Progress in Wax Deposition Characteristics and Prediction Methods for High Pour Point and Viscous Crude Oil Water System

Abstract

With the continuous growth of global energy demand, the exploitation of deepwater oil and gas resources has become an important part of national energy strategies. The high-viscosity crude oil in deepwater areas such as the South China Sea poses severe challenges to oil and gas pipeline transportation due to its high pour point and high viscosity characteristics. Wax deposition, particularly significant under low temperature and high viscosity conditions, can lead to reduced pipeline flow rates, decreased transportation efficiency, and even potential safety hazards. Therefore, in-depth research on the wax deposition characteristics and mechanisms in high-viscosity systems holds significant theoretical and engineering application value. Current research primarily focuses on the influencing factors of wax deposition, deposition mechanisms, and the establishment of prediction models. Studies have shown that external factors such as temperature, shear intensity, operating time, and water content have significant effects on the wax deposition process. Specifically, increased temperature differences accelerate the deposition of wax molecules, while the presence of the aqueous phase inhibits wax crystallization and deposition. Furthermore, the formation mechanisms of wax deposition mainly include molecular diffusion, shear stripping, and aging effects. Researchers have explored the dynamic changes and influencing laws of wax deposition by establishing mathematical models combined with experimental data. In summary, although some progress has been made in studying the wax deposition characteristics in high-viscosity systems, research on wax deposition characteristics in mixtures, especially under the combined action of pour point depressants and flow improvers, is still inadequate. Future research should strengthen the systematic exploration of wax deposition mechanisms, quantify the effects of different external factors, and develop wax deposition prediction models suitable for practical engineering to ensure the safe and stable operation of deepwater oil and gas pipelines. Through in depth theoretical and experimental research, robust technical support can be provided for the efficient development of deepwater oil and gas resources.

1. Introduction

The deepwater areas of China are abundant in petroleum resources, with the South China Sea in particular boasting geological reserves of petroleum estimated at around 30 billion tons [1]. In the context of the gradual depletion of onshore oil and gas resources, deepwater areas have become a strategic alternative region to ensure the security of China’s energy supply. China is also accelerating the development of oil and gas resources in the South China Sea and other deepwater areas from a national strategic level.

Subsea pipelines are the lifelines of deepwater oil and gas field development, serving as the crucial link for transporting offshore oil and gas resources to onshore facilities (Figure 1). They also serve as crucial links between subsea wellheads, offshore platforms, and onshore terminals [2]. Currently, the total mileage of subsea pipelines in China has exceeded 9000 km, with the oil and gas transmission capacity approaching 25% of the country’s total oil and gas transmission volume [3]. The crude oil produced in China’s deepwater areas, such as the South China Sea, is primarily distinguished by its high content of wax, high temperature at which it becomes pourable, and high degree of viscosity and is referred to as easy-to-solidify high-viscosity crude oil (Figure 2). The current development model for deepwater oil and gas fields adopts the approach of connecting subsea wellheads to manifolds for recovery. Under the harsh conditions of deep-sea low temperatures (2~4 °C) and strong heat exchange, the wax molecules in the easy-to-solidify high-viscosity crude oil are highly likely to precipitate and deposit on the inner walls of the subsea pipelines during transportation, forming a wax deposition layer [4]. This will cause a reduction in the transmission capacity of the pipeline and a diminished transportation efficiency and in severe cases, can even lead to blockages, causing safety accidents and significant economic losses (Figure 3). In the process of deepwater oil and gas resource extraction and transportation, wax deposition has become a non-negligible safety hazard in the oil and gas transportation process of subsea pipelines. With the frequent utilization of marine resources in recent years, oil spills at sea have also occurred frequently, severely impacting marine animals and ecosystems [5] At the same time, drilling and fracturing technologies are often employed for reservoir reconstruction in heavy oil reservoirs and waxy deposit reservoirs, which is also an issue of concern for us [6,7].

For single-phase crude oil pipeline flow, the mechanisms of wax deposition can be summarized into four types: molecular diffusion, shear dispersion, Brownian motion, and gravitational settling. Among them, molecular diffusion is the primary cause of wax molecule deposition and the main theoretical support for studying wax deposition models. The influence of Brownian motion and gravitational settling on wax deposition can generally be neglected [8]. Additionally, shear stripping and aging effects are significant factors that lead to changes in the thickness and hardness of the deposited layer [9]. With the use of polymer-based oil displacement agents in offshore oil fields and the application of depressant additives for crude oil transportation, two aspects of engineering issues urgently require attention. On the one hand, the use of drive agents increases the oil recovery rate, but due to the presence of these agents, the emulsion formed by oil and water in two phases becomes more stable, making separation more difficult. On the other hand, the mixed transportation method still occupies a dominant position among subsea pipeline transportation methods. The wax deposition characteristics of the mixture modified by depressants are more complex and unpredictable than the characteristics of wax deposition behavior exhibited by single-phase crude oil. Generally speaking, adding depressants can significantly alter the properties of wax deposits, and different types of depressants have varying effects on reducing wax deposition [10,11].

At present, both domestic and international scholars have extensively explored the issue of wax deposition. However, a comprehensive research system has yet to be established in regards to the challenges associated with wax deposition in mixtures system under the influence of depressants, and there is even less attention to the phenomena of wax accumulation in mixtures system under the combined action of drive agents and depressants, with no research results published in this area. Some domestic scholars [12,13] are also actively utilizing AI models and big data to conduct research activities, and have achieved relatively substantial results, providing directions for subsequent research. Therefore, conducting research on the wax deposition characteristics of the mixture system under the combined action of drive agents and depressants possesses considerable scientific significance for the development of theory of wax deposition in biphasic systems. Furthermore, it has significant engineering implications for guaranteeing the stable operation of subsea pipelines transporting easily solidifying and highly viscous crude oil, and it holds important application prospects in the development of deepwater oil and gas fields.

This paper mainly explores the wax deposition characteristics and prediction methods in high-viscosity crude systems. The Introduction section elaborates on the significance of deepwater oil and gas resource development and the challenges posed by wax deposition. It analyzes the influencing factors of wax deposition, including temperature, shear strength, operation time, and water content, and discusses the formation mechanisms of wax deposition. The paper introduces wax deposition prediction models for single-phase crude oil and crude oil emulsions. It summarizes the limitations of existing research, particularly in the systematic study of wax deposition in mixtures, and proposes directions for future research, aiming to provide theoretical support and technical assurance for the safe and stable operation of deepwater oil and gas pipelines.

2. Wax Deposition Characteristics for High Pour Point and Viscous Crude Oil Water System

2.1. The Influence of Factors on Wax Deposition

Wax deposition in pipelines is a complex thermodynamic and kinetic process that is significantly influenced by external factors. These external factors primarily include temperature, shear strength, operating time, pipe wall material, water content, droplet size, and so on.

2.1.1. The Influence of Temperature

Temperature is a crucial factor influencing wax deposition. When the temperature drops below the wax precipitation temperature, wax will crystallize and deposit. The larger the temperature difference between the oil and the wall, the higher the deposition rate. Temperature also affects the characteristics of the wax deposition layer, the rheological properties of crude oil, and the effectiveness of depressants [14]. Reasonable temperature control can mitigate wax deposition and enhance the efficiency and safety of pipeline transportation.

The wax precipitation characteristics of crude oil are most intuitively reflected by the oil’s intrinsic differential temperature observed between the oil and the pipeline wall. Researches [15,16] indicates that when the oil temperature drops to the wax precipitation peak temperature, the wax deposition rate is significantly higher. To the contrary, crude oil at the wax appearance temperature or temperature exhibits a relatively lower rate of wax deposition. Some studies [17,18] have shown that during isothermal pipeline transportation, the wax deposition rate increases as the oil temperature rises; conversely, during isothermal crude oil transportation, the wax deposition rate increases as the pipeline wall temperature decreases. It is evident that increasing the temperature difference between the crude oil and the pipeline wall will lead to an increased wax deposition rate. More recent studies [19,20] have provided deeper insights, stating that when the pipe wall temperature is fixed, the deposition rate of wax molecules is correlated with their solubility. The solubility of wax molecules increases as the temperature difference between the oil and the wall widens. Consequently, as the temperature of the oil flow decreases, the rate of wax deposition accelerates. Detailed data can be found in Figure 4.

2.1.2. The Influence of Shear Strength

Shear strength influences multiple aspects of wax deposition: A high shear strength can break up wax crystals, increasing their active centers and promoting aggregation and deposition; at the same time, it enhances the shear strength and aging rate of the wax deposition layer. Under low shear strength conditions, the wax deposition rate is higher, whereas under high shear strength, shear forces can disrupt wax crystal aggregation, reducing deposition [21]. Additionally, strong shear may disrupt the molecular chains of depressants, reducing their effectiveness, while weak shear aids in the function of depressants. A reasonable control of shear strength can effectively mitigate wax deposition, improving pipeline transportation efficiency and safety.

During the transportation of crude oil through pipelines, the shear effect strengthens with increasing flow velocity and the wax deposition layer will be thinned to a certain extent due to the shear stripping effect [22]. Compared to turbulent flow, laminar flow exhibits a lower crude oil velocity, resulting in relatively weaker shear effects within the pipe flow and a thinner peeled wax layer. Therefore, under laminar flow conditions, the quantity of wax deposition on the pipe wall is greater [23]. Research of Li et al. [21].indicates that in the mass transfer process, increasing the flow rate can effectively reduce deposition, there is no apparent connection between change in flow rate and the thickness of the deposit under laminar flow conditions. Zhu et al. [24] indicates that the most notable change resulting from increased fluid shear force within the pipeline is the enhancement of the shear effect at the interface between the deposits and crude oil. Additionally, the temperature gradient, concentration gradient of wax molecules, solubility coefficient of wax crystals, and diffusion coefficient of wax molecules at this interface are all affected, which will reduce the thickness of the deposits.

2.1.3. The Influence of Operating Time

The longer the crude oil runs in the pipeline, the thicker the wax deposition layer will gradually increase. At the same time, as caliper of the wax deposition layer increases, the heat dissipation capacity of the crude oil to the surrounding environment relatively decreases. Decreasing the thermal disparity between the crude oil and the pipe wall lead to a slow decrease in the pace of wax deposition [25]. Kiyingi et al. [26] noted that when the oil temperature drops to a specific point, wax starts to precipitate inside the pipeline, gradually forming a solid layer over time. If left unchecked, this layer will continue to accumulate with each passing moment of transportation, ultimately leading to the complete blockage of the pipeline. Research of Eskin et al. [27] indicates that as the duration of crude oil flowing through the pipeline increases, there is a positive correlation between the thickness of wax deposition within the pipeline and the time elapsed.

2.1.4. The Influence of Pipe Wall Material

The material of the pipe wall also has a certain impact on the flow of crude oil in pipelines. Research [28] has shown that the material of the pipe wall can affect the flow state of crude oil and its emulsions, thereby influencing wax deposition. In some studies [29,30], the processes of wax deposition and ice deposition were compared, and both were found to have very similar processes under testing environments. The volume of wax deposition increases with the temperature decreases and the mixing speed slows down. In the experiment conducted by Burmaster et al. [31], six types of alloy materials with differing carbon contents were utilized. When wax comes into contact with oil, each type of deposition system forms a cooled layer on the surface of the wax, resulting in wax deposition. Different materials exhibit different physical properties, thus showing significant differences in deposition behavior. The experimental results indicated that different steel grades resulted in a deposition mass variation ranging from 5% to 15%, a thickness variation exceeding 30%, and a corresponding tendency in the composition of the deposits.

The surface roughness, chemical properties, thermal conductivity, and coating technology of pipe wall materials affect wax deposition [32,33]. Rough surfaces promote wax crystal deposition, while smooth surfaces reduce deposition; chemical properties and coatings can decrease the adhesion force of wax crystals; materials with good thermal conductivity reduce temperature differences, thereby lowering the deposition rate.

2.1.5. Influence of Water Content

Water content is a significant influencing factor for wax deposition in crude oil emulsions. Studies by Adeyanju et al. [34], Li et al. [35], and Fan et al. [36] have indicated that the presence of the aqueous phase hinders the crystallization and deposition of wax molecules. As the percentage of water rises, both the wax deposition rate and the volume of wax deposition in the crude oil emulsion decrease. The influence of the aqueous phase also exhibits a certain threshold. Some studies [37,38] have found that a higher aqueous phase ratio can lead to significant differences in the behavior of emulsions towards crude oil. Specifically, a decrease in water content is accompanied by a simultaneous increase in wax precipitation temperature, but beyond a certain threshold, the wax precipitation temperature and the water content goes up, a certain value drops correspondingly.

Water content significantly affects the wax deposition process by reducing the wax deposition rate, altering the properties of the deposits, influencing deposition mechanisms, and decreasing the wax formation area [39]. A high water content forms a water film, which decreases the diffusion rate of wax molecules and reduces wax deposition. At the same time, it changes the composition of the deposits, weakens molecular diffusion, enhances gelation, narrows the wax formation area, and improves pipeline transportation conditions. The following figure (Figure 5) [40] shows the change of the apparent viscosity (μ) of the oil–water mixture with the water cut (φ_w).

2.1.6. The Influence of Droplet Size

The droplet size and distribution of crude oil emulsions also make a notable difference to wax deposition in emulsions. Studies [41,42] indicate that emulsion droplets have a defined blocking action on the diffusion of wax molecules. As the droplets become smaller and their number increases, the degree of obstruction to the diffusion of wax molecules will correspondingly intensify, bringing about a diminished rate of wax deposition. The research of Xue et al. [43] also indicates that in emulsions containing 15 weight percent paraffin wax, there are numerous water droplets of extremely small size. These droplets can efficiently serve as nucleation points for wax crystal formation, significantly refining the precipitated wax crystals into tiny particles and aiding in the formation of a layer of wax crystal film around them. The research of Chen et al. [44] shows that an increase in the number of droplets strengthens the setup of the deposit, facilitating the precipitation of wax crystals.

2.1.7. The Influence of Pour-Point Depressants

The usage of modifying transportation technology by adding depressants (PPDs) to crude oil has gradually garnered attention regarding the influence of these additives on wax deposition in crude oil. Several studies [45,46,47] indicate that the addition of depressants can effectively inhibit wax deposition, resulting in a significant reduction within the depth of the wax deposit layer and the mass of the deposits. Furthermore, the mass fraction of heavy components with higher carbon numbers in the deposits increases notably. The novel material prepared by Sharma [48] has also experimentally confirmed that depressants significantly decrease the viscosity of crude oil and enhance its fluidity. Through experiments, it was also found to have a significant effect: a marked decrease in cutting down wax deposition. Through experiments, Savulescu [49] found that depressants can successfully prevent the development of wax crystals and indicated in the results that they can successfully reduce the thickness of wax deposition layers.

Depressants effectively inhibit wax deposition by altering the morphology and structure of wax crystals, reducing the wax deposition rate and the thickness of the deposition layer, improving the low-temperature fluidity of crude oil, changing the chemical composition of the deposits, and inhibiting the precipitation and aggregation of wax crystals [50].

2.2. Aging Characteristics of the Wax Deposition Layer

During the wax deposition process, aging occurs within the deposited layer. As deposition time increases, wax molecules will gradually penetrate from the surface of the oil flow and the deposit into the underlying layers. Conversely, oil molecules containing fewer light hydrocarbons and wax diffuse in the opposite direction, from the bottom layer of the deposit towards the surface of the deposit and back into the oil flow. This brings about a continuous augmentation in the hardness of the deposited layer [51,52,53].

Hsu et al. [54] investigated the wax deposition patterns under turbulent flow conditions and found that, under turbulent flow, the deposition layer on the pipe wall undergoes shear forces exerted by the flowing fluid. This causes a gradual increase in both the stiffness of the deposited layer and the mean carbon chain length of the deposited materials undergo alterations as deposition time progresses. Singh et al. [55,56] observed through microscopic examination and gas chromatography analysis that there exists a critical carbon number within the deposited layer. As deposition time increases, wax molecules exceeding the critical carbon number threshold tend to separate and accumulate as deposits on the pipeline surfaces in the oil flow or the substances present on the outer surface of the deposited layer gradually penetrate into its inner layers. Conversely, wax molecules with a carbon number lower than this critical value within the deposited layer diffuse in the opposite direction, moving from the interior towards the surface of the deposited layer or back into the oil flow. This process leads to the occurrence of aging effects. Haj-Shafiei et al. [57], Quan et al. [58,59] and Gao et al. [60] have conducted research indicating that as deposition time increases, the aging effect becomes more pronounced. This leads to a steady rise in content of high carbon number wax molecules in the deposit layer and a continuous reduce in the content of low carbon number wax molecules. Additionally, the diffusion and counter diffusion effects are more intense in higher temperature ranges, which affects both the wax content and the hardness of the deposit layer. In subsea pipeline transportation, the aging process may cause a decrease in the amount of offshore oil reserves and gas resources transported, ultimately resulting in a decrease in the physical lifespan of the pipeline.

2.3. The Influence of Asphaltenes on Wax Deposition

Asphaltenes are highly polar, multi-aromatic ring heavy components in crude oil, characterized by their solubility in benzene/toluene but insolubility in low carbon number alkanes. An analysis of field pipeline deposits reveals that they mainly consist not only of wax but also a significant proportion of asphaltenes. Asphaltenes and wax molecules co-deposit on the pipe wall through a synergistic interaction, significantly affecting both the rate of wax substantially impacting both the speed at which wax accumulates and the total quantity of wax deposited.

Tinsley et al. [61], Alnaimat et al. [62] and Li et al. [63] have conducted investigation into the impact of asphaltenes on wax deposition from the perspectives of deposit thickness, composition of the deposit, and the microstructure of the deposit. They found that asphaltenes in crude oil have the effect of reducing the rate of wax deposition and increasing the fraction of high-carbon wax constituents in the deposit. Additionally, the concentration of asphaltenes in the deposit ranges from 3 to 10 times greater than in the crude oil. Lei et al. [64] pointed out that there is a critical concentration (approximately 0.30 wt%) of asphaltenes that affects the wax deposition rate under static situations. Below this threshold concentration, the rate at which wax deposits is at its peak. If the asphaltene concentration falls below this critical level, the asphaltenes will dissolve and become dispersed throughout the crude oil, thereby impeding the wax molecules’ ability to diffuse and migrate, ultimately resulting in a diminished wax deposition rate. Research by Ali et al. [65] found that during shearing, the crystallization of wax deposit molecular aggregates is inhibited by changes in asphaltene molecular weight, which leads to a higher diffusion rate of wax molecules and has an impact on the makeup of the solid materials that settle down, resulting in an increase in deposition amount. Yang et al. [66] and Li et al. [67] found that asphaltenes cause stratification in the deposits, with asphaltene particles diffusing from the outer layer to the inner layer of the deposit under the action of Brownian motion, and the content of asphaltenes in the deposit gradually increases as the deposition time extends. Research by Lei et al. [68] found that as the amount of dispersed asphaltenes increases, both the wax precipitation temperature and the wax content in crude oil tend to increase. However, as the dispersed asphaltenes continue to increase, an opposite trend emerges. Zhai et al. [69] analyzed the deposition pattern of wax molecules, modulated by asphaltenes. Through molecular dynamics simulations, they found that the growth of wax crystalline clusters is hindered by the aggregation form of asphaltene molecules in the crude oil system. When the asphaltene concentration is 0.2 wt% or below, there exists a direct relationship between the asphaltene concentration and the wax molecule aggregation rate.

Some commonly used types of pour point depressants and their functions are shown in

Table 1. Common types and functions of pour-point depressants.

Type of Pour-Point Depressant	Mechanism of Action	Common Pour-Point Depressants
Surfactant type Pour-Point Depressant	Adsorb on the surface of wax crystals, change the growth morphology and aggregation behavior of wax crystals, inhibit the formation of the three-dimensional network structure of wax crystals, reduce wax deposition, and reduce the adhesion force of wax crystals to prevent aggregation on the pipe wall.	Allyl Glycidyl Ether [70]
Copolymer type Pour-Point Depressant	Such as ethylene vinyl acetate copolymer (EVA) and its modified derivatives, effectively reduce the viscosity of crude oil, improve low-temperature fluidity, reduce wax deposition, change the size and shape of wax crystals to inhibit the formation of the three-dimensional network structure of wax crystals.	EVAM CNT\EVAM g NSiO2 [71,72]
Compound Pour-Point Depressant	The synergy of multiple depressants further enhances the depression effect, more effectively changes the morphology and aggregation behavior of wax crystals, reduces wax deposition, has a better depression effect at low temperatures, and reduces the precipitation of wax crystals. Commonly seen are PPD and PPDA.	PPD\PPDA [73,74]
Other Types of Pour-Point Depressants (Poly (meth) acrylate Series)	Change the size and shape of wax crystals, inhibit the formation of the network structure of wax crystals, and reduce wax deposition.	Tetradecyl Methacrylate Pour Point Depressant [70]

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3. Wax Deposition Prediction Model

3.1. Wax Deposition Model for Single-Phase Crude Oil

The single-phase crude oil wax deposition model is primarily established based on mechanisms such as molecular diffusion, shear detachment, or aging. It utilizes theories from disciplines such as fluid mechanics and heat and mass transfer to determine the amount of wax deposited under different hydraulic and thermal conditions. A multitude of scholars and researchers have established several kinetic models for wax deposition in single-phase crude oil, derived from the foundation of laboratory experiments, field tests, and theoretical analysis.

Burger et al. [75] summarized the mechanisms of wax deposition as molecular diffusion, Brownian motion, and shear dispersion, and pointed out that, compared to molecular diffusion and shear dispersion, the impact of Brownian motion is fairly insignificant, resulting in a significantly smaller amount of wax deposition, which can be neglected. Based on this, they established for the first time a wax deposition model that takes into account the effects of molecular diffusion and shear dispersion. Research by Hamouda et al. [76] points out that although shear dispersion has an impact on wax deposition, molecular diffusion plays a dominant role in the process. In the process of developing the wax deposition model, the effect of shear dispersion was ignored, and instead, researchers formulated a kinetic model for wax deposition. Hsu et al. [54] established a wax deposition model that considers both molecular diffusion and shear effects. The wax deposition on the pipe wall caused by molecular diffusion in their model is essentially the same as the wax deposition model proposed by Burger, but the forms differ. In developing their model, they introduced the concept of wax deposition tendency and, founded upon the critical wax strength, scaled up the model to actual pipelines to obtain a scaled-up model of wax deposition rate for practical applications.

During crude oil transportation via pipelines not all wax molecules precipitate and deposit on the pipe wall due to the scouring action of the oil flow. Therefore, considering the impact of oil flow scouring on wax deposition rate, Huang [77] developed a kinetic wax deposition model utilizing Fick’s diffusion law as the foundational principle. Within this framework, the concept of wax deposition tendency coefficient was presented as a new concept and the main factors influencing this coefficient were analyzed. Based on the mechanism of wax deposition and the conservation of mass transfer during the deposition process, Singh et al. [78,79] proposed a calculation model for wax deposition rate that is applicable to laminar flow conditions. Based on the wax deposition model proposed by Singh, Hernandez et al. [80] considered the impact of shear detachment on wax deposition and proposed an improved model for wax deposition rate as well as a model for calculating the wax content in the deposits.

Similarly, research by Liu et al. [81] further proved the process of wax diffusing from the initial stage to the surface of the container and forming aggregates that detach under shear flow conditions. They pointed out that an increase in shear rate enhances wax precipitation: it prompts more wax to adhere to the already formed aggregated precipitate layer. In the latest research, Sousa et al. [37] emphasized the dominant role of molecular diffusion in the mechanism of wax deposition. By incorporating the mechanism of shear dispersion, Sousa derived a wax deposition model that is primarily dominated by molecular diffusion. With technological advancements, Wang et al. [82] have incorporated the effects of magnetic fields and depressants on wax deposition into their considerations. They have found that as the frequency and intensity of the magnetic field increase, the deposited mass of wax decreases significantly. Based on these findings, they have proposed a computational model for inhibiting the number of wax crystal particles using magnetic fields. Obaseki [83], by introducing the wax deposition mass flux equation, comprehensively considered the relationship between fluid velocity and time, and determined the wax molecule mass flux equation. He studied the combined equation that describes wax molecule deposition and hardening in shear environments. His team made more precise measurements, revealing that the thickness of the deposition layer gradually decreases over time as it accumulates. Based on these findings, he reformulated the calculation model to validate his hypotheses.

The equations of these wax deposition models for single-phase crude oil are shown in

Table 2. Wax deposition models for single-phase crude oil.

Models	Equations	Notes
Burger	$W={\rho }_{\mathrm{L}}{D}_{\mathrm{m}}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}+k^{*}{C}_{\mathrm{w}}^{*}{\left({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}\right)}_{\mathrm{w}}$	w $\mathrm{is} \mathrm{wax} \mathrm{deposition} \mathrm{rate}; {\rho }_{\mathrm{L}}$ $\mathrm{represents} \mathrm{wax} \mathrm{density}; D_{\mathrm{m}}$ $\mathrm{represents} \mathrm{coefficient} \mathrm{of} \mathrm{diffusion} \mathrm{for} \mathrm{wax} \mathrm{molecules}; \mathrm{C} \mathrm{represents} \mathrm{wax} \mathrm{volume} \mathrm{fraction}; \mathrm{r} \mathrm{is} \mathrm{pipeline} \mathrm{radius}; k^{*}$ $\mathrm{represents} \mathrm{the} \mathrm{constant} \mathrm{for} \mathrm{wax} \mathrm{deposition} \mathrm{rate}, \mathrm{determined} \mathrm{experimentally}; C_{\mathrm{w}}^{*}$ represents the percentage of volume occupied by wax crystals at the pipe wall; u represents the speed at which oil is flowing at the pipe wall.
Hamouda	$W={\displaystyle \frac{M_{\mathrm{w}}}{N_{\mathrm{A}}}}{\displaystyle \frac{B}{\mu }}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\left({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}\right)}_{\mathrm{w}}$	$M_{\mathrm{w}}$ $\mathrm{is} \mathrm{molar} \mathrm{mass} \mathrm{of} \mathrm{wax} \mathrm{molecules}; N_{\mathrm{A}}$ is Avogadro’s constant; $\mu$ is crude oil viscosity; B is undetermined parameter.
Hsu	$W_{\mathrm{F}}={\Omega }_{\mathrm{M}}{\left({\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\right)}_{\mathrm{F}}{\left(vD{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}L}}\right)}_{\mathrm{F}}$	$\mathrm{y} \mathrm{is} \mathrm{wax} \mathrm{build}\text{-}\mathrm{up} \mathrm{extent};{\Omega }_{\mathrm{M}}$ represents wax deposition tendency coefficient; D represents inner diameter of pipeline; L represents pipeline length.
Huang	$W=k{\tau }_{\mathrm{w}}^m{\displaystyle \frac{1}{\mu }}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\left({\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}\right)}^{n+1}$	${\tau }_{\mathrm{w}}$ is shear stress at pipe wall; k, m, n are undetermined coefficients that need to be determined through experiments.
Singh	$W={\displaystyle \frac{D_{\mathrm{m}}}{\left(1-C\right)\zeta }}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\displaystyle \frac{Nu}{R}}\left(T_{\mathrm{h}}-T_{\mathrm{w}}\right)$	$\zeta$ $\mathrm{is} \mathrm{the} \mathrm{thickness} \mathrm{of} \mathrm{the} \mathrm{cloud} \mathrm{point} \mathrm{layer}; \mathrm{Nu} \mathrm{is} \mathrm{Nusselt} \mathrm{number}; \mathrm{R} \mathrm{is} \mathrm{circulating} \mathrm{flow} \mathrm{radius}; T_{\mathrm{h}}$ $\mathrm{is} \mathrm{the} \mathrm{temperature} \mathrm{at} \mathrm{the} \mathrm{center} \mathrm{of} \mathrm{pipeline}; T_{\mathrm{w}}$ is the temperature of pipeline wall.
Hernandez	$W={\displaystyle \frac{J_{\mathrm{c}}\left[1-\varphi \left(F_{\mathrm{w}}\right)\right]-J_{\mathrm{s}}}{{\rho }_{\mathrm{L}}{F}_{\mathrm{w}}}}$ ${\displaystyle \frac{\mathrm{d}{F}_{\mathrm{w}}}{\mathrm{d}t}}={\displaystyle \frac{2\left[J_{\mathrm{c}}\varphi \left(F_{\mathrm{w}}\right)\right]\left(r-y\right)}{{\rho }_{\mathrm{L}}y\left(2r-y\right)}}$ $J_{\mathrm{c}}=k_{\mathrm{c}}\left(C_{\mathrm{b}}-C_{\mathrm{i}}\right)$ $\varphi \left(F_{\mathrm{w}}\right)=\left(1+{\displaystyle \frac{{\alpha }^2{F}_{\mathrm{w}}^2}{1-F_{\mathrm{w}}}}\right)$	$J_{\mathrm{c}}$ $\mathrm{is} \mathrm{mass} \mathrm{flow} \mathrm{rate} \mathrm{of} \mathrm{wax} \mathrm{from} \mathrm{oil} \mathrm{flow} \mathrm{to} \mathrm{deposition} \mathrm{interface}; F_{\mathrm{w}}$ $\mathrm{is} \mathrm{mass} \mathrm{fraction} \mathrm{of} \mathrm{wax} \mathrm{in} \mathrm{sediments}; J_{\mathrm{c}}$ $\mathrm{is} \mathrm{mass} \mathrm{flow} \mathrm{rate} \mathrm{of} \mathrm{wax} \mathrm{that} \mathrm{is} \mathrm{sheared} \mathrm{off}; k_{\mathrm{c}}$ $\mathrm{is} \mathrm{convective} \mathrm{mass} \mathrm{transfer} \mathrm{coefficient}; C_{\mathrm{b}}$ $\mathrm{is} \mathrm{mass} \mathrm{fraction} \mathrm{of} \mathrm{wax} \mathrm{in} \mathrm{oil} \mathrm{flow}; C_{\mathrm{i}}$ is mass fraction of wax at deposition interface; $\alpha$ is shape parameter of wax crystal particles.
Sousa	${\displaystyle \frac{d^2}{D_M}}={\displaystyle \frac{D^2\mu \ast {10}^3\ast {(V_{SOULTE}\ast {10}^6)}^{0.6}}{7.4\ast {10}^{-12}\left(MW\ast {10}^3\right)\frac{1}{2}T}}$	$\mathrm{M}\mathrm{W}$, molecular weight of solvent;
Martins	$\begin{array}{l}{\displaystyle \frac{\mathrm{d}m}{DT}}=2{\pi }_{d}L(-D_{12}{\displaystyle \frac{\partial c}{\partial r_d}}{|}_{12})\\ {}^{•}j=-D_{23}{\displaystyle \frac{\partial c}{\partial r_d}}\\ ({\displaystyle \frac{\partial m}{\partial \eta }}·{\displaystyle \frac{d\eta }{DT}})=2\pi r_d{L}^{•}j=2\pi r_{d}L(-D_{23}{\displaystyle \frac{\partial c}{\partial r_d}}{|}_{23})\end{array}$	${}^{•}j$ $\mathrm{represents} \mathrm{the} \mathrm{mass} \mathrm{flux} \mathrm{of} \mathrm{internal} \mathrm{diffusion};$
Wang	$\log [-\ln (1-X(t))]=\log K+n\log (t)$	X(t) represents the crystallinity; t represents time.

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3.2. Wax Deposition Model for Crude Oil Emulsion

The presence of the aqueous phase and microdroplets can affect the diffusion pathways and heat transfer characteristics of wax molecules, making the mechanism and process of wax deposition in crude oil emulsions more complex. Currently, the construction of wax deposition models for crude oil emulsions, both domestically and internationally, is typically based on improvements made to single-phase crude oil wax deposition models.

Drawing upon the Singh single-phase wax deposition model for crude oil, Couto et al. [84] replaced the oil parameters with those of crude oil emulsions to develop a predictive model for wax deposition specifically tailored to water-in-oil emulsion. Based on the Couto emulsion wax deposition model, Bruno et al. [85] introduced the role of water content into the relationship for the molecular diffusion coefficient and proposed a modified emulsion wax deposition model. Assuming that the water-in-oil emulsion is a pseudo-single-phase fluid, Wang [86] believes that the primary factors controlling the wax deposition process in water-in-oil emulsions are the combined actions of molecular diffusion and gelation adhesion mechanisms. According to this model, a modeling approach for growth rate of wax deposition thickness in water-in-oil emulsion was established. Fan et al. [87,88] believed that molecular diffusion and aging serve as the primary drivers of wax deposition. Taking into account the impact of pipe flow erosion, but neglecting effects like shear dispersion, gravitational settling, and Brownian diffusion, a wax deposition model for water-in-oil emulsion in pipeline flow was established. Liu [19] found that the wax deposition rate is relatively high in the initial stage. However, as the deposition thickness increases, the thermal insulation properties of the wax deposits enhance, resulting in a decrease in the deposition rate. Furthermore, the driving force for mass deposition diminishes. Once the oil reservoir interface reaches the wax appearance temperature (WAT), the growth of the wax deposit will cease. Li [89] integrated the gelation effect and comprehensively considered molecular diffusion and gelation phenomena, leading to the discovery of a wax deposition model that differs from the one solely based on molecular diffusion. He found that the thickness of deposits in emulsions will first rise to a certain peak and then decrease an increase in water content. This model fills the gap in understanding wax deposition in emulsions under conditions of lower water content.

The equations of these wax deposition models for crude oil emulsion are shown in

Table 3. Wax deposition models for crude oil emulsion.

Models	Equations	Notes
Liu	$\begin{array}{l}{\rho }_{dep}\overline{F}(t){\displaystyle \frac{d{\delta }_{\xi }}{dt}}=J_{wax,\xi }=J_{A,\xi }-J_{B,\xi }\\ {\rho }_{dep}\overline{F}(t){\displaystyle \frac{d{\delta }_{\eta }}{dt}}=J_{wax,\eta }=J_{A,\eta }-J_{B,\eta }\\ {\rho }_{dep}{\displaystyle \frac{d\overline{F}}{dt}}={\displaystyle \frac{J_{B,\xi }}{{\delta }_{\xi }}}+{\displaystyle \frac{J_{B,\eta }}{{\delta }_{\eta }}}\end{array}$	${\displaystyle \frac{d{\delta }_{\xi }}{dt}}$ $\mathrm{is} \mathrm{the} \mathrm{rate} \mathrm{of} \mathrm{thickness} \mathrm{growth}; J_{A,\xi }$ $\mathrm{represents} \mathrm{the} \mathrm{difference} \mathrm{in} \mathrm{convective} \mathrm{diffusive} \mathrm{mass} \mathrm{flux} \mathrm{from} \mathrm{the} \mathrm{pipeline} \mathrm{center} \mathrm{to} \mathrm{the} \mathrm{oil} \mathrm{bed} \mathrm{interface} \mathrm{direction}; J_{B,\xi }$ is the diffusive mass flux entering wax deposition.
Bruno	$f_{\mathrm{w}}^{\mathrm{d}}=0.0283{\mathrm{e}}^{2.4184f_{\mathrm{w}}^{\mathrm{b}}}$ $D_{\mathrm{o}/\mathrm{w}}=D_{\mathrm{w}/\mathrm{o}}\left(1-f_{\mathrm{w}}^{\mathrm{b}}\right)$	$f_{\mathrm{w}}^{\mathrm{d}}$ $\mathrm{is} \mathrm{water} \mathrm{content} \mathrm{of} \mathrm{sediments}; f_{\mathrm{w}}^{\mathrm{b}}$ $\mathrm{is} \mathrm{water} \mathrm{content} \mathrm{of} \mathrm{mixture}; D_{\mathrm{o}/\mathrm{w}}$ $\mathrm{is} \mathrm{molecular} \mathrm{diffusion} \mathrm{coefficient} \mathrm{of} \mathrm{O}/\mathrm{W} \mathrm{emulsion}; D_{\mathrm{w}/\mathrm{o}}$ is molecular diffusion coefficient of W/O emulsion.
Wang	$W={\displaystyle \frac{{\rho }_{\mathrm{oil}}}{{\rho }_{\mathrm{dep}}}}\left[{\displaystyle \frac{D_{\mathrm{wo}}\frac{\mathrm{d}{C}_{\mathrm{wax}}}{\mathrm{d}T}\frac{\mathrm{d}T}{\mathrm{d}r}}{{F^{\prime }}_{\mathrm{w}}}}+{\displaystyle \frac{k\left(C-C_{\mathrm{int}}\right)}{{F^{\prime }}_{\mathrm{w}}\left(1-{\varphi }_{\mathrm{wt}}\right)}}\exp \left(T_{\mathrm{pp}}/T_{\mathrm{int}}\right)f\right]$	${\rho }_{\mathrm{dep}}$ $\mathrm{is} \mathrm{sediments} \mathrm{density}; D_{\mathrm{wo}}$ $\mathrm{is} \mathrm{molecular} \mathrm{diffusion} \mathrm{rate} \mathrm{of} \mathrm{wax} \mathrm{molecules}; C_{\mathrm{wax}}$ $\mathrm{is} \mathrm{solubility} \mathrm{of} \mathrm{wax} \mathrm{molecules}; \mathrm{f} \mathrm{is} \mathrm{Fanning} \mathrm{friction} \mathrm{coefficient}; C_{\mathrm{int}}$ $\mathrm{is} \mathrm{solubility} \mathrm{of} \mathrm{wax} \mathrm{molecules} \mathrm{at} \mathrm{oil} \mathrm{sediment} \mathrm{interface}; T_{\mathrm{pp}}$ $\mathrm{is} \mathrm{pour} \mathrm{point} \mathrm{of} \mathrm{crude} \mathrm{oil}; T_{\mathrm{int}}$ $\mathrm{is} \mathrm{temperature} \mathrm{at} \mathrm{oil} \mathrm{sediment} \mathrm{interface}; {F^{\prime }}_{\mathrm{w}}$ $\mathrm{is} \mathrm{wax} \mathrm{content} \mathrm{of} \mathrm{sediments} \mathrm{after} \mathrm{removal} \mathrm{of} \mathrm{the} \mathrm{water} \mathrm{phase}; {\varphi }_{\mathrm{wt}}$ is mass water content in oil flow.
Fan	$W={\displaystyle \frac{k\left(D_{\mathrm{wo}}-D_{\mathrm{e}}\right){\left(\frac{{\mu }_{\mathrm{e}}}{{\mu }_{\mathrm{o}}}\right)}^f\frac{\mathrm{d}{C}_{\mathrm{w}}}{\mathrm{d}r}\exp \left(\frac{{\tau }_{\mathrm{cv}}-\tau }{{\tau }_{\mathrm{cv}}}\right)}{{\rho }_{\mathrm{dep}}{\dot{\phi }}_{\mathrm{w}}}}$	$D_{\mathrm{e}}$ $\mathrm{is} \mathrm{effective} \mathrm{wax} \mathrm{molecule} \mathrm{diffusion} \mathrm{coefficient} \mathrm{in} \mathrm{sediments}; {\mu }_{\mathrm{o}}$ $\mathrm{is} \mathrm{crude} \mathrm{oil} \mathrm{viscosity}; {\mu }_{\mathrm{e}}$ $\mathrm{is} \mathrm{emulsion} \mathrm{viscosity}; {\dot{\phi }}_{\mathrm{w}}$ $\mathrm{is} \mathrm{net} \mathrm{increase} \mathrm{in} \mathrm{wax} \mathrm{content} \mathrm{of} \mathrm{sediments}; {\tau }_{\mathrm{cv}}$ is critical shear stress.
Couto	${\mu }_{\mathrm{e}}={\mu }_{\mathrm{c}}{\left(1-{\varphi }_{\mathrm{i}}\right)}^{-2.5}$ $k_{\mathrm{m}}=k_{\mathrm{o}}\left[1+{\displaystyle \frac{3F_{\mathrm{c}}\left(k_{\mathrm{w}}-k_{\mathrm{o}}\right)}{k_{\mathrm{w}}+2k_{\mathrm{o}}}}-F_{\mathrm{c}}\right]$ $k_{\mathrm{d}}=\left[{\displaystyle \frac{2k_{\mathrm{w}}+k_{\mathrm{m}}+\left(k_{\mathrm{w}}-k_{\mathrm{m}}\right)F_{\mathrm{w}}}{2k_{\mathrm{w}}+k_{\mathrm{m}}-2\left(k_{\mathrm{w}}-k_{\mathrm{m}}\right)F_{\mathrm{w}}}}\right]k_{\mathrm{m}}$	${\mu }_{\mathrm{e}}$ $\mathrm{is} \mathrm{emulsion} \mathrm{viscosity}; {\mu }_{\mathrm{c}}$ $\mathrm{is} \mathrm{the} \mathrm{apparent} \mathrm{viscosity} \mathrm{refers} \mathrm{to} \mathrm{the} \mathrm{viscosity} \mathrm{characteristic} \mathrm{of} \mathrm{the} \mathrm{continuous} \mathrm{phase}; {\varphi }_{\mathrm{i}}$ $\mathrm{is} \mathrm{volume} \mathrm{fraction} \mathrm{of} \mathrm{dispersed} \mathrm{phase}; k_{\mathrm{m}}$ $\mathrm{is} \mathrm{thermal} \mathrm{conductivity} \mathrm{of} \mathrm{emulsion}; k_{\mathrm{d}}$ $\mathrm{is} \mathrm{the} \mathrm{thermal} \mathrm{conductivity} \mathrm{of} \mathrm{dispersed} \mathrm{phase} \mathrm{within} \mathrm{sediments}; k_{\mathrm{o}}$ $k_{\mathrm{w}}$ $\mathrm{are} \mathrm{thermal} \mathrm{conductivity} \mathrm{of} \mathrm{the} \mathrm{oil} \mathrm{phase}, \mathrm{water} \mathrm{phase} \mathrm{in} \mathrm{deposited} \mathrm{layer} \mathrm{of} \mathrm{emulsion}; F_{\mathrm{c}}$ is water content of emulsion.
Li	${\displaystyle \frac{d\delta }{dt}}={\displaystyle \frac{{\rho }_{oil}}{{\rho }_{dcp}}}[{\displaystyle \frac{D_{ow}\frac{d{C}_{wax}}{dT}\frac{dT}{dr}}{F}}+{\displaystyle \frac{k(C-C_{\mathrm{int}})}{F_w(1-{\varphi }_{wt}\exp (\frac{T_{PP}}{T_{\mathrm{int}}})f)}}]$	C represents the wax molecule concentration in the bulk of the oil; k represents the gelation deposition coefficient; f represents the Fanning friction factor.

.

3.3. The Application of Wax Deposition Prediction Models in Practice

These models are developed based on different physical properties, but from the key parameters, we can deduce that temperature variation plays the most direct role in the wax deposition process. In actual flow conditions, shear forces predominantly influence the rate of wax deposition. Additionally, the moisture content of the oil itself affects the crystallization rate of waxy substances. Therefore, these models can guide various aspects of oil and gas storage and transportation.

(1) They can inform the optimization of pipeline design and material selection. By evaluating the impact of different design schemes and materials on wax deposition, engineers can formulate more reasonable pipeline layouts and material selections, thereby reducing the risk of wax deposition.

(2) Combined with real-time monitoring technologies, as well as the latest large databases and artificial intelligence-assisted computations, field workers can monitor parameters such as temperature, pressure, and flow rate within the pipeline during transportation. This enables timely prediction of wax deposition and the establishment of early warning systems, allowing operators to take corresponding maintenance measures.

(3) These models also serve as important indicators for optimizing operating conditions, such as adjusting fluid temperature, selecting appropriate depressants, and flow improvers to effectively suppress wax deposition and ensure the normal operation of the pipeline.

(4) In terms of maintenance and cleaning strategies, based on the wax deposition predictions from the models, operators can formulate scientific maintenance plans, reasonably arrange pipeline inspection and cleaning cycles, and reduce maintenance costs. An economic benefit analysis can also be achieved through these models, helping enterprises assess the impact of wax deposition on transportation efficiency, formulate more cost-effective operational strategies, and implement corresponding risk management measures to reduce safety hazards and economic losses caused by wax deposition.

(5) The research and application of wax deposition prediction models will drive the development of new technologies, such as the research and development of new depressants, improvements in pipeline insulation technologies, and the development of new pipeline materials, providing strong support for the sustainable development of the oil and gas industry.

Therefore, the practical application of wax deposition prediction models will provide important safeguards for the safe and stable operation of deepwater oil and gas pipelines, promoting technological progress and economic benefit enhancement in the industry.

4. Research Outlook and Recommendations

Currently, both domestic and international investigations of wax deposition in single-phase crude oil, as well as the mechanisms of wax deposition in crude oil emulsions, is relatively mature. Additionally, there is a significant amount of research on the wax deposition of single-phase crude oil that has been modified by using depressants. Yet, there is still insufficient systematic research on the wax deposition of mixtures under the action of depressants, and even more so for the wax deposition of mixtures under the combined action of drive agents and depressants. There has been little focus on this area, and no research results have been published to date.

Research on the impact of external factors on wax deposition and the aging pattern of wax deposit layers has largely remained at the stage of wax deposition in single-phase crude oil or crude oil emulsions without additives, and most of the findings are limited to qualitative conclusions. On the one hand, there is a lack of discussion on the wax deposition issues in mixtures under the combined action of drive agents and depressants; on the other hand, there is a lack of research on the quantitative relationship between wax deposition, the aging of wax deposition layers, and their influencing factors, making it difficult to quantitatively characterize the wax deposition characteristics of mixtures. Research on the impact of asphaltenes on wax deposition is still largely focused on the deposition characteristics of single-component waxes, with a lack of discussion on the synergistic action patterns between asphaltenes and wax. This situation hinders in-depth research on the mechanisms by which asphaltenes affect wax deposition. There is still a lack of attention on the synergistic deposition of asphaltenes and wax under the combined effects of oil-displacement agents and depressants.

It is essential to delve deeply into research on the influence patterns of various external factors (such as the content of oil-displacement agents, depressants, temperature, shear strength, operating time, water content, droplet size, emulsion viscosity, etc.) on wax deposition and deposit aging in mixtures before and after the addition of additives. Additionally, in-depth research should be carried out on the synergistic mechanism and patterns of asphaltenes and wax while deposition is occurring. Furthermore, it is necessary to explore the quantitative relationships between the aforementioned multi-dimensional influencing factors and wax deposition characteristics, so as to fully understand the wax deposition characteristics of mixed systems under the combined action of drive agents and depressants.

Current research on wax deposition prediction models has several limitations: (1) Domestically and internationally, research on single-phase crude oil wax deposition models started earlier, and these models are relatively mature. In contrast, there are fewer research results on wax deposition models for mixed systems, and they have not yet formed a systematic body of knowledge. (2) There is still a lack of attention on the impact of the combined effects of oil-displacement agents and depressants on wax deposition, and this has not been reflected in the models. (3) The established wax deposition models all contain undetermined parameters, which means that every time these models are applied to actual pipeline process calculations, relevant undetermined parameters still need to be ascertained through necessary experiments. This undoubtedly weakens the convenience of applying these models. (4) Previous researchers have been limited in the types of crude oil used in their modeling (mostly only types of crude oil), and they have not fully considered the diversity of crude oil properties. The models they developed have not been quantitatively linked to the properties of crude oil, resulting in a lack of universal applicability of the models across different types of crude oil. (5) The established wax deposition models only consider the deposition process of single-component wax molecules, and do not account for the synergistic deposition between asphaltenes and wax. This does not align with the actual situation where pipeline wax deposits contain a high proportion of asphaltenes.

In future research, it is recommended to conduct in-depth studies on the wax deposition characteristics of crude mixtures and provide quantitative descriptions. Based on these findings, a comprehensive consideration should be given to the diversity of crude oil physical properties, the synergistic effects between asphaltenes and wax, as well as the combined effects of oil displacement agents and depressants. A universal wax deposition prediction model for crude mixtures, utilizing quantitative analysis of crude oil’s physical attributes, and framework, should be developed to offer theoretical support for ensuring the safe operation of deepwater subsea pipelines.