The Analytical Model for the Impact Assessment of the Magnetic Treatment of Oil on the Wax Deposition Rate on the Tubing Wall

Abstract

There has been a large amount of experience in recent decades in the use of magnetic fields on reservoir fluids. This paper discusses the effect of a magnetic field on wax precipitation. An analytical model is developed to quantify the wax deposition rate on the tubing surface during the magnetic treatment of reservoir oil. It has been established that the passage of the oil flow through a non-uniform magnetic field causes a high-intensity electric field for a sufficiently long period of time, the effect of which decreases the solubility of wax in oil, increases the intensity of wax precipitation in oil, and reduces the wax deposition on the tubing surface. The model accounts for the fact that the wax deposits present on the tubing surface are a highly efficient heat insulator that changes the temperature regime of the flow and the temperature of the tubing wall. This circumstance changes the rate of deposits but does not make these deposits less harmful to wells’ operation. A method for calculating the equilibrium wax concentration and changing the solubility of wax in oil under a constant electric field has been developed. We show that the effect of magnetic treatments on wax deposition rises with the increase in the concentration of asphaltenes in the oil and water cut.

1. Introduction

In practice in the field, mechanical, thermal, magnetic, and adhesion methods to control wax deposition are used to prevent the formation of plugs in producing wells. It has been empirically established that in a number of cases, magnetic treatment can significantly improve the technical and economic features of oil production by increasing the time between overhauls (TBO) to eliminate the accumulation of wax on the tubing wall. Currently, a number of oil companies use downhole magnetic dewaxing units and continue to study the effect of magnetic fields on the rheological properties of oil. The application of such technologies requires an adequate understanding of the influence of a physical field on wax deposition. The kinetics of wax accumulation is determined by the wax precipitation rate in the flow and on the tubing surface. These rates are influenced by oil viscosity, flow rate, size of wax crystals, their quantity, flowing temperature, and tubing wall temperature. In fact, the increase in the wax precipitation in the oil flow is the main goal of prevention methods of wax deposition, as has been noted above. The mechanism of action of the magnetic field during MT is primarily associated with a decrease in oil viscosity due to the destruction of super-molecular structures formed by asphaltenes, polar resins, and aggregates of iron oxides contained in oils. Indeed, the influence of the magnetic field on the TBO is most clearly manifested itself during the MT of high-viscosity and waxy oils. Nevertheless, magnetic dewaxing units have not been recommended in the fields of Western Siberia with low-viscosity oils with an asphaltene content of less than 1%. However, the data on their effectiveness are very contradictory. The viscosity of high-viscosity and waxy oils after MT can both decrease and increase. The article [1] showed that it depends on the oil composition, including the contents of asphaltenes, polar, and non-polar resins. A comparable positive effect from the use of magnetic devices is no exception for offshore oil fields.

The authors of [2] showed that the published data on the effect of a magnetic field on the crystallization of diamagnetic substances are very contradictory. Calculations have shown that the variation in the parameters of the phase changes in any substances in a magnetic field is very small. Given the extremely low magnetic susceptibility of paraffin hydrocarbons, it can be assumed that there are no physical causes that would lead to the formation of a large number of wax crystals in a magnetic field of 104–105 A/m or higher. At the same time, numerous field observations show the significant influence of MT on the intensity of wax deposition on downhole equipment. This is stimulating researchers to find an explanation for this phenomenon and to use it more effectively and purposefully in oil production.

2. Materials and Methods

Analytical Model of the Wax Deposition

The wax deposition on the tubing surface is determined by the dependence of the equilibrium concentration of paraffin fractions in oil on temperature, the temperature distribution in the oil flow, and the intensity of mass transfer between the flow and the tubing surface. The intensity of mass transfer depends on the diffusion coefficient of dissolved wax in oil and is mainly determined by the dynamics of the fluid flow in the tubing. Wax precipitation occurs when the oil in the flow reaches the temperature of the wax appearance. In this case, wax precipitates both in the oil flow and on the tubing surface. The appearance of nuclei of a solid phase and the subsequent growth of wax crystals is presented as a series of stepwise processes to form a straight-line configuration to paraffin molecules, in order to align their preliminary orientation relative to each other and to the surface of a growing solid phase, and subsequently, the deposition of molecules occurs. The difference between the wax precipitation on the tubing surface and in the volume of the liquid phase lies in the presence, in the first case, of the conditions for the forced growth of the solid phase on the tubing surface.

The authors of [3] provide a comprehensive review of wax deposition models in pipelines, which include the main mechanisms such as molecular diffusion, Soret diffusion, Brownian diffusion, gravity settling, shear dispersion, and shear stripping. Thermodynamic and flow dynamics models of wax deposition, as well as wax aging models, have also been considered. Methods are also considered for inhibiting wax deposits and types of chemicals, as well as various methods for removing deposits. It has been shown that wax-aging mechanisms are very important to understanding the wax deposition in pipelines. Understanding these mechanisms would be helpful in selecting the most effective chemicals for preventing wax deposition in pipelines. The article [4] obtained an empirical correlation of the wax deposition for two real tieback systems in the Gulf of Mexico, which takes into account the radial temperature gradient, shear stress, shear rate, molecular diffusion of wax (based on the cold finger experiment), and the formation rate of wax crystals. The results are based on field and laboratory data. The issue resides in modeling the field production conditions of pipelines in laboratory experiments. The author showed that the diffusion of wax molecules through the laminar boundary layer is largely responsible for wax deposition in pipelines. The article [5] experimentally showed that the magnetic treatment reduces the wax deposition rate by more than 80% and oil viscosity by 50%. It also showed that the effect of the magnetic treatment on both wax deposition and viscosity reduction could remain stable only for a certain duration after the removal of the magnetic field and then decline gradually towards the level of untreated oil. The article [6] obtained results for an unheated oil pipeline in China. It was shown that oil viscosity depends on temperature. The conditions of pipeline operation are also presented in this article, and the wax deposition model in the pipeline based on the loop test data was developed. Additionally, the authors showed the ranges of thickness of wax deposits in the pipeline caused by seasonal temperature fluctuations, and they also showed that the wax deposition in the non-heated pipeline in the winter achieves a more intensive rate than that in the summer, and they identified the conditions for the most favorable procedure for pigging of the pipeline. According to this research, the flowability of oil in the pipeline will become poor when the scraped deposit has heavily contaminated oil in front of the pig. In the article [7], the existing wax deposition model with a non-Newtonian constitutive equation for the wax-in-oil suspension (the suspension of fractal aggregates model) was improved. The proposed improved model allows one to investigate the influence of non-Newtonian oil properties on heat and mass transfer and the influence of shear stress and paraffin inhibitor on wax deposition. The authors showed that the wax deposition rate decreases with an increasing oil flow rate. Heat and mass transfer analysis carried out alone cannot explain the decrease in deposit thickness with increasing flow rate. The authors also showed that this phenomenon can be predicted when the non-Newtonian characteristics are incorporated and the wax deposition is modeled as a gelation process. Puente et al. in 2018 [8] created models for the deposition of wax and hydrates and validated a non-insulated oil pipeline offshore at low temperatures. Input parameters in models can be derived from laboratory data and can be selected as a default. Models can be used to estimate the thickness of wax deposits on the pipeline wall to plan the pigging schedule. The estimated pressure loss in the pipeline based on the models allows the operating pressure for the equipment to be evaluated. The effects of wax deposition and hydrate transport were studied individually. However, for field production where both these problems co-exist, it will be important to model these phenomena together.

In this study, we propose describing the wax deposition on the tubing surface by a system of differential equations, neglecting the thermal effects associated with the liberation of gas, oil expansion, and friction:

$$\{\begin{array}{}\mathrm{(1)}& \hfill \frac{\partial {\mathrm{G}}_{\mathrm{o}}}{\partial \mathrm{x}}\mathrm{C}\left(\mathrm{x}\right)+{\mathrm{G}}_{\mathrm{o}}\frac{\partial \mathrm{C}\left(\mathrm{x}\right)}{\partial \mathrm{x}}& =& -{\mathrm{K}}_{\mathrm{wax}}\left(\mathrm{C}\left(\mathrm{x}\right)-{\mathrm{C}}_{\mathrm{e}}\right)-{\mathrm{K}}_{\mathrm{wall}}{\mathrm{K}}_{\mathrm{a}}\left(\mathrm{C}\left(\mathrm{x}\right)-{\mathrm{C}}_{\mathrm{ew}}\right),\hfill \mathrm{(2)}& \hfill \frac{\partial \mathrm{T}}{\partial \mathrm{x}}& =& -\frac{{\mathrm{K}}_{\mathrm{ef}}}{\mathrm{G}\cdot {\mathrm{c}}_{\mathrm{p}}}\cdot \left(\mathrm{T}\left(\mathrm{x}\right)-{\mathrm{T}}_{\mathrm{rock}}\left(\mathrm{x}\right)\right),\hfill \mathrm{(3)}& \hfill \frac{\mathrm{dr}}{\mathrm{dt}}& =& -\frac{{\mathrm{K}}_{\mathrm{wall}}{\mathrm{K}}_{\mathrm{a}}}{2{\mathsf{\pi }\mathrm{r}\mathsf{\rho }}_{\mathrm{wax}}}\cdot \left(\mathrm{C}\left(\mathrm{x}\right)-{\mathrm{C}}_{\mathrm{ew}}\right),\hfill \end{array}$$

The heat transfer and mass transfer coefficients can be calculated using the following equations:

$${\mathrm{K}}_{\mathrm{ef}}=\frac{1}{\frac{1}{{\mathsf{\pi }\mathrm{d}}_{\mathrm{t}}{\mathrm{K}}_{\mathsf{\tau }}}+\frac{1}{2{\mathsf{\pi }\mathsf{\lambda }}_{\mathrm{wax}}}\cdot \ln \frac{{\mathrm{d}}_{\mathrm{t}}}{2\mathrm{r}}},$$

(4)

$${\mathrm{K}}_{\mathrm{wax}}=0.5{\mathsf{\pi }}^2{\mathrm{Sh}}_{\mathrm{w}}{\overline{\mathrm{a}}\mathrm{D}}_{\mathrm{l}}{\mathrm{Nd}}_{\mathrm{t}}{}^2{\mathsf{\rho }}_{\mathrm{l}},$$

(5)

$${\mathrm{K}}_{\mathrm{wall}}={\mathsf{\pi }\mathrm{Sh}}_{\mathrm{e}}{\mathrm{D}}_{\mathrm{l}}{\mathsf{\rho }}_{\mathrm{l}},$$

(6)

$${\mathrm{D}}_{\mathrm{l}}=\frac{2{\mathrm{D}}_{\mathrm{o}}\left(1-{\mathsf{\phi }}_{\mathrm{w}}\right)}{2+{\mathsf{\phi }}_{\mathrm{w}}}.$$

(7)

The following assumptions were made when deriving the equations above.

The well operates in steady-state conditions, and the liquid flow rate in the tubing section changes only due to the liberation of gas;

The depth along the well bore at which pressure and temperature conditions correspond to the beginning of wax precipitation is assumed as zero. Above this depth, the wellbore is assumed to be vertical.

Equations (1)–(3) take into account the equilibrium of the paraffin fractions (1), the cooling of the fluid flow (2), the change in the radius of the tubing section due to the wax deposition on the tubing wall (3), the change in the heat transfer coefficient due to the heat insulation of the tubing with wax deposits (4), the change in the mass transfer of paraffin fractions in the fluid flow (5), and the surface of the tubing wall (6).

Crystallization of paraffin fractions on the tubing wall leads to the formation of a skeleton of deposits with a porosity of 40–50%, the voids of which are filled with heavy oil components (resins, asphaltenes, etc.).

For laminar flow (Re ≤ 2000), ${\mathrm{Sh}}_{\mathrm{e}}\approx 4,$ and for turbulent flow (Re > 2000), ${\mathrm{Sh}}_{\mathrm{e}}$ takes the following form:

$${\mathrm{Sh}}_{\mathrm{e}}=0.021{\mathrm{Re}}^{0.8}{\mathrm{S}}_{\mathrm{c}}{}^{0.43}{\left(\frac{\mathrm{Sc}}{\mathrm{Scw}}\right)}^{0.25}.$$

(8)

The Sherwood number caused by shear stress can be calculated (neglecting the sliding velocity of the solid phase relative to the liquid) in a wide range of Peclet numbers by the equation:

$${\mathrm{Sh}}_{\mathrm{w}}=1+\frac{0.26{\mathrm{Pe}}^{0.5}}{1+0.074{\mathrm{Pe}}^{0.5}},$$

(9)

To estimate the average radius of wax crystals and their number, under the assumption that the precipitated wax crystals are immobile and grow only due to diffusion from the surrounding solution, it can be shown that the average crystal radius is determined by the following expression:

$$\overline{a}={\left[\frac{8\mathsf{\sigma }{\mathrm{v}}^{\prime }{\mathsf{\rho }}_{\mathrm{o}}{\mathrm{D}}_{\mathrm{l}}\left({\mathrm{C}}_{\mathrm{o}}-{\mathrm{C}}_{\mathrm{e}}\right)}{9{\mathrm{RT}}_{\mathrm{a}}{\mathrm{M}}_{\mathrm{wax}}{\mathrm{Bw}}_{\mathrm{l}}{\mathsf{\Gamma }}_{\mathrm{o}}}\right]}^{\frac{1}{3}},$$

(10)

$$\mathrm{B}=\frac{1}{{\mathrm{C}}_{\mathrm{e}}}\cdot \frac{\partial {\mathrm{C}}_{\mathrm{e}}}{\partial {\mathrm{T}}_{\mathrm{a}}}.$$

(11)

The total number of wax crystals per unit volume decreases with position (x) according to the equation:

$$\mathrm{N}=\frac{\left({\mathrm{C}}_{\mathrm{o}}-{\mathrm{C}}_{\mathrm{e}}\right){\mathrm{RT}}_{\mathrm{a}}{\mathrm{M}}_{\mathrm{wax}}{\mathrm{w}}_{\mathrm{l}}}{4{\mathrm{D}}_{\mathrm{l}}\mathsf{\sigma }{\mathrm{v}}^{\prime }{\mathrm{C}}_{\mathrm{e}}{\mathsf{\rho }}_{\mathrm{o}}\mathrm{X}}.$$

(12)

The solution to Equations (1)–(3) allows the change in the inner diameter of the tubing to be calculated over time and depth. The wax deposits present on the inner surface of the tubing wall are a highly efficient heat insulator that changes the temperature regime of the flow and the temperature of the tubing walls. This circumstance is taken into account in expression (4) with the dependence of the value of the current heat transfer coefficient on the radius of the flow section of the tubing. We validated the proposed model on the field data for three producing wells. The profile of wax deposits along wellbores № 546, 504, and 6 of one of the Western Siberian oilfields (Russia) is presented in Figure 1.

A study of the wax deposition shows that the wax precipitation rate varies with depth, and the wax deposits profile along the wellbore, depending on the oil flow characteristics, can both be steadily increasing (there is maximum thickness of deposits at the wellhead) and have a maximum along the wellbore. Figure 1 shows that the proposed wax deposition model is consistent with the field data, allowing the wax deposits profile and the depth of the formation of wax plugs to be calculated. The prediction of the profile of wax deposits over time also allows one to calculate the TBO, depending on the proposed method for controlling wax deposition.

3. Results

It is obvious that the main effect of the alternating electric field is manifested in a decrease in the oil viscosity due to the destruction of the super-molecular structures formed by heavy polar components. As a result, the particle size of the solid phase changes, and the wax crystals grow and their number decreases. Indeed, from Equations (8) and (9), it follows that the number of wax crystals is proportional to the oil viscosity $\mathrm{N}~{\mathsf{\mu }}_{\mathrm{o}}$, and the crystals’ size is inversely proportional to the oil viscosity $\overline{\mathrm{a}}~\frac{1}{{\mathsf{\mu }}_{\mathrm{o}}}$.

The influence of a constant electric field on the wax precipitation consists in the orientation of long-chain molecules of paraffin fractions in a certain direction (by a given vector of the electric field strength), thereby providing a more effective structure-forming contact between them. As a result, a larger number of nuclei are formed at a lower degree of super-saturation of the solution with paraffin fractions. An additional factor is the change in the wax solubility under the influence of a constant electric field. The wax equilibrium concentration is determined by the equality of the chemical potentials of the solute, both in the solid and in the dissolved phase. A change in the chemical potential of a solute will be associated with a change in the entropy of the system; for polar dielectric substance in an electric field, the change in entropy depends on the orientation of the molecules over the field. The effect of a constant electric field on the equilibrium concentration of wax is described by the following expression:

$${{\mathrm{C}}_{\mathrm{e}}}^{\prime }={\mathrm{C}}_{\mathrm{e}}\exp \left[-A\cdot {\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{o}}}\right)}^2\right].$$

(13)

Lei in 2014 [9] showed that asphaltenes have a dispersing effect on wax crystals. It can be argued that this is the result of the influence of local electric fields of elementary charges (charges of polar groups of asphaltene molecules). Punase in 2016 [10] showed that the polarity of asphaltenes is much higher than the polar fractions of resins, so their role in the orientation of paraffin molecules has secondary importance. Near the ends of the dipoles of the polar groups of asphaltenes, the electric field strength reaches 10¹⁰–10¹¹ V/m, which results in a strong orientation of the molecules of non-polar wax. Therefore, in the presence of asphaltenes, paraffin molecules can be considered pseudopolar, and their orientation will depend on the orientation of asphaltene molecules, which can be found according to the following considerations. Provided that PoE/RT < 1, the Langevin function, meaning the orientation of polar molecules, can be taken in the linear approximation; in this case, we can put down that ${\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{o}}}\right)}^2=\mathrm{const}\cdot {\left(\frac{{\mathrm{P}}_{\mathrm{o}}\mathrm{E}}{\mathrm{T}}\right)}^2$. Since the Po value is directly proportional to the concentration of polar components in oil, it can be shown that in the presence of a constant electric field, the equilibrium concentration of paraffin fractions in oil is described by the following expression:

$${{\mathrm{C}}_{\mathrm{e}}}^{\prime }={\mathrm{C}}_{\mathrm{e}}\exp \left[-\mathsf{\beta }{\left(\frac{{\mathrm{C}}_{\mathrm{asph}}E}{{\mathrm{T}}_{\mathrm{a}}}\right)}^2\right].$$

(14)

Using dependencies (10), (12), and (14), it can be shown that $\mathsf{\beta }=2.76{\left(\frac{\mathrm{K}\cdot \mathrm{m}}{\mathrm{V}}\right)}^2$. Thus, according to expression (14), the electric field can significantly reduce the equilibrium concentration of wax in oil, thereby changing the kinetics of wax precipitation both in the flow and on the tubing surface. For example, for oil with a viscosity of 100 mPa∙s at E = 2100 V/m, the ratio ${{\mathrm{C}}_{\mathrm{E}}}^{\prime }/{\mathrm{C}}_{\mathrm{E}}$ = 0.71, E = 3900 V/m, and ${{\mathrm{C}}_{\mathrm{E}}}^{\prime }/{\mathrm{C}}_{\mathrm{E}}$ = 0.31 with a weight content of asphaltenes in oil of 0.05.

Therefore, since the magnetic field cannot have a direct effect on the wax precipitation, it remains to assume that after the passage of the magnetic apparatus, a sufficiently powerful electric field arises in the fluid flow. The appearance of an electric field is due to the fact that the oil flow moving in the reservoir and in the well acquires an electric charge (i.e., it is charged). However, when moving in the well, these charges are concentrated in a narrow region near the surface of the walls of the tubing (diffusion layer). When the charged liquid passes through the area of the non-uniform magnetic field created by the magnets of the device for magnetic processing of the flow, the diffusion layer of ions is scattered. The charge is uniformly mixed over the volume of the flow, and an electric field arises (and is at a maximum near the walls of the tubing). The theory of a double-electric layer makes it possible to explain and calculate the appearance of a high-intensity field in the fluid flow in the tubing. When a solid surface is in contact with a liquid, an inequivalent exchange of charges is observed between them, which occur both in the reservoir and on the surface of the downhole equipment. The magnitude of the resulting potential depends on the composition of the solid and liquid phases. A tubing string in a producing well can be considered as a grounded metal pipeline through which oil flows in a mixture with gas, water, and a solid phase (wax, sand, etc.).

The charges of the diffusion part of the double-electric layer completely enter the moving fluid flow, and the value of the charge corresponds to the total charge of the ions of the diffusion part of the double-electric layer. The intensity of the charging of the liquids depends on the flow rate, its conductivity, the roughness of the tubing, the contact area, the presence of solid particles in the fluid, water, gas, etc. As the conductivity of the liquid increases, the electrical current increases at a constant flow rate and then reaches a higher value, after which it decreases (the conductivity of the produced liquid increases with increasing water cut).

In a gross way, the strength of an electric current J caused by this phenomenon is determined by the expression:

$$\mathrm{J}=2.74·{10}^{-4}{\mathsf{\phi }}_{\mathrm{o}}{\mathsf{\epsilon }}_{\mathrm{o}}\mathsf{\epsilon }\frac{{\mathrm{q}}^{1.875}{\mathrm{T}}_{\mathrm{a}}}{{\mathsf{\nu }}^{0.625}{\mathrm{r}}_{\mathrm{o}}^{2.875}}.$$

(15)

For example, at q = 30 m³/day, ${\mathrm{T}}_{\mathrm{a}}$ = 293 K, $\mathsf{\nu }$ = 1.25∙10⁻⁵ m²/s, r_o = 0.031 m, we obtain, for clean tubing, the string J = 2.1∙10⁻¹⁰ A by Formula (15). At the same time, the density of charges ρ_ch (if they are evenly distributed cross-sectionally inside the tubing string) will be ${\mathsf{\rho }}_{\mathrm{ch}}=\frac{\mathrm{J}}{\mathrm{q}}=6.0·{10}^{-7}$C/m³.

However, when the liquid phase moves, these charges are concentrated in a narrow area near the tubing walls in a diffuse layer. When a charged liquid passes through a region of a non-uniform magnetic field inside a magnetic device, the diffusion layer of ions is scattered, and the charge is uniformly mixed throughout the volume of oil with a bulk density ρ_ch = J/q, while an electric field appears with an intensity depending on the distance from the center of the tubing (maximum near the tubing wall), which is calculated using Gauss’s law as follows:

$$\mathrm{E}=\frac{{\mathsf{\rho }}_{\mathrm{ch}\cdot \mathrm{r}}}{2\cdot \mathsf{\epsilon }\cdot {\mathsf{\epsilon }}_{\mathrm{o}}}=\frac{\mathrm{J}\cdot {\mathrm{r}}_{\mathrm{o}}}{2\cdot \mathrm{q}\cdot \mathsf{\epsilon }\cdot {\mathsf{\epsilon }}_{\mathrm{o}}}.$$

(16)

Thus, with a volume charge density of 6.0∙10⁻⁷ C/m³, the intensity of the resulting field is ~10³ V/m. The resulting field has a double effect on the wax deposition. Firstly, since it is variable when oil passes through a magnetic unit, it destroys the super-molecular structure in oil and reduces its viscosity. Secondly, after oil passes through the device, the distribution of charges should remain for some time (the relaxation time is from several hours to a day), determined by the rate of ion diffusion; therefore, the electric field acts directly on the wax precipitation. This leads to more intensive wax precipitation in the volume of oil and, consequently, to a decrease in the rate of wax crystals growth on the tubing walls. These assumptions are confirmed by the established fact of the effect of magnetic treatment of oil when installing magnetic units in the tubing near the wax precipitation depth. In this case, the above-mentioned mechanism of the influence of physical fields on the oil flow is carried out, and considerable changes are absent in the well-production conditions with installed magnetic units on the tubing shoe or electrical submersible pumps at great depths . The appearance of an insoluble suspended phase (water droplets, gas bubbles, sand, wax, or iron scale) in the oil flow increases the number of active surfaces capable of creating double-electric layers and, therefore, increases the diffusion-scattered charge of the flow.

The solutions to Equations (1)–(3) using dependencies (14), (15), and (16) shows that the efficiency of the MT of oil in all cases is affected by the flow rate, the amount of asphaltenes in oil, water cut, and oil viscosity. The relative wax deposition rate (i.e., the efficiency of magnetic treatment) shown in Figure 2, Figure 3 and Figure 4 is the ratio of the wax accumulation rate (after MT) to the basic wax accumulation rate without MT for waterless oil with a viscosity of ${\mathsf{\mu }}_{\mathrm{o}}=$ 10 mPa∙s, ${\mathrm{C}}_{\mathrm{o}}=0.0546$, and ${\mathrm{C}}_{\mathrm{asph}}=0.05$. The base wax accumulation rates are 0.071, 0.101, and 0.177 mm/d, at well flow rate of 10, 30, and 100 ton/d, respectively. The influence of the content of polar components in oil on the relative wax deposition rate is presented in Figure 2.

Figure 2 shows that with an increase in the content of polar components in oil at a constant flow rate, the relative wax deposition rate decreases. In addition, with a decrease in the flow rate at a constant content of polar components in oil, the relative wax deposition rate increases.

The influence of the water cut on the relative wax deposition rate is presented in Figure 3.

Figure 3 shows that with an increase in the water cut at a constant flow rate, the relative wax deposition rate decreases. Moreover, with a decrease in the flow rate at a constant water cut, the relative wax deposition rate increases.

The influence of oil viscosity on the relative wax deposition rate for different wells’ waterless flow rates is presented in Figure 4.

Figure 4 shows that with a decrease in oil viscosity at a constant flow rate, the relative wax deposition rate decreases. In addition, with a decrease in the flow rate at a constant oil viscosity, the relative wax deposition rate increases. The presented calculations take into account the change in the diffusion of paraffin fractions, the viscosity of the liquid, and the dielectric constant, depending on the water cut of the oil. The calculations of the dielectric constant of a mixture of oil and water, depending on their volume ratio, were carried out using the Lichtenecker equation. The initial content of paraffin fractions was 0.0546 (wax appearance temperature is 30 °C). The calculations have shown that the use of magnetic units is more effective for oils with a high content of asphaltene, but with an increase in viscosity, the MT efficiency naturally decreases. The efficiency of MT increases with an increase in flow rate and water cut, as long as oil remains in the external phase in the fluid flow (otherwise, the wax deposition on the tubing walls is suspended).

The considered mechanism of the influence of the magnetic field on the wax precipitation does not take into account the mechanism of change in the oil viscosity due to the destruction of super-molecular structures; however, the magnitude of this change is 15–20% of the initial one, and this changes the TBO by the same values.

4. Discussion

Tung et al. in 2001 [11] showed that the effects of magnetic treatment on oil viscosity were strongly dependent on temperature, magnetic intensity, and time of exposition. Magnetic treatment reduces the wax deposition by approximately 20–25%. Tao and Tang in 2014 [12] found that the electrical treatment of oil aiming to reduce its viscosity is most effective at low temperatures, and it is a low-energy-consuming technology. Ma et al. in 2017 [13] showed that lower treatment temperatures, lower shear rates, and higher electric field strengths provided a greater reduction in oil viscosity. The above studies found that electrical treatment leads to the extension of wax crystal size distribution in oil and, consequently, to a reduction in oil viscosity. Chen et al. in 2018 [14] demonstrated that with an increase in the electric field intensity, the wax aggregation extent increases. An electric field leads to an increase in the diffusion coefficient of paraffin molecules. The main cause of the electrical viscosity-reducing effect, according to the above studies, is the increase in the diffusion coefficient. Jing et al. in 2019 [15] proved that magnetic treatment has an orienting effect on wax crystals, disaggregates them, and reduces the temperature of wax appearance.

Based on laboratory studies, the following statements were established (Ma et al., 2019 [16]):

• Electrical treatment reduces the number of wax crystals in the oil volume and increases their size.
• Electrical treatment on waxy oil containing no asphaltenes is effective to reduce viscosity.
• With an increase in the concentration of polar components (asphaltenes) added to the waxy oils, the effect of the electric field on viscosity reduction decreases, the number of precipitated wax increases, and the difference between the particle size distribution of wax crystals in treated and untreated oil becomes less pronounced.
• The oil viscosity reduction increased with the increasing polarity of asphaltenes.

The presented calculations take into account the change in the diffusion of paraffin fractions, the viscosity of the liquid, and the dielectric constant, depending on the water cut of oil. The calculations of the dielectric constant of a mixture of oil and water, depending on their volume ratio, were carried out using the Lichtenecker equation. The initial content of paraffin fractions was 0.0546 (wax appearance temperature is 30 °C). The calculations have shown that the use of magnetic units is more effective for oils with a high content of asphaltenes; however, with an increase in the viscosity, the MT efficiency naturally decreases. The efficiency of MT increases with an increase in the flow rate and water cut, as long as oil remains in the external phase in the fluid flow (otherwise, the wax deposition on the tubing walls is suspended).

The considered mechanism of the influence of the magnetic field on the wax precipitation does not take into account the mechanism of change in the oil viscosity due to the destruction of super-molecular structures; however, the magnitude of this change is 15–20% of the initial one, and this changes the TBO by the same values.

5. Conclusions

• An analytical wax deposition model has been developed based on a detailed consideration of the mechanisms of mass transfer of paraffin fractions dissolved in oil with a growing solid phase both in the oil flow and on the tubing wall. The model accounts for the fact that wax deposits on the tubing surface are a highly efficient heat insulator that changes the temperature regime of the oil flow and the temperature of the tubing walls. The proposed wax deposition model is consistent with the field data, allowing for the calculation of the wax deposits’ profile and the depth of the formation of wax plugs.
• The influence of the magnetic field on wax deposition has been studied. It was found that the passage of the oil flow through a non-uniform magnetic field causes the appearance of a high-intensity electric field for a sufficiently long period. The electric field reduces the solubility of wax in oil, increases the intensity of wax precipitation in the volume of oil, and reduces the wax accumulation on the tubing surface.
• Forecasting the wax deposition rate in time using the developed analytical model allows one to calculate the time between overhauls depending on any methods of controlling wax deposition, including the application of magnetic devices.