International Conference on Gas Hydrates (ICGH) (6th : 2008)

A MODELING APPROACH TO HYDRATE WALL GROWTH AND SLOUGHING IN A WATER SATURATED GAS PIPELINE Nicholas, Joseph W.; Inman, Ryan R.; Steele, John P.H.; Koh, Carolyn A.; Sloan, E. Dendy Jul 31, 2008

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A MODELING APPROACH TO HYDRATE WALL GROWTH AND SLOUGHING IN A WATER SATURATED GAS PIPELINE  Joseph W. Nicholas, Ryan R. Inman, John P.H. Steele, Carolyn A. Koh, and E. Dendy Sloan*  Center for Hydrate Research Department of Chemical Engineering Colorado School of Mines 1500 Illinois Street Golden, CO 80401 USA  ABSTRACT A hydrate plugging and formation model for oil and gas pipelines is becoming increasingly important as producers continue to push flow assurance boundaries.  A key input for any hydrate plugging model is the rate of hydrate growth and the volume fraction of hydrate at a given time. This work investigates a fundamental approach toward predicting hydrate growth and volume fraction in a water saturated gas pipeline. This works suggests that, in the absence of free water, hydrate volume fraction can be predicted using a wall growth and sloughing model.  Wall growth can be predicted using a one- dimensional, moving boundary, heat and mass transfer model.  It is hypothesized that hydrate sloughing can be predicted when a coincident frequency exists between hydrate natural frequency and flow induced vibrations over the hydrate surface. Keywords: Flow assurance, deposition, wall growth, sloughing, natural frequency   NOMENCLATURE nt Diffusive water flux through the deposit nt Total water flux from fluid to deposit wTemperature of pipe wall sSurface temperature of deposit ∞Temperature of bulk fluid qconv Convective heat transfer from fluid to deposit qcond Conductive heat transfer through the deposit qlat Conductive heat transfer through the deposit due to latent heat svh Density water vapor or dissolved water ∞ Density of bulk fluid Hydrate thickness fw  Wake Shedding Frequency S  Strouhal Number Lc Cavity Length U  Flow Velocity b  Cavity Width h  Cavity Height Lh Length of Hydrate hh  Height of Hydrate T   Temperature of Gas P   Pressure of Gas f10  Radial Acoustic Resonance Frequency f1    Fundamental Natural Frequency    INTRODUCTION Clathrate hydrates have hindered the oil and gas industry since 1934, when Hammerschmidt discovered hydrates were capable of plugging pipelines.  Clathrate hydrates are crystalline inclusion compounds wherein hydrogen bonded water molecules form cages containing guest molecules.  Hydrates form at high pressures * Corresponding author: Phone: +1 303 273 3723 Fax: +1 303 273 3730 E-mail: Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. and low temperatures, similar to conditions on the seafloor; because hydrocarbons act as guest molecules, offshore pipelines are prime candidates for hydrate formation and plugging.  In addition to lost production and revenue, hydrates pose a significant safety hazard[1].  In recent years, several research groups have investigated hydrate plugging mechanisms in pipelines, with an emphasis on oil systems.  In a separate paper in these proceedings (Nicholas et al.) our experimental evidence suggests gas- dominated hydrate formation and plugging mechanisms may differ from an oil pipeline[2].  This lab recently investigated hydrate deposition in a water saturated condensate flowloop and determined that hydrates will form on a pipe wall when the fluid cools below the hydrate equilibrium temperature[2].  This work presents a conceptual approach to modeling hydrate deposition and sloughing in water saturated gas pipelines.  MODELING APROACH Because pipe walls are the coldest point in the system and are widely believed to be water wet, they provide an excellent nucleation and growth site if hydrate formation conditions are met.  The current conceptual model is divided into three steps as shown in Figure 1.  Saturated V/L HC Hydrate Wall growth model Sloughing model Plugging model  Figure 1. Diagram of the three steps that occur once wall growth begins.  This modeling approach is designed to predict the volume fraction of hydrates in a water saturated hydrocarbon (HC) pipeline. The model is applicable in either a vapor (V) or liquid (L) system.  The wall growth portion of the model predicts how fast the hydrate deposit grows and matures.  This information then feeds into a sloughing model, which predicts when the hydrate deposit will collapse or be removed from the pipe wall.  The sloughing model yields a volume fraction of hydrate which will serve as a key input to any hydrate plugging model. Wall Growth Model The wall growth model applied in this work is similar to previous frost growth studies[3- 5].  The model accounts for the change in hydrate height, hydrate temperature profile, and hydrate volume fraction (h).  Figure 2 shows the mass and energy balances applied in this model.  xw s  nd(w)=0 nt nd ∞ vsh(Th) ∞ s T(w)=Tw qconv qlat qtotal Control Volume h hydrate vol frac g gas vol frac  Figure 2. Diagram of mass and energy balances across hydrate layer and a sample control volume. This approach is similar to the Le Gall et al. frost model[4]. Figure 2 illustrates the frost height (mass transfer (n), water vapor density (vsh), temperature (T), and heat transfer (q).  The bulk frost layer is divided into control volumes containing a hydrate volume fraction (h) and a gas volume fraction (g). In this model, liquid condensate and gas are used interchangeable with each other; additionally dissolved water and water vapor are used for their respective systems. Total mass transfer of water vapor to the hydrate slab is based on convective mass transfer (nt).  Within the hydrate itself, water vapor diffuses (nd) through the slab and converts to hydrate, thereby decreasing gas volume fraction and increasing hydrate volume fraction.  Dissolved water concentration is used as the mass transfer driving force.  These vapor concentrations are a function of temperature and assumed to be in thermodynamic equilibrium with the surrounding hydrate.  The difference between convective mass flux and diffusive mass flux at the hydrate surface (s) is responsible for the increase in frost height. The energy balance is analogous to the mass balance, where energy is transferred from the bulk fluid (∞) to the surface via convection (qconv).  Formation of hydrate is accompanied by a latent energy term (qlat) – due to exothermic hydrate formation.  This convective and latent energy must be conducted through the hydrate deposit or else it results in a temperature increase within the hydrate or fluid outside the hydrate. The mass and energy balances in Figure 2 result in a set of coupled, second order differential equations, which can be solved numerically. Hydrate sloughing model Our model assumes that sloughing will occur when there is a coincidence between the wake shedding frequency and hydrate natural frequency.  If the coincidence happens near an acoustic resonance frequency, the process would be more likely to occur.  The inputs to this model are hydrate density, elastic modulus, flow velocity, gas properties and pipeline dimensions.  The final model will output the volume of hydrate present at the time of sloughing for incorporation into a jamming model.   Figure 3. Schematic of hydrate sloughing concepts: acoustic resonance, wake shedding, and hydrate natural frequency.  Previous literature indicates that the acoustic resonance is the most important vibration excitation mechanism in gas pipelines[6]. Acoustic resonances occur when sound waves constructively interfere in a cavity to make a standing wave.  These resonances can occur in both the longitudinal and radial directions.  The frequency of the resonance depends on the gas properties and pipe dimensions.  Amplification occurs for vibrations at frequencies near the acoustic resonances[7]. This model focuses on the radial resonance frequencies because the longitudinal frequencies are too low to be excited by fluid vibrations.  The source of vibrations in the model is a phenomenon known as wake shedding.  Wake shedding occurs downstream of cavities, or objects, in cross flow as illustrated in Figure 4 and has been shown to follow Equation 4. c w L SU f   (4)   U L c  h U b L h  Figure 4.  Diagram of Wake Shedding where: fw is the shedding frequency, U is the fluid velocity, Lc is the cavity length, and S is the Strouhal number.  The Strouhal number is an experimentally-determined parameter based on the dimensions of an obstruction or a cavity.  In this case, the Strouhal number ranges between 0.6 and 0.8, and is a function of the cavity width b and the cavity depth h as indicated in Figure 4 [8]. As the hydrate accumulates along the wall of the pipe, local imperfections allow wake shedding to occur as the fluid flows past the hydrate.  When this wake shedding frequency equals the acoustic resonance of the pipe, amplification of the wake occurs. Initial calculations have shown that a coincidence between wake shedding frequency and acoustic resonance can occur for reasonable values of cavity length (see Figure 5). As the fissures propagate through the hydrate, unique macroscopic structures U “Cantilevered Hydrate”Lc Lh dh fws Pipe Wall Hydrate pipe d (macro-structures) of hydrate become isolated from the whole of the annulus, as in the circled “cantilevered hydrate” in Figure 3.  These macro-structures have a natural frequency of vibration dependent on geometry, and the density and elastic modulus of the hydrate.  If these macro- structures are excited at their natural frequency, they will vibrate.  If the isolated macro-structures vibrate long enough, then they will break off from the large structure (i.e. slough). As a first approach, these macro-structures are modeled as cantilevered beams so their frequencies can be found by mathematical analysis or a computer model[9].  For a given frequency, there are a variety of macroscopic hydrate geometries that have the same natural frequency.  Thus, sloughing can occur for a variety of macro-structures. Figure 5 is an example of conditions that would predict sloughing under the current model.  Figure 5. Example of sloughing condition with current model.  The fluid flows pass the cavity and experiences wake shedding near the acoustic resonance, which causes the cantilevered hydrate to vibrate (dotted lines). In a 12” diameter pipe at 3000 psi and 4 ºC, with a 1.1” thick layer of hydrate, the radial acoustic resonance in free methane gas is 1292 Hz from calculation.  A matching wake shedding frequency requires a cavity length of about 0.195 in.  This is a possible size for a cavity that could form in the pipe as fissures propagate through the hydrate. Figure 9 shows a cantilevered square cross- section hydrate (fn = 1292 Hz) with a length of 3.81” and a side height of 1.0”.  Other geometric combinations that would result in sloughing, using the previously listed pipeline conditions, are (Lh, hh) = (2.72”, 0.5”) and (1.93”, 0.25”).  Again, these dimensions are possible inside a pipe of this size.  EXPERIMENTAL We conducted experiments aimed at measuring the hydrate natural frequency. An ice/hydrate (IH) annulus is frozen into a 1.5” diameter schedule 40 pipe.  An excitation hammer was used to provide an impulse-like excitation to the system and the resulting vibrations were measured by accelerometers. The accelerometer’s time domain signals were then transformed to the frequency domain to find the natural frequencies of the system.  The natural frequencies of the IH annulus system were compared to the natural frequencies of pipe without IH annulus.  This analysis is expected to identify the effect of IH growth on the natural frequencies of the pipe.  These natural frequencies can then be compared to the theoretical acoustic resonance frequencies of the pipe. The experimental focus of the sloughing model was to predict the natural frequencies of the hydrate structures that form in the pipeline.  The experimental setup is shown in Figure 6.  It consists of a 1.5 inch schedule 40 pipe that is about 20 inches long. Two accelerometers were attached to the pipe, one was oriented in the radial direction and the other in the axial direction. The brass excitation hammer was positioned to hit the pipe parallel to the radial accelerometer and provide an impulse-like input.  U = 30 ft/s fw=1292 Hz Lh = 3.81” hh =1” f1(h,Lh)=1292 Hz P=3000 psi, T = 4ºC For 12” pipe with 1.1” hydrate, f10 = 1292 Hz Lc = 0.195”       1.1”  Figure 6. Natural frequency experimental setup. The hammer is mounted on a tripod.  The red rectangles represent accelerometers.   Preliminary results COSMOSWorks ®  is a CAD package that can predict the modes of a system. Modes occur when all parts of a system move together in a sinusoidal fashion.  The natural frequencies of a system are associated with these modes.  A model of the experimental setup was made in COSMOSWorks ®  to see how accurately it could predict the natural frequencies of a real world system.  Table 1 shows a comparison of the experimental and theoretical results for an empty pipe of the dimensions stated previously. For an empty pipe, the package predicted natural frequencies at 183 Hz, 1069 Hz, 1620 Hz, 2593 Hz, 2724 Hz, and seven different frequencies between 4500 and 5500 Hz as shown in Table 1.  Additional natural frequencies were predicted at high frequencies, but they are not pertinent to this work. Figure 7 shows an actual spectrum of the empty pipe.  Figure 7. Frequency spectrum of the empty schedule 40 1.5” cantilevered 20” long pipe Peaks in this experimental spectrum are located at 180 Hz, 926 Hz, 2305 Hz, and 2491 Hz.  There are also 6 peaks visible between 3900 and 5500 Hz.  The missing peaks are most likely torsional modes, which are difficult for the accelerometers to detect.  While the theoretical prediction overpredicted the frequency of each mode, it does predict which modes will occur and their general position in the spectrum as shown in Table 1. Mode Type Theoretical Frequency (Hz) Experimental Frequency (Hz) 1st Bending 183 180 2nd Bending 1069 926 1st Torsional 1620 Not Seen 1st Longitudinal 2593 2305 3rd Bending 2724 2491 Miscellaneous 4500 to 5500 3900 to 5500 Hz Table 1. A Comparison of theoretical predictions and experimental results for a schedule 40 1.5” 20 inch long cantilevered pipe.  The next step is to create an ice annulus and compare the experimental results with the COSMOSWorks ®  predictions.  These results will help determine whether COSMOSWorks ®  can be used to predict the natural frequencies of hydrate structures in a pipeline.  This technology is also being investigated as a means to monitor hydrate deposition thickness in both lab and field applications.  CONCLUSIONS A key input for any hydrate plugging model is the rate of hydrate growth and the volume fraction of hydrate at a given time.  This work investigates a fundamental approach toward predicting hydrate growth and volume fraction in a water saturated gas pipeline. This works suggests that, in the absence of free water, hydrate volume fraction can be predicted using a wall growth and sloughing model.  Wall growth can be predicted using a one-dimensional, moving boundary, heat and mass transfer model.  It is hypothesized that hydrate sloughing can be predicted when a coincidence frequency exists between pipeline acoustic resonance, hydrate natural frequency, and flow induced vibrations over the hydrate surface.  REFERENCES 1. Sloan, E.D. and C.A. Koh, Clathrate Hydrates of Natural Gases. 3rd ed. Chemical Industries, ed. J.G. Speight. 2008: Taylor & Francis Group, LLC  2. Nicholas, J., et al. Experimental Investigation of Deposition and Wall Growth in Water Saturated Hydrocarbon Pipelines in the Absence of Free Water. in 6th International Gas Hydrate Conference 2008. 3. Hayashi, Y., et al., Study of Frost Properties Correlating With Frost Formation Types. Journal of Heat Transfer, 1977. 99: p. 239 - 245. 4. Le Gall, R.G., J.M., Modelling of frost growth and densification. International Journal of Heat and Mass Transfer, 1997. 40(13): p. 3177 - 3187. 5. Lee, K.-S.J., Sung; Yang, Dong- Keun, Prediction of the frost formation on a cold flat surface. International Journal of Heat and Mass Transfer, 2003. 46: p. 3789 - 3796. 6. Pettigrew, M.J., et al., Flow- induced vibration: recent findings and open questions. Nuclear Engineering and Design, 1998. 185: p. 249-276. 7. Jungbauer, D.E. and L.L. Eckhardt, Flow-indcued noise and vibration in turbocompressors and piping systems: Plant safety and reliability. Hydrocarbon Processing, 2000. 79(10): p. 65- 75. 8. Howe, M.S., Low Strouhal number instabilities of flow over apertures and wall cavities. Journal of the Acoustical Society of America, 1997. 102(2): p. 772-780. 9. Gere, J.M. and S.P. Timoshenko, Mechanics of Materials. 4th ed. 1997, Boston: PWS Publishing Company.  


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