Depth-Dependent Density and Effective Stress
Self-compaction reflects the dependency between effective stress
and sediment density
. In this section, an asymptotically correct compaction model is used and the evolution of effective stress
, and sediment density
is investigated for a wide range of sediments in the upper 1,000 m of the sediment column.
Fig. 1 presents compaction trends for a wide range of soils, from high-plasticity clays to clean sands. Clearly, the sediment void ratio
decreases as a function of the vertical effective stress
, and compressibility increases with sediment plasticity. The compaction trend is nonlinear in the conventional semi-log space, so the classical Terzaghi model
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
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
at low effective stress (
at very high effective stress (
); then the void ratio
is a function of the vertical effective stress
where the model parameter
captures the void ratio sensitivity to effective stress and is often
is the characteristic effective stress so that
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
to test the wide stress range validity of the exponential
model [Eq. (1)].
Table 1. Select reference sediments. Constitutive parameters for effective stress-dependent compaction model [Eq. (1): model parameter
] 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
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
The void ratio of sandy sediments at low effective stress
, the value of
can be estimated from the coefficient of uniformity
and particle roundness
; for example,
for a sandy sediment with
On the other hand, mineralogy and depositional environment determine the void ratio
is a function of the particle slenderness,
where the slenderness ratio
is equal to the particle length divided by its thickness; for example, kaolinite has
, while montmorillonite has
Consider a thin seabed layer of thickness
. Force equilibrium combines with gravimetric-volumetric relations to predict the effective stress gradient
as a function of the void ratio
(note that gravity
where the mineral-specific gravity
is the ratio between the mineral and water densities
predicts the following effective stress profile
(for η = 1/3—See related example for methane hydrate-bearing sediments in Terzariol et al. 2020):
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
and the sediment density
as a function of depth
Fig. 2(b) shows the change in sediment density
. At the sediment–water boundary
, sediments experience zero effective stress and the sediment density is the asymptotic value
[Eq. (4)]; conversely, the sediment density
increases toward the value defined by
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
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
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
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.
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:
in diameter and
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)].
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
). 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
|Material||Diameter (mm)||Extreme void ratios,||Mean grain size (mm)||Specific gravity,||Liquid limit ( )||Specific surface ( )||RSCS—fines classification|
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.
Fig. 6 plots the measured and collected shear velocity
data set. The shear wave velocity increases with depth for all sediments. The effective stress
determines the sediment shear stiffness
in the absence of diagenetic cementation. In fact, the shear wave velocity
follows a Hertzian-type power relation with effective stress (Roesler 1979):
) is the shear wave velocity at
, and the
-exponent represents the sensitivity of the shear wave velocity to effective stress. These
-exponent reflect the sediment type; in general, higher-specific-surface sediments are more compressible, exhibit a lower
-factor, and a more pronounced increase in shear stiffness with stress, i.e., a higher
-exponent. Thus, there is an inverse relationship between the
-exponent, as observed for a wide range of soils:
[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
Table 1 lists
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
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)].
The sediment shear modulus
of the sediment granular skeleton is related to the shear modulus as
where the small-strain Poisson’s ratio of the soil skeleton is
. The bulk modulus of the water-saturated sediment
depends on its porosity
and is a function of the bulk stiffness of the skeleton
, and mineral
On the other hand, the sediment density
) is a function of the water
densities and the sediment porosity
Finally, the low-frequency Biot-Gassmann P-wave velocity
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
controls the P-wave velocity in saturated soft sediments near the seafloor (pure water
), 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
and the low-frequency P-wave velocity
is a tortuosity factor. The velocity ratio is approximately
[Eq. (10), porosity
, and tortuosity factor
]. Fig. 8 shows the effect of porosity
on P-wave velocities
. Data points are values measured for “undisturbed samples” recovered from the seafloor (Richardson and Briggs 1993). The low-frequency P-wave velocity
[red trend—Eq. (9)] and the high-frequency
(red line) provide lower and upper trends for most of the data shown in Fig. 8 when the skeletal stiffness is null,
. The central trend is best predicted with Eq. (9) for a shear wave velocity
, which could result from either diagenesis or suction effects when recovered specimens are tested in air (dashed line).
Data plots in the global depth
-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
Discussion and Implications
In the previous two sections, a formulation and supportive database were constructed for depth-dependent geotechnical properties (density
, shear stiffness
) and geoacoustic properties (
) 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.
The asymptotic conditions at the water–sediment interface
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
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
near the water–sediment interface
. The mismatch in acoustic impedance
between the water column and the reflector defines the energy
in the reflected signal, i.e., the reflection coefficient. Then the relative reflection coefficient
between a sediment bed and the steel target is
where the subscripts
indicate sediment, water, and steel, respectively. The “1.14” factor in the last mathematical expression corresponds to the water–steel interface (steel:
). Fig. 11 summarizes the energy-based
measured for all sediments versus their mean grain size
. 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
and P-wave velocity profiles for the six reference sediments explored earlier make it possible to estimate the theoretically computed
]. Fig. 11 shows the theoretical reflection coefficient computed at different sediment depths from
to 1,000 mm. The depth
relates to the wavelength
, while the impedance at depth
may be more relevant for the operating frequency of subbottom profilers
. Data trends suggest a slope of approximately
. The theoretically computed
tracks the experimental data when the mean grain size
is smaller than
of the wavelength
. 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
in log-log scale.
The previously discussed bathymetric study shows that the mass density
at very low effective stress
is the critical parameter for high-frequency seafloor reflection studies (leveled seafloor). The
corresponds to the asymptotic void ratio
Furthermore, the shear stiffness vanishes at
. Then the P-wave velocity
becomes [Eq. (9)]
is a function of the asymptotic void ratio
[After Eq. (11)]:
Eq. (14) corresponds to the line for
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
for the six reference sediments successfully anticipates the lower bound for
Clearly, the asymptotic void ratio
emerges as an important parameter, both for self-compaction and for the stiffness evolution with depth. The asymptotic void ratio
is inversely proportional to the median grain size
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
is geometry-controlled by granular packings in sands and gravels. In fact, data in Table 1 for the six selected reference sediments suggest
]. For comparison, this equation agrees well with the central trend in Jackson and Richardson (2007), though their data are limited to
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
values used in this paper. Overall, this analysis suggests the potential use of reflection data to estimate the asymptotic void ratio
and to infer the sediment type.
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
and become clay-controlled when the fines fraction exceeds
. Transition boundaries shift to lower fines contents
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).