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PEH:Oil Reservoir Primary Drive Mechanisms

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Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume V – Reservoir Engineering and Petrophysics

Edward D. Holstein, Editor

Chapter 9 – Oil Reservoir Primary Drive Mechanisms

Mark P. Walsh, Consultant

Pgs. 895-980

ISBN 978-1-55563-120-8
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Muskat defines primary recovery as the production period "beginning with the initial field discovery and continuing until the original energy sources for oil expulsion are no longer alone able to sustain profitable producing rates." [1] Primary recovery is also sometimes referred to as pressure depletion because it necessarily involves the decline of the reservoir pressure. Primary recovery should be distinguished clearly from secondary recovery. Muskat defines secondary recovery as "the injection of (fluids) after the reservoir has reached a state of substantially complete depletion of its initial content of energy available for (fluid) expulsion or where the production rates have approached the limits of profitable operation." [1] One of the most popular secondary-recovery methods is waterflooding. Because primary recovery invariably results in pressure depletion, secondary recovery requires "repressuring" or increasing the reservoir pressure.

Primary recovery includes pressure-maintenance methods. Muskat defines pressure maintenance as "the operation of (fluid) injection into a reservoir during the course of its primary-production history." [1] The main effect of pressure maintenance is to mitigate the reservoir’s pressure decline and conserve its energy. The purpose of pressure maintenance is ultimately to improve oil recovery. The most common injected fluids for pressure maintenance are water and separator or residue gas. "Partial" and "complete" pressure maintenance describe the general effectiveness of a given pressure-maintenance operation to retard the rate of pressure decline. Partial pressure maintenance refers to fluid injection while a general state of pressure decline still exists. Full or complete pressure maintenance refers to fluid injection while the reservoir pressure remains essentially constant.

According to Muskat’s definition of pressure maintenance, secondary-recovery methods such as waterflooding are not strictly pressure-maintenance operations because they begin after pressure depletion. However, if water injection takes place before the end of pressure depletion, which is not uncommon, it is considered a pressure-maintenance method. If water is injected before the end of primary recovery, the reservoir is classified as an artificial waterdrive. Since Muskat first proposed his definition, others have loosely applied the term pressure maintenance to include any fluid-injection strategy at any stage in the reservoir’s production.

Oil reservoirs are classified according to their fluid type. There are three broad oil classes. In order of increasing molecular weight, they are volatile oil, black oil, and heavy oil. Heavy-oil reservoirs are of minor interest during pressure depletion because they typically yield only marginal amounts of oil because of their low dissolved-gas contents and high fluid viscosities. Although heavy-oil reservoirs are not addressed specifically in this chapter, the methods of analysis presented here are equally applicable to them. The distinguishing characteristic between volatile and black oils is the stock-tank-oil content of their equilibrium gases. Equilibrium gases liberated from volatile oils contain appreciable stock-tank or condensable liquids whereas the gases from black oils contain negligible stock-tank liquids. While this distinction leads to only slightly different recovery strategies, it leads to very different methods of analysis and mathematical modeling requirements.

Volatile- and Black-Oil Fluid Characteristics

The petroleum fluid spectrum is gradational. There is no strict definition of volatile and black oils; there are only general guidelines and characteristics. Despite this lack of precision and the occasional confusion it brings, classification is quite useful and popular.

Molecular weight is a useful yardstick. Black oils typically range from 70 to 150 in molecular weight but may range as high as 190 to 210. In contrast, volatile oils are lower in molecular weight than black oils and typically range from 43 to 70. Oils with molecular weights greater than 210 usually are classified as heavy oils. Fluids with molecular weights of less than 43 are generally gases, which include gas condensates, wet gases, and dry gases. A molecular weight of 43 marks the lower molecular-weight limit of volatile oils.

Black and volatile oils are sometimes subdivided into different fluid types. For instance, volatile oils include near-critical fluids and high-shrinkage oils. Near-critical fluids represent light volatile oils and can include some very rich condensates. High-shrinkage oils represent the high-molecular-weight end of volatile oils and can include some light black oils.

Volatile and black oils are characterized in terms of a number of different properties. Table 9.1 summarizes their characteristics. This table includes the properties of the full range of petroleum fluids, including gases.

The defining property that distinguishes black and volatile oils is the volatilized-oil content of their equilibrium gases. The volatilized-oil content of a gas represents its condensable liquid portion. Condensable refers to the portion that condenses or "drops out" during pressure reduction and ultimately results as stock-tank liquid. Condensation may take place within the reservoir as the gas passes through the lease separators. Physically, intermediate-hydrocarbon components, typically C2 through C7, dominate this fraction. Volatilized oil also is called lease condensate or distillate. Gas condensates and wet gases also contain volatilized oil. Volatilized oil is reported conventionally as part of the crude-oil reserves and production. It should not be confused with and is distinctly different from natural-gas liquids. Natural-gas liquids are derived from the gas-processing plant and are called plant products.

The volatilized-oil content of gases is quantified in terms of their volatilized-oil/gas ratio, typically expressed in units of STB/MMscf or stock-tank m3 per std m3 of separator gas. The volatilized-oil/gas ratio of equilibrium gases of black oils is usually less than 1 to 10 STB/MMscf (approximately 0.04 to 0.4 gal/Mscf).The volatilized-oil content of these gases is so low that it usually is ignored. In contrast, the volatilized-oil content of gases from volatile oils is much greater. Their volatilized-oil/gas ratio typically ranges from 10 to 300 STB/MMscf or 0.4 to 8 gal/Mscf.

Several benchmark properties can be correlated with the reservoir fluid’s initial molecular weight. Fig. 9.1 plots the initial formation volume factor (FVF) and initial dissolved gas/oil ratio (GOR) as a function of reservoir-fluid molecular weight for 36 reservoir fluids. The abscissa in Fig. 9.1 spans from a molecular weight of 15 to 180. This range of molecular weights covers the full spectrum of petroleum fluids ranging from dry gases to heavy oils.

Volatile oils exhibit an initial oil FVF in the range of 1.5 to 3.0. Black oils exhibit an initial oil FVF in the range of 1.1 to 1.5. Volatile oils exhibit an initial GOR in the range of 900 to 3,500 scf/STB. Black oils exhibit an initial GOR in the range of 200 to 900 scf/STB. These relations establish molecular weight as a credible correlating parameter. McCain[2] has found success in the use of the heptanes-plus content as a correlating parameter.

The inverse of the oil FVF yields a measure of the original oil in place (OOIP) per unit volume of reservoir pore space. Because the oil FVF is greater for volatile oils than black oils, the latter yield greater OOIP per unit volume. Black-oil reservoirs contain 850 to 1130 STB/acre-ft (bulk) while volatile-oil reservoirs contain less, typically 400 to 850 STB/acre-ft.

Although volatile-oil reservoirs contain less oil per unit volume, they typically yield slightly higher oil recoveries than black-oil reservoirs because of their higher dissolved-gas content and lower oil viscosity. Ultimately, volatile-oil reservoirs may yield greater oil reserves than black-oil reservoirs. Light black oils and heavy volatile oils are among the most economically attractive reservoir fluids.

There has been no systematic study to determine the relative percentage of black-oil and volatile-oil reservoirs; however, an examination of the world’s 500 largest reservoirs reveals that black-oil reservoirs overwhelmingly dominate the group. [3] One reason there are more black-oil than volatile-oil reservoirs is that the latter are characteristically located at greater depths than the former. As exploration continues to go deeper, more volatile-oil reservoirs can be expected to be discovered.

Types of Reservoir Energy

The following list outlines the major types of energy available for petroleum production.

  • The energy of compression of the water and rock within the reservoir.
  • The energy of compression of oil within the reservoir.
  • The energy of compression of gas within the reservoir.
  • The energy of compression of waters contiguous to and in communication with the petroleum reservoir.
  • The gravitational energy that causes the oil and gas to segregate within the reservoir.

Water within the reservoir refers to the water that is originally present within the reservoir at the time of discovery. Oil within the reservoir refers to the oil phase that is originally present at discovery or that may form from the condensation of volatilized oil upon pressure release. Likewise, gas within the reservoir refers to the gas phase that is originally present at discovery or that may form subsequently from the liberation of dissolved gas upon pressure release.

As mechanisms of energy release are provided by the drilling and operation of wells, reservoir pressure declines, fluids expand, flow is induced, and fluids are produced. The net volume of expansion of rock and fluids within the reservoir results in an equal volume of expulsed fluids. The water-bearing reservoirs that adjoin petroleum reservoirs are called aquifers. The expansion of water from the aquifer results in an overflow of water from the aquifer into the petroleum reservoir. The net overflow of water into the petroleum reservoir, in turn, results in an equal volume of fluid expulsion from the petroleum reservoir. Gravity segregation does not directly result in fluid expulsion but causes oil to settle to the bottom and gas to migrate to the top of the reservoir. By producing from only the lower reaches of the reservoir, this process affords a skilled operator a means to recover oil selectively and possibly recover more oil than would otherwise be recovered.

In ranking the types of energy in order of least importance to oil recovery, the energy of the compressed water and rock originally within the reservoir is probably the least important because of the relatively low compressibilities of water and rock. Of equal unimportance is the energy of the compressed oil, although the effects of compressed oil are slightly greater than the effects of compressed water and rock, as evidenced by the slightly greater compressibility of oil (10–5 per psi) than water (3 × 10–6 per psi) and rock (6 × 10–6 per psi). Of the energies of the compressed fluids, the effects of compressed gas are undoubtedly the most important because of the greater compressibility of gas. The effects of compressed gas are important even if there is not much free gas initially present, as in the case of an initially undersaturated oil reservoir. In these cases, gas will appear naturally during the course of pressure depletion because of the release of dissolved gas from the oil once the pressure falls below the bubblepoint pressure.

Gravitational forces can be a major factor in oil recovery if the reservoir has sufficient vertical relief and vertical permeability. The effectiveness of gravitational forces will be limited by the rate at which fluids are withdrawn from the reservoir. If the rate of withdrawal is appreciably greater than the rate of fluid segregation, then the effects of gravitational forces will be minimized.

The energy from the compressed waters of aquifers also can be a major factor even though the water has a low compressibility because the size of most aquifers tends to be much larger than the petroleum reservoir. Most oil fields have areas of less than 10 sq mile (6,400 acres), whereas aquifers often have areas of more than 1,000 sq mile. [1]

The energies discussed thus far represent "internal" reservoir energies (i.e., energies originally present within the reservoir and its adjoining geological units at the time of discovery). In addition to these energies, there may be important "external" energies (i.e., energies that originate from outside the reservoir).External energies imply the practice of injecting fluids into the reservoir to augment the reservoir’s natural energies. This practice is called pressure maintenance. The two most important injection fluids are compressed water and gas. The resultant action of injected fluids once inside the reservoir is much the same as the fluids originally present. The overall intention of injecting fluids is to add energy to the reservoir to recover more oil or gas than would otherwise be recovered. If gas is injected, it is clear that the intention is to recover more oil than otherwise would be recovered. In addition, the economic attractiveness of this practice relies on the expectation that the additional income derived from the increased oil production will more than offset the additional expenditures and lost or deferred revenues incurred by gas injection. The most common source of gas for gas injection is the gas produced from the reservoir. The chapters on Water Injection and Immiscible Gas Injection in this volume discuss these subjects further.

Producing Mechanisms

The general performance characteristics of hydrocarbon-producing reservoirs are largely dependent on the types of energy available for moving the hydrocarbon fluids to the wellbore. The predominate energy forms give rise to distinct producing mechanisms. These producing mechanisms are used to help classify petroleum reservoirs.

In this section, these producing mechanisms are defined and delineated, although there is not a well-established consensus for some of these definitions. A petroleum reservoir rarely can be characterized throughout its pressure-depletion life by any single producing mechanism. A petroleum reservoir usually is subject to several producing mechanisms over its lifetime; nevertheless, the practice of describing a petroleum reservoir by its predominant producing mechanism is helpful.

Broadly, all commercially productive petroleum reservoirs are divided into either expansion-drive, compaction-drive, or waterdrive reservoirs. An expansion- or compaction-drive reservoir is a predominantly sealed reservoir in which the expansion of fluids and rock originally within the reservoir is responsible for petroleum expulsion from the reservoir. Fig. 9.2 shows the producing-mechanism system of classification.

In contrast, a waterdrive reservoir is an unsealed petroleum reservoir in communication with water-bearing reservoirs and in which there is appreciable movement of water from the water-bearing reservoir to the petroleum reservoir. If the rate of water intrusion into the reservoir is equal to the volumetric rate of fluid withdrawal from the reservoir, then the reservoir is more descriptively referred to as a complete-waterdrive reservoir. A complete-waterdrive reservoir often experiences, but does not necessarily imply, very little pressure decline. Complete-waterdrive reservoirs may require substantial pressure decline before the water-delivery rate can balance the production rate.

If the rate of water intrusion into the reservoir is substantial but substantially less than the volumetric rate of fluid withdrawal from the reservoir, then the reservoir is referred to as a partial-waterdrive reservoir. In all cases, when a waterdrive is the major producing mechanism, the reservoir pressure will be sensitive to the producing rate. If the reservoir-producing rate is too high relative to the water-influx rate, the waterdrive will lose its effectiveness and the reservoir pressure will decline.

Waterdrives are also classified as edgewater or bottomwater drives, depending on the nature and location of the water encroachment into the reservoir. Fig. 9.3 shows a schematic of a bottomwater-drive reservoir. Because waterdrive reservoirs experience increasing water content and decreasing hydrocarbon content, they are referred to as nonvolumetric reservoirs. More generally, nonvolumetric reservoirs are reservoirs in which hydrocarbon pore volume (PV) changes during pressure depletion. Conversely, volumetric reservoirs are reservoirs in which hydrocarbon PV does not change during pressure depletion. Because waterdrive reservoirs involve water influx into the reservoir, they also are referred to as water-influx reservoirs.

Pressure depletion causes the internal stress within the reservoir rock to increase. This change produces changes in the grain arrangement and other phenomena that ultimately cause the pore volume of the rock to decrease. The contraction of the reservoir pore volume aids in expelling fluids from the reservoir. The terms "pore-volume contraction" and "rock expansion" are used interchangeably in this chapter to describe this phenomenon, even though very little grain expansion usually takes place. If this phenomenon is a major producing mechanism, the reservoir is a compaction-drive reservoir. Compaction-drive reservoirs are rare because the PV compressibility is usually less than the oil compressibility.

Expansion-drive reservoirs are further classified as oil- or gas-expansion-drive reservoirs depending on whether the oil or gas expansion is the predominant producing mechanism. Dry- and wet-gas reservoirs are gas-expansion-drive reservoirs because they do not contain any free oil at reservoir conditions. More descriptively, a gas-drive reservoir is one in which the expansion of free gas is the predominant producing mechanism. The expanding free gas may originate as initial free gas or as dissolved gas. An oil-drive reservoir, on the other hand, is one in which the expansion of free oil is the predominant producing mechanism. [1] According to these definitions, black-oil and volatile-oil reservoirs are not likely to be oil-drive reservoirs but gas-drive reservoirs because the expansion of gas is ultimately much greater than the expansion of oil. The oil in saturated, black-oil and volatile-oil reservoirs does not expand but contracts during pressure depletion because of the release of dissolved gas. Because the overwhelming majority of expansion-drive reservoirs are gas-drive reservoirs, the term oil-drive reservoir is rarely used. An oil-drive producing mechanism dominates in oil reservoirs only while they are undersaturated.

Gas-drive reservoirs are further subdivided into either solution-gas-drive or gas-cap expansion-drive reservoirs. A gas-cap expansion-drive reservoir is a gas-cap reservoir in which the expanding gas cap is responsible for the majority of the gas expansion. A gas cap is a free-gas zone that overlies an oil zone. The free-gas zone may be pre-existing or may form during the depletion process. Pre-existing gas caps are called primary gas caps. Gas caps that are not originally present but that develop during the depletion process are called secondary or developed gas caps. Secondary gas caps can form from the upward migration of either liberated dissolved gas or from reinjected gas. Fig. 9.4 shows a schematic of a gas-cap expansion-drive reservoir.

Gas caps are also classified according to their displacement efficiency. At the most favorable extreme, the expanding gas displaces oil in a piston-like manner. At the other limit, the expanding gas displaces oil in a totally diffuse manner. The former are segregation-drive or gravity-drainage gas caps; the latter are nonsegregation-drive gas caps. The boundary between the gas-cap zone and oil zone is the gas/oil contact (GOC). Segregation-drive gas caps exhibit a GOC that moves progressively downward during depletion. In contrast, nonsegregation-drive gas caps exhibit a GOC that appears stationary. The gas-cap displacement efficiency depends on the producing rate and vertical permeability. Segregation-drive gas caps tend to have high vertical permeability, while nonsegregation-drive gas caps tend to have low vertical permeability. These two types of gas caps represent limiting cases. In reality, there is a continuum of character between these limits. The exact gas-cap character depends on the actual conditions.

Gas-drive reservoirs that are not gas-cap reservoirs but are dominated by the expansion of solution gas are called solution-gas-drive or dissolved-gas-drive reservoirs. Fig. 9.5 shows a schematic of a solution-gas-drive reservoir. Gas-drive reservoirs that are neither gas-cap nor solution-gas-drive reservoirs are called gas-drive reservoirs. For example, dry-gas reservoirs are gas-drive reservoirs because they do not qualify as solution-gas-drive or as gas-cap reservoirs. The practice of reinjecting dry gas into and producing wet gas from gas/condensate reservoirs is called gas cycling or cycling.

Recovery Ranges

Table 9.2 lists the approximate primary-recovery range for the different producing mechanisms. The ranges reflect the rank of the reservoir energies. Black-oil reservoirs that exclusively produce by solution-gas-drive mechanism typically recover 10 to 25% of the OOIP by pressure depletion. The American Petroleum Inst. reports an average primary oil recovery of 20.9% for 307 solution-gas-drive reservoirs. [4] In contrast, primary oil recovery from waterdrive, black-oil reservoirs typically ranges from 15 to 50% or higher of the OOIP. Waterdrive, black-oil reservoirs have yielded some of the highest recoveries ever recorded. The primary oil recovery from gas-cap, black-oil reservoirs varies widely depending on whether there is significant gravity drainage. The primary oil recovery from nongravity-drainage, gas-cap, black-oil reservoirs ranges from 15 to 40% of the OOIP. In contrast, the primary oil recovery from gravity-drainage, gas-cap, black-oil reservoirs ranges from 15 to 80% of the OOIP. Primary oil recoveries from gravity-drainage, black-oil reservoirs are among the highest of any black-oil reservoir. Pressure maintenance by gas reinjection is practiced commonly in black-oil reservoirs to improve oil recovery. Black-oil reservoirs subject to gas reinjection without gravity drainage typically recover 15 to 45% of the OOIP. If gas is reinjected in a reservoir with active gravity drainage, the primary oil recovery typically ranges from 15 to 80%.

Fluid Properties

Black and volatile oils, as well as other petroleum fluids, are characterized routinely in terms of their standard pressure/volume/temperature (PVT) parameters: oil FVF (Bo'), gas FVF (Bg''), dissolved GOR (Rs'), and volatilized oil/gas ratio (Rv''). These fluid properties, in addition to some others, are prerequisites for a wide variety of reservoir-engineering calculations, including estimating the OOIP and original gas in place (OGIP) and material-balance calculations.

Table 9.3 tabulates and Fig. 9.6 plots the standard PVT parameters as a function of pressure for a black oil from a west Texas reservoir located at a depth of 6,700 ft with an initial pressure of 3,100 psia and a temperature of 131°F. Only the PVT properties below 2,000 psia are listed. The fluid exhibited a bubblepoint at approximately 1,688 psia and had a molecular weight of 81. Table 9.4 summarizes its compositional analysis. The fluid has an initial oil FVF of 1.467 RB/STB and dissolved GOR of 838 scf/STB. The equilibrium gas contains negligible volatilized oil. Fig. 9.7 plots the oil and gas viscosities as a function of pressure.

Table 9.5 tabulates and Fig. 9.8 plots the standard PVT parameters for a volatile oil from a north-central Louisiana reservoir located at a depth of approximately 10,000 ft with an initial pressure of 5,070 psia and a temperature of 246°F. [5][6] The fluid exhibited a bubblepoint at approximately 4,677 psia and had a molecular weight of 47. Table 9.6 summarizes the initial fluid composition. The fluid has an initial oil FVF of 2.704 RB/STB and dissolved GOR of 2,909 scf/STB. The bubblepoint gas had a volatilized-oil/gas ratio of approximately 120 STB/MMscf. The volatilized-oil/gas ratio decreases with pressure until a pressure of 998 psia is reached. At pressures between 998 and 598 psia, the volatilized-oil/gas ratio increases slightly.

The standard PVT parameters of volatile and black oils are determined experimentally with different laboratory procedures. Black oils are evaluated with a differential-vaporization (DV) experiment; [7][8] in contrast, volatile oils are evaluated with constant volume depletion (CVD). [9][10] Sometimes, however, volatile oils use a specialized DV experiment[11] instead of a CVD experiment. The specialized DV experiment includes a step to measure the volatilized-oil content of equilibrium gases.

The standard PVT parameters for black oils are specified routinely in commercial PVT reports. McCain provides some example PVT reports. [12] The reported PVT parameters, however, may or may not be adjusted for the effects of surface separators. Surface separators maximize the stock-tank liquid yield as fluids pass through them. The oil FVF and dissolved GOR of adjusted properties are characteristically less than unadjusted properties. If the PVT report specifies the adjusted parameters, then no further adjustment is required. If only the raw parameters are specified, then adjustment is needed.

Various empirical methods are used to correct the standard PVT parameters for the effects of separators. [13][14][7] Generally, correction is very important. For example, the unadjusted bubblepoint oil FVF and the dissolved GOR for the example black oil in Table 9.3 are 1.584 RB/STB and 1,007 scf/STB, respectively. On adjustment for separators at 100 psia, the corresponding oil FVF and dissolved GOR are 1.467 RB/STB and 838.5 scf/STB, reflecting increased stock-tank-liquid recovery. Failure to correct the standard PVT parameters for separators can lead to substantial errors in subsequent reservoir-engineering calculations including the volumetric OOIP and OGIP calculations. Volatile oils are even more sensitive to the effects of separators than black oils. Volatile oils, however, are subjected to an entirely different laboratory procedure for measurement.

The standard PVT parameters for volatile oils rarely are given in commercial PVT reports. They must be calculated from CVD measurements. In order of increasing complexity, the three methods to calculate standard PVT parameters are the Walsh-Towler algorithm, [15] the Whitson-Torp method, [9] and the equation-of-state (EOS) method. [16][17] The Walsh-Towler algorithm uses recovery data directly from the CVD measurement and computes the corresponding properties. This method is suited for spreadsheet calculation and is fast and simple. The Whitson-Torp method, in contrast, uses equilibrium gas-composition data and computes the properties with Standing’s[18] low-pressure K- values and a stock-tank-liquid density correlation such as the Alani-Kennedy EOS. [19] This method requires iterative, K -value flash calculations. Although this method is more computationally intensive than the Walsh-Towler algorithm, it is more versatile because it allows for arbitrary separator conditions. The EOS method is much more computationally intensive than the other methods. This method tunes a cubic EOS to the attending phase behavior and then uses the EOS to simulate the CVD numerically and estimate the PVT parameters. This method regularly uses commercial software. The methods yield virtually identical results despite their differences.

Material Balance

The material-balance equation is the simplest expression of the conservation of mass in a reservoir. The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal. The applicable equation for initially saturated volatile- and black-oil reservoirs is[20][21][22][23]


where Gfgi, Nfoi, and W are the initial free gas, oil, and water in place, respectively; Gp, Np, and Wp are the cumulative produced gas, oil, and water, respectively; GI and WI are the cumulative injected gas and water, respectively; and Eg, Eo, Ew, and Ef are the gas, oil, water, and rock (formation) expansivities. Most of the equations in this chapter apply to any consistent set of units. A few equations, however, are written assuming English or customary units. Those equations are expressed in SI units in the Appendix.

Nfoi and Gfgi are related to the total OOIP and OGIP, N and G, according to N = Nfoi + Gfgi Rvi and G = Gfgi + Nfoi Rsi.

The expansivities are defined as




and RTENOTITLE, where B to and B tg are the two-phase FVFs,


and RTENOTITLE....................(9.6)

The rock expansivity is obtained from direct measurement. See Sec. 9.10 for a greater discussion.

Physically, the two-phase FVF is the total hydrocarbon volume per unit volume of oil or gas at standard conditions. The two-phase FVF mimics the observations noted during a constant-composition expansion test. For instance, the two-phase oil FVF is the total hydrocarbon (oil + gas) volume of a saturated oil sample per unit volume of oil at standard conditions. In contrast, the two-phase gas FVF is the total hydrocarbon volume of a saturated gas sample per unit volume of gas at standard conditions. Bto and Btg typically are expressed in units of RB/STB and RB/Mscf, respectively. For undersaturated oils, the two-phase oil FVF is equal to the oil FVF; for undersaturated gases, the two-phase gas FVF is equal to the gas FVF.

Eqs. 9.5 and 9.6 account for volatilized oil in the equilibrium gas phase. If volatilized oil is negligible, these equations are simplified. For instance, Bto = Bo + Bg (RsiRs) and Btg = Bg. These equations apply for black oils. Eq. 9.4 ignores dissolved gas in the aqueous phase.

Eq. 9.1 broadly states that net expansion equals net withdrawal. More specifically, it shows the different forms of expansion and withdrawal. The different forms of expansion such as gas expansion are responsible for the different producing mechanisms.

For the sake of simplicity, Eq. 9.1 is often written in the abbreviated form of


where F = total net fluid withdrawal or production, Egwf = composite gas expansivity, and Eowf = composite oil expansivities. F, Egwf, and Eowf are defined in



and RTENOTITLE....................(9.10)

The composite expansivities include the connate-water and rock expansivities. Eq. 9.8 includes Gps, which is the cumulative produced sales gas and is defined as (GpGI). F is expressed in reservoir volume units (e.g., RB or res m3), Egwf is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf), and Eowf is expressed in reservoir volume units per standard unit volume of oil (e.g., RB/STB).

For strictly undersaturated oil reservoirs, no free gas exists (i.e., Gfgi = 0) and the initial free oil in place is equal to the OOIP (i.e., Nfoi = N) and Eqs. 9.1 , 9.7, and 9.8 simplify, respectively, to[20][23][24]





Eqs. 9.11 through 9.13 ignore gas reinjection.

The material-balance equation also helps explain most oil-recovery strategies. If the material-balance equation is solved for the produced fraction of the original free oil in place, then


Eq. 9.14 succinctly shows that oil recovery increases with water influx (We), initial free-gas-cap volume (which is proportional to Gfgi), surface water injection (WI), and surface gas injection (by minimizing gas sales through Gps). It also shows that oil recovery increases by minimizing water production (Wp).

The material-balance equation and its many different forms have many uses including confirming the producing mechanism, estimating the OOIP and OGIP, estimating gas-cap sizes, estimating water-influx volumes, identifying water-influx model parameters, and estimating producing indices.

Solution-Gas Drives

Oil reservoirs that do not initially contain free gas but develop free gas on pressure depletion are classified as solution-gas drives. The solution-gas-drive mechanism applies once the pressure falls below the bubblepoint. Both black- and volatile-oil reservoirs are amenable to solution-gas drive. Other producing mechanisms may, and often do, augment the solution-gas drive. Solution-gas-drive reservoir performance is used as a benchmark to compare other producing mechanisms.

Stages of Production

Pure solution-gas-drive reservoirs are subject to four stages of idealized production. In chronological order, the four stages are: (1) production while undersaturated; (2) production while saturated but the free gas is immobile; (3) production while saturated and the free gas is mobile and the producing GOR is increasing; and (4) production while saturated and the free gas is mobile and the producing GOR is decreasing. Not all these stages are necessarily realized. For instance, Stage 4 may not be realized if primary recovery is terminated during Stage 3.

The key characteristics of each stage are outlined here.

Stage 1

  • No free gas.
  • Producing GOR is equal to initial dissolved GOR.
  • Fractional oil and gas recoveries are small and approximately equal.
  • Reservoir pressure drops rapidly.
  • Duration of stage depends on degree of initial undersaturation. The greater the initial undersaturation, the longer the duration of Stage 1. The stage duration is generally short.

Stage 2

  • Reservoir pressure is less than bubblepoint.
  • Free gas appears, but the saturation is small and immobile.
  • Producing GOR is slightly less than initial dissolved GOR.
  • Rate of pressure decline is mitigated.

Stage 3

  • Free gas becomes mobile.
  • Producing GOR increases steadily.
  • Fractional gas recovery exceeds fractional oil recovery.
  • Longest of all stages; typically consumes 85 to 95% of primary recovery.
  • Primary recovery may be terminated during this stage.

Stage 4

  • Reservoir pressure is very low, typically less than 100 to 400 psia.
  • Producing GOR decreases.
  • Primary recovery often terminated before this stage is realized.

This chronology and these characteristics are an idealization and oversimplification of actual behavior; nevertheless, they are instructive and provide a preliminary basis for understanding scenarios that are more complicated.


To illustrate solution-gas-drive performance, tank model predictions of a west Texas black-oil reservoir are presented. Though idealized, these simulations, which are from a commercial simulator, [25][26] capture the main features and establish the theory of solution-gas drives.

For the sake of simplicity, the simulations consider the depletion of only a single well in an 80-acre closed area. Table 9.7 summarizes some of the reservoir and fluid properties. The simulations assume that the PVT parameters in Table 9.3 apply.

Fig. 9.9 shows the performance in terms of pressure, producing GOR, and gas saturation as a function of cumulative produced oil. The four stages of production are depicted. During Stage 1, less than 1% of the OOIP is produced. The pressure decreases from the initial pressure of 2,000 psia to the bubblepoint pressure of 1,688 psia. The producing GOR remains constant and equal to the initial dissolved GOR of 838 scf/STB; no free gas evolves.

During Stage 2, the pressure falls below the bubblepoint; solution gas is liberated; and low, immobile gas saturations form. The cumulative oil recovery reaches approximately 4.5% of the OOIP. The pressure decreases from the bubblepoint to approximately 1,550 psia. The gas saturation increases to approximately 5% PV. The producing GOR actually decreases slightly, but this change is barely noticeable.

During Stage 3, the gas saturation increases to the point at which gas is mobile. Free-gas production begins, and the producing GOR rises steadily. By the end of Stage 3, the cumulative oil recovery is 28% of the OOIP, the pressure has decreased to 200 psia, the gas saturation reaches approximately 35% PV, and the producing GOR reaches approximately 6,700 scf/STB.

During Stage 4, the pressure has reached such a low level that the expansion of gas from reservoir to surface conditions is minimal. Consequently, the producing GOR decreases. By the time the pressure reaches 50 psia, the GOR is only 2,000 scf/STB and the total oil recovery is 32% of the OOIP.

Fig. 9.10 shows reservoir performance as a function of time. This figure plots the pressure, instantaneous producing GOR, cumulative producing GOR, gas saturation, oil rate, gas rate, and fraction of OOIP and OGIP recovered as a function of time. Stage 1 is very short and lasts less than one month. The oil and gas producing rates and pressure decline sharply. The producing rates decline if the bottomhole pressure (BHP) is restricted. The producing rates can remain constant, but only if the minimum BHP is not yet reached.

Stage 2 also is relatively brief, lasting only several months. The reservoir pressure and producing rates also decline sharply but not as quickly as during Stage 1. The decline rate dampens because solution gas is liberated. The producing rates decline if the BHP is restricted. Constant producing rates can be realized only if the minimum BHP is not yet reached.

Stage 3 starts before one year of depletion and continues until the economic limit is reached. In this example, the limit is reached after 13.5 years when the oil-producing rate reaches 20 STB/D. The length of pressure depletion depends strongly on the reservoir permeability and on the prevailing economic conditions. For instance, lower permeabilities will decelerate recovery and protract depletion. The final pressure is 613 psia. This pressure is not low enough to realize Stage 4; therefore, this stage of depletion is not portrayed in Fig. 9.10. The absence of Stage 4 in field cases is not uncommon. The marked increase in the GOR from 838 to 4,506 scf/STB during Stage 3 coincides with marked increase in the gas saturation from 5 to 28.7% PV. At the economic limit, these simulations predict final oil and gas recoveries of 24.2% of the OOIP and 53.1% of the OGIP. Stage 3 clearly dominates the depletion life of a solution-gas-drive reservoir.

The results of this simulation are an oversimplification and idealization of actual performance. Oversimplification stems from the fact that the tank model ignores many important secondary phenomena. For example, the simulations ignore reservoir heterogeneity, which can be expected to reduce the recoveries by approximately 20 to 50% depending on the degree of heterogeneity. For instance, if a volumetric recovery efficiency of 80% is applied, then the idealized oil recovery of 24.2% corresponds to an adjusted oil recovery of 19.4%. Also, the simulations ignore spatial effects.

Qualitatively, solution-gas-drive, volatile-oil reservoirs act very similarly to their black-oil counterparts. One pronounced quantitative difference, however, is that volatile-oil reservoirs exhibit much greater peak producing GORs. The field example in Sec. 9.7.3 illustrates this difference. This example considers a volatile-oil reservoir that exhibits a peak GOR of approximately 29,000 to 32,000 scf/STB. This GOR is considerably greater than the peak GOR for the example black oil of 6,700 scf/STB. Another difference between volatile- and black-oil reservoirs is that the former often exhibit slightly greater oil recoveries; however, there are numerous exceptions to this trend.

Field Example

Cordell and Ebert[5] report the performance of a volatile-oil reservoir located in north-central Louisiana. Table 9.8 summarizes some of the pertinent reservoir data. This reservoir produced from the Smackover lime located at an approximate depth of 10,000 ft. The field was discovered in 1953 and was developed with 11 wells on 160-acre spacing. Jacoby and Berry[6] report on the fluid properties of this volatile oil. The standard PVT parameters in Table 9.5, which were developed from laboratory data with the EOS method, are applicable.

Table 9.9 summarizes the reservoir performance in terms of the cumulative oil production, cumulative gas production, and instantaneous producing GOR as a function of pressure. Table 9.9 includes the oil and gas recoveries as a percent of the OOIP and OGIP. These recoveries were based on the volumetric OOIP and OGIP estimates of 10.7 million STB and 31.1 Bscf, respectively.

Fig. 9.11 shows the reservoir performance in terms of pressure and producing GOR as a function of cumulative oil recovery. This figure qualitatively agrees with the theoretical results for the black oil in Fig. 9.9. A comparison confirms that volatile-oil reservoirs experience much higher producing GORs. For example, this volatile-oil reservoir reaches a peak GOR of 29,000 scf/STB (compared with 6,700 scf/STB for the black oil). This reservoir was produced to a low enough pressure to exhibit Stage 4, which is the period of declining GOR. The GOR began to decline at a reservoir pressure of approximately 800 psia.

Material-Balance Analysis

A material-balance analysis is performed routinely to confirm the suspected producing mechanism and to estimate the OOIP independently. The applicable material-balance equation for a solution-gas-drive reservoir is[20][23][24][27]


Eq. 9.15 is a simplification of Eq. 9.7 and assumes no initial free gas (Gfgi = 0). Because there is no initial free gas, Nfoi = N. If free gas is present initially, the material-balance methods for gas-cap reservoirs should be applied (see Section 9.8.2). Eq. 9.15 also applies to waterdrives; however, if the following methods are applied to waterdrives, the water-influx history must be reliably known. If the water-influx history is not known, then the methods in Section 9.9.8 regarding waterdrives must be applied. If there is no water influx, then We = 0.

If a reservoir produces exclusively by solution-gas drive with only supplemental connate-water expansion and pore-volume contraction, then Eq. 9.15 dictates that a plot of F vs. Eowf is a straight line, emanates from the origin, and has a slope equal to N. This observation is used to confirm the producing mechanism. If water influx exists and if We is known, then an F-vs.-Eowf plot is replaced by a (FWe)-vs.-Eowf plot. Fig. 9.12 shows a (FWe)-vs.-Eowf plot for a volatile-oil reservoir. Once the OOIP is determined, the OGIP is given by G = RsiN. If an F-vs.-Eowf plot is not a straight line, then another producing mechanism, such as a waterdrive or an initial gas cap, exists. The shape of the nonlinearity is important in diagnosing the true producing mechanisms. For instance, if the F-vs.-Eowf plot curves upward, this suggests that a waterdrive or an initial gas cap exists. Fig. 9.13 shows the effect of water influx or an initial gas cap on an F-vs.-Eowf plot.

The number of data points in an F-vs.-Eowf plot is usually limited by the number of average-reservoir-pressure measurements. Recall that F and Eowf are functions of pressure by means of the standard PVT parameters. If two or more data points (other than the origin) exist, then a mathematical criterion must be adopted to determine the "best" line though the data or the "best" estimate of N. If a least-squares criterion is adopted, then the OOIP estimate is[28]


where subscript j denotes the value at pressure pj and n is the total number of data points. Eq. 9.16 offers a strictly mathematical means to estimate the OOIP without constructing an F-vs.-Eowf plot. In general, however, a plot is recommended because it provides a visual means to assess the scatter of the data. The straightness of the data points is a measure of material balance and confirmation of the solution-gas-drive mechanism.

The composite expansivity Eowf implicitly includes and accounts for rock and connate-water expansion. Thus, the methods offered here are applicable equally to reservoirs in which rock and connate-water expansion are important. In practice, rock and connate-water expansion cannot be neglected unless the reservoir is saturated and the pressure is less than approximately 1,500 psia. These phenomena cannot be neglected while the reservoir is undersaturated because their combined effects are not negligible compared to oil expansion. For instance, the relative expansion of oil, rock, and water in an undersaturated west Texas black-oil reservoir was 72, 25, and 3%, respectively. This example also demonstrates that the connate-water expansion is normally insignificant and can be ignored. Not until the pressure falls below the bubblepoint and approximately 1,500 psia will the rock expansion be negligible compared to the net hydrocarbon expansion. If doubt persists as to whether it is safe to ignore rock and connate-water expansion, the safest approach is to include them. To include these phenomena, the rock and connate-water expansivities, Ef and Ew, must be calculated. Sec. 9.10 discusses experimental and empirical methods to estimate Ef. The connate-water expansivity is calculated from Eq. 9.4. This equation ignores dissolved hydrocarbon gases in the water. To include dissolved gases, the water expansivity is calculated from


where Btw is the two-phase water FVF and is given by


where Rsw is the dissolved gas/water ratio. [27]

Two common errors occur when applying a material-balance analysis to volatile-oil reservoirs. First, an incorrect set of PVT parameters is used. This occurs if the volatile oil is subjected to a conventional DV test instead of a CVD or a specialized DV experiment that measures volatilized oil. The resulting set of PVT parameters will not reflect the true phase behavior. If this mistake occurs, the volatilized oil/gas ratio, Rv, will be omitted altogether and the resulting values of Bo and Rs will be erroneous and overestimated. Significant errors in these fluid properties will occur if appreciable volatilized oil exists. For example, the volatile oil in Table 9.5 yielded an erroneous initial oil FVF of 3.379 RB/STB and a dissolved GOR of 3,636 scf/STB (errors of approximately 25%) when it was subjected to a standard DV instead of a CVD. The second error commonly occurs if the conventional or black-oil material-balance equation[29][30] is applied instead of the generalized equation in Eq. 9.1. The conventional material balance inherently ignores Rv. Both of these errors will cause the OOIP to be underestimated, which can be quite serious if the volatilized-oil content is appreciable.

Example 9.1: Material-Balance Analysis of a Volatile-Oil Reservoir Perform a material-balance analysis on the Louisiana volatile-oil reservoir in Sec. 9.7.3. Use the production data in Table 9.9 and the PVT data in Table 9.5 as necessary. Estimate the OOIP (million STB) and confirm the suspected solution-gas-drive producing mechanism if possible. Compare your OOIP estimate to the volumetric estimate of 10.7 million STB reported by Cordell and Ebert. [5]

Solution. To confirm the producing mechanism and estimate the OOIP, construct an F-vs.-Eowf plot. Because the lower pressure in Table 9.9 is less than 1,500 psia and below the bubblepoint, connate-water expansion and pore-volume contraction can be ignored. Thus, Eowf can be replaced by Eo, where Eo = BtoBoi and Bto is given by Eq. 9.5. Table 9.10 tabulates the results of Bto and Eo as a function of pressure. For example, at p = 4,398 psia, evaluating Eqs. 9.5 and 9.2 yields


and Eo = 2.864 – 2.704 = 0.160 RB/STB.

F is given by Eq. 9.8 when the reservoir is saturated and by Eq. 9.13 when the reservoir is undersaturated. For example, at p = 4,398 psia, evaluating Eq. 9.8 yields


Table 9.10 tabulates the results at other pressures and the cumulative GOR, Rps = Gps/Np.

Fig. 9.12 shows a plot of F vs. Eo. The slope of this plot is 10.2 million STB, which is an estimate of the OOIP. This estimate agrees closely with the volumetric estimate of 10.7 million STB. The agreement of the volumetric and material-balance OOIP estimates, together with the straightness of the F-vs.-Eo plot, is strong evidence that this reservoir is producing exclusively by a solution-gas-drive mechanism.

If the volatilized oil was ignored and standard PVT parameters based on a conventional DV test were used, the material balance would yield an OOIP estimate of 8.2 million STB, or an error of 23%. Alternatively, if the volatilized oil was ignored and the conventional (black oil) material-balance equation were used instead of the generalized equation defined by Eqs. 9.7 through 9.10 , the material balance would yield an OOIP of 9.09 million STB, or an error of 15%.

Gas-Cap Drives

In some instances, oil reservoirs are discovered with a segregated-gas zone overlying an oil column. The overlying gas zone is referred to as a primary gas cap. In addition to free gas, gas caps usually contain connate water and residual oil. The underlying oil column is sometimes referred to as an oil leg. In other instances, as reservoir pressure declines with production, gas evolves in the reservoir (see Sec. 9.7) and migrates to the top of the structure to add to an existing primary gas cap or to form a gas cap. If properly harnessed, gas caps can enhance oil recovery considerably. The degree with which they improve recovery depends mainly on their size and on the vertical permeability and/or formation dip. Producing wells usually are completed only in the oil leg to minimize gas production. Broadly, gas caps are classified as segregating or nonsegregating. Table 9.11 summarizes the distinguishing characteristics of each.

Segregating gas caps are gas caps that grow and form an enlarged gas-cap zone. Fig. 9.4 shows a schematic of a segregation-drive reservoir. Two different segregation mechanisms are possible: (1) expansion of and frontal displacement by pre-existing gas-cap gas and (2) upward migration of oil-column gas as solution gas is liberated and a free-gas phase forms. The second mechanism involves the simultaneous downward movement of oil to balance the upward flow of gas. This diametric flow pattern is referred to as counterflow. [31] Pirson[32] refers to the first mechanism as passive segregation and the latter mechanism as active segregation. Hall[31] refers to the first mechanism as segregation drive without counterflow and the second mechanism as segregation drive with counterflow. Both mechanisms are time-dependent, and their displacement efficiency depends on the gas/oil density difference, the producing rate, and the vertical permeability.

Both segregation mechanisms yield a progressively descending GOC. The segregation-drive mechanisms can be augmented by crestal gas injection.

If neither of these segregation mechanisms is present, the gas cap is called a nonsegregating gas cap. Nonsegregating gas caps do not form an enlarged gas-cap zone, and their GOC appears stationary. The gas-cap gas expands but the displacement efficiency is so poor that the expanding gas appears to merely diffuse into the oil column. Fig. 9.14 illustrates the distribution of water, oil, and gas in a nonsegregation-drive gas-cap reservoir.

Broadly, gas caps act to mitigate the pressure decline, extend the life of the reservoir, and ultimately improve the oil recovery. The degree of oil-recovery improvement depends on the size of the gas cap and whether it is a segregation-drive or nonsegregation-drive gas cap.

To understand the mechanics of gas-cap reservoirs, numerical simulation results of segregating and nonsegregating gas caps are presented. Each example uses the fluid-property data in Table 9.3. Each example also uses the reservoir data summarized in Table 9.7, except that the initial pressure is 1,640 psia instead of 2,000 psia and the gas-cap thickness is 10 ft. The gas, oil, and water saturations in the gas cap are 60, 20, and 20%, respectively. The gas cap initially contains 270,000 STB of oil and 816 MMscf of gas; the oil leg initially contains 210 million STB of oil and 1.718 Bscf of gas. The total OOIP = 2.37 million STB, and OGIP = 2.534 Bscf, and m = 0.33. For reference, the segregating- and nonsegregating-gas-cap cases are compared with an identical reservoir without a gas cap (base case).


Non-Segregation-Drive Gas Caps. Fig. 9.15 plots pressure as a function of cumulative oil recovery for a nonsegregation-drive gas-cap reservoir. For comparison, this figure includes the results of the no-gas-cap (base) case. This figure also includes the results of other cases, which are discussed later in this section. All recoveries are reported as a fraction of the original oil-leg OOIP to make direct comparisons valid. The nonsegregation-drive gas-cap case consistently yields higher oil recoveries at a given pressure than the no-gas-cap case, which illustrates the superior recovery performance of gas caps. Viewed another way, the nonsegregation-drive gas-cap case consistently yields a higher pressure at a given oil recovery than the no-gas-cap case, which illustrates the superior pressure-maintenance ability of gas caps.

Fig. 9.16 is a composite figure and shows the performance as a function of time. This figure includes the GOR, gas saturation, oil rate, and oil-recovery histories. The GOR history shows that nonsegregating gas caps eventually yield higher producing GORs than the no-gas-cap reservoir. The higher GOR is caused by higher gas saturation in the oil leg. The higher gas saturation is caused by the gas-cap gas migrating from the gas cap into the oil leg as the pressure declines.

Fig. 9.16 also shows the effect of a nonsegregating gas cap on the oil-rate history. The nonsegregating gas cap consistently yields higher oil rates than without the gas cap. If an economic limit corresponding to a minimum oil rate of 20 STB/D is arbitrarily assumed, then the no-gas-cap case is terminated after 13.8 years while the nonsegregation-drive gas-cap case is terminated after 15.2 years. This comparison shows that the presence of a gas cap extends the primary-recovery life of the reservoir. The curve endpoints denote the time of termination. The no-gas-cap case is slightly different from the black oil case discussed earlier due to the assumed lower original pressure.

Fig. 9.16 includes the fractional oil-recovery history; Fig. 9.17 shows the gas-recovery history. The curve endpoints denote the time of the economic limit. Table 9.12 summarizes the conditions at the economic limit. The no-gas-cap and nonsegregation-drive gas-cap cases recover 23.7% and 26.8% of the oil-leg OOIP, respectively. Thus, the nonsegregating gas cap recovers more oil than without the gas cap. The nonsegregating gas cap also is terminated at a higher pressure, producing GOR, gas saturation, and gas rate than without the gas cap. The nonsegregating gas cap recovers 74.9% of the oil-leg OGIP while the no-gas-cap case recovers 52.3% of the oil-leg OGIP. The nonsegregating gas cap recovers more gas because some of the gas-cap gas infiltrates the oil leg and is produced. In conclusion, the presence of a nonsegregating gas cap yields higher ultimate oil and gas recoveries, accelerates recovery, and extends the primary-recovery life of a reservoir.

The effect of a gas cap on oil recovery is related directly to its size relative to the size of the oil leg. The size of the gas cap is described effectively in terms of the dimensionless variable m, which is defined as the ratio of the initial free-gas and free-oil phase volumes (see Eq. 9.23). If all the free gas is located in the gas cap, all the free oil is located in the oil leg, and the oil-leg and gas-cap porosities and connate-water saturations are the same, then m represents the ratio of the gas-cap and oil-leg PV. Fig. 9.18 shows the effect of m on the final fractional oil recovery for an nonexpanding-gas-cap reservoir. The results in Fig. 9.18 use the same reservoir data as in the previous simulations except different gas-cap sizes are considered. Other reservoir conditions may yield slightly different results. The most noticeable improvement in oil recovery comes as m increases from 0 to 2.0.

The gas-cap size also affects the peak GOR. As the gas cap increases, the peak GOR increases. Fig. 9.18 shows the peak GOR as a function of m for the west Texas reservoir properties. The peak GOR increases with the gas-cap size because more gas-cap gas migrates into the oil column as the gas cap increases. In summary, nonsegregation-drive gas-cap reservoirs tend to yield final fractional oil recoveries in the range of 15 to 40% of the OOIP. Segregation-drive gas-cap reservoirs tend to yield even higher final oil recoveries.

Segregation-Drive Gas Caps. Segregating gas caps are characterized by progressively descending GOCs. The movement of the GOC is caused by active or passive gravity segregation. Active gravity segregation is the simultaneous migration of gas upward and drainage of oil downward. Passive segregation is the natural expansion of the gas-cap gas. Both of these processes involve frontal displacement of oil at the GOC. Frontal displacement helps drive oil to the producing wells. Frontal displacement does not dominate in nonsegregation-drive gas-cap reservoirs. The extent to which gravity segregation occurs depends on the vertical permeability and the rate at which fluids are withdrawn from the reservoir. The greater the vertical permeability and slower the fluid withdrawal, the more pronounced the effects of gravity segregation.

Figs. 9.15 through 9.17 include simulation results of a segregation-drive gas-cap reservoir. These simulations assume properties identical to those of the nonsegregation-drive gas-cap simulations except gravity segregation is included. The simulations assume no free-gas production from the gas cap.

Fig. 9.15 shows the pressure as a function of cumulative oil recovery. This figure shows that oil recovery in a segregation-drive gas-cap reservoir at a given pressure is consistently greater than that in a nonsegregation-drive gas-cap or nongas-cap reservoir, especially at low pressures when the effects of gas expansion become pronounced. The oil recovery performance is discussed below.

Fig. 9.16 shows the effect of a segregating gas cap on the GOR history. Only a marginal increase in the GOR is noted; after 15 years, the GOR actually decreases slightly. This type of GOR behavior is characteristic of segregation-drive gas-cap reservoirs. [1][33][34] The segregating gas cap effectively drives and concentrates oil into the shrinking oil leg. The oil leg shrinks as the GOC descends; thus, the segregating gas cap minimizes the gas saturation in the oil leg. The GOR reversal coincides with a reversal in the gas saturation. Fig. 9.16 includes the gas saturation history. The gas saturation steadily increases until it peaks at approximately 0.25 PV; then it decreases. The GOR and gas saturation reversals occur at a moderate to low pressure when the expansion of the gas-cap gas becomes pronounced. The change in the position of the GOC yields a measure of the oil-leg shrinkage. At termination, the GOC has descended approximately 9.3 ft into the original 20-ft oil column.

Fig. 9.16 includes the oil-rate history. The oil rate for the segregating gas cap is consistently higher than for the nonsegregating gas cap or without the gas cap. The oil rate eventually flattens out to between 20 and 50 STB/D and stays within this range for 15 to 31 years. This moderate but steady oil rate explains the superior performance and long life of segregation-drive gas-cap reservoirs. Table 9.12 summarizes and compares the primary-recovery lifetimes of the various cases: the segregating gas cap has a life of 31.3 years; the nonsegregating gas cap has a life of 15.2 years; and the solution-gas drive (base case) has a life of 13.8 years.

Fig. 9.16 includes the cumulative-oil-recovery history. The segregating-gas-cap reservoir recovers 38.7% of the oil-leg OOIP while the nonsegregating-gas-cap and solution-gas-drive reservoirs recover 26.8 and 23.7% of the OOIP, respectively. Such a high recovery level for a segregation-drive reservoir is not uncommon. It is not uncommon for gravity-drainage reservoirs to realize recoveries as high as 60 to 70% of the OOIP; however, they generally require a long time to do so. The curve endpoints in Fig. 9.16 denote the time of the economic limit. The segregating-gas-cap reservoir terminates at a pressure of 508 psia.

Fig. 9.17 shows the gas-recovery history. The segregating-gas-cap reservoir recovers 91.1% of the oil-leg OGIP. This recovery level is considerably greater than the nonsegregating-gas-cap or solution-gas-drive reservoirs (74.9 and 52.3%, respectively). One reason segregating-gas-cap reservoirs tend to yield such high gas recoveries is that they often recover some of the original gas-cap gas, which migrates into the oil leg. In addition, they generally realize lower termination pressures.

The final fractional oil recovery in a segregating-gas-cap reservoir is a strong function of the vertical communication within the reservoir. Vertical communication dictates the extent of segregation. If vertical communication is good, then most of the gas-cap gas will be available for segregation. It will also be available to help drive oil through frontal displacement to the producing wells. If vertical communication is poor, then very little, if any, of the gas-cap gas will segregate. In summary, segregation is controlled principally by three variables: the vertical reservoir permeability, the producing rate, and well spacing. As well spacing and vertical permeability increase and as the producing rate decreases, the effect of gravity segregation increases. For the effects of gravity segregation to be important, however, the well spacing may need to be prohibitively large or the producing rate may need to be prohibitively low. In such reservoirs, the vertical permeability is not high enough to permit much gravity segregation.

The likely role of gravity segregation can be measured in terms of a gravity number, Ng. Ng is defined as the ratio of the time it takes a fluid to move from the drainage radius to the wellbore to the time it takes a fluid to move from the bottom of the reservoir to the top. In oilfield units, the gravity number is


where kv = vertical permeability, md; Δρ = density difference, lbm/ft3; re = drainage radius, ft; q = producing rate at reservoir conditions, RB/D; and μo = oil viscosity, cp.

Gravity segregation is likely pronounced if Ng > 10; gravity segregation is likely unimportant if Ng < 0.10.For example, if kv = 10 md, Δρ = 50 lbm/ft3, re = 930 ft, q = 500 RB/D, and μo = 1 cp, then Ng = 21.3 and the effects of gravity segregation are likely important. If the vertical permeability is kv = 0.10 md instead of 10.0 md, then Ng = 0.21 and the effects of gravity segregation are relatively unimportant.

Gas Reinjection. One method for improving oil recovery is to reinject a portion of the produced gas. The reinjected gas helps maintain reservoir pressure. One obvious drawback of gas reinjection is that gas sales revenues are reduced or delayed. The overall intention of gas reinjection is to increase the net profit despite lower gas sales. When there is no sales outlet for produced gas, reinjection can improve oil recovery until a sales outlet is established. Regulations may require reinjection until sales are possible. Inert gases such as nitrogen or carbon dioxide also could be used to supplement or replace natural-gas reinjection.

Figs. 9.15 through 9.17 present simulation results of a gas-reinjection scenario. In this scenario, 70% of the produced wellhead gas is reinjected into the gas cap, and the gas cap is nonsegregating. This means that only 30% of the produced wellhead gas is available for sales. The non-reinjected gas is referred to as sales gas. This term is sometimes a misnomer because not all of the non-reinjected gas is necessarily sold. In practice, some of the sales gas is used routinely as fuel for power or utility requirements.

Fig. 9.15 shows the effect of gas reinjection on pressure as a function of oil recovery. Oil recovery at a given pressure is consistently higher for the gas-reinjection case than for the other cases in Fig. 9.15, except at very low pressures at which the segregating-gas-cap case yields superior performance. Gas reinjection leads to higher oil recoveries because the compressed reinjected gas effectively adds extra energy to the reservoir.

Fig. 9.16 shows the effect of gas reinjection on the GOR history. Gas reinjection leads to very high producing GORs, significantly higher than the other cases. The GOR is higher because the gas saturation is higher. The gas saturation is higher because reinjected gas and initial gas-cap gas migrate into the oil leg during pressure depletion. This occurs because the gas cap is nonsegregating. High producing GORs are a characteristic feature of reservoirs subject to gas reinjection if there is little or no active gravity drainage. High producing GORs mean that large volumes of produced gas will have to be handled and processed at the surface.

Fig. 9.16 includes the effect of gas reinjection on the oil-rate history. This figure shows that the oil rate is higher for the first 8 1/2 years for the gas-reinjection case than for any of the other cases. After 8 1/2 years, the oil rate for the segregating-gas-cap case is slightly greater than the oil rate for the gas-reinjection case. These results demonstrate that gas reinjection is an effective means to arrest the normal oil-rate decline dramatically.

Fig. 9.16 also shows the effect of gas reinjection on the fractional oil-recovery history and that the gas-reinjection case is superior to the other cases. The gas-reinjection case recovers 36.7% of the original oil-leg OOIP at its economic limit of 18 1/2 years. Only the segregating-gas-cap reservoir recovers more oil (38.7%); however, the segregating-gas-cap reservoir requires more time to recover the additional oil.

Fig. 9.17 shows the effect of gas reinjection on the fractional gas-recovery history. The fractional gas recovery is the cumulative produced wellhead gas normalized by the original oil-leg OGIP. The gas-reinjection case recovers 177% of the oil-leg OGIP (see Table 9.12). More than 100% of the oil-leg OGIP is produced because some of the reinjected gas is produced. Because 30% of the produced gas is not reinjected, 0.30 × 177 or 53.1% of the oil-leg OGIP is available for gas sales. This sales-gas recovery is comparable to the case without gas reinjection (52% OGIP).

Reservoirs subject to gravity drainage are especially attractive for gas reinjection. Crestal gas injection into the developing gas cap is the preferred strategy because gravity drainage helps control the movement of the injected gas. Excellent sweep and displacement efficiencies and high oil recoveries can be realized. The Tensleep pool in the Elk Basin field in Wyoming is a good example. [35][36][37] This pool was projected to recover approximately 64% of the OOIP. See the chapter on Immiscible Gas Injection in Oil Reservoirs in this volume of the Handbook for more information on gravity drainage.

Material-Balance Analysis

The purpose of a material-balance analysis includes confirming the producing mechanism and estimating the OOIP, OGIP, and size of the gas cap. The applicable material-balance equation for initially saturated oil reservoirs is[20][22][23]


This equation is applicable to all initially saturated reservoirs regardless of the distribution of the initial free gas. For example, this equation is applicable to reservoirs whether the initial free gas is segregated into a gas cap or uniformly dispersed throughout the reservoir. Eq. 9.20 also applies to waterdrives; however, if the following methods are applied to waterdrives, the water-influx history must be reliably known. If the water-influx history is unknown, then the methods Section 9.9.8 must be applied.

The quantities Gfgi and Nfoi are related to N (OOIP) and G (OGIP) by the following equations:


and RTENOTITLE....................(9.22)

where m is the ratio of the free-gas-phase and free-oil-phase volumes and is defined by


The dimensionless variable m is sometimes called the dimensionless gas-cap volume.

Because Gfgi and Nfoi are independent, they must be determined simultaneously. At least two sets of the independent variables (F, We, Egwf, Eowf) must be known at two or more pressures (other than the initial pressure) to determine the set (Gfgi, Nfoi). If three or more sets (F, We, Egwf, Eowf) are known, then multiple sets (Gfgi, Nfoi) can be determined. The optimal set is determined by one of two least-squares solution techniques: iterative or direct methods.

In the iterative method, Eq. 9.20 is expressed as


where Et is the total expansivity expressed per unit volume of stock-tank oil and is defined by


The solution procedure to estimate the OOIP and OGIP involves the following steps:

  1. Compute F, Egwf, and Eowf for each data point (i.e., average reservoir-pressure measurement).
  2. Guess m.
  3. Compute Et(m) with Eq. 9.25.
  4. Estimate Nfoi with a least-squares analysis using Eq. 9.26.
    where j denotes the data point index and n is the total number of data points.
  5. Compute the residual R for each data point with
  6. Compute sum of the squares of residual, Rss, as
  7. Return to Step 2 and repeat until Rss is minimized.
  8. Compute G, N, and Gfgi from Eqs. 9.21 through 9.23.

Minimization algorithms speed solution. This procedure is ideally suited for spreadsheet calculation, especially spreadsheet programs that contain minimization algorithms.

The use of Eq. 9.26 in Step 4 to determine Nfoi is equivalent to the slope of a (FWe)-vs.-Et(m) plot. This graphical solution method can be substituted for Eq. 9.26 in Step 4 if desired. Overall, Steps 2 through 7 are equivalent to the graphical procedure of varying m until the straightest possible (FWe)-vs.-Et(m) plot is realized. [38] Fig. 9.19 shows the qualitative effect of m on the shape of the (FWe)-vs.-Et plot. If m is too small, the plot curves upward slightly; if m is too large, the plot curves downward slightly.

Once m is determined, the final (FWe)-vs.-Et plot is used to confirm the producing mechanism. The linearity of the plot is a measure of material balance and the applicability of the presumed producing mechanism. If the plot exhibits considerable curvature, then either the presumed mechanism is incorrect or additional producing mechanisms are active. If curvature exists, the shape of the curvature provides insight into the true producing mechanism. For instance, if the plot curves upward, this indicates that net withdrawal exceeds net expansion and that water influx, for example, has been ignored or is possibly underestimated.

As an alternative to the iterative method, Walsh[23][28] presented a direct method. This method is based on least-squares multivariate regression. The least-squares equations are simple but lengthy. The technique is ideally suited for spreadsheet calculation. Walsh’s method is especially attractive because it avoids iteration and the complications of attaining and judging convergence.

Havlena and Odeh[38] proposed another solution method in which (FWe)/Eowf is plotted vs. (Egwf /Eowf); the slope of the plot is equal to Gfgi and the y-intercept is equal to Nfoi. This method is popular and attractive because it yields a direct solution. In theory, this method is perfectly acceptable. In practice, however, it has shown to be unreliable because it suffers from hypersensitivity to pressure uncertainty. [28][39] The method has been shown to yield highly erroneous Gfgi and Nfoi estimates in the presence of only small amounts of uncertainty. For instance, Walsh[28] shows that only a 5-psi pressure uncertainty yielded an error of more than 150% in Nfoi and an error of more than 250% in Gfgi. The hypersensitivity is caused by the fact that the divisor (Eowf) approaches zero as the pressure approaches the initial pressure. Small errors in Eowf, in turn, produce large errors in the quotients (FWe)/Eowf and (Egwf /Eowf). Tehrani[40] calls this problem a "loss in resolving power." Because of this hypersensitivity, this method should be used cautiously.

Walsh[28] tested the direct and iterative methods for their tolerance to uncertainty. He observed sensitivity, but the degree of sensitivity was less than the method of plotting (FWe)/Eowf vs. (Egwf /Eowf). He concluded that material-balance methods for gas-cap reservoirs should be used cautiously.


Waterdrive petroleum reservoirs are characteristically bounded by and in communication with aquifers. As pressure decreases during pressure depletion, the compressed waters within the aquifers expand and overflow into the petroleum reservoir. The invading water helps drive the oil to the producing wells, leading to improved oil recoveries. Like gas reinjection and gas-cap expansion, water influx also acts to mitigate the pressure decline. The degree with which water influx improves oil recovery depends on the size of the adjoining aquifer, the degree of communication between the aquifer and petroleum reservoir, and ultimately the amount of water that encroaches into the reservoir. Some of the most prolific oil fields in the world are waterdrive reservoirs. Perhaps the most celebrated example is the East Texas field. The final oil recovery in the East Texas field is projected to be approximately 79%.[41] As this example shows, water influx has the potential to improve oil recovery considerably.

Once a water-influx mechanism has been identified, it is important to monitor the producing wells closely and to minimize water production. Minimizing water production in "edgewater drives" may require systematically shutting in flank wells once the advancing water reaches them. Minimizing water production in "bottomwater drives" may require systematically cementing in lower perforations as the bottom water slowly rises.

An integral part of reservoir surveillance for waterdrives is an active assessment program. The first phase of assessment includes diagnosis, classification, and characterization. The second phase identifies mathematical models that effectively simulate the aquifer, especially its deliverability. This phase includes reliably estimating aquifer model parameters. The third and final phase includes combining aquifer and reservoir models into a common model that can be used to forecast future recovery effectively and to identify optimal depletion strategies. The success of the third phase depends heavily on the success of the preceding two phases.

Waterdrive and Aquifer Classification

Waterdrives are classified in several ways. First, they are classified according to the location of the aquifer relative to the reservoir. If the aquifer areally encircles the reservoir, either partially or wholly, the waterdrive is called a peripheral waterdrive. If the aquifer exclusively feeds one side or flank of the reservoir, the waterdrive is called an edgewater drive. If the aquifer underlays the reservoir and feeds it from beneath, the waterdrive is called a bottomwater drive.

Waterdrives also are classified according to the aquifer’s strength and to how well the aquifer delivers recharge water to the reservoir. The aquifer strength also refers to how well the aquifer mitigates the reservoir’s normal pressure decline. A strong aquifer refers to one in which the water-influx rate approaches the reservoir’s fluid withdrawal rate at reservoir conditions. These reservoirs also are called complete waterdrives and are characterized by minimal pressure decline. Strong aquifers are generally very large in size and highly conductive. A moderate or weak aquifer is one in which the water recharge rate is appreciably less than the reservoir’s fluid withdrawal rate. These reservoirs are called partial waterdrives and they are characterized by pressure declines greater than a complete waterdrive but less than a volumetric reservoir. An aquifer’s weakness is related directly to its lack in size or conductivity.

Waterdrive Diagnosis

There are several diagnostic indicators to help identify or discount a possible active aquifer.

First, an understanding of the reservoir’s geology is important. The entire outer surface of the reservoir must be scrutinized carefully to identify communicating and noncommunicating pathways; communicating pathways represent possible water entry points. Geological maps should be consulted to identify the type of reservoir trap and the trapping surfaces. Trapping surfaces represent impenetrable surfaces and are discounted automatically as possible water-entry points. The remaining outer surfaces need to be evaluated and classified. If no communicating pathways exist, then the reservoir can be confidently discounted as a possible waterdrive; however, if communicating pathways exist, then the reservoir remains a candidate waterdrive.

Second, and perhaps most importantly, the water-cut history of all producing wells should be recorded and regularly monitored. A steady rise in a well’s water cut is a good indicator of an active aquifer. Although this is among the best indicators, it is not foolproof. For instance, an increasing water cut might be caused by water coning instead of an active waterdrive. Special precautions need to be exercised to avoid water coning. A rising water/oil contact (WOC) is a good indicator of a bottomwater drive. Special attention should be paid to the location of high-water-cut wells. Their location will help define the position of the reservoir/aquifer boundary in peripheral and edgewater drives.

Third, the change in reservoir pressure also can be a helpful indicator. Strong-waterdrive reservoirs are characterized by a slow or negligible pressure decline. Thus, a slower-than-expected pressure decline can help indicate a waterdrive. Material-balance calculations are important to help identify and confirm a slower-than-expected pressure decline.

The reservoir pressure distribution also can help diagnose an active aquifer. For peripheral-water and edgewater drives, higher pressures tend to exist along the reservoir/aquifer boundary while lower pressures tend to exist at locations that are more distant. A pressure contour map is sometimes helpful to identify pressure disparities.

Fourth, the producing GOR can be a helpful indicator. Strong waterdrives are characterized by small changes in the producing GOR. The small GOR change is directly related to the small pressure decline. Sec. 9.9.3 discusses this and other performance features characteristic of waterdrive reservoirs.

Fifth, a material-balance analysis can help diagnose water influx. Several different types of material-balance analyses such as the McEwen[42] analysis can identify water influx. Sec. 9.9.8 discusses these methods.


To illustrate the performance of waterdrives, simulation results of an 80-acre segment of a west Texas black-oil reservoir are presented. The segment is assumed to be surrounded by an infinite radial-flow aquifer. The reservoir properties in Table 9.7 apply. The aquifer permeability and porosity are 37 md and 27%, respectively.

Fig. 9.20 shows the effect of water influx on a plot of pressure vs. fractional oil recovery. The initial reservoir pressure is 2,000 psia. Waterdrive and solution-gas-drive performances are compared. This figure shows that water influx consistently improves the fractional oil recovery at a given pressure. Alternatively, the waterdrive maintains a higher pressure at a given recovery.

Fig. 9.21 shows the reservoir performance with time. This figure contains four separate plots: GOR, saturation, oil-recovery, and pressure histories. Waterdrives typically yield a characteristic GOR history. After a brief increase, the GOR typically levels off. This behavior is explained by the gas-saturation history. The gas saturation increases as soon as the pressure falls below the bubblepoint. After a brief increase, the gas saturation also levels off. The gas saturation levels off because the invading water drives the oil toward the producers. The oil is concentrated in a nonswept region or shrinking oil zone. Without oil displacement, the gas saturation and GOR would rise unabated. The solution-gas drive yields a final GOR of 4,506 scf/STB and gas saturation of 0.287 PV, while the waterdrive yields 1,323 scf/STB and 0.19 PV, respectively. The GOR for the water drive in Fig. 9.21 actually peaks after approximately 10 years and then decreases slightly. A decrease in the GOR reflects good displacement efficiency of oil by water.

As expected, the waterdrive yields a substantially higher recovery. The waterdrive also lengthens the productive life of the reservoir considerably. In this example, the waterdrive recovers 53.2% of the OOIP after 32.6 years, while the solution-gas drive recovers 24.2% of the OOIP after 13.5 years. Both cases assume a terminal oil rate of 20 STB/D. This recovery level indicates a relatively moderate to strong waterdrive. The waterdrive also yields a higher gas recovery (80.5 vs. 53.1%). The water-influx history basically mimics the incremental oil-recovery history. The cumulative encroached water is 58% hydrocarbon pore volume (HCPV) or 0.46 PV. This translates to approximately 1% OOIP incremental recovery for each 0.16 PV (or 2.0% HCPV) of encroached water. The waterdrive in Fig. 9.21 consistently yields higher producing rates than solution-gas drive.

Also as expected, the waterdrive consistently yields a higher pressure at a given time. The waterdrive yields a lower terminal pressure because lower gas saturations are realized at a given pressure. The example water and solution-gas drives yield final pressures of 471 and 613 psia, respectively.

The performance trends noted in Figs. 9.20 and 9.21 are not without exception. Waterdrive performance is strongly influenced by the displacement efficiency of oil by water. Figs. 9.20 and 9.21 are representative of moderate-to-good displacement efficiency. If displacement efficiency is poor, lower oil recovery will occur. A less obvious result, however, is that the GOR history will exhibit much different character than already discussed. Instead of rising slightly and leveling off, the GOR acts much like solution-gas drive; namely, the GOR steadily and monotonically increases. This difference occurs because the invading water bypasses substantial oil and fails to drive enough oil toward the producers to arrest the natural increase in the gas saturation. The GOR of a waterdrive can even exceed the GOR of the solution-gas drive if the displacement efficiency is poor enough. Peripheral waterdrives do not tend to be as efficient as bottomwater drives.

In summary, water influx can markedly improve oil recovery in oil reservoirs. The final oil recovery in a waterdrive reservoir depends largely on the net volume of influxed water. The net volume of influxed water is defined as the volume of influxed water less the volume of produced water. As the net volume of influxed water increases, the oil recovery increases. The volume of the influxed water depends mainly on the size of the aquifer and the communication between the aquifer and the reservoir. The maximum possible net volume of influxed water expressed as a fraction of the reservoir PV is


where Sorw is the residual oil saturation to water, Sgrw is the residual gas saturation to water, and Swi is the initial water saturation. As this equation shows, the residual saturations directly affect oil recovery by limiting the net volume of water that can influx into the reservoir. The residual saturations are a direct measure of the displacement efficiency of water. Lower residual oil saturations are preferred over lower residual gas saturations to promote oil production over gas production. The maximum fractional oil recovery for an initially undersaturated black-oil reservoir is


where Ev is the volumetric sweep efficiency of the invading water. This equation assumes a complete waterdrive (i.e., no pressure depletion).

Water-Influx Models

Water-influx models are mathematical models that simulate and predict aquifer performance. Most importantly, they predict the cumulative water-influx history. When successfully integrated with a reservoir simulator, the net result is a model that effectively simulates waterdrive performance.

There are several popular aquifer models: the van Everdingen-Hurst (VEH) model, [43] the Carter-Tracy model, [44] the Fetkovich model, [45] the Schilthuis model, [29] and the small- or pot-aquifer model. [46] The first three models are unsteady-state models and are the most realistic. They attempt to simulate the complex pressure changes that gradually occur within the aquifer and between the aquifer and reservoir. As pressure depletion proceeds, the pressure difference between the reservoir and aquifer grows rapidly and then abates as the aquifer and reservoir eventually equilibrate. This pressure interaction causes the water-influx rate to start at zero, grow steadily, reach a maximum, and then dissipate. This particular water-influx-rate history behavior applies to initially saturated oil reservoirs; the behavior for initially undersaturated oil reservoirs is often slightly but distinctly different. The effects of undersaturation on the water-influx performance are discussed in Sec. 9.9.7. The unsteady-state models are far more successful at capturing the real dynamics than other models. In contrast, Schilthuis’ steady-state model assumes the aquifer pressure remains constant. The small-aquifer model, however, assumes the aquifer and reservoir pressures are equal.

The VEH model is the most sophisticated of all these models. Its main advantage is its realism. Originally, its main disadvantage was its cumbersome nature. Charts or tables had to be consulted repeatedly to execute a single calculation. To address this limitation, the Carter-Tracy and Fetkovich models were alternatives that were free of tables and charts. These models, however, were only approximations to and simplifications of the VEH model. Since the VEH charts and tables were digitized, [47][48][49] the need for alternatives has diminished.

Allard and Chen[50] proposed an aquifer model specifically for bottomwater drives. This model included 2D flow. In comparison, the VEH model considered only 1D flow. Simulation practitioners, however, have found that the VEH model is satisfactory in simulating bottomwater drives. [51]

van Everdingen-Hurst (VEH) Model

van Everdingen and Hurst considered two geometries: radial- and linear-flow systems. The radial model assumes that the reservoir is a right cylinder and that the aquifer surrounds the reservoir. Fig. 9.22 illustrates the radial aquifer model, where ro = reservoir radius and ra = aquifer radius. Flow between the aquifer and reservoir is strictly radial. This model is especially effective in simulating peripheral and edgewater drives but also has been successful in simulating bottomwater drives. [50]

In contrast, the linear model assumes the reservoir and aquifer are juxtaposed rectangular brllelepipeds. Fig. 9.23 shows examples. Flow between the aquifer and reservoir is strictly linear. This model is intended to simulate edgewater and bottomwater drives. The model definition depends on the application. For edgewater drives, the thicknesses of the reservoir and aquifer are identical; the widths of the reservoir and aquifer are also the same, and the aquifer and reservoir lengths are La and Lr, respectively (Fig. 9.23a). For bottomwater drives, the width of the reservoir and aquifer are identical; the length (L) of the reservoir and aquifer are also the same; the aquifer depth is La, and the reservoir thickness is h (Fig. 9.23b).

van Everdingen and Hurst solved the applicable differential equations analytically to determine the water-influx history for the case of a constant pressure differential at the aquifer/reservoir boundary. This case assumes the reservoir pressure is constant. They called this case the "constant terminal pressure" and reported their results in terms of tables and charts. This solution is not immediately applicable to actual reservoirs because it does not consider a declining reservoir pressure. To address this limitation, van Everdingen and Hurst applied the superposition theorem to a specific reservoir pressure history. This adaptation usually requires that the reservoir’s pressure history be known. The first step in applying their model is to discretize the time and pressure domains.

Discretization. The time domain is discretized into (n + 1) points (t0, t1, t2, ...., tn), where t0 < t1 < t2 < .... < tn and t0 corresponds to t = 0. The average reservoir pressure domain also is discretized into (n + 1) points RTENOTITLE, where RTENOTITLE is the initial pressure pi. The time-averaged pressure between levels j and j – 1 is


The time-averaged pressure at level j = 0 is defined as the initial pressure pi. Table 9.13 shows discretization of t, RTENOTITLE, and RTENOTITLE. The time-averaged pressure decrement between levels j and j – 1 is


No value is defined for j = 0. Table 9.13 shows the complete discretization of t, RTENOTITLE, RTENOTITLE, and Δp.

Cumulative Water Influx. The cumulative water influx at kth level is


where U is the aquifer constant and WD is the dimensionless cumulative water influx. This equation is based on the superposition theorem. The term WD (tDktDj) is not a product but refers to the evaluation of WD at a dimensionless time difference of (tDktDj). If we apply Eq. 9.33 for k = 1, 2, and 3, we obtain


The length of the equation grows with the time. The aquifer constant, U, and the dimensionless cumulative water influx, WD(tD), depend on whether the radial or linear model is applied.

Radial Model. The radial model is based on the following equations. The effective reservoir radius is a function of the reservoir PV and is


where ro is expressed in ft, Vpr is the reservoir PV expressed in RB, ϕr is the reservoir porosity (fraction), and h is the pay thickness in ft. The constant f is θ/360, where θ is the angle that defines the portion of the right cylinder. Fig. 9.24 illustrates the definition of θ for a radial aquifer model. The dimensionless time is


where ka = aquifer permeability (md), μw = water viscosity (cp), ct = total aquifer compressibility (psi–1), ϕa = aquifer porosity (fraction), and t is expressed in years. The total aquifer compressibility is the sum of the aquifer and rock compressibilities. The aquifer constant is


where U is in units of RB/psi if h is in ft, ro is in ft, and ct is in psi–1.

The dimensionless aquifer radius is


The dimensionless water influx, WD, is a function of tD and reD and depends on whether the aquifer is infinite acting or finite.

Infinite Radial Aquifer. The aquifer is infinite acting if r e approaches infinity or if the pressure disturbance within the aquifer never reaches the aquifer’s external boundary. If either of these conditions is met, then WD is



where a7 = 4.8534 × 10–12, a6 = –1.8436 × 10–9, a5 = 2.8354 × 10–7, a4 = –2.2740 × 10–5, a3 = 1.0284 × 10–3, a2 = –2.7455 × 10–2, a1 = 8.5373 × 10–1, a0 = 8.1638 × 10–1, or


Marsal[48] presented Eqs. 9.38 and 9.40. Walsh[23][49] presented Eq. 9.39.

Finite Radial Aquifer. For finite aquifers, Eqs. 9.38 through 9.40 apply if tD < tD*, where


and RTENOTITLE....................(9.37)

If tD > tD*, then


where RTENOTITLE....................(9.43)

Marsal[48] gave Eqs. 9.41 through 9.43. These equations are effective in approximating the charts and tables by van Everdingen and Hurst. Minor discontinuities exist at some of the equation boundaries. A slightly more accurate but much more lengthy set of equations has been offered by Klins et al.[47] Fig. 9.25 shows WD as a function of tD for reD = 5, 7.5, 10, 20, and ∞. These equations simplify the application of the VEH model enormously.

A finite aquifer can be treated effectively as an infinite aquifer if


or RTENOTITLE....................(9.45)

where tDmax is the maximum value of tD. These equations follow from Eq. 9.41. For example, if tDmax is 540 and corresponds to a time of 8 years, then Eq. 9.45 yields ≥ reD = 38. Therefore, if the aquifer has a dimensionless radius greater than 38, then the aquifer acts indistinguishably from and equivalent to an infinite aquifer at all times less than 8 years.

Linear Aquifer. The aquifer size in the linear model is given in terms of the aquifer/reservoir pore-volume ratio, Vpa/Vpr.


The aquifer constant is


For edgewater drives, the aquifer length is


where La and Lr are defined in Fig. 9.23a. For bottomwater drives, the aquifer depth is


where La is defined in Fig. 9.23b. The dimensionless time is


Eqs. 9.50 and 9.35 use the same units except La is given in ft. One difference between the linear and radial models is that tD is a function of the aquifer size for the linear model, whereas tD is independent of the aquifer size for the radial model. This difference forces a recalculation of tD in the linear model if the aquifer size is changed. The dimensionless cumulative water influx is


and RTENOTITLE....................(9.52)

Eq. 9.51 is by Marsal, [48] and Eq. 9.52 is by Walsh. [23][49] Fig. 9.26 shows WD as a function of tD. The aquifer can be treated as infinite if the aquifer length is greater than the critical length.


where tmax is the maximum time expressed in years and Lac is in units of ft. Eqs. 9.50 and 9.53 use the same units. Alternatively, the aquifer is infinite-acting if tD ≤ 0.50. If infinite-acting and an edgewater drive, We can be evaluated directly without computing WD and is


where the units in Eq. 9.35 apply, and We is in units of RB and h and w are in units of ft.

Calculation Procedure. The calculation requires historical average reservoir-pressure data. Given this data, the following procedure is used to compute the water-influx history. This method applies for radial and linear models, except where noted:

  1. Discretize the time and average reservoir-pressure domains and define tj and RTENOTITLE for (j = 0, 1, ..., n) according to Table 9.13.
  2. Compute the time-averaged reservoir pressure RTENOTITLE for (j = 1, 2, ..., n) with Eq. 9.31. Note that RTENOTITLE = pi.
  3. Compute the time-averaged incremental pressure differential Δpj for (j = 1, 2, ...., n) with Eq. 9.32.
  4. Compute tDj for (j = 0, 1, ..., n) with Eq. 9.35 for radial aquifers or with Eq. 9.50 for linear aquifers.
  5. Steps 5 through 9 create a computational loop that is repeated n times. The loop index is k, where k = 1, ..., n. For the kth time level, compute (tDktDj) for (j = 0, ..., k – 1).
  6. For the kth time level, compute WD(tDktDj) for (j = 0, ..., k – 1).
  7. For the kth time level, compute Δpj + 1 WD(tDktDj) for (j = 0, ..., k – 1).
  8. For the kth time level, compute Wek with Eq. 9.33.
  9. Increment the time from level k to k + 1, and return to Step 5 until k > n.

This procedure is highly repetitive and well suited for spreadsheet calculation. Example 9.2 illustrates the procedure.

Determining Water-Influx Model Parameters

The minimum parameters that need to be specified in the radial model are the aquifer constant, U, the time constant, kt, and the dimensionless aquifer radius, reD. The time constant combines a number of constants, is the proportionality constant between the dimensionless and real time, and is defined by


Physically, the time constant represents the aquifer conductivity. In summary, U and kt are defined as



and RTENOTITLE....................(9.37)

Eq. 9.56 assumes the same units as Eq. 9.36, and Vpr is given in res bbl. Eq. 9.57 assumes the same units as Eq. 9.35.

The minimum parameters that need to be specified in the linear model are the aquifer constant, time constant, and aquifer/reservoir PV ratio (Vpa/Vpr). The aquifer constant and time constant are


and RTENOTITLE....................(9.59)

Eq. 9.59 assumes the same units as Eq. 9.35.

There are three common methods to estimate model parameters: direct measurement, history matching, and material balance. The first two methods are described in the following sections. The material-balance method, through the McEwen method, [42] is described in Section 9.9.8.

Direct Measurement. This method estimates model parameters from direct measurement of the independent constants. Though ideally preferred, this method is rarely possible because of the uncertainty of some of the constants.

For the radial model, the model parameters (U, kt, and reD) are a function of the following constants: re, ro, ka, h, f, ϕa, ct, and μw. These constants follow from inspection of Eqs. 9.36, 9.37, and 9.57. The uncertainty among these constants varies. Of these constants, re, ro, and ka are perhaps the most uncertain. Qualitatively, these constants are related to the aquifer size, reservoir size, and aquifer conductivity (i.e., Vpa, Vpr, and kt).

For the linear model, the model parameters (U, kt, and Vpa/Vpr) are a function of the following constants: Vpa, Vpr, ka, ϕa, ct, μw, and La. This list follows from inspection of Eqs. 9.47 and 9.59. Of these constants, Vpa, Vpr, La, and ka are the most uncertain. If h is approximately known, then Vpa and La are not independent but related through Eqs. 9.48 or 9.49. Thus, Vpa, Vpr, and ka are the most uncertain independent constants. Qualitatively, these constants are related to the aquifer size, reservoir size, and aquifer conductivity (i.e., Vpa, Vpr, and kt, respectively). Note the similarity between the radial and linear models.

In summary, because of the uncertainty of the aquifer size and conductivity and reservoir size, it is difficult to estimate reliably the water-influx model parameters. Nevertheless, every attempt should be made to estimate the median, variance, and range of each constant and the model parameters. This information is helpful in the history-matching method.

History Matching. If the water-influx history can be estimated, then model parameters can be estimated from history matching. When history matching is used, only the most uncertain constants should be treated as adjustable parameters: preferably only re, ro, and ka for the radial model or Vpa, Vpr, and ka for the linear model. Unless only one adjustable parameter exists, history matching is usually complicated by nonuniqueness. [40][42][52][53][54] Nonuniqueness, however, can be minimized by limiting the range of parameter adjustment to realistic ranges. Example 9.2 illustrates the history-matching procedure.

Example 9.2: History Matching Water Influx Table 9.14 summarizes the cumulative water influx and average reservoir pressure as a function of time for an initially saturated, black-oil reservoir. Areally, the reservoir is approximately semicircular, bounded on one side by a sealing fault and the other side by an aquifer. Fig. 9.27a shows a schematic representation of the reservoir. Assume the reservoir and aquifer properties in Table 9.15 apply.

Assume a radial-flow aquifer with f = 0.50. Find the optimal aquifer size (Vpa and reD) that best simulates the water-influx performance. Plot and compare the actual and predicted water-influx histories.

Find the optimal aquifer size (Vpa/Vpr and La) that best matches the water-influx performance assuming a linear-flow aquifer. Assume the reservoir width is w = 2ro and length is Lr = πro/4, where ro is given by Eq. 9.34. Fig. 9.27b schematically shows the areal interpretation. Plot and compare the actual and predicted water-influx histories. Which model (linear or radial) best matches the data?

Solution. Compute the effective reservoir radius from


where f = 0.50. The total compressibility is the sum of the rock and water compressibilities or ct = 5.88 × 10–6 psi–1. The time constant, kt, is given by Eq. 9.57 and is kt = 0.8682 years–1. U is given by Eq. 9.56 and is 3,955 RB/psi. Table 9.16 tabulates tD, RTENOTITLE, and Δp.

The solution procedure varies reD until the best match is obtained. Tables 9.17 through 9.19 give the details of the calculation for reD = 5.0. Table 9.17 tabulates (tDktDj) for k and j = 0, 1, ..., 7. Table 9.18 tabulates WD(tDktDj) for k and j = 0, 1, ..., 7. Table 9.19 tabulates Δpj + 1WD(tDktDj) for k and j = 0, 1, ..., 7. Table 9.16 tabulates ΣΔpj+ 1WD(tDktDj) for k = 1, 2, ..., 7 for reD = 5 and Wej for j = 0, 1, ..., 7 for reD = 5. Fig. 9.28 plots We vs. t and compares model and actual results; excellent agreement is noted. These calculations were repeated for other values of reD but reD = 5.0 was found to give the best fit of the actual and model water-influx histories. Notice how well the radial model matches the history. A dimensionless aquifer radius of reD = 5.0 corresponds to Vpa/Vpr = 20.8. This means that the aquifer is 20.8 times larger than the reservoir.

To determine whether this aquifer can be treated as infinite acting, we evaluate Eq. 9.45 with tDmax = 26.08. This calculation yields reD = 8.1. Because this value of reD is greater than the history-matched value of reD = 5.0, this aquifer cannot be treated as infinite.

For the linear aquifer model, the geometry dictates that Lr = 11,696 ft if Lr = πro/4 = π (14,892)/4.With the same trial-and-error procedure as used for the radial aquifer, the linear aquifer yields Vpa/Vpr = 12 for the best match between the actual and predicted water-influx data. This value of Vpa/Vpr yields La = 161,145 ft, U = 27,271 RB/psi, and kt = 0.0074 years–1. Fig. 9.28 compares the predicted and actual data and shows that the match is poor. This comparison reveals that the linear model is not preferable to simulate water influx for this reservoir.

Aquifer Performance

The aquifer performance is described in terms of the delivery rate, average aquifer pressure, and cumulative water-influx volume as a function of time. The aquifer pressure characteristically lags behind the reservoir pressure and is estimated by


The aquifer delivery rate is qw = ∂We(t)/∂t, which is determined from the slope of the We vs. t curve.

Fig. 9.29 shows the aquifer pressure and delivery-rate history for the data in Example 9.2. This figure includes the reservoir pressure history for comparison. The qualitative results in Fig. 9.29 are representative of many aquifers. The water-delivery rate is initially zero and increases rapidly. It peaks after approximately 12 to 14 years and then slowly decreases. The aquifer and reservoir pressures start at equivalent values. The reservoir pressure declines more quickly than the aquifer pressure. The pressure differential between the aquifer and reservoir grows and is approximately 250, 350, and 500 psia, respectively, after 2, 5, and 10 years. The pressure differential peaks after 12 to 14 years and then begins to dissipate. The pressure differential and delivery rate decline together.

The aquifer performance noted in Fig. 9.29 is not without exception. The qualitative performance in Fig. 9.29 is characteristic of an initially saturated reservoir. Aquifers feeding initially undersaturated reservoirs may behave quite differently. The difference stems from the difference in the reservoir pressure histories. The reservoir pressure in initially undersaturated oil reservoirs initially declines much more quickly than in initially saturated reservoirs. Consequently, initially undersaturated reservoirs create a substantial pressure differential between the reservoir and aquifer much sooner than initially saturated reservoirs. Of course, this distinction depends on the degree of undersaturation. If the reservoir is significantly undersaturated, a large pressure differential between the reservoir and aquifer is quickly established. This large pressure differential, in turn, promotes water influx; consequently, the water-influx rate increases more rapidly in initially undersaturated reservoirs than initially saturated reservoirs. Once the bubblepoint is reached, the pressure differential between the aquifer and reservoir may decline temporarily. Later, the pressure differential may increase, reminiscent of an initially saturated reservoir, as in Fig. 9.29. The net effect is that water recharge rate may oscillate in an initially undersaturated oil reservoir.

Material-Balance Analysis

The objectives of a material-balance analysis include confirming the producing mechanism, estimating the OOIP, estimating the water-influx history, and estimating the water-influx model parameters. The water-influx model parameters are needed to forecast future water influx and oil recovery.

The analysis depends on the known and unknown constants and variables. Three scenarios are considered: the water-influx history is known but the OOIP is unknown, the water-influx history is unknown but the OOIP is known, and both the water-influx history and the OOIP are unknown.

Water Influx Known, OOIP Unknown. If the water-influx history is known and the OOIP unknown, the material-balance methods in the previous sections are directly applicable. For instance, if the reservoir is initially undersaturated, then an (FWe)-vs.-E owf plot can be used to confirm the mechanism and estimate the OOIP. The water-influx model parameters can be determined by history matching.

Water Influx Unknown, OOIP Known. If the OOIP is known and the water-influx history is unknown, then the material-balance equation can be used to estimate the water-influx history. Solving Eq. 9.7 for We yields


If the OOIP (N) and OGIP (G) are known, Gfgi and Nfoi are computed from Eqs. 9.21 and 9.22. Eq. 9.61 is applied for each historical average pressure measurement to compute the cumulative water influx. Example 9.3 illustrates this method. Once the water-influx history is estimated, the aquifer parameters can be estimated from history matching.

Water Influx Unknown, OOIP Unknown. This case simultaneously determines the OOIP, water-influx history, and water-influx model parameters. This is a challenging problem. Woods and Muskat[52] were among the first to study this problem and they noted that the solution was complicated by nonuniqueness. Others, too, have noted nonuniqueness. [40][42][53][54] Despite these complications, certain techniques have proved useful and some approaches are better than others are. The solution method is based on the work of McEwen[42] and depends on whether the radial or linear version of the VEH model is applied.

Radial Aquifer. This method simultaneously determines the OOIP, water-influx history, and model parameters reD and kt. The aquifer constant U is then subsequently determined. The water influx is


This equation is an abbreviation of Eq. 9.33. The summation ΣΔpWD is a function of only reD and kt. McEwen noted that U is related to Nfoi and Gfgi through


Substituting this equation into Eq. 9.62 and substituting this result into Eq. 9.7 yields




and RTENOTITLE....................(9.66)

For the case of an initially undersaturated oil reservoir,


This equation shows that a plot of F vs. Eow is a straight line, emanates from the origin, and has a slope equal to N, the OOIP. This observation provides a means to confirm the producing mechanism.

Eow, however, is a function of kt and reD, and these parameters are unknown a priori. Thus, the problem reduces to one of finding the optimal kt and reD that minimizes the material-balance error. Graphically, this is equivalent to varying kt and reD until the straightest possible line is realized. The slope of the line equals the OOIP, and the OGIP is the product NRsi if the reservoir is initially undersaturated. Mathematically, the material-balance error is minimized when the sum of the squares of the residual is minimized. The residual for point i is


The sum of the squares of the residual is


where n is the total number of data points.

In summary, the McEwen method to simultaneously estimate N, reD, and kt is as follows:

  1. Estimate a limited range of realistic values for kt if possible.
  2. Compute F for each data point with Eq. 9.8 if saturated or Eq. 9.13 if undersaturated.
  3. Guess kt and reD.
  4. Compute Eow for each data point with Eq. 9.65.
  5. Compute N with least-squares linear regression or graphically from the slope of an F-vs.-Eow plot.
  6. Compute Ri for each data point with Eq. 9.68.
  7. Compute Rss with Eq. 9.69.
  8. If the Rss is minimized, then go to Step 9; otherwise, return to Step 3.
  9. Compute the aquifer constant with Eq. 9.63.
  10. Compute the water influx for each data point with Eq. 9.62.

If least-squares linear regression is used to compute N in Step 5, an equation analogous to Eq. 9.16 is used (where Eow is substituted for Eowf). This solution method is iterative because the material-balance error must be minimized. This calculation is carried out with a trial-and-error method or a minimization algorithm. Least-squares linear regression and minimization algorithms have become standard features in commercial spreadsheets.

McEwen’s method also can be applied to initially saturated reservoirs; however, the solution procedure must be expanded and modified slightly. More specifically, the solution procedure is the same as for initially undersaturated reservoirs except Steps 5 and 6. Step 5 must be modified to include the simultaneous calculation of Nfoi and Gfgi by multivariate, least-squares (planar) regression. [23][28] Step 6 must be modified, and the residual for point i is computed with Ri = (F)iGfgi (Egw)iNfoi (Eow)i.

Linear Aquifer. McEwen’s method for linear aquifers is very similar to the radial model. The method simultaneously determines the OOIP, water-influx history, and the model parameters [(Vpa/Vpr) and kt].

The aquifer constant U is related to Vpa through


Substituting this equation into Eq. 9.62 and substituting the result into Eq. 9.7 gives Eq. 9.64, where



and ΣΔpWD is a function of only kt and not (Vpa/Vpr). Eqs. 9.71 and 9.72 are analogous to Eqs. 9.65 and 9.66 in the radial model.

The solution procedure for the linear model is identical to that of the radial model except that kt and Vpa/Vpr are optimized to minimize the material-balance error. Once Vpa/Vpr is determined, the aquifer constant U is determined from Eq. 9.70 and We is determined from Eq. 9.62. Example 9.4 illustrates an example of the McEwen method.

Numerous alternative material-balance methods have been proposed to analyze waterdrive reservoirs. Some are very popular and widely used. While most are theoretically valid, most are also unreliable. van Everdingen et al., [55] for instance, proposed plotting F/Eo vs. (ΣΔpWD)/Eo. The slope of this plot equals U and the y-intercept equals N. van Everdingen et al. proposed varying kt until the straightest possible line was obtained. Later, Havlena and Odeh[38] popularized this method and modified it to include the aquifer size (reD) as an additional unknown and determinable parameter. Dake[30] also advocated this method. Chierici et al.[53] proposed a variation of this method with a F/(ΣΔpWD) vs. Eo/(ΣΔpWD) plot. McEwen[42] studied the method of van Everdingen et al. and noted hypersensivity to pressure uncertainty. He observed unacceptably large errors and deemed the method unreliable. Later, Tehrani[40] presented a systematic analysis of these methods and confirmed McEwen’s conclusions. Wang and Hwan[39] confirmed Tehrani’s findings. Sills[54] presents a review and comparison of the McEwen, Havlena-Odeh, and van Everdingen-Timmerman-McMahon methods.

To help diagnose waterdrives, Campbell[56] proposed plotting F/Eowf vs. Np for initially undersaturated oil reservoirs. This method is analogous to Cole’s[57] popular method of plotting F/Egwf vs. Gp for gas reservoirs. In theory, an active waterdrive is indicated if the plot varies appreciably from a horizontal line. The degree of curvature is a qualitative measure of the waterdrive strength. The curve emanates from a y-intercept equal to the OOIP. The shape of the curve mimics and is related to the attending water recharge rate history. Cole and Campbell plots are attractive because of their simplicity and are widely reported. Unfortunately, in practice, they are not always reliable because of hypersensitivity caused by uncertainty. The origin of the hypersensitivity is analogous to the problems noted by McEwen, [42] Tehrani, [40] Wang and Hwan, [39] and Walsh[28] for other types of material-balance plots. The quotient F/Eowf approaches infinity initially because the Eowf approaches zero. A systematic reservoir pressure error of only 1 to 2%, for instance, can lead to erroneous conclusions regarding water-influx diagnosis. These facts complicate the interpretation. For these reasons, Campbell plots should be used cautiously to diagnose water influx and used very cautiously to estimate the OOIP. If they are used, the early-time data should be weighted minimally. The reliability of these plots increases with pressure depletion; however, water influx mitigates pressure depletion and delays reliability. Unfortunately, water-influx diagnoses are sought as early as possible, which further complicates and compromises the use of these plots.

Example 9.3: Estimating Water Influx With Material Balance Table 9.20 summarizes the cumulative oil and gas production as a function of time and average reservoir pressure for a black-oil reservoir. The discovery (initial) pressure is 1,640 psia, and production data are tabulated through a pressure of 800 psia.

Volumetric measurements estimate an OOIP of 210,420 thousand STB with no initial free gas. Estimate the cumulative water influx (RB) at each pressure in Table 9.20 with material balance. Assume that the standard PVT parameters in Table 9.3 apply.

Solution. Eq. 9.61 gives the cumulative water influx. Because there is no initial free gas, Gfgi = 0, Nfoi = N, and Eq. 9.61 simplifies to


where rock and water expansion are ignored and F and Eo are given by Eqs. 9.8 and 9.2, respectively. Eo is a function of Bto, which is given by Eq. 9.5.

Table 9.21 tabulates the results. Fig. 9.30 plots the water-influx history.

Example 9.4: Determining Water-Influx Parameters and OOIP

van Everdingen et al.[55] studied water influx in an initially undersaturated oil reservoir located in the Wilcox formation at a depth of 8,100 ft subsea. The accumulation covered approximately 1,830 acres. The maximum gross and net thicknesses were 37 and 26 ft, respectively. The reservoir fluid exhibited an initial oil FVF of 1.538 RB/STB and a GOR of 900 scf/STB. Table 9.22 reports the reservoir and aquifer properties.

van Everdingen et al.[55] estimated the OOIP at 24 to 25 million STB from volumetric measurement. From the oil, gas, and water production data and standard PVT properties, they computed F and Eo as a function of pressure. Production took place from 1942 through 1950. Approximately 6.965 million STB of oil and 7.1 Bscf of gas were produced during this period. Table 9.23 gives the average reservoir pressure, total fluid withdrawal, and oil-phase expansivity as a function of time.

Use McEwen’s method to find the optimal dimensionless aquifer radius, reD, and the aquifer time constant, kt (years–1). Plot F vs. Eow. Also, compute the aquifer constant, U (RB/psi). Estimate the OOIP (million STB). Estimate the water delivery rate (RB/D) and average aquifer pressure (psia), and plot the histories.

Solution. Though the time constant, kt, is unknown, first estimate a realistic range of values based on reservoir and aquifer properties. The time constant is given by Eq. 9.57. The total compressibility is the sum of the rock and water compressibility or 6.8 × 10–6 psi–1. The aquifer porosity is 20.9%. The water viscosity is 0.25 cp. The aquifer permeability is 275 md. The only unspecified quantity on the right side of Eq. 9.57 is the effective reservoir radius. Although this quantity is unknown because the size of the reservoir is uncertain, it can be estimated from


To use this equation, estimate the reservoir PV, which is given by


Evaluating this equation on the basis of Nfoi = 25 million STB yields Vpr = 45.23 million RB. Evaluating Eq. 9.34 for f = 1, h = 26 ft, and ϕr = 19.9% yields ro = 3,951 ft. Evaluating Eq. 9.57 yields kt = 116 years–1. Thus, a liberal range for kt is 10 to 1,000 years–1.

Next, compute Eow from Eq. 9.65. This equation requires computing ΣΔpWD, which is a function of reD and kt. The overall solution procedure contains the following steps:

  1. Assume values of reD and kt.
  2. Compute tD for each historical data point with Eq. 9.35.
  3. Compute ΣΔpWD for each historical data point with the VEH model.
  4. Compute Eow for each historical data point with Eq. 9.65.
  5. Plot F vs. Eow.
  6. Determine the OOIP from the slope.
  7. Compute the residual, Ri, for each data point with Eq. 9.68 and compute Rss with Eq. 9.69.
  8. Change reD and kt and return to Step 1 until a minimum Rss is obtained.

Table 9.24 summarizes the results for the case of reD = 20 and kt = 17 years–1. The step-by-step calculations to compute ΣΔpWD are omitted for the sake of brevity. Example 9.2 illustrates these calculations.

Fig. 9.31 plots F vs. Eow for reD = 20 and kt = 17 years–1. The slope of this plot is 18.7 million STB, which is an estimate of the OOIP. The material-balance error is given by the Rss, which is 0.015 million res bbl.

The values of reD and kt were varied until a minimum Rss was realized. Table 9.25 summarizes the results for the following values of reD: 10, 15, 20, and 30. No values greater than reD = 30 were considered because they yielded identical results and were equivalent to an infinite-acting aquifer.

The results in Table 9.25 are summarized graphically in Fig. 9.32. This figure plots the standard error vs. the time constant for reD = 10, 20, and 30. The standard error is defined as the square root of Rss/n, where n is the total number of data points. These results show that a minimum error is realized for the following range of properties: reD = 20 to 30 and kt = 16 to 17 years–1. Table 9.26 summarizes the range of constants yielded by this range of values.

The material-balance OOIP estimate (18.7 to 20.3 million STB) approximately agrees with the volumetric estimate (24 to 25 million STB). Technically, reD = 20 and kt = 17 years–1 yields a minimum error; however, comparable errors are realized for the stated range of reD and kt. The boundary between infinite- and noninfinite-acting aquifers occurs at reD = 20.1. Thus, an aquifer with reD = 20 is not quite but very nearly an infinite-acting aquifer. These observations help explain why values of reD in the range 20 to 30 yield virtually identical errors and similar results. On the basis of simulating reservoir performance through 9 years, selecting any value of reD greater than 20 is practically acceptable. A value of reD = 30, however, yields an OOIP estimate (20.2 million STB) slightly closer to the volumetric OOIP estimate than the OOIP estimate (18.8 million STB) for reD = 20.On this basis, reD = 30 may be preferable. Without additional historical data and history matching, a more conclusive determination of reD beyond the stated range is not possible.

Fig. 9.31 includes an F-vs.-Eow plot for reD = 10 and kt = 11 years–1. The F vs. Eow plots for reD = 20 and reD = 30 are virtually indistinguishable. The plot for reD = 10 exhibits appreciable curvature; in contrast, the plot for reD = 20 is linear. The degree of curvature is a measure of the lack of material balance. The upward curvature of the former plot indicates that water influx is underpredicted.

Fig. 9.33 shows the predicted water-influx history. This figure assumes reD = 20. A cumulative water-influx volume of 14.83 million res bbl is predicted after 9 years. This amount of water influx equates to 46.0% of the (reservoir) HCPV (assuming an OOIP of 18.7 million STB and a HCPV of 33.83 million res bbl). This substantial amount of water influx, together with the relatively small pressure decline (from 3,800 to 3,060 psia), suggests a moderate to strong waterdrive.

The pressure data in Table 9.23 show that the reservoir pressure actually increased after 7 years. This example illustrates another consequence of the VEH model; namely, the model can treat pressure increases as well as pressure decreases. The operators offered no explanation regarding why the reservoir pressure began to increase after seven years. It was noted that the pressure remained approximately constant and increased during periods of marked GOR decline. It was suspected that a secondary gas cap formed as the reservoir pressure declined below the bubblepoint. Upstructure wells reportedly began producing free gas as the gas cap formed and grew. The GOR grew as the free-gas production grew. The GOR declined when production from the high-GOR wells was curtailed. Though this scenario explains the GOR behavior and some of the pressure behavior, it does not explain why the reservoir pressure actually increased slightly during the last two years of production.

Compaction Drives

If pore-volume contraction contributes prominently to overall expansion while the reservoir is saturated, then the reservoir is classified as a compaction drive. Compaction-drive oil reservoirs are supplemented by solution-gas drive if the reservoir falls below the bubblepoint; they may or may not be supplemented by a water or gas-cap drive.

Compaction drives characteristically exhibit elevated rock compressibilities, often 10 to 50 times greater than normal. Rock compressibility is called PV, or pore, compressibility and is expressed in units of PV change per unit PV per unit pressure change. Rock compressibility is a function of pressure. Normal compressibilities range from 3 to 8 × 10–6 psi–1 at pressures greater than approximately 1,000 psia. In contrast, elevated rock compressibilities can reach as high as 150 × 10–6 psi–1 or higher at comparable pressures. [58]

In general, compaction-drive reservoirs are rare; however, strong compaction drives do exist. The Ekofisk field in the Norwegian sector of the North Sea, with reserves in excess of 1.7 billion bbl, lies at a depth of 9,300 ft below sea level in 235 ft of water. The reservoir is a chalk formation that exhibits porosities in the range of 25 to 48%.[59] The operators reported rock compressibilities as high as 50 to 100 × 10–6 psi–1.[60] Extreme compressibilities such as these can account for 70 to 80% of the expansion above the bubblepoint and 20 to 50% or more of the expansion below the bubblepoint.


Compaction-drive oil reservoirs act like their noncompaction counterparts except that they exhibit enhanced recoveries. For instance, a solution-gas-drive, compaction-drive reservoir will act qualitatively like a normal solution-gas-drive reservoir except the oil recovery will be greater. The enhanced recoveries are a direct consequence of the extra rock expansion that compaction-drive reservoirs naturally possess.

The excessive compaction noted in compaction-drive reservoirs has contributed to some production problems. For example, the compaction has been linked to a decline in reservoir permeability, fracture closure, and subsidence. [58][59][60] In most cases, however, these problems have been manageable, and the net result of compaction has been very favorable.

Material-Balance Analysis

The material-balance methods discussed in the previous sections are equally applicable to compaction drives. The only difference is that rock expansion cannot be ignored. Including rock expansion requires evaluating the rock expansivity, Ef.

The most accurate and reliable method is to measure Ef as a function of pressure. This method is strongly recommended if a compaction-drive mechanism is suspected because of the sensitivity of the analysis to Ef. Table 9.27 summarizes the experimental results for a high-pressure Gulf Coast gas reservoir. [61] The initial reservoir and hydrostatic pressure was 9,800 psia. The rock expansivity ranged between 0 and 8.07%. The porosity decreased from 16.7 to 15.5% over the course of the test. This particular sample exhibited higher than normal expansion.

The rock expansivity and compressibility are related through


where cf is the rock compressibility. This equation assumes that the fractional change in PV is small. Physically, the rock expansivity represents the fractional change in PV while, in contrast, the rock compressibility represents the rate of change in fractional PV with pressure. While the former is more pertinent to material-balance calculation, experimental data are often reported in terms of the latter. Table 9.27 includes rock-compressibility measurements. If cf is known as a function of pressure, then the integral on the right side of Eq. 9.75 can be evaluated numerically to determine Ef(p). If cf is relatively independent of pressure, then Eq. 9.75 can be simplified to


This method of estimating Ef is not usually preferable because cf is rarely constant. Fig. 9.34 illustrates a case and plots the rock compressibility as a function of pressure from the data in Table 9.27. Several features are worth noting, and many of these features are characteristic of compaction drives. First, the rock compressibility ranges between 4 to 21 × 10–6 psi–1, which is a greater-than-normal range. Second, the rock compressibility clearly is not independent of pressure. Third, the compressibility declines sharply as the pressure first declines below the initial pressure. This phenomenon is largely attributed to grain rearrangement. Fourth, the rock compressibility increases at pressures below 4,000 psia. This phenomenon is attributed to pore collapse.

Once Ef(p) is estimated, the material-balance methods in the previous sections can be applied to estimate the OOIP and confirm the producing mechanism.

Water and Gas Coning

Coning is a production problem in which gas-cap gas or bottomwater infiltrates the perforation zone in the near-wellbore area and reduces oil production. Gas coning is distinctly different from, and should not be confused with, free-gas production caused by a naturally expanding gas cap. Likewise, water coning should not be confused with water production caused by a rising WOC from water influx. Coning is a rate-sensitive phenomenon generally associated with high producing rates. Strictly a near-wellbore phenomenon, it only develops once the pressure forces drawing fluids toward the wellbore overcome the natural buoyancy forces that segregate gas and water from oil.

Under ideal conditions in which no coning exists, flow is principally horizontal and mainly oil is produced. Fig. 9.35 illustrates a producing well with no coning. When coning exists, however, the overlying gas is drawn downward or bottomwater is drawn upward and into the oil column. Coning trades oil production for gas or water production. Fig. 9.36 illustrates a producing well subject to gas and water coning.

Two strategies commonly are used to minimize coning. One approach is partial perforation or penetration. In this approach, only a limited portion of the pay thickness is perforated. If gas coning is anticipated, the pay thickness near the GOC is not perforated. If water coning is anticipated, the pay thickness near the WOC is not perforated. In instances in which severe coning is expected, only a small portion of the pay thickness may be perforated. The variables in Fig. 9.37 define the length of the perforation interval, b, and its position within the pay thickness, h. The distance Lg is the distance between the top of the pay and the uppermost perforation, and the distance Lw is the distance between the bottom of the pay and the lowest perforation. The quotient b/h is the partial perforation fraction. Although this strategy will reduce and can eliminate coning problems, it suffers an obvious drawback; namely, it temporarily reduces oil production in the hope of eventually avoiding coning.

A second remedial strategy is based on the observation that there is a critical producing rate below which the cone stabilizes and will not reach the perforations. This critical rate is a function of the perforation length. As the perforation length increases, the critical producing rate decreases. Often, the critical producing rate is much less than the possible producing rate. This difference creates an operational decision: produce at a rate greater than the critical and eventually risk coning, or produce at a rate less than the critical and temporarily sacrifice oil production. If the critical rate is less than the minimum economic rate, then the operator has no choice but to produce above the critical rate or abandon the well.

To combat coning, a hybrid strategy is often used whereby a combination of partial perforation and a reduced producing rate is used. One especially unattractive consequence of gas coning is that it prematurely depletes the gas-cap gas and diminishes the gas-cap producing mechanism. Fortunately, gas coning is not as problematic as water coning because the density difference between oil and gas is greater than the difference between water and oil. This density difference through gravity segregation helps mitigate coning.

To develop an effective remedial strategy against coning, certain theoretical aspects regarding coning must be understood. Mathematically, coning is a challenging problem because of its complexity. To develop tractable analytical solutions, tenuous assumptions must be invoked. These assumptions limit the practical applicability of these solutions. The most reliable way to study coning is with a specially designed finite-difference simulator. [61][62][63] Nevertheless, certain analytical solutions and empirical correlations can be helpful and serve as a preliminary guide.

Muskat and Wyckoff [64] and Chaney et al.[65] were among the first to contribute substantively to this problem. Since their efforts, several other authors have contributed to the body of literature. [66][67][68][69][70][71][72][73] Many of these works have led to similar correlations. Wheatley[72] presented a comparison of some popular correlations. As a representative sample, the correlations of Schols[71] and Chierici et al.[66] are presented here. Both works apply to both water and gas coning. Both efforts also use the following equation to compute the critical producing rate:


where Δρ = density difference (g/cm3), Bo = average oil FVF, μo = average oil viscosity (cp), ko = oil permeability (md), qDc = dimensionless critical producing rate, h = pay thickness (ft), and qc is given in STB/D. The oil permeability, ko, is the product of the horizontal permeability and the oil relative permeability. The dimensionless critical rate, qDc, is specified by correlation.

Schols’ Correlation

Schols’ correlation[71] is based on a numerical simulation study. The dimensionless critical producing rate is


where b is the length of the perforation interval and re is the drainage radius.

This correlation applies to both water and gas coning; however, it directly applies only to cases in which water or gas coning exist separately. In other words, it does not directly apply to cases in which water and gas coning act simultaneously. The correlation can be used to predict the critical rate for a pre-existing completion or to predict the optimum perforation length for a future completion. In this latter application, the optimum perforation length is defined as the length at which the critical and theoretical producing rates are equal. The theoretical producing rate of a partially penetrating well is computed in Section 9.11.3. Example 9.5 illustrates the former application while Example 9.6 illustrates the latter application.

Chierici et al. Correlation

The correlation by Chierici et al.[66] was based on a potentiometric study. This was one of the most sophisticated correlations. It allows the vertical permeability to differ from the horizontal permeability. This can be an important factor because coning vanishes as the vertical permeability approaches zero. This correlation also treats the problem of simultaneous gas/water coning. This is important in situations in which a gas cap and bottomwater coexist.

The correlation of Chierici et al. was specified in terms of a series of charts. The charts used the following nomenclature. The dimensionless critical rate is denoted as ψ (previously defined as qDc) and the following dimensionless variables are defined as




and RTENOTITLE....................(9.82)

The dimensionless critical rate, ψ, is a function of rDe, ε, and δ. Figs. 9.38 through 9.44 show the charts. Each chart corresponds to a different value of rDe. Specific charts exist for rDe = 5, 10, 20, 30, 40, 60, and 80.

The chats are used differently depending on whether they are used to compute the critical rates or optimize the perforation length. The problem of optimizing the perforation length for simultaneous gas and water coning is more complicated than separate water or gas coning. The charts simplify the problem; however, the solution procedure depends on the application.

Calculating Critical Rates. For a pre-existing perforation length, b, the critical rates are computed with the following procedure:

  1. Compute δg, δw, ε, and rDe with Eqs. 9.79 through 9.82.
  2. Locate the correct chart or charts, depending on the value of rDe. Interpolation between two charts may be required.
  3. Compute ψg from the charts based on rDe, ε, and δg; then, compute ψw from the charts based on rDe, ε, and δw.
  4. Compute the critical rates to avoid water and gas coning with Eq. 9.77.

This procedure ignores the curves labeled Δρogρwo. The calculation procedure is simplified if only bottomwater or a gas cap exists. If no bottomwater exists, then the calculation of ψw and δw can be ignored. Conversely, if no gas cap exists, then the calculation of ψg and δg can be ignored. Example 9.5 illustrates an application.

Calculating Optimum Perforation Length. For a bottomwater, gas-cap reservoir, the procedure to calculate the optimum perforation (length and position) uses the curves labeled Δρogρwo and is as follows:

  1. Compute rDe with Eq. 9.82. Also, compute Δρogρwo.
  2. Assume a value of ε.
  3. Compute ψ and δ with the charts. These values correspond to ψg and δg.
  4. Compute Lg = g and Lw = h(1 – ε) – Lg.
  5. Compute the critical rate from Eq. 9.77. Only one critical rate is needed because the procedure assumes equal critical rates for gas and water coning.
  6. Return to Step 2. Assume a new value of ε, and repeat the calculation until an adequate range of ε is covered.
  7. Compute the theoretical producing rate of a partially penetrating well as a function of ε with Eq. 9.83. (See Section 9.11.3 below).
  8. Plot the critical and theoretical producing rates as a function of ε . The value of ε at which the critical and producing rates intersect yields the optimal perforation interval.

This procedure is simplified if only a gas cap or bottomwater exists. The same procedure applies except Δρogρwo is not calculated and δg = 1 – ε (if there is no bottomwater and only gas coning is a problem) or δw = 1 – ε (if there is no gas cap and only water coning is a problem). ψ is computed for a range of ε until an optimal value of ε is identified. Example 9.6 illustrates this procedure for a gas-cap reservoir. Example 9.7 illustrates an application for a gas-cap, bottomwater reservoir.

Partially Penetrating Wells

The theoretical producing rate of a partially penetrating well is needed to compute the optimum perforation length. Partially penetrating wells are wells that do not fully penetrate or are not fully perforated throughout the pay thickness. If vertical permeability exists, these wells will produce fluids from above and below the perforations. Fig. 9.45 illustrates fluid delivery into a partially penetrating producing well. Under these circumstances, fluid flow is obviously not strictly horizontal. The producing rate in partially penetrating wells with nonzero vertical permeability is greater than the rate with no vertical permeability.

Partially penetrating wells commonly are used to minimize coning. The critical rate gives a producing rate below which no coning will occur. Often, however, the critical flow rate is much less than maximum possible flow rate. To judge the difference, estimates for the flow rate of a partially penetrating well are needed. Several authors have offered analytical expressions to estimate the flow rate. [1][74][75] Most efforts yield estimates within a few percent of one another. The Kozeny expression, [75] for example, is


where kH = horizontal permeability, kv = vertical permeability, pe = pressure at drainage radius, pw = wellbore pressure, s = skin factor, q = oil producing rate, and the argument of the cosine assumes radians. This equation assumes units of md, ft, psi, cp, and STB/D. This equation also assumes steady-state flow in a circular drainage area, where re and rw are the drainage and wellbore radii, respectively. Eq. 9.83 gives the Kozeny equation in SI units.

Variables Affecting Coning

The ratio of qc/q is a measure of the tendency not to cone. As q c increases or q decreases, the likeliness to avoid increases. According to Eqs. 9.77 and 9.83, the ratio qc/q is proportional to


This expression shows that the likeliness to control coning increases as the penetration interval b decreases. Eq. 9.84 also shows that the likeliness to control coning increases as the pay thickness increases, density difference increases, well spacing increases, and perforation length decreases. Horizontal permeability does not affect the likelihood of success. This expression also suggests that controlling coning in a thin reservoir may be difficult.

Additional Measures To Control Coning

Other techniques have been applied to control coning. These include placing an artificial barrier above or below the pay to suppress vertical flow, [76] injecting oil to control gas coning, [77] or the use of horizontal wells. Barriers composed of cement and high-molecular-weight polymers have been tried. Another, although expensive, technique is to drill additional wells and produce them at the critical rate.

Example 9.5: Computing Critical Rate To Prevent Coning Compute the critical rate (STB/D) for a well in a gas-cap reservoir with the following characteristics: kH = 60 md, kv = 39 md, h = 150 ft, re = 933 ft (80-acre well spacing), rw = 0.5 ft, Bo = 1.25 RB/STB, μo = 1.11 cp, ρo = 0.714 g/cm3, and ρg = 0.098 g/cm3. The well is completed in only the lower 60 ft of the pay thickness.

Solution. Schols’ Correlation. With Eq. 9.78, compute qDc. This yields


The critical rate is then computed with Eq. 9.77, which yields


Chierici et al. Correlation. First, evaluate Lg = 90 ft, δg = 90/150 = 0.60, ε = 60/150 = 0.40, and rDe = 5.01. With the chart for rDe = 5, obtain ψ = 0.120. Evaluating Eq. 9.77 yields


Example 9.6: Computing Optimum Perforation Length To Prevent Coning

For an uncompleted well in the gas-cap reservoir in Example 9.5, compute the optimum perforation length at a reservoir pressure of 1,800 psia. Assume a wellbore pressure of 1,500 psia and a skin factor of 10.

Solution. Schols’ Correlation. Compute the dimensionless critical rate and critical rate as a function of b/h. Table 9.28 summarizes the results for b/h = 0 to 1.

Next, estimate the theoretical producing rate for a partially penetrating well. With Eq. 9.83 for b = 60 ft yields


Table 9.28 summarizes the results at other values of b. The rate varies from 0 to 808 STB/D, depending on the length of the perforation interval. The optimum perforation length corresponds to the value of b at which the critical and theoretical producing rates are equal. Fig. 9.46 shows a plot of q and qc vs. b/h. The curves intersect at approximately b/h = 0.15. This corresponds to b = 22.5 ft, Lg = 127.5 ft, and q = 187 STB/D. In conclusion, only the lower 22.5 ft of the pay thickness should be perforated to avoid gas coning.

Chierici et al. Correlation. First, compute δg = 1– b/h for a range of b/h. Next, compute rDe. From Example 9.5, rDe = 5. With the appropriate chart, compute ψ for the range of b/h. The charts use b/h as ε. Then compute the critical rate with Eq. 9.77. Table 9.29 summarizes the results. Fig. 9.46 plots qc vs. b/h.

The optimal perforation length corresponds to the value of b/h at which the critical and producing rates are equal. This approximately occurs for b/h = 0.25. This corresponds to b = 37.5 ft, Lg = 112.5 ft, and q = 275 STB/D. In conclusion, only the lower 37.5 ft of the pay should be perforated. The method of Chierici et al. yields a wider perforation interval than the method of Schols (37.5 vs. 22.5 ft). The Chierici et al. method is consistently more liberal than Schols’ method.

Example 9.7: Optimum Perforation Length To Prevent Coning in a Bottom-Water Gas-Cap Reservoir Assume the gas-cap reservoir in Examples 9.5 and 9.6 is underlain by water. The water density is 1.092 g/cm3. Compute the optimum perforation length and position of the perforation interval with the Chierici et al. correlation.

Solution. First, compute Δρogρwo. This yields (0.741 – 0.098)/(1.092 – 0.741) = 1.83. Next, determine δg and ψ from the charts for a range of ε. If ε = 0.40, for example, use the chart corresponding to rDe = 5 to determine that δg = 0.24 and ψ = 0.040. Table 9.30 summarizes the results for a range of ε values from 0.05 to 0.40. Next, compute the critical rate for each value of ψ with Eq. 9.77. Finally, compute the theoretical producing rate with Eq. 9.83.

An examination of Table 9.30 shows that the critical and producing rates are equal at ε = b/h = 0.075. This value of b/h corresponds to Lg = (0.385) (150) = 57.8 ft, b = (0.075) (150) = 11.3 ft, and Lw = 150 – 57.8 – 11.3 = 81 ft. In conclusion, the well should be perforated with an 11.3-ft interval located 57.8 ft below the GOC and 81 ft above the WOC.


a = constant
b = perforation length, L, ft
Bg = gas FVF, RB/scf
Bo = oil FVF, RB/STB
Btg = two-phase gas FVF, RB/scf
Bto = two-phase oil FVF, RB/STB
Btw = two-phase water/gas FVF, RB/STB
Bw = water FVF, RB/STB
cf = rock compressibility, Lt2/m, 1/psi
ct = total aquifer compressibility, Lt2/m, 1/psi
Ef = rock (formation) expansivity
Eg = gas expansivity, RB/scf
Egw = expansivity for McEwen method, RB/scf
Egwf = composite gas/water/rock FVF, RB/scf
Eo = oil expansivity, RB/STB
Eow = expansivity for McEwen method, RB/STB
Eowf = composite oil/water/rock FVF, RB/STB
Et = total expansivity, RB/STB
Ev = volumetric sweep efficiency, fraction
Ew = water expansivity, RB/STB
f = reservoir angle
F = total fluid withdrawal, L3, RB
G = total original gas in place, L3, scf
Gfgi = initial free gas in place, L3, scf
Gi = cumulative gas injected, L3, scf
Gp = cumulative produced gas, L3, scf
Gps = cumulative produced sales gas, L3, scf
h = pay thickness, L, ft
J* = constant
k = permeability, L2, md
ka = aquifer permeability, L2, md
kH = horizontal permeability, L2, md
kt = time constant, 1/t, 1/years
kv = vertical permeability, L2, md
K = gas/oil equilibrium ratio or constant
La = aquifer length, L, ft
Lg = perforation gap between uppermost perforation and GOC, L, ft
Lr = reservoir length, L, ft
Lw = perforation gap between lowermost perforation and WOC, ft, L
m = initial free-gas-phase to free-oil-phase volume ratio (gas-cap size)
n = total number of data points
N = total original oil in place, L3, STB
Nfoi = initial free oil in place, L3, STB
Ng = dimensionless gravity number
Np = cumulative produced oil, L3, STB
p = pressure, m/Lt2, psi
RTENOTITLE = time-averaged pressure, m/Lt2, psi
RTENOTITLE = average reservoir pressure, m/Lt2, psi
RTENOTITLE = average aquifer pressure, m/Lt2, psi
pe = pressure at drainage radius, m/Lt2, psi
pw = wellbore pressure, m/Lt2, psi
q = producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L3/t
qc = critical coning rate, STB/D, L3/t
qDc = dimensionless critical coning rate
qsg = surface producing gas rate, L3/t, scf/D
qso = surface producing oil rate, L3/t, STB/D
qw = aquifer delivery rate, L3/t, RB/D
ra = aquifer radius, L, ft
rDe = dimensionless radius including vertical permeability
re = reservoir drainage radius
reD = ratio of reservoir to aquifer radii
ro = effective reservoir radius, L, ft
rw = wellbore radius, L, ft
R = instantaneous producing GOR, scf/STB
Rj = residual for point j, L3, RB
Rp = cumulative GOR, scf/STB
Rps = cumulative sales GOR, scf/STB
Rs = dissolved GOR, scf/STB
Rss = sum of squares of the residual, L6, RB2
Rsw = dissolved-gas/water ratio, scf/STB
Rv = volatilized-oil/gas ratio, STB/MMscf
s = skin factor
Sg = gas saturation, fraction
Sgrw = residual gas saturation to water flow, fraction
Sorw = residual oil saturation to water flow, fraction
Swi = initial water saturation, fraction
t = time, t, years
tmax = maximum time, t, years
tD = dimensionless time
tD* = dimensionless time boundary between infinite-acting and finite aquifers
tDmax = maximum dimensionless time
U = aquifer constant, L4t2/m, RB/psi
Vfgi = initial volume of free gas, L3, RB
Vfoi = initial volume of free oil, L3, RB
Vp = reservoir PV, L3, RB
Vpa = aquifer PV, L3, RB
Vpi = initial reservoir PV, L3, RB
Vpr = reservoir PV, L3, RB
w = reservoir width, L, ft
W = initial water in place, L3, STB
WD = dimensionless cumulative water influx
We = cumulative water influx, L3, RB
WI = cumulative injected water, L3, STB
Wp = cumulative produced water, L3, STB
δg = dimensionless perforation gap below GOC
δw = dimensionless perforation gap above WOC
Δp = difference of time-averaged pressure, m/Lt2, psi
Δρ = density difference, m/L3, lbm/ft3 and g/cm3
Δρog = oil/gas density difference, m/L3, g/cm3
Δρwo = water/oil density difference, m/L3, g/cm3
ε = b/h
ϕa = aquifer porosity, fraction
ϕr = reservoir porosity, fraction
θ = angle
μg = gas viscosity, m/Lt, cp
μo = oil viscosity, m/Lt, cp
μw = water viscosity, m/Lt, cp
ρg = gas density, m/L3, g/cm3
ρo = oil density, m/L3, g/cm3
ρw = water density, m/L3, g/cm3
ψ = dimensionless critical rate
ψg = dimensionless critical gas coning rate
ψw = dimensionless critical water coning rate


i = initial condition
j = index
k = index


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General References

Amyx, J.W., Bass, D.M., and Whiting, R.L. 1960. Petroleum Reservoir Engineering—Physical Properties. New York City: McGraw-Hill Book Co. Inc.

Archer, J.S. and Wall, P.G. 1992. Petroleum Engineering: Principles and Practice. London: Graham and Trotman.

Bradley, H.B. 1987. Petroleum Engineering Handbook. Richardson, Texas: SPE.

Calhoun, J.C. Jr. 1951. Fundamentals of Reservoir Engineering. Norman, Oklahoma: U. of Oklahoma Press.

Caudle, B.H. 1967. Fundamentals of Reservoir Engineering—Parts 1 and 2. Richardson, Texas: SPE.

Clark, N.J. 1960. Elements of Petroleum Reservoirs. Dallas, Texas: Society of Petroleum Engineers of AIME.

Craft, B.C. and Hawkins, M.F. Jr. 1959. Applied Petroleum Reservoir Engineering. Upper Saddle River, New Jersey: Prentice-Hall Inc.

Craft, B.C., Hawkins, M.F. Jr., and Terry, R.E. 1991. Applied Petroleum Reservoir Engineering, second edition. Upper Saddle River, New Jersey: Prentice-Hall Inc.

Dake, L.P. 1978. Fundamentals of Reservoir Engineering. New York City: Elsevier Scientific Publishing Co.

Guerrero, E.T. 1951. Practical Reservoir Engineering. Tulsa, Oklahoma: Petroleum Publishing Co.

Hagoort, J. 1988. Fundamentals of Gas Reservoir Engineering. New York City: Elsevier.

Katz, D.L . et al. 1959. Handbook of Natural Gas Engineering. New York City: McGraw-Hill Book Co. Inc.

Koederitz, L.F., Harvey, A.H., and Honarpour, M. 1989. Introduction to Petroleum Reservoir Analysis. Houston, Texas: Gulf Publishing Co.

Muskat, M. 1949. Physical Principles of Oil Production. New York City: McGraw-Hill Book Co. Inc. National Human Resources Development Corp., reprinted (1981).

Pirson, S.J. 1958. Oil Reservoir Engineering. New York City: McGraw-Hill Book Co. Inc.

Smith, C.R., Tracy, G.W., and Farrar, R.L. 1992. Applied Reservoir Engineering—Vols. 1 and 2. Tulsa, Oklahoma: Oil and Gas Consultants Intl.

Standing, M.B. 1951. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems. Dallas, Texas: SPE.

Timmerman, E.H. 1982. Practical Reservoir Engineering—Parts 1 and 2. Tulsa, Oklahoma: PennWell Books.

Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.

Appendix-Key Equations in SI Units







and RTENOTITLE....................(9.83)

where the following units apply: k is given md; Δρ is in g/cm3; b, h, r, and La are in m; q is in std m3/day; μ is in mPa-sec or cp; t is in years; c is in 1/kPa; U is in m3/kPa; and kt is in 1/years.

SI Metric Conversion Factors

acre × 4.046 856 E + 03 = m2
acre-ft × 1.233 489 E + 03 = m3
bbl × 1.589 873 E – 01 = m3
cp × 1.0* E – 03 = Pa•s
ft × 3.048* E – 01 = m
°F (°F − 32)/1.8 = °C
gal × 3.785 412 E – 03 = m3
lbm × 4.535 924 E – 01 = kg
psi × 6.894 757 E + 00 = kPa
sq mile × 2.589 988 E + 00 = km2
std ft3 × 2.831 685 E – 02 = m3


Conversion factor is exact.