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Thermodynamic models for asphaltene precipitation

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Thermodynamic models for predicting asphaltene precipitation behavior fall into two general categories: activity models and equation-of-state (EOS) models. This page provides the mathematics underlying the most commonly used models of each type.

Thermodynamic equilibrium

With the precipitated asphaltene treated as a single-component or multicomponent solid, the condition for thermodynamic equilibrium between the oil (liquid) and solid phase is the equality of component chemical potentials in the oil and solid phases. That is, ....................(1)

where μio and μ is are the chemical potential of component i in the oil and solid phases, respectively, and nc is the number of components. The application of activity coefficient models or EOS models gives different expressions for the chemical potential. In addition, not all components in the oil phase undergo precipitation; therefore, Eq. 1 applies only to those components that precipitate.

Activity models

Activity coefficients

Because asphaltenes are a solubility class that can be precipitated from petroleum by the addition of solvent, activity coefficient models have been applied to model the phase equilibrium phenomena. The introduction of activity coefficients in Eq. 1 yields ....................(2)

where:

Several approaches that use the activity coefficient model assume the oil and asphaltene as two pseudocomponents: one component representing the deasphalted oil and the other the asphaltenes. Andersen and Speight provided a review of activity models in this category. Other approaches represent the precipitate as a multicomponent solid. Chung, Yarranton and Masliyah, and Zhou et al. gave detailed descriptions of these models.

Flory-Huggins model

The solubility model used most in the literature is the Flory-Huggins solubility model introduced by Hirschberg et al. Vapor/liquid equilibrium calculations with the Soave-Redlich-Kwong EOS are performed to split the petroleum mixture into a liquid phase and a vapor phase. The liquid phase then is divided into two components: a component that corresponds to the asphaltene and a component that represents the remaining oil (deasphalted oil). When solvent is added into the oil, the second component represents the mixture of deasphalted oil and solvent. These two components are for modeling purposes and do not correspond to any EOS components used in the vapor/liquid calculations. It also is assumed that asphaltene precipitation does not affect vapor/liquid equilibrium.

Application of the Flory-Huggins solution theory gives the following expression for the chemical potential of the asphaltene component in the oil phase. ....................(3)

with ....................(4)

where:

• subscripts a, o, and m are used to denote the asphaltene component, the deasphalted oil, and the oil phase mixture, respectively
• va = molar volume of pure asphaltene,
• vm = molar volume of mixture,
• δi = solubility parameter of component i,
• δm = solubility parameter of mixture,
• Φa = volume fraction of asphaltene in the mixture, μam = chemical potential of asphaltene in the mixture
• = reference chemical potential of asphaltene component

Because the precipitated asphaltene is pure asphaltene, μs = . From the equality of chemical potential μam = μs, Eq. 3 gives ....................(5)

The molar volume, vm, of the oil mixture is calculated from the composition of the liquid phase obtained from vapor/liquid calculations that use the Soave-Redlich-Kwong EOS. The solubility parameter, δm, is calculated from ....................(6)

where ΔUv is the molar internal energy of vaporization at the system temperature, which also can be calculated from the Soave-Redlich-Kwong EOS. The remaining parameters are the molar volume of asphaltene, va, and the solubility parameter of asphaltene, δa, which are essential to the performance of this model. The molar volume of asphaltene can only be speculated on. Hirschberg et al. used values of v a in the range of 1 to 4 m3/kmol. The solubility parameter of asphaltene can be estimated by measuring the solubility of asphaltene in different solvents of increasing solubility parameters. The asphaltene is assumed to have the solubility parameter of the best solvent. Hirschberg et al. suggests that the solubility parameter of asphaltene is close to that of naphthalene. Eq. 5 gives the amount (volume fraction) of asphaltene soluble in the oil mixture. The amount of precipitation is determined by the difference between the total amount of asphaltenes present in the initial oil and the solubility of asphaltene under given conditions.

The solubility parameter can be correlated as a linear equation with respect to temperature as ....................(7)

where a and b are constants. parameter b is negative as the solubility parameter decreases with increasing temperature. Buckley et al. and Wang and Buckley showed that the measurements of the refractive index of crude oils can be used to determine the solubility parameters required for the Flory-Huggins model.

The Hirschberg et al. approach also has been used with some degree of success by:

• Burke et al.,
• Kokal and Sayegh,
• Rassamdana et al.

de Boer et al. used this model to screen crude oils for their tendency to precipitate asphaltene. They compared properties of some crudes in which asphaltene problems were encountered and properties of crudes that operated trouble free. They found that asphaltene problems were encountered with light crudes with high C1 to C3 contents, high bubblepoint pressures, large differences between reservoir pressure and bubblepoint pressure, and high compressibility. With an asphaltene molar volume of 1 m3/kmol, de Boer et al. showed that the solubility of asphaltene in a light crude oil with Eq. 5 follows the curve shown in Fig. 1. Above the bubblepoint, the decrease in asphaltene solubility is caused by pressure effects. Below the bubblepoint, the increase in asphaltene solubility is caused by the variation in the oil composition. Clearly, a minimum asphaltene solubility occurs around the bubblepoint.

de Boer et al. calculated the solubility of asphaltene with Eq. 5 for different values of in-situ crude oil densities and asphaltene-solubility parameters. They also introduced a maximum supersaturation at bubblepoint defined as ....................(8)

where pr and pb are, respectively, the reservoir pressure and the bubblepoint pressure at the reservoir temperature. Fig. 2 shows the maximum supersaturation at the bubblepoint as a function of the difference between reservoir and bubblepoint pressure, the in-situ oil density, and the asphaltene-solubility parameter. The influence of the asphaltene-solubility parameter is very small. Supersaturations are larger for lighter crudes. The boundary between problem and nonproblem areas lies at a maximum supersaturation of approximately 1. Although these results were derived with North Sea and Kuwait crudes, Hammami et al. showed that they also are applicable to crudes from the Gulf of Mexico.

Extension of Flory-Huggins model

The Flory-Huggins model initially was developed for polymer solutions. The Hirschberg et al. approach is based on the representation of asphaltene as a homogeneous polymer. Novosad and Constain used an extension of the model that includes asphaltene polymerization and asphaltene-resin association in the solid phase. Kawanaka et al. proposed an improvement whereby the precipitated asphaltene is treated as a heterogeneous polymer (i.e., a mixture of polymers of different molecular weights). The Scott-Magat theory was used to obtain a solubility model for a given molecular-weight distribution for asphaltene. Cimino et al. also used the Flory-Huggins model with two components (solvent and asphaltene) but considered the solid phase to be a mixture of solvent and asphaltene instead of pure asphaltene as in Hirschberg et al. ‘s approach. Yang et al. proposed a multicomponent Flory-Huggins model in which components are the same as the EOS components used in the oil/gas flash calculations.

Multicomponent activity coefficient models

These models are derived from methods for modeling wax precipitation. Multicomponent solid/liquid K values are derived from Eq. 2 and then used with an EOS in a three-phase oil/gas/solid flash calculation. The solid/liquid K values are defined as ....................(9)

Eq. 2 gives: ....................(10)

with ....................(11)

Eq. 11 is equivalent to: ....................(12)

where:

• Tif = fusion temperature of component i
• Δ Cpi = Cpo,iCps,i, heat capacity change of fusion
• ΔHif = heat of fusion of component i

ΔCpi is assumed to be independent of temperature in Eq. 12.

Starting with Eq. 12, methods were derived through the use of different models for activity coefficients. The earliest approach is from Won in the modeling of wax precipitation. Won suggested that the term involving ΔCpi and the integral involving Δvi are negligible and used regular solution theory to calculate the activity coefficients in Eq. 12 as follows. ....................(13) ....................(14) ....................(15)

where:

• δik is the solubility parameter of component i in phase k (k = o, s)
• vik is the molar volume of component i in phase k
• Φik is the volume fraction of component i in phase k

Won gave solubility parameter values, δio and δis, for normal paraffins up to C40. Correlations also are provided to calculate ΔHif, vio, and vis. Although Won’s model was developed for wax precipitation, Thomas et al. have applied it with some success in predicting asphaltene precipitation. However, they have developed their own correlations for solubility parameters. MacMillan et al. also used Won’s model but kept all the terms in Eq. 12 instead of neglecting the terms involving ΔCpi and Δvi as Won did. They also included additional multiplication factors to the different terms in Eq. 12 to facilitate phase-behavior matching.

Hansen et al. and Yarranton and Masliyah used the Flory-Huggins model to calculate the activity coefficients in Eq. 12. Hansen et al. applied their method to the modeling of wax precipitation, while Yarranton and Masliyah modeled precipitation of Athabasca asphaltenes. Yarranton and Masliyah proposed an approach for calculating the molar volumes and solubility parameters from experimental measurements of molar mass and density. Asphaltene density, molar volume, and solubility parameter are correlated with molar mass. Zhou et al. used the Flory-Huggins polymer-solution theory with a modification to account for the colloidal suspension effect of asphaltenes and resins.

Equation of state models

These approaches model the oil, gas, and precipitate by an EOS, which is used to calculate the component fugacities in different phases. Cubic EOSs have been used to model petroleum reservoir fluids that exhibit vapor/liquid 1/liquid 2 behavior (see Fussell, Nghiem and Li, or Godbole et al.). Godbole et al. observed that the apparent second liquid phase could be approximated as a mixture of aggregated asphaltenes (solid phase) entrained in a portion of the other liquid phase in the modeling of mixtures of crude oil from the North Slope of Alaska and enriched gas. Under certain conditions, a phase-behavior program that includes a three-phase calculation with an EOS could be used to model some aspects of asphaltene precipitation; however, the prevailing approach consists of the use of a cubic EOS (e.g., Soave-Redlich-Kwong EOS or Peng-Robinson EOS) for the oil and gas phases and a solid model for the precipitate.

The simplest model for precipitated asphaltene is the single-component solid model. The precipitated asphaltene is represented as a pure solid, while the oil and gas phases are modeled with a cubic EOS. The fugacity of the pure solid is given by ....................(16)

where:

The following fugacity equality equations are solved to obtain oil/gas/solid equilibrium. ....................(17a)

and ....................(17b)

The oil and gas fugacities, fio and fig, for component i are calculated from an EOS. In Eq. 17b, subscript a denotes the asphaltene component in solution. Normally, this asphaltene component is the heaviest and last component of the oil (i.e., a = nc). The following simple stability test can be used to determine whether there is asphaltene precipitation: if faofs, asphaltene precipitation occurs, and if fao < fs, there is no precipitation.

Earlier applications of the single-component solid model for asphaltene precipitation were not successful. Nghiem et al. introduced a method for representing the asphaltene component in the oil that improves the capabilities of the single-component solid model to predict asphaltene precipitation. The method was subsequently refined by Nghiem et al. The key to the approach is the split of the heaviest fraction of the oil into two pseudocomponents:

• One that does not precipitate (nonprecipitating component)
• One that can precipitate (precipitating component)

These two pseudocomponents have identical critical temperatures, critical pressures, acentric factors, and molecular weights. The differences are in the interaction coefficients. The interaction coefficients between the precipitating components and the light components are larger than those between the nonprecipitating component and the light components. The parameters of the model are the reference fugacity and the solid molar volume. The reference fugacity could be estimated from a data point on the asphatene precipitation envelope (APE), and a value for solid molar volume slightly larger than the EOS value for the pure component a is adequate.

The following application of the model to a North Sea fluid from Nghiem et al. illustrates the procedure. Table 1 shows the pseudocomponent representation of the reservoir fluid with the separator gas and separator oil compositions. The reservoir oil corresponds to a combination of 65.3 mol% separator oil and 34.7 mol% separator gas. The crucial step in the modeling of asphaltene is the split of the heaviest component in the oil (e.g., C32+) into:

• A nonprecipitating component (C32A+)
• A precipitating component (C32B+)

These two components have identical critical properties and acentric factors but different interaction coefficients with the light components. The precipitating component has larger interaction coefficients with the light components. With larger interaction coefficients, the precipitating component becomes more "incompatible" with the light components and tends to precipitate as the amount of light component in solution increases. Although C32B+ is called the precipitating component, the amount that precipitates is governed by Eq. 16. Normally, only a portion of the total amount of C32B+ will precipitate during a calculation. Hirschberg et al. reports that the asphalt precipitate from a tank oil consists mainly (90%) of C30 to C60 compounds. For the purpose of modeling asphaltene precipitation, a heaviest component in the vicinity of C30+ is adequate. For this example, C32+ is used.

The Peng-Robinson EOS was used to model the oil and gas phases. The critical properties and acentric factors of the pseudocomponents in Table 1 are calculated as described in Li et al. The interaction coefficients are calculated from ....................(18)

where:

• dij = the interaction coefficient between component i and j
• vci = the critical volume of component i,
• e = an adjustable parameter

A value of e(C32A+) = 0.84 and a value of e(C32B+) = 1.57 were found to provide a good match of the saturation and onset pressure. The reference solid fugacity was obtained by calculating the fugacity of oil at one point on the APE (recombined oil with 69.9 mol% separator gas and 30 000 kPa) with the Peng-Robinson EOS and equating it to . The molar volume of the asphaltene precipitate was assumed equal to 0.9 m3/kmol.

Fig. 3 shows a good match of the experimental and calculated APE and saturation pressure curves at the reservoir temperature of 90°C. The model was able to predict precipitation conditions that are far from the reference conditions used to determine . Fig. 3 shows the amounts of precipitation calculated as constant weight percent of precipitate (similar to "quality lines" in oil/gas phase diagrams). As pressure decreases below the APE, the amount of precipitation increases and reaches a maximum at the saturation pressure. Below the saturation pressure, the amount of precipitation decreases with decreasing pressure. The results are consistent with the laboratory observations described here.

For nonisothermal conditions, Eq. 19 can be used to calculate the solid fugacity at (p, T) from the solid fugacity at a reference condition (p*, T*). ....................(19)

where:

• f = fugacity of the asphaltene component in the pure liquid state
• Tf = melting point temperature
• v = molar volume of liquid
• ΔCp = heat capacity of fusion
• ΔHf = enthalpy of fusion

Kohse et al. used Eq. 19 to model the precipitation behavior of a crude oil with changes in pressure and temperature. Fig. 4 shows good agreements between the experimental and calculated APE and saturation-pressure curves. The measured data point of 1.6 wt% of precipitate also is close to the predictions.

The previous two examples illustrate the application of the single-component solid model to the modeling of precipitation behavior of crudes with changes in:

• Pressure
• Temperature
• Composition

From a mechanistic point of view, the nonprecipitating component can be related to:

• Resins
• Asphaltene/resin micelles that do not dissociate
• Heavy paraffins

The precipitating component corresponds to both the asphaltenes that dissociate and the asphaltene/resin micelles that precipitate unaltered. Because of identical critical properties and acentric factors, the nonprecipitating and precipitating components behave as a single component in solution. The larger interaction coefficients between the precipitating and the solvent components cause the precipitation of the former with the addition of solvent. The amount of precipitation depends on the solution of Eqs. 17a and 17b. Normally, only a portion of the precipitating component actually precipitates.

Solid precipitation with the previous model is reversible. Nghiem et al. proposed an enhancement to the approach to obtain partial irreversibility. A second solid (Solid 2) is introduced that is obtained from the reversible solid (Solid 1) through a chemical reaction:

If the forward reaction rate k12 is much larger than the backward reaction rate k21, Solid 2 behaves as a partially irreversible solid.

Thermodynamic models

Thermodynamic-colloidal model

Leontaritis and Mansoori proposed a more mechanistic approach based on the assumption that asphaltenes exist in the oil as solid particles in colloidal suspension stabilized by resins adsorbed on their surface. This thermodynamic-colloidal model assumes thermodynamic equilibrium between the resins in the oil phase and the resins adsorbed on the surface of colloidal asphaltene (asphaltene micelle). The corresponding equilibrium equation is ....................(20)

Assuming that resins behave as monodisperse polymers and applying the Flory-Huggins polymer-solution theory gives the volume fraction of dissolved resins as ....................(21)

which is analogous to Eq. 5 for the asphaltene component in Hirschberg et al.‘s approach. In Hirschberg et al.‘s approach, the asphaltene component contains both resins and asphaltene, whereas Eq. 21 applies to the resins only. As in Hirschberg et al.‘s approach, EOS flash calculations with a multicomponent system are performed to obtain an oil/gas split and oil properties from which Φr is calculated. This value of Φr is compared with a critical resin concentration, Φcr, which is given as a function of pressure, temperature, molar volume, and solubility parameters.

Φcr is the key parameter of the model. If Φr > Φcr, the system is stable and no precipitation occurs. If ΦrΦcr, asphaltene precipitation occurs. The amount of precipitated asphaltene can be made a function of the asphaltene particle sizes.

Thermodynamic-micellization model

Pan and Firoozabadi proposed the most mechanistic approach to model asphaltene precipitation by calculating the Gibbs free energy of formation of the asphaltene micelle and including it in the phase-equilibrium calculations. Details of the approach can be found in Firoozabadi. Fig. 5 portrays schematically the system to be modeled. The species in the liquid phase (L1) are monomeric asphaltenes, monomeric resins, micelles, and asphalt-free oil species. The micelle consists of a core of n1 asphaltene molecules surrounded by a shell containing n2 resins molecules. The precipitate phase is considered as a liquid mixture (L2) of asphaltene and resin molecules. An expression for Gibbs free energy of formation of the micelle, , which includes n1, n2, and the shell thickness, D, was proposed. The Gibbs free energy of the liquid phase, L1, then is derived with:

Similarly, the Gibbs free energy of the precipitated phase, L2, which is a binary mixture of monomeric asphaltenes and resins, also is derived with the use of an EOS. The total Gibbs free energy of the system, ....................(22)

then is minimized with respect to:

The minimization requires a robust numerical procedure.

The model was applied to predict precipitation from a tank oil with propane, Weyburn oil with CO2, and a North Sea oil with separator gas. Fig. 6 shows the predictions of Weyburn oil with CO2 obtained with the thermodynamic-micellization model. For comparison, the match obtained with the pure solid model also is shown.

Nomenclature

 a = constant b = constant dij = interaction coefficient between component i and j D = shell thickness of the micelle, L fao = fugacity of asphaltene component in oil phase, m/Lt2 fi = fugacity of component i, m/Lt2 = standard state fugacity of component i, m/Lt2 fig = fugacity of component i in the oil phase, m/Lt2 fik = fugacity of component i in phase k (k = o, s), m/Lt2 = fugacity of pure component i in phase state k (k = o, s), m/Lt2 = standard state fugacity of component i in phase k (k = o, s), m/Lt2 fio = fugacity of component i in the oil phase, m/Lt2 = fugacity of pure component i in oil phase, m/Lt2 = standard state fugacity of component i in oil phase, m/Lt2 fis = fugacity of component i in the solid phase, m/Lt2 = fugacity of pure component i in solid phase, m/Lt2 = standard state fugacity of component i in solid phase, m/Lt2 fℓ = fugacity of the asphaltene component in the pure liquid state, m/Lt2 fp = porous medium particle transport efficiency factor fs = solid fugacity, m/Lt2 = reference solid fugacity, m/Lt2 G = total Gibbs free energy of the system, m/L2t2 GE = total excess Gibbs energy for a phase, m/L2t2 Kis = solid/liquid K value for component i nc = number of components = number of asphaltene monomers in phase L1 = number of asphaltene monomers in phase L2 = number of micelles in phase L1 = number of resin monomers in phase L1 = number of resin monomers in phase L2 p = pressure, m/Lt2 p* = reference pressure, m/Lt2 pb = bubblepoint pressure, m/Lt2 pr = reservoir pressure, m/Lt2 R = gas constant T = temperature, T T* = reference temperature, T Tc = temperature-dependent parameter Tf = melting point temperature, T Tif = temperature of fusion (melting temperature) of component i, T uc = critical speed required to mobilize surface deposit asphaltene, L/t uo = oil velocity, L/t va = molar volume of pure asphaltene, L3/n vci = critical volume of component i, L3/n vcj = critical volume of component j, L3/n vik = partial molar volume of component i in phase k (k = o, s), L3/n vio = partial molar volume of component i in oil phase, L3/n vis = partial molar volume of component i in solid phase, L3/n vjk = partial molar volume of component j in phase k (k = o, s), L3/n vjo = partial molar volume of component j in oil phase, L3/n vjs = partial molar volume of component j in solid phase, L3/n vℓ = molar volume of liquid, L3/n vm = molar volume of mixture, L3/n vr = molar volume of resins, L3/n vs = solid molar volume, L3/n wi = weight fraction of component i, m/m W = weight percent of precipitated asphaltene, m/m yi = mole fraction of component i, n/n yik = mole fraction of component i in phase k (k = o, s), n/n yio = mole fraction of component i in oil phase, n/n yis = mole fraction of component i in solid phase, n/n yjo = mole fraction of component j in oil phase, n/n yjs = mole fraction of component j in solid phase, n/n γ = shear rate, L/t γi = activity coefficient of component i in a mixture γik = activity coefficient of component i in phase k (k = o, s) γio = activity coefficient of component i in oil phase = combinatorial free volume contribution = residual contribution γis = activity coefficient of component i in solid phase δa = solubility parameter of asphaltene δi = solubility parameter for component i δik = solubility parameter for pure component i in phase k (k = o, s) = volume fraction average solubility parameter for phase k δm = solubility parameter of mixture δo = solubility parameter of oil phase = volume fraction average solubility parameter of oil phase δs = solubility parameter of solid phase = volume fraction average solubility parameter of solid phase ΔCp = heat capacity of fusion, mL2/nt2T ΔCpi = heat capacity of fusion of component i, mL2/nt2T ΔCpij,tr = heat capacity of jth solid state transition of component i, mL2/nt2T = expression for Gibbs free energy of formation of the micelle, mL2/nt2 ΔHf = enthalpy of fusion, mL2/nt2 ΔHif = enthalpy of fusion of component i, mL2/nt2 ΔUv = molar internal energy of vaporization at the system temperature, mL2/nt2 Δvi = change of molar volume caused by fusion of component i, L3 = reference chemical potential of asphaltene component μam = chemical potential of asphaltene in the mixture μc = critical speed required to mobilize surface deposit asphaltene μi = chemical potential of component i, m/L2t2 μik = chemical potential of component i in phase k, m/L2t2 μio = chemical potential of component i in the oil phase, m/L2t2 μis = chemical potential of component i in the solid phase, m/L2t2 μpi,k = chemical potential of pure component i in phase k (k = o, s), m/L2t2 μpi,o = chemical potential of pure component i in oil phase, m/L2t2 μpi,s = chemical potential of pure component i in solid phase, m/L2t2 μr,m = chemical potential of resins on the surface of the asphaltene micelle, m/L2t2 μr,o = chemical potential of resins in the oil phase, m/L2t2 μs = chemical potential of the solid, m/L2t2 ρo = mass density of oil, m/L3 Φik = fugacity coefficient of component i in phase k = fugacity coefficient of pure component i in phase state k (k = o, s) Φio = fugacity coefficient of component i in oil phase Φa = volume fraction of asphaltene in the mixture Φcr = critical volume fraction of resins in the mixture Φik = volume fraction of component i in phase state k (k = o, s) Φr = volume fraction of resins in the mixture