Progress on Wax Deposition Characteristics and Prediction Methods for Crude Oil Pipelines

Abstract

During the pipeline transport of high-wax crude oil, paraffin precipitation often results in deposit formation, and a thorough investigation into the issue of wax deposition is crucial for ensuring the safe operation and economic benefits of the pipeline. This work critically reviews the latest research progress focusing on the mechanisms, the factors influencing it, and the kinetic models used to describe it. Although research on single-phase crude oil wax deposition has made certain progress both domestically and internationally, existing studies have limitations in terms of the diversity of crude oil types. In previous studies, the types of crude oil used to explore influencing factors were relatively singular. When modeling, the diversity of crude oil properties was not fully considered, leading to a lack of general applicability of the established models across different types of crude oil. To overcome this limitation, future research should place greater emphasis on the diversity of crude oil properties. Specifically, it is necessary to collect and analyze wax deposition data from a wider variety of crude oils and delve into the mechanisms by which different crude oil properties influence the wax deposition process. Therefore, future research needs to further take into account the diversity of crude oil properties and establish kinetic models that are quantitatively correlated with these properties. This will contribute to more accurate prediction and assessment of wax deposition risks during pipeline transportation for various types of crude oil, thereby providing robust assurance for the safe operation of pipelines.

1. Introduction

In view of the increasingly dwindling onshore crude oil resources, China is actively adjusting its strategic direction and enhancing efforts to exploit marine petroleum reserves [1]. In the transportation of oil and gas resources, especially for the long-distance and large-scale transportation of crude oil, China relies heavily on efficient and safe pipeline transportation systems (Figure 1).

Oil pipelines serve as the lifeblood for energy transportation and the hub for the long-distance allocation of oil and gas resources [2]. Currently, China’s onshore crude oil pipelines have exceeded 33,000 km, equivalent to 20% of the country’s total trunk pipeline mileage for energy transportation. The total mileage of submarine pipelines has surpassed 9000 km, and these marine transmission systems account for approximately 25% of domestic oil and gas volume movements [3]. By 2025, the global cumulative length of oil and gas transmission pipelines has reached 2.8 million kilometers, expanding at an annual growth rate of approximately 50,000 km. The United States leads the world in pipeline infrastructure, with over 290,000 km of crude oil pipelines and more than 300,000 km of natural gas pipelines, collectively constituting the most extensive pipeline network globally [4]. Over 80% of China’s crude oil production has a high pour point, high viscosity, high wax content, complex rheological properties, and poor low-temperature fluidity. During transportation, wax deposition inevitably occurs. Paraffin accumulation along the pipe interior reduces the effective flow area, resulting in both elevated system pressure and reduced throughput capacity. In severe cases, it may even threaten the secure operation of pipelines, resulting in economic losses for the enterprises engaged in crude oil extraction and distribution, and undermining the effective utilization of energy [5,6,7] (Figure 2).

Therefore, conducting research on crude oil wax deposition is crucial. Currently, many have achieved comprehensive and in-depth research results in the exploration of crude oil wax deposition mechanisms, analysis of influencing factors, and model establishment [8].

Crude oil is a complex mixture mainly of carbon, hydrogen, sulfur, nitrogen, and oxygen, with carbon and hydrogen being dominant [9]. Physically, its density is 0.75–0.95 and viscosity varies widely from light to heavy crude, being sensitive to temperature and pressure. Chemically, it has hydrocarbons and non-hydrocarbons. Properties like acidity, pour point, and sulfur content vary by origin. These properties are important for crude oil’s extraction, transport, refining, and model development, especially for wax deposition kinetic models [10]. For example, viscosity and composition affect pyrolysis and processing, and sulfur content relates to environmental and product quality [11]. In wax deposition models, crude oil properties influence wax molecule behavior. In the Huang Qiyu model, parameters are related to crude oil properties. So, knowing crude oil properties accurately is key for precise wax deposition models, helping predict and reduce wax deposition’s impact on pipeline transportation.

2. Mechanism of Wax Deposition in Crude Oil Pipelines

The wax deposition is a complex and multi-factorial phenomenon that occurs during the process of transporting waxy crude oil through pipelines. Early researchers proposed that molecular diffusion, shear dispersion, Brownian motion, and gravitational settlement were the primary mechanisms of wax deposition. With the continuous advancement and deepening of technology and theory, scholars have discovered a series of mechanisms such as shear stripping effect, aging mechanism, and gelation.

Molecular diffusion refers to the process where as crude oil flows through a pipeline, a radial thermal gradient develops across the pipeline cross-section. This gradient causes differences in solubility within the crude oil, leading to the formation of a concentration differential of wax molecules. This gradient drives the wax molecules to diffuse and deposit from high-concentration areas (pipeline center) to low-concentration areas (pipeline wall) [12]. Burger et al. [13] found in their research that molecular diffusion is the predominant factor responsible for wax deposition under high temperature conditions. Correra et al. [14] proposed that the flow regimes of crude oil have a certain impact on generation and formation of wax deposition. They suggested that when the fluid is in a turbulent state, the formation of the paraffin deposits near the pipeline wall is related to molecular diffusion. After conducting a large number of experiments and data analysis comparisons, Akbarzadeh et al. [15] ultimately proposed that among the various mechanisms contributing to wax deposition, molecular diffusion stands out as the predominant mechanism. Yang et al. [16] based on extensive experiments, verified molecular diffusion as the primary factor in wax deposition. These researchers additionally emphasized the necessity of incorporating waxy crude’s non-Newtonian behavior when applying diffusive transport models to predict wax deposition patterns.

Shear dispersion occurs during the flow of crude oil where under shear stress influence, wax molecules migrate from higher-shear regions to lower-shear regions under the influence of velocity gradients, ultimately depositing within the pipeline. In numerous experiments, Burger et al. [13] found that shear dispersion plays a minor role in the process of wax deposition. However, under low-temperature and low-heat-flux conditions, shear dispersion becomes the primary deposition mechanism. Bern et al. [17] emphasized that shear dispersion significantly affects the radial diffusion dynamics of wax molecules. Cai et al. [18], using wax deposition loop experiments, confirmed that the contribution of shear dispersion in the process of wax deposition can be neglected. Through comparisons between field and laboratory experiments, Huang et al. [19] analyzed the impact of shear on wax deposition under conditions with or without temperature differences. They observed that the outcome caused by shear dispersion is fairly insignificant under certain conditions. Wang et al. [20], through experiments, believed that shear dispersion dominates when the mass of oil in the pipeline is below a critical flow rate. Due to differences in research methods and experimental conditions among different scholars, there is still controversy regarding how shear dispersion influences wax deposition. This controversy reflects the complexity of the shear dispersion mechanism under different conditions and the uncertainty of its impact on wax deposition.

Brownian motion refers to the irregular motion of wax molecules or wax crystal particles in a pipeline, which is caused by continuous collisions with surrounding fluid molecules due to the thermal motion of these molecules. When there is a temperature difference within the pipeline, wax crystal particles, driven by Brownian motion, gradually relocates from the middle of the oil flow and adheres to the pipeline wall [21]. Gravitational settlement occurs due to the fact that the precipitated wax crystals are denser than the crude oil, causing them to settle under the action of gravity. In low-flow or static conditions, the shear force of the fluid is insufficient to prevent the wax crystal particles from sinking [22]. In current experiments, it is generally believed that the effects of Brownian motion and gravitational settlement can be neglected during the wax deposition [23].

Beyond the earlier-mentioned mechanisms related to wax deposition, there is also a phenomenon termed aging. Based on previous research, Hoteit et al. [24] defined the aging process as the gradual hardening of wax deposits on the pipeline’s inner surface over time. This phenomenon occurs because wax already on the pipeline wall encapsulates oil within a wax-based network structure. Within the formed wax deposition layer, wax molecules diffuse towards the bottom of the deposition, while oil molecules diffuse in the opposite direction, starting at the base of the deposition layer and extending towards the exterior and the core region of the oil flow [25]. Wang et al. [26] conducted research using a cold finger experimental setup to investigate crude oil temperature, temperature gradient across the oil–wall interface, different temperature ranges when the crude oil temperature difference is the same, carbon number distribution, and waxy crude oil. Their study reveals that wax deposition is more strongly affected by the oil–wall thermal gradient compared to other variables such as crude oil temperature, carbon number distribution, and wax concentration in the oil. Quan et al. [27] employed a cold finger experimental setup to delve into the aging characteristics of wax deposition and the role that temperature plays in this aging process. The experiments revealed that the aging process increases the content of a high number of carbon atoms in the deposition layer and in the abundance of wax molecules with a low number of carbon atoms. Additionally, diffusion and counter-diffusion intensify at higher temperatures, leading to more significant changes in carbon number. The narrower the temperature span, the earlier the maximum wax content in the deposition layer appears.

3. Influencing Factors of Wax Deposition

The process of wax deposition in waxy crude oil pipeline transportation is extremely complex and is influenced by a combination of multiple factors, for instance, the temperature gradient between the oil and the pipeline wall and the temperature of crude oil, shear action, deposition time, and crude oil properties.

3.1. Influence of Crude Oil Temperature on Wax Deposition

Current research generally believes that temperature is a crucial factor influencing the crude oil wax deposition rate [28]. Theyab [29] through experimental investigations discovered that wax deposition takes place during the transportation of crude oil at reduced temperatures, with the issue intensifying when the oil’s temperature falls beneath the wax appearance temperature (WAT). Kouhi et al. [30] found that wax deposition takes place as the crude oil temperature declines below the WAT. They also deem the WAT as a vital component that plays a crucial role in wax deposition modeling, development, testing, and selection of suitable wax inhibitors. Huang et al. [31] discovered that the wax deposition rate exhibits fluctuations across various temperature intervals, and there is a specific temperature range where wax deposition reaches its maximum level. With the wall temperature being held constant, an incremental rise in the oil temperature leads to a corresponding gradual increase in the wax deposition rate. Through experiments with various formulated artificial oil samples, Bai [32] believes that wax deposition is not significant when the artificial oil sample’s temperature is above the WAT but below the pour point. Wax deposition occurs within the temperature interval from the WAT to the pour point, and wax formation is more severe at the peak WAT.

3.2. Influence of Oil–Wall Temperature Difference on Wax Deposition

When the oil temperature drops to the peak wax-recipitation temperature, the wax—deposition rate rises significantly and is relatively low near the wax-appearance temperature. During isothermal transportation, the wax-deposition rate is related to the oil temperature; during non-isothermal transportation, it increases as the pipe wall temperature drops. A larger temperature difference between the crude oil and the pipe wall accelerates the wax-deposition rate. When the pipe wall temperature is fixed, the deposition rate of wax molecules is related to their solubility. As the oil temperature drops, the change in solubility speeds up the wax-deposition rate.

The general consensus is that an elevation of the oil–wall temperature difference promotes an increase in wax deposition in pipelines. When the pipe wall temperature is lower than the oil temperature and below the WAT, the thermal driving force generated by the temperature gradient enhances effects such as molecular diffusion. As the oil–wall temperature difference goes up further, the pipe wall experiences a more significant temperature variation resulting in an increased wax molecule concentration gradient, which in turn strengthens molecular diffusion and speeds up the process of wax deposition [33]. Janamatti et al. [34] conducted experiments over a period of 72 h by associating with the oil tank’s initial wall temperature and modifying the crude oil’s thermal condition. They identified that there was a disparity in the oil–wall temperature, and this led to the formation of deposits slowing down while the wax content within the deposits increased. Junyi et al. [35] found that the key determinant of wax deposition in pipelines is the temperature variance between the pipeline and the fluid inside it. They also discovered that reducing this temperature difference results in a reduction in the wax quantity deposited on the pipeline wall. Quan et al. [36] conducted an experimental study to analyze the relationship between wax deposition patterns and the temperature variance (including the coolant’s temperature, the oil’s flow temperature, and the overall temperature difference). They pointed out that during pipeline transportation, an inlet oil temperature that is approximately 5 °C above the pour point but below the WAT can effectively reduce wax deposition and facilitate cleaning. Zhang et al. [37] undertook experimental investigations by maintaining a stable crude oil temperature and progressively modifying the thermal disparity between the crude oil and the inner surface of the pipeline. They found that the wax deposition rate decreased sharply, which also indicated that the oil–wall temperature difference plays a pivotal role in the process of wax deposition.

3.3. Influence of Shearing on Wax Deposition

Shear action is also an important factor affecting the wax deposition rate. Scholars have experimentally proven that, under constant other conditions and with only an increase in flow velocity, the wax deposition layer’s thickness will gradually decrease [38,39]. Under different flow states, the effect of pipe flow erosion on the quantity and rate of wax deposition varies: An increase in flow velocity, in both laminar and turbulent flows, will lead to a rise in the rate of wax deposition. However, at critical flow velocity, the wax deposition rate decreases. Helsper et al. [40] conducted research on various oil types and found that waxy solids formed at higher shear rates are more prone to fracture. It implies that, when transporting waxy crude oil, sustaining a greater shear rate can enhance its fluidity. Lu et al. [41] highlighted that changes in flow velocity affect the boundary layer thickness and interface temperature, thereby modifying the growth dynamics of the wax deposition layer’s thickness and the migration rate of wax molecules towards the interface. Practical operations have found that when the flow velocity exceeds 1.5 m/s, there is a marked reduction in the quantity of wax deposition inside the pipeline. However, in the transition zone between laminar and turbulent flow, due to unstable flow conditions, the speed of wax deposition is greater. In the case where the oil transportation volume remains unchanged and exceeds a given boundary, the wax deposit layer’s thickness rises sharply in the initial phase after pigging, and subsequently, the growth rate declines gradually, finally attaining a relatively stable state. However, for large-diameter pipelines, this process is much slower [42].

3.4. Influence of Deposition Time on Wax Deposition

As time progresses, the thickness of the wax deposition layer continues to increase, acting like an insulating layer that reduces thermal energy dissipation from the crude oil. This insulating effect decreases the oil–wall temperature difference, which in turn leads to a reduction in the wax deposition rate [8,38]. Mahir et al. [43] found through experiments that as deposition time increases, the wax layer’s thickness, which accumulates over time, experiences significant growth. However, with the augmentation of wax content in the sediment and a concurrent reduction in wax concentration within the surrounding oil, the thickness of the sediment gradually diminishes. Mansourpoor et al. [44] put forward the argument that, within the wax deposition process, the reduction in deposition mass as time progresses might stem from either the creation of an insulating layer on the cold finger’s surface or a decrease in the available wax quantity for deposition. Furthermore, as time advances, the accumulation of deposition thickness generates a thermal barrier, subsequently causing a decrease in heat exchange between the oil and the cold finger’s surface. Jin et al. [45] constructed a model capturing the time evolution of wax deposition thickness. They believe that while wax deposition initially rises during the early deposition phase, it subsequently declines as time progresses.

3.5. Influence of Crude Oil Properties on Wax Deposition

The essential cause of wax deposition lies in the characteristics inherent to waxy crude oil, particularly reflected in its composition and intrinsic properties. The temperature-dependent solubility characteristics of waxy constituents within crude oil are primarily responsible for its influence on wax deposition. All waxy components in the crude oil dissolve completely, demonstrating Newtonian fluid behavior, once the crude oil temperature rises above its WAT. However, upon the crude oil temperature falling beneath the WAT, waxy components gradually start to deposit on the inner wall of the pipeline. This deposition subsequently causes a decrease in the pipeline’s flow area and an elevation in flow resistance [6]. Maqbool et al. [46] not only studied the impact of paraffin wax on wax deposition but also considered the combined effects of resins and asphaltenes on wax deposition. Xue [47], through experimental research, believed that the morphology and microstructure of wax crystals can be influenced by the interaction between asphaltenes and wax molecules, thereby influencing the precipitation and deposition characteristics of wax. Studies have demonstrated that when colloids exist independently, they can hinder the ongoing growth and aggregation of wax crystals, thus diminishing the extent of wax deposition, and their influence on the wax deposition process is not substantial [48,49]. Asphaltenes, when present, have the ability to alter the size and distribution of wax crystals, which in turn affects the compactness and the rate at which the wax deposition layer ages. Colloids can adsorb on the surface of asphaltenes, promoting the aggregation of asphaltenes. Asphaltenes, in turn, accelerate the shaping of a dense wax deposition layer through strong interactions with wax molecules. Colloids and asphaltenes have a complex dual impact on the formation of the wax deposition layer. Specifically, their individual presence has limited impact, but their synergistic effect significantly influences the genesis and attributes of the waxy deposit layer. In addition, the genesis and attributes of the waxy deposit layer are profoundly affected by the quantities and ratios of resins and asphaltenes within crude oil [50,51]. Azizollah Khormali’s [52] research indicates that the water content in crude oil can also interact with inhibitors. By reducing surface adsorption, it can reduce the generation of sediments.

4. Model for Forecasting Crude Oil Wax Deposition

Currently, the calculation of wax deposition is mainly divided into two categories: thermodynamic and kinetic model distinction. Thermodynamic models focus predominantly on investigating the thermodynamic equilibrium state of crude oil systems and estimating the quantity of wax that precipitates out of crude oil under varying temperature, pressure, and composition circumstances. The fundamental essence of these models lies in ascertaining the WAT of crude oil as well as the solubility of wax within the crude oil. Kinetic models, on the other hand, focus on the movement and deposition process of wax within pipelines after it precipitates from crude oil. These models predict the deposition rate, deposition amount, and structural changes in the accumulated deposit layer on the inner pipe surface. The wax precipitation amounts and WAT predicted by thermodynamic models are important input parameters for kinetic models. Kinetic models can further analyze the dynamic mechanism of wax precipitation and the physical properties of the deposition layer, providing more comprehensive guidance for practical applications.

Kinetic models quantitatively analyze the process of wax molecules separating out of crude oil, undergoing migration, and accumulating on the internal surface of pipelines by integrating fluid mechanics, heat transfer, and mass transfer theories. These models typically focus on key factors that influence wax deposition and require relatively fewer parameters, enabling them to accurately predict wax deposition amounts in actual pipelines.

Burger et al. [13] constructed a model that takes into account the impacts of molecular diffusion and shear dispersion on wax deposition, considering that the role of Brownian motion is negligible and can be omitted. Therefore, it is possible to state that the Burger model is equivalent to the addition of wax deposition arising from molecular diffusion and wax deposition stemming from shear dispersion. Although this model has a clear and straightforward structure, there are areas for improvement: it does not consider the impact of oil-flow erosion on wax deposits, and it isolates the calculations of molecular diffusion and shear dispersion without considering their mutual interactions. Homouda et al. [53] developed a model in accordance with the molecular diffusion mechanism. This model, while ignoring the consequence of shear dispersion regarding wax deposition, introduces the concept of “wax deposition tendency coefficient”. On this basis, they further took into account the erosive impact of oil flow on the deposition layer and derived a formula for the actual wax deposition rate. This model is concise and clear, but it does not take into account the differences in wax deposition thickness at different locations or the differences in wax content among different deposits. The model developed by Hsu et al. [54,55] primarily describes the wax deposition process drawing upon the two mechanisms involving molecular diffusion and shear dispersion. It puts forward the idea of the “wax deposition tendency coefficient” with the aim of quantifying the wax deposition tendency and also introduces a “critical wax strength” as an amplification parameter, enabling the model to more accurately predict wax deposition in actual pipelines. The model has certain limitations when predicting wax deposition rates, which mainly stem from issues such as flow rate matching, flow state differences, inaccurate temperature calculations, reliance on experimental data, and model simplification. Therefore, in practical applications, it is necessary to revise and improve the model based on actual conditions to enhance prediction accuracy and reliability. Singh et al. [56,57] developed a model utilizing principles like molecular diffusion and the conservation law of mass transfer during deposition. This model comprehensively considers various factors and proposes the concept of variations in wax content within the deposited layer. Although this model has a certain accuracy in predicting wax deposition, it still has limitations, such as potential errors when predicting wax deposition in complex flow states. For example, some high-viscosity and high-pour-point crude oils are viscoelastic, having both viscosity (dissipating energy) and elasticity (storing and releasing energy). When they flow, there is not only viscous resistance, but also complex mechanical behaviors due to elasticity, such as elastic stress relaxation and creep, which make the flow laws more complicated.

Based on the core concepts of Fick’s law of diffusion, Huang [58] put forward a model. This model meticulously takes into consideration not only the influence exerted by molecular diffusion on the process of wax deposition but also the erosion effect induced by oil flow on wax deposition. This model can offer a more precise and comprehensive forecast of wax deposition within crude oil transportation pipelines and provide important guidance for pipeline maintenance and management. Although the model has high prediction accuracy, it still requires a large amount of experimental data for support. In addition, this model may have certain errors when predicting wax deposition in complex flow states. Hernandez et al. [59,60], building on Singh’s model, took into account the effect of shear stripping occurring during the wax accumulation process and then put forward a novel model for computing the wax content in the deposits. Peng [61] established a model based on molecular diffusion theory that comprehensively considers the shear influence of oil flow within the pipeline and the aging characteristics of wax deposits. Not only can this model accurately estimate the extent of deposition and the concentration of wax within the wax layers in experimental loops but it can also effectively simulate wax deposition phenomena in production pipelines in industrial applications. It provides a solid theoretical foundation for optimizing pipeline operation strategies and formulating pigging plans. Liu et al. [62] considered the phenomenon of wax deposition occurring during the transportation of waxy crude oil to be an unalterable process. They utilized the theories and methods of non-equilibrium thermal dynamics and heat and mass transfer to derive a linear phenomenological equation for wax molecule diffusion. By integrating the fundamental principles of mass and energy conservation, differential equations for heat and mass transfer were derived to predict the real-world wax deposition rate. The crude oil loop experimental model was validated through simulations and laboratory wax deposition rate tests. On this foundation, by considering the processes of wax molecule attachment and erosion, a new-type mathematical model was formulated to predict wax deposition.

The calculation models relevant to wax deposition within crude oil are presented in

Table 1. Crude oil wax deposition calculation model.

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 is wax deposition rate, kg/(m2·s); Dm is molecular diffusion coefficient of wax, m2/s; ${\rho }_{\mathrm{L}}$ is density of wax, kg/m3; C is wax’s volume fraction; T is temperature of the oil flow, °C; r is radius of the pipeline, m; k* is wax deposition rate constant determined by experiment; u is flow velocity of the oil at the pipe wall, m/s; $C_{\mathrm{w}}^{*}$ is volume fraction of wax crystals at the pipe wall.
Homouda	$W={\displaystyle \frac{M_{\mathrm{w}}}{N_{\mathrm{A}}}}{\displaystyle \frac{B}{\mu }}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\left({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}\right)}_{\mathrm{w}}$	Mw is molar mass of wax molecules; NA is Avogadro constant; B is a constant, calculated using a theoretical formula with the average volume of wax molecules as its basis; $\mu$ is crude oil viscosity, Pa·s.
Hsu	$W_{\mathrm{F}}={\mathsf{\Omega }}_{\mathrm{M}}{\left({\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\right)}_{\mathrm{F}}{\left(vD{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}L}}\right)}_{\mathrm{F}}$	WF is wax deposition rate after enlargement, kg/(m2·s); ${\mathsf{\Omega }}_{\mathrm{M}}$ is coefficient of inclination; y is thickness of the deposit, mm; L is pipe length, m; v is oil flow velocity, m/s; D is internal diameter of the pipe, m.
Singh	$W={\displaystyle \frac{D_{\mathrm{m}}}{\left(1-C\right)\zeta }}{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}T}}{\displaystyle \frac{Nu}{R}}\left(T_{\mathrm{h}}-T_{\mathrm{w}}\right)$	$\zeta$ is layer’s thickness in the cloud-point scenario, m; $Nu$ is nusselt number; R is radius of recirculation, m; Th is oil temperature at the core of the pipe flow, °C; Tw is Wall temperature, °C.
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 wall shear stress of the pipe, Pa; k, m, n are undetermined coefficients that need to be determined through experiments.
Hernandez	$W={\displaystyle \frac{J_{\mathrm{c}}\left[1-\varphi \left(F_{\mathrm{w}}\right)\right]-J_{\mathrm{s}}}{{\rho }_{\mathrm{L}}{F}_{\mathrm{w}}}}$ ${\displaystyle \frac{\mathrm{d}{F}_{\mathrm{w}}}{\mathrm{d}t}}={\displaystyle \frac{2\left[J_{\mathrm{c}}\varphi \left(F_{\mathrm{w}}\right)\right]\left(r-y\right)}{{\rho }_{\mathrm{L}}y\left(2r-y\right)}}$ $J_{\mathrm{c}}=k_{\mathrm{c}}\left(C_{\mathrm{b}}-C_{\mathrm{i}}\right)$ $\varphi \left(F_{\mathrm{w}}\right)=\left(1+{\displaystyle \frac{{\alpha }^2{F}_{\mathrm{w}}^2}{1-F_{\mathrm{w}}}}\right)$	Jc is the mass flux of wax from oil to the sediment interface, kg/(m2·s); Js is mass flow rate of wax that has been sheared off, kg/(m2·s); Fw is the wax mass fraction present in the sediment; kc is coefficient regarding convective mass transfer; Cb is the wax mass fraction in the oil stream; Ci wax mass fraction at the sediment-contacting interface; $\alpha$ is the shape characteristic parameter for wax crystal particles.
Peng	$C_{\mathrm{ws}}\left(T_{\mathrm{i}}\right)=a{\left(T_{\mathrm{i}}+b\right)}^c$ $V_R\left(C_{\mathrm{wb}0}-C_{\mathrm{w}b}\right)={\displaystyle {\int }_0^L\left(R^2-r_{\mathrm{i}}^2\right)}\pi F_W{\rho }_{\mathrm{gel}}\mathrm{d}L$ $\begin{array}{c}{F}_{\mathrm{W}}\left(t\right){\displaystyle \frac{\mathrm{d}\delta }{\mathrm{d}t}}={\displaystyle \frac{k_{\mathrm{l}}}{R{\rho }_{\mathrm{gel}}}}\left[C_{\mathrm{wb}}-C_{\mathrm{ws}}\left(T_{\mathrm{i}}\right)\right]+\\ {\displaystyle \frac{D_{\mathrm{e}}}{R{\rho }_{\mathrm{gel}}}}{\displaystyle \frac{\mathrm{d}{C}_{\mathrm{ws}}}{\mathrm{d}T}}{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}{|}_{\mathrm{i}}\end{array}$ ${\displaystyle \frac{\mathrm{d}\delta }{\mathrm{d}t}}={\displaystyle \frac{J_{\mathrm{c}}-J_{\mathrm{s}}}{{\rho }_{\mathrm{dep}}{F}_{\mathrm{w}}}}$ $J_{\mathrm{s}}=k{\displaystyle \frac{\mu \sigma }{Q\rho \mathsf{\gamma }}}$	Cws is solubility of wax molecules, kg/m3; Ti is critical temperature of the deposit layer inside the pipe, K; a, b, c are empirical constants, VR is volume of crude oil inside the pipe, m3; Cwb0 is initial mass concentration of wax in the crude oil within the pipeline, kg/m3; Cwb is wax mass concentration, kg/m3; ${\rho }_{\mathrm{gel}}$ is density of deposited layer, kg/m3; De is diffusion coefficient of wax molecules within the deposit layer; Q is transportation flow rate of oil, m3/s; $\sigma$ is shear stress of the crude oil, Pa; $\rho$ is density of the crude oil, kg/m3; $\mathsf{\gamma }$ is shear rate of the crude oil, s−1; k is an empirical parameter.
Liu	$\mathrm{W}={\displaystyle \frac{K}{P}}{\rho }_f\left(1+\beta \mathsf{\Delta }T\right)d_p{\left({\rho }^2\eta g\right)}^{1/3}{x}_f{w}^2$ ${\displaystyle \frac{P}{K}}=83.2\cdot w^{0.54}$ ${\rho }_f={\displaystyle \frac{m_f}{x_f}}$ $\mathsf{\Delta }T=T_w-T_f=R_f\cdot \stackrel{\cdot }{q}=\left({\displaystyle \frac{1}{k_f}}-{\displaystyle \frac{1}{k_0}}\right)\cdot \stackrel{\cdot }{q}={\displaystyle \frac{x_f}{{\lambda }_f}}\stackrel{\cdot }{q}$	P is intergranular adhesion force; $\mathsf{\Delta }T$ is temperature difference; K is number of defects in the deposit layer; β is expansion coefficient; dp is crystal grain size; ${\rho }_f$ is wax deposition density; xf is average thickness of the deposit layer; mf is total mass per unit area; $1+\beta \cdot \mathsf{\Delta }T$ is thermal stress of the deposit layer.

, as mentioned earlier.

5. Research Outlook and Recommendations

Wax deposition in crude oil poses a substantial challenge to the petroleum industry, having a notable impact on pipeline transportation capacity and flow assurance. Currently, research outcomes regarding wax deposition in single-phase crude oil have been relatively mature both domestically and internationally. This paper summarizes the main mechanisms, influencing factors, and existing models specifically designed for wax deposition prediction in crude oil.

Within the wax-deposition scenario, it is widely recognized that molecular diffusion functions as the key deposition mechanism, it cannot be overlooked that various other mechanisms are also simultaneously at play, including shear dispersion effects, Brownian motion phenomena, and gravitational settlement. These mechanisms collectively influence the deposition process of wax crystals within pipelines. Concurrently, the traits of wax deposition within oil pipelines are shaped by a complex interplay of various factors, including crude oil temperature, thermal disparity between the oil and the inner surface of the pipeline, shear effects, deposition time, and the characteristics of the crude oil in terms of physical aspects.

Currently, research on wax deposition within petroleum pipelines is relatively mature globally. However, most findings focus on qualitative and regular conclusions, lacking studies on the quantitative relationships between wax deposition and its numerous influencing factors. It is difficult to effectively quantify and characterize the wax deposition characteristics of crude oil under the combined influence of multiple factors. Furthermore, investigation and analysis have revealed two limitations in the current research on wax deposition prediction models within crude-based systems: On the one hand, the established wax deposition models all contain undetermined parameters, which means that when these models are applied to actual pipeline process calculations, some relevant yet undetermined parameters still require determination by means of experiments. This undoubtedly affects the convenience of using these models. On the other hand, previous researchers used limited types of crude oil (mostly only 1–2 types) when building their models and did not fully consider the diversity of crude oil properties. The models they built were not quantitatively correlated with crude oil properties, resulting in a lack of universal applicability among different types of crude oil.

In summary, based on an assessment of current research on wax deposition characteristics and prediction methods in oil pipelines, as well as an analysis of existing problems, the following suggestions for future research in this field are proposed: On the one hand, it is necessary to strengthen and deepen the research on the quantitative relationships between wax deposition in crude oil and influencing factors; for instance, the temperature gradient between the oil and the pipeline wall and the temperature of crude oil, shear effects, deposition time, and the characteristics of the crude oil in terms of physical aspects. There is a necessity to exert efforts to create quantitative characterization techniques for the wax deposition related features of crude oil under the combined influence of numerous factors. On the other hand, crude oil from different producing areas exhibits significant differences in physical properties. During the investigation into wax deposition prediction models for crude oil, it is necessary to fully consider the diversity of crude oil properties, with the aim of establishing a universal wax deposition prediction model that is quantitatively correlated with crude oil properties.