PEH:Asphaltenes and Waxes

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Copyright 2007, Society of Petroleum Engineers

Chapter 9 – Asphaltenes and Waxes

Long X. Nghiem and Bruce F. Kohse, Computer Modelling Group

Pgs. 397-464

ISBN 978-1-55563-108-6
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Introduction

Deposition of the high-molecular-weight components of petroleum fluids as solid precipitates in surface facilities, pipelines, downhole tubulars, and within the reservoir are well-recognized production problems. Depending on the reservoir fluid and the type of recovery process, the deposited solid may consist of asphaltenes, waxes, or a mixture of these materials. The deposits also can contain resins, crude oil, fines, scales, and water.

This chapter discusses the experimental measurement and thermodynamic modeling of the phase behavior of solid waxes and asphaltenes in equilibrium with fluid hydrocarbon phases. Models for solid deposition in the reservoir and in pipelines also are presented. Although some of the laboratory techniques for determining solid precipitation are applicable to both waxes and asphaltenes, the characteristic behaviors of these materials can be very different; therefore, wax and asphaltene topics are treated separately.

Sec. 9.1 presents some commonly observed behaviors of asphaltenic and waxy crudes. Sec. 9.2 gives the chemical characteristics of asphaltenes and waxes. Secs. 9.3 through 9.6 discuss asphaltene measurement, precipitation modeling, and deposition modeling and provide a brief review of remediation methods. Secs. 9.7 through 9.10 present the same sequence of topics for waxes. See the chapters on phase behavior and phase diagrams in this section of the handbook for additional information on hydrocarbon phase behavior. Information on water/hydrocarbon systems is given in the chapters on water hydration and crude oil emulsions.

Asphaltene-Precipitation Behavior

Asphaltenes precipitation is caused by a number of factors including changes in pressure, temperature, and composition. The two most prevalent causes of asphaltene precipitation in the reservoir are decreasing pressure and mixing of oil with injected solvent in improved-oil-recovery (IOR) processes. Drilling, completion, acid stimulation, and hydraulic fracturing also can induce precipitation in the near-wellbore region. As oil flows up the wellbore, asphaltene can precipitate as a result of pressure and temperature changes. A summary of the different field and laboratory observations associated with asphaltene precipitation during primary depletion and IOR gas injection follows.

Aphaltene Precipitation During Primary Depletion. In normal pressure depletion, reservoirs that experience asphaltene precipitation usually have the following characteristics:

• The fluid in place is light to medium oil with small asphaltene content.
• The initial reservoir pressure is much larger than the saturation pressure. That is, the fluid is highly undersaturated.
• Maximum precipitation occurs around the saturation pressure.

Heavier crudes that contain a larger amount of asphaltene have very few asphaltene-precipitation problems because they can dissolve more asphaltene. Leontaritis and Mansoori and Kokal and Sayegh compiled field cases with asphaltene-precipitation problems during primary depletion. Extreme cases include the Venezuelan Boscan crude with 17 wt% asphaltene produced nearly without precipitation, whereas the Venezuelan Mata-Acema crude with 0.4 to 9.8 wt% asphaltene and the Algerian Hassi Messaoud crude with 0.062 wt% encountered serious precipitation problems during production.

Asphaltene Precipitation During IOR Gas Injection. The injection of hydrocarbon gases or CO₂ for IOR promotes asphaltene precipitation. Numerous field reports and laboratory studies on this phenomenon have been published. Although it frequently manifests itself at the production wellbore at solvent breakthrough, precipitation can occur anywhere in the reservoir.

Asphaltene precipitation also may occur during solvent injection into heavy-oil reservoirs. Butler and Mokrys proposed an in-situ solvent-extraction process for heavy oils and tar sands called VAPEX. This process uses two horizontal wells (one injector and one producer). The injection of solvent (e.g., propane) creates a solvent chamber in which oil is mobilized and drained toward the producer. In addition to the mobilization process, the solvent may induce asphaltene precipitation, which provides an in-situ upgrading of the oil.

Asphaltene Precipitation and Deposition. Sec. 9.2.2 discusses the chemistry of asphaltenes and nature of the thermodynamic equilibrium of asphaltenes in petroleum fluids. Changes in pressure, temperature, and composition may alter the initial equilibrium state and cause asphaltene precipitation.

The region in which precipitation occurs is bounded by the asphaltene precipitation envelope (APE). Fig. 9.1 shows a typical pressure-composition APE and a pressure-temperature APE. The APEs also are referred to as asphaltene deposition envelopes. In this chapter, precipitation refers to the formation of the asphaltene precipitate as a result of thermodynamic equilibrium and deposition refers to the settling of the precipitated asphaltene onto the rock surface in a porous medium. The onset conditions correspond to points on the APE. Within the APE, the amount of precipitated asphaltene increases as pressure decreases from the upper onset pressure to the saturation pressure of the oil. The precipitation reaches a maximum value at the saturation pressure and decreases as pressure decreases below the saturation pressure.

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  Fig. 9.1 – Pressure-composition and pressure-temperature APEs (after Leontaritis).

Inside the reservoir, after precipitation has occurred, the asphaltene precipitate can remain in suspension and flow within the oil phase or can deposit onto the rock surface. The main deposition mechanisms are adsorption and mechanical entrapment. The deposited asphaltene may plug the formation and alter rock wettability from water-wet to oil-wet.

Wax-Precipitation Behavior

Wax components can precipitate from petroleum fluids when the original equilibrium conditions of the reservoir are changed so that the solubility of the waxes is reduced; however, wax precipitation does not necessarily lead to deposition. Individual wax crystals tend to disperse in the fluid instead of depositing on a surface. If the number of wax crystals becomes large enough or if other nucleating materials such as asphaltenes, formation fines, clay, or corrosion products are present, the crystals may agglomerate into larger particles. These larger particles then may separate out of the fluid and form solid deposits.

Fig. 9.2 shows a typical wax-precipitation envelope on a pressure-temperature diagram. In contrast to the APE, the solid/liquid-phase boundary is nearly vertical for waxes, illustrating wax precipitation’s strong dependence on temperature and weak dependence on pressure.

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  Fig. 9.2 – Pressure-temperature wax precipitation envelopes (after Leontaritis).

Temperature reduction is the most common cause of wax deposition because wax solubility in hydrocarbon fluids decreases as the temperature is lowered. Reservoir fluid cooling occurs throughout the producing fluid system. Cooling can be caused by oil and gas expansion at the formation face, through casing perforations, or through other orifices or restrictions; by dissolved gas being liberated from solution; by radiation of heat from the fluid to the surrounding formation as it flows up the wellbore; by transfer of the fluid through low-temperature surface facilities; and by injection of water or other fluids at temperatures below the reservoir temperature.

Pressure changes usually have a very small effect on wax-precipitation temperatures and amounts; however, changes in the original equilibrium composition of the fluids can result in a loss of wax solubility. A fairly consistent trend is that the lightest components in a crude oil act as good solvents for waxes. Liberation of solution gas from a crude oil as pressure decreases below the bubblepoint of the fluid has been shown to increase the cloud-point temperature of the oil. This effect also has been observed in synthetic mixtures of methane, decane, and heavy n-alkanes with carbon numbers from 18 to 30 and for stock-tank oils mixed with methane and carbon dioxide. This trend has been shown to be reversed in a study of two gas-condensate fluids in which the cloud-point temperature decreases as pressure is reduced below the vapor/liquid-phase boundary and may increase only at very low pressures. The addition of intermediate paraffinic, naphthenic, and aromatic components with carbon numbers from 5 to 10 has been shown experimentally to decrease the cloud-point temperature for two crude oils. Some model predictions contradict these results, indicating an increase in cloud-point temperature when pentane, hexane, or nonane were mixed with stock-tank oils.

Characteristics of Asphaltenes and Waxes

Chemical Classification of Petroleum Fluids

Petroleum-reservoir fluids are complex multicomponent mixtures. The chemical constituents of petroleum may be classified broadly as belonging either to the C_6- or the C₆₊ fraction. The light end, or C_6- fraction, of petroleum fluids is composed of well-defined pure hydrocarbon components with carbon numbers up to 5 and the light gases N₂, CO₂ and H₂S. The hydrocarbons in the light end primarily are straight-chain normal alkanes (n-alkanes) and their branched isomers (i-alkanes). The heavy end, or C₆₊ fraction, consists of all the components with carbon numbers of 6 or greater.

Classification of Petroleum Constituents. A classification system and nomenclature commonly used in the petroleum industry describes components as belonging to the paraffinic (P), naphthenic (N), or aromatic (A) fractions. These are often referred to jointly as PNA.

Paraffins. This class includes n-alkanes and i-alkanes that consist of chains of hydrocarbon segments (-CH_2-, -CH₃) connected by single bonds. Methane (CH₄) is the simplest paraffin and the most common compound in petroleum-reservoir fluids. The majority of components present in solid-wax deposits are high-molecular-weight paraffins.

Naphthenes. This class includes the cycloalkanes, which are hydrocarbons similar to paraffins but contain one or more cyclic structures. The elements of the cyclic structures are joined by single bonds. Naphthenes make up a large part of microcrystalline waxes.

Aromatics. This class includes all compounds that contain one or more ring structures similar to benzene (C₆H₆). The carbon atoms in the ring structure are connected by six identical bonds that are intermediate between single and double bonds, which are referred to as hybrid bonds, aromatic double bonds, or benzene bonds.

Resins and Asphaltenes. Resins and asphaltenes primarily are a subclass of the aromatics, although some resins may contain only naphthenic rings. They are large molecules consisting primarily of hydrogen and carbon, with one to three sulfur, oxygen, or nitrogen atoms per molecule. The basic structure is composed of rings, mainly aromatics, with three to ten or more rings per molecule.

SARA Classification of Petroleum Constituents. The components of the heavy fraction of a petroleum fluid can be separated into four groups: saturates, aromatics, resins, and asphaltenes (SARA).

• Saturates include all hydrocarbon components with saturated (single-bonded) carbon atoms. These are the n-alkanes, i-alkanes, and cycloalkanes (naphthenes).
• Aromatics include benzene and all the derivatives composed of one or more benzene rings.
• Resins are components with a highly polar end group and long alkane tails. The polar end group is composed of aromatic and naphthenic rings and often contains heteroatoms such as oxygen, sulfur, and nitrogen. Pure resins are heavy liquids or sticky solids.
• Asphaltenes are large highly polar components made up of condensed aromatic and naphthenic rings, which also contain heteroatoms. Pure asphaltenes are black, nonvolatile powders.

The experimental method used to determine the weight fractions of these groups is called SARA analysis.

Asphaltene Characteristics

Nature of Asphaltenes. Asphaltenes are a solubility class that is soluble in light aromatics such as benzene and toluene but is insoluble in lighter paraffins. They normally are classified by the particular paraffin used to precipitate them from crude (e.g., n-pentane or n-heptane). Fig. 9.3 from Mitchell and Speight shows that different alkane solvents yield different amounts of precipitates. Fig. 9.4 from Speight et al. shows dependence of the aromacity (hydrogen/carbon atomic ratio) and molecular weight of asphaltene on the precipitating solvent. These figures also indicate that the amounts and natures of asphaltenes precipitated with n-heptane or heavier alkanes are very similar. Speight, Long and Trowbridge provides a summary of standard analytical methods for asphaltene separation with either n-pentane or n-heptane.

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  Fig. 9.3 – Relation of amount precipitated to carbon-numbering nonaromatic solvents. (Reprinted from Fuel, Vol. 52, D.L. Mitchell and J.G. Speight, “The Solubility of Asphaltenes in Hydrocarbon Solvents,” pages 149-152, Copyright 1973, with permission from Elsevier Science.)

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  Fig. 9.4 – Relationship of asphaltene aromaticity to carbon number of the paraffin. (Reprinted from Fuel, J.G. Speight, R.B. Long, and T.D. Trowbridge, “Factors Influencing the Separation of Asphaltenes From Heavy Petroleum Feedstocks,” pages 616-620, Copyright 1984, with permission from Elsevier Science.)

Although the exact nature of the original state of equilibrium of asphaltenes in petroleum fluids is still under investigation, one characteristic is the tendency of asphaltenes to form aggregates in hydrocarbon solutions. These aggregates are called micelles. The micelles and the hydrocarbon medium form a colloidal system. One commonly held view is that the colloids are stabilized by resins adsorbed on their surface, and the dispersion of colloids in the fluid form a two-phase system. Fig. 9.5 from Leontaritis schematically shows asphaltene-resin micelles that are suspended in the oil. Colloids also may be solvated by the surrounding medium, forming a true single-phase solution. Thermodynamic models (e.g., the solubility-parameter model of Hirschberg et al.) inherently assume the single-phase view. The role of resins in the single-phase or two-phase solution models may be quite different. Changes in pressure, temperature, and composition may alter the solubility parameter of the oil and/or the asphaltene-resin association and cause precipitation.

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  Fig. 9.5 – Asphaltene-resin micelles.

The definition of asphaltenes as compounds that are soluble in aromatics such as toluene and insoluble in light alkanes are referred to as laboratory asphaltenes by Joshi et al. Asphaltenes that precipitate in the field from a depressurization process are called field asphaltenes and contain different constituents. Laboratory and field precipitates contain combinations of asphaltenes and resins. Speight referred to them as asphalts, but that distinction is not made here.

Stability of Asphaltenic Crudes. SARA ratios play an important role in the solubility of asphaltenes. Avila et al. performed SARA analyses on 30 Venezuelan oil samples and attempted to associate the SARA contents with asphaltene precipitation observed in the field. Fig. 9.6 shows the SARA contents of crude oils that experience asphaltene precipitation in the field and those that do not. Crude oils with a high content of saturates and low contents of aromatics and resins clearly are more prone to asphaltene precipitation.

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  Fig. 9.6 – SARA effect on the stability of several crude oils regarding asphaltene precipitation.

Correlation for Asphaltene Precipitation With Alkanes. Asphaltene precipitation at laboratory and field conditions can be predicted with thermodynamic models. Sec. 9.4 discusses this topic in detail. For precipitation with alkanes at atmospheric conditions, a simple correlation from Rassamdana et al. and Sahimi et al. can be used.

Fig. 9.7 from Sahimi et al. shows the experimental weight percents of precipitated asphaltene, W (g of precipitated asphaltene/g of crude oil × 100%), as a function of the solvent to crude oil ratio, R (cm³ of solvent /g of crude oil), from precipitation experiments of an Iranian crude oil with n-C₅, n-C₆, n-C₇, n-C₈, and n-C₁₀ at 26°C and atmospheric pressure. As expected, the amount of precipitates decreases with increasing solvent carbon number. Rassamdana et al. and Sahimi et al. found that the experimental points in Fig. 9.7 could be collapsed onto a scaling curve of Y vs. X with

....................(9.1)

and ....................(9.2)

where M_a is the molecular weight of the alkane solvent. Fig. 9.8 shows the resulting scaling curve. This curve can be represented accurately by a cubic order polynomial:

....................(9.3)

The critical solvent ratio, R_c, where precipitation starts to occur obeys the correlation

....................(9.4)

The factor 0.275 corresponds to a temperature of 26°C. For other temperatures, the following correlation is proposed.

....................(9.5)

where T_c is a temperature-dependent parameter.

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  Fig. 9.7 – Experimental data for the weight percent, W, of precipitated asphaltene as a function of the solvent to crude oil ratio, R, in cm³/g. The results are, from top to bottom, for n-C₅, n-C₇, n-C₈, and n-C₁₀ as the precipitating agent.

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  Fig. 9.8 – Scaling curve for data in Fig. 9.7.

Characteristics of Petroleum Waxes

This section discusses the phase behavior and properties of wax-forming components, primarily normal alkanes, relevant to understanding and modeling wax-phase behavior.

Types of Petroleum Waxes. Petroleum waxes are complex mixtures of n-alkanes, i-alkanes, and cycloalkanes with carbon numbers ranging approximately from 18 to 65. The minimum energy-chain structure of alkanes is a flat zig-zag of carbon atoms with the hydrogen atoms located in planes passing through the carbon atoms perpendicular to the chain axes. Fig. 9.9 shows this structure schematically for typical petroleum-wax components.

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  Fig. 9.9 – Example structure of wax-forming components. (Reprinted from Asphaltenes and Asphalts, 2, C. Lira-Galeana and A. Hammami, “Wax Precipitation from Petroleum Fluids: A Review,” pages 557-608, Copyright 2000, with permission from Elsevier Science.)

There are two general classes of petroleum waxes. Waxes composed primarily of normal alkanes crystallize in large flat plates (macrocrystalline structures) and are referred to as paraffin waxes. Waxes composed primarily of cycloalkanes and i-alkanes crystallize as small needle structures and are referred to as microcrystalline waxes. Table 9.1 shows a comparison of the properties of paraffin and microcrystalline waxes as given by Gilby. Musser and Kilpatrick isolated waxes from sixteen different crude oils and found that paraffinic waxes had molecular weight ranges of 350 to 600, while microcrystalline waxes had large molecular weight ranges of 300 to 2,500. Of the 16 oils analyzed, five exhibited microcrystalline wax deposition, six precipitated paraffinic waxes, and the remaining five showed a mixture of paraffinic and microcrystalline waxes.

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  Table 9.1

In addition to the possibility of precipitating mixtures of the two different types of waxes, the crystal structures in solid-wax deposits will be malformed to some degree because of the complex precipitation environment encountered in petroleum production. Crystal imperfections may occur when the temperature of the solution is decreased rapidly or when heavy aromatic components of the oil are incorporated into the lattice structure. The presence of molecules that hinder the lattice formation result in a wax phase composed of many small, independent crystal lattices.

Precipitation of Petroleum Waxes. Solid-wax formation consists of two distinct stages: nucleation and crystal growth. As the temperature of a liquid solution is lowered to the wax-appearance temperature (WAT), the wax molecules form clusters. Wax molecules continue to attach and detach from these clusters until they reach a critical size and become stable. These clusters are called nuclei and the process of cluster formation is called nucleation. Once the nuclei are formed and the temperature remains below the WAT, the crystal-growth process occurs as further molecules are laid down in a lamellar or plate-like structure.

Nucleation is described as either homogeneous or heterogeneous. Homogeneous nucleation occurs in liquids that are not contaminated with other nucleating materials. In this case, the development of nucleation sites is time dependent. Heterogeneous nucleation occurs when there is a distribution of nucleating material throughout the liquid. If there is sufficient nucleating material, heterogeneous nucleation can be nearly instantaneous. Pure hydrocarbon mixtures in laboratories rarely undergo heterogeneous nucleation, whereas crude oil in the reservoir and production tubing will most likely nucleate this way because of the presence of asphaltenes, formation fines, clay, and corrosion products.

Solidification Behavior of Normal Alkanes. Turner reviewed the properties of normal alkanes found in petroleum waxes, including solid-phase transitions, crystal structures, and phase behavior of binary mixtures. Fig. 9.10 shows experimental data and correlation predictions for normal alkane-melting temperatures at atmospheric pressure as a function of carbon number. In addition to the solid/liquid-phase transition indicated in this figure, many normal alkanes undergo solid/solid-phase transitions within a few degrees below the melting point.

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  Fig. 9.10 - Experimental and correlated normal melting points of n-alkanes.

Normal alkanes can assume four different crystal structures: hexagonal, orthorhombic, triclinic, and monoclinic. For normal alkanes with odd carbon numbers from 11 to 43 and even carbon numbers from 22 to 42, the crystal structure formed on cooling from a melt is hexagonal. This structure has a high degree of molecular-rotational freedom and is characteristically plastic and translucent. All the other crystal structures are restricted rotationally, resulting in a hard deposit and opaque appearance. The even-carbon-number alkanes from 12 to 20 form a triclinic structure on cooling from the melt, whereas all alkanes with carbon numbers 43 or greater form an orthorhombic structure on cooling from the melt. This is also the stable low-temperature form of the alkanes with odd carbon numbers less than 43, which is achieved by further cooling from the hexagonal structure. The monoclinic structure is never attained directly from the melt but is assumed by the even-carbon-number alkanes on cooling from the hexagonal or orthorhombic structures.

Solidification Behavior of Alkane Mixtures. Binary mixtures of wax-forming n-alkanes are completely miscible in the liquid state. In general, these binary mixtures form continuous-solid solutions if both molecules are similar in form and dimension and exhibit the same crystal structure in their pure state. Practically, this means that single-phase-solid solutions form when the molecular length difference is less than 6%. For n-alkanes with carbon numbers 18 to 35, the critical length difference is 2 to 6 carbon atoms. The behavior of binary mixtures depends on whether the constituents are both odd-numbered alkanes, both even-numbered alkanes, or a mixture of odd- and even-numbered alkanes because of the different pure component crystal structures.

The solid-phase behavior of binary mixtures also has been observed to be time and temperature dependent. Dorset shows that some mixtures, such as C₃₀ with C₃₆, form metastable continuous-solid solutions that separate into eutectics with complete fractionation of the constituents over a period of days. Other mixtures, such as C₃₀ with C₄₀, show complete immiscibility immediately on cooling.

For binary mixtures that form continuous-solid solutions, the stable low-temperature configuration is an orthorhombic structure, which is slightly different from the pure component orthorhombic crystal. This occurs for systems in which one alkane is contaminated with even 1 or 2% neighboring alkanes. This same structure has been observed for synthetic ternary and higher mixtures, as well as for diesel fuels. The diesel fuels exhibited an amorphous (microcrystalline) solid phase in addition to the orthorhombic macrocrystalline phase. Pedersen et al. and Hansen et al. also noted the probable existence of solid/solid-phase transitions with variations in temperature in their studies on a number of North Sea crude oils.

In contrast with the phase separations observed in binary mixtures of alkanes with significant length differences, Dirand et al. and Chevallier et al. found that commercial paraffin waxes with continuous distributions of 20 to 33 consecutive n-alkanes formed single-phase orthorhombic-solid solutions at room temperature. The wax deposit from one crude oil also showed the same single-phase macrocrystalline structure; however, an amorphous solid was also present. Increasing the temperature of the commercial waxes to their melting points of 55 to 65°C showed the existence of several different two-phase solid domains for these mixtures.

Significance of Experimental Solidification Behavior for Model Development. As indicated in the previous discussion, solidification behavior of petroleum-mixture components can range from the relatively simple crystallization of pure n-alkanes into well-defined solid structures to the very complex precipitation of solids from live reservoir fluids into multiphase microcrystalline and imperfect macrocrystalline domains. Development of thermodynamic models for predicting the equilibrium-phase behavior of solid waxes depends on which phenomena are to be modeled and on the availability of experimental data for estimating parameters and testing models. The determination of the properties and phase behavior of solid waxes is an area of active research.

The simplest models are written for a single-component single-phase solid. Models of this type may be applied to pure-component-solidification cases or as an approximation in which a multicomponent wax is treated as one lumped component. More common is the solid-solution model in which a single-phase multicomponent-solid deposit is assumed. Some researchers have extended the experimental evidence of immiscible pure solid phases for binary mixtures to the multicomponent case. Lira-Galeana et al. proposed a multisolid wax model in which the solid deposit is assumed to consist of a number of immiscible solid phases, each of which is composed of a single pure component. Generally, the solid deposit is considered to be made up of a number of multicomponent phases, as in the work of Coutinho.

The experimental work discussed generally supports the assumption of multiple solid phases, although Dirand et al. and Chevallier et al. have shown that commercial waxes with a large number of consecutive n-alkanes can form a single multicomponent solid solution at room temperature. As Sec. 9.8 discusses, the models currently available are able to operate in predictive mode for some well-defined systems, but reservoir-fluid modeling still relies heavily on the availability of experimental data.

Experimental Measurements of Asphaltene Precipitation

Measurements of APE

As previously discussed, the APE defines the region in which asphaltene precipitation occurs. Accurate measurements of the APE and the amounts of precipitate within the APE are required for design purposes and for tuning existing models. The upper pressure on the APE is denoted by p_Au and the lower pressure on the APE is denoted by p_Aℓ. Several techniques are available for determining the onset of precipitation with various degrees of accuracy.

Gravimetric Technique. This technique is conducted in a conventional pressure/volume/temperature (PVT) cell. For a pressure below the p_Au, precipitation occurs and larger particles segregate and settle at the bottom of the cell because of gravity. Asphaltene analysis (titration with n-pentane or n-heptane) of the oil shows a decrease in asphaltene content compared with the original oil. Pressure steps must be chosen carefully to capture the inflection point at p_Au and p_Aℓ.

Acoustic-Resonance Technique. The acoustic-resonance technique has been used effectively to define p_Au. The live oil is charged at a high pressure (e.g., 8,500 psia) into a resonator cell maintained at the reservoir temperature. The resonator pressure then is decreased at a very low rate (e.g., 50 psia/min) by changing the volume. The depressurization rate decreases with time to a typical rate of 5 psia/min toward the end of the experiment. Acoustic data exhibit sharp changes at p_Au and at the oil saturation pressure, p_s.

Light-Scattering Technique. Light-scattering techniques also have been successfully used to measure the APE. For dark-colored oil, a near-infrared laser light system (800×10^-9 m to 2200×10^-9 m wavelength) is required to detect asphaltene-precipitation conditions. The principle behind the measurements is based on the transmittance of a laser light through the test fluid in a high-pressure, high-temperature visual PVT cell undergoing pressure, temperature, and composition changes. A receiver captures the amount of light that passes through the oil sample. The power of transmitted light (PTL) is inversely proportional to the oil mass density, to the particle size of the precipitate, and to the number of particles per unit volume of fluid. The PTL curve exhibits sharp jumps at p_Au, p_s, and p_Aℓ.

Flirtation Technique. In this method, the cell contents during a depressurization test are mixed in a magnetic mixer, and small amounts of the well-mixed reservoir fluid are removed through a hydrophobic filter at various pressures. The material retained on the filter is analyzed for SARA contents.

Electrical-Conductance Technique. This technique measures the change in the fluid conductivity with changes in concentration and mobility of charged components. Asphaltenes have large dipole moments, and, therefore, the conductivity curve exhibits a change in the slope when precipitation occurs.

Viscometric Technique. The key point of this method is the detection of a marked change in the viscosity curve at the onset of precipitation because the viscosity of oil with suspended solids is higher than that of the oil itself.

Other Techniques. Asphaltene precipitation has been detected through visual observations with a microscope. Measurements of interfacial tension between oil and water also can be used to detect the onset. A technique based on pressure-drop measurements across a capillary tube was discussed by Broseta et al.

Comparison of Different Methods. Fig. 9.11 shows the results of Jamaluddin et al.‘s comprehensive comparison of measurements with the gravimetric, acoustic-resonance, light-scattering, and filtration techniques on the same oil. These methods, except for the acoustic-resonance technique, determine both the upper and lower APE pressure. The acoustic-resonance technique normally provides only the upper onset pressure. In addition to APE pressures, the gravimetric and filtration techniques also give the amount of precipitated asphaltene within the precipitation region. The gravimetric and filtration techniques are more time consuming than the acoustic-resonance and light-scattering techniques. Fotland et al. showed that the electrical-conductance technique can determine both precipitation onset and amounts of precipitate that are consistent with the gravimetric technique. The advantage of the viscometric technique is in its applicability to heavy crude oil, which may give some difficulties to light-scattering techniques, and in the low-cost equipment. In many cases, two measurement techniques are applied to the same oil to enhance data interpretation. MacMillan et al. recommended the combination of light-scattering and electrical-conductance techniques, while Jamaluddin et al. suggested the simultaneous application of light-scattering and filtration techniques.

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  Fig. 9.11 – APE pressure measurements with different methods (after Jamaluddin et al.).

Reversibility

The reversibility of asphaltene precipitation is a subject of some controversy. Fotland and Wang et al. suggested that asphaltene precipitation is less likely to be reversible for crude oils subjected to conditions beyond those of the precipitation onset. Hirschberg et al. speculated that asphaltene precipitation is reversible but that the dissolution process is very slow. Hammami et al. reported experimental measurements that seem to support this conjecture. They observed that asphaltene is generally reversible but that the kinetics of the redissolution vary significantly depending on the physical state of the system. Hammami et al. shows the laser-power signal (light-scattering technique) from a depressurizing and repressurizing experiment on a light oil that exhibits strong precipitation behavior. [Refer to the source for the figure. Figure is shown in printed version; ACS did not provide permission for its use in PetroWiki.] The laser-power signal increased linearly as the pressure decreased from 76 to 56 MPa. This increase results from the continuous decrease of oil density above the bubblepoint as the pressure is reduced. With further depletion between 56 and 52 MPa, a large drop (one order of magnitude) in the laser-power signal occurred. The onset of asphaltene precipitation was estimated to be 55.7 MPa and the laser-power signal dropped to a very low level at 45 MPa. The bubblepoint pressure for this oil is 33.5 MPa. On repressurization of this oil from 27 MPa (7 MPa below the bubblepoint), almost the entire laser-power signal was recovered, but the signal followed a slightly different curve. The figure shows that the repressurization laser-power curve lags the depressurization curve, which is an indication that the kinetics of redissolution is slower than the kinetics of precipitation. The figure also shows that the ultimate laser-power value reached from repressurization is higher than the predepletion value. Hammami et al. suggested that a large fraction of the precipitated asphaltene (the suspended solid) could easily go back into solution while a smaller fraction exhibits partial irreversibility or slow dissolution rate. The oil at the end of the repressurization process is partially deasphalted and is slightly lighter that the original oil.

Joshi et al. performed further experiments to study the reversibility process. Their results corroborate the observations of Hammami et al. for depressurization and repressurization experiments at field conditions; however, they observed that the precipitation caused by the addition of alkane at atmospheric conditions is partially irreversible. They explained that asphaltene precipitation with pressure depletion at field conditions (field asphaltenes) results from the destabilization but not the destruction of asphaltene micelles. On the other hand, asphaltene precipitation caused by the addition of an alkane solvent in the laboratory under atmospheric conditions (laboratory asphaltenes) strips the asphaltene micelles of their resin components, and the restoration of reformed micelles is a very difficult process.

Similar experimental results on partial irreversibility were obtained by Rassamdana et al. with an Iranian oil and different alkane solvents at atmospheric conditions. The laboratory asphaltenes from Joshi et al. and Rassamdana et al. were precipitated from light oils. Peramanu et al. performed precipitation experiments with Athabasca and Cold Lake bitumens and n-heptane solvent and found the process completely reversible. It could be argued that in heavy oils and bitumens, larger amounts of resins and asphaltenes facilitate the reversibility of asphaltene precipitation with alkane solvents; thus, precipitation behaviors for light oils and heavy oils/bitumens are quite different and need to be examined separately.

Many of the references in Secs. 9.3.1 and 9.3.2 contain data on precipitation caused by pressure depletion. Additional data can be found in Hirschberg et al. and Burke et al. Asphaltene precipitation also occurs during rich-gas and CO₂-flooding processes. Sec. 9.3.3 discusses the experimental results on these processes.

Hirschberg et al. presented static precipitation data of a recombined crude oil with the separator gas, a lean gas, and a rich gas. The results show that precipitation is more pronounced with rich gas and that the injection of separator gas could induce asphaltene precipitation at reservoir conditions. Burke et al. reported comprehensive static precipitation data for six recombined reservoir oils and different hydrocarbon gases. Their results indicate that precipitation depends on the composition of the crude oil, the added solvent, and the concentration of asphaltene in the crude. They also observed that for oil/solvent mixtures that exhibit a critical point on the p-x diagram, maximum precipitation occurred at the critical point.

Monger and Fu and Monger and Trujillo provided extensive data on asphaltene precipitation in CO₂ flooding. In Monger and Trujillo, 17 stock-tank oils with gravity ranging from 19.5 to 46.5°API were used in a variable-volume circulating cell that could reproduce multiple-contact experiments. The temperature was set to 114°F (319 K) and run pressures were set above the minimum miscibility pressures. Fig. 9.13 shows the amounts of precipitation induced by CO₂ in the variable-volume circulating cell vs. the n-C₅ asphaltene content of the stock-tank oil. This figure shows that the CO₂-induced precipitate is not the same as the n-pentane precipitate from the stock-tank oil. For Samples 1, 2, and 8, the extent of precipitation is substantially less than the asphaltene content. For Samples 3 and 9, the extent of precipitation exceeds the asphaltene content. They concluded that the precipitation of asphaltene by CO₂ was neither complete nor exclusive. Some asphaltenes can remain suspended, and other heavy organic compounds can precipitate. It also was observed that precipitation usually occurs in the development of miscibility. Srivastava et al. studied asphaltene precipitation for Saskatchewan Weyburn’s oil with CO₂ and found that precipitation started to occur at 42 mol% CO₂ concentration in a static test. After that, there was a linear increase in asphaltene precipitate with CO₂.

• 

  Fig. 9.13 – Relationship between CO₂-induced solid precipitation an n-C₅ asphaltene content in stock-tank oil.

It has been reported that asphaltene precipitation from static tests may be quite different from dynamic tests. Parra-Ramirez et al. performed static and multiple-contact precipitation experiments with a crude oil from the Rangely field and CO₂. They observed that live oils yielded significantly higher amounts of precipitates than the corresponding dead oil and that multiple-contact experiments gave rise to more precipitation than single-contact experiments.

This discussion shows that field asphaltene precipitates resulting from a rich-gas or CO₂-injection process are different from laboratory asphaltenes induced by the addition of alkane. This field asphaltene is also different from the field asphaltene resulting from pressure depletion, and its nature also varies with the composition of the injection fluid.

Thermodynamic Models for Asphaltene Precipitation

Thermodynamic Equilibrium

Thermodynamic models for predicting asphaltene-precipitation behavior fall into two general categories: activity models and equation-of-state (EOS) models. 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,

....................(9.6)

where μ_io and μ is are the chemical potential of component i in the oil and solid phases, respectively, and n_c 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. 9.6 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. 9.6 yields

....................(9.7)

where = standard state fugacity of component i in phase k (k = o, s), v_ik = partial molar volume of component i in phase k (k = o, s), y_ik = mole fraction of component i in phase k (k = o, s), and γ_ik = activity coefficient of component i in phase k (k = o, s).

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.

....................(9.8)

with ....................(9.9)

where subscripts a, o, and m are used to denote the asphaltene component, the deasphalted oil, and the oil phase mixture, respectively, and where v_a = molar volume of pure asphaltene, v_m = molar volume of mixture, δ_i = solubility parameter of component i, δ_m = solubility parameter of mixture, Φ_a = volume fraction of asphaltene in the mixture, μ_am = chemical potential of asphaltene in the mixture, and = reference chemical potential of asphaltene component.

Because the precipitated asphaltene is pure asphaltene, μ_s = . From the equality of chemical potential μ_am = μ_s, Eq. 9.8 gives

....................(9.10)

The molar volume, v_m, 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, δ_m, is calculated from

....................(9.11)

where ΔU_v 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, v_a, and the solubility parameter of asphaltene, δ_a, 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. 9.10 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

....................(9.12)

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, Novosad and Costain, Nor-Azian and Adewumi, and 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. 9.10 follows the curve shown in Fig. 9.14. 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.

• 

  Fig. 9.14 – Asphaltene solubility for a light crude oil.

de Boer et al. calculated the solubility of asphaltene with Eq. 9.10 for different values of in-situ crude oil densities and asphaltene-solubility parameters. They also introduced a maximum supersaturation at bubblepoint defined as

....................(9.13)

where p_r and p_b are, respectively, the reservoir pressure and the bubblepoint pressure at the reservoir temperature. Fig. 9.15 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.

• 

  Fig. 9.15 – Maximum supersaturation of asphaltene at saturation pressure.

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. 9.7 and then used with an EOS in a three-phase oil/gas/solid flash calculation. The solid/liquid K values are defined as

....................(9.14)

Eq. 9.7 gives

....................(9.15)

with ....................(9.16)

Eq. 9.16 is equivalent to:

....................(9.17)

where T_if = fusion temperature of component i, Δ C_pi = C_po,i – C_ps,i, heat capacity change of fusion, and ΔH_if = heat of fusion of component i. ΔC_pi is assumed to be independent of temperature in Eq. 9.17.

Starting with Eq. 9.17, 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 ΔC_pi and the integral involving Δv_i are negligible and used regular solution theory to calculate the activity coefficients in Eq. 9.17 as follows.

....................(9.18)

....................(9.19)

....................(9.20)

where δ_ik is the solubility parameter of component i in phase k (k = o, s), v_ik is the molar volume of component i in phase k, and Φ_ik is the volume fraction of component i in phase k. Won gave solubility parameter values, δ_io and δ_is, for normal paraffins up to C₄₀. Correlations also are provided to calculate ΔH_if, v_io, and v_is. 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. 9.17 instead of neglecting the terms involving ΔC_pi and Δv_i as Won did. They also included additional multiplication factors to the different terms in Eq. 9.17 to facilitate phase-behavior matching.

Hansen et al. and Yarranton and Masliyah used the Flory-Huggins model to calculate the activity coefficients in Eq. 9.17. 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.

EOS 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

....................(9.21)

where f_s = solid fugacity, = reference solid fugacity, p = pressure, p* = reference pressure, R = gas constant, v_s = solid molar volume, and T = temperature. The following fugacity-equality equations are solved to obtain oil/gas/solid equilibrium.

....................(9.22a)

and ....................(9.22b)

The oil and gas fugacities, f_io and f_ig, for component i are calculated from an EOS. In Eq. 9.22b, subscript a denotes the asphaltene component in solution. Normally, this asphaltene component is the heaviest and last component of the oil (i.e., a = n_c). The following simple stability test can be used to determine whether there is asphaltene precipitation: if f_ao ≥ f_s, asphaltene precipitation occurs, and if f_ao < f_s, 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) and 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 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 9.2 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 (C_32A+) and a precipitating component (C_32B+). 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 C_32B+ is called the precipitating component, the amount that precipitates is governed by Eq. 9.21. Normally, only a portion of the total amount of C_32B+ 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 9.2 are calculated as described in Li et al. The interaction coefficients are calculated from

....................(9.23)

where d_ij = the interaction coefficient between component i and j, v_ci = the critical volume of component i, and e = an adjustable parameter. A value of e(C_32A+) = 0.84 and a value of e(C_32B+) = 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.

• 

  Table 9.2

Fig. 9.16 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. 9.16 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 in Sec. 9.3.

• 

  Fig. 9.16 – PX asphaltene precipitation and saturation curves for a North Sea fluid at 90°C.

For nonisothermal conditions, Eq. 9.24 can be used to calculate the solid fugacity at (p, T) from the solid fugacity at a reference condition (p*, T*).

....................(9.24)

where f_ℓ = fugacity of the asphaltene component in the pure liquid state, T_f = melting point temperature, v_ℓ = molar volume of liquid, ΔC_p = heat capacity of fusion, and ΔH_f = enthalpy of fusion.

Kohse et al. used Eq. 9.24 to model the precipitation behavior of a crude oil with changes in pressure and temperature. Fig. 9.17 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.

• 

  Fig. 9.17 – PT asphaltene and saturation curves for a South American fluid.

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, and composition. From a mechanistic point of view, the nonprecipitating component can be related to resins, asphaltene/resin micelles that do not dissociate, and 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. 9.22a and 9.22b. 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-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

....................(9.25)

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

....................(9.26)

which is analogous to Eq. 9.10 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. 9.26 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 Φ_r is calculated. This value of Φ_r is compared with a critical resin concentration, Φ_cr, which is given as a function of pressure, temperature, molar volume, and solubility parameters. Φ_cr is the key parameter of the model. If Φ_r > Φ_cr, the system is stable and no precipitation occurs. If Φ_r ≤ Φ_cr, 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. 9.18 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 an EOS for the asphalt-free oil species, activity models for the monomeric asphaltenes and resins, and the Gibbs free energy of formation of the micelle, . 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,

....................(9.27)

then is minimized with respect to n₁ = number of asphaltene molecules in the micellar core, n₂ = number of resin molecules in the micellar cell, D = shell thickness of the micelle, = number of asphaltene monomers in liquid phase L₁), = number of resin monomers in phase = number of micelles in phase = number of asphaltene monomers in precipitated phase L₂, and = number of resin monomers in phase L₂. The minimization requires a robust numerical procedure.

• 

  Fig. 9.19 – Schematic representation of crude oil with asphaltene micelles.

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. 9.19 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.

• 

  Fig. 9.19 – Experimental and calculated precipitation with thermodynamics-micellization model and solid model of Weyburn reservoir fluid and CO₂ mixtures at 160 bar and 332 K. Experimental data from Srivastava et al.

Asphaltene Deposition and Plugging

Outline

The measurements and modeling of the phase behavior aspect of asphaltene precipitation were described in the previous sections. After precipitation, asphaltene can remain as a suspended solid in the oil or deposit onto the rock. Here, the term precipitation corresponds to the formation of a solid phase from thermodynamic equilibrium and deposition means the settling of solid particles onto the rock surface. Deposition will induce alteration of wettability (from water-wet to oil-wet) of the rock and plugging of the formation. These aspects have been known for a long time and are the subject of many recent investigations. This section reviews the investigations and laboratory observations of these aspects.

Experiments

Measurements of the deposition and plugging effects were performed by Piro et al. in sand packs and by Turta et al., Minssieux, and Ali and Islam in cores to study asphaltene deposition and the subsequent effect of permeability reduction. Yeh et al., Kamath et al., and Yan et al. performed core displacements to investigate the effect of wettability alteration caused by deposition and its subsequent effect on the recovery. The following sections discuss the results from these experiments with techniques for modeling the observed phenomena.

Deposition and Plugging

Asphaltene deposition in porous media exhibits similarities with the deposition of fines. The main phenomena are adsorption, surface deposition, and plugging deposition.

Piro et al. used asphaltene precipitates collected from a field in northern Italy or induced by diluting two crude oils with n-heptane. The diluted mixture of crude oil with a given concentration of precipitate was injected into sand packs, and the concentration of asphaltene precipitate at the outlet was measured. The deposited amounts were calculated by difference.

Minssieux performed comprehensive core experiments for three crude oils from different parts of the world (France, North Africa, and North America) and four types of cores (three sandstone cores with different permeabilities and clay contents and a core from the Algerian Hassi Messaoud field, which suffers strong asphaltene-precipitation problems). Asphaltene precipitates were obtained by diluting crude oils with n-heptane. Pressure drops across the core were measured to determine the permeability reduction caused by asphaltene deposition. The amounts of deposited asphaltene along the core were estimated with a pyrolysis technique.

Ali and Islam performed core tests with crude oils from the United Arab Emirates. Crude oil with 3 wt% of asphaltene precipitate (induced by n-heptane) was injected into carbonate cores at four different rates. The pressure drops across the core were measured to determine the permeability reduction.

Turta et al. performed high-pressure core-displacement experiments with propane. Crude oils from west-central and northwestern Alberta were used. Asphaltene precipitation occurred within the core when propane mixed with the oil in the displacement process. Permeability reduction was inferred by measuring pressure drops across the core.

Adsorption. The first step in the deposition is the adsorption of asphaltene onto the rock surface. The adsorption of asphaltene onto different rocks has been measured extensively in static experiments that showed that the asphaltene adsorption onto different rocks can be modeled with Langmuir isotherms. Fig. 9.20 from Dubey and Waxman shows typical Langmuir isotherms for asphaltene adsorption on different rocks. The Langmuir isotherm equation is

....................(9.28)

where C_sf = concentration of suspended solid in the oil phase, w_sa = mass of adsorbed asphaltene per mass of rock, (w_sa)_max = maximum adsorbed mass fraction (the plateau in Fig. 9.20), and K_a = ratio of rate constants of the adsorption/desorption reactions. Adsorption is higher for rock containing a higher content of shales. Because adsorption is a surface phenomenon, its main effect is the alteration of the rock wettability from water-wet to oil-wet.

• 

  Fig. 9.20 – Langmuir isotherms for asphaltene adsorption.

General Deposition Process. In addition to adsorption, Minssieux showed that deposition occurs because of mechanical entrapment similar to the deposition of fines in porous media. Pressure drops across the core were measured for several experiments to assess the deposition and plugging effects caused by asphaltene. Minssieux reported that the most noticeable plugging occurred in sandstones containing clays and in tight sandstones. Fig. 9.21 shows the reduction of oil permeability as a function of pore volume injected for sandstones with and without clay. Fig. 9.22 shows the permeability reduction for tight sandstone. Minssieux also used the pore-blocking model of Wojtanowicz et al. to analyze the experimental results.

• 

  Fig. 9.21 – Clay influence on plugging evolution.

• 

  Fig. 9.22 – Plugging evolution in a tight sandstone.

Ali and Islam combined a model for adsorption with the model of Gruesbeck and Collins for the entrainment and deposition of fines in porous media to analyze their experimental results. Gruesbeck and Collins assumed that the porous medium could be divided into two parallel pathways: small pore sizes, in which plug-type deposits occur and can eventually be plugged completely, and larger pore sizes, in which surface nonplugging deposits occur. Particles could be mobilized from the surface deposits if the fluid velocity exceeds a critical value. Fig. 9.23 illustrates this concept.

• 

  Fig. 9.23 – Parallel pathway model for deposition.

For nonpluggable pathways,

....................(9.29)

whereas for pluggable pathways,

....................(9.30)

where C_a = concentration of precipitated asphaltene in weight percent, u_c = critical speed required to mobilize surface deposit asphaltene, u_np = fluid velocity in nonpluggable pathways, σ_np = volume fraction of deposited asphaltene in nonpluggable pathway, σ_p = volume fraction of deposited asphaltene in pluggable pathway, and α, β, χ, and γ = model parameters.

Gruesbeck and Collins gave empirical correlations for calculating u_p and u_np from u, as well as the permeabilities of pluggable and nonpluggable pathways as functions of the volumes of deposited asphaltene. Eq. 9.29 implies that the deposited asphaltene in nonpluggable pathways is mobilized if the velocity, u_np, is greater than the critical velocity, u_c. Ali and Islam developed a 1D, single-phase flow simulator with the Gruesbeck and Collins deposition model. They identified three regimes for asphaltene deposition and plugging depending on the flow rate: monotonous steady state, quasisteady state, and continuous plugging. Fig. 9.24 shows the experimental results and the match obtained with the model described in Eqs. 9.29 and 9.30. At low flow rates (monotonous steady-state regime), the permeability reduction took place in a monotonous fashion. At intermediate flow rates (quasisteady-state regime), initial reduction in permeability was observed until a minimum was reached. After reaching this minimal value, the trend was reversed with an increase in permeability. Ali and Islam attributed this increase to the mobilization of asphaltene deposited in nonpluggable pathways. At higher flow rates (continuous-plugging regime), the permeability reduction began late in the injection process but was very rapid once begun.

• 

  Fig. 9.24 – Effect of flow rates on permeability reduction.

Wang and Civan modified the Gruesbeck and Collins model to obtain

....................(9.31)

where E_a = volume of deposited asphaltene per bulk volume of rock, v_c = critical interstitial velocity for surface deposition, v_o = interstitial oil velocity ( = u_o/Φ), and α, β, γ, η are model parameters. The separation of pathways into pluggable and nonpluggable has been eliminated. The last term in Eq. 9.31 represents the plugging deposit and is set to zero if the average pore throat diameter is greater than a critical pore throat diameter (i.e., there is no plugging deposit if the pore throat is large).

The porosity occupied by the fluid is

....................(9.32)

where Φ₀ is the initial porosity. The reduction in permeability is calculated from

....................(9.33)

where k₀ is the initial permeability and f_p is the porous medium particle transport efficiency factor. Wang and Civan developed a 1D, three-phase, four-pseudocomponent simulator that incorporates the previous deposition and plugging model. They showed that their model could match some of the core deposition experiments by Minssieux and Ali and Islam.

Kocabas and Islam extended the model of Ali and Islam to the analysis of deposition and plugging in the near-wellbore region. Leontaritis also developed a single-phase radial model to analyze the near-well pressure behavior when asphaltene deposition and plugging occur. Ring et al. described a three-component, thermal reservoir simulator for the deposition of waxes in which only surface deposition is considered. Nghiem et al. have incorporated in a 3D compositional simulator both a thermodynamic single-component solid model for asphaltene precipitation and a deposition model based on adsorption and plugging deposit. A resistance factor approach was used to model permeability reduction caused by asphaltene deposition. Qin et al. proposed a method for compositional simulation based on similar approaches.

Wettability Alteration

The alteration of formation wettability caused by asphaltene deposition has been the subject of numerous investigations. Asphaltene adsorption onto the rock surface is the main factor for wettability alteration from water-wet to oil-wet. Collins and Melrose, Kamath et al., Clementz, Crocker and Marchin, and Buckley et al. described the change of formation wettability from water-wet to mixed-wet or oil-wet on adsorption of asphaltene onto the rock surface. Clementz discussed the permanent alteration of core properties after asphaltene adsorption. Collins and Melrose showed that asphaltene adsorption is reduced but not eliminated by the presence of water films on water-wet rock. Crocker and Marchin and Buckley et al. studied asphaltene adsorption for different oil compositions and the corresponding degree of wettability alteration. Yan et al. performed injection of asphaltenes (obtained for diluting crude oils from Wyoming and Prudhoe Bay with n-hexane) into Berea core. After the displacements, imbibition tests were performed to determine changes in core wettability. They showed that the amount of adsorbed asphaltene is dependent on the ions present in the brines (in this case Na⁺, Ca²⁺, and Al³⁺) and that adsorption increases with an increase in ion valency. The highest adsorption occurred with Al³⁺ in the brine. Significant changes in wettability of the sandstone core were observed after asphaltene adsorption.

Morrow reviewed the effect of wettability on oil recovery. Wettability has been shown to affect relative permeabilities, irreducible water saturation, residual oil saturation, capillary pressures, dispersion, and electrical properties. The alteration of relative permeabilities and endpoints has the strongest influence on displacement processes. Morrow reviewed results for core waterfloods showing that the shift toward a less water-wet condition can range from being highly adverse to highly beneficial to oil recovery. Huang and Holm, Lin and Huang, and Yeh et al. presented results on the implication of wettability changes on water-alternating-gas (WAG) processes. Typical results for CO₂ WAG processes indicate that the amount of oil trapped in water-wet cores (45%) was much higher than that trapped in either mixed-wet (15 to 20%) or oil-wet cores (5%).

Yeh et al. performed experiments in a capillary-tube visual cell showing the change in wettability on asphaltene precipitation by mixing a west Texas oil with CO₂ and a Canadian Mitsue crude oil with hydrocarbon gas at reservoir conditions. They also carried out WAG coreflood experiments under reservoir conditions in which asphaltene precipitation occurred. The residual oil saturation after each flood was measured and compared with the value obtained in displacements with refined oils in which there were essentially no changes in wettability. For some experiments, they observed substantial reduction in residual oil saturations when wettability was altered. A wettability change from water-wet to oil-wet conditions increases the contact between oil and solvent and is responsible for a decrease in residual oil saturation.

Kamath et al. performed injection of a precipitating solvent (n-pentane or n-heptane) in cores saturated with crude oil. The plugging caused by asphaltene was assessed by measuring pressure drops across the cores. After the injection of solvent, water was injected and recovery and relative permeabilities were measured to study the effect of deposition on displacement efficiency. Three cores were used. Core 1 is a Berea sandstone core with permeability of 236 md and porosity of 27.9%. Cores 2 and 3 are unconsolidated sandpack cores with permeability of 2380 and 1520 md and porosity of 32.7 and 31.3%, respectively. Fig. 9.25 shows the reduction in permeability with respect to the degree of asphaltene deposition. As expected, permeability reduction was highest for the least permeable core (Core 1) and smallest for the most permeable core (Core 2). Fig. 9.26 shows cumulative fractional recovery for Core 1 vs. pore volume of water injected for various degrees of asphaltene deposition. The results show an improved displacement efficiency with an increase in the deposited amounts. Similar results were obtained for Cores 2 and 3. Kamath et al. concluded from their experiments that although deposition causes permeability reduction, it may improve the sweep efficiency through the alteration of relative permeability curves and flow-diverting effects. Shedid performed similar displacement experiments on low-permeability carbonate cores instead of sandstone cores and observed substantial permeability damage with deposition.

• 

  Fig. 9.25 – Permeability reduction for different cores.

• 

  Fig. 9.26 – Cumulative fractional-oil recovery vs. PV water injected for various degree of asphaltene deposition.

The wettability alteration caused by asphaltene deposition is a complex process that is still a subject of many investigations. The degree of wettability change may not be uniform, as discussed in Al-Maamari and Buckley. The subsequent effect of wettability on relative permeabilities and oil recovery is also a complex subject. There are still unexplored areas, and the whole process is not completely understood at this time. Although the change from water-wet to oil-wet conditions caused by asphaltene precipitation may favor sweep efficiency of waterflood or WAG processes inside the reservoir, the plugging effect near the wellbore remains detrimental to oil production. Inside the reservoir, fluids can find their way around regions of deposition, but, around the wellbore, plugging will prevent flow of oil from converging to the wellbore. Remedial actions then are required to increase production.

Remedial Treatment for Asphaltene Precipitation

Asphaltene precipitation and its subsequent deposition in the wellbore and near-well region are detrimental to oil production. The most effective preventive method is to operate at conditions outside the APE. This is not always possible because of the large drawdown in the vicinity of the wellbore, which lowers the reservoir pressure below the onset pressure. For precipitation in the wellbore, mechanical methods, such as rod and wireline scrapers, can be used to remove asphaltene deposits. Although these methods provide good cleaning and minimal formation damage, their application is limited to the wellbore and does not resolve the problem associated with near-wellbore formation plugging.

Because the solubility of asphaltene increases with an increase in aromatic contents, solvents such as xylene and toluene commonly are used to dissolve asphaltene deposits in both the wellbore and formation. Stricter regulations governing disposals, volatile-emission limits, and flammability concerns have made the use of xylene and toluene less attractive, and alternate solvents have been investigated. Cosolvents for asphaltene removal also have been studied. Cosolvents are xylene-enriched materials with water-wetting properties that use moderate-length carbon-chain alcohols. Production restoration is comparable to that obtained with xylene, but the treatment lasts longer (average of 6 to 8 months). Polymeric dispersants also have been used as alternatives to aromatic solvents. These dispersants inhibit the deposition of asphaltene by breaking the precipitate into smaller particle sizes, which can remain in suspension in the oil phase. Solubility-parameter models have been used to evaluate and screen solvents and inhibitors. Jamaluddin et al. performed experiments that showed that deasphaltened oil is a strong solvent for asphaltene because of its native resin and aromatic contents; however, the cost of producing large amounts of deasphaltened oil to be used as solvent is not viable economically.

Experimental Analysis of Wax Characteristics of Petroleum Fluids

There are a number of experimental measurements performed on petroleum fluids to define their tendency to precipitate wax. Measurements of the temperature at which wax precipitation occurs and the amount of wax precipitated are done with stabilized (stock tank) oils and live reservoir fluids. Compositional analysis of the fluids is performed to determine the concentrations of chemical species that can precipitate as waxes. This section describes these types of analyses.

Compositional Analysis of Petroleum Fluids

As discussed in Sec. 9.2, petroleum constituents may be broadly classified as belonging to the C_6- or the C₆₊ fraction. The heavy end may be further classified with SARA analysis. Various chromatography methods allow the determination of the mass fractions of single carbon number (SCN) fractions of a fluid. One SCN is composed of all the components with boiling points between consecutive n-alkane boiling points. For example, the C₇ SCN is composed of all the components with boiling points between the boiling point of n-C₇ and n-C₈. These analyses routinely extend up to carbon number 30 and may be done up to a carbon number of 45 or more.

Detailed PNA analyses also can be performed. Depending on the details of the analysis, the aromatic fraction may or may not include the resins and asphaltenes. It is also possible to determine the amounts of individual n-alkanes. These types of analyses, although expensive, are especially valuable for wax-precipitation modeling because they very accurately define the components of a fluid that will precipitate as wax.

Measurement of Wax-Precipitation Data

There are a few basic measurements that characterize a fluid’s tendency to precipitate wax. Lira-Galeana and Hammami reviewed the experimental techniques used to obtain these measurements.

Wax-Appearance Temperature or Cloud Point. When a liquid solution or melt is lowered to the WAT, the wax molecules form clusters of aligned chains. Once these nuclei reach a critical size, they become stable and further attachment of molecules leads to growth of the crystal. Formation of these nuclei causes the fluid to take on a cloudy appearance, hence the name cloud point. This also is referred to as the wax-crystallization temperature or wax-appearance point. Determination of a WAT significantly higher than the temperatures expected to be encountered during production indicates the potential for wax-deposition problems.

The WAT depends on which technique is used for the analysis. For example, a microscopy method allows for observation of much smaller wax crystals than a visual technique with the unaided eye. The following techniques are used to determine the WAT.

• American Soc. for Testing and Materials (ASTM) visual methods. Oil in a glass jar is submerged in a cooling bath. As the temperature of the bath is lowered, the temperature at which the fluid’s cloudiness is first observed is determined to be the cloud point.
• Cold finger. A temperature-controlled rod is inserted in a gently heated oil sample. The WAT is determined as the temperature at which wax begins to adhere to the rod.
• Viscometry methods. Viscometric techniques rely on detection of changes in rheological properties of an oil as wax precipitates. A break in the curve of viscosity plotted vs. temperature is taken as the WAT.
• Differential-scanning calorimetry. This method detects the latent heat of fusion released on crystallization. Although there can be some uncertainty in interpretation of the results, differential-scanning calorimetry has been widely used for WAT determination and also can provide data on the heat capacities and heats of fusion or transition associated with liquid/solid and solid/solid phase transitions.
• Cross-polarized microscopy. In this technique, a microscope with a temperature-controlled "hot stage" is used to view an oil sample that is being cooled at a constant rate. The use of a polarized light source and polarized objectives on the microscope allow the wax crystals to show up as bright spots on a black background. This technique usually provides the highest WAT value for dead oils.
• Light transmittance. The experimental apparatus for this method consists of a PVT cell with a light source and a light power receiver mounted on opposite sides of the cell. When wax crystals appear in the fluid, the amount of light transmitted is reduced dramatically, and the WAT can be seen as a sharp drop in a plot of light power received vs. temperature. This method can be used at high pressure and, therefore, can be applied to live reservoir fluids as well as stock-tank oils.
• Ultrasonics. Similar to the light-transmittance technique, an ultrasonic signal is sent through the fluid sample and received at a transducer. The velocity of the ultrasonic wave depends on the density of the medium; thus, the transit time for the wave will change at the WAT.

Wax-Dissolution Temperature. The wax-dissolution temperature is defined as the temperature at which all precipitated wax has been dissolved on heating the oil. The experimental techniques most often used for determining wax-dissolution temperature are differential-scanning calorimetry and cross-polar microscopy.

Pour-Point Temperature. The pour-point temperature is the lowest temperature at which the oil is mobile. This is usually identified as the stock-tank-oil gelation temperature. The ASTM pour-point test, similar to the ASTM cloud point tests, involves placing a sample of the fluid in a jar and cooling it in a temperature-controlled bath. At each 3°C temperature step, the sample is tested by tipping the jar to determine if the oil is still mobile.

Quantification of Wax Precipitation. None of the tests used to determine the WAT provide data on the amount of solid precipitated at a temperature below the WAT. Experimental techniques to determine the amount of precipitated wax are described next.

Bulk-Filtration Apparatus. In this simple experiment, oil in a cylinder is equilibrated at the desired conditions of pressure, temperature, and, possibly, solvent concentration. The entire contents of the cylinder, including oil and any solids that may have precipitated, are ejected through a filter. The solids collected in the filter then may be analyzed for amount and chemical composition. This technique is time consuming and expensive but has the advantage of providing samples of the precipitated solid for analysis.

Pulsed Nuclear Magnetic Resonance (NMR). Pedersen et al. used an NMR apparatus to determine the amount of precipitated solids as a function of temperature for 17 crude oils. The experimental NMR signals for each oil were compared with calibrated samples of polyethylene in wax-free oil. Although this technique does not allow for chemical analysis of the deposited solids, results are obtained much more quickly than with the bulk-filtration apparatus.

Thermodynamic Models for Wax Precipitation

The thermodynamic basis of solid/liquid equilibrium of components in a melt or dissolved in a solution is well established and is described in many standard texts (e.g., Prausnitz et al.). The basic principles continue to be applied to more complex systems as researchers attempt to develop more accurate models of solid-wax precipitation. Lira-Galeana and Hammami reviewed experimental techniques and thermodynamic models for studying wax precipitation in petroleum fluids.

The predictive capability of the thermodynamic models is affected both by the form and assumptions of the models themselves and the characterization procedures used to quantify the number and properties of wax-forming components present in a fluid. This section presents a general form of the thermodynamic relation used to define the K values for solid and liquid phases in equilibrium, and the effect of different simplifying assumptions and thermodynamic descriptions of the phases involved on the model results are examined.

Thermodynamics of Solid/Liquid Equilibrium

Thermodynamic Equilibrium. Thermodynamic models for predicting wax precipitation may be derived assuming single-component or multicomponent, single-phase or multiphase solid deposits. Regardless of which set of assumptions is chosen, the condition of thermodynamic equilibrium between phases is expressed as the equality of chemical potential for each component in all phases. For one solid phase in equilibrium with an oil, this condition is given by

....................(9.34)

where μ_io and μ_is are the chemical potentials of component i in the oil and solid phases, respectively, and n_c is the number of components. With the fundamental relation between chemical potential and fugacity of component i (n_c) ,

....................(9.35)

the equilibrium relation also may be expressed in terms of fugacities:

....................(9.36)

where f_io and f_is are the fugacities of component i in the oil and solid phases, respectively.

Calculation of Pure Solid Component Fugacity. CalculationEOSs are not available to describe the volumetric behavior of the solid phase as a general function of temperature and pressure; therefore, thermodynamic solid-precipitation models are derived by relating the chemical potential of a pure solid to the chemical potential of the pure liquid at the same pressure and temperature in terms of experimentally known melting properties. Derivation of this expression is discussed in standard thermodynamics texts such as Prausnitz et al. The most general form of this relationship, including multiple solid-phase transitions, is

....................(9.37)

where

μ_pi,k = chemical potential of pure component i in phase k (k = o, s),

ΔH_if = enthalpy of fusion of component i,

T_if = temperature of fusion (melting temperature) of component i,

n_tr = number of solid state transitions,

ΔH_ij,tr = enthalpy of the jth solid state transition of component i,

T_ij,tr = jth solid state transition temperature of component i,

ΔC_pi = (C_Po,i - C_Ps,i), heat capacity of fusion of component i,

P_if = pressure of fusion (corresponding to T_if) of component i,

ΔC_pij,tr = heat capacity of jth solid state transition of component i, and

Δv_i = (v_o,i-v_s,i), change of molar volume caused by fusion of component i.

In the majority of wax-precipitation models, multiple-solid-state transitions are not considered, or the effects are lumped into the enthalpy of fusion and heat capacity of fusion terms. Removing these terms and applying the relation between chemical potential and fugacity given in Eq. 9.35, Eq. 9.37 can be written in terms of fugacities as

....................(9.38)

where is the fugacity of pure component i in phase state k (k = o, s).

K-Value Equations. Eq. 9.38 may be used directly to determine pure-solid-component fugacities, or it may be combined with activity- or fugacity-coefficient models to derive expressions for solid/liquid K values. This section gives the fundamental forms of these K-value equations. These equations then are used with various assumptions to perform solid/liquid or solid/liquid/vapor equilibrium calculations.

Activity-Coefficient Models. Activity coefficients can be defined in terms of fugacities as

....................(9.39)

where γ_i = activity coefficient of component i in a mixture, f_i = fugacity of component i in the mixture, x_i = mole fraction of component i in the mixture, and = standard state fugacity of component i. The standard state fugacity is the fugacity of component i in the same state and at the same temperature as the mixture and at an arbitrarily chosen pressure and composition. If the activity coefficients are defined with reference to an ideal solution in the sense of Raoult’s law, then the pressure is chosen as the system pressure and the composition is chosen as pure component i. The development of the equations presented here uses this definition.

An expression for solid/liquid K values in terms of activity coefficients can be derived with the use of the definition of Eq. 9.39 as

....................(9.40)

where K_is = solid/liquid K value for component i, x_ik = mole fraction of component i in phase k (k = o, s), γ_ik = activity coefficient of component i in phase k (k = o, s), and = fugacity of pure component i in phase k (k = o, s). For use with activity-coefficient models, the condition of equilibrium between the solid and liquid phases given in Eq. 9.36 can be substituted into Eq. 9.38 to yield the following relation in terms of pure component fugacities.

....................(9.41)

Substituting Eq. 9.41 into Eq. 9.40 then gives the general relationship for solid/liquid K values in terms of activity coefficients and melting properties:

....................(9.42)

For use with EOSs, it is convenient to write the solid/liquid K-value equation in terms of the wax-melting properties and fugacity coefficients, as opposed to activity coefficients. Fugacity coefficients are defined as

....................(9.43)

....................(9.44)

where = fugacity coefficient of pure component i in phase state k and Φ_ik = fugacity coefficient of component i in phase k. Substituting the fugacity-coefficient definitions, Eq. 9.38 can be rearranged to give the solid/liquid K-value expression

....................(9.45)

The use of the fugacity coefficient as defined in Eq. 9.44 for the liquid phase and the activity coefficient as defined in Eq. 9.39 for the solid phase leads to the following equation for the solid/liquid K values when the equality of fugacity condition is applied.

....................(9.46)

This formula is convenient when the fluid-phase fugacities are determined with an EOS and the solid-phase activity coefficient is determined with another model.

Pure Ideal Solid Model

In Eq. 9.42, the last term in the exponential accounting for the difference in molar volume between the solid and liquid as a function of pressure is usually the smallest and is most often neglected. The heat-capacity term is of larger magnitude but also is assumed negligible in many applications. If the nonidealities of the oil and solid phases also are considered to be small (i.e., γ_io/γ_is = 1) and the solid phase is assumed to be a pure component, the equation of ideal solubility results in

....................(9.47)

This equation may be regarded as being based on the Clausius-Clapeyron or van’t Hoff equations.

Reddy reported one application of the ideal solubility equation. Eq. 9.47 was used to determine the cloud points and amounts of precipitated wax for synthetic fuels and diesels. In this case, only n-paraffins were assumed to precipitate. For the synthetic fuels, measured quantities of n-paraffins were combined with a solvent. For the diesel fuels, the amounts of n-paraffins up to C₂₇ were determined experimentally. The ideal solubility equation was used to convert the amounts of all n-paraffins in a system to an equivalent amount of reference paraffin. The solubility behavior of the reference component, n-eicosane, was determined experimentally. The mixtures then were treated as binary solute/solvent systems for computation. The predicted amount of wax precipitated at one temperature below the cloud point is compared with the experimental values in Fig. 9.27. These results illustrate the ability of the ideal solubility equation to correlate correctly experimentally observed trends, provided the distribution of wax-forming components is well defined.

• 

  Fig. 9.27 – Ideal solubility model predictions compared with measured values for composition of wax formed in a synthetic fuel blend. (Reprinted from Fuel, Vol. 65, S.R. Reddy, “A Thermodynamic Model for Predicting n-Paraffin Crystallization in Diesel Fuels,” pages 1647-1652, Copyright 1986, with permission from Elsevier Science).

The ideal solubility equation also was used by Weingarten and Euchner for predicting wax precipitation from live reservoir fluids. Experimental determination of wax-crystallization temperatures (cloud points) for two reservoir fluids was performed at 10 different bubblepoint pressures during differential-liberation experiments. Constants relating the enthalpy of fusion and temperature of fusion were determined by linear regression to the experimental data. Fig. 9.28 shows a comparison of the crystallization temperature predicted by the model to the experimental values. In this implementation, no characterization of the feed is necessary. The precipitated wax is treated as a single component. The model is able to only approximately reproduce an important trend in the data as a function of pressure: at high pressures, at which only small amounts of the lightest gases are being liberated, the crystallization temperature increases slowly with decreasing pressure; at lower pressures, at which more gas and heavier gas components are liberated, the crystallization temperature increases more rapidly with decreasing pressure.

• 

  Fig. 9.28 – Crystallization temperatures fitted to ideal solution theory.

Solid-Solution Models

Wax models describing the precipitated solid as a single-phase multicomponent solution have been used in a large number of studies. The solid phase has most often been modeled as an ideal or regular solution. The fluid phases are modeled with the regular solution theory, Flory-Huggins theory, or EOSs. This section describes some of the variations of the solid-solution model.

Regular Solid-Solution Models. Regular solution theory, as developed by Scatchard and Hildebrand, refers to mixtures with zero-excess entropy provided that there is no volume change of mixing. The Scatchard-Hildebrand equation for activity coefficients is

....................(9.48)

where δ_ik = solubility parameter for pure component i in phase k and = volume fraction average solubility parameter for phase k. The volume fraction average solubility parameter for a phase is given by

....................(9.49)

At conditions far removed from the critical point, the solubility parameter for a component in the oil phase may be expressed in terms of the enthalpy of vaporization and the molar volume of the component.

....................(9.50)

Won proposed a modified regular solution theory in which the solubility parameter for a component in the solid phase is given by

....................(9.51)

With Eq. 9.48 and assuming that v_is = v_io, the activity-coefficient ratio can be described by

....................(9.52)

Substituting Eq. 9.52 into Eq. 9.42 and assuming the pressure and heat-capacity terms are negligible gives the final equation used by Won for the solid/liquid K values as

....................(9.53)

Won also presented correlations for the heat of fusion, temperature of fusion, and molar volume as functions of molecular weight and tabulates values of the solubility parameters for the liquid and solid phases. The correlations are applicable to normal paraffins. The heat of fusion is given by

....................(9.54)

where M_i is the molecular weight of component i. The heat of fusion from Eq. 9.54 is approximately equal to the sum of the heat of fusion and one-half the heat of transition for molecules heavier than C₂₂ and approximately equal to the heat of fusion for odd carbon number molecules lighter than C₂₂. The temperature of fusion is given by

....................(9.55)

and the molar volume is given by

....................(9.56)

In Won’s model, solid/liquid/vapor equilibrium is determined. Liquid/vapor K values are calculated with the Soave-Redlich-Kwong EOS. These K values are used with the solid/liquid K values in a three-phase flash algorithm to determine the solid/liquid/vapor-phase split as a function of temperature and pressure. There is an inconsistency in this technique in that the liquid properties are calculated from an activity-coefficient model for the solid/liquid K values and from an EOS for the vapor/liquid K values. Despite this inconsistency, Won’s technique has some important advantages over the ideal solubility models presented previously. These advantages include accounting for nonidealities in the solid and liquid phases and accounting for the simultaneous effects of pressure, temperature, and vaporization or solution of gas in the liquid on solid precipitation.

Won applied this method to a hydrocarbon gas defined as a mixture of SCN fractions from C₁ to C₄₀. These SCN fractions are assumed to have paraffinic properties as given by Eqs. 9.54 through 9.56. The feed composition is determined by extrapolating the measured mole fractions of C₁₅ through C₁₉. Fig. 9.29 shows the effect of temperature on the molar-phase splits for this feed gas. The cloud-point temperature can be seen as the highest temperature at which the solid phase exists, just below 310°K. The amount of solid increases rapidly as the temperature is decreased below this point. Fig. 9.30 shows the effect of pressure on the phase equilibrium.

• 

  Fig. 9.29 – Effect of temperature on the phase equilibria of a gas condensate predicted with the activity-coefficient model with regular-solution theory. (Reprinted from Fluid Phase Equilibria, Vol. 30, K.W. Won, “Thermodynamics for Solid-Liquid-Vapor Equilibria: Wax Phase Formation From Heavy Hydrocarbon Mixtures,” pages 265-279, Copyright 1986, with permission from Elsevier Science.)

• 

  Fig. 9.30 – Effect of pressure on the phase equilibria of a gas condensate predicted with the activity-coefficient model with regular-solution theory. (Reprinted from Fluid Phase Equilibria, Vol. 30, K.W. Won, “Thermodynamics for Solid-Liquid-Vapor Equilibria: Wax Phase Formation From Heavy Hydrocarbon Mixtures,” pages 265-279, Copyright 1986, with permission from Elsevier Science.)

Regular Solution Theory Model for Liquid Phase. Pedersen et al. use the general form of the solid/liquid K-value relation as given in Eq. 9.42, including the heat-capacity term but neglecting the pressure term. This results in the following equation for the K values:

....................(9.57)

The activity-coefficient ratio is calculated with the regular solution theory (Eq. 9.52), as in Won’s model. Correlations are given for the solubility parameters of paraffins in the oil and solid phases as

....................(9.58)

and ....................(9.59)

where C_i is the carbon number of component i. Won’s correlation for the enthalpy of formation (Eq. 9.55) is modified as

....................(9.60)

and the model is completed by defining a relation for the heat-capacity difference as

....................(9.61)

Constants a₁ through a₅ were determined by a least-squares fit to the data of Pedersen et al. as a₁ = 0.5914 (cal/cm³)^0.5, a₂ = 5.763 (cal/cm³)^0.5, a₃ = 0.5148, a₄ = 0.3033 cal/(g•K), and a₅ = 0.635×10^-4 cal/(g•K²).

The oils were characterized on the basis of experimentally determined SCN fraction distributions. The fractions are subdivided into a paraffinic part and a naphthenic plus aromatic (NA) part. The NA fractions are given solubility parameters 20% higher than those obtained from Eqs. 9.8 and 9.59, while the enthalpy of formation for the NA fractions is set to 50% of the value calculated from Eq. 9.60.

Pedersen et al. compared experimental wax precipitation as a function of temperature with model predictions for 16 crude oils. Only liquid/solid equilibrium was calculated. A figure from that source shows typical predictions illustrating the effect of various model assumptions. [Note: Figure is shown in printed version; ACS did not provide permission for its use in PetroWiki.] The squares indicate the experimental results, while the solid line indicates the full model predictions as given by Eqs. 9.57 through 9.61. The asterisks show the calculation results obtained when the heat-capacity difference is neglected. The triangles show the results of the use of pure component enthalpies of fusion of n-alkanes instead of those obtained with Eq. 9.60, and the crosses show the results of the use of the liquid- and solid-solubility parameters of Won, as opposed to those given by Eqs. 9.58 and 9.59.
Internally Consistent Model With EOS for Fluid Phases. Mei et al. applied the mixed activity/fugacity coefficient model given in Eq. 9.46 with a three-phase flash algorithm, in conjunction with liquid/vapor K values obtained from the Peng-Robinson EOS. As opposed to Won’s model, this form maintains internal consistency with the use of the EOS for all fluid phase calculations and uses regular-solution theory only for the solid solution. The fugacity of the pure solid is calculated with Eq. 9.41, neglecting the pressure effect. Solid-solubility parameters required for regular-solution theory are calculated with a correlation given by Thomas et al. Won’s correlations for enthalpy of fusion, temperature of fusion, and molar volume are used with additional adjustable coefficients. A heat capacity of fusion correlation of the form given by Pedersen completes the model.

Fluids used in the study were characterized on the basis of experimental SCN analysis to C₄₀. No further subdivision of the components into P, N, and A subfractions was performed. A good match to experimental cloud points and wax precipitation amounts as a function of temperature was attained through the adjustment of five correlation coefficients.

Ideal Solid-Solution Models. Applying the assumptions that the solid phase may be considered an ideal solution, the heat capacity terms are negligible, and the pressure terms are negligible, the K-value expression from Eq. 9.42 can be written as

....................(9.62)

Flory-Huggins Model for the Liquid Phase. Flory and Huggins derived expressions for the thermodynamic properties of polymer solutions. A key parameter in determining the properties of these mixtures was found to be the large difference in molecular size between the polymer and the solvent species. The same situation is found in petroleum fluids, in which the large molecules of the heavy end are in solution with much smaller hydrocarbons. Flory-Huggins theory has been applied to asphaltene-precipitation modeling as discussed in Sec. 9.4.

Hansen et al. used the generalized polymer-solution theory given by Flory to derive an expression for the activity coefficient of a component in the liquid phase. Eq. 9.62 then was applied to liquid/solid equilibrium calculations. Characterization of the oils is done on the basis of experimental determination of the SCN fraction distribution to at least C₂₀₊. Each of the SCN fractions then is divided into two subfractions: the aromatic part and the combined paraffinic and naphthenic part. Flory interaction parameters are calculated between the subfractions with a group-contribution method. Although good results were obtained, the resulting expression is complicated and the model has not been used by other researchers.

Ideal Solution Model for the Liquid Phase. Erickson et al. used Eq. 9.62 with the additional assumption that the liquid phase is also an ideal solution. These authors note that the heat of fusion and melting-temperature terms are of much greater importance than the activity-coefficient terms for prediction of liquid/solid equilibria of stabilized liquids, justifying the use of the ideal solubility equation. Won’s correlation for melting temperature as given in Eq. 9.55 is used for n-alkanes. A modification of this expression is used for all other species in the fluid. A single constant multiplying Won’s enthalpy of fusion correlation is used as an adjustable parameter to enable a better fit of the experimental data.

Erickson et al. applied the model to stabilized oils with detailed experimental compositional analysis, which allows a direct determination of the amount of n-alkanes in each SCN fraction up to carbon numbers of 35 or 40. Extrapolation to C₅₀ or higher is then performed. They also apply a "staged" equilibrium flash, which assumes that once a solid forms, it does not remix with additional solid that precipitates at lower temperatures. Fig. 9.32 compares model results with experimental data.

• 

  Fig. 9.32 – Comparison of multicomponent ideal solubility model predictions and experimental wax-precipitation amounts as a function of temperature.

EOS Models for Liquid and Vapor Phases. Brown et al. used a simplification of the fugacity coefficient form of the solid/liquid K-value expression (Eq. 9.45) to study the effects of pressure and light components on wax formation. The assumptions used are that the heat capacity difference is negligible, the solid phase can be considered an ideal solution, Δv_i is constant, and p_if is small compared with p. Applying these conditions leads to the final K-value expression:

....................(9.63)

The melting temperature and heat of fusion terms are calculated with the correlations given by Erickson et al., and the molar-volume difference is correlated as function of molecular weight. Brown et al. used the simplified perturbed-hard-chain theory EOS to calculate the fugacity coefficients. A correlation was developed for binary-interaction parameters of the paraffin components. The fluid-characterization method is the same as that described for the model of Erickson et al.

Model predictions are compared with experimental data in Fig. 9.33 for a live fluid with a bubblepoint of 285 bar. The model predictions show that increasing the pressure from atmospheric (dead) oil causes a decrease in the cloud-point temperature as light ends dissolve in the oil phase. The light ends increase the solubility of heavy-wax components in the oil. A minimum in the cloud point is achieved at the bubblepoint of the oil. Further pressure increase in the single-phase region causes an increase in the cloud-point temperature.

• 

  Fig. 9.33 – Ideal solubility model with SPHCT EOS; effect of pressure on cloud-point temperatures for a live oil.

Pedersen used the fugacity-coefficient model of Eq. 9.63 with the additional simplification that the pressure effects were neglected, resulting in the following expression for the solid/liquid K values:

....................(9.64)

The Soave-Redlich-Kwong EOS is used to determine the fugacity coefficients for liquid and vapor phases. The characterization of the fluid is performed on the basis of a standard extended compositional analysis in which the mole fraction, molecular weight, and density of each SCN fraction is given. Pedersen developed an empirical expression to calculate the mole fraction of the potentially wax-forming part of any SCN fraction. This approach is in contrast to the characterization procedure of Erickson et al., in which all the subfractions of a SCN fraction may potentially enter the wax phase, but the parameters of the nonnormal alkane fractions are defined such that they will enter the solid phase in lesser amounts.

Three adjustable parameters are used in the expression to determine the mole fraction of the wax-forming part of each SCN fraction. Fig. 9.34 shows example results for the model comparing the predicted and experimental amount of wax precipitated as a function of temperature.

• 

  Fig. 9.34 – EOS fluid/ideal solid model: comparison of predicted and experimental wax precipitation amounts as a function of temperature.

Multiple-Pure-Solid-Phase (Multisolid) Models

All the models discussed up to this point treat the deposited wax as a single phase, consisting of either a pure component or a mixture of components as a solid solution. As discussed in Sec. 9.2, experimental work on binary-alkane mixtures shows that the components can separate into two immiscible solid phases, where each phase is essentially a pure component. Solid-phase transitions also are observed in crude oils.

Lira-Galeana et al. developed a thermodynamic model for wax precipitation based on the concept that the precipitated wax is made up of several solid phases, at which each phase consists of a single component or pseudocomponent. From stability considerations, a component may exist as a pure solid if the following inequality is satisfied:

....................(9.65)

The number of solid-forming components and the number of solid phases, n_s , is determined from Eq. 9.65. Once the number of solid phases is known, the phase-equilibrium relationships for vapor, liquid, and solid are given by

....................(9.66)

and ....................(9.67)

Eq. 9.41 is used, neglecting pressure effects, to obtain the pure-solid fugacity. The pure-liquid fugacity is obtained from the Peng-Robinson EOS, as are the component fugacities in the liquid and vapor phases.

In the original multisolid-wax model presented by Lira-Galeana et al., the fluids are characterized by splitting the C₇₊ fraction into 7 to 12 pseudocomponents. No further subdivision of the pseudocomponents into P, N, or A fractions is performed. Instead, melting temperature and enthalpy of fusion correlations are developed to define properties that represent an average of the three subgroups. These correlations weight the aromatic properties more heavily for heavier molecular-weight pseudocomponents. The heat capacity of fusion is given by the correlation of Pedersen et al., as shown in Eq. 9.61. Fig. 9.35 shows experimental data and predicted results of the model.

• 

  Fig. 9.35 - Multisolid-wax model: comparison of predicted and experimental wax-precipitation amounts as a function of temperature. [After Lira-Galeana, C., Firoozabadi, A., and Prausnitz, J.M.: “Thermodynamics of Wax Precipitation in Petroleum Mixtures,” AlChE J. (1996) V. 42, 239. Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1996 AlChE. All rights reserved.]

Pan et al. also have used the multisolid-wax model but with a different fluid-characterization procedure. The characterization is based on experimental SCN analysis. Every five consecutive carbon number fractions are lumped together. The relative amounts of the P, N, and A subfractions are determined experimentally or with correlations. Melting temperature and enthalpy of fusion properties are assigned to the paraffinic subfractions with Won’s correlations. For naphthenes and aromatics, the correlations of Lira-Galeana et al. were used, with the exception of the enthalpy of fusion for aromatics, which was fit with a new correlation independent of the molecular weight. For the heat capacity of fusion, the correlation of Pedersen et al. was used for all components. Results of the model for a synthetic oil at 110 bar are shown in Fig. 9.36, illustrating the reduction in cloud point and also the reduction in amount of wax precipitated with the addition of methane to the system.

• 

  Fig. 9.36 – Multisolid-wax model with PNA characterization: effect of C₁ on amount of precipitated wax.

Multisolid-Wax Model Including Enthalpies of Transition. Nichita et al. used Eq. 9.37 to derive an expression for the ratio between the pure-solid and pure-liquid fugacities including the effect of multiple solid-state transitions. Assumptions used in the derivation are that the heat capacity of fusion is constant, the heat capacities of transition are negligible, the solid/liquid molar-volume difference is constant, and the terms for the enthalpies of transition are all evaluated at the temperature of the first transition. The authors state that this treatment of the enthalpies of transition may lead to relative differences in results of up to 10% compared with lumping the enthalpies of transition in with the enthalpy of fusion. Applying these considerations results in

....................(9.68)

Ungerer et al. derived a similar expression with multiple enthalpies of transition; how-ever, the enthalpy terms are evaluated at the fusion temperature rather than the first transition temperature. The model is applied with a single pure-component-solid phase. Nichita et al. used Eq. 9.68 with the modified multisolid-wax model presented in Pan et al., including the correlations for all component properties except enthalpies of fusion and enthalpies of transition; new correlations are presented for these properties. The model of Nichita et al. is used with the Peng-Robinson EOS to calculate a pressure-temperature phase diagram, shown in Fig. 9.37, for a synthetic fluid with phenanthrene as the precipitating component. The binary-interaction parameter between methane and phenanthrene was adjusted to match the vapor/liquid dewpoint.

• 

  Fig. 9.37 – Multisolid-wax model with multiple-phase transitions: phase diagram for a synthetic gas mixture with phenanthrene.

Excess Gibbs Energy Models

Activity coefficients are related to the partial molar excess Gibbs energy for a component i, , and the total excess Gibbs energy for a phase, G^E, by

....................(9.69)

and ....................(9.70)

Excess-free-energy models thus can be used with the solid/liquid K-value equation expressed in terms of activity coefficients for wax-precipitation modeling.

Coutinho and coworkers published a number of studies on modeling paraffin-wax formation from synthetic and real petroleum fluids. In these works, an equation is used for the pure-component solid to liquid-fugacity ratio similar to that given in Eq. 9.68, with the additional assumption that only a single enthalpy of transition term is used and the pressure effect is neglected. This results in the following solid/liquid K- value expression in terms of activity coefficients:

....................(9.71)

The liquid-phase activity coefficient is given by

....................(9.72)

where the combinatorial free-volume contribution, , is obtained from a Flory free-volume model, and the residual contribution, lnγ^r_io, is obtained from the UNIFAC model, which is based on the universal quasichemical (UNIQUAC) equation. Coutinho et al. contains more detail and references on these models.

Excess Gibbs energy models are used for the solid phase. A modified Wilson’s equation with one adjustable parameter was used initially. Then, a predictive version of the UNIQUAC equation was developed, which incorporates multiple-mixed-solid phases and is used to predict wax formation in jet and diesel fuels. An analysis of the amounts of the individual n-alkanes is required for the fluid characterization. The N and A subfractions of a SCN fraction can be treated separately or lumped as a single pseudocomponent. Coutinho et al. presents model results for the amount of wax precipitated as a function of temperature compared with experimental data for a number of fuels and the change in composition of the solid phase as a function of temperature. The accuracy of the model is very good. [Note: The two figures from Coutinho et al. are shown in the printed version; ACS did not provide permission for their use in PetroWiki.]
Pauly et al. presented further development of the excess Gibbs energy model. In this model, the modified Wilson equation, as given by Coutinho and Stenby, is used for the activity coefficients in the solid phase at atmospheric pressure. The Poynting factor is used to determine the high-pressure solid fugacity from the fugacity determined at atmospheric pressure. The liquid phase is modeled with an EOS/G^E model. This combination of fluid and solid treatments yields good results for prediction of solid/liquid and solid/liquid/vapor phase boundaries up to 200 Mpa for binary and multicomponent systems of n-alkanes. The quality of the predictions is a result of the treatment of the pressure effect on the solid phase and the EOS/G^E model, which guarantees continuity between fugacities of the fluid and solid phases.

Comparison of Models

Pauly et al. compared the models of Won, Pedersen et al., Hansen et al., Coutinho and Stenby, Ungerer et al., and the ideal solution model. The models are tested on systems composed of n-decane and a heavy fraction of normal alkanes from C₁₈ to C₃₀. Fig. 9.40 compares the total amount of solid precipitate as a function of temperature for the models with experimental data. The solid-solution models overpredict the cloud-point temperature and the amount of solid precipitated, while the multisolid model gives better results for the cloud point but underpredicts the amount of wax precipitated, at least for the higher temperature region. Coutinho and Stenby’s model gives a very good match of the data.

• 

  Fig. 9.40 – Amount of solids precipitated as a function of temperature for a mixture of n-alkanes predicted with various models. (Reprinted from Fluid Phase Equilibria, Vol. 149, J. Pauly, C. Dauphin, and J.L. Daridon, “Liquid-Solid Equilibria in a Decane + Multi-Paraffins System,” pages 191-207, Copyright 1998, with permission from Elsevier Science.)

Nichita et al. also compared their modification of the multisolid-wax model with a solid-solution model. The solid phase is assumed to be ideal, and the liquid phase is described by the EOS. Results for the two models are compared in Fig. 9.41 for three synthetic mixtures of n-decane with n-alkanes from C₁₈ to C₃₀. As in the comparison performed by Pauly et al., the solid-solution model overpredicts the cloud-point temperature and the amount of wax precipitated, while the multisolid-wax model gives good estimates of the cloud-point temperature yet underpredicts the amount of wax precipitated.

• 

  Fig. 9.41 – Comparison of multisolid- and solid-solution wax precipitation models.

Wax-Deposition Models

A number of thermodynamic models were described in the previous section to calculate the amount of solid wax precipitated as a function of pressure, temperature, and fluid composition. As discussed in Sec. 9.1, wax precipitation does not necessarily lead to solid deposition. Thermodynamic models for solid/liquid K values have been coupled with models for wax deposition in pipelines. The form of these models is discussed briefly in this section. For deposition to occur in pipelines, the following conditions must be fulfilled.

• The pipeline wall temperature must be below the WAT for the fluid.
• A negative radial temperature gradient must be present in the flow. That is, the wall temperature must be lower than the centerline temperature. A zero gradient implies that no deposition will occur.
• Wall friction must be large enough so that wax crystals can stick to the wall.

Burger et al. investigated the significant physical processes leading to wax deposition in pipelines. These processes are molecular diffusion, Brownian diffusion, shear dispersion, and gravity settling. Brownian movement of small solid-wax crystals will result in diffusion-like transport of these particles when a concentration gradient exists. This effect is normally neglected in pipeline-deposition models. Gravity settling can occur because precipitated wax crystals are denser than the surrounding liquid. Again, this effect is usually neglected in flow models. Molecular diffusion and shear dispersion are described next, assuming that the three deposition conditions have been satisfied.

Molecular Diffusion

Flow in pipes will be laminar or will have a laminar sublayer adjacent to the pipe wall. There will be a temperature gradient across this sublayer with the lower temperature at the pipe wall. When the temperature is below the WAT, the flowing oil will contain precipitated solid wax, which is in equilibrium with the liquid. Because the temperature is colder toward the wall, more of the wax components will exist in the solid phase at equilibrium. This results in a concentration gradient in the liquid phase with a lower concentration of wax-forming components at the pipe wall. Wax molecules will be transported toward the wall by molecular diffusion. Once these molecules reach the solid/liquid interface, they are available to be added to the solid deposit by the mechanisms of crystal growth. The equation describing the rate of mass transport caused by molecular diffusion is

....................(9.73)

where m_i = mass of component i, t = time, ρ_oil = mass density of oil, D_i = effective diffusion coefficient for component i, A = deposition area, w_i = weight fraction of component i, and r = radial distance.

Because the radial-concentration gradient is not readily available, the chain rule is used in Eq. 9.73 to express this as the product of the mass-concentration (weight fraction) gradient with respect to temperature and the temperature gradient. The mass-concentration gradient is derived from the solubility limit as a function of temperature obtained from a thermodynamic model.

Shear Dispersion

When suspended solid particles are being transported in a fluid in the laminar-flow regime, they tend to travel with the mean speed and direction of the fluid. Particles have higher velocities at greater distances from the pipe wall, and the particles also rotate as they flow. These rotating particles will exert drag forces, causing displacement of the flow paths of any neighboring particles. When the particle concentration is high, these interactions result in net transport of particles toward the low-velocity region at the pipe wall.

Considering all the wax-forming components together as a single wax pseudocomponent, the rate of mass transport of wax caused by shear dispersion takes the form

....................(9.74)

where m_w = mass of wax, k* = empirical constant, C_w = concentration of precipitated wax at the wall, and γ = shear rate. The form of this equation shows that the deposition rate increases linearly with increasing shear rate.

Weingarten and Euchner reported results of diffusion and shear-deposition experiments and modeling with Eqs. 9.73 and 9.74. They note that shear rate also has an important effect that is not related to shear transport. Pieces of deposited wax can be dislodged from the pipe wall in a process called sloughing. Sloughing will be dependent on the shear rate, the nature of the deposit, and the nature of the wall surface. Sloughing occurs when the wall shear rate exceeds the shear strength of the deposit and may occur both in the laminar and turbulent flow regimes.

Keating and Wattenbarger also have used the diffusion and shear-deposition equations in conjunction with a wellbore simulator to model wax deposition and removal in wellbores. Wax removal is caused by equilibrium conditions, not explicit modeling of the sloughing process. A study isolating and comparing the relative effects of molecular diffusion and shear dispersion on wax deposition concludes that molecular diffusion is the dominant effect. Majeed et al. obtained good results modeling wax deposition in pipelines considering only the diffusive transport.

A detailed compositional wax-deposition model for pipelines has been derived by combining the differential equations of mass and energy conservation and the laws of diffusion with a thermodynamic model for solid/liquid K values of the form given in Eq. 9.54. These mass and heat-transfer relations also have been applied with the multisolid-wax model by Ramirez-Jaramillo et al.

Prevention and Remediation of Wax Precipitation

Crystallization of waxes in crude oils leads to non-Newtonian flow characteristics, including very high yield stresses that are dependent on time and the shear and temperature histories of the fluid. This crystallization may cause three problems: high viscosity, which leads to pressure losses; high-yield stress for restarting flow; and deposition of wax crystals on surfaces.

Wax-precipitation-induced viscosity increases and wax deposition on pipes are the primary causes of high flowline pressure drops. In turn, these pressure losses lead to low flow rates that make conditions for wax deposition more favorable. In extreme cases, pumping pressure can exceed the limits of the system and stop flow entirely. A related problem is the high-yield stress for restarting flow. When oil is allowed to stand in a pipeline at temperatures below its pour point, a certain pressure is required to break the gel and resume flow. Again, this pressure may be higher than the pressure limits of the pumps and pipelines.

Wax can deposit on surfaces in the production system and in the formation. Wax deposition can be prevented or removed by a number of different methods. These methods fall into three main categories: thermal, chemical, and mechanical.

Thermal

Because precipitation is highly temperature dependent, thermal methods can be highly effective both for preventing and removing wax-precipitation problems. Prevention methods include steam- and electrical-heat tracing of flowlines, in conjunction with thermal insulation. Thermal methods for removing wax deposition include hot oiling and hot watering. Hot-water treatments cannot provide the solvency effects that hot oiling can, so surfactants are often added to aid in dispersion of wax in the water phase. Surfactants are discussed under chemical methods.

Hot oiling is one of the most popular methods of deposited wax removal. Wax is melted and dissolved by hot oil, which allows it to be circulated from the well and the surface-producing system. Hot oil is normally pumped down the casing and up the tubing; however, in flowing wells, the oil may be circulated down the tubing and up the casing. There is evidence that hot oiling can cause permeability damage if melted wax enters the formation.

Higher molecular-weight waxes tend to deposit at the high-temperature bottom end of the well. Lower molecular-weight fractions deposit as the temperature decreases up the wellbore. The upper parts of the well receive the most heat during hot oiling. As the oil proceeds down the well, its temperature decreases and the carrying capacity for wax is diminished. Thus, sufficient oil must be used to dissolve and melt the wax at the necessary depths.

Chemical

The types of chemicals available for paraffin treatment include solvents, crystal modifiers, dispersants, and surfactants.

Solvents can be used to treat deposition in production strings and also may be applied to remediate formation damage. Although chlorinated hydrocarbons are excellent solvents for waxes, they generally are not used because of safety and processing difficulties they create in the produced fluid. Hydrocarbon fluids consisting primarily of normal alkanes such as condensate and diesel oil can be used, provided the deposits have low asphaltene content. Aromatic solvents such as toluene and xylene are good solvents for both waxes and asphaltenes.

Crystal modifiers act at the molecular level to reduce the tendency of wax molecules to network and form lattice structures within the oil. Operating at the molecular level makes them effective in concentrations of parts per million, as opposed to hot oil and solvents, which must be applied in large volumes. Crystal modifiers have relatively high molecular weights to allow them to interact with high-molecular-weight waxes. Because they have high melting points, their use is limited in cold climates.

Dispersants are chemicals that break deposited wax into particles small enough to be reabsorbed into the oil stream. These chemicals are used in low concentrations in aqueous solutions, making them relatively safe and inexpensive.

Surfactants may be used as deposition inhibitors or can act as solubilizing agents for nucleating agents in an oil. Surfactants are not used as generally as the other chemical types.

Mechanical

Scrapers and cutters are used extensively to remove wax deposits from tubing because they can be economical and result in minimal formation damage. Scrapers may be attached to wireline units, or they may be attached to sucker rods to remove wax as the well is pumped. Deposits in surface pipelines can be removed by forcing soluble or insoluble plugs through the lines. Soluble plugs may be composed of naphthalene or microcrystalline wax. Insoluble plugs are made of plastic or hard rubber.

Another method of mechanical intervention to prevent deposition is the use of plastic or coated pipe. Low-friction surfaces make it more difficult for wax crystals to adhere to the pipe walls. Deposition will still occur if conditions are highly favorable for wax precipitation, and deposits will grow at the same rate as for other pipes once an initial layer of material has been laid down; therefore, the pipe and coating system must be capable of withstanding one of the other methods of wax removal.

Nomenclature

a	=	constant
a1-5	=	constants
A	=	deposition area, L2
b	=	constant
Ci	=	carbon number of component i
Ca	=	concentration of precipitated asphaltene in wt %, m/m
Csf	=	concentration of suspended solid in the oil phase [ppm (μg/g)]
Cw	=	concentration of precipitated wax at the wall, m/m
dij	=	interaction coefficient between component i and j
D	=	shell thickness of the micelle, L
Di	=	effective diffusion coefficient for component i, L2/t
e	=	adjustable parameter in Eq. 9.23
Ea	=	volume of deposited asphaltene per bulk volume of rock, L3/L3
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
	=	partial molar excess Gibbs energy for a component i, m/L2t2
k	=	permeability, L2
k*	=	empirical constant for mass transport of wax caused by shear dispersion
k0	=	initial permeability, L2
Ka	=	ratio of rate constants of the adsorption/desorption reactions
Kis	=	solid/liquid K value for component i
mi	=	mass of component i, m
mw	=	mass of wax, m
Ma	=	molecular weight of alkane solvent, m
Mi	=	molecular weight of component i, m
nc	=	number of components
ns	=	number of solid phases
ntr	=	number of solid state transitions
n1	=	number of asphaltene molecules in the micellar core
n2	=	number of resin molecules in the micellar cell
	=	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
pAℓ	=	lower pressure on the APE, m/Lt2
pAu	=	upper pressure on the APE, m/Lt2
pb	=	bubblepoint pressure, m/Lt2
pr	=	reservoir pressure, m/Lt2
ps	=	oil saturation pressure, m/Lt2
Pif	=	pressure of fusion (corresponding to Tif) of component i, m/Lt2
r	=	radial distance, L
R	=	gas constant
R	=	solvent to crude oil ratio
Rc	=	critical solvent ratio
t	=	time, t
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
Tij,tr	=	jth solid state transition temperature of component i, T
Til,tr	=	lth solid state transition temperature of component i, T
uc	=	critical speed required to mobilize surface deposit asphaltene, L/t
unp	=	fluid velocity in nonpluggable pathways, L/t
uo	=	oil velocity, L/t
va	=	molar volume of pure asphaltene, L3/n
vc	=	critical interstitial velocity for surface deposition, L/t
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
vo	=	interstitial oil velocity ( = uo / Φ), L/t
vr	=	molar volume of resins, L3/n
vs	=	solid molar volume, L3/n
wi	=	weight fraction of component i, m/m
wsa	=	mass of adsorbed asphaltene per mass of rock, m/m
(wsa)max	=	maximum adsorbed mass fraction (the plateau in Fig. 9.20), m/m
W	=	weight percent of precipitated asphaltene, m/m
x	=	mole fraction, n/n
xi	=	mole fraction of component i, n/n
xik	=	mole fraction of component i in phase k (k = o, s), n/n
xio	=	mole fraction of component i in oil phase, n/n
xis	=	mole fraction of component i in solid phase, n/n
xjk	=	mole fraction of component j in phase k (k = o, s), n/n
X	=	defined in Eq. 9.1
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
Y	=	defined in Eq. 9.2
α	=	asphaltene-deposition model parameters (Eqs. 9.29 and 9.31)
β	=	asphaltene-deposition model parameters (Eqs. 9.29 and 9.31)
γ	=	shear rate, L/t
γ	=	asphaltene-deposition model parameters (Eqs. 9.30 and 9.31)
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
ΔHij,tr	=	enthalpy of the jth solid state transition of component i, mL2/nt2
ΔHil,tr	=	enthalpy of the lth solid state transition of component i, mL2/nt2
ΔHiv	=	enthalpy of vaporization 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
η	=	asphaltene deposition model parameters (Eq. 9.31)
	=	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
σnp	=	volume fraction of deposited asphaltene in nonpluggable pathway
σp	=	volume fraction of deposited asphaltene in pluggable pathway
Φ	=	porosity
Φ0	=	initial porosity
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
χ	=	asphaltene deposition model parameters (Eq. 9.30)

References

SI Metric Conversion Factors

°API		141.5/(131.5+°API)		=	g/cm3
bar	×	1.0*	E + 05	=	Pa
cal	×	4.184*	E + 03	=	J
ft	×	3.048*	E − 01	=	m
ft3	×	2.831 685	E − 02	=	m3
°F		(°F − 32)/1.8		=	°C
°F		(°F + 459.67)/1.8		=	K
psi	×	6.894 757	E + 00	=	kPa

*

Conversion factor is exact.