Equations: Reservoir oil properties

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

Holds when: At pressure $p$ and reservoir temperature the oil is single-phase and the gas in it is in equilibrium. Both volumes come from the same method of degassing (the separation conditions). The result depends on this method.

m³/m³, which means reservoir cubic metres per cubic metre of stock-tank oil. In oilfield units it is rb/STB (reservoir barrels per stock-tank barrel), and the number is the same

Holds when: At $p$ and $T$ the oil is single-phase, so free gas is not part of $V_o$. The stock-tank oil is the oil left after separation under the stated conditions.

consistent units of pressure and compressibility: MPa and MPa⁻¹, or psi and psi⁻¹

Holds when: The pressure is not below the bubble point pressure and the oil is single-phase. The compressibility is constant over the range considered (over a wide range, the average value is used).

m³/d; in oilfield units, bbl/d with $q_o$ in STB/d, $R$ in scf/STB and $B_g$ in rb/scf

Holds when: $R_s$, $B_o$ and $B_g$ are taken at the average pressure in the drainage area of the well (the part of the reservoir that feeds the well). All the gas in excess of the dissolved gas came from the reservoir as free gas. Gas injected into the well for gas lift (a way of lifting oil by injecting gas) is subtracted from the measurement. Above the bubble point pressure, $R = R_s$, and the free-gas term is zero.

absolute pressures and temperatures in consistent units

Holds when: The gas is single-phase. The black oil model does not track its composition as the pressure falls. $Z$ has been found for this gas at reservoir conditions (Ideal and real gas: the gas deviation factor (Z)).

m³/m³; in oilfield units, rb/STB with $B_g$ in rb/scf and $R_s$ in scf/STB

Holds when: The freed gas stays with its oil (a closed system). $B_o$, $B_g$ and $R_s$ refer to the same method of degassing.

MPa, m³/m³

Holds when: This is an approximation for one assumed oil, not a correlation. The exponent 0.9 and the linear link between $B_o$ and $R_s$ were chosen to reproduce the typical shape of laboratory curves. A constant $Z = 0.87$ is taken for $B_g$. This is a simplification that is acceptable for illustration in the range 5–18 MPa. The standard conditions are 0.101325 MPa and 20 °C.

SI (densities in kg/m³)

Holds when: Every component of the reservoir oil goes either into the stock-tank oil or into the gas that is counted; there are no losses.

kg/m³ and °C (or g/cm³ and °C)

Holds when: The pressure is atmospheric and the oil is degassed. The temperature difference is a few degrees to a few tens of degrees. The formula is linear in $\beta$, and $\beta$ itself changes a little with temperature. In commercial measurement (oil sales), oil volumes and densities are converted with standard tables, not with a single coefficient.

SI: densities in kg/m³, $R_s$ and $B_o$ in m³/m³

Holds when: $R_s$, $B_o$, $\rho_{o,\text{sc}}$ and $\rho_{g,\text{sc}}$ refer to the same separation scheme and the same standard conditions. All the gas freed in separation is counted in $R_s$, and the oil is counted in $\rho_{o,\text{sc}}$. Below the bubble point pressure the heavy components of the gas stay in the oil, so the dissolved gas is heavier than the total separator gas. If we use $\gamma_g$ of the total gas, the density at low pressures comes out slightly too low.

SI; in practice $G_o$ is in kPa/m: $\rho_o\,[\text{g/cm}^3] = G_o\,[\text{kPa/m}]/9.81$

Holds when: The oil in the measurement zone is stationary and uniform in composition, and the measurements are of good quality. The producing gas–oil ratio was measured at a bottomhole pressure above the bubble point pressure. Otherwise it contains free gas (Reservoir fluid sampling). The relative error of $B_o$ equals the relative error of the gradient.

MPa⁻¹ with pressure in MPa; in oilfield units, psi⁻¹

Holds when: The pressure is not below the bubble point pressure. The oil is single-phase and its composition does not change, so $V$, $B_o$ and $1/\rho_o$ change in the same proportion. The temperature is constant. The minus sign makes $c_o$ positive, because the volume decreases as the pressure rises.

consistent units of pressure and compressibility

Holds when: $p \ge p_b$, the oil is single-phase, and the temperature is constant.

oilfield units: scf/STB, °F, psia, psi⁻¹

Holds when: The oil is undersaturated, $p \ge p_b$. The input data lie within the database of the correlation (roughly 15–60 °API). The separator gas is corrected to 100 psig, and the pressure is absolute.

any pressure units (the ratio and the logarithm are dimensionless)

Holds when: The same as for the compressibility correlation.

consistent: pressure in MPa, compressibility in MPa⁻¹

Holds when: $p < p_b$. The gas freed as the pressure falls stays within the volume of the reservoir that we are considering. The expansion of gas that was already free before is not included here. The term $S_gc_g$ of the total compressibility accounts for it. $B_o$ and $R_s$ come from a table for one degassing scheme. We take the derivatives from a smoothed curve or from differences between neighbouring points.

1/°C (the same as 1/K); the second formula is in MPa per °C

Holds when: The oil is single-phase and its composition does not change. For reservoir oil the coefficient is $(0.7\text{–}1.2)\cdot 10^{-3}$ 1/°C, and for stock-tank oil around 20 °C it is $(0.6\text{–}1.0)\cdot 10^{-3}$ 1/°C. The second formula is for oil trapped in a rigid volume. Heating by one degree raises the pressure by $\beta_o/c_o$.

oilfield units: psia, scf/STB, °F, °API. To get MPa, multiply the result by 0.006895

Holds when: The data behind the correlation: bubble point pressure 130–7000 psia (0.9–48 MPa), temperature 100–258 °F (38–126 °C), $R_{sb}$ 20–1425 scf/STB (4–258 m³/m³), 16.5–63.8 °API, $\gamma_g$ 0.59–0.95. The gas came from two-stage separation (first stage 265–465 psia at 100 °F). With another scheme $R_{sb}$ and $\gamma_g$ are different, and the result shifts. The oil has no appreciable admixture of carbon dioxide, nitrogen and hydrogen sulfide.

oilfield units

Holds when: The same data as for the bubble point pressure equation. The exponent 1.2048 = 1/0.83.

oilfield units; $B_o$ is a dimensionless ratio of volumes

Holds when: The oil is saturated ($p \le p_b$). The data have $B_{ob}$ of 1.024–2.15 and the same ranges as above. Above the bubble point pressure, $B_o = B_{ob}\,\e^{-c_o(p - p_b)}$ with the oil compressibility (Density and compressibility of reservoir oil).

per cent

Holds when: The samples are representative and were studied in one laboratory or by one method. There are enough samples for the statistics to mean something. With three or four samples the conclusions are preliminary.

any pressure units; the factor is dimensionless

Holds when: The oils belong to one system: a common source and a similar composition of the gas and the heavy fraction. For an oil of another type the factor does not transfer. The input data of the new oil lie within the range of the correlation and are close to those of sample 1.

gas at 0.101325 MPa and 20 °C; $B_g = p_{\text{sc}}Z_wT/(pT_{\text{sc}})$

Holds when: The reservoir pressure is above the dew point, so the reservoir mixture is single-phase, and all the production came from the reservoir as gas. The producing gas–oil ratio was measured at steady-state conditions, and the condensate droplets carried in the gas are counted (Reservoir fluid sampling).

oilfield units; $T$ is the reservoir temperature, °F

Holds when: The data: $R_s$ 0–2199 scf/STB, 15.3–59.5 °API. Different reviews give slightly different limits. The formation volume factor of saturated oil: $B_o = 1 + A_1R_s + (T - 60)\,(\mathrm{API}/\gamma_{g100})\,(A_2 + A_3R_s)$ with $A_1 = 4.677\cdot 10^{-4}$, $A_2 = 1.751\cdot 10^{-5}$, $A_3 = -1.811\cdot 10^{-8}$ (up to 30 °API) and $4.670\cdot 10^{-4}$, $1.100\cdot 10^{-5}$, $1.337\cdot 10^{-9}$ (lighter oils).

oilfield units

Holds when: The data: bubble point pressure 165–7142 psia (1.1–49 MPa), 80–280 °F (27–138 °C), $R_{sb}$ 90–2637 scf/STB, 22.3–48.1 °API, $\gamma_g$ 0.65–1.276. Formation volume factor: $\lg(B_{ob} - 1) = -6.58511 + 2.91329\lg B^{*} - 0.27683\,(\lg B^{*})^2$, $B^{*} = R_{sb}(\gamma_g/\gamma_o)^{0.526} + 0.968\,T$.

oilfield units

Holds when: The data: bubble point pressure 130–3573 psia (0.9–24.6 MPa), 74–240 °F (23–116 °C), $R_{sb}$ 26–1602 scf/STB, 19.4–44.6 °API, $\gamma_g$ 0.752–1.367.

oilfield units

Holds when: The data: bubble point pressure 1574–6523 psia (10.9–45 MPa), 114–288 °F (46–142 °C), $R_{sb}$ 217–1406 scf/STB, 16.3–45.0 °API, $\gamma_g$ 0.578–0.852.

SI: Pa·s and m²/s

Holds when: The fluid is Newtonian: its viscosity does not depend on the shear rate. Oil above the wax appearance temperature (the temperature at which wax crystals start to form) usually is. The flow is laminar (smooth, in layers, with no turbulence).

oilfield units: °F; the result is in cP, that is, mPa·s

Holds when: The data behind the correlation cover 16–58 °API and 70–295 °F (21–146 °C). At low temperatures (near the lower limit of its data, and below) the correlation tends to overestimate the viscosity. The oil is Newtonian, which holds above the wax appearance temperature.

oilfield units: $R_s$ in scf/STB (from m³/m³ with the gas at 20 °C, multiply by 5.53), viscosity in cP

Holds when: It is built on data for several hundred oils. At $R_s = 0$ it gives $a \approx b \approx 1$, that is, $\mu_o \approx \mu_{od}$. $\mu_{od}$ is the viscosity of the same oil at the same temperature. A measured value is better than a calculated one. Below the bubble point the formula takes the solution gas–oil ratio at the current pressure, that is, the gas that remains in the oil.

oilfield units: pressure in psia ($p\,[\text{psia}] = p\,[\text{MPa}]/0.006895$), viscosity in cP

Holds when: The oil is undersaturated (above the bubble point), $p \ge p_b$. $\mu_{ob}$ is the measured or calculated viscosity at the bubble point. The database is the same as for the solution gas–oil ratio and compressibility correlations of the same authors (PVT correlations for oil and gas condensate, compressibility).

the units must be exactly these: $\sigma$ in mN/m (dyn/cm), densities in g/cm³, molar masses in g/mol

Holds when: The phases are in equilibrium, and their compositions and densities are known, from an experiment or from an equation-of-state calculation (a thermodynamic model of the fluids) (Vapour–liquid equilibrium). For a pure substance the sum reduces to a single term: $\sigma^{1/4} = \mathcal{P}(\rho_L - \rho_V)/M$. The parachors of heavy fractions have not been measured. They are estimated from the molar mass, for example $\mathcal{P} = -11.4 + 3.23M - 0.0022M^2$ ($M$ in g/mol). Near the critical point, and in systems close to it, the method is less accurate.

SI, dimensionless

Holds when: This is the simplest form. Definitions with $\cos\theta$ in the denominator, or with the pore velocity instead of the Darcy velocity, are also found. Only numbers from the same definition can be compared. The thresholds that follow are rough guides. Residual oil starts to decrease from about $N_c \sim 10^{-5}\text{–}10^{-4}$ (higher in oil-wet rocks, where oil clings to the rock surface), and almost all of it is displaced at about $N_c \sim 10^{-2}$.

oilfield units

Holds when: It approximates a chart built for oils at 100 and 200 °F. At high temperatures it tends to underestimate the viscosity.

oilfield units

Holds when: It was built on North Sea oils, like the same author's bubble point correlation (correlation equations).

oilfield units

Holds when: It has the same form as the Beggs–Robinson correlation, with coefficients fitted to a different database.

oilfield units

Holds when: It started as a chart, and the formulas approximate that chart. At $R_s = 0$ it gives $\mu_o \approx \mu_{od}$ ($a = 0$, $b = 0.992$).

oilfield units

Holds when: $p \ge p_b$. Viscosity grows linearly with pressure.