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A Compositional Model Simulating Wax Deposition in Pipeline Systems Niels Lindeloff*,† and Kristian Krejbjerg‡ Calsep A/S, Gl. Lundtoftevej 1C, DK-2800 Kgs. Lyngby, Denmark, and Calsep Inc., 11490 Westheimer Road, Suite 610, Houston, Texas 77077 Received February 6, 2001. Revised Manuscript Received February 6, 2002
A new algorithm for calculating deposition of wax in pipeline systems is presented. The algorithm considers steady-state conditions and handles multiphase flow. Pressure drop and liquid holdup along the pipeline is calculated using a standard steady-state point model, while the energy balances and thermodynamic equilibria are solved consecutively using a pressure-enthalphy (PH) flash algorithm which has been modified to account for wax formation. The approach allows heat exchanged in connection with phase transitions to be accounted properly for in the simulations. The algorithm is fully compositional. The model accounts for deposition due to both molecular diffusion and shear dispersion. To ensure a fast and efficient solution of the problem, the program generates an optimized discretization of the pipeline, based on an analytically derived temperature profile.
Introduction Flow assurance has become a hot topic within the petroleum industry, particularly as off-shore fields at ever increasing water depths are being considered. It is usual for such fields to be developed with wellheads located at a subsea template from which the produced fluids are transported to a platform or FPSO through a multiphase pipeline. In this context, wax deposition in pipelines is a problem which must be dealt with at an early stage in the design and planning of such field developments. Previous efforts to model wax deposition in pipelines described in the literature include the works by Burger et al.,1 Svendsen,2 Brown et al.,3 Dawson,4 and Rygg et al.5 The mathematical formulation of the problem must describe a complex interaction between effects from phase equilibrium and flow conditions, and for that reason a number of assumptions are traditionally made to simplify the problem. One such assumption is that the heat balances are not affected by the heat consumed or liberated in connection with changes in the phase amounts. As will be discussed in the present paper, this assumption is often a poor one when considering wax deposition in a multiphase pipeline. Model Description The model is fundamentally a steady-state compositional pipeline simulator, in which wax deposition on * Corresponding author. E-mail:
[email protected]. † Calsep A/S. ‡ Calsep Inc. (1) Burger, E. D.; Perkins, T. K.; Striegler I. H. J. Pet. Technol. 1981, June, 1075-1086. (2) Svendsen, J. A. AIChE J. 1993, 39 (8), 1377-1388. (3) Brown, T. S.; Niesen, V. G.; Erickson, D. P. SPE 26548. 68th ATCE SPE, Houston TX, 1993, 3-6 October. (4) Dawson, S. G. B. Paper presented at the IBC Conference on Controlling Hydrates, Waxes and Asphaltenes, 1996, Sept. 16-17, Aberdeen, U.K. (5) Rygg, O. B.; Rydahl, A. K.; Rønningsen, H. P. Proc. 1st North Am. Conf. Multiphase Technol. 1998, 10-11 June, Banff, AB, Canada, pp 193-205.
the pipe wall is overlaid on the steady-state results. The steady-state approach is a good approximation to the problem because wax deposition is a very slow process relative to typical residence times. The simulator is based on an approach where the mathematical problem is discretized by dividing the pipeline into a number of cells. Since the temperature of the fluid as it enters into the pipeline is generally higher than that of the surroundings, the bulk fluid temperature will generally exhibit an exponential decline as the fluid passes through the pipeline. Assuming single phase flow and steady state in the simulation, a temperature profile may be estimated analytically from eq 1.
(
Tx ) Tamb + (Tin - Tamb) exp
-πDUtot x Cp m ˘
)
(1)
The equation states that under the above assumptions, the temperature Tx at a given distance x can be calculated on the basis of the mass flowrate m ˘ , the heat capacities Cp, the pipe diameter D, and the overall heat transfer coefficient Utot. Tamb is the ambient temperature while Tin is the fluid temperature at the inlet to the pipe. This expression may be exploited to optimize the discretization of the problem by assigning cell lengths in such a way that the temperature declines only a predefined amount in each cell. This results in short cell lengths near the inlet, while cells are longer further down the pipeline where the temperature changes less. Having set up the pipeline configuration, the problem is solved by a cell-to-cell approach in each time step. The principles of the algorithm are illustrated in Figure 1. Known inlet conditions to a section, mass flowrate, temperature, pressure, and composition, and known pipe specifications, insulation, and temperature of the surroundings, allows the program to calculate heat loss from the pipe, enthalpy of the fluid exiting, and pipewall temperatures. A flow model, OLGAS2000, calculates
10.1021/ef010025z CCC: $22.00 © 2002 American Chemical Society Published on Web 04/04/2002
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Prandtl numbers are calculated on the basis of thermal conductivities, densities, and superficial velocities for the phases present. The simulation is quite sensitive toward the calculation of the film heat transfer coefficient, since this is used to determine the wall temperature and consequently the driving force for diffusion of wax molecules toward the wall:
Figure 1. Illustration of the mathematical problem in a single cell.
pressure drop, flow regime, and liquid holdup, on the basis of phase equilibria and viscosity information passed from the thermodynamic models. The wax model6 is now used to calculate wax deposition on the pipe walls by determining the equilibrium concentration difference between bulk and wall. In turn, knowing pressure, enthalpy, and feed composition at the outlet of the section, an integrated wax-PH flash is used to calculate the temperaturr.he and phase compositions. These values are then used as inlet conditions for the next section. This proceeds until the calculation has been completed for the entire pipeline in the current time step. Subsequent time steps are calculated in the same way, the only change from the last tim devalues are then used as inlet conditions for the next section.. This proceeds until the calculation has been completed for the entire pipeline in the current time step. Subsequent time steps are calculated in the same way, the only change from the last time step being that the pipe diameter and pipe insulation have changed due to a layer of wax deposited on the pipe wall. The structure of the algorithm for a single cell, as described above, can be summarized by the following four points: Heat balance, Ho ) Hi - (Q + W) Pressure drop and flow regime, OLGA-S w Po PT-wax flash at wall and deposition PH-wax flash, (Po, Ho) w To The heat balance is made by exploiting the fact that the enthalpy of the fluid entering into a cell is known from the last cell. At the inlet, a PT-flash is made to determine these properties. The heat loss can be calculated from the temperature difference between the bulk and the surroundings, and the overall heat transfer coefficient Utot. The latter is calculated from the sum of the individual wall layer resistances and the film heat transfer coefficients at the inside and outside of the pipe wall.7 W is the work due to gravity, a term which in particular becomes significant in a riser. The inside film heat transfer coefficient hin varies with the flow regime in the pipe and can for 104 < NRe < 2 × 105 be derived from the Dittus-Boelter equation:8 0.3 NNu ) 0.023N0.8 Re NPr
(2)
For flow regimes outside this range of Reynolds numbers, various forms of the Sieder-Tate equation is used.9 To account for multiphase flow, mixed Reynolds and (6) Rønningsen, H. P.; Sømme, B. F.; Pedersen, K. S. Proc. 8th Int. Conf. Multiphase Flow ’97, Cannes, France, 18-20 June, 1997. (7) Bird, R. B.; Steward, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; pp 286-28. (8) Dittus, F. W.; Boelter, L. M. K. University of California Publications on Engineering; Berkeley: CA, 1930; Vol. 2, p 443. (9) Szilas, A. P. Production and Transport of Oil and Gas, Developments in Petroleum Science 18B; Elsevier: Amsterdam, 1986.
Twall ) Tbulk -
Q hinA
(3)
In this calculation it is exploited that the same amount of heat passes through the film layer as through the entire system of layers. Q is therefore determined from the overall heat transfer coefficient and the overall temperature difference between the bulk fluid and the surroundings. The multiphase steady-state pressure drop model OLGAS2000 is used in the simulations.10 The correlations suggested by Bendiksen and co-workers10 are used in cells where single phase flow occurs. Besides pressure drop, the OLGAS2000 model also returns information on flow regime and liquid holdup. Simple geometric considerations allow the liquid holdup to be used to arrive at a wetted perimeter.This, in turn, may be used to determine the section of the pipe wall surface area in contact with the oil phase and thereby available for wax deposition.5 Deposition of wax may occur due to diffusion and shear dispersion. It is generally believed that diffusion is the dominating mechanism when there is a significant amount of wax in solution in the bulk oil. The concentration difference between the bulk oil and the region at the wall is determined by performing a PT-wax flash at both sets of conditions. Wax already precipitated in the bulk oil will not participate in the diffusion, but may redissolve as the diffusion removes wax component from the liquid phase. nw
Vdiff wax
)
∑ i)1
Di∆ciSwetMWi δFi
(4)
In this equation, ∆ci is the concentration difference between bulk fluid and the wall surface, Swet is the wetter perimeter, δ is the film layer thickness, Fi is the density, MWi the molecular weight of each wax component, and nw is the total number of wax components. The diffusion coefficients Di of the wax components are calculated using a correlation by Hayduk and Minhas.11 Shear dispersion of solidified wax from the bulk fluid is accounted for in the way described by Rygg et al.5 The concentration difference between dissolved wax in the bulk and by the wall, ∆ci, is obtained from the thermodynamic wax model by Rønningsen et al.6 The model is based on a previous approach by Pedersen12 and treats the solid wax phase as an ideal solid solution. Gas and liquid phases are modeled using a cubic Equation of State.13 The model deals with paraffinic pseudo-compounds from the characterization of the fluid rather than the pure paraffinic molecules. Fugacities for the solid-phase constituents are obtained from eq 5: (10) Bendiksen, K. H.; Maines, D.; Moe, R.; Nuland, S. SPE 19451. SPE Prod. Eng. 1991, May, 171-180. (11) Hayduk, W.; Minhas, B. S. Can. J. Chem. Eng. 1982, 60, 295299. (12) Pedersen, K. S. SPE Prod. Facil. 1995, February, 46. (13) Soave, G. Chem. Eng. Sci. 1972, 27, 1195.
A Model for Wax Deposition in Pipeline Systems
f Si ) xSi ΦoL i P exp
( ( )) ( ) ∆Hfi T 1- f RT T
exp
i
P∆vi RT
Energy & Fuels, Vol. 16, No. 4, 2002 889 Table 1. Pipeline Configuration and Fluid Description
(5)
Pipeline Configuration
S i
where f is the solid-phase fugacity of component i, xSi the mole fraction of component i in the solid,ΦoL i a reference state fugacity coefficient for component i in the liquid obtained from the Equation of State, P the pressure, T the temperature, ∆Hfi the heat of fusion for component i, Tfi the temperature of fusion, and ∆vi the change in molar volume upon the phase change from liquid to solid. The model deals with the paraffinic fraction of the pseudo-components resulting from a standard characterization of the fluid. The paraffin content of each pseudo-component is estimated from eq 6:
[
( )]
(1 - A + B × MWi) zSi ) ztot i
Fi - Fpi Fpi
C
(6)
y (m)
0 8000 12500 12500
-80 -80 -90 20 materials
pipeline
14 mm steel 1 mm paint 6 mm concrete 750 mm soil as pipeline, but no soil
riser
flow data inlet pressure inlet temperature flowrate ambient temperature pipe inner diameter
30 bar 70 °C 15.55 kg/s 4 °C 0.1905 m
Fluid Composition
ztot i
is the overall molfraction of the pseudoIn eq 6, component, zSi is the mole fraction of the paraffinic part of the pseudocomponent, A, B, and C are empirical constants determined from experimental wax precipitation data. The paraffin densities Fpi are correlated with the molecular weight of the corresponding carbon number fraction using eq 7:
Fpi ) 0.3915 + 0.0675 × ln MWi
x (m)
(7)
To reflect the different nature of the paraffinic fraction relative to the remainder of the pseudo-component, the critical pressures of the paraffin fractions were adjusted systematically to make them resemble those of the corresponding pure paraffins. The PH-flash algorithm solves the phase equilibrium problem at specified pressure and overall enthalpy for the system. The result of the flash is the temperature and the compositions and phase amounts of all phases present. The algorithm is designed to include the effects of all phase transitions between gas, oil, and solid wax, while water is handled as an inert phase. Simulation Results The fluid composition and pipeline configuration considered are shown in Table 1. The case considered is the same as Case A discussed by Rygg and coauthors,5 and treats a semi-stabilized oil flowing through a pipeline and riser system with poor insulation. Simulations were made on a case in which the pipeline is lying directly on the seabed and a case in which the pipeline in buried on the seabed, covered by 0.75 m of soil. The experimental information available on wax appearance temperatures and viscosity data were exploited to adjust the characterization of the fluid and the parameters of the CSP viscosity model.14 Nonnewtonian flow behavior (below the wax appearance temperature) may also be accounted for in the viscosity model, provided that data points of pressure, temperature, and shear rate vs viscosity are available.15 The fluid composition given in Table 1 was characterized to 22 components, of which the 12 are pseudo(14) Pedersen, K. S.; et al. Chem. Eng. Sci. 1984, 39, 1011-1016. (15) Pedersen, K. S.; Rønningsen, H. P. Energy Fuels 2000, 14, 4351.
component
mol %
MW (g/mol)
density (g/cm3)
N2 CO2 CH4 C2H6 C3H8 iC4H10 NC4H10 iC5H12 NC5H12 C6 C7 C8 C9 C10+
1.030 1.200 16.180 4.550 8.260 1.520 5.370 2.290 3.020 3.980 6.000 5.790 3.860 36.950
28.07 44.01 16.04 30.07 44.10 58.12 58.12 72.15 72.15 85.10 93.20 107.10 119.70 286.00
0.732 0.750 0.770 0.884
components. The characterization method applied may be found elsewhere.16 The paraffinic fractions were identified by the algorithm according to eq 6. The case described in Table 1 was selected from a large set of scenarios considered for a North Sea production system. The first calculations were made on the unburied pipeline case. The wax deposition profile along the pipeline is plotted at different times in Figure 2, along with temperature and pressure profiles. The initial simulation was performed at an inlet pressure of 100 atm, which means that the simulation takes place above the saturation pressure of the fluid. This simulation represents a worst case scenario in terms of wax deposition. The plots show how wax deposition initiates close to the pipeline inlet. Less wax is deposited further down the line where the bulk temperature approaches the ambient temperature, thus reducing the driving force for diffusion of wax molecules from the bulk fluid toward the wall. As a layer of wax builds up on the wall, the insulation properties of the deposit cause a lower heat flux and as a result a higher bulk temperature. This means that at later times, more wax will deposit further down the pipeline. The pressure profile is also affected by the deposited wax layer, which causes the effective diameter of the pipe to decrease over time. The large pressure decrease near the outlet is caused by the hydrostatic pressure drop in the vertical riser section. The corresponding simulation performed on the buried pipeline is shown in Figure 3. In this case, the inlet pressure was 30 atm, which means that most of the (16) Pedersen, K. S.; Blilie, A. L.; Meisingset, K. K. I&EC Res. 1992, 31, 1378-1383.
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Figure 2. Simulation results showing the effects of deposited wax in a pipeline exposed to surrounding seawater.
Figure 3. Plots showing the effects of deposited wax. Buried pipeline case.
Figure 4. Plots of compositional variations in the deposited wax along the length of the pipeline, the second plot shows component distribution after 5 days of simulation.
simulation takes place below the bubble point pressure of the fluid. The plots clearly demonstrate, that while the wax is deposited further away from the pipeline inlet and to a lesser extent, a large amount of wax deposit is to be expected in the uninsulated riser system. The reason for this is, that while the lower heat loss and resulting higher temperatures on the pipe wall causes less deposition, the amount of dissolved wax in the bulk oil is still very high when the oil enters the cold riser. The result is that a large amount of wax will deposit in the riser.
The model allows for an analysis of the compositional variations in the deposited wax as a function of time or as a function of the distance from the inlet of the pipeline. Figure 4 shows snapshots of the avearage molecular weight distribution and a plot of the component distribution in the deposit after 5 days. As the first plot in Figure 4 demonstrates, the average molecular weight decreases along the length of the pipeline. This is what would be expected, the first wax molecules to deposit are likely to be heavy, with a high melting points, while gradually lighter paraffins and naphthenes deposit as the temperature drops further.
A Model for Wax Deposition in Pipeline Systems
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Figure 5. Comparisons of temperature profiles and heat flux obtained using the PH-flash approach described in the paper versus a traditional PT-flash approach.
It can also be seen from the plot, that the heavy wax molecules are carried further and further down the pipeline as time passes. This is because the insulating effect of the early deposits cause the bulk temperature to increase over time as shown in the second plot in Figure 3. This is confirmed by the second plot in Figure 4, in which the composition of the deposited wax at three different positions in the pipeline is compared. As the plot shows the composition taken closest to the inlet is richer in the heaviest wax components, while the composition becomes dominated by the lighter wax components further down toward the outlet. Comparative runs were made in order to investigate the effect of applying the PH-flash approach in the simulations, rather than using the traditional approach with a PT-flash and the assumption of negligible heat production as a result of phase transitions. The resulting temperature profiles at two different time steps are plotted in Figure 5, along with a plot of the heat flux variation over time at two different positions in the pipeline. The plots in Figure 5 demonstrate that a lower temperature profile is obtained when energy consumed or released in connection with phase transitions is neglected in the energy balances. In the present case, the simulated outlet temperatures are 3 K lower when the phase transitions are neglected. As shown by the second plot in Figure 5, the actual heat flux throughout the simulation is at all times higher than that found with a traditional approach, and consequently the driving force for wax deposition is not calculated correctly if phase transitions are not accounted for in the energy balance. Note that the temperature profiles deviate before wax starts forming around 7000 m. This is caused by energy contributions from vapor-liquid phase transfer. The effect will be more pronounced for a live fluid with a higher light-end content. Conclusions The present work clearly demonstrates that the heat released in connection with phase transfer contributes significantly to the energy balance in wax deposition simulations. Both vapor-liquid transitions and liquidsolid transitions contribute. This will further affect the wax deposition process, since the driving force for deposition is directly related to the temperature difference between the bulk fluid and the wall.
The compositional variations in the deposited wax along the pipeline are directly related to the shape of the temperature profile. The composition of the wax deposited near the inlet of the pipeline will be dominated by the heaviest wax molecules, while wax deposited closer to the outlet is richer in the lighter wax molecules. Nomenclature A ) Surface area c ) Concentration Cp ) Heat capacity A, B, C ) Empirical constants D ) Diameter Di ) Diffusion coefficient f ) Fugacity h ) Film layer heat transfer coefficient H ) Enthalpy m ) Mass flow rate MW ) Molecular weight N ) Dimensionless number P ) Pressure Q ) Heat R ) Gas constant S ) Wetted perimeter T ) Temperature U ) Overall heat transfer coefficient v ) Molar volume V ) Volume x ) Distance or Mol fraction Greek δ ) Film layer thickness φ ) Fugacity coefficient F ) Density Subscripts amb ) Ambient i, in ) Inlet i ) Component number o ) Outlet Nu ) Nusselt number Re ) Reynolds number Pr ) Prandtl number Superscripts oL ) Liquid reference state p ) Paraffin S ) Solid EF010025Z
