Release of multiple bubbles from cohesive sediments



[1] Methane is a strong greenhouse gas, and marine and wetland sediments constitute significant sources to the atmosphere. This flux is dominated by the release of bubbles, and quantitative prediction of this bubble flux has been elusive because of the lack of a mechanistic model. Our previous work has shown that sediments behave as elastic fracturing solids during bubble growth and rise. We now further argue that bubbles can open previously formed, partially annealed, rise tracts (fractures) and that this mechanism can account for the observed preferential release at low tides in marine settings. When this mechanical model is applied to data from Cape Lookout Bight, NC (USA), the results indicate that methanogenic bubbles released at this site do indeed follow previously formed rise tracts and that the calculated release rates are entirely consistent with the rise of multiple bubbles on tidal time scales. Our model forms a basis for making predictions of future bubble fluxes from warming sediments under the influence of climate change.

1. Introduction

[2] Methane is generated in sediments during the later stages of anoxic microbial organic matter decay (methanogenesis), the melting of shallow gas hydrates, or the thermal decomposition (catagenesis) of organic compounds. Near-surface methane profiles in sediment porewaters indicate that the dissolved methane flux is effectively intercepted by aerobic and anaerobic oxidation [Boetius et al., 2000; Dale et al., 2006; Caldwell et al., 2008]. Bubble formation and rise (ebullition) constitutes, therefore, the primary pathway for gas release from sediments [Hovland et al., 1993; Joyce and Jewell, 2003]. While the flux of methane as bubbles from sediments is probably modest, i.e., <10 Tg yr−1 [Chappellaz et al., 1993], the potency of methane as a greenhouse gas makes any flux of interest to climate modellers. Furthermore, as the temperature of mid- and high-latitude sediments increase with global warming, methane fluxes will also increase, due to thawing of sediments [Shakhova et al., 2010], thermal effects on diagenetic sources, and melting of shallow gas hydrates.

[3] Prediction of future fluxes of methane from sediments has been stymied by a lack of a mechanistic model for bubble release. Recent studies [e.g., Johnson et al., 2002; van Kessel and van Kesteren, 2002; Barry et al., 2010], show that bubbles grow in sediments via elastic expansion and fracture of this medium. X-ray imaging [e.g., Boudreau et al., 2005] shows that bubbles in cohesive sediments are thin, cornflake-like bodies, as a result of non-fluid sediment mechanics. Gardiner et al. [2003], Algar and Boudreau [2009, 2010], Jain and Juanes [2009], and Algar et al. [2011] have used elastic fracture mechanics and an idealized oblate spheroidal geometry to create an overall model of the initial growth and rise of bubbles in cohesive sediments. Such a 3D elastic-fracture process is extremely difficult to include in standard diagenetic transport-reaction models [Berner, 1980; Boudreau, 1997; Burdige, 2006], which are typically used to calculate fluxes from sediments to the overlaying waters or the atmosphere. The present paper offers a model for the calculation of bubble release rates.

[4] In addition, the release of bubbles in marine sediments is often tied to decreases in overlying pressure, e.g., low tides for shallow coastal sediments [Martens and Klump, 1980; Chanton et al., 1989]. Preferential release of bubbles at low tide has been linked to expansion of the gas when the hydrostatic pressure drops [e.g., Chanton et al., 1989]. Such statements are undoubtedly true, but at the same time, they do not explain quantitatively the mechanics of the observed release.

[5] Algar and Boudreau [2009] calculated the effects of tidal pressure variations on the growth rate of the first bubble to form in sediment similar to that at Cape Lookout Bight [Martens and Klump, 1980]. Figure 1 illustrates an expansion of those calculations for an initial bubble in the same sediment. We assume 8-m mean water depth and impose semi-diurnal tides of ±1 m, ±3 m, ±5 m, and ±7 m, the last to produce a tidal environment similar to the Bay of Fundy, Canada. Notice that while the volume of a bubble changes in proportion to the tidal amplitude, for tides ≤ ±3 m, the growth curve always returns to the no-tide growth curve. This means that tides have no net effect on the mass of gas within a bubble, and Figure 1 illustrates perfect gas expansion and contraction. Tides greater than ±5 m in amplitude do produce a net increase in growth; however, the effect is apparent only as the bubbles grow large (>500 mm3), and it is not clear what natural environments would have such large tides coupled to strong sediment methanogenesis.

Figure 1.

Model simulations that illustrate the effects of tidal height on the volume of a growing bubble started at high tide. The model was run with tidal amplitudes of 1, 3, 5 and 7 m. The dashed line shows bubble growth in the absence of tidal forcing. The influence of tidal height on bubble growth is not evident until tides exceed three meters in height.

[6] This prediction of a modest effect of tidal variations on initial growth rates appears to fly in the face of the observed tide-bubble flux correlation at various locations, including Cape Lookout Bight and White Oak River [Martens and Klump, 1980; Chanton et al., 1989]. The present communication offers a simple solution to this apparent contradiction and reveals more details of the unfamiliar physics of bubbles in sediments.

2. Model

[7] The elastic-fracture model of bubble growth and rise is detailed by Johnson et al. [2002], Gardiner et al. [2003], Algar and Boudreau [2009, 2010], Barry et al. [2010], and Algar et al. [2011]. The gas exerts a pressure on the bubble walls, which opposes the total external pressure (load) and prevents the bubble from closing. When gas diffuses into a bubble, the internal pressure mounts and the bubble grows elastically in thickness 2b (Figure S1 of the auxiliary material). If the gas source is sufficiently strong, then the internal force can grow to exceed the sum of the external load and the fracture toughness of the sediment. The bubble length, 2a in Figure S1, then grows by opening a crack, thus reducing the stresses in the surrounding sediment as the width 2b decreases elastically. These processes can be repeated many times as a bubble grows. (Note bubble, fracture and crack are interchangeable terms in this paper.)

[8] Growing bubbles begin to rise in sediments once the crack half-length, a, reaches a critical value, ar, where [Algar et al., 2011]

equation image

in which K1C is the tensile fracture toughness, ρs is the bulk density of the sediment, and g is the acceleration due to gravity.

[9] The volume of a bubble, Vb, when a = ar is determined by the amount of linear elastic expansion [Algar et al., 2011]:

equation image

where ν is Poisson's ratio and E is Young's modulus.

[10] We can now examine how a bubble reforms at an initial rise point and what are the effects of tides. To this end, we employ the finite-element model of bubble growth, LEFM-RD, of Algar and Boudreau [2009, 2010]. This model couples a reaction-diffusion equation, which describes gas production and diffusional transport through sediment porewater, to a linear elastic fracture mechanical model (LEFM) for the physics of the sediment.

[11] Fracture occurs when the stress intensity factor at the upper tip of the bubble, K1(+), exceeds the fracture toughness of the sediment, K1C, at which point the crack length is extended upward, and the internal bubble pressure drops. In a linear depth-dependent pressure field, K1(+) is [Algar et al., 2011]

equation image

where σa is the internal bubble loading at the crack tail (bottom of the bubble) in excess of the ambient total pressure. When a bubble rises σa = 0. We need not consider fracture at the bottom of the bubble because the stress at the top of the bubble always exceeds K1C before it can be exceeded at the bottom. This preference for growth at the top is illustrated in Animation S1 of the auxiliary material, which shows a growing bubble in gelatin, a mechanical analog for sediment [Barry et al., 2010]. (Note we recognize that the stress field perturbation that is caused by the container bottom and sides affects bubble growth direction in this experiment.)

[12] If a = ar, the bubble will release from its growth point and rise. When a bubble rises, the crack closes behind the bubble. If this crack now takes time to heal, i.e., to rebuild the broken bonds and the sediment strength to a pre-fracture level [e.g., see Boudreau et al., 2005, Figure 4c], the partially annealed crack will have a lower K1C than the surrounding sediment. As K1C is lower in a partially annealed crack, equation (1) informs us that ar will be smaller; therefore, subsequent bubbles will also be smaller than the initial bubble when the former rise. Assuming that the specific growth rate, i.e., mass delivered by diffusion to a new bubble per unit time, remains roughly constant, subsequent bubble release will be more frequent. It now remains to be seen if rise tracts with lower KIC are preserved and if the frequency of release can correspond to tidal frequencies.

3. Application and Results

[13] Our hypothesis is that subsequent bubbles re-open pre-existing bubble-created fracture paths and that this re-opening is aided by tidal pressure variations. To test this conjecture we utilize bubble and methanogenesis data for Cape Lookout Bight, as an example, and apply equations (1)(3) and LEFM-RD to calculate the apparent KIC of the re-opened paths and the potential release rates. Parameter values for this site can be found in Table S1 of the auxiliary material.

[14] K1C of the bubble paths can be estimated because Martens and Klump [1980] report sizes and numbers of bubbles released from sediments at Cape Lookout Bight during low tide (Table 1). Using these volumes and equation (2), the critical rise size, ar, is calculated for each volume class, with an assumed E value for these sediments (Table S1). Then, assuming that these bubbles all rose when K1(+) = K1C (σa = 0), equation (3) provides K1C. These K1C values (Table 1) are 10–20% of the limited values measured in situ in other sediments, i.e., 500–1500 N m−3/2 [Johnson et al., 2002]. The escaping bubbles appear to have followed paths with significantly lower strength than undisturbed sediment.

Table 1. Time Required to Grow a Bubble in a Previously Formed, but Partially Annealed Fracture, From an Initial Flaw Size of a0 to the Size at Which It Will Leave the Sediment, ara
Percent of Total BubblesMean Volume (mm3)K1C (Nm−3/2)ar (mm)Minimum Growth Time a0 = arMaximum Growth Time a0 = 0.001 m
6113752 min3.6 hr
32727143.3 hr11 hr
393851186.5 hr31 hr
13101742314 hr56 hr
10210972717 hr88 hr

[15] Next we need to estimate the time required to grow the bubbles in Table 1 using the LEFM-RD model. This model demands, however, an estimate of the initial flaw (crack) size, a0, from which the bubble will grow before release. In the case at hand, we are unaware of the typical flaw size for the pre-existing rise path. Fortunately, we can set upper and lower bounds on a0 and, thus, obtain upper and lower bounds on the formation times.

[16] As an upper bound, we use a0 = ar, as calculated by equation (2). This value means that subsequent bubbles do not go through a sequence of fracture events to reach their release volume. Instead, methanogenesis will supply gas to the crack and open it up purely by elastic expansion. When the volume becomes critical, i.e., Vb, the bubble rises. This produces the shortest possible growth time and a lower bound. The other possibility is that there is a minimum a0 for sediments. We do not know this value, but our previous work indicates that initial growth times (i.e., minutes) for bubbles with a0 ≤ 0.001 m are negligible compared to the time for growth from a0 to ar. Thus, if we start with this value, we will obtain a maximum growth time for the bubbles. The minimum and maximum growth times are given as the last two columns of Table 1.

[17] The results in Table 1 show that observed bubbles of a volume of 38 mm3 or smaller can be easily created on tidal time scales by re-opening a rise path. These bubbles account for 77% of the bubbles collected at Cape Lookout Bight.

[18] The larger bubbles in Table 1 cannot be grown on tidal scales, but these larger bubbles could be, in part, the result of the collision and union of smaller bubbles, both in the sediment and in the water column. An example of such collisions within sediment is provided in Animation 2; this movie shows a clear box filled with gelatin containing a pre-established bubble rise path. Each subsequent bubble follows the initial path, but there is a natural constriction, i.e., a point with a higher K1C, about mid-way along the path. A rising bubble will stall at this point because it is not large enough to force the constriction open. The next bubble will rise to that point, merge with the previous bubble and continue to rise, as the larger bubble can force the gap open. The result is a bubble with twice the volume leaving the seabed.

[19] Figure 2 shows the growth history of several bubbles due to the reopening of a previously formed fracture with an initial length of 0.007 m. A particular bubble's history starts at a point on the abscissa, e.g., point A. Once each bubble reaches a critical size, it is removed by rise, e.g., point B, and another bubble starts to form in its place, e.g., point C.

Figure 2.

The growth of multiple bubbles due to the re-opening of an initial flaw of length a0 = 0.007 m. When a bubble that starts at A reaches its critical size B, as determined by equation (1), it is removed and another one starts to grow in its place C. Bubbles only grow to critical sizes during low tidal periods and start to dissolve during high tide D, and then recommence growth on the falling tide E. Tidal height was taken to be 1 m.

[20] Figure 2 shows all the bubbles formed from a single site during a period of 24 hours. Between two and four bubbles form during each low tide. During high tide bubble growth ceases (point D) and reverses as the increase in ambient pressure compresses the bubble, which also causes it to dissolve. In this simulation, bubbles do not dissolve appreciably and growth restarts on the falling tide (point E). These results demonstrate multiple bubble release on tidal cycles.

4. Conclusions

[21] Once a rise path is formed, subsequent bubbles can easily form and rise by re-opening this fracture. Drops in pressure, e.g., low tides, promote this process. Results from our LEFM-RD model show that over 75% of the bubbles observed to release from the sediments at Cape Lookout Bight, NC, USA can easily be generated by this fracture-reopening mechanism.

[22] With our model, we are one step closer to predicting bubble fluxes from sediments. The missing elements are observations of the areal density of release points, a geographical and temporal tabulation of methane source strengths in relevant sediments, and better documentation of KIC from these same sediments. If we can estimate the effects of warming on the source strengths, then we could predict bubble fluxes of the future.


[23] This research was funded by the U.S. Office of Naval research through grants N00014-08-0818 and N00014-05-1-0175 (project managers J. Eckman and T. Drake), the Natural Sciences and Engineering Council of Canada, and the Killam Trust (Dalhousie University). We also thank our reviewers for their efforts. The authors thank Ilia Ostrovsky and an anonymous reviewer for their assistance in evaluating this paper.

[24] The Editor thanks Ilia Ostrovsky and an anonymous reviewer for their assistance in evaluating this paper.