Prediction of the apparent viscosity of non-Newtonian water-in-crude oil emulsions

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 40, Issue 1, February 2013 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2013, 40(1): 130–133.

RESEARCH PAPER

Prediction of the apparent viscosity of non-Newtonian water-in-crude oil emulsions WANG Wei, WANG Pengyu, Li Kai, DUAN Jimiao, WU Kunyi, GONG Jing* Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, China

Abstract: On the basis of the effective medium theory and the classical viscosity prediction model proposed by Pal et al, the apparent viscosity and rheological properties of the water in crude oil emulsion with natural surface active components such as asphaltene and resin are predicted. The adsorption and entrainment of continuous phase by the surfactants molecular, which leads to the increment of effective water fraction, is the inducement of Non-Newtonian behavior in the bulk emulsion. The two actions of promoting and counteracting effects on the effective water fraction are considered separately and represented by Water Fraction Factor and Particle Reynolds Number Factor respectively. The Water Fraction Factor takes into account the positive effect of water fraction and drop size distribution in microscope, while the Particle Reynolds Number Factor reflects the negative effect of shear rate at different drop size. A new prediction model taking the influence of nearly all factors such as drop size distribution, shear rate, water fraction into consideration was built. Measured viscosity of emulsion of the crude oil and water from an oilfield was used to verify the predicted results of the model, which proves that the new model is high in accuracy with a maximum relative error of 11.44%. Key words: water-in-crude oil emulsion; non-Newtonian property; apparent viscosity; drop size distribution; prediction model

Introduction It is easy for crude oil to form stable emulsions with water due to the existence of certain amount of surface active components such as asphaltenes and resins. These formed emulsions exhibit remarkable increase in apparent viscosity and non-Newtonian shear-thinning behavior, which are encountered at nearly every step of the petroleum exploitation, gathering and transportation. Notably due to the internal microstructure differences, it is difficult to accurately predict the viscosities of these emulsions and their rheological properties [1−10] . A number of factors affect rheological behavior of emulsions interactively, such as volume fraction, continuous-phase viscosity, dispersed-phase viscosity, shear rate, temperature, drop size distribution and emulsifiers. In recent years, the effect of internal droplet distribution on apparent viscosity of emulsion has caught much attention. Otsubo, Pal, and others[2, 11−14] have researched the influence of drop size on the apparent viscosity of emulsion and found that the apparent viscosity of emulsion is related not only to mean drop diameter but also drop size distribution. Yet there are still not enough prediction models which properly describe the effect

of drop size distribution on apparent viscosity. A new viscosity prediction model is put forward in this paper taking into account the mutual effects of drop size distribution, shear rate, and water fraction etc.

1

Development of new apparent viscosity model

The prediction model is developed on the idea that the non-Newtonian behavior of an emulsion can be manifested by the difference between effective and real water fraction. Relationship of the two and their effects on the apparent viscosity of the emulsion can be represented according to the effective medium theory. Surface active components like asphaltanes and resins are amphiphilic and will induce the adherence of oil molecules to the surface of water droplets. Moreover, fine water droplets have a tendency to form flocculation under the effect of Van der Waals forces thus further promotes the entrainment of oil molecules. The actual volume fraction of water in emulsion is named real water fraction, then the enlarged water fraction due to absorption/flocculation effects is defined as effective water fraction. When the emulsions are sheared, the absorbed oil molecules become partly stripped and finally totally desorbed as the shear rate continues to increase, which shows an

Received date: 23 Apr. 2012; Revised date: 24 Oct. 2012. * Corresponding author. E-mail: [email protected] Foundation item: Supported by National Natural Science Foundation of China (51104167), CNPC Innovation Foundation (2010D-5006-0604), and the Foundation for the Author of National Excellent Doctoral Dissertation of China (201254). Copyright © 2013, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

WANG Wei et al. / Petroleum Exploration and Development, 2013, 40(1): 130–133

externally non-Newtonian shear-thinning behavior and is represented by non-Newtonian factor kN. Based on the effective medium theory [15−16], the desired real water fraction can be reached by the following steps: (1) minute amount of water is added into the continuous oil phase; (2) certain amount of water is added at any stage i, thus stage i+1 is reached; (3) the emulsion of stage i is regarded as a continuous phase in relation to the added droplets at stage i+1. By applying Taylor’s equation [16], one obtains ⎛ η + 2.5ηd ΔVe ⎞⎟ ηi +1 = ηi ⎜⎜⎜1 + i (1) ⎟⎟ ⎜⎝ η + η V + ΔV ⎠⎟ i

d

ti

d

Between stage i and stage i+1, the added effective water volume is ΔVe, thus the total effective water volume is Vei+ΔVe and the total volume of the emulsion is Vti +ΔVd. The added water with the volume of ΔVd will be dispersed as new droplets, and a certain amount of oil molecules will be adhered by the newly formed droplets, thus the change in effective water fraction Δfwe can be expressed as: V + ΔVe Δf w e = ei − f w ei (2) Vti + ΔVd Substituting the equation Vei = fwei Vti to Equation (2), one obtains f V + ΔVe − f wei (Vt i + ΔVd ) (3) Δf we = w i t i Vt i + ΔVd Substituting Vt i + ΔVd ≈ Vt i + ΔVe in the numerator of Equation (3), one obtains ΔVe Δf we (4) = Vt i + ΔVd 1 − f wei

Substituting Equation (4) and kN =fwei /fwi to Equation (1), the differential equation can be written as dη k N η η + 2.5ηd (5) = df w 1 − k N f w η + ηd Upon integration with the limit η→ηc at fw→0, Equation (5) gives: 1.5

η ⎜⎛ η + 2.5ηd ⎞⎟ ⎟⎟ = (1 − k N f w )−2.5 ⎜ ηc ⎜⎝⎜ ηc + 2.5ηd ⎠⎟

(6)

Or 1.5

⎛ η + 2.5k ⎞⎟ = (1 − k N f w )−2.5 ηr ⎜⎜ r ⎜⎝ 1 + 2.5k ⎠⎟⎟

(7)

where, ηr=η/ηc k=ηd/ηc Three key factors, namely water fraction, shear rate and Sauter mean diameter of droplets are considered in the calculation of non-Newtonian factor (kN), and particle Reynolds Number NRe,p(NRe,p = ρ c γd 322/4ηc) is introduced. Therefore, (8) kN=K(fw γ d32)= K(fw)K(NRe,p) Thus the apparent viscosity equation is given as: 3

⎛ η + 2.5k ⎞⎟5 −1 (ηr ) ⎜⎜⎜ r ⎟⎟ = ⎡⎢⎣1 − K ( f w ) K ( N Re ,p ) f w ⎤⎥⎦ ⎝ 1 + 2.5k ⎠ 2 5

2

(9)

fects on the effective water fraction are considered separately and characterized by water fraction factor K(fw) and particle Reynolds number factor K(NRe,p) respectively. 2.1

Characterizing the effect of promoting the effective water fraction, water fraction factor is influenced by water fraction and drop size distribution. The average surface area (A) of the dispersed droplets is regarded proportional to the number of absorbed oil molecules. The average volume V0 of absorbed water droplets is equal to the sum of average volume V1 for water droplets and volume V2 for absorbed oil phase: AΔd32 V0 = V1 + V2 = V1 + (10) 2 Taking the dispersed droplets as regular spheres, then n n n 1 πd j 2 πd j 2 ∑ πd j 3 ∑ ∑ 6 j =1 6 = nj =1 = (11) A = j =1 V1 1 n n d 32 3 π d ∑ j j =1 6 Water fraction factor is expressed as V 3Δd32 K ( fw ) = 0 = 1 + V1 d32 where,

n

n

j =1

j =1

(12)

1

d 32 = ∑ πd j 2 / ∑ πd j 3 6

For concentrated non-Newtonian water-in-oil emulsion, the thickness of the adsorption layer is approximately equal to the liquid film between droplets [16−17], as given in Equation 13. 1/3 ⎡⎛ ⎤ f ⎞ (13) Δd32 ≈ am = ⎢⎢⎜⎜⎜ w m ⎟⎟⎟ −1⎥⎥ d 32 ⎢⎣⎝⎜ f w ⎠⎟ ⎥⎦ 2.2

Particle Reynolds number factor

Particle Reynolds number factor expresses the action counteracting the rise of effective water fraction, displaying as the drop degree of effective water fraction when an emulsion is subject to external shear. This factor is mainly affected by shear rate and drop size distribution. Since crude oils forming emulsions are different in terms of composition and physical properties, surface active components and their contents, it’s recommended that the particle Reynolds number factor is fitted to two experiments at one drop size but different shear rates. Fig. 1 shows a comparison between the predicted and measured Particle Reynolds Number Factor at different mean drop sizes at water fraction of 40%, 50% and 60%. The prediction is fitted to the experiments with mean diameter of 4.68 μm, shear rates at 10 s−1 and 100 s−1. It’s clear that the predictions are in good agreement with the measurements, which indicates that the proposed method can accurately predict the particle Reynolds number factor at different drop sizes and shear rates.

3

Calculation of non-Newtonian factor

Water fraction factor

In this paper, the roles of promoting and counteracting ef− 131 −

Validation of the new viscosity model Taking the influence of drop size distribution, shear rate,

WANG Wei et al. / Petroleum Exploration and Development, 2013, 40(1): 130–133

Fig. 1 Comparisons between the predicted and measured particle Reynolds number factor (The solid points are experimental data used for prediction)

water fraction and etc. on the apparent viscosity into consideration, the proposed model better explains the physical conTable 1 Water fraction/%

40

50

60

Fig. 2

Sample

notation of the constitutive relation of emulsion viscosity. It was then verified using a large number of experimental data of emulsions prepared by several kinds of crude oil and formation water collected from oilfield [18]. To give a further illustration, data from one set of crude oil is shown as an example. The viscosity and density of the crude oil are 475.12 mPa⋅s and 954.16 kg/m3 at 50 °C, while the viscosity and density of the formation water are 0.89 mPa⋅s and 928.18 kg/m3 at the same temperature. Four emulsion samples were prepared under two different stirring rates and two different temperatures: sample 1#, 600 r/min, 50 °C; sample 2#, 600 r/min, 60 °C; sample 3#, 1 000 r/min, 50 °C; and sample 4#, 1 000 r/min, 60 °C. Apparent viscosities of emulsion samples 1—4# at different shear rates are listed in Table 1. It is obvious that the emulsion presents a shear thinning behavior. Fig. 2

Apparent viscosities of 1#—4# emulsion samples Apparent viscosity under different shear rates/(mPa⋅s)

20 s−1

30 s−1

40 s−1

60 s−1

80 s−1

1

1 867

1 829

1 801

1 758

1 720

2

1 045

1 031

1 021

1 005

991

3

1 891

1 862

1 834

1 796

1 767

4

1 055

1 042

1 034

1 018

1 005

1

2 908

2 794

2 680

2 528

2 423

2

1 715

1 657

1 609

1 534

1 472

3

3 221

3 117

3 046

2 931

2 827

4

1 809

1 718

1 662

1 590

1 542

1

5 260

4 889

4 632

4 214

3 867

2

3 348

3 137

2 977

2 731

2 571

3

6 533

6 101

5 782

5 217

5 055

4

3 997

3 722

3 516

3 215

2 953

Comparisons between the predictions of new viscosity model and measurements from 1#—4# emulsion samples.

− 132 −

WANG Wei et al. / Petroleum Exploration and Development, 2013, 40(1): 130–133

References [1]

[2] [3]

[4] Fig. 3

Comparison of predicted and measured viscosities.

shows the relative viscosities of sample 1#—4# predicted by the proposed model along with experimental data. As can be seen from Fig. 2 and Fig. 3, the viscosity predicted from the proposed model are in good agreement with the experimental results, with the maximum relative error of 11.44%.

4

[5]

[6]

Conclusions

Based on the effective medium theory, a new viscosity model for non-Newtonian water-in-crude oil emulsions was developed. The model takes into consideration the effect of drop size distribution, shear rate, and water fraction etc, which better represents the constitutive relation of emulsion viscosity. The proposed model was verified with a large number of experimental data of emulsions prepared by crude oil and formation water collected from the oilfield. Results indicate that the new model is high in prediction accuracy.

[7]

[8]

[9]

[10]

Nomenclature kN—Non-Newtonian factor; fwe— Effective water fraction, %; fw— Real water fraction, %; η— Emulsion viscosity, mPa⋅s; ηd— Dispersed-phase viscosity, mPa⋅s; ΔVe— Change in effective volume after water addition, mL; Vt— Total volume of emulsion, mL; ΔVd—Volume of water added in each step, mL; Δfwe—Change in effective water fraction, %; ηc— Continuous-phase viscosity, mPa⋅s; ηr—Relative viscosity of emulsion; k—Ratio of dispersed-phase viscosity to continuous-phase viscosity; γ—Shear rate, s−1; ρc—Continuous-phase density, kg/m3; K(fw)—Water fraction factor; K(NRe,p)— Particle Reynolds number factor; dj—Size of the drop of No. j, μm; n—Total number of droplets in the dispersion; d32—Sauter mean diameter, μm; fwm—Maximum packing fraction of the dispersed phase, %; am—Thickness of the liquid film, μm; Δd32—Thickness of the absorbed layer at droplet interface, μm. Subscript: i —the stage in the process of dispersed-phase addition.

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