# Shallow Seafloor Sediments: Density and Shear Wave Velocity

## Abstract

## Introduction

## In-Situ Density—Asymptotic Void Ratio

### Coarse-Grained Sediments—Gravels, Sands, and Silts

### Fine-Grained Sediments—Clays

*e*

_{L}for low plasticity kaolinite ranges from ${e}_{L}=4$-to-9 [Fig. 2(a)] and is more sensitive to pH than to ionic concentration: clearly, fabric formation reflects pH-dependent edge and surface charges, i.e., face-to-face aggregation versus dispersed fabric.

*e*

_{L}for high-plasticity bentonite is in the range of ${e}_{L}=5$-to-40 [Fig. 2(b)]. In contrast to kaolinite, the asymptotic void ratio does depend on ionic strength, i.e., the concentration co and valance z of prevailing cations, in line with the diffused double layer thickness ${\vartheta \propto ({\mathrm{c}}_{\mathrm{o}}\xb7{\mathrm{z}}_{2})}_{-1/2}$ (Santamarina et al. 2001; Mitchell and Soga 2005).

### Asymptotic Void Ratio and Compressibility—Caution

*LL*(Burland 1990). Schematic trends in Fig. 3 compare a clay response when consolidation starts from its true asymptotic void ratio ${e}_{L}$ as ${\sigma}_{z}^{\prime}\to 0$ and from a paste prepared at a water content $\omega \%>LL$. For reference, the liquid limit corresponds to $\sim 6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kPa}$ suction (Russell and Mickle 1970; Hong et al. 2010); conversely, the void ratio at 1 kPa effective stress correlates with the void ratio at the liquid limit as ${\mathrm{e}}_{1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kPa}}=1.25\xb7{\mathrm{e}}_{\mathrm{LL}}$ (based on 28 datapoints with ${\mathrm{R}}^{2}=0.92$ - Chong and Santamarina 2016). Clearly, suction preconsolidates pastes, and consequently, typical consolidation tests with remolded specimens underpredict the true asymptotic void ratio needed for seafloor sediment analyses (Fig. 3).

### Trends as a Function of the Specific Surface Area

## Shear Wave Velocity

### S-Wave Velocity Probe

### Experimental Study: Laboratory and Field

### Results—Prevalent Trends

### Effect of Interparticle Electrical Forces

## Discussion: Stiffness Profiles and Pressure Diffusion

### Data Analysis: Ray Bending

*V*

_{h}is the horizontal velocity

### Small-Strain Stiffness ${\mathrm{G}}_{\mathrm{max}}$ in Depth

### Insertion Effects: Fabric Change and Excess Pore Pressure

*Fabric effects*. To investigate fabric effects, we prepared a very dilute silt slurry (silica flour, initial water content $\omega >\mathrm{2,000}\%$), deployed the shear wave probe and fixed it 5 cm above the bottom of the large-diameter plexiglass column, and monitored the shear wave signal during self-weight consolidation without displacing the probe. The cascade of shear wave signals in Fig. 11 captures the sediment evolution with elapsed time: the first arrival continuously decreases as effective stress increases during consolidation. At the same sediment depth, the shear wave velocity obtained during probe insertion tests is ${V}_{\mathrm{s}}=24.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$, compared to ${V}_{s}=23.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$ measured for the undisturbed fabric using the preburied probe. This observation suggests that potential fabric changes caused by probe insertion have minimal effects on the measured ${V}_{s}$ profiles at $z=20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{cm}$ (at least in silty sediments). In addition, ray bending ahead of the probe may hide any disturbance effects in the upper few centimeters (Supplemental Materials).

*Excess pore pressure generation.*We observed consistent time-dependent changes in travel time after probe insertion (Fig. 12 - Note: the vertical separation between signals scales with the logarithm of time following insertion). In particular, the time to first arrival decreases, and the signal amplitude increases after insertion in clayey and silty sediments, yet opposite transients are observed in sandy sediments (albeit very small). These results suggest that probe insertion causes positive excess pore fluid pressure generation in contractive sediments, and negative excess pressure in dilative soils (note: the generation of excess pore-water pressure was anticipated in both the inner zones of the probe and surrounding areas, then dissipated).

*Coefficient of consolidation*. The sediment-dependent time-varying shear wave velocity trends suggest the possibility of estimating the coefficient of consolidation from these measurements. We followed four steps:

*S*

_{s}) and can range from a few minutes in clean sands to several hours in clays.

Governing equation | Solution | Method/parameters | References |
---|---|---|---|

$\frac{\partial u}{\partial t}={c}_{h}\frac{{\partial}^{2}u}{{\partial}^{2}r}+\frac{{c}_{h}}{r}\frac{\partial u}{\partial r}$ | $U={u}_{o}\frac{{n}^{2}\mathrm{ln}p-0.5({\rho}^{2}-1)}{{n}^{2}{F}_{n}}\mathrm{exp}\left(\frac{-2{T}_{r}}{{F}_{n}}\right)$ | Equal strain consolidation for instantaneous loading | Barron (1948) |

${F}_{n}=\frac{{n}^{2}\mathrm{ln}n}{{n}^{2}-1}-\frac{3{n}^{2}-1}{4{n}^{2}}$ | ${u}_{o}$: initial excess water pressure | ||

${T}_{r}=\frac{{c}_{h}t}{{r}_{e}^{2}}n=\frac{{r}_{e}}{{r}_{w}}$ | ${c}_{h}$: consolidation coefficient | ||

$\rho =r/{r}_{w}$ | ${r}_{e}$: equivalent radius drainage | ||

${r}_{w}$: radius of drain well | |||

$\left\{\begin{array}{ll}U=\sum {B}_{n}\mathrm{exp}(-{a}_{n}^{2}t){C}_{o}({\lambda}_{n}r)& {r}_{o}\le r\le {r}^{*}\\ U=0& r>{r}^{*}\end{array}\right\}$ | Radial consolidation around a cylindrical cavity/ pore elasticity/ cavity expansion | Randolph and Wroth (1979) | |

${B}_{n}=\frac{4{c}_{u}}{{\lambda}_{n}^{2}}\xb7\frac{{C}_{o}({\lambda}_{n}{r}_{o})-{C}_{o}({\lambda}_{n}R)}{{r}^{*2}{C}_{1}^{2}({\lambda}_{n}{r}^{*})-{r}_{o}^{2}{C}_{o}^{2}({\lambda}_{n}{r}_{o})}$ | ${r}_{o}$: radius of the pile | ||

${C}_{i}({\lambda}_{n}r)={J}_{i}(\lambda r)+\mu {Y}_{i}(\lambda r)$ | ${r}^{*}$: radius u is negligibly small | ||

${S}_{u}$: undrained shear strength | |||

${C}_{o}({\lambda}_{n}r)$: cylindrical function, where ${J}_{i}$ and ${Y}_{i}$ are Bessel functions and $\lambda r$ is a zero of the Bessel function | |||

$U=\frac{\mathrm{\Delta}u}{\mathrm{\Delta}{u}_{\mathrm{ref}}}\approx \frac{1}{1+{(T/{T}_{50})}^{b}}$ | Strain path method | Teh and Houlsby (1991) | |

$T={c}_{h}t/{d}^{2}$ | ${c}_{h}$: consolidation coefficient | ||

${T}_{50}=0.061{I}_{r}^{0.5}$ | $d$: diameter of the pile | ||

$b\approx 0.75$ | ${S}_{u}$: undrained shear strength | ||

${I}_{r}=G/{S}_{u}$ | $G$: shear modulus | ||

$b$: exponent |

## Conclusions

*e*

_{L}is determined by the grain size distribution and the particle shape in coarse-grained soils (sands and silts) and by the mineralogy and pore fluid pH and ionic strength in fine-grained sediments. Overall, the sediment-specific surface area

*S*

_{s}provides a first-order estimate of the asymptotic void ratio

*e*

_{L}, in agreement with simple geometric fabric analyses.

*e*

_{L}and the sediment compressibility are correlated. Together, they determine the sediment self-compaction, effective stress gradient, and stiffness profile with depth. The asymptotic void ratio

*e*

_{L}obtained from consolidation tests using remolded specimens prepared as soft pastes is smaller than the asymptotic void ratio observed at the seafloor surface. Sedimentation tests with low solid content slurries provide a more realistic estimate of the asymptotic void ratio including the effects of pore fluid chemistry on fabric formation.

*V*

_{s}changes with time after probe penetration can be used to obtain a first-order estimate of the coefficient of consolidation

*c*

_{h}. Together, shear wave velocity and diffusion time provide valuable information for sediment preclassification and analyses.

## Notation

*The following symbols are used in this paper:*

- $a$
- parameter in linear velocity fitting ($\mathrm{m}/\mathrm{s}$);
- $b$
- parameter in linear velocity fitting ($1/\mathrm{s}$);
- ${C}_{c}$
- coefficient of consolidation;
- ${C}_{u}$
- coefficient of uniformity;
- $c$
- velocity anisotropy;
- ${c}_{h}$
- coefficient of consolidation (subscripts: h = radial, v = vertical) (${\mathrm{m}}^{2}/\mathrm{s}$);
- ${c}^{*}$
- parameter in sigmoidal model;
- $d$
- parameter sigmoidal model;
- ${d}_{50}$
- mean particle size (mm);
- $e$
- void ratio (subscripts: at 1 kPa, H: at high effective stress, L: at low effective stress);
- ${e}_{\mathrm{max}}$
- maximum void ratio;
- ${G}_{\mathrm{max}}$
- small-strain shear modulus (Pa);
- ${G}_{s}$
- specific gravity;
- ${k}_{o}$
- coefficient of earth pressure at rest;
- $L$
- travel distance (mm);
- $LL$
- liquid limit (%);
- $R$
- roundness;
- $r$
- radius (m);
- ${S}_{s}$
- specific surface area (${\mathrm{m}}^{2}/\mathrm{g}$);
- $t$
- time (subscript: o = arrival time) (s);
- $u$
- pore fluid pressure (subscript: o = hydrostatic) (kPa);
- ${V}_{s}$
- shear wave velocity ($\mathrm{m}/\mathrm{s}$);
- $w$
- water content (%);
- $z$
- depth (m);
- $\alpha \text{-}\mathrm{factor}$
- shear wave velocity at 1 kPa;
- $\beta \text{-}\mathrm{exponent}$
- sensitivity to changes in the effective stress;
- $\eta $
- exponent used in consolidation model;
- $\rho $
- density (subscript: w = water) ($\mathrm{kg}/{\mathrm{m}}^{3}$);
- ${\sigma}^{\prime}$
- effective stress (subscripts: c = characteristic, h = horizontal, m = mean, v = vertical) (kPa);
- ${\sigma}_{A}^{\prime}$
- van der Waals attraction equivalent effective stress (kPa);
- $\mathrm{\Delta}{\sigma}^{\prime}$
- change in effective stress (kPa); and
- $\mathrm{\Delta}u$
- excess pore fluid pressure (kPa).

## Supplemental Materials

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## Data Availability Statement

## Acknowledgments

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#### History

**Received**: Jan 29, 2022

**Accepted**: Oct 25, 2022

**Published online**: Feb 28, 2023

**Published in print**: May 1, 2023

**Discussion open until**: Jul 28, 2023

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- Longde Jin, Andrew Fuggle, Haley Roberts, Christian P. Armstrong, Lina-Maria Pua, Estimating In Situ Shear Wave Velocity Using Machine Learning Techniques, Geo-Congress 2024, 10.1061/9780784485347.044, (436-444), (2024).