Polymer Viscosity

The polymer viscosity is a key parameter in improving the mobility ratio. Darcy‘s law describes the flow of fluid through porous media as
where q is the flow rate, A is the cross-sectional area, L is the length of the sample, ∆P is the pressure drop, and µ is the Newtonian viscosity of the flowing fluid.Polymer is a non-Newtonian fluid; the apparent viscosity (𝜂app) is not a constant.The apparent viscosity can be defined by rearranging the above Equation as follows:

Factors affecting polymer viscosity

Polymer viscosity is affected by a number of factors, such as the salinity, temperature, concentration, molecular weight, and shear rate. Polymer viscosity is affected by water salinity and divalent ions such as calcium (Ca²⁺) and magnesium (Mg²⁺), which decrease the viscosity of the polymer solution. As the salinity increases, the distance between the polymer chain and the molecules decreases. The repulsive forces are shielded by a double layer of electrolytes when salt is added to a polymer solution. Figure p25 illustrates the polymer viscosity reduction when the salinity increases. Divalent ions can more effectively neutralize charges than monovalent ions, such as Na⁺ and K⁺. The viscosity of the polymer is dominated by Ca²⁺ in brine.

Figure p25. Polymer viscosity as a function of salinity (Dong et al., 2008)

In general, the viscosity of a polymer solution decreases as the temperature increases, as shown in Figure p26. However, this relationship does not always follow the same pattern; it depends on other factors, such as the polymer concentration, molecular weight, salt, and hydrolyzation (Nouri & Root, 1971).

Figure ‎p26. Polymer viscosity versus temperature (Nouri & Root, 1971)

The polymer concentration has a direct relationship with the polymer viscosity. The polymer viscosity increases when the polymer concentration increases, as shown in Figure p27.

Figure ‎p27. Polymer viscosity versus polymer concentration (Dong et al., 2008)

The molecular weight of polymer is related to its molecular size, which means that polymers with a larger molecular size have a higher molecular weight. Polymers with a higher molecular weight provide higher viscosity, as shown in Figure p28.

Figure ‎p28. Polymer viscosity as a function of molecular weight (Dong et al., 2008)

Hydrolysis also affects the viscosity of a polymer solution. The viscosity increases with the degree of hydrolysis. A high degree of hydrolysis (35%) increases the viscosity in fresh water and sodium chloride brines (Martin & Sherwood, 1975). The apparent viscosity depends on the shear rate. Newtonian fluid exhibits a linear relationship between the shear stress (τ) and the shear rate, as shown in Figure p29, which can be expressed as
Shear stress is defined by


Figure ‎p29. Shear stress versus shear rate for a Newtonian fluid (Bourgoyne et al., 1986)

Non-Newtonian fluid is considered pseudoplastic if the apparent viscosity decreases as the shear rate increases.

Figure p30. Pseudoplastic behavior of a non-Newtonian fluid (Norrise, 2011)

Polymers behave as pseudoplastic non-Newtonian fluids at higher shear rates (Nouri & Root, 1971; Gogarty, 1967). This behavior has been described by various models, including the Bingham Plastic and Power Law models (Bourgoyne et al., 1986). The Power Law model and the Carreau model are most commonly used to describe the rheological behavior of the apparent viscosity decreasing as the shear rate increases (shear thinning) (Green & Willhite, 1998). The Power Law model is defined by

where k is the power-law constant and n is the power-law exponent. For polymer flooding, both k and n can be correlated as a function of polymer concentration, and
where C is the polymer concentration in the unit of ppm, and k is in the unit of mpa.sn.

Hirasaki and Pope (1974) modeled the pseudoplastic flow of a biopolymer solution through a porous media using the Blake–Kozeny model for power-law model fluids. The apparent viscosity is expressed as
and
where H is the Blake–Kozeny model coefficient (cp), u is the superficial velocity (cm/sec), n and k are the power-law exponent and coefficient respectively, and ϕ is the porosity and k_p is the particle permeability.

Shear Rate

Polymers exhibit non-Newtonian fluid rheological behavior when the viscosity decreases and the shear rate increases (Hirasaki & Pope, 1974). Polymers flowing through porous media may display one of the following four flow regimes: Newtonian, shear thinning, shear thickening, and degradation.
Newtonian and shear thinning Figure p29 illustrates a typical rheology for a shear thinning fluid. This figure shows the relationship between the apparent viscosity and the shear rate. The first region is the lower Newtonian region which is characterized by a constant apparent viscosity. The apparent viscosity does not change with the shear rate (Stahl & Schulz, 1988).
The polymer will maintain a constant viscosity in the lower Newtonian region until it reaches the second region, which represents the shear thinning behavior, as shown in Figure p31. As the shear rate increases, the apparent viscosity decreases. Shear thinning can also be appear as a decrease in the resistance factor and an increase in the flow velocity (flux rate), as shown in Figure p32.

Figure ‎p31. Rheological behavior of a typical polymer within a shear thinning fluid (Modified from Stahl & Schulz, 1988)

Figure ‎p32. Viscosity versus shear rate of a biopolymer (xanthan) (Seright, 2009a)

Figure ‎p33. Viscosity versus shear rate of a synthetic polymer (HPAM) (Norris, 2011)

The Carreau model is used to describe the rheological behavior of shear thinning, which is expressed as where ƞ_pN is the viscosity in the lower Newtonian region, ƞ_∞ is the polymer viscosity in the upper Newtonian region, is the shear rate associated with the viscosity of interest, n is the shear thinning index of the power law region, α is typically 2, and τ_r is the Carreau relaxation time (sec-1), which is defined as
and