Systematic Investigation of Asphaltene Deposition in Wellbore and

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Systematic Investigation of Asphaltene Deposition in Wellbore and NearWellbore Region of a Deepwater Oil Reservoir under Gas Injection. Part 2: Computational Fluid Dynamics Modeling of Asphaltene Deposition. Narmadha Rajan Babu, Pei-Hsuan Lin, Mohammed I. L. Abutaqiya, Caleb J. Sisco, Jianxin Wang, and Francisco M. Vargas Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b03239 • Publication Date (Web): 04 Dec 2018 Downloaded from http://pubs.acs.org on December 5, 2018

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Systematic Investigation of Asphaltene Deposition in Wellbore and Near-Wellbore Region of a Deepwater Oil Reservoir under Gas Injection. Part 2: Computational Fluid Dynamics Modeling of Asphaltene Deposition. Narmadha Rajan Babu1, Pei-Hsuan Lin1, Mohammed I. L. Abutaqiya1, Caleb J. Sisco1, Jianxin Wang2, Francisco M. Vargas1* 1

Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas-77005, USA 2

Chevron Energy Technology Company, Houston, Texas, 77002, USA

Authors Email Address: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] *

Corresponding Author: email- [email protected], Phone- +1 (713) 348-2384

Abstract Asphaltene deposition during oil production is a major flow assurance problem. Asphaltene deposit layer reduces the pipe cross-section leading to a significant reduction in flow rate and eventually plugging the pipeline. This flow assurance problem caused during oil production has motivated the development of several experimental and modeling techniques to investigate the asphaltene behavior. This study proposes an integrated approach to simultaneously model asphaltene precipitation, aggregation, and deposition on a single platform. It focuses on the development of a deposition simulator that performs thermodynamic modeling using the Perturbed Chain version of the Statistical Associating Fluid Theory Equation of State (PC-SAFT EOS) and depicts the deposition profile by means of a Computational Fluid Dynamics (CFD) model based on finite element method. In this work, asphaltene deposition risk was assessed in the near-wellbore region and the production tubing as a result of gas-breakthrough. To achieve this goal, a sample of crude C2 was analyzed to determine its properties, and also the tendency of the asphaltenes contained in this sample to precipitate and deposit under various conditions. ACS Paragon Plus Environment

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Laboratory scale experiments were performed to analyze the rates of asphaltene precipitation, aggregation, and deposition. With the results obtained from the various experiments, advanced modeling methods based on PC-SAFT EOS and CFD models were calibrated and used to predict asphaltene precipitation and deposition under field conditions. Simulation methods for oil flow and asphaltene precipitation in the near-wellbore region of the reservoir and inside the production tubing were coupled to provide the most rigorous modeling approach ever developed to understand and predict this complex flow assurance problem. The results show low to moderate asphaltene deposition rate produced by crude C2 as the gas breaks through. Nevertheless, further investigation is recommended to analyze the effect of other fluids that may be co-produced to enhance our ability to understand and predict asphaltene deposition under different conditions. Graphical Abstract

Keywords: CFD Modeling, Wellbore simulation, Near-wellbore region simulation, Finite Element Method, Surface Deposition Mechanism, PC-SAFT EOS, Friction Theory. ACS Paragon Plus Environment

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1. Introduction The potential for asphaltene to precipitate and deposit in wellbore and flowlines is a major concern for the oil and gas industry. Asphaltene precipitation and subsequent deposition can occur as a result of changes in pressure, temperature and composition of crude oil during its production from the reservoir to the wellhead. Asphaltene deposit removal from onshore and offshore facilities is an expensive operation which can lead to extended shutdown time for oil producing wells. In order to properly assess the risk of asphaltene deposition, experimental and modeling techniques have been developed to predict the asphaltene deposition profile and rate. There have been only a few established works on modeling asphaltene deposition in wellbores and pipelines as well as in porous media compared to the number of work dedicated to the development of thermodynamic models to investigate asphaltene phase behavior. The phase stability of asphaltenes can be well described by thermodynamic models. Although the prediction and analyses of possible asphaltene precipitation is extremely critical, it is equally important to predict the transport of precipitated asphaltenes and identify their deposition tendencies. Therefore, a predictive tool to estimate the amount and profile of asphaltene deposited along the wellbore is required. It is important to establish an efficient and reliable modeling technique to validate the mechanism of asphaltene deposition in the oil field based on the various laboratory-scale experimental results. Hence, a reliable simulator that can model and predict precipitation, aggregation, and deposition of asphaltene, along with an understanding of the mechanisms and tendencies of asphaltene deposition can prove to be extremely helpful for the oil and gas industry. Perturbed Chain version of the Statistical Associating Fluid Theory Equation of State (PC-SAFT EOS) has been successfully applied to model asphaltene phase behavior under both ambient and reservoir conditions.1–3 There are a few studies published in the literature focusing on the

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prediction of asphaltene deposition in pipelines, clearly illustrating the importance of this problem in the oil industry.4–8 The model developed by Vargas et al.6 had sub-models describing particle precipitation, aggregation, transport and deposition on the wall. The aggregation and deposition phenomena were modeled using pseudo-first-order reactions. The model contained three parameters which were estimated from capillary tube deposition experiments. The work done by Kurup et al.7 was a continuation of the deposition simulator developed by Vargas et al.6 It involved the development of an Asphaltene Deposition Tool (ADEPT) that can predict the occurrence and calculate the magnitude and profile of asphaltene deposition in a wellbore. Similar to the model developed by Vargas et al.6, this deposition model also consisted of three tuning parameters. Instead of a two-dimensional convection-diffusion transport equation as in Vargas et al.6, this model was simplified to a one-dimensional axial dispersion equation. The model was validated by comparing the simulation results against the experimentally measured asphaltene deposition flux in a capillary deposition experiment.

9,10

A proper choice of the

kinetic parameters helped the deposition simulator in predicting the deposition profile. Since there is a lack of understanding of the rich physics involved in the deposition process, the transport model is frequently of a mechanistic nature and it relies heavily on experimental data. 11 A more rigorous analysis of the asphaltene deposition model can be done using a Computational Fluid Dynamics (CFD) approach. Ge et al.12 presented a general framework for modeling the asphaltene deposition process which focuses on a CFD based transport model with an evolving depositing front coupled to the associated fluid, mass, and energy transport. Along similar lines, a one-dimensional model for asphaltene deposition in wellbores or pipelines was presented by Guan et al.13 This model consisted of a Thermodynamic Module and a Transport Module. The Thermodynamic Module focused on the modeling of asphaltene precipitation using the PengRobinson (PR) EOS. The Transport Module included the modeling of fluid transport, particle


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transport, and asphaltene deposition while tracking the deposition front as well. The numerical model was one-dimensional in nature. In this work, the risk of asphaltene deposition problems in the near-wellbore region and the production tubing as a result of gas-breakthrough has been assessed. This project required the development and implementation of advanced experimental methods and simulation tools to forecast the occurrence and the magnitude of asphaltene deposition in porous media and production tubing at high temperatures and under dynamic conditions. The study illuminates the development of a simple, yet comprehensive, CFD model that depicts the precipitation, aggregation, and deposition of asphaltenes, along with their transport with oil as oil flows up the wellbore and encounters varying conditions of temperature, pressure and flow regimes. The deposition model is based on convection-diffusion-reaction equations. Enhanced performance of the thermodynamic modeling of asphaltene phase behavior is attained by incorporating the PCSAFT EOS. Furthermore, this work illustrates the development and implementation of a model to simulate the oil flow in porous media including the potential permeability reduction caused by asphaltene deposition and hence its integration with the wellbore deposition tool. The model developed in this work has the scope to be sufficiently sophisticated to capture the physics of the system, yet simple enough to allow implementation in a programming environment, utilizing thermodynamic principles and state-of-the-art algorithms and heuristics to increase the speed, accuracy, and reliability of results generated. Systematic modeling of asphaltene deposition has been performed, by making sure that the model relies only on as few tunable parameters as possible. The significance of this work is that transient simulations have been performed using the CFD model based on Finite Element Method (FEM). An indigenous CFD model has been developed in MATLAB platform for this work. A two-dimensional FEM is used to numerically solve the


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momentum and mass transfer equations for the asphaltene precipitation, aggregation and deposition phenomena taking place in the wellbore. Turbulent fluid flow simulations have been performed, while simultaneously considering the change of density and viscosity of the oil flowing up the wellbore as a function of pressure and temperature. A surface deposition mechanism for asphaltenes has been introduced in this work, which is different from the pseudofirst order or first order reaction mechanisms used in previous models found in the literature.6,7,13 Although the deposition model is based on convection-diffusion-reaction mechanism, unlike the previous models, the asphaltene precipitation and deposition kinetic parameters are no longer a single value that is obtained from experiments and used along the entire length of the wellbore. Rather the kinetic parameters obtained from the experiments are scaled as a function of temperature, pressure, flow field and driving force for precipitation. They vary along the length of the wellbore. Generating a finite element mesh facilitates in incorporating the different values of kinetic parameters, oil density, and viscosity at different lengths of the wellbore, as oil flows up and encounters different conditions of pressure, temperature, and composition. The unique contribution of this work is that asphaltene deposition simulations have also been performed for the near-wellbore region and then the wellbore and near-wellbore asphaltene deposition models have been integrated, making this modeling technique a very rigorous approach in understanding this complex flow assurance problem. 2. Asphaltene Deposition Simulator To accurately capture the process of asphaltene deposition that takes place in the wellbore and near-wellbore region, a comprehensive Asphaltene Deposition Simulator has been developed. The structure of the simulator will be discussed in detail in this section. At reservoir conditions, the asphaltenes are mostly stable and soluble in oil. However, a change in pressure, temperature, and composition may destabilize the asphaltenes that were originally ACS Paragon Plus Environment

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dissolved in the oil, leading to phase separation of asphaltenes from the oil. The precipitated asphaltenes are known as primary particles. These primary particles then combine with each other to form aggregates. It is proposed that these aggregates are susceptible to deposit on the walls of the wellbore. Hence, the three steps that take place in the wellbore, namely, precipitation, aggregation and deposition of asphaltenes can be modeled using a transport (convection-diffusion) equation. The developed simulator is based on the notion proposed by Vargas et al.6 model. The following are the assumptions involved in the development of the simulator: 

Asphaltene precipitation is dominated by dispersion forces and, therefore, PC-SAFT is used without the association term to represent the thermodynamics of precipitation.



Oil flow in the wellbore is modeled by assuming that the fluid is a single-phase multicomponent mixture.



Oil density and viscosity depend only on given temperature and pressure conditions.



It is assumed that only precipitated asphaltene particles can deposit at the given temperature and pressure conditions and once deposited, these asphaltenes do not redissolve in oil.



Only the asphaltene aggregates, which are smaller than the critical particle size, participate in the deposition process. The large aggregated particles are considered to be carried with the flow because of inertia and do not tend to deposit. In this work, 0.2 μm is assumed as the critical particle size. This concept of critical particle size is in agreement with the model proposed by Vargas et al.6



The deposition process is not dominated by transport occurring in the core flow but dominated by transport and kinetics occurring in the laminar boundary layer adjacent to the wall of the wellbore or pipeline. The laminar flow inside a packed bed column mimics the laminar boundary layer of the turbulent flow in a wellbore. Hence, the ACS Paragon Plus Environment

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deposition kinetic constant is obtained from deposition experiments using a packed bed column and scaled to wellbore conditions. 

It is assumed that the asphaltene deposited is solid in nature, with constant density. Shear removal of deposit, erosion of deposit from location and its deposition on another location further down the flow, have not been considered.

The structure of the developed simulator is shown in Figure 1. The simulator consists of a thermodynamic module and a deposition module. The thermodynamic module describes the phase behavior of the oil at the varying temperature and pressure conditions. PC-SAFT EOS has been used to model asphaltene phase behavior. The Friction Theory for viscosity modeling is used to account for the variation of viscosity with temperature and pressure. The deposition module uses an integrated approach and performs asphaltene deposition simulations for both the wellbore and near-wellbore region. The turbulent flow of fluid in the wellbore is modeled and the spatial distribution of free-flowing and deposited asphaltenes have been determined using a CFD model based on transient FEM simulations. The simulation results of the near-wellbore region deposition model are used as a starting point for the asphaltene deposition simulations in the wellbore. The deposition module receives inputs from the thermodynamic module, which include, asphaltene solubility in the oil phase (𝐶𝑒𝑞 ), density (𝜌), viscosity (𝜇), solubility parameter of asphaltenes (𝛿𝑎𝑠𝑝ℎ ) and the solubility parameter of solution (𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ) as a function of temperature and pressure along the length of the wellbore. Both these modules require information from field data, if available, and experimental data.


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Oil and Asphaltene characterization

Field data Pressure and Temperature Experimental data • • • • •

PVT properties of oil Asphaltene precipitation Precipitation and aggregation kinetics Packed bed column Microfluidics

Pipeline Profile and production rate Precipitation, Aggregation and Deposition Rates

Asphaltene Deposition in Wellbore (Fluid and Particle Transport) CFD Model (FEM) Thermodynamics & Viscosity

PC-SAFT EOS Friction Theory

Asphaltene deposition rate, deposit profile and thickness along length of wellbore and across near-wellbore region

𝐶𝑒𝑞 , 𝜌 𝜇

Asphaltene Deposition in Near-wellbore Region (Fluid and Particle Transport) CFD Model (LBM)

Figure 1. Structure of Asphaltene Deposition Simulator The kinetic parameters for precipitation and aggregation are obtained by performing a series of asphaltene precipitation kinetic experiments. The deposition parameters for modeling asphaltene deposition in wellbore and near-wellbore are obtained based on packed bed column and microfluidic experiments respectively. The thermodynamic module is integrated with the deposition module using a look-up table approach. 𝐶𝑒𝑞 , 𝜇 and 𝜌 are generated for specific pressure and temperature conditions using the thermodynamic module and the deposition module linearly interpolates them for the intermediate pressure and temperature conditions along the length of the wellbore, if and when required. Section 3 describes the modeling of asphaltene deposition in the wellbore, calibration of required model parameters based on laboratory scale experiments and their scale up to field conditions. In Section 4, the modeling of asphaltene deposition in the near-wellbore region based on model ACS Paragon Plus Environment

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parameters obtained from microfluidic experiments, have been presented Finally, Section 5 demonstrates the integration of both these approaches to obtain a comprehensive technique to model asphaltene deposition in the wellbore and the near-wellbore region. 3. Modeling of Asphaltene Deposition in Wellbore 3.1. Thermodynamic Module 3.1.1. Modeling asphaltene phase behavior using PC-SAFT EOS The thermodynamics of asphaltene precipitation is modeled using the PC-SAFT EOS. In this work, crude C2 is considered. The crude oil is characterized based on the Saturates-AromaticsResins-Asphaltenes (SARA) analysis. Simulation parameters are optimized to match asphaltene precipitation data on dead oil sample from Indirect Method experiment.14 The characterization and thermodynamic modeling of crude C2, including details such as crude C2 composition, API gravity, GOR, SARA and composition of the injected gas, can be found in the first part of this series of publications.3 Figure 2 shows the asphaltene phase envelope for crude C2 with different amounts of gas injection along with the P-T trace for oil production from bottomhole to wellhead. 40000 UAOP PC-SAFT

BP PC-SAFT

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P-T Trace

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Figure 2. Asphaltene phase envelope predicted from PC-SAFT for crude C2 with 50, 55, and 60% gas injection ACS Paragon Plus Environment

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The thermodynamic properties required for the deposition model are: 1. Equilibrium concentration (𝐶𝑒𝑞 ) of asphaltenes in the continuous phase (𝐿1 ) expressed as weight of asphaltenes in the continuous phase (𝐿1 ) per total weight of asphaltenes in the system. 2. Weight fraction of asphaltenes in the precipitated phase (𝐿2 ) expressed as weight of asphaltenes in the precipitated phase (𝐿2 ) per total weight of precipitated phase. 3. Density of asphaltenes in g/cm3. 4. Difference between solubility parameter of asphaltenes (𝛿𝑎𝑠𝑝ℎ ) and solubility parameter of solution (𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ). The solubility parameter 𝛿 of a substance was introduced by Hildebrand15 and is defined as: ⁄2

−𝑢𝑟𝑒𝑠 1 𝛿=( ) 𝑣

(1)

where 𝑢𝑟𝑒𝑠 is the residual internal energy and 𝑣 is the molar volume. The solubility parameter is a measure of the intermolecular interactions of a given substance and is a function of pressure, temperature, and composition. Substances with similar solubility parameters are likely to be miscible in each other whereas substances with vastly different solubility parameters are likely to be immiscible. In this study, we quantify the driving force for asphaltene precipitation as the difference of solubility parameters between the asphaltenes and the oil phase. The difference of solubility parameters between the asphaltenes and the oil phase is maximum at the bubble point, which is consistent with known notion that the maximum driving force for asphaltene precipitation occurs at the bubble pressure.16 In this work, solubility parameters are calculated using PC-SAFT EOS. Plots of the properties, required for deposition modeling, as predicted by PC-SAFT along the P-T trace are shown in Figure 3. ACS Paragon Plus Environment

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Figure 3. Predictions of thermodynamic properties along the P-T trace for crude C2 with 60% injection using PC-SAFT: (a) equilibrium asphaltenes concentration, (b) concentration of asphaltenes in the precipitated phase, (c) density of asphaltenes, and (d) difference of solubility parameters of asphaltenes and the solution 3.1.2. Viscosity modeling using Friction Theory Viscosity is a key element in modeling fluid transport and asphaltene deposition during oil production. Oil viscosity changes due to changes in temperature and pressure as oil flows from the reservoir to wellhead. If the production conditions are such that the pressure is below the asphaltene onset pressure (AOP) or the bubble point (BP), the composition of the bulk phase changes which causes a variation in viscosity. In order to capture the effect of temperature, pressure, and composition on oil viscosity in the deposition simulator developed in this work, a compositional viscosity model is needed. Amongst the several viscosity models available in the literature, the Friction Theory17,18 has shown promising capability in modeling the viscosity of hydrocarbon systems.19–24


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In Friction Theory (FT), the viscosity is modeled as a summation of a dilute gas contribution (𝜇0 ) and a friction contribution (𝜇𝑓 ). The dilute gas contribution is generally negligible as compared to the friction contribution for liquids. The friction contribution is correlated to repulsive and attractive pressures which can be obtained from an equation of state. The general expression for the friction viscosity contribution is18:

𝜇̂ 𝑓 =

𝜇𝑓 2 = 𝑘̂𝑟 𝑃̂𝑟 + 𝑘̂𝑎 𝑃̂𝑎 + 𝑘̂𝑟𝑟 𝑃̂𝑟 𝜇𝑐

(2)

where 𝜇̂ 𝑓 is the dimensionless friction viscosity contribution, 𝜇𝑐 is the characteristic critical viscosity, 𝑃̂𝑟 is the reduced repulsive pressure, 𝑃̂𝑎 is the reduced attractive pressure, 𝑘̂ is the reduced viscous friction coefficient whose subscript 𝑟 means repulsive, subscript 𝑎 means attractive, and 𝑟𝑟 means second order repulsive. Abutaqiya et al.23 modeled the viscosity of 10 oil samples from the Middle East and showed that using Peng-Robinson (PR) EOS to calculate repulsive and attractive pressures in the Friction Theory is sufficient to reproduce experimental data. The authors concluded that although PC-SAFT yields improved predictions of the phase behavior of the petroleum fluid, it does not have a significant advantage as compared to PR when used to calculate repulsive and attractive pressures for viscosity modeling using the Friction Theory. Therefore, in this work the PR EOS is used, to calculated repulsive and attractive pressures. The viscous friction coefficients 𝑘̂𝑟 , 𝑘̂𝑎 , 𝑘̂𝑟𝑟 are functions of critical temperature (𝑇𝑐 ), critical pressure (𝑃𝑐 ), and acentric factor (𝜔) for each component in the mixture, in addition to temperature (𝑇) and 16 universal constants that are dependent on the EOS used.18 The critical properties for the non-asphaltenic pseudo-components (i.e. Heavy Gas, Saturates, and Aromatics) are calculated using the critical property correlations developed from the One-Third rule25,26: ACS Paragon Plus Environment

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2 0.1674𝑀𝑊 0.613 (𝑀𝑊 + ) + 24.85 𝑆𝐺 𝑇𝑐 = 𝑇𝑏 ( ) 0.1674𝑀𝑊 0.577 (𝑀𝑊 + ) + 11.12 𝑆𝐺

(3)

2 0.1674𝑀𝑊 (0.613 (𝑀𝑊 + ) + 24.85) 𝑆𝐺 𝑃𝑐 = 𝑇𝑏2 4 0.1674𝑀𝑊 (0.577 (𝑀𝑊 + ) + 11.12) 𝑆𝐺

(4)

where 𝑇𝑐 and 𝑃𝑐 are in units of K and bar, respectively, 𝑇𝑏 is the normal boiling point in K, 𝑆𝐺 is liquid specific gravity at 20 °C. The normal boiling point and density of each pseudo-fraction are calculated from PC-SAFT. The acentric factor is calculated using Lee-Kesler27 correlation:

𝜔=

6.09649 6 ) 𝑇𝑏𝑟 + 1.28862 ln(𝑇𝑏𝑟 − 0.169347𝑇𝑏𝑟 15.6875 6 15.2518 − 𝑇 − 13.4721 ln(𝑇𝑏𝑟 ) + 0.43577𝑇𝑏𝑟 𝑏𝑟

𝑠 ) − 5.92714 + ln(𝑃𝑏𝑟

(5)

𝑓𝑜𝑟 𝑇𝑏𝑟 < 0.8 𝜔 = −7.904 + 0.1352 ∗ 𝐾𝑤 − 0.007465𝐾𝑤2 + 8.359𝑇𝑏𝑟 + 1.408 −

0.01063𝐾𝑤 𝑇𝑏𝑟

(6)

𝑓𝑜𝑟 𝑇𝑏𝑟 ≥ 0.8 𝑠 where 𝑃𝑏𝑟 is the ratio of atmospheric pressure to 𝑃𝑐 , 𝑇𝑏𝑟 is the ratio of 𝑇𝑏 to 𝑇𝑐 , and 𝐾𝑤 is the

Watson characterization factor defined by: 1

(1.8𝑇𝑏 )3 𝐾𝑤 = 𝑆𝐺

(7)

It is found that when equations (3) and (4) are used to calculate critical properties for Resins and Asphaltenes, the predictions of viscosity from the Friction Theory show unusual trends. The repulsive friction coefficients 𝑘̂𝑟 and 𝑘̂𝑟𝑟 become very high when the critical pressure of a component in the system is too low (i.e. < 4 bar), leading to very high values of viscosity and unrealistic optimized values of 𝐾𝑐 for the pseudo-fractions. Therefore, for Resins and Asphaltenes, the Lee-Kesler correlations for 𝑇𝑐 and 𝑃𝑐 are used: ACS Paragon Plus Environment

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Energy & Fuels

105 𝑇𝑐 = 189.9 + 450.6𝑆𝐺 + (0.4244 + 0.1174𝑆𝐺)𝑇𝑏 + (0.1441 − 1.0069𝑆𝐺) 𝑇𝑏 𝑃𝑐 = exp [5.689 −

(8)

0.0566 4.1216 0.21343 − (0.43639 + + ) 10−3 𝑇𝑏 𝑆𝐺 𝑆𝐺 𝑆𝐺 2 (9)

1.182 0.15302 9.9099 −6 2 + (0.47579 + + ) 10 𝑇 − (2.4505 + ) 10−10 𝑇𝑏3 ] 𝑏 2 2 𝑆𝐺 𝑆𝐺 𝑆𝐺 with true boiling points (𝑇𝑏 ) for the heavy fractions calculated from the Soreide Correlation28 similar to the approach used by Johnston et al.29: 𝑇𝑏 = 1928.3 − (1.69 (10) 5 )𝑀𝑊 −0.03522

× 10

𝑆𝐺

3.266 [−0.004922∗𝑀𝑊−4.7685∗𝑆𝐺+0.003462∗𝑀𝑊∗𝑆𝐺]

𝑒

The characteristic viscosities (𝜇𝑐 ) for pure components are reported can be found elsewhere.18 For pseudo-components, Quiñones-Cisneros et al.18 suggested to estimate 𝜇𝑐 using: 2/3

𝜇𝑐,𝑖 = 𝐾𝑐

√𝑀𝑊𝑖 𝑃𝑐,𝑖 1/6

𝑇𝑐,𝑖

(11)

where the constant 𝐾𝑐 is 7.95 for n-alkanes. For the pseudo-components, 𝐾𝑐 can be treated as a tuning parameter to match available experimental viscosity data.23,24 In this work, Heavy Gas and Saturates pseudo-components are assumed to consist of n-alkanes and a value of 7.95 is used for 𝐾𝑐 . For Aromatics, Resins, and the four Asphaltene fractions, a single value of 𝐾𝑐 is tuned to match experimental live oil viscosity data at temperatures of 259, 160, and 90 °F. The optimized value of 𝐾𝑐 is found to be 7.97. Figure 4 shows a comparison between Friction Theory model and experimental data. The set of critical properties for crude C2 used in this work is shown in Table 1. The variation of oil viscosity along the P-T trace for crude C2 with 60% gas injection is shown in Figure 5. These viscosity modeling results, along with the thermodynamic properties


15

Energy & Fuels

shown in Figure 3 are used to generate lookup tables which are used in the deposition simulator to provide necessary information to construct the deposition profile. 15.0

40.0

T = 259 °F

Experimental

T = 160 °F

PR_FT

30.0 10.0

Viscosity / cP

Viscosity / cP

5.0

0.0

20.0

10.0

0.0 0

10000

20000

Pressure / psi

30000

0

10000

20000

Pressure / psi

120.0

30000

T = 90 °F

100.0

Viscosity / cP

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 16 of 50

80.0 60.0 40.0

20.0 0.0 0

10000

20000

30000

Pressure / psi

Figure 4. Comparison between experimental data and modeling results for viscosity of live crude C2 at different temperatures. Table 1. Critical properties for different components in Crude C2. Component

𝑴𝑾 [g/mol]

𝝁𝒄 [cP]

𝑻𝒄 [K]

𝑷𝒄 [bar]

𝝎 [-]

H 2S

34.1

300.0

373.2

89.4

0.081

N2

28.0

174.2

126.2

34.0

0.037

CO2

44.0

376.9

304.1

73.8

0.223

C1

16.0

152.9

190.6

46.0

0.010

C2

30.1

217.6

305.3

48.7

0.099

C3

44.1

249.7

369.9

42.5

0.152

Heavy Gas

69.5

269.6

491.6

38.6

0.096

Saturates

202.6

237.4

697.7

15.7

0.647

Aromatics

322.9

286.7

1028.0

16.2

1.119

Resins

649.6

354.1

1008.0

13.1

1.202

Asphaltenes1

999.0

378.2

1077.3

10.6

1.292

Asphaltenes2

1026.7

379.8

1081.3

10.5

1.296

Asphaltenes3

1085.7

383.3

1089.5

10.2

1.305

Asphaltenes4

1708.1

416.8

1147.9

8.4

1.359


16

Page 17 of 50

Pressure (psi) 6000 2.50

Oil Viscosity (cP)

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

Energy & Fuels

8000

10000

12000

14000

16000

18000

20000

2.00 1.50 1.00 0.50 0.00 140

160

180

200

220

240

260

Temperature ( F)

Figure 5. Variation of oil viscosity along the P-T trace for crude C2 with 60% injection. 3.2.

Deposition Module

Transport of asphaltenes in the wellbore follows a multistep process, including precipitation, aggregation, advection, diffusion, and deposition. The rate of asphaltene precipitation is assumed to be proportional to the supersaturation degree of asphaltenes, which is defined as the difference between the actual concentration of asphaltenes dissolved in the oil and the concentration of asphaltene at equilibrium. In other words, the precipitation rate is determined by the degree of supersaturation. The higher the supersaturation degree of asphaltenes in the oil phase, the greater is the precipitation rate. The precipitated asphaltenes further aggregate with each other, tend to diffuse either along the radial or axial directions and deposit on the pipe walls or get carried away along with the flow of the fluid. The kinetics of aggregation is assumed to be a secondorder kinetic process. Deposition of asphaltene is modeled using a surface deposition mechanism. This is schematically demonstrated in Figure 6.


17

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Page 18 of 50

Figure 6. Schematic of asphaltene deposition mechanism in the wellbore Taking into account the assumptions already stated in Section 2 and the mechanism shown in Figure 6, a mass balance is performed for a control volume of the wellbore. A two-dimensional deposition model that accounts for asphaltene precipitation, aggregation and deposition has been developed to predict asphaltene deposition in the wellbore. The mass balance for asphaltene in the transient state can, therefore, be written as, 𝜕𝐶 ′ ⏟ 𝜕𝑡 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛

′ ′ (𝒖. 𝜵)𝐶 ′ = 𝐷 +⏟ ⏟𝑒 𝛻 2 𝐶 ′ + 𝑘 ⏟𝑝 (𝐶𝑓 − 𝐶𝑒𝑞 ) − 𝐴𝑑𝑣𝑒𝑐𝑡𝑖𝑜𝑛

𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛

𝑃𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑎𝑡𝑖𝑜𝑛

′2 𝑘⏟ 𝑎𝑔 𝐶 𝐴𝑔𝑔𝑟𝑒𝑔𝑎𝑡𝑖𝑜𝑛

−𝑘 ⏟𝑑 (𝒏𝒏𝑻 : 𝜵. 𝒖)𝐶 ′

(12)

𝐷𝑒𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛

where, 𝐶 ′ is the dimensional concentration of the primary particles, 𝐶𝑓 ′ is the dimensional concentration of dissolved asphaltene in the oil phase at that particular time 𝑡, and 𝐶𝑒𝑞 ′ is the dimensional thermodynamic equilibrium concentration of asphaltene, which can be regarded as the solubility of asphaltene at the given pressure, temperature and composition. 𝑘𝑝 is the precipitation kinetic parameter, 𝑘𝑎𝑔 is the aggregation kinetic parameter and 𝑘𝑑 is the deposition kinetic parameter for wellbore. 𝒖 refers to the velocity field and 𝒏 refers to a unit normal vector. 𝒏𝑻 is the transpose of the unit normal vector. For a two-dimensional system, with coordinates 𝑟 and 𝑧 as in Figure 6, (𝒏𝒏𝑻 : 𝜵𝒖) =

𝜕𝑢𝑟 𝜕𝑟

+

𝜕𝑢𝑧 𝜕𝑟

+

𝜕𝑢𝑟 𝜕𝑧

+

𝜕𝑢𝑧 𝜕𝑧

. Let us denote (𝒏𝒏𝑻 : 𝜵𝒖) = 𝐺𝑢 for

simplicity from here on. ACS Paragon Plus Environment

18

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Energy & Fuels

𝐶′

𝐶𝑓 ′

0

𝐶0

Introducing, 𝐶 = 𝐶 , 𝐶𝑓 =

, 𝐶𝑒𝑞 =

𝐶𝑒𝑞 ′ 𝐶0

, with 𝐶𝑜 as the concentration of asphaltene in solution

at bottomhole conditions, equation (11) becomes, 𝜕𝐶 + (𝒖. 𝜵)𝐶 = 𝐷𝑒 𝛻 2 𝐶 + 𝑘𝑝 (𝐶𝑓 − 𝐶𝑒𝑞 ) − 𝐾𝑎𝑔 𝐶 2 − 𝑘𝑑 𝐺𝑢 𝐶 𝜕𝑡

(13)

where, 𝐾𝑎𝑔 = 𝑘ag 𝐶𝑜 , is the aggregation kinetic parameter in our asphaltene deposition model. 𝐶𝑒𝑞 is obtained as a function of temperature and pressure using the thermodynamic module. The initial and boundary conditions are given as follows: Initial condition : 𝐶(𝑡 = 0) = 0, ∀ 𝑟, 𝑧 Boundary conditions: for all 𝑡 > 0 𝒏. 𝜵𝐶(𝑟 = 𝑅) = 0, ∀ 𝑧

(14)

𝐶(𝑧 = 0) = 𝐶𝑖𝑛 , ∀ 𝑟 −𝒏. 𝐷𝑒 𝜵𝐶(𝑧 = 𝐿) = 0, ∀ 𝑟 where, 𝐶𝑖𝑛 is the dimensionless concentration of available asphaltene primary particles at inlet conditions, 𝒏 refers to a unit normal vector, 𝑅 is the radius of the wellbore, 𝐿 is the length of the wellbore and 𝐷𝑒 is the diffusivity of asphaltene particles. We need the velocity field 𝒖, before we proceed to evaluate the amount of asphaltene deposited. When Reynolds number is very high, the inertial forces are much larger than the viscous forces. Performing such simulations using the Navier-Stokes equations is computationally intensive. Instead Reynolds-Averaged Navier-Stokes (RANS) formulation of the NavierStokes equations can be used. RANS equations are time-averaged equations of motion for fluid ̅ ) and flow, where an instantaneous quantity (𝒖) is decomposed into its time-averaged (𝒖 fluctuating quantities (𝒖′ ), to simulate turbulent flows. RANS equations are as follows, ′ 𝒖′ ) + 𝜌𝒈 ̅̅̅̅̅̅ ̅ . 𝛁)𝒖 ̅ = 𝛁. (−𝑝̅ 𝐼 + 𝜇(𝛁𝒖 ̅ + (𝛁𝒖 ̅ )𝑇 ) − 𝜌𝒖 𝜌(𝒖


(15) 19

Energy & Fuels 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 50

̅ and 𝑝̅ are the time-averaged velocity and pressure, respectively. The left hand side of Here, 𝒖 this equation represents the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and ′ 𝒖′ ) owing to the fluctuating velocity field, generally referred to as ̅̅̅̅̅̅ apparent stress (−𝜌𝒖 ′ 𝒖′ = 2𝜇 𝜀(𝒖) = 𝜇 (𝛁𝒖 ̅̅̅̅̅̅ ̅ + (𝛁𝒖 ̅ )𝑇 ), where 𝜀(𝒖) is the Reynolds stress which is given as, −𝜌𝒖 𝑇 𝑇

the strain rate tensor and the term 𝜇 𝑇 represents the turbulent viscosity, that is, the effects of the small-scale time-dependent velocity fluctuations. 𝜇 𝑇 requires additional modeling to solve the RANS equations, which is evaluated using the k-ε turbulence model in this work. This model is often used in industrial applications because it is both robust and computationally inexpensive. It is a two equation model which gives a general description of turbulence by means of two transport equations (partial differential equations), which are, for the transport of turbulent kinetic energy k and turbulent dissipation rate ε. The equations used for modeling turbulent flow in the wellbore and hence, discretized using finite element method are shown in equations (16). ̅ . 𝛁)𝒖 ̅ = 𝛁. (−𝑝̅ 𝐼 + (𝜇 + 𝜇 𝑇 )(𝛁𝒖 ̅ + (𝛁𝒖 ̅ )𝑇 )) + 𝜌𝒈 𝜌(𝒖 𝜕𝑘 𝜇𝑇 ̅ . 𝛁)𝑘 = 𝛁. ((𝜇 + ) 𝛁𝑘) + 𝑃𝑘 − 𝜌𝜀 + ρ(𝒖 𝜕𝑡 𝜎𝑘

(16b)

𝜕𝜀 𝜇𝑇 𝜀 𝜀2 ̅ . 𝛁)𝜀 = 𝛁. ((𝜇 + ) 𝛁𝜀) + 𝐶𝜀1 𝑃𝑘 − 𝐶𝜀2 𝜌 + ρ(𝒖 𝜕𝑡 𝜎𝜀 𝑘 𝑘

(16c)

𝜌

𝜌

(16a)

𝜇 𝑇 = 𝜌𝐶𝜇

𝑘2 𝜀

̅ : (𝛁𝒖 ̅ + (𝛁𝒖 ̅ )𝑇 )) 𝑃𝑘 = 𝜇 𝑇 (𝛁𝒖

(16d) (16e)

The turbulence model parameters have been obtained by numerous iterations of data fitting for a wide range of turbulent flows and are given as: 𝐶𝜀1 = 1.44, 𝐶𝜀2 = 1.92, 𝐶𝜇 = 0.09, 𝜎𝑘 = 1.00, and 𝜎𝜀 = 1.3. ACS Paragon Plus Environment

20

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Energy & Fuels

For turbulent flows, the eddy diffusion due to the fluctuating velocity components governs the deposition of aggregated asphaltene on the pipeline wall surface. Thus, the particle diffusivity 𝐷𝑒 in a turbulent flow can be calculated as: 𝐷𝑒 = 𝐷𝑇 + 𝐷𝐵 where 𝐷𝑇 is the turbulent particle diffusivity and 𝐷𝐵 is the Brownian diffusivity.8 Since, the particles are small the particle diffusivity is approximately equal to the eddy diffusivity (𝐷𝑒 = 𝐷𝑇 ), and the Brownian diffusivity is neglected. The eddy diffusivity at the laminar sub-layer surface is given as, 𝐷𝑇 = 𝜈𝑇 /𝑆𝑐𝑇 , where 𝜈𝑇 = 𝜇 𝑇 /𝜌 is the eddy diffusivity (m2/s) and 𝑆𝑐𝑇 is the turbulent Schmidt number. It is taken as 0.8 for this work.8,30 𝜇 𝑇 is calculated using the k-ε turbulence model. 𝜌 and 𝜇 are the density and viscosity of the fluid respectively. They are obtained from the thermodynamic module as a function of pressure and temperature along the length of the wellbore. No slip boundary conditions are applied on the wellbore wall surface in order to obtain the velocity field for the given oil production rate. The in-house CFD model (using FEM) has been developed in MATLAB to simulate the fluid flow and predict asphaltene precipitation, aggregation and deposition in the production tubing. Fluid flow simulation is performed for the wellbore. The velocity profile thus obtained, along with the initial and boundary conditions specified in equation (14), are used to solve equation (13) and obtain the amount of asphaltene deposited and the asphaltene deposit profile along the length of the wellbore. 3.3. Calibration of model parameters 3.3.1. Asphaltene Precipitation and Aggregation Kinetics Kinetics of asphaltene precipitation and aggregation of crude C2 were investigated using a direct kinetics spectroscopy method. In this method, mixtures of crude oil and heptane with different concentrations

were

prepared

in

a

quartz

cuvette.

A

stir

bar

made

of

PTFE

(Polytetrafluoroethylene) was added to the mixture. To avoid evaporation of the volatile 21 ACS Paragon Plus Environment

Energy & Fuels 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 50

components of the mixture, a PTFE cap was used to seal the cuvette. The light intensity of the sample is measured over time using a Shimadzu UV-Vis-NIR spectrophotometer Model UV3600, in which the stirring rate and the temperature of the sample inside the cell can be controlled. Overall, thirteen experiments using crude C2 were conducted to investigate the kinetics of asphaltene precipitation at various precipitation driving forces, using n-heptane and npentane as the precipitants, at two different temperatures (20 °C and 70 °C).31 The rate of asphaltene precipitation and aggregation are modeled as per the mechanism described in Section 3.2 and shown as follows: 𝑑𝐶 = 𝑘𝑝 (𝐶𝑓 − 𝐶𝑒𝑞 ) − 𝐾𝑎𝑔 𝐶 2 𝑑𝑡

(17)

𝐶𝑒𝑞 is the solubility of asphaltene at the given pressure, temperature and composition, which is obtained using the PC-SAFT EOS. 𝑘𝑝 (Precipitation kinetic parameter) and 𝐾𝑎𝑔 = 𝑘ag 𝐶𝑜 (Aggregation kinetic parameter) are the two tuning parameters. The value of the precipitation and aggregation kinetic parameter are obtained by solving the equation (17) by minimizing the difference between the experimental data and the modeling results, with the initial condition as the initial concentration of asphaltene particles solubilized in the oil phase. A few examples of the modeling of asphaltene precipitation and aggregation kinetics are shown in Figure 7.


22

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Energy & Fuels

Figure 7. Determination of Precipitation and Aggregation kinetic constants In order to estimate the value of 𝑘𝑝 (Precipitation kinetic parameter) for the crude oil at reservoir conditions (High Temperature and High Pressure), it was required to investigate the relationship between 𝑘𝑝 (Precipitation kinetic parameter) and the driving force for precipitation, (𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ), where 𝛿 is the solubility parameter. Linear regression was performed between − log(𝑘𝑝 ) and

1⁄(𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 )

2

values for different experiments performed at a

particular temperature (20oC or 70oC in this case) and the corresponding dependence of 𝑘𝑝 on the driving force for precipitation, (𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ) was obtained. An Arrhenius type of equation has been used to understand the effect of temperature. Hence, 𝑘𝑝 (Precipitation kinetic parameter) is given as,

𝑘𝑝 = 𝑒𝑥𝑝 (𝑎𝑜 𝑒𝑥𝑝(−𝑎1 /𝑇) −

𝑏𝑜 𝑒𝑥𝑝(−𝑏1 /𝑇) (𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 )

2)

(18)

where 𝑎𝑜 , 𝑎1 , 𝑏𝑜 and 𝑏1 are constants for a particular crude. For crude C2, based on the several precipitation and aggregation experiments performed, the values of the constants are, 𝑎𝑜 = 30.5, ACS Paragon Plus Environment

23

Energy & Fuels 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 24 of 50

𝑎1 = 21.0, 𝑏𝑜 = 1049.4 and 𝑏1 = 27.7. The 𝑘𝑝 (Precipitation kinetic parameter) value for the crude oil at reservoir and wellbore conditions can be estimated, provided, the values of the solubility parameters of the solution gas and pure asphaltene are known at those corresponding high temperature and high pressure conditions. The thermodynamic module is used to generate these solubility parameter values as a function of temperature and pressure. The value of 𝐾𝑎𝑔 was obtained equivalent to 5x10-6 s-1 for all the experiments performed using the crude C2. 3.3.2. Asphaltene deposition using packed bed column In order to investigate the deposition of asphaltenes at high temperature and under dynamic conditions, a stainless-steel packed bed column deposition setup with carbon steel spheres was designed and constructed.32,33 Experiments were performed in the packed bed column setup with different inlet flow rates, driving forces and run time.31 CFD modeling has been performed to simulate the asphaltene deposition phenomena taking place in a packed bed column.34 The three-dimensional geometry of the packed bed column, with uniform packing arrangement, was constructed. The flow in the packed bed column is laminar. Fluid flow simulation was performed for the constructed geometry by solving the threedimensional Navier-Stokes equations, with boundary conditions such that the fluid does not slip at the surface of the spheres and the cylinder. A change in the composition of the oil, due to the addition of a precipitant, can make the asphaltenes that are originally dissolved in the oil unstable, causing phase separation of asphaltenes and hence, precipitation of asphaltenes. These precipitated particles are then susceptible to further aggregate with each other to form largersized aggregates and deposit on the surface of the spheres in the packed bed column. The mathematical model for this process can be written as the material balance of the primary particles in the transient state over a control volume of the packed bed column.


24

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Energy & Fuels

𝜕𝐶 + (𝒖. 𝜵)𝐶 = 𝐷𝑒 𝛻 2 𝐶 − 𝐾𝑎𝑔 𝐶 2 − (𝑘𝑑 )𝑝𝑏 (𝐺𝑢 )𝑝𝑏 𝐶 𝜕𝑡

(19)

where 𝐶 is the dimensionless concentration of asphaltene primary particles, 𝐷𝑒 is the diffusivity of depositing asphaltene particles, the velocity field 𝒖 is obtained by solving the Navier-Stokes equation for flow in the required packed bed column geometry and (𝐺𝑢 )𝑝𝑏 = (𝒏𝒏𝑻 : 𝜵𝒖)𝑝𝑏 for the packed bed column. Asphaltene precipitation is considered to happen instantaneously, when the precipitant is added to the oil sample. Aggregation is modeled as a second order reaction. Deposition of asphaltene is modeled using the surface deposition mechanism. 𝐾𝑎𝑔 = 𝑘ag 𝐶𝑜 (Aggregation kinetic parameter) is obtained by modeling asphaltene precipitation and aggregation kinetics as shown in Section 3.3.1. Hence, (𝑘𝑑 )𝑝𝑏 (Deposition kinetic parameter for packed bed) is the only tuning parameter of this asphaltene deposition model for packed bed column. It is tuned such that the difference between the mass of asphaltenes deposited on the surface of the spheres, obtained from the experiment and the simulation is a minimum. The concentration of the available asphaltene primary particles is specified at the inlet and the concentration flux on the surface of the column and the spheres is considered to be zero. In order to model asphaltene deposition in the wellbore for crude C2 with gas injection, the deposition kinetic parameter has been calibrated with respect to the deposition tests in packed bed column for crude C2 using n-pentane (60 vol%) as the precipitant. Figure 8 shows the cumulative mass of asphaltenes deposited on the surface of the spheres along the length of the packed bed column. Results from two tests are shown in this figure. Both the tests have been performed at the same temperature (176 °F), flow rate (9 mL/h), driving force (crude C2+60 vol% n-pentane) and for the same runtime (6 hours), but the size of the spheres (3/32 inches and 5/32 inches) used in each of these experiments were different. It was found that the (𝑘𝑑 )𝑝𝑏 = 1.5x10-4 yields CFD modeling results which represent the available experimental data the best.


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The calibrated deposition kinetic parameter from packed bed column deposition tests ((𝑘𝑑 )𝑝𝑏 ) can be further scaled to that of asphaltene deposition in the wellbore (𝑘𝑑 ).

Figure 8. Mass of asphaltene deposited along the length of the packed bed column Two competing processes occur in the boundary layer, which includes, transport of asphaltene particles into the boundary layer and depletion of asphaltene because of deposition kinetics. The rate of mass transfer of particles to the wellbore boundary layer is given as

𝐷𝑒 𝛿𝑚𝑜𝑚 2

(𝐶 − 𝐶𝑏𝑙 ),

where 𝛿𝑚𝑜𝑚 is the momentum boundary layer thickness, 𝐶 is the dimensionless concentration of asphaltenes in the bulk and 𝐶𝑏𝑙 is the dimensionless concentration in the wellbore boundary layer.

10

The rate of deposition in a packed bed column is known from equation (17). The

required scaling factor is obtained by establishing a correlation such that the fluid flow in a packed bed column is similar to that of boundary layer flow in the wellbore. Hence, by equating the rate of mass transfer of asphaltenes to the boundary layer in the wellbore to the rate of deposition we see in packed bed column (similar to boundary layer flow in wellbore), the asphaltene deposition simulation predictions are extended to wellbore with the help of equation (20).


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𝑘𝑑 = (𝑘𝑑 )𝑝𝑏

2𝛿𝑚𝑜𝑚 𝜙 = (𝑘𝑑 )𝑝𝑏 𝑆𝑐𝐹 𝑅 𝜙+1

𝑤ℎ𝑒𝑟𝑒, 𝜙 =

𝐷𝑒 𝛿𝑚𝑜𝑚 2 (𝑘𝑑 )𝑝𝑏 (𝐺𝑢 )𝑝𝑏

𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝒇𝒂𝒄𝒕𝒐𝒓: 𝑆𝑐𝐹 =

(20)

2𝛿𝑚𝑜𝑚 𝜙 𝑅 𝜙+1

where 𝑅 is the radius of the wellbore. The thickness of the momentum boundary layer can be calculated using the Prandtl boundary layer theory, 𝛿𝑚𝑜𝑚 = 62.7 𝐷 𝑅𝑒 −7/8 , where 𝐷 is the diameter of the wellbore and 𝑅𝑒 (= 𝑢𝐷𝜌/𝜇) is the Reynolds number. 3.4. Wellbore Simulation Results The results from precipitation and aggregation kinetics experiments and deposition results from the packed bed column were used to calibrate our asphaltene deposition model for wellbore, which is shown in equation (13). The precipitation kinetic parameter is scaled to the wellbore conditions using equation (18) and the deposition kinetic parameter is scaled using equation (20). The P-T trace along which the simulations are performed is shown in Figure 2. The thermodynamic properties required for the deposition model are calculated along this P-T trace. CFD modeling using FEM has been performed in order to obtain the asphaltene deposit profile along the length of wellbore. The current asphaltene deposition model developed for wellbore has the sophistication to predict asphaltene deposit thickness as well as total deposited material along the length of the wellbore. The asphaltene deposit profile is obtained by solving equation (13) using FEM. The fraction of asphaltene in the asphaltene-rich phase (precipitated phase) is used to obtain the thickness of total deposited material, which can be obtained from the thermodynamic module as a function of pressure and temperature along the length of the wellbore.


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The effect of time on the asphaltene deposit thickness in the wellbore and the corresponding asphaltene deposition flux have been analyzed. Figure 9 shows the growth of asphaltene deposit thickness over time, assuming no removal of deposits during that time period. The simulations are performed for a Gas-Oil Ratio (GOR) of 60 mol % (1.6 MSCF/STB) and oil production rate of 7500 STB/Day. 𝛿𝑑𝑚 refers to the maximum thickness of the total deposited material and its value for each case is also indicated in Figure 9. Asphaltene deposition starts at a length where the pressure is less than the Asphaltene Onset Pressure (AOP). For the given P-T trace, the pressure does not fall below the bubble pressure, hence, a decrease in the asphaltene deposit profile is not seen. It should be noted that the precipitation kinetic parameter is a function of pressure, temperature and the driving force. And, the deposition kinetic parameter is a function of the fluid flow field, viscosity, density and hence, the pressure and temperature conditions. As the pressure and temperature varies along the wellbore, the values of 𝑘𝑝 and 𝑘𝑑 are different at different nodes of the finite element mesh generated. Hence, the CFD model gives a more realistic deposit thickness profile. This is a unique feature of the developed asphaltene deposition simulator which is not seen in earlier models described in the literature.


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Figure 9. CFD Results - Total deposited material thickness as a function of time The deposition flux of total deposited material in the wellbore during a particular period of time, corresponding to the results shown in Figure 9, have also been calculated and shown in Figure 10. It is seen that the deposition flux decreases with increase in the time period, indicating that the amount by which asphaltene deposit thickness increases is not linear with respect to time.


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Figure 10. Deposition flux (corresponding to maximum thickness of total deposited material) as a function of time Generally, it is extremely difficult to measure the deposit profile along the wellbore using a caliper. However, pressure drop measurements are usually made available. Hence, with the deposition thickness profile predicted by the asphaltene deposition model for wellbore, the frictional pressure drop values can be calculated and compared with the field data. The Darcy−Weisbach formula is used to calculate the frictional pressure drop, which is given as,

∆𝑃𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛

𝐿 𝑈2 =𝑓 𝜌 𝐷 2

(21)

where, 𝐿 is the length of the wellbore, 𝐷 is the diameter of the wellbore, 𝑈 is the average velocity of the fluid, 𝜌 is the density of the fluid and 𝑓 is the friction factor, which can be calculated using the Colebrook-White equation for a given Reynolds number. It was found that, for a GOR of 60 mol % (1.6 MSCF/STB), oil production rate of 7500 STB/Day and a time period of 25 days, the frictional pressure drop is 815 psi for a smooth deposit. Pressure drop calculations have been performed based on the results shown in Figure 9. In order to understand the effect of different parameters on asphaltene deposition, several sensitivity analyses were performed. Deposition simulations were also performed for different GOR. The simulations are performed for a time period of one year and oil production rate of ACS Paragon Plus Environment

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7500 STB/Day. It was seen that with an increase in GOR the deposit thickness increases. When the GOR increases, the amount of primary particles available for precipitation and consequent deposition increases, hence, the deposit thickness also proportionately increases. This phenomenon is seen very clearly in Figure 11. Higher the GOR, higher is the driving force for precipitation. Hence, asphaltene deposition is most likely to occur, due to the high GOR.

Figure 11. CFD Results - Effect of GOR on the total deposited material thickness Figure 12 shows the sensitivity analysis for the effect of the kinetic constant for asphaltene deposition, (𝑘𝑑 )𝑝𝑏 , on the location and the magnitude of asphaltene deposition. (𝑘𝑑 )𝑝𝑏 is scaled to 𝑘𝑑 using the scaling factor shown in equation (20). The value of 𝑘𝑑 varies along the length of the wellbore, as it is a function of velocity, density and viscosity of oil which vary due to changes in pressure and temperature conditions. The results show the predictions of the asphaltene deposition profile in the wellbore after three years, assuming that the asphaltene ACS Paragon Plus Environment

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deposits are never removed in those three years. The simulations are performed for a GOR of 60 mol% (1.6 MSCF/STB) and oil production rate of 7500 STB/Day.

Figure 12. CFD Results - Effect of (𝑘𝑑 )𝑝𝑏 on the total deposited material thickness Wellbore simulations have been performed with the precipitation kinetic parameter, 𝑘𝑝 being a function of temperature and the driving force for asphaltene precipitation (𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ), as shown in equation (18). 𝑘𝑝 values have been scaled to wellbore conditions based on the selected P-T trace and applied to wellbore simulations. Hence, 𝑘𝑝 varies along the length of the wellbore. Figure 13 shows the comparison between total deposited material thicknesses simulated without and with scaling 𝑘𝑝 as a function of temperature and driving force for asphaltene precipitation, using equation (18). The simulations are performed for a GOR of 60 mol % (1.6 MSCF/STB), oil production rate of 7500 STB/Day and a time period of one year. The value of 𝑘𝑝 increases as we move up the wellbore, as the driving force for asphaltene precipitation (𝛿𝑎𝑠𝑝ℎ − 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ) ACS Paragon Plus Environment

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increases and the temperature decreases. Hence, for a given production rate, the rate of asphaltene precipitation increases as we move towards the wellhead and hence, the corresponding amount of asphaltene deposition increases as well.

Figure 13. CFD Results - Asphaltene deposition thickness without and with scaling of k p as a function of temperature and the driving force for asphaltene precipitation 4.

Modeling of Asphaltene Deposition in the Near-Wellbore Region

4.1. Deposition Module The velocity field in the near-wellbore region has been calculated by using Darcy’s law for a given pressure gradient in that region. The corresponding velocity gradient is calculated by using equation (22).35 𝐺 = 13.4

(1 − 𝜀𝑝 )𝑢 𝜀𝑝 𝑑𝑝


(22) 33

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where 𝐺 is the velocity gradient in porous media, 𝜀𝑝 is the average porosity, 𝑑𝑝 is the average particle size, 𝑢 is the velocity. For a given pressure and temperature condition, thermodynamic module facilitates in the calculation of the 𝐶𝑒𝑞 , which is the solubility of asphaltenes at the given conditions. Further, using equation (17), the concentration of asphaltenes dissolved in the oil phase and the concentration of precipitated asphaltenes can be calculated. The precipitated asphaltenes tend to deposit in the porous media. The rate of asphaltene deposition is given as; 𝑅𝑑 = −

𝜕𝑚𝐷 = 𝑎𝑚𝐺 + 𝑏𝑢 𝜕𝑡

(23)

where 𝑚 and 𝑚𝐷 represent the mass fraction of the precipitated and deposited asphaltene in the crude oil, respectively. 𝑎 is called the surface deposition coefficient and 𝑏 is the entrainment coefficient. The results of velocity and velocity gradient are coupled into the developed asphaltene deposition model for the near-wellbore region to obtain the profile of deposited asphaltenes in the near-wellbore region. A continuous mass balance is performed as oil proceeds from the near-wellbore region to the wellbore and encounters varying pressure conditions. Permeability reduction is correlated to the amount of deposited asphaltene using the equation proposed by Gruesbeck and Collins (1982)36: 𝐾 = 𝑒𝑥𝑝(−ϒ𝜎 4 ) 𝐾0

(24)

where, 𝐾 represents the permeability when there is asphaltene deposition, 𝐾0 represents the permeability when there is no asphaltene deposition, 𝜎 is the volume of the deposited asphaltene with respect to the pore volume and ϒ is a non-dimensional characteristic constant for a given porous medium. 4.2. Calibration of Model Parameters


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Microfluidic experiments were performed in order to analyze the asphaltene deposition in porous media. Experiments were performed by injecting the crude C2 with 75 vol% n-heptane, at room temperature and ambient pressure, in a microchannel with clay. A modeling method for the transport and deposition of asphaltenes in porous media based on the Lattice-Boltzmann Method (LBM) was adapted to reproduce the results obtained using microfluidic experiments. The fluid flow simulation in the microchannel geometry is performed by solving the two-dimensional Navier-Stokes equations. Equation (23) facilitates in the calculation of the deposited asphaltene mass fraction. The detailed description of performing asphaltene deposition simulation in a microchannel by applying LBM can be found elsewhere.11,37,38 The deposition model parameters have been calibrated with respect to the microfluidic experiments and the calibrated surface deposition coefficient is 1.7×10-4, the entrainment coefficient is 3.7×10-3 and porous medium non-dimensional characteristic constant is 22. Figure 14 shows the permeability reduction versus time. The blue line represents the data from micthe rochannel, and the simulation result is shown in orange.


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Figure 14. Comparison of permeability reduction obtained in the microfluidic experiments and simulation results using LBM 4.3. Near-Wellbore Region Simulation Results The simulation parameters obtained by modeling asphaltene deposition in a microchannel, as shown in Section 4.2, can be used to model asphaltene deposition in the near-wellbore region. The pressure variation in the near-wellbore region is shown in Figure 15. The temperature of the near-wellbore region is assumed to be constant and equal to 248oF.

Figure 15. Pressure profile in near wellbore region PC-SAFT EOS is used to estimate the amount of asphaltene dissolved in the oil for different pressure and temperature conditions. The amount of deposited asphaltene is modeled using equation (23). Figure 16 shows profiles of (a) dissolved (b) precipitated (c) deposited asphaltenes, assuming that the near-wellbore region extends up to 400 feet away from the wellbore and oil flows only towards the wellbore. The results are reported as the mass of the asphaltene in their different states, with respect to the total amount of asphaltenes initially present in the oil.


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Figure 16. Profiles in the near-wellbore region of (a) dissolved, (b) precipitated (c) deposited asphaltenes with respect to the total amount of asphaltenes initially present in the oil The deposited asphaltene based on the pore volumes of the porous medium is also calculated and shown in Figure 17. The permeability reduction is calculated using Gruesbeck and Collins36 model and the results are shown in Figure 17. From the simulation results, it is seen that the asphaltene deposition does not cause serious permeability reduction for the given conditions.

Figure 17. (a) Variation of deposited asphaltene (with respect to pore volume) along the distance from the wellbore, (b) Permeability reduction along the distance from the wellbore 5. Integration of Wellbore and Near-Wellbore Region Deposition Modeling


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In Section 3 and 4, the modeling of asphaltene deposition in the wellbore and near-wellbore region were discussed respectively. It is of prime interest to integrate these two simulations so that a comprehensive idea of the asphaltene deposition flow assurance problem can be obtained. The simulations for the near-wellbore region facilitated in the calculation of the amount of asphaltene deposited in that region. The amount of asphaltene precipitated and the amount of asphaltene still soluble in the oil were also calculated. The amount of precipitated asphaltene thus obtained can then be used as the initial amount of available asphaltene primary particles for the wellbore asphaltene deposition simulations. With this initial condition, following the simulation technique illustrated in Section 3, asphaltene deposition profile in the wellbore can be predicted. The P-T trace selected for this simulation in the near wellbore region and in the wellbore are shown in Figure 18. For the selected P-T trace, the pressure at the wellbore inlet is below the asphaltene onset pressure and hence, there is asphaltene precipitation and consequent deposition. The corresponding total deposited material thickness profile has been shown in Figure 19. The simulations have been performed for a GOR = 1.6 MSCF/STB (60 mol%), production rate = 7500 STB/day and for a time period of one year, with the assumption that deposits were not cleaned from the wellbore in that time period.


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Figure 18. (a) Pressure profile in the near-wellbore region, and (b) P-T trace for wellbore

Figure 19. CFD Results - Asphaltene deposition prediction in the wellbore (a) without integrating with the near-wellbore region simulation, (b) on integration with the near-wellbore region simulation ACS Paragon Plus Environment

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Figure 19 (a) shows the asphaltene deposition predictions in the wellbore, with the assumption that no asphaltene is deposited in the near-wellbore region and asphaltene precipitation and deposition start only at the inlet of the wellbore. Whereas, Figure 19 (b) is obtained as a result of the integration of near-wellbore simulations with the wellbore simulations. Here, a certain amount of asphaltene has already been deposited in the near-wellbore region and there are asphaltene primary particles available at the inlet of the wellbore which further undergo aggregation and deposition as they move up the wellbore under varying conditions of pressure and temperature. It is seen from the simulation results that there is not much change in the maximum thickness of the total deposited material. But, at the wellbore inlet, it is seen that there is an increase in the deposit thickness, indicating the fact that, the available asphaltene primary particles at the wellbore inlet have undergone aggregation and hence deposited on wellbore surface. Further upstream, asphaltene deposition is caused due to changes in pressure, temperature, and composition of the oil. 6. Conclusions In this work, a deposition simulator has been developed to conduct a systematic investigation of the asphaltene deposition in the wellbore and near-wellbore region. The simulator makes use of PC-SAFT EOS to generate thermodynamic properties such as the onset of asphaltene precipitation, the amount of precipitated asphaltenes, the density of oil, and the equilibrium asphaltene concentration in the bulk phase under different gas injection scenarios and as a function of temperature and pressure. The variation of viscosity as a function of temperature and pressure is accounted for using the Friction Theory. The rate of asphaltene deposition has been predicted using a CFD model which aids in investigating the occurrence and the magnitude of asphaltene deposition and the deposit profile along the production tubing after gas breakthrough. The CFD model developed performs transient simulation in order to calculate the asphaltene


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deposition rate as a function of time. Also, the model captures the turbulence effects, by rigorously modeling the turbulent flow of oil in the wellbore. In contrast to conventional modeling techniques based on experimental data from capillary deposition tests, modeling of deposition of asphaltenes has been performed based on experimental results from deposition tests in a packed bed column. The unique contribution of this work is to develop a new model for asphaltene deposition in the wellbore and near-wellbore region by using CFD simulation. There are models in the available in the literature to simulate the asphaltene deposition in the wellbore, and there are a few models to describe the process of asphaltene deposition in the near-wellbore region. However, the integration of the two models has not been developed yet. We have developed a novel integrated asphaltene deposition model for wellbore and near-wellbore region. This has been achieved by combining an in-house wellbore deposition model with an in-house near-wellbore region deposition model. It should be noted that the overall emphasis of this study is to develop computationally efficient deposition simulator that can be integrated with the thermodynamic model, such that a single comprehensive platform is available to model, simulate and predict asphaltene precipitation, aggregation and deposition. According to the extensive experimental and modeling investigation conducted on crude C2, there is a relatively low risk of asphaltene deposition as the gas breaks through. However, it should be noted that the investigation only included crude C2. For this reason, it is recommended that the investigation is extended to include fluids from other sands, and enhance the capabilities of the model and the risk assessment methods by incorporating other important effects, such water-oil interactions, corrosion and its effect on asphaltene deposition. Acknowledgments The authors are grateful for the financial support provided by Chevron Energy Technology Company. ACS Paragon Plus Environment

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Narmadha Rajan Babu acknowledges the Ken Kennedy Institute for Information Technology for the 2017/18 ExxonMobil Graduate Fellowship. Nomenclature Symbols 𝑎 = surface deposition coefficient 𝑏 = entrainment coefficient 𝐶 ′ = dimensional concentration of the primary particles 𝐶𝑓 ′ = dimensional concentration of dissolved asphaltene in the oil phase at a particular time 𝐶𝑒𝑞 ′ = dimensional thermodynamic equilibrium concentration of asphaltene 𝐶0 = initial concentration of asphaltene primary particles solubilized in the oil phase 𝐶 = dimensionless concentration of the precipitated asphaltene particle 𝐶𝑒𝑞 = dimensionless thermodynamic equilibrium concentration of asphaltene 𝐶𝑓 = dimensionless concentration of asphaltene in the oil−precipitant mixture 𝐶𝑖𝑛 = dimensionless concentration of asphaltene primary particles at wellbore inlet 𝐶𝑏𝑙 = dimensionless concentration in the wellbore boundary layer 𝐷𝐵 = Brownian particle diffusivity 𝐷𝑇 = turbulent particle diffusivity 𝐷𝑒 = asphaltene particle diffusivity/diffusion coefficient 𝐷 = diameter of the wellbore ACS Paragon Plus Environment

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𝑑𝑝 = average particle size 𝑓 = friction factor 𝐺 = velocity gradient in porous media 𝒈 = acceleration due to gravity 𝐾𝑤 = Watson characterization factor 𝑘̂𝑎 = reduced viscous friction coefficient due to attraction 𝑘̂𝑟 = reduced viscous friction coefficient due to repulsion 𝑘̂𝑟𝑟 = reduced viscous friction coefficient due to repulsion (2nd order) 𝑘𝑝 = precipitation kinetic parameter 𝐾𝑎𝑔 = 𝑘ag 𝐶𝑜 = aggregation kinetic parameter 𝑘𝑑 = deposition kinetic parameter (𝑘𝑑 )𝑝𝑏 = deposition kinetic parameter for packed bed column 𝑘 = turbulent kinetic energy 𝐾 = permeability of porous media when there is asphaltene deposition 𝐾0 = permeability of porous media when there is no asphaltene deposition 𝐿 = axial length of the wellbore 𝑚𝐷 = mass fraction of the deposited asphaltene

𝒏 = unit normal vector ACS Paragon Plus Environment

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𝑃 = pressure 𝑃𝑐 = critical pressure 𝑃̂𝑎 = reduced attractive pressure 𝑃̂𝑟 = reduced repulsive pressure 𝑝̅ = time-averaged pressure ∆𝑃𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = frictional pressure drop 𝑈 = average velocity of the fluid 𝑢𝑟𝑒𝑠 = residual internal energy 𝒖 = velocity field ̅ = time-averaged velocity 𝒖 𝒖′ = fluctuating velocity 𝑣 = molar volume 𝑅 = radius of wellbore 𝑅𝑑 = rate of asphaltene deposition 𝑅𝑒 = Reynolds number (𝑅𝑒 =

𝑢𝐷𝜌 𝜇

)

𝜈

𝑆𝑐𝑇 = turbulent Schmidt number (𝑆𝑐𝑇 = 𝐷𝑇 ) 𝑇

𝑇 = temperature 𝑇𝑏 = boiling point ACS Paragon Plus Environment

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𝑇𝑐 = critical temperature 𝑉 = volume Greek symbols 𝜇 = viscosity of the medium 𝜇0 = dilute gas viscosity contribution 𝜇 𝑇 = turbulent viscosity 𝜇𝑐 = characteristic critical viscosity 𝜇𝑓 = friction viscosity contribution 𝜈𝑇 = eddy diffusivity 𝜀 = turbulent dissipation rate 𝜀𝑝 = average porosity of porous media 𝛿𝑎𝑠𝑝ℎ = solubility parameter of asphaltenes 𝛿𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = solubility parameter of solution 𝛿𝑚𝑜𝑚 = momentum boundary layer thickness 𝛿𝑑𝑚 = maximum thickness of total deposited material 𝜌 = mass density (mass / volume) 𝜎 = volume of the deposited asphaltene with respect to the pore volume ϒ = non-dimensional characteristic constant for a given porous medium


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𝜔 = acentric factor Acronyms AOP = asphaltene onset pressure BP = bubble point CFD = Computational Fluid Dynamics EOS = Equation of State FEM = Finite Element Method FT = Friction Theory GOR = Gas-Oil Ratio LBM = Lattice Boltzmann Modeling MSCF = Thousand Standard Cubic Feet 𝑀𝑊 = molecular weight PC-SAFT = Perturbed Chain version of the Statistical Associating Fluid Theory PR = Peng Robinson 𝑆𝑐𝐹 = Scaling Factor 𝑆𝐺 = specific gravity STB = Stock Tank Barrel 𝑆𝑐𝐹 = Scaling Factor


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Systematic Investigation of Asphaltene Deposition in Wellbore and

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