Subsurface Trapping of Oil Plumes in Stratification: Laboratory Investigations

Transcription

1 Subsurface Trapping of Oil Plumes in Stratification: Laboratory Investigations David Adalsteinsson, 1,2,3 Roberto Camassa, 1,2,3 Steven Harenberg, 1 Zhi Lin, 4 Richard M. McLaughlin, 1,2,3 Keith Mertens, 1,2,3 Jonathan Reis, 1 William Schlieper, 1 and Brian White 1,5 Laboratory experiments demonstrating how the addition of surfactants creates the possibility of trapping buoyant immiscible fluids are presented. In particular, these experiments demonstrate that buoyant immiscible plumes like those which occurred during the Deepwater Horizon Gulf oil spill can be trapped as they rise through an ambient, stratified fluid. The addition of surfactants is an important mechanism by which trapping can occur. In this paper, we describe experiments and theory on trapping/escape of plumes containing an oil/water/surfactant mixture released into nonlinear stratification. We also present results on the timescale for trapping and for destabilization and release of trapped subsurface plumes. This timescale is shown to be a function of the oil to surfactant ratio. 1. INTRODUCTION The Deepwater Horizon (DWH) oil spill in the Gulf of Mexico has raised several questions about the behavior of oil plumes released into deep ocean environments. Intuition might suggest that because oil is less dense than water, it should rise to form a slick on the surface. In DWH case, the deep water source, combined with the use of chemical dispersants to break up the oil into fine droplets, produced a deepwater hydrocarbon plume approximately 100 m thick trapped at about Joint Fluids Laboratory, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA. 2 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA. 3 Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA. 4 Department of Mathematics, Zhejiang University, Hangzhou, China. 5 Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA. Monitoring and Modeling the Deepwater Horizon Oil Spill: A Record-Breaking Enterprise Geophysical Monograph Series 195 Copyright 2011 by the American Geophysical Union /2011GM depth [Camilli et al., 2010; Joye et al., 2011]. The processes leading to the formation of such internal plumes are highly dependent on the ocean density stratification, oil and dispersant chemical characteristics, the oil flow rate, the temperature at the source and in the ambient water, and the presence of gases mixed with the oil. The dynamics of multiphase plumes in stratification have been explored in laboratory and numerical models, which have primarily focused on the plume intrusion height in uniform stratification [Socolofsky and Adams, 2002, 2003, 2005]. Plume behavior in nonlinear stratification is more complex. For example, McLaughlin [2005] showed that a buoyant miscible plume can become trapped at the interface between layers, while an immiscible plume can pass through the interface and reach the surface. Recently, Socolofsky et al. [2011] predicted the DWH intrusion depth by adapting their multiphase plume model to the nonlinear density profile measured in the Gulf near the DHW wellhead. Here we focus on the trapping and escape of multiphase plumes in highly nonlinear stratification by combined modeling and experiments. Specifically, we look at the effect oil/ surfactant ratio and ambient stratification on determining whether a plume will trap or escape. In Figure 1 we show an example which documents the dramatic effect of adding surfactants. As one can see, the surfactant/mixture is trapped in the subsurface, whereas the pure oil plume (red gauge oil) escapes to the free surface. This figure also serves as a general schematic for the experiments we describe. While the other experiments are conducted with a vertically-oriented jet, this

2 258 SUBSURFACE TRAPPING OF OIL PLUMES IN STRATIFICATION Figure 1. Comparison of a (left) pure oil plume and a (right) mixed oil/surfactant/water plume released into an ambient fluid with sharp stratification. The oil plume rises to the surface, whereas the plume with added surfactant traps at the density interface (marked by the black line). particular scenario with a horizontal jet mimics the field conditions in the DHW release (prior to cutting the riser pipe). It has been shown that plumes can trap at a critical depth in uniform stratification [Morton et al., 1956], and some approaches for nonlinear stratification have been explored [Socolofsky et al., 2011; Lemckert and Imberger, 1993], however, a simple mathematical formula for trapping/escape for arbitrary, nonlinear stratification is not generally available. Recent work has developed theory for trapping/escape in sharp stratification (R. Camassa et al., Underwater oil plumes and examples of universal trapping phenomena in strong stratification, manuscript in preparation, 2011, hereinafter referred to as Camassa et al., manuscript in preparation, 2011). In addition, a major outstanding question concerns the timescale for trapping and retention. Are trapped plumes stable, or do they eventually rise to the surface? In this paper, we present laboratory results for multiphase plumes in sharply stratified fluids. These experiments document trapping/escape regimes, which depend on the properties of the plume and ambient stratification. Additionally, we find that the residence time of a trapped plume depends highly upon its oil to dispersant ratio. Depending on plume composition, the trapped plumes can rapidly undergo a spontaneous destabilization and rise to the surface or, in other cases, can be stable for an indefinite duration along a neutral density interface. These regimes of trapping/escape and retention/destabilization are crucial to predicting the shortterm and long-term fate of multiphase plumes. 2. EXPERIMENTAL METHODS A glass aquarium of dimensions 61 cm 31 cm 63 cm (W D H) is half filled with 1.03 g cm 3 salt water, with density measured accurately to g cm 3 using an Anton-Parr DMA 4500 densitometer. Fresh water is poured slowly through a diffuser to create a sharply stratified transition layer of thickness approximately 1.0 cm. Red gauge oil of density g cm 3, liquid dishwashing soap (density 1.02 g cm 3 ), and freshwater (density g cm 3 ) are mixed together in different proportions (shown below), first adding the oil with the soap, and then adding the water. These mixtures are stirred with a magnetic stir bar (3.5 cm, 700 rpm) for approximately 30 min prior to injecting in a 1000 ml beaker, with a total liquid volume of 300 ml. This created emulsions that were highly stable through the time of injection. The characteristic droplet size of the emulsions was measured optically with a confocal microscope and found to be approximately 1 10 μm (therewasslightvariationin droplet size with the applied coverslip pressure) and appeared homogeneously distributed. The emulsions are then injected into the stratified tanks from either the tank bottom, by a Cole- Parmer Gear Pump through a jet nozzle of diameter 0.2 cm or using a syringe of radius 1.7 mm to inject the jet near the density transition. The flow rates are regulated to approximately 10 ml s 1, and the images are recorded with a Nikon 7000 digital camera. The density of the mixtures is measured in the densitometer. Measured densities corresponding to oil/ surfactant/water (OSW) mixtures of 8:1:5, 4:3:2, and 4:3:17, were 0.91, 0.94, and 0.98 g cm 3, respectively. 3. RESULTS We first explore the effect of varying the distance between the plume release and the background density transition. When this distance is smaller than a critical value, theory

3 ADALSTEINSSON ET AL. 259 predicts the plume will escape, but when the distance is larger, theory predicts the plume will initially trap (Camassa et al., manuscript in preparation, 2011). That theory predicts a critical escape height, L, for a plume rising into sharp stratification given by L ¼ L 0 A ds p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1þ 1 s 5=4 þ ε 1 where the bottom top and plume densities are > ρ t > ρ p, the (nondimensional) Richardson number is e ¼ 5ð1 þ λ 2 ÞðΔ ρþr 0 g p 16 ffiffiffi, with gravitational acceleration g, an initial plume density anomaly of Δ ρ ¼ ρ p 2 αw 2 0, initial plume outfall radius and velocity r 0, w 0, entrainment coefficient α, and mixing coefficient λ, analogous to a turbulent Prandtl number. 1 þ λ 2 Further, A ¼ 1 þ ε θ2 0 θ 2 f 1!! 4=5 with θ 0 ¼ λ 2 Δ ρ, θ f ¼ ρ t, and L 0 ¼! 1=2. 5r 0 w 2 0 p 16 ffiffi 2 g 1 þ λ 2 α Δ ρ The details of this derivation are given in the Appendix and have been validated experimentally in the fully miscible limit (Camassa et al., manuscript in preparation, 2011). This formula is a novel exact solution of the canonical integral model for miscible plumes originally introduced by Morton et al. [1956]. Using the values α = and λ = 1.2, and our experimental parameters corresponding to the OSW 8:1:5 plume, the theory predicts an escape a height of 3.6 cm. Shown in Figure 2 are two time series. The top series demonstrates a plume released 1 cm below the density transition every 2.5 s and clearly demonstrates the plume escaping. The bottom time series shows a plume release 4 cm below the ambient density transition and clearly shows the plume trapped. These values bracket the critical escape height of 3.6 cm predicted by theory, demonstrating that the Figure 2. Time series showing critical escape height for a plume with initial mixture: eight parts oil, one part surfactant, five parts water (ρ p = g cm 3, Q = 10 ml s 1 ). (a) Plume released 1 cm below background density transition, demonstrating escape. (b) Plume released 4 cm below transition, demonstrating trapping. Time interval between frames is 2.5 s. Bottom and top layer densities are 1.03 and g cm 3, respectively. Black line represents the density interface. Yellow dots mark the plume center of mass within the region marked by the green contour. Theory predicts a critical escape height of 3.6 cm.

4 260 SUBSURFACE TRAPPING OF OIL PLUMES IN STRATIFICATION critical escape height of these plumes is consistent with the miscible plume model. In Figure 3, two time series are presented which demonstrate novel behavior in which the plume is initially trapped and then spontaneously destabilizes. The multiphase oil/surfactant plume is initially trapped at the density interface, consistent with the theory above, but subsequently undergoes an instability by which the oil spontaneously rises out of the layer to the free surface. This instability, evident in the top time series with OSW of 4:3:2, begins around t = 870 s and is characterized by a moving front with vertical fingering. This front eventually occupies the entire top layer. The bottom series with OSW 4:3:17 does not exhibit the instability, and the plume remains trapped at the interface indefinitely. Similar experiments conducted in linear stratification (N = 0.6 s 1 ) with the same OSW as the first case above (4:3:2) result in a thicker trapped plume that also destabilizes on a comparable but longer timescale (t = 1400 s). Here the instability does not exhibit a clear fingering pattern, but rises diffusively. In general, our experiments suggest that multiphase plumes with lower oil content (and higher density) have a delayed onset of instability. The mechanisms for this erosion of the subsurface plume appear to be a combination of buoyancy effects, diffusion of entrained lower layer fluid, and coalescence of the oil microdroplets. When initially trapped, the plume carries with it fluid entrained from the lower layer. A completely miscible plume, after entrainment, would trap at a height of neutral buoyancy and remain indefinitely. However, an immiscible plume can exchange lower-layer fluid trapped in the interstitial space between oil droplets with fluid from above the interface. This occurs over a diffusive timescale that depends on the characteristics of the emulsion (droplet size and concentration and OSW ratio). 4. CONCLUSIONS We have presented laboratory results for the trapping and escape of multiphase oil/surfactant plumes in nonlinear sharp stratification. We have documented regimes of trapping and escape and explored the long-term stability and residence times of trapped plumes. We see a distinct dependence of residence time on plume composition, as plumes with high oil content can rapidly destabilize, while plumes with higher water/surfactant ratios can remain trapped indefinitely. The long-term persistence of the underwater trapped plume following the DHW spill demonstrates that long residence times may be common for such plumes, for sufficiently small oil droplets, and when surfactants are employed. These residence times are very important for determining the rates of bacterial degradation and long-term contaminant fate. We have demonstrated that escape/trapping of oil in the nonlinear, sharp stratification limit. Integral plume models have been successful in predicting initial trapping in uniform or weakly nonuniform stratification, but the long-term residence time and potential destabilization of multiphase plumes in generic stratification, which are more typical of the ocean, is less clear. A detailed description of residence time will require an understanding of the hydrodynamics of many-particle systems interacting with stratification. Recently, a complete theory for single particles interacting with sharp stratification has been developed [Camassa et al., 2009, 2010] and demonstrates dramatically enhanced Figure 3. Time series showing timescale of plume instability. (a) OSW 4:3:2, t = 30, 450, 870, 900, 1800, 3600, 7200 s. (b) OSW 4:3:17, t = 30, 450, 900, 1800, 3600, 7200, s. Notice the onset of instability in the top row, first evident at t = 870 s.

5 ADALSTEINSSON ET AL. 261 residence times. We are currently exploring the extension of these theories to multiphase many-body systems. APPENDIX Consider the system of reduced differential equations originally presented by Morton et al. [1956] for the behavior of a turbulent jet/plume in a constant density ambient fluid: d dz ðb2 wθþ ¼0; d dz ðb2 w 2 Þ¼2gλ 2 b 2 θ; d dz ðb2 wþ¼2αbw; ða1þ ða2þ ða3þ where b(z) is the jet width as a function of z, θðzþ ¼ ρðzþ is the density anomaly, and w(z) is the vertical jet velocity. Using equation (A1) implies b 2 wθ = constant = γ. Introducing new variables M ¼ w θ and N ¼ 1, equations (A2) and (A3) θ become dm dz ¼ð2gλ2 Þ N M dn dz ¼ 2α p ffiffiffiffiffiffi pffiffi M: γ Now we change variables again, letting Y = M 2 results in dy dz ¼ 4gλ2 N dn dz ¼ 2α pffiffi Y 1=4 : γ ða4þ ða5þ ða6þ ða7þ Dividing equation (A7) by equation (A6) and separating variables yields the conserved quantity N 2 ¼! 4α pffiffiffiffiffiffiffiffiffi Y 5=4 þ A: 5 γgλ 2 Imposing initial conditions determines A: ða8þ A ¼ θ 2 0 4α w 2 0 5gλ 2 b 0 θ 3 ; ða9þ 0 where w 0 ; θ 0 ; b 0 are the initial conditions for the system (A1) (A3). Using the conserved quantity, we immediately arrive at a formula for Z as a function of the density anomaly: z ¼ 1 4gλ 2 Y Y 0 ds p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ða10þ as 5=4 þ A where a ¼ 4α 5gλ 2 p ffiffi. The upper limit of integration will be γ chosen so that the density of the mixture matches the upper layer density, i.e., we need to find the value of z* such that ρ(z*) = ρ t <. This implies θðz Þ¼ ρ t Nðz Þ¼ ρ t : Further from equation (A10), we know y f ¼ N 2 4=5 A ¼ a ρ 2 b að ρ t Þ 2 A a! 4=5 : ða11þ The lower limit of integration is selected in terms of the initial condition, θ 0. Lastly, these models are derived under the assumption of Gaussian profiles, and to properly implement the physical initial conditions requires introducing a virtual source. The result of this sets θ 0 ¼ ð1 þ λ2 Þð ρ j Þ λ 2, w 0 ¼ 2w 0, b 0 ¼ p r0 ffiffiffi, where r 0 is the physical jet radius, w 0 is 2 the physical jet speed, and ρ j is the physical jet initial density. Rescaling variables on the integral results in the formula presented in the main text. Acknowledgments. PIs Camassa, McLaughlin, and White are supported by the National Science Foundation, through NSF RTG DMS , NSF RTG DMS , NSF RAPID CBET , NSF CMG ARC , and NSF DMS , and the Office of Naval Research through ONR DURIP N Mertens was supported by a Postdoctoral Fellowship provided by NSF RTG DMS Harenberg, Reis, Schlieper are supported as Undergraduate Assistants through NSF RAPID CBET Adalsteinsson provided data analysis and image processing with his software DataTank. REFERENCES Camassa, R., C. Falcon, Z. Lin, R. M. McLaughlin, and R. Parker (2009), Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime, Phys. Fluids, 21, , doi: /

6 262 SUBSURFACE TRAPPING OF OIL PLUMES IN STRATIFICATION Camassa, R., C. Falcon, Z. Lin, R. M. McLaughlin, and N. Mykins (2010), A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number, J. Fluid Mech., 664, Camilli, R., C. M. Reddy, D. R. Yoerger, B. A. S. Van Mooy, M. V. Jakuba, J. C. Kinsey, C. P. McIntyre, S. P. Sylva, and J. V. Maloney (2010), Tracking hydrocarbon plume transport and biodegradation at Deepwater Horizon, Science, 330(6001), , doi: /science Joye, S. B., I. R. MacDonald, I. Leifer, and V. Asper (2011), Magnitude and oxidation potential of hydrocarbon gases released from the BP oil well blowout, Nat. Geosci., 4(3), Lemckert, C. J., and J. Imberger (1993), Energetic bubble plumes in arbitrary stratification, J. Hydraul. Eng., 119(6), , doi: /(ASCE) (1993)119:6(680). McLaughlin, R. (2005), Plume dynamics, in Encyclopedia of Nonlinear Science,editedbyA.Scott,pp , Routledge, New York. Morton, B. R., G. I. Taylor, and J. S. Turner (1956), Turbulent gravitational convection from maintained and instantaneous sources, Proc. R. Soc. London, Ser. A, 234, Socolofsky, S. A., and E. E. Adams (2002), Multiphase plumes in uniform and stratified crossflow, J. Hydraul. Res., 40(6), , doi: / Socolofsky, S. A., and E. E. Adams (2003), Liquid volume fluxes in stratified multiphase plumes, J. Hydraul. Res., 129, , doi: /(asce) (2003)129:11(905). Socolofsky, S. A., and E. E. Adams (2005), Role of slip velocity in the behavior of stratified multiphase plumes, J. Hydraul. Res., 131, , doi: /(asce) (2005)131:4(273). Socolofsky, S. A., E. E. Adams, and C. R. Sherwood (2011), Formation dynamics of subsurface hydrocarbon intrusions following the Deepwater Horizon blowout, Geophys. Res. Lett., 38, L09602, doi: /2011gl D. Adalsteinsson, R. Camassa, S. Harenberg, R. M. McLaughlin, K. Mertens, J. Reis, W. Schlieper and B. White, Joint Fluids Laboratory, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. (bwhite@unc.edu) Z. Lin, Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang Province , China.