Impact of various aggregation kinetics on thermophoretic velocity of asphaltene deposition

Abstract

Asphaltene deposition may pose serious challenges to flow assurance of crude oil in well columns. Different aggregation kinetics would partly be responsible for asphaltene particle growth ending in deposition on the surface of well columns. This work primarily investigates the thermophoretic deposition velocity caused by temperature gradients inside well columns for various asphaltene aggregation kinetics, including crossover behaviour, sedimentation, reaction-limited aggregation (RLA), and diffusion-limited aggregation (DLA). To do so, the experimental observations of size distribution for a live crude oil was performed at 80 °C and pressure range of 4500–5500 psia. Moreover, various patterns of different size distributions were gathered from the literature for the sake of comparison. Next, a well column in southern Iran was selected to study the kinetic behaviour of thermophoretic velocity of deposition, with a difference between geothermal and static temperatures of around 5 to 50 °C. The non-isothermal deposition velocity was shown to decrease from the top to the bottom of the well column, according to the findings of the study. The results also revealed that the thermophoretic velocity decreases as particle size increases and vice versa. This was confirmed by examining a comparably large range of asphaltene particle sizes, ranging from approximately 100 nm to roughly 9 µm. Practical implications of these findings were also discussed which would provide guidance for mitigation of asphaltene deposition in well columns.

Introduction

Asphaltenes are complex, high-molecular-weight compounds in crude oil¹. They are a fraction of primarily heavy organic materials which may even remain after the distillation of crude oil². Asphaltenes are typically black or dark brown in colour, and they have a solid-like shape at ambient conditions. Furthermore, asphaltene refers to any organic black field deposit that is not wax and is completely distinct from laboratory asphaltenes made from solvents, mostly normal alkanes.

The deposition of asphaltene in well columns can lead to various problems across the oil industry. The operational issues would include, but not limited to, clogging/plugging of the wellbore, flowlines, and surface facilities, which would lead to flow restrictions and reduced production rates³. In the perforation region, it can also reduce the permeability, hampering the flow of oil and negatively impacting the production^4,5. Moreover, asphaltenes can cause damage to pumps, valves, and other production equipment by accumulating on surfaces, impeding their functionality, and potentially leading to mechanical failures^6,7. The economic implications of asphaltene deposition in well columns include lower overall economic returns from the well, increased operating costs as well as shutdowns and workovers^8–10. Fluid disposal and soil or water contamination risks are potential detrimental environmental impacts¹¹. In addition to what was stated, safety concerns including well integrity and blowouts as well as challenges during refining processes are among other problems associated with asphaltene deposition.

The prevention of asphaltene-related problems requires full recognition of different aspects of asphaltene deposition including its stability/instability, precipitation, aggregation, transport, and removal as schematically shown in Fig. 1.

Figure 1.

Process of asphaltene deposition in well column.

Asphaltene instability is a complex phenomenon, and the exact causes can vary depending on the specific crude oil composition and operating conditions. Pressure changes, temperature variations, changes in reservoir fluid composition, as well as the presence of incompatible fluids, contaminations, and impurities are some key factors that can contribute to asphaltene instability^12,13. Asphaltene will start precipitation as a heavy phase from crude oil once it becomes unstable in crude oil, according to Fig. 1. Afterwards, aggregation is referred to as the process by which the precipitated particles may come together and form larger aggregates¹⁴, insoluble structures for which different aggregation kinetics would play key role as it will be described later in “Kinetics of asphaltene aggregation” section. To better understand how the presence of nanoparticles affects the formation and aggregation of asphaltene particles in crude oil, Najjar et al.¹⁵ reported that the presence of nanoparticles in the range of 300 to 700 ppm might potentially regulate the aggregation process. Stated differently, metallic (iron, zinc, and magnesium)-based nanoparticles can be used as a dispersant or inhibitor of asphaltene aggregation. The modified Yen model, also called the Yen–Mullins model, takes into account the fairly large polycyclic aromatic hydrocarbon with peripheral alkanes as the major single molecular structure. Approximately six asphaltene molecules come together to produce asphaltene nanoaggregates. Clusters consisting of asphaltene nanoaggregates have an estimated aggregation number of around eight^16,17.

After aggregation, asphaltenes may be transported from the bulk of crude oil to the interface between crude oil and surface. Two different isothermal and non-isothermal mechanisms would dominate this stage. Asphaltene particles may be carried by inertia¹⁸, eddy diffusion¹⁹, turbulent impaction¹⁹, and gravitational deposition¹⁸ under constant temperature, as clearly is demonstrated in Fig. 1. Moreover, as stated before, the application of these mechanisms is not a function of temperature gradient. However, when there is a temperature gradient and charge difference within the well column, thermophoresis²⁰ and electrophoresis²¹ may act to move the particles, respectively. As soon as the asphaltene particles are placed close to the crude oil/surface interfaces, then adhesion and cohesion forces would cause them to deposit on the surface²². The deposited particles may simultaneously be driven to dislodge from the surface by different removal mechanisms, the most frequent of which are shear force, rolling, and rebound as shown in Fig. 1²³. It should be noted that the processes shown in Fig. 1 could occur simultaneously rather than always one after the other.

In the present study, the thermophoretic velocity of asphaltene deposition comes into focus for a measured asphaltene particle size distribution and some aggregation kinetics collated from the literature. The impact of different aggregation kinetics of asphaltene on the thermophoretic velocity of deposition can be complex and highly dependent on various factors, such as:

• Aggregation rate: The rate, at which asphaltene particles aggregate and form clusters, may influence the deposition process. Faster aggregation kinetics can often lead to the formation of larger and denser aggregates, which are less likely to deposit depending on the nano/micro size-scale of the particles. In other words, this would decrease the thermophoretic velocity of deposition as it will be discussed later in “Results and discussion” section.

• Particle size distribution: The size distribution of asphaltene particles can affect their deposition propensity. Aggregation kinetics can influence the particle size distribution, with faster aggregation resulting in larger particle sizes. Larger particles would have lower thermophoretic velocities.

• Particle stability: The stability of asphaltene particles is influenced by their aggregation kinetics. As it will be discussed later in “Results and discussion” section, slower aggregation kinetics may result in smaller, more stable particles that are more likely to deposit. On the other hand, rapid aggregation can lead to the formation of unstable aggregates, which may readily be removed and contribute to lower thermophoretic velocities.

• Temperature gradient: The magnitude and direction of the temperature gradient in well columns can also impact thermophoretic velocity as shown later in “Thermophoretic deposition velocity” section. Based on operating conditions, a higher temperature gradient may induce stronger thermophoretic forces, promoting faster deposition of asphaltene particles.

Asphaltene deposition is a complicated interfacial phenomenon influenced by several variables, as illustrated in Fig. 1. Nonetheless, there has not been a thorough investigation of the influence of aggregation kinetics on the deposition velocities, exclusively with respect to thermophoretic deposition velocity²⁴. The literature lacked explanations on how thermophoresis affects the rate of deposition. To the best of our knowledge, almost none of the previous studies have looked into how the thermophoretic phenomenon would influence the speed at which asphaltene particles would be deposited²³. Only one pertinent study has dynamically modelled asphaltene deposition in pipelines and concluded that particle size increases deposition velocity, which included thermophoretic effects²⁵. However, as this investigation will demonstrate, this conclusion needs to be re-examined. To sum up, the influence of particle size, after asphaltene precipitation, as a result of various aggregation kinetics on the thermophoretic deposition velocity for an oil well, is the focus of the current work.

Methodology

To proceed, “Kinetics of asphaltene aggregation” section explains different kinetics of asphaltene aggregation including crossover behavior, sedimentation, RLA, and DLA to describe the growth of asphaltene precipitated aggregates and presents their asphaltene particle size distribution collected from the literature. In “Experimental setup and measurements” section, the experimental setup of flow assurance system (FLASS) and the measurement of asphaltene particle size distribution for a live oil sample at 80 °C and pressure range of 4500–5500 psia is described. The kinetic behavior of thermophoretic velocity of deposition was then investigated in a well column in southern Iran, where there was a temperature differential of around 5 to 50 °C between geothermal and static temperatures. After that, “Thermophoretic deposition velocity” section presents the definition and correlations of the thermophoretic deposition velocity. The obtained findings of this study including (i) thermophoretic effects within well column and (ii) the impact of particle size on the thermophoretic velocity will be presented and discussed in “Results and discussion” section.

Kinetics of asphaltene aggregation

Asphaltene particles have a propensity to self-associate and form aggregates under certain conditions. Reaction-limited aggregation (RLA) and diffusion-limited aggregation (DLA) are two common irreversible models used to describe the kinetics of asphaltene aggregation. Aggregation depends on how long it takes for two asphaltene particles to disperse and come into contact with one another as well as how long it takes for smaller particles to interact with each other and grow into bigger ones²⁶. Here, the first and second are referred to as diffusion time and reaction time, respectively.

In the context of asphaltene, the DLA kinetics model assumes that asphaltene particles move randomly in the crude oil and aggregate when they come into contact with other particles²⁷. In the initial stages of asphaltene aggregation, individual asphaltene molecules would diffuse in the crude oil, undergoing Brownian motion due to the thermal energy of the system²⁸. As these asphaltene molecules collide and interact, they can form weak bonds or associations, leading to the formation of small aggregates. At this stage, the DLVO theory^29,30 has extensively been applied to understand the bahavior of larger colloids. As the aggregates grow, they become increasingly capable of capturing and binding additional asphaltene molecules that come into their vicinity. This process is driven by diffusion, where asphaltene molecules diffuse in the surrounding fluid and are captured by the growing aggregate through attractive forces. It should be noted that these forces, i.e. Brownian motion, electrostatic double-layer, Lewis acid–base, and Lifshitz-van der Waals interactive forces may be estimated using the extended DLVO (XDLVO) theory^31–33 since crude oil has only a low degree of polarization owing to the presence of asphaltene.

The DLA kinetics imply that the growth of asphaltene aggregates is primarily controlled by the rate of diffusion. In a system with slower diffusion, asphaltene particles may have reduced mobility and a higher probability of aggregation, leading to more significant aggregate growth. Conversely, a system with faster diffusion may result in smaller aggregates due to a lower probability of particle capture. Temperature, pressure, composition, shear force, are expected to influence the DLA kinetics³⁴.

The DLA model considers both the time and length scales of the diffusion process. Over time, asphaltene aggregates continue to grow by capturing more asphaltene molecules, resulting in larger and more complex structures³⁵. The final morphology of the aggregates formed under the DLA kinetics would exhibit fractal-like patterns, characterized by intricate branching structures and irregular shapes²⁶.

Unlike DLA, where the growth of aggregates is primarily controlled by diffusion and particle capture, RLA kinetics emphasize the importance of chemical reactions or reactions between asphaltene particles during the aggregation process. In the case of asphaltene, these chemical reactions can involve various mechanisms such as van der Waals forces, hydrogen bonding, π-π stacking, and other intermolecular interactions²². The interactions between asphaltene molecules can lead to the formation of bonds or associations that contribute to the growth of aggregates. Similar to the DLA mechanism, the growth of asphaltene aggregates under RLA kinetics is influenced by factors such as the concentration of asphaltene molecules, the presence of catalysts or mediators, temperature, pressure, and medium composition^36,37.

The morphology of asphaltene aggregates formed under RLA kinetics can vary depending on the specific reaction mechanisms. The aggregates may exhibit different shapes and structures compared to those formed under DLA. The final morphology can range from irregularly shaped clusters to more organized and compact structures^38,39.

It is also worth noting that asphaltene aggregation kinetics in real systems can involve a combination of DLA and RLA models. The relative contributions of these models may vary depending on the specific conditions, such as the oil composition, temperature, pressure, and other factors present in the system. To examine the particle-by-particle development of meso-aggregates made using sulfur-rich asphaltene, Hammond et al.⁴⁰ used a combination of in-situ microscopy and molecular modelling. The growth rate of aggregates made from sulfur-rich asphaltene underwent a crossover aggregation pattern between the classic RLA and DLA mechanisms, according to their experimental findings. Salehzadeh et al.⁴¹ investigated the connection between asphaltenes' structure and aggregation process. They found that there would be a clear relationship between the aggregation rates of asphaltenes and their structure for three types of oils: light, medium, and heavy. The results showed that when normal heptane was added as a precipitant agent, the aggregation pattern was DLA followed by RLA. As illustrated in Fig. 2, four distinct size distribution patterns, collected from the literature, were employed in the current study. In Fig. 2a and c, the growth of asphaltene particles is controlled by RLA and DLA. However, the combination of both RLA and DLA is seen for Fig. 2b and d. The crossover behavior can be a function of particle size, where the aggregation rate increases initially with time exponentially (called RLA) and then power law-like at the end (called DLA). It should be noted that the acronym ASR denotes asphaltene settling region in Fig. 2d. Table 1 tabulates more details about the conditions upon which the patterns of Fig. 2 were obtained.

Figure 2.

Asphaltene particle radius vs. time for (a) RLA, (b) RLA-crossover-DLA, (c) DLA, and (d) RLA-DLA-ASR aggregation kinetics.

Table 1.

Specifications of studies investigating the aggregation kinetics of asphaltene in Fig. 2.

Study [Aggregation kinetics]	Oil type	Precipitant	Temperature	Pressure	Observation method
37 [RLA-crossover-DLA]	Oil model (asphaltene in toluene)	52% n-heptane	Not mentioned	Not mentioned	Light scattering
42 [RLA]	Oil model (asphaltene in toluene)	54% n-heptane	Not mentioned	Not mentioned	Dynamic light scattering
43 [DLA]	Boscan and Furrial crude oil	88% n-heptane	Not mentioned	Not mentioned	Confocal microscopy
44 [RLA-DLA-ASR]	Oil model (asphaltene in toluene)	42% n-hexane	Room temperature	Ambient pressure	Image processing using high pressure microscope

Asphaltene aggregation can also be described by several different kinetic models, including population balance modeling and percolation theory^45,46. Nevertheless, it is generally known that RLA and DLA provide a reliable framework for the kinetics of asphaltene aggregation⁴⁷. They are frequently used to examine experimental data and predict aggregation behavior, and they offer insights into the statistical characteristics of asphaltene aggregation^41,48. Thus, the current study included crossover and ASR patterns in addition to these primary kinetics.

Experimental setup and measurements

Live oil sample

A reservoir in south of Iran was used to obtain the dead crude oil. 200 cc of dead crude oil and 90 cc of methane gas were combined to make live crude oil, which had a gas to oil ratio of 630 scf/STB at 3000 psia. The characteristics of the oil sample utilized in this investigation are listed in Table 2. The crude oil contains a significant portion of saturates and aromatics, moderate resin content, but low asphaltene content (see Table 2). However, there are serious operational challenges with respect to its asphaltene.

Table 2.

Physicochemical properties of crude oil sample.

Property	Value
API gravity	31̊
Kinematic viscosity	8.51 cSt (40 °C), 2.56 cSt (100 °C)
Acid number	0.06 mg K/g
Base number	1.11 mg K/g
SARA analysis	Saturate: 61.42%, aromatic: 31.95%, resin: 6.07%, asphaltene: 0.56%

FLASS apparatus

One of the suitable techniques for monitoring the precipitation and also for tracking the aggregation kinetics of asphaltene particles is the FLASS apparatus (FLASS, VINCI TECHNOLOGIES, France). The device may be used to, among other things, measure the onset of asphaltene precipitation of a live oil sample using a laser beam and assess the asphaltene particle size distribution via a high-pressure microscope (HPM). Before the crude oil sample enters the HPM to be examined for asphaltene particle size distribution, it is screened using a solid filtering system with a pore size of 0.45 micron. Filtration assists firstly to verify the asphaltene particle precipitation caused by pressure changes and secondly to prevent pre-precipitated particles from entering the HPM. Following filtering, the sample is introduced into the main cell with a volume of 100 cc. In the main cell, there is a mixer with controllable rotation to prevent the possible asphaltene particles from settling during high pressure–temperature measurements. Into a circular joint cross-section of two solid sapphire glasses with strong heat and pressure resistance, the oil sample is pumped from the main cell. To monitor the development and aggregation evolution of asphaltene particles online, a HPM and a light source are in contact with the two free cross-sections of the sapphire glasses. The FLASS-related software provides the average particle diameter, total number of detected particles, and average area of those particles in a two-dimensional coordinate. Some of the FLASS apparatus's parameters are provided in Table 3. With the help of HPM, particle sizes up to 1 micron may be detected. Three variables are provided by HPM software: (i) the total particle area; (ii) total number of particles; and (iii) mean particle diameter. A portion of the HPM's sapphire lenses was not completely cleaned during the trials, and as a result, certain stains persisted which then were found by the program. Consequently, using the total area and number of particles, assumed to be spherical, we had to compute the mean asphaltene particle size. Accordingly, asphaltene particles smaller than one micron were identified (see Fig. 3b).

Table 3.

FLASS specifications for measuring particle size distribution.

Parameter	Value/range
Maximum allowable pressure	Up to 20,000 psia
Maximum allowable temperature	Up to 200 °C
Flow rate	Dependent on process time
Material exposed to sample	Stainless steel and sapphire glass
Being able to rotate or discharge sample	Rotational only
Magnification	Up to 500 times
Particle size detection	0.2 µm
Laser power	250 mWatts

Figure 3.

(a) live crude oil pressure and (b) asphaltene particle radius vs. time.

Size distribution of asphaltene particles

The impact of pressure and flow rate decrease was investigated at 80 °C and 30% of full power mixer since the asphaltene particle size distribution may be influenced by pressure, flow rate, temperature, and mixer power. The temperature was kept constant at 80 °C since pressure has a higher impact on live oil sample. Over a period of around 260 min, the asphaltene particle size distribution was examined. In order to guarantee that an equilibrium state was attained, 60 cc of the live oil from the main cell was pumped to the HPM and left there for around 50 min at 5500 psia and 80 °C as shown in Fig. 3a. To ascertain the impact of pressure reduction on the kinetics of particle formation, the live oil pressure was subsequently decreased from 5500 to 4500 psia over a period of 50 to 100 min at a constant temperature of 80 °C (see Fig. 3a). As it is explicitly demonstrated in Fig. 3a, to see how much the size of the asphaltene particles would alter with the flow rate, the sample was then left for a further 160 min with the flow rate reduced from 0.2 to 0.05 cc/min. It should also be pointed out that in the HPM, isothermal depressurization was carried out using specified equilibrium steps of 5 min and a designated depressurization ramp of 208 kPa/min. To inject new fluid from the PVT cell to the HPM, a flow rate was also set simultaneously by a syringe pump^49,50.

Figure 3b shows changes in asphaltene particle radius over time. According to this graph, the mean radius of the particles linearly decreases with time by about 25%, from around 0.3 µm to 0.225 µm since the system is in equilibrium for the first 50 min. Over the period of 50 to 100 min with a 1000 psi pressure reduction, the mean radius once more decreases by around 11%, from 0.225 to 0.2 µm. On the other hand, the mean particle size remained basically unchanged after additional 160 min of lowering the flow rate from 0.2 cc/min to 0.05 cc/min. In conclusion, the operating parameter with the most impact on the kinetics of particle size distribution is the pressure, as shown in Fig. 3. Notably, it would never be possible to collect the particle size distribution from the tests using FLASS through the literature. This argument is supported by the fact that, although the tests conducted for the study used real oil samples in which precipitation happened as a result of pressure changes, the literature mostly deals with oil models and normal alkanes as precipitant agents. Moreover, the attained particle size distributions using FLASS were used to obtain an experiment pattern for the evolution of asphaltene particle size over time.

Temperature distribution within well column

The static temperature of the crude oil sample (see Table 2), the geothermal temperature of the well column, and the difference between the two temperatures changes in the well column, as shown in Fig. 4. As a result, the static temperature of crude oil rises through the well column and eventually achieves an approximately asymptotic behavior. However, as depth increases, the geothermal temperature rises linearly. More specifically, as depth increases, the static and geothermal temperatures nearly rise in a parallel manner before nearing one another. For the present case study, the relaxation distance is approximately 1750 m.d.d. In contrast to what is often expected, the static temperature profile in Fig. 4 shows a variation for the data point (100.1 °C–1327.8 m.d.d.). This serves as an indication implying that, for instance, the entry of gas into the well column or a potential leak. It sounds that the dynamic temperature of the crude oil sample should be greater than the static temperature since the fluid would move through the well column more quickly in a dynamic state. As a result, the time of heat transfer between well column and fluid would reduce. The difference in temperatures is crucial for the thermophoresis process. Having considered this, Fig. 4 also illustrates the variation in temperatures. It should be mentioned that Eq. (1) is used to compute this difference. Moreover, as the fluid is in a dynamic condition rather than static, Eq. (1) is only an approximation and will not accurately represent the temperature difference in the well column. Furthermore, the surface temperature of the well column is not the same as the geothermal temperature. Despite this, the necessary data is not accessible, hence Eq. (1) would be used to approximatively estimate the temperature difference.

$$\mathrm{\Delta }T=T_{\mathit{Geothermal}}-T_{\mathit{Static}}$$ 1

Figure 4.

Static, geothermal, and difference temperature profiles within the investigated well column.

The temperature difference should be expressed in the form of Eq. (1) as thermophoresis will operate as a transport mechanism to transfer the asphaltene particles from the bulk media to the surface of the well column due to temperature gradient. As a result, the velocity of thermophoretic deposition will be positive when the temperature difference is negative (see Eq. (2)), which indicates that the particles would have to move towards the well column's surface. While the thermophoretic effects would force the particles to move away from the well column's surface for the positive values of temperature difference, the negative thermophoretic velocity would be achieved. Figure 4 shows both negative and positive temperature differential readings. The temperature difference reduces from the top of the well column, about − 50 °C, to its bottom, nearly + 5 °C, which is located close to the production zone. Thus, it may be ascertained that as well column depth lowers, the possibility for particle transfer by thermophoresis increases. The temperature profiles in Fig. 4 are thought to be in a steady-state situation, which is another fact that has to be highlighted. Their temporal fluctuations are therefore not taken into consideration.

Thermophoretic deposition velocity

Thermophoresis is defined as the movement of primarily fine particles due to a temperature gradient, during which the particles are carried from one area of higher temperature to another area of lower temperature. According to the kinetic theory, which states that molecules present at higher temperatures have proportionately larger kinetic energies than those present at lower temperatures, a net thermophoretic force would be established^51,52. The thermophoretic velocity of deposition for asphaltene particles in a well column, or any system, can be influenced by several factors including temperature gradient, particle size, fluid and medium properties as well as chemical environment. The following presents the more popular Brock-Talbot and MCMW empirical correlations that have been developed for the thermophoretic velocity of deposition. The crude oil fluid rises in an axial direction. However, the radial transportation of asphaltene particles results in their deposition on the well column's surface. Thus, the particle velocity has to be considered. Finally, the concentration driving force, sticking probability of particles, and transport velocity, in this case, thermophoretic velocity, are multiplied to determine the deposition rate.

Brock-Talbot correlation

The thermophoretic velocity of a particle in a fluid flow is defined as⁵³:

$$v_{\mathit{th}}=-K_{\mathit{th}}\frac{{\nu }_F}{T}\mathrm{\nabla }T$$ 2

The coefficient $K_{\mathit{th}}$ in Eq. (2) is expressed as:

$$K_{\mathit{th}}=\frac{2C_s{C}_c}{\left(1+3C_{m}Kn\right)}\times \frac{\left(k_F/k_A\right)}{1+2\left(k_F/k_A\right)+2C_{t}Kn}$$ 3

where, $C_c$ is Cunningham correction factor which is calculated using Eq. (4).

$$C_c=1+Kn\left[A+Bexp\left(-C/Kn\right)\right]$$ 4

In these equations, constants A, B, C, C_m, C_s, and C_t are equal to 1.257, 0.4, 1.1, 1.17, 1.14, and 2.18, respectively. The temperature gradient in Eq. (2) is given by Eq. (5).

$$\mathrm{\nabla }T=\left(T_m-T_w\right)/\delta$$ 5

In Eq. (5), $\delta$ stands for the viscous sub-layer thickness. Provided that the temperature distribution in the viscous sub-layer is linear, $\delta$ is then calculated using Eq. (6)⁵⁴.

$$\delta =\frac{{\nu }_F\left(U_m-5u_{}^{\ast }\right)}{\left(u_{}^{\ast }\right)_{}^2}$$ 6

In Eq. (6), $u_{}^{\ast }$ denotes friction velocity and is yielded from Eq. (6).

$$u_{}^{\ast }=\sqrt{{\tau }_w/{\rho }_F}$$ 7

Equation (8) gives the wall shear stress, ${\tau }_w$, in Eq. (7).

$${\tau }_W=\frac{1}{8}f{\rho }_F{U}_m^2$$ 8

In equation above, $f$ is Darcy friction factor and is given as a function of Reynolds number.

$$f={\left(0.79ln\text{Re}-1.64\right)}^{-2}$$ 9

MCMW correlation

According to this empirical correlation, the thermophoretic force is defined as⁵⁴:

$$F_{\mathit{th}}=\frac{1.15Kn}{4\sqrt{2}\alpha \left(1+\frac{{\pi }_1}{2}Kn\right)}\left[1-exp\left(-\frac{\alpha }{\mathit{Kn}}\right)\right]\sqrt{\left(\frac{4}{3\pi }\varphi {\pi }_{1}Kn\right)}\frac{k_b}{{\left(d_m\right)}^2}\mathrm{\nabla }T{\left(d_A\right)}^2$$ 10

where, $K_b$ and $d_m$ indicate Boltzmann’s constant and molecular diameter, respectively. The rest of variables in Eq. (10) are given as the following.

$${\pi }_1=\frac{36/\pi }{4/\pi \left(2-S_n+S_t\right)+S_n}$$ 11

$$\varphi =0.25\left(9\varpi -5\right)\frac{C_v}{R}$$ 12

$$\alpha =\sqrt{\frac{\frac{\pi }{6}\varphi }{1+\frac{{\pi }_1}{2}Kn}}$$ 13

In Eqs. (11–13), $\varpi$ is the heat capacity ratio of fluid, $C_v$ the heat capacity at constant volume, R the universal gas constant as well as $S_n$ and $S_t$ are the normal and tangential momentum accommodation coefficients.

The thermophoretic velocity is then determined using Eq. (14).

$$v_{\mathit{th}}=\frac{F_{\mathit{th}}{t}_A}{m_A}=\frac{F_{\mathit{th}}{t}_A}{\frac{\pi }{6}{\rho }_A{\left(d_A\right)}^3}$$ 14

Which of the correlations of Brock-Talbot or MCMW may be employed to calculate the thermophoretic velocity depends on the range of Knudsen number⁵⁴. The Brock-Talbot correlation may be applicable when $Kn<2$ and the MCMW correlation for $Kn>2$. Equation (15) presents the mathematical description of Knudsen number⁵⁵.

$$Kn=\frac{2\psi }{d_A},\psi =\frac{2{\mu }_F}{{\rho }_F}\sqrt{\frac{\pi M{W}_F}{8RT}}$$ 15

Figure 5 provides a flowchart for the calculation of thermophoretic velocity. It should be noted that $\psi$ denotes the mean free path of medium molecules. As the values of Knudsen number were all smaller than two, then the empirical Brock-Talbot correlation was used in the current work to quantify the changes of thermophoretic deposition velocity for varied aggregation kinetics. It should also be pointed out that during the calculations of deposition velocity, the other parameters such as crude oil flow rate, well column diameter, and the physical characteristics of oil density and viscosity, remained unchanged.

Figure 5.

Flowchart for the calculation of thermophoretic velocity.

Results and discussion

Thermophoretic effects in well column as a function of depth

Based on the aggregation patterns shown in Figs. 2 and 3b, the temporal variation of the thermophoretic velocity at various depths has been examined for the aforementioned well column with unique static and geothermal temperature profiles as given in Figs. 4. Consequently, Fig. 6 demonstrates the thermophoretic velocity as time elapses when the reaction limits the particles’ aggregation. This indicates that the reaction time between asphaltene particles takes too much time compared to the diffusion time (see “Kinetics of asphaltene aggregation” section). The positive thermophoretic velocities are seen at a depth of 213.1 m.d.d, as illustrated in Fig. 6a, where the static temperature profile is located above the geothermal temperature. This implies that asphaltene particles would be moving from the bulk of crude oil to the well column's surface. It can be inferred from a comparison of Figs. 2a and 6a that the thermophoretic velocity decreases for larger asphaltene particle sizes. In other words, thermophoretic effects cause the smaller particles to travel at a faster rate. The kinetic behavior of thermophoretic velocity in Fig. 6b at depth of 3156.6 m.d.d., is similar to what was indicated for Fig. 6a. As a result, the effects of thermophoresis diminish as well column depth increases from the surface to the perforation zone. This is because as depth increases, the thermophoretic velocity's driving force, or temperature difference, reduces. Figure 6a and b show that the thermophoretic effects would diminish by almost 73% with a 14-fold increase in depth. Negative thermophoretic values (see Fig. 6c) are noticeable when the thermophoretic velocity is examined close to the production zone because the geothermal temperature is higher than the static temperature. In other words, the temperature differential would be such that the asphaltene particles would be removed from the well column's surface via thermophoretic processes.

Figure 6.

Thermophoretic deposition velocity vs. time for RLA aggregation kinetics at (a) 213.1, (b) 3156.6, and (c) 4010 m.d.d.

Figure 7 shows the velocity of thermophoretic deposition over time for the RLA-crossover-DLA aggregation kinetics at various well column depths. As a result, the velocity of thermophoresis generally decreases with time due to an increase in particle size, as seen in Fig. 2b. The thermophoretic velocity also decreases as one descends from the well column's surface toward its bottom because the driving force of thermophoresis is weaker. For instance, based on what was shown in Fig. 7a and b, the velocity of asphaltene transportation drops by about 72% with a 14-times rise in depth. Additionally, a 19% increase in depth from 3156.6 to 3766.2 m.d.d. would result in a nearly 100% decrease in thermophoretic velocity. The rationale for Fig. 7c, in which the thermophoretic velocity is negative close to the production zone, would be similar to what was described earlier for Fig. 6c.

Figure 7.

Thermophoretic deposition velocity vs. time for RLA-crossover-DLA aggregation kinetics at (a) 213.1, (b) 3156.6, and (c) 4010 m.d.d.

When the asphaltene particles follow the DLA aggregation kinetics, the effect of time on the deposition velocity of thermophoresis is shown in Fig. 8 at 213.1 m.d.d, 3156.6 m.d.d, and 4010 m.d.d. Here, the process of aggregation would be controlled by the period of asphaltene particle diffusion (see “Kinetics of asphaltene aggregation” section). This figure shows that over time, the thermophoretic velocity decreases while the size of the asphaltene particles rises, as seen in Fig. 2c. The patterns in Fig. 8 would be explained similar to what was previously shown for the effect of depth on the velocity of thermophoresis. One may discover that about 14-fold increase in the well column depth from 213.1 to 3156.6 m.d.d would result in a reduction in the velocity of deposition up to 73%. The thermophoretic velocity of deposition is negative in Fig. 8c in contrast to Fig. 8a,b. This indicates that the velocity of asphaltene particle transfer from the surface of the well column to bulk of the fluid reduces with time at this depth, which is close to the production zone.

Figure 8.

Thermophoretic deposition velocity vs. time for DLA aggregation kinetics at (a) 213.1, (b) 3156.6, and (c) 4010 m.d.d.

On the basis of the RLA, DLA, and finally ASR aggregation kinetics depicted in Figs. 2d, 9 shows how the thermophoretic deposition velocity decreases over time. This is due to the fact that, as seen in Fig. 2b, the asphaltene particle radius first increases with time with a strong slope. Thereafter, the thermophoretic velocity roughly behaves as an asymptotic behavior since the aggregation kinetics in Fig. 2d show no discernible changes after the reduction. Needless to say again, from the production zone to the top of the well column, the significance of thermophoretic effects would increase. This behavior would be linked to a wider differential between static and geothermal temperatures (see Fig. 4), which would range from around 5 °C in the production zone to about 50 °C at 213.1 m.d.d. Similar results are obtained in Fig. 9c, where the thermophoretic-driven transport velocity of asphaltene particles towards the bulk medium profoundly decreases firstly and eventually exhibits an asymptotic behavior.

Figure 9.

Thermophoretic deposition velocity vs. time for RLA-DLA-ASR aggregation kinetics at (a) 213.1, (b) 3156.6, and (c) 4010 m.d.d.

Figure 10 illustrates how thermophoretic deposition velocity evolves over time when the aggregation kinetics of RLA followed by an asymptotic behavior would describe the status of asphaltene particles. According to this figure, the transport velocity firstly rises and eventually replaces with an asymptotic thermophoretic velocity. According to Fig. 10, as the depth of the well column increases from 213.1 to 3156.6 m.d.d, the thermophoretic velocity of deposition decreases by ~ 73%. The pattern seen in Fig. 3b for the relationship of particle radius-time contrasts the stated tendency, which is the increasing of velocity with time. As shown in Fig. 10c, the thermophoretic velocity of deposition increases in negative values because the mean radius of asphaltene particles diminish with time according to Fig. 3b.

Figure 10.

Thermophoretic deposition velocity vs. time for RLA-asymptotic behavior aggregation kinetics at (a) 213.1, (b) 3156.6, and (c) 4010 m.d.d.

Using a Couette device, Mohammadi et al.⁵⁶ examined the effect of temperature on the deposition of asphaltene particles from a reservoir in southwest of Iran. They reported that a 45 °C rise in temperature would result in an approximately 46.84% decrease in the overall mass of asphaltene deposits. Moreover, Li et al.⁵⁷ used gravimetric techniques and microscopic analysis to examine the impact of temperature on the precipitation of asphaltene for two crude oils from Xinjiang, China. They showed that the solubility of asphaltene increased with temperature in the range of 25 to 120 °C. This is consistent with the findings of the current investigation, which suggest that decreasing the temperature difference with increasing depth would cause lower thermophoretic velocities.

Finally, it should be noted that with depth reduction in well columns, a lower temperature should be expected so the crude oil fluid becomes more viscous. This might impede the mobility of the asphaltene particles, potentially retarding the kinetics of aggregation. Moreover, due to lower mobility and higher chance of collision between asphaltene particles, bigger aggregates of asphaltene would evolve. Regarding the viscosity-thermophoresis relationship, it can be asserted that, based on Eq. (2), the thermophoretic deposition velocity rises with fluid viscosity. Contrariwise, Hosseini-Moghadam et al.⁵⁸ experimentally assessed the effect of temperature on the kinetic behaviour of asphaltene particles in the presence of three inhibitors of salicylic acid, benzoic acid, and dodecyl benzene sulfonic acid for a crude oil from the southwest of Iran. They reported that as the temperature increased from 25 to 65 °C at constant concentration, there were more collisions between the asphaltene particles casing lower efficacy of the investigated inhibitors.

Challenges and future work

The findings of this study underline the influence of thermophoretic transport velocity on the modeling of asphaltene deposition flux inside the well column. This is because several factors, such as axial advection, radial transport, precipitation, and particle aggregation, would dominate how much asphaltene deposition would occur. The influence of thermophoretic velocity is imperative thus must be taken into account if there is even a small temperature difference, here around $\left|T_{\mathit{Geo}}-T_m\right|=5\text{K}$, between the casing and the bulk of the fluid, which is typically the case. To materialize this goal, there would be a number of challenges. The main challenge is the impact of various types of aggregation kinetics on the thermophoretic velocity. Particle size distribution and temperature gradient are two crucial input variables used to compute thermophoretic velocity.

The most common types of field data are static/dynamic and geothermal temperature profiles. Having said so, particle size distribution resulting from different aggregation kinetics is often modeled using an equation like the population balance equation or evaluated using lab apparatus that may be quite distinct from field circumstances. Therefore, this has to be resolved by matching the field circumstances during experimental observations and/or modeling procedures. It is important to note that in the present investigation, the size distribution was measured using FLASS close to field conditions.

The present investigation did not address the thermodynamics of oil and asphaltene solubility parameters. The presented experimental data points have identified the aggregation mechanisms for four particle size-time patterns collected from the literature (Fig. 2). Moreover, the depth changes cannot be included by the FLASS apparatus for the particle size distribution acquired experimentally. Future research that simulates the flow of crude oil in a well column using a closed loop has to take this fact into consideration. Another challenge is to incorporate the thermal resistances present in the well column's structure, such as cement, to utilize casing temperature instead of geothermal one.

Conclusions

The present study aimed at investigating how various aggregation kinetics affected the thermophoretic velocity of asphaltene deposition. In order to do this, four distinct kinetic patterns were gathered from the literature that show how asphaltene particle size changes over time. Additionally, when measuring asphaltene aggregation in a live oil medium using the FLASS apparatus, the effects of pressure and flow rate decrease were carefully examined. After that, both the measured and collected aggregation patterns were examined for the kinetic change of the thermophoretic velocity of deposition. The main results are as follows:

• The RLA, DLA, asymptotic pattern, and/or a combination of these would govern the process of particle aggregation.

• The thermophoretic velocity is influenced by a number of interrelated parameters, including temperature gradient, physical fluid properties, flow regime, and chemical environment. As a result, it is a complicated phenomenon when asphaltene particles behave thermophoretically in a well column.

• The formation of aggregates is a challenging interfacial phenomenon that is influenced by several variables, including fluid composition, temperature, pressure, and intermolecular interactions.

• Due to the growing difference in static and geothermal temperatures as one moves from the perforation zone to the surface, the thermophoretic effects would become more pronounced.

• To determine how the change in particle size affects the thermophoretic velocity of deposition, a comparatively wide range of asphaltene particle radii were studied in the current work, ranging from around 100 nm to almost 9 µm. The results showed that the thermophoretic velocity of deposition increases as asphaltene particle size decreases and vice versa. This proves that the impact of particle size on the thermophoretic velocity would not alter around a temporary threshold for asphaltene particle size. It is important to emphasize that the conclusion would be general and case-independent based on the criteria utilized in this study.

• Finally, the findings of this study showed that better management of asphaltene particle size inside well columns may serve as a promising strategy to mitigate the deposition of asphaltene. From an industrial perspective, a tube can be inserted into a well column to (i) introduce a flocculent and (ii) act as a venturi to raise the average size of asphaltene particles and accelerate the pace at which asphaltene deposits can be dislodged.

List of symbols

$A$

Constant (–)

$B$

Constant (–)

$C$

Constant (–)

$C_c$

Cunningham correction factor (–)

$C_m$

Constant (–)

$C_s$

Constant (–)

$C_t$

Constant (–)

$C_v$

Heat capacity at constant volume (J/kg K)

$d_A$

Asphaltene particle diameter (m)

$d_m$

Molecular diameter (m)

$f$

Darcy friction factor (–)

$\mathit{Kn}$

Knudsen number (–)

$K_{\mathit{th}}$

Thermophoretic coefficient (–)

$k_A$

Asphaltene thermal conductivity (W/m K)

$k_b$

Boltzmann constant (J/K)

$k_F$

Fluid thermal conductivity (W/m K)

$M{W}_F$

Fluid molecular weight (g/mole)

$m_A$

Mass of asphaltene particle (kg)

$R$

Gas constant (J/kg K)

$S_n$

Normal momentum accommodation coefficient (–)

$S_t$

Tangential momentum accommodation coefficient (–)

$T$

Temperature (K)

$T_m$

Fluid mean temperature (K)

$T_w$

Wall temperature (K)

$t_A$

Relaxation time of asphaltene particle (s)

$U_m$

Mean fluid velocity (m/s)

$u_{}^{\ast }$

Friction velocity (m/s)

$v_{\mathit{th}}$

Thermophoretic velocity (m/s)

Greek symbols

$\alpha$

Constant (–)

$\delta$

Viscous sub-layer thickness (m)

$\mu$

Fluid dynamic viscosity (kg/m s)

${\nu }_F$

Fluid kinematic viscosity (m²/s)

${\pi }_1$

Constant (–)

$\varpi$

Heat capacity ratio (–)

${\rho }_F$

Fluid density (kg/m³)

${\rho }_A$

Asphaltene density (kg/m³)

${\tau }_w$

Wall shear stress (Pa)

$\varphi$

Constant (–)

$\mathrm{\nabla }T$

Temperature gradient (K/m)

Abbreviations

API

The American petroleum institute gravity

ASR

Asphaltene settling region

DLA

Diffusion-limited aggregation

DLVO

Derjaguin-Landau-Verwey-Overbeek

FLASS

Flow assurance system

Geo

Geothermal

HPM

High pressure microscope

m.d.d.

Meters drilled depth

Re

Reynolds number

RLA

Reaction-limited aggregation

Author contributions

A.H.N. is responsible for conceptualization, methodology, data curation, investigation, software, analysis, and writing the main manuscript. M.G. supervised the manuscript. M.R.M. is responsible for conceptualization, supervision, validation, as well as review and editing the manuscript.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.