Detrainment Fluxes for Multi-Phase Plumes in Quiescent Stratification

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1 Environmental Hydraulics: Jets, Plumes, and Wakes Detrainment Fluxes for Multi-Phase Plumes in Quiescent Stratification S. A. Socolofsky 1 & E. E. Adams 2 1 Inst. for Hydromechanics, University of Karlsruhe, Karlsruhe, Germany 2 Dept. Civil & Environ. Engrg., Massachusetts Inst. of Tech., Cambridge, MA 02139, USA socolofs@alum.mit.edu, eeadams@mit.edu Abstract: This paper presents laboratory experiments to determine the detrainment fluxes of passive tracers and entrained fluid for multi-phase plumes in quiescent stratification. The fraction of passive tracer which detrains differs, in general, from the fraction of plume water which detrains due to internal plume recirculation. Total passive tracer fluxes of Rhodamine 6G dye were measured using an in-situ fluorescence profiler; net plume fluid fluxes were measured by comparing pre- and post-experiment profiles of density. To distinguish between upward- and downward-flowing plume fluxes, a control-volume model of the plume was developed, and the corresponding fluxes were computed using a Bayesian estimation technique. Results correlate with the non-dimensional slip velocity U = u s /(B) 1/4, where u s is the dispersed phase terminal rise velocity, B is the plume buoyancy flux, and is the buoyancy frequency of the stratification. As an illustration, the results are applied to the case of dispersed oil droplets resulting from an accidental oil/gas blowout in deep water. Introduction Multi-phase plumes occur in a wide range of natural and engineered systems, including reservoir mixing and aeration, industrial processing, and contaminant containment [9]. Of particular interest to the authors are CO 2 plumes for deep-ocean carbon sequestration and oil and gas plumes released from rare, accidental deep-sea oil well blowouts [10, 8, 3]. These latter plumes are characterized by modest depths of order 1000 m, typical ambient density gradients on the order of 10-1 kg/m 4, and buoyancy fluxes ranging from 10-2 to 10 0 m 4 /s 2. Because the characteristic length scale of these plumes is 1 to 10% of the total depth, the analysis is simplified by ignoring the effects of pressure changes (which would result in dispersed phase expansion and changes in the buoyancy flux). This paper focuses on the fluxes of entrained fluid and dissolved contaminants that detrain from the rising dispersed phase core due to the affects of stratification. Analysis General multi-phase plumes in stratification are affected by the strength of the stratification, the strength of the plume, and the characteristics of the dispersed phase. For a continuously stratified ambient, the governing parameter describing the stratification is the Brunt-Vaisälä

2 stratification frequency,, given by 2 = -(g/ρ 0 )(dρ/dz), where ρ(z) is the ambient density profile, ρ 0 is a reference density, and z is the positive upward vertical coordinate. The strength of the plume is given by the buoyancy flux of the dispersed phase, B = gq ρ/ρ 0 where ρ is the density difference between the dispersed phase and the ambient and Q is the volume flow rate of dispersed phase at the source. In general, the effect of the dispersed phase depends on size, density, shape, cohesion, surface tension, and expansion effects [9]. Since the terminal rise velocity, or slip velocity u s, is itself a function of these characteristics, u s is commonly used as the parameter describing the characteristics of the dispersed phase [1, 6, 10]. Since expansion also affects the buoyancy flux, expansion must be considered separately, but as mentioned above, expansion can be neglected for most deep ocean plumes. To compare results across a range of prototype dimensions, the parameters described above are combined to form non-dimensional groups. To do this, we first define characteristic length, velocity, and flow rate scales, given by l C = (B/ 3 ) 1/4, u C = (B) 1/4, and Q C = (B 3 / 5 ) 1/4, respectively. Experimental results are then compared by normalizing with these characteristic scales. For instance, the relative importance of dispersed phase expansion for affecting local plume properties can be determined by comparing to l C. For ideal gas law behavior, expansion is proportional to the depth, H, and is important for H/l C of order 1. Applying the Buckingham Π-Theorem, we can form one dimensionless variable from the three governing parameters, giving U = u s /u C, a dimensionless slip velocity. This variable is, thus, the governing parameter controlling the plume behavior and we expect plume characteristics, χ, to be proportional to U, i.e. χ = f (U ). Methods Experiments were conducted in the R. M. Parsons Laboratory at MIT using a 1.4 m square by 2.4 m tall, glass-walled experimental tank. The tank was stratified with salt (acl) using the two-tank method [1], creating a linear density profile between 1027 kg/m 3 and 1003 kg/m 3 ( = 0.3 s -1 ). Salt concentration (salinity) profiles were recorded using a Head micro-scale conductivity and temperature (CT) probe and an Ocean Sensors OS300 CT probe, both mounted to a linear actuator, providing a resolution of less than 1 cm. Plumes were created from a range of dispersed phases (air, oil, and glass beads) to investigate a broad spectrum of slip velocities. Rhodamine 6G fluorescent dye tracer was injected at the base of each plume using a collar diffuser at a rate of about 0.1 mg/s. Dye concentration profiles were recorded using a Chelsea in-situ field fluorometer connected to an Ocean Sensors OS200 conductivity, temperature, and depth (CTD) probe. A 6 W argon-ion LASER was used with a 2.5 in progressive-scanning CCD camera and various filters to obtain LASER-Induced Fluorescence (LIF) images of the dye, the bubbles, or both. Complete details are reported in Socolofsky [10]. The flux of passive dye tracer (Rhodamine 6G) was measured by integration of the fluorometer profiles. Profiles were taken about 6 hours after each experiment which gave adequate time for the dye distribution to homogenize in the horizontal through the action of residual small-scale density currents. Profiles taken at several intermediate times showed that, because of the strong stratification, the dye distribution remained unchanged in the vertical during this homogenization

3 Q oe Q 2 h P Q ol Q p Q i Q r Q, S, C Figure 1. Schematic of conceptual plume flow model showing each of the model flow rates, Q, and indicating the associated salinities, S, and dye concentrations, C. h P is the peel height. period. The fraction of dye that peeled, f*, was calculated as the mass of dye found below the peeling zone, M i, to the total mass of dye recovered by the profiles, M T. et liquid flux measurements were made by comparing pre- and post-experiment salinity profiles. Post-experiment salinity profiles were taken one hour after each experiment, after the internal waves had damped out. Following the method of Baines & Leitch [2, 4, 5], the net upward plume liquid flux, Q net, was calculated from Q net = A( ρ/ t)/( ρ/ z), where A is the tank cross-section and t is the time. To distinguish the upward- and downward-flowing components of Q net, a control-volume model of the plume was developed [see also 10, and 9]. Shown in Figure 1, this model defines seven liquid fluxes, Q, and their related salinities, S, and average passive tracer concentrations, C. To solve for the 21 unknowns, 10 equations can be extracted as direct measurements from the salinity and dye profiles. Four additional constraints are obtained by assuming a top-hat concentration profile in the conceptual model (i.e. S 2 = S p, C 2 = C p, S i = S r, C i = C r ). Applying mass conservation to the conceptual model supplies an additional 8 constraint equations, giving a total of 22 measurement and constraint equations. To solve this system of equations, a constrained Bayesian estimation method was used [7, 10]. This method optimizes the model flux quantities (Q s, C s, and S s) in a least-squares sense while matching the constraints exactly. The Bayesian term of the objective function accounts for our prior knowledge based on the measurements and our uncertainty in the experimental method.

4 Figure 2. Dimensionless correlations of the peeling fraction of fluid and passive tracer, the bubble spreading ratio, and the buoyancy flux through the first peel with U. Results Figure 2 shows the results for the first peeling event in 11 experiments for four major variables as functions of U (f is the percent of plume fluid that peels, f* is the percent of passive tracer that peels, λ is the ratio of the bubble column width to the width of passive tracer, and B 2 is the dimensionless buoyancy flux of entrained fluid escaping the peel). The dashed correlation lines are given by the following regressions: 2 f = U f* = U λ = U B / B= 0.082U The behavior of these variables as functions of U is responsible for the different plume types observed in the literature. For U between 1.5 and 2.4, Type 2 plumes with distinct, nonoverlapping intrusions are observed. Intrusions do not overlap because over 90% of the entrained fluid peels, sending a small negative buoyancy flux to the subsequent peels. This high value of f occurs because the bubble column is well-distributed (λ 70%), pushing entrained (1)

5 fluid high into regions of negative buoyancy, causing sudden and efficient detrainment events. For U > 2.5, Type 3 plume are observed with intrusions that occur randomly and overlap. The intrusions overlap because of inefficient peeling (f < 90%), sending a larger negative buoyancy flux into subsequent peels. The peeling becomes inefficient for these plumes because the bubble column is narrowly distributed (λ < 60%), allowing entrained fluid near the edge of the plume to be easily lost in a series of random, unsteady intrusions. For U < 1.5, the slip velocity of the dispersed phase is small compared to the plume fluid velocity, the dispersed phase peels with the detraining fluid, and in the limit of U = 0, the behavior approaches that of a single phase buoyant plume. Application The laboratory results are applied in this section to predict the behavior of fine oil droplets released from accidental oil-well blowouts in deep water. Typical blowouts involve the release of oil together with natural gas. The chemical behavior of these phases is complicated in itself, and can include emulsion formation of the oil phase and clathrate hydrate formation of the gas phase [11]. We will assume here that a non-emulsion-forming oil is released with a gas-oil ratio (GOR) at STP of 100 and that possible hydrate formation is limited to the bubble surface and does not affect the gas phase slip velocity. Assuming typical conditions for the Gulf of Mexico, Table 1 summarizes an analysis of three possible blowout sizes. Spill Size Oil flow rate [m 3 /s] et buoyancy flux [m 4 /s 3 ] Bulk U [--] h P [m] f* [%] Small Medium Large Table 1. Parameter values for oil-well blowout plumes. GOR is 100, is 10-3 s -1, gas bubble slip velocity is 30 cm/s, and U is calculated from the gas slip velocity and the net buoyancy flux for the combined oil and gas plume. h P is the predicted peel height from Socolofsky [10]. For all three cases, the bulk U is greater than 1.5, and we do not expect the plume behavior to approach that of a single-phase plume. However, because the slip velocity of the oil phase can be small, it is typically assumed that all the oil intrudes with the first peeling event (from results in Socolofsky (2001), this height, h P, would be as in Table 1). Comparing this assumption with the previous section, the peeling efficiency for these plumes can be low (up to 15% of the entrained fluid does not intrude with the first intrusion), and the detrainment flux of passive tracer, which gives the result for the limit of oil slip velocity equal to zero, is even further reduced due to internal plume circulation. For instance, for the medium spill, over 26% of the fine oil droplets would continue to rise with the gas plume above the first intrusion formation. For the large spill, which has the greatest potential for oil to detrain (f* = 87%), the peel height is near the water surface, and 13% of the fine oil droplets would be expected to reach the surface. Hence, for these plumes we expect a significant portion of oil to reach the surface with the main blowout plume.

6 This result is significant for two reasons. First, oil that remains with the gas plume should reach the surface in a narrow region near the blowout location, where it could subsequently be cleaned up. If all the oil had intruded, as generally assumed, the oil rises slowly in a dispersed cloud and the surface slick would be much more distributed, thus, less likely to be cleaned up. Second, because the oil rises with the plume, the time for oil to reach the surface is significantly less than the time predicted by assuming oil droplets simply rise with their own slip velocity. Hence, cleanup must respond quickly and locally to accidental oil-well blowouts even in deep water. Summary Laboratory results for the peeling characteristics of multi-phase plumes in linear stratification are presented. The results correlate with the dimensionless slip velocity U = u s / (B) 1/4, which allows laboratory results to be applied at the field scale of plumes where bubble expansion is negligible over the characteristic plume length scale (B/ 3 ) 1/4. Results applied to field-scale oil-well blowouts indicate that a significant portion of released oil continues to rise with the gas-portion of the blowout plume above the first intrusion. This has the effect that oil accompanies the gas to the surface and would need to be subsequently cleaned up. This contrasts with previous assumptions that all oil would be trapped below the surface in the first intrusion layer. Acknowledgements This study was supported by the MIT Sea Grant College Program, the ational Energy Technology Laboratory of the U.S. Department of Energy, and the Deep Spills Task Force, comprised of the Minerals Management Service of the U.S. Department of Interior and a consortium of 12 member oil companies of the Offshore Operator s Committee. 1. Asaeda, T. & Imberger, J. (1993), Structure of bubble plumes in linearly stratified environments, J. Fluid Mech. 249, Baines, W. D. & Leitch, A. M. (1992), Destruction of stratification by bubble plumes, J. Hydr. Engrg. 118, Crounse, B. C., Socolofsky, S. A. & Adams, E. E. (2000), Bubble and droplet plumes in stratification 2: umerical studies, in Proc. IAHR 5 th Int. Symp. Strat. Flow, Vancouver, BC, July Leitch, A. M. & Baines, W. D. (1989), Liquid volume flux in a weak bubble plume, J. Fluid Mech. 205, Lemckert, C. J. & Imberger, J. (1993), Energetic bubble plumes in arbitrary stratification, J. Hydr. Engrg. 119, McDougall, T. J. (1978), Bubble plumes in stratified environments, J. Fluid Mech. 85, Schweppe, F. C. (1973), Uncertain Dynamic Systems, Prentice-Hall, Inc., Englewood Cliffs, ew Jersey. 8. Socolofsky, S. A., Crounse, B. C. & Adams, E. E. (2000), Bubble and droplet plumes in stratification 1: Laboratory studies, in Proc. IAHR 5th Int. Symp. Strat. Flow, Vancouver, BC, July Socolofsky, S. A., Crounse, B. C. & Adams, E. E. (2001), Multi-phase plumes in uniform, stratified and flowing environments, in H. Shen, A. Cheng, K.-H. Wang & M. H. Teng, eds, Environmental Fluid Mechanics Theories and Applications, ASCE/Fluids Committee. 10. Socolofsky, S. A. (2001), Laboratory experiments of multi-phase plumes in stratification and crossflow, Ph.D. Thesis, Dept. Civil & Environ. Engrg., MIT, Cambridge, MA. 11. Yappa, P. D. & Zheng, L. (1997), Simulation of oil spills from underwater accidents I: Model development, J. Hydr. Res. 35(5),

PIV measurements of turbulence in an inertial particle plume in an unstratified ambient

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