Suction dynamics‐induced bubbly sand under groundwater table fluctuations

Bubbly sand is found in intertidal zones and has been considered to be closely related to swash and tides. Unlike bubble structures that arise from in situ methane gas production and bubble growth in river and marine sediments due to organic matter decomposition, the formation mechanism and conditions of bubbly sand remain unresolved. The present study aims to resolve them on the basis of a conceptual model and controlled laboratory experiments. The study demonstrates that bubbly sand is a consequence of the cyclic expansion of sand under groundwater table fluctuations. The varying intensity of dynamics of suction, which is sand moisture tension defined by negative pore water pressure relative to atmospheric pressure, controls the manifestation of a full spectrum of sand behaviour relevant to the formation of bubbly sand. Namely, with increasing intensity of suction dynamics under larger groundwater table fluctuations, the sand behaviour varies from enhanced cyclic contraction, to weak contraction, loosening and expansion resulting in bubbly sand. The suction dynamics‐induced cyclic expansion of sand occurs under conditions where the maximum suction developed exceeds the air‐entry suction of sand such that the degree of saturation becomes lower than 60%, while the groundwater tables cyclically rise to and fall from the sediment surface. Accordingly, the sand porosity changes remarkably from dense to super‐loose states of bubbly sand. These findings account for previously unanswered questions with respect to bubbly sand, both qualitatively and quantitatively, and will lead to a renewed understanding of the geological record and morphological features at waterfronts that are subject to groundwater table fluctuations.


INTRODUCTION
Bubble formation and growth in river, lake and marine sediments occur as a result of in situ methane gas production and release from organic matter decomposition under submerged conditions (Boudreau et al., 2005;Liu et al., 2016Liu et al., , 2018. By contrast, bubbly sand is found in intertidal zones, and has been considered to be closely related to the entrapment of air due to swash and tides (Kindle, 1936;Emery, 1945;Stewart, 1956;Hoyt & Henry, 1964;Reineck & Singh, 1973;Harris, 1974;Jago, 1980;Allen, 1982;Pilkey et al., 2011). However, these previous studies did not provide a concrete mechanism for the formation and conditions of bubbly sand. Accordingly, despite the widespread interest in the phenomenon, many aspects of bubbly sand remain uncertain and unresolved. Indeed, Pilkey et al. (2011) showed that there is no apparent reason why bubbly sands form in some places but not in others, and adding to the mystery is the bubbly sand found along some sand shores above normal water level. The present study aims to resolve this mystery both qualitatively and quantitatively.
Bubbly sand has been observed across the globe (Fig. 1), such as bubbly sand in which deep footprints are formed (Pilkey et al., 2011;Fig. 1A), bubbly sand structures in beaches (Reineck & Singh, 1973;Fig. 1B), and bubbly sand that was sampled for this study in sandy beaches and a sandflat in South Korea (May 2017; Australia (December, 2009;Fig. 1E). Bubbly sand has an essential feature such that a number of bubbles of various sizes ranging from 1 to 10 mm develop within the surficial layer of 100 mm thickness ( Fig. 1B to E). Here, on the basis of a conceptual model and controlled laboratory experiments, this study demonstrates that bubbly sand is a consequence of the cyclic expansion of sand, a novel process that occurs under varying intensities of suction dynamics due to groundwater table fluctuations.

Conceptual model
Sand in intertidal zones is cyclically exposed and submerged under the semi-diurnal fluctuations of the tides. Accordingly, there are temporal changes in the groundwater level, causing dynamic changes in the suction state of the sediments . Suction, s, is the tension of moisture in sand (Bear, 1979) and is defined by: where u a is the atmospheric air pressure and u w is the pore water pressure in the sand. By definition, suction is zero at the groundwater level. The sand void state is represented by the void ratio, e, which is related to the sand porosity, n: e ¼ n 1Àn (2) The state of sand packing, such as dense or loose, can be denoted by the relative density in percent, D r : For a given sand, the maximum void ratio e max represents the loosest possible packing, and the minimum void ratio e min represents the densest possible packing (Lambe & Whitman, 1979). Thus, D r is a normalized index by which to assess the packing states of sands. A conceptual model concerning the suction dynamics-induced porosity changes in sand is shown in Fig. 2. Here, the suction dynamics denote the dynamic changes in the suction state of the sediments in association with the temporal changes in the groundwater level. With reference to Fig. 2, the intensity of suction dynamics, s i , represents the maximum suction developed during a given cycle where the groundwater table rises to and falls from the sediment surface in the cyclically exposed and submerged sediments. The air-entry suction, s aev , is the suction at which the sand starts to become unsaturated with pore-air entrainment. The air-entry suction, s aev , is higher in finer and denser sands (Fredlund & Rahardjo, 1993). It is closely related to capillary height, which is an inverse function of the effective grain size at 10% finer by mass, D 10 , and the sand void ratio, e (Bear, 1979). Accordingly, the air-entry suction of intertidal sand can be expressed as: where γ w is the unit weight of seawater, and ψ is a material coefficient that may depend on the grain characteristics such as grain shape and grain-size distributions. A ψ value of 20 mm 2 has been shown to conform to laboratory and field observations on intertidal sands with D 10 ranging from 0.1 to 0.45 mm . Under conditions where the developed suction, s i , is below or equal to the air-entry suction, s aev , the sand is essentially saturated with a degree of saturation S r ≅ 100% ( Fig. 2A). The suction dynamics in this region where s i ≤ s aev (Fig. 2) bring about a significant cyclic contraction of the sand, the magnitudes of which depend strongly on the intensity of the suction dynamics ensuing there , 2009. Accordingly, the sand becomes densest at s i = s aev (Fig. 2B). The suction dynamicsinduced cyclic contraction becomes less pronounced and weaker as the degree of saturation decreases at s i > s aev , and eventually vanishes; in other words, no densification ensues (as denoted by the dotted line in Fig. 2B). A further increase in the intensity of suction dynamics brings about loosening of the sand (Fig. 2B). In this region where such suction dynamicsinduced loosening of sand occurs, a sufficient amount of pore air is entrapped in the sand due to a high developed suction and a deeper groundwater table. The rise in the groundwater table to the sediment surface pushes the entrapped pore air upward, which spreads the sand void under declining sand-confining pressure due to decreasing suction, causing loosening of the sand. An even greater increase in the intensity of suction dynamics results in expansion of the sand such that the sand relative density becomes negative D r < 0 (Fig. 2B), a situation that would never ensue in a depositional process but could only be achieved by a cyclic expansion of the sand. Such a cyclic expansion of the sand gives rise to bubbly sand.

Laboratory experiment
A series of controlled laboratory experiments under semi-diurnal fluctuations of the tides was conducted to test the conceptual model described above and to observe a full spectrum of sand behaviour relevant to the formation of bubbly sand. The median grain size of the sand used was 0.232 mm. The maximum void ratio of the sand, e max , was 0.925 and the minimum void ratio, e min , was 0.593. The initial sand relative density, D r , was set at 40%. This may represent a loosely deposited sand under submerged conditions . The corresponding sand void ratio was e = 0.792. The specific gravity of the sand particles, G s , was 2.65.
The apparatus used had two transparent acrylic cylindrical chambers and larger water tanks connected to an elevating table with a water reservoir and a motor controlled by an actuator (Fig. 3). A sand bed with a diameter of 0.2 m and a depth of 0.45 m was formed inside a transparent acrylic cylindrical chamber with a height of 0.7 m, by carefully sprinkling sand into water according to the procedure in . These chambers had an open end at the surface and a porous bottom through which only water was allowed to flow. The difference between the chambers was that one chamber had a porous lateral boundary, as in the field, whereas the other chamber had a closed lateral boundary for the purpose of comparison. The porous lateral boundary was set by installing a total of 14 porous units in a staggered way with each unit having a hole with a length of 100 mm, where the highest two units had a length of 50 mm and a width of 5 mm as filled with a polyester non-woven fabric (TC-600, TESHION Corp., Saitama, Japan) at an interval of 90 degrees in the chamber (Fig. 3C). This realized a situation such that the water and groundwater levels inside the chamber and the water levels outside the chamber were essentially the same, with reference to Fig. 3. This aspect of behaviour can also be seen in Fig. 4A and B.
Each chamber was set in a larger water tank. The water level in the tank was controlled so as to match the water level above the sand surface or the groundwater level in the sand deposit formed inside the chamber. A pore water pressure transducer (P310A-02, SSK Co. Ltd, Tokyo, Japan) was set at the bottom of the sand deposit, and a water pressure transducer was set in the tank to measure the water and groundwater table fluctuations. The state of suction, s, at the level of the sand surface was varied by changing the water/groundwater levels. Suctions were measured using tensiometers (ML-2600AES, MOL, Inc, Tokyo, Japan). In the essentially saturated states where the developed suction is below the air-entry suction, a linear relationship holds between suction, s, and groundwater level, G.W.L., such that s = −γ w ÁG.W.L., where γ w is the unit weight of water . In the unsaturated states, however, the above relationship deviates from the linear relationship, yielding s = 3.45 kPa against G.W.L. = −0.35 m and s = 3.75 kPa against G.W.L. = −0.4 m for the sand beds concerned.   Table 1. The black dotted lines in the top panels of Fig. 4A and B denote the water levels measured outside the chamber (Fig. 3A).
The sand deposits were subjected to a varying intensity of suction dynamics under semidiurnal water/groundwater table fluctuations with a period T = 12 h. The experimental conditions are summarized in Table 1. A total of 40 cycles (20 days) was imposed for each series of experiments. With reference to Table 1, the lowest groundwater tables were set to be −0.1 m (Series 1), −0.2 m (Series 2), −0.3 m (Series 3), −0.35 m (Series 4 and 4*) and − 0.4 m (Series 5) below the sand surface. The series with an asterisk, Series 4*, corresponds to a closed lateral boundary (Table 1). The submerged water depth was set to be 0.05 m. For each series of experiments, the sand vertical displacement was continuously measured using a laser displacement meter (LB-300, KEYENCE Corp., Osaka, Japan), while continuously measuring suctions and groundwater levels in parallel throughout the course of the experiments. After each of the experiments, undisturbed samples of surficial sand with 30 mm diameter and 10 mm thickness were taken in duplicate. The sand samples were subjected to a series of laboratory soil tests to determine the water content in percent, w, void ratio, e, sand relative density, D r , and degree of saturation in percent, S r = G sÁ w/e.

RESULTS AND DISCUSSION
The results of the series of controlled laboratory experiments are summarized in Fig. 4. When the groundwater level advanced downward during the period of exposure, suction increased, and the vertical displacement decreased, indicating contraction of the sand (Fig. 4A, Series 2). By contrast, when the groundwater level advanced upward during the period of exposure, suction decreased, and the vertical displacement increased, indicating expansion of the sand (Fig. 4A, Series 2). No vertical displacement ensued during the period of submergence. Since the magnitude of the contraction exceeded the magnitude of the expansion, densification proceeded, showing the occurrence of the suction dynamics-induced contraction of the sand.
The above-described phenomena changed distinctly when the intensity of suction dynamics became substantially more severe with larger groundwater table fluctuations (Fig. 4B, Series 5); specifically, the magnitude of the expansion exceeded the magnitude of the contraction, and hence loosening proceeded, showing the occurrence of the suction dynamics-induced expansion of the sand.
The variations of the vertical displacements in a cycle were an order of magnitude larger in Series 5 (Fig. 4B) than in Series 2 (Fig. 4A). Such enhanced contraction and expansion within a cycle ensued due to the substantially more severe suction dynamics at which a sufficient amount of the pore air was entrained in the sand under larger groundwater table fluctuations. A close examination of the sand behaviour in Series 5 tells us that a rapid sand expansion occurred when the groundwater table was shallower than G.W.L. = −100 mm and approached the sand surface (Fig. 4B). This means that the rising groundwater table pushed the pore air upward, which spread the sand void as the sand-confining pressure declined due to vanishing suction. This resulted in the cyclic expansion of the sand.
The time histories of the vertical displacements with tidal cycles demonstrate that the sand behaviour changed markedly depending on the intensity of the suction dynamics with different groundwater table fluctuations (Fig. 4C, Series 1-5 and 4*). Series 1, 2 and 3 showed the suction dynamics-induced cyclic contraction of the sand that proceeded under continued tideinduced groundwater table fluctuations. In contrast, Series 4 and 5, in which the intensity of the suction dynamics was greater, showed contraction at early stages of the tidal cycles; however, they subsequently exhibited cyclic expansion of the sand. Since the groundwater table rises more rapidly when air is encapsulated than when it is not (Fayer & Hillel, 1986;Horn, 2006), the temporal accumulation of the pore air led to the suction dynamics-induced expansion of the sand that proceeded under continued tide-induced groundwater table fluctuations. Both of the sands under the same intensity of suction dynamics, s i = 3.45 kPa, showed substantial cyclic expansion behaviour irrespective of the porous and closed lateral boundaries (Fig. 4C, Series 4 and 4*). An even greater intensity of suction dynamics with larger groundwater table fluctuations resulted in a more substantial cyclic expansion of the sand (Fig. 4C, Series 5). These aspects of the sand behaviour with varying intensity of suction dynamics with different groundwater table fluctuations can be more clearly seen in Fig. 4D (Initial conditions and Series 1-5 and 4*). The air-entry suction, s aev , shown is based on an effective grain size D 10 = 0.130 mm and a void ratio e = 0.652 at the state of densest packing achieved during the course of the experiments, according to Eq. 4. The observed full spectrum of the sand behaviour ranging from enhanced cyclic contraction, to weak contraction, loosening and expansion resulting in bubbly sand was found to be consistent with the conceptual model presented above (Fig. 2). The experimentally observed bubbly sand structure (Fig. 4E, s i = 3.75 kPa for Series 5) reproduced the essential features, such that a number of bubbles of various sizes ranging from 1 to 10 mm develop within the surficial layer of 100 mm thickness, of those observed in the field ( Fig. 1B to E).
The above results demonstrate that bubbly sand occurred as a result of the suction dynamics-induced cyclic expansion of the sand under tide-induced groundwater table fluctuations. Essentially the same intensity of suction dynamics can ensue due to swash in intertidal zones (Sassa & Yang, 2019) in association with the swash-induced groundwater table fluctuations (Longuet-Higgins, 1983;Cartwright et al., 2006). Hence, the swash-induced suction dynamics can also bring about the formation of bubbly sand by satisfying the condition that the intensity of suction dynamics, s i , exceeds the air-entry suction, s aev , and the lowest degree of saturation, S r , is below 60% in the cyclically exposed and submerged sand. As a consequence, bubbly sand may not occur in places where the sand is exposed, yet the developed suction is below or equal to the air-entry suction of the sand. Bubbly sand may also not occur in places where the degree of saturation is substantially low due to a high suction, but the groundwater table does not rise to and fall from the sand surface. These bases for bubbly sand formation explain the reason why bubbly sands form in some places but not in others (Pilkey et al., 2011). Since the air-entry suction decreases in coarser and looser sands, bubbly sand can occur above normal groundwater level where s i sufficiently exceeds s aev . The groundwater table fluctuations that satisfy the conditions for the formation of bubbly sand can manifest in a variety of ways arising not only from tides and swash but also from floods, river discharges, storms, ship waves and long-term sea-level rises driven by climate change (Taylor et al., 2013). Bubbly sand should thus be relevant to the interpretation of the geological record and morphological features at such waterfronts that are subject to groundwater table fluctuations.

CONCLUSION
The present study has demonstrated that bubbly sands, whose formation mechanisms and conditions have so far remained elusive, are in fact the consequence of the cyclic expansion of sand as induced by the suction dynamics under groundwater table fluctuations. This is based on a conceptual model and the results from a series of controlled laboratory experiments. The varying intensity of the suction dynamics under different groundwater table fluctuations gave rise to a full spectrum of sand behaviour that ranged from enhanced contraction, to weak contraction, loosening and expansion resulting in the formation of bubbly sand. A remarkably wide range of sand void states manifested, from dense to super-loose states, yielding variations of 200% in terms of the sand relative density, which overwhelmed the maximum possible range of variation in a depositional process. Since the sand porosity affects the form and function of the Earth's surface, the full spectrum of the sand porosity changes induced by the suction dynamics may serve as a rational basis by which to understand such waterfront Earth surface evolution, particularly in rivers, estuaries and coastal seas. However, linking the suction dynamicsinduced sand behaviour and the associated morphodynamics will be a subject for a future study. Importantly, the present findings may account for previously unanswered questions with respect to bubbly sand, both qualitatively and quantitatively. Our results resolving the ambiguities in bubble sand therefore highlight the important role of the varying intensity of suction dynamics at waterfronts in causing diverse sand behaviour under groundwater table fluctuations.