Viscosity models for mixtures

The shear viscosity (or viscosity, in short) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces (or sheets) flowing with different fluid velocities. This friction is the effect of (linear) momentum exchange caused by molecules with sufficient energy to move (or "to jump") between these fluid sheets due to fluctuations in their motion. The viscosity is not a material constant, but a material property that depends on temperature, pressure, fluid mixture composition, and local velocity variations. This functional relationship is described by a mathematical viscosity model called a constitutive equation which is usually far more complex than the defining equation of shear viscosity. One such complicating feature is the relation between the viscosity model for a pure fluid and the model for a fluid mixture which is called mixing rules. When scientists and engineers use new arguments or theories to develop a new viscosity model, instead of improving the reigning model, it may lead to the first model in a new class of models. This article will display one or two representative models for different classes of viscosity models, and these classes are:

• Elementary kinetic theory and simple empirical models^[1][2][3] - viscosity for dilute gas with nearly spherical molecules
• Power series^[2][3] - simplest approach after dilute gas
• Equation of state analogy^[3] between PVT and T${\displaystyle \eta }$P
• Corresponding state^[2][3] model - scaling a variable with its value at the critical point
• Friction force theory^[3] - internal sliding surface analogy to a sliding box on an inclined surface
  • Multi- and one-parameter version of friction force theory
• Transition state analogy - molecular energy needed to squeeze into a vacancy analogous to molecules locking into each other in a chemical reaction
  • Free volume theory^[3] - molecular energy needed to jump into a vacant position in the neighboring surface
  • Significant structure theory^[3] - based on Eyring's concept of liquid as a blend of solid-like and gas-like behavior / features

Selected contributions from these development directions is displayed in the following sections. This means that some known contributions of research and development directions are not included. For example, is the group contribution method applied to a shear viscosity model not displayed. Even though it is an important method, it is thought to be a method for parameterization of a selected viscosity model, rather than a viscosity model in itself.

The microscopic or molecular origin of fluids means that transport coefficients like viscosity can be calculated by time correlations which are valid for both gases and liquids, but it is computer intensive calculations. Another approach is the Boltzmann equation which describes the statistical behaviour of a thermodynamic system not in a state of equilibrium. It can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport, but it is computer intensive simulations.

From Boltzmann's equation one may also analytically derive (analytical) mathematical models for properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection–diffusion equation. The mathematics is so complicated for polar and non-spherical molecules that it is very difficult to get practical models for viscosity. The purely theoretical approach will therefore be left out for the rest of this article, except for some visits related to dilute gas and significant structure theory.

The classic Navier-Stokes equation is the balance equation for momentum density for an isotropic, compressional and viscous fluid that is used in fluid mechanics in general and fluid dynamics in particular:

${\displaystyle \rho \left[{\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right]=-\nabla P+\nabla [\zeta (\nabla \cdot \mathbf {u} )]+\nabla \cdot \left[\eta \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{T}-{\frac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right)\right]+\rho \mathbf {g} }$

On the right hand side is (the divergence of) the total stress tensor ${\displaystyle {\boldsymbol {\sigma }}}$ which consists of a pressure tensor ${\displaystyle \left(-P\mathbf {I} \right)}$ and a dissipative (or viscous or deviatoric) stress tensor ${\displaystyle {\boldsymbol {\tau }}_{d}}$. The dissipative stress consists of a compression stress tensor ${\displaystyle {\boldsymbol {\tau }}_{c}}$ (term no. 2) and a shear stress tensor ${\displaystyle {\boldsymbol {\tau }}_{s}}$ (term no. 3). The rightmost term ${\displaystyle \rho \mathbf {g} }$ is the gravitational force which is the body force contribution, and ${\displaystyle \rho }$ is the mass density, and ${\displaystyle \mathbf {u} }$ is the fluid velocity.

${\displaystyle {\boldsymbol {\sigma }}=-P\mathbf {I} +{\boldsymbol {\tau }}_{d}=-P\mathbf {I} +{\boldsymbol {\tau }}_{c}+{\boldsymbol {\tau }}_{s}}$

For fluids, the spatial or Eularian form of the governing equations is preferred to the material or Lagrangian form, and the concept of velocity gradient is preferred to the equivalent concept of strain rate tensor. Stokes assumptions for a wide class of fluids therefore says that for an isotropic fluid the compression and shear stresses are proportional to their velocity gradients, ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {S} _{0}}$ respectively, and named this class of fluids for Newtonian fluids. The classic defining equation for volume viscosity ${\displaystyle \zeta }$ and shear viscosity ${\displaystyle \eta }$ are respectively:

${\displaystyle {\boldsymbol {\tau }}_{c}=3\zeta \mathbf {C} }$

${\displaystyle {\boldsymbol {\tau }}_{s}=2\eta \mathbf {S} _{0}}$

The classic compression velocity "gradient" is a diagonal tensor that describes a compressing (alt. expanding) flow or attenuating sound waves:

${\displaystyle \mathbf {C} ={\frac {1}{3}}\left(\nabla \!\cdot \!\mathbf {u} \right)\mathbf {I} }$

The classic Cauchy shear velocity gradient, is a symmetric and traceless tensor that describes a pure shear flow (where pure means excluding normal outflow which in mathematical terms means a traceless matrix) around e.g. a wing, propeller, ship hull or in e.g. a river, pipe or vein with or without bends and boundary skin:

${\displaystyle \mathbf {S} _{0}=\mathbf {S} -{\frac {1}{3}}\left(\nabla \!\cdot \!\mathbf {u} \right)\mathbf {I} }$

where the symmetric gradient matrix with non-zero trace is

${\displaystyle \mathbf {S} ={\frac {1}{2}}\left[\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right]}$

How much the volume viscosity contributes to the flow characteristics in e.g. a choked flow such as convergent-divergent nozzle or valve flow is not well known, but the shear viscosity is by far the most utilized viscosity coefficient. The volume viscosity will now be abandoned, and the rest of the article will focus on the shear viscosity.

Another application of shear viscosity models is Darcy's law for multiphase flow.

${\displaystyle \mathbf {u} _{a}=-\eta _{a}^{-1}\mathbf {K} _{ra}\cdot \mathbf {K} \cdot \left(\nabla P_{a}-\rho _{a}\mathbf {g} \right)}$  where a = water, oil, gas

and ${\displaystyle \mathbf {K} }$ and ${\displaystyle \mathbf {K} _{ra}}$ are absolute and relative permeability, respectively. These 3 (vector) equations models flow of water, oil and natural gas in subsurface oil and gas reservoirs in porous rocks. Although the pressures changes are big, the fluid phases will flow slowly through the reservoir due to the flow restriction caused by the porous rock.

The above definition is based on a shear-driven fluid motion that in its most general form is modelled by a shear stress tensor and a velocity gradient tensor. The fluid dynamics of a shear flow is, however, very well illustrated by the simple Couette flow. In this experimental layout, the shear stress ${\displaystyle {\boldsymbol {\tau }}_{s}}$ and the shear velocity gradient ${\displaystyle \mathbf {S} _{0}}$ (where now ${\displaystyle \mathbf {S} _{0}=\mathbf {S} }$) takes the simple form:

${\displaystyle \tau =\eta S\quad {\text{where}}\quad \tau ={\frac {F}{A}}\quad {\text{and}}\quad S={du_{} \over dy}={u_{max} \over y_{max}}}$

Inserting these simplifications gives us a defining equation that can be used to interpret experimental measurements:

${\displaystyle {\frac {F}{A}}=\eta {du_{} \over dy}=\eta {u_{max} \over y_{max}}}$

where ${\displaystyle A}$ is the area of the moving plate and the stagnant plate, ${\displaystyle y}$ is the spatial coordinate normal to the plates. In this experimental setup, value for the force ${\displaystyle F}$ is first selected. Then a maximum velocity ${\displaystyle u_{max}}$ is measured, and finally both values are entered in the equation to calculate viscosity. This gives one value for the viscosity of the selected fluid. If another value of the force is selected, another maximum velocity will be measured. This will result in another viscosity value if the fluid is a non-Newtonian fluid such as paint, but it will give the same viscosity value for a Newtonian fluid such as water, petroleum oil or gas. If another parameter like temperature, ${\displaystyle T}$, is changed, and the experiment is repeated with the same force, a new value for viscosity will be the calculated, for both non-Newtonian and Newtonian fluids. The great majority of material properties varies as a function of temperature, and this goes for viscosity also. The viscosity is also a function of pressure and, of course, the material itself. For a fluid mixture, this means that the shear viscosity will also vary according to the fluid composition. To map the viscosity as a function of all these variables require a large sequence of experiments that generates an even larger set of numbers called measured data, observed data or observations. Prior to, or at the same time as, the experiments is a material property model (or short material model) proposed to describe or explain the observations. This mathematical model is called the constitutive equation for shear viscosity. It is usually an explicit function that contains some empirical parameters that is adjusted in order to match the observations as good as the mathematical function is capable to do.

For a Newtonian fluid, the constitutive equation for shear viscosity is generally a function of temperature, pressure, fluid composition:

${\displaystyle \eta =f(T,P,\mathbf {w} )\quad {\text{where}}\quad \mathbf {w} =\mathbf {x} ,\mathbf {y} ,\mathbf {z} ,1_{purefluid}}$

where ${\displaystyle \mathbf {x} }$ is the liquid phase composition with molfraction ${\displaystyle x_{i}}$ for fluid component i, and ${\displaystyle \mathbf {y} }$ and ${\displaystyle \mathbf {z} }$ are the gas phase and total fluid compositions, respectively. For a non-Newtonian fluid (in the sense of a generalized Newtonian fluid), the constitutive equation for shear viscosity is also a function of the shear velocity gradient:

${\displaystyle \eta =f(T,P,\mathbf {w} ,\mathbf {S} _{0})\quad {\text{where}}\quad \mathbf {w} =\mathbf {x} ,\mathbf {y} ,\mathbf {z} ,1_{purefluid}}$

The existence of the velocity gradient in the functional relationship for non-Newtonian fluids says that viscosity is generally not an equation of state, so the term constitutional equation will in general be used for viscosity equations (or functions). The free variables in the two equations above, also indicates that specific constitutive equations for shear viscosity will be quite different from the simple defining equation for shear viscosity that is shown further up. The rest of this article will show that this is certainly true. Non-Newtonian fluids will therefore be abandoned, and the rest of this article will focus on Newtonian fluids.

In textbooks on elementary kinetic theory^[1] one can find results for dilute gas modeling that have widespread use. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. This non-equilibrium flow is superimposed on a Maxwell–Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle \sigma }$ be the collision cross section of one molecule colliding with another. The number density ${\displaystyle C}$ is defined as the number of molecules per (extensive) volume ${\displaystyle C=N/V}$. The collision cross section per volume or collision cross section density is ${\displaystyle C\sigma }$, and it is related to the mean free path ${\displaystyle l}$ by

${\displaystyle l={\frac {1}{{\sqrt {2}}C\sigma }}}$

Combining the kinetic equations for molecular motion with the defining equation of shear viscosity gives the well known equation for shear viscosity for dilute gases:

${\displaystyle \eta _{0}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {mk_{B}T}}{\sigma }}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {MRT}}{\sigma N_{A}}}}$

where

${\displaystyle k_{B}\cdot N_{A}=R\quad {\text{and}}\quad M=m\cdot N_{A}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle R}$ is the gas constant, ${\displaystyle M}$ is the molar mass and ${\displaystyle m}$ is the molecular mass. The equation above presupposes that the gas density is low (i.e. the pressure is low), hence the subscript zero in the variable ${\displaystyle \eta _{0}}$. This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation displayed above further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic hard core particles of spherical shape. This assumption of particles being like billiard balls with radius ${\displaystyle r}$, implies that the collision cross section of one molecule can be estimated by

${\displaystyle \sigma =\pi \left(2r_{}\right)^{2}=\pi d^{2}\qquad \qquad \qquad \,\quad {\text{for monomolecular gases and monoparticle beam experiments }}}$

${\displaystyle \sigma _{ij}=\pi \left(r_{i}+r_{j}\right)^{2}={\frac {\pi }{4}}\left(d_{i}+d_{j}\right)^{2}\quad {\text{for binary collision in gas mixtures and dissimilar bullet / target particles}}}$

But molecules are not hard particles. For a reasonably spherical molecule the interaction potential is more like the Lennard-Jones potential or even more like the Morse potential. Both have a negative part that attracts the other molecule from distances much longer than the hard core radius, and thus models the van der Waals forces. The positive part models the repulsive forces as the electron clouds of the two molecules overlap. The radius for zero interaction potential is therefore appropriate for estimating (or defining) the collision cross section in kinetic gas theory, and the r-parameter (conf. ${\displaystyle r,r_{i}}$) is therefore called kinetic radius. The d-parameter (where ${\displaystyle d=2r,d_{i}=2r_{i}}$) is called kinetic diameter.

The macroscopic collision cross section ${\displaystyle \sigma \cdot N_{A}}$ is often associated with the critical molar volume ${\displaystyle V_{c}}$, and often without further proof or supporting arguments, by

${\displaystyle \sigma N_{A}\propto V_{c}^{2/3}\quad {\text{or}}\quad \sigma N_{A}={\frac {2}{3{\sqrt {\pi }}}}\cdot K_{rv}^{-1}V_{c}^{2/3}}$

where ${\displaystyle K_{rv}}$ is molecular shape parameter that is taken as an empirical tuning parameter, and the pure numerical part is included in order to make the final viscosity formula more suitably for practical use. Inserting this interpretation of ${\displaystyle \sigma N_{A}}$, and use of reduced temperature ${\displaystyle T_{r}}$, gives

${\displaystyle \eta _{0}={\sqrt {T_{r}}}K_{rv}D_{rv}\quad {\text{where}}\quad T_{r}=T/T_{c}\quad {\text{and}}}$

${\displaystyle D_{rv}=\left(MRT_{c}\right)^{1/2}V_{c}^{-2/3}=R^{1/2}D_{v}}$

which implies that the empirical parameter ${\displaystyle K_{rv}}$ is dimensionless, and that ${\displaystyle D_{rv}}$ and ${\displaystyle \eta _{0}}$ have the same units. The parameter ${\displaystyle D_{rv}}$ is a scaling parameter that involves the gas constant ${\displaystyle R}$ and the critical molar volume ${\displaystyle V_{c}}$, and it used to scale the viscosity. In this article the viscosity scaling parameter will frequently be denoted by ${\displaystyle D_{xyz}}$ which involve one or more of the parameters ${\displaystyle R}$, ${\displaystyle V_{c}}$, ${\displaystyle P_{c}}$ in addition to critical temperature ${\displaystyle T_{c}}$ and molar mass ${\displaystyle M}$. Incomplete scaling parameters, such as the parameter ${\displaystyle D_{v}}$ above where the gas constant ${\displaystyle R}$ is absorbed into the empirical constant, will often be encountered in practice. In this case the viscosity equation becomes

${\displaystyle \eta _{0}={\sqrt {T_{r}}}K_{v}D_{v}}$

where the empirical parameter ${\displaystyle K_{v}}$ is not dimensionless, and a proposed viscosity model for dense fluid will not be dimensionless if ${\displaystyle D_{v}}$ is the common scaling factor. Notice that

${\displaystyle \eta _{0}={\sqrt {T_{r}}}K_{rv}D_{rv}={\sqrt {T_{r}}}K_{v}D_{v}\implies K_{v}=R^{1/2}K_{rv}}$

Inserting the critical temperature in the equation for dilute viscosity gives

${\displaystyle \eta _{0c}=K_{rv}D_{rv}=K_{v}D_{v}}$

The default values of the parameters ${\displaystyle K_{rv}}$ and ${\displaystyle K_{v}}$ should be fairly universal values, although ${\displaystyle K_{v}}$ depends on the unit system. However, the critical molar volume in the scaling parameters ${\displaystyle D_{rv}}$ and ${\displaystyle D_{v}}$ is not easily accessible from experimental measurements, and that is a significant disadvantage. The general equation of state for a real gas is usually written as

${\displaystyle PV=ZRT\implies P_{c}V_{c}=Z_{c}RT_{c}}$

where the critical compressibility factor ${\displaystyle Z_{c}}$, which reflects the volumetric deviation of the real gases from the ideal gas, is also not easily accessible from laboratory experiments. However, critical pressure and critical temperature are more accessible from measurements. It should be added that critical viscosity is also not readily available from experiments.

Uyehara and Watson (1944)^[4] proposed to absorb a universal average value of ${\displaystyle Z_{c}}$ (and the gas constant ${\displaystyle R}$) into a default value of the tuning parameter ${\displaystyle K_{p}}$ as a practical solution of the difficulties of getting experimental values for ${\displaystyle V_{c}}$ and/or ${\displaystyle Z_{c}}$. The visocity model for a dilute gas is then

${\displaystyle \eta _{0}={\sqrt {T_{r}}}K_{p}D_{p}\quad {\text{where}}\quad T_{r}=T/T_{c}\quad {\text{and}}}$

${\displaystyle D_{p}=T_{c}^{-1/6}P_{c}^{2/3}M^{1/2}}$

By inserting the critical temperature in the formula above, the critical viscosity is calculated as

${\displaystyle \eta _{0c}=K_{p}D_{p}}$

Based on an average critical compressibility factor of ${\displaystyle {\bar {Z}}_{c}=0.275}$ and measured critical viscosity values of 60 different molecule types, Uyehara and Watson (1944)^[4] determined an average value of ${\displaystyle K_{p}}$ to be

${\displaystyle {\bar {K}}_{p}=7.7\cdot 1.01325^{2/3}\approx 7.77\quad {\text{for}}\quad \left[\eta _{0}\right]=\mu P\quad {\text{and}}\quad \left[P_{c}\right]=bar}$

The cubic equation of state (EOS) are very popular equations that are sufficiently accurate for most industrial computations both in vapor-liquid equilibrium and molar volume. Their weakest points are perhaps molar volum in the liquid region and in the critical region. Accepting the cubic EOS, the molar hard core volume ${\displaystyle b}$ can be calculated from the turning point constraint at the critical point. This gives

${\displaystyle b=\Omega _{b}{\frac {RT_{c}}{P_{c}}}\quad {\text{which is similar to}}\quad V_{c}={\bar {Z}}_{c}{\frac {RT_{c}}{P_{c}}}}$

where the constant ${\displaystyle \Omega _{b}}$ is a universal constant that is specific for the selected variant of the cubic EOS. This says that using ${\displaystyle D_{p}}$, and disregarding fluid component variations of ${\displaystyle Z_{c}}$, is in practice equivalent to say that the macroscopic collision cross section is proportional to the hard core molar volume rather than the critical molar volume.

In a fluid mixture like a petroleum gas or oil there are lots of molecule types, and within this mixture there are families of molecule types (i.e. groups of fluid components). The simplest group is the n-alkanes which are long chains of CH₂-elements. The more CH₂-elements, or carbon atoms, the longer molecule. Critical viscosity and critical thermodynamic properties of n-alkanes therefore show a trend, or functional behaviour, when plotted against molecular mass or number of carbon atoms in the molecule (i.e. carbon number). Parameters in equations for properties like viscosity usually also show such trend behaviour. This means that

${\displaystyle \eta _{0cj}=K_{pj}D_{pj}\neq {\bar {K}}_{p}D_{pj}\quad {\text{for many or most fluid components j }}}$

This says that the scaling parameter ${\displaystyle D_{p}}$ alone is not a true or complete scaling factor unless all fluid components have a fairly similar (and preferably spherical) shape.

The most important result of this kinetic derivation is perhaps not the viscosity formula, but the semi-empirical parameter ${\displaystyle D_{p}}$ that is used extensively throughout the industry and applied science communities as a scaling factor for (shear) viscosity. The literature often reports the reciprocal parameter and denotes it as ${\displaystyle \xi }$.

The dilute gas viscosity contribution to the total viscosity of a fluid will only be important when predicting the viscosity of vapors at low pressures or the viscosity of dense fluids at high temperatures. The viscosity model for dilute gas, that is shown above, is widely used throughout the industry and applied science communities. Therefore, many researchers do not specify a dilute gas viscosity model when they propose a total viscosity model, but leave it to the user to select and include the dilute gas contribution. Some researchers do not include a separate dilute gas model term, but propose an overall gas viscosity model that cover the entire pressure and temperature range they investigated.

In this section our central macroscopic variables and parameters and their units are temperature ${\displaystyle T}$ [K], pressure ${\displaystyle P}$ [bar], molar mass ${\displaystyle M}$ [g/mol], low density (low pressure or dilute) gas viscosity ${\displaystyle \eta _{0}}$ [μP]. It is, however, common in the industry to use another unit for liquid and high density gas viscosity ${\displaystyle \eta }$ [cP].

From Boltzmann's equation Chapman and Enskog derived a viscosity model for a dilute gas.

${\displaystyle \eta _{0}\times 10^{6}=2.6693{\frac {\sqrt {MT}}{\sigma ^{2}\Omega \left(T^{*}\right)}}\quad {\text{where}}\quad T^{*}=k_{B}T/\varepsilon }$

where ${\displaystyle \varepsilon }$ is (the absolute value of) the energy-depth of the potential well (see e.g. Lennard-Jones interaction potential). The term ${\displaystyle \Omega (T^{*})}$ is called the collision integral, and it occurs as a general function of temperature that the user must specify, and that is not a simple task. This illustrates the situation for the molecular or statistical approach: The (analytical) mathematics gets incredible complex for polar and non-spherical molecules making it very difficult to achieve practical models for viscosity based on a statistical approach. The purely statistical approach will therefore be left out in the rest of this article.

Zéberg-Mikkelsen (2001)^[3] proposed empirical models for gas viscosity of fairly spherical molecules that is displayed in the section on Friction Force theory and its models for dilute gases and simple light gases. These simple empirical correlations illustrate that empirical methods competes with the statistical approach with respect to gas viscosity models for simple fluids (simple molecules).

The gas viscosity model of Chung et alios (1988)^[5] is combination of the Chapman–Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et alios (1972)^[6] for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range. This viscosity model illustrates a successful combination of kinetic theory and empiricism, and it is displayed in the section of Significant structure theory and its model for the gas-like contribution to the total fluid viscosity.

In the section with models based on elementary kinetic theory, several variants of scaling the viscosity equation was discussed, and they are displayed below for fluid component i, as a service to the reader.

${\displaystyle \eta _{0i}={\sqrt {T_{ri}}}K_{rvi}D_{rvi}\quad {\text{where}}\quad D_{rvi}={\sqrt {M_{i}RT_{ci}}}\cdot V_{ci}^{-2/3}}$

${\displaystyle \eta _{0i}={\sqrt {T_{ri}}}K_{vi}D_{vi}\ \ \,\quad {\text{where}}\quad D_{vi}\ \ ={\sqrt {M_{i}T_{ci}}}\cdot V_{ci}^{-2/3}}$

${\displaystyle \eta _{0i}={\sqrt {T_{ri}}}K_{pi}D_{pi}\ \ \,\quad {\text{where}}\quad D_{pi}\ =M_{i}^{1/2}P_{ci}^{2/3}\cdot T_{ci}^{-1/6}}$

Zéberg-Mikkelsen (2001)^[3] proposed an empirical correlation for the ${\displaystyle V_{ci}}$ parameter for n-alkanes, which is

${\displaystyle V_{ci}^{-1}=A+B\cdot {\frac {P_{ci}}{RT_{ci}}}\iff V_{ci}={\frac {RT_{ci}}{ART_{ci}+BP_{ci}}}}$

${\displaystyle A=0.000235751\ mol/cm^{3}\quad {\text{and}}\quad B=3.42770}$

The critical molar volume of component i ${\displaystyle V_{ci}}$ is related to the critical mole density ${\displaystyle \rho _{nci}}$ and critical mole concentration ${\displaystyle c_{ci}}$ by the equation ${\displaystyle V_{ci}^{-1}=\rho _{nci}=c_{ci}}$. From the above equation for ${\displaystyle V_{ci}^{-1}}$ it follows that

${\displaystyle Z_{ci}={\frac {P_{ci}}{ART_{ci}+BP_{ci}}}\iff {\frac {Z_{ci}RT_{ci}}{P_{ci}V_{ci}}}=1}$

where ${\displaystyle Z_{ci}}$ is the compressibility factor for component i, which is often used as an alternative to ${\displaystyle V_{ci}}$. By establishing a trend function for the parameter ${\displaystyle V_{ci}}$ for a homologous series, groups or families of molecules, parameter values for unknown fluid components in the homologous group can be found by interpolation and extrapolation, and parameter values can easily re-generateat at later need. Use of trend functions for parameters of homologous groups of molecules have greatly enhanced the usefulness of viscosity equations (and thermodynamic EOSs) for fluid mixtures such as petroleum gas and oil.^[2]

Uyehara and Watson (1944)^[4] proposed a correlation for critical viscosity (for fluid component i) for n-alkanes using their average parameter ${\displaystyle {\bar {K}}_{p}}$ and the classical pressure dominated scaling parameter ${\displaystyle D_{pi}}$ :

${\displaystyle \eta _{ci}={\bar {K}}_{p}D_{pi}}$

${\displaystyle \ \ {\bar {K}}_{p}\,=7.7\cdot 1.01325^{2/3}\approx 7.77\quad {\text{for}}\quad \left[\eta _{0}\right]=\mu P\quad {\text{and}}\quad \left[P_{c}\right]=bar}$

Zéberg-Mikkelsen (2001)^[3] proposed an empirical correlation for critical viscosity η_ci parameter for n-alkanes, which is

${\displaystyle \eta _{ci}=C\cdot P_{ci}M_{i}^{D}}$

${\displaystyle \ C=0.597556\ \mu P/bar\cdot (g/mol)^{-D}\quad {\text{and}}\quad D=0.601652}$

The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are

${\displaystyle [P_{c}]=bar\quad {\text{and}}\quad [V_{c}]=[RT_{c}/P_{c}]=cm^{3}/mol\quad {\text{and}}\quad [T]=K\quad {\text{and}}\quad [Z_{c}]=1\quad {\text{and}}\quad [\eta _{c}]=\mu P}$

Inserting the critical temperature in the three viscosity equations from elementary kinetic theory gives three parameter equations.

${\displaystyle \eta _{ci}=K_{rvi}D_{rvi}=K_{vi}D_{vi}=K_{pi}D_{pi}\quad {\text{or}}\quad }$

${\displaystyle K_{rvi}={\frac {\eta _{ci}}{D_{rvi}}}\quad {\text{and}}\quad K_{vi}={\frac {\eta _{ci}}{D_{vi}}}\quad {\text{and}}\quad K_{pi}={\frac {\eta _{ci}}{D_{pi}}}}$

The three viscosity equations now coalesce to a single viscosity equation

${\displaystyle \eta _{0i}={\sqrt {T_{ri}}}\eta _{ci}={\sqrt {T}}{\frac {\eta _{ci}}{\sqrt {T_{ci}}}}}$

because a nondimensional scaling is used for the entire viscosity equation. The standard nondimensionality reasoning goes like this: Creating nondimensional variables (with subscript D) by scaling gives

${\displaystyle \eta _{Di}={\frac {\eta _{0i}}{\eta _{ci}}}\quad {\text{and}}\quad T_{Di}={\frac {T}{T_{ci}}}=T_{ri}\implies \eta _{Di}\eta _{ci}={\sqrt {T_{Di}}}K_{pi}D_{pi}}$

Claiming nondimensionality gives

${\displaystyle {\frac {K_{pi}D_{pi}}{\eta _{ci}}}=1\iff K_{pi}={\frac {\eta _{ci}}{D_{pi}}}\implies \eta _{Di}={\sqrt {T_{Di}}}}$

The collision cross section and the critical molar volume which are both difficult to access experimentally, are avoided or circumvented. On the other hand, the critical viscosity has appeared as a new parameter, and critical viscosity is just as difficult to access experimentally as the other two parameters. Fortunately, the best viscosity equations have become so accurate that they justify calculation in the critical point, especially if the equation is matched to surrounding experimental data points.

Wilke (1950)^[7] derived a mixing rule based on kinetic gas theory

${\displaystyle \eta _{gmix}=\sum _{i=1}^{N}{\frac {\eta _{gi}}{1+{\frac {1}{y_{i}}}\sum _{j=1,j\neq i}^{N}y_{j}\varphi _{ij}}}}$

${\displaystyle \varphi _{ij}={\frac {\left[1+{\sqrt[{2}]{\frac {\eta _{0i}}{\eta _{0j}}}}\cdot {\sqrt[{4}]{\frac {M_{j}}{M_{i}}}}\right]^{2}}{{\frac {4}{\sqrt[{2}]{2}}}{\sqrt[{2}]{1+{\frac {M_{i}}{M_{j}}}}}}}}$

The Wilke mixing rule is capable of describing the correct viscosity behavior of gas mixtures showing a nonlinear and non-monotonical behavior, or showing a characteristic bump shape, when the viscosity is plotted versus mass density at critical temperature, for mixtures containing molecules of very different sizes. Due to its complexity, it has not gained widespread use. Instead, the slightly simpler mixing rule proposed by Herning and Zipperer (1936),^[8] is found to be suitable for gases of hydrocarbon mixtures.

The classic Arrhenius (1887).^[9] mixing rule for liquid mixtures is

${\displaystyle \ln \eta _{lmix}=\sum _{i=1}^{N}x_{i}\ln \eta _{li}}$

where ${\displaystyle \eta _{lmix}}$ is the viscosity of the liquid mixture, ${\displaystyle \eta _{li}}$ is the viscosity (equation) for fluid component i when flowing as a pure fluid, and ${\displaystyle x_{i}}$ is the molfraction of component i in the liquid mixture.

The Grunberg-Nissan (1949)^[10] mixing rule extends the Arrhenius rule to

${\displaystyle \ln \eta _{lmix}=\sum _{i=1}^{N}x_{i}\ln \eta _{li}+\sum _{i=1}^{N}\sum _{j=1}^{N}x_{i}x_{j}d_{ij}}$

where ${\displaystyle d_{ij}}$ are empiric binary interaction coefficients that are special for the Grunberg-Nissan theory. Binary interaction coefficients are widely used in cubic EOS where they often are used as tuning parameters, especially if component j is an uncertain component (i.e. have uncertain parameter values).

Katti-Chaudhri (1964)^[11] mixing rule is

${\displaystyle \ln \left(\eta _{lmix}V_{lmix}\right)=\sum _{i=1}^{N}x_{i}\ln \left(\eta _{li}V_{li}\right)}$

where ${\displaystyle V_{li}}$ is the partial molar volume of component i, and ${\displaystyle V_{lmix}}$ is the molar volume of the liquid phase and comes from the vapor-liquid equilibrium (VLE) calculation or the EOS for single phase liquid.

A modification of the Katti-Chaudhri mixing rule is

${\displaystyle \ln \left(\eta _{lmix}V\right)=\sum _{i=1}^{N}z_{i}\ln \left(\eta _{li}V_{li}\right)+{\frac {\Delta G^{E}}{RT}}}$

${\displaystyle \Delta G^{E}=\sum _{i=1}^{N}\sum _{j=1}^{N}z_{i}z_{j}E_{ij}}$

where ${\displaystyle G^{E}}$ is the excess activation energy of the viscous flow, and ${\displaystyle E_{ij}}$ is the energy that is characteristic of intermolecular interactions between component i and component j, and therefore is responsible for the excess energy of activation for viscous flow. This mixing rule is theoretically justified by Eyring's representation of the viscosity of a pure fluid according to Glasstone et alios (1941).^[12] The quantity ${\displaystyle \eta _{li}V_{li}}$ has been obtained from the time-correlation expression for shear viscosity by Zwanzig (1965).^[13]

Very often one simply selects a known correlation for the dilute gas viscosity ${\displaystyle \eta _{0}}$, and subtracts this contribution from the total viscosity which is measured in the laboratory. This gives a residual viscosity term, often denoted ${\displaystyle \Delta \eta }$, which represents the contribution of the dense fluid, ${\displaystyle \eta _{df}}$.

${\displaystyle \eta _{df}=\eta -\eta _{0}\quad \iff \quad \eta =\eta _{0}+\eta _{df}}$

The dense fluid viscosity is thus defined as the viscosity in excess of the dilute gas viscosity. This technique is often used in developing mathematical models for both purely empirical correlations and models with a theoretical support. The dilute gas viscosity contribution becomes important when the zero density limit (i.e. zero pressure limit) is approached. It is also very common to scale the dense fluid viscosity by the critical viscosity, or by an estimate of the critical viscosity, which is a characteristic point far into the dense fluid region. The simplest model of the dense fluid viscosity is a (truncated) power series of reduced mole density or pressure. Jossi et al. (1962)^[14] presented such a model based on reduced mole density, but its most widespread form is the version proposed by Lohrenz et al. (1964)^[15] which is displayed below.

${\displaystyle \left[{\frac {\eta _{df}}{D_{p}}}+10^{-4}\right]^{1/4}=LBC_{}}$

The LBC-function is then expanded in a (truncated) power series with empirical coefficients as displayed below.

${\displaystyle LBC_{}=LBC_{}\left(\rho _{nr}\right)=\sum _{i=1}^{5}a_{i}\rho _{nr}^{i-1}}$

The final viscosity equation is thus

${\displaystyle \eta =\eta _{0}-10^{-4}D_{p}+D_{p}L_{}^{4}}$

${\displaystyle \eta _{0}=\eta _{0}\left(T\right)}$

${\displaystyle D_{p}=T_{c}^{-1/6}P_{c}^{2/3}M_{n}^{1/2}}$

Local nomenclature list:

${\displaystyle \eta _{mix}=\eta _{0mix}-10^{-4}D_{pmix}+D_{pmix}L_{mix}^{4}}$

${\displaystyle LBC_{mix}=LBC_{mix}\left(c_{rmix}\right)=\sum _{i=1}^{5}a_{i}c_{rmix}^{i-1}}$

${\displaystyle D_{pmix}=T_{cmix}^{-1/6}P_{cmix}^{2/3}M_{mix}^{1/2}}$

${\displaystyle \eta _{0mix}=\eta _{0mix}\left(T\right)}$

The formula for ${\displaystyle \eta _{0}}$ that was chosen by LBC, is displayed in the section called Dilute gas contribution.

LBC constants

i	${\displaystyle a_{i}}$
1	0.10230
2	0.023364
3	0.058533
4	−0.040758
5	0.0093324

${\displaystyle T_{cmix}=\sum _{i}z_{i}T_{ci}}$

${\displaystyle M_{mix}=M_{n}=\sum _{i}z_{i}M_{i}}$

${\displaystyle P_{cmix}=\sum _{i}z_{i}P_{ci}}$

${\displaystyle \rho _{ncmix}^{-1}=V_{cmix}=\sum _{i}z_{i}V_{ci}+z_{C7+}\cdot V_{cC7+}\quad i<C7+}$

The subscript C7+ refers to the collection of hydrocarbon molecules in a reservoir fluid with oil and/or gas that have 7 or more carbon atoms in the molecule. The critical volume of C7+ fraction has unit ft³/lb mole, and it is calculated by

${\displaystyle V_{cC7+}=21.573+0.015122\cdot M_{C7+}-27.656\cdot SG_{C7+}+0.070615\cdot M_{C7+}SG_{C7+}}$

where ${\displaystyle SG_{C7+}}$ is the specific gravity of the C7+ fraction.

${\displaystyle T_{ci}\quad {\text{for}}\quad i\geq C7+\quad {\text{or}}\quad T_{cC7+}\quad {\text{is taken from EOS characterization}}}$

${\displaystyle M_{i}\quad {\text{for}}\quad i\geq C7+\quad {\text{or}}\quad M_{C7+}\quad {\text{is taken from EOS characterization}}}$

${\displaystyle P_{ci}\quad {\text{for}}\quad i\geq C7+\quad {\text{or}}\quad P_{cC7+}\quad {\text{is taken from EOS characterization}}}$

The molar mass ${\displaystyle M_{i}}$ (or molecular mass) is normally not included in the EOS formula, but it usually enters the characterization of the EOS parameters.

From the equation of state the molar volume of the reservoir fluid (mixture) is calculated.

${\displaystyle V_{mix}=V_{mix}(T,P)\quad {\text{for 1 mole fluid}}}$

The molar volume ${\displaystyle V}$ is converted to mole density ${\displaystyle \rho _{n}}$ (also called mole concentration and denoted ${\displaystyle c}$), and then scaled to be reduced mole density ${\displaystyle \rho _{nr}}$.

${\displaystyle \rho _{nmix}=1/V_{mix}\quad and\quad \rho _{ncmix}=1/V_{cmix}\quad and\quad \rho _{nrmix}=V_{cmix}/V_{mix}=\rho _{nmix}/\rho _{ncmix}}$

The correlation for dilute gas viscosity of a mixture is taken from Herning and Zipperer (1936)^[8] and is

${\displaystyle \eta _{0mix}\left(T\right)={\frac {\sum _{i}z_{i}\eta _{0i}\left(T_{ri}\right)M_{i}^{1/2}}{\sum _{j}z_{j}M_{j}^{1/2}}}\quad i,j<C7+}$

The correlation for dilute gas viscosity of the individual components is taken from Stiel and Thodos (1961)^[16] and is

${\displaystyle \eta _{0i}\left(T_{ri}\right)={\begin{cases}34\times 10^{-5}\cdot D_{pi}T_{ri}^{0.94}&{\text{if}}\quad T_{ri}\leqslant 1.5\\17.78\times 10^{-5}\cdot D_{pi}\left(4.58\cdot T_{ri}-1.67\right)^{5/8}&{\text{if}}\quad T_{ri}>1.5\end{cases}}}$

where

${\displaystyle D_{pi}=T_{ci}^{-1/6}P_{ci}^{2/3}M_{i}^{1/2}\quad i<C7+}$

${\displaystyle T_{ri}={\frac {T}{T_{ci}}}\quad i<C7+}$

The principle of corresponding states (CS principle or CSP) was first formulated by van der Waals, and it says that two fluids (subscript a and z) of a group (e.g. fluids of non-polar molecules) have approximately the same reduced molar volume (or reduced compressibility factor) when compared at the same reduced temperature and reduced pressure. In mathematical terms this is

${\displaystyle {\frac {V_{a}\left(P_{ra},T_{ra}\right)}{V_{ca}}}={\frac {V_{z}\left(P_{rz},T_{rz}\right)}{V_{cz}}}\iff V_{a}\left(P_{a},T_{a}\right)={\frac {V_{ca}}{V_{cz}}}\cdot V_{z}\left(P_{z}={\frac {P_{a}P_{cz}}{P_{ca}}},T_{z}={\frac {T_{a}T_{cz}}{T_{ca}}}\right)}$

When the common CS principle above is applied to viscosity, it reads

${\displaystyle \eta \left(P,T\right)={\frac {\eta _{c}}{\eta _{cz}}}\cdot \eta _{z}\left(P_{z},T_{z}\right)\approx {\frac {K_{p}D_{p}}{K_{pz}D_{pz}}}\cdot \eta _{z}\left(P_{z},T_{z}\right)}$

Note that the CS principle was originally formulated for equilibrium states, but it is now applied on a transport property - viscosity, and this tells us that another CS formula may be needed for viscosity.

In order to increase the calculation speed for viscosity calculations based on CS theory, which is important in e.g. compositional reservoir simulations, while keeping the accuracy of the CS method, Pedersen et al. (1984, 1987, 1989)^[17][18][2] proposed a CS method that uses a simple (or conventional) CS formula when calculating the reduced mass density that is used in the rotational coupling constants (displayed in the sections below), and a more complex CS formula, involving the rotational coupling constants, elsewhere.

The simple corresponding state principle is extended by including a rotational coupling coefficient ${\displaystyle \alpha }$ as suggested by Tham and Gubbins (1970).^[19] The reference fluid is methane, and it is given the subscript z.

${\displaystyle \eta _{mix}\left(P,T\right)=\left({\frac {T_{cmix}}{T_{cz}}}\right)^{-1/6}\cdot \left({\frac {P_{cmix}}{P_{cz}}}\right)^{2/3}\cdot \left({\frac {M_{mix}}{M_{z}}}\right)^{1/2}\cdot {\frac {\alpha _{cmix}}{\alpha _{cz}}}\cdot \eta _{z}\left(P_{z},T_{z}\right)}$

${\displaystyle P_{z}={\frac {P\cdot P_{cz}\alpha _{z}}{P_{cmix}\alpha _{mix}}}}$

${\displaystyle T_{z}={\frac {T\cdot T_{cz}\alpha _{z}}{T_{cmix}\alpha _{mix}}}}$

The interaction terms for critical temperature and critical volume are

${\displaystyle T_{cij}=\left(T_{ci}T_{cj}\right)^{1/2}}$

${\displaystyle V_{cij}={\frac {1}{8}}\left(V_{ci}^{1/3}+V_{cj}^{1/3}\right)^{3}}$

The parameter ${\displaystyle V_{ci}}$ is usually uncertain or not available. One therefore wants to avoid this parameter. Replacing ${\displaystyle Z_{ci}}$ with the generic average parameter ${\displaystyle {\bar {Z}}_{c}}$ for all components, gives

${\displaystyle V_{ci}=RZ_{ci}T_{ci}/P_{ci}={\bar {R}}_{zc}T_{ci}/P_{ci}\quad {\text{where}}\quad {\bar {R}}_{zc}=R{\bar {Z}}_{c}}$

${\displaystyle V_{cij}={\frac {1}{8}}R_{zc}\left(\left({\frac {T_{ci}}{P_{ci}}}\right)^{1/3}+\left({\frac {T_{cj}}{P_{cj}}}\right)^{1/3}\right)^{3}}$

${\displaystyle T_{cmix}={\frac {\sum _{i}\sum _{j}z_{i}z_{j}V_{cij}T_{cij}}{\sum _{i}\sum _{j}z_{i}z_{j}V_{cij}}}}$

The above expression for ${\displaystyle V_{cij}}$ is now inserted into the equation for ${\displaystyle T_{cmix}}$. This gives the following mixing rule

${\displaystyle T_{cmix}={\frac {\sum _{i}\sum _{j}z_{i}z_{j}\left(\left({\frac {T_{ci}}{P_{ci}}}\right)^{1/3}+\left({\frac {T_{cj}}{P_{cj}}}\right)^{1/3}\right)^{3}\left(T_{ci}T_{cj}\right)^{1/2}}{\sum _{i}\sum _{j}z_{i}z_{j}\left(\left({\frac {T_{ci}}{P_{ci}}}\right)^{1/3}+\left({\frac {T_{cj}}{P_{cj}}}\right)^{1/3}\right)^{3}}}}$

Mixing rule for the critical pressure of the mixture is established in a similar way.

${\displaystyle P_{cmix}=R_{zc}T_{cmix}/V_{cmix}}$

${\displaystyle V_{cmix}=\sum _{i}\sum _{j}z_{i}z_{j}V_{cij}}$

${\displaystyle P_{cmix}={\frac {8\sum _{i}\sum _{j}z_{i}z_{j}\left(\left({\frac {T_{ci}}{P_{ci}}}\right)^{1/3}+\left({\frac {T_{cj}}{P_{cj}}}\right)^{1/3}\right)^{3}\left(T_{ci}T_{cj}\right)^{1/2}}{\left(\sum _{i}\sum _{j}z_{i}z_{j}\left(\left({\frac {T_{ci}}{P_{ci}}}\right)^{1/3}+\left({\frac {T_{cj}}{P_{cj}}}\right)^{1/3}\right)^{3}\right)^{2}}}}$

The mixing rule for molecular weight is much simpler, but it is not entirely intuitive. It is an empirical combination of the more intuitive formulas with mass weighting ${\displaystyle {\overline {M}}_{w}}$ and mole weighting ${\displaystyle {\overline {M}}_{n}}$.

${\displaystyle M_{mix}=1.304\times 10^{-4}\left({\overline {M}}_{w}^{2.303}-{\overline {M}}_{n}^{2.303}\right)+{\overline {M}}_{n}}$

${\displaystyle {\overline {M}}_{w}={\frac {\sum _{i}z_{i}M_{i}^{2}}{\sum _{j}z_{j}M_{j}}}\quad and\quad {\overline {M}}_{n}=\sum _{i}z_{i}M_{i}}$

The rotational coupling parameter for the mixture is

${\displaystyle \alpha _{mix}=1+7.378\times 10^{-3}\rho _{rz\alpha }^{1.847}M_{mix}^{0.5173}}$

The accuracy of the final viscosity of the CS method needs a very accurate density prediction of the reference fluid. The molar volume of the reference fluid methane is therefore calculated by a special EOS, and the Benedict-Webb-Rubin (1940)^[20] equation of state variant suggested by McCarty (1974),^[21] and abbreviated BWRM, is recommended by Pedersen et al. (1987) for this purpose. This means that the fluid mass density in a grid cell of the reservoir model may be calculated via e.g. a cubic EOS or by an input table with unknown establishment. In order to avoid iterative calculations, the reference (mass) density used in the rotational coupling parameters is therefore calculated using a simpler corresponding state principle which says that

${\displaystyle P_{z\alpha }={\frac {P\cdot P_{cz}}{P_{cmix}}}\quad {\text{and}}\quad T_{z\alpha }={\frac {T\cdot T_{cz}}{T_{cmix}}}\quad \Rightarrow \quad V_{z\alpha }=V(T_{z\alpha },P_{z\alpha })\quad {\text{for 1 mole methane}}}$

The molar volume is used to calculate the mass concentration, which is called (mass) density, and then scaled to be reduced density which is equal to reciprocal of reduced molar volume because there is only on component (molecule type). In mathematical terms this is

${\displaystyle \rho _{z\alpha }=M_{z}/V_{z\alpha }\quad and\quad \rho _{cz}=M_{z}/V_{cz}\quad \Rightarrow \quad \rho _{rz\alpha }=\rho _{z\alpha }/\rho _{cz}=V_{cz}/V_{z\alpha }}$

The formula for the rotational coupling parameter of the mixture is shown further up, and the rotational coupling parameter for the reference fluid (methane) is

${\displaystyle \alpha _{z}=1+0.031\rho _{rz\alpha }^{1.847}}$

The methane mass density used in viscosity formulas is based on the extended corresponding state, shown at the beginning of this chapter on CS-methods. Using the BWRM EOS, the molar volume of the reference fluid is calculated as

${\displaystyle V_{z}=V(T_{z},P_{z})\quad {\text{for 1 mole methane}}}$

Once again, the molar volume is used to calculate the mass concentration, or mass density, but the reference fluid is a single component fluid, and the reduced density is independent of the relative molar mass. In mathematical terms this is

${\displaystyle \rho _{z}=M_{z}/V_{z}\quad and\quad \rho _{cz}=M_{z}/V_{cz}\quad \Rightarrow \quad \rho _{rz}=\rho _{z}/\rho _{cz}=V_{cz}/V_{z}}$

The effect of a changing composition of e.g. the liquid phase is related to the scaling factors for viscosity, temperature and pressure, and that is the corresponding state principle.

The reference viscosity correlation of Pedersen et al. (1987)^[18] is

${\displaystyle \eta _{z}\left(\rho _{z},T_{z}\right)=\eta _{0}(T_{z})+{\hat {\eta }}_{1}(T_{z})\rho _{z}+F_{1}\Delta \eta '(\rho _{z},T_{z})+F_{2}\Delta \eta ''(\rho _{z},T_{z})}$

The formulas for ${\displaystyle \eta _{0}(T_{z})}$, ${\displaystyle {\hat {\eta }}_{1}(T_{z})}$, ${\displaystyle \Delta \eta '(\rho _{z},T_{z})}$ are taken from Hanley et al. (1975).^[22]

The dilute gas contribution is

${\displaystyle \eta _{0}\left(T_{z}\right)=\textstyle \sum _{i=1}^{9}g_{i}T_{z}^{\frac {i-3}{4}}}$

The temperature dependent factor of the first density contribution is

${\displaystyle {\hat {\eta }}_{1}\left(T_{z}\right)=h_{1}-h_{2}\left\lbrack h_{3}-ln\left({\frac {T_{z}}{h_{4}}}\right)\right\rbrack ^{2}}$

The dense fluid term is

${\displaystyle \Delta \eta '\left(\rho _{z},T_{z}\right)=e^{j_{1}+j_{4}/T_{z}}\times \lbrack exp{\lbrack \rho _{z}^{0.1}(j_{2}+j_{3}/T_{z}^{3/2})+\theta _{rz}\rho _{z}^{0.5}\left(j_{5}+j_{6}/T_{z}+j_{7}/T_{z}^{2}\right)\rbrack }-1\rbrack }$

where exponential function is written both as ${\displaystyle e^{x}}$ and as ${\displaystyle exp{\lbrack x\rbrack }}$. The molar volume of the reference fluid methane, which is used to calculate the mass density in the viscosity formulas above, is calculated at a reduced temperature that is proportional to the reduced temperature of the mixture. Due to the high critical temperatures of heavier hydrocarbon molecules, the reduced temperature of heavier reservoir oils (i.e. mixtures) can give a transferred reduced methane temperature that is in the neighborhood of the freezing temperature of methane. This is illustrated using two fairly heavy hydrocarbon molecules, in the table below. The selected temperatures are a typical oil or gas reservoir temperature, the reference temperature of the International Standard Metric Conditions for Natural Gas (and similar fluids) and the freezing temperature of methane (${\displaystyle T_{fz}}$).

Transfer of reduced temperature

	Temp.	Temp.	n-decane	n-eicosane	methane	${\displaystyle T_{r\,c_{10}h_{22}}}$	${\displaystyle T_{r\,c_{20}h_{42}}}$
Temperature	deg C	K	${\displaystyle T_{r\,c_{10}h_{22}}}$	${\displaystyle T_{r\,c_{20}h_{42}}}$	${\displaystyle T_{r\,c_{1}h_{4}}}$	${\displaystyle \rightarrow T_{r\,c_{1}h_{4}}}$	${\displaystyle \rightarrow T_{r\,c_{1}h_{4}}}$
${\displaystyle T_{c}\,\,(K)}$			617.6	767	190.6
reservoir	95	368.15	0.60	0.48	1.93	0.60	0.48
ISMC-NG	15	288.15	0.47	0.38	1.51	0.47	0.38
${\displaystyle T_{f\,c_{1}h_{4}}}$	−182.45	90.7	0.15	0.12	0.48

Pedersen et al. (1987) added a fourth term, that is correcting the reference viscosity formula at low reduced temperatures. The temperature functions ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are weight factors. Their correction term is

${\displaystyle \Delta \eta ''\left(\rho _{z},T_{z}\right)=e^{k_{1}+k_{4}/T_{z}}\times \lbrack exp{\lbrack \rho _{z}^{0.1}(k_{2}+k_{3}/T_{z}^{3/2})+\theta _{rz}\rho _{z}^{0.5}\left(k_{5}+k_{6}/T_{z}+k_{7}/T_{z}^{2}\right)\rbrack }-1\rbrack }$

HCH and PF constants

i	${\displaystyle g_{i}}$	${\displaystyle h_{i}}$	${\displaystyle j_{i}}$	${\displaystyle k_{i}}$
1	−2.090975E5	1.696985927	−10.3506	−9.74602
2	2.647269E5	0.133372346	17.5716	18.0834
3	−1.472818E5	1.4	−3019.39	−4126.66
4	4.716740E4	168.0	188.730	44.6055
5	−9.481872E3		0.0429036	0.976544
6	1.219979E3		145.290	81.8134
7	−9.627993E1		6127.68	15649.9
8	4.274152E0
9	−8.141531E-2

${\displaystyle \theta _{rz}=\left(\rho _{z}-\rho _{cz}\right)/\rho _{cz}=\rho _{rz}-1}$

${\displaystyle F_{1}={\frac {HTAN+1}{2}}}$

${\displaystyle F_{2}={\frac {1-HTAN}{2}}}$

${\displaystyle HTAN=tanh\left(\Delta T_{z}\right)={\frac {e^{\left(\Delta T_{z}\right)}-e^{\left(-\Delta T_{z}\right)}}{e^{\left(\Delta T_{z}\right)}+e^{\left(-\Delta T_{z}\right)}}}}$

${\displaystyle \Delta T_{z}=T_{z}-T_{fz}}$

Phillips (1912)^[23] plotted temperature ${\displaystyle T}$ versus viscosity ${\displaystyle \eta }$ for different isobars for propane, and observed a similarity between these isobaric curves and the classic isothermal curves of the ${\displaystyle PVT}$ surface. Later, Little and Kennedy (1968)^[24] developed the first viscosity model based on analogy between ${\displaystyle T\eta P}$ and ${\displaystyle PVT}$ using van der Waals EOS. Van der Waals EOS was the first cubic EOS, but the cubic EOS has over the years been improved and now make up a widely used class of EOS. Therefore, Guo et al. (1997)^[25] developed two new analogy models for viscosity based on PR EOS (Peng and Robinson 1976) and PRPT EOS (Patel and Teja 1982)^[26] respectively. The following year T.-M. Guo (1998)^{[27] [3]} modified the PR based viscosity model slightly, and it is this version that will be presented below as a representative of EOS analogy models for viscosity.

PR EOS is displayed on the next line.

${\displaystyle P={\frac {RT}{V-b_{eos}}}-{\frac {a_{eos}}{V(V+b_{eos})+b_{eos}(V-b_{eos})}}}$

The viscosity equation of Guo (1998) is displayed on the next line.

${\displaystyle T={\frac {rP}{\eta -d}}-{\frac {a}{\eta \left(\eta +b\right)+b\left(\eta -b\right)}}}$

To prepare for the mixing rules, the viscosity equation is re-written for a single fluid component i.

${\displaystyle T={\frac {r_{i}P}{\eta _{i}-d_{i}}}-{\frac {a_{i}}{\eta _{i}\left(\eta _{i}+b_{i}\right)+b_{i}\left(\eta _{i}-b_{i}\right)}}}$

Details of how the composite elements of the equation are related to basic parameters and variables, is displayed below.

${\displaystyle a_{i}=0.45724{\frac {r_{ci}^{2}P_{ci}^{2}}{T_{ci}}}}$

${\displaystyle b_{i}=0.07780{\frac {r_{ci}P_{ci}}{T_{ci}}}}$

${\displaystyle r_{i}=r_{ci}\tau _{i}\left(T_{ri},P_{ri}\right)}$

${\displaystyle d_{i}=b_{i}\phi _{i}\left(T_{ri},P_{ri}\right)}$

${\displaystyle r_{ci}={\frac {\eta _{ci}T_{ci}}{P_{ci}Z_{ci}}}}$

${\displaystyle \eta _{ci}=K_{p}D_{pi}\quad {\text{where}}\quad K_{p}=7.7\cdot 10^{4}\quad {\text{and}}\quad D_{pi}=T_{ci}^{-1/6}M_{i}^{1/2}P_{ci}^{2/3}}$

${\displaystyle \tau _{i}=\tau _{i}\left(T_{ri},P_{ri}\right)=\left(1+Q_{1i}\left({\sqrt {T_{ri}P_{ri}}}-1\right)\right)^{-2}}$

${\displaystyle \phi _{i}=\phi _{i}\left(T_{ri},P_{ri}\right)=\exp \left[Q_{2i}\left({\sqrt {T_{ri}}}-1\right)\right]+Q_{3i}\left({\sqrt {P_{ri}}}-1\right)^{2}}$

${\displaystyle Q_{1i}={\begin{cases}0.829599+0.350857\,\omega _{i}-0.747680\,\omega _{i}^{2},&{\text{if }}&\omega _{i}<0.3\\0.956763+0.192829\,\omega _{i}-0.303189\,\omega _{i}^{2},&{\text{if }}&\omega _{i}\geq 0.3\end{cases}}}$