FLOW LIQUEFACTION FAILURE OF SUBMARINE SLOPES DUE TO
MONOTONIC LOADINGS - AN EFFECTIVE STRESS APPROACH
E. ATIGH and P. M. BYRNE
Department of Civil Engineering University of British Columbia, 2324 Main Mall,
Vancouver, B.C., V6T 1Z4, Canada.
Abstract
Current liquefaction analyses of slopes are generally based on undrained soil response.
Recent experimental studies have shown that small net flow of water into an element
will result in additional pore pressure generation and further reduce its strength. Many
flow slides have occurred in submarine slopes, most of which were induced by
monotonic loadings. Tidal variations can cause unequal pore pressure generation with
depth in unsaturated seabed soils. An effective stress approach is presented to model
flow liquefaction of sand under a range of drainage conditions to evaluate the triggering
of liquefaction during low tides for an unsaturated underwater slope.
Keywords: Submarine slide, flow liquefaction, gassy sand, partial drainage,
liquefaction analysis.
1. Introduction
Flow liquefaction failure of submarine slopes has become a major concern due to its
frequent occurrence and its effect on the safety of coastal structures. Many flow slides
have occurred in underwater slopes consisting of cohesionless sediments (Chillarige et
al., 1997a). In the Fraser River delta on the west coast of Canada, five liquefaction flow
slides have been reported between 1970 and 1985 (McKenna et. al 1992). A flow slide
event occurred near Sand Heads at the delta front in 1985 and resulted in the loss of at
least 106 m3 of sediments (Christian et al, 1997). McKenna et al, 1992 postulated that
this event was related to rapid sedimentation, the presence of interstitial gas, tidal
currents, waves and seismic activities.
In situ test results including cone penetration resistance showed that fresh deposits of
Fraser River sand are very loose and highly susceptible to liquefaction (Chillarige et al
1997a). Vaid et al (1998) carried out a comprehensive experimental investigation into
the potential of instability and liquefaction of saturated Fraser River sand under
conditions other than completely undrained. They have shown that partially drained
conditions that result in even very small expansive volumetric strain could trigger
liquefaction at constant shear stress that would not develop if conditions remained
undrained.
Tidal variations on gassy seabed soil result in a reduced pore pressures response with
depths and time due to the compressibility of the pore fluid. During low tides the
reduction in pore pressure does not follow the total seabed pressure changes. This
results in a reduction in effective stresses and may lead to flow liquefaction failure of
3
4
Atigh and Byrne
submarine slopes. This hypothesis will be examined here using an effective stress
approach based on an elastic plastic stress strain model.
2. Instability and flow liquefaction of sands
Loose saturated granular materials that strain soften if loaded undrained under
controlled monotonic strain conditions, are referred to as liquefiable soils. The
fundamental behaviour of sands is governed by the skeleton response obtained from
drained test results. The undrained response is controlled by the skeleton response and
pore fluid stiffness. The tendency of sand to change volume during loading; positive or
negative dilatancy, is constrained by the pore fluid and results in pore pressure changes.
Under controlled loading conditions, as commonly occur in the field, soils may become
unstable due to a reduction in effective stresses associated with pore pressure rise.
Vaid et al, 1998 and Elliadorani, 2000 investigated the potential for instability under
conditions other than undrained. They accomplished this by injecting or removing small
volumes of water from the sample as it was being sheared and referred to this as a
partially drained condition. The results of partially drained tests on Fraser River sand for
various dεv/dε1 are shown in Fig. 1 as well as drained and undrained tests results. As
may be seen in these tests the sand becomes extremely strain softening for a small ratio
of injection corresponding to dεv/dε1 =-1. The outflow tests indicate a strain hardening
response similar to a drained test for dεv/dε1 =+0.4.
600
Outflow , dεv /dε1=+0.4
( 1 - 3), kPa
500
Drained
400
300
Outflow dεv /dε1=+0.2
200
Undrained, dεv /dε1=0
100
Inflow , dε /dε =-1
0
0
1
2
3
Axial strain ε1 , %
4
5
Figure 1. Stress strain response of loose Fraser River Sand for different drainage conditions
(Eliadorani, 2000).
In an unsaturated soil, gas compressibility can cause additional expansive volumetric
strains to develop during falling tides that further reduce the pore pressure response.
This could trigger instability and flow liquefaction under constant shear stress and could
be responsible for the numerous slope failures that have occurred.
3. Stress strain model for sands
The constitutive model is based on the elastic-plastic stress strain model proposed by
Byrne et al (1995) and has been further developed by Puebla (1999). It is an incremental
Flow liquefaction of submarine slopes
5
Shear stress, t
elastic-plastic model in which the yield loci are lines of constant stress ratio or
developed friction angle. The flow rule relating the plastic strain increment directions is
non-associated and leads to a plastic potential defined in terms of dilation angle ψ as
shown in Fig. 2. Plastic shear strain is the hardener that allows the yield locus to expand
or stress ratio, ηd=sinφd, to increase.
sinφ > sinφ
Plastic
potentials
1
∆εps
Yield
loci
sinφcv
sinφd < sinφcv
sinψ
Mean stress, s’
Figure 2. Yield loci, plastic potentials, and plastic strain increment vectors.
The plastic shear strain increment for any increase in stress ratio ∆η is obtained from the
normalized tangent plastic shear modulus G* as follows:
∆γp = ∆η/G*
(1)
Where G* is given by:
 s’ 

G * = K GP 

 PA 
np −1
 η
1 −  d
  η f

R f





2
(2)
In which, KGP is plastic shear modulus number, np is plastic shear modulus exponent, ηf
is stress ratio at failure, Rf is failure ratio=ηf /ηult; ηult is ultimate stress ratio from the
best-fit hyperbola, ηd is developed stress ratio = sinφd; φd is developed friction angle, t is
(σ1−σ3)/2, s′ is (σ1′+σ3′)/2∆γp is given by Eq. 1 and PA is atmospheric pressure.
The plastic volumetric strain increment ∆εvp is obtained from the flow rule:
∆εvp=sinψ ⋅ ∆γp
(3)
Where ψ is the dilation angle, which is related to the constant volume friction angle φcv
and the developed friction angle φd by:
6
Atigh and Byrne
sinψ=(sinφcv-sinφd) α
(4)
In which, α=2 for φd < φcv and α=0.5 for φd > φcv.
The elastic response is assumed to be incremental linear and isotropic and is specified
by two elastic parameters; the elastic shear modulus Ge and Poisson’s ratio ν, and is
stress level dependant. The total response is the sum of elastic and plastic components.
4. Effective stresses in gassy soils
Effective stress in saturated soil may be written in incremental form (Terzaghi et al,
1948) as:
∆σ′=∆σ-∆uw
(5)
In the field, pore pressure changes are caused by: 1) seepage through soils, and/or 2)
volume changes. Darcy’s law governs flow of water through saturated soil. In an
undrained state pore water pressure response due to volume changes are obtained from a
volumetric constraint; ∆εv, caused by fluid stiffness as follows:
∆uw = (Bw / n) ∆εv
(6)
In which, Bw is water bulk modulus, n is porosity of soil skeleton, ∆εv is volumetric
strain of the soil element, and (Bw / n) represents an equivalent fluid stiffness.
The mechanical behaviour of unsaturated soil is directly affected by changes in pore gas
and pore water pressures (Fredlund et al, 1993). Pore gas pressure and volume change
respond in accordance with Boyle’s law, and Henry’s law governs dissolving of free gas
into the water. For a gas-water mix, assuming no gas goes into or out of solution the
compressibility of the mix is given by Eq. 7.
Cgw =
1
1 − Sr
+
Bw u g + PA
(7)
According to Fredlund et al (1993), for degrees of saturation between 80% and 100%,
air bubbles are of spherical form in an occluded zone within the pore fluid. It is
suggested that pore gas and pore water pressures may be assumed equal in an occluded
zone. Hence the same definition of effective stress; stated in Eq. 5, for saturated soil can
be used for gassy soil. Using Bgw from Eq. 7 will allow gas pressure changes as well as
the gas volume changes to be predicted.
5. Model calibration and verification
In this study the analyses were carried out using the computer code FLAC, version 4.0
(Cundall, 2000). The stress strain model described in Section 3 has been implemented in
FLAC and is used in this study. Shear behaviour of Fraser River sand has been modelled
in a previous study (Atigh and Byrne, 2000) for drained, undrained, and partially
drained conditions based on the comprehensive laboratory test results of Eliadorani
(2000). The model is calibrated against drained triaxial compression tests for both stress
strain and volumetric strain response. The calibrated model with properties presented in
Flow liquefaction of submarine slopes
7
Table 1 is used to predict the response in undrained and partially drained tests as shown
in Fig. 3. Both stress strain; Fig. 3a and stress paths; Fig. 3b, are in good agreement with
the tests results. Grozic, (1999) reported drained and undrained triaxial compression
tests on saturated and gassy samples of Ottawa sand. The details of the laboratory
program are presented in Grozic et al, (1998). Behaviour of gassy Ottawa sand was
modelled by Atigh and Byrne, 2001. Model properties were selected to give a best fit to
the dry or drained test; Sr=0%, result on Ottawa sand for both shear stress-strain and
volume change responses shown in Fig 4. The calibrated model was then applied to
predict the undrained triaxial compression tests of saturated and gassy samples of
Ottawa sands. Predicted stress strain response and volumetric strains shown in Fig. 4 are
in good agreement with test results. The model parameters for both Ottawa sand and
loose Fraser River sand is shown in table 1.
a) Stress Strain Response
b) Stress Paths
500
300
Outflow , dεv /dε1=+0.4
250
Drained
Outflow dεv /dε1=+0.2
300
( 1- 3)/2, kPa
( 1 - 3), kPa
400
200
Undrained, dε v /dε1=0
100
Undrained, dεv /dε 1=0
200
Outflow dεv /dε1=+0.2
150
Inflow , dε /dε =-1
100
Drained
50
Inflow , dε /dε =-1
Outflow , dεv /dε1=+0.4
0
0
0
0.5
1
1.5
2
2.5
0
3
100
200
Axial strain ε 1 , %
300
400
500
(σ1+σ3)/2, kpa
Figure 3. Loose Fraser River sand undrained and partially undrained responses, a) Stress Strain response,
b) Stress Paths.
Table 1. Model parameters used in analyses.
Model Parameters
Fraser River
sand
200
0.50
0.125
200
0.50
35.5°
33.0°
0.97
0.333
KGe:
Elastic shear modulus number
ne: Elastic shear modulus exponent
ν: Elastic Poisson’s ratio
KGp: Plastic shear modulus number
np: Plastic shear modulus exponent
φf: Peak friction angle
φcv: Constant volume friction angle
Rf: Failure ratio
F: Factor of anisotropy
a) Stress strain
800
Ottawa
Sand
150
0.50
0.125
150
0.25
32.5°
32.0°
1
0.333
b) Volumetric Strains
0
5
ε 1, %
0
Sr=0%
10
Sr=%100, εv =0
Sr=%98
Sr=83%
%
Sr=%97
400
v,
( ’1- ’3) , kPa
600
Sr=97%
200
2
Sr=%83
Sr=98%
4
Sr =%0
Sr=100
0
0
5
ε 1, %
10
15
Figure 4. Simulation of element test results, behaviour of loose Ottawa sand for different
degrees of saturation: a) stress strain, b) Volumetric Strains.
15
8
Atigh and Byrne
Since there is no laboratory test on gassy Fraser River sand, only the model predictions
are presented in Fig. 5 in terms of stress strain and stress paths. As expected, the
response is bounded by drained and undrained responses.
a)Stress Strain
b) Strss Paths
300
Sr=0%
( ’1- ’3)/2 , kPa
( ’1- ’3), kPa
600
Sr=80%
400
Sr=90%
Sr=95%
Sr=98%
Sr=100%
200
0
0
2
4
6
ε1, %
8
10
Sr=80%
Sr=90%
200
100
Sr=95%
Sr=98%
Sr=0%
Sr=100
0
100
200
300
(σ’1+σ’3)/2 , kPa
400
500
Figure 5. Predictions of behaviour of loose gassy Fraser River sand in undrained triaxial
compression tests: a) stress strain, b) stress paths.
6. Coupled stress flow analysis
Analyses of submarine flow slides in the Fraser delta are carried out here in coupled
stress flow mode using the constitutive model and soil parameters for Fraser River sand
described above. The purpose of this analysis was to simulate the potential of flow
liquefaction under falling tides for an underwater gassy slope. The analysis procedures
and results are presented here.
A degree of saturation in the range of 90% to 100% is considered in the analysis. The
pore fluid stiffness of the gassy soil is obtained using Eq. (7), which is variable
according to gas pressure and degree of saturation. Analysis is performed on a 20° slope
consisting of loose gassy Fraser River sand. The sea level is assumed to be 10 m above
the crest of the slope and is varied sinusoidally with time.
The 2-D analysis was first performed for the normal tides and pore pressure response at
depth compared with the seabed pressure. The results are shown in Fig. 6, and agree
well with the measurements.
The analysis was then carried out for the maximum tides with amplitude of 2.5m and
period of 16h, for various degree of saturation. Liquefaction flow failure is predicted
during falling tide for the slope shown in Fig. 7 for a degree of saturation approximately
95% and soil hydraulic conductivity of 10-6m/sec. The deformed mesh after triggering
of liquefaction is shown in Fig. 7. Maximum horizontal displacement of 12.5m is
predicted near the toe area, which reduces to 2.5m toward the crest of the slope.
Stress paths at different levels below the seabed are shown in Fig. 8a, for this slope
angle. Very small amount of expansive volumetric strains; Fig 8b and 8c, due to gas
expansion and inflow triggers liquefaction during low tides. Predicted stress paths, and
runaway strains indicate that flow liquefaction of the gassy sediment to a depth of 8m
during tidal variations could occur for loose Fraser delta sand at this slope angle during
tidal variations.
Flow liquefaction of submarine slopes
9
7. Conclusions
The behaviour of submarine slopes in the Fraser River delta during tidal variations is
examined using a numerical modelling effective stress approach. The model ability in
predicting soil response under various drainage conditions and saturation is verified
using triaxial test results on Fraser River Sand. It is shown that a small amount of
expansive volumetric strains due to gas compressibility can cause unequal pore pressure
generation with depths. This can lead to a reduction in effective stresses on tidal
drawdown that may result in flow liquefaction and instability of gassy submarine slopes.
Predicted failure pattern suggest a retrogressing flow slide starting from the toe of the
slope and moving towards the shore.
20
pore pressure variations, kPa
15
seabed pressure
10
5
0
-5
0
5
10
15
20
Predicted
-10
Measured
-15
-20
Time, hour
100m
20m
5m
40m
35m
20m
10m
Figure 6. Predicted and measured pore pressure response at depth of 5m.
120m
Figure 7. Flow liquefaction failure of submarine slope under maximum tide of 2.5m.
Atigh and Byrne
5
4
3
2
1
0
Stress Path, D=2m
1,
Stress Path, D=2.25m
and
%
50
45
40
35
30
25
20
15
10
5
0
v
Stress Path, D=3.5m
ε1
8
%
Initial stress
0
a)
20
40
60
(σ’1+σ’3)/2, kPa
80
εv
Liquefaction Triggered during
low tide
9
9
10
Time, hour
b)
v,
(σ’1-σ’3)/2, kPa
10
-0.5
-1.5
-2.5
100
0
c)
10
20
Axial Strain ε1, %
Figure 8. 2-D analysis: a) stress paths leading to liquefaction following one cycle of tides, b) Axial and
volumetric strains.
8. References
Atigh, E., Byrne, P.M., 2000, The effects of drainage conditions on liquefaction response of slopes and the
inference for lifelines, Proceedings of the 14th Vancouver Geotechnical Symposium, Vancouver,
British Columbia.
Atigh, E., Byrne, P.M., 2001, Flow liquefaction of loose gassy sand under monotonic loadings-An effective
stress approach, 54th Canadian Geotechnical Conference, 2nd Joint IAH and CGS Groundwater
Conference, Calgary, Alberta, Canada.
Bishop, A. W. (1959). ‘‘The principle of effective stress.’’ Teknisk Ukeblad I Samarbeide Med Teknikk, Oslo,
Norway, 106(39), 859–863.
Bishop, A. W. (1961). ‘‘The measurement of pore pressure in the triaxial test.’’ Conf. British Nat. Soc. of Int.
Soil Mech. and Found. Engrg., Butterworth’s, London, 38–46.
Byrne, P.M., Phillips, R., and Zang, Y., 1995, Centrifuge tests and analysis of CANLEX field event, 48th
Canadian Geotechnical Conference, Vancouver, British Columbia.
Chillarige, A.R.V., Robertson, P.K., Morgenstern, N.R., and Christian, H.A. 1997a, Evaluation of the in situ
state of Fraser River sand. Canadian Geotechnical Journal, 34: 510-519.
Chillarige, A.R.V., Morgenstern, N.R., Robertson, P.K., and Christian, H.A. 1997b, Seabed instability due to
liquefaction in the Fraser River delta. Canadian Geotechnical Journal, 34:520-533.
Christian, H.A., Woeller, D.J., Robertson, P.K., and Courtney, R.C., 1997. Site investigation to evaluate flow
liquefaction slides at Sand Heads, Fraser River delta. Canadian Geotechnical Journal, 34: 384-397.
Cundall, P.A., 1998, FLAC user’s manual, Version 3.4. ITASCA Consulting Group Inc., Minneapolis, Minn.
Eliadorani, A. A., 2000. The response of sands under partially drained states with emphasis on liquefaction,
Ph.D. Thesis, University of British Columbia, Vancouver, Canada.
Fredlund, D. G., Morgenstern, N. R., and Widger, R. A. (1978). ‘‘The shear strength of unsaturated soils.’’
Can Geotech. J., Ottawa, 15(3), 313–321.
Fredlund, D.G., and Rahardjo, H., 1993. Soil mechanics for unsaturated soils. John Wiley & Sons, Inc., New
York.
Grozic, J.L., Robertson, P.K., and Morgenstern, N.R., 1998. The behavior of loose gassy sand, Canadian
Geotechnical Journal, 36: 482-492.
Grozic, J.L., 1999, The behaviour of loose gassy sand and its susceptibility to liquefaction, Ph.D. Thesis, The
University Alberta, Edmonton, Alberta, Canada.
Grozic, J.L.H., Robertson, P.K., and Morgenstern N.R., 2000, Cyclic liquefaction of loose gassy sand,
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McKenna, P.A., Luternauer, J.L., and Kostaschuk, R.A, 1992. Large scale mass-wasting events on the Fraser
River delta front near Sand Heads, British Columbia. Canadian Geotechnical Journal, 29: 151-156.
Puebla, H., Byrne, P.M., Phillips, R., 1997, Analysis of CANLEX liquefaction embankment: prototype and
centrifuge models, Canadian Geotechnical Journal, 34: 641-657.
Puebla, H. 1999. A constitutive model for sand and analysis of the CANLEX embankment, Ph.D. Thesis,
University of British Columbia, Vancouver, Canada.
Terzaghi, K. V. and Peck, R.B. (1948). Soil mechanics and engineering practice, Wiley, New York.
Vaid, Y.P., Eliadorani, A. 1998, Instability and liquefaction of granular soils under undrained and partially
drained states. Canadian Geotechnical Journal, 35: 1053-1062.