# 19.3: Viscosity

- Page ID
- 597

What other properties are we interested in? We are interested in flow properties. Whether you are interested in flow in pipes or in porous media, one of the most important transport properties is *viscosity*. Fluid viscosity is a measure of its internal resistance to flow. The most commonly used unit of viscosity is the centipoise, which is related to other units as follows:

1 c_{p} = 0.01 poise = 0.000672 lbm/ft-s = 0.001 Pa-s

**Natural gas viscosity** is usually expected to increase both with pressure and temperature. A number of methods have been developed to calculate gas viscosity. The method of Lee, Gonzalez and Eakin is a simple relation which gives quite accurate results for typical natural gas mixtures with low non-hydrocarbon content. Lee, Gonzalez and Eakin (1966) presented the following correlation for the calculation of the viscosity of a natural gas:

\[\mu_{g}=1 \cdot 10^{-4} k_{v} \exp\left(x_{v}\left(\frac{\rho_{g}}{62.4}\right)^{y_{v}}\right) \label{19.25a}\]

where:

\[k_{v}=\frac{\left(9.4+0.02 M W_{g}\right) T^{1.5}}{209+19 M W_{g}+T} \label{19.25b}\]

\[y_{v}=2.4-0.2 x_{v} \label{19.25c}\]

In this expression, temperature is given in (°R), the density of the fluid (\(\rho_{g}\)) in lbm/ft^{3} (calculated at the pressure and temperature of the system), and the resulting viscosity is expressed in centipoises (c_{p}).

The most commonly used **oil viscosity** correlations are those of Beggs-Robinson and Vasquez-Beggs. Corrections must be applied for under-saturated systems and for systems where dissolved gas is present in the oil. However, in compositional simulation, where both gas and condensate compositions are known at every point of the reservoir, it is customary to calculate ** condensate viscosity **using Lohrenz, Bray & Clark correlation Clark correlation. It this type of simulation, it is usual to calculate

*gas viscosities*based on Lohrenz, Bray & Clark correlation as well. This serves the purpose of guaranteeing that the gas phase and condensate phase converge to the same value of viscosity as they approach near-critical conditions.

Lohrenz, Bray and Clark (1964) proposed an empirical correlation for the prediction of the viscosity of a liquid hydrocarbon mixture from its composition. Such expression, originally proposed by Jossi, Stiel and Thodos (1962) for the prediction of the viscosity of dense-gas mixtures, is given below:

\[\mu=\mu+\xi_{m}^{-1}\left(0.1023+0.023364 \rho_{r}+0.058533 \rho_{r}^{2}-0.040758 \rho_{r}^{3}+0.0093724 \rho_{r}^{4}\right)^{4}-1 \times 10^{-4} \label{19.26}\]

where

- \(\mu\)= fluid viscosity (c
_{p}), - \(\mu^{*}\) = viscosity at atmospheric pressure (c
_{p}), - \(\xi_{m}\) = mixture viscosity parameter (cp
^{-1}), - \(\rho_{T}\) = reduced liquid density (unitless),

Lohrentz *et al.* original paper presents a typographical error in Equation \ref{19.26}. Here it is written as originally proposed by Jossi, Stiel and Thodos (1962). All four parameters listed above have to be calculated as a function of critical properties in order to apply Equation \ref{19.26}. Lohrentz *et al.* original paper uses scientific units, here we present the equivalent equations in field (English) units.

For the *viscosity of the mixture at atmospheric pressure* (\(\mu^{*}\)), Lohrentz *et al.* suggested using the following **Herning & Zipperer equation**:

\[\mu^{i}=\frac{\displaystyle \sum_{i} z_{i} \mu_{i}^* \sqrt{M W_{i}}}{ \displaystyle \sum_{i} z_{i} \sqrt{M W_{i}}} \label{19.27}\]

where:

- \(z_j\) = mole composition of the i-th component in the mixture,
- \(MW_i\) = molecular weight of the i-th component (lbm/lbmol)
- \(\mu_{i}^{*}\) = viscosity of the i-th component at low pressure (c
_{p}):

Moreover,

\[\mu_{i}^*=\frac{34 \cdot 10^{-5} T_{r i}^{0.94}}{\xi_{i}} \]

if \(T_{ri} ≤ 1.5\) and

\[\mu_{i}^*=\frac{17.78 \cdot 10^{-5}\left(4.5 T_{r i}-1.67\right)^{0.625}}{\xi_{i}}\]

if \(T_{ri} > 1.5\).

where:

\(T_{ri}\) is the reduced temperature for the i-th component (T/Tci) and \(MW_i\) is the viscosity parameter of the i-th component, given by:

\[\xi_{i}=\frac{5.4402 T_{c i}^{1 / 6}}{\sqrt{M W_{i}} P_{c i}^{2 / 3}}\]

For the *mixture viscosity parameter *(\(\xi m\)), Lohrentz *et al.* applied an equivalent expression to that shown above but using pseudo-properties for the mixture:

\[\xi m=\frac{5.4402 T_{p c}^{1 / 6}}{\sqrt{M W_{l}} P_{p c}^{2 / 3}} \label{19.28}\]

where

- \(T_{pc}\) = pseudocritical temperature (
^{o}R), - \(P_{pc}\) = pseudocritical pressure (psia),
- \(MW_l\) = liquid mixture molecular weight (lbm/lbmol).

The *reduced density of the liquid mixture* (\(\rho_{r}\)) is calculated as:

\[\rho_{r}=\frac{\rho_{l}}{\rho_{p c}}=\left(\frac{\rho_{l}}{M W_{l}}\right) V_{p c} \label{19.29}\]

where

- \(\rho_{p c}\) is the mixture pseudocritical density (lbm/ft
^{3}), - \(V_{pc}\) is the mixture pseudocritical molar volume (ft
^{3}/lbmol),

All mixture pseudocritical properties are calculated using Kay’s mixing rule, as shown:

\[T_{p c}=\sum z_{i} T_{c i} \label{19.30a}\]

\[P_{p c}=\sum z_{i} P_{c i} \label{19.30b}\]

\[V_{p c}=\sum z_{i} V_{c i} \label{19.30c}\]

“\(z_i\)” pertains to the fluid molar composition, \(T_{ci}\) is given in ^{o}R, P_{ci} in psia, and V_{ci} in ft^{3}/lbmol. When the critical volumes are known in a mass basis (ft^{3}/lbm), each of them is to be multiplied by the corresponding molecular weight. In the case of lumped C_{7+} heavy fractions, Lorentz *et al.* (1969) presented a correlation for the estimation C_{7+} critical volumes.

## References:

Lee, A., Gonzalez, M., Eakin, B. (1966), “The Viscosity of Natural Gases”, SPE Paper 1340, Journal of Petroleum Technology, vol. 18, p. 997-1000.

Lohrenz, J., Bray, B.G., Clark, C.R. (1964), “Calculating Viscosities of Reservoir Fluids from their compositions”, SPE Paper 915, Journal of Petroleum Technology, p. 1171-1176.

## Contributors and Attributions

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.