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gas spills
Journal of Marine Systems 31 (2002) 299 – 309 www.elsevier.com/locate/jmarsys
Modeling gas dissolution in deepwater oil/gas spills Li Zheng 1, Poojitha D. Yapa * Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699-5710, USA Accepted 25 September 2001
Abstract Gases in deepwater oil/gas spills can lose considerable amounts of the gas phase due to dissolution in water. Gas dissolution has a significant impact on the behavior of the oil/gas jet/plume because of its impact on the buoyancy. A method is presented in this paper for computing gas dissolution that covers a broad range of water depth, from shallow water where gases behave as ideal ones under low pressure to deepwater where gases behave as non-ideal ones under high pressures. The method presented also accounts for the spherical and non-spherical shapes of gas bubbles. The gas dissolution computations are validated by comparing the computed results with observed data from previously conducted laboratory experiments. The gas dissolution computation module is then integrated with a model for underwater oil/gas jets/plumes by Yapa and Zheng [J. Hydraul. Res. 35 (5) (1997) 673]. Scenario simulations are presented to show the impacts of gas dissolution on the behavior of jets/plumes. These scenarios show the impact of dissolution on the behavior of the jet/plume. The comparison of results using ideal gas conditions and non-ideal gas conditions is also shown. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Gas dissolution; Oil/gas spills; Deep water
1. Introduction Modern technology has made it economically feasible to explore and produce oil and gas from ultra deepwater wells. For example, in the Gulf of Mexico (GOM), the number of exploratory wells has increased by 70% in the 2 years from 1996 to 1998 (Lane and Labelle, 2000). Current estimates show that future production from installations deeper than 800 m will be 69% by year 2007 (Lane and Labelle, 2000). As the production increases, the potential for an oil/gas spill increases. Major concerns from a deepwater oil/gas spill are fire and toxic hazard to *
Corresponding author. E-mail address: [email protected] (P.D. Yapa). 1 Now at Quantitative Environmental Analysis, LLC, Montvale, NJ 07645, USA.
the people working on the surface installations and loss of buoyancy of ships and any floating installations. For this purpose, it is important to know when and where the gas will surface and how much. To meet these new challenges, spill response plans need to be upgraded. An important component of such a plan would be a model to simulate the behavior of oil and gases, if accidentally released, in the deepwater. Many underwater oil/gas spills can be described as a two-phase plume that consists of gas (e.g., CH4, C2H6, and C3H8) and a liquid mixture of oil and water. In this paper, a plume that consists of gas and liquids will be referred to as a gas plume for simplicity. In deepwater, the high pressure and low temperature makes the behavior of oil/gas spills different from those that occur in shallow water in two respects: (i) gas may be converted to an ice-like solid compound called gas hydrate; (ii) gas bubbles
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may lose considerable amounts of the gas phase through dissolution during their long journey. These two processes alter the buoyancy of the plume significantly. Therefore, knowing the state of gases as they rise with the plume when predicting the fate of an oil/gas plume released from a deepwater location is very important. Yapa and Zheng (1997) developed a gas plume model to simulate oil/gas spills from underwater accidents. This model was subjected to extensive comparison with data from analytical solutions, laboratory and field experiments (Zheng and Yapa, 1998). The above model was integrated with a farfield oil spill model by Yapa et al. (1999) and used to compare the model results with two field experiments of oil/gas release. Recently, modules to simulate gas hydrate formation and decomposition were integrated with the gas plume model (Zheng and Yapa, 2000b). However, gas dissolution was not taken into account in the above oil/gas spill models. Johansen (2000) developed a comprehensive gas plume model for oil/gas spills in deepwater, in which gas dissolution into seawater was included. The calculation of mass transfer coefficient in Johansen (2000) was based on a correlation for rigid spherical bubbles. Johansen (2000) did not provide details of the gas solubility calculation. Even outside the area of oil/gas spill modeling, research related to gas dissolution in a gas plume is scarce because the residence time of gas bubbles in most gas plume problems is so short that the effect of gas dissolution can be neglected. Wu¨est et al. (1992) developed a gas plume model (including gas dissolution) for the restoration of deep stratified lakes. The gas solubilities in Wu¨est et al. (1992) were calculated from a simple Henry’s law, which would break down if the pressure is raised sufficiently (King, 1969). The mass transfer coefficients in the same paper were calculated from a correlation for small size O2 bubbles, which cannot serve as a general formulation for calculating mass transfer coefficients. Some research on CO2 sequestration in the deep ocean concerns dissolution calculations in plumes (Liro et al., 1992; Morishita et al., 1993; Haugan et al., 1995). However, these dissolution calculations were for liquid CO2 droplets instead of gaseous CO2 bubbles because CO2 in general is
in the liquid phase below 500 m (Liro et al., 1992). As a result, there is no gas bubble expansion mechanism due to pressure decrease with height in CO2 plumes if CO2 is in the liquid phase. Haugan and Drange (1992) proposed a gravity current model, in which gaseous CO2 dissolution was considered, to simulate CO2 disposal at 200 – 400-m depth. A modified Henry’s law capable of handling high pressure was used to calculate the CO2 solubility by Haugan and Drange (1992). The mass transfer coefficient in the same paper was calculated by a correlation for fluid spherical particles from Clift et al. (1978). In deepwater, the behavior of gas is non-ideal due to high ambient pressure. Gas bubbles may experience significant variations in sizes and shapes, due to gas expansion and dissolution. These factors need to be taken into account when estimating solubility and the mass transfer coefficient of gas in water. This paper presents an improved method to calculate solubility and the mass transfer coefficient for gases in water. The formulation for gas solubility is valid for both low and high pressures. The general formulation presented for calculating the mass transfer coefficient can be applied to different gases and different shapes of gas bubbles. Although the methods used here have been used in other disciplines (e.g., chemical engineering), they were not integrated in a way that is easy to use in engineering applications and were rarely applied to modeling environmental hydraulics problems. The possible formation and decomposition of gas hydrates within a plume are not considered in this paper. The presence of salt can slightly decrease the gas solubility in water (Yamamoto et al., 1976) but the effect of salinity on the mass transfer coefficient is not clear. Therefore, salinity effects on the gas dissolution are not considered in this paper. The improved method to calculate gas dissolution is integrated with the model for underwater oil/gas jets/plumes by Yapa and Zheng (1997). Scenario simulations are presented to show the impacts of gas dissolution on the behavior of jets/plumes. Although the work developed in this paper originated in connection with simulating oil/gas spills, it can be useful to other environmental problems, such as CO2 sequestration in the ocean, restoration of eutrophic lakes, and ice control in harbors of cold regions.
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2. Computing gas dissolution In general, the dissolution rate for a gas bubble is calculated by dn ¼ KAðCs C0 Þ dt
ð1Þ
where n = number of moles of gas in a bubble [mol]; K = mass transfer coefficient [m/s]; A = surface area of a gas bubble [m2]; C0 = concentration of dissolved gas [mol/m3]; Cs = saturated value of C0 (i.e., solubility). The key parameters in Eq. (1) are the solubility, Cs, and mass transfer coefficient, K. 2.1. Solubility of gas in water The solubility of gas in water is commonly calculated by the simple Henry’s law: P ¼ Hxl
ð2Þ
where P = gas pressure [MPa]; H = Henry’s law constant [MPa]; xl = mole fraction of dissolved gas in water at equilibrium condition, which can be transferred to solubility, Cs, by Cs xlqw/Mw; qw = water density [kg/m3]; Mw = molecular weight of water [kg/ mol]. The applicability of the simple form of Henry’s law given in Eq. (2) is limited to an ideal gas or low pressure conditions. If the pressure is raised sufficiently (e.g., in the case of deepwater), Eq. (2) is found to break down (King, 1969; Weiss, 1974; Lekvam and Bishnoi, 1997). King (1969) presented a modified form of Henry’s law for the solubility of slightly soluble gases at elevated pressures: f g ¼ Hxl expð106 Pvl =RT Þ
ð3Þ
where f g = fugacity of gas in gas phase [MPa]; vl = partial molar volume of gas in solution [m3/mol]; R = universal gas constant = 8.31 [J/molk]; T = water temperature [K]. Eq. (3) reduces to Eq. (2) as P ! 0. 2.2. Mass transfer coefficient of gas bubbles According to Clift et al. (1978), the mass transfer coefficient of gas bubbles in liquids is dependent on the size and shape of bubbles as well as gas dif-
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fusivity in liquids. Gas bubbles in liquid are generally approximated as spheres for small size bubbles, ellipsoids for intermediate size bubbles, and spherical-caps for large size bubbles. For small bubbles, the system purity also affects the value of the mass transfer coefficient. Therefore, the formulation for a single shape of bubbles, such as that for spherical bubbles, is unable to cover the mass transfer coefficient for a broad range of bubble sizes in deepwater scenarios. We combine the following equations originally developed by Johnson et al. (1969) and Clift et al. (1978), respectively, to form a general formulation for the mass transfer coefficient of bubbles in contaminated liquids. The validity of this integrated approach will be examined later by comparing the computed results with experimental observations. (i) For bubbles with spherical shape (small size range)
UD K ¼ 0:0113 0:45 þ 0:2de
12 ð4Þ
where de = equivalent diameter of bubbles [cm]; D = molecular diffusivity of a gas in a liquid [cm2/s]; U = terminal velocity of bubbles [cm/s], which is determined by using the method described by Zheng and Yapa (2000a). (ii) For bubbles with ellipsoidal shape (intermediate size range) 1
K ¼ 0:065D2
ð5Þ
(iii) For bubbles with spherical-cap shape (large size range) 1
1
K ¼ 0:0694de 4 D2
ð6Þ
Eq. (5) is recommended for diameters greater than 5 mm (Clift et al., 1978). Therefore, the critical diameter for transition from small to intermediate size is taken to be 5 mm. The critical diameter for transition from intermediate to large size and is obtained by solving Eqs. (5) and (6) simultaneously and is about 13 mm.
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2.3. Verification of solubility and mass transfer coefficient of gas bubbles in water In this paper, H and f g are calculated by a correlation for gas in water and the Peng – Robinson equation-of-state as given by Sloan (1997). vl is taken to be 33 and 36.5 cm 3/mol for CO 2 and CH 4, respectively, as suggested by King (1969). Fig. 1 shows the computed CO2 solubility in water at 40 C by using Eq. (2) (simple Henry’s law), computed CO2 solubility by using Eq. (3) (modified Henry’s law), and experimental data (King, 1969), respectively. The comparison shows that the simple Henry’s law approximates the CO2 solubility well until about 20 atm (approximate water depths of 200 m) but deviates significantly after that. Modified Henry’s law calculates solubility well up to 200 atm (approximate water depths of 2000 m). Fig. 2 shows the comparisons between the observed and computed CH4 solubility in water using a simple Henry’s law and a modified Henry’s law. The data are presented in three different blocks because they correspond to different temperatures. The available experimental data (Lekvam and Bishnoi, 1997) are such that at higher temperatures, the pressure range is also higher. Nevertheless, it shows that at higher pressures, the discrepancy between the experimental observations and solubility computed using a simple Henry’s law is larger. In fact, we can see that as the pressure increases, the discrepancy grows. Therefore,
Fig. 1. Comparison between computed and observed solubility of CO2 in water at 40 C.
Fig. 2. Comparison between computed and observed solubility of CH4 in water.
at higher pressures a modified Henry’s law provides better computational values for solubility. The higher pressures in this plot correspond to what is expected in deepwater spills. When all the data are taken together, the fractional standard error for all cases is 0.0625 when computed using a simple Henry’s law and is 0.0049 when computed using a modified Henry’s law. Figs. 3– 5 show the comparison between the values of mass transfer coefficient computed by the present method (i.e., using Eqs. (4) –(6)) and the observed data for different gases and ambient liquids. Fig. 3 shows the comparison between computed values and observed data for O2 in tap water at 25 C. Fig. 4 shows the comparison between the computed values and the observed data for CO2 in an aqueous glycerol solution at 25 C. Fig. 5 shows the comparison between the computed values and observed data for CO2 in tap water. Considering that the same set of equations is used for three cases involving three different combinations of gases and ambient liquids, the results
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Fig. 3. O2 bubbles in tap water at 25 C.
of Eqs. (4) – (6) are in reasonably good agreement with the experimental data. The values of the diffusion coefficient, D, listed in Figs. 3 – 5 are the values suggested by the respective authors who obtained the experimental data.
3. Integration of gas dissolution with jet /plume model The computation of gas dissolution described in the previous section is integrated with the jet/plume hydrodynamic model developed by Yapa and Zheng (1997). This section briefly describes the jet/plume hydrodynamics of the model. The following assumptions relate to the jet/plume hydrodynamics.
Fig. 5. CO2 bubbles in tap water.
(i) The flux of the number of bubbles, N [1/s], is kept constant with height, i.e., bubble coalescence is neglected. (ii) Bubbles are all of uniform size at the opening. If a bubble size spectrum becomes available based on field experiments at a latter time, it is possible to replace the number flux of bubbles N by an array N (r1,. . . ,rn). Then the fate of the individually sized bubbles can be calculated independently. 3.1. Governing equations In a jet or plume that consists of a mixture of oil and gas released vertically, the gas portion can be expected to occupy the inner core (Yapa and Zheng, 1997). Based on assumption (i), the number of bubbles per unit plume height is N/(w + wb), where, w = vertical velocity of liquid part of plume; wb = slip velocity of gas, which is a function of gas bubble size. Zheng and Yapa (2000a) give details of computing wb. Fig. 6 shows a schematic diagram of the oil/gas plume. The number of bubbles in a control volume is given by Nh/(w + wb), where h = height of a control volume. (i) Volume fraction of gas bubbles The volume fraction of gas (e) within the inner gas bubble core can be defined by e¼
Fig. 4. CO2 bubbles in 90.6% aqueous glycerol solution at 25 C.
303
l l b
ð7Þ
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acting on the liquid part. The last term is related to the vertical force acting on the gas bubbles. (v) Conservation of heat, salinity and oil mass dðml IÞ dml ¼ Ia dt dt
ð11Þ
where I = a symbol representing a property such as heat content (CpT ), salinity (S), or oil concentration by mass (C). The subscript ‘a’ refers to the conditions in the ambient water. Eq. (11) states that the change of heat, salinity or oil mass in the control volume is due to the input contributed by the entrained mass. (vi) Non-ideal gas law PMg ¼ qb ZRT
Fig. 6. Schematic diagram for an oil/gas plume.
where ql = density of the liquid in a plume [kg/m3]; q = density of the plume bubble-water mixture [kg/ m3]; qb = density of the gas [kg/m3]. (ii) Conservation of liquid mass dml ¼ a Q e dt
Nh dn Mg Dt w þ wb dt
4. Model application
ð9Þ
where Dmb = loss of gas mass due to gas dissolution in the control volume [kg]; Mg = molecular weight of gas [kg/mol]; Dt = time step [s]. (iv) Conservation of momentum d ½ml w þ mb ðw þ wb Þ
dt ¼ wa qa Qe þ ðqa ql Þgpb2 ð1 b2 eÞh þðqa qb Þgpb2 b2 eh
where Z = compressibility factor. In deep water scenarios, the gas can deviate from the ideal gas behavior because of high pressure. Compressibility factor Z is calculated using the method described by Sloan (1997).
ð8Þ
where ml = liquid mass of a control volume [kg] = qlpb2(1 b2e)h; qa = density of the ambient fluid [kg/m3]; Qe = entrainment rate for the ambient water [m3/s]. Details of computing Qe were given in Yapa and Zheng (1997) (iii) Loss of gas mass due to gas dissolution Dmb ¼
ð12Þ
ð10Þ
where wa = vertical velocity of ambient fluid [m/s]. The first term of the right-hand side of Eq. (10) represents the momentum from the entrained liquid mass. The second term is related to the vertical force
In Figs. 1– 5, the dissolution computations were compared with available relevant data. It is desirable to compare the overall dissolution from a plume under field conditions. An extensive search for relevant data in the open literature that can be used for model comparison did not yield any data. There have been a number of observational studies on gas seeps in deepwater (e.g., Sassen and MacDonald, 1997). However, the information found in these papers is qualitative in nature and cannot be compared with the model results. In 2000, the Minerals Management Service and a consortium of oil companies sponsored a large-scale field experiment (DeepSpill), at a cost of US$ 2.5 million, in the North Sea. Although this is the best known study available yet (the data was not in the open literature at the time the paper was written), no direct measurements were taken on gas dissolution that can be used to compare the results of the model presented here. In this section, the model is applied to simulate several deepwater jet/plume scenarios of practical interest. The ambient condition selected has a constant salinity of 35% and linear distributions of temperature
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Table 1 Parameters used in the simulations
as shown in Fig. 7. This simpler ambient condition is preferred over a more complex salinity and temperature distribution from a field location (e.g., Gulf of Mexico) because it is easier to identify the impact of gas dissolution. When a complex temperature or salinity profile is used, the results are indirectly affected by other parametric variations and, hence, the contribution of dissolution becomes unclear. CH4 is common in these situations and is chosen for the application. The phase equilibrium condition for CH4 hydrate is shown by a dotted curve in Fig. 7. In the region below the dotted curve, the pressure and temperature of water meet the thermodynamic conditions necessary for hydrate formation. The crossing point between the dotted curve and the solid line (temperature distribution) in Fig. 7 approximately corresponds to a depth of 700 m. All simulations in this scenario are kept to less than 700 m to avoid possible hydrate formation. If the hydrate formation is included as part of the simulation, the results become more complex and the analysis of the effects of dissolution is harder. The values of model parameters used in the simulations correspond to a medium wellhead blowout and are listed in Table 1. The initial bubble radius was based on the information by Barbosa et al. (1996). A jet or plume of oil/gas mix under water is lighter than the ambient water. As it travels upwards, ambient water is entrained into the jet/plume and the density is increased. Ambient water density decreases upwards in a density-stratified environment. Simulations by Yapa et al. (1999), Zheng and Yapa (1999), Johansen (2000), and field experiments (Rye and Brandvik,
1997) have shown that for deep-water cases, it is most likely that there will be a point at which the density of the jet/plume is approximately equal to the ambient density. This level is referred to as the neutral buoyancy level (NBL). The dynamics of the jet/plume is assumed to cease at this level. Oil and gas will travel beyond this level under their own buoyancy. The validity of this argument was verified in Yapa et al. (1999). Fig. 8 shows the NBL for jets/plumes released at different depths. The NBLs when the gas dissolution is included are lower compared to those when the gas dissolution is excluded. The gas dissolution reduces the mass of free gas, resulting in reduced buoyancy. This results in lowering the NBL. A lower NBL for a plume means that the oil droplets have to travel a longer distance to the water surface under their own buoyancy. The slower rise velocity of oil droplets compared to a plume velocity means that oil will be subjected to dispersion by ambient currents during its journey to the water surface for a longer period. Therefore, a lower NBL can be expected to result in a wider cloud of oil with a lower concentration as compared to a higher NBL. Fig. 8 also shows that
Fig. 7. Water temperature and CH4 hydrate phase diagram.
Fig. 8. Neutral buoyancy levels vs. release depth.
Oil discharge rate — m3/s (bopd) Gas discharge rate — normal m3/s (scfd) GOR — N m3 s 1/m3 s 1 (scfd/bopd) Diameter of the opening — m (in.) Temperature of discharged oil — C (F) Density of discharged oil (kg/m3) Initial gas bubble radius (m)
0.0184 (10,000) 3.2752 (107) 178 (1000) 0.0889 (3.5) 80 (176) 893 6.0 10 3
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Fig. 9. Variation of relative bubble diameter with depth.
Fig. 11. Variation of momentum flux with depth.
simulating gas dissolution based on non-ideal gas (modified Henry’s law, Eq. (3)) or ideal gas (simple Henry’s law, Eq. (2)) does not make a significant difference for the NBL of the jet/plume. Figs. 9 –12 illustrate the different perspectives of the impact of gas dissolution on a gas plume discharged at 700 m. Fig. 9 shows the variation of relative bubble diameter respective to depth. If there is no gas dissolution, the bubble diameter increases during most of its journey due to the drop in hydrostatic pressure, except at the very beginning when the bubble diameter decreases because of the decrease in plume temperature caused by the entrainment. However, if the gas dissolution is included, the bubble diameter decreases as the plume rises because the effect due to gas loss caused by dissolution is more dominant than the increase in bubble diameter due to
hydrostatic pressure drop. Furthermore, the bubble diameter decreases faster with the ideal gas assumption compared to the non-ideal gas assumption because the solubility of ideal gas is greater than that of non-ideal gas under high pressure. Similar behavior can be observed in Fig. 10 that shows the mass fraction of dissolved gas in the gas plume, with and without the ideal gas assumptions, as it moves up from the release point at 700 m. Fig. 11 shows the variation of momentum flux of the gas plume with depth. The momentum flux at a given depth with gas dissolution is less than that without gas dissolution. This is because the gas dissolution consumes free gas and thus reduces the buoyancy force. Since the ideal gas dissolves faster than the non-ideal gas, the momentum flux at a given depth with an ideal gas assumption is less than that with non-ideal gas
Fig. 10. Variation of mass fraction of dissolved gas with depth.
Fig. 12. Variation of buoyancy flux with depth.
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assumption. Fig. 12 shows the variation of buoyancy flux of the gas plume with depth. In a densitystratified environment, the buoyancy flux of a plume decreases as the plume rises due to the entrainment. With the gas dissolution included, the buoyancy flux at a given depth decreases even faster. Ideal gas dissolves faster than the non-ideal gas, the buoyancy flux at a given depth with an ideal gas assumption is therefore less than that with a non-ideal gas assumption. The expressions used for computing the buoyancy flux and the momentum flux are given in the Appendix A.
5. Conclusions In this paper, a method was presented, by combining previous work, to compute the dissolution of gases into water. The formulation for gas solubility covers a broad range of water depth, from shallow water where gases behave as ideal ones under low pressure to deep water where gases behave as non-ideal ones under high pressures. The general formulations for calculating mass transfer coefficient can be applied to different gases and different shapes (e.g., the spherical and nonspherical shapes) of gas bubbles. The calculations were compared with available experimental data. The comparison between the calculations and the observed values was good. The module for computing dissolution was then integrated with the hydrodynamics of a jet/plume that consist of oil and gas by modifying the model by Yapa and Zheng (1997). The purpose of the model was to investigate the effect of dissolution on the fate of the jet/plume, especially when released in deepwater. The possible formation and decomposition of gas hydrates within a plume are beyond the scope of this paper and not addressed. Scenario simulations are presented to show the impacts of gas dissolution on the behavior of plumes, such as neutral buoyancy level (NBL) of the plume, relative bubble diameter, mass fraction of dissolved gas, plume momentum flux, and plume buoyancy flux. Dissolution affects the gas bubble diameter. The simulations show the mass loss due to dissolution affects the gas bubble size more than the expansion due to reduction in pressure as gas travel upwards. The change in gas bubble size is somewhat depen-
307
dent on the choice of ideal gas equations or nonideal gas conditions. The momentum flux (MF), buoyancy flux (BF), and NBL are affected by noticeable amounts due to dissolution. However, momentum flux, buoyancy flux, and NBL do not depend in a significant way on the use of ideal gas equations or non-ideal gas conditions. This is because the changes caused to the plume by the loss of gas mass are somewhat compensated by the changes in ambient water entrainment brought upon by the change in plume velocity. When mass is reduced due to dissolution, it changes the velocity of the jet/plume. This in turn changes the entrainment of ambient water. Therefore, how the dissolution affect MF, BF, and NBL are complex processes that are difficult to visualize. Therefore, the model results are valuable in understanding the impact of dissolution. The work developed here can be applied to other environmental problems related to gas plumes, such as CO2 sequestration in oceans, restoration of eutrophic lakes, and ice control in harbors of cold climate regions.
Acknowledgements The work presented in this paper originated during the development of a numerical model to simulate the oil and gas spills from ultra deepwater. The project was supported by the Minerals Management Service (MMS) of the United States Department of the Interior. The authors would also like to thank the Deep Sea Task Force (DSTF) for the sponsorship of the project. The results represent the views of the authors and not the Minerals Management Service.
Appendix A. Expressions for momentum and buoyancy flux Momentum flux ðMFÞ ¼ pb2 w2 Pe ð1 k2 eÞ þ pb2 ðw þ wb Þ2 Pb k2 e
Buoyance flux ðBFÞ ¼ pb2 wgðPa Pmix Þ
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Appendix B. Notation The following symbols are used in this paper: A C0 Cs de D fg H I K Mg ml Dmb n P Qe R T Dt U vl w wa wb xl Z e q qa qb ql
surface area of a gas bubble [m2] concentration of dissolved gas [mol/m3] saturated value of C0 (i.e., solubility) equivalent diameter of bubbles [cm] molecular diffusivity of a gas in a liquid [cm2/s] fugacity of gas in gas phase [MPa] Henry’s law constant [MPa] a symbol representing heat content, salinity, or oil concentration by mass a mass transfer coefficient [m/s] molecular weight of gas [kg/mol] liquid mass of a control volume [kg] loss of gas mass due to gas dissolution in the control volume [kg] number of moles of gas in a bubble [mol] gas pressure [MPa] entrainment rate for ambient water [m3/s] universal gas constant = 8.31 [J/molk] water temperature [K] time step [s] terminal velocity of bubbles [cm/s] partial molar volume of gas in solution [m3/mol] vertical velocity of liquid part of plume vertical velocity of ambient fluid [m/s] bubble slip velocity mole fraction of dissolved gas in water at equilibrium condition compressibility factor volume fraction within the bubble core density of plume bubble-water mixture [kg/m3] the density of ambient fluid [kg/m3] density of gas [kg/m3] density of the liquid part of a plume [kg/m3]
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Zheng, L., Yapa, P.D., 2000a. Buoyant velocity of spherical and non-spherical bubbles/droplets. Journal of Hydraulic Engineering, ASCE 126 (11), 852 – 854. Zheng, L., Yapa, P.D., 2000b. Modeling a deepwater oil/gas spill under conditions of gas hydrate formation and decomposition. Proceedings of the 23rd Arctic and Marine Oil Spill Program (AMOP) Technical Seminar, Vancouver, BC, Canada, June 2000, vol. 2. Environment, Canada, Ottawa, pp. 541 – 560.
Modeling gas dissolution in deepwater oil/gas spills Li Zheng 1, Poojitha D. Yapa * Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699-5710, USA Accepted 25 September 2001
Abstract Gases in deepwater oil/gas spills can lose considerable amounts of the gas phase due to dissolution in water. Gas dissolution has a significant impact on the behavior of the oil/gas jet/plume because of its impact on the buoyancy. A method is presented in this paper for computing gas dissolution that covers a broad range of water depth, from shallow water where gases behave as ideal ones under low pressure to deepwater where gases behave as non-ideal ones under high pressures. The method presented also accounts for the spherical and non-spherical shapes of gas bubbles. The gas dissolution computations are validated by comparing the computed results with observed data from previously conducted laboratory experiments. The gas dissolution computation module is then integrated with a model for underwater oil/gas jets/plumes by Yapa and Zheng [J. Hydraul. Res. 35 (5) (1997) 673]. Scenario simulations are presented to show the impacts of gas dissolution on the behavior of jets/plumes. These scenarios show the impact of dissolution on the behavior of the jet/plume. The comparison of results using ideal gas conditions and non-ideal gas conditions is also shown. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Gas dissolution; Oil/gas spills; Deep water
1. Introduction Modern technology has made it economically feasible to explore and produce oil and gas from ultra deepwater wells. For example, in the Gulf of Mexico (GOM), the number of exploratory wells has increased by 70% in the 2 years from 1996 to 1998 (Lane and Labelle, 2000). Current estimates show that future production from installations deeper than 800 m will be 69% by year 2007 (Lane and Labelle, 2000). As the production increases, the potential for an oil/gas spill increases. Major concerns from a deepwater oil/gas spill are fire and toxic hazard to *
Corresponding author. E-mail address: [email protected] (P.D. Yapa). 1 Now at Quantitative Environmental Analysis, LLC, Montvale, NJ 07645, USA.
the people working on the surface installations and loss of buoyancy of ships and any floating installations. For this purpose, it is important to know when and where the gas will surface and how much. To meet these new challenges, spill response plans need to be upgraded. An important component of such a plan would be a model to simulate the behavior of oil and gases, if accidentally released, in the deepwater. Many underwater oil/gas spills can be described as a two-phase plume that consists of gas (e.g., CH4, C2H6, and C3H8) and a liquid mixture of oil and water. In this paper, a plume that consists of gas and liquids will be referred to as a gas plume for simplicity. In deepwater, the high pressure and low temperature makes the behavior of oil/gas spills different from those that occur in shallow water in two respects: (i) gas may be converted to an ice-like solid compound called gas hydrate; (ii) gas bubbles
0924-7963/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 7 9 6 3 ( 0 1 ) 0 0 0 6 7 - 7
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may lose considerable amounts of the gas phase through dissolution during their long journey. These two processes alter the buoyancy of the plume significantly. Therefore, knowing the state of gases as they rise with the plume when predicting the fate of an oil/gas plume released from a deepwater location is very important. Yapa and Zheng (1997) developed a gas plume model to simulate oil/gas spills from underwater accidents. This model was subjected to extensive comparison with data from analytical solutions, laboratory and field experiments (Zheng and Yapa, 1998). The above model was integrated with a farfield oil spill model by Yapa et al. (1999) and used to compare the model results with two field experiments of oil/gas release. Recently, modules to simulate gas hydrate formation and decomposition were integrated with the gas plume model (Zheng and Yapa, 2000b). However, gas dissolution was not taken into account in the above oil/gas spill models. Johansen (2000) developed a comprehensive gas plume model for oil/gas spills in deepwater, in which gas dissolution into seawater was included. The calculation of mass transfer coefficient in Johansen (2000) was based on a correlation for rigid spherical bubbles. Johansen (2000) did not provide details of the gas solubility calculation. Even outside the area of oil/gas spill modeling, research related to gas dissolution in a gas plume is scarce because the residence time of gas bubbles in most gas plume problems is so short that the effect of gas dissolution can be neglected. Wu¨est et al. (1992) developed a gas plume model (including gas dissolution) for the restoration of deep stratified lakes. The gas solubilities in Wu¨est et al. (1992) were calculated from a simple Henry’s law, which would break down if the pressure is raised sufficiently (King, 1969). The mass transfer coefficients in the same paper were calculated from a correlation for small size O2 bubbles, which cannot serve as a general formulation for calculating mass transfer coefficients. Some research on CO2 sequestration in the deep ocean concerns dissolution calculations in plumes (Liro et al., 1992; Morishita et al., 1993; Haugan et al., 1995). However, these dissolution calculations were for liquid CO2 droplets instead of gaseous CO2 bubbles because CO2 in general is
in the liquid phase below 500 m (Liro et al., 1992). As a result, there is no gas bubble expansion mechanism due to pressure decrease with height in CO2 plumes if CO2 is in the liquid phase. Haugan and Drange (1992) proposed a gravity current model, in which gaseous CO2 dissolution was considered, to simulate CO2 disposal at 200 – 400-m depth. A modified Henry’s law capable of handling high pressure was used to calculate the CO2 solubility by Haugan and Drange (1992). The mass transfer coefficient in the same paper was calculated by a correlation for fluid spherical particles from Clift et al. (1978). In deepwater, the behavior of gas is non-ideal due to high ambient pressure. Gas bubbles may experience significant variations in sizes and shapes, due to gas expansion and dissolution. These factors need to be taken into account when estimating solubility and the mass transfer coefficient of gas in water. This paper presents an improved method to calculate solubility and the mass transfer coefficient for gases in water. The formulation for gas solubility is valid for both low and high pressures. The general formulation presented for calculating the mass transfer coefficient can be applied to different gases and different shapes of gas bubbles. Although the methods used here have been used in other disciplines (e.g., chemical engineering), they were not integrated in a way that is easy to use in engineering applications and were rarely applied to modeling environmental hydraulics problems. The possible formation and decomposition of gas hydrates within a plume are not considered in this paper. The presence of salt can slightly decrease the gas solubility in water (Yamamoto et al., 1976) but the effect of salinity on the mass transfer coefficient is not clear. Therefore, salinity effects on the gas dissolution are not considered in this paper. The improved method to calculate gas dissolution is integrated with the model for underwater oil/gas jets/plumes by Yapa and Zheng (1997). Scenario simulations are presented to show the impacts of gas dissolution on the behavior of jets/plumes. Although the work developed in this paper originated in connection with simulating oil/gas spills, it can be useful to other environmental problems, such as CO2 sequestration in the ocean, restoration of eutrophic lakes, and ice control in harbors of cold regions.
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2. Computing gas dissolution In general, the dissolution rate for a gas bubble is calculated by dn ¼ KAðCs C0 Þ dt
ð1Þ
where n = number of moles of gas in a bubble [mol]; K = mass transfer coefficient [m/s]; A = surface area of a gas bubble [m2]; C0 = concentration of dissolved gas [mol/m3]; Cs = saturated value of C0 (i.e., solubility). The key parameters in Eq. (1) are the solubility, Cs, and mass transfer coefficient, K. 2.1. Solubility of gas in water The solubility of gas in water is commonly calculated by the simple Henry’s law: P ¼ Hxl
ð2Þ
where P = gas pressure [MPa]; H = Henry’s law constant [MPa]; xl = mole fraction of dissolved gas in water at equilibrium condition, which can be transferred to solubility, Cs, by Cs xlqw/Mw; qw = water density [kg/m3]; Mw = molecular weight of water [kg/ mol]. The applicability of the simple form of Henry’s law given in Eq. (2) is limited to an ideal gas or low pressure conditions. If the pressure is raised sufficiently (e.g., in the case of deepwater), Eq. (2) is found to break down (King, 1969; Weiss, 1974; Lekvam and Bishnoi, 1997). King (1969) presented a modified form of Henry’s law for the solubility of slightly soluble gases at elevated pressures: f g ¼ Hxl expð106 Pvl =RT Þ
ð3Þ
where f g = fugacity of gas in gas phase [MPa]; vl = partial molar volume of gas in solution [m3/mol]; R = universal gas constant = 8.31 [J/molk]; T = water temperature [K]. Eq. (3) reduces to Eq. (2) as P ! 0. 2.2. Mass transfer coefficient of gas bubbles According to Clift et al. (1978), the mass transfer coefficient of gas bubbles in liquids is dependent on the size and shape of bubbles as well as gas dif-
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fusivity in liquids. Gas bubbles in liquid are generally approximated as spheres for small size bubbles, ellipsoids for intermediate size bubbles, and spherical-caps for large size bubbles. For small bubbles, the system purity also affects the value of the mass transfer coefficient. Therefore, the formulation for a single shape of bubbles, such as that for spherical bubbles, is unable to cover the mass transfer coefficient for a broad range of bubble sizes in deepwater scenarios. We combine the following equations originally developed by Johnson et al. (1969) and Clift et al. (1978), respectively, to form a general formulation for the mass transfer coefficient of bubbles in contaminated liquids. The validity of this integrated approach will be examined later by comparing the computed results with experimental observations. (i) For bubbles with spherical shape (small size range)
UD K ¼ 0:0113 0:45 þ 0:2de
12 ð4Þ
where de = equivalent diameter of bubbles [cm]; D = molecular diffusivity of a gas in a liquid [cm2/s]; U = terminal velocity of bubbles [cm/s], which is determined by using the method described by Zheng and Yapa (2000a). (ii) For bubbles with ellipsoidal shape (intermediate size range) 1
K ¼ 0:065D2
ð5Þ
(iii) For bubbles with spherical-cap shape (large size range) 1
1
K ¼ 0:0694de 4 D2
ð6Þ
Eq. (5) is recommended for diameters greater than 5 mm (Clift et al., 1978). Therefore, the critical diameter for transition from small to intermediate size is taken to be 5 mm. The critical diameter for transition from intermediate to large size and is obtained by solving Eqs. (5) and (6) simultaneously and is about 13 mm.
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2.3. Verification of solubility and mass transfer coefficient of gas bubbles in water In this paper, H and f g are calculated by a correlation for gas in water and the Peng – Robinson equation-of-state as given by Sloan (1997). vl is taken to be 33 and 36.5 cm 3/mol for CO 2 and CH 4, respectively, as suggested by King (1969). Fig. 1 shows the computed CO2 solubility in water at 40 C by using Eq. (2) (simple Henry’s law), computed CO2 solubility by using Eq. (3) (modified Henry’s law), and experimental data (King, 1969), respectively. The comparison shows that the simple Henry’s law approximates the CO2 solubility well until about 20 atm (approximate water depths of 200 m) but deviates significantly after that. Modified Henry’s law calculates solubility well up to 200 atm (approximate water depths of 2000 m). Fig. 2 shows the comparisons between the observed and computed CH4 solubility in water using a simple Henry’s law and a modified Henry’s law. The data are presented in three different blocks because they correspond to different temperatures. The available experimental data (Lekvam and Bishnoi, 1997) are such that at higher temperatures, the pressure range is also higher. Nevertheless, it shows that at higher pressures, the discrepancy between the experimental observations and solubility computed using a simple Henry’s law is larger. In fact, we can see that as the pressure increases, the discrepancy grows. Therefore,
Fig. 1. Comparison between computed and observed solubility of CO2 in water at 40 C.
Fig. 2. Comparison between computed and observed solubility of CH4 in water.
at higher pressures a modified Henry’s law provides better computational values for solubility. The higher pressures in this plot correspond to what is expected in deepwater spills. When all the data are taken together, the fractional standard error for all cases is 0.0625 when computed using a simple Henry’s law and is 0.0049 when computed using a modified Henry’s law. Figs. 3– 5 show the comparison between the values of mass transfer coefficient computed by the present method (i.e., using Eqs. (4) –(6)) and the observed data for different gases and ambient liquids. Fig. 3 shows the comparison between computed values and observed data for O2 in tap water at 25 C. Fig. 4 shows the comparison between the computed values and the observed data for CO2 in an aqueous glycerol solution at 25 C. Fig. 5 shows the comparison between the computed values and observed data for CO2 in tap water. Considering that the same set of equations is used for three cases involving three different combinations of gases and ambient liquids, the results
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Fig. 3. O2 bubbles in tap water at 25 C.
of Eqs. (4) – (6) are in reasonably good agreement with the experimental data. The values of the diffusion coefficient, D, listed in Figs. 3 – 5 are the values suggested by the respective authors who obtained the experimental data.
3. Integration of gas dissolution with jet /plume model The computation of gas dissolution described in the previous section is integrated with the jet/plume hydrodynamic model developed by Yapa and Zheng (1997). This section briefly describes the jet/plume hydrodynamics of the model. The following assumptions relate to the jet/plume hydrodynamics.
Fig. 5. CO2 bubbles in tap water.
(i) The flux of the number of bubbles, N [1/s], is kept constant with height, i.e., bubble coalescence is neglected. (ii) Bubbles are all of uniform size at the opening. If a bubble size spectrum becomes available based on field experiments at a latter time, it is possible to replace the number flux of bubbles N by an array N (r1,. . . ,rn). Then the fate of the individually sized bubbles can be calculated independently. 3.1. Governing equations In a jet or plume that consists of a mixture of oil and gas released vertically, the gas portion can be expected to occupy the inner core (Yapa and Zheng, 1997). Based on assumption (i), the number of bubbles per unit plume height is N/(w + wb), where, w = vertical velocity of liquid part of plume; wb = slip velocity of gas, which is a function of gas bubble size. Zheng and Yapa (2000a) give details of computing wb. Fig. 6 shows a schematic diagram of the oil/gas plume. The number of bubbles in a control volume is given by Nh/(w + wb), where h = height of a control volume. (i) Volume fraction of gas bubbles The volume fraction of gas (e) within the inner gas bubble core can be defined by e¼
Fig. 4. CO2 bubbles in 90.6% aqueous glycerol solution at 25 C.
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l l b
ð7Þ
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acting on the liquid part. The last term is related to the vertical force acting on the gas bubbles. (v) Conservation of heat, salinity and oil mass dðml IÞ dml ¼ Ia dt dt
ð11Þ
where I = a symbol representing a property such as heat content (CpT ), salinity (S), or oil concentration by mass (C). The subscript ‘a’ refers to the conditions in the ambient water. Eq. (11) states that the change of heat, salinity or oil mass in the control volume is due to the input contributed by the entrained mass. (vi) Non-ideal gas law PMg ¼ qb ZRT
Fig. 6. Schematic diagram for an oil/gas plume.
where ql = density of the liquid in a plume [kg/m3]; q = density of the plume bubble-water mixture [kg/ m3]; qb = density of the gas [kg/m3]. (ii) Conservation of liquid mass dml ¼ a Q e dt
Nh dn Mg Dt w þ wb dt
4. Model application
ð9Þ
where Dmb = loss of gas mass due to gas dissolution in the control volume [kg]; Mg = molecular weight of gas [kg/mol]; Dt = time step [s]. (iv) Conservation of momentum d ½ml w þ mb ðw þ wb Þ
dt ¼ wa qa Qe þ ðqa ql Þgpb2 ð1 b2 eÞh þðqa qb Þgpb2 b2 eh
where Z = compressibility factor. In deep water scenarios, the gas can deviate from the ideal gas behavior because of high pressure. Compressibility factor Z is calculated using the method described by Sloan (1997).
ð8Þ
where ml = liquid mass of a control volume [kg] = qlpb2(1 b2e)h; qa = density of the ambient fluid [kg/m3]; Qe = entrainment rate for the ambient water [m3/s]. Details of computing Qe were given in Yapa and Zheng (1997) (iii) Loss of gas mass due to gas dissolution Dmb ¼
ð12Þ
ð10Þ
where wa = vertical velocity of ambient fluid [m/s]. The first term of the right-hand side of Eq. (10) represents the momentum from the entrained liquid mass. The second term is related to the vertical force
In Figs. 1– 5, the dissolution computations were compared with available relevant data. It is desirable to compare the overall dissolution from a plume under field conditions. An extensive search for relevant data in the open literature that can be used for model comparison did not yield any data. There have been a number of observational studies on gas seeps in deepwater (e.g., Sassen and MacDonald, 1997). However, the information found in these papers is qualitative in nature and cannot be compared with the model results. In 2000, the Minerals Management Service and a consortium of oil companies sponsored a large-scale field experiment (DeepSpill), at a cost of US$ 2.5 million, in the North Sea. Although this is the best known study available yet (the data was not in the open literature at the time the paper was written), no direct measurements were taken on gas dissolution that can be used to compare the results of the model presented here. In this section, the model is applied to simulate several deepwater jet/plume scenarios of practical interest. The ambient condition selected has a constant salinity of 35% and linear distributions of temperature
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Table 1 Parameters used in the simulations
as shown in Fig. 7. This simpler ambient condition is preferred over a more complex salinity and temperature distribution from a field location (e.g., Gulf of Mexico) because it is easier to identify the impact of gas dissolution. When a complex temperature or salinity profile is used, the results are indirectly affected by other parametric variations and, hence, the contribution of dissolution becomes unclear. CH4 is common in these situations and is chosen for the application. The phase equilibrium condition for CH4 hydrate is shown by a dotted curve in Fig. 7. In the region below the dotted curve, the pressure and temperature of water meet the thermodynamic conditions necessary for hydrate formation. The crossing point between the dotted curve and the solid line (temperature distribution) in Fig. 7 approximately corresponds to a depth of 700 m. All simulations in this scenario are kept to less than 700 m to avoid possible hydrate formation. If the hydrate formation is included as part of the simulation, the results become more complex and the analysis of the effects of dissolution is harder. The values of model parameters used in the simulations correspond to a medium wellhead blowout and are listed in Table 1. The initial bubble radius was based on the information by Barbosa et al. (1996). A jet or plume of oil/gas mix under water is lighter than the ambient water. As it travels upwards, ambient water is entrained into the jet/plume and the density is increased. Ambient water density decreases upwards in a density-stratified environment. Simulations by Yapa et al. (1999), Zheng and Yapa (1999), Johansen (2000), and field experiments (Rye and Brandvik,
1997) have shown that for deep-water cases, it is most likely that there will be a point at which the density of the jet/plume is approximately equal to the ambient density. This level is referred to as the neutral buoyancy level (NBL). The dynamics of the jet/plume is assumed to cease at this level. Oil and gas will travel beyond this level under their own buoyancy. The validity of this argument was verified in Yapa et al. (1999). Fig. 8 shows the NBL for jets/plumes released at different depths. The NBLs when the gas dissolution is included are lower compared to those when the gas dissolution is excluded. The gas dissolution reduces the mass of free gas, resulting in reduced buoyancy. This results in lowering the NBL. A lower NBL for a plume means that the oil droplets have to travel a longer distance to the water surface under their own buoyancy. The slower rise velocity of oil droplets compared to a plume velocity means that oil will be subjected to dispersion by ambient currents during its journey to the water surface for a longer period. Therefore, a lower NBL can be expected to result in a wider cloud of oil with a lower concentration as compared to a higher NBL. Fig. 8 also shows that
Fig. 7. Water temperature and CH4 hydrate phase diagram.
Fig. 8. Neutral buoyancy levels vs. release depth.
Oil discharge rate — m3/s (bopd) Gas discharge rate — normal m3/s (scfd) GOR — N m3 s 1/m3 s 1 (scfd/bopd) Diameter of the opening — m (in.) Temperature of discharged oil — C (F) Density of discharged oil (kg/m3) Initial gas bubble radius (m)
0.0184 (10,000) 3.2752 (107) 178 (1000) 0.0889 (3.5) 80 (176) 893 6.0 10 3
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Fig. 9. Variation of relative bubble diameter with depth.
Fig. 11. Variation of momentum flux with depth.
simulating gas dissolution based on non-ideal gas (modified Henry’s law, Eq. (3)) or ideal gas (simple Henry’s law, Eq. (2)) does not make a significant difference for the NBL of the jet/plume. Figs. 9 –12 illustrate the different perspectives of the impact of gas dissolution on a gas plume discharged at 700 m. Fig. 9 shows the variation of relative bubble diameter respective to depth. If there is no gas dissolution, the bubble diameter increases during most of its journey due to the drop in hydrostatic pressure, except at the very beginning when the bubble diameter decreases because of the decrease in plume temperature caused by the entrainment. However, if the gas dissolution is included, the bubble diameter decreases as the plume rises because the effect due to gas loss caused by dissolution is more dominant than the increase in bubble diameter due to
hydrostatic pressure drop. Furthermore, the bubble diameter decreases faster with the ideal gas assumption compared to the non-ideal gas assumption because the solubility of ideal gas is greater than that of non-ideal gas under high pressure. Similar behavior can be observed in Fig. 10 that shows the mass fraction of dissolved gas in the gas plume, with and without the ideal gas assumptions, as it moves up from the release point at 700 m. Fig. 11 shows the variation of momentum flux of the gas plume with depth. The momentum flux at a given depth with gas dissolution is less than that without gas dissolution. This is because the gas dissolution consumes free gas and thus reduces the buoyancy force. Since the ideal gas dissolves faster than the non-ideal gas, the momentum flux at a given depth with an ideal gas assumption is less than that with non-ideal gas
Fig. 10. Variation of mass fraction of dissolved gas with depth.
Fig. 12. Variation of buoyancy flux with depth.
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assumption. Fig. 12 shows the variation of buoyancy flux of the gas plume with depth. In a densitystratified environment, the buoyancy flux of a plume decreases as the plume rises due to the entrainment. With the gas dissolution included, the buoyancy flux at a given depth decreases even faster. Ideal gas dissolves faster than the non-ideal gas, the buoyancy flux at a given depth with an ideal gas assumption is therefore less than that with a non-ideal gas assumption. The expressions used for computing the buoyancy flux and the momentum flux are given in the Appendix A.
5. Conclusions In this paper, a method was presented, by combining previous work, to compute the dissolution of gases into water. The formulation for gas solubility covers a broad range of water depth, from shallow water where gases behave as ideal ones under low pressure to deep water where gases behave as non-ideal ones under high pressures. The general formulations for calculating mass transfer coefficient can be applied to different gases and different shapes (e.g., the spherical and nonspherical shapes) of gas bubbles. The calculations were compared with available experimental data. The comparison between the calculations and the observed values was good. The module for computing dissolution was then integrated with the hydrodynamics of a jet/plume that consist of oil and gas by modifying the model by Yapa and Zheng (1997). The purpose of the model was to investigate the effect of dissolution on the fate of the jet/plume, especially when released in deepwater. The possible formation and decomposition of gas hydrates within a plume are beyond the scope of this paper and not addressed. Scenario simulations are presented to show the impacts of gas dissolution on the behavior of plumes, such as neutral buoyancy level (NBL) of the plume, relative bubble diameter, mass fraction of dissolved gas, plume momentum flux, and plume buoyancy flux. Dissolution affects the gas bubble diameter. The simulations show the mass loss due to dissolution affects the gas bubble size more than the expansion due to reduction in pressure as gas travel upwards. The change in gas bubble size is somewhat depen-
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dent on the choice of ideal gas equations or nonideal gas conditions. The momentum flux (MF), buoyancy flux (BF), and NBL are affected by noticeable amounts due to dissolution. However, momentum flux, buoyancy flux, and NBL do not depend in a significant way on the use of ideal gas equations or non-ideal gas conditions. This is because the changes caused to the plume by the loss of gas mass are somewhat compensated by the changes in ambient water entrainment brought upon by the change in plume velocity. When mass is reduced due to dissolution, it changes the velocity of the jet/plume. This in turn changes the entrainment of ambient water. Therefore, how the dissolution affect MF, BF, and NBL are complex processes that are difficult to visualize. Therefore, the model results are valuable in understanding the impact of dissolution. The work developed here can be applied to other environmental problems related to gas plumes, such as CO2 sequestration in oceans, restoration of eutrophic lakes, and ice control in harbors of cold climate regions.
Acknowledgements The work presented in this paper originated during the development of a numerical model to simulate the oil and gas spills from ultra deepwater. The project was supported by the Minerals Management Service (MMS) of the United States Department of the Interior. The authors would also like to thank the Deep Sea Task Force (DSTF) for the sponsorship of the project. The results represent the views of the authors and not the Minerals Management Service.
Appendix A. Expressions for momentum and buoyancy flux Momentum flux ðMFÞ ¼ pb2 w2 Pe ð1 k2 eÞ þ pb2 ðw þ wb Þ2 Pb k2 e
Buoyance flux ðBFÞ ¼ pb2 wgðPa Pmix Þ
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Appendix B. Notation The following symbols are used in this paper: A C0 Cs de D fg H I K Mg ml Dmb n P Qe R T Dt U vl w wa wb xl Z e q qa qb ql
surface area of a gas bubble [m2] concentration of dissolved gas [mol/m3] saturated value of C0 (i.e., solubility) equivalent diameter of bubbles [cm] molecular diffusivity of a gas in a liquid [cm2/s] fugacity of gas in gas phase [MPa] Henry’s law constant [MPa] a symbol representing heat content, salinity, or oil concentration by mass a mass transfer coefficient [m/s] molecular weight of gas [kg/mol] liquid mass of a control volume [kg] loss of gas mass due to gas dissolution in the control volume [kg] number of moles of gas in a bubble [mol] gas pressure [MPa] entrainment rate for ambient water [m3/s] universal gas constant = 8.31 [J/molk] water temperature [K] time step [s] terminal velocity of bubbles [cm/s] partial molar volume of gas in solution [m3/mol] vertical velocity of liquid part of plume vertical velocity of ambient fluid [m/s] bubble slip velocity mole fraction of dissolved gas in water at equilibrium condition compressibility factor volume fraction within the bubble core density of plume bubble-water mixture [kg/m3] the density of ambient fluid [kg/m3] density of gas [kg/m3] density of the liquid part of a plume [kg/m3]
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