SoFA - American Chemical Society

Article pubs.acs.org/IECR

Viscosity Prediction of Waxy Oils: Suspension of Fractal Aggregates (SoFA) Model Thierry Palermo* and Eric Tournis TOTAL Exploration & Production, Avenue Larribau, 64018 Pau Cedex, France S Supporting Information *

ABSTRACT: A new viscosity model, called the Suspension of Fractal Aggregates (or SoFA) model, is presented. It has been developed by considering waxy oil systems as suspensions of wax crystals that can interact and form fractal aggregates whose size is limited by the shear stress τ. The viscosity μ can be expressed as a function of the viscosity of the suspending liquid phase μL and a function of the volume fraction of wax crystals ϕ. The constitutive law has the form μ = μL(1 − Aϕτ−X)/[1 − (Aϕτ−X/ ϕM)2] if τ > τy = (Aϕ/ϕM)1/X, where ϕM is the maximal packing fraction (ϕM = 4/7) and A and X are parameters related to the structure and properties of the aggregates. If τ ≤ τy, then μ = +∞. Application of the SoFA constitutive law to experimental flow curves has shown very good agreement by matching the two model parameters A and X. Good results have also been obtained by using the Herschel−Bulkley, Li and Zhang, and Pedersen and Rønningsen models. The capability of the SoFA model to predict the viscosity of different systems at a given volume fraction of wax crystals ϕ has been successfully investigated. In this case, the nparaffins distribution has been preserved. Only the physical properties of the dispersing liquid phase have been changed. This first result gives us hope that such a model should enable the viscosity of live oil systems (presence of a gas phase) to be predicted. However, to be definitively accepted for live oil application, comparison between SoFA model predictions and viscosity measurements in the presence of a gas phase under pressure will have to be carried out. In the “liquid” state, the waxy oil behavior can be approximated by nonthixotropic shear-thinning models, which are much more simple and suitable for engineering applications. Thus, the viscosity of waxy oils transported in pipelines below the WAT is expected to be evaluated from a limited number of rheological characterizations. Most of viscosity models reported in the literature have been developed by combining theory of dispersion viscosity with constitutive non-Newtonian laws18,19 or by introducing, in dispersion viscosity models, a flocculation factor that depends on the shear rate.20,21 Another approach, based on an extension to non-Newtonian fluids of the Friction Theory (FT) that relates the viscosity of the fluid to its equation of state, has also been reported.22 The objective of this study is to propose a new model for predicting the viscosity of petroleum fluids as waxy crude oils or condensates that can be considered in the “liquid” state for temperatures below the WAT. This model, called the Suspension of Fractal Aggregates (or SoFA) model, is based on the idea that the oil phase can be considered as a suspension of wax crystals that form fractal aggregates whose size is limited by the shear stress. It is very similar to the approach presented by Kane et al.3 to discuss the evolution of the viscosity when wax crystallization occurs under imposed shear rates. Similarly, in other models, such as those developed by Pedersen and Rønningsen18 and by Li and Zhang,20 the viscosity μ is expressed as:

1. INTRODUCTION Below the wax appearance temperature (WAT), n-paraffins contained in petroleum fluids crystallize and generally form discotic particles either isolated or assembled into clusters under flowing conditions.1 In the case of the predominance of noncrystallizable aromatic and isoparaffins and cycloparaffins, small needles and amorphous wax solids are also observed.2 Wax clusters can be viewed as fractal objects.1,3−5 Below the pour point, a gelation process, associated with the buildup of a waxy crystal interlocking network, can take place under quiescent or low shear stress conditions.3,6 Gelled oils can be characterized as colloidal gels1,4 with a fractal structure.3 It has also been proposed that wax−oil networks are structured as an ensemble of close-packed spherical cells filled with oil.7 Because of the presence of wax precipitates, waxy oils undergo strong changes in their rheological properties: from a Newtonian to a shear-thinning fluid and ultimately to an elastoviscoplastic material. Rheological characterization of waxy crude oils has been the object of many studies, with a particular focus on the thixotropic and yielding behavior of gelled oils,3,4,6,8−13 as well as on the impact of the thermomechanical history on the rheological response of the system.3,4,6,9 In order to predict the restart pressure in the case of an oil gelation process in pipelines during shutdown, thixotropic rheological models are required. The time dependency is generally introduced by expressing the rheological parameters (apparent yield stress, plastic viscosity) as functions of time9,13,14 or by incorporating a structure parameter (usually denoted λ) that is dependent on time and shear rate.15−17 Recently, Dimitriou et al.12 proposed a new constitutive model in which a second structure parameter has been introduced to capture the nonisotropic resistance of the material. © 2015 American Chemical Society

Special Issue: Scott Fogler Festschrift Received: Revised: Accepted: Published: 4526

October 23, 2014 February 28, 2015 March 2, 2015 March 2, 2015 DOI: 10.1021/ie504166n Ind. Eng. Chem. Res. 2015, 54, 4526−4534

Article

Industrial & Engineering Chemistry Research

μ = μL (T )f (ϕ)

(1)

where μL is the viscosity of the suspending liquid phase (solvent), which is dependent only on the temperature T (and, to a lesser extent, the pressure P). The term f(ϕ), which is dependent on the shear conditions, is a function of the volume fraction ϕ of wax crystals in the liquid phase. Such an expression should allow us to predict the change in oil viscosity with a change in thermophysical properties of the solvent (viscosity μL, density ρL, and wax solubility curve ϕ(T)), as long as it can be assumed that the microstructure of wax aggregates evolves in the same way with shear and wax crystal concentration. This point has been investigated by modifying the solvent properties of a condensate with the addition of silicone oil.

(7)

BPR C + PR γ̇ γ̇

(8)

The non-shear-dependent parameter APR in eq 8 is expressed as a function of the volume fraction of wax crystals ϕ by using an analogy from the water/oil emulsions:

(2)

APR = μL exp(DPR ϕ)

(9)

The parameters BPR and CPR in eq 8 are assumed to be functions of ϕ. Based on experimental results that represent the shear dependency of the viscosity for high and low solid wax contents, the authors proposed to make the parameters BPR and CPR proportional to μLϕ and μLϕ4, respectively. Finally, the viscosity as a function of both the shear rate and the solid wax fraction is given by: ⎡ E ϕ F ϕ4 ⎤ μ = μL ⎢exp(DPR ϕ) + PR + PR ⎥ γ̇ γ̇ ⎦ ⎣

where γ̇ is the shear rate, and μ0 and μ∞ are the limiting values of viscosity at low and high shear rates, respectively. The term γ̇C is a critical shear rate that is associated with the sudden onset of shear thinning, and n′ is an exponent that characterized the rate of thinning. It should be noted that (i) Rønningsen9 also used the Cross model for the modeling of viscosity data and (ii) Visintin et al.4 used a similar model (RBC model) in which a critical shear stress is used instead of the critical shear rate. Dimitriou et al.11 showed that, beyond the point of yield (shear stress larger than ∼μ0γ̇C), the flow curves can be wellrepresented by the Herschel−Bulkley model (HB model):

(10)

The viscosity of the Newtonian dispersing liquid μL, as well as the solid wax concentration ϕ, can be calculated thanks to viscosity and thermodynamic models. Moreover, by matching eq 10 on 713 measured viscosity data points, numerical value for DPR, EPR, and FPR were given, with an average absolute deviation of 48%: • DPR = 37.82 • EPR = 83.96 • FPR = 8.559 × 106 2.2.3. Ghanaei and Mowla Model (G&M Model). A similar approach to the P&R model was proposed by Ghanaei and Mowla,19 by considering an expression of the viscosity based on the Herschel−Bulkley model, instead of the Casson model:

(4)

where τ is the shear stress, τy the apparent yield stress, k the plastic viscosity, and n the flow index (n ≤ 1). Other models have also been proposed as the power law and Bingham models6,9 that are particular cases of the HB model (τy = 0 and n = 1, respectively), or the Casson model: τ = acas + bcas γ ̇

ln(μ) = A TM + BTM T −1 + γC ̇ TM −0.5 + DTM W

μ = APR +

where μ* is a viscosity constant and θ a temperature (in Kelvin) related to the activation energy. Below the WAT, waxy oils exhibit shear-thinning properties with a more or less pronounced apparent yield stress behavior. According to Dimitriou et al.,11 the variation of the viscosity with applied shear stress can be well-described by the Cross model: μ0 − μ∞ μ = μ∞ + 1 + (γ /̇ γĊ )n ′ (3)

τ = τy + kγ ṅ

(6)

The terms A, B, C, and D (with the subscript “AZ” referring to the Al-Zahrani and Al-Fariss model and the subscript “TM” referring to that of Tiwary and Mehrotra) are constants. The weakness of these models is that the viscosity is related to the wax content and not to the concentration of precipitated wax, which is expected to be more relevant. 2.2.2. Pedersen and Rønningsen Model (P&R Model). Pedersen and Rønningsen18 argued that the non-Newtonian character of waxy oils below the WAT can be well-represented by the Casson law (eq 5). They deduced a general expression for the viscosity:

2. BACKGROUND 2.1. Generalities. Above the WAT, waxy crude oils and condensates are Newtonian and, at a given pressure, the evolution of the viscosity μ with the temperature is given by the Arrhenius equation: ⎛θ⎞ μ = μ* exp⎜ ⎟ ⎝T ⎠

nAZ ⎤1/ nAZ ⎞ ⎛C BAZ ⎡⎛ γ ̇ + AAZ ⎞ ⎢ exp⎜ AZ +DAZW ⎟ μ= ⎜ ⎟ − 1⎥ ⎠ ⎝ ⎢ ⎥ γ ̇ ⎣⎝ AAZ ⎠ T ⎦

μ=

A GM + BGM γ Ċ GM − 1 + μL exp(DGM ) γ̇

(11)

By regressing 162 experimental data, the authors obtained the following relationships for the parameters AGM, BGM, CGM, and DGM, with an average absolute deviation of 18.45%. • AGM = 0.04Cwax exp(1.37849Cwax) • BGM = 43.54937Cwax exp(0.19348Cwax) • CGM = 0.7 • DGM = 0.225Cwax0.93446 The coefficients given above have been obtained with consideration of units of mPa·s for the viscosity and wt % for the wax crystal concentration Cwax.

(5)

Here, acas and bcas are constants. 2.2. Viscosity Models Related to Wax Concentration. 2.2.1. Al-Zahrani−Al-Fariss and Tiwary−Mehrotra Models. By fitting experimental viscosity data obtained for different oils at different temperatures and different wax contents, Al-Zahrani and Al-Fariss,23 as well as Tiwary and Mehrotra,24 expressed the viscosity as a function of the shear rate, temperature T, and mass fraction of wax W. Expressions are, respectively, 4527

DOI: 10.1021/ie504166n Ind. Eng. Chem. Res. 2015, 54, 4526−4534

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According to eq 14, the effective volume fraction ϕeff of the aggregates can be related to the actual volume fraction of wax crystals ϕ through the following relation:

2.2.4. Li and Zhang and Adeyanju and Oyekunle Models. By applying theory of suspension viscosity and by introducing a flocculation factor kLZ that is dependent on the shear rate, Li and Zhang20 proposed an equation of the form: μ = μL [1 − kLZ(γ )̇ ϕ]−2.5

ϕeff =

(12)

No theoretical expression of the flocculation factor as a function of the shear rate has been given by the authors. However, comparison between experimental data and model prediction was performed by fitting kLZ as a power law function of the shear rate. Good results have been obtained at temperatures above the gel point (absolute average deviation of 7.43% on 3458 viscosity data points). On the other hand, as indicated by the authors, deviations were much greater for wax crystal concentrations corresponding to temperatures close to and below the gel point. A same approach has been used by Adeyanju and Oyekunle.21 Only the suspension viscosity law has been changed: −1 ⎡ ⎛ 2.5ϕ − ϕ2 ⎞⎤ μ = μL ⎢1 − kAO(γ )̇ ⎜ ⎟⎥ ⎢⎣ ⎝ (1 − ϕ)2 ⎠⎥⎦

m ⎧ for τ ≤ τ0 ⎪(τ0 / τ ) R wax =⎨ ⎪ a wax for τ > τ0 ⎩1

(16)

τ0 is the critical shear stress below which aggregates can form and depends on the strength of the particle−particle interaction. The exponent m characterizes the breakage mechanism and is dependent on the nature on the particle interactions. Potanin30 distinguished two different cases: “rigid” aggregates and “soft” aggregates for which m is in the range of 0.23−0.29 and 0.4−0.5, respectively. Note that, in the original paper by Snabre and Mills,25 the right-hand term of eq 16 was written as 1 + (τ0/τ)m. We think that there is no physical justification for such an expression. By combining eqs 15 and 16, we obtain the relationship between the effective volume fraction, the actual volume fraction, and the shear stress:

(13)

ϕeff

3. SOFA MODEL The SoFA model has been developed by considering waxy oil systems as suspensions of wax crystals that can interact and form fractal aggregates whose size is limited by the shear stress. The concept of fractal aggregates was already used by Kane et al.3 to discuss the evolution of the viscosity when wax crystallization occurs under imposed shear rates. The model presented below is mainly based on the theoretical approach proposed by Snabre and Mills.25 A consequence of the fractal morphology is that the number of primary particles (monomers) N in an aggregate scales as N ∝ RD, where D is the fractal dimension and (R/a) is a characteristic radius of the aggregate normalized by the monomer radius. To replace the proportionality by an identity, a structure factor S is introduced as N = S(R/a)D.26 Different characteristic aggregate sizes can be considered: radius of gyration, dynamic radius, and geometrical radius.27 For our purposes, only the dynamic radius is relevant. As defined by Gmachowski,28 it is the radius of an impermeable sphere of the same mass having the same dynamic properties. In the case of aggregates composed of spherical primary particles in contact, the author argued that S = 1. Perry et al.29 showed that the morphology of fractal aggregates can be impacted by the monomer shape. Since wax crystals have a rather disklike morphology, we will keep the prefactor in the scaling relationship applied to wax aggregates: Nwax

(15)

Under shear conditions, the size of aggregates is controlled by the shear stress τ:3,25

However, there is no clear evidence of the advantage of this new formulation, compared to the previous one.

⎛ R ⎞D = Swax ⎜ wax ⎟ ⎝ a wax ⎠

⎛ R ⎞3 − D ϕ⎜ wax ⎟ Swax ⎝ a wax ⎠ 1

ϕ

=

(3 − D)m 1 ⎛⎜ τ0 ⎞⎟ = Aτ −X Swax ⎝ τ ⎠

(17)

The exponent X = (3 − D)m is related to the structure of aggregates via the fractal dimension D and to the breakage mechanism of aggregates via m. The parameter A = τ0X/Swax is related to the particle interactions via τ0 and also to the structure and breakage mechanism via Swax and X. Finally, the viscosity can be expressed as a function of the effective volume fraction, according to the equation proposed by Snabre and Mills:25 μ = μL

1 − ϕeff 2

(1 − ) ϕeff

for ϕeff < ϕM

ϕM

(18)

with ϕM being the maximal packing fraction (ϕM = 4/7). Equations 17 and 18 allow us to give a physical meaning of the yield stress τy, as explicitly introduced in the Herschel− Bulkley equation (defined by eq 4). The yield stress corresponds to the shear stress for which the effective volume fraction tends to the maximum packing fraction. We can deduce the following: ⎛ Aϕ ⎞1/ X ⎟⎟ ϕeff → ϕM ⇒ τ → τy ⇒ τy = ⎜⎜ ⎝ ϕM ⎠

(19)

Finally, the general constitutive law can be expressed as: ⎧ 1 − Aϕτ −X ⎪ for τ > τy ⎪ μL 2 −X μ = ⎨ ⎡⎣1 − (Aϕτ /ϕM)⎤⎦ ⎪ ⎪+∞ for τ ≤ τy ⎩

(14)

Nwax is the number of wax crystals in the aggregates, Rwax the dynamic radius of the aggregates, and awax a characteristic size of wax crystals. Both the fractal dimension D and the prefactor Swax give information on the compactness of the aggregates. However, D also indicates the radial variation of the particle density inside the aggregates.

(20)

The approach proposed by Kane et al. mainly differs from the SoFA model over the following points: (i) no prefactor has been introduced (equivalent to Swax = 1), (ii) the value of the 4528


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Industrial & Engineering Chemistry Research exponent m has been set to 0.3, and (iii) the Krieger− Dougherty equation has been chosen to express the viscosity. As long as the viscosity of the suspending liquid phase μL and the actual volume fraction ϕ are known, only the determination of the two parameters A and X is required for a complete rheological characterization of the wax aggregated suspension. A priori, A and X might vary with ϕ. Indeed, there is no reason to argue that aggregates maintain the same structure, regardless of the temperature and, therefore, the quantity of wax precipitates. Similar to the other viscosity models mentioned in section 2 of this paper, the SoFA model cannot predict time-dependent properties. However, as it is based on an aggregation mechanism of wax crystals, the time-dependent properties could be handled by accounting for the kinetics of aggregation and the kinetics of breakage. A general analysis of aggregation/ breakage kinetics should be based on a population balance equation (PBE) in which the so-called aggregation kernel and breakage kernel are terms that are dependent on conditions (Brownian motion, laminar shear flow, turbulent flow, settling) as well as on fractal structure of aggregates. Particularly, probability of collision between particles, collision efficiency, breakage rate, type of aggregation (particle−cluster and/or cluster−cluster aggregation), must be considered. The account for these phenomena would make the aggregation-related viscosity approach much more complex. The reader is referred to the literature for more details.31−33

h under shear conditions (imposed shear rate or imposed shear stress) or quiescent conditions. Flow curves have then been recorded at the final working temperature by increasing the shear rate after a holding time of 30 min. The experimental protocol is schematically illustrated in Figure 1.

Figure 1. Experimental protocol used for rheological characterization.

Comparison between data recorded in viscometers and in the Anton Paar apparatus (for T > WAT) as well as repeatability tests in viscometers have shown a relative variation of the viscosity in the range of ±2%. Larger deviations can be achieved in case the samples were subjected to different thermomechanical histories. Consequently, particular attention was paid to controlling experimental conditions, in terms of temperature, cooling rate, and shear conditions.

4. SAMPLES AND EXPERIMENTAL METHODS The SoFA model has been applied to crude oils (Oil#1 and Oil#2) and condensates (Cond#1 and Cond#2). For some experimental needs, condensates have been modified by adding silicone oil (grade N4000) provided by Paragon Scientific. These blends are referred as Cond#1a, Cond#1b, Cond#1c, Cond#1d, and Cond#2a; the first four blends correspond to different fractions of silicone oil in Cond#1 and the last blend corresponds to a different fraction of silicone oil in Cond#2. Before characterizations and viscosity measurements, the samples were heated to an initial temperature Tinit 15−20 °C above the WAT for several hours. The pour point (PP) was determined by a modified ASTM standard method (ASTM Standard D97). The wax appearance temperature (WAT), as well as the weight fraction of wax crystals Cwax was obtained by differential scanning calorimetry (DSC) in a Perkin−Elmer Model DSC8000 apparatus. The heat flux was recorded while the sample was cooled at a cooling rate of 2 °C/min. The mass of solid phase was determined by considering an enthalpy of crystallization of 200 J/g. Above the WAT, the density and dynamic viscosity were measured in an Anton Paar Model SVM3000 apparatus. The density ρL and viscosity μL of the suspending liquid phase were calculated for temperatures below the WAT by extrapolating the density as a linear function of the temperature (ρL = aρT + bρ) and the viscosity as an Arrhenius function (according to eq 2). The volume fraction of wax crystals ϕ is deduced from ϕ = Cwax ρL/ρwax by setting the density of wax crystals to ρwax = 900 kg/m3. Rheological characterizations were carried out either in a Thermo Mars III viscometer, using a single-cylinder Couette geometry (Oil#1, Cond#1, and Cond#2), or in an AR2000 viscometer from TA Instruments, by using a plate−plate geometry (Oil#2). In the viscometer, samples were cooled from Tinit to the final working temperature at a cooling rate of 10 °C/

5. RESULTS 5.1. Application to Flow Curve Measurements. A comparison of the flow curve measurements obtained with Oil#1 and Oil#2 is presented below. Prior to recording flow curves at a fixed working temperature T, systems were subjected to a cooling step either under shear conditions at a constant shear rate of 100 s−1 (Oil#1 and Oil#2) or under quiescent conditions (Oil#2). The working temperatures and the corresponding solid wax fractions, as well as density and viscosity of the suspending liquid phase, are reported in Table 1. In order to find the parameters A and X of the SoFA constitutive law, experimental flow curves are first represented as the evolution of the normalized effective volume fraction ϕeff/ϕ, with respect to the shear stress τ. The effective volume fraction ϕeff is calculated from viscosity data by resolving the quadratic equation (eq 18) and by keeping the root lower than ϕM. The parameters A and X are then obtained by fitting experimental data with a power law expression (eq 17). An example of such a determination is shown Figure 2. The A and X values for all of the systems are given in Table 1. Representations of the experimental flow curves by the SoFA model are shown in Figure 3 for Oil#1 with ϕ = 0.035 and in Figure 4 for Oil#2 with ϕ = 0.0212. The HB model (eq 4) with fitted parameters (τy, k, and n, reported in Table 1) also gives a good representation of the experimental results. In this case, the yield stress is first calculated by using eq 19. On the other hand, the P&R model (eq 10), as well as the G&M model (eq 11) with default values of the parameters DPR, EPR, FPR, and AGM, BGM, CGM, DGM, respectively, generally underpredict the magnitude of the shear stress. Nevertheless, very good 4529


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Industrial & Engineering Chemistry Research Table 1. Values of the Different Parametersa Oil#1 (cooling under a shear rate of 100 s−1) T (°C) ρL (kg/m3) μL (Pa s) Cwax (wt %) ϕ SoFA A (PaX) X HB τy (Pa) k (Pa sn) n P&R fit DPR EPR (s−1/2) FPR (s−1)

Oil#2 (cooling with no shear)

Oil#2 (cooling under a shear rate of 100 s−1)

20.4 844

45 847

45 847

0.00146 3.74

0.0246 2.25

0.0246 2.25

0.0350

0.0212

0.0212

17.44

43.26

27

0.085

0.148

0.068

2.18 1.41

24.47 2.59

1.02 1.34

0.567

0.6

0.68

61 2.20 × 103

57.4 5.79 × 103

77.7 2.71 × 103

1.17 × 108

5.52 × 109

7.9 × 108

Figure 3. Flow curve recorded at T = 20.4 °C (ϕ = 0.035) with Oil#1 after cooling under shear rate conditions at 100 s−1. Exp, experimental data; SOFA, SoFA model; HB, Herschel−Bulkley model; P&R, Pedersen and Rønningsen model with default parameters; G&M, Ghanaei and Mowla model (Cwax = 3.74 wt %); P&R fit, Pedersen and Rønningsen model with fitted parameters.

SOFA = SOFA model; HB = Herschel−Bulkley model; P&R fit = Pedersen and Rønningsen model with corrected constant values. a

Figure 4. Flow curves recorded at T = 45 °C (ϕ = 0.0212) with Oil#2 after cooling under shear rate conditions at 100 s−1 and quiescent conditions. Exp, experimental data; SOFA, SoFA model.

mean that aggregates resulting from a wax crystallization process under quiescent conditions exhibit a more open structure associated with the smaller fractal dimension D = (3 − X)/m. 5.2. Variation of SoFA Parameters with Temperature. This section addresses the application of the SoFA model to predict the temperature evolution of the viscosity of Cond#2a. During the rheological characterization of the condensate Cond#2, we experienced difficulties in measuring the viscosity due to a settling phenomenon of wax crystals in the viscometer during measurement. Consequently, in order to avoid this phenomenon, the condensate was made more viscous by adding silicone oil and the viscosity was recorded during the cooling process under shear conditions at 10 s−1 and 100 s−1 from 50 °C to 0 °C. For every temperature, two independent couples (μ,ϕ) can be obtained. Resolution of eq 20 then leads to expressions of A(ϕ) and X(ϕ). Note that this manner of determination, for which only two viscosity values are used for a given temperature, is less accurate than that consisting in matching A and X on a flow curve. Results are presented Figure 5. For practical reasons, A and X have been expressed analytically. The yield stress defined by eq 19 has been first calculated and fitted by a second-order polynomial forced to

Figure 2. Evolution of the effective volume fraction with the shear stress for Oil#1 at ϕ = 0.035.

agreement can be achieved by adjusting their value (see the P&R fit section in Table 1 for the P&R model). An example of application of the different models is illustrated in Figure 3. As already mentioned, the impact of the thermomechanical history on the resulting rheological behavior of waxy oils is crucial. This point is illustrated Figure 4. Data obtained with Oil#2 with ϕ = 0.0212 for two different cooling scenarios (quiescent and shear conditions) show a significant gap between flow curves. The change in the rheological behavior, depending on the cooling scenario, can be interpreted as the result of a change in morphology of wax aggregates. After the yielding process, the gelled oil behaves like a suspension composed of wax aggregates characterized by a higher value of X (0.148 instead of 0.068). If we suppose that the aggregates are subjected to a similar breaking mechanism under shear conditions (same value of m), the higher value of X would 4530


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increases with wax crystal concentration while the fractal structure does not significantly evolve. For comparison, results reported by Kane et al.3 for a crude oil subjected to a cooling process under imposed shear rate, shown a slight increase in D (from 2.2 to 2.4) and a decrease in A (from 32 to 17) when the system was cooled. In this last case, D corresponds to m = 0.3 and A has been calculated by us, according to the values of the critical shear stress τ0 indicated in the paper (denoted as σC by the authors) and by setting Swax = 1. For volume fractions ϕ below 0.005, the rise of X beyond 1 when ϕ tends to zero has no longer physical meanings. The fractal dimension of an aggregate in a three-dimensional space is larger than 1 and generally reported to be ∼2 or above.34 Therefore, the highest value of X, by considering m = 0.5, should be ∼1. The SoFA model is consistent only if the evolution of the viscosity is the result of an aggregation process between wax crystals. On one hand, Kane et al.1 reported that, at low concentration, wax crystals are present in the oil as mainly isolated particles. On the other hand, Visintin et al.4 suggested that high shear relative viscosities can be ascribed to dilute dispersions of anisotropic particles for sufficiently high shape factors. If we consider isolated anisotropic particles, eqs 15 and 17 must be replaced by:

Figure 5. Yield stress, X, and A expressed as a function of ϕ for the system Cond#2a (yield stress given in units of Pa and A is expressed in units of PaX).

cross the x- and y-axis at zero. Then, a power law expression for X has been found to match A data. Finally, we have: τy = 4093ϕ2 + 7.79ϕ X = 2.758 × 10−5ϕ−1.636 + 0.211 A=

4/7 (τy)X ϕ

ϕeff ϕ (21)

=

[η]wax [η]wax = [η]sphere 2.5

(22)

where [η]wax is the intrinsic viscosity of wax crystals and [η]sphere = 2.5 is the intrinsic viscosity of spherical particles. Determination of dϕeff/dϕ from viscosity data at low volume fraction (ϕ < 0.005) for both 10 s−1 and 100 s−1 yields [η]wax ≈ 25, that is to say, ∼10 times larger than the intrinsic viscosity of spheres. Even if the formalism developed in the SoFA approach may lose its physical meaning at very low concentration of solid wax particles, it can be used as a mathematical empirical relationship to ensure continuity between the low- and high-concentration domains. 5.3. Viscosity Predictions. The main advantage of writing the viscosity as an explicit function of both the suspending liquid phase viscosity and the wax crystal concentration (eq 1) is the potential capability of predicting the viscosity of different systems from the characterization of a reference system. As reminded in the Introduction, the morphology of wax precipitates can be impacted by the oil composition. The structure of wax aggregates might also be dependent on the oil composition. However, it can be expected that such differences are negligible in case the composition of the systems are kept close to each other, particularly if the n-paraffins distribution is preserved. This point has been investigated by changing the solvent properties of the condensate Cond#1 by adding silicone oil at different concentrations (blends referred as Cond#1a, Cond#1b, Cond#1c, and Cond#1d). The different systems were cooled under an imposed shear stress of 0.5 Pa and flow curves were recorded at different working temperatures, which were chosen in order to set the volume fraction ϕ at about the same value (ϕ ≈ 0.0155). Conditions are gathered in Table 2. The parameters A and X of the SoFA model should only be dependent on the actual volume fraction ϕ. Therefore, the determination of A and X for a reference system should allow us to predict the rheological behavior of similar systems at the same volume fraction ϕ. The values of A (A = 24.31) and X (X

The simulated viscosity is reported in Figure 6. The slight gap between experimental data and calculated data comes from the use of analytical expressions for A and X.

Figure 6. Viscosity for the system Cond#2a (Cond#2 + silicone oil) at different shear rates; Exp, experimental data; SoFA, calculated using the SoFA model.

It is interesting to note that, for volume fractions ϕ above 0.005, the parameter X = (3 − D)m remains roughly constant at a value of ∼0.23. With respect to the physical meaning of D and m, it can thus be deduced that wax aggregates roughly keep the same fractal structure and are subjected to the same breakage mechanism with an increase in wax crystal concentration. By assuming m in the range of 0.25−0.5, the fractal dimension D is found in the range of 2.1−2.5. On the other hand, the parameter A slightly decreases for ϕ > 0.005, from ∼55 to 30. Such a variation may be explained by an increase of the prefactor Swax (Swax = τ0X/A, where X is a constant), which means that the compactness of wax aggregates 4531


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been developed by considering waxy oil systems as suspensions of wax crystals that can interact and form fractal aggregates whose size is limited by the shear stress. Similar to other models reported in the literature, the viscosity μ can be expressed as a function of the viscosity of the suspending liquid phase μL and a function of the volume fraction of wax crystals ϕ (see eq 1). Application of the SoFA constitutive law to experimental flow curves has shown a very good agreement by matching the two model parameters A and X. Good results have also been obtained by using the Herschel−Bulkley, the Li and Zhang, and the Pedersen and Rønningsen models. However, in this last case, a good agreement was achieved by setting the three parameters DPR, EPR, and FPR at different values than those proposed by the authors. It is expected that all parameters introduced in the waxy oil viscosity models proposed in the literature are dependent on the oil, the thermomechanical history, and the volume fraction of wax crystals. The capability of the SoFA model to predict the viscosity of different systems at a given volume fraction of wax crystals ϕ has been successfully investigated. In this case, the distribution of n-paraffins has been preserved. Only the physical properties of the dispersing liquid phase have been changed. The parameters A and X were obtained from the regression of an experimental flow curve recorded for a reference system. This first result gives us hope that such a model should enable the viscosity of live oil systems (presence of a gas phase) to be predicted by deploying the following procedure: • Determination of the parameters A and X of the SoFA constitutive law, as functions of the wax crystal volume fraction on a reference system (i.e., dead oil). Rheological characterizations must be chosen in such a way thermal and shear conditions are as close as possible to those undergone by the fluid under real conditions. • Calculation of the amount of solid wax present at any temperature and pressure, using a thermodynamic wax model. • Calculation of the viscosity and the density of the Newtonian suspending liquid phase, using corresponding states viscosity and density models. However, to be definitively accepted for live oil application, a comparison between SoFA model predictions and viscosity measurements in the presence of a gas phase under pressure will need be performed.

Table 2. Conditions for Flow Curve Comparison with Cond#1 Systems system

T (°C)

Cwax (wt %)

ϕ

μL (cP)

Cond#1 Cond#1a Cond#1b Cond#1c Cond#1d

−7 −10.9 −11.9 −14.6 −14.54

1.76 1.76 1.71 1.72 1.73

0.0154 0.0156 0.0153 0.0156 0.0158

1.72 2.63 4.68 8.57 11.18

= 0.072) found with Cond#1 for ϕ = 0.0154 have been used to predict the flow curves of systems Cond#1a, Cond#1b, Cond#1c, and Cond#1d. The good agreement between experiments and predicted flow curves is shown Figure 7. It confirms the capability to predict the rheological behavior of systems that differ from a reference system in the physical properties of the suspending liquid phase.

Figure 7. Comparison between experiments and predicted flow curves for Cond#1 systems at ϕ ≈ 0.0155. Dots represent experimental data, solid lines represent predicted flow curves with the SoFA model, the L&Z model, and the P&R model with fitted parameters. The curve denoted as “L&Z (Stress)” refers to the L&Z model with the flocculation factor kLZ expressed as a function of the shear stress.

The Pedersen and Rønningsen model with fitted parameters (P&R fit) and the Li and Zhang model (L&Z) have also been evaluated by first regressing the flow curve obtained with Cond#1. The parameters of the P&R fit model have been found to be DPR = 85.5, EPR = 935, FPR = 2.47 × 108. The flocculation factor kLZ in the L&Z equation has been expressed as a power function of the shear rate: kLZ = 45.9γ̇−0.08. The two models show similar results with an over prediction of the shear stress for Cond#1d. As previously discussed in the presentation of the SoFA model, the size of aggregates is expected to be controlled by the shear stress and not by the shear rate. By relating the flocculation factor kLZ to the shear stress instead of the shear rate in the L&Z equation, we obtain kLZ = 31.5τ −0.1. When this new relationship for the flocculation factor is applied to Cond#1d, a much better agreement with the experimental flow curve is obtained.

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ASSOCIATED CONTENT

S Supporting Information *

General properties of systems used in this work (Tables S1− S3). Solid wax fraction as a function of temperature of systems used in this work (Figures S1 and S2). Example of repeatability test (Figure S3). Example of impact of thermomechanical history (Figure S4). This material is available free of charge via the Internet at http://pubs.acs.org/.

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6. CONCLUSIONS A new viscosity model, called the Suspension of Fractal Aggregates (or SoFA), has been presented. The approach is similar to the procedure previously used by Kane et al. to interpret the viscosity evolution of a crude oil that was subjected to a cooling process under imposed shear rates. It has

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 4532


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ACKNOWLEDGMENTS The authors are grateful to TOTAL for the permission to publish this article. NOTATION P = pressure T = temperature Tinit = initial temperature τ = shear stress γ̇ = shear rate μ = viscosity of the waxy oil μL = viscosity of the suspending liquid phase ρL = density of the suspending liquid phase aρ, bρ = constants in ρL expressed as a function of the temperature μ*, θ = constants in the Arrhenius equation μ0 , μ∞ = limiting values of the viscosity in the Cross equation γ̇C = critical shear rate in the Cross equation n′ = exponent in the Cross equation τy = yield stress in HB model k = plastic viscosity in HB model n = flow index in HB model acas , bcas = constants in Casson’s law W = mass fraction of wax AAZ, BAZ, CAZ, DAZ = constants in the Al-Zahrani and AlFariss model ATM, BTM, CTM, DTM = constants in the Tiwary and Mehrotra model APR, BPR, CPR = constants in the Pedersen and Rønningsen model (eq 8) DPR, EPR, FPR = constants in the Pedersen and Rønningsen model (eq 10) kLZ = flocculation factor in the Li and Zhang model kAO = flocculation factor in the Adeyanju and Oyekunle model ρwax = density of wax crystals Cwax = weight fraction of wax crystals Nwax = number of wax crystals in aggregates Rwax, awax = dynamic radius of wax aggregates and wax crystals D = fractal dimension of aggregates Swax = prefactor ϕ = volume fraction of wax crystals ϕeff = effective volume fraction of aggregates ϕM = maximal packing fraction; ϕM = 4/7 τ0 = critical shear stress below which aggregates can form m = exponent that is dependent on the breakage mechanism of aggregates A, X = parameters of the SoFA constitutive law [η]wax = intrinsic viscosity of wax crystals [η]sphere = intrinsic viscosity of spheres

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SoFA - American Chemical Society

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