PNG 520
Phase Relations in Reservoir Engineering

Viscosity

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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 centi-poise, which is related to other units as follows:

1 cp = 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:

μ g =1 10 4 k v EXP( x v ( ρ g 62.4 ) y v ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.25a)
 

where:

k v = ( 9.4+0.02M W g ) T 1.5 209+19M W g +T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.25b)
 
y v =2.40.2 x v This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.25c)
 
x v =3.5+ 986 T +0.01M W g This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.25d)
 

In this expression, temperature is given in (°R), the density of the fluid ( ρ g This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ) in lbm/ft3 (calculated at the pressure and temperature of the system), and the resulting viscosity is expressed in centipoises (cp).

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. 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:

μ= μ · + ξ m 1 ( 0.1023+0.023364 ρ r +0.058533 ρ r 2 0.040758 ρ r 3 +0.0093724 ρ r 4 ) 4 1 10 4 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.26)
 

where:
μ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = fluid viscosity (cp),
μ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = viscosity at atmospheric pressure (cp),
ξ m This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = mixture viscosity parameter (cp-1),
ρ r This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = reduced liquid density (unitless),

Lohrenz et al. original paper presents a typographical error in Equation (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 (19.26). Lohrenz 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 ( μ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ), Lohrenz et al. suggested using the following Herning & Zipperer equation:

μ · = i z i μ i · M W i i z i M W i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.27)
 

where:
zj = mole composition of the i-th component in the mixture,
MWi = molecular weight of the i-th component (lbm/lbmol)
μ i · This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = viscosity of the i-th component at low pressure (cp):
μ i · = 34 10 5 T ri 0.94 ξ i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. [ if Tri ≤ 1.5 ]
μ i · = 17.78 10 5 ( 4.5 T ri 1.67 ) 0.625 ξ i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. [ if Tri > 1.5
where:
Tri = reduced temperature for the i-th component (T/Tci),
MWi = viscosity parameter of the i-th component, given by: ξ i = 5.4402 T ci 1/6 M W i P ci 2/3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
For the mixture viscosity parameter ( ξm This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ), Lohrenz et al. applied an equivalent expression to that shown above but using pseudo-properties for the mixture:

ξm= 5.4402 T pc 1/6 M W l P pc 2/3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.28)
 

where:

Tpc = pseudocritical temperature (oR),
Ppc = pseudocritical pressure (psia),
MWl = liquid mixture molecular weight (lbm/lbmol).

The reduced density of the liquid mixture ( ρ r This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ) is calculated as:

ρ r = ρ l ρ pc =( ρ l M W l ) V pc This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.29)
 
where:
ρ pc This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. = mixture pseudocritical density (lbm/ft3),
Vpc = mixture pseudocritical molar volume (ft3/lbmol),

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

T pc = z i T ci This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.30a)
 
P pc = z i P ci This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.30b)
 
V pc = z i V ci This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(19.30c)
 

“zi” pertains to the fluid molar composition, Tci is given in oR, Pci in psia, and Vci in ft3/lbmol. When the critical volumes are known in a mass basis (ft3/lbm), each of them is to be multiplied by the corresponding molecular weight. In the case of lumped C7+ heavy fractions, Lorentz et al. (1969) presented a correlation for the estimation C7+ 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.