6th International Conference on Gas Hydrates


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* Corresponding author: Phone: +1 303 273 3237 Fax +1 303 273 3730 E-mail: ckoh@mines.edu   1 A NOVEL APPROACH TO MEASURING METHANE DIFFUSIVITY THROUGH A HYDRATE FILM USING DIFFERENTIAL SCANNING CALORIMETRY  Simon R. Davies, Jason W. Lachance, E. Dendy Sloan, Carolyn A. Koh* Center for Hydrate Research Colorado School of Mines, CO 80401 USA  ABSTRACT The  avoidance  of  hydrate  blockages  in  deepwater  subsea  tiebacks  presents  a  major  technical challenge with severe implications for production, safety and cost. The successful prediction of when and where hydrate plugs form could lead to substantial reductions in the use of chemical inhibitors,  and  to  corresponding  savings  in  operational  expenditure. The diffusivity  of  the  gas hydrate former (methane) or the host molecule (water), through a hydrate film is a key property for such predictions of hydrate plug formation.  In this paper, a novel application of Differential Scanning Calorimetry is described in which a hydrate film was allowed to grow at a hydrocarbon-water interface for different hold-times. By determining  the  change  in  mass  of  the  hydrate  film  as  a  function  of  hold-time,  an  effective diffusivity  could  be  inferred.  The  effect  of  the  subcooling,  and  of  the  addition  of  a  liquid hydrocarbon layer were also investigated. Finally, the transferability of these results to hydrate growth from water-in-oil emulsions is discussed.   Keywords: Diffusivity, DSC, Hydrate Film Growth   NOMENCLATURE C  Dimensionless methane concentration in the aqueous phase [-] CA Methane concentration in the aqueous phase [kg m-3] CAi Methane concentration in the aqueous phase in equilibrium with hydrate [kg m-3] CA0 Initial concentration of methane in the aqueous phase before hydrate formation [kg m-3] DA Diffusivity of methane in water [m2 s-1] Film Thickness of the hydrate film [m] iend Total number of grid points [-] m Mass of methane in the hydrate film [kg] n An integer [-] t Time [s] x Distance [m] X Dimensionless distance [-] Z A function depending only on x  ?  Thickness of the hydrate film and water layer [m] ?  Porosity of hydrate layer [m3m-3] ? 0 Initial porosity of hydrate layer [m3m-3] ?  A function depending only on t ?  A constant [-] ?  Pi [-] ? Dummy variable for integration [s]   INTRODUCTION As  the  oil  and  gas  industry  moves  into  deeper water  and  corresponding  higher  pressures  from larger liquid heads, the risk of hydrate formation is increasing. Recent research has focused on the rate of formation of a hydrate plug, in addition to the traditional  research  on  hydrate  avoidance.  The    2 conceptual picture for hydrate formation in water-in-oil (W/O) emulsions is shown in Figure 1 [1].  There  are  two  critical  interrelated  steps  in  the formation  of  a  plug:  hydrate  growth  and  hydrate agglomeration. Hydrate growth is the focus of this paper.    Figure 1: A Conceptual Picture for Hydrate Formation in Water-in-Oil (W/O) Emulsions  Upon  nucleation,  a  hydrate  film  rapidly  forms around  the  water  droplets;  this  process  is  limited by heat transfer, both for CO2-water systems  [2],  [3] and for methane-water systems [4]. The various heat transfer  models  have  been  summarized  by Mochizuki and Mori [5]. After the film has formed completely, the growth rate transitions from a heat transfer  limited  process  to  a  process  limited  by mass transfer [2].   Guest molecules initially dissolved inside the water droplet  can  migrate  to  the  hydrate  film,  as  can water initially dissolved in the oil phase. At longer timescales the hydrate formation rate is limited by the mass transfer of the water [6] or guest molecule [7] through the hydrate film.  In this paper High Pressure Differential Scanning Calorimetry  (HP-DSC)  is  applied  to  study  mass transfer limitations to hydrate formation. A hydrate film was allowed to grow at a hydrocarbon-water interface  for  different  hold-times.  Assuming  that the growth rate was controlled by the transport of methane across the  film and using a quasi-steady state  approximation (neglecting  accumulation  and hydrate growth inside pores or fissures within the film), the change of hydrate mass with time could be  measured  and  used  to  infer  an  effective diffusivity.  The effect of the subcooling, and of the addition of a liquid hydrocarbon layer were also investigated. The  transferability  of  these  results  was  tested  by comparison  to  the  hydrate  growth  in  a  similar experiment with water-in-oil emulsions.  EXPERIMENTAL METHODOLOGY Differential  Scanning  Calorimetry  (DSC)  is  a widely  used  technique  to  study  the  thermal properties of a sample [8]. A Differential Scanning Calorimeter  consists  of  a  sample  cell  and  a reference cell.   During  an  experiment,  both  cells  are  heated  or cooled according to a pre-programmed temperature profile.  The  difference  in  heat  flux  required  to obtain  a  zero  temperature  difference  between  the sample  and  reference  cells  can  be  analyzed  to determine  the  thermal  properties  of  the  sample, such  as  specific  heat  capacity,  or  the  latent  heat associated with phase changes  In this paper, a High Pressure DSC was used (HP-micro  DSC  VII,  Setaram  Instruments,  France). Ultra high purity methane (> 99.99 %,Airgas Inc.) and  chromatography  grade  water  (Aldrich  Inc.) were used.  The  experimental  procedure  is  summarized  in Figure  2.  The  DSC  cell  was  first  loaded  with  a known  mass  of  water.  The  cell  was  then  sealed, placed in the DSC and pressurized with methane. Pressures  ranging  from  30  to  150  bar  have  been investigated.  A  saturation  period  of  3  hours  at 30?C was used to ensure that the water was fully saturated before hydrate formation occurred.   The sample was then cooled to an isotherm (-5 or -10?C),  and  hydrate  was  allowed  to  nucleate  as indicated  by  an  exotherm.  Following  a  pre-determined  hold  period,  the  sample  was  then heated to dissociate the hydrate, as indicated by an endotherm. The exothermic and endothermic peaks could both be integrated to determine the mass of hydrate  after  the  initial  growth  period  and  at  the point of dissociation.  In order to understand the time dependence on the effective  diffusivity,  a  series  of  identical experiments  were  performed  with  different  hold times. The  mass  of  hydrate  as  a  function of  time could  then  be  determined  from  the  full  series  of experiments.       3    Figure 2: A Schematic Diagram Showing the Experimental Procedure for the HP-DSC Experiments   A typical thermogram for the DSC experiments is shown  in  Figure  3.  The  mass  of  hydrate  initially formed,  was  found  by  integrating  the  exothermic formation  heat-flux  peak,  and  then  dividing  the energy by the latent heat of methane hydrate (446 J/g).  The  same  procedure  was  followed  for  the dissociation peak. See the thesis by Lachance [9] for a  more  detailed  explanation  of  the  experimental procedure.  Since the formation conditions were below the ice point, ice formed instead of hydrate in some of the experiments.  Ice  nucleation  was  indicated  by  a much  larger  formation  exotherm;  all  of the  water phase converted to ice compared to only the gas-water interface  for  hydrate  formation.  In  addition the  melting  endotherm  would  occur  at  0?C  on reheat. These results were omitted from this study. If ice did not nucleate before the hydrate then no ice  formation  occurred  during  the  growth  period. The presence of ice would have been indicated at the end of the experiment by an endotherm at 0?C during the reheat stage.  The hydrate nucleation at -10?C generally occurred earlier than at -5?C due to the larger subcooling [10]. In some experiments, the hydrate nucleated during the  cooling  step.  The  results  from  these experiments  were  omitted  from  this  study  due  to the  poorly  defined  formation  temperature  which may have influenced mass transfer across the film.     Figure 3: A Typical Thermogram from a Diffusivity Experiment in the HP-DSC   THEORY A conceptual picture for the growth and annealing of the hydrate film is presented in Figure 4. From the experimental data it is apparent that the mass transfer resistance varies non-linearly with the film thickness.  This  suggests  that  mass  transfer resistance increases as the hydrate ages.   Initially it is proposed that the hydrate forms as a homogeneous  layer  with  a  given  porosity  (? 0). Over time the film porosity decreases as the pores fill.     Figure 4: A Conceptual Picture of Hydrate Film Growth and Annealing (blue shading: water/gas system, grey shading: hydrate film)   There  are  two  sources  of  methane  for  hydrate formation: the free water phase and the gas phase. Dissolved  methane  will  diffuse  from  the  water phase,  initially  saturated  with  methane,  to  the hydrate film over time. At the same time, methane from  the  free  gas  phase  and  water  from  the aqueous  phase  will  diffuse  through  the  hydrate film. All of these processes will cause pore filling and film growth. A schematic representation of the free  methane  concentration  profile  is  shown  in Figure 5.    4    Figure 5: A Schematic Representation of the Free Methane Concentration Profile   The  diffusivity  of  methane  in  water  at  4?C  is approximately 8.5 x 10-10 m2/s [11]. The equilibrium methane concentrations in the aqueous phase with and without a hydrate phase were estimated using Multiflash  with  the  Association  Model  (CPA-Infochem); the estimated concentrations are listed in Table 1.     Table 1: Methane Concentrations in the Aqueous Phase as Estimated by Multiflash   As  already  mentioned,  there  are  two  sources  of hydrate formers to be considered: methane already dissolved in the water phase, and hydrate formers being transported across the hydrate film. In order to de-convolute the contribution to the film growth of  dissolved  methane  from  the  mass  transfer  of water  molecules  or  gas  molecules  across  the hydrate  film  it  is  necessary  to  first  model  the methane  diffusion  in  the  free  water  phase.  One-dimensional  Fick?s  Law  in  rectilinear  coordinates applies  (Equation  1).  Two  solution  procedures were  applied:  an  analytical  solution,  and  a numerical solution.               Equation 1  Boundary Conditions:      Initial Condition:    Analytical Solution for the Free Water Phase The partial differential equation (PDE) in Equation 1  can  be  split  into  two  ordinary  differential equations (ODE?s)  using  the  product  method; the ODE?s  can then be solved with a Fourier series [12].   Since  the  closed  boundary  condition  is problematic,  it  is  convenient  to  solve  the symmetrical  problem  (twice  the  distance): . Only half of the solution is utilized. Using the dimensionless variable C, the revised initial conditions and boundary conditions are as follows:      Where:       The resulting ODEs from the product method are shown in Equation 2 and 3 where  the  term  ?  is  a constant  and  Z  and  ?  are  functions  depending  on only one variable.                Equation 2                 Equation 3     5 The solution of the PDE is provided by the Fourier series as shown in Equation 4.                      Equation 4   Numerical Solution for the Free Water Phase The  numerical  solution  of  the  PDE  was  found using the forward difference method [13]. The PDE is expressed in standard explicit form in Equation 5.        Equation 5      The  closed  boundary  condition  was  imposed  by equating the two grid points at the bottom of the cell:   The  comparison  of  the  analytical  solutions  for 1000 steps to the numerical solutions is shown in Figure  6.  The  agreement  between  both  models  is excellent. The Gibbs Phenomenon [14] is evident in the analytical solution at 0 minutes.     Figure 6: Comparison of the Analytical Solutions for 1000 Steps to the Numerical Solutions  RESULTS AND DISCUSSION Before  initiating  experiments  in  the  DSC  it  was necessary  to  pre-saturate  the  water  phase  with methane. The saturation was achieved by holding the pressurized DSC cell at 30?C for three hours in a  methane  atmosphere  prior  to  cooling  to  the hydrate formation conditions.  Before the saturation procedure was implemented the  mass  of  hydrate  formed  in  the  nucleation exotherm was stochastic. However, when the water had  been  pre-saturated,  the  results  were reproducible: ?   Standard  Deviation.  =  2  J/g  for  different samples ?   Standard  Deviation  =    0.56  J/g  for  the same sample  Table  2  compares  the  experimental  data  obtained for various saturation periods. Less hydrate formed in the experiments before the saturation period of three hours was implemented. For the experiments given in Table 2, hydrate was formed at -5?C.     Table 2: Comparison of Experimental Results Before and After the Pre-Saturation Procedure was Implemented  In order to study the mass of hydrate as a function of  time,  a  series  of  repeat  experiments  were performed  for  different  hold-times.  The  mass  of hydrate at the end of each experiment  was found from  the  dissociation  endotherm.  Two  such experimental  series  are  plotted  in  Figure  7  for isotherms of -5?C and -10?C.   The predicted mass of hydrate versus time for an impermeable  hydrate  layer  (methane  diffusing  to the hydrate from the bulk water only) is plotted in the  same  figure  for  comparison.  The  mass  of hydrate  formed  from  dissolved  methane  was calculated  by  integrating  the  mass  transfer  rate Time    6 (Equation 6) over time at the hydrate boundary, as shown  Equation  7.  The  concentration  gradient  in the  x  direction  was  found  by  differentiating Equation 4.               Equation 6               Equation 7  In  the  case  of  the  experimental  series  at  -10?C most of the hydrate formation could be attributed to methane that was initially dissolved in the water phase;  little  methane  appears  to  have  diffused through the hydrate film in the timescale of these experiments. However for the experimental series at  -5?C,  the  mass  of  hydrate  formed  was significantly  higher  than  could  be  attributed  to dissolved  methane  alone;  the  film  was  more permeable to hydrate formers in this case.   Figure 7: Mass of Hydrate Formed Versus Time at Various  Subcoolings  Compared  to  the  Mass  of Hydrate Formed from Dissolved Methane in Water   The addition of a liquid hydrocarbon layer on top of the water layer was found to greatly reduce the rate of hydrate formation, either as a result of the additional  mass  transfer  resistance  to  gas molecules, or due to a less permeable hydrate film. In  the  experiments  with  a  hydrocarbon  layer,  the hydrate  growth  could  again  be  explained  by methane  dissolved  in the water layer  (See  Figure 8).     Figure 8: Mass of Hydrate Formed Versus Time for a West African Crude Layer above the Water Phase Compared to the Mass of Hydrate Formed from Dissolved Methane   A  greater  mass  of  hydrate  formed  at  -5?C compared to -10?C. This is at first thought counter-intuitive  and  in  contrast  to  previous  work  that shows  increased  hydrate  growth  with  higher subcooling  [15].    Figure  9  illustrates  the  hydrate growth at several subcoolings using the DSC.  Figure 9: Hydrate Conversion with Various Subcoolings at Different Time Periods   The  DSC  results  in  Figure  9  confirm  previously reported [15] trends, but additionally illustrate that at subcoolings greater than 20?C, the hydrate growth is impeded possibly due to the faster annealing of the  film.    This  phenomenon  was  also  found  to occur in water-in-crude oil emulsions (Figure 10).       7  Figure 10: Hydrate Conversion in Water-In-Conroe Crude at Different Isotherms. Inset: Microscopic image of water-in-oil emulsion.   Conroe  crude  oil  was  used  in  the  emulsion  tests due to the large droplet sizes (Figure 10 inset) that did  not  fully  convert  to  hydrate  upon  nucleation.  Figure  10  shows the  hydrate  growth  for  different isotherms. Again, higher conversion was achieved at the -5?C isotherm than at the -10?C isotherm, as in Figure 7.  However, the 0?C isotherm begins to follow the expected literature [15]  trend that higher subcoolings  result  in  more  hydrate  growth  (cf. Figure 9 up to subcoolings of 20?C).  Estimating the Effective Diffusivity In the film growth experiments (Figures 7 and 8), hydrate was allowed to form at the water interface. In the case of the experiment performed at -10?C, the mass of hydrate formed could be explained by methane  initially  dissolved  in  the  water  phase. However at -5?C, the hydrate growth was in excess of  that  which  could  be  explained  by  dissolved methane.  Assuming  that  the  difference  in  the hydrate  growth  in  these  experiments  could  be attributed  to  the  transport  of  methane  across  the film, an effective diffusivity could be regressed to the experimental data.  A quasi-steady state approximation  was made for these calculations; accumulation of methane in the film was neglected, the thickening of the film and hydrate growth inside pores or fissures within the film was expected to be slow in comparison to the mass  transport  rate  across  the  film.  The  one-dimensional  steady  state  Fick?s  Law  (Equation  6) therefore applied.  The  mass  of  hydrate  formed  is  the  sum  of  that formed  from  dissolved  methane  and  that  formed from  the  mass  transfer  of  methane  across  the hydrate film. The effective diffusivity of methane across  the  film  was  unknown  and  could  be regressed to the experimental data using the least squares  method  (See  Figure  11).  The  regressed effective  diffusivity  for  the  experiments  at  -5?C was  7.6x10-13  m2/s;  for  the  experiments  at  -10?C the effective diffusivity was 3.4 x 10-13 m2/s.   However,  it  is  apparent  that  the  experimental observations  and  the  predicted  hydrate  growth rates  show  markedly  different  trends.  At  short timescales  the  predictions  under-predict  and  at longer  timescales  the  predictions  over-predict  the hydrate  growth  rate.  It  appears  that  the  effective diffusivity  is  time  dependent,  dropping  off  over time as the film anneals. This result coincides with the  conceptual  picture  for  hydrate  film  formation presented in Figure 4.     Figure 11: Mass of Hydrate Formed Versus Time for Various Subcoolings Compared to the Mass of Hydrate Formed from Dissolved Methane and the Best Fit Diffusivity through the Shell   CONCLUSIONS Less permeable hydrate films form in the presence of a liquid hydrocarbon layer. The hydrate growth measured  in  these  DSC  experiments  could  be explained  by  methane  initially  dissolved  in  the aqueous phase.  The growth rate of hydrate was found to increase with sub-cooling until the temperature was reduced below  approximately  -5?C  (20?C  subcooling);  at lower  temperatures  the  growth  rate  was  reduced. -10?C  -5?C    8 This  effect  has  been  observed  for  pure  water systems and for emulsified water-in-oil emulsions.  The  effective  diffusivity  of  methane  through  the hydrate shell was shown to decrease with time as the hydrate layer anneals and pores and fissures in the film fill.   ACKNOWLEDGEMENTS The  authors  wish  to  acknowledge  the  financial support  received  from  the  CSM  Hydrate Consortium of energy companies: BP, Champion, Chevron,  ConocoPhillips,  ExxonMobil, Halliburton,  Petrobras,  Schlumberger,  Shell,  and StatoilHydro.   REFERENCES [1]  Sloan  E.D.  and  Koh  C.A.  Clathrate Hydrates  of  Natural  Gasses,  3rd  Ed., Chemical  Industries  119,  CRC  Press, Taylor  and  Francis  Group,  Boca  Raton, 2008. [2]  Uchida  T.,  Ebinuma  T.,  Kawabata  J., Narita  H.    Microscopic  Observations  of Formation Processes of Clathrate-Hydrate Films  at  an  Interface  between  Water  and Carbon  Dioxide.  Journal  of  Crystal Growth, 1999; 204: 348-356. [3]  Mori  Y.H.  Estimating  the  Thickness  of Hydrate Films from Their Lateral Growth Rate  Using  a  Simplified  Heat  Transfer Model.  Journal  of  Crystal  Growth,  2001; 223: 206-212. [4]  Freer  E.M.,  Selim  M.S.  and  Sloan  E.D. Methane  Hydrate  Film  Growth  Kinetics. Fluid Phase Equilibria, 2001; 185: 65-75.  [5]  Mochizuki  T.  and  Mori  Y.H.  Clathrate Hydrate  Film  Growth  Along  Water  / Hydrate  -  Former  Phase  Boundaries  ?   A Conductive  Heat-Transfer  Model. Proceedings  of  the  5th  International Conference on Gas Hydrates, 1009, Vol. 1, pp64-74, Trondheim, Norway, 2005.      [6]  Mori  Y.  and  Mochizuki  T.  Modeling  of Mass  Transport  across  a  Hydrate  Layer Intervening  Between  Liquid  Water  and ?Guest?  Fluid  Phases.  Proceedings of the 2nd  International  Conference  on  Gas Hydrates,  pp267-8274,  Toulouse,  France, 1996. [7]  Turner D.J. Clathrate Hydrate Formation in Water-in-Oil Dispersions.  PhD Thesis, Colorado School of Mines, 2005. [8]  Dean,  John  A.  The  Analytical  Chemistry Handbook. New York. McGraw Hill, Inc. pp. 15.1? 15.5, 1995. [9]  Lachance  J.W.  Investigation  of  Gas Hydrates  Using  Differential  Scanning Calorimetry With Water-In-Oil Emulsions. Masters  Thesis,  Colorado  School  of Mines, 2008. [10]  Hester K.C., Davies S.R., Lachance J.W., Sloan  E.D.  and  Koh  C.A.  Hydrate Nucleation  Measurements  using  High Pressure  Differential  Scanning Calorimetry.  Proceedings  of  the  6th International Conference on Gas Hydrates, Vancouver, Canada, 2008. [11]  Witherspoon  P.I.  and  Bonoli  L. Correlation  of  Diffusion  Coefficients  for Parafin,  Aromatic  and  Cycloparafin Hydrocarbons  in  Water.  I  &  EC Fundamentals. 1969; 8(3): 589-591. [12]  Kreyszig  E.  Advanced  Engineering Mathematics.  8th  Edition,  New  York. Wiley Inc. p587-593. 1997. [13]  Von  Rosenberg  D.U.  Methods  for  the Numerical Solution of Partial Differential Equations.  Modern  Analytic  and Computational  Methods  in  Science  and Mathematics.  No.  16,  American  Elsevier Publishing Company Inc. New York, p16-20, 1969. [14]  Thompson  W.J.  Fourier  Series  and  the Gibbs  Phenomenon.  American  Journal  of Physics, 1992; 60(5): 425-429. [15]  Taylor  C.J.  Adhesion  Force  Between Hydrate  Particles  And  Macroscopic Investigation  Of  Hydrate Film  Growth  At The  Hydrocarbon  /  Water  Interface.  Masters  Thesis,  Colorado  School  of Mines, 2006.  


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