Waveinduced soil liquefaction
Article under review
Definition of Soil liquefaction:
When a watersaturated soil starts behaving as a fluid, losing stiffness and bearing capacity.
This is the common definition for Soil liquefaction, other definitions can be discussed in the article

Seabed soils which are not well consolidated are prone to liquefaction when subjected to pressure fluctuations. Soil liquefaction can be produced for example by earthquakes, but here we concentrate on submarine soils which are subjected to cyclic loading by waves. Structures built on freshly deposited submarine soils can collapse or sink when the seabed is liquified^{[1]}. A simplified mathematical introduction to the theory of soil liquefaction is given in the Appendix.
Contents
Two types of soil liquefaction
Two types of waveinduced soil liquefaction can be distinguished: Transient liquefaction (also called instantaneous or momentary liquefaction) and residual liquefaction. Both types can occur in poorly drained loosely packed soils, for example soils containing a high percentage of fine silty material. In welldrained soils, for example a medium/coarsesand seabed, pore pressure by cyclic loading cannot build up, but transient liquefaction is possible.
Transient liquefaction
The upper part of the seabed can be liquefied if the soil pore pressure near the bed surface is higher than the pressure exerted by the overlying water mass. This occurs when the pore pressure in the seabed follows (with some delay, depending on the soil permeability) the fluctuating water pressure exerted by surface waves. The soil top layer then experiences an upward force during the passage of the wave trough, which lifts sediment particles near the seabed surface out of the soil skeleton, see Fig. 1. The soil top layer (typically ten or a few tens of centimeters) becomes liquefied over a short period of time during which this condition prevails. One consequence is that objects deposited on the seabed can sink and become buried^{[3]}^{[1]} or float up if the density of the object is less than that of the liquified soil.
Most observations of waveinduced transient liquefaction come from laboratory experiments. There are also some studies based on field observations, in the nearshore subtidal zone and in the intertidal zone. In the latter case it was found that even a small volume fraction of air in the soil can greatly dampen the transmission of waveinduced water pressure to the pore fluid^{[4]}.
Residual liquefaction
Soils prone to liquefaction are typically watersaturated, consisting of loosely packed fine sediments – fine sand, silt and some clay – with poor drainage capacity (low hydraulic conductivity). When such a soil is subjected to cyclic waveinduced loading and associated shear deformations, the soil grains tend to rearrange such that the soil skeleton is progressively compressed. (The reverse happens for a densely compacted soil.) The resulting reduction of the pore volume is associated with an increase of the pore water pressure. When the pore pressure exceeds the pressure exerted by the load of the overlying water and soil (pore pressure higher than the initial mean normal effective stress, taking soil cohesion into account), the contact friction points between soil grains are broken and the soil skeleton collapses. The soil grains become unbound and free to move, carried by the pore water (Fig. 2). The soil begins to act like a liquid, losing its stiffness and bearing capacity. Any structure built on this soil will sink or collapse^{[1]}.
Experiments in a wave flume have shown that the residual pore pressure generated by random waves can be much larger than that by regular waves with equivalent mean wave height and period. Lowfrequency waves in particular can make a significant contribution to the residual pressure^{[5]}. Other flume experiments have shown that soil liquefaction through excess pore pressure enhances sediment resuspension^{[6]}. Seabed liquefaction and seabed scour reinforce each other, as removal of top soil strengthens the upward pore pression gradient in the soil underneath^{[7]}
Liquefactionconsolidation process
The onset of liquefaction occurs first at the surface of the bed, and rapidly spreads out across the soil depth, causing the entire soil to behave like a liquid. When the liquefaction reaches the impermeable base, the soil begins to compact. The large pores between particles in the liquefied state allow the expel (drainage) of pore water and waveinduced compaction. Soil grains fall out of the liquid state, settle through the pore water until they come into contact with each other. The compaction gradually progresses from the impermeable base in the upward direction, and the entire sequence of the liquefaction/compaction process comes to an end when the compaction reaches the soil surface^{[8]}^{[9]}. After liquefaction, a loosely packed soil will not recover its initial state. The soil is compacted and much less susceptible to renewed liquefaction. The liquefactioncompaction process is depicted schematically in Fig. 3.
Influence of soil composition
Flume experiments show that the liquefaction potential strongly depends on the soil composition. Experiments by Gratchev et al. (2006^{[11]}) revealed the significant influence of plasticity on soil liquefaction resistance. Soil vulnerable to liquefaction had an open microfabric in which clay aggregates mainly gathered at the silt particle contact points, forming lowstrength 'clay bridges' that were easily destroyed during cyclic loading (Fig. 2). The contact friction between particles that maintained the soil skeleton structure was reduced by the clay aggregates, so grains could easily slip and rotate once the seabed was subjected to wave loading. Clay minerals thus enhance the plasticity and compressibility of the claysilt seabed and increases the sensitivity of the seabed to liquefaction.
Experiments with kaolinitefine sand and illitefine sand showed that the susceptibility of a soil to liquefaction increases with increasing clay content if the clay content is low (0.55%), but decreases with a higher clay content^{[12]}. At a clay content of more than 1030%, depending on the type of clay, the siltclay mixture was not susceptible to liquefaction. This can be explained by the fact that in case of high clay content the silt grains are completely encapsulated in the clay matrix and therefore cannot rearrange under cyclic shear stresses, causing resistance to liquefaction. Experiments by Kirca et al. (2014^{[13]}) showed that the sensitivity of a sandclay mixture to waveinduced liquefaction not only depends on the clay content but also on the sand grainsize. Liquefaction occurred at a much larger clay content for medium sand than for fine sand or silt. The presence of shell fragments also decreases the susceptibility of silt to liquefaction^{[10]}. Experiments for a mediumsandclay (kaolinite) seabed supporting a submerged breakwater showed that waveinduced liquefaction required a clay content greater than 40%^{[14]}. These experiment also showed that the initial water content in the soil is relevant to the liquefaction potential as it affects the pressure buildup process. Higher pore water fractions allow the residual pore pressure to accumulate faster.
The soil in an actual field situation usually has a long history of wave action, and hence it normally is a consolidated stiff soil, with low sensitivity to waveinduced liquefaction. This does not hold for freshly deposited soils. This may be the case, for example, where a pipeline is laid in a trench, and the trench is then backfilled^{[1]}.
Fluid mud
Fluid mud – a dense colloidal suspension of fine, cohesive sediments (mainly clay minerals) – can be considered liquified soil. However, cohesive soils are not liquefied by a similar process of pore pressure buildup as noncohesive or slightly cohesive soils. Cohesive soil (mud) behaves like a viscoelastic material whose viscosity decreases when a stress is applied that is strong enough (fluctuates) to break the electrochemical bounds that hold the constituent particles together^{[15]}^{[16]}. Cohesive soil has thixotropic properties: after stress removal, it gradually returns to its original state. A fluid mud layer is formed when flocforming particles settle to the seabed in sufficiently high concentration. High concentrations of suspended fine sediment often originate from previous mud bed erosion by strong waves or currents or from convergence of residual fine sediment transport related to tidal asymmetry and estuarine circulation (see Estuarine turbidity maximum). Stirring of unconsolidated mud deposits by waveinduced pressure variations is another process that can contribute to the formation of fluid mud^{[17]}. The high stresses needed in case of advanced consolidation will more likely lead to sediment erosion and suspension rather than to fluidization. A fluid mud layer can be formed when the suspended sediment settles^{[18]}. When a fluid mud layer flows along the seabed, the density (and viscosity) can increase by bed erosion or decrease by uptake (entrainment) of overlying water^{[19]}. See Fluid mud and Dynamics of mud transport.
Preventing soil liquefaction
The susceptibility of a soil to liquefaction can be checked through Standard Penetration Tests or Cone Penetration Tests. Such tests also enable the determination of soil characteristics needed to model soil behavior (see Appendix). Liquefaction risks can be reduced by measures such as replacing soft soil (expensive) or by consolidation measures. Soil consolidation can be done for example by vibro compaction or by preloading the soft bed with coarse sediment^{[20]}. Monitoring of the consolidation process should ensure that a stable situation is reached.
Related articles
Further reading
Sumer, B.M. 2009. Liquefaction Around Marine Structures. Advanced Series on Ocean Engineering : Volume 39. World Scientific Publ. Co.
Appendix
This appendix provides a mathematical introduction to the phenomena of transient and residual liquefaction. It is not intended as a model for estimating liquefaction conditions in practice.
Transient liquefaction
According to linear wave theory (see Shallowwater wave theory) the fluctuating pressure (vertical stress) exerted on the seabed by a propagating sinusoidal wave is given by
[math]p_{seafloor} (x,t) = \rho g h + p_0 \cos \Big( 2 \pi (\large\frac{x}{L}\frac{t}{T}\normalsize)\Big) , \quad p_0 = \Large\frac{ \rho g a}{\cosh(2 \pi h/L)} \normalsize, \qquad (1) [/math]
where [math]x=[/math] coordinate wave propagation direction, [math]t=[/math] time, [math]g=[/math] gravitational acceleration, [math]\rho=[/math] seawater density, [math]a=[/math] wave amplitude, [math]L=[/math] wavelength, [math]T=[/math] wave period, [math]h=[/math] water depth.
We consider loosepacked homogeneous soil consisting of fine sand, with fractions of silt and clay. The soil characteristics are:
 [math]n[/math] porosity of the soil, i.e. the volume not occupied by solids, thus left for pore water and gaz (air)
 [math]k[/math] [m/s] soil permeability (= hydraulic conductivity), relating pore water flow and pressure gradient according to Darcy's law
 [math]G[/math] [Pa] shear modulus of soil (= modulus of rigidity = the ratio of shear stress and shear strain)
 [math]\nu[/math] Poisson ratio of soil (= amount of transversal elongation divided by the amount of vertical compression, related to the friction angle [math]\phi[/math])
 [math]\beta[/math] [Pa] bulk modulus of pore water and included air (= the ratio of pressure increase and relative volume decrease = inverse of compressibility). The water densitypressure relationship is [math]\rho = \rho_0 \exp((pp_0)/\beta)[/math] and the dependence of [math]\beta[/math] on the air fraction [math]C[/math] is [math]1/\beta = 5 \, 10^{9} \, (1C) + C/p \;[/math].
Typical values for loosely packed soil of fine sand with a small fraction of silt an clay, saturated with water and including a small fraction (order 1‰) of included air, are^{[21]}: [math]n=0.4, \; k=10^{6} m/s, \; G=10^7 Pa, \; \beta=10^8 Pa, \; \nu=0.45[/math]. As an example, we consider sea surface waves with period [math]T = 10 \, s[/math] and wavelength [math]L = 70 \, m[/math].
The fluctuating wave load applies over a large stretch of order [math]L/(2 \pi)[/math] length, which precludes substantial lateral soil dilatation. We therefore assume horizontal soil uniformity and focus in the following on the vertical compression [math]dV/dt[/math] of a small soil volume element [math]V[/math] resulting from the waveinduced varying pressure load at the seabed surface Eq. (1). The compression [math]dV/dt[/math] is due to:
 compression of the pore fluid, [math]\Large\frac{dV_1}{dt}\normalsize = V \Large\frac{n}{\beta}\frac{\partial p}{\partial t}\normalsize ,[/math] where [math]p(z,t)[/math] is the pore fluid pressure at depth [math]z[/math] below the seafloor, [math]n[/math] is the porosity and [math]\beta[/math] is the bulk modulus of the pore water (including gaz).
 pore water outflow, [math]\Large\frac{dV_2}{dt}\normalsize = V \Large\frac{\partial u}{\partial z}\normalsize = V \Large\frac{k}{\rho g}\frac{\partial^2 p}{\partial z^2}\normalsize , [/math] where [math]u[/math] is the pore water vertical flow velocity, and [math]k[/math] is the soil permeability. The second equation follows from Darcy's law.
The volume compression implies a change in the volumetric strain (also called 'bulk strain') [math]d\epsilon=\Large\frac{1}{V}\normalsize (dV_1+dV_2)[/math],
[math]\Large\frac{d \epsilon}{dt}\normalsize = \Large\frac{n}{\beta}\frac{\partial p}{\partial t}\normalsize + \Large\frac{k}{\rho g}\frac{\partial^2 p}{\partial^2 z}\normalsize . \qquad (2)[/math]
The soil is considered a poroelastic medium for which Hooke's law applies. The volumetric strain [math]\epsilon[/math] (degree of soil compression) is then linearly related to the effective stress (intergrain stress) [math]\sigma'[/math] by the formula
[math]\epsilon = m_v \sigma' = m_v (\sigma  p) , \qquad (3)[/math]
where [math]p[/math] is the pore pressure and where [math]\sigma \approx p_{seafloor} + \alpha (\rho_{soil}  \rho) g z \;[/math] is the total stress applied to a soil volume [math]V[/math] at depth [math]z[/math] near the seabed surface ([math]z \lt \lt L [/math]). The factor [math]\alpha = (1+2K_0)/3[/math], where [math]K_0 = \nu / (1  \nu) \approx 1 \sin \phi \;[/math] expresses the partial transfer of soil weight to horizontal stresses ([math]\phi =[/math] friction angle) ^{[22]}. Liquefaction occurs when the effective stress vanishes, [math]\sigma' = 0 [/math]. The compressibility modulus [math]m_v[/math] can be expressed as a function of the soil shear modulus [math]G[/math] and the Poisson ratio [math]\nu[/math], [math]\; m_v = \Large\frac{12 \nu}{2 G (1\nu)}\normalsize \approx \Large\frac{\sin \phi}{2 G}\normalsize .[/math]
Using Hooke's law, Eq. (2) can be rewritten as a differential equation from which the pore pressure [math]p(z,t)[/math] can be solved:
[math]\Large\frac{\partial p}{\partial t}\normalsize = c_s \Large\frac{\partial \sigma}{\partial t}\normalsize + c_v \Large\frac{\partial^2 p}{\partial z^2}\normalsize , \quad c_s = \Large\frac{1}{1+m}\normalsize , \quad c_v = \Large\frac{k \beta}{\rho g n}\frac{m}{1+m}\normalsize , \qquad m = \Large\frac{n}{\beta m_v}\normalsize = 2 n \Large\frac{G}{\beta}\frac{1\nu}{12 \nu}\normalsize . \qquad (4)[/math]
The time variation [math]\partial \sigma / \partial t [/math] of the total stress is due to the fluctuating pressure transmitted by the surface wave to the pore fluid. The solution of Eq. (4) for a homogeneous soil layer with pore pressure [math]p=p_{seafloor}[/math] at the seabed surface [math]z=0[/math] is
[math]p(x,z,t) = p_{seafloor} + \large\frac{m}{1+m}\normalsize p_0 \Bigg[ \cos \Big( 2 \pi (\large\frac{x}{L}\frac{t}{T}\normalsize)\Big) + e^{\Large\frac{z}{d}\normalsize} \cos \Big( 2 \pi \large (\frac{x}{L}\frac{t}{T})+\frac{z}{d}\normalsize\Big) \Bigg] .\qquad (5)[/math]
This expression is only valid near the soil surface, as the amplitude [math]p_0[/math] of the waveinduced pressure fluctuation is a gradually decreasing function of depth. The depth [math]d[/math] is given by
[math]d = \Big( \Large\frac{T \, c_v}{\pi}\normalsize \Big)^{1/2} = \Big( \Large\frac{T \, k \, \beta \, m}{\pi \, \rho \, n \, g \, (1+m)}\normalsize \Big)^{1/2} . \qquad (6)[/math]
Substitution of the values of the numerical example gives [math]m \approx 0.4 , \; d \approx[/math] 16 cm. The pore pressure gradient just below the seafloor ([math]z \lt \lt d[/math]) can be approximated by
[math]\Large\frac{\partial p(x,z,t)}{\partial z}\normalsize \approx  p_0 \Large\frac{m}{m+1}\frac{\sqrt{2}}{d}\normalsize \cos \Big( 2 \pi \large ( \frac{x}{L}\frac{t}{T} )\frac{\pi}{4} \normalsize\Big) . \qquad (7)[/math]
For liquefaction, the waveinduced pore pressure gradient near the soil surface must be larger than the submerged weight of the soil grains. This condition reads
[math]\Large\frac{m \sqrt{2}}{1+m}\frac{p_0}{d}\normalsize \gt g (\rho_{soil}  \rho) . \qquad (8)[/math]
In the numerical example, considering a water depth of 5 m, liquefaction occurs if the wave height exceeds 1.3 m. The maximum depth of liquefaction can be found from Eq. (5) by equating [math]p(z,t) = p_{seafloor} + (\rho_{soil}\rho) g z \;[/math]. Liquefaction of the top soil layer or formation of a sheet flow layer can occur already for lower waves due to lift forces generated by wave orbital flow along the seabed.
Equation (5) does not take into account the gradual buildup of pressure below the soil surface under sustained cyclic loading. The resulting residual liquefaction process is discussed below.
Residual liquefaction
For transient liquefaction, it has been assumed that the volumetric soil strain [math]\epsilon[/math] under wave loading is cyclic. According to the soil mechanics theory of Terzaghi (1925^{[23]}), the strain [math]\epsilon[/math] can gradually increase under cyclic loading if the soil is saturated but poorly drained. We assume for simplicity that the total stress [math]\sigma[/math] increases throughout the soil in proportion to the number [math]N[/math] of wave cycles,
[math]\quad \Large\frac{d \sigma}{dt}\normalsize \propto \Large\frac{p_0}{T}\normalsize , \qquad (9)[/math].
where [math]p_0[/math] is the amplitude of the fluctuating vertical stress on the seafloor given in Eq. (1). We call [math]\overline{p}(z,t)[/math] the waveaveraged pore pressure component related to the fluctuating surface water pressure. Instead of Eq. (3) we now have the following equation for [math]\overline{p}(z,t)[/math] :
[math]\Large\frac{\partial \overline{p}}{\partial t}\normalsize= B \Large\frac{p_0}{T} + c_v \Large\frac{\partial^2 \overline{p}}{\partial z^2}\normalsize , \qquad (10)[/math]
where [math]B[/math] is a proportionality constant dependent on the soil characteristics, that must be determined experimentally. An approximate solution of Eq. (8) for a homogeneous soil layer of thickness [math]0.1 L \lt D \lt 0.3 L[/math] can be found by assuming a parabolic depth dependence of the pore pressure^{[21]}. The waveaveraged boundary conditions are: for [math]z=0: \; \overline{p}(z,t)=0 \;[/math] (no pore pressure builds up at the seabed surface because pore water can flow out here) and for [math] \; z=D: \; \Large\frac{\partial \overline{p}}{\partial z}\normalsize =0 \;[/math]. The approximate solution is (exact when depthaveraged):
[math]\overline{p}(z,t) \approx z(2Dz)\Large\frac{B p_0}{2 c_v T}\normalsize \Big[ 1  e^{\Large\frac{3 c_v t}{D^2}}\normalsize \Big] . \qquad (11)[/math]
The dimensionless coefficient [math]B[/math] represents the rate of pore pressure increase due to soil compaction generated in one wave cycle by a unit applied maximum pressure load. For loosely packed fine sand this rate is of the order [math]B \approx 10^{3}[/math]; for densely packed soil [math]B[/math] is much smaller^{[21]}. The coefficient [math]B[/math] has been assumed constant in Eq. (10), but in reality it will gradually decrease during soil compaction. The formula (11) therefore overestimates values of the pore pressure derived from measurements (Fig. 3).
The condition for liquefaction near the seabed surface is
[math]\Large\frac{\partial p}{\partial z} _{z=0} \normalsize = \Large\frac{BDp_0}{c_v T}\normalsize \gt g(\rho_{soil} \rho) . \qquad (12)[/math]
In the numerical example, the coefficient of consolidation (Eq. 4) takes the value [math]c_v \approx 0.007 \, m^2/s[/math]. Taking [math]B = 10^{3}[/math] and [math]D = 10 \, m[/math], residual liquefaction occurs for wave heights exceeding 1.6 m.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Sumer, B.M., Fredsøe, J., Christensen, S. and Lind, M.T. 1999. Sinking/floatation of pipelines and other objects in liquefied soil under waves. Coast. Eng. 38: 53–90 Cite error: Invalid
<ref>
tag; name "S99" defined multiple times with different content  ↑ de Groot, M. B., Bolton, M. D., Foray P., Meijers, P., Palmer, A. C., Sandven, R., Sawicki, A. and The, T. C. 2006 Physics of Liquefaction Phenomena around Marine Structures. Journal of Waterway, Port, Coastal, and Ocean Engineering 132: 227243
 ↑ Klammler, H., Penko, A.M., Staples, T., Sheremet, A. and Calantoni, J. 2021. Observations and modeling of waveinduced burial and sediment entrainment: Likely importance of degree of liquefaction. Journal of Geophysical Research: Oceans 126, e2021JC017378
 ↑ Michallet, H., Mory, M. and PiedraCueva, I. 2009. Waveinduced pore pressure measurements near a coastal structure, J. Geophys. Res. 114, C06019
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 ↑ Sui, T., Sumer, B.M., Kirca, V.S.O., Carstensen, S., Zheng, J. and Fuhrman, D.R. 2023. Effect of history of wave exposure on seabed liquefaction. Coastal Engineering 183 (2023) 104307
 ↑ ^{10.0} ^{10.1} Sui, T., Kirca, V.S.O, Sumer, B.M., Carstensen, S. and Fuhrman, D.R. 2022. Waveinduced liquefaction in a silt and seashell mixture. Coastal Engineering 178, 104215
 ↑ Gratchev, I.B., Sassa, K., Osipov, V.I. and Sokolov, V.N. 2006. The liquefaction of clayey soils under cyclic loading. Eng. Geol. 86: 70–84
 ↑ Zang, J., Jiang, Q., Jeng, D., Zhang, C., Chen, X. and Wang, L. 2020. Experimental Study on Mechanism of WaveInduced Liquefaction of SandClay Seabed. J. Mar. Sci. Eng. 8, 66
 ↑ Kirca, V., Sumer, B.M. and Fredsøe, J. 2014. Influence of clay content on waveinduced liquefaction, J. Waterw. Port Coast. Ocean Eng. ASCE 140, 04014024
 ↑ Chavez, V., Mendoza, E., Silva, R., Silva, A. and Losada, M.A. 2017. An experimental method to verify the failure of coastal structures by wave induced liquefaction of clayey soils. Coastal Engineering 123: 1–10
 ↑ McAnally, W.H., Friedrichs, C., Hamilton, D., Hayter, E., Shrestha, P., Rodriguez, H., Sheremet, A. and Teeter, A. 2007. Management of Fluid Mud in Estuaries, Bays, and Lakes. I: Present State of Understanding on Character and Behavior. J. Hydr. Eng. 133: 922
 ↑ Robillard, D.J. Mehta, A.J. and Safak, I. 2023. Comments on waveinduced behavior of a coastal mud. Coastal Engineering 186, 104400
 ↑ McAnally, W.H., Friedrichs, C., Hamilton, D., Hayter, E., Shrestha, P., Rodriguez, H., Sheremet, A. and Teeter, A. 2007. Management of Fluid Mud in Estuaries, Bays, and Lakes. I: Present State of Understanding on Character and Behavior. J. Hydr. Eng. 133: 922
 ↑ Winterwerp, J.C., de Boer, G.J., Greeuw, G. and van Maren, D.S. 2012. Mudinduced wave damping and waveinduced liquefaction. Coastal Engineering 64: 102–112
 ↑ Winterwerp, J.C., Vroom, J., Wang, ZB., Krebs, M., Hendriks, E.C.M., van Maren, D.S., Schrottke, K., Borgsmüller, C. and Schol, A. 2017. SPM response to tide and river flow in the hyperturbid Ems River. Ocean Dynamics 67: 559–583
 ↑ Van der Meer, J. and Sigurdarson, S. 2017. Design and construction of berm breakwaters. Advanced Series on Ocean Engineering vol. 40, World Scientific Publ. Co., Singapore
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 ↑ Terzaghi, K. 1925. Principles of Soil Mechanics: IV—Settlement and consolidation of clay. Engineering NewsRecord 95(22): 874878
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