

-4

Brazilian creep test

-602

4log t (s)68Fig. 13. Relationship between the volume of rock failure and time from critical point to failure.

10
Vaiont landslide

8Cruden & Masoumzadeh (1987)

6log volume (m3)Asamushi landslide

4

Wyllie & Munn. (1979) & MacRae (1982)

2

Rock mass failure
Zavodni & Broadbent (1980)

0|r| = 0.95-2-4-6-6

Uniaxial creep test
Brazilian creep test

-4-20log critical deformation rate (mm/min)

2Fig. 14. Relationship between the volume of rock failure and deformation rate at the critical point.

8Vaiont landslide

6Asamushi landslidelog t (s)|r| = 0.89

4Rock mass failure

2Lab. uniaxial creep test
Brazilian creep test

0-6-4-20log critical deformation rate (mm/min)Fig. 15. Relationship between the critical deformation rate and time from critical point to failure.

2