Equations: Facilities and processing of production

any consistent units; volumes at standard conditions

Holds when: The wells can give more liquid than the facilities can take, so the rate is limited by the facilities, not by the reservoir. The water cut depends only on the share of the reserves already produced, not on the rate of production. This is a simplification: in practice fast production can make the waterflood sweep (the part of the reservoir reached by the water) worse. In the late years the economic limit also restricts production: when the revenue no longer covers the costs, the field is shut down.

SI; to get thousand m³/d, multiply the result by 86.4

Holds when: The efficiency is constant. In gas turbines it falls at part load and in hot weather, and then the fuel consumption is higher. The heating value of associated gas depends on its composition: the more ethane, propane and butanes, the higher it is.

volumes at standard conditions

Holds when: The measure is handy for comparing fields of different size. A sensible share depends on the reservoir, the development method and the costs, and it is not a norm.

time in years; $L$ in m³/d, $N_r$ in m³

Holds when: The water cut follows this lesson's curve $\text{WC} = \sqrt{(R - R_{\text{bt}})/(1 - R_{\text{bt}})}$. The facilities are full of liquid all the time.

any consistent units: $K$ and $v_{\max}$ in the same units of velocity, the densities in the same units

Holds when: The form follows from the balance of the weight and the drag of a droplet (the derivation is in the advanced part of the lesson). Everything that depends on the droplet size and on the way the gas flows around it is collected in $K$. Rough guides for an estimate: for a vertical vessel with a mesh mist extractor $K$ is about 0.05–0.1 m/s, and for a horizontal one about 0.12–0.15 m/s (0.40–0.50 feet per second). The manufacturers give exact values for their internal parts. At high pressure, with foaming oil and with a heavy liquid load on the mist extractor, we reduce $K$. Without a mist extractor the allowed velocity is clearly lower: everything that has not settled leaves with the gas.

absolute pressures, and temperatures in kelvins

Holds when: The gas at standard conditions is treated as ideal. $Z$ at separator conditions comes from a correlation or an equation of state (Ideal and real gas: the gas deviation factor (Z)).

any consistent units

Holds when: The flow goes through the whole volume evenly. Dead zones (places where the liquid hardly moves) and short paths of the flow (where part of the liquid runs straight to the outlet) reduce the real time, which is why the inlet and distribution devices are so important. Guides from industry practice: in two-phase separators from 1 to 3–4 min; in three-phase separators 3–10 min for light oil and 10–30 min or more for heavy and cold oil. For a particular oil, a laboratory settling test of a sample gives a better value.

geometry; angles in radians

Holds when: The cylindrical part of the vessel. The volume of the heads (the rounded ends of the vessel) is counted separately or included in the margin.

SI

Holds when: Slow (creeping) flow around the droplet: the Reynolds number of the droplet, $\mathrm{Re} = \rho_c v d/\mu_c$ (a number that shows how turbulent the flow around it is), is less than about 1. At larger $\mathrm{Re}$ the drag grows faster, and the formula overestimates the velocity. The droplet is a rigid sphere, and the droplets are far from each other (at a high concentration they hinder each other and settle more slowly). The flow in the vessel is calm. Turbulence and convection from uneven heating hinder settling.

mg/dm³; with a small fraction of water, the volume of the mixture is almost equal to the volume of the oil

Holds when: The salts are only dissolved in the water. Salt crystals and salts on solid particles are not counted.

fractions and mg/dm³

Holds when: The wash water is fresh, and the droplets that have mixed merge and average their salinity. The residual water after washing has the salinity of the mixture. In practice, small droplets of produced water that have not mixed are harder to remove from the oil than the mixed ones. So a model with a single fraction $E$ is a simplification that underestimates the wash water needed. $E$ is chosen from the performance of a particular unit: the mixing valve, the pressure drop across it, the number of washing stages.

t, %, mg/dm³, kg/m³

Holds when: The gross mass is measured by a mass meter, or worked out from the volume and the density at the same conditions. The factor 0.1 converts units: mg/dm³ divided by kg/m³ (that is, by g/dm³) gives thousandths of a fraction, which is tenths of a per cent.

SI

Holds when: A single spherical droplet; the gas is at rest on average. With creeping flow $C_D = 24/\mathrm{Re}$, and the formula turns into Stokes' law.

absolute pressure in MPa (in the ratio $p_s/p$ any identical units), temperature in kelvins

Holds when: Sweet natural gas, without noticeable hydrogen sulfide and carbon dioxide (they increase the water content), in equilibrium with fresh water (salty water gives a little less). The tested range of the correlation is from 15.5 to 238 °C and from 0.1 to 70 MPa. Below 15.5 °C, and still more in the hydrate region, the result is an estimate. We take $p_s(T)$ from an accurate equation for water (in the models of the lesson, the international equation for the vapour pressure of water). Any error in it goes straight into the result. 748.9 g/m³ is the mass of a kilomole of water (18.015 kg) divided by the volume of a kilomole of gas at 20 °C and 0.101325 MPa (24.055 m³).

$\Delta T$ in °C, $W$ in per cent by mass

Holds when: A dilute (weak) solution: for methanol up to about 20–25% by mass. At higher concentrations the formula overestimates $\Delta T$, that is, it gives too little inhibitor (the advanced part of the lesson has a logarithmic form for methanol). For monoethylene glycol, reference books give its own constant, a few per cent smaller. With the same constant, about twice as much MEG by mass is needed as methanol, because its molar mass is twice as large. The hydrate formation temperature without inhibitor must be known in advance, from the gas composition, laboratory tests or an estimate from the specific gravity.

any identical units of pressure

Holds when: An ideal mixture of gases. At high pressures it is more accurate to use fugacity (a corrected pressure that accounts for non-ideal behaviour), but the standards for materials use this simple measure.

g/m³, g/mol; standard conditions 20 °C and 101.325 kPa

Holds when: The gas at standard conditions is ideal.

absolute pressures in identical units

Holds when: The gas is cooled after each stage to the same temperature. The pressure losses in coolers and separators are small (in practice the ratios are raised slightly to cover them).

SI; the power of a stage is $w$ multiplied by the mass flow rate of the gas

Holds when: An ideal gas with a constant $k$ and the correction $Z$ at suction. At high pressures, and for rich gas, it is more accurate to calculate with an equation of state. The losses are covered by one efficiency, referred to adiabatic compression. Mechanical losses and losses in the driver are added separately. The gas is single-phase: the liquid is separated before the stage.

°C (in °F the constant is 129.6)

Holds when: It was proposed for methanol. Reference books recommend it for concentrations from about 25 to 50% by mass, where the Hammerschmidt formula is already inaccurate. The solution is treated as ideal. For glycols, and for mixed inhibitors with salts, a calculation with an equation of state and a water activity model is more accurate.

SI

Holds when: The droplet is a sphere and moves like a solid ball. For oil, which is more viscous than water, the circulation inside the droplet speeds up the rise by only a few per cent. The flow around the droplet is laminar (smooth): the Reynolds number of the droplet, $\rho_w v_t d/\mu_w$, is less than about 1. For oil droplets in water this holds up to sizes of the order of 150 µm. The droplets do not hinder each other and do not merge, and the water is at rest. In a vessel, flow and mixing hinder the rise, while the merging of droplets helps it.

SI

Holds when: An ideal settling tank: the water moves evenly over the whole area, the droplets enter evenly over the height, and they neither break up nor merge. With a horizontal flow, droplets smaller than $d_{\min}$ are caught in part: those that entered close enough to the surface. Their share is $(d/d_{\min})^2$. In a vertical tank, where the water moves down against the droplets, all the small droplets leave with the water (the derivation is in the section for advanced study). In a real vessel part of the flow moves faster than the average (“short-circuiting”), and thermal currents and eddies interfere, so it catches less than the ideal one. The calculation gives the best case: the real smallest droplet that is caught is larger than the calculated one.

any units of pressure, the same for $p_i$ and $p$

Holds when: The gas is treated as a mixture of ideal gases (Dalton's law). At high pressures the standards also allow more accurate methods. The pressure is absolute: the gauge pressure from a pressure gauge gives too low a value, especially at low pressures (Systems of units: SI, oilfield and metric oilfield). For the water, we take the gas that is in equilibrium with it: in the separator or in the vapour space of the tank.

rates at standard conditions, m³/d

Holds when: The pressure is above the bubble point pressure (the pressure at which gas starts to come out of the oil), and there is no free gas in the reservoir. Otherwise the gas is also counted in the voidage (Why water is injected: pressure maintenance and oil displacement). All the produced water is returned to the reservoir. Losses in treatment (water that left with the oil, the reject of hydrocyclones, the washing of filters) increase the water from the source a little. The produced water is compatible with the reservoir into which it is injected.

SI

Holds when: The rock is represented by a bundle of identical straight tubes. With tortuosity $\tau$ (the winding of the flow paths) the radius is $\tau$ times larger, so the formula gives a lower estimate, and the allowed particle sizes derived from it have a margin. This is an order-of-magnitude estimate. Real throats have a wide distribution of sizes, and the design is based on the large ones, which carry the main flow.

concentrations in identical units, pressures in identical units

Holds when: Equilibrium: the water is in contact with the gas long enough. In a column it is approached gradually, stage by stage, so in practice more oxygen is left. The temperature and salinity are the same as for $C_{\text{sat}}$. With heating, the solubility of oxygen falls.

metric oilfield units: m³/d, MPa, kW

Holds when: $86.4$ is the same constant as in the formula for the hydraulic power of a pump: the seconds in a day divided by a thousand. The losses in the water lines from the station to the block are the same for all wells and are not included here. The electric power is $1/\eta$ times the hydraulic power, where $\eta$ is the efficiency of the pump unit.

metric oilfield units

Holds when: The injection is close to steady state. Right after a change of the rate, the pressure has not settled yet. The reservoir pressure in the injection area is known, from a falloff test (pressure falloff after shut-in) or from neighbouring wells. The fracture has not opened. After it opens, the rate grows with almost no rise in pressure, and the injectivity stops being a constant.

SI; to get MPa, divide by $10^6$

Holds when: A single-phase flow of water. The viscosity and density are at the mean temperature in the tubing. Friction grows roughly as the square of the rate, so at a low rate it can be neglected, and at a high rate it cannot. Injection down the annulus (the space between the tubing and the casing) or into a deviated well needs its own lengths and diameters.

metric oilfield units

Holds when: The injection in each period is close to steady state, and the reservoir pressure in the injection area hardly changes. Its rise also increases the slope. We calculate the bottomhole pressure from the wellhead pressure, counting the hydrostatic pressure and the friction at the actual rate. The slope over a segment is an average. If the injectivity changed within the segment, the slope is the average value of $1/I$, weighted by the injection.