Models for wax deposition in pipelines

Thermodynamic models for wax precipitation describes a number of models to calculate the amount of solid wax precipitated as a function of pressure, temperature, and fluid composition. Wax precipitation does not necessarily lead to solid deposition. Thermodynamic models for solid/liquid K values have been coupled with models for wax deposition in pipelines. The form of these models is discussed briefly in this section.

Conditions for wax deposition

For deposition to occur in pipelines, the following conditions must be fulfilled.

• Pipeline wall temperature must be below the wax appearance temperature (WAT) for the fluid.
• Negative radial temperature gradient must be present in the flow. That is, the wall temperature must be lower than the centerline temperature. A zero gradient implies that no deposition will occur.
• Wall friction must be large enough so that wax crystals can stick to the wall.

Physical processes

Burger et al. investigated the significant physical processes leading to wax deposition in pipelines. These processes are:

• Molecular diffusion
• Brownian diffusion
• Shear dispersion
• Gravity settling

Brownian movement of small solid-wax crystals will result in diffusion-like transport of these particles when a concentration gradient exists. This effect is normally neglected in pipeline deposition models. Gravity settling can occur because precipitated wax crystals are denser than the surrounding liquid. Again, this effect is usually neglected in flow models. Molecular diffusion and shear dispersion are described below, assuming that the three deposition conditions have been satisfied.

Molecular diffusion

Flow in pipes will be laminar or will have a laminar sublayer adjacent to the pipe wall. There will be a temperature gradient across this sublayer with the lower temperature at the pipe wall. When the temperature is below the WAT, the flowing oil will contain precipitated solid wax, which is in equilibrium with the liquid. Because the temperature is colder toward the wall, more of the wax components will exist in the solid phase at equilibrium. This results in a concentration gradient in the liquid phase with a lower concentration of wax-forming components at the pipe wall. Wax molecules will be transported toward the wall by molecular diffusion. Once these molecules reach the solid/liquid interface, they are available to be added to the solid deposit by the mechanisms of crystal growth. The equation describing the rate of mass transport caused by molecular diffusion is

....................(1)

where:

• m_i = mass of component i
• t = time
• ρ_oil = mass density of oil
• D_i = effective diffusion coefficient for component i
• A = deposition area
• w_i = weight fraction of component i
• r = radial distance

Because the radial-concentration gradient is not readily available, the chain rule is used in Eq. 1 to express this as the product of the mass-concentration (weight fraction) gradient with respect to temperature and the temperature gradient. The mass-concentration gradient is derived from the solubility limit as a function of temperature obtained from a thermodynamic model.

Shear dispersion

When suspended solid particles are being transported in a fluid in the laminar-flow regime, they tend to travel with the mean speed and direction of the fluid. Particles have higher velocities at greater distances from the pipe wall, and the particles also rotate as they flow. These rotating particles will exert drag forces, causing displacement of the flow paths of any neighboring particles. When the particle concentration is high, these interactions result in net transport of particles toward the low-velocity region at the pipe wall.

Considering all the wax forming components together as a single wax pseudocomponent, the rate of mass transport of wax caused by shear dispersion takes the form

....................(2)

where:

• m_w = mass of wax
• k* = empirical constant
• C_w = concentration of precipitated wax at the wall
• γ = shear rate

The form of this equation shows that the deposition rate increases linearly with increasing shear rate.

Weingarten and Euchner reported results of diffusion and shear-deposition experiments and modeling with Eqs. 1 and 2. They note that shear rate also has an important effect that is not related to shear transport. Pieces of deposited wax can be dislodged from the pipe wall in a process called sloughing. Sloughing will be dependent on the shear rate, the nature of the deposit, and the nature of the wall surface. Sloughing occurs when the wall shear rate exceeds the shear strength of the deposit and may occur both in the laminar and turbulent flow regimes.

Modeling wax deposition

Keating and Wattenbarger also have used the diffusion and shear-deposition equations in conjunction with a wellbore simulator to model wax deposition and removal in wellbores. Wax removal is caused by equilibrium conditions, not explicit modeling of the sloughing process. A study isolating and comparing the relative effects of molecular diffusion and shear dispersion on wax deposition concludes that molecular diffusion is the dominant effect. Majeed et al. obtained good results modeling wax deposition in pipelines considering only the diffusive transport.

A detailed compositional wax-deposition model for pipelines has been derived by combining the differential equations of mass and energy conservation and the laws of diffusion with a thermodynamic model for solid/liquid K values of the form given in Eq. 2. These mass and heat-transfer relations also have been applied with the multisolid-wax model by Ramirez-Jaramillo et al.

Nomenclature

A	=	deposition area, L2
Cw	=	concentration of precipitated wax at the wall, m/m
Di	=	effective diffusion coefficient for component i, L2/t
k*	=	empirical constant for mass transport of wax caused by shear dispersion
mi	=	mass of component i, m
mw	=	mass of wax, m
r	=	radial distance, L
t	=	time, t
wi	=	weight fraction of component i, m/m
ρo	=	mass density of oil, m/L3
γ	=	shear rate, L/t

Wax Deposition in Oil-gas Flow in Pipes

In two-phase flow, wax deposition is flow pattern specific and dependent on the flow velocities of the two-phase fluids. Matzain (1999) experimentally studied wax deposition for different flow patterns in horizontal, vertical pipes and slightly upward inclined pipes. The flow pattern map for horizontal flow overlayed by diagrams of pipe cross-section that shows the radial distribution of wax deposits can be found in Matzain's work. In stratified flow, no wax deposition is anticipated at the upper pipe wall since the wall is not in contact with oil. However, wax forms a crescent-shaped radial deposition at the bottom wall. In stratified wavy flow, additional wax deposits along the edge of the wave boundary due to the wave-induced cooling along the edge. In slug flow, deposition occurs at the entire circumference of the pipe, however, a thicker layer is observed at the top. This thicker layer results from the cooling effect of the gas in the Taylor bubble since the liquid in the film region drains at the top of the inner pipe wall. The same behavior for slug flow is observed by Rittirong (2014). In annular flow, a uniform distribution is observed at all circumferential locations. This uniform distribution is observed for all flow patterns in vertical flow.

The thickness and hardness of the deposited layer are dependent on flow patterns and superficial liquid and gas velocities. Since the layer is a mixture of oil and wax molecules, the hardness of the layer is a property related to the fraction of wax molecules present in the layer, while the thickness is a representation of the entire total entire components. The variation of thickness and hardness of the wax deposits as a function of flow patterns and flow conditions can be seen in Matzain's work (Matzain 1999).

Noteworthy papers in OnePetro

Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read

1. Flow Assurance: Validation of Wax Deposition Models Using Field Data from a Subsea Pipeline Singh, Amrinder, U. of Tulsa, Lee, Hyun Su, ConocoPhillips Co., Singh, Probjot, ConocoPhillips Co., Sarica, Cem, U. of Tulsa 21641-MS OTC Conference Paper - 2011

2. A Case Study of Scale-Up of Wax Deposition Model Predictions Using Flow Loop Wax Deposition Data for Pipeline Design Gonççalves, M.A.L., Petrobras, Pinho, S.P.G., Petrobras, Montesanti, J.R.T., Petrobras , Shang, W., University of Tulsa, Sarica, C., University of Tulsa 2011-B1 BHR Conference Paper - 2011

External links

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See also

Asphaltenes and waxes

Thermodynamic models for wax precipitation

Wax precipitation

Flow assurance for offshore and subsea facilities

PEH:Asphaltenes_and_Waxes