## Reynolds Number Is Important in Understanding Wax Deposition

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Comprehension of the mechanisms that influence wax deposition in oil-production systems has not yet been achieved fully. Given the absence of a theory able to explain the evolution and characteristics of these deposits, the resulting production limitation is one of the main issues in flow assurance. This paper investigates the influence of the Reynolds number on wax deposition.

## Introduction

Several numerical models and experiments have been conducted to understand wax deposition in flowlines. A typical experiment consists of injecting a heated solution (wax dissolved in a solvent) through a test section with one of its surfaces kept at a colder temperature than the wax appearance temperature (WAT). As the fluid temperature is reduced below the WAT, the first wax crystals appear, initiating deposit growth.

Typically, oil flow in production lines is turbulent. As the Reynolds number increases, deposit thickness decreases. Experiments have shown that composition changes with time, with the increase of shear stress (higher Reynolds numbers) resulting in deposits with higher percentages of solids.

In this work, an enthalpy/porosity model was coupled with a kinetic energy/specific dissipation (*κ/w*) turbulence model to allow an investigation of this flow regime. The outer pipe is maintained at a constant high temperature equal to the inlet temperature, and the inner pipe is cooled.

## Mathematical Model

The enthalpy/porosity model was based on applying conservation equations in the liquid region of the domain, identified by its volume liquid fraction, and using a Darcy term at the linear momentum equation to account for the solid interactions in the porous region. To take turbulence into account, the Reynolds average/Navier-Stokes model was applied to determine the time average value of all relevant quantities. The fluctuation in the average flow was modeled on the basis of the Boussinesq approximation, and the *κ/w* model was used because it usually presents good results for low-Reynolds-number flow. In addition to these characteristics, the *κ/w* model does not need a wall function, which is convenient in this case because the liquid/deposit interface moved as time progressed. The equations necessary for building the mathematical model, as well as the methodology of determining initial states and boundaries, are provided in the complete paper.

To complete the model, a thermodynamic equilibrium was included to determine which components were and were not precipitating and the new composition was provided by the equilibrium. This was achieved by comparing the component’s fugacity in a mixture with the pure component’s fugacity in an interactive method.

## Numerical Model

To solve the required equations numerically, a finite-volume method was applied, consisting of dividing the computational domain into control volumes and integrating the conservation equation at each one. Staggered mesh was used, where scalar variables were stored at the control volume center and velocities were stored at the faces. The temporal integration was performed considering the implicit first-order Euler scheme, and, for the spatial integration, the power-law scheme was applied. The velocity/pressure coupling was solved with an algorithm often used to solve the Navier-Stokes equation (the semi-implicit method for pressure-linked equation-consistent algorithms). The systems of algebraic equations obtained by discretization of the differential equations were solved by the iterative tridiagonal-matrix algorithm line by line, and the block-correction algorithm was used to accelerate the convergence.

A grid test was performed, and a nonuniform mesh distribution of 77×62 concentrated near the walls was defined because mesh convergence of 1% was obtained. The timestep was set to guarantee a Courant number less than 0.2.

## Results

A solution of 80 wt% kerosene with 20 wt% commercial paraffin with a WAT of 33°C was investigated in this study. The solution entered the test section at 37°C, while the inner pipe wall was cooled to 12°C, taking 12 seconds to be reduced from 37 to 20°C and finally attaining 12°C after 10 minutes. Three Reynolds numbers were investigated: 1,260, 3,500, and 6,000. The first Reynolds number corresponds to the laminar regime, and the other two correspond to the turbulent regime.

For the first case, a comparison with measurements made in other experiments could be performed, allowing an evaluation of the model considered. For instance, a comparison of the deposit distribution along the test section for three time instants, obtained with the present model and experimental data, reveals excellent agreement for a time of 30 seconds, followed by a slight deterioration as time progresses. However, a good agreement is obtained in general.

The thickness of the permanent deposit decreases with an increase in the Reynolds number, agreeing with several experiments recorded in the literature. The thickness of the deposit becomes more uniform axially as the Reynolds number increases.

The behavior of the thickness variation with the flow regime can be understood by observing the thermal field for each case corresponding with the steady state. The corresponding WAT isotherm of this fluid (approximately 33°C) is closer to the copper surface for larger Reynolds-number tests (i.e., the region of the cooled domain is smallest for the largest Reynolds number, resulting in thinner deposits).

The composition and porosity provide inputs for understanding the aging behavior of the deposit. Observing the steady-state solid-saturation fields for the three Reynolds numbers reveals that the deposit is more compact near the cold wall where there is a small region with solid saturation greater than 40%. The thickness of this solid-saturation region is reduced with an increasing Reynolds number significantly. For the three cases, the bulk of the deposit thickness has a solid saturation between 10 and 30%. Also, a reduction of the solid-saturation value in the test section’s exit region is observed for the highest Reynolds number (6,000). Further tests are needed to understand this phenomenon better.

Total molar fraction contours for four species in the three cases were investigated, from light to heavy (Species 3, 6, 9, and 12). The concentration of light and heavy species for Reynolds numbers of 3,500 and 1,260 are very similar. As expected, the lighter species (3, 6, and 9) have a higher concentration in the region outside the deposit. For the heaviest species (12), however, the situation is reversed, with the highest concentration being within the deposit.

For the turbulent case, with a Reynolds number of 6,000, it is verified that, in the region near the exit, which presents lower values of solid saturation, all the light species present a region of high concentration near the copper surface. Except for this small region near the exit, the concentration distribution of Species 3, 6, and 9 is analogous to that of the other two cases (i.e., higher concentration in the outer region of the reservoir). However, for Species 12 (heavier), a high-concentration nucleus is observed at the beginning of the deposit, with a low concentration near the copper in the exit region.

In cases of turbulent flow, two more variables should be evaluated: *κ* and *w*. The *κ* is zero on the solid surfaces, and *w* has a value that depends on the Reynolds number and surface roughness. The turbulence was neglected in the deposit; *κ*, furthermore, was arbitrated as zero in this region. Regarding the specific dissipation, the interface between the gel and the liquid is treated as a wall, with arbitrated roughness (equal to copper) imposed on the interface.

In **Fig. 1,** the results for the two turbulent cases are presented, where the kinetic energy is minimal in the region of the gel and increases as it moves away from the gel. Note that, for the modeling of the turbulence used, because the deposit is treated as a solid on which the solvent flows, the dissipation inside the deposit is maximum, reflecting its estimated values for the roughness of the deposit. This value decays rapidly to the middle of the channel where the flow is free and the w is almost null, growing again as it approaches the acrylic wall. The dissipation imposed by the acrylic is lower than that generated by the deposition, which is coherent, because higher shear stress leads to higher dissipation. The derivative of the velocity profile is more pronounced near the deposit interface, corresponding to higher shear stress.

## Conclusion

In this work, the influence of the Reynolds number in wax deposition was studied. The model combines the enthalpy/porosity approach with the *κ/w* turbulence model. The influence of the Reynolds number on the evolution of paraffin deposits was investigated. The main behaviors observed with respect to the shape of the paraffin deposit for turbulence regimes were verified with the proposed model. The model also allowed observation that harder paraffin is found near the cold surface. Further, the deposition of each species occurs at a different temperature, resulting in a strongly nonuniform deposit composition.

*JPT*Technology Editor Chris Carpenter, contains highlights of paper OTC 28053, “Reynolds Number Influence on Wax Deposition,” by

**R.C. Albagli,**Petrobras, and

**L.B. Souza**and

**A.O. Nieckele,**Pontifical Catholic University of Rio de Janeiro, prepared for the 2017 Offshore Technology Conference Brasil, Rio de Janeiro, 24–26 October. The paper has not been peer reviewed. Copyright 2017 Offshore Technology Conference. Reproduced by permission.

Reynolds Number Is Important in Understanding Wax Deposition

01 November 2018

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