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Numerical analysis of liquefaction potential of partially drained seafloors
Coastal Engineering, 13 (1989) 117-128 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
117
Numerical Analysis of Liquefaction Potential of Partially Drained Seafloors STEFANOS TSOTSOS 1, MICHAEL GEORGIADIS1 and ANASTASIA DAMASKINIDOU2
1Department of Civil Engineering, University of Thessaloniki, 540 06 Thessaloniki (Greece) 2Department of Rural Engineering, University of Thessaloniki, 540 06 Thessaloniki (Greece) (Received June 27, 1988; revised and accepted November 29, 1988)
ABSTRACT Tsotsos, S., Georgiadis, M. and Damaskinidou, A., 1989. Numerical analysis of liquefaction potential of partially drained seafloors. Coastal Eng., 13: 117-128. The cyclic normal and shear stresses induced in the seafloor sands by sea waves, generate transient and residual pore-water pressures. When the pore pressure becomes equal to the total normal stress the soil liquefies and loses its strength. This paper presents a numerical method for evaluating pore-water pressure build-up and liquefaction potential of offshore sites, in which partial drainage is implemented through a two-dimensionalconsolidation model. The effect of soil permeability, relative density and deformability on pore-pressure generation is studied and an application of the method to evaluate the liquefaction potential of a North Sea site is examined.
INTRODUCTION
The liquefaction potential of a non-cohesive seafloor under the action of storm waves has received increasing attention in recent years with respect to the design of offshore pipelines, anchors and offshore platforms. The waves induce a sequence of cyclic shear stresses in the soil below the seafloor and progressively increase the pore-water pressure. This, depending on a number of soil and wave characteristics, may lead to the liquefaction of the seabed and therefore create problems to any installations founded in the area. The prediction of pore-water pressures and shear stresses generated in the seabed is a complex problem and all computation methods are based on simplifying assumptions (Madsen, 1978; Seed and Rahman, 1978; Finn et al., 1983; Ishihara and Yamazaki, 1984; Rahman and Jaber, 1986). Storm waves are considered as a series of harmonic waves, while linear wave theory and the assumption of a rigid and impermeable seafloor are usually used in the computation of the harmonic variation of the seabed pressure p (see Fig. 1 ). The pore-water pressure generated by cyclic wave loading consists of a tran0378-3839/89/$03.50
© 1989 Elsevier Science Publishers B.V.
118
k./?" kL/
/st,, : . r
:..-. ( . : - . S e a b e d
k . / lokJ:/_k_g
x2/
]- : : . ..:.'.'.J: : . ..:..$ v.........::..
:........
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:":':'.:-'.'.]:'ii:':"~ vh:'.'.'.'.'.'-'.'-'.'-'." :"'":"'::::::::.":.':::'.'.'.'.~ h . ".'.'.'."i :'.:.:.:.:. . . . . . . . . . . . . . . . . . . . . .".'.'.'.'.'.'.'.'.'::ii::.: .... z ~ " " .': : .'. .......... $~o".".':':
• -':'.'.'.: " ' . : : : ' "
.
..................
.
.
.
.
.
.
.
.
.
.
.
Z .......................
Fig. 1. Wave characteristics a n d seabed stresses.
Sea be d
CroPU O P
(7 i , u ~
zi
Z~
C ¢ q~
tl
tT
Fig. 2. Seabed pore pressures a n d stresses vs. d e p t h below seabed a n d time.
sient and a residual part (see Fig. 2). Madsen (1978) and Yamamoto (1978) analyzed the development of transient pore-water pressure and effective stresses using the conventional two-dimensional consolidation equation for a compressible pore fluid in a compressible porous medium. For every point of the porous bed there is a unique harmonic variation of the transient pore-water pressure with time, depending on the wave loads. The residual pore-water pressures are generated by the cyclic shear stresses developed by the waves. A method for determining residual pore-water pressures, based on the liquefaction potential analysis for earthquake loading of Seed et al. (1976), has been developed by Seed and Rahman (1978). This method considers a simultaneous build-up and partial dissipation of residual
119 pore-water pressures at several depths under one-dimensional conditions. It provides the net increase of pore-water pressures with increasing number of wave cycles. Generation of residual pore-water pressure is based on a semiempirical approach which uses cyclic test results, while the partial pore-pressure dissipation follows a one-dimensional consolidation theory. The method of analysis presented in the following sections provides the variation of the total excess pore-water pressures, i.e. the combined transient and residual pressures, with increasing number of wave cycles using a two-dimensional model. This method can be used to investigate the liquefaction potential of offshore sites. The liquefaction potential of a North Sea offshore site is analyzed as an example to study the effect of several parameters such as water depth, relative density of the seabed, vertical and horizontal permeabilities, compressibility and depth below the seabed. The approach adopted in the description of the pore-water pressure generation was checked by means of cyclic triaxial tests. PORE PRESSURE GENERATION-DISSIPATION
Wave characteristics Although design storms contain large numbers of waves of several different heights, the analysis is simplified by treating the storm as a sequence of an equivalent number of uniform waves. The intensity and the number of waves of this equivalent uniform storm are established following a procedure similar to that applied to earthquake waves (Seed and Idriss, 1971; Lee and Focht, 1975). This procedure utilizes the shear stress induced by the different components of the storm at the surface of the seabed and an experimental curve for the soil which relates the applied cyclic stresses to the number of cycles required to cause liquefaction. An equivalent design storm is described by the number of waves Neq, the wave height H, the wave length L and the wave period T (see Fig. 1 ). The wave profile is described by the following linear wave theory equation:
y(x,t)= Hcos f2~x
)
(1)
Stresses induced by waves The seafloor soils which are susceptible to liquefaction are in general finer than gravel and the pressures exerted by the waves on the surface of the seabed can be accurately determined assuming that the seabottom is impermeable (Madsen, 1978). For constant water depth, d, the profile of wave pressures on the seabottom is described by the following equation:
120
p(x,t)-2
cosh
(2dTr/L) cos
--
(2)
or:
p(x,t) =Po cos(
x
(3)
wherepo is the amplitude of the seabottom pressure. The vertical normal stress av, the horizontal normal stress ah and the shear stress Zvh, induced by this seabed pressure at a depth z below seabed, are determined from the following equations (Ishihara and Yamazaki, 1984):
av(X,z,t)=po(1 +--~--)e 2rez~ - ~2/z'L cos~f27CXL ~t)
(4)
h,XZ , o(1
(5)
Zvh(X,Z,t)=po--~-e
° "'\ L
(6)
From Eqs. 4 and 5 we find that the mean normal stress, a, is:
a(x,z,t)=Poe_2~z/Lcos(2Lx ~t)
(7)
Similarly, Eq. 6 shows that the amplitude of the shear stress, Zvh,which is equal to the deviatoric stress ( a l - a 3 )/2 is given by the equation: "Cvh( Z ) =po e-2nz/L
(8)
Porepressure build-up and dissipation The application of cyclic normal and shear stresses to a soil element develops both transient and residual pore-water pressures. Cyclic shear stresses tend to decrease the soil volume by losing some of the pore water. If the permeability of the soil is very low this drainage procedure is prevented, resulting in buildup of residual pore-water pressure. The residual pore pressure increases with increasing number of stress cycles to a level at which the effective normal stress becomes zero, i.e. the non-cohesive soil loses its strength and liquefies. If the soil has some permeability, the large period of sea-waves (compared to earthquake waves) allows some drainage to take place between consecutive stress cycles and delays the development of residual pore-water pressure and liquefaction. The progressive pore-water pressure build-up at a depth zi below the seabed is schematically demonstrated in Fig. 2. The mean normal total pres-
121
sure az (Eq. 7 ) fluctuates about an initial, O'oi, value which corresponds to still water. Similarly, the pore-water pressure, ui, fluctuates with time showing a transient and a residual part until finally, the effective stress (a~ =ai-ui) becomes zero. The two-dimensional problem of migration of pore fluid in a porous medium is described by the following equation: KxO2U
- 02U
~ x 2 + K Z ~ z2=yw
O(~.x-[-ez)
Ot
(9)
where Kx and Kz are the principal permeabilities in the x and z directions, respectively, ?w is the unit weight of the water and ex and ez are the normal strains. Following the reasoning presented by Seed and Rahman (1978) for onedimensional consolidation in which ~ represents a function giving the rate of pore-pressure generation caused by the wave-induced cyclic shear stresses under undrained conditions, the dissipated pore pressure during a time interval At will be (Aa+ ~.At-zlu), where Aa is the change in pore pressure due to the change in mean total stress and Au is the net pore pressure change. If we consider that the pore fluid is incompressible and that the normal strains ex and ez are developed by the change in effective bulk stress (Aa' =Au-~,.At-Aa), the change in strain during the time interval At becomes:
A(ex +ez) =my (Au-v/.At-Aa)
(10)
where mv is the coefficient of volume compressibility which increases when the pore pressure increases (Martin, 1975; Finn et al., 1983). The assumption of incompressible fluid, compared to the grain skeleton, adopted in the analysis, is considered satisfactory for the relatively recent and not extremely dense soil deposits which are susceptible to liquefaction. But even for denser soil deposits this assumption is still reasonable. As pointed out by Madsen {1978) the porefluid compressibility is mainly due to gas bubbles present in the soil skeleton and unless the water is very shallow the gas in the water is forced into solution. Equations 9 and 10 are combined to the following:
02u 02u [ Ou Oa'~ Kx-~x2+ Kz~z2=Ywmv~-[-~'--~) The term
(11)
Oa/Ot can be obtained by differentiating Eq. 7 i.e.:
-~=po-~e-OCT 27r 2~z'L . { 21rX , sm[. Z 2~t) _
(12)
The function ~ can be determined from undrained cyclic tests and in the absence of such tests, from the following relationship between generated pore
122
pressure under undrained conditions Ug, and the number of stress cycles, N, proposed by Seed et al. (1976): Ug=
2a~ 7~
arcsin (N IN1) °n
(13)
where N] is the number of stress cycles which cause liquefaction under undrained conditions. The above relationship is plotted in Fig. 3 together with the results of four cyclic triaxial tests, performed on samples of a fine uniform silty sand. Differentiation of Eq. 13 yields:
Oug a'o N~q 1 ~u= Ot--0.7rTD N1 sin°4(~Ug/2a'o)COS(~Ug/2a'o)
(14)
where TD is the storm duration and N~q is the equivalent number of uniform stress cycles. The number of stress cycles N1 which causes liquefaction can be determined from cyclic laboratory tests or alternatively if such tests are not available, from the following empirical equation (Faccioli, 1973): T
aG- D r a N { -b
(15)
where (T/a'o) is the cyclic stress ratio, Dr the relative density of the non-cohesive soil and a and b empirical coefficients with typical values a = 0.48 and b = 0.2. Equation 15 is plotted in Fig. 6 together with results of the cyclic tests 1.o
,
"b°
,
]
~ .
D r - l O 0 % , ~ / d ' o = 0.3 Dr= 55"/., ~/o~)=0.2
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*
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--'-Dr= 8 0 % , t / ( ~ = 0 . 3 ........ Dr= 80"/. , ~ / o ~ =0.2
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L~
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I
// / /
/'/
.
W
,~;" /,/
/
"/
/
//
/
w
oQ.
,( 0.2 &U n
/ 0
, 0.2
0.4 0.6 0.8 CYCLE RATIO , NINt
F i g . 3. R a t e o f p o r e - p r e s s u r e g e n e r a t i o n .
1.O
123
reported earlier in Fig. 3. It is noted that the agreement between Eqs. 13 and 15 and the experimental results is very satisfactory. Incorporating Eqs. 12 and 14 into Eq. 11, the problem reduces to the solution of this latter equation, by means of a finite-difference computer program. To achieve this, a two-dimensional consolidation computer program was adapted to the liquefaction problem requirements, so that the pore-water pressure variation with time under cyclic loading could be determined. APPLICATION
The method described in the previous section was used to evaluate the liquefaction potential of an offshore North Sea site. A geotechnical site investigation revealed that the soil profile consists of a 12 m thick layer of fine uniform sand which overlies a stiff clay layer. Grain-size distribution curves for the sand are presented in Fig. 4. Cone penetration tests conducted at the site indicated that the 12 m thick sand layer is compacted to a relative density higher than 60%. In order to study the effect of relative density on the liquefaction potential, three different relative densities (50, 60 and 70% ) were examined. Similarly, the effect of soil compressibility on liquefaction was studied by examining two values of the coefficient of volume compressibility, my (1.2 × 10 -3 m 2 (KN) - 1 and 3.0 × 10 -3 m 2 (KN) - 1). The value of my was considered to increase with increasing pore pressure as recommended by Martin (1975). The water depth at the site, d, was found to be about 100 m and the analysis was performed for the typical 100 year North Sea storm of Table 1 in which the maximum wave height is 25 m (Bjerrum, 1973). The non-uniform stresses developed by the waves of this storm were converted to an equivalent number
':o°lllL[ll ,11111 ~1~
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illlill
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ililill
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PARTICLE
I
SAND
Fig. 4. Particle-size distribution.
D I A M E T E R IN MM I SILT I
c~,Y
I
124 TABLE 1 Typical North Sea storm wave data Wave height (m)
Waveperiod (s)
Numberof waves
25.0 22.0 18.0 14.1 10.1 6.1
13.5 13.4 13.2 12.5 11.5 10.0
3 32 121 282 471 481
o_
100
I
I
I
l
I
Cyclic Stress Ratio 0.2
e~
Relative
Density
100
.II j
P
80%
hi
50
0
Q.
bJ I.) X bd
0
O 0
1 TIME
2 (MINUTES)
3
4
Fig. 5. Typical cyclictriaxial test results. of uniform stress cycles using the stresses created by 22 m waves as a reference. To study the effect of water depth on pore-water pressure build-up and liquefaction potential, water depths of 50, 65, 80 and 100 m were used in the analysis. The equivalent number of uniform 22 m waves computed for these water depths were 119, 106, 97 and 90 respectively. The pore-water pressure generation under undrained conditions was studied by means of cyclic triaxial tests which were conducted on sand samples compacted to three different relative densities, using several cyclic stress ratios. Typical results obtained from one of these tests, corresponding to a relative density of 80% and a cyclic shear stress ratio of 0.2 appear in Fig. 5. The results of all the cyclic tests are presented in Fig. 6 in the form of three relationships (Dr--55, 80 and 100% ) between cyclic shear stress ratio and number of cycles required to cause liquefaction. On the same figure are also plotted the three curves obtained with Eq. 15 for a = 0 . 6 5 and b=0.2. Cyclic tests conducted by Ishihara and Yamazaki (1984) in which the principal stresses were rotated continuously, simulating real wave loading, demonstrated t h a t the shear strength obtained from cyclic triaxial tests is overestimated by about 30%.
125 0.5
]
[
I
I t[lll
E q u a t i o n (15) Triaxial tests
,,,
0° . 43
.
._%
-55
1
10 NUMBER
OF
100 CYCLES,
lOoO NL
Fig. 6. Number of stress cycles to cause liquefaction.
Based on these findings the coefficient a was reduced by 30% (i.e. a = 0.45 and b=0.2). RESULTS
Typical results of pore-water pressure evolution with respect to number of wave cycles are presented in Fig. 7. These results correspond to 50 m water depth, relative density of 50%, isotropic permeability of 1 m day -1 and to a point 5 m below seabed. It is clearly demonstrated that the excess pore-water pressure has a transient and a residual part from which the latter increases progressively until, after 14 cycles, the pore-water pressure becomes equal to the total stress determined by Eq. 7. The effective stress at that time is zero and therefore the soil has no strength and liquefies. To illustrate the effect of permeability, relative density, water depth, depth below seabed, and soil compressibility on pore-water pressure build-up and liquefaction potential, a parameter called Effective Stress Ratio (ESR) is plotted vs. the number of wave cycles in Figs. 8, 9 and 10. This parameter shows the reduction in effective stress and is defined as: !
ESR=a_~t _ a-u, fro
(16)
(7o
where a~ is the initial effective stress and a[ is the current effective stress. The parameter ESR is equal to one at the beginning of the storm and equal to zero when liquefaction occurs. The effect of permeability and relative density on the liquefaction potential at a depth of one meter below seabed for 50 m water depth, is illustrated in Fig. 8. Relative densities of 50, 60 and 70% and permeabilities of 0.1 and 5 m d a y - 1
126
t3
E
d =50m Dr-Q5 K x . K y = l m d a y -1 mv=l.2xl(} 3 m2 (K N71 z=Sm
w b.J
700
0 650 600
'i//..,'
hi n, 5 5 0 v) ~9 I,, S O 0 n~ n Ld rr
,, ~,
, ;
0
1
IIj
,I,
l
2
0 fi_
; II
: ~rI
l.[' ll it '..:
,,
v
;!
V
3 4 5 6 7 8 N U M B E R OF C Y C L E S
9 10 (N)
F! -
v
11
v
12
13
14
Fig. 7. Typical pore-water pressure build-up.
U3
hi
cl '=50m mv =1.2x1(j3 m2(KN)-1 z =lm
' Kx ='Ky=C; . . . . Kx=Ky =1 m d a y -1 . . . . . K x = K y = 5 m d a y -1
o
F< ¢r Oq 1.0 t,O
'" 0.8
~:~._
D_r=0.7
D~=O.6
~%,,'~'.. , . _ -
m 0.6 w
V-o.4 U w
'\
\D~-O.O
\
... u. 0.2 LL Dr= ?.5/ ' ~:)r =0~.~' - : _ _ , ,~Dr 0"7 ", UJ (~C 0 4 8 12 16 20 24 2 8 32 3 6 4 0 4 4 N U M B E R OF C Y C L E S (N)
48
52
56
Fig. 8. Effect of relative density and permeability on liquefaction. were examined and it is found that both parameters have a very significant effect on pore-water pressure generation. Although large permeabilities (Kx = Ky = 5 m d a y - 1 ) prevented liquefaction whatever the relative density of the soil was, the effective stress reduction was higher in the lower relative densities. For an isotropic permeability of 1 m day -1 (typical for a fine silty sand), liquefaction occurs after 11 cycles if D r = 0 . 5 0 and after 44 cycles if Dr=0.6, while for Dr=0.7 the storm examined here is unable to cause liquefaction. It should be noted that the importance of horizontal permeability was found to be rather insignificant on pore-water pressure generation and liquefaction. The effect of depth below seabed on pore-water pressure build-up and liquefaction potential is shown in Fig. 9. Although the rate of the downward excess pore-water pressure propagation is higher for the lower permeabilities, as
127 I
i
,
,
,
,
0
,
i
K'=Ky=O
d =50 m rnv=l.2xl~3 m2(KN)-I D r =0.6
.... .....
Kx =Ky =1 m day -1 Kx =Ky =5 m clay -1
a: 1.0 ~
t/)
',' 0.8 n~
.
.
:
8
12
...__,~..~ ~.z.==_gm
.= z = S m
t~ 0.6 bl
_> o4 ~ 0.2 ~ 0.0 0
U.I
4
16
20
24
28
32
36 4 0
44
40
52
56
NUMBER OF CYCLES (N)
Fig. 9. Effect of depth below seabed and permeability on liquefaction.
,
t~
,
,
,
Dr =0,6 Kx=Ky=lm day -1 z =lm
o
.
.... .....
.
.
.
.
.
my =1.2x1(~3 m2 (KN) -1 mv=3.0x163 m=(KN) -1
b-
< n-
==~,(~=100 m 1.0 ~ . ~ '~'~"~=,~:=.d = SO m
o8
o6[
" ~ - ' : .... ~a.5o m
0, F ~0
ul
. =65m
. . . .
0
..... .... ,
.
4
8
,
,
,
x.
,
,
,
12 16 20 24 28 32 36 40 44 NUMBER OF: CYCLES ( N )
48
52
56
Fig. 10. Effect of water depth and soil compressibility on liquefaction.
soon as the first point liquefies, liquefaction advances faster to the deeper points when the permeability is high. Liquefaction reaches the depth of 5 m, 11 cycles after the liquefaction of the first meter when Kx=Ky=O,while only 6 cycles are required, after the first meter is liquefied, for the liquefaction to reach the depth of 5 m when Kx = Ky = 1 m d a y - 1. Figure 10 demonstrates the importance of water depth and coefficient of volume compressibility. It is clearly shown that the water depth has an extremely significant effect on pore-water pressure generation and liquefaction. For d = 50 m liquefaction occurs for both values of my in contrast to the greater water depths in which the stresses developed by the waves are unable to develop significant excess pore-water pressures and liquefaction. Finally, it is noted that the results presented in Figs. 8, 9 and 10 show clearly that there is no liquefaction danger for the site examined in this study, i.e. 100
128 m w a t e r depth, relative d e n s i t y above 60% a n d a n isotropic p e r m e a b i l i t y of about 1 m day-1. CONCLUSIONS A n u m e r i c a l m e t h o d has b e e n d e v e l o p e d for t h e e v a l u a t i o n o f p o r e - w a t e r pressure g e n e r a t i o n a n d liquefaction p o t e n t i a l in t h e seafloor due to cyclic wave action. T h e analysis includes t h e d e v e l o p m e n t o f b o t h t r a n s i e n t a n d residual p o r e - w a t e r p r e s s u r e s due to n o r m a l a n d s h e a r stresses, respectively, a n d t h e s i m u l t a n e o u s p a r t i a l - p r e s s u r e dissipation. T h e m e t h o d was applied to d e t e r m i n e t h e liquefaction p o t e n t i a l of a n offs h o r e site for w h i c h in-situ a n d cyclic l a b o r a t o r y soil t e s t s were p e r f o r m e d . T h e i m p o r t a n c e o f several p a r a m e t e r s such as t h e soil relative density, the soil p e r m e a b i l i t y , t h e soil d e f o r m a b i l i t y a n d t h e w a t e r d e p t h on p o r e - w a t e r pressure g e n e r a t i o n were i n v e s t i g a t e d b y m e a n s of a p a r a m e t r i c study. T h i s s t u d y revealed t h a t soil p e r m e a b i l i t y has a v e r y significant influence on p o r e - w a t e r p r e s s u r e g e n e r a t i o n a n d l i q u e f a c t i o n because high p e r m e a b i l i t i e s p r e v e n t the d e v e l o p m e n t of excess p o r e - w a t e r pressure. Very i m p o r t a n t is also the influence of w a t e r d e p t h followed by t h e i n f l u e n c e o f relative density. REFERENCES Biot, M.A., 1941. General theory of three dimensional consolidation. J. Appl. Phys., 12: 155-164. Bjerrum, L., 1973. Geotechnical problems involved in foundations of structures in the North Sea. Geotechnique, 23 (3): 319-358. Faccioli, E., 1973. A stochastic model for predicting seismic failure in a soil deposit. Earthquake Eng. Struct. Dyn., 1: 293-307. Finn, W.D.L., Siddharthan, R. and Martin, G.R., 1983. Response of seafloor to ocean waves. J. Geotech. Eng. Div., ASCE, 109 (4): 556-572. Ishihara, K. and Yamazaki, A., 1984. Analysis of wave-induced liquefaction in seabed deposits of sand. Soils Found., 24(3): 85-100. Lee, K.L. and Focht, J.A., 1975. Liquefaction potential at Ekofisk tank in North Sea. J. Geotech. Eng. Div., ASCE, 101(1): 1-18. Madsen, O.S., 1978. Wave-induced pore pressures and effective stresses in a porous bed. Geotechnique, 28(4): 377-393. Martin, P.P., 1975. Non-linear methods for dynamic analysis of ground response. Ph.D. thesis, Univ. California, Berkeley, Calif., pp. 307-310. Rahman, M.S. and Jaber, W.Y., 1986. A simplified drained analysis for wave-induced liquefaction in ocean floor sands. Soils Found., 26 (1): 57-68. Seed, H.B. and Idriss, I.M., 1971. Analysis of soil liquefaction: Niigata earthquake. J. Soil. Mech. Found. Div., ASCE, 97(9): 1249-1273. Seed, H.B. and Rahman, M.S., 1978. Wave-induced pore pressure in relation to ocean floor stability of cohesionless soils. Mar. Geotechnol., 3 (2): 123-150. Seed, H.B., Martin, P.O. and Lysmer, J., 1976. Pore-water pressure changes during soil liquefaction. J. Geotech. Eng. Div., ASCE, 102 (4): 323-346. Yamamoto, T., 1978. Sea bed instability from waves. Proc. 10th Offshore Tech. Conference, Houston, OTC3262: 1819-1824.
117
Numerical Analysis of Liquefaction Potential of Partially Drained Seafloors STEFANOS TSOTSOS 1, MICHAEL GEORGIADIS1 and ANASTASIA DAMASKINIDOU2
1Department of Civil Engineering, University of Thessaloniki, 540 06 Thessaloniki (Greece) 2Department of Rural Engineering, University of Thessaloniki, 540 06 Thessaloniki (Greece) (Received June 27, 1988; revised and accepted November 29, 1988)
ABSTRACT Tsotsos, S., Georgiadis, M. and Damaskinidou, A., 1989. Numerical analysis of liquefaction potential of partially drained seafloors. Coastal Eng., 13: 117-128. The cyclic normal and shear stresses induced in the seafloor sands by sea waves, generate transient and residual pore-water pressures. When the pore pressure becomes equal to the total normal stress the soil liquefies and loses its strength. This paper presents a numerical method for evaluating pore-water pressure build-up and liquefaction potential of offshore sites, in which partial drainage is implemented through a two-dimensionalconsolidation model. The effect of soil permeability, relative density and deformability on pore-pressure generation is studied and an application of the method to evaluate the liquefaction potential of a North Sea site is examined.
INTRODUCTION
The liquefaction potential of a non-cohesive seafloor under the action of storm waves has received increasing attention in recent years with respect to the design of offshore pipelines, anchors and offshore platforms. The waves induce a sequence of cyclic shear stresses in the soil below the seafloor and progressively increase the pore-water pressure. This, depending on a number of soil and wave characteristics, may lead to the liquefaction of the seabed and therefore create problems to any installations founded in the area. The prediction of pore-water pressures and shear stresses generated in the seabed is a complex problem and all computation methods are based on simplifying assumptions (Madsen, 1978; Seed and Rahman, 1978; Finn et al., 1983; Ishihara and Yamazaki, 1984; Rahman and Jaber, 1986). Storm waves are considered as a series of harmonic waves, while linear wave theory and the assumption of a rigid and impermeable seafloor are usually used in the computation of the harmonic variation of the seabed pressure p (see Fig. 1 ). The pore-water pressure generated by cyclic wave loading consists of a tran0378-3839/89/$03.50
© 1989 Elsevier Science Publishers B.V.
118
k./?" kL/
/st,, : . r
:..-. ( . : - . S e a b e d
k . / lokJ:/_k_g
x2/
]- : : . ..:.'.'.J: : . ..:..$ v.........::..
:........
::
:":':'.:-'.'.]:'ii:':"~ vh:'.'.'.'.'.'-'.'-'.'-'." :"'":"'::::::::.":.':::'.'.'.'.~ h . ".'.'.'."i :'.:.:.:.:. . . . . . . . . . . . . . . . . . . . . .".'.'.'.'.'.'.'.'.'::ii::.: .... z ~ " " .': : .'. .......... $~o".".':':
• -':'.'.'.: " ' . : : : ' "
.
..................
.
.
.
.
.
.
.
.
.
.
.
Z .......................
Fig. 1. Wave characteristics a n d seabed stresses.
Sea be d
CroPU O P
(7 i , u ~
zi
Z~
C ¢ q~
tl
tT
Fig. 2. Seabed pore pressures a n d stresses vs. d e p t h below seabed a n d time.
sient and a residual part (see Fig. 2). Madsen (1978) and Yamamoto (1978) analyzed the development of transient pore-water pressure and effective stresses using the conventional two-dimensional consolidation equation for a compressible pore fluid in a compressible porous medium. For every point of the porous bed there is a unique harmonic variation of the transient pore-water pressure with time, depending on the wave loads. The residual pore-water pressures are generated by the cyclic shear stresses developed by the waves. A method for determining residual pore-water pressures, based on the liquefaction potential analysis for earthquake loading of Seed et al. (1976), has been developed by Seed and Rahman (1978). This method considers a simultaneous build-up and partial dissipation of residual
119 pore-water pressures at several depths under one-dimensional conditions. It provides the net increase of pore-water pressures with increasing number of wave cycles. Generation of residual pore-water pressure is based on a semiempirical approach which uses cyclic test results, while the partial pore-pressure dissipation follows a one-dimensional consolidation theory. The method of analysis presented in the following sections provides the variation of the total excess pore-water pressures, i.e. the combined transient and residual pressures, with increasing number of wave cycles using a two-dimensional model. This method can be used to investigate the liquefaction potential of offshore sites. The liquefaction potential of a North Sea offshore site is analyzed as an example to study the effect of several parameters such as water depth, relative density of the seabed, vertical and horizontal permeabilities, compressibility and depth below the seabed. The approach adopted in the description of the pore-water pressure generation was checked by means of cyclic triaxial tests. PORE PRESSURE GENERATION-DISSIPATION
Wave characteristics Although design storms contain large numbers of waves of several different heights, the analysis is simplified by treating the storm as a sequence of an equivalent number of uniform waves. The intensity and the number of waves of this equivalent uniform storm are established following a procedure similar to that applied to earthquake waves (Seed and Idriss, 1971; Lee and Focht, 1975). This procedure utilizes the shear stress induced by the different components of the storm at the surface of the seabed and an experimental curve for the soil which relates the applied cyclic stresses to the number of cycles required to cause liquefaction. An equivalent design storm is described by the number of waves Neq, the wave height H, the wave length L and the wave period T (see Fig. 1 ). The wave profile is described by the following linear wave theory equation:
y(x,t)= Hcos f2~x
)
(1)
Stresses induced by waves The seafloor soils which are susceptible to liquefaction are in general finer than gravel and the pressures exerted by the waves on the surface of the seabed can be accurately determined assuming that the seabottom is impermeable (Madsen, 1978). For constant water depth, d, the profile of wave pressures on the seabottom is described by the following equation:
120
p(x,t)-2
cosh
(2dTr/L) cos
--
(2)
or:
p(x,t) =Po cos(
x
(3)
wherepo is the amplitude of the seabottom pressure. The vertical normal stress av, the horizontal normal stress ah and the shear stress Zvh, induced by this seabed pressure at a depth z below seabed, are determined from the following equations (Ishihara and Yamazaki, 1984):
av(X,z,t)=po(1 +--~--)e 2rez~ - ~2/z'L cos~f27CXL ~t)
(4)
h,XZ , o(1
(5)
Zvh(X,Z,t)=po--~-e
° "'\ L
(6)
From Eqs. 4 and 5 we find that the mean normal stress, a, is:
a(x,z,t)=Poe_2~z/Lcos(2Lx ~t)
(7)
Similarly, Eq. 6 shows that the amplitude of the shear stress, Zvh,which is equal to the deviatoric stress ( a l - a 3 )/2 is given by the equation: "Cvh( Z ) =po e-2nz/L
(8)
Porepressure build-up and dissipation The application of cyclic normal and shear stresses to a soil element develops both transient and residual pore-water pressures. Cyclic shear stresses tend to decrease the soil volume by losing some of the pore water. If the permeability of the soil is very low this drainage procedure is prevented, resulting in buildup of residual pore-water pressure. The residual pore pressure increases with increasing number of stress cycles to a level at which the effective normal stress becomes zero, i.e. the non-cohesive soil loses its strength and liquefies. If the soil has some permeability, the large period of sea-waves (compared to earthquake waves) allows some drainage to take place between consecutive stress cycles and delays the development of residual pore-water pressure and liquefaction. The progressive pore-water pressure build-up at a depth zi below the seabed is schematically demonstrated in Fig. 2. The mean normal total pres-
121
sure az (Eq. 7 ) fluctuates about an initial, O'oi, value which corresponds to still water. Similarly, the pore-water pressure, ui, fluctuates with time showing a transient and a residual part until finally, the effective stress (a~ =ai-ui) becomes zero. The two-dimensional problem of migration of pore fluid in a porous medium is described by the following equation: KxO2U
- 02U
~ x 2 + K Z ~ z2=yw
O(~.x-[-ez)
Ot
(9)
where Kx and Kz are the principal permeabilities in the x and z directions, respectively, ?w is the unit weight of the water and ex and ez are the normal strains. Following the reasoning presented by Seed and Rahman (1978) for onedimensional consolidation in which ~ represents a function giving the rate of pore-pressure generation caused by the wave-induced cyclic shear stresses under undrained conditions, the dissipated pore pressure during a time interval At will be (Aa+ ~.At-zlu), where Aa is the change in pore pressure due to the change in mean total stress and Au is the net pore pressure change. If we consider that the pore fluid is incompressible and that the normal strains ex and ez are developed by the change in effective bulk stress (Aa' =Au-~,.At-Aa), the change in strain during the time interval At becomes:
A(ex +ez) =my (Au-v/.At-Aa)
(10)
where mv is the coefficient of volume compressibility which increases when the pore pressure increases (Martin, 1975; Finn et al., 1983). The assumption of incompressible fluid, compared to the grain skeleton, adopted in the analysis, is considered satisfactory for the relatively recent and not extremely dense soil deposits which are susceptible to liquefaction. But even for denser soil deposits this assumption is still reasonable. As pointed out by Madsen {1978) the porefluid compressibility is mainly due to gas bubbles present in the soil skeleton and unless the water is very shallow the gas in the water is forced into solution. Equations 9 and 10 are combined to the following:
02u 02u [ Ou Oa'~ Kx-~x2+ Kz~z2=Ywmv~-[-~'--~) The term
(11)
Oa/Ot can be obtained by differentiating Eq. 7 i.e.:
-~=po-~e-OCT 27r 2~z'L . { 21rX , sm[. Z 2~t) _
(12)
The function ~ can be determined from undrained cyclic tests and in the absence of such tests, from the following relationship between generated pore
122
pressure under undrained conditions Ug, and the number of stress cycles, N, proposed by Seed et al. (1976): Ug=
2a~ 7~
arcsin (N IN1) °n
(13)
where N] is the number of stress cycles which cause liquefaction under undrained conditions. The above relationship is plotted in Fig. 3 together with the results of four cyclic triaxial tests, performed on samples of a fine uniform silty sand. Differentiation of Eq. 13 yields:
Oug a'o N~q 1 ~u= Ot--0.7rTD N1 sin°4(~Ug/2a'o)COS(~Ug/2a'o)
(14)
where TD is the storm duration and N~q is the equivalent number of uniform stress cycles. The number of stress cycles N1 which causes liquefaction can be determined from cyclic laboratory tests or alternatively if such tests are not available, from the following empirical equation (Faccioli, 1973): T
aG- D r a N { -b
(15)
where (T/a'o) is the cyclic stress ratio, Dr the relative density of the non-cohesive soil and a and b empirical coefficients with typical values a = 0.48 and b = 0.2. Equation 15 is plotted in Fig. 6 together with results of the cyclic tests 1.o
,
"b°
,
]
~ .
D r - l O 0 % , ~ / d ' o = 0.3 Dr= 55"/., ~/o~)=0.2
C) 0 . 8 ' -'
*
,("
--'-Dr= 8 0 % , t / ( ~ = 0 . 3 ........ Dr= 80"/. , ~ / o ~ =0.2
,//~'25/// / I
L~
"
/.4.." ../ .:
I
// / /
/'/
.
W
,~;" /,/
/
"/
/
//
/
w
oQ.
,( 0.2 &U n
/ 0
, 0.2
0.4 0.6 0.8 CYCLE RATIO , NINt
F i g . 3. R a t e o f p o r e - p r e s s u r e g e n e r a t i o n .
1.O
123
reported earlier in Fig. 3. It is noted that the agreement between Eqs. 13 and 15 and the experimental results is very satisfactory. Incorporating Eqs. 12 and 14 into Eq. 11, the problem reduces to the solution of this latter equation, by means of a finite-difference computer program. To achieve this, a two-dimensional consolidation computer program was adapted to the liquefaction problem requirements, so that the pore-water pressure variation with time under cyclic loading could be determined. APPLICATION
The method described in the previous section was used to evaluate the liquefaction potential of an offshore North Sea site. A geotechnical site investigation revealed that the soil profile consists of a 12 m thick layer of fine uniform sand which overlies a stiff clay layer. Grain-size distribution curves for the sand are presented in Fig. 4. Cone penetration tests conducted at the site indicated that the 12 m thick sand layer is compacted to a relative density higher than 60%. In order to study the effect of relative density on the liquefaction potential, three different relative densities (50, 60 and 70% ) were examined. Similarly, the effect of soil compressibility on liquefaction was studied by examining two values of the coefficient of volume compressibility, my (1.2 × 10 -3 m 2 (KN) - 1 and 3.0 × 10 -3 m 2 (KN) - 1). The value of my was considered to increase with increasing pore pressure as recommended by Martin (1975). The water depth at the site, d, was found to be about 100 m and the analysis was performed for the typical 100 year North Sea storm of Table 1 in which the maximum wave height is 25 m (Bjerrum, 1973). The non-uniform stresses developed by the waves of this storm were converted to an equivalent number
':o°lllL[ll ,11111 ~1~
so ILllli] 7oillllll oo I1111] so[liITI z 4oilllil 3ollllli 2o II1111 ,o illlli ~llil 100 I ~°""'-
I
,.,~,~
illlill
IINIII [lillll
IIlllli
[llltll IlL[Ill Illllll ]lll]ll
ililill
10
]11111 Iil111[I IIIIII]1 IIIIIII T~,Illlll II1[1111 IIIIIIII II1[111 ~11[11] IIL[IIII I1[11111 IIIIIII lllllll 11111111 11111111 IIIIIII ,~111111 11111111 IIIIII IIIIIII i~lllll illlilVl IIIIIII i~JIII IIIlilil [1111111 Illllll llNII iiliilli [liilill IIII]11 YI~{.LI liilllIi [llllil[ l I 0.001 I111111 0.1II~tl"Ft-~[I illlllii 1.o 0.Ol
PARTICLE
I
SAND
Fig. 4. Particle-size distribution.
D I A M E T E R IN MM I SILT I
c~,Y
I
124 TABLE 1 Typical North Sea storm wave data Wave height (m)
Waveperiod (s)
Numberof waves
25.0 22.0 18.0 14.1 10.1 6.1
13.5 13.4 13.2 12.5 11.5 10.0
3 32 121 282 471 481
o_
100
I
I
I
l
I
Cyclic Stress Ratio 0.2
e~
Relative
Density
100
.II j
P
80%
hi
50
0
Q.
bJ I.) X bd
0
O 0
1 TIME
2 (MINUTES)
3
4
Fig. 5. Typical cyclictriaxial test results. of uniform stress cycles using the stresses created by 22 m waves as a reference. To study the effect of water depth on pore-water pressure build-up and liquefaction potential, water depths of 50, 65, 80 and 100 m were used in the analysis. The equivalent number of uniform 22 m waves computed for these water depths were 119, 106, 97 and 90 respectively. The pore-water pressure generation under undrained conditions was studied by means of cyclic triaxial tests which were conducted on sand samples compacted to three different relative densities, using several cyclic stress ratios. Typical results obtained from one of these tests, corresponding to a relative density of 80% and a cyclic shear stress ratio of 0.2 appear in Fig. 5. The results of all the cyclic tests are presented in Fig. 6 in the form of three relationships (Dr--55, 80 and 100% ) between cyclic shear stress ratio and number of cycles required to cause liquefaction. On the same figure are also plotted the three curves obtained with Eq. 15 for a = 0 . 6 5 and b=0.2. Cyclic tests conducted by Ishihara and Yamazaki (1984) in which the principal stresses were rotated continuously, simulating real wave loading, demonstrated t h a t the shear strength obtained from cyclic triaxial tests is overestimated by about 30%.
125 0.5
]
[
I
I t[lll
E q u a t i o n (15) Triaxial tests
,,,
0° . 43
.
._%
-55
1
10 NUMBER
OF
100 CYCLES,
lOoO NL
Fig. 6. Number of stress cycles to cause liquefaction.
Based on these findings the coefficient a was reduced by 30% (i.e. a = 0.45 and b=0.2). RESULTS
Typical results of pore-water pressure evolution with respect to number of wave cycles are presented in Fig. 7. These results correspond to 50 m water depth, relative density of 50%, isotropic permeability of 1 m day -1 and to a point 5 m below seabed. It is clearly demonstrated that the excess pore-water pressure has a transient and a residual part from which the latter increases progressively until, after 14 cycles, the pore-water pressure becomes equal to the total stress determined by Eq. 7. The effective stress at that time is zero and therefore the soil has no strength and liquefies. To illustrate the effect of permeability, relative density, water depth, depth below seabed, and soil compressibility on pore-water pressure build-up and liquefaction potential, a parameter called Effective Stress Ratio (ESR) is plotted vs. the number of wave cycles in Figs. 8, 9 and 10. This parameter shows the reduction in effective stress and is defined as: !
ESR=a_~t _ a-u, fro
(16)
(7o
where a~ is the initial effective stress and a[ is the current effective stress. The parameter ESR is equal to one at the beginning of the storm and equal to zero when liquefaction occurs. The effect of permeability and relative density on the liquefaction potential at a depth of one meter below seabed for 50 m water depth, is illustrated in Fig. 8. Relative densities of 50, 60 and 70% and permeabilities of 0.1 and 5 m d a y - 1
126
t3
E
d =50m Dr-Q5 K x . K y = l m d a y -1 mv=l.2xl(} 3 m2 (K N71 z=Sm
w b.J
700
0 650 600
'i//..,'
hi n, 5 5 0 v) ~9 I,, S O 0 n~ n Ld rr
,, ~,
, ;
0
1
IIj
,I,
l
2
0 fi_
; II
: ~rI
l.[' ll it '..:
,,
v
;!
V
3 4 5 6 7 8 N U M B E R OF C Y C L E S
9 10 (N)
F! -
v
11
v
12
13
14
Fig. 7. Typical pore-water pressure build-up.
U3
hi
cl '=50m mv =1.2x1(j3 m2(KN)-1 z =lm
' Kx ='Ky=C; . . . . Kx=Ky =1 m d a y -1 . . . . . K x = K y = 5 m d a y -1
o
F< ¢r Oq 1.0 t,O
'" 0.8
~:~._
D_r=0.7
D~=O.6
~%,,'~'.. , . _ -
m 0.6 w
V-o.4 U w
'\
\D~-O.O
\
... u. 0.2 LL Dr= ?.5/ ' ~:)r =0~.~' - : _ _ , ,~Dr 0"7 ", UJ (~C 0 4 8 12 16 20 24 2 8 32 3 6 4 0 4 4 N U M B E R OF C Y C L E S (N)
48
52
56
Fig. 8. Effect of relative density and permeability on liquefaction. were examined and it is found that both parameters have a very significant effect on pore-water pressure generation. Although large permeabilities (Kx = Ky = 5 m d a y - 1 ) prevented liquefaction whatever the relative density of the soil was, the effective stress reduction was higher in the lower relative densities. For an isotropic permeability of 1 m day -1 (typical for a fine silty sand), liquefaction occurs after 11 cycles if D r = 0 . 5 0 and after 44 cycles if Dr=0.6, while for Dr=0.7 the storm examined here is unable to cause liquefaction. It should be noted that the importance of horizontal permeability was found to be rather insignificant on pore-water pressure generation and liquefaction. The effect of depth below seabed on pore-water pressure build-up and liquefaction potential is shown in Fig. 9. Although the rate of the downward excess pore-water pressure propagation is higher for the lower permeabilities, as
127 I
i
,
,
,
,
0
,
i
K'=Ky=O
d =50 m rnv=l.2xl~3 m2(KN)-I D r =0.6
.... .....
Kx =Ky =1 m day -1 Kx =Ky =5 m clay -1
a: 1.0 ~
t/)
',' 0.8 n~
.
.
:
8
12
...__,~..~ ~.z.==_gm
.= z = S m
t~ 0.6 bl
_> o4 ~ 0.2 ~ 0.0 0
U.I
4
16
20
24
28
32
36 4 0
44
40
52
56
NUMBER OF CYCLES (N)
Fig. 9. Effect of depth below seabed and permeability on liquefaction.
,
t~
,
,
,
Dr =0,6 Kx=Ky=lm day -1 z =lm
o
.
.... .....
.
.
.
.
.
my =1.2x1(~3 m2 (KN) -1 mv=3.0x163 m=(KN) -1
b-
< n-
==~,(~=100 m 1.0 ~ . ~ '~'~"~=,~:=.d = SO m
o8
o6[
" ~ - ' : .... ~a.5o m
0, F ~0
ul
. =65m
. . . .
0
..... .... ,
.
4
8
,
,
,
x.
,
,
,
12 16 20 24 28 32 36 40 44 NUMBER OF: CYCLES ( N )
48
52
56
Fig. 10. Effect of water depth and soil compressibility on liquefaction.
soon as the first point liquefies, liquefaction advances faster to the deeper points when the permeability is high. Liquefaction reaches the depth of 5 m, 11 cycles after the liquefaction of the first meter when Kx=Ky=O,while only 6 cycles are required, after the first meter is liquefied, for the liquefaction to reach the depth of 5 m when Kx = Ky = 1 m d a y - 1. Figure 10 demonstrates the importance of water depth and coefficient of volume compressibility. It is clearly shown that the water depth has an extremely significant effect on pore-water pressure generation and liquefaction. For d = 50 m liquefaction occurs for both values of my in contrast to the greater water depths in which the stresses developed by the waves are unable to develop significant excess pore-water pressures and liquefaction. Finally, it is noted that the results presented in Figs. 8, 9 and 10 show clearly that there is no liquefaction danger for the site examined in this study, i.e. 100
128 m w a t e r depth, relative d e n s i t y above 60% a n d a n isotropic p e r m e a b i l i t y of about 1 m day-1. CONCLUSIONS A n u m e r i c a l m e t h o d has b e e n d e v e l o p e d for t h e e v a l u a t i o n o f p o r e - w a t e r pressure g e n e r a t i o n a n d liquefaction p o t e n t i a l in t h e seafloor due to cyclic wave action. T h e analysis includes t h e d e v e l o p m e n t o f b o t h t r a n s i e n t a n d residual p o r e - w a t e r p r e s s u r e s due to n o r m a l a n d s h e a r stresses, respectively, a n d t h e s i m u l t a n e o u s p a r t i a l - p r e s s u r e dissipation. T h e m e t h o d was applied to d e t e r m i n e t h e liquefaction p o t e n t i a l of a n offs h o r e site for w h i c h in-situ a n d cyclic l a b o r a t o r y soil t e s t s were p e r f o r m e d . T h e i m p o r t a n c e o f several p a r a m e t e r s such as t h e soil relative density, the soil p e r m e a b i l i t y , t h e soil d e f o r m a b i l i t y a n d t h e w a t e r d e p t h on p o r e - w a t e r pressure g e n e r a t i o n were i n v e s t i g a t e d b y m e a n s of a p a r a m e t r i c study. T h i s s t u d y revealed t h a t soil p e r m e a b i l i t y has a v e r y significant influence on p o r e - w a t e r p r e s s u r e g e n e r a t i o n a n d l i q u e f a c t i o n because high p e r m e a b i l i t i e s p r e v e n t the d e v e l o p m e n t of excess p o r e - w a t e r pressure. Very i m p o r t a n t is also the influence of w a t e r d e p t h followed by t h e i n f l u e n c e o f relative density. REFERENCES Biot, M.A., 1941. General theory of three dimensional consolidation. J. Appl. Phys., 12: 155-164. Bjerrum, L., 1973. Geotechnical problems involved in foundations of structures in the North Sea. Geotechnique, 23 (3): 319-358. Faccioli, E., 1973. A stochastic model for predicting seismic failure in a soil deposit. Earthquake Eng. Struct. Dyn., 1: 293-307. Finn, W.D.L., Siddharthan, R. and Martin, G.R., 1983. Response of seafloor to ocean waves. J. Geotech. Eng. Div., ASCE, 109 (4): 556-572. Ishihara, K. and Yamazaki, A., 1984. Analysis of wave-induced liquefaction in seabed deposits of sand. Soils Found., 24(3): 85-100. Lee, K.L. and Focht, J.A., 1975. Liquefaction potential at Ekofisk tank in North Sea. J. Geotech. Eng. Div., ASCE, 101(1): 1-18. Madsen, O.S., 1978. Wave-induced pore pressures and effective stresses in a porous bed. Geotechnique, 28(4): 377-393. Martin, P.P., 1975. Non-linear methods for dynamic analysis of ground response. Ph.D. thesis, Univ. California, Berkeley, Calif., pp. 307-310. Rahman, M.S. and Jaber, W.Y., 1986. A simplified drained analysis for wave-induced liquefaction in ocean floor sands. Soils Found., 26 (1): 57-68. Seed, H.B. and Idriss, I.M., 1971. Analysis of soil liquefaction: Niigata earthquake. J. Soil. Mech. Found. Div., ASCE, 97(9): 1249-1273. Seed, H.B. and Rahman, M.S., 1978. Wave-induced pore pressure in relation to ocean floor stability of cohesionless soils. Mar. Geotechnol., 3 (2): 123-150. Seed, H.B., Martin, P.O. and Lysmer, J., 1976. Pore-water pressure changes during soil liquefaction. J. Geotech. Eng. Div., ASCE, 102 (4): 323-346. Yamamoto, T., 1978. Sea bed instability from waves. Proc. 10th Offshore Tech. Conference, Houston, OTC3262: 1819-1824.