Predicting the Viscosity of Oil Emulsions Depending on the Degree of Water Cut

1. Introduction

The real media encountered in oil production are usually multiphase, multicomponent, and dispersed systems. They are at least two-phase systems, in which one of the phases is dispersed in the other. Due to the water breakthrough of oil reservoirs, technological processes more often involve not pure hydrocarbon products, but their mixtures with water, in most cases in the form of emulsions. During these processes, the viscosity of the formed emulsions often undergoes sharp changes.

Dispersed water, mineral salts, and solid-phase particles present in oil, as well as high-molecular-weight chemical compounds dissolved in it, such as asphaltene-resin-paraffin substances (ARP), which act as “black” emulsifiers, can significantly change the rheophysical properties and structure of oil emulsions. It should be noted that oil emulsions are considered polydisperse systems and, in most cases, are classified as non-Newtonian systems. As a rule, these systems have size and property characteristics that change with time. In other words, the rheology of oil emulsions depends on many factors. These factors include the following:

• formation of a coagulation structure in emulsions as a result of the interaction of water, ARP substances, and solid-phase particles contained in the oil mass;
• significant influence of deformation of water droplets and loss of their spherical shape on the effective viscosity and shear stress of the emulsion;
• reduction in the water-cut degree of the emulsion due to separation of water from the oil emulsion;
• degree of water dispersion in the emulsion.

Since the above-mentioned factors can considerably change the structure and properties of oil emulsions, they have a significant effect on their flow behavior, that is, their rheology.

2. Methodology

It should be noted that taking into account the above-mentioned factors, as well as droplet coalescence, breakup and sedimentation, several studies have been conducted to systematize the individual effects of droplet size and the evolution of their time-dependent distribution functions on the properties of oil emulsions [1,2,3,4,5,6,7]. In this study, a comprehensive methodology based on laboratory experiments and mathematical modeling was applied in order to predict the effective viscosity of stable water-in-oil emulsions depending on the degree of water cut. Under laboratory conditions, water-oil emulsions with water cut varying within the range of 35–65% were prepared. To ensure the stability of the emulsions, the mixing process was carried out in a high-speed laboratory disperser at a constant rotational speed and within a predetermined time interval. The rheological properties of the prepared stable emulsions were investigated, and their effective viscosities were measured at different temperatures and shear-rate gradients using a rotational viscometer. Based on the measurement results, regularities between emulsion viscosity and water cut were established. Existing studies and empirical expressions proposed by different authors for determining the viscosity of oil emulsions depending on water content were analyzed.

$${\mathsf{\eta }}_{ef}={\mathsf{\eta }}_0\left(1+2.5\phi \right),\phi <0.01$$

(1)

At present, numerous empirical formulas are available for calculating viscosity for disperse media. When the volume concentration of particles in the dispersed phase is high, taking into account their hydrodynamic interaction, a number of authors have proposed the use of the following modified expressions of the Einstein equation:

$${\mathsf{\eta }}_{ef}={\mathsf{\eta }}_0\left(1+2.5\phi -{\alpha }_0{\phi }^2+{\alpha }_1{\phi }^3+\cdots \right)$$

(2)

Where, ${\mathsf{\eta }}_{ef}$− effective viscosity of the emulsion;

${\eta }_0$- viscosity of the dispersed medium (oil);

$\phi$− volume fraction of the dispersed phase (water);

${\alpha }_1$ - a coefficient taking into account physical phenomena in dispersed flows (varying between different works)

The following formula is also used to determine the viscosity of emulsions under high water cut conditions:

$${\mathsf{\eta }}_{ef}={\mathsf{\eta }}_0{\left(1+k\phi \right)}^{-2.5}$$

(3)

Here, k is the coefficient that accounts for the dispersity and properties of the emulsifiers. For imported systems, this coefficient is adopted.

It is known that the viscosity of emulsions can also be assessed using the following expression, taking into account the effect of the temperature factor.

$${\mathsf{\eta }}_{ef}={\mathsf{\eta }}_0{e}^{a\phi }$$

(4)

Where, being an empirical coefficient a varies in the range of a=2-10 depending on the type of oil and its temperature.

Being used for specific additives, the existing empirical models are the formulae that are adequate for certain experimental data. It is very complex to develop a generalised model for various systems. Given that the volume fraction of particles in unit volume $\phi ={\displaystyle \frac{N_0\pi {\alpha }^3}{6}}$ (N₀ is the number of particles in the unit volume), then any expression that allows for the determination of the effective viscosity will, indirectly, express the dependence of the viscosity of the dispersed medium on the particle size. Indeed, the stability and durability of the above mentioned protective oil emulsions being dependent on their viscosity, which in turn dependent on the diameter of the water droplets, i.e., the degree of dispersion have been confirmed once again.

The following empirical models are also available for different sizes of solid particles, respectively, to determine the effective viscosity of suspensions based on experimental data:

$${\eta }_{ef}=1+2.5\phi +1.5\phi \mathit{exp}\left[{\displaystyle \frac{0.45\phi }{(\phi -{\phi }_{\mathrm{\infty }}{)}^2}}\right]$$

(5)

$${\eta }_{ef}=1+2.5\phi +{\displaystyle \frac{3}{4}}\phi \mathit{exp}\left[{\displaystyle \frac{m\phi }{(\phi -{\phi }_{\mathrm{\infty }}{)}^2}}\right]$$

(6)

$$m=2.2+0.03d$$

(7)

d – particle diameter.

As can be seen from the preceding statements, the effective viscosity of a dispersion is highly dependent on the volume fraction and size of the particles. In this case, the effective viscosity increases with the growth in particle size. The effective viscosity of the dispersion system increases to its maximum critical value, which in turn affects the velocity and property of the flow.

The results of the rheological analysis of dispersed systems show that, despite the multitude and variety of rheological models, the main research has been dedicated to the development of empirical models, which describe experimental data with a certain accuracy without taking into account the mechanism of the phenomena. It is known that the nature and properties of coagulation structures significantly affect the main properties of the dispersed medium.

On the other hand, the application of the above mentioned rheological models in oilfield practice for determining the viscosity of various emulsions is associated with numerous difficulties and, in most cases, is not suitable for solving engineering problems. This is mainly due to the determination of the size and volume fraction of water particles dispersed in the oils. Field practice shows that as fields are exploited over time, the rheological and physical-chemical properties of oil emulsions also vary over a wide range owing to the increase in the water content and change in thermobaric conditions. As such changes likely affect the degree of water dispersion, carrying out any rheological experiment necessitates the determination of the particle sizes.

Currently, it is proposed in researches to use the following formula to calculate the viscosity of various types of emulsions [8,9,10]:

$${\displaystyle \frac{{\eta }_{ef}}{{\eta }_0}}\approx \mathit{exp}(5\phi )(1-3\phi +b\phi )$$

(8)

Where, coefficient b varies in the range of 3-7.3 depending on the concentration of the emulsion.

As it is known leading oil production companies and oil extraction enterprises are currently actively implementing progressive technologies based on the mathematical hydrodynamic model of the formation, the parameters of exploitation wells, and the controlled injection of various mixtures into the formation.

One of the most important and complex issues is measuring the volume of mechanical mixtures, including individual components of emulsions, extracted from the well. Thus, as thermodynamically unstable disperse medium, the emulsions that are formed, in most cases, consist of multitude components of both organic and mineral. When measuring the consumption of such emulsions, the most difficult parameter to consider is their apparent viscosity. As oil emulsions consist of oil and water droplets of various diameters, they are classified as polydisperse systems. The viscosity of these systems is not an additive property but depends on the type of mixture, the viscosity of the oil, temperature, moisture, the dispersion of the system, and the velocity gradient in the flow. It is precisely these factors that make the use of the above mentioned well-known empirical expressions for determining viscosity considerably more difficult. Thus, it is considered appropriate to use the above mentioned formulas in order to determine viscosity only for a specific range of conditions for the dispersed particles (water). In this regard, a more progressive method for considering the true value of apparent viscosity is the direct measurement of viscosity.

Recent studies show that as in suspensions, the effective viscosity of oil emulsions is significantly impacted not only by the size of the particles in the dispersed medium, but also by their shape, in other words, their deformability [11,12,13,14]. Thus, as a coagulation structure can form in the system at a high droplet concentration, it can lead to a significant change in the rheological properties. On the other hand, in heavy and stable oil emulsions, they should also be considered partially degassed, owing to the presence of occluded gas in most cases. Therefore, it is also possible for such emulsions to contain bubbles and have various shapes.

The formation of oil emulsions, as a rule, occurs frequently in the wellbore, at the wellhead, and in the gathering-transportation pipelines. Depending on the chemical composition of the produced oils, the resulting emulsions are stable in most cases, and since chemical or thermal methods seperately are insufficient for their dehydration, their decomposition is carried out by a combined, i.e., thermochemical method [15,16,17,18].

Based on field practice, it can be stated that according to rheological experiments determining the viscosity of the above mentioned stable systems is difficult or impossible. On the other hand, the rheological properties of these emulsions are progressively deteriorated due to the mixing of oils of different grades and watering degree during their gathering and transportation in the fields. The above mentioned factors necessitate the regular monitoring of the increase in the viscosity of emulsions with increasing water content, in order to control the technological processes and increase their efficiency during the extraction, collection, and transportation of hydrocarbons. In this regard, the development of analytical methods and empirical expressions for the operational determination of the viscosities of oil emulsions is of particular importance.

Despite the current use of above mentioned empirical expressions (e.g., the Einstein and Taylor formulas, etc.) for determining the viscosities of oil emulsions, there is a need to develop reliable and high-quality mathematical expressions. Since the assessment of the viscosity properties of oil emulsions belonging to dispersion systems is accompanied by large errors based on the principle of additivity, it is almost practically inapplicable. Even the application of the linear dependencies proposed for binary systems by various researchers, at very low concentrations of one of the components, is completely unacceptable in many cases. Therefore, in any case, the calculation of the viscosity of heterogeneous systems such as water-oil mixtures must be approached with caution, particularly when three or more components are present in the mixture [19,20,21,22].

3. Results & Discussion

Taking the above into consideration, oil samples were selected and rheological tests were carried out in order to determine how the viscosity of stable oil emulsions changes depending on the degree of water cut and temperature. The rheological tests were performed using a rotational viscometer to establish and analyze the relationships between shear stress (τ) and shear-rate gradient ($\dot{\gamma }$). The rheological tests were conducted for both dehydrated oil and oil samples with different degrees of water cut, namely 35, 40, 45, 50, 55, 60, 65, and 67%, at temperatures of 20°C and 40°C. For a temperature of 20°C and a dehydrated oil viscosity of η = 0.1 Pa·s, the flow curves of the emulsions at different degrees of water cut, expressed as τ=f$\dot{\left(\gamma \right)}$, are shown in Figure 1.

The experiments were continued using oil samples with viscosities of η_n= 0.1 and 0.3 Pa·s up to the water cut limit at which the oil emulsions reached their maximum viscosity. The rheological analysis showed that, as the water cut of the oil increased, the viscosity of the emulsions increased significantly. The results of the rheological studies carried out on the example of oil from the Muradkhanli field, with an initial water cut of 29%, density of ρ_n= 0.909 g/cm³, and viscosity of η_n= 0.1 Pa·s, are presented in Table 1 and Figure 2, Figure 3, Figure 4 and Figure 5 for water-cut values of 35, 40, 45, 50, 55, 60, and 65%.

Viscosities determined at $\dot{\gamma }$= 0.33; 1.0; 3.0 s^-1 in accordance with mathematical models 1-3.

Viscosities determined at $\dot{\gamma }$= 9.0; 48.6; 145.8 s^-1 in accordance with mathematical models 1-3

The comparison of the results obtained at relatively low and high shear rates with the viscosity values calculated for the emulsions using the mathematical model showed that the results were generally in good agreement. The difference between the experimental and predicted values was observed to vary within the range of 3–5%, which is considered an acceptable error margin for engineering calculations (Figure 2, Figure 3, Figure 4 and Figure 5). As can be seen from Figure 2, Figure 3, Figure 4 and Figure 5, with increasing water cut, the viscosity of oil emulsions increases significantly at both temperatures, and this increase follows the same regular pattern.

The value of the water-cut limit corresponding to the maximum viscosity of the emulsions was determined based on the flow curves τ=f$\dot{\left(\gamma \right).}$ Analysis of the laboratory test results shows that, up to a certain limiting value of the amount of water as the dispersed phase in the oil-water mixture, the viscosity of the emulsion increases and reaches its maximum value [5,6,9]. The analysis indicates that, depending on the physicochemical and rheological properties of the oil, this condition for Azerbaijani oils occurs within the water-cut range of β_water=50-85%. In other words, at the indicated maximum water-cut limit, the maximum dispersion of water in oil is completed, and this state is reflected in the flow curve by a sharp decrease in shear stress.

The interpretation of this phenomenon as a change in the emulsion type, accompanied by a sharp decrease in viscosity, has been shown by researchers to be incorrect. They explained this behavior as being related to the improper interpretation of viscosity determination using a viscometer, particularly due to the slip effect. Therefore, the viscosity value corresponding to the indicated water-cut limit was considered as the maximum viscosity [5].

In the present case, according to Figure 1, the maximum water-cut limit can be accepted as the corresponding critical value. As can also be seen from the figure, further increase in water cut is accompanied by a sharp decrease in shear stress. The maximum viscosity values of the emulsions at the maximum water cut for different shear-rate gradients at t =20°C and 40°C are given in Table 2.

Our research has determined that this regularity can be expressed by the following mathematical model:

$${\eta }_{em}={\eta }_n({\displaystyle \frac{{\eta }_{\mathrm{em}}^{*}}{{\eta }_n}}{)}^{{\displaystyle \frac{{\beta }_{\mathrm{water}}}{{\beta }_{\mathrm{water}}^{*}}}}$$

(9)

Where, β_water – current water cut,%;

η_n - the viscosity of dehydrated oil, Pa⋅s;

${\beta }_{\mathrm{water}}^{*}$ and ${\eta }_{\mathrm{em}}^{*}$- correspondingly, the maximum water cut (%) and the maximum value of the emulsion viscosity (Pa⋅s) corresponding to that water cut. As can be seen from (7), when β water =${\beta }_{water}^{*}$, then ${\eta }_{\mathrm{em}}={\eta }_{\mathrm{em}}^{*}$, whereas when the water cut state (β_water =0) does not occur ${\eta }_{\mathrm{em}}={\eta }_n$ are obtained. According to the mathematical model established to determine the viscosity of emulsions, the value of viscosities ${\eta }_{\mathrm{em}}^{mod}$ for oil emulsions tested at various water cut degrees were calculated, and the results are given in Table 1. The calculated viscosity ${\eta }_{\mathrm{em}}^{mod}$ of oil emulsions for various degrees of water cut and velocity gradients was compared with the experimentally obtained values ${\eta }_{\mathrm{em}}^{eks}$ of emulsions, and a graph of the dependence on the number of measurements was constructed for the cases of the parameter ${\eta }_{\mathrm{em}}^{mo{d}_{\mathrm{em}}^{eks}}$ under consideration (Figure 6).

As can be seen from Figure 6, according to the proposed prediction model, the calculated viscosity values for the emulsions differ from the values determined by laboratory tests by an average of 3–5%. Such a deviation is acceptable for engineering calculations, including the determination of the flow rate of oil emulsions.

Thus, for any oil sample, whether taken from a well, tank or pipeline, the proposed mathematical model can be used to predict how the viscosity of oil emulsions changes depending on water cut. For this purpose, it is sufficient to conduct rheological tests of the given oil sample under laboratory conditions at different water-cut values, construct the τ=f$\dot{\left(\gamma \right)}$ and ${\eta }_{\mathrm{em}}=f\left({\beta }_{\mathrm{water}}\right)$ dependencies and determine the relevant parameters from these relationships, including ${\beta }_{\mathrm{water}}^{*}$ and η_n. Thereafter, the variation in the viscosity of oil emulsions depending on water cut can be predicted using the proposed mathematical model without conducting additional laboratory tests.

5. Conclusion

1. The conducted studies show that the viscosity of oil emulsions significantly depends on the degree of water cut, temperature, and shear-rate gradient. As the water cut increases, an increase in viscosity is observed due to the rise in the internal resistance of the emulsion. In emulsions with a high water cut, the viscosity changes more sharply, which indicates enhanced structuring of the system.

2. At the same time, an increase in temperature from 20°C to 40°C leads to a decrease in the viscosity of the emulsions. This confirms that increasing temperature weakens intermolecular interactions and improves the flowability of the emulsion.

3. According to the experimental results, at shear-rate gradient values of 0.3333, 1.0, 3.0, 9.0, 48.6, and 145.8 s⁻¹, the viscosity of oil with 65% water cut at 20°C decreased from 183.8 Pa·s to 2.31 Pa·s, corresponding to a reduction of 79.57%. At 40°C, the viscosity decreased from 83.4 Pa·s to 1.62 Pa·s, corresponding to a reduction of 51.48%.

4. According to the results calculated using the proposed mathematical model, at the same shear-rate gradient values, the viscosity of oil with 65% water cut at 20°C decreased from 183.38 Pa·s to 2.30 Pa·s, corresponding to a reduction of 79.73%. At 40°C, the viscosity decreased from 83.47 Pa·s to 1.62 Pa·s, corresponding to a reduction of 51.48%.

5. According to the proposed prediction model, the calculated viscosity values for the emulsions differ from the laboratory-measured values by an average of 3–5%. Such a deviation is acceptable for engineering calculations, including the determination of oil-emulsion flow rates.

Based on the comparative analysis, the experimental results obtained at low and high shear rates are in good agreement with the viscosity values calculated using the mathematical model. The fact that the difference between the experimental and calculated results does not exceed 3–5% demonstrates the adequacy of the proposed model and confirms the feasibility of its application in engineering practice.