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Reservoir Fluid Properties Course (2nd Ed.)

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mailto:[email protected]

http://www.about.me/AlamiNia

1. Gas Behavior

2. Gas Properties: A. Z Factor:

a. Calculation for pure components

b. Calculation for mixture componentsI. Mixing rules for calculating pseudocritical properties

II. Correlations for calculating pseudocritical properties

c. Nonhydrocarbon adjustment

d. High molecular weight gases adjustment

1. empirical correlations for calculating z-factors

2. Gas Properties: A. isothermal gas compressibility (Cg)

B. gas formation volume factor (Bg) and gas expansion factor (Eg)

C. Gas Viscosity correlations

Direct Calculation of Compressibility FactorsAfter four decades of existence, the Standing-Katz

z-factor chart is still widely used as a practical source of natural gas compressibility factors.As a result, there has been an apparent need for a

simple mathematical description of that chart. Several empirical correlations for calculating z-factors

have been developed over the years including:Hall-Yarborough

• It is not recommended for application if Tpr is less than one.

Dranchuk-Abu-Kassem (DAK)• is applicable over the ranges: 0.2 < ppr < 15 and 1.0 < Tpr < 3.0

Dranchuk-Purvis-Robinson• is valid within the ranges: 1.05 < Tpr < 3.0 and 0.2 < ppr < 3.0

Fall 13 H. AlamiNia Reservoir Fluid Properties Course: Gas Properties: Z, Cg, Bg, Eg, μg 5

The Hall-Yarborough (1973) Method

They presented an equation-of-state that accurately represents the Standing and Katz z-factor chart. The proposed expression is based on the Starling-Carnahan

equation-of-state.

where t = reciprocal of the pseudo-reduced temperature, i.e., Tpc/TY = the reduced density and obtained as the solution of:

• X1 = −0.06125 ppr t exp [−1.2 (1 − t)2]• X2 = (14.76 t − 9.76 t2 + 4.58 t3)• X3 = (90.7 t − 242.2 t2 + 42.4 t3)• X4 = (2.18 + 2.82 t)

It is a nonlinear equation and can be solved for the reduced density Y by using the Newton-Raphson iteration technique.


The computational procedure of solving F(Y) at any specified Ppr & TprStep 1. an initial guess of the unknown parameter, Yk,

where k is an iteration counter.

Step 2. Substitute this initial value in F(Y) and evaluate the nonlinear function.

Step 3. A new improved estimate of Y, i.e., Yk+1, from:

Step 4. Steps 2–3 are repeated n times, until the error, i.e., abs(Yk − Yk+1), becomes smaller than a preset tolerance, e.g., 10^−12.

Step 5. The correct value of Y is then used for the compressibility factor.


The Dranchuk-Abu-Kassem Method prerequisiteDranchuk and Abu-Kassem (1975)

derived an analytical expression for calculating the reduced gas density ρrthat used to estimate the gas compressibility factor.

ρr is defined as the ratio of the gas density at a specified pressure and temperature to that of the gas at its critical pressure or temperature, or:

The critical gas compressibility factor zc is approximately 0.27, which leads to:


Calculation of the reduced gas density

The authors proposed the eleven-constant equation-of-state for calculating the reduced gas density:

The coefficients have the following values:


The Dranchuk-Abu-Kassem Method (DAK)The eleven-constant equation-of-state for

calculating the reduced gas density ρr can be solved by applying the Newton-Raphson iteration technique.

The correct value of ρr is then used to evaluate the compressibility factor, i.e.:

The proposed correlation was reported to duplicate compressibility factors from the Standing and Katz chart with an average absolute error of 0.585 percent.


The Dranchuk-Purvis-Robinson MethodDranchuk, Purvis, and Robinson (1974) developed a

correlation based on the Benedict-Webb-Rubin type of equation-of-state. Fitting the equation to 1,500 data points from the

Standing and Katz z-factor chart optimized the eight coefficients of the proposed equations. The equation has the following form:


The Dranchuk-Purvis-Robinson Method (Cont.)where ρr is defined by

(the same as The Dranchuk-Abu-Kassem Method)

the coefficients A1 through A8 have the following values:

The solution procedure is similar to that of Dranchuk and Abu-Kassem.


Compressibility of Natural Gases

Knowledge of the variability of fluid compressibility with pressure and temperature is essential in performing many reservoir engineering calculations. For a liquid phase,

the compressibility is small and usually assumed to be constant.

For a gas phase, the compressibility is neither small nor constant.

By definition, the isothermal gas compressibility is the change in volume per unit volume for a unit change in pressure or, in equation form:

Where cg = isothermal gas compressibility, 1/psi.


Compressibility of Natural Gases (Cont.)From the real gas equation-of-state:

Differentiating with respect to p at constant T gives:

Substituting, produces the generalized relationship of:

For an ideal gas, z = 1 and (∂z/∂p) T = 0, so:

It is useful in determining the expected order of magnitude of the isothermal gas compressibility.


Cg In Terms ofthe Pseudoreduced PropertiesThe Equation can be conveniently expressed in terms of

the ppr and Tpr by simply replacing p with (Ppc Ppr), or:

The term cpr is called the isothermal pseudo-reduced compressibility and is defined by the relationship: cpr = cg Ppc,

Values of (∂z/∂ppr) Tpr can be calculated from the slope of the Tpr isotherm on the Standing and Katz z-factor chart.


Trube (1957) graphs (the isothermal compressibility of natural gases)

Trube’s pseudo-reduced compressibility Trube’s pseudo-reduced compressibilityFall 13 H. AlamiNia Reservoir Fluid Properties Course: Gas Properties: Z, Cg, Bg, Eg, μg 17

analytical technique for calculating the isothermal gas compressibilityMatter, Brar, and Aziz (1975) presented

an analytical technique for calculating the isothermal gas compressibility. The authors expressed cpr as a function of ∂p/∂ρr rather

than ∂p/∂ppr.

Where:

where the coefficients T1 through T4 and A1 through A8 are defined previously by the Dranchuk-Purvis-Robinson Method.


Gas Formation Volume Factor

The gas formation volume factor is used to relate the volume of gas,

as measured at reservoir conditions,

to the volume of the gas as measured at standard conditions, i.e., 60°F and 14.7 psia.

This gas property is then defined as the actual volume occupied by a certain amount of gas

at a specified pressure and temperature,

divided by the volume occupied by the same amount of gas at standard conditions.


Bg Calculation

Applying the real gas equation-of-state, and substituting for the volume V, gives:

Assuming that the standard conditions:psc =14.7 psia and

Tsc = 520, zsc = 1.0

Bg = gas formation volume factor, ft3/scf, z=gas compressibility factor, T=temperature, °R

It can be expressed in terms of the gas density ρg:

Where: 𝜌𝑔[lb/ft3]


Bg & Eg Calculation

In other field units, the gas formation volume factor can be expressed in bbl/scf to give:

The reciprocal of the gas formation volume factor is called the gas expansion factor

is designated by the symbol Eg, or:

or in terms of the gas density ρg:

In other units:


Viscosity

The viscosity of a fluid is a measure of the internal fluid friction (resistance) to flow.

If the friction between layers of the fluid is small, i.e., low viscosity, an applied shearing force will result in a large velocity gradient. As the viscosity increases,

each fluid layer exerts a larger frictional drag on the adjacent layers and velocity gradient decreases.

The viscosity of a fluid is generally defined as the ratio of the shear force per unit area

to the local velocity gradient.

Viscosities are expressed in terms of centipoise.


Gas Viscosity

The gas viscosity is not commonly measured in the laboratory because it can be estimated precisely from empirical correlations.

Like all intensive properties, viscosity of a natural gas is completely described by the following function: μg = (p, T, yi)

Where μg = the viscosity of the gas phase.

The above relationship simply states that the viscosity is a function of pressure, temperature, and composition.

Many of the widely used gas viscosity correlations may be viewed as modifications of that expression.


Methods Of Calculating The Viscosity Of Natural GasesTwo popular methods that are commonly used in

the petroleum industry are the:Carr-Kobayashi-Burrows Correlation Method

Carr, Kobayashi, and Burrows (1954) developed graphical correlations for estimating the viscosity of natural gas • as a function of temperature, pressure, and gas gravity.

Lee-Gonzalez-Eakin Methodstandard deviation of 2.7% and a maximum deviation of 8.99%.

less accurate for gases with higher specific gravities

the method cannot be used for sour gases


Carr-Kobayashi-Burrows Correlation MethodThe computational procedure of applying the

proposed correlations is summarized in the following steps:Step 1. Calculate the Ppc, Tpc and apparent molecular

weight from the specific gravity or the composition of the natural gas. Corrections to these pseudocritical properties for the presence

of the nonhydrocarbon gases (CO2, N2, and H2S) should be made if their concentrations are greater than 5 mole percent.

Step 2. Obtain the viscosity of the natural gas at one atmosphere and the temperature of interest (μ1) from next slide.


Carr’s Atmospheric Gas Viscosity Correlation


The Carr’s Method (Cont.)

μ1, must be corrected for the presence of nonhydrocarbon components by using the inserts of previous slide. • nonhydrocarbon fractions increases the viscosity of the gas phase

• The effect of nonhydrocarbon components on the viscosity of the natural gas can be expressed mathematically by:

o μ1 = (μ1) uncorrected + (Δμ) N2 + (Δμ) CO2 + (Δμ) H2S

Step 3. Calculate the Ppr and Tpr.

Step 4. From the Ppr and Tpr, obtain the viscosity ratio (μg/μ1) from next slide.

Step 5. The gas viscosity, μg, at the pressure and temperature of interest is calculated by multiplying the viscosity at one atmosphere and system temperature, μ1, by the viscosity ratio.


Carr’s Viscosity Ratio Correlation

The term μg represents the viscosity of the gas at the required conditions.

Carr’s viscosity ratio correlationFall 13 H. AlamiNia Reservoir Fluid Properties Course: Gas Properties: Z, Cg, Bg, Eg, μg 31

The Lee-Gonzalez-Eakin Method

Lee, Gonzalez, and Eakin(1966) presented a semi-empirical relationship for calculating the viscosity of natural gases. 𝜇𝑔in terms of

reservoir temperature, gas density, and the molecular weight of the gas.

Where

ρg = gas density at reservoir pressure and temperature, lb/ft3

T = reservoir temperature, °R

Ma = apparent molecular weight of the gas mixture


1. Ahmed, T. (2010). Reservoir engineering handbook (Gulf Professional Publishing). Chapter 2

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