Equations: Phase equilibrium and PVT studies

dimensionless

Holds when: The gas and the liquid are in equilibrium at the same pressure and temperature. The name “constant” is conventional: $K_i$ changes with pressure and temperature, and at high pressures also with the compositions of the phases.

absolute pressures, in the same units

Holds when: An ideal solution (Raoult's law) and an ideal gas (Dalton's law): low pressures, and molecules close in size and structure. The temperature is below the critical temperature of the component (the highest temperature at which it can be a liquid). Above it there is no vapour pressure.

pressures in the same units, temperatures in kelvins

Holds when: This is the ideal ratio $p_{s,i}/p$, with the vapour pressure found from two reference points. The compositions of the phases are not taken into account. For components above their critical temperature (methane and nitrogen at reservoir temperature), the “vapour pressure” is only the straight line continued beyond the critical point. It has no physical meaning. It works as a first guess for iterations, and as a rough estimate at low pressures. The more accurate empirical correlations are in the lesson Equilibrium ratios: correlations and equations of state.

dimensionless

Holds when: The phases are in equilibrium and the $K_i$ are known. There are two hydrocarbon phases; formation water does not take part in the hydrocarbon equilibrium and is accounted for separately. The balances are written per 1 mole of the mixture; the mole fraction of the liquid phase is $L = 1 - V$.

dimensionless

Holds when: The equilibrium ratios at the given $p$ and $T$ are known and do not depend on $V$ (in an equation-of-state calculation they are refined in an outer loop).

absolute pressures

Holds when: The $K_i$ are calculated for the pair “mixture and new phase”: at the bubble point pressure, for a liquid of composition $z_i$ and a bubble of composition $K_iz_i$. At one temperature a gas condensate can have two dew points: an upper one (retrograde) and a lower one.

dimensionless

Holds when: The iterations are repeated until $Y_i$ stops changing. Both trials are checked: the gas-like one and the liquid-like one. If the trial converges to the composition of the mixture itself ($y_i = z_i$, the trivial solution), it tells us nothing. The mixture is unstable (two-phase) if $\sum Y_i > 1$ for at least one non-trivial trial.

dimensionless

Holds when: The fugacities are calculated with one equation of state for both phases at the same $p$ and $T$.

dimensionless

Holds when: At least one equilibrium ratio is greater than one and at least one is less than one. Otherwise there are no poles on both sides, and the mixture is single-phase for any $V$. If a Newton step goes outside the current interval in which $g$ changes sign, take a bisection step instead.

dimensionless

Holds when: The fugacity coefficients come from the equation of state. For the original mixture, take the root with the lower Gibbs energy.

kmol per 1 kmol of reservoir oil

Holds when: At each stage the gas and the liquid are in equilibrium and all the gas is removed. The feed of the first stage is single-phase reservoir oil (for $B_{ob}$, at the bubble point pressure); water is not included in the calculation. The ratios $K_{i,j}$ of each stage are calculated for its pressure, temperature and feed composition (here from an equation of state, Equilibrium ratios: correlations and equations of state).

m³/m³: standard cubic metres of gas per cubic metre of stock-tank oil

Holds when: We treat gas at standard conditions as ideal. Its $Z$ (the compressibility factor, which is 1 for an ideal gas) is 0.99–1.00, and about 0.99 for the heavy stock-tank gas. The assumption overestimates the solution gas–oil ratio by less than one per cent: by 0.5% in the lesson example. The density of the stock-tank oil is known. It is measured, or calculated from an equation of state with volume translation (a correction that improves the density of liquids), or calculated from the densities of the components, assuming that volumes add up. The stock-tank gas is included in the sum: it too was dissolved in the reservoir oil.

m³/m³ (reservoir cubic metres per cubic metre of stock-tank oil)

Holds when: The feed is single-phase oil at the bubble point pressure; at a higher pressure the numerator is its molar volume at that pressure. The density of the reservoir oil is measured or calculated from an equation of state with volume translation; such an estimate of liquid densities is off by a few per cent, and $B_{ob}$ by as much.

absolute pressures in the same units

Holds when: It is an empirical rule for the first choice, not a result of theory. The separator calculation and the economics refine the final pressures. The temperatures of the stages are close. If the stages differ greatly in temperature, the rule is less reliable.

absolute temperatures in the same units (in Standing's correlation, °R); pressures in the same units

Holds when: A vapour pressure line through two points is an approximation: the real curve is slightly bent in these coordinates. For components above their critical temperature (nitrogen and methane at separator temperature; CO₂ and ethane too, above 31–32 °C), $F_i$ is only the line continued beyond the critical point.

oilfield units only: $p$ in psia, temperatures in °R ($p\,[\text{psia}] = 145.04\,p\,[\text{MPa}]$, $T\,[{}^{\circ}\text{R}] = 1.8\,T\,[\text{K}]$)

Holds when: The pressure is up to 6.9 MPa (1000 psia) and the temperature up to 93 °C (200 °F): the range of the data to which the correlation was fitted. For nitrogen, CO₂, hydrogen sulfide, methane and ethane, $b$ and $T_b$ were fitted to measurements and differ from those calculated by Hoffman's formula. The composition of the mixture does not enter the formula. The correlation describes an “average” oil, and for unusual mixtures the error grows.

$T_F$ in °F ($T_F = 1.8\,t + 32$, $t$ in °C), $p$ in psia, $b$ and $T_b$ in °R

Holds when: The C7+ fraction is represented by an imaginary normal alkane (a straight-chain hydrocarbon) with $n$ carbon atoms. The molar mass and density of the fraction do not enter the formula. The estimate is rough: in the lesson example the ratio of C7+ at atmospheric pressure comes out several times greater than from the equation of state (see the table below).

$p_k$ in psia; $p_k\,[\text{MPa}] = p_k\,[\text{psia}]/145.04$

Holds when: An empirical estimate for oils that gives only the order of magnitude; temperature does not enter it.

absolute pressures, in the same units (their differences are gauge pressures); temperatures in kelvins

Holds when: At $p = p_{\text{sc}}$ the formula coincides with Wilson's formula, and at $p = p_k$ it gives $K_i = 1$. In between it is an interpolation, not a physical model. Composition enters only through $p_k$. The exponent 0.6 is empirical. In the original, as with Wilson, the factor is 5.37. Here it is 5.373, as in the lesson on phase equilibrium.

pressures in the same units, absolute temperatures

Holds when: The vapour pressure follows a straight line in the coordinates $\log_{10} p_s$ versus $1/T$ between $T_b$ and $T_c$.

metric oilfield units

Holds when: The inflow is single-phase and proportional to the drawdown: $q = J(\bar p - p_{wf})$ above the bubble point pressure (Well inflow: the well performance diagram). $J$ was measured on this well after cleaning. While the skin factor is changing, $J$ changes with it. (The skin factor is a number that shows how much the zone around the well helps or hinders flow.)

MPa and °C

Holds when: The chamber is rigid. The thermal expansion of the steel, a few per cent of that of the oil, slightly softens the pressure drop. The oil is single-phase. When gas comes out, the pressure falls more slowly, because the gas phase supports it.

optical density is dimensionless; volume in any units

Holds when: The filtrate is transparent in the chosen channel; if not, $\eta = (\mathrm{OD}_\infty - \mathrm{OD})/(\mathrm{OD}_\infty - \mathrm{OD}_{\text{f}})$, where $\mathrm{OD}_{\text{f}}$ is the optical density of the filtrate. Mixing is linear by volume. The exponent 5/12 is empirical and depends on the geometry of the inflow and on how uneven the reservoir is. At the start of pumping the law does not hold. For gas condensates and light oils the colour contrast is small, so the gas–oil ratio, density and methane bands are used instead.

dimensionless

Holds when: The volume of the filtrate hardly changes on going to standard conditions, and there is no gas in it.

gas at standard conditions (here 0.101325 MPa and 20 °C)

Holds when: The gas is measured by an orifice (differential-pressure) flow meter, whose reading is inversely proportional to $\sqrt{\gamma_g Z}$. For other types of meter the correction is different. Both volumes are referred to the same standard conditions.

gas at standard conditions

Holds when: The stock-tank gas is already dissolved in the separator oil and is not added separately in recombination. The gas and oil samples were taken at the same time, in a steady flow, at a bottomhole pressure above the bubble point pressure.

MPa, cm³, K (the temperature difference in °C or K)

Holds when: Nitrogen is an ideal gas, the piston is frictionless, and the chamber is rigid. The changes in the volume of the oil are small and linear in temperature and pressure. The equations are solved for $p_1$ by trial and error.

MPa and 1/MPa; the volumes can be in any units, as long as both are in the same unit (relative volumes work too)

Holds when: The temperature is constant (reservoir temperature) and the oil is single-phase. Between the two readings the compressibility changes little. Reports sometimes divide the change in volume not by the mean volume but by the volume at the higher pressure. The difference is about 1%, but you need to know it when you compare numbers.

m³ of reservoir oil per m³ of stock-tank oil

Holds when: Above the bubble point pressure the composition of the oil does not change, so the ratio of the volumes at $p$ and at $p_b$ is the same in the cell and in the reservoir. $B_{ob}$ must be the value for the separation scheme to which the result should refer.

absolute pressures in any identical units; $Y$ is dimensionless

Holds when: Only for readings below the bubble point pressure. For black oils $Y$ is almost linear in pressure. This is an empirical observation, not a law. For volatile oils, and close to the critical point, the line is curved.

gas at 0.101325 MPa and 20 °C; volume ratios in m³/m³

Holds when: The gas is let out completely and at the constant pressure of the step. At the bubble point pressure $R_{sD} = R_{sDb}$: all the gas that came out during the test. At the last step $R_{sD} = 0$ by definition. The results refer to the residual oil, not to the stock-tank oil of separation.

absolute pressures and temperatures; volumes in identical units

Holds when: At standard conditions we take the gas as ideal ($Z_{\text{sc}} = 1$). Only gas leaves the cell, with no drops of oil.

gas at 0.101325 MPa and 20 °C; m³/m³

Holds when: While it is pushed out, the oil in the cell stays single-phase: the pressure is kept at or above the bubble point pressure. The pressures and temperatures of the stages are the same as in the field. Otherwise the result refers to another scheme.

all quantities in m³/m³; in oilfield units (rb/STB, scf/STB) the formulas are the same

Holds when: We take the ratio $B_{ob}/B_{oDb}$ (the cubic metres of residual oil per cubic metre of stock-tank oil obtained from the same reservoir oil) to be the same at all pressures. This is an approximation, and close to atmospheric pressure it breaks down (advanced section). We convert the gas that came out in the reservoir ($R_{sDb} - R_{sD}$ per 1 m³ of residual oil) to stock-tank oil with the same factor. Above the bubble point pressure $R_s = R_{sb}$, and $B_o = B_{ob}\,V/V_b$ from the constant-composition test.

SI

Holds when: The flow is laminar (smooth, in layers) and steady: the Reynolds number $\mathrm{Re} = 4\rho q/(\pi d\mu)$, which shows whether a flow is smooth or turbulent, is much less than 2000. The liquid is single-phase and Newtonian (its viscosity does not depend on the rate of shear). The effects of the inlet and the outlet of the capillary are small or are included in the calibration. The pressure in the capillary is above the bubble point pressure of the liquid, otherwise gas comes out.

MPa·cm³ = J, so pressure in MPa and volume in cm³ give the amount of substance in moles

Holds when: After the gas is let out, the volume of the cell is again equal to $V_d$, like the pore volume of a reservoir without water influx. $n_0 - n_p$ includes both the gas and the liquid that remain in the cell.

grams on both sides: g/cm³·cm³ and g/L·L

Holds when: Only gas leaves the cell, and all the gas of each stage is collected and measured. The density of the gas at standard conditions is $\rho_{\text{air}}\,\gamma_j$, which means that we take the gas as ideal at 0.1 MPa. For the heavy gas of the last stages this slightly underestimates the mass.

pressures in MPa, formation volume factors and solution gas–oil ratio in m³/m³

Holds when: This is the apparent compressibility $-\frac{1}{B_o}\frac{\partial B_o}{\partial p} + \frac{B_g}{B_o}\frac{\partial R_s}{\partial p}$ written for a finite interval: the derivatives are replaced by differences between two rows (apparent compressibility). We take $R_s$, $B_o$ and $B_g$ from one table, on one base.

dimensionless ratios; viscosities in the same units

Holds when: Steady flow of a mixture of constant composition: as much liquid flows out into the well as drops out. Capillary and gravity forces are neglected. Gas and condensate are in equilibrium.

pressures in any common units; radii in common units

Holds when: Steady-state radial inflow, with the gas flow proportional to the pressure difference (high pressures). In a wet (rich) gas, $p/(\mu Z)$ rises slightly towards low pressures. So the drawdown below the dew point weighs a little more in the real inflow than in the estimate, and for this reason the estimate is slightly optimistic. The mobility in the zone is constant, which makes the estimate conservative, because the saturation is smaller at the boundary of the zone. The gas viscosity is taken to be the same in the zone and outside it. But the gas in the zone is leaner and less viscous, so this assumption too slightly overstates the losses. Inertial losses and the improvement of gas permeability at high velocities near the wellbore are not taken into account.

SI; heat capacities in the same units

Holds when: The injected water gives up all its “cold” to the rock: behind the front the temperature equals the injection temperature, and ahead of it the reservoir temperature. Heat exchange with the cap rock and the base rock (the rocks above and below the reservoir) is not taken into account. For thin reservoirs and long injection the radius comes out too large. The flow is radial and the reservoir is homogeneous.

dimensionless; fractions in the same units

Holds when: It is an empirical screening of stock-tank oil: pressure and dissolved gas are not taken into account. The accepted limits: below 0.7 the asphaltenes are stable, 0.7–0.9 is uncertain, above 0.9 they are unstable.