Viscosity Correlations

In the following sections, a detailed description of three viscosity models are given. Viscosity models are used in equation of state modeling and black-oil PVT.

Lee Correlation

The viscosity model proposed by Lee in 1966¹ is a relatively simple model for describing reservoir gas and oil viscosities. This correlation is not used much in the industry any more, but is a simple correlation to give first order estimates.

The correlation is given by

Note

The units for the temperature: (R) and density: (lb/ft3).

$\begin{equation}\label{eq:Lee_visc} \mu = 10^{-4}k_v \exp{\{x_v(\frac{\rho}{62.4})^{y_v}\}} \end{equation}$

where

$\begin{equation}\label{eq:kv} k_v = \frac{(9.4 + 0.02MW)T^{1.5}}{209 + 19MW + T} \end{equation}$

$\begin{equation}\label{eq:xv} x_v = 3.5 + \frac{986}{T}+0.01MW \end{equation}$

$\begin{equation}\label{eq:yv} y_v = 2.4 - 0.2x_v \end{equation}$

Corresponding States Principle Correlation

whitson comment

Note from Markus Hays Nielsen: Be cautious if you are moving your EOS model from PVTp to Eclipse, as PVTp has an option for tuning the CSP model that Eclipse does not have. This might give inconsistent results between your PVT model in the two softwares.

The procedure of the corresponding states principle (CSP) for viscosity modeling as described bt Pedersen in 1982² is given below and is adapted from "Phase Behavior of Petroleum Reservoir Fluids"³.

Calculate $T_c$ and $p_c$ from equation \eqref{eq:Tc} and \eqref{eq:pc}
Estimate Methane density from equation \eqref{eq:dens} which is the McCarty modified BWR EOS and the reduced densities from equation \eqref{eq:red_dens} where $a_n$ are empirical constants.
Calculate mixture molecular weight ($MW$) from equation \eqref{eq:MW_mix} and equations \eqref{eq:MW_w} and \eqref{eq:MW_n}.
Calculate $\alpha_{mix}$ factor from equations \eqref{eq:alpha}.
Calculate corresponding state methane $T_{methane}$ and $p_{methane}$ from equations \eqref{eq:To} and \eqref{eq:po} respectively also using equation \eqref{eq:Tcij} where $\alpha_{methane}$ is defined by using equation \eqref{eq:alpha} but with the molecular weight instead of the mixture molecular weight ($MW_{mix}$).
Calculate mixture viscosity from equation \eqref{eq:mix_visc} where $\mu_{methane}$ is calculated using equation \eqref{eq:mix_methane} and $c_1$ to $c_6$ are constants.

Note

The units for this model are given in pressure:(atm), temperature: (K), density: (mol/L) and gas constant: (L.atm/mol.K) and viscosity: (1 E-4 cP)

$\begin{equation}\label{eq:Tc} T_{c,mix} = \frac{\sum_{i=1}^N\sum_{j=1}^Nz_iz_j[(\frac{T_{ci}}{p_{ci}})^{1/3}+(\frac{T_{cj}}{p_{cj}}) ^{1/3}]^3\sqrt{T_{ci}T_{cj}}}{\sum_{i=1}^N\sum_{j=1}^Nz_iz_j[(\frac{T_{ci}}{p_{ci}})^{1/3}+(\frac{T_{cj}}{p_{cj}}) ^{1/3}]^3} \end{equation}$

$\begin{equation}\label{eq:pc} p_{c,mix} = \frac{8\sum_{i=1}^N\sum_{j=1}^Nz_iz_j[(\frac{T_{ci}}{p_{ci}})^{1/3}+(\frac{T_{cj}}{p_{cj}}) ^{1/3}]^3\sqrt{T_{ci}T_{cj}}}{(\sum_{i=1}^N\sum_{j=1}^Nz_iz_j[(\frac{T_{ci}}{p_{ci}})^{1/3}+(\frac{T_{cj}}{p_{cj}}) ^{1/3}]^3)^2} \end{equation}$

$\begin{equation}\label{eq:dens} p = \sum_{n=1}^9a_n(T)\rho^n + \sum_{n=10}^{15}a_n(T)\rho^{2n-17}\exp{\{-\gamma\rho^2\}} \end{equation}$

$\begin{equation}\label{eq:red_dens} \rho_r = \frac{\rho_{methane}(T\frac{T_{c,methane}}{T_{c,mix}},p\frac{p_{c,methane}}{p_{c,mix}}) }{\rho_{c,methane}} \end{equation}$

$\begin{equation}\label{eq:MW_mix} MW = 1.304\cdot 10^{-4}(MW_w^{2.303}-MW_n^{2.303}) \end{equation}$

$\begin{equation}\label{eq:MW_w} MW_w = \frac{\sum_{i=1}^Nz_iMW_i^2}{\sum_{i=1}^Nz_iMW_i} \end{equation}$

$\begin{equation}\label{eq:MW_n} MW_n = \sum_{i=1}^Nz_iMW_i \end{equation}$

$\begin{equation}\label{eq:alpha} \alpha_{mix} = 1+7.378\cdot 10^{-3}\rho_r^{1.847}MW_{mix}^{0.5173} \end{equation}$

$\begin{equation}\label{eq:To} T_{methane} = T\frac{T_{c,methane}}{T_{c,mix}}\frac{\alpha_{methane}}{\alpha_{mix}} \end{equation}$

$\begin{equation}\label{eq:po} p_{methane} = p\frac{p_{c,methane}}{p_{c,mix}}\frac{\alpha_{methane}}{\alpha_{mix}} \end{equation}$

$\begin{equation}\label{eq:Tcij} T_{cij} = \sqrt{T_{ci}T_{cj}} \end{equation}$

$\begin{equation}\label{eq:mix_visc} \mu_{mix} = (\frac{T_{c,mix}}{T_{c,methane}})^{-1/6}(\frac{p_{c,mix}}{p_{c,methane}})^{2/3} (\frac{MW_{mix}}{MW_{methane}})^{1/5}(\frac{\alpha_{mix}}{\alpha_{methane}}) \mu_{methane}(p_{methane},T_{methane}) \end{equation}$

$\begin{equation}\label{eq:mix_methane} \mu_{methane} = \sum_{n=-3}^3 c_{n+3}T^{n/3} \end{equation}$

Lohrenz-Bray-Clark Correlation

The Lohrenz-Bray-Clark viscosity correlation (LBC) is an adaptation of the original work by Jossi, Stiel and Thodos⁴. The details for the correlation are given below.

$\begin{equation}\label{eq:visc_lbx} [(\mu-\mu^*)\xi+10^{-4}]^{1/4} = a_0 + a_1\rho_r + a_2\rho_r^2 + a_3\rho_r^3 + a_4\rho_r^4 \end{equation}$

where $\rho_r=\rho/\rho_c$, $\mu^*$ is the low-pressure gas mixture viscosity and $\xi$ is the viscosity-reducing parameter. The constants $a_n$ we originally treated as universal constants by LBC, but are typically used as tuning parameters for a given set of viscosity data.

The viscosity-reducing parameter ($\xi$) is defined by

$\begin{equation}\label{eq:visc_red} \xi = \frac{[\sum_{i=1}^Nz_iT_{ci}]^{1/6}}{[\sum_{i=1}^Nz_iMW_i]^{1/2}[\sum_{i=1}^Nz_ip_{ci}]^{2/3}} \end{equation}$

and the critical density is defined by

$\begin{equation}\label{eq:crit_dens} \rho_c = \frac{1}{v_{c,mix}} = \frac{1}{\sum_{i=1}^Nz_iv_{ci}} \end{equation}$

The critical volumes for heptane plus components ($C_{7+}$) are either defined by correlation or tuning. One example of a critical volume correlation is given by ³

Note

The units for the critical volume is ft3/lb and the density is in g/cm3.

$\begin{equation}\label{eq:crit_vol} v_{ci} = 21.573 + 0.015122\cdot MW_i - 27.656\cdot \rho_i + 0.070615\cdot MW_i \cdot \rho_i \end{equation}$

Finally, the dilute gas mixture viscosity ($\mu^*$) is given by Herning and Zippener ⁵ as

$\begin{equation}\label{eq:visc_dilute} \mu^* = \frac{\sum_{i=1}^Nz_i\mu_i^*\sqrt{MW_i}}{\sum_{i=1}^Nz_i\sqrt{MW_i}} \end{equation}$

where $\mu_i^*$ is given by Stiel and Thodos⁶ as

$\begin{equation}\label{eq:visc_comp_dilute_1} \mu_i^* = 34\cdot 10^{-5} \frac{T_{ri}^{0.94}}{\xi_i} \end{equation}$

for $T_{ri}<1.5$ and

$\begin{equation}\label{eq:visc_comp_dilute_2} \mu_i^* = 17.78\cdot 10^{-5} \frac{(4.58T_{ri}-1.67)^{5/8}}{\xi_i} \end{equation}$

for $T_{ri} \geq 1.5$

For equations \eqref{eq:visc_comp_dilute_1} and \eqref{eq:visc_comp_dilute_2} the component viscosity-reducing parameter is given by

$\begin{equation}\label{eq:visc_comp_red} \xi = \frac{T_{ci}^{1/6}}{MW_{i}^{1/3}\cdot p_{ci}^{2/3}} \end{equation}$

A. L. Lee, M. H. Gonzalez, and B. E. Eakin. The viscosity of natural gases. Journal of Petroleum Technology, 18:paper SPE–1340–PA, 1966. doi:https://doi.org/10.2118/1340-PA. ↩
K. S. Pedersen and Aa. Fredenslund. An improved corresponding states model for the prediction of oil and gas viscosities and thermal conductivities. Chemical Engineering Science, 42:182–186, 1987. doi:https://doi.org/10.1016/0009-2509 $87$ 80225-7. ↩
K. S. Pedersen, P. L. Christensen, and J. A. Shaikh. Phase behavior of petroleum reservoir fluids. CRC press, 2014. ↩↩
J. A. Jossi, L. I. Stiel, and G. Thodos. The viscosity of pure substances in the dense gaseous and liquid phases. AIChE Journal, 8:59–63, 1962. doi:https://doi.org/10.1002/aic.690080116. ↩
F. Herning and L. Zipperer. Calculation of the viscosity of technical gas mixtures from the viscosity of the individual gases. Gas u. Wasserfach, 79:69, 1936. ↩
L. I. Stiel and G. Thodos. The viscosity of nonpolar gases at normal pressures. AIChE Journal, 7:611–615, 1961. doi:https://doi.org/10.1002/aic.690070416. ↩