Equations: Fluid composition and phase behaviour

The equations of the “Fluid composition and phase behaviour” lessons in “Reservoir engineering”, with their conditions and units, and what each symbol means in them.

Molar mass of a mixture

M = \sum_i z_i M_i

g/mol (kg/kmol)

Holds when: The mole fractions of all the components, including the non-hydrocarbon ones and the C7+ fraction, add up to one.

Composition of reservoir fluids →

Conversion from mole fractions to mass fractions

w_i = \frac{z_i M_i}{\sum_j z_j M_j}

fraction

Holds when: The composition is given completely, including the C7+ fraction with its molar mass.

Composition of reservoir fluids →

Specific gravity of a gas relative to air

\gamma_g = \frac{M_g}{M_{\text{air}}} = \frac{M_g}{28.97}

dimensionless

Holds when: We compare the gas and air at the same pressure and temperature, as ideal gases. At standard conditions this is accurate to better than 1%.

Composition of reservoir fluids →

Partial pressure of a gas component (Dalton's law)

p_i = y_i\,p

any pressure units; the pressure is absolute

Holds when: The gas is close to ideal. At high pressures the partial pressure remains a convenient rule-of-thumb measure, but for phase-equilibrium calculations we use fugacity (an “effective pressure” for real fluids) instead (Vapour–liquid equilibrium).

Composition of reservoir fluids →

Oil density in degrees API

{}^{\circ}\mathrm{API} = \frac{141.5}{\gamma_o} - 131.5

°API is a conventional scale; $\gamma_o$ is dimensionless

Holds when: The oil and water densities are at 60 °F and atmospheric pressure. We first bring a density measured at another temperature to 60 °F, using standard tables. The formula applies to stock-tank oil and condensate, not to reservoir oil with dissolved gas.

Composition of reservoir fluids →

Watson characterisation factor

K_w = \frac{T_b^{1/3}}{\gamma}

empirical: $T_b$ only in degrees Rankine

Holds when: This is an empirical index. For narrow fractions we take their mean boiling point. For wide fractions we use a specially averaged boiling point of the whole fraction.

Composition of reservoir fluids →

Vapour pressure from two reference points (Clapeyron–Clausius approximation)

\ln\frac{p_s}{p_{\text{atm}}} = \ln\frac{p_c}{p_{\text{atm}}}\cdot\frac{1/T_b - 1/T}{1/T_b - 1/T_c}

absolute pressures in the same units, absolute temperatures in kelvins

Holds when: The formula rests on the Clapeyron–Clausius equation. We treat the vapour as an ideal gas, the volume of the liquid as small compared with the volume of the vapour, and the heat of vaporisation (the heat needed to turn liquid into vapour) as constant. Then $\ln p_s$ depends linearly on $1/T$ . We draw the straight line through two known points, the normal boiling point and the critical point. The errors of the assumptions partly cancel, and for light hydrocarbons the error is usually no more than a few per cent. The formula describes only the liquid–gas curve between the triple point and the critical point; above $T_c$ there is no vapour pressure.

Phase behaviour: pure substances and binary mixtures →

Bubble point pressure and first-gas composition of an ideal mixture

p_b = \sum_i x_i\,p_{s,i}(T),\qquad y_i = \frac{x_i\,p_{s,i}(T)}{p_b}

absolute pressures in the same units

Holds when: An ideal solution: the molecules of the components are close in size and structure (for example propane and butane), and Raoult's law holds for each component. The gas is ideal, so Dalton's law holds. The pressures are low, and the temperature is below the critical temperatures of all the components. For reservoir mixtures at high pressures these laws do not work. There we calculate phase equilibrium from equilibrium ratios (the ratio of a component's share in the gas to its share in the liquid) or from equations of state (Vapour–liquid equilibrium, Cubic equations of state).

Phase behaviour: pure substances and binary mixtures →

Dew point pressure and last-drop composition of an ideal mixture

\frac{1}{p_d} = \sum_i \frac{y_i}{p_{s,i}(T)},\qquad x_i = \frac{y_i\,p_d}{p_{s,i}(T)}

absolute pressures in the same units

Holds when: The same as for the bubble point pressure.

Phase behaviour: pure substances and binary mixtures →

Gibbs phase rule

F = C - P + 2

integers

Holds when: The system is in equilibrium, there are no chemical reactions, and every component can be present in every phase.

Phase behaviour: pure substances and binary mixtures →

Clapeyron–Clausius equation

\frac{\dd p_s}{\dd T} = \frac{\Delta H_v}{T\,\Delta v}\quad\Rightarrow\quad \frac{\dd \ln p_s}{\dd (1/T)} = -\frac{\Delta H_v}{R}

Holds when: The second form follows from the first if we take $\Delta v \approx v_{\text{g}} = RT/p_s$ . This means we neglect the volume of the liquid and treat the vapour as an ideal gas.

Phase behaviour: pure substances and binary mixtures →

Composition of the reservoir mixture from gas and liquid samples (recombination)

z_i = \frac{n_g\,y_i + n_o\,x_i}{n_g + n_o},\qquad n_g = \frac{R}{V_m},\qquad n_o = \frac{\rho_o}{M_o}

mole fractions as fractions of one; the gas volume is at standard conditions

Holds when: The gas and liquid were taken at the same time, at steady flow, with the bottomhole pressure above the bubble point pressure. Otherwise the flow at the surface does not match the reservoir mixture. The gas–oil ratio and the liquid properties refer to the same liquid. This is either the separator liquid or the stock-tank oil. In the second case the gas is all the gas of the separator and the tank, and its composition is the average over both streams. We treat the gas at standard conditions as ideal. The error is less than 1%.

Types of reservoir fluids →

Assumed approximation of condensate dropout

S_L(p) = S_{\max}\,\frac{\Delta p}{\Delta p_m}\,\e^{\,1 - \Delta p/\Delta p_m},\qquad \Delta p = p_d - p \ge 0

MPa, %

Holds when: This is an approximation for one imaginary fluid, not a correlation. We chose the shape of the function to reproduce the typical form of laboratory curves: rapid growth below the dew point, a maximum and slow re-vaporisation. Above the dew point $S_L = 0$ . Real curves come from the laboratory, from a constant volume depletion experiment (a test in which gas is withdrawn from a cell of constant volume as the pressure falls) (Laboratory PVT tests).

Types of reservoir fluids →

Symbols

Symbol	Meaning and context
$\Delta v$	difference of the molar volumes of gas and liquid, m³/molPhase behaviour: pure substances and binary mixtures
${}^{\circ}\mathrm{API}$	density in degrees API, °APIComposition of reservoir fluids
$C$	number of components, —Phase behaviour: pure substances and binary mixtures
$F$	number of degrees of freedom — independent intensive parameters (pressure, temperature, phase compositions), —Phase behaviour: pure substances and binary mixtures
$K_w$	characterisation factor, —Composition of reservoir fluids
$\Delta p_m$	pressure reduction below the dew point at which the dropout is greatest; taken as 16 MPa (the maximum is at 20 MPa), MPaTypes of reservoir fluids
$M$	molar (average) mass of the mixture, g/mol = kg/kmolComposition of reservoir fluids
$M_{\text{air}}$	molar mass of dry air, 28.97 g/mol, g/molComposition of reservoir fluids
$M_g$	molar mass of the gas, g/molComposition of reservoir fluids
$M_i$	molar mass of the component; for the C7+ fraction, its measured molar mass, g/molComposition of reservoir fluids
$n_g,\ n_o$	amounts of gas and liquid per 1 m³ of liquid, kmolTypes of reservoir fluids
$p$	absolute pressure of the gas, MPa or kPaComposition of reservoir fluids
$P$	number of phases in equilibrium, —Phase behaviour: pure substances and binary mixtures
$p_{\text{atm}}$	normal atmospheric pressure, 0.101325 MPa, at which the substance boils at $T_b$ , MPaPhase behaviour: pure substances and binary mixtures
$p_b$	bubble point pressure of the mixture at temperature $T$ , MPaPhase behaviour: pure substances and binary mixtures
$p_d$	dew point pressure of the mixture at temperature $T$ , MPaPhase behaviour: pure substances and binary mixtures dew point pressure; taken as 36 MPa, MPaTypes of reservoir fluids
$p_i$	partial pressure of component $i$ , MPa or kPaComposition of reservoir fluids
$p_s$	vapour pressure at temperature $T$ , absolute, MPaPhase behaviour: pure substances and binary mixtures
$p_{s,i}(T)$	vapour pressure of pure component $i$ at $T$ , MPaPhase behaviour: pure substances and binary mixtures vapour pressure of the pure component, MPaPhase behaviour: pure substances and binary mixtures
$R$	universal gas constant, 8.314 J/(mol·K), J/(mol·K)Phase behaviour: pure substances and binary mixtures gas–oil ratio: the standard volume of gas per 1 m³ of liquid, m³/m³Types of reservoir fluids
$S_L$	liquid fraction of the hydrocarbon pore volume, %Types of reservoir fluids
$S_{\max}$	maximum liquid fraction; taken as 12%, %Types of reservoir fluids
$T$	temperature at which $p_s$ is sought; the formula is intended for $T_b \le T \le T_c$ , KPhase behaviour: pure substances and binary mixtures absolute temperature, KPhase behaviour: pure substances and binary mixtures
$T_b$	mean boiling point of the fraction in degrees Rankine: $T_b\,[{}^{\circ}\text{R}] = 1.8\,T_b\,[\text{K}]$ , °RComposition of reservoir fluids normal boiling point, KPhase behaviour: pure substances and binary mixtures
$T_c,\ p_c$	critical temperature and pressure of the substance, K, MPaPhase behaviour: pure substances and binary mixtures
$\Delta H_v$	molar heat of vaporisation (the heat needed to vaporise one mole), J/molPhase behaviour: pure substances and binary mixtures
$V_m$	molar volume of an ideal gas at standard conditions: 24.06 m³/kmol at 0.101325 MPa and 20 °C (23.64 at 15 °C), m³/kmolTypes of reservoir fluids
$w_i$	mass fraction of component $i$ , fractionComposition of reservoir fluids
$x_i$	mole fraction of component $i$ in the liquid; at the bubble point, in the mixture, —Phase behaviour: pure substances and binary mixtures mole fraction of the component in the last drop of liquid, —Phase behaviour: pure substances and binary mixtures
$y_i$	mole fraction of the component in the gas, fractionComposition of reservoir fluids mole fraction of the component in the first gas bubbles, —Phase behaviour: pure substances and binary mixtures mole fraction of the component in the gas; at the dew point, in the mixture, —Phase behaviour: pure substances and binary mixtures
$y_i,\ x_i$	mole fractions of the component in the gas and in the liquid, —Types of reservoir fluids
$z_i$	mole fraction of component $i$ , fractionComposition of reservoir fluids mole fraction of component $i$ in the reservoir mixture, —Types of reservoir fluids
$z_i,\ M_i$	mole fraction and molar mass of the component, —, g/molComposition of reservoir fluids
$\gamma$	specific gravity of the fraction relative to water at 60 °F, —Composition of reservoir fluids
$\gamma_g$	gas specific gravity — the ratio of its density to the density of air at the same pressure and temperature, —Composition of reservoir fluids
$\gamma_o$	specific gravity of the stock-tank oil: $\rho_o/\rho_w$ , both densities at 60 °F (15.6 °C); the density of water at this temperature is 999.0 kg/m³, —Composition of reservoir fluids
$\rho_o,\ M_o$	density and molar mass of the liquid at the conditions to which the gas–oil ratio refers, kg/m³, kg/kmolTypes of reservoir fluids