# Depth-Dependent Seabed Properties: Geoacoustic Assessment | Journal of Geotechnical and Geoenvironmental Engineering

Depth-Dependent Density and Effective Stress

Self-compaction reflects the dependency between effective stress

$σz′$

and sediment density

$ρs$

. In this section, an asymptotically correct compaction model is used and the evolution of effective stress

$σz′$

, porosity

$n$

, and sediment density

$ρs$

is investigated for a wide range of sediments in the upper 1,000 m of the sediment column.

Compaction Model: Void Ratio versus Effective Stress

Fig. 1 presents compaction trends for a wide range of soils, from high-plasticity clays to clean sands. Clearly, the sediment void ratio

$e$

decreases as a function of the vertical effective stress

$σz′$

, and compressibility increases with sediment plasticity. The compaction trend is nonlinear in the conventional semi-log space, so the classical Terzaghi model

$e=e1 kPa−Cc·log(σz′/1 kPa)$

fails to satisfy asymptotic conditions at very low and high effective stresses (see also Mesri and Olson 1971; Johns 1986; Burland 1990; Pestana and Whittle 1995; Gregory et al. 2006). In particular, the low effective stress asymptote void ratio

$eL$

must be properly captured because it plays a critical role in high-frequency bathymetric data analysis.

Therefore, this study uses an asymptotically correct exponential compaction model in terms of void ratios

$eL$

at low effective stress (

$σz′→0$

) and

$eH$

at very high effective stress (

$σz′→∞$

); then the void ratio

$ez$

at depth

$z$

is a function of the vertical effective stress

$σz′$

$ez=eH+(eL−eH)exp[−(σz′σc′)η]$

where the model parameter

$η$

captures the void ratio sensitivity to effective stress and is often

$η≈1/3$

, and

$σc′$

is the characteristic effective stress so that

$(ez−eH)=0.37(eL−eH)$

when

$σz′=σc′$

.

Fig. 1 shows exponential compaction trends for six reference sediments selected to bound the experimental data. The model parameters correlate with the sediment specific surface, plasticity, and grain shape (Table 1 summarizes the model parameters for the six reference sediments). The scale used in this figure reaches

$σz′=10 MPa$

to test the wide stress range validity of the exponential

$ez−σz′$

model [Eq. (1)].

Select reference sediments. Constitutive parameters for effective stress-dependent compaction model [Eq. (1): model parameter

$η=1/3$

] and shear stiffness in terms of shear wave velocity [Eq. (5)]. Parameters are self-consistent and satisfy published correlations in terms of specific surface, plasticity, and grain shape

Table 1. Select reference sediments. Constitutive parameters for effective stress-dependent compaction model [Eq. (1): model parameter η=1/3] and shear stiffness in terms of shear wave velocity [Eq. (5)]. Parameters are self-consistent and satisfy published correlations in terms of specific surface, plasticity, and grain shape

Reference sediment

$eL$

$eH$

$σc′$

(kPa)

$α$

(

$m/s$

)

$β$

1 8 0.1 500 22 0.38
2 3.2 0.3 500 33 0.32
3 1.80 0.3 700 58 0.25
4 1.35 0.3 1,000 75 0.23
5 0.91 0.2 2,000 110 0.19
6 0.60 0.2 3,000 146 0.17

The void ratio of sandy sediments at low effective stress

$eL$

reflects the grain size distribution and particle shape (Youd 1973; Cho et al. 2006). At a given relative density

$Dr$

, the value of

$eL$

can be estimated from the coefficient of uniformity

$Cu$

and particle roundness

$R*$

as

$eL=(0.02−0.032·Dr)+(0.893−0.522·Dr)/Cu+(0.236−0.154·Dr)/R*$

; for example,

$eL=0.89$

for a sandy sediment with

$Cu=3$

, and

$R*=0.3$

at

$Dr=30%$

.

On the other hand, mineralogy and depositional environment determine the void ratio

$eL$

and the compressibility of clay-rich sediments (Palomino and Santamarina 2005; Wang and Siu 2006; Wang and Xu 2007). Assuming an edge-to-face fabric, the value of

$eL$

is a function of the particle slenderness,

$eL=(a−1)/2$

where the slenderness ratio

$a$

is equal to the particle length divided by its thickness; for example, kaolinite has

$a≈10$

and

$eL=4.5$

, while montmorillonite has

$a≈100$

and

$eL=50$

(Santamarina et al. 2001). Remolding disturbs the initial fabric and any subsequent diagenesis (Burland 1990; Leroueil 1996; Hong et al. 2012).

Depth-Dependent Density and Porosity

Consider a thin seabed layer of thickness

$dz$

at depth

$z$

. Force equilibrium combines with gravimetric-volumetric relations to predict the effective stress gradient

$dσz′/dz$

as a function of the void ratio

$ez$

at depth

$z$

(note that gravity

$g=9.81 m/s2$

):

$dσz′dz=ρw[SG−11+ez]g$

where the mineral-specific gravity

$SG=ρm/ρw$

is the ratio between the mineral and water densities

$ρm$

and

$ρw$

. Eq. (1) is inserted in Eq. (2), and the differential equation is solved. The closed-form solution for

$η=1/3$

predicts the following effective stress profile

$σz′$

with depth

$z$

(for η = 1/3—See related example for methane hydrate-bearing sediments in Terzariol et al. 2020):

$z=(1+eH)(SG−1)ρwgσz′+3(eL−eH)(SG−1)ρwgσc′{[(σz′σc′)23+2(σz′σc′)13+2]·exp[−(σz′σc′)13]−2}$

Fig. 2(a) shows the effective stress trends for the six reference sediments identified earlier (Fig. 1; Table 1). Effective stress gradients decrease markedly as sediments change from sands to high-plasticity clays.

Finally, Eq. (1) is used to compute the void ratio

$ez$

and the sediment density

$ρs(z)$

as a function of depth

$z$

:

Fig. 2(b) shows the change in sediment density

$ρs$

with depth

$z$

. At the sediment–water boundary

$z=0$

, sediments experience zero effective stress and the sediment density is the asymptotic value

$ρo$

defined by

$e(z=0)=eL$

[Eq. (4)]; conversely, the sediment density

$ρs$

increases toward the value defined by

$eH$

as the sediment depth increases. Once again, the six trends in Fig. 2(b) correspond to the six reference sediments identified in Fig. 1 (parameters in Table 1) and show marked differences in density for sand, silts, and clays in the upper 1,000 m. While remolding by sediment transport can affect the sediment density in shallow accumulations, eventually effective stress-dependent compaction prevails as sediment burial progresses.

Fig. 3 shows the evolution of sediment compaction with depth in terms of porosity

$n=e/(1+e)$

for data gathered from published references and the Ocean Drilling Program (ODP) projects. The data set involves normally consolidated sediments (i.e., no overpressure) and relatively homogeneous deposits. Once again, trends correspond to the reference sediments; the 1,000-m depth makes it possible to corroborate the wide stress range validity of the exponential

$ez−σz′$

model [Eq. (1)]. Some reported so-called sands exhibit high porosities; this may reflect grains with intraparticle porosity [e.g., carbonate sands (Goldhammer 1997)] or incorrect reporting of sandy sediments with high fines content so that fabric formation is fines-dominant. In general, computed trends for the six reference sediments selected earlier follow the field data for the different sediment types. (Note: the six reference sediment trends reflect self-compaction properties according to sediment type; these profiles are not intended to fit any particular data set.) The logarithm of depth used in Fig. 3 helps with exploring near-surface conditions in detail, but trends can be misleading: in linear scale, porosity decreases at a faster rate near the sediment–water interface.

Depth-Dependent P- and S-Wave Velocities

Effective stress determines not only the sediment density

$ρs$

but also the stiffness of the sediment granular skeleton in the absence of cementation. In turn, stiffness and mass density define P- and S-wave velocities. This section brings together P- and S-wave velocity data compiled from the literature and new experimental data gathered in this study to overcome the scarcity of published data at very shallow depths. Then, the P- and S-wave velocity data are analyzed in the context of effective stress-dependent density, shear stiffness, and bulk modulus.

Experimental Study: P- and S-Wave Velocities at Very Low

$σz′$

Devices

The miniature P- and S-wave measurement probes built for this study involve split tubes to minimize the transmission of vibrations around the tube and boundary-reflected waves (dimension:

$ϕ=34 mm$

in diameter and

$H=500 mm$

in height). We mount the parallel-type bender elements for S-waves and the piezo disk elements for P-waves at the tip of the split tubes to reduce disturbance effects caused by probe penetration (insets in Figs. 4 and 5). Crystals are coated with silver paint and grounded to prevent electrical crosstalk [details in Lee and Santamarina (2005a)].

Test Procedure

The experimental study involves sediment beds prepared with fine sand, silt, and kaolinite in large-diameter columns to minimize boundary effects during the penetration of the P- and S-wave probes (columns are 300 mm in diameter and 600 mm in height). Table 2 summarizes the sediment index properties. These seafloor analogs form by sedimentation from diluted slurries to prevent gas entrapment and layering (initial water content

$ω>2,000%$

). All sediments consolidate for 2 weeks. Then P- and S-waves are recorded every 1 or 2 cm (note that signals are stable within a few minutes after each insertion, as soon as any excess pore-water pressure dissipates).

Tested sediments—index properties

Table 2. Tested sediments—index properties

Material Diameter (mm) Extreme void ratios,

$emax/emin$

Mean grain size

$d50$

(mm)

Specific gravity,

$SG$

Liquid limit (

$W/B/K$

)

Specific surface (

$m2/g$

)

RSCS—fines classification
Cobbles 10–20 15 2.65 N/A N/A N/A
Coarse sand 1.0–2.0

$0.82/0.59$

1.5 2.65 N/A N/A N/A
Fine sand 0.075–0.25

$0.81/0.45$

0.16 2.63 N/A N/A N/A
Silt

$<0.075$

$1.50/0.73$

$1.0×10−2$

2.65 N/A N/A N/A
Kaolinite (RP2)

$<0.075$

N/A

$3.0×10−4$

2.67

$67/52/82$

33 I-I
Bentonite

$<0.075$

N/A

$5.0×10−6$

2.65

$302/92/39$

544 H-H
P- and S-Wave Signatures

Figs. 4 and 5 present P- and S-wave signatures gathered at very shallow depths and low effective stress in the sandy, silty, and clayey sediments [field data gathered with a velocity-resistivity probe can be found in Lee et al. (2010) and Yoon and Lee (2010)]. The time to first arrival decreases with depth for S-waves, while the travel time for P-waves is almost constant with depth.

Shear Wave Velocity versus Depth

Fig. 6 plots the measured and collected shear velocity

$VS$

data set. The shear wave velocity increases with depth for all sediments. The effective stress

$σz′$

determines the sediment shear stiffness

$Gs$

in the absence of diagenetic cementation. In fact, the shear wave velocity

$Vs=Gs/ρs$

follows a Hertzian-type power relation with effective stress (Roesler 1979):

$Vs=α(σz′1 kPa)β$

The

$α$

-factor (

$m/s$

) is the shear wave velocity at

$σz′=1 kPa$

, and the

$β$

-exponent represents the sensitivity of the shear wave velocity to effective stress. These

$α$

-factor and

$β$

-exponent reflect the sediment type; in general, higher-specific-surface sediments are more compressible, exhibit a lower

$α$

-factor, and a more pronounced increase in shear stiffness with stress, i.e., a higher

$β$

-exponent. Thus, there is an inverse relationship between the

$α$

-factor and

$β$

-exponent, as observed for a wide range of soils:

$β=0.7−0.25log[α/(m/s)]$

[see database in Cha et al. (2014)].

The shear wave velocity for fines-controlled sediments is affected by pore fluid chemistry (Klein and Santamarina 2005; Kang et al. 2014). Furthermore, diagenesis may affect both fine- and coarse-grained sediments. In particular, synsedimentary diagenesis makes deeper and older sediments stiffer than remolded specimens at the same effective stress; in this case, the shear wave velocity increases with depth owing to the effective stress and diagenesis and the field

$β$

-exponent will be higher than for remolded specimens in the laboratory [see data in Ku (2012) and Ku et al. (2017)].

Table 1 lists

$α$

and

$β$

values for the six reference sediments, inferred from their compressibility (refer to Part 1). Effective stress profiles from Eq. (3) combine with Eq. (5) to predict the shear wave velocity trends shown in Fig. 6. The computed

$VS$

profiles for the six reference sediments exhibit trends similar to those in the data and successfully bound laboratory and field measurements. Once again, the shear wave velocity increases rapidly with depth in linear-linear scale [power trend in Eq. (5)].

P-Wave Velocity versus Depth

The sediment shear modulus

$Gs=ρs·VS2$

is a function of the shear wave velocity and the sediment mass density [Figs. 6 and 2(b)]. At small strains, the bulk modulus

$Bsk$

of the sediment granular skeleton is related to the shear modulus as

$Bsk=Gs2(1+νsk)3(1−2νsk)$

where the small-strain Poisson’s ratio of the soil skeleton is

$νsk≈0.15$

. The bulk modulus of the water-saturated sediment

$Bs$

depends on its porosity

$n$

and is a function of the bulk stiffness of the skeleton

$Bsk$

, water

$Bw$

, and mineral

$Bm$

:

On the other hand, the sediment density

$ρs$

(

$kg/m3$

) is a function of the water

$ρw$

and mineral

$ρm$

densities and the sediment porosity

$n$

:

Finally, the low-frequency Biot-Gassmann P-wave velocity

$VP$

(

$m/s$

) for soft marine sediments is [from Eqs. (6)–(8)]

$Vp=Bs+43Gsρs=[2(1+νsk)3(1−2νsk)+43](nρw+(1−n)ρm)Vs2+(nBw+1−nBm)−1nρw+(1−n)ρm$

Fig. 7 shows the estimated P-wave velocity profiles versus depth for the six selected reference sediments. These trends take into consideration the depth-dependent shear wave velocity and porosity profiles computed earlier (Figs. 3 and 6). The figure includes a comprehensive data set gathered from published studies and our own experimental data (Fig. 5). Trends for the reference sediments adequately resemble the data for different sediments (not intended to fit any specific data set).

The water bulk modulus

$Bw$

controls the P-wave velocity in saturated soft sediments near the seafloor (pure water

$VP=1,480 m/s$

; seawater

$VP=1,531 m/s$

), and changes in P-wave velocity are relatively minor in the upper tenths of meters in clayey deposits. In fact, the P-wave velocity may fall below that of water when the increase in mass density is faster than the increase in skeletal stiffness in plastic sediments at shallow depth. (Note: This explains the crossing of the two trends for clayey sediments in Fig. 7.)

The P-wave velocity relaxation in sediments leads to the following ratio between the Biot high-frequency P-wave velocity

$VP∞$

and the low-frequency P-wave velocity

$VPo$

computed in Eq. (9) (Santamarina et al. 2001):

$VP∞VP0=[nξ+n2(SG−2)−n3(SG−1)][SG(1−n)+n]ξnSG(1−n)+n2(ξ−1)$

where

$ξ$

is a tortuosity factor. The velocity ratio is approximately

$VP∞/VPo≈1.05$

[Eq. (10), porosity

$n≈0.4$

, and tortuosity factor

$ξ=2$

]. Fig. 8 shows the effect of porosity

$n$

on P-wave velocities

$VPo$

and

$VP∞$

. Data points are values measured for “undisturbed samples” recovered from the seafloor (Richardson and Briggs 1993). The low-frequency P-wave velocity

$VPo$

[red trend—Eq. (9)] and the high-frequency

$VP∞$

(red line) provide lower and upper trends for most of the data shown in Fig. 8 when the skeletal stiffness is null,

$VS=0 m/s$

. The central trend is best predicted with Eq. (9) for a shear wave velocity

$VS≈250 m/s$

, which could result from either diagenesis or suction effects when recovered specimens are tested in air (dashed line).

Heterogeneity

Data plots in the global depth

$z$

-scale include changes in stratigraphy with depth [e.g., ODP holes in Leg 139–Juan de Fuca Ridge (Mottl et al. 1994)]. Compaction models make it possible to convert depth into effective stress [Eq. (3)] in order to analyze geoacoustic properties in terms of local effective stress

$σz′$

[Eqs. (1), (4), (5), and (9)].

Discussion and Implications

In the previous two sections, a formulation and supportive database were constructed for depth-dependent geotechnical properties (density

$ρs$

, porosity

$n$

, shear stiffness

$Gs=ρs·VS2$

) and geoacoustic properties (

$VS$

and

$VP$

) as functions of sediment type. The model parameters selected for the reference soils satisfy well-proven correlations with specific surface, plasticity, and grain shape. Here, the implications for bathymetric studies and seafloor sediment characterization are explored and crucial parameters that require further investigation are identified.

High-Frequency Reflection—Enhanced Bathymetric Analysis

The asymptotic conditions at the water–sediment interface

$z→0$

are of particular interest because they determine the reflectivity in high-frequency bathymetric studies. A well-controlled reflection data set was created from sediment beds formed with bentonite, kaolinite, silica flour, fine sand, coarse sands, and cobbles. Table 2 summarizes the selected sediments and their index properties. The chamber is 300 mm in diameter and 600 mm in height. All tests are repeated with two crystal pairs to measure reflections at frequencies

$fr=160$

and 500 kHz [see details on transducer performance in Lee and Santamarina (2005b)]. The spacing between the source and receiver crystals is 20 mm in both cases.

Fig. 9 presents P-wave signatures reflected from a submerged steel plate placed at different depths. The multiple reflection events confirm phase inversion at the air–water interface and the combined effects of geometric spreading and material attenuation. The energy in the first reflection is used as a reference.

Fig. 10 presents the first reflection from the steel plate in comparison to reflections from the fine sand, silt, kaolinite, and bentonite beds, all at the same water depth. Clearly, the reflection amplitude is strongly related to sediment type and the ensuing asymptotic properties at very low effective stress

$σz′→0$

near the water–sediment interface

$z→0$

. The mismatch in acoustic impedance

$Z=ρ·VP$

between the water column and the reflector defines the energy

$E$

in the reflected signal, i.e., the reflection coefficient. Then the relative reflection coefficient

$RR$

between a sediment bed and the steel target is

$RR=EsEst=[(Zw+Zst)(Zw−Zs)(Zw−Zst)(Zw+Zs)]2=1.14×(1−ZsZw1+ZsZw)2$

where the subscripts

$s$

,

$w$

, and

$st$

indicate sediment, water, and steel, respectively. The “1.14” factor in the last mathematical expression corresponds to the water–steel interface (steel:

$ρ=7,900 kg/m3$

,

$VP=5,900 m/s$

; water:

$ρ=1,024 kg/m3$

,

$VP=1,531 m/s$

). Fig. 11 summarizes the energy-based

$RR$

measured for all sediments versus their mean grain size

$d50$

. Sandy sediments exhibit the highest reflection coefficient, while the very soft bentonite bed produces the smallest coefficient. [Note that published results corroborate the importance of sediment type on reflectivity, for example, van Walree et al. (2005, 2006) and Snellen et al. (2011); attenuation in Panda et al. (1994); spectral strength in Sternlicht and de Moustier (2003).]

The sediment density

$ρs$

and P-wave velocity profiles for the six reference sediments explored earlier make it possible to estimate the theoretically computed

$RR$

[Eq. (11); refer to Figs. 3(b) and 7 for the sediment

$ρs$

and

$VP$

]. Fig. 11 shows the theoretical reflection coefficient computed at different sediment depths from

$z*=3$

to 1,000 mm. The depth

$z*=3 mm$

relates to the wavelength

$λ$

at

$fr=500 kHz$

, while the impedance at depth

$z*=150 mm$

may be more relevant for the operating frequency of subbottom profilers

$fr=10 kHz$

. Data trends suggest a slope of approximately

$1/4$

for the

$log(RR)$

versus the

$log(d50)$

. The theoretically computed

$RR$

at depth

$z*=3 mm$

tracks the experimental data when the mean grain size

$d50$

is smaller than

$1/10$

of the wavelength

$λ=3 mm$

. Thereafter, the reflection coefficient decreases rapidly (Brillouin zone). Field data gathered with a single-beam echosounder (Snellen et al. 2011) show good agreement with the results presented here for silts and sands where the slope between the reflectivity and grain size is about

$1/4$

in log-log scale.

Asymptotic Void Ratio

The previously discussed bathymetric study shows that the mass density

$ρo$

at very low effective stress

$σz′→0$

is the critical parameter for high-frequency seafloor reflection studies (leveled seafloor). The

$ρo$

corresponds to the asymptotic void ratio

$eL$

[Eq. (4)]:

$ρo=ρw[SG+eL1+eL] (at depth z=0)$

Furthermore, the shear stiffness vanishes at

$σz′=0$

and

$VS=0$

. Then the P-wave velocity

$VP$

at

$z=0$

becomes [Eq. (9)]

$Vp(z=0)=Bsρs=[(ρm+ρweL1+eL)·(eL1+eLBw+1−eL1+eLBm)]−1$

Finally,

$RR$

is a function of the asymptotic void ratio

$eL$

[After Eq. (11)]:

$RR=1.14×[1−2·(SG+eLBwBm+eL+1)−1]2$

Eq. (14) corresponds to the line for

$z*=0$

in Fig. 11. The relative reflection coefficients measured with the low-frequency and low-damping transducer are lower owing to the longer wavelength. In all cases, the asymptotic void ratio

$eL$

for the six reference sediments successfully anticipates the lower bound for

$RR$

.

Clearly, the asymptotic void ratio

$eL$

emerges as an important parameter, both for self-compaction and for the stiffness evolution with depth. The asymptotic void ratio

$eL$

is inversely proportional to the median grain size

$d50$

in clays and silts (due to the prevalent role of interparticle electrical forces) and is affected by pore fluid chemistry [see data in Mesri and Olson (1971), Studds et al. (1998), and Stewart et al. (2003)]. On the other hand, the asymptotic void ratio

$eL$

is geometry-controlled by granular packings in sands and gravels. In fact, data in Table 1 for the six selected reference sediments suggest

$eL=0.4·[1+(d50/mm)−0.28$

]. For comparison, this equation agrees well with the central trend in Jackson and Richardson (2007), though their data are limited to

$d50=10−3$

to 1 mm). Admittedly, very high-plasticity clays can form stable slurries at higher void ratios in freshwater (Liu and Santamarina 2018); however, a skeleton capable of shear wave transmission is first detected at void ratios similar to

$eL$

values used in this paper. Overall, this analysis suggests the potential use of reflection data to estimate the asymptotic void ratio

$eL$

and to infer the sediment type.

Sediment Classification—Nomenclature

Semiempirical seabed classification methods that rely on low-perturbation acoustic reflection measurements exhibit large variations (Bachman 1985; Leblanc 1992; Panda et al. 1994). Factors such as surface roughness, bioturbation, and variability contribute to data scatter (Jackson and Briggs 1992; Jackson et al. 1996; Clarke 1994; Lyons and Orsi 1998; Jackson and Richardson 2007). Several commercially available software packages attempt to create seafloor backscatter mosaics (e.g., Fledermaus by QPS). Analyses advanced in this study support and extend these efforts by providing mutually compatible geotechnical and geoacoustic properties.

Sediment classification requires more information than what can be extracted from acoustic data. Therefore, acoustic seafloor surveys and sediment sampling combine to provide robust spatial distributions of seafloor sediment types (Goff et al. 2004). Sediment classification for engineering purposes helps engineers anticipate the sediment properties and behavior by grouping them into similar engineering response categories (Casagrande 1948; Kulhawy and Chen 2009). Unfortunately, current marine sediment classifications use a 50% fraction to separate sand from silt or clay (Shepard 1954; Folk et al. 1970; Flemming 2000). Indeed, the 50% boundary does not capture the transition from coarse-controlled to fines-controlled sediment behavior.

The newly revised soil classification system (RSCS) overcomes this limitation and classifies sediments as sand-controlled, transitional, or fines-controlled (Park and Santamarina 2017; Park et al. 2018); in particular, the RSCS captures the critical role of the fines fraction on geoacoustic properties. This is demonstrated with sand–silt mixtures in Fig. 12, where porosity, S- and P-wave velocities, and acoustic impedance data support the transition boundaries predicted by the RSCS: geoacoustic properties are sand-controlled up to a fines fraction of

$FF≈18%$

and become clay-controlled when the fines fraction exceeds

$FF≈37%$

. Transition boundaries shift to lower fines contents

$FF$

for higher-plasticity fines. Furthermore, sediment analyses must consider the salt concentration in pore fluid when sediments are classified with a letter F in the RSCS triangular textural chart (Jang and Santamarina 2016, 2017).

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