PEH:Gas Reservoirs

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume V – Reservoir Engineering and Petrophysics

Edward D. Holstein, Editor

Copyright 2007, Society of Petroleum Engineers

Chapter 10 – Gas Reservoirs

Mark A. Miller, Consultant Edward D. Holstein, Consultant

Pgs. 981-1036

ISBN 978-1-55563-120-8
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This chapter addresses the flow characteristics and depletion strategies for gas reservoirs. The focus will be primarily on nonassociated accumulations, but much of the fluid behavior, flow regimes, and recovery aspects are also applicable to gas caps associated with oil columns.

In this chapter, gas reservoirs have been divided into three groups; dry gas, wet gas, and retrograde-condensate gas. A dry-gas reservoir is defined as producing a single composition of gas that is constant in the reservoir, wellbore, and lease-separation equipment throughout the life of a field. Some liquids may be recovered by processing in a gas plant. A wet-gas reservoir is defined as producing a single gas composition to the producing well perforations throughout its life. Condensate will form either while flowing to the surface or in lease-separation equipment. A retrograde-condensate gas reservoir initially contains a single-phase fluid, which changes to two phases (condensate and gas) in the reservoir when the reservoir pressure decreases. Additional condensate forms with changes in pressure and temperature in the tubing and during lease separation. From a reservoir standpoint, dry and wet gas can be treated similarly in terms of producing characteristics, pressure behavior, and recovery potential. Wellbore hydraulics may be different. Studies of retrograde-condensate gas reservoirs must consider changes in condensate yield as reservoir pressure declines, the potential for decreased well deliverability as liquid saturations increase near the wellbore, and the effects of two-phase flow on wellbore hydraulics.

A comprehensive discussion of gas and condensate properties and phase behavior can be found in several chapters of the General Engineering section of this Handbook. Aspects of predicting wellbore hydraulics are covered in the Production Operations Engineering section of this Handbook . Lease equipment for processing gas and pipelining considerations are covered in several chapters of the Facilities Engineering section of this Handbook. The reader may want to refer to these chapters to understand some of the nomenclature and concepts referred to in the present chapter.

Properties of Natural Gases

Natural petroleum gases contain varying amounts of different (primarily alkane) hydrocarbon compounds and one or more inorganic compounds, such as hydrogen sulfide, carbon dioxide, nitrogen (N₂), and water. Characterizing, measuring, and correlating the physical properties of natural gases must take into account this variety of constituents.

Phase Behavior of Natural-Gas Reservoirs

A widely accepted system for categorizing petroleum reservoir fluids is based on five classes: low-shrinkage (crude) oils, high-shrinkage (volatile) oils, retrograde-condensate gases, wet gases, and dry gases. Typical phase diagrams for the gas categories are shown in Figs. 10.1 through 10.3.

• 

  Fig. 10.1 – Phase diagram of a retrograde-condensate gas. (Source: The Properties of Petroleum Fluids, second edition, by William D. McCain Jr. Copyright Pennwell Books, 1990.)

• 

  Fig. 10.2 – Phase diagram of a wet gas.(Source: The Properties of Petroleum Fluids, second edition, by William D. McCain Jr. Copyright Pennwell Books, 1990.)

• 

  Fig. 10.3 – Phase diagram of a dry gas. (Source: The Properties of Petroleum Fluids, second edition, by William D. McCain Jr. Copyright Pennwell Books, 1990.)

A retrograde-condensate fluid has a phase envelope such that reservoir temperature lies between the critical temperature and the cricondentherm (Fig. 10.1). As a result, a liquid phase will form in the reservoir as pressure declines, and the amount and gravity of produced liquids will change with time. Condensate liquids are generally "water white" or light in color (brown, orange, or greenish), with gravities typically between 40 and 60°API. Producing-liquid yields can be as high as 300 STB/MMscf. McCain suggests that when yields are below approximately 20 STB/MMscf, even though phase-behavior considerations may show retrograde behavior, the amount of liquid dropout in the reservoir is insignificant. The primary difficulties in producing condensate reservoirs are as follows: (a) liquid deposition near the wellbore causes a decrease in gas deliverability that can approach 100% in a reservoir with less than 50-md permeability, and (b) a large amount of the most valuable hydrocarbon components is left in the reservoir rather than produced.

In a wet-gas reservoir, temperature is higher than the cricondentherm (Fig. 10.2). Therefore, a liquid phase never forms in the reservoir. Considerable liquid can still form (condense) at surface conditions or even in the wellbore. The term "condensate" is often applied to any light hydrocarbon liquid produced from a gas well. However, the term "condensate reservoir" should be applied only to situations in which condensate is actually formed in the reservoir because of retrograde behavior. Wet-gas reservoirs can always be treated as containing single-phase gas in the reservoir, while retrograde-condensate reservoirs may not. Wet-gas reservoirs generally produce liquids with gravities similar to those for retrograde condensates, but with yields less than approximately 20 STB/MMscf.

In a dry-gas reservoir, the cricondentherm is much lower than the reservoir temperature (Fig. 10.3), resulting in little or no liquid production at the surface. A somewhat arbitrary cutoff liquid yield of 10 STB/MMscf is sometimes used to distinguish dry-gas reservoirs.

While the difference between retrograde-condensate and wet gases is notable, there is much less distinction between wet and dry gases. For both wet and dry gases, reservoir engineering calculations are based on a single-phase reservoir gas. The only issue is whether there is a sufficient volume of produced liquid to be considered in such calculations as material balance or wellbore hydraulics. Retrograde systems require more-complex calculations using equations of state (EOSs) and other advanced engineering methods.

Pressure/Volume/Temperature (PVT) Behavior

The chapters on Fluid Sampling, Gas Properties, and Thermodynamics and Phase Behavior in the General Engineering section of this Handbook contain a detailed explanation of PVT and thermodynamic relationships for ideal and real gases. Some fundamental relations are repeated here as background for the remainder of this chapter.

The basis of gas PVT behavior is the ideal gas law, and by extension the real gas law:

....................(10.1)

The universal gas constant in practical units is



....................(10.2)

For economic reasons, most (but not all) pressure gauges read zero pressure when pressure is equal to the ambient atmospheric pressure. Therefore, atmospheric pressure must be added to gauge pressures to convert them to an absolute basis. For most engineering purposes, atmospheric pressure is usually taken to be 14.7 psia (101 kPa). For precise scientific and engineering applications, actual atmospheric pressure (i.e., barometric pressure, which varies with both location and time) should be used. Standard temperature and pressure are set by different governmental agencies and should be determined for a specific field or reservoir to be sure that reserves and produced quantities are reported with the correct bases. The SPE standard temperature is 59°F (15°C), and the standard pressure is 14.696 psia (101.325 kPa).

Application of Eq. 10.1, in a practical sense, must consider how to determine the various factors for petroleum gases that are mixtures of several components. Such determinations would include apparent molecular weight and z (using pseudoreduced pressure and temperature and corrections for nonhydrocarbon components).

Gas Density and Formation Volume Factor

The density of a gas can be calculated from the real-gas law once a z factor has been determined. For pressure in psia and temperature in °R, density in lbm/ft³ is given by

....................(10.3)

For pressure in kPa, temperature in K, and density in kg/m³,

....................(10.4)

The gas formation volume factor is defined as the volume occupied by a gas at reservoir conditions divided by the volume at standard conditions:

....................(10.5)

The second and third lines of Eq. 10.5 give B_g using standard pressure of 14.696 psia and standard temperature of 60°F.

In SI units using SPE standard pressure and temperature,

....................(10.6)

Viscosity

Most gas viscosities range from 0.01 to 0.03 cp, making them difficult to measure accurately. Accurate determination of gas viscosities has low economic value. Instead, values are normally determined from one of two correlations.

The first one in common use today, from Lee et al., is given in equation form as

....................(10.7)

where , and

The gas density in Eq. 10.7 is in g/cm³ when p and T are in oilfield units (psia, °R). The equivalent formula for SI units (pressure and temperature in kPa and K, respectively) is

....................(10.8)

Fig. 10.4 shows gas viscosities generated from this correlation for a 0.80-gravity natural gas.

• 

  Fig. 10.4 – Viscosity of 0.80-gravity natural gas vs. pressure and temperature according to the Lee et al. correlation.

Another common correlation entails a two-step graphical process and is cumbersome for computer applications. Because gas viscosities are seldom needed with great accuracy, the Lee et al. correlation is most applicable for modern reservoir-engineering practice.

Determining Reservoir-Fluid Properties

Condensation of liquids from wet-gas and retrograde-condensate fluids in the production system means that gas produced from separation equipment may be significantly different from the gas that flows into the wellbore from the reservoir. In general, separator gas will be lower in gravity and will have fewer high-molecular-weight hydrocarbons present in the mixture.

For proper laboratory measurements, a bottomhole sample should be collected. An alternative is a reconstituted sample that is created by mixing the separator-produced gas and liquid in proportion to their relative production rates. When the compositions of liquid and gaseous streams have both been measured, the composition of the mixture can be determined from

....................(10.9)

Note that

....................(10.10)

....................(10.11)

Relative molar amounts can be determined by converting measured produced volumes either to moles or to equivalent standard cubic feet. For a gas phase, conversion of a produced volume referenced to standard conditions to moles is

....................(10.12)

For a liquid, conversion of a volume to moles is

....................(10.13)

If liquid volume is measured at standard conditions, density can be calculated from specific gravity or API gravity. If liquid molecular weight is not measured, it can be approximated with the Gold et al. correlation:

....................(10.14)

An alternative to converting measured volumes to moles is to convert all measured volumes to equivalent standard volumes because standard volume is directly proportional to moles (through the value of the standard molar volume). This procedure has the advantage that measured gas volumes need not be converted (this is necessary only for liquid volumes).

Liquid volumes are converted to equivalent gas standard volumes using a parameter called the gas equivalent of oil. This parameter represents the effective standard volume occupied by hydrocarbons that are liquid at surface conditions but are in the gas phase at downhole conditions. This parameter is calculated by

....................(10.15)

GE_o is calculated with oilfield units in Mscf/STB (the second term below uses standard conditions of 14.696 psia and 60°F) by

....................(10.16)

In SI units, GE_o in std m³/std m³ is (using standard conditions of 101.325 kPa and 15°C)

....................(10.17)

Liquid-production volumes are multiplied by GE o to determine the equivalent standard volume as gas in the reservoir; that is,

....................(10.18)

where is the actual oil volume measured at stock-tank conditions, and is the gas-equivalent volume of the oil.

These equations also can be used to determine the equivalent gas production of pure hydrocarbons separated from gas in a processing plant. Values of specific gravity and molecular weight for pure components can be found from standard sources such as the Gas Processors Suppliers Assn. (GPSA) Handbook. In addition, if liquid production is measured at separator, rather than stock-tank, conditions, Eq. 10.16 or Eq. 10.17 can be used with separator temperature and pressure rather than standard temperature and pressure.

Relative volumes of gas and liquid phases can then be calculated as

....................(10.19)

When wellstream composition is unavailable, correlations must be used to determine gas properties, requiring the calculation of the wellstream (mixture) gravity on the basis of the gravity of the separator gas (often called dry gas) and the specific gravity of the produced liquid (condensate or oil):

....................(10.20)

where the subscript g refers to the separator-gas gravity, wg refers to the wellstream gas, and o refers to the produced condensate (oil). Y is the produced condensate yield. Gold’s correlation can be used to estimate condensate molecular weight.

In oilfield units with yield in STB/MMscf,

....................(10.21)

In SI units with yield in std m³/std m³,

....................(10.22)

Measuring Retrograde Behavior

When a liquid phase begins to form in the reservoir, the produced stream is no longer representative of the reservoir-fluid composition, but rather only the composition of the fluids entering the wellbore. Situations in which the liquid content of gases is high require the use of advanced laboratory tests and/or equation-of-state modeling to measure and predict these multiphase effects.

Laboratory measurements of the PVT behavior of condensate systems are similar to tests used for black oils; however, the primary interest becomes the measurement of relatively small amounts of condensed liquid. In general, systems with producing gas/oil ratios of 15,000 scf/STB (67 STB/scf) have a liquid dropout of approximately 4 to 6% by volume, while reservoirs with ratios around 50,000 scf/STB ordinarily have liquid dropouts of less than 1% by volume.

Two types of tests are generally run on retrograde fluids: constant-composition expansion (CCE) and constant-volume depletion. For examples refer to Tables 10.1 through 10.3.

• 

  Table 10.1

• 

  Table 10.2

• 

  Table 10.3

Table 10.1 gives the compositions of separator gas and liquid streams and other data used in making a recombined sample for analysis of reservoir-fluid composition.

A CCE using a visual cell furnishes the dewpoint of the reservoir fluid at reservoir temperature and the total volume of the reservoir fluid as a function of pressure. The volume of liquid formed at pressures below the dewpoint can also be measured. Table 10.2 shows the results of such a test. The term "relative volume" refers to the volume of gas plus liquid compared to the dewpoint volume. Retrograde-liquid volume is given as a percent of pore space, which essentially shows how the average condensate saturation changes with average reservoir pressure. Fig. 10.5 is a graphical representation of the relative condensate volume.

• 

  Fig. 10.5 – Example retrograde-liquid volume vs. pressure.

Visual cells also can be used to simulate pressure depletion. The validity of these tests is based on the assumption that the retrograde liquid that condenses in the reservoir will not be mobile. This assumption is valid except for very rich gas/condensate reservoirs. If significant retrograde liquid becomes mobile and migrates to producing wells, gas/liquid relative permeability data should be measured and used to adjust the predicted recovery.

Table 10.3 is an example of a visual-cell depletion study on the same retrograde gas for which properties are shown in Tables 10.1 and 10.2.

The depletion study begins by expanding the reservoir fluid in the cell until the first depletion pressure is reached (5,000 psig in this example). The fluid in the cell is brought to equilibrium, and the volume of retrograde liquid is observed. Gas is removed from the top of the cell while a constant pressure is maintained until the hydrocarbon volume of the cell is the same as when the test began. The gas volume removed is measured at the depletion pressure and reservoir temperature, analyzed for composition, and measured at atmospheric pressure and temperature.

The ideal-gas law can be used to calculate the "ideal volume" at the depletion pressure and reservoir temperature of the gas withdrawn from the cell. Dividing the ideal volume by the actual volume yields the deviation factor, z, for the produced gas. This is listed in Table 10.4 under z for the equilibrium gas. The actual volume of gas remaining in the cell at this point is the gas originally in the cell at the dewpoint pressure minus the gas produced at the first depletion level. Dividing the actual volume remaining in the cell into the calculated ideal volume remaining in the cell at this first depletion pressure yields the two-phase deviation factor shown. The two-phase z factor is an equivalent z factor that includes the total volume of gas plus liquid:

....................(10.23)

The two-phase z factor is the correct value to apply to such things as p/z analysis of retrograde-condensate reservoirs.

• 

  Table 10.4

A series of expansions and constant-pressure displacements is repeated at each depletion pressure until an arbitrary abandonment pressure is reached. The abandonment pressure is considered arbitrary because no engineering or economic calculations have been made to determine this pressure for the purpose of the reservoir-fluid study.

At the final depletion pressure, the compositions of both the produced well stream and the retrograde liquid are measured. These data are included as a control composition in the event that the study is used for compositional material-balance purposes.

The composition data can be used with equilibrium constants (determined by either laboratory measurements or general correlations) to determine recovery at the various stages of pressure depletion represented by the laboratory measurements. In this case, initial condensate content was 181.74 STB/Mrcf (213 STB/Mscf), and the amount recovered from the dewpoint to 700 psig was 51.91 STB/Mrcf. The gas formation volume factor was determined to be 0.6472 RB/Mscf at initial conditions and 0.6798 RB/Mscf at the dewpoint. If a hydrocarbon pore space of 500 × 10 6 ft³ were determined from volumetric calculations, then from these data and those presented in Table 10.3, recoveries by pressure depletion would be

....................(10.24)

These calculations indicate the large amount of liquid remaining in the reservoir at depletion even with excellent drainage to the wells. Further reductions in recovery would be expected because of areas of the reservoir inadequately drained with existing wells.

To deal with such phase-behavior effects in more than an empirical manner requires the use of PVT simulators. These simulators are based on EOSs that describe the phase volumes and compositions of liquid and gaseous phases as functions of pressure and temperature. Because hydrocarbon molecules interact with each other in solution, the coefficients in an EOS are not always adequately known. PVT tests such as those described in this section, along with the known composition of the original fluid, can be used to "tune" an appropriate EOS to achieve results that nearly match the measurements. Once this tuning process is complete, those coefficients can then be used to make predictions under differing operating conditions with some degree of reliability.

EOSs

When the effects of complex phase behavior on phase compositions and physical properties cannot be calculated accurately with simple approaches, it is often desirable to use an EOS. An EOS approach is often necessary when dealing with volatile oils and retrograde-condensate gases.

EOSs provide a numerical method for calculating both composition and relative amount for each phase present in the system. In reservoir simulation, EOS calculations are typically restricted to two hydrocarbon phases: a liquid (oleic) phase and a gaseous phase. However, there are situations in which an aqueous phase is included in the EOS calculations, or even in which a third hydrocarbon-containing phase may be present (e.g., in CO₂ flooding). These are generally done in more advanced compositional simulators.

The two most common EOSs used in petroleum engineering applications are the Peng-Robinson and Soave-Redlich-Kwong equations, which historically were derived from van der Waals’ equation. These three equations are called "cubic" because they result in a cubic representation for the molar volume. The basic equations are as follows:

Ideal Gas

....................(10.25)

van der Waals

....................(10.26)

Soave-Redlich-Kwong

....................(10.27)

Peng-Robinson

....................(10.28)

The parameters a_c, α(T), and b are determined empirically from experimental data (for pure components, the data are critical temperature and pressure and a specified point on the vapor-pressure curve), α(T) being a function of temperature and having a value of 1 at the critical temperature. Note that the parameters have different values depending on the equation.

The reader is referred to texts such as those by Ahmed, Pedersen et al., McCain, and Whitson and Brule.

Petrophysical Properties

Petrophysical properties required for typical reservoir engineering purposes include porosity, pore-volume compressibility, permeability, relative-permeability-vs.-saturation curves, capillary-pressure-vs.-saturation curves, and liquid saturations. Additional data are sometimes required as well, but typically not for natural-gas reservoirs.

An extensive discussion of the methods for measuring, analyzing, and interpreting petrophysical data on these and other petrophysical properties can be found in several chapters of the General Engineering section of this Handbook and in subchapters of this section.

Two petrophysical properties of interest in gas engineering work are the Klinkenberg effect and non-Darcy flow.

Klinkenberg Effect

Low-pressure (i.e., laboratory) measurements give rise to what is termed the Klinkenberg or "slippage" effect because the mean free path of gas molecules is approximately the same size as the pores in a reservoir rock, meaning that gas molecules are so far apart that the gas does not behave as a continuum fluid, resulting in erroneously high apparent permeability. At low pressures, measured gas permeabilities can be empirically related to effective liquid (or high-pressure gas) permeabilities by

....................(10.29)

The effective liquid permeability can be determined in the laboratory by measuring gas permeabilities at different average core pressures. A plot of yields an intercept equal to k_l (Fig. 10.6) of 24 md compared with 48 md at low pressure. The Klinkenberg effect is unimportant at reservoir pressures.

• 

  Fig. 10.6 – Example of Klinkenberg permeability correction.

Non-Darcy Flow

At high fluid velocities, Darcy’s law may not always be accurate. An additional energy loss is often apparent above that predicted from the laminar-flow relationship suggested by Darcy’s law. This effect has sometimes been called turbulence or inertial turbulence based on analogies with pipe flow. The effect, however, is probably caused by multiple factors, including pore-scale as well as reservoir-scale phenomena. Because of the lack of understanding of the fundamental nature of such phenomena, it is usually simply referred to as non-Darcy flow.

The most common expression of the non-Darcy effect is through the Forchheimer equation:

....................(10.30)

where β is called the non-Darcy velocity coefficient, having units of L^–1, and u is the volumetric flux (q/A) through the rock.

Note that Eq. 10.30 introduces a velocity-squared term into Darcy’s law. This effect shows up as a flow-rate-squared term in flow relationships (e.g., well-deliverability equations) that involve Darcy’s law.

Although non-Darcy flow can occur at all points in a reservoir, in practice it is only significant in the near-well region, where gas velocities are highest owing to radial-flow effects and expansion of gas volume at low pressure. For this reason, non-Darcy flow is incorporated primarily as a flow-rate-dependent skin factor and is seldom, if ever, incorporated to calculate flow away from the wellbore. The magnitude of the non-Darcy effect must generally be measured empirically at reservoir conditions using well tests.

Well Performance

This section deals with methods for analyzing and predicting the performance of producing natural-gas wells. Steady-state-, pseudosteady-state-, and transient-flow concepts are developed, resulting in a variety of specific techniques and empirical relationships for both testing wells and predicting their future performance under different operating conditions. The information included here is a condensation of the concepts and equations developed in detail in the chapter on fluid flow and well analysis in this section as they apply to gas reservoirs.

Basic Equations

The basis for all well-performance relationships is Darcy’s law, which in its fundamental differential form applies to any fluid—gas or liquid. However, different forms of Darcy’s law arise for different fluids when flow rates are measured at standard conditions. The different forms of the equations are based on appropriate equations of state (i.e., density as a function of pressure) for a particular fluid. In the resulting equations, presented next, flow rate is taken as being positive in the direction opposite to the pressure gradient, thus dropping the minus sign from Darcy’s law. When multiple-line equations are presented, the first will be in fundamental units, the second in oilfield units, and the third in SI units.

Four different fluid representations are considered: (1) liquid (small and constant compressibility), (2) real gas, (3) approximate high-pressure gas, and (4) approximate low-pressure gas. Because this chapter is devoted to well performance, radial forms of the flow equations are presented. Linear and spherical forms follow a similar development.

Steady-State Radial Horizontal Liquid Flow.

....................(10.31)

Steady-State Radial Horizontal Gas Flow.

....................(10.32)

For liquids, the product of B_oμ_o is approximately constant over a fairly wide pressure range so that for practical purposes, Eq. 10.31 can be written as

....................(10.33)

where Δp = p₂ - p₁ and is evaluated at some average pressure between p₁ and p₂. The exact pressure at which the oil formation volume factor and viscosity are evaluated is not critical because the product of is approximately constant.

Because this approximation is not generally valid for gases, the steady-state radial gas-flow equation is written as

....................(10.34)

where the real-gas potential Δm is defined by

....................(10.35)

Fig. 10.7 shows a typical plot of p/μz vs. p. Twice the area under the curve between any two pressures represents the real-gas potential difference. Note that at high pressures, p/μz is approximately constant. Also, although it is not readily apparent from the plot, at low pressures the product μz is approximately constant.

• 

  Fig. 10.7 – Typical plot of p/μz vs. p.

Values for Δm are usually determined by numerically integrating Eq. 10.35 using gas viscosity and z-factor data from measurements or correlations. Typical results are shown in Fig. 10.8.

• 

  Fig. 10.8 – Example graph of real-gas potential vs. pressure.

Note that for ideal gases, z = 1 and does not vary with pressure, resulting in the identity . The steady-state radial-flow equation for an ideal gas is thus

....................(10.36)

where .

To avoid numerical evaluation of Δm, which can be time-consuming if done by hand, it is sometimes useful to have two other approximate real-gas forms of each of the flow equations. These approximate forms are derived by noting that for most natural gases at low pressures (i.e., less than approximately 2,000 psia, or 14 MPa), the product μ_gz is approximately constant. Under these conditions,

....................(10.37)

Both μ_g and z should be evaluated at some average pressure between the two pressures. The specific value of average pressure used is not very significant because the product μ_gz is relatively constant, as demonstrated in Fig. 10.9.

• 

  Fig. 10.9 – Example showing that μ_gz is approximately constant at < 2,000 psia.

At high pressures (i.e., greater than approximately 2,000 psia, or 14 MPa), for most natural gases the product of μ_gβ_g is relatively constant (see Fig. 10.10 for an example).

• 

  Fig. 10.10 – Example graph showing that the product μ_gB_g is approximately constant for gases at pressures greater than approximately 2,000 psia.

This means that

....................(10.38)

Note that the conversion between the real-gas potential in oilfield vs. SI units is

....................(10.39)

So that one simplified set of equations can be used throughout the remainder of the chapter, some additional parameters will be defined. First, a "generic" potential difference, Δψ, can be expressed for each of the fluid cases according to Table 10.4.

A general radial-flow equation can then be expressed for all cases as

....................(10.40)

where β is given by the following expressions, which include the unit conversions necessary to apply Eq. 10.40. The first line of each equation is in fundamental units, the second in oilfield units, and the third in SI units.

Liquids (β units are psi-md-ft-D/STB, kPa-m²-m-d/std m³).

....................(10.41)

Real Gases (β units are psi²-md-ft-D/Mscf/cp, kPa²•m²•m•d/std m³/mPa•s).

....................(10.42)

High-Pressure Gases (approximate) (β units are psi-md-ft•D/Mscf, kPa•m²•m•d/std m³).

....................(10.43)

Low-Pressure Gases (approximate) (β units are psia²-md-ft-D/Mscf, kPa²•m²•m•d/std m³).

....................(10.44)

To concentrate on the specifics of well flow, in the remainder of the chapter the subscript e will refer to an external drainage radius, and the subscript wf will refer to pressure at the inlet sandface of a flowing well.

Skin

Sometimes wells experience near-wellbore phenomena (e.g., fractures and mud-filtrate damage) that cause production to be different from that calculated by Darcy’s law. These near-wellbore effects are often very complex. Their total effect is normally characterized with the use of a skin factor, S, which appears in the steady-state radial-flow equation as

....................(10.45)

Skin is a dimensionless parameter treated mathematically as an infinitely thin damaged or stimulated zone, regardless of the actual dimensions of the altered zone. Positive values indicate well damage (decreased productivity). Negative values indicate well stimulation (increased productivity). Fig. 10.11 shows a typical pressure profile for a well with a positive skin. Wells with formation damage, partially penetrating wells, and wells with significant pressure drops in their completions have positive skins. Hydraulically fractured wells have negative skins. In general, skin must be determined empirically, usually from pressure-transient tests. Further discussion of the physical meaning of skin is given in the section on pressure-transient analysis.

• 

  Fig. 10.11 – Pressure distribution in a reservoir with a positive skin.

Non-Darcy Flow

In gas wells, there may be a significant non-Darcy component of flow that results in an additional potential difference that depends on the square of the flow rate. The non-Darcy effect appears in well-deliverability equations as a flow-rate-dependent skin,

....................(10.46)

where S is the fixed or "mechanical" skin, and S′ represents the total apparent skin including non-Darcy effects. D is called the non-Darcy-flow coefficient, having units of D/Mscf or d/std m³. Although the non-Darcy coefficient may be calculated from laboratory measurements of the non-Darcy coefficient, it is typically determined in the field from well tests.

Transient Flow

At early times after a well has been put on production and at early times after a well has been shut in, flow occurs in a transient mode, making the steady-state forms of Darcy’s law inappropriate. To mathematically represent transient flow, the relationship between density and pressure and material-balance (continuity) relationships must also be considered. When combined with Darcy’s law, the result is the diffusivity equation, which in radial coordinates is

....................(10.47)

Note that the generic potential ψ is used in Eq. 10.47. This will be discussed further with regard to transient solutions for different fluid systems.

It is useful to distinguish between four different time periods when dealing with solutions to Eq. 10.47: (1) early time, (2) infinite-acting time, (3) transition time, and (4) stabilized time (Fig. 10.12).

• 

  Fig. 10.12 – Pressure/time history for a well produced at constant rate. Stabilized time occurs after all drainage-volume boundaries are fully “felt” and the well enters either steady-state or pseudosteady-state flow.

Early time is dominated by wellbore, rather than reservoir, effects. During this time, little can be determined about the reservoir. This time may last from a few minutes to a few days.

During infinite-acting time, however, well response is the same as a well being produced from an infinite reservoir. Most pressure-transient tests analyze data during this time period. Because all reservoirs are finite, however, this time must end. It does so when the well response is affected by a part of the outer boundary of the well’s drainage volume.

Steady-state flow is characterized by pressures being constant with time, requiring that the outer boundary of the system be maintained at constant pressure and the well be kept at either constant pressure or constant rate. This flow regime applies to certain water-influx situations or fluid-injection projects.

Pseudosteady-state flow occurs at late time in closed systems with a well produced at constant rate. Although pressures still change with time in pseudosteady state, all pressures everywhere in the reservoir decline at the same rate. This means that the pressure profile reduces uniformly throughout the reservoir.

Transition time occurs between infinite-acting and late time. During the transition period, the outer drainage boundaries are being felt in succession, causing the shift from infinite-acting to late time to occur over some length of time. In regularly shaped drainage areas (e.g., circles and squares), transition time may not exist. In irregularly shaped drainage areas, particularly with a well placed off-center, transition time can be quite long.

Infinite-Acting Flow. The time of primary interest in pressure-transient testing is the infinite-acting period, the mathematical solution for which comes from the diffusivity equation, expressed with the following dimensionless variables.

....................(10.48)

....................(10.49)

....................(10.50)

The subscript i refers to the initial conditions of the well’s drainage volume. Time is in hours for both dimensionless-time equations.

At sufficiently large values of , which are typical for most reservoir conditions, the solution can be mathematically approximated by

....................(10.51)

This approximation is good to within 2% accuracy for > 5 , within 1% accuracy for > 8.5 , and essentially identical to > 100.

Noting that the dimensionless potential at the well differs from the dimensionless potential at the wellbore radius (r_D = 1) by the skin factor, the basic radial well flow equation used for most well-testing purposes is

....................(10.52)

A graph of ψ_wf vs. the logarithm of time is a straight line. This slope is the basis for much pressure-drawdown and -buildup testing.

When predicting well performance, it is important to recognize when data are being collected during transient flow and to take into account the continuing decline in well deliverability until a steady-state or pseudosteady-state condition is established.

Pseudosteady State. In a closed drainage volume, once all the outer boundaries have been fully felt, a constant-rate well will experience pseudosteady-state flow. Because all pressures in the reservoir decline at the same rate during pseudosteady-state flow, the difference between reservoir pressures and the well pressure remains constant, even though both individually are changing with time. Because the resulting equation does not explicitly show a time dependence, the term pseudosteady state is used. Some authors also refer to this time period as "semisteady state."

Because pressure differences remain constant during pseudosteady-state flow, the following equations can be written to represent well performance during this period.

....................(10.53)

where C_A is the Dietz shape factor, dimensionless.

Because , Eq. 10.53 can also be written in a more usable engineering form as

....................(10.54)

where .

The shape factor can be determined empirically from mathematical solutions to the diffusivity equation in closed systems. Tables 10.5 and 10.6 provide a list of shape factors for different drainage shapes and well placements.

• 

  Table 10.5

• 

  Table 10.6

For a well in a closed circle, the pseudosteady-state equation can also be written as follows, which is equivalent to having a shape factor of 31.62.

....................(10.55)

This equation also works for a well in the center of a closed square or other regular shape by calculating an equivalent-radius circle:

....................(10.56)

Transient Drainage Radius. To simplify the flow equations, it is sometimes useful to use what has been called a transient drainage radius, defined by

....................(10.57)

This drainage radius is defined so that it represents the radius out to which there is a significant pressure drop.

During infinite-acting time, because there has been little withdrawal from the reservoir, which means that

....................(10.58)

or, equivalently,

....................(10.59)

Note that the product is simply dimensionless time without the wellbore radius in the denominator.

During pseudosteady-state time,

....................(10.60)

For a circular or equivalent square drainage area, it can be shown that

....................(10.61)

The deliverability equation for wells can then be written in the following simplified form for all times:

....................(10.62)

Note that pressures decline in proportion to the logarithm of time during infinite-acting time, while in pseudosteady state, pressures decline approximately in direct proportion to time.

Another way to determine the onset of pseudosteady state is through knowledge of the drainage shape and reservoir parameters. Tables 10.5 and 10.6 give times for the end of the infinite-acting period and for the beginning of time when the pseudosteady-state equation can be used. These tables use a dimensionless time based on drainage area,

....................(10.63)

Estimating Drainage Shapes

With little effort, it is possible to make a reasonable approximation of well drainage volumes and shapes. The process is based on the following assumptions:

1. The volume drained by an individual well is proportional to its flow rate.
2. Distance to a "no-flow" boundary between pairs of wells is proportional to each competing well’s flow rate.

The following technique can then be used to estimate the drainage area at a given time. Assign a flow rate to each well based on an average production over some reasonable time period.

1. Using Assumption 2 above, assign no-flow points between pairs of wells.
2. Sketch no-flow lines by connecting up no-flow points.
3. Adjust the lines near reservoir boundaries to ensure that Assumption 1 is satisfied. Make adjustments for variations in thickness and known geologic features (e.g., faults).

Although this process is fairly rough, it can give a reasonable estimate of the drainage areas of each well in a reservoir.

Radius of Investigation

Tables 10.5 and 10.6 list times for the end of the infinite-acting-flow period and the beginning of the pseudosteady-state-flow period. These times can be used to define a "radius of investigation." The physical meaning of the radius of investigation is that it represents the minimum radius at which a boundary could exist, but which has not yet been "felt" at a given time (usually meaning the time at the end of a drawdown test that has not yet been affected by a reservoir boundary).

Thus, if is the area-based dimensionless time that defines the end of the infinite-acting period for a circular drainage area,

....................(10.64)

Depending on the level of precision that would define the end of the infinite-acting period, the radius of investigation would have different values. If a 1% criterion is used, is equal to 0.06, and

....................(10.65)

Another common cutoff criterion is a of 0.1, after which the pseudosteady-state solution is listed by Earlougher as being "exact" (probably to within the number of significant digits of one’s computer). With this criterion,

....................(10.66)

For practical purposes, it is reasonable to use what many authors recommend for the radius of investigation,

....................(10.67)

Pressure-Transient Testing of Gas Wells

Refer to the chapter on well testing in this section of the Handbook for detailed discussions of interpreting pressure-transient tests of gas wells.

Deliverability Testing

Gas wells have historically been tested at a series of bottomhole pressures and rates to develop stabilized-deliverability relationships. One of the reasons for this is the importance of the non-Darcy flow contribution to well performance. A single-rate test cannot address this contribution. This section describes the types of tests typically run, along with their appropriate analysis techniques.

Stabilized-Deliverability Test. A stabilized deliverability, sometimes called a four-point or backpressure test, is conducted by producing a well at four rates (Fig. 10.13). In this test, each rate is run long enough to reach stabilized conditions. To be theoretically valid, "stabilized" should mean to pseudosteady state, although in practice tests are sometimes run only until little variation in well flowing pressure is observed.

• 

  Fig. 10.13 – Rate vs. time history of a stabilized deliverability test.

The most widely used method of presenting such data was first suggested by Rawlins and Schellhardt. This method is not based on the pseudosteady-state-flow equations but is based on an empirical observation that Δp² vs. q_g plotted on a log-log graph typically lies on a straight line (Fig. 10.14). The equation for a straight line on a log-log graph is

....................(10.68)

The slope of the straight line on a log-log graph is 1/n. The easiest way to determine slopes on a log-log graph is to recall that differences in the run and rise of the line must be taken in terms of logarithms, so that if and are two points on the straight line,

....................(10.69)

A value of n = 1 corresponds to Darcy (laminar) flow, while n = 0.5 (slope= 2) corresponds to completely turbulent flow.

• 

  Fig. 10.14 – Typical gas-well stabilized-deliverability plot.

A parameter usually presented with this data is the absolute open flow (AOF) potential. The AOF may be found by extrapolating the log-log plot and reading the flow rate at a value of . The value of C is calculated from any point on the straight line but is perhaps easiest to calculate as

....................(10.70)

An alternative is to write the deliverability equation in terms of AOF instead of the parameter C:

....................(10.71)

The AOF is sometimes used as a measure of a well’s potential flow capacity for regulatory and other purposes.

A more accurate way to analyze the stabilized deliverability test is to use the following equation, which is simply an alternative algebraic form of Eq. 10.62.

....................(10.72)

A Cartesian plot of Δψ/q_g vs. q_g will yield a slope of b and an intercept of a (Fig. 10.15). The advantage of this approach is that a and b have physical meaning based on reservoir parameters and thus can be compared to what is known about well and reservoir properties. C and n, on the other hand, have no such physical meanings. One disadvantage of this approach is that it is often more difficult to find a good straight line. However, this is a vagary of using Cartesian rather than log-log plots and does not represent any actual degradation of the data.

• 

  Fig. 10.15 – Theoretical gas-well stabilized-deliverability plot.

Once a and b are determined from the graph, the AOF using this approach is calculated by

....................(10.73)

The second factor in Eq. 10.73 represents the reduction in the AOF caused by non-Darcy effects. The AOF determined from the a/b approach is usually lower than that calculated from the C/n approach.

Isochronal Tests. Another type of test often run on gas wells is the isochronal test. The difference between an isochronal test and a stabilized-deliverability test is that the flow periods are not run long enough to reach stabilized flow. This is done to shorten testing time and to conserve gas, particularly where no pipeline is available. Fig. 10.16 shows a rate-vs.-time history for a typical isochronal test. Note that although the shut-in times are sufficiently long to approach initial reservoir pressure, the producing times are not long enough to reach pseudosteady state.

• 

  Fig. 10.16 – Rate-vs.-time history for a typical isochronal test.

Lines are drawn through a Δp²-vs.-q_g (Fig. 10.17) or Δψ/q_sc-vs.-q_sc plot (Fig. 10.18) at common producing times; that is, during each flowing period, pressures are read at fixed times since the initiation of flow.

• 

  Fig. 10.17 – Results of a typical isochronal test.

• 

  Fig. 10.18 – Modified isochronal test analyzed by a/b analysis.

For a given producing time, a line through the data should have a constant slope. This slope should be the same no matter what the producing time because the transient drainage radius does not depend on flow rate. Thus, from each flow period, data are plotted for identical producing times (Fig. 10.17 or Fig. 10.18).

Finally, an extended flow period (to stabilization) is run at one rate. A line is passed through this single stabilized point, with the same slope as the other isochronal lines. If properly conducted, this test has been shown to give results combrble to a stabilized-deliverability test.

A modified isochronal test is also sometimes run in which the shut-in times are also shortened (Fig. 10.19). This type of test also works well if the value of pressure at the end of the last shut-in period is used in place of the average reservoir pressure.

• 

  Fig. 10.19 – Rate vs. time history for a typical modified isochronal test.

Using Gas-Well Deliverability Relationships

Single-well deliverability equations can be used for a variety of purposes, including the following:

• Prediction of flow-rate changes caused by changing reservoir pressure (i.e., during reservoir depletion over time).
• Prediction of flow-rate changes caused by changing well flowing pressure resulting from production-equipment changes (e.g., compression).
• Prediction of bottomhole-flowing-pressure changes caused by changing well rates.

In general, the most theoretically valid deliverability equation should be used:

....................(10.74)

The use of Eq. 10.74 to solve for well flowing pressure is straightforward, except for the requirement to convert the generic potential ψ to actual pressure when using the real-gas potential m(p). The conversion of m(p) to p can be done either graphically or numerically with a computational algorithm (preferred).

Eq. 10.74 is a quadratic equation in flow rate, so when flow rate is being calculated from known pressures, the following can be used.

....................(10.75)

In some circumstances, it may be desirable to use either the C/n equation,

....................(10.76)

or a productivity index (PI) equation,

....................(10.77)

for deliverability calculations. Both can be used in a similar manner, as described above.

Effects of Skin

There are several ways to look at the physical effects attributable to skin. The first is in terms of a flow efficiency (FE), which is defined as the well productivity with skin compared to the no-skin case:

....................(10.78)

The effective potential drop caused by skin also can be calculated as

....................(10.79)

Some prefer to consider skin in terms of an effective wellbore radius,

....................(10.80)

It should be remembered that the skin in these equations is rate-dependent because of the non-Darcy effect. To determine the two different components of skin, mechanical-vs.-non-Darcy pressure-buildup or -drawdown tests must be run at more than one rate. If multirate transient tests are run, a simple plot of S′ vs. q_g will yield a slope of D and an intercept of S.

Volumes of Recovery

This section discusses various aspects of gas reservoir performance, primarily to determine initial gas in place and how much is recoverable. The equations developed in this section also will be used to form the basis of forecasting future production rates by capturing the relationship between cumulative fluid production and average reservoir pressure.

Gas in Place

Volumetric Determination. Original gas in place (OGIP) can be estimated volumetrically with geological and petrophysical data:

....................(10.81)

In oilfield units with gas measured in Bscf and B_gi in ft³/Mscf,

....................(10.82)

In SI units with gas measured in std m³ and A in m²,

....................(10.83)

Material-Balance Determination of OGIP. Material-balance equations provide a relationship between original fluids in place, cumulative fluid production, and average reservoir pressure. For many gas reservoirs, a simple material-balance equation can be derived on the basis of the following assumptions:

• The gas-filled pore volume is constant.
• Gas dissolved in water or liberated from the rock is negligible.
• Reservoir temperature is uniform and constant.

With these assumptions, the real-gas law can be used to derive

....................(10.84)

This equation can be rearranged to get the usual volumetric gas material-balance equation,

....................(10.85)

This equation is the basis for the p/z-vs.-G_p graph used to analyze gas reservoirs.

Determining Average Reservoir Pressure

Reservoir engineers have often used pressure contour maps or some approximate methods to determine field average reservoir pressure for p/z analysis. Usually, however, individual well pressures are based on extrapolation of pressure buildup tests or from long shut-in periods. In either case, the average pressure measured does not represent a point value, but rather is the average value within the well’s effective drainage volume (see Sec. 10.4.5).

By combining the assumptions used to assign drainage shapes and considerations of the gas law, the following procedure could be used for developing an average reservoir pressure at any point in time.

1. Be certain to determine average reservoir pressure accurately. Sometimes, shut-in times are inadequate to achieve complete buildup. When this occurs, one way to approximate reservoir pressure from a long shut-in is to use the Matthew-Brons-Hazebroek method estimating the semilog straight-line slope from reservoir properties rather than a buildup test. Of course, buildup tests are the preferable way to determine average reservoir pressure when economically feasible. An alternative way to ramp up an incomplete buildup is to run a buildup test early in the life of a well, noting the time to complete buildup and the percentage buildup at shorter times. Then, these percentages can be used in subsequent shut-in tests of shorter times than those required for full buildup.
2. For each well, make a graph of p/z vs. G_p. In general, these graphs will not necessarily yield a straight line. If the well’s drainage volume is changing with time, these will be curves. Either way, pass a smooth curve (not necessarily straight) through the data points.
3. To estimate the average p/z for a given well’s drainage volume at a given point in time, first determine the cumulative gas produced for that well at the desired time, then use the value of G_p to get a value of well p/z from the graphs created in Step 2.
4. Estimate the average production rate for each well at the desired time. This should be some reasonable average "eye-balled" from production curves, and not necessarily a specific daily rate.
5. Determine the reservoir average p/z as the average of the individual-well values (Step 2), weighted by their production rate:

....................(10.86)

where n_w is the number of active producing wells. This procedure works reasonably well and is straightforward.

Accurate determination of average reservoir pressure is particularly difficult in tight gas sands. Shut-in pressures may not be near average reservoir pressure for several months or years, obviously too long to be of any value. In addition, low-permeability reservoirs can have significant pressure differences across the field because certain areas can be drained more effectively than others.

Poston and Berg discuss methods for adjusting p/z plots for the lack of sufficient buildup time in determining average reservoir pressures. Although these methods have some validity, they also are prone to large errors because of data uncertainties. A recommended practice is, where feasible, to perform advanced pressure-transient-analysis methods on pressure-buildup tests to provide the means to extrapolate to expected values of average reservoir pressure. Such methods rely on the extrapolation of buildup pressures followed by a correction that incorporates drainage shape and volume. The problem with these techniques is that the correction for drainage shape and volume can be very significant (because of low reservoir permeabilities), and these techniques are highly uncertain, given the extent of heterogeneities and compartmentalization in typical tight gas reservoirs. Calibration of pressure-buildup analyses against actual well responses and reservoir simulation history-match studies can be helpful.

Another problem in tight reservoirs is the variation in average reservoir pressure across the field, both because of reservoir compartmentalization and because of low permeabilities. In higher-permeability gas reservoirs, this problem is generally not so severe, meaning that the average reservoir pressure needs to be measured only in a few wells to generate accurate p/z analyses. When variations in average reservoir pressure are large, however, methods need to be used to account for differences across the field.

Although reservoir simulation is one possibility to deal with this problem, it is also possible to use a "compartmentalized-reservoir model" to incorporate these effects. This model treats the reservoir as a set of communicating "tanks." The technique is basically a history-matching process that uses compartment volumes and compartment-to-compartment transmissibilities as tuning parameters. This method is important and has technical value, although it does not address the problem of required long shut-in times. The method has been applied to several reservoirs with very good results; hence, its use is recommended.

Volumetric Reservoirs

In volumetric dry- and wet-gas reservoirs, p/z vs. cumulative gas production will be a straight line intercepting the gas-production axis at the OGIP. An example is given in Fig. 10.20. The intercept (G_p = 0) on the p/z axis is p_i/z_i, and the intercept on the G_p axis (p/z = 0) is G. This graph provides a convenient method of using average-reservoir-pressure data to estimate OGIP and recoverable reserves once an abandonment p/z is established. When these plots are applicable, results for OGIP are generally considered very accurate after approximately 10% of gas reserves have been produced (sometimes a bit earlier).

• 

  Fig. 10.20 – Example p/z-vs.- G_p curve for a volumetric gas reservoir.

When only a small amount of early data is available, the OGIP can be determined from any point (G_p, p/z) by

....................(10.87)

When placing the straight line through p/z data, it is usually prudent to consider the first point (i.e., at field discovery pressure) as more accurate than others. Because there has been little field depletion and there has been sufficient time for pressures to return to stabilized conditions, discovery-pressure measurements are generally reliable. Regression approaches to placing the p/z-s.-G_p line should take this into consideration.

Volumetric estimates based on cores, well logs, fluid analyses, and geological estimates of reservoir size provide a "rock-based" estimate of gas in place, while material-balance relationships provide a "fluids-based" or "pressure-based" estimate. These two types of estimates are essentially independent (except for the use of consistent values of B_gi). Thus, when the two estimates are combrble, there is greater certainty in the OGIP estimate. It is now a common practice to develop geologic and simulation models of a reservoir to determine reserves and depletion strategies and to evaluate alternative development scenarios. Acquisition of p/z data can provide another measure of the volume being drained by a well or set of wells in what is thought to be a common reservoir. Differences between models and p/z data can be a valuable tool in managing a reservoir and detecting opportunities for additional development or deferral of expenditures that become unnecessary because of a change in the size of a resource.

Well-deliverability forecasts can be used to predict the economic limit of production for a field (income= costs) and the resulting p_a/z_a. Recoverable reserves then become

....................(10.88)

Highly Compressive Reservoirs

For some high-pressure gas reservoirs (e.g., geopressured or abnormally pressured reservoirs), the combined rock and water compressibility can result in a nonlinear p/z plot (Fig. 10.21). Ignoring this effect can lead to large overestimates of the OGIP. Local knowledge is the best source of information about whether these effects should be considered. Such performances usually should be suspected for geopressured reservoirs.

• 

  Fig. 10.21 - p/z vs. cumulative production, North Ossum field, Lafayette Parish, Louisiana, NS2B reservoir.

If the pore volume and water can be considered to have constant compressibility, then the change of gas-filled pore volume with pressure is

....................(10.89)

Using the first two terms in a Taylor series expansion for the exponential function,

....................(10.90)

where ....................(10.91)

The material-balance equation for a compressive reservoir then becomes

....................(10.92)

Eq. 10.92 suggests that if the effective reservoir compressibility, c_e, can be estimated, then the p/z plot for such reservoirs can be linearized by multiplying the p/z values by 1 - c_eΔp. However, there is typically very little knowledge of the effective system compressibility, meaning that this relationship is of limited practical use for reservoir engineering purposes. In addition, the effective system compressibility may even change with time, typically becoming smaller as reservoir pressure declines and the reservoir rocks compact.

There is sometimes a change in the slope of the p/z plot when an abnormally pressured gas reservoir reaches normal pressure, as shown in Fig. 10.21. Approaches suggested for analyzing geopressured gas reservoirs include methods to account for some unusually high apparent values of the effective pore-volume compressibility. Soft-sediment compaction, shale dewatering, and limited aquifer influx are among the physical effects proposed by various authors. For further information on this topic, the reader is referred to papers by Hammerlindl, Roach, Prasad and Rogers, Bernard, Fetkovich et al., Ambastha, Yale et al., El Sharkawy, and Gan and Blasingame. Poston and Berg also provide an evaluation of different methods of accounting for pressure support experienced in geopressured reservoirs.

Many overpressured reservoirs, however, do not demonstrate the change in slope, as illustrated by Fig. 10.22. These data are from four wells in a common reservoir with an initial pressure gradient > 0.65 psi/ft. As indicated, there was no change in slope when the reservoir pressure reached a normal gradient. The reservoir consists of a competent sandstone that may have a low effective compressibility.

• 

  Fig. 10.22 – Actual vs. expected p/z curves for an overpressured gas reservoir.

Considering the previous discussions, using early p/z has many uncertainties. Best practices would suggest that early-time analyses use ranges for effective pore-volume compressibility based (where possible) on analogous or similar regionally located reservoirs to reduce the high uncertainty in early data and its potential dramatic effect on estimates of OGIP.

Abandonment conditions for highly compressive reservoirs are determined in the same manner as those for volumetric reservoirs.

Waterdrive Reservoirs

Fig. 10.23 shows a typical p/z plot for a gas reservoir with an active waterdrive. Note that for a given value of cumulative gas production, pressures are higher than for a volumetric reservoir.

• 

  Fig. 10.23 – Volumetric-vs.-waterdrive p/z trends.

The material-balance equation for a waterdrive reservoir is

....................(10.93)

If it can be assumed that volumetric estimates of G are accurate, then Eq. 10.93 can be rearranged to calculate a water-influx history for comparison against contact mapping.

....................(10.94)

This can be used with different aquifer models to determine how to predict future water influx. This will be discussed further in the next section.

Using Eq. 10.93 to estimate G requires information about the cumulative water influx. Estimates may be obtained by mapping water movement using watered-out wells, logging surveys, or data from infill wells.

By writing Eq. 10.93 in terms of a volumetric sweep efficiency, E_v, and the residual gas saturation to water displacement, S_gr, the material-balance calculation can then be written as

....................(10.95)

If E_v is taken to be the estimated volumetric sweep efficiency at abandonment, then this equation represents all possible abandonment conditions regardless of the rate of water influx. An abandonment line can then be drawn on the p/z plot, the bottom point of which is at p/z = 0, G_p = G, and the top point of which is at p/z = p_i/z_i. A straight line connecting these two points is the locus of all possible abandonment points. The intersection of this abandonment line and the actual p/z-vs.-G_p line, as illustrated in Fig. 10.24, gives an estimate of ultimate recovery.

• 

  Fig. 10.24 – Typical p/z plot for a waterdrive gas reservoir.

Many factors affect gas-recovery potential when a waterdrive is active. The main considerations are how much of the reservoir will be invaded by water, what the pressure decline will be, and what the trapped-gas saturation will be behind the waterfront. Simple 3D simulations can be used to study the strength of an aquifer based on its size relevant to the gas volume and the variation in rock quality. A good reservoir description is the key to a successful prediction. How the reservoir is affected is also a function of withdrawal rates. Strong aquifers can sustain reservoir pressure and result in low recoveries. Gas saturations trapped behind the invading water are typically about one-third of the initial hydrocarbon saturation. If volumetric sweep is high and there is little pressure depletion (B_ga ≈ B_gi), then a reservoir with S_wi equal to 25% of pore volume would experience a recovery of

....................(10.96)

Recoveries as low as 50% of OGIP can occur in adverse circumstances, but more often recoveries exceed 70% of OGIP owing to partial pressure depletion. There are documented cases in which a significant increase in offtake rate has resulted in pressure depletion of a waterdrive reservoir when withdrawal rates were sufficient to outrun invading water. Recovery from reservoirs that exceed 1 Tcf OGIP with permeabilities greater than 250 md will normally be unaffected by an aquifer of any size if field depletion occurs over a 20-year period or less. Again, a simple simulation model will confirm how potentially effective an aquifer may be.

The effect of a weak to moderate waterdrive is often difficult to detect with a simple p/z plot. Often, a straight-line plot will occur (Fig. 10.23) and will lead to incorrect estimates. Cole has suggested an improved method. If the expansibility of water is small compared to gas expansibility, then the material balance can be arranged as

....................(10.97)

Cole’s methodology is to plot the left side of the equation against G_p. The shape of the resulting plot will vary depending on the existence and strength of a waterdrive, as illustrated by Fig. 10.25.

• 

  Fig. 10.25 – Plot for volumetric and waterdrive gas reservoirs (after Cole).

Data from a volumetric reservoir will plot as a horizontal line. A weak waterdrive yields an early increase in ordinate values followed by a negative slope. The initial increase may not be detected because many pressure measurements are needed very early in the producing life of the reservoir. Moderate to strong waterdrives give overstated OGIP values. This plot is very sensitive to the effects of water influx and is a good qualitative tool. Back extrapolating the plot to OGIP has been suggested. In practice, the slope usually changes with each pressure measurement, and extrapolation is difficult to impossible.

Pletcher has suggested a further modification to the Cole plot to account for rock and water compressibility. In doing so, Eq. 10.97 becomes

....................(10.98)

where F = G(E_g + E_fw) + W_e,

....................(10.99)

The shapes of the resulting plots are the same as those in Fig. 10.25, but they do avoid the negative slope of a Cole plot that results from an abnormally pressured reservoir with no water influx.

Retrograde-Condensate Reservoirs

Depletion behavior of retrograde-condensate reservoirs can be handled through the p/z analyses discussed previously, with the caveat that the z factor must be the two-phase z factor (see Sec. 10.2). Two-phase z factors either may be obtained from laboratory tests or predicted from composition with an EOS. In wet-gas and retrograde-condensate reservoirs, cumulative gas produced must include both gas and liquid (as equivalent gas) production. This is particularly important for high-liquid-yield gases.

In the calculation of future reserves for planning purposes, it is usually necessary to break out gas and liquid reserves separately, perhaps even by individual gas component. For wet-gas reservoirs, liquid yields from a particular gas can be expected to remain constant with time, so long as the gas is processed in the same manner. Changes in separator conditions and/or gas-processing facilities could result in changing liquid yields, however.

Retrograde-condensate reservoirs, on the other hand, will produce at a variable yield as the reservoir pressure declines. Determination of the expected yields can be based on laboratory tests and/or EOS calculations. The PVT test presented in Sec. 10.2 shows how yields of the various gas components can vary over time.

Pressure Maintenance and Cycling Operations

Pressure maintenance of a retrograde-condensate gas reservoir can exist by virtue of an active waterdrive, water-injection operations, gas-injection operations, or combinations of all of these. Certain reservoirs may contain fluids near their critical points and are thereby candidates for special recovery methods, such as the injection of specially tailored gas compositions to provide miscibility and phase-change processes that could improve recovery efficiency. These usually are not regarded as gas/condensate cases. All these improved-recovery methods are best studied with simple-to-complex computer models. Simple models can be used initially to screen prospects, and then more-detailed studies including compositional considerations can be conducted.

Waterdrive and Water-Injection Pressure Maintenance. Recovery from retrograde-condensate gas reservoirs with a waterdrive or water injection is subject to the same considerations as described in the chapter on Waterflooding (in the Reservoir Engineering and Petrophysics section of this Handbook) for water injection into oil reservoirs. To make a recovery assessment, the first requirement is a good description of the rock and fluid characteristics of the reservoir and the aquifer. Variations in the permeability of various strata, mobility ratios, and gravity-stable advance of the water front will affect the volumetric sweep. S_gr should be approximately the same as described earlier for dry- and wet-gas displacements by water. The favorable mobility ratio can result in a high volumetric sweep. There is strong evidence, however, that displacement efficiency by water is not high. While Buckley et al. indicated that the displacement efficiency of water displacement of gas can be as high as 80 to 85%, experiments and field observations by Geffen et al. indicate that it may be as low as 50%. All things considered, the recovery of gas condensate in the vapor phase by water injection is likely to be appreciably lower than by cycling, and any consideration of water injection for gas/condensate recovery should be accompanied by detailed experimental work on cores from the specific reservoir involved. This will help to determine whether water can, in fact, accomplish a high-enough displacement efficiency to justify its use.

Premature water breakthrough can, and often does, result in "load up" and loss of the ability of a well to flow. It is difficult to obtain economical flow rates by artificial lift. This loss of productivity may result in premature abandonment of the project. The problems would be particularly serious for deeper reservoirs in which the cost of removing water would be a significant factor. Yuster discusses possible remedial methods for drowned gas wells. Bennett and Auvenshine discuss dewatering gas wells. Dunning and Eakin describe an inexpensive method to remove water from drowned gas wells with foaming agents.

Generally, the use of water injection for maintaining pressure in a gas/condensate reservoir will be unattractive where a wide range of permeabilities exists in a layered reservoir and selective breakthrough of large volumes occurs early in the life of the reservoir.

Dry-Gas Injection. Comparative economics determine whether a gas/condensate reservoir should be produced by pressure depletion or by pressure maintenance (i.e., does the additional condensate recovery justify the cost of compressing, injecting, and processing the injected gas?). Delayed gas sales also may be a factor. The objective of using dry-gas injection is to maintain the reservoir pressure usually above or near the dewpoint to minimize the amount of retrograde condensation. Dry field gases are miscible with nearly all reservoir gas/condensate systems; methane normally is the primary constituent of dry field gas. Dry-gas cycling of gas/condensate reservoirs is a special case of miscible-phase displacement of hydrocarbon fluids for improving recovery. Experimentation has shown that the displacement of one miscible fluid by another that is miscible is highly efficient on a microscopic scale; usually, the efficiency is considered 100% or very nearly so. Cycling does result in liquid recoveries at economical rates while avoiding waste of the produced gas when a market for that gas is not available.

Inert-Gas Injection. The use of inert gas to replace voidage during cycling of gas/condensate reservoirs can be an economical alternative to dry natural gas. One of the first successful inert-gas-injection projects was in 1949 at Elk Basin, Wyoming, where stack gas from steam boilers was used for injection. In 1959, the first successful use of internal-combustion-engine exhaust was seen in a Louisiana oil field. The first use of pure cryogenically produced N₂ to prevent the retrograde loss of liquids was in the Wilcox 5 sand in the Fordoche field located in Pointe Coupee Parish, Louisiana. In the Fordoche field, the N₂ amounted to approximately 30% of the natural-gas/N₂ mixture injected.

Studies by Moses and Wilson confirmed that the mixing of N₂ with a gas/condensate fluid elevated the dewpoint pressure. Moses and Wilson also presented data to show that the mixing of a lean gas with a rich-gas condensate would result in a fluid with a higher dewpoint pressure. The increase in dewpoint pressure was greater with N₂ than with the lean gas. In the same study, results are presented from slimtube displacement tests of the same gas/condensate fluid both by pure nitrogen and by a lean gas. In both displacements, more than 98% recovery of reservoir liquid was achieved. These results also were observed by Peterson using gas-cap gas material from the Painter field located in southwest Wyoming. The authors concluded that the observed results were obtained because of multiple-contact miscibility.

Cryogenic-produced N₂ possesses many desirable physical properties. Those that make nitrogen most useful for a cycling fluid are that it is totally inert (noncorrosive) and that it has a higher compressibility factor than lean gas (requires less volume). The latter advantage is partially offset by increased compression requirements when compared with lean gas.

The use of inert gas as a cycling fluid offers both advantages and disadvantages. The major advantages are early sale of residue gas and liquids and a higher recovery of total hydrocarbons because the reservoir contains large volumes of inert gas rather than hydrocarbons at abandonment. Disadvantages are production problems and increased operating costs caused by corrosion if combustion or flue gas is used, possible additional capital investments and operating costs to remove inert gas from the sales gas (a condition aggravated by early breakthrough of inerts), and potential costs to pretreat before compression and/or to fund reinjection facilities.

Forecasting

This section explores the fundamental relationships underlying gas-reservoir performance and presents some simple techniques for forecasting production rate vs. time.

System Performance

One way to envision the different factors affecting the performance of a gas reservoir is to define the production "system" with three components: (1) well deliverability (developed in Sec. 10.4), (2) wellbore hydraulics, and (3) production-equipment constraints. Rate-vs.-time behavior is governed by the combined effect of these three parts, which in turn have performance characteristics that vary with pressure and production rate. Wellbore and system constraints include

• Tubing and choke sizes.
• Amount of entrained liquids (condensate and water).
• Accumulation of sand or debris in the wellbore.
• Flowline pressure drops.
• Compression ratios across compressors.
• Pressure losses in separators.

If these relationships are plotted on the same presentation, the resulting graph will look like Fig. 10.26.

• 

  Fig. 10.26 – Example well- and equipment-performance curves.

At low flow rates, the equipment-performance curve is nearly horizontal, reflecting the small flowing frictional pressure drops in the system. If there is liquid holdup in the production tubing, multiphase-flow calculations can show the curve bending upward at low production rates.

The curves represent maximum performance with existing completion, well, and equipment configurations. At any time, of course, it is possible to operate a well below its maximum performance characteristics by adjusting such things as a choke size at the wellhead. Because this is effectively a "zero-expense" change, it represents how a system is "tuned" to operate at some predetermined rate or other operating condition. Perforating additional producing intervals, using stimulation treatments, lowering separator pressure, or installing compressors, larger flowlines, and/or production tubing can change the performance curves.

The concave downward lines (flow rate increasing with decreasing well pressure) are the well-performance curves for different average reservoir pressures. Note that the value of well flowing pressure at zero flow rate is the average reservoir pressure and that the value of flow rate at zero well flowing pressure is the AOF. As reservoir pressure declines, of course, the well-performance curves move down and to the left. These curves may be altered by operational changes that affect well deliverability. Such processes as hydraulic fracturing, acidizing, or reperforating will increase well productivity and move up the AOF point.

Reservoir Deliverability

The intersection of the equipment- and well-performance curves represents the operating point of the well at a given value of average reservoir pressure. Note that if the equipment-performance curve represents maximum (full-open choke) performance, then the intersection point also represents maximum rate performance.

The intersection points of the well- and equipment-performance curves can be used to construct a relationship between average reservoir p/z and maximum well deliverability such as that shown in Fig. 10.27. This curve represents the rate that the combined well and equipment design is capable of delivering at any particular average reservoir pressure. Note that this curve can be used to determine the abandonment p/z by knowing the economic limit (i.e., the minimum economic rate).

• 

  Fig. 10.27 – Example reservoir-deliverability curve.

For the purposes of forecasting total reservoir performance, it is necessary to develop a graph such as Fig. 10.27 for the reservoir and not just for an individual well. One way to do this is to construct individual graphs for each well and then to add up the flow rates for each well at a given value of average reservoir p/z. Alternatively, one could construct the graph for an average well in the reservoir and then simply multiply by the total number of wells.

Another method would be to first determine reservoir deliverability constants a and b (that is, a and b should represent the following deliverability equation for the reservoir):

....................(10.100)

where q_R is the total reservoir production rate.

There are many ways to develop this sort of relationship, including an analysis of reservoir performance in the same manner as well performance with techniques explained in Sec. 10.4. A simple way would be to obtain and for the individual wells in the reservoir, and then, by assuming that the average well produces at the total reservoir rate divided by the number of wells, n_w,

....................(10.101)

This equation could then be used with an average well-equipment-performance curve.

Forecasting Methods

It was seen in Sec. 10.5 that through material-balance relationships, it is possible to evaluate expected reservoir pressures knowing cumulative production. The material-balance equation combined with the reservoir-deliverability relationship thus leads to a set of two relationships that must hold simultaneously:

....................(10.102)

....................(10.103)

For simple gas-reservoir situations, it is possible to solve these two relationships either graphically or computationally.

Consider, for example, a typical gas-reservoir production scenario in which the reservoir is first put on production at some fixed reservoir rate for an extended period of time. This fixed rate may result from sales-contract considerations or limitations on processing equipment and pipelines. Let q_c denote the fixed initial reservoir production rate and t_c denote the time of the fixed-rate period.

It is possible to use the reservoir-deliverability and p/z material-balance curves to determine the length of the constant-rate period given the rate or vice versa. First, consider how to determine the time.

For a given rate, the reservoir-deliverability curve can be used to determine the lowest reservoir p/z that will deliver this rate. Because the reservoir-deliverability curve represents the maximum reservoir deliverability before the end of the constant-rate period, the reservoir will be produced at less-than-maximum rates. The value of p/z at the end of the constant-rate period can be entered into the p/z material-balance plot to determine G_pc at the end of the constant-rate period. The constant-rate time period is then

....................(10.104)

For instance, refer to Fig. 10.28, where reservoir deliverability and G_p vs. p/z are plotted. If rates from the field were limited to 40 Bscf/yr by contract, equipment, or pipeline limitations, then the lowest p/z that will support this rate is equivalent to a cumulative production of 400 Bscf. The period of constant production would be 10 years.

• 

  Fig. 10.28 – Example reservoir-deliverability and material-balance curves.

Similarly, other rates can be selected and equivalent periods of production calculated to develop the inset curve in Fig. 10.28.

These procedures should be based on average sustainable rates that account for down time and other factors that reduce the composite well deliverability. Care also should be taken to account for the normal "ramp-up" of production that occurs during development drilling.

At the end of the constant-rate period, the reservoir will, barring well or equipment changes, go on decline. At this point, the reservoir will produce at its maximum rate according to the reservoir-deliverability curve. A simple procedure can be used to forecast this period as well by discretizing future production into increments.

Consider that a prediction will be made by discretizing the future cumulative gas produced in an increment ΔG_p. Over the time period that this amount of gas is to be produced, there will be some average flow rate . The time to produce the incremental gas production can be approximated by

....................(10.105)

Assume that q_j is the flow rate at the end of the time increment, and q_j–1 is the flow rate at the beginning of the time increment. Approximating the average flow rate during flow period j as

....................(10.106)

then ....................(10.107)

Total time since the beginning of the decline period is determined by

....................(10.108)

A plot of q_j (not ) vs. t_j is the desired forecast. Improved accuracy may be achieved by using smaller values of ΔG_p. An example decline graph can be seen in Fig. 10.29.

• 

  Fig. 10.29 – Example production rate-vs.-time curve for a gas reservoir.

The above procedures can be used to evaluate different reservoir-development scenarios. For example, infill drilling or adding compressors would improve the reservoir-deliverability relationship (and thus change the forecast) by sustaining or increasing early-time production but causing the reservoir to have a steeper decline. Whether this would be desirable would depend on economic considerations.

These procedures also suggest ways that past reservoir performance may be evaluated. For example, a historical reservoir-deliverability curve can be generated by plotting reservoir p/z vs. q_R during the decline period. This curve could then be adjusted for changes in operating equipment to determine future performance characteristics.

Water Influx

If a gas reservoir is under waterdrive conditions, there is an additional requirement to forecast the amount of water influx to be expected. The difficulty in performing water-influx calculations for most reservoir situations is in knowing the performance characteristics of the aquifer. As mentioned before, computer models ranging from simple to very complex are now available for general use and represent the best method of predicting the amount and timing of water influx. The models also allow investigation of the effects of varying aquifer size and rock characteristics. Matching of performance data will significantly improve the reliability of model projections.

Retrograde-Condensate Reservoirs

The options discussed before for depleting retrograde-condensate reservoirs are:

• Produce by depletion.
• Recycle processed produced gas.
• Recycle processed produced gas plus other makeup gas.
• Inject N₂ or other inert gas.
• Waterdrive or injected water.

With straight depletion, a considerable amount of the heaviest and most valuable hydrocarbons will be left in the reservoir. Also, reservoir fluids passing through the low-pressure region around the wellbore experience retrograde condensation, resulting in a large liquid saturation buildup and a significant decrease in gas permeability. This is important from at least two standpoints. First, the composition history of produced fluids early in the life of the reservoir may diverge from predictions that assume uniform pressure in the reservoir at any instant of time. Second, and more importantly, well deliverability will be decreased significantly, affecting both timing and ultimate recovery. Wells can even cease to produce in reservoirs with permeabilities less than 10 md. The effects of liquid buildup on well deliverability need special consideration in low-permeability environments.

Recycling of gas aids recovery from condensate reservoirs in two ways. First, reservoir pressure is maintained, if not above the dewpoint, at least at pressures that minimize liquid deposition. If makeup gas is injected, it is, of course, possible to never go below the dewpoint pressure. The economics of recycling produced gas must weigh the additional recovery benefits against the additional handling and injection costs, the deferred revenue of recovered injected gas, and the lost revenue of injected gas that will remain in the reservoir. These factors make most hydrocarbon gas-injection schemes uneconomical if a gas sale is possible. But where gas sales are to be delayed for many years (e.g., Prudhoe Bay), the resulting economics support a cycling program.

If, however, reservoir pressure has dipped below the dewpoint pressure, injected lean gas can also serve to revaporize a significant part of the deposited condensate because of phase-behavior effects. There has been some interest in the use of N₂ as an injected gas for condensate reservoirs. There is evidence that phase behavior is nearly equivalent to methane injection, and costs may be somewhat lower.

These effects, however, are complex. Injected gas may sweep inefficiently between injection and production wells, causing poor economics. Because of the complexities of this process, cycling operations should not be undertaken without the aid of reservoir-simulation studies. Compositional reservoir-simulation models are available that can easily handle the thermodynamics and fluid-flow characteristics of a recycling project. Characterization of phase behavior is generally done through an EOS, which can be tuned to laboratory PVT tests.

Gas Requirements in Cycling Operations. Miller and Lents expected to cycle the equivalent of approximately 115% of the gas in place to recover some 85% of the wet-gas reserves of the Cotton Valley Bodcaw reservoir. Brinkley indicated cycling-gas volumes of as much as 130% of original wet gas in place for various reservoirs. The requirements for a given reservoir will be driven mostly by economic considerations. The makeup gas needed for constant-pressure cycling is mainly the volume required to replace shrinkage by liquid recovery and the amount consumed for various fuel needs. For some composition, temperature, and pressure ranges, the removal of high-molecular-weight constituents from the produced wet gas may result in a higher compressibility factor for the injected dry gas; hence, the greater volume per mole injected may require little or no makeup gas for constant-pressure cycling.

The amount of gas not available for injection because of consumption for operating needs should be taken into account when determining makeup-gas requirements if pressure is to be maintained. The amount of fuel for compression and treatment plants depends mainly on the total amount of gas to be returned to the reservoir and the discharge pressure for the plant. Discharge pressure, in turn, depends on the total rate of injection demanded, the number of injection wells, and their intake capacities throughout the life of the operation. Other factors affecting the amount of gas required for overall operations are type of plant, type of liquid-recovery system used, and auxiliary field requirements (such as for drilling, completion, and well testing; camp fuel and power for maintenance shops, general service facilities, and employee housing; and other factors that vary from one case to another).

Moore reports that fuel consumption for the compression plant alone varies from 7 to 12 ft³/bhp-hr; this is probably for gases with heat values of approximately 1,000 Btu/scf.

If the fuel consumption is 8 ft 3 /bhp-hr and the compression ratio is 15 (compressing from, say, 461 to 7,000 psia), fuel requirements would be 34.4 Mscf/MMscf injected. For an example reservoir originally containing 131 Bscf of wet gas, which might be cycled the equivalent of 1.25 times, the approximate compressor fuel consumption would be 5.6 Bscf, or approximately 3% of the gas handled through the plant.

Treatment-plant fuel and other plant needs added to compressor fuel bring the range of consumption inside the plant fence to 3 to 7% of the gas handled by a cycling plant. In addition to these needs and others mentioned earlier, possible gas losses that can occur in a cycling operation are: gas used in "blowing down" wells, should this be necessary for cleaning or treating purposes; small gas leaks at compressor plants and in field lines; and gas leaks resulting from imperfect seals or corrosion in well tubing, casing, and cement jobs. Remedial workover operations should be planned immediately when there is evidence of appreciable loss of gas between the compression plant and the reservoir sandface or between the outflow-well sandface and the plant intake.

Summary

The study of gas reservoirs must consider the unique nature of gaseous fluids and their behavior in the reservoir and during production. This chapter has provided methods for characterizing gas reservoirs and their contents, including deliverability calculations, OGIP determination, and recovery potential. Differences in phase behavior between dry, wet, and retrograde-condensate gases result in different depletion schemes that may need additional altering if a strong waterdrive exists. Recycling of produced hydrocarbon gases or injection of nonhydrocarbon fluids may be justified in retrograde-condensate gas reservoirs, particularly if gas sales will be delayed for several years.

Nomenclature

a	=	empirical constant
A	=	drainage area, reservoir area, L2
AOF	=	absolute open flow potential, std L3/t
b	=	empirical constant
B	=	formation volume factor, L3/std L3
Bgi	=	initial gas formation volume factor, L3/std L3
c	=	compressibility, Lt2/m
cf	=	pore-volume compressibility, Lt2/m
cw	=	water compressibility, Lt2/m
C	=	constant in gas-deliverability equation
CA	=	Dietz shape factor, dimensionless
D	=	non-Darcy-flow coefficient, t/std L3
Efw	=	cumulative formation and water expansion, L3
Eg	=	cumulative gas expansion, L3
ER	=	recovery efficiency, fraction
Et	=	total cumulative expansion, L3
Ev	=	volumetric sweep efficiency, fraction
F	=	cumulative reservoir voidage, L3
G	=	original gas in place, std L3
GE	=	gas equivalent, std L3/std L3
Gpc	=	cumulative gas production during a period of constant rate, std L3
h	=	average reservoir thickness, L
kg	=	measured gas permeability, L2
kl	=	effective liquid permeability, L2
K	=	parameter in Lee et al.2 viscosity correlation
m	=	real-gas potential, m/Lt2
M	=	molecular weight
n	=	number of moles of gas or exponent in gas-deliverability equation
nc	=	total number of components in gas mixture
nw	=	number of wells
	=	relative number of total moles in gaseous phase, fraction
	=	relative number of total moles in oil phase, fraction
Np	=	cumulative condensate production, std L3
p	=	pressure, m/Lt2
	=	average pressure, m/Lt2
	=	variable of integration in real-gas potential equation, m/Lt2
PI	=	productivity index, std L3/t/m/Lt2
q	=	production rate, std L3/t
qc	=	production rate during period of constant rate, std L3/t
qR	=	total reservoir gas production rate, std L3/t
r1	=	radial distance at which pressure p1 is measured, L
r2	=	radial distance at which pressure p2 is measured, L
R	=	universal gas constant, mL2/nt2T
S	=	mechanical skin, dimensionless
S′	=	total skin, dimensionless
Sgi	=	initial average gas saturation, fraction
Swi	=	initial water saturation, fraction
t	=	time, t
tc	=	time of constant-rate production, t
T	=	temperature, T
u	=	volumetric flux ( q/A ), L3/t/L2
V	=	volume, L3
Vm	=	molar volume, L3/n
We	=	cumulative water influx, L3
Wp	=	cumulative water produced, std L3
xj	=	mole fraction of component j in liquid phase
X	=	parameter in Lee et al.2 viscosity correlation
yj	=	mole fraction of component j in gaseous phase
Y	=	produced condensate yield, std L3/std L3
z	=	gas deviation factor, dimensionless
zj	=	mole fraction of component j in mixture
α	=	cubic equation-of-state parameter
αc	=	empirical constant
β	=	defined in Eq. 10.41
ρ	=	density, m/L3
ϕ	=	porosity, fraction
γ	=	specific gravity (air = 1.0 for gas)
μ	=	viscosity, cp
ψ	=	generic potential
Δψ	=	generic-potential difference

References

SI Metric Conversion Factors

°API		141.5/(131.5 + °API)		=	g/cm3
bbl	×	1.589 873	E − 01	=	m3
Btu	×	1.055 056	E + 00	=	kJ
cp	×	1.0*	E − 03	=	Pa•s
ft	×	3.048*	E − 01	=	m
ft3	×	2.831 685	E − 02	=	m3
°F		(°F + 459.67)/1.8		=	K
in.3	×	1.638 706	E + 01	=	cm3
lbm	×	4.535 924	E − 01	=	kg
psi	×	6.894 757	E + 00	=	kPa
psi2	×	4.753 8	E + 01	=	kPa2

*

Conversion factor is exact.