According to the advancement of technology, population growth, and increasing energy needs in the world, enhanced oil recovery (EOR) from oil reservoirs is essential. One of these methods is the injection of W/O emulsions. Injection of water in oil emulsions due to the low interfacial tension and also having a higher viscosity than water and oil, increases oil production from reservoirs. In the following, the properties of emulsions and mathematical models for predicting viscosity are investigated.
1,1) Emulsion properties
Emulsions are dispersive systems in which liquid droplets are dispersed in one immiscible liquid and are kinetically stable but thermodynamically unstable and become two-phase over time (Binks and Lumsdon 2000). Basically there are two types of emulsions that depend on the type of continuous phase liquid (Kokal 2005):
- Oil in water emulsions (O/W): water is continuous phase and oil dispersed in water
- Water in oil emulsions (W/O): oil is continuous phase and water dispersed in oil
In the classical classification, emulsions consist of two non-miscible phases of water and oil, one of which disperses into the other. Dispersed phases are usually referred to as internal phases and continuous phases as external phases. Depending on the location of the phases, their physical properties will be different. According to Winsor theory, from a microscopic point of view, water and oil emulsions are divided into three main categories (Maaref and Ayatollahi 2018):
- Oil-in-water emulsion (O/W) in which oil droplets are dispersed in water.
- Water-in-oil (W/O) emulsions in which water droplets are dispersed in oil.
- Multiple emulsions, such as oil-in-water-in-oil (O/W/O) and water-in-oil-in-water (W/O/W) emulsions, in which the dispersed droplets themselves contain very fine particles dispersed from the continuous phase. Much more complex multiple emulsions may also occur.
Stability in emulsions is achieved by surface active agents. Surfactants are molecules that migrate to surfaces between two physical phases and therefore have a higher concentration in that region than other parts of the two phases. Emulsions have dual properties, lipophilic and hydrophilic. By adding these materials to two immiscible liquids, the surface tension between the two phases is reduced and prevented biphasic; thus, two liquids will be able to form an emulsion. These substances are also found in crude oil (Shi et al. 2018).
Resins and asphaltenes have a similar chemical structure that tends to be adsorbed on the surface between water and oil, leading to their stability (Zaki, Schoriing, and Rahimian 2007). For this reason, emulsion formation is very common in the oil industry. Emulsion formation occurs spontaneously in the processes of production, refining, distribution and storage of crude oil because of natural surfactant presence in crude oil phase so that the amount of water in these emulsions can reach 60% (Zaki, Schoriing, and Rahimian 2007). The emulsions can cause a lot of production issues such as higher oil viscosity flow in pipes, need for more heat to reduce the viscosity and separation of crude oil and water to meet marketing specifications, more time to separate in the storage tank caused more corrosion. These problems lead to higher costs for oil refining. The stability and viscosity of the emulsions have an important role in transfer and separation to meet marketing specifications (Maneeintr, Sasaki, and Sugai 2013). Between all these features, viscosity is the most major parameter because of direct effects on pressure drop in oil systems (C. Li et al. 2016).
W/O emulsions viscosity is a complex function of water content, temperature, dispersed and continuous phase viscosity, shear rate, particle size and water salinity. These parameters are manifold and can affect each other. For example, a change in shear rate causes a change in droplet sizes, or as the temperature increases, the viscosity of the emulsion decreases because the viscosity of the continuous and dispersed phase decreases with increasing temperature. In general, if the temperature is kept constant in a specific crude oil, the main variables involved in its viscosity are emulsion pressure and dispersed phase volume (dispersed phase concentration) (De Oliveira et al. 2018).
Many researchers have studied the prediction of W/O emulsion viscosity, and different relationship are presented based on experiments. The studies on water in oil emulsions have mostly shown that viscosity increases with increasing water content and decreases with increasing temperature and shear stress (Krieger and Dougherty 1959). When the volume fraction of water is more than 0.2, increasing the shear rate results in a decrease in viscosity of the emulsions, and with increasing water content, this phenomenon will be more pronounced; Therefore, as the water content increases, the rheological behavior of the emulsion shifts to non-Newtonian fluids, and the emulsions relative and apparent viscosity increase spontaneously (C. Li et al. 2016).
In 2020, Zhang et al. conducted a study on a water emulsion in paraffin oil containing a large amount of chitosan polysaccharide and was stabilized by span80 oil-soluble surfactant. In this study, they investigated the effect of various parameters such as water content in the system, the molecular properties of chitosan, the amount of span80 and the ultrasonic process on the average particle size. They showed that reducing the negative zeta potential by increasing the amount of water from 5–35% causes the emulsion particle size distribution to become more uniform, droplets to become smaller and the viscosity increased. They observed that as the water content increased, the viscosity of emulsion gradually increased. From this results it can be concluded that the viscosity of such water in oil emulsions depends significantly on the water content in the emulsions system (Zhang et al. 2020). In 2021, Peña and Ghosh studied the effect of volume fraction of water and pectin on improving the viscoelastic properties of water -in-oil emulsions. They increased the water content in the range of 20 to 50 percent and observed that the emulsion viscosity and gel strength increased with increasing water content (Romero-Peña and Ghosh 2021). In 2021, Du et al. Conducted experimental research to demonstrate the effect of W/O emulsions by nitrogen/carbon dioxide to enhance oil recovery. They found that as the water content increased, the viscosity significantly increased, but this increase was observed up to a water content of 70%. The viscosity then decreased with increasing water content, which was attributed to the conversion of simple water-in-oil emulsion to an oil-in-water-in-oil multiple emulsion. They explained that as the volume of water increased, the amount of dispersed phase droplets increased in emulsion and the distance between the droplets decreased. Under the influence of molecular force and IFT of water and oil, the viscosity increased and the emulsion showed a stable state but when the volume fraction of water increased to 70%, the amounts of water droplets decreased and the size of the droplets significantly increased and the emulsion becomes more unstable and the viscosity decreased (Du et al. 2021). In 2021, Li et al. Studied the deposition of water in oil emulsions and developed a model for predicting it. In this model, they considered parameters such as water content, viscosity prediction, droplet displacement models and the probability of droplet failure to predict the deposition process of W / O emulsion in different conditions. In part of their simulation, they stated that the viscosity increases with increasing amount of water in the early times, but because the particle size increases, the emulsion stability decreases and the viscosity decreases (Y. B. Li et al. 2021).
Viscosity models in constant temperature
Most models used to predict emulsion viscosity estimate the relative viscosity, which is the ratio of emulsion viscosity (µ) to viscosity of continuous phase (µc) and is defined as follows:
\({\mu }_{r}=\frac{\mu }{{\mu }_{c}}\) ( 1
Einstein (1906, 1911) developed a thermodynamic model for very low-concentration suspensions, assuming that all particles were spherical. He suggested that µr is a linear function of ϕ (volume content of the dispersed phase) and relative viscosity increases with increasing ϕ (Farah et al. 2005).
\({\mu }_{r}=1+2.5\varphi\) ( 2
Years after Einstein, Taylor was trying to improve Einstein's equation. Taylor (1932) believed that both dispersed and continuous phases affect viscosity, and again repeated Einstein's assumptions, considering particles completely spherical, and the equation that Taylor proposed could be used only for low concentrations. To improve the previous equations, Taylor defined the parameter α, which is the ratio of the dispersed phase viscosity (µD) to continuous phase viscosity (µc) (Taylor and A 1932).
\({\mu }_{r}=1+\left[2.5\left(\frac{\alpha +0.4}{\alpha +1}\right)\right]\varphi\) ( 3
\(\alpha =\frac{{\mu }_{D}}{{\mu }_{c}}\) ( 4
When the coefficient α tends to infinity (for the dispersing of spherical solid particles), Taylor's equation becomes Einstein's equation.
In 1975, Choi Schowalter attempted to reduce the assumptions of the Einstein and Taylor equations. They stated that with increasing concentration, the emulsion particles are not spherical, so to solve this problem, they considered a function of the volumetric content of the dispersed phase to modified the viscosity of emulsions. Refer to the reference provided for more details on the correction factor (Choi and Schowalter 1975).
\({\mu }_{r}=1+f\left({\varphi }^{\frac{1}{3}}\right)\varphi\) ( 5
Thien and Pham in 1997 proposed another model for estimating viscosity. At low concentrations the viscosity increases linearly with increasing water content and when Φ tend to unity, \({{\mu }}_{\mathbf{r}}\) asymptotically grows. In this model, as in the previous model, there is no limit to the concentration and it will be suitable for low capillary numbers ( Thien and Pham 1997).
\({\mu }_{r}^{2.5}{\left(\frac{2{\mu }_{r}+5\alpha }{2+5\alpha }\right)}^{\frac{3}{5}}=\left(\frac{1}{1-\varphi }\right)\) ( 6
Krieger and Dougherty (1959) suggested an experimental model that is credible for the dispersed phase at high-level concentrations. They introduced the parameter ϕm which is known as the maximum dispersed phase concentration and it occurs when the emulsions' viscosity tends to infinity. To estimate ϕm, the values ϕ must be plotted in terms of \(\frac{1}{{\mu }_{r}-1}\) and by extrapolating, can find a point whose coordinates are zero. When µr tends to infinity, ϕ tends to ϕm and the value ϕm is obtained [10].
\({\mu }_{r}={\left[1-\left(\frac{\varphi }{{\varphi }_{m}}\right)\right]}^{-\left[\mu \right]{\varphi }_{m}}\) ( 7
\(\left[\mu \right]=\frac{{\mu }_{D}}{{\mu }_{C}}-1\) ( 8
In 1998, Pal proposed a model based on experimental studies and dimensional analysis for single-dispersion emulsions with assumptions such as similar phase density, low surface tension, constant flow, and ignoring Brownian motion (Pal 1998).
\({\varphi }_{m}^{\frac{1}{2}}\left(1-{\mu }_{r}^{-\frac{1}{\left[\mu \right]{\varphi }_{m}}}\right)={c}_{0}+{c}_{1}{log}\left({N}_{Re,p}\right)+{c}_{1}{{log}\left({N}_{Re,p}\right)}^{2}\) ( 9
Where c0, c1 and c2 are constant and NRe, p is the Reynolds number of the particles.
\({N}_{Re,p}=\left(\frac{{\rho }_{c}\gamma {r}^{2}}{{\mu }_{c}}\right)\) ( 10
Where ρc and µc are the density and viscosity of the continuous phase, r and γ are radius of particle and share rate respectively.
Mendoza and Holek used a spherical rigid particle model to calculate the emulsion viscosity. The semi-empirical correlation obtained is as follows (Mendoza and Santamaría-Holek 2009):
\({\eta }_{r}=(1-\frac{\varphi }{c\varphi })\) ( 11
\(c=\frac{(1-{\varphi }_{m})}{{\varphi }_{m}}\) ( 12
Faroughi and Huber used a method similar to Mendoza and Hulk to calculate the viscosity of the emulsion using an effective medium model (Faroughi and Huber 2015).
\({\eta }_{r}={\left[\frac{{\varphi }_{m}-\varphi }{{\varphi }_{m}\left(1-\varphi \right)}\right]}^{-\frac{2.5{\varphi }_{m}}{1-{\varphi }_{m}}}\) ( 13
The main purpose of this investigation is providing a method for estimating the viscosity of W/O emulsions as a function of water content and pressure. Emulsion properties and emulsion viscosity models were described as a function of water content or dispersed phase volume content in this Section. The method of work is explained in Section 2. In Section 3, the results were discussed. The proposed relationship was investigated using the various errors in Section 4 and eventually, the results and conclusions of this work is given in section 5.