Analyzing the Effects of Produced Water on Asphaltene Deposition in

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Process Systems Engineering

Analyzing the Effects of Produced Water on Asphaltene Deposition in a Vertical Production Tubing Roozbeh Mollaabbasi, Dmitry Eskin, and Seyed Mohammad Taghavi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03336 • Publication Date (Web): 19 Oct 2018 Downloaded from http://pubs.acs.org on October 19, 2018

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Industrial & Engineering Chemistry Research

Analyzing the Effects of Produced Water on Asphaltene Deposition in a Vertical Production Tubing Roozbeh Mollaabbasi,∗,† Dmitry Eskin,‡,¶ and Seyed Mohammad Taghavi† Department of Chemical Engineering, Université Laval, Québec, G1V 0A6, Canada, and Schlumberger-Doll Research, Cambridge, MA, 02139, USA E-mail: [email protected] Phone: +1 418 656-2131. Fax: +1 418 656-5993

Abstract The reduction of asphaltene deposition flux in a production tubing, caused by produced water, is studied by an engineering modeling approach. Two phenomena, which according to our analysis have the most significant potential to reduce the asphaltene deposition, are considered. First, decreasing the asphaltene concentration in a hydrocarbon fluid due to the asphaltene absorption on droplet surfaces is estimated. Second, it is considered that droplets colliding with a pipe wall partially block the surface area available for asphaltene deposition, which in return reduces the deposition flux. The fraction of the pipe wall surface dynamically coated with water droplets is calculated from analyzing the wall bombardment by droplets fluctuating in a turbulent flow. The ∗

To whom correspondence should be addressed Université Laval ‡ Schlumberger-Doll Research ¶ Skolkovo Institute of Science and Technology †

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

results reveal that the asphaltene deposition flux is not much influenced by the decrease due to the asphaltene absorption on droplet surfaces or the droplet-pipe wall interaction.

Introduction Asphaltenes are molecular substances present in crude oil along with resins, aromatic hydrocarbons and saturated hydrocarbons. Asphaltenes precipitate from a hydrocarbon fluid when the local pressure drops below the so-called asphaltene precipitation onset pressure. 1,2 Asphaltene deposition represents a significant concern mainly for a vertical production tubing, because namely in this case a substantial pressure drop is obtained due to a hydrostatic pressure difference. 1,3 The deposit layer, formed over a considerable time (months, years), can eventually become thick enough to cause significant blockage of the tube cross-section. This blockage may lead to a considerable reduction in production rate; therefore, the asphaltene deposition must be prevented. 4,5 For this purpose, asphaltene inhibitors are usually injected downhole. To determine whether inhibitors are needed, the asphaltene deposition process should be studied. Numerous articles are dedicated to the problem of asphaltene deposition in reservoirs, 6–8 in pipelines 1,2,9–11 as well as in a laboratory Couette device. 9,12,13 A few of the known asphaltene deposition models (i.e., Eskin et al. 1,2 ) demonstrate reasonable agreement with experimental data. Due to the complexity of the asphaltene deposition phenomena, any known deposition model contains several empirical parameters, which are identified from laboratory experiments. These empirical parameters indirectly account for asphaltene/oil system properties determining the asphaltene deposition rate. The asphaltene deposition process includes four major phenomena: 1) precipitation of primary asphaltene particles; 2) particle agglomeration; 3) particle transport to the pipe wall; 4) particle interaction with the wall. Let us briefly describe an asphaltene deposition model from the literature, 1 which ex2


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plicitly considers the phenomena explained above. Based on Couette device experiments, Eskin et al. 1,2 conclude that the asphaltene deposition rate is a strong function of the particle size; therefore, modeling of the particle size distribution evolution along a production tubing is a key element of the deposition model. Initially, Eskin et al. 1,2 assume that the agglomeration of asphaltene particles is governed by both Brownian motion and turbulent diffusion. Similarly, the particle deposition in a turbulent flow should be also controlled by the same mechanisms. However, Eskin et al. 2 demonstrate that turbulent diffusion can be neglected because only fine (sub-micron) particles are able to deposit on the pipe wall. For such small particles, Brownian diffusivity significantly exceeds turbulent diffusivity. They modelled the particle size distribution evolution along a production tubing using a population balance equation. Here, the two major model parameters are: the particle-particle collision efficiency, α, determining the probability of particle sticking with one another, and the particle-wall sticking efficiency (probability) γ. They assume that the pipe wall becomes coated with a layer of asphaltene particles as soon as the deposition process begins. Therefore, their deposition model considers that the particles deposit on the previously formed deposit layer. This assumption leads to the conclusion that the parameters α and γ are of the same order of magnitude. Eskin et al. 1 use a high pressure-high temperature laboratory Couette device to identify these parameters from experiments. The experimental device consists of two coaxial cylinders with inner cylinder rotating and the outer cylinder being immobile. The experiments show that the identified model parameters α and γ are always in the range of 10−6 -10−5 . Thus, the probability of asphaltene particles to stick to one another is extremely low. Therefore, the concentration gradient of asphaltene particles at the wall is very small. Based on these, Eskin et al. 2 calculate the deposition flux to the wall by a simple relation based on the particle fluctuation frequency caused by Brownian motion. Once the deposition model parameters are determined from the Couette device experiments, modeling of asphaltenes deposition in a production tubing is straightforward. A few other known deposition models, 14–17 recently developed by different research groups, are based on 3



rather similar ideas. Let us emphasize that the described deposition model as well as other models known from the open literature are developed for a single phase (hydrocarbon) flow, whereas, in practice, oil production is almost always accompanied with production of water. To the best of our knowledge, there are no publications in which the effects of produced water on asphaltene deposition are considered. The influences of produced water (water droplets in turbulent flows) on asphaltene deposition in a production tubing turbulent flow are not addressed due to the complexity of the phenomena involved. Produced water often contains dissolved salts of Sodium, Calcium, Magnesium, Strontium and Barium. 18 The effects of salts dissolved in water on asphaltene deposition are small and can be excluded from consideration. Arla et al. 19 and Andersen et al. 20 explained that at a high water pH, divalent ions of these salts interact with naphthenic acid, usually present in oil in small amounts, and create naphthenates (organic carboxylate salts). The latter are sticky particles, which can deposit on the pipe wall and, therefore, represent a significant flow assurance issue. However, deposition of naphthanates is a completely different problem from asphaltene deposition being considered in this work. Let us briefly analyze the possible migration of the mentioned salt ions through the water/oil interface. As it will be explained in the next section, asphaltene nano-aggregates migrate to the water/oil interface (droplet surfaces) and absorb on it. Asphaltene particles can potentially transport naphthenic acid molecules to water droplet surfaces, where this acid can interact with the metal ions at the interface forming naphthenates that, in this case, significantly contribute into droplet stabilization. 19 This interface process significantly reduces the rates of ion migration from the water to the oil phase. Hence, if the water pH is low or the concentration of naphthenic acid is low, the metal salts dissolved in water do not represent any flow assurance issue. At any level of acid concentration, the salts may contribute into water droplet stabilization causing some droplet size reduction, which according to the developed model in the present work, influences the water effects on asphaltene deposition. Note that for the sake of simplicity, 4


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in the present work we assume that the emulsion is stable because asphaltene particles themselves are rather efficient stabilizers. In addition, at a low acid concentration, metal ions transported into the oil phase do not affect the properties of depositing asphaltene particles. At a high naphthenic acid concentration, deposition of naphtenates may occur, but this problem is beyond the scope of the present research. Our literature analysis reveals that practically any known aspheltene deposition model is reduced to the calculation of the deposition flux of asphaltene particles on the wall. Therefore, accounting for the effects of water on asphaltene deposition can be reduced to determining a decrease in the deposition flux caused by the presence of water in an oil flow. Since this decrease does not depend on the deposition model, instead of analysing of the asphaltene deposition process itself, our effort will be focused only on modeling the mechanisms leading to the reduction in the asphaltene deposition flux due to the presence of water droplets suspended in oil. It is worth mentioning that studying the size evolution of asphaltene particles is not the aim of this paper as it has been meticulously studied in previous works. 1,2,6–13 In addition, there is a significant difference between the fluctuation scales of asphaltene particles and water droplets, 21 which allows neglecting the direct effect of droplets on asphaltene transport to the wall. Let us remind that we are currently unable to confirm our modeling results with experimental data, due to extreme complexities (high pressure, high temperature) and expenses (very large fluid volumes and experiment duration) associated with conducting asphaltene deposition experiments in a turbulent pipe flow even without water. To our best knowledge, such experiments have never been performed. Deposition profiles have been measured in the field (e.g. Haskett 12 ) but the data provided are not sufficient even for modeling a regular deposition case (no produced water). Such measurements are quite expensive, which is perhaps the reason why there are no publications in the open literature. On the other hand, deposition experiments in the presence of water in a small-size Couette device, recently used for asphaltene deposition studies, cannot provide the desired data, due to centrifugal 5



force effects. Eskin et al. 22 showed that in a Couette flow water droplets, whose density is higher than that of oil, are pressed to the outer Couette device wall by the centrifugal force and a water film is formed on this wall. Film formation entirely excludes deposition so to suppress droplet coalescence and provide a dispersed flow, Eskin et al. 22 used a surfactant. However, surfactants significantly modify asphaltene properties and, therefore, cannot be applied to deposition experiments. Thus, the infeasibility of the experimental studies of the effects of water on asphaltene deposition under realistic conditions underlines a necessity of an engineering modeling of this phenomenon. The outline of the paper is as follows. In the modeling section, the basic concept of accounting for the effect of dispersed water on asphaltene deposition and the model for estimation of the wall pipe area contaminated by droplets are discussed. The results and discussion section presents the results of the model in general and practical forms. The paper ends in the conclusions section, with a summary and conclusions.

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Modeling Basic concept of accounting for the effect of dispersed water on asphaltene deposition

D

x

U Figure 1: The schematic of the geometry and the flow. Diagram illustrating asphaltene/droplets deposition on the wall in a turbulent flow in a vertical pipe. The gray bullets represent the asphaltene particles and the blue circles are the droplets. A conventional oil flows through a production tubing, usually in a regime of developed turbulence; therefore, water in such a flow is represented by droplets, almost uniformly dispersed across a tubing cross-section. Figure 1 illustrates asphaltene deposition on the tubing wall in the presence of water. The smaller particles and the larger spheres denote asphaltene particles and water droplets, respectively. There are the two major phenomena associated with the presence of water in oil which may affect the asphaltene deposition: 1. Absorption of asphaltene molecules on the surface of water droplets. 2. Bombardment of the pipe wall with water droplets. Let us start with analyzing of the first phenomenon. Asphaltene molecules are absorbed on the surface of water droplets, which leads to a reduction in the mobility of these surfaces causing the coalescence suppression; i.e., droplet stabilization. Non-coalescing droplets are 7



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intensely fragmented by turbulence and can reach rather small sizes; therefore, at a high water hold up, the droplet specific surface may reach very high values. It is important to evaluate how significantly the absorption of asphaltenes on droplets may reduce the asphaltene content in a hydrocarbon fluid, as this reduction obviously can lead to a decrease in the deposition rate. It is assumed that asphaltene molecules absorbed on a droplet surface form a monolayer. Let us first estimate the specific droplet surface dispersed in oil. Fortunately, a rather accurate estimation is possible because droplets are uniformly distributed over a vertical tubing operating in a turbulent regime. 21 For the sake of simplicity, it is assumed that droplets are monodispersed. Then, the specific droplet surface is calculated as: 23

sw =

6φ , d

(1)

where d and φ are the droplet size and water hold up, respectively. The volume concentration of asphaltene molecules which form the monolayer on droplet surfaces is calculated as follows: δ ξl ≈ sw δ = 6φ , d

(2)

where δ is the asphaltene monolayer thickness. Then, the ratio of the volume concentration of asphaltenes absorbed on droplets to that of precipitated asphaltene particles (which potentially could deposit on the pipe wall) is evaluated as: ξl φδ =6 . ξp ξp d

(3)

A reasonable estimation of the mean droplet diameter is a droplet Sauter mean diameter. 24 The equations to calculate the Sauter mean diameter are given in the modeling deposition rate reduction caused by droplets-pipe wall collisions section, where the second phenomenon linked to the deposition reduction (due to presence of dispersed water droplets in oil) is analyzed. The thickness of the asphaltene monolayer is equal to the diameter of asphaltene

8


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molecule roughly evaluated as the size of a newly precipitated asphaltene nano-aggregate. 25 In this case, the asphaltene monolayer thickness can be estimated to be δ ' 5 nm, 2 which is greatly conservative (upper limit). The volume concentration of precipitated asphaltenes causing an intense asphaltene deposition can be roughly evaluated as ξp ∼ 0.01. 26 For the sake of obtaining a conservative (upper limit) estimation, it can be assumed that the water hold up is high, i.e., φ = 0.5. Also, it is considered that the deposition occurs in a production tubing of a standard diameter, D = 0.1 m. The flow velocity is not expected to be higher than U = 2 m s−1 in this case. Based on the computations of Eskin et al. 21 for similar flow conditions, it can be assumed that the Sauter mean diameter will not be smaller than dS = 50 µm. Then, the ratio of the concentration of asphaltenes absorbed by droplets to the concentration of all precipitated asphaltenes in a fluid ( ξξpl ) calculated by Eq. 3 is equal to 0.03. Thus, even this conservative (upper) estimate shows that the absorption of asphaltenes on water droplet surfaces cannot reduce asphaltene content in a hydrocarbon fluid significantly enough to noticeably affect the asphaltene deposition on the pipe wall. Evaluation of the second phenomenon mentioned (see the introduction) affecting the asphaltene deposition due to the presence of water is a more complicated problem. Therefore, the rest of the present paper will be dedicated to a detailed analysis of the effects of water droplets colliding with the pipe wall on the asphaltene deposition. Droplets, dispersed in an oil flow fluctuate being driven by turbulence eddies. Fluctuating droplets collide with the wall (or the surface of deposit layer formed on the wall) and bounce from it. During a droplet-wall interaction, a deformed droplet covers some surface area of the wall. This area varies during the bouncing process. Evidently, the area coated by a droplet is unavailable for depositing aspahltene particles. Therefore, the problem of accounting for the effect of water droplets colliding with the wall on asphaltene deposition can be reduced to determining the mean fraction of the pipe wall surface that is dynamically coated with water droplets bombarding the wall. Note that the deposition flux is linearly proportional to surface that is available

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for deposition. 1,21 The deposition flux in the presence of dispersed water is calculated as:

qav = qa (1 − S),

(4)

where, qa is the asphaltene deposition flux and S is the mean fraction of the pipe wall surface dynamically coated by water droplets. Thus, determining the effects of dropletswall interactions on asphaltene deposition can be reduced to calculating of the coefficient of the asphaltene deposition rate reduction, AR = 1 − S. The smaller this coefficient is, the stronger the deposition rate will be decreased due to the presence of water.

Modeling deposition rate reduction caused by droplets–pipe wall collisions To describe the process of droplets-wall interaction in a tubing flow, the evolution of droplet size distribution along a tubing should be simulated. This problem can be solved, for example, by considering an advection-diffusion-population balance equation. 21 However, solving this equation is relatively complicated, which can be fulfilled only numerically. Moreover, there are no reliable models of both droplet breakup and coalescence, which are required for using the population balance approach. Since we prefer to limit our work only reliable engineering estimations we do not consider a population balance modeling approach, and instead employ a droplet Sauter mean diameter, mentioned in the previous section. This characteristic parameter represents the volume/surface area ratio of a droplet ensemble, characterized by a certain size distribution. The Sauter mean diameter is defined as: 27

dS =

N P

, N P

zi d3i

i=1

zi d2i

(5)

i=1

where di is the diameter of the i-th size fraction droplet and zi is the fraction by number of the i-th droplets. This characteristic can be rather accurately evaluated by a semi-empirical

10


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correlation for a stable steady-state emulsion formed in a pipe flow. 24 Note, we will assume our emulsion is stable because our primary interest is a system with a rather high content of asphaltenes, which reduce mobility of water droplet surfaces suppressing coalescence. A steady-state droplet size distribution is usually log-normal. There are different semi-empirical equations for the calculation of the maximum steady-state droplet diameter of an emulsion in a pipe flow. 28,29 By analyzing experimental data, Hesketh et al. 24 have shown that for a steady-state pipe flow, the ratio of the Sauter mean diameter (dS ) to the maximum droplet size is approximately equal to Cn = 0.62. As the maximum droplet size, these authors have employed the characteristic droplet diameter d99 indicating that 99 % of droplets in a system are smaller than a droplet of this size. Brauner 30 have derived known equations for the estimation of the maximum droplet size in a turbulent pipeline flow. For a low concentration of a dispersed phase, the maximum size is evaluated from the balance between the dynamic pressure difference exerted on opposite droplet sides and the droplet capillary pressure. The dynamic pressure difference is estimated using the turbulence kinetic energy of a turbulent eddy of a scale equal to a droplet size. Then, the maximum droplet size is calculated as follows: 30

(d˜max )dilute

" #−0.4 (1 − φ) dmax )dilute = (1.88 φ W e−0.6 Re0.08 ), =( D + (1 − φ) 2Λ

(6)

where, Re = ρo DU/µo and W e = ρo DU 2 /σ are the Reynolds and Weber numbers based on the continuous phase parameters, respectively. Λ = ρo /2ρd is the density ratio of the fluids. σ, ρo and ρd are the interfacial surface tension, the density of oil and the density of droplet, respectively. If the dispersed phase concentration is high, the maximum droplet size is determined from the balance of the generation rate of the surface energy of a dispersed phase flowing through a pipe and the turbulent energy supplied by the continuous phase. The maximum

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droplet size, in this case, is calculated as: 30 ) = (d˜max )dense = ( dmax D dense h φ 0.6 ) 1+ (7.61CH 3/5 W e−0.6 Re0.08 ( 1−φ

φ 1 2Λ (1−φ)

i−0.4

(7) ),

where, CH is a constant which may require some tuning (in this work we assume use CH = 1 considering that the turbulent kinetic energy of a continuous phase is entirely available for dispersion of another phase 30 ). Because a dispersed phase concentration, differentiating diluted and dense emulsions, are not specified within the calculation method formulated, it is recommended 30 to determine the maximum diameter using both the equations as follows: n o ˜ ˜ ˜ dmax = M ax (dmax )dilute , (dmax )dense .

(8)

The largest size should be employed as the maximum droplet size for further calculations. A collision of a single droplet with the wall should be considered in order to evaluate the cumulative wall coating effect caused by multiple droplets-wall interactions. In reality, the pipe wall is often characterized by a rather significant roughness. This circumstance complicates the interaction analysis. However, a reasonable estimate of the interaction phenomenon could be obtained assuming that the wall is smooth. This assumption becomes more realistic as the ratio of the droplet diameter to the wall roughness increases. The droplet fluctuation created by a turbulent eddy is the only significant mechanism causing a droplet-wall collision. Based on Kuboi’s method, 31 we assume that eddies whose scales are equal to droplet sizes mainly control the droplet fluctuations. In addition, we assume that the droplet deformation during the droplet-wall interaction process is small as the droplet fluctuation velocity is relatively low. Obviously, after a collision with the wall, the droplet should ricochet. Thus, the droplet-wall interaction process consists of the two stages: droplet compression (deformation) and relaxation (shape restoration). We assume that a hydrodynamic flow field inside the droplet, caused by deformation, is weak and its

12


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effect on the ricochet can be ignored due to the smallness of droplet deformation. Now, we can formulate the momentum equation for the droplet-wall interaction assuming that all fluid elements inside the droplet have the same velocity (see Figure 2). In addition, we assume that the droplet interacting with the wall maintains a shape of a sphere with a cut segment. We neglect the volume of this segment due to the deformation smallness. We also assume that the wall reaction force acting on a droplet is comprised of the two components during the compression stage. First, a hydrodynamic reaction force appears as the droplet interacting with the wall loses its momentum component normal to the wall. We evaluate this force by considering an interaction of an imaginary jet of the radius a (see Figure 2) with the wall. Note that this is absent during the relaxation stage. The second component of the reaction force is the capillary force applied also to the same droplet-wall contact area. For the sake of simplicity, we neglect the drag force acting on the droplet during its interaction with the wall. Because a turbulent eddy transporting the droplet is also decelerated in the wall vicinity, this assumption should not significantly affect the droplet-wall interaction process. Hence, the momentum conservation equation for a droplet interacting with the wall during

Vy

R

a

R

y

Figure 2: Schematic of a droplet-wall interaction. From left to right each subfigure shows a droplet close to the wall and a stagnant droplet on the wall.

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the compression stage is: dVyd 2σ 2 1 = −ρd Vy2d πa2 − πa , (ρd + ρo )v 2 dt R

(9)

where a is the radius of the droplet-wall interaction area. Vyd is the normal-to-the-wall velocity component of the droplet. The left-hand side of this equation represents the inertial force that is equilibrated by the forces expressed by the terms on the right-hand side, i.e., the hydrodynamic reaction force and the capillary force, respectively. Note that the term expressing the inertial force accounts for the virtual mass force. We can formulate Eq. 9 in the dimensionless form as follows:

where y˜ = y/R, V˜yd

3 1 d˜ y d2 y˜ =− (1 − y˜2 )(( )2 + 1), (10) 2 dτ 4 (1 + Λ) dτ p = Vyd /Ω, Ω= 2σ/ρd R and τ = Ωt/R. We can solve this equation

numerically using the following boundary conditions:

at τ = 0

   y˜ = 1,   d˜ y /dτ = V˜y0d .

(11)

So do so, however, we need to first calculate the velocity of the droplet at the moment of touching the wall (V˜y0d ), as described further below. We assume that the distribution of the fluid velocity fluctuation is Maxwellian. Also, we assume that the droplet exactly follows the fluid in a fluctuation motion. It is also reasonable to consider that the initial value of normal-to-the-wall droplet fluctuation velocity is equal to the normal-to-the-wall component of the most probable fluid fluctuation velocity at y = 0.5d. r s 2 d νd = Vy0 2+ ( ), π 2

14


(12)

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where Vy0 2+ is the mean-square fluid fluctuation velocity. The root-mean-square fluid fluctuation velocity can be calculated by the following empirical correlation: 1 q Vy0 2+ = Here,

0.005y +2 . 1 + 0.002923y +2.128

r y + = R/δ + =

f Re2 ˜ ds = 32

r

0.046 1.8 ˜ Re ds 32

(13)

(14)

is the normalized distance from the wall at the moment when a droplet touches the wall. p δ + = νo /u∗ is the wall layer thickness (νo is the kinematic viscosity of the oil). u∗ = (τw /ρo ) is the wall friction velocity, τw = ρo f U 2 /8 is the shear stress at the pipe wall, f = 0.046/Re0.2 is the Fanning friction factor 32,33 and d˜s = ds /D. The normal-to-the-wall component of r q 0 2+ 0 Vy the root-mean-square fluid fluctuation velocity is Vy 2y+ = . 34 With respect to the 3 velocity scales of the fluid (u∗ ) and droplets (Ω), the dimensionless velocity of the droplet at the moment of touching the wall is s V˜y0d =

u∗ 2

s

0 2+

Vy 2 Ω 3

=

0.046 W e d˜s 0 2+ V . 192 Re0.2 Λ y

(15)

We have used a Rungee-Kutta method implemented in Matlab (with absolute and relative error tolerances equal to 1×10−14 ) in order to solve Eq. 10 numerically, up to the moment when V˜yd = 0, indicating the end of the compression stage. At this moment, the distance from the droplet center to the wall is yf . Then, the relaxation stage is calculated using the same momentum equation (Eq. 9) where the first right-hand term is zero. However, the expected compression-relaxation stage sequence can be distorted if the characteristic period of the turbulence eddy transporting the droplet is smaller than the total time of the compressionrelaxation stages. The characteristic period is evaluated as T ∼ d/Vy0d . 35 It is reasonable to assume that the droplet-wall interaction time is limited to this period. However, this

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scenario can be excluded in the case of small droplet deformations considered here. The specific pipe-wall area dynamically covered by droplets due to their multiple collisions with the wall is:

(16)

S = f rςA,

where A is the mean droplet-wall contact area during a single collision; ς is the mean droplet wall contact time; f r is the frequency of droplet-wall collisions per surface area unity. The frequency of droplet-wall collisions per surface area unity is determined, assuming the Maxwellian droplet fluctuation velocity distribution, as: 36 q

fr =

0

Vy 2+ u∗ √ . md 6π

ρd φ

(17)

The dimensionless form of the frequency of droplet-wall collisions is: q fr f˜r = Ω = D3

φ

0

Vy 2+ 6 1 √ 6π π d˜3S

s

0.046 W e d˜s . 64 Re0.2 Λ

(18)

Then, the mean specific (per pipe surface unity) surface area dynamically covered with droplets is calculated as: 



Z R˜ Z y˜f d˜S ˜2 d˜ y d˜S ˜2 d˜ y   2 2 ˜ ˜ ˜ S = fr  π(R − ( )y ) + π(R − ( )y ) , ˜ ˜ 2 2  R˜ Vyd Vy d  y˜f | {z } | {z } compression

when tint =

Ryf R

dy Vyd

+

RR yf

dy Vyd

(19)

relaxation

< T . If the interaction time of a droplet with the wall is shorter

than the eddy characteristic period, then the corresponding integration limit in Eq. 10 is adjusted to provide the actual interaction time to be equal to the characteristic period.

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Results and discussion General results The Reynolds number, Weber number, density ratio and water hold up are four dimensionless groups that play considerable roles in results of the developed model. Here, the dimensionless distance from the droplet center to the wall surface (˜ y ) and the total specific area covered by droplets are studied for a wide range of the Reynolds and Weber numbers. During touching the wall

Before touching the wall

Vy 0 d  0.1224

Vyd  0.1063

  0.0

Vyd  0.0637

  0.5

  1.0

Vyd  0.0000

  1.5

Figure 3: Top view of droplet-wall interaction with time. A droplet (the blue sphere) with radius R touches the pipe wall at τ = 0 with the initial velocity of V˜y0d . The droplet stops on the wall after τ = 1.5. W e = 1 × 10 4 , Re = 1 × 10 5 , Λ = 0.35 and φ = 0.4. Figure 3 shows an example of the droplet-wall interaction with time for a droplet with radius of R, W e = 1 × 10 4 , Re = 1 × 10 5 , Λ = 0.35 and φ = 0.4. The droplet touches the wall at τ =0. The initial dimensionless velocity of the droplet at the moment of touching the wall is V˜y0d =0.1224. The droplet penetrates to the wall and its inertial energy is dissipated with time. The droplet stops at τ =1.5. The overlapping area between the droplet and wall (indicated with the white arrow) is occupied by the droplet and is not available for the asphaltene particles to deposit ant more. Figure 4 demonstrates the effects of the density ratio (Λ) and the water hold up (φ) on y˜f (when V˜yd = 0) at different W e and Re numbers. First let us look at one of the panels individually (i.e., Figure 4d) at constant Λ and φ. The results show the W e and Re numbers have an opposite effect on y˜f . At a fixed W e number, y˜f decreases by increasing Re, as the inertial force increases so that the droplets penetrate more in to the wall. In addition, at 17



(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4: The dimensionless distance from the droplet center to the wall surface at different Re and We numbers. The density ratio (Λ) is fixed in each column and it is 0.35, 0.4 and 0.45 from left to the right. The water hold up is fixed in each row and it is 0.1, 0.25 and 0.4 from top to the bottom. a fixed Re number, y˜f increases by increasing W e because the interfacial tension decreases, preventing the deformation of droplets. Comparing the panels of each row explains the effects of Λ at a fixed φ. The results show a slight reduction in y˜f by increasing Λ. This phenomenon is more clear while comparing Figure 4g and Figure 4i at the Re = 9×104 and W e = 1 × 104 when Λ changes from 0.35 to 0.45. Finally, in each column, y˜f decreases by increasing φ (compare Figure 4b and Figure 4h). Figure 5 presents the effects of W e, Re, Λ and φ on the mean specific surface area dynamically covered with droplets. First, let us consider one of the panels when Λ and φ are fixed (Figure 5g). At a constant Re number, increasing W e increases the total specific area covered by droplets because the droplet-wall collision frequency increases (see Eq. 18). In addition, S is enhanced as Re increases at a fixed W e number because the inertial force 18


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increases and consequently the droplet size decreases. Comparing the results of the panels in each row shows that S decreases by increasing Λ at a fixed φ. Finally, enhancing φ increases S at a constant Λ. (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 5: The total specific area contours covered by droplets at different Re and We numbers. The density ratio (Λ) is fixed in each column and it is 0.35, 0.4 and 0.45 from left to the right. The water hold up is fixed in each row and it is 0.1, 0.25 and 0.4 from top to the bottom.

Typical practical results To evaluate the model from an industrial point of view, let us solve the model equations for a few practical examples. We assume a dispersion of water in oil flowing in a production tubing of the diameter D = 0.0625 m with the mean velocity U . The oil density and viscosity are ρo = 760 kg m−3 and µo = 1.3 mPa s, respectively. The water hold up is φ = 0.1 and the interfacial tension is σ = 0.025 N m−1 . Figure 6 illustrates the droplet-wall interaction process in dimensionless coordinates. Figure 6a demonstrates how the normal-to-the-wall 19



components of the velocities of droplets of different sizes, obtained at the different flow velocities (U = 0.5, 1, 1.5 and 2 m s−1 ), change during the droplet compression stage. Figure 6b shows how the distance from the droplet center to the wall changes during the same stage. The largest deformation is observed for the droplet of the largest size. However, even for the largest droplet considered, the maximum deformation value does not exceed 4% of the droplet radius. Thus, the assumption on the small droplet deformation accepted for modeling is satisfactory. Moreover, the fraction of larger droplets in the entire population of droplets dispersed in a pipe flow is, usually, relatively small, 22 whereas the deformation rapidly decreases with a reduction in droplet size. This circumstance confirms the model validity. (a) 0.05

(b) 1

0.04

0.99

0.03

y˜

V˜yd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 31

0.98

0.02 0.97

0.01 0

0.96 0

0.5

1

1.5

0

τ

0.5

1

1.5

τ

Figure 6: The dimensionless velocity (a) and the dimensionless distance from the droplet center to the wall surface (b) versus dimensionless time during the droplet compression stage at different fluid velocities for a practical case (Re varies between 1.826 ×104 and 7.307 ×104 and W e is between 475 and 7600). U = 0.5 m s−1 (black line), 1 m s−1 (red dashed line), 1.5 m s−1 (blue dotted line) and 2 m s−1 (green dash-dotted line). The continuous phase is oil (µo = 1.3 mPa s, ρo = 760 kg m−3 ) and the dispersed phase is water; φ = 0.1, σ = 0.025 N m−1 . The tubing diameter is D = 0.0625 m. The calculated dimensionless Sauter mean diameter and the dimensionless droplet-wall collisions frequency versus the Re number are shown in Figure 7 at different water hold ups. An increase in the flow velocity leads to a decrease in droplet sizes and an increase in the frequency of droplets-wall bombardment. The dimensionless mean Sauter diameter increases 20


Page 21 of 31

by increasing φ at constant Re number and consequently the dimensionless frequency of the droplet-wall collision decreases. (a)

(b) 1500

0.2 0.15

f˜r

1000

d˜S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

500

0.05

0

0 2

4

2

6

Re

×10

4

6

Re

4

×10 4

Figure 7: The dimensionless mean Sauter diameter (a) and the dimensionless frequency of droplets-wall collisions (b) versus Re at different water hold ups for a practical case (U varies between 0.5 and 2 m s−1 ). φ = 0.1 (black line), 0.2 (red dashed line), 0.3 (blue dotted line) and 0.4 (green dash-dotted line). The continuous phase is oil (µo = 1.3 mPa s, ρo = 760 kg m−3 ) and the dispersed phase is water; σ = 0.025 N m−1 (W e is between 475 and 7600). The tubing diameter is D = 0.0625 m. Figure 8 shows the effects of water hold up on the dimensionless mean Sauter diameter (d˜S ) and the dimensionless distance from droplet center to the wall surface (˜ yf ) when V˜yd = 0, for different mean flow velocities. At the constant mean flow velocity, d˜S increases with an increase in the water hold up. Increasing the mean flow velocity decreases the mean Sauter diameter at a fixed water hold up. These results are compatible with Figure 7a. Figure 8b shows, at a fixed mean flow velocity, y˜f decreases by increasing the water hold up. In addition, at a fixed water hold up, an increase in the mean flow velocity increases the inertial force and consequently leads to a decrease in droplet sizes. Therefore, the droplet deformation decreases. The results are compatible with those in Figure 6b.

21



(a) 0.2

(b)

1

0.15

y˜f

0.95

d˜S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 31

0.1

0.9 0.05 0 0.1

0.2

0.3

0.4

0.85 0.1

0.5

φ

0.2

0.3

0.4

0.5

φ

Figure 8: The dimensionless mean Sauter diameter (a) and dimensionless distance from droplet center to the wall surface (˜ yf ) when V˜yd = 0 (b) versus φ at different mean flow velocities. U = 0.5 m s−1 (black line), 1 m s−1 (red dashed line), 1.5 m s−1 (blue dotted line) and 2 m s−1 (green dash-dotted line). The continuous phase is oil (µo = 1.3 mPa s, ρo = 760 kg m−3 ) and the dispersed phase is water; σ = 0.025 N m−1 (W e is between 475 and 7600). The tubing diameter is D = 0.0625 m. Figure 9 represents the fraction of a pipe surface area dynamically coated by droplets versus the droplet volume fraction for different mean flow velocities. The fraction of a coated pipe surface indicates a reduction in the asphaltene deposition flux due to the presence of water droplets in the flow.

22


Page 23 of 31

10 -2

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 -3

10 -4 0.1

0.2

0.3

0.4

φ Figure 9: The total specific area covered by droplets versus water hold up at different fluid velocities. U = 0.5 m s−1 (black line), 1 m s−1 (red dashed line), 1.5 m s−1 (blue dotted line) and 2 m s−1 (green dash-dotted line). The continuous phase is oil (µo = 1.3 mPa s, ρo = 760 kg m−3 ) and the dispersed phase is water; σ = 0.025 N m−1 (W e is between 475 and 7600). The tubing diameter is D = 0.0625 m. Based on the results obtained, one can see that the flux reduction increases with an increase in the mean flow velocity as well as with an increase in the water volume fraction. The higher the mean flow velocity, the higher the energy dissipation rate leading to the smaller Sauter mean diameter will be. A reduction in the Sauter mean diameter along with an increase in the turbulence fluctuation velocities, caused by an increase in the energy dissipation rate, leads to an increase in the frequency of droplet-wall collisions. This in return causes an increase in the wall surface area which is dynamically coated by droplets. The effects of the water volume fraction on the deposition flux reduction is explained straightforwardly. The higher the droplet concentration is, the larger the surface area dynamically coated by droplets will be. A major result of our calculations, illustrated in Figure 9, shows that the reduction in the deposition flux caused by the dynamic coating of a pipe surface with water droplets is obviously small. It is worth mentioning that the variations in the coefficient of

23



the deposition rate reduction due to the presence of water (AR), introduced in the basic concept of accounting for the effect of dispersed water on asphaltene deposition section, are not graphically presented here, as our results show that this coefficient is only slightly different from unity. In addition, the phenomena associated with the presence of water (analyzed in the basic concept of accounting for the effect of dispersed water on asphaltene deposition section), such as asphaltene absorption on water droplets, cannot significantly reduce the asphaltene deposition flux. Thus, the calculations by the developed model reveals that the effects of produced water on asphaltene deposition in a turbulent production tubing flow are negligible. Let us emphasize again that according to our analysis, the proposed engineering model accounts for the major phenomena associated with asphaltene deposition in a pipeline in the presence of water. Further investigations of this phenomenon could be focused on a challenging problem of the validation of this conclusion.

Conclusions The effects of water dispersed in a turbulent pipe oil flow on asphaltene deposition was investigated. The two phenomena, which could potentially affect the deposition rate, were considered: 1) the reduction of asphaltene content in a hydrocarbon fluid due to the absorption of asphaltene molecules on droplet surfaces; 2) the reduction of a pipe wall surface area available for deposition, due to the bombardment of this surface by water droplets suspended in a turbulent flow. The first phenomenon was evaluated assuming monodispersed droplets, coated by a monlolayer of asphaltenes. A simple model was also developed to consider the second phenomena. Our calculations showed that effects of both these phenomena on the asphaltene deposition are negligible (considering our assumptions) and, therefore, can be ignored in engineering calculations.

24


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Acknowledgement This research has been carried out at Université Laval, supported financially by an NSERC Engage grant. The authors acknowledge Schlumberger Ltd. for the support of this work. The authors also wish to thank J. Solsvik (Norwegian Institute of Science and Technology) for the constructive discussions.

Nomenclature a

radius of the droplet-wall contact area (m)

A

mean droplet-wall contact area (m2 )

AR

coefficient of the asphaltene deposition rate reduction

d

particle or droplet size (m)

dS

mean Sauter diameter (m)

d˜S

dimensionless mean Sauter diameter

D

pipe diameter (m)

f

Fanning friction factor

fr

frequency of droplet-wall collisions per surface area unit (s−1 m−2 )

f˜r

dimensionless frequency of droplet-wall collisions

q

total depositing particle flux (kg m2 s−1 )

Re

Reynolds number

R

droplet radius (m)

sw

specific droplet surface (m2 m−3 )

S

mean fraction of the pipe wall surface coated by droplets

t

time (s)

T

characteristic period of the turbulence eddy transporting the droplet (s)

u∗

wall friction velocity (m s−1 )

U

mean flow velocity in a pipe (m s−1 ) 25



Vyd

velocity of droplet (m s−1 )

V˜yd q Vy0 2+ q 0 Vy 2y+

dimensionless velocity of droplet root-mean-square fluid fluctuation velocity

We

Weber number

x

coordinate along a pipeline (m)

y+

normalized coordinate

y˜

dimensionless distance from the droplet center to the wall surface

z

fraction by number of droplet

Page 26 of 31

normal-to-the-wall component of the root-mean-square fluid fluctuation velocity

Greek letters α

particle-particle collision efficiency

σ

interfacial tension (N m−1 )

δ+

wall layer thickness (m)

Λ

density ratio

φ

water hold up

Ω

characteristic velocity (m s−1 )

µ

dynamic viscosity (Pa s)

ρ

density (kg m−3 )

ξl

volume concentration of asphaltenes absorbed on droplets

ξp

volume concentration of precipitated asphaltenes

τ

dimensionless time

τw

shear stress at the pipe wall (Pa)

ς

mean droplet wall contact time (s)

ν

kinematic viscosity (m2 s−1 )

νd

initial value of normal-to-the-wall droplet fluctuation velocity

υ

droplet volume (m3 ) 26


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Subscripts a

asphaltene

av

available

d

droplet

f

final

i

droplet size number

int

interaction

m

mixture

max

maximum

o

oil

0

initial

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(35) Nijdam, J. J.; Guo, B.; Fletcher, D. F.; Langrish, T. A. Lagrangian and Eulerian models for simulating turbulent dispersion and coalescence of droplets within a spray. Appl Math Model. 2006, 30, 1196–1211. (36) Abramovich, G. Applied gas dynamics. Moscow Izdatel Nauka 1976, Table of Contents

Schematic

Mean fraction of the pipe wall surface Coated by droplets

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Result

U = 0.5 m/s U = 1.0 m/s U = 1.5 m/s U = 2.0 m/s

Water hold up

Droplet-wall interaction

31


Analyzing the Effects of Produced Water on Asphaltene Deposition in

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