Wave-induced pore pressure measurements near a coastal structure

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008jc005071, 2009 Wave-induced pore pressure measurements near a coastal structure H. Michallet, 1 M. Mory, 2 and I. Piedra-Cueva 3 Received 6 August 2008; revised 23 February 2009; accepted 3 April 2009; published 19 June [1] Wave-induced pore pressures were measured at various depths below the sandy bed in front of a coastal structure. The results of a weeklong field experiment carried out in September 2003 on the beach of Capbreton, in southwest France, are presented. The coastal structure was located in the intertidal zone of the beach. The transmission of pressure variations inside the soil, as compared to the pressure variations produced by the waves in the water layer, are analyzed in terms of both amplitude decay and phase shifts and compared to theoretical models. The gas content inside the soil was also measured. The results confirm that the gas content is a key parameter affecting the transmission of pressure inside the soil. It is shown that a significant upward pressure gradient is generated during a wave period, which can liquefy a 30 cm deep superficial layer of the soil. This is interpreted in terms of momentary liquefaction events. The manner in which the phenomena change during the different tidal periods investigated is described. The dependence of the results on wave height and bed level is discussed. Whereas the soil properties were not modified over a tidal period when the wave activity was sufficiently low, a significant change in the transmission of pressure variations inside the soil was observed when the structure was subjected to larger waves. This is interpreted in terms of a change in the gas content in the superficial layer. Citation: Michallet, H., M. Mory, and I. Piedra-Cueva (2009), Wave-induced pore pressure measurements near a coastal structure, J. Geophys. Res., 114,, doi: /2008jc Introduction [2] For more than 30 years the physics of soil liquefaction has been extensively studied by geotechnical professionals and researchers, geophysicists interested in earthquake phenomena, and researchers in fluid mechanics. Liquefaction can occur by different mechanisms but in all cases it occurs when a sufficiently high vertical pressure gradient is produced inside the soil to overcome the gravity static equilibrium of the bed. The first mechanism which can result in liquefaction is the buildup of pore pressure induced by waves, seismic action, or impulsive loads. This process is more likely to occur in silty soils where pore pressure may increase over time because of the lower drainage potential. A second phenomenon, called momentary liquefaction, can be produced during the passage of a surface wave trough. In loose sandy soil, pore pressure cannot accumulate, but significant vertical pressure gradients can be generated during a wave period, resulting in liquefaction. [3] The literature of theoretical studies on liquefaction is especially rich. The pioneering studies [Madsen, 1978; Yamamoto et al., 1978; Mei and Foda, 1981] showed how complex it is to model pore pressure in the soil. Soil 1 LEGI, UMR5519, UJF, INPG, CNRS, Grenoble, France. 2 LASAGEC2, Université de Pau et des Pays de l Adour, ENSGTI, Pau, France. 3 Faculty of Engineering, University of the Republic, Montevideo, Uruguay. Copyright 2009 by the American Geophysical Union /09/2008JC permeability is far from sufficient to describe the flow in the porous medium. Soil deformation due to the stresses applied on the soil skeleton has to be taken into account. The presence of gas in the soil was found to be a key parameter for the occurrence of liquefaction. [4] The physics of liquefaction has attracted the attention of coastal engineers in the past decade. An analysis related to coastal structures design was given by de Groot et al. [2006a]. A wide class of marine structures and armoring systems are used to protect the coastal community and to provide essential support to human life. To achieve reliable and well-functioning coastal protection, a comprehensive knowledge of the behavior of the soil foundation supporting this protection is required in relation to sea bed stability. A theory, developed by Hsu et al. [1993] and Hsu and Jeng [1994], extends the work of Yamamoto et al. [1978] and Mei and Foda [1981] by describing pore pressure transmission in homogeneous soils of infinite or finite thickness in front of a vertical wall. Jeng [2003] presented a thorough review of recent developments in the interaction between water waves, sea bed and coastal structures. [5] Several observations of structure failure resulting from earthquake events [Sumer et al., 2007] or from a storm event [Zen et al., 1986] have been made. The relevance of a liquefaction phenomenon to explain such failures was discussed by de Groot et al. [2006b]. Detailed field observations of pore pressure in the surf zone near coastal structures are scarce, partly as a result of the difficulties of taking measurements in the zone where waves are breaking against structures. Measurements have been taken in the Mississippi 1of18

2 Delta clayey sediments [Bennett and Faris, 1979], in Central Japan around Shimizu Harbor [Okusa, 1985a] in a silty sand and sand-gravel sediments, in a sandy beach [Raubenheimer et al., 1998], in the Zeebrugge breakwater (Belgium) [de Rouck and Troch, 2002], and in a large-scale facility [Kudella et al., 2006]. All these studies have shown that pressure fluctuations due to tidal and surface wave activity produced significant pore pressure response in these soils. An alternative approach to field measurements is to reproduce the wave action in the laboratory by cyclically moving a structure on a sand bed. Such experiments performed by Foray et al. [2006] have shown that momentary liquefaction can be reached locally around the structure at each cycle. [6] Liquefaction (sometimes called fluidization) is an extreme event for which a full layer of soil becomes unstable. To a lesser extent, pore pressure transmission is an important characteristic for estimating erosion and sediment transport. Vertical pressure gradients induce exfiltration and infiltration through the sand bed that modify sediment mobility [Turner and Masselink, 1998; Butt et al., 2001]. In recent years, several studies have attempted to take accurate pore pressure measurements in the swash zone. Cartwright et al. [2006] discussed pore pressure measurements in the swash zone in relation to the water table exit point on the beach. Saturation is indeed a key feature for pore pressure transmission. As pointed out by Baldock et al. [2001], very large pressure gradients have the potential to fluidize the top 15 mm of the bed material at some point on the beach face. In contrast to these experiments conducted in the swash zone, coupled bed stability and pore pressure measurements are scarce in the breaking zone. [7] This paper presents the analysis of an extensive set of data measured during an in situ study carried out in 2003 as part of the LIMAS research program (Liquefaction around Marine Structures) supported by the European Union ( ). Partial and preliminary results of this field experiment have already been presented by Bonjean et al. [2004] and Mory et al. [2007]. Pore pressure measurements were taken inside the sand soil in front of a coastal structure located in the intertidal area of a beach. Soil properties were measured. Bed level displacements due to scour and sand deposition were recorded over the time interval of the tidal periods during which the coastal structure was partially submerged. Hydrodynamic measurements in the water layer were carried out as well. Because the bed was made of sand, no buildup of pore pressure was observed. The primary objective of this field experiment was to observe in situ the occurrence of momentary liquefaction events. Such events were indeed observed as already reported by Mory et al. [2007]. The full set of data is presented in this paper together with a discussion of the various wave conditions observed during the field experiment and the two locations investigated (vertical wall and corner of the structure). The paper focuses on the pressure transmission inside the soil. It is interpreted in terms of the likelihood of liquefaction. The role of saturation in the transmission of pressure variations is analyzed in detail with gas content data obtained using a geoendoscopic camera. The field experiment and measurement methods are described in section 2. The observations of pressure variation transmission are described in the two following sections. Section 3 focuses on the tidal period of 28 September which is taken as a reference case to explain in detail the experimental data analysis principle. The results obtained during different tidal periods are then discussed in section 4. An overall analysis and interpretation of the set of data is presented in section 5. The last section summarizes the conclusions. 2. Description of the Field Experiment and Measurement Techniques [8] This section describes how the field experiment was conducted and the nature of measurements taken. The field experiment was carried out in September 2003 and October 2003 on the Atlantic coast of Aquitaine, in south west France. Investigations were carried out in the vicinity of a Second World War bunker which is now in the intertidal zone of the beach at Capbreton (30 km north of Biarritz), because of the significant beach retreat that has occurred. The coast is subjected to strong wave forcing. The main data set obtained during the field experiment consists of time series of pressure measurements taken at five different levels along a vertical axis. The pressure sensors were deployed with a spacing of 30 cm in a vertical bar that was fixed on the bunker. The bar was partially embedded in the sandy bottom. The properties of waves impacting on the structure were determined using the pressure sensors in the water layer and the transmission of pressure variations inside the soil was analyzed by comparing the pressure variations measured at different levels inside the soil to those measured in the water layer. The general aim of this experiment was to investigate the possible occurrence of liquefaction of the sandy bed caused by waves. The bed level variations along the bar equipped with pressure sensors were measured by a system consisting of eight fibre optic sensors which detect (from the backscattering of light) whether a fibre tip is inside the soil, in the water layer, or in a sediment suspension. The system, similar to that used by Conley and Beach [2003], has been more extensively described by Michallet et al. [2005]. A fibre optic sensor was placed near each pressure sensor and the three remaining ones were located in between the four uppermost pressure sensors. The system was used to determine when the pressure sensors remained inside the soil layer. The more detailed description of the field experiment and of the measurement techniques employed, given by Mory et al. [2007], is not repeated here. The position of the bed level was also regularly measured using a graduated pole put manually on the bottom. Additional hydrodynamics measurements of pressure and the three velocity components were recorded using a Nortek Vector Acoustic Doppler Velocimeter. [9] The deployment of the various instruments described above on the bunker is shown in Figure 1. The photograph was taken from offshore. Figure 1 shows the configuration studied between 24 and 28 September, for which the pressure and fibre optics bar was located in the middle of the bunker wall facing the ocean. Between the afternoon of 28 September and 2 October, measurements were obtained 2of18

3 Figure 1. Deployment of measurement systems for investigation in the middle of the bunker wall facing the ocean. Measurements were taken at the corner indicated from 28 September in the afternoon until the end of the field experiment. ADV, Acoustic Doppler Velocimeter. in the vicinity of the bunker corner. The second position is also indicated in Figure 1. [10] Soil properties were also characterized during the field experiment. The grain size distribution in the vicinity of the bunker consisted of a narrow size band with mean sediment size d 50 = 0.35 mm. Penetrometer tests were carried out around the bunker at each neap tide in order to evaluate the variation in soil properties with regard to erosion and deposition that occurred. The geotechnical properties were described partly by Mory et al. [2007], and more extensively by Bonjean et al. [2004]. For the purpose of the present paper, the most significant observation was that the effects of disturbances caused when setting up or displacing instruments could no longer be detected after one tidal period. Investigations inside the soil were additionally made on 25 and 26 September using a geoendoscopic camera placed inside a Perspex tube fixed to the bunker. The vertical variation in soil properties was analyzed by monitoring the grains and their displacement as well as the gas content in the soil. Figure 1 shows the location of the Perspex tube for the geoendoscopic camera. The principles and results of the geoendoscopic camera investigations were given by Mory et al. [2007] and Bonjean et al. [2004]. [11] Table 1 summarizes the conditions observed during the field campaign, conducted during a week with high tide conditions. The significant wave height at full tide is given for each tidal period. The sea was relatively calm. The most energetic wave conditions were experienced on 24 September. The wave activity was lower during the following days. However, the magnitude of waves impacting on the bunker was quite significant compared to the mean water depth. This paper focuses on the tides of 24, 25, and 28 September (morning tide) as well as 1 October (evening tide). The tide of 28 September is taken as a reference case. Measurements are analyzed in detail in section 3. On 28 September, the sea was relatively calm, the soil variations were limited, and the pressure bar was deeply embedded in the soil. The pressure transmission phenomena inside the soil could be described precisely. Section 4 also focuses on the tides of 24 and 25 September and 1 October. Intense phenomena were observed during the tides of 24 September and 1 October. On the first day, the wave activity was strong and measurements were taken at the middle of the bunker wall facing the ocean. On the second day, measurements were taken at the bunker corner. Although the wave height was smaller, significant phenomena were observed because of the intense flow produced when a wave passed at the corner. The tide of 25 September does not really add any further insight compared to the reference case of 28 September. It is briefly discussed in section 3 for completeness because the geoendoscopic camera observations were made on that day. [12] An aggregated analysis of the damping of pore pressure variations and changes in soil properties for all tidal periods investigated during the field campaign is given in section 5. The effect of gas content is discussed. In order to compare the different cases, the position of the upper Table 1. Summary of Tidal Periods Analyzed During the Field Experiment Date Time a Case P 1 Absolute Position b (cm) Bed Position c (cm) Mean Water Level d (cm) H s e (cm) P 1 Relative Position f (cm) P 5 Relative Position f (cm) Symbol in Figures 17, 18, 19, and Sep 0405 Wall Solid circle 25 Sep 1648 Wall Solid square 26 Sep 1731 Wall Solid triangle down 27 Sep 1814 Wall Solid triangle left 28 Sep 0643 Wall Solid triangle right 28 Sep 1829 Corner Open circle 29 Sep 0741 Corner Open square 29 Sep 1926 Corner Open triangle down 30 Sep 0810 Corner Open triangle left 30 Sep 2024 Corner Open triangle right 1 Oct 0846 Corner Open triangle up 1 Oct 2122 Corner Open diamond 2 Oct 0950 Corner Open star a Time of full tide. b Measured from the top of the bunker (precision ±1 cm). c Measured at full tide from the top of the bunker (precision ±10 cm). d At full tide (precision ±10 cm). e Significant wave height at full tide (precision ±5 cm). f Taken from the bed surface at full tide (precision ±10 cm). 3of18

4 Figure 2. Variation in the mean water level (solid line) and the maximum water level (dash-dotted line) during the morning tide of 28 September and state of pressure sensors P 1 P 5 with regard to their location in static soil (squares), in unstable soil (asterisks), in a sediment suspension (crosses), and in clear water or in air (dots). Diamonds indicate the position of the deepest pressure sensor above which the soil layer can be liquefied according to criterion (1). pressure sensor P 1 and bed position at full tide, measured from the top of the bunker, are given in Table 1 for each tidal period. The bed position was determined from optical probe measurements. The precision is basically given by the spacing between two neighboring optical fibre optics probes (15 cm) but the interpretation of the optical sensor response in terms of soil mobility can be used to refine the precision in the bed level position to about ±10 cm. The mean water depth is the difference between the mean water level measured by pressure sensors and the position of the bed level. The precision on the mean water level is therefore ±10 cm as well. [13] Mory et al. [2007] have already showed the occurrence of momentary liquefaction events during their field experiment. A significant upward pressure gradient can be produced in the sediment bed when the damping of pressure variations in the soil is sufficiently high. Below the wave trough, the pressure is less reduced in the bed than it is at the bed surface if pressure variations are not well transmitted in the soil. A sediment layer is in a transient state not in static equilibrium when the upward pressure gradient overcomes the effective weight of the soil. Because of the upward seepage flow [e.g., Zen et al., 1998], the sediment layer is dilated, the porosity in the soil increases, but the pore pressure is rapidly adjusted to the value on top of the soil because of the increase in permeability. This is a momentary liquefaction event. The soil around the bunker is a sandy bed and no pore pressure buildup phenomenon was observed in the course of the field study. [14] From the measurement of pressure at different levels, it can be inferred when the sediment layer between two pressure sensors is not in static equilibrium. The pressure difference DP i,i 1 = P i P i 1 required to overcome the effective weight of the soil between neighboring pressure sensors is DP i;i 1 > rgh r s r ð1 Þþ C gas 1 r s r ; ð1þ where r is the fluid density, r s is the density of sand, C gas is the volumic gas content and h = 30 cm is the distance between the sensors. The gas content is unimportant in this estimate which varies essentially with the soil porosity. The occurrence of momentary liquefaction events is analyzed systematically in this paper for all tidal periods of the field experiment, using the criterion (1). 3. Reference Case of the Morning Tide of 28 September Conditions [15] The sea was calm on 28 September. Because pressure variations were measured deep inside the soil, the analysis of the morning tidal period of 28 September describes well the procedures used and the nature of results obtained. [16] Bed level changes and pressure variations inside the soil were measured at the middle of the bunker wall facing the ocean. The instruments had been set up on 23 September. The beam supporting the pressure sensors was lowered 39 cm on 27 September, but this created an insignificant disturbance in the soil. Geotechnical measurements have shown [Bonjean et al., 2004] that soil properties result almost entirely from processes occurring during the preceding tide. Figure 2 shows the change in mean and maximum 4of18

5 Figure 3. Power spectral density (estimated with Welch method) of water pressure variations measured simultaneously by pressure sensors P 1 P 5 during the morning tide of 28 September at high water (from 0618 to 0638). water level during the tide. The diagram also indicates the status of the five pressure sensors with regard to the position of the bed surface and to the state of the soil. Four cases are distinguished: (1) the sensor is inside the soil, and no motion is detected (soil in static equilibrium); (2) the sensor is inside the soil, but displacements are detected; (3) the sensor is in the fluid layer containing sediment in suspension; and (4) the sensor is in clear water or in air. These different states are estimated over 10 min sequences and they do not therefore reveal all the wealth of soil and bed dynamics. In the bed region in particular, a sensor may record several changes from static soil to suspension within a 10 min sequence. Little sediment erosion and deposition occurred during the tide of 28 September. The upper pressure sensor P 1 was always above soil level while pressure sensors P 3,P 4 and P 5 remained inside the soil. Pressure sensor P 2 was very close to the bed surface during the entire tide, sometimes in the water layer and sometimes inside the soil. Because of this, the position z = 0 in Figure 2 is taken as the position of P 2 in order to help the interpretation. The origin Z = 0 at the top of the bunker is also sometimes used for interpreting results. Although the sea was calm, Figure 2 clearly shows that water depth variations caused by waves impacting the bunker wall were quite significant compared to the mean water depth, especially during the rising tide. [17] The possible occurrence of momentary liquefaction events is indicated in Figure 2 using diamonds. A diamond positioned at the level of a pressure sensor P i means that the pressure difference DP i,i 1 = P i P i 1 between P i and its above neighboring pressure sensor P i 1 was found to verify the inequality (1). The criterion was computed for a porosity = 0.5 corresponding to loose sand. This underestimates by about 18% the pressure difference required to break the static equilibrium in more compact sand ( = 0.4). Although it was clear that the bed was not made of loose sand at neap tide, the value of the porosity during the tide is unknown. Criterion (1) assumes that the entire layer of sediment contained between the two sensors is deposited. Figure 2 shows that this is not always the case. All liquefaction events are predicted between P 2 and P 3, but the sediment state at the level of P 2 is either unstable soil or sediment suspension. The value = 0.5 for the porosity was chosen as an average value for estimating the quantity of sediment contained between the two sensors. No buildup of pore Figure 4. Analysis of pressure transmission inside the soil during the morning tide of 28 September (from 0618 to 0638) as a function of frequency. Ratio of the energy density spectra of (a) pressure variations measured by two neighboring pressure sensors P 2 2 i ( f )/P i 1 ( f ) and (b) pressure variations to the pressure sensor in the water column P 2 i ( f )/P 2 1 ( f ), for i =2,..., 5. Phase shift of (c) pressure variations measured between neighboring sensors and (d) pressure variations to pressure sensor P 1. Line symbols refer to i = 2 (dotted), i = 3 (solid), i = 4 (dashed), and i = 5 (dash-dotted). 5of18

6 Figure 5. Variations during the morning tide of 28 September. (a) Significant wave height measured by the five pressure sensors: P 1 (sharp), P 2 (dotted), P 3 (solid), P 4 (dashed), and P 5 (dash-dotted). (b) Damping ratio 2 P 2 i (f p )/P i 1 (f p ) of pressure variations measured at neighboring depths and (c) phase shift between P i (f p ) and P i 1 (f p )for i = 2 (dotted), i = 3 (solid), i = 4 (dashed), and i =5 (dash-dotted). pressure is observed in the pressure time series. When occurring, criterion (1) is verified over a short period of time during a wave period (see examples of liquefaction events in the paper by Michallet et al. [2005] and Mory et al. [2007]). Consequently, the breaking of the static equilibrium given by (1) is called a momentary liquefaction event. When the pressure difference DP i,i 1 is found to verify criterion (1), it is actually observed that the whole layer above P i can also be liquefied. Diamonds therefore indicate the position of the deepest pressure sensor above which the soil layer could be liquefied at a given time during the tide, considering the pressure variations measured. Although the criterion (1) is reached many times during the tide, the soil is most of the time evaluated as stable by the optical sensor in between P 2 and P 3. Once liquefaction has ceased, it is expected that the sand grains are left in a more compact arrangement. No motion of the grains is then recorded by the optical sensor until the next liquefaction event Analysis of the Damping Pressure Variations Using Fourier Time Series [18] Considering typical time series of pressure variations measured at different levels in the water layer and inside the soil, Mory et al. [2007] have shown that a Fourier frequency analysis can be used to quantify the damping of pressure variations in the soil. This work has been complemented by also considering the phase shift of pressure variations inside the soil. Figure 3 shows the energy density spectra of 1200 s recordings of water pressure variations measured by the 5 pressure sensors P 1 P 5. The spectra measured by sensors P 1 and P 2 almost overlap because both sensors are located in the water layer. The five spectra display similar shapes with well identified frequency bands, but the magnitude of the spectrum decreases at all frequencies with increasing depth inside the soil. The damping of pressure variations with increasing distance inside the soil is quantified by comparing the value of the power spectrum measured at the different levels for each frequency. In Figure 3 the spectrum measured by sensor P 3 is significantly below the spectra measured by sensors P 1 and P 2, indicating that P 3 is inside the soil. [19] Figure 4a displays the ratio P 2 2 i ( f )/P i 1 ( f ) of the pressure energy spectra measured at different neighboring depths along the vertical direction (i = 2,..., 5). The variation with frequency is limited to the band [0.02 Hz, 0.4 Hz] which contains almost all the pressure variation energy. Phase lags between the pressure variations measured at different depths were also determined by computing the cross-power spectra density of P i ( f ) with P i 1 ( f ). The results are shown in Figure 4c. The distance between neighboring pressure sensors is 30 cm. Figure 4a shows that the damping varies along the vertical direction. Because P 1 and P 2 are both in the water layer, no damping or phase shift is noted between the variations measured by the two upper sensors. At all frequencies, the most significant damping is obtained in the upper part of the soil between P 2 and P 3. The damping decreases with increasing depth. Similar trends are observed for the phase shift. The greater the damping between two pressure sensors, the greater the phase shift between the pressure signals. A particular case is observed between pressure sensors P 4 and P 5. The phase shift is almost zero while the damping remains significant. The same observations can be deduced from the alternate representation in Figures 4b and 4d where the damping and phase shift for each sensor in the vertical is computed relatively to the sensor in the water column P 1. [20] Figure 5 presents the variation in the ratio P 2 i (f p )/ 2 P i 1 (f p ) and phase shift between P i (f p ) and P i 1 (f p ) during the morning tide of 28 September for the four lowest pressure sensors (for i =2,..., 5). From images similar to Figure 3 plotted at different times during the tide, the damping ratio 2 P 2 i ( f )/P i 1 (f) and the phase shift were averaged over the frequency band [0.08 Hz, 0.12 Hz] centered at the peak frequency f p 0.1 Hz. The results were checked to make sure they did not depend on the frequency band width by comparing them with estimates obtained for a wider band width ([0.05 Hz, 0.15 Hz]). The interesting result of Figure 5 is that the damping and phase shift change only slightly, although wave conditions were maintained at a significant level during the tide (Figure 5a). Pressure variations were transmitted a little better at the end of the tide than at the beginning, but this is not very significant compared to what is observed for other tidal periods discussed later. Because Figure 5 compares the conditions at neighboring heights, it is evident that most of the damping of pressure variations 6of18

7 Figure 6. Wavelet analysis at the wave peak frequency f p = 0.1 Hz of pressure records measured in the soil during the morning tide of 28 September at (a) Raw data record measured by P 2 (solid) and P 4 (dashed), (b) real parts of the wavelet transforms, (c) corresponding phase shift, and (d) ratio of the wavelet norm between P 2 and P 4. occurs between pressure sensors P 2 and P 3. The largest phase shift is also obtained between these two sensors. Deeper inside the soil, the decay is significantly weaker Analysis of Damping Pressure Variations Using a Wavelet Approach [21] In essence, a wavelet analysis provides a detailed glimpse of the characteristics of the data in the timefrequency domain [Torrence and Compo, 1998]. It is especially helpful for analyzing waves propagating in wave groups. For a selected frequency, the time-dependent modulation of the wave envelope is determined. The pressure data time series was analyzed by a continuous wavelet transform based on the Morlet mother wavelet [Liu, 2000; Liu and Hawley, 2002] which is a complex-valued wavelet derived from a plane wave modulated by a Gaussian envelope. It is a natural extension of the familiar and conventional Fourier spectrum analysis. [22] Figure 6a shows a 250 s raw data record of pressure variations measured by sensors P 2 and P 4. The wave group structure of the signal is clearly visible. In Figure 6b, the real parts of the wavelet transforms at the peak frequency of the spectrum (f p = 0.1 Hz) are superimposed. The correlation between the two signals is clearly apparent. The phase shift computed from the cross correlation of the two wavelet transforms of P 2 and P 4 is a continuous function of time plotted in Figure 6c. It varies around 45 degrees, in agreement with the result of the Fourier analysis (Figure 4d). The wavelet norm corresponds to the square of the envelope of the real parts plotted in Figure 6b. The ratio between norms is plotted in Figure 6d. This is an estimate of the damping between P 2 and P 4, and is in good agreement with the value (about 0.3) deduced at the wave peak frequency in Figure 4b. [23] From the real part of the wavelet transforms (Figure 6b), a wave-to-wave correspondence can be established showing both the damping with increasing depth of pressure variation amplitude and the phase shift between signals. The pressure variation amplitude was determined adopting the zero down-crossing technique. Figure 7 shows the near-linear correlation between the amplitudes of the wavelet transforms of the pressure variations measured by sensors P 2 and P 4 for all waves recorded during the morning tide of 28 September. Different symbols are used for three successive tide sequences indicating that damping is greater at the beginning (from 4:30 till 5:10, P 4 /P ) than for the following 250 waves (from 5:10 till 5:50, P 4 /P ) and the rest of the tide. The ratio P 4 /P at high tide given by Figure 7 is consistent with the value obtained from the Fourier analysis at the wave peak frequency (Figure 5b). The variation in wave height ratio during the tide is plotted in Figure 8a while the corresponding phase shift variation is shown in Figure 8b. The tendency of damping to decrease with time, as already shown in Figure 5, is confirmed. 7of18

8 Figure 7. Correlation between the wave heights deduced from a wave-by-wave analysis of the real parts of the wavelet transforms (at the wave peak frequency f p = 0.1 Hz) of pressure measurements P 2 and P 4. The whole set of waves recorded during the morning tide of 28 September is taken into account: first 250 waves (plus signs), from wave 251 to wave 500 (dots), and last 860 waves (multiplication signs). phase shift in the pore pressure response also increases with depth. The theoretical work by Hsu et al. [1993] and Hsu and Jeng [1994] is best suited for a comparison with the conditions of our field experiment. Their analytical solution predicts the wave-induced soil response for a seabed of infinite or finite thickness subjected to a three-dimensional wave system impacting a vertical wall. The solution has been further applied to the cases with random wave loading by Liu and Jeng [2007]. However, the calculation is very lengthy because of the numerous coefficients which have to be computed. We compared our experimental results with the predictions from the set of equations presented by Sakai et al. [1992]. They describe the time-dependent change in pore pressure in the seabed when a monochromatic wave passes over an elastic plane sandy bed. These equations actually originate directly from the work of Mei and Foda [1981]. Although the geometry and the conditions considered by Sakai et al. [1992] are different from ours, the comparison is meaningful for several reasons. First, the difference between the solution proposed by Hsu and Jeng [1994] and that of Mei and Foda [1981] is almost insignificant in the upper part of the soil, as shown by their Figures 5, 6, 7, and 8. Penetrometer tests have been performed on the [24] The conclusions from the Fourier analysis and from the wavelet analysis are fully consistent. Nevertheless, the wavelet analysis is better at highlighting the linearity of the pressure transmission process inside the soil. The pressure variation amplitude measured inside the soil varies almost linearly with the pressure variation amplitude measured in the water layer, although the ratio between the two varies with the properties of the soil between the two positions, as discussed later. The clustering of data around different straight lines in Figure 7 indicates that the soil properties were modified during the morning tide of 28 September. This is also confirmed by the Fourier analysis of pressure data. Figure 9 compares the spectra measured by the five pressure sensors at the beginning and at the end of the tide. Two time series were chosen during which the wave climate was roughly the same, as shown by the pressure spectra measured by P 1 (Figure 9a). Pressure is transmitted better at the end of the tide than at the beginning (Figure 9b). The phase shift decreases a little (Figure 9c). Figure 9 is a detailed account of the overall variations quantified in Figure 5 for the whole tide. Because the soil properties were not modified to any great extent, the morning tide was chosen as a reference case for analyzing pressure transmission inside the soil. This behavior is however specific and should not be considered as representative of the whole field experiment. Observations made on different days will display significant variations during the tidal periods, as discussed in section Comparison With Theory and Estimation of Gas Content in the Soil [25] The damping of pore pressure variations inside the soil produced by water waves has been considered theoretically by various authors. The pioneering work by Yamamoto et al. [1978] and Mei and Foda [1981] were mentioned in section 1. The pressure is found to attenuate more rapidly in partially saturated dense sand than in the saturated case. The Figure 8. Variations in (a) wave height ratio and (b) phase shift during the morning tide of 28 September between the real parts of the wavelet transforms (at the wave peak frequency f p = 0.1 Hz) of pressure variations measured by P 2 and P 4. 8of18

9 model. The pore pressure variation inside the bed is given by equation (3) of Sakai et al. [1992] Pz ðþ 1 ¼ 1 þ m e lz þ m 1 þ m e i 1 P 0 ffiffi ð Þz= p 2 d e i lx st ð Þ : ð2þ l and s =2pf are the wave number and angular frequency of the wave propagating in the water layer, respectively. The z axis is oriented downward and z = 0 is at the bed surface. The pressure decay inside the soil is given by the wave length 2p/l of water waves and the length scale d defined as d ¼ k G 1=2 ng 1 2n 1=2 þ : ð3þ rg s b 21 ð nþ d depends on the bed porosity n, on the bed permeability k, and on the elastic properties of the soil given by the shear modulus of the solid skeleton G and the Poisson modulus n. An important issue of the Sakai et al. [1992] model is the very significant sensitivity of the results to the gas content in the soil C gas, as shown by Gratiot and Mory [2000] [see also Okusa, 1985b]. The gas content enters in the effective bulk modulus of pore water b [Verruijt, 1969] 1 b ¼ 1 þ C gas b w P ref ð4þ and in the parameter m in (2) Figure 9. Comparison of pressure transmission inside the soil at the beginning (from 0438 to 0458, sharp lines) and at the end (from 0808 to 0828, bold lines) of the morning tide of 28 September. (a) Pressure spectra P 1. (b) Ratio of the energy density spectra of pressure variations measured by two neighboring pressure sensors P 2 2 i ( f )/P i 1 ( f ) and (c) phase shift of pressure variations measured by neighboring sensors as a function of frequency: P 2 to P 1 (dotted), P 3 to P 2 (solid), P 4 to P 3 (dashed), and P 5 to P 4 (dash-dotted). site as deep as 3 m without reaching a rocky substrate [Bonjean et al., 2004]. The soil depth is therefore at least 3 times the water depth and our pressure measurements are confined in the upper soil layer. In such conditions, the effect of the seabed thickness is weak compared to the effect of a potentially varying degree of saturation. Second, the presence of the wall does not modify the pressure transmission process as long the reference pressure on the bed surface is correctly specified. Finally, the relative simplicity of the Sakai equations will facilitate discussion of the dependence of the results on the elastic properties of the bed. [26] The equations of Sakai et al. [1992] can be used for a direct comparison with the experimental data analyzed by Fourier series and for a discussion of the experimental observations in terms of gas content inside the soil for any frequency of the wave spectrum. Our experimental results were therefore compared with the results of Sakai et al. [1992] in spite of the inherent shortcomings of the m ¼ n G 1 2n b : P ref is the atmospheric pressure level and b w the effective bulk modulus of water. In the absence of gas inside the soil, b G for classical soil parameters, m 1, and the pressure decay is driven by the first term in (2) so that Pz ðþp* ðþ z ¼ e 2lz ; P 2 0 where P*(z) is the complex conjugate of P(z). The pressure variations are transmitted deep inside the soil. When present, the gas has a significant effect on the compressibility of the fluid. A range of soil parameters was considered, for soils made of loose sand and compact sand (Table 2), in order to estimate the variations in the parameter m and length scale d when the gas content varies in the range [2 10 3, ]. The length scale d decreases with increasing gas content approximately in the same manner for compact and loose sand, from about 1.8 m to 0.4 m. Estimates of m show that the damping of pressure variations inside the soil is mainly caused by the second term in (2) Pz ðþp* ðþ z P 2 0 ffiffi ¼ e p 2 z=d : The damping of pressure variations in the soil is greatly enhanced by the presence of gas. The value of d ð5þ ð6þ ð7þ 9of18

10 Table 2. Soil Properties Used for Analyzing Pressure Damping in the Soil With the Sakai Equations (3), (4), and (5) a Elastic Properties of Skeleton Porosity G (Pa) n Gas Content d (m) m Loose Sand Compact Sand a Computations were with bed permeability k = ms 1, effective bulk modulus of water b w =210 9 Pa, and wave frequency f =0.1Hz (s =2pf ). characterizing the length for damping is found to be almost independent of the values of G and n of the elastic soil properties given in Table 2 for loose sand and for compact sand containing a sufficient amount of gas. [27] The significant damping of pressure variations inside the soil shown by Figure 5 suggests that the soil contains a nonnegligible amount of gas. Note that the damping ratio P 2 2 i (f p )/P i 1 (f p ) is significantly smaller for i = 3 compared to i = 4 and i = 5. If the gas content were uniform in the soil, equation P 2 i P 2 i 1 ðf Þ p ffiffi ðf Þ ¼ e 2 ð zi z i 1 Þ=d would imply that this ratio is the same at all depths because the distance between neighboring pressure sensors is constant (z i z i 1 = 30 cm). Figure 5 obviously indicates a different behavior. Considering a possible decrease from the top to the bottom of the soil layer, the value of gas content was adjusted in the effective bulk modulus of pore water b w (4) in order for the length scale d (3) to fit the decay values given by (8) to the experimental spectra of amplitude decay and phase shift. The comparison of the spectra in Figure 4 with the spectra predicted by the theory of Sakai et al. [1992] is shown in Figure 10. A good fit is obtained when the gas content decreases from 0.04 in the upper layer to deeper inside the soil. In spite of the various unknowns of our field conditions and the differences between our configuration and the Sakai et al. [1992] hypotheses, the comparison is meaningful for several reasons. First, the comparison between the estimates of the Sakai et al. [1992] equations and field data is in quantitative agreement for the whole frequency range. Both amplitude and phase shift decrease with increasing frequency. The only surprise is in Figure 10 where significant damping is found in the experiment between P 4 and P 5 whereas the phase shift is small. Most of the wave energy is dissipated in the upper soil layers, in which case the estimation of damping in the lowest layer may be less accurate because of larger relative errors. Second, the model equations predict a significant amount of gas inside the soil. This is an important result in itself. For the values of gas content obtained the length scale of decay d is not particularly sensitive to the elastic properties of the soil G and n. The ð8þ fact that these were not measured does not really weaken the conclusion. As mentioned by Baldock et al. [2001], matrix compressibility may be neglected when compared to compressibility of encapsulated gas in the swash zone. Our measurements in water depths of 1 to 2 m lead to the same conclusion. 4. Analysis of Field Conditions and Results for Different Tidal Periods 4.1. Measurement in Front of the Bunker: The Tides of 24 and 25 September 2003 [28] Further conclusions are obtained from observations made during the tides of 24 and 25 September. The wave activity was weak on 25 September and limited erosion and deposition occurred. A much higher wave level was established on 24 September Significant soil erosion occurred at the beginning of the tide and sand deposition occurred toward the end. Figure 11 displays the variation in mean water level, wave height and status of pressure sensors for the tides of 24 and 25 September using the same presentation as in Figure 2. Figure 12 shows the variation in damping in the soil during both tides (same presentation as Figure 5). Although the experimental conditions on 25 and 28 September were similar, the results shown in Figures 5 and 12 look different because the pressure sensors were deeper in the soil on the latter day. On 25 September the pressure sensor P 3 is close to the soil surface. Pressure damping occurs mainly between P 3 /P 4 and Figure 10. Comparison of pressure transmission inside the soil during the morning tide of 28 September (bold lines: P 2 to P 1 (dotted), P 3 to P 2 (solid), P 4 to P 3 (dashed), and P 5 to P 4 (dash-dotted)) with the prediction of Sakai et al. [1992] (sharp lines: C gas = 0.04 (solid), C gas = 0.01 (dashed), and C gas = (dash-dotted)). Gas content is adjusted to fit prediction to experimental data. (a) Ratio of the energy density spectra of pressure variations measured by two neighboring pressure sensors P 2 2 i ( f )/P i 1 ( f ) and (b) phase shift of pressure variations measured between neighboring sensors as a function of frequency. 10 of 18

11 Figure 11. Variation during the tides of (a) 24 September and (b) 25 September in mean water level (solid line), maximum water level (dash-dotted line), and state of the pressure sensors P 1 P 5 with regard to their location: in static soil (squares), in unstable soil (asterisks), in a sediment suspension (crosses), and in clear water or in air (dots). Diamonds indicate the position of the deepest pressure sensor above which the soil layer can be liquefied according to criterion (1). P 4 /P 5. Almost the same damping ratio is observed between P 3 /P 4 and P 4 /P 5 while the phase shift is clearly different when comparing P 3 /P 4 and P 4 /P 5. As mentioned above, the phase shift may be closer to zero for smaller pressure variations as experienced here in the lowest layer. On 28 September the pressure sensors were embedded 39 cm deeper in the soil, P 2 is close to the soil surface. The same level of damping is measured between P 2 /P 3 as was measured between P 3 /P 4 and P 4 /P 5 on 25 September. On 28 September the damping decreases deeper in the soil. This could not be seen on 25 September because there was no pressure sensor at similar depths below the bed surface. Whereas the case of 28 September displays a slight reduction in damping (Figure 5) during the tide, the damping is seen to be fairly constant on 25 September except at the very beginning and the very end for the upper pressure sensors P 2 /P 3. Although the tide of 25 September does not really provide any further insight than that of 28 September, its results are presented because an investigation of gas content inside the soil was conducted on 25 September using a geoendoscopic camera (see the next section). It is useful to comment on how the results of 25 September compare to those obtained during other tidal periods. [29] Different features were observed during the tide of 24 September. The wave activity was much stronger (Figure 11a) than during the two tides discussed before. Sand erosion of more than 80 cm occurred during the first hour of the tide and a 60 cm deep layer of sand was deposited at the end. Only the pressure sensor P 1 was outside the sediment bed at the beginning of the tide. At about the midway point, the pressure sensor P 4 was very close to the bed surface and P 3 was in the water layer with 11 of 18

12 Figure 12. Variations measured during the tides of (a, c, e) 24 September and (b, d, f) 25 September. Significant wave height measured by the five pressure sensors: P 1 (sharp), P 2 (dotted), P 3 (solid), P 4 (dashed), and P 5 (dash-dotted) (Figures 12a and 12b). Damping ratio P 2 2 i (f p )/P i 1 (f p ) of pressure variations measured at neighboring depths (Figures 12c and 12d) and phase shift between P i (f p ) and P i 1 (f p ), for i = 3 (solid), i = 4 (dashed), and i = 5 (dash-dotted) (Figures 12e and 12f). suspension. At the end, P 2 was in the water layer with suspension and P 3 was inside the soil. The consequences of this are visible in the pressure transmission data plotted in Figure 12. In the first place, both the ratio of pressure variations P 5 2 (f p )/P 4 2 (f p ) and phase shift between P 5 (f p ) and P 4 (f p ) remain almost constant during the tide, indicating that the soil properties were not modified at this level. Between P 3 and P 4 the damping of pressure variations is rapidly reduced since part of the sediment in between the two sensors is in suspension. The damping increases again toward the end of the tide but it does not reach the initial level, suggesting that the air content was reduced in the redeposited sand. In the upper layer, between P 2 and P 3,the sediment is entirely in suspension during most of the tidal period. It is not possible to estimate whether the reduction in damping observed at the end of the tide compared to the damping at the beginning is due to a change in properties in the layer or to the fact that the sediment is not fully deposited between the two sensors at the end of the tide. [30] Figure 11b does not display the occurrence of liquefaction events, as predicted by the criterion (1). This is not surprising since the sea was fairly calm that day and P 3 and P 4 were quite deep in the soil. For the tide of 24 September, Figure 11a shows liquefaction events between P 3 and P 4. While the pressure sensor P 4 is located in deposited soil, the pressure sensor P 3 is in the suspension layer with the optical sensor in between Measurement at the Corner of the Bunker: The Tide of 1 October 2003 [31] The flow around the corner is more complicated than in front of the bunker. Vortices are produced at the corner by waves impacting the structure and scour holes are generated in the sand bed [Sumer and Fredsøe, 2002]. Intense sediment transport occurs there within a wave period. The consequences of this were observed during the field experiment. For a wave forcing of similar level, the variations in bed surface and the changes inside the soil were observed to be more intense at the corner than in front of the bunker. Horizontal velocities are much higher in the corner compared to that on the wall front. Liquefaction events are more likely to appear because of shear under the wave trough instead of the rapid decrease in water level when the wave breaks on the wall [Michallet et al., 2005]. The results of the evening tide of 1 October are presented here. The maximum significant wave height H s = 69 cm was obtained during this tide with the measurement devices set up at the corner (Table 1). Although this is not a high wave forcing, the intensity of the phenomena observed during this tide is comparable with the observations made in front of the bunker on 24 September for the highest wave conditions (H s = 123 cm). 12 of 18