# Fluid Properties

The fluid properties described here are a set of bulk propertied that are used to describe hydrocarbon fluids. Note that these propertied do not contain the volumetric properties like the formation volume factors ($$B_o$$ and $$B_g$$) or the solution surface volume ratios (GOR | $$R_s$$ and OGR | $$r_s$$).

## Density

Density ($$\rho$$) is the measure of mass per unit volume. A substance is said to be more dense if the weight per unit volume is greater (i.e. heavier objects with the same volume are more dense). The formal definition of density is given by

where $$\rho$$ is the density, $$m$$ is the mass and $$V$$ is the bulk volume. The most common units for density are in SI $$\frac{kg}{m^3}$$ (or in $$\frac{g}{cm^3}$$ which is just 1/1000 of the SI unit) or field units $$\frac{lb}{ft^3}$$.

Density is a so-called bulk property, as this quantity is defined for some bulk volume and is not concerned with the microscopic irregularities with respect to the scale in question. As an example, a porous rock is said to have a given density based on the weight of the rock and the bulk volume of the rock. The bulk volume can either be the entire rock (containing small pores) or only the non-porous rock.

### Specific Gravity

Another useful term related to density is the specific gravity ($$\gamma$$). This property is in essence the same as density, but is scaled by some reference density to make value unitless. The definition of specific gravity is given by

where $$\gamma$$ is the specific gravity, $$\rho$$ is the density and $$\rho_{\mathrm{ref}}$$ is the reference density.

The reference density for vapors and liquids are given below.

Unit Vapor Liquid
SI-Units ($$\frac{kg}{m^3}$$) 1.2232 999.014
Field-Units ($$\frac{lb}{ft^3}$$) 0.076362 62.3664

### API Gravity

API gravity is a another special unit for density for oils that is commonly used in the petroleum industry. The definition of API gravity is given by

### Molar density

Molar density ($$\hat{\rho}$$) given in units mole per unit volume, is an equivalent metric for measuring density. Molar density is often used in describing virial EOS models, like the BWR EOS models. The relationship between "normal" density (or mass density) and molar density is the following: $$\rho = \hat{\rho} \cdot M$$, where $$M$$ is the molecular weight.

## Compressibility

Compressibility ($$c$$) is a material's ability to compress under stress caused by some force. In reservoir engineering, a sub-class of compressibility are typically considered, namely isothermal compressibility. This type compressibility is limited by constant temperature (i.e. the term isotherm).

The definition of isothermal compressibility is given by the differential equation

where $$c$$ is the compressibility, $$V$$ is the volume and $$p$$ is the pressure.

## Viscosity

Viscosity ($$\mu$$ or sometimes $$\eta$$) is a measure of the internal friction of fluid flow. If a fluid has a high viscosity (is highly viscous), then there is a lot of internal friction and the fluid will flow more "slowly". The unit for viscosity is centi-Poise (cP) or Pascal second (Pa.s). As an example, in increasing order with respect to viscosity you have: air, water and honey.

Viscosity is primarily used for describing flow in porous media though Darcy's equation (or the Forchheimer equation) or as part of pipe-flow equations for production engineering. It is therefore important to accurately describe the pressure and temperature dependence of the viscosity as part of the equation of state model or the black-oil model.

## Mole

A mole ($$n$$) is a method used to stochiometrically count particles in a fluid based on a number of particles in some unit mass. Similar metrics are used in everyday life, such as a pair (i.e. 2) or a dusin (i.e. 12). A general definition of a unit-mole is

where $$m$$ is some mass of a particle with particle mass $$u$$. The SI unit definition of gram-moles (also called mol) is defined by the number of Carbon-12 atoms in 12 grams of Carbon-12. Equivalently, a lb-mole is the number of Carbon-12 atoms there are in 12 lbs of Carbon-12.

## Molecular Weight

The ratio between some mass of a substance and the amount of moles is a useful ratio and is defined as the molecular weight ($$MW$$). The definition of the molecular weight is given by

and is constant and independent of mass unit given that the mole unit and mass unit are the same (e.g. lb-mole and lbs).

## Composition

A composition, whether it be mass or molar based, gives the fractional amount of a given component (e.g. molecule or lumped isomers) with respect to the entire mixture. The two most used types of compositions are mass and molar compositions.

### Mass Composition

Given a set of component masses ($$m_i$$) for components $$i \in [ N_2,H_2S, C_1, ... ]$$, then the mass composition is defined by

where $$w_i$$ is the mass composition for component $$i$$, $$m_i$$ is the mass amount of component $$i$$ and $$N$$ is the total number of components in the mixture.

Mass compositions are often measured using extended gas chromatography.

### Molar Composition

Given a set of component molar amounts ($$n_i$$) for components $$i \in [ N_2,H_2S, C_1, ... ]$$, then the molar composition is defined by

where $$z_i$$ is the molar composition for component $$i$$, $$n_i$$ is the molar amount of component $$i$$ and $$N$$ is the total number of components in the mixture.

Typically the molar amounts are not measured directly, but the masses are measured and a set of molecular weights are chosen to calculate the molar amount.

whitson comment

Note from Markus Hays Nielsen: Be careful when reading PVT reports as they typically report both the measured mass fractions and the calculated molar fractions. However, the molecular weights (if given at all) are not necessarily the correct values. For most reservoir fluids, the variation of molecular weight for the heptane plus fraction is different from field to field.

Typically, if no molecular weight is specified, then the Katz-Firoozabadi values are used and a quality check should be performed to compare the molecular weights used in the PVT report with the Katz-Firoozabadi values1.

## Component Properties

When developing a fluid model for a field or a basin using a cubic equation of state (EOS), there are a set of component parameters that must be defined. These component properties are used directly in the cubic EOS and are described below with the exception of the molecular weights which are described above.

### Critical Properties

The critical properties are typically defined by the critical pressure ($$p_c$$) and temperature ($$T_c$$), however, there are also critical properties for the molar volume ($$v_c$$) and Z-factor ($$Z_c$$).

The component critical pressure and temperature are defined as the point on the vapour pressure line where all the vapor and liquid phase properties are equal. Similar to the critical point of a multi-component system, the component critical point can be defined as the value where the vapor fraction ($$F_V$$) can take any value between 0 and 1.

### Acentric Factor

Pitzer defines the acentric factor ($$\omega$$) as

where $$p_v^*$$ is the vapor pressure at $$T=0.7\cdot T_c$$ and $$p_c$$ is the critical pressure. Alternatively, the acentric factor can be thought if as the size or the shape of the molecule.

Whitson describes the practical description of the acentric factor as a measure of the steepness of the vapor pressure curve from $$T_r=0.7$$ to $$T_r=1$$2 (i.e. the chord).

An example correlation for the acentric factor is the Edmister correlation3 given by

### Volume Shifts

The volume shifts ($$s$$ or $$c$$) also referred to as volume translations are an additional parameter in the EOS model that was introduced by Peneloux in 19824. The definition of the volume shift is given by

where $$c$$ is the volume shift (with units), $$v^{EOS}$$ is the calculated molar volume from the EOS (without the volume shift) and $$v^{new}$$ is the volume shifted molar volume.

A re-basing of the volume shift can be applied to make it unitless by

where $$b$$ is the EOS parameter.

Volume shifts were introduced to correct the inaccuracies in the EOS prediction of the vapor and liquid densities without affecting certain other phase behavior. A cubic EOS model with volume shifts can typically predict both vapor and liquid densities to within 1-3%. A detailed review on the effect of the volume shifts is given by Jaubert et al.5.

### Binary Interaction Parameters

Note

The main article is: Binary Interaction Parameters

Binary interaction parameters (BIPs) are a set correction terms for the mixing rule of the "a" term in a cubic EOS model between each pair of components. The BIPs have a significant impact on the phase behavior predictions as has been shown by a wide range of authors dating back to the 1960's with Chueh and Prausnitz6.

The matrix of BIPs has certain features. First, the BIP matrix is said to be symmetric - i.e. $$k_{ij}=k_{ji}$$. This means that the interaction between component $$i$$ and $$j$$ is the same as between component $$j$$ and $$i$$. The second property is that the BIP between component $$i$$ and itself is zero by default - i.e. $$k_{ii}=0$$.

1. D. L. Katz and A. Firoozabadi. Predicting phase behavior of condensate/crude-oil systems using methane interaction coefficients. Journal of Petroleum Technology, 30:paper SPE–6721–PA, 1978. doi:https://doi.org/10.2118/6721-PA

2. C. H. Whitson and M. R. Brulé. Phase behavior. Volume 20. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers …, 2000.

3. W. C. Edminster. Applied hydrocarbon thermodynamics, part 4: compressibility factors and equations of state. 1958.

4. A. Péneloux, E. Rauzy, and R. Fréze. A consistent correction for redlich-kwong-soave volumes. Fluid phase equilibria, 8:7–23, 1982. doi:https://doi.org/10.1016/0378-3812$82$80002-2

5. J.N. Jaubert, R. Privat, Y. Le Guennec, and L. Coniglio. Note on the properties altered by application of a péneloux–type volume translation to an equation of state. Fluid Phase Equilibria, 419:88–95, 2016. doi:https://doi.org/10.1016/j.fluid.2016.03.012

6. P.L. Chueh and J.M. Prausnitz. Vapor-liquid equilibria at high pressures: calculation of partial molar volumes in nonpolar liquid mixtures. AIChE journal, 13:1099–1107, 1967. doi:https://doi.org/10.1002/aic.690130612