Nearshore Vertical Pore Pressure Gradients and Onshore Sediment Transport under Tropical Storm Forcing

Two pressure gauges sampling at 16 Hz were inserted into the beach face shoreward of the water line during low tide on October 2, 2019, just north of the pier at the United States Army Corps of Engineers Field Research Facility (USACE-FRF) in Duck, North Carolina (Fig. 1) attached to a steel pipe approximately 0.20 m apart in the vertical [Fig. 2(a)]. The steel pipe was buried such that the topmost sensor (P₁) was approximately 0.02 m below the sand surface at the time of installation. Changes to the elevation of the sand surface owing to erosion and accretion were small during the first week of the field experiment. However, approximately 0.3 m accretion occurred during the peak of tropical storm Melissa early on October 11, 2019. Data were collected for 335 h (approximately 2 weeks) starting October 5, 2019.

Water velocity measurements were made with an acoustic doppler velocimeter (ADV) sampling at 8 Hz about 0.15 m above the bed surface [Fig. 2(a), the ADV sensor position was adjusted during daylight low tides, when possible, to maintain a roughly constant distance above the seafloor]. Velocity values with high noise-to-signal ratios were removed (Elgar et al. 2005), and cross-shore (u, positive is onshore-directed) and vertical (v, positive is upward) velocities were analyzed. The ADV measurements were upsampled via interpolation to increase the frequency to 16 Hz to compare with the pressure data.

Photographs of the area surrounding the sensors were collected at 2 Hz for 9 min beginning at 15 min after each daylight hour between October 10 and 17, 2019, from a nearby five-megapixel camera. During these time periods, the images were used to corroborate wave crest arrival and the direction of surface water movement (e.g., shoreward with crest arrival). Image quality was not sufficient to check wave heights or other hydrodynamic parameters, nor could they be used to determine (subaqueous) sediment levels around the sensors as the seafloor eroded and accreted.

Both attenuation and lags of the wave-induced pressure signal through porous seafloors have been observed in the field and can contribute to vertical pressure gradients and momentary liquefaction (Yamamoto et al. 1978; Sakai et al. 1992; Raubenheimer et al. 1998; Mory et al. 2007; Yeh and Mason 2014; Guest and Hay 2017; Guo et al. 2019; Zhai et al. 2021; Stark et al. 2022). Small changes in the timing of pressure signals (fractions of a second) at different depths can result in large changes to the resulting vertical pressure gradients. Accurate measurements of these signal lags can be obtained with sensors synched to a common clock.

The pressure sensors used here had individual clocks that may have drifted relative to one another. Prior studies have shown the near-surface pressure signal occurs before (leads) the signal at depth (Fig. 4 in Yamamoto et al. 1978 and Fig. 10 in Guest and Hay 2017). Here, before accounting for possible clock drifts, the crests and troughs measured at the deeper location (P₂) occur before those near the bed surface (P₁) (Fig. 3).

To examine the signal relationship further, cross spectra between the two pressure time series were calculated using 55,296 points (57.6 min) starting at the top of each hour with overlapping window lengths of 8,192 points and merging of three adjacent frequencies, resulting in 40 degrees of freedom. In contrast to the frequency-dependent phase shifts expected for partially saturated porous media (Yamamoto et al. 1978; Raubenheimer et al. 1998; Guest and Hay 2017), for frequencies (f) above those of infragravity waves (f < 0.05 Hz), phase lags increase linearly with frequency (Fig. 4), consistent with a constant time shift within each hour period. Furthermore, the phase lag at a fixed frequency increases with time [compare different curves during the month of October in Fig. 4(b)], consistent with a slowly increasing clock drift (time shift between sensors). Surge and setup were increasing as tropical storm Melissa impacted the area, and it is unlikely that saturation decreased or that the sediment properties changed to cause the increasing phase lags.

Cross-correlation was used to identify the hourly relative time shift between P₁ and P₂, which increased from 0.7 s at deployment to 1.8 s at recovery. The hour-long time series from the two pressure sensors (P₁ and P₂) were aligned (phase-corrected) to minimize the phase differences between them. The estimated vertical pressure gradients may have been affected by the removal of the estimated clock drift phase lag, which represents a source of error and uncertainty. All discussion regarding the pressure sensors and the derived results are for the phase-corrected pressure signals.

The ADV clock was synced with a different computer clock than the pressure sensors, resulting in some uncertainty when cross referencing times between the ADV and the pore pressure clocks. Therefore, the maximum velocity associated with a possible liquefaction event is the maximum velocity (either horizontal or vertical) within ±1 s of the time given by the pore pressure sensor clock. The camera clock was also set independently and was synced approximately to the pressure and ADV measurements.

Sediment samples were collected at three cross-shore transects (Fig. 1) on October 7, 9, 10, and 11, 2019, to determine porosity and relative density. Waves and swash at the location of the sensors prevented the acquisition of sediment samples with intact particle packing arrangements, and therefore, the most offshore samples (properties given in Table 1) were collected approximately 25 m onshore of the sensors. Sediment samples at A8 and C8 were collected only on October 7, 2019, due to their proximity to the water line at the time of sample collection. The values reported for B6 (Table 1) represent an average of four different samples taken on October 7, 9, 10, and 11, 2019. Properties derived from the sediment samples for all stations in the transect and for all times can be found in Table S1.

Stations A7, A8, B6, C5, C6, C7, and C8 were submerged during high tide (Fig. 1, high tide line at A7 and C5). Sediment properties derived from samples at Stations A8, B6, and C8 are considered reasonable representations of those at the sensor location (Heathershaw et al. 1981; Kirchner et al. 1990). However, owing to the deeper water depths and longer periods of submergence, the sediment at the sensor location is expected to have a higher average moisture content and a higher unit weight (i.e., denser) compared with the estimated sediment properties (Heathershaw et al. 1981). Differences in sediment properties, water depths, and wave conditions may result in different levels of risk with respect to mobilization and transportation (Yamamoto et al. 1978; Sakai et al. 1992; Mory et al. 2007; Guo et al. 2019; Zhai et al. 2021).

Laboratory tests were not performed to estimate Poisson’s ratio, sediment permeability, or the shear modulus, which are often needed in numerical models of pore pressures within the sediment bed (Yamamoto et al. 1978). However, common values of Poisson’s ratio (0.33), shear modulus (4 × 10⁻⁸ Pa), and permeability (ranging between 10⁻³ and 10⁵ m/s) can be used. In the absence of laboratory tests, there are more robust calculations of sediment permeability (Carrier 2003).

Three criteria were used to assess the potential for liquefaction owing to the upward pore pressure gradients that developed within the beach, including using the Mory et al. (2007) liquefaction criterion, a seepage force–derived criterion (Terzaghi 1943; Duncan et al. 2011), and a criterion for the exceedance of the sediment buoyant weight. These criteria provide calculations of liquefaction thresholds with minimal assumptions and require sediment properties (e.g., porosity, saturated, and buoyant unit weight) that are relatively straightforward to measure.

The liquefaction equation criterion (Mory et al. 2007) has been modified to compare with the values measured by the pressure sensors (i.e., a pressure head with units of meters):where ΔP_crit = critical pressure head difference (m) required to induce liquefaction (Mory et al. 2007); Δz = vertical distance (m) between the sensors (0.2 m); ρ_s,f,g = density (kg/m³) values of the sediment (2,650 kg/m³), fluid (seawater, 1,030 kg/m³), and gas (air, 1.225 kg/m³), assumed to be constant; n = porosity; and C_gas = amount of gas within the sediment as a decimal value.

A mean ΔP_crit was calculated based on the mean porosity of the three closest stations, n = 0.428 (Table 1), the typical density values listed previously (assumed constant), a Δz of 0.2 m (assumed constant), and a range of gas contents from 0 (fully saturated) to 0.3 (30% gas content and 70% saturated) based on rounded values from Table 1, and the assumption that the offshore sediments at the sensors were at one point fully saturated. Direct measurements of gas content were not made for the sediments and were assumed based on the degree of saturation (C_gas = 1 − degree of saturation). The corresponding mean ΔP_crit from Eq. (1) is 0.35 m.

The seepage force liquefaction criterion (Terzaghi 1943; Duncan et al. 2011) uses a factor of safety to assess whether a given hydraulic gradient (i) induces failure of the sediment. Here, this method of failure is momentary liquefaction induced by upward vertical pressure gradients. This criterion is determined using

where F_e = factor of safety where failure occurs at F_e < 1; γ_b = sediment buoyant weight (kN/m³); γ_sat = saturated sediment unit weight (kN/m³); γ_sw = unit weight of seawater (10.1 kN/m³); i = unitless hydraulic gradient; h_1,2 = corresponding hydraulic head determined via Bernoulli’s equation (4) at the two sensor locations; L = unsigned distance (m) between the sensors (0.2 m, constant); P_1,2, z_1,2, and v_1,2 = respective pressure (kN/m²), vertical position (m) with respect to a common datum, and velocity (m/s), respectively, at the measurement locations; and g = acceleration due to gravity (9.81 m/s²).

Rearranging Eqs. (2)–(5) yields Eq. (6), which can be used to solve for the critical pressure head difference (ΔP_crit = (P₂/γ_sw) − (P₁/γ_sw)) that induces failure of the sediment:

The distance between the sensors in this study was 0.2 m (L), and the elevations of the two sensors with respect to the sand bed were z₁ = −0.02 and z₂ = −0.22 m (negative indicating below sand bed surface). The pore fluid velocity at the P₂ sensor was assumed to be small, resulting in a small velocity head (v₂ = 0). An increasing velocity head at the upper sensor will increase the required pressure head difference to induce liquefaction in Eq. (6). Assuming that a v₁ of 0 m/s yields a smaller ΔP_crit for a worst-case scenario (i.e., the smallest ΔP required for sediment failure), the resulting ΔP_crit value using Eq. (6) will be 0.36 m.

The final liquefaction criterion is based on the vertical pore pressure gradient exceeding the sediment buoyant weight:

Multiplying by the distance between the sensors and the unit weight of seawater provides units consistent with the pore pressure sensor units (i.e., meters):

From the mean bulk unit weight, constant distance between sensors, and constant unit weight of seawater, the corresponding ΔP_crit value is 0.16 m.

From Eqs. (1), (6), and (8), the corresponding threshold values are 0.35, 0.36, and 0.16 m, respectively. The threshold (Mory et al. 2007) used in the “Results” section was developed under similar conditions to, and may be a better representation of, the processes at this site relative to the seepage-driven liquefaction criterion (Duncan et al. 2011; Terzaghi 1943). However, the difference between the two criteria is small (2.8% difference).

Data collected from October 8 to 11, 2019 (long enough after the deployment time to reduce associated sediment disturbance), were analyzed to investigate the effects of tropical storm Melissa. Pressure data were analyzed in hour-long segments, each of which was detrended to remove atmospheric and still water height pressures [including the 0.20 m vertical distance between sensors, differences in the y axes between Figs. 2(b and c)], leaving only the excess pressure head [Fig. 2(c)]. The liquefaction criteria assume that the pressure gauges P₁ and P₂ are 0.02 and 0.22 m below the sediment bed surface, respectively [Fig. 2(a)], which is roughly valid prior to October 11. Data sets were organized into three categories based on each hour-long maximum upward (positive) excess pressure head difference (ΔP = P₂ − P₁) with respect to a liquefaction criterion [ΔP > 0.35 m, Eq. (1) from Mory et al. 2007].

Maximum pressure head differences that meet or exceed 0.35 m (Mory et al. 2007) theoretically induce momentary liquefaction and are labeled GM. Data for which 0.28 (80% of 0.35) < ΔP < 0.35 are labeled LM, indicating ΔP values that may increase the risk of momentary liquefaction but are less than the 0.35 m criterion. The 80% cutoff was chosen because it delineates between the larger (GM) ΔP values and the much smaller and more numerous ΔP values [Fig. 5(a)]. Hour-long data sets that had ΔP < 0.28 m theoretically result in no liquefaction and are labeled NL. Of the 335 hour-long data sets, 18 (5.4%) were classified as GM, 9 (2.7%) were classified as LM, and 308 (91.9%) were classified as NL. The GM and LM data occurred between October 8 and 11, 2019, when local (and offshore, not shown) water levels and wave heights were increasing due to tropical storm Melissa (Fig. 5). It is possible that liquefaction was not observed following the peak of the storm owing to accretion, which resulted in sensors being buried by 0.3 to 0.5 m of sand.

In contrast to previous studies (Sakai et al. 1992; Conley and Inman 1994; Mory et al. 2007; Stark et al. 2022), excess pore pressure heads [Fig. 2(c), approximately 910 s] did not occur during wave troughs or on the back of wave crests, but instead usually occurred on the rise and peaks of wave crests. These wave crests [similar to Fig. 2(c) at approximately 910 s] occurred shortly after a significant drop in the excess pressure head [Fig. 2(c), 895–905 s, approximately 1.5 m of pressure head over 10 s], often followed by a deep trough [Fig. 2(c), approximately 907 s]. The images at the sensor location (Video S1) indicate that the water receded so that the beach surface emerged (decreasing the excess pore pressure) within about a meter of the sensors, followed by the arrival of a wave (rapidly increasing the excess pore pressure), thus causing a rapid fluctuation of the excess pore pressure within the sand column [Fig. 2(c)]. Coupled with the known attenuation of high-frequency waves at shallow depths (Yamamoto et al. 1978; Raubenheimer et al. 1998; Guest and Hay 2017), this sudden decrease and then increase of the excess pressure often caused a noticeable attenuation of the P₁ pressure signal compared with that of P₂ [Fig. 2(c), at approximately 910 s, the P₁ curve is below the P₂ curve], resulting in an upward excess pressure head difference that exceeded the failure criteria.

Large excess pore pressure head differences were not correlated with water depth, significant wave height of sea-swell or longer period (infragravity) waves, or average wave period (Fig. 5), suggesting that other wave or sediment parameters (e.g., density, porosity, and gas content) (Yamamoto et al. 1978; Mory et al. 2007) may be important for the development of large excess pore pressure head differences. Observations gathered on a beach with similar sediment properties, but more moderate wave conditions (wave heights less than about 0.5 m) indicated small excess pore pressures following the passage of the wave crest, but did not show momentary liquefaction (Stark et al. 2022). Therefore, it is possible that high wave energy is necessary for liquefaction, but is not sufficient on its own. Time histories of geotechnical properties to determine what sediment parameters, if any, were important to the occurrence of the large excess differences were not gathered, although it is unlikely that sediment properties changed from wave to wave. In contrast, the shapes of waves did change from wave to wave. The shapes of waves and pressure gradients can be quantified by their third moments, skewness (sharp crests and flat troughs), and asymmetry (steep front faces and gently sloping rear faces) (Elgar and Guza 1985). During tropical storm Melissa (October 8–11, 2019), the maximum ΔP was not correlated with the skewness or asymmetry of P₁ or P₂ (not shown), or with the asymmetry of ΔP [Fig. 6(a), R² = 0.29], but was correlated with the skewness of ΔP [Fig. 6(b), R² = 0.74]. A ΔP signal with a large positive skewness value indicates that P₂ records a larger excess pressure head than P₁ before quickly returning to values similar to those of P₁, and that the ΔP signal exhibits rapid increases and decreases followed by more quiescent periods [Figs. 2(b) and 1(c), t = 910 s], similar to a skewed wave with a sharp crest and broad trough.

For the 3-day period (October 8–11, 2019) when the larger pressure gradients (LM and GM) occurred, the cross-shore and vertical velocities most often were negative (offshore and downward, Fig. 7, histogram). However, for most of the theoretical liquefaction events (Fig. 7, triangular and x symbols), both horizontal and vertical velocities observed approximately 0.15 m above the sand bed were positive (onshore- and upward-directed, respectively), suggesting that the events coincide with a shoreward propagating wave crest.