Retrograde condensation, wax, asphaltenes and high-pressure fluids

Quick refresher

Condensate drops out near the wellbore as soon as the bottomhole flowing pressure is below the dew point. The flow keeps bringing in fresh gas, so the condensate saturation grows until the liquid starts to move. In steady flow, the properties of the gas set the ratio $k_{rg}/k_{ro}$ (the ratio of the relative permeabilities to gas and to condensate). The saturation then comes out several times higher than in a constant volume depletion experiment.
The part of the drawdown that lies below the dew point “works” with a reducing factor: the mobility of the mixture in the condensate zone divided by the mobility of the gas. In the lesson's example the factor is about 0.23, and the well loses almost two thirds of its rate.
Wax crystallises below the wax appearance temperature (WAT) and grows on cold walls. Below the pour point the oil stops flowing. In the reservoir, wax drops out if the reservoir is cooled, for example by the injection of cold water.
Asphaltenes become unstable as the pressure falls above the bubble point pressure, and they are most unstable close to it. Below it they dissolve again. Light, strongly undersaturated oils with a small asphaltene content suffer more often than heavy ones.
Hydrates form from free water and light gases at high pressure and low temperature, above all downstream of the choke (a flow restriction) and in flowlines.
Close to the critical point the properties change sharply. At very high pressures the gas is dense, $Z$ is greater than one, and the density difference between gas and oil is small. In tall accumulations the gas–oil ratio and the saturation pressure change with depth.

Key relationships

\frac{k_{rg}}{k_{ro}} = \frac{\mu_g}{\mu_o}\left(\frac{1}{V_{ro}} - 1\right)

the zone near the well where both gas and condensate flow

\frac{q}{q_0} = \frac{(p_e - p_d) + m\,(p_d - p_{wf})}{p_e - p_{wf}}

share of the rate that is kept when there is condensate banking (two-zone estimate)

r_T = \sqrt{\frac{V_{\text{inj}}\,(\rho c)_w}{\pi h\,(\rho c)_{\text{rock}}}}

radius of the zone cooled by water injection

\text{CII} = \frac{w_{\text{sat}} + w_{\text{asph}}}{w_{\text{arom}} + w_{\text{resin}}}

colloidal instability index from the group composition

Common pitfalls

Estimating the condensate saturation near the wellbore from the dropout curve of the constant volume depletion experiment: near the wellbore it is several times higher.
Assuming that a gas-condensate accumulation is safe while the reservoir pressure is above the dew point: banking appears at the well as soon as the bottomhole flowing pressure falls below it.
Confusing the wax appearance temperature with the pour point: the former is noticeably higher, and deposits begin long before the oil solidifies.
Treating asphaltenes as a problem of heavy oils only.
Taking $Z \approx 1$ for gas at pressures of 70–100 MPa: there it is noticeably greater than one.

After this lesson you can

explain why the condensate saturation (the share of the pore space filled with condensate) near the wellbore of a gas-condensate well is much higher than in the laboratory experiment, and estimate the loss of productivity from condensate banking
describe where and when wax drops out, distinguish the wax appearance temperature from the pour point, and estimate the zone of the reservoir cooled by the injection of cold water
explain why asphaltenes drop out above the bubble point pressure and why light undersaturated oils are more at risk than heavy ones; find the interval of the tubing where they are unstable, and calculate the colloidal instability index
name the conditions for the formation of hydrates and the ways of studying and preventing them
describe the features of near-critical and high-pressure fluids and estimate the error caused by neglecting the gas deviation factor ( $Z$ , which shows how far a real gas departs from an ideal gas)
explain how and why the composition of a fluid changes with depth and what this means for hydrocarbons in place and for sampling

Where the fluid causes trouble

On its way from the reservoir to the separator, the produced fluid loses pressure and, from the bottom of the well upwards, loses temperature. Along this path the fluid crosses the boundaries of phase regions: the bubble point and dew point pressures (Phase behaviour: pure substances and binary mixtures) and, for solid phases, the boundaries where wax, asphaltenes and hydrates drop out. Where exactly the path crosses them depends on the composition of the fluid and on how the well is operated. That is why the same phenomena cause trouble in one field and not in another.

SchematicMap of complications on a pressure–temperature diagram (a schematic, not to scale; the pressure scale is non-linear): the oil bubble point pressure line, the region of asphaltene instability around it, the wax appearance temperature (WAT), the hydrate formation curve and the production path from the reservoir through the tubing and wellhead to the separator.

Phenomena of this lesson

Phenomenon: Retrograde condensation
What causes it: gas-condensate pressure below the dew point
Where it shows up: near the wellbore, later in the reservoir
Consequences: loss of productivity, loss of condensate

Phenomenon: Wax
What causes it: temperature below the wax appearance temperature
Where it shows up: tubing, flowlines, pipelines; the reservoir when it is cooled
Consequences: narrowing of pipes, solidification during a shutdown

Phenomenon: Asphaltenes
What causes it: pressure decline above the bubble point pressure, mixing with gas or light oil, acid
Where it shows up: tubing, near-wellbore zone, equipment
Consequences: deposits that are hard to dissolve

Phenomenon: Hydrates
What causes it: water and light gas at high pressure and low temperature
Where it shows up: downstream of the choke, flowlines, wellhead equipment
Consequences: plugs, stoppage of flow

Phenomenon: Near-critical fluids and high pressures
What causes it: composition and conditions close to critical; pressures of 70 MPa and above
Where it shows up: reservoir, samples, calculations
Consequences: errors in the fluid type and in the hydrocarbons in place

Phenomenon	What causes it	Where it shows up	Consequences
Retrograde condensation	gas-condensate pressure below the dew point	near the wellbore, later in the reservoir	loss of productivity, loss of condensate
Wax	temperature below the wax appearance temperature	tubing, flowlines, pipelines; the reservoir when it is cooled	narrowing of pipes, solidification during a shutdown
Asphaltenes	pressure decline above the bubble point pressure, mixing with gas or light oil, acid	tubing, near-wellbore zone, equipment	deposits that are hard to dissolve
Hydrates	water and light gas at high pressure and low temperature	downstream of the choke, flowlines, wellhead equipment	plugs, stoppage of flow
Near-critical fluids and high pressures	composition and conditions close to critical; pressures of 70 MPa and above	reservoir, samples, calculations	errors in the fluid type and in the hydrocarbons in place

The change of composition with depth is not a flow problem but a property of the accumulation; it is covered at the end of the lesson.

The laboratory studies all these phenomena on the same samples as the ordinary PVT properties, but with special experiments that are ordered separately: the wax appearance temperature and the pour point, the group composition (the shares of the main chemical groups in the oil), the asphaltene onset pressure and the hydrate formation curve. If the report contains no such experiments, record this as a gap. Do not read the silence as “no problems” (the order in which to read a report).

Retrograde condensation near the well: condensate banking

The lesson on retrograde condensation explains why liquid drops out of gas as the pressure falls. The constant volume depletion experiment (gas-condensate experiments) shows how much liquid drops out in the reservoir as a whole when it is produced by pressure depletion. (In this experiment the pressure of a gas sample is lowered in a cell of fixed volume, and gas is removed at each step.) Near the well the picture is different. The pressure is lowest at the bottom of the well. While the reservoir pressure is still above the dew point, condensate drops out only in a small zone around the wellbore. But all the gas that the well produces passes through this zone, and each portion of it leaves a little liquid behind.

The zone is called a condensate bank or a condensate plug, and its effect is condensate banking (condensate blockage). In the zone, the liquid saturation grows until the condensate becomes mobile and starts to flow into the well as fast as it drops out. The flows balance: the well receives exactly the gas that enters the zone, only part of it as liquid. A model of three zones around a gas-condensate well describes this picture (Fevang and Whitson, 1990s):

Zone 1 — at the wellbore: Both gas and condensate flow. The composition of the flowing mixture is constant and equal to the composition of the gas entering the zone. The outer boundary is where the pressure equals the dew point of that gas.
Zone 2 — accumulation: The pressure is below the dew point. Condensate drops out but is immobile, and the saturation is close to the dropout curve of the constant volume depletion experiment. This zone appears when the reservoir pressure also falls below the dew point.
Zone 3 — single-phase gas: The pressure is above the dew point and there is no liquid; the fluid is the original reservoir gas.

Ratio of relative permeabilities in the zone near the well

\frac{k_{rg}}{k_{ro}} = \frac{\mu_g}{\mu_o}\left(\frac{1}{V_{ro}} - 1\right)

where:

$k_{rg},\ k_{ro}$ relative permeabilities to gas and to condensate at the saturation in the zone, —
$\mu_g,\ \mu_o$ viscosities of gas and condensate at reservoir conditions in the zone, mPa·s
$V_{ro}$ liquid volume fraction of the mixture of the original composition at the local pressure, from the constant-composition experiment (the sample expands in a closed cell and nothing is removed), —

Units: 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.

The volume fraction of liquid in the flow equals its fraction in the mixture, so $\dfrac{k_{ro}/\mu_o}{k_{ro}/\mu_o + k_{rg}/\mu_g} = V_{ro}$ . The formula follows from this. The relative permeability curves set the saturation, not the dropout curve: in zone 1 it is the saturation at which $k_{rg}/k_{ro}$ equals the right-hand side. The richer the gas (the larger $V_{ro}$ ), the smaller this ratio, and the more condensate there is near the wellbore.

Worked exampleCondensate saturation and gas permeability near the wellbore

Given (assumed data)

Gas condensate: the same as in the lesson How to read a PVT report: dew point 36.0 MPa at 95 °C; the maximum liquid fraction in the constant volume depletion experiment is 11.9% at 18 MPa
In the zone near the wellbore: $V_{ro} = 0.10$ ; $\mu_g = 0.028$ mPa·s, $\mu_o = 0.25$ mPa·s
Relative permeabilities (Corey model): connate water 0.20, critical condensate saturation 0.10; $k_{rg} = 0.90\,\left(\frac{0.80 - S_o}{0.80}\right)^2$ , $k_{ro} = 0.80\,\left(\frac{S_o - 0.10}{0.70}\right)^2$

Solution

Required ratio: $\dfrac{k_{rg}}{k_{ro}} = \dfrac{0.028}{0.25}\left(\dfrac{1}{0.10} - 1\right) = 0.112\cdot 9 = 1.008$ .
We find the saturation by trial and error: at $S_o = 0.4362$ , $k_{rg} = 0.90\cdot(0.45475)^2 = 0.1861$ and $k_{ro} = 0.80\cdot(0.48029)^2 = 0.1845$ , which gives the ratio 1.009.
As a fraction of the hydrocarbon pore volume (the pore space not taken by connate water): $0.4362/0.80 = 0.545$ , that is 54.5%, which is 4.6 times the maximum liquid fraction in the constant volume depletion experiment.
The relative permeability to gas has fallen from 0.90 to 0.1861, that is, to 20.7% of the value without condensate.

Answer: Near the wellbore the condensate occupies about 54.5% of the hydrocarbon pore volume, and the relative permeability to gas falls almost fivefold, to 20.7% of its original value.

What it means. The flow sets the saturation near the wellbore; it is not tied to the maximum of the dropout curve. At a saturation like that of the constant volume depletion experiment (about 9.5% of the pores), the condensate would be immobile, because it is below the critical saturation of 0.10, and it would keep accumulating. The saturation in the zone is constant only in this estimate. At the outer boundary of the zone, where the pressure is close to the dew point, $V_{ro}$ and the saturation are smaller. The values of $V_{ro}$ , the viscosities and the curves are assumed. For a real reservoir, the relative permeability curves of the gas–condensate system are measured on core (rock samples) at conditions close to reservoir conditions. The data in this example are assumed for illustration.

CalculatedRelative permeabilities to gas and condensate from the example, and the state of the zone near the wellbore. The ratio

k_{rg}/k_{ro}

equals 1.008 at a saturation of about 0.44 of the pore volume. The vertical dashed line is the saturation at the maximum dropout in the constant volume depletion experiment. At that saturation the condensate would be immobile.

A convenient way to estimate how banking reduces the rate is a two-zone model of steady-state radial inflow. Inside the radius $r_b$ , where the pressure is below the dew point, a mixture of gas and condensate flows. Outside it, single-phase gas flows. At high pressures (here from 26 to 38 MPa) the quantity $p/(\mu Z)$ , on which the gas flow depends, changes by only a few per cent. So we can take the flow as proportional to the pressure difference, as for a liquid. The mobility of the mixture in the zone is $k_{rg}/\mu_g + k_{ro}/\mu_o$ , and by the formula above $k_{ro}/\mu_o = (k_{rg}/\mu_g)\,V_{ro}/(1 - V_{ro})$ . Relative to the mobility of the gas without condensate, it therefore equals $m = (k_{rg}/k_{rg}')/(1 - V_{ro})$ , where $k_{rg}'$ is the relative permeability to gas at connate water (the water that stays in the pores).

Share of the rate and radius of the condensate zone (two-zone estimate)

\frac{q}{q_0} = \frac{(p_e - p_d) + m\,(p_d - p_{wf})}{p_e - p_{wf}},\qquad \ln\frac{r_b}{r_w} = \frac{m\,(p_d - p_{wf})\,\ln(r_e/r_w)}{(p_e - p_d) + m\,(p_d - p_{wf})}

where:

$q,\ q_0$ rate with and without the condensate zone at the same $p_e$ and $p_{wf}$ , m³/d
$p_e,\ p_d,\ p_{wf}$ pressure at the boundary of the drainage area, dew point pressure, bottomhole flowing pressure, MPa
$m$ mobility of the mixture in the condensate zone relative to the mobility of the gas outside it, —
$r_b,\ r_e,\ r_w$ radii of the condensate zone, the drainage area and the well, m

Units: 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.

The part of the drawdown above the dew point works in full, and the part below it works with the factor $m$ . The share of the rate does not depend on the radii; only the size of the zone does. The equivalent skin factor of the zone (a number that shows how much the zone hinders flow) follows Hawkins' formula: $S = (1/m - 1)\ln(r_b/r_w)$ (Hawkins' formula).

Worked exampleHow much of the rate condensate banking takes away

Given (assumed data)

Well: reservoir pressure at the boundary of the drainage area 38.0 MPa, dew point 36.0 MPa; bottomhole flowing pressure 26.0 MPa; $r_e = 500$ m, $r_w = 0.1$ m
Zone near the wellbore (previous example): $k_{rg}/k_{rg}' = 0.2068$ , $V_{ro} = 0.10$

Solution

Mobility of the mixture in the zone: $m = 0.2068/(1 - 0.10) = 0.2298$ .
Share of the rate: $\dfrac{q}{q_0} = \dfrac{2.0 + 0.2298\cdot 10.0}{12.0} = 0.358$ , so the well delivers 36% of the rate it would give without banking.
Size of the zone: $\ln(500/0.1) = 8.517$ ; $\ln\dfrac{r_b}{r_w} = \dfrac{0.2298\cdot 10.0\cdot 8.517}{2.0 + 0.2298\cdot 10.0} = 4.554$ , $r_b = 0.1\,e^{4.554} = 9.5$ m.
Equivalent skin factor: $S = (1/0.2298 - 1)\cdot 4.554 = 15.3$ . Check: $8.517/(8.517 + 15.3) = 0.358$ .
Other bottomhole flowing pressures: at 30 MPa the share of the rate is 0.422 (zone 3.2 m), and at 34 MPa it is 0.615 (0.49 m). But the rate itself at 26 MPa is still 1.27 times higher than at 30 MPa.

Answer: A condensate zone with a radius of about 9.5 m acts like a skin factor of 15.3 and takes away 64% of the rate. The well delivers 36% of what it would give without banking.

What it means. A dozen metres or so around the wellbore decide the fate of the well, as with any damage to the near-wellbore zone. A smaller drawdown reduces the share of the rate that is lost, but it does not raise the rate itself: 2 MPa of drawdown above the dew point delivers as much as 8.7 MPa below it. The estimate is pessimistic in one way: the saturation at the boundary of the zone is smaller, and at high velocities near the wellbore the permeability to gas grows. It is optimistic in another way: inertial losses at high velocities (Inflow at high rates: non-Darcy flow) add resistance. When the reservoir pressure also falls below the dew point, an accumulation zone is added to zone 1 and the losses grow. The data in this example are assumed for illustration.

CalculatedPressure around the well of the example, at a bottomhole flowing pressure of 26 MPa: without condensate (dashed line) and with a condensate zone near the wellbore (solid line). The distance scale is logarithmic. In the zone, which has a radius of about 9.5 m, the pressure drops steeply, and at the same drawdown the rate is almost three times lower.

Well operating mode. Keeping the bottomhole flowing pressure above the dew point is reliable, but it usually costs too much rate. A compromise is chosen, knowing the price of each megapascal of drawdown (see the example above).
A larger inflow area. A hydraulic fracture (a crack propped open to carry flow) or a horizontal wellbore delivers the same rate at a smaller drawdown and lower velocities, and the condensate zone is spread over a larger area (Hydraulic fracturing: purpose and well selection, Horizontal wells).
Pressure maintenance. Injecting dry gas keeps the reservoir pressure above the dew point (gas cycling). Banking remains near the wells, but the accumulation zone does not grow.
Near-wellbore treatments. Injecting a solvent or dry gas into a producing well temporarily vaporises or displaces the condensate. The effect lasts until the zone builds up liquid again.

Wax: crystallisation, deposits and solidification

Wax, in this sense, means heavy alkanes, mostly straight-chain (normal) ones, with long chains of about twenty carbon atoms or more (Composition of reservoir fluids). At reservoir temperature they are dissolved in the oil. On cooling, their solubility falls, and at the wax appearance temperature (WAT; in Russian-language literature, the temperature of saturation of the oil with wax) the first crystals appear. As the oil cools further, more crystals form and grow together into a network. The oil thickens and stops behaving as a Newtonian fluid (Oil viscosity and interfacial tension), which means that its viscosity now depends on the shear rate. At the pour point it stops flowing.

Wax appearance temperature (WAT): The temperature at which the first crystals appear on cooling. It can be measured in several ways: under a microscope in polarised light, from the heat given off by crystallisation (calorimetry), from a kink in the viscosity curve, or from a rise in the pressure drop across a filter. The method and the cooling rate change the result by a few degrees, so the report states the method.
Pour point: The temperature at which oil, cooled by a standard procedure, stops flowing. This is the solidification temperature of Russian-language practice. The closely related flow temperature is the lowest temperature at which the oil still flows, and English-language practice calls it the pour point. Both are usually noticeably below the wax appearance temperature, often by tens of degrees, and both matter for pipeline shutdowns and for the storage of oil.
Wax content: The mass fraction of wax in the stock-tank oil. Oils with a large wax content are called high-wax oils.

Deposits grow on a wall that is colder than the flow: wax molecules diffuse towards the cold wall and crystallise on it. So we expect deposits above the depth at which the temperature of the flow in the tubing falls to the wax appearance temperature. This depth is found from the temperature profile of the well (temperature margin). Deposits are also expected in flowlines and pipelines. In Russian-language practice, deposits of this kind, in which resins and asphaltenes settle out together with the wax, are called asphaltene–resin–paraffin deposits (ASPO, from the Russian abbreviation). Dissolved gas usually lowers the wax appearance temperature a little, so it is higher for stock-tank oil than for reservoir oil. For accurate calculations, measure it on a reservoir sample.

In the reservoir itself, wax drops out only if the temperature there falls to the wax appearance temperature, or if the wax appearance temperature rises, for example when the oil loses dissolved gas below the bubble point pressure. This is dangerous for oils whose wax appearance temperature is close to the reservoir temperature. For such oils, injecting cold water, flushing with cold fluids, or simply operating an injection well for a long time is enough to create a zone around it where wax crystallises in the pores (damage during production).

Radius of the zone cooled by water injection

r_T = \sqrt{\frac{V_{\text{inj}}\,(\rho c)_w}{\pi\,h\,(\rho c)_{\text{rock}}}},\qquad (\rho c)_{\text{rock}} = (1 - \phi)(\rho c)_{\text{ma}} + \phi\,\big[S_w(\rho c)_w + S_o(\rho c)_o\big]

where:

$r_T$ radius of the cooled zone around the injection well, m
$V_{\text{inj}}$ volume of injected water, m³
$(\rho c)_w,\ (\rho c)_o,\ (\rho c)_{\text{ma}}$ volumetric heat capacities of water, oil and the mineral matrix of the rock, MJ/(m³·K)
$(\rho c)_{\text{rock}}$ heat capacity of a unit volume of the saturated rock behind the front, MJ/(m³·K)
$h,\ \phi,\ S_w,\ S_o$ reservoir thickness, porosity, saturations behind the front, m, —, —, —

Units: 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.

The heat capacity of a unit volume of the reservoir, which is the mineral matrix together with the fluids, is greater than that of the water in its pores, so the cooled zone lags behind the water front. Even so, over years of injection it extends a hundred metres or more.

Worked exampleHow far the cold from an injection well reaches

Given (assumed data)

Injection: 400 m³/d of cold water for 3 years
Reservoir: $h = 12$ m, $\phi = 0.22$ ; initial water saturation 0.25, residual oil saturation 0.25
Heat capacities, MJ/(m³·K): water 4.18; mineral matrix 2.25; oil 1.75 (typical values assumed)

Solution

Injected volume $V = 400\cdot 365.25\cdot 3 = 438{,}300$ m³.
Water front for piston-like displacement (the water pushes the oil ahead of it like a piston): $r_{\text{front}} = \sqrt{\dfrac{438{,}300}{\pi\cdot 12\cdot 0.22\cdot 0.50}} = 325$ m.
Heat capacity of the reservoir behind the front: $(\rho c)_{\text{rock}} = 0.78\cdot 2.25 + 0.22\,(0.75\cdot 4.18 + 0.25\cdot 1.75) = 2.541$ MJ/(m³·K).
Cooled zone: $r_T = \sqrt{\dfrac{438{,}300\cdot 4.18}{\pi\cdot 12\cdot 2.541}} = 138$ m. In the ratio of the radii the injected volume cancels out: $r_T/r_{\text{front}} = \sqrt{0.22\cdot 0.50\cdot 4.18/2.541} = 0.43$ . So the cooled zone occupies 18% of the swept area.

Answer: In three years the water has swept the reservoir to 325 m around the well and cooled it to 138 m.

What it means. The cold travels behind the water, not ahead of it, so the oil that the water displaces stays warm. The area around the injection well is cooled: the residual oil and the near-wellbore zone, where wax can plug the pores and reduce injectivity (how easily the well takes water). But a field has many injection wells, injection goes on for decades, and the cooled zones expand and merge, so the reservoir temperature falls as a whole. If the water breaks through to a producing well, the cold comes to it as well. Heat exchange with the cap rock and the base rock slows the cooling, so the estimate errs on the safe side. The remedy is to inject heated water so that its temperature in the reservoir stays above the wax appearance temperature. The data in this example are assumed for illustration.

High-wax oils of Mangyshlak

Uzen and Zhetybai, discovered in 1961, produce oil with a high wax content. It solidifies at temperatures above zero, so it is heated during gathering and transport. A study of the oil of the Yu-13 horizon (a producing layer) of Uzen, published in 2026, found 17–19% wax by mass (in the 1960s–1980s the figure was 18–29%), and the oil solidifies at 32–36 °C. After decades of cold water injection, the reservoir temperature in the field has fallen. The wax appearance temperature is now comparable to the present reservoir temperature, and in places higher than it, so wax can drop out already in the reservoir and the near-wellbore zone. According to the same study, dissolved gas has almost no effect on these temperatures. To stop wax dropping out in the reservoir, Uzen has also begun to inject heated water (reservoir temperature).

Heat. Thermal insulation of the tubing and pipelines, heating of the produced fluid, and flushes with hot oil or water. Pipelines for high-wax oil are built with heating.
Mechanical methods. Scrapers lowered on wireline in the tubing, and cleaning pigs (scraper pistons pushed along by the flow) in pipelines (Wireline operations).
Chemicals. Wax deposition inhibitors, and pour point depressants, which lower the pour point. Solvents remove the deposits.
Operating mode. Avoid long shutdowns: a stopped line cools, and at start-up it needs pressure to move the solidified oil.

Asphaltenes: onset pressure and stability

Asphaltenes are the heaviest and most polar part of the oil (their molecules have electrically charged regions). They are defined by their solubility: they do not dissolve in light alkanes (pentane, heptane) and they do dissolve in aromatic solvents such as toluene (Composition of reservoir fluids). In the oil they are held as a colloidal suspension (tiny particles spread through the liquid), stabilised by resins (mechanism). The stability of the suspension depends on how good a solvent the oil is, and that depends on its density and composition.

As the pressure falls above the bubble point pressure, the oil expands, becomes less dense, and dissolves asphaltenes less well. At the asphaltene onset pressure the first particles appear. (It is abbreviated AOP, it is also called the precipitation onset pressure, and it is the upper boundary of the precipitation region.) The instability is greatest near the bubble point pressure. Below the bubble point pressure, light components leave the oil, so the oil becomes heavier and dissolves asphaltenes better again, and the particles partly dissolve. On a pressure–temperature diagram the precipitation is bounded by the asphaltene precipitation envelope, a band around the bubble point pressure line.

Where they drop out. In the tubing, in the interval where the flowing pressure lies between the onset pressure and the bubble point pressure (see the example below). They also drop out in the near-wellbore zone, when the bottomhole flowing pressure falls below the onset pressure.
What else triggers precipitation. Gas injection, especially of carbon dioxide or rich gas, and mixing with light oil or condensate (miscible displacement). Hydrochloric acid does too, especially with iron ions (Acid systems and treatment design).
The paradox. Heavy oils with a large asphaltene content are usually stable, because they contain a lot of resins and aromatics. Trouble more often comes from light, strongly undersaturated oils with a fraction of a per cent of asphaltenes. These oils lose their dissolving power most as the pressure falls.

The onset pressure is measured on a reservoir sample in a PVT cell. The pressure is lowered in steps at reservoir temperature, while the transmission of near-infrared light through the oil is followed: particles scatter the light, so the transmission falls. The result is confirmed by microscopy under pressure and by filtration. The experiment is repeated at several temperatures to construct the precipitation envelope. Without a reservoir sample, screening is used. One screening tool is the de Boer plot, which shows the oil density in the reservoir against undersaturation (the difference between the reservoir pressure and the bubble point pressure). It has regions marked “serious problems”, “problems possible” and “no problems”, and light, strongly undersaturated oils fall into the problem regions. The other is the colloidal instability index, found from the group composition (the shares of saturated hydrocarbons, aromatics, resins and asphaltenes, from SARA analysis).

Colloidal instability index

\text{CII} = \frac{w_{\text{sat}} + w_{\text{asph}}}{w_{\text{arom}} + w_{\text{resin}}}

where:

$w_{\text{sat}},\ w_{\text{arom}}$ mass fractions of saturated and aromatic hydrocarbons of the stock-tank oil (SARA analysis), % by mass
$w_{\text{resin}},\ w_{\text{asph}}$ mass fractions of resins and asphaltenes, % by mass

Units: 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.

The numerator contains what pushes asphaltenes out (the saturated hydrocarbons) and the asphaltenes themselves. The denominator contains what keeps them in solution. The index answers the question “is the oil prone to it?”, but not “where and at what pressure?”. Only an experiment on a reservoir sample answers that.

Worked exampleTwo oils and the interval of the tubing where asphaltenes are unstable

Given (assumed data)

Group composition, % by mass (saturates / aromatics / resins / asphaltenes): light oil: 62 / 24 / 11 / 3; heavy oil: 35 / 35 / 22 / 8
Light oil in the reservoir: 4600 m, 60 MPa, 110 °C; bubble point pressure 22 MPa; asphaltene onset pressure 38 MPa (at 110 °C)
Well: bottomhole flowing pressure 46 MPa; mean pressure gradient of the flow in the tubing above the bubble point 6.8 kPa/m

Solution

The index of the light oil is $\text{CII} = (62 + 3)/(24 + 11) = 1.86$ : above 0.9, so it is unstable. The index of the heavy oil is $(35 + 8)/(35 + 22) = 0.75$ : in the uncertain range, although this oil contains almost three times as many asphaltenes.
Undersaturation of the light oil: $60 - 22 = 38$ MPa. On the de Boer plot, it is precisely such light, strongly undersaturated oils that form the risk group.
The onset pressure is reached at a depth of $4600 - (46 - 38)/0.0068 = 3424$ m, and the bubble point pressure at a depth of $4600 - (46 - 22)/0.0068 = 1071$ m.
The interval of instability is from 3420 to 1070 m, that is 2350 m of tubing. Most deposits are expected in its upper part, closer to the bubble point pressure.

Answer: The light oil is prone to asphaltene precipitation (CII 1.86), and in this well the asphaltenes are unstable in the tubing interval from about 3420 to 1070 m. There is no precipitation in the reservoir yet, because the bottomhole flowing pressure is above the onset pressure.

What it means. This points to the measures. Deliver the inhibitor (a chemical that keeps the asphaltenes from dropping out) below the interval, through a capillary string (a thin tube) to the lower part of the tubing, or squeeze it into the reservoir. Flush the interval periodically with an aromatic solvent, or clean it with coiled tubing (a continuous steel tube run into the well). When choosing the operating mode, remember that precipitation will begin in the near-wellbore zone as soon as the bottomhole flowing pressure falls below 38 MPa, whether through a larger drawdown or because the reservoir pressure declines. The calculation is simplified. The oil cools in the wellbore, and the onset pressure depends on temperature (in which direction differs from oil to oil), so the interval is refined with the precipitation envelope measured at several temperatures. The pressure gradient in the tubing is taken as constant. The data in this example are assumed for illustration.

CalculatedFlowing pressure in the tubing of the well of the example (the inclined line; it is not continued above the depth at which it falls to the bubble point pressure), and the depths at which it equals the asphaltene onset pressure (38 MPa) and the bubble point pressure (22 MPa). The interval between these depths is where deposits are expected.

Hydrates

Gas hydrates are ice-like solid crystals in which a framework of water molecules holds molecules of light gases: methane, ethane, propane, carbon dioxide or hydrogen sulfide. Hydrates need free water, gas, high pressure and low temperature to form. Each gas has its own hydrate formation curve on the pressure–temperature diagram. To the left of the curve and above it, the hydrate is stable. The heavier the gas, and the more hydrogen sulfide and carbon dioxide it contains, the higher the temperature at which the hydrate can still exist.

The curve is measured in a high-pressure cell. A mixture of gas and water is cooled until a hydrate forms, then slowly heated, and the temperature at which the last crystals disappear is recorded, either by eye or from the kink in the curve of pressure against temperature at constant volume. For calculations, the curve is built from the gas composition with correlations and thermodynamic models, which are checked against such experiments. In the same cell, we check how much inhibitor is needed to shift the curve.

In the reservoir, hydrates usually do not form, because it is warm there. The dangerous places are downstream of the choke of a gas well (the choke is a flow restriction), where the gas cools sharply while the pressure is still high; flowlines and wellhead equipment in winter; and subsea pipelines. The lesson on chokes and temperature covers the temperature margin and the cooling on throttling. Hydrates are prevented by heating and thermal insulation, by drying the gas, and by inhibitors. Methanol and glycols shift the curve to lower temperatures. Small doses of kinetic inhibitors and anti-agglomerants slow the growth of the crystals, or keep them from sticking together into a plug.

Near-critical fluids and high-pressure, high-temperature fluids

In a near-critical fluid, the reservoir temperature is close to the critical temperature of the mixture (more detail). Then small changes of pressure, temperature or composition change the properties sharply. At the same temperature, a slightly lighter mixture has a dew point and a slightly heavier one has a bubble point. The densities and compositions of gas and liquid are close, and the interfacial tension is small. A near-critical gas condensate drops out a lot of liquid immediately below the dew point. A near-critical (volatile) oil gives off a lot of gas immediately below the bubble point pressure.

In the laboratory, the most reliable way to tell the type of saturation is to see what appears in the window of the cell: gas bubbles or liquid droplets. The kink in the pressure–volume curve is almost invisible.
In calculations, the black oil model does not work. You need a compositional model with an equation of state tuned to many experiments, and close to the critical point even a tuned equation makes its largest errors (accuracy of the equations).
In the reservoir, at a low interfacial tension the liquid is more mobile, and gas injection can lead to miscible displacement (miscibility).

High pressure, high temperature fluids (HPHT) have no generally accepted boundary. Following British regulations, wells are often classed as HPHT if the bottomhole temperature is above 149 °C (300 °F) and the pore pressure gradient is at least 18.1 kPa/m (0.8 psi/ft), or if blowout preventer equipment (the safety valves that close the well) rated for more than 69 MPa (10,000 psi) is needed. A technical report of the American Petroleum Institute (API) on equipment for such wells calls pressures above 103 MPa (15,000 psi) high, and temperatures above 177 °C (350 °F) high. For fluids this has three consequences. The gas is dense, and its gas deviation factor is noticeably greater than one. The density difference between gas and oil is small, so contacts are harder to tell apart from pressure lines (Pressure gradients, fluid contacts and capillary equilibrium). And samplers, PVT cells and pumps must withstand these pressures and temperatures. If such fluids contain hydrogen sulfide, as in the subsalt accumulations of the Pre-Caspian Basin, working with their samples requires special safety measures (transport of samples).

Worked exampleGas at 75 MPa: the gas deviation factor is greater than one

Given (assumed data)

Gas: specific gravity 0.70; pseudo-critical properties from the correlation for natural gas (Ideal and real gas: the gas deviation factor (Z)): $T_{pc} = 216.3$ K, $p_{pc} = 4.613$ MPa
Reservoir: 75 MPa, 155 °C (428.15 K); for comparison, the same gas at 30 MPa

Solution

Pseudo-reduced properties: $T_{pr} = 428.15/216.3 = 1.979$ , $p_{pr} = 75/4.613 = 16.26$ ; from the Dranchuk–Abou-Kassem equation $Z = 1.457$ . At 30 MPa ( $p_{pr} = 6.50$ ), $Z = 1.00$ .
Formation volume factor at 75 MPa: $B_g = \dfrac{0.101325}{75}\cdot\dfrac{1.457\cdot 428.15}{293.15} = 0.00287$ m³/m³; with $Z = 1$ it would be 0.00197.
Gas in place $G = V_{\text{pore, gas}}/B_g$ : with $Z = 1$ it would be overestimated by a factor of 1.457, that is by 46%.
Gas density in the reservoir: $\rho = \dfrac{75\cdot 10^3\cdot 0.70\cdot 28.97}{1.457\cdot 8.314\cdot 428.15} = 293$ kg/m³; the pressure gradient in the gas is 2.88 kPa/m.

Answer: At 75 MPa

Z \approx 1.457

: the gas occupies almost one and a half times as much space as an ideal gas, and its density of 293 kg/m³ is of the order of half the density of a light oil.

What it means. At moderate pressures $Z < 1$ , and ignoring it underestimates the gas in place. At very high pressures the opposite is true. For this gas, $Z$ passes through one at about 30 MPa, where the error of the “ideal gas” happens to be small. Correlations for $Z$ are also less accurate in this region than at ordinary pressures, so for HPHT gases $Z$ is taken from the constant-composition experiment. The data in this example are assumed for illustration.

Change of composition with depth

In an accumulation at rest, gravity acts on the components: heavy molecules concentrate at the bottom and light ones at the top. In equilibrium this produces compositional grading. The gas–oil ratio and the oil formation volume factor decrease with depth, the saturation pressure changes, and the fluid density increases. The temperature difference over the height of the accumulation, and thermal diffusion, can weaken or strengthen the effect. Fluids that entered the trap at different times can also leave differences that are not in equilibrium.

In black oils, in accumulations of ordinary height, the changes are small. They are large in tall accumulations of near-critical fluids: a gas condensate at the top can pass smoothly, without a contact, into a volatile oil at the base (near-critical fluids). Then a sample from one depth does not describe the accumulation, and the gas–oil contact found from the saturation pressure of a single sample is wrong (GOC from the saturation pressure). So samples are taken from several depths with a wireline formation tester, and an equation of state that includes gravity is built.

Worked exampleOil in place of an accumulation with compositional grading, from one sample and from two

Given (assumed data)

Oil accumulation: from 2800 to 3000 m; the pore volume occupied by oil is 30.0 million m³, uniformly over the height
Samples: at the top (2800 m): $B_o = 1.70$ , $R_s = 200$ m³/m³; at the base (3000 m): $B_o = 1.44$ , $R_s = 128$ m³/m³; between them the properties change linearly

Solution

With grading: the mean value of $1/B_o$ for a linear $B_o$ is $\dfrac{\ln(1.70/1.44)}{1.70 - 1.44} = 0.6384$ , and $N = 30.0\cdot 0.6384 = 19.152$ million m³.
From the sample at the top: $30.0/1.70 = 17.647$ million m³ (−7.9%); at the base: $30.0/1.44 = 20.833$ (8.8%); from the middle, $B_o = 1.57$ : 19.108 (−0.2%).
Dissolved gas: with grading 3.12 billion m³; from the sample at the top $30.0\cdot 200/1.70 = 3.53$ billion m³ (13%).

Answer: The oil in place is 19.15 million m³. The sample at the top would underestimate it by 7.9%, the one at the base would overestimate it by 8.8%, and the dissolved gas calculated from the sample at the top is overestimated by 13%.

What it means. A sample from the middle of the accumulation gives an almost correct answer, but only because the properties change linearly here. In near-critical systems the change speeds up towards the transition zone, and a single sample is wrong by more. Something else matters even more than the volume in place: wells at the top will give a higher producing gas–oil ratio, the oil at the top is closer to saturation, and with gas injection from the top the miscibility will differ over the height. The data in this example are assumed for illustration.

Terms and symbols

Condensate banking: The zone near the bottom of a gas-condensate well where the pressure is below the dew point and the condensate saturation is high (the condensate bank). It reduces the permeability to gas and the productivity.
Zones 1, 2, 3: At the wellbore, a mixture of gas and condensate of constant composition flows. Farther out, condensate accumulates and is immobile. Farther still, there is single-phase gas.
$V_{ro}$: The liquid volume fraction of a mixture of the original composition at a given pressure, from the constant-composition experiment.
Wax appearance temperature (WAT): The temperature at which the first wax crystals appear on cooling.
Pour point: The temperature at which oil, cooled by a standard procedure, stops flowing. The closely related flow temperature (the pour point of English-language practice) is the lowest temperature at which it still flows.
Asphaltene onset pressure (AOP): The pressure, above the bubble point pressure, at which the first asphaltene particles appear as the pressure falls.
CII: Colloidal instability index: the ratio of the saturated hydrocarbons plus asphaltenes to the aromatics plus resins.
HPHT: High pressure, high temperature. There is no generally accepted boundary. Often it means a bottomhole temperature above 149 °C with a high pore pressure, or equipment rated for pressures above 69 MPa.

Common mistakes and misconceptions

Wrong

“The dropout curve shows that there will be no more than 12% liquid in the pores, so the banking is small.”

In fact

The dropout curve describes the reservoir under depletion. Near the wellbore all the gas of the well passes through the zone, and the saturation grows until the condensate becomes mobile. In the lesson's example it reaches 54.5% of the hydrocarbon pore volume.

Wrong

“The reservoir pressure is above the dew point, so there is no condensate in the reservoir.”

In fact

There is none far from the wells. If the bottomhole flowing pressure is below the dew point, there is a condensate zone near the wellbore, and it reduces the productivity.

Wrong

“Wax is no problem until the pour point.”

In fact

Crystallisation begins at the wax appearance temperature, which is noticeably higher, often by tens of degrees. Deposits on cold walls grow long before the oil solidifies.

Wrong

“The oil has only 0.5% asphaltenes, so there will be no problems.”

In fact

What decides the danger is not the amount of asphaltenes but how much the oil loses its dissolving power as the pressure falls. Light undersaturated oils with a fraction of a per cent of asphaltenes suffer more often than heavy ones.

Wrong

“Below the bubble point pressure the asphaltenes drop out even more.”

In fact

On the contrary: below the bubble point pressure the oil loses light components and becomes a better solvent, so the asphaltenes partly dissolve. The most unstable region is near the bubble point pressure.

In depth: advanced detail

Derivation of the two-zone estimate

In steady-state radial inflow, the pressure difference across the ring from $r_1$ to $r_2$ is $\dfrac{q}{2\pi k h\lambda}\ln\dfrac{r_2}{r_1}$ , where $\lambda$ is the mobility. Outside the condensate zone the mobility is that of the gas, $\lambda_0$ . Inside it is $m\lambda_0$ . The rate is the same in both parts, so $p_e - p_d = A\ln\dfrac{r_e}{r_b}$ and $p_d - p_{wf} = \dfrac{A}{m}\ln\dfrac{r_b}{r_w}$ , with a common $A = q/(2\pi k h\lambda_0)$ .

Let $L = \ln(r_e/r_w)$ and $x = \ln(r_b/r_w)$ . The two equalities give $(p_e - p_d)\,x = m\,(p_d - p_{wf})\,(L - x)$ , and this gives the formula for $x$ . Substituting it, we obtain $A = \big[(p_e - p_d) + m(p_d - p_{wf})\big]/L$ . Without the zone, $A_0 = (p_e - p_{wf})/L$ . The ratio of the two is the share of the rate. The logarithm of the radii cancels: with a constant mobility in the zone, the share of the rate depends only on what part of the drawdown falls at pressures below the dew point.

A rigorous calculation replaces the pressure differences with a pseudo-pressure integral (a way to handle gas whose properties change with pressure). In zone 1 this integral contains $k_{rg}$ and $k_{ro}$ at the saturation that belongs to the local pressure, and it accounts for the gas dissolved in the flowing condensate. The two-zone estimate is a crude copy of it, with a single “average” saturation.

Which experiments to order for a complex fluid

Special experiments and the questions they answer

Fluid or risk: Gas condensate
Experiments: constant-composition experiment with liquid volumes; constant volume depletion; gas–condensate relative permeabilities on core
What they give: $V_{ro}(p)$ , the dropout curve, data for the banking calculation

Fluid or risk: High-wax oil
Experiments: wax appearance temperature on stock-tank and reservoir oil, wax content, pour point, viscosity on cooling, deposition experiments on a cooled surface
What they give: temperature margins, start-up and shutdown procedures

Fluid or risk: Asphaltenes
Experiments: group composition, onset pressure at several temperatures, experiments on mixing with the injected gas and with inhibitors
What they give: the precipitation envelope, the interval of deposits, the choice of inhibitor

Fluid or risk: Gas with water
Experiments: hydrate formation curve, its shift by inhibitors
What they give: minimum temperature and inhibitor dosage

Fluid or risk: Tall accumulation
Experiments: samples from several depths
What they give: compositional grading, contacts

Fluid or risk	Experiments	What they give
Gas condensate	constant-composition experiment with liquid volumes; constant volume depletion; gas–condensate relative permeabilities on core	$V_{ro}(p)$ , the dropout curve, data for the banking calculation
High-wax oil	wax appearance temperature on stock-tank and reservoir oil, wax content, pour point, viscosity on cooling, deposition experiments on a cooled surface	temperature margins, start-up and shutdown procedures
Asphaltenes	group composition, onset pressure at several temperatures, experiments on mixing with the injected gas and with inhibitors	the precipitation envelope, the interval of deposits, the choice of inhibitor
Gas with water	hydrate formation curve, its shift by inhibitors	minimum temperature and inhibitor dosage
Tall accumulation	samples from several depths	compositional grading, contacts

The list is only a guide. The study programme is drawn up for the particular accumulation and the way it is developed.

Connections to other topics

Phase behaviour: pure substances and binary mixtures — the phase envelope, the dew point and retrograde condensation.
Types of reservoir fluids — gas condensate, volatile oil, near-critical fluids and cycling.
How to read a PVT report — how to read a report on a gas condensate and where the special experiments are in it.
Skin factor — Hawkins' formula and two-phase flow near the wellbore as a component of the skin factor.
Chokes, surface pressure losses and well temperature — the temperature profile of the well, the margin to the wax appearance temperature and to hydrates.
Sources of formation damage — wax and asphaltenes as mechanisms of near-wellbore damage.
Displacement of oil by gas — gas injection: miscibility and gas reinjection in the fields of the Pre-Caspian Basin.
Pressure gradients, fluid contacts and capillary equilibrium — contacts from pressure lines and from the saturation pressure in the presence of compositional grading.

Check yourself

1
Why can the condensate saturation near the wellbore of a gas-condensate well be much higher than the maximum liquid fraction in the constant volume depletion experiment?
1. A.The temperature near the wellbore is higher than the reservoir temperature
2. B.The laboratory underestimates the dropout
3. C.All the gas of the well passes through the zone near the wellbore; liquid accumulates until it becomes mobile, and the saturation is set by the relative permeabilities, not by the dropout curve
4. D.The condensate drains to the bottom of the well under gravity
Show the answer
Correct answer: C. All the gas of the well passes through the zone near the wellbore; liquid accumulates until it becomes mobile, and the saturation is set by the relative permeabilities, not by the dropout curve
The constant volume depletion experiment describes a volume of the reservoir from which gas is withdrawn. Near the wellbore each portion of gas leaves liquid behind, and the balance sets in only when the condensate starts to flow into the well at the same rate: $k_{rg}/k_{ro} = (\mu_g/\mu_o)(1/V_{ro} - 1)$ . The saturation is then several times higher than the maximum of the dropout curve.
2
In the zone near the wellbore $V_{ro} = 0.15$ , $\mu_g = 0.030$ mPa·s, $\mu_o = 0.30$ mPa·s. What must the ratio $k_{rg}/k_{ro}$ be? Will the saturation be higher or lower than at $V_{ro} = 0.10$ ?
Show the answer
$k_{rg}/k_{ro} = (0.030/0.30)\cdot(1/0.15 - 1) = 0.1\cdot 5.67 = 0.57$ . At $V_{ro} = 0.10$ it would be $0.1\cdot 9 = 0.9$ . A smaller ratio $k_{rg}/k_{ro}$ is reached at a higher condensate saturation: with a richer gas the banking is stronger.
3
The reservoir pressure is 1 MPa above the dew point, the bottomhole flowing pressure is 9 MPa below it; the mobility of the mixture in the condensate zone is 0.30 of the gas mobility. What share of the rate without banking does the well deliver, by the two-zone estimate?
Show the answer
$q/q_0 = (1 + 0.30\cdot 9)/(1 + 9) = 3.7/10 = 0.37$ . Almost the whole drawdown falls at pressures below the dew point, so the well loses about two thirds of its rate.
4
Which of the oils is more likely to produce asphaltene deposits in the tubing?
1. A.An oil whose reservoir pressure is below the bubble point pressure
2. B.A light oil with 0.5% asphaltenes, with a reservoir pressure 35 MPa above the bubble point pressure
3. C.Any stock-tank oil containing asphaltenes
4. D.A heavy oil with 10% asphaltenes, with a bubble point pressure close to the reservoir pressure
Show the answer
Correct answer: B. A light oil with 0.5% asphaltenes, with a reservoir pressure 35 MPa above the bubble point pressure
Asphaltenes drop out when the oil loses its dissolving power as the pressure falls above the bubble point pressure. For a light, strongly undersaturated oil this fall is large, and there are few resins and aromatics to hold the asphaltenes. A heavy oil close to saturation passes through only a small range of pressures above the bubble point pressure.
5
The group composition of an oil: saturates 55%, aromatics 28%, resins 14%, asphaltenes 3%. What is the colloidal instability index and what does it say?
Show the answer
$\text{CII} = (55 + 3)/(28 + 14) = 58/42 = 1.38$ , which is above 0.9. By this screening the oil is prone to asphaltene precipitation. The index does not say where and at what pressure this will happen. For that you need the onset pressure measured on a reservoir sample.
6
A gas at 80 MPa has $Z = 1.5$ . How will the estimated gas in place change if $Z = 1$ is taken by mistake?
1. A.It will increase by 5%
2. B.It will not change
3. C.It will increase by a factor of 1.5
4. D.It will decrease by a factor of 1.5
Show the answer
Correct answer: C. It will increase by a factor of 1.5
$G = V_{\text{pore, gas}}/B_g$ , and $B_g$ is proportional to $Z$ . With $Z = 1$ instead of 1.5, the formation volume factor comes out 1.5 times too small and the gas in place 1.5 times too large. At moderate pressures, where $Z < 1$ , the error would go the other way.

Updated: 17 October 2026