Water-Injection Pressure Maintenance in Oil Reservoirs

Chapter 43

Gas-Injection Pressure Maintenance In Oil Reservoirs I.F. Roebuck Jr., Roebuck-Walton Inc.*

Introduction The first recorded deliberate attempt to stimulate recovery from an oil reservoir by hydrocarbon gas injection was in the Macksburg field, Washington County, OH, ’ long before water injection was used for secondary recovery purposes. For almost 60 years, most secondary recovery projects included some form of immiscible gas injection, and its use continued even after the advent of new methods and materials. In spite of this, it was the late 1940’s before serious attempts were made to develop quantitative techniques for describing reservoir performance under gas-injection operations, especially with regard to depleted oil reservoirs. Before then, such efforts were directed primarily toward describing the water displacement process.

As a result, techniques used to describe the performance characteristics of immiscible gas injection consist of modifications to methods originally developed for describing performance of water-injection operations, even though there is a fundamental difference in the basic displacement mechanisms of the two fluids. Such modifications, therefore, include the effects of gas solution in the reservoir oil, vaporization of lighter hydrocarbons from oil, or both.

Physical criteria for successful gas-injection operations are basically the same as for other types of fluid injection: the same physical and thermodynamic variables control the displacement process. As in all engineering investigations, pertinent variables must be defined, evaluated, and applied by the investigative techniques available and with a knowledge and awareness of the limitations of the techniques and the accuracy and reliability of the data and information at hand.

Gas injection has been used to maintain reservoir pressure at some selected level or to supplement natural

‘Or!gmal chapter m 1962 edmon. Part 1. Gas-Injection Pressure Maintenance, was wnnen by I F. Roebuck Jr. and Kenneth M Garms. Part 2, Msclble D!splacement, IS now a separate chapter (see Chap 45).

reservoir energy to a lesser degree by reinjection of a portion of the produced gas. Complete or partial pressure-maintenance operations can result in increased hydrocarbon recovery and improved reservoir production characteristics.

The quantity of additional liquid hydrocarbons that can be recovered from a reservoir is influenced by several characteristics of the particular reservoir, including reservoir rock properties, reservoir temperature and pressure, physical and compositional properties of the reservoir fluids, type of reservoir drive mechanism, reservoir geometry, sand continuity, structural relief, rates of production, and fluid saturation conditions.

Basically, increased hydrocarbon recovery can be at- tributed to the oil displacement and vaporization action of the injected gas and, in some cases. to the prevention of losses in recovery that would occur if pressure were not maintained. The conservation aspects of gas- injection pressure-maintenance operations can be particularly important with reservoirs containing volatile high-shrinkage crude oils and with gas-cap reservoirs containing large quantities of retrograde condensate gas. Gas injection has also been employed frequently to pre- vent migration of oil into a gas cap in oil reservoirs with natural water drives, with downdip water injection, or both. Other uses of gas injection in high relief reservoirs have been to enhance gravity drainage processes and to recover so-called attic oil residing above the uppermost oil-zone perforations.

Improvements in reservoir producing characteristics may, in some cases, be sufficient justification to initiate gas-injection operations even though a competitive recovery process might be used to achieve greater ultimate hydrocarbon recovery. Decreased depletion time resulting from pressure-maintenance operations can have a significant influence on the economic justification for gas injection. Decreased reservoir oil viscosity and gas saturation in the vicinity of the wellbore tend to

43-2 PETROLEUM ENGINEERING HANDBOOK

maintain individual well productivities, and producing wells are generally more able to maintain their desired producing rates or allowables. Further advantages can be obtained by elimination of penalties imposed by regulatory agencies for excessive net gas production where produced gas is not reinjected. Thus, many times it is possible to maintain full-field allowables over most of the producing life of the project, thereby reducing the depletion time of the reservoir, with attendant savings in operating costs and increased present value of future revenues.

Since 1978 and the passage of the Natural Gas Policy Act. the increasing value of sales gas has resulted in a decline in the numbers of new gas-injection projects. However, some opportunities still exist in remote areas where recovery considerations are augmented by the storage aspects of such projects and by specialized applications in connection with gravity drainage systems and attic oil recovery projects.

Concurrent with this, CO2 and nitrogen injection for miscible displacement of crude oil have been of increasing interest and application. On the basis of both economic and technical considerations, it is not unreasonable to expect that immiscible nitrogen- injection projects will see increasing application in many oil reservoirs that in the past would have been subjected to hydrocarbon-gas injection. In general, calculation techniques previously developed for hydrocarbon-gas injection and displacement can be used for the design and application of nitrogen-injection projects under conditions of immiscible displacement.

It is the purpose of this chapter to point out the physical criteria for successful gas-injection operations, to describe the variables that must be defined and evaluated, and to demonstrate some of the techniques available for the prediction and evaluation of field performance under immiscible gas-injection operations.

Most of the calculations described are now ac- complished with hand calculators or digital computers; many of them can be applied with relatively basic varieties of today’s generation of microcomputers. At the same time, the physical and mathematical relationships described have been incorporated into a wide varie- ty of mathematical reservoir simulation models. The for- mulation and application of such models is beyond the intended scope of this chapter, but a few selected references to technical articles describing models for gas-injection processes are included in Appendix B.

The calculation techniques described here are the classical methods for describing immiscible displacement with complete pre-equilibrium between the injected and displaced phases, gas and oil, while accounting for the effects of reservoir heterogeneities, injectioniproduc- tion well configurations, and differing physical characteristics of the fluids. The reservoir is treated in terms of the average properties of a unit volume of rock, and production performance is described on the basis of an average well.

The simplest types of so-called reservoir simulation models employ essentially these same techniques but, by means of one-, two-, or three-dimensional cell arrays, account for area1 and vertical variations in rock and fluid properties, well-to-well gravity effects, and individual well characteristics.

More complex component or compositional models allow also for nonequilibrium conditions between injected and displaced fluids and can be used to describe individual well streams in terms of the compositions of the produced fluids.

The accuracy and reliability of the results obtained generally increase with each of these methods, or models, in the order described, depending on the quantity and quality of the reservoir and fluid data available, the internal variations in reservoir properties, the fluid characteristics, and the ability to describe the overall physical system. The time and worker requirements, and hence the cost of the study, also increase in the same order.

Therefore, the choice of a method for describing project performance is a matter of judgment, considering economics, the time available, and the requirements for accuracy in a practical sense. Obviously, these requirements will vary with the phase of work undertaken and the overall purpose of the study at hand. Certainly, early feasibility studies usually can and should be made with nothing more than the simple, classical techniques. Such is also the case for many detailed studies where the effects of gravity and phase equilibrium are negligible or when the quantity and quality of data are inadequate to support more complex full-scale simulation studies.

Types of Gas-Injection Operations Gas-injection pressure-maintenance operations are generally classified into two distinct types depending on where in the reservoir, relative to the oil zone, the gas is introduced. Basically, the same physical principles of oil displacement apply to either type of operation; however, the analytical procedures for predicting reservoir performance, the overall objectives, and the field applications of each type of operation may vary considerably.

Dispersed Gas Injection Dispersed gas-injection operations, frequently referred to as internal or pattern injection, normally use some geometric arrangement of injection wells for the purpose of uniformly distributing the injected gas throughout the oil-productive portions of the reservoir. In practice, injection-well/production-well arrays vary from the conventional regular pattern configurations (e.g., five-spot, seven-spot, nine-spot) to patterns seemingly haphazard in arrangement with relatively little uniformity over the injection area. The selection of an injection arrangement is usually based on considerations of reservoir configura- tion with respect to structure, sand continuity, permeability and porosity variations, and the number and relative positions of existing wells.

This method of injection has been found adaptable to reservoirs having low structural relief and to relatively homogeneous reservoirs having low specific permeabilities. Because of greater injection-well density, dispersed gas injection provides rapid pressure and production response-thereby reducing the time necessary to deplete the reservoir. Dispersed injection can be used where an entire reservoir is not under one ownership, particularly if the reservoir cannot be conveniently unitized.

Some limitations to dispersed-type gas injection are: (1) little or no improvement in recovery efficiency is derived from structural position or gravity drainage, (2)

GAS-INJECTION PRESSURE MAINTENANCE IN OIL RESERVOIRS 43-3

area1 sweep efficiencies are generally lower than for external gas-injection operations, (3) gas “lingering” caused by high flow velocities generally tends to reduce the recovery efficiency over that which could be ex- pected from external injection, and (4) higher injection- well density contributes to greater installation and operating costs.

External Gas Injection External gas-injection operations, frequently referred to as crestal or gas-cap injection, use injection wells in the the structurally higher positions of the reservoir-usually in the primary or secondary gas cap. This manner of injection is generally employed in reservoirs having significant structural relief and average to high specific permeabilities. Injection wells are positioned to provide good area1 distribution of the injected gas and to obtain maximum benefit of gravity drainage. The number of injection wells required for a specific reservoir will generally depend on the injectivity of each well and the number of wells necessary to obtain adequate area1 distribution.

External injection is generally considered superior to dispersed-type injection, since full advantage can usually be obtained from gravity drainage benefits. In addition, external injection ordinarily will result in greater area1 sweep and conformance efficiencies than will similar dispersed injection operations.

Optimal Time to Initiate Gas Pressure- Maintenance Operations Generalizations as to the optimal time to initiate gas pressure maintenance are of limited practical value because of the exceedingly large number of variables that must be considered from an economic and reservoir mechanics standpoint. Obviously, there is no method of calculating directly the optimal time from an economic standpoint: instead, several calculations of future performance, assuming initiation of injection at various stages of reservoir depletion, must be made and compared on an economic basis.

Considering only hydrocarbon recovery and improvements in producing characteristics, it can be stated that generally more favorable reservoir conditions for gas-injection operations are present when the reservoir is at or slightly below the reservoir fluid saturation pressure. Within this range of reservoir pressures, the initial free-gas saturation in the oil zone is at a minimum-a condition favorable to obtaining maximum recovery efficiency from the gas displacement process.

Efficiencies of Oil Recovery by Gas Displacement It is convenient to analyze and evaluate the recovery efficiency obtainable by gas displacement operations in terms of three efficiency factors, generally referred to as (1) unit-displacement efficiency, (2) conformance efti- ciency, and (3) area1 sweep efficiency. Each recovery efficiency may be considered as one component element that accounts for the influence of certain parameters on the overall recovery efficiency of the displacement process. The product of the three efficiency factors provides an estimate of the percentage oil recovery that can be ex- pected with this recovery process in a particular reservoir

under specified conditions. Analytical procedures are available for evaluating each efficiency factor in- dividually. In certain instances, such analytical procedures are combined to determine two or more of the factors as a unit; for example, the term “volumetric efti- ciency ” is sometimes employed where the conformance and area1 sweep efficiencies are combined into one factor. Similarly, the term “displacement efficiency” is sometimes used where the unit displacement and conformance efficiencies are evaluated in combination. For the purpose of this chapter, the three components describing the overall recovery process are defined as follows.

1. Unit displacement eficiency is the percentage of oil in place within a totally swept reservoir-rock volume that is recovered as a result of the displacement process.

2. Conformance eficiency is the percentage of the total rock or pore volume within the swept area that is contacted by the displacing fluid.

3. Areal sweep e$iciency is the percentage of the total reservoir or pore volume that is within the swept area, the area contacted by the displacing fluid.

Each of the three efficiencies increases with continued displacement; therefore, each is a function of the number of displacement volumes injected. The rate of increase in recovery efficiency in a given portion of a reservoir diminishes as gas breakthrough occurs. Therefore, the maximum value of each component efficiency and, consequently, the ultimate recovery efficiency is limited by economic considerations.

Methods of Evaluating Unit-Displacement Efficiency Equations Unit-displacement efficiency is normally determined by analytical procedures developed from the two fundamental equations reported by Buckley and Leverett. * These equations essentially characterize the mechanics of steady-state, two-phase fluid flow encountered in oil displacement by an immiscible fluid. These equations were developed by means of relative-permeability concepts and are based on Darcy’s law describing steady- state fluid flow through porous media.

The so-calledfi-ucrional-flow equation describes quantitatively the fraction of gas flowing in terms of the physical characteristics of a unit element of porous media. In customary units, using a unit area, this equation is as follows.

fg =

1 + l.l27[k,A/(~,q,)][(aP,./aL)-0.433(p,, -p,<)sina]

1 +(~&,)(PL,k,)

. . . . . . . . . . . . . . . . (1)

where fs = fractional flow of gas, q, = total flow rate, B/D, A = cross-sectional area, sq ft,

P, = oil/gas capillary pressure (p, -p,), psi L = distance, ft,

PETROLEUM ENGINEERING HANDBOOK

-c---------- TERSTITIAL WATER

DISTANCE, L

Fig. 43.1-Schematic representation of saturation distribution during gas-displacement process.

ti a 50 % L 840

NOTE: V =VOLUME OF GAS (MEASURED UNDER RkERVOIR CONDITIONS) WHICH HAS INVADED UNIT CROSS SECTION OF OIL SAND.

OO I I I I I I

100 200 300 400 500 600 700 800 900 1000 DISTANCE, FT

Fig.43.2-Fluid saturation distribution at four time periods during gas displacement process.

PO = oil density, g/cm’, P,q = gas density, g/cm’,

CY= angle of dip, positive down-dip, degrees, Ii,, = effective permeability to oil, darcies,

k n, = relative permeability to oil, fraction, k,, = relative permeability to gas, fraction, PC = oil viscosity. cp, and PCS = gas viscosity, cp.

To relate the fraction of gas flowing to time, Buckley and Leverett developed the following material-balance equation.

L= 5.615 y,t aj-&!

~A (-1

, . . as,s (2)

where t = time, days,

4 = porosity, fraction, and S, = gas saturation, fraction.

The value of the derivative d(f,)/&S,) may be obtained for any value of gas saturation by plotting j, from Eq. 1 vs. S, and determining slopes at various points on the resulting curve. 3*4 This graphical procedure is generally considered to be sufficiently precise for most reservoir engineering calculations. It is especially suited where the calculations are to be made by hand calculators. A more precise mathematical procedure for evaluating the function a&,)/a(S,) was presented by Kern5 and is particularly adaptable for use with digital computers.

Figs. 43.1 and 43.2 illustrate the displacement process described by Eqs. 1 and 2. Calculated oil- and gas- saturation distributions for a hypothetical example of gas displacement after successive periods of injection are shown in Fig. 43.2. The area beneath any curve represents the gas-invaded zone, whereas the area to the right of the “gas front” at any time represents the uninvaded zone.

Modifications of Displacement Equations Eqs. 1 and 2 were developed on the basis of the following simplifying assumptions.

1. Steady-state flow conditions prevail. 2. Displacement takes place at constant pressure. 3. The displacing and displaced phases are in com-

positional equilibrium. 4. None of the injected gas is dissolved in the oil. 5. There is no production of fluids from behind the

gas front. 6. The advancing gas moves parallel to the bedding

planes of the formation. 7. The gas front moves uniformly through laminated

sands. 8. The interstitial water present is immobile. The applicability of the basic displacement equations

to a given reservoir is, of course, governed to a large extent by the restrictions imposed by the basic assumptions. Several authors have reported modifications to the displacement equations that eliminate the need for making certain of the assumptions. Modifications that take into consideration the swelling effects experienced from injection into an undersaturated reservoir and production of fluids from behind the gas front have been presented by Welge,3 Kern,’ Shreve and Welch,6 and others. Jacoby and Berry,’ Attra,8 and others have presented equations and analytical procedures for calculating per-

GAS-INJECTION PRESSURE MAINTENANCE IN OIL RESERVOIRS

formance where there is significant compositional inter- change of components between the displacing gas phase and the reservoir oil. The influence of deviations from the conditions described in Assumptions 6 and 7 is generally taken into consideration in the determination of conformance efficiencies.

Influencing Factors

Eqs. 1 and 2 provide a means for investigating the relative influence of the various parameters affecting unit-displacement efficiency. These factors are (1) initial saturation conditions, (2) fluid viscosity ratios, (3) relative-permeability ratios, (4) rate and formation dip, (5) capillary pressure, and (6) reservoir pressure and fluid properties.

Initial Saturation Conditions. Frequently, gas- injection operations are initiated after reservoir pressures have declined to such an extent as to permit the ac- cumulation of free gas released from solution in the oil. If the free-gas saturation exceeds the breakthrough or critical saturation determined from the fractional-flow curve, an oil bank ahead of the front will not be formed; consequently, oil production will be accompanied by im- mediate and continually increasing free-gas production. 2 This influence of initial mobile gas saturation on gas displacement performance has been demonstrated by laboratory investigations and mathematical analyses. 9 Fig. 43.3 shows a comparison of calculated and ex- perimentally determined gas displacement performance. It will be noted that approximately 10% oil recovery was attained prior to gas breakthrough where the initial gas saturation was zero, whereas with an initial gas saturation of 18.1% PV, a period of gas-free production was not observed.

The magnitude of the interstitial water saturation present in a reservoir, of course, influences the quantity of oil subject to gas displacement. It apparently does not have an influence on the breakthrough unit-displacement efficiency as determined by the fractional-flow equations, however. lo If the interstitial water saturation is a mobile phase, the displacement equations are not directly applicable since they were developed from concepts of two-phase flow. Approximations of gas displacement performance can usually be made where three phases are mobile by treating the water and oil phases as a single liquid phase. Displacement calculations can then be made with k,/k, data determined from core samples containing interstitial water saturation. Oil recovery can be differentiated from total liquid recovery on the basis of k,/k, data or by material-balance calculations incorporating an estimated minimum interstitial water saturation.

Fluid Viscosity Ratios. The effects of variations in oil viscosity on calculated unit-displacement efficiency can be seen from an examination of the curves presented in Fig. 43.4. Note that the oil recovery is significantly improved as the viscosity of the oil approaches that of the displacing gas. This indicates that the most efficient displacement will occur where the oil-to-gas viscosity ratio is unity or less.

“EXPERIMENTAL DATA -PREDICTED PERFORMANCE

IO-

10 -

o-

O-

/

0 IITI

Fig. 43.3-Comparison of calculated and experimental gas- injection performance for two conditions of initial gas saturation.

Rate and Formation Dip. Note from Eq. 1 that several factors influence the magnitude of the gravity term. Since the fractional flow of gas decreases as the magnitude of the gravity term increases, maximum benefits from gravity segregation are obtained when the following occur.

1. Specific permeabilities and relative permeabilities to oil are high.

2. Reservoir oil viscosities are low and densities are high.

3. The cross-sectional area to flow is large. 4. The angle of dip is high (Fig. 43.5). 5. Injection and production rates are low. Frequently, the design of a gas-injection program can

have an appreciable effect on whether maximum advantage is obtained from gravity drainage in a given reservoir. For example, proper location and distribution of injection wells along the structurally high portions of the reservoir may in some cases increase the cross-sectional area to flow and take full advantage of maximum reservoir dip. Cap oil viscosities and relative oil permeabilities are favorable when pressures are highest. In addition, injection and production rates, in terms of reservoir withdrawals, are generally lowest at high reservoir pressures, indicating that maximum benefits from gravity drainage can be achieved by initiating gas- injection operations early in the life of a reservoir.

Relative-Permeability Ratios. It has been shown that the concepts of relative permeability can be applied equally well to complete or partial pressure-maintenance operations. t ’ Since relative-permeability ratio, along


o-

‘5-

80 -

15 -

OO 1

10 20 30 40 50 60 70 80 90 too GAS SATURATION-PER CENT

Fig. 43.4-Effect of oil viscosity on fractional flow of gas.

with viscosity ratio, fixes the relative portions of gas and oil flowing at any given saturation condition, it is one of the more important factors influencing unit-displacement efficiency. Relative permeability is a characteristic of the reservoir rock and is a function of fluid-saturation conditions; therefore, an operator has no control over the relative-permeability characteristics of a given reservoir. However, because of the significant influence that this factor has on the performance of gas-displacement operations, it is important that calculations be based on dependable data obtained from laboratory analyses of core samples. If possible, the laboratory-determined data should be supplemented by relative permeabilities calculated from field performance data.

Capillary Pressure. Capillary-pressure forces tend to oppose the forces of gravity drainage and, as a result, tend to decrease unit gas displacement efficiency. At ex- tremely low rates of displacement where frictional factors become negligible, the saturation distribution may be controlled to a large extent by the balance between capillary and gravitational forces. However, at the rates of displacement normally employed in practice, it is generally considered that in most cases capillary forces, or capillary-pressure gradients, can be neglected without seriously detracting from the utility of the analysis.

Reservoir Pressures and Fluid Properties. In certain highly undersaturated reservoirs, particularly those containing high-gravity crude oils that are to some degree volatile, the unit-displacement efficiency can be increased by initiating pressure-maintenance operations at the highest pressure possible. Under the proper conditions of pressure and fluid composition and at the proper degree of undersaturation, a miscible-fluid displacement can be achieved by use of relatively “dry” injection gas. The mechanics of this process, which reportedly achieves unit-displacement efficiencies approaching

SA;:RAT::N - P6: 80 90

Fig. 43.5-Effect of formation dip on fractional flow of gas.

lOO%, will be considered more in detail in Chap. 45. Recovery efficiency often can be improved by gas injection at high reservoir pressures even though miscibility is not achieved. This improvement in recovery may be a result of (1) swelling or expansion of the undersaturated reservoir oil resulting from addition of dissolved gas, (2) reduction of the oil viscosity from addition of dissolved gas, and (3) vaporization of the residual oil and subse- quent recovery from the produced gas. I2

Laboratory data obtained from tests using samples of reservoir fluid and injection gas are necessary to evaluate quantitatively the degree of swelling and vaporization that will take place under specified reservoir conditions. These data may be used in conjunction with conventional material-balance, compositional-balance, and displacement equations to arrive at an estimate of unit- displacement efficiency.

Calculation Procedures Example procedures for calculating displacement efficiency are included in Appendix A for the cases of horizontal and vertical (downdip) flow of displacing gas.

Methods of Evaluating Conformance Efficiency Several methods have been advanced for evaluating the conformance efficiency for a given reservoir. Generally, all the methods are somewhat empirical and are based on either comparisons of calculated and observed past displacement performance or statistical analyses of core- analysis data.

If a displacement process such as gas-cap expansion or pilot injection operations has been operative in a reservoir long enough to yield sufficient and reliable data concerning the position of the gas front and recovery as a function of time, past reservoir performance can be used to calculate conformance efficiency. The basic premise


for this type of analysis is that the conformance efficiency is the predominant factor responsible for deviations between actual displacement performance and the ideal or theoretical. On this basis, the conformance efficiency is calculated by dividing the observed recovery at various time intervals by theoretical recovery for correspond- ing time periods. Theoretical recovery may be determined from unit-displacement-efficiency calculations including an appropriate areal sweep efficiency. The conformance efficiencies thus determined may then be empirically cor- related with either rate of production or percent recovery to determine an average value or trend for use in making future performance predictions.

Several authors have presented methods for determining conformance efficiencies based on statistical treatments of core-analysis data. Perhaps the most frequently used is an adaptation of the method presented by Stiles I3 for evaluating the effect of permeability variations on waterflood performance (see Chap. 44). Conformance-efficiency calculations for miscible-fluid displacement using this analytical technique are presented in Chap. 45. The same calculation procedures may be used when immiscible gas displacement is considered, except that the relative-permeability ratio k,/k, must be considered for immiscible gas displacement, whereas it is not applicable to miscible displacement. The relative-permeability ratio used in such calculations is considered to be constant and is generally taken to be the relative permeability to gas at residual oil saturation divided by the relative permeability to oil at initial gas saturation.

Influencing Factors

The conformance efficiency for a given reservoir is largely controlled by the influence of (1) variations in rock properties, (2) mobility ratios, and (3) gravity segregation.

Variations in Rock Properties. Reservoir-rock porosity and permeability vary from one pore channel to the next. In addition, reservoir rock almost universally is formed in layers-stratified-either to a small extent or over large distances. Stratification can be merely differences in porosity and permeability of layers in capillary equilibrium or can be separations caused by im- permeable shale or other rock streaks. Variations in porosity and permeability can be both vertical and horizontal. All these rock heterogeneities tend to decrease the effective size of the reservoir as far as displacement operations are concerned. Therefore, the degree of heterogeneity controls to a large extent the conformance efficiency attainable from gas-injection operations in a given reservoir.

Mobility Ratios. The mobility of a fluid is an index of the ease with which the fluid will flow under specified conditions. Herein, mobility is defined as the relative permeability to a fluid at a given saturation divided by the fluid viscosity. Mobility ratio, M, is an index of the ease with which one fluid will flow relative to another fluid. It is defined herein as the ratio of the gas mobility to the oil mobility or, in equation form,

A4=p “0, . . . . . . . . . . . . . . . . ..I...... (3) m Fg

with permeabilities and viscosities as before. If the mobility ratio is equal to unity, it indicates that,

for a given pressure differential, oil and gas will flow with equal ease; values greater than unity indicate that gas will be the more mobile fluid, etc. During the gas displacement process, mobility ratio can vary from essentially zero during periods of low gas saturation to values approaching infinity during the periods of high gas saturation.

In heterogeneous reservoir-rock systems, relati<e- permeability characteristics may be extensively variable both laterally and vertically. As a result, the displacing gas will not form a uniform front as it advances but will tend to “finger” ahead in the layers or areas having higher mobility ratios. As the displacement progresses, the mobility ratio continues to increase in the portions of the reservoir previously contacted by displacing gas. As a result, there is a decreasing tendency for gas to enter regions of low permeability or regions of low gas saturation. These volumes are therefore bypassed and little or rio oil is recovered from them. It can be seen that the factors tending to increase the mobility ratio also tend to ac- centuate the detrimental effects of sand heterogeneity on conformance efficiency.

High localized injection and production rates in the presence of adverse mobility ratios and sand heterogeneity can add to the severity of gas channeling and resultant bypassing of oil. The possibility of creating this adverse effect frequently can be reduced through proper selection of the number and location of injection wells and proper scheduling of fluid withdrawals so that minimum pressure drawdown is created in the vicinity of the advancing gas front.

Gravity Segregation. As was previously mentioned, gravity forces tend to improve unit-displacement efficiency. Gravity drainage has essentially the same influence on conformance efficiency, and its effectiveness is controlled by the same factors-i.e., rate, angle of dip, vertical permeability, etc. Under favorable conditions, gravity drainage tends to maintain a more uniform gas front and therefore tends to offset the effects of adverse mobility ratios and permeability variations.

Under certain conditions, gravity segregation of the displacing and displaced fluids has an adverse effect on the conformance efficiency. In reservoirs having relatively good vertical communication, low formation dips, and slow displacement rates, the gas tends to segregate to the top of the formation, bypassing oil in the lower portions and creating a so-called umbrella effect, which causes premature breakthrough of the gas and a lowering of conformance efficiency.

Methods of Evaluating Area1 Sweep Efficiency Several investigators have shown that the area1 sweep efficiency for a given reservoir is controlled to a large extent by (1) injection/production well arrangements with respect to reservoir geometry, (2) mobility ratio of the fluids involved, and (3) number of displacement volumes injected.

PETROLEUM ENGINEERING HANDBOOK
REClkOCAL MOBILITY RATIO, I/M
Fig. 43.6-Sweep efficiency as a function of mobility ratio.

6.0 80 IO

Applied mathematical techniques have been used to investigate the influence of these factors on regular geometrical reservoir units of constant thickness. On the other hand, various types of laboratory and numerical models have been used to study the effects on area1 sweep efficiencies of irregular reservoir boundaries, irregular well arrangements, variable formation thick- nesses, and variable mobility ratios. From these investigations, it generally can be concluded that the areal sweep efficiency at gas breakthrough will bc a maximum in a given reservoir when the mobility ratio is low and when the distance from injection to production well is large. After gas breakthrough, areal sweep efficiencies are improved as the number of injected displacement volumes increase. The influence of mobility ratio and displacement volumes injected on the area1 sweep efficiency of a regular five-spot reservoir unit may be seen in Fig. 43.6. The data presented in this illustration were obtained from model studies that used miscible fluids of various viscosities to study the influence of various mobility ratios. These data are generally considered to be applicable to reservoir analyses for either water or gas displacement when actual model studies for a given reservoir are not available.

Areal sweep efficiencies, calculated at gas breakthrough and at successive periods thereafter until the economic limit is reached, are required for estimating reservoir performance under pressure-maintenance operations. If the injection/production well arrangements and the fluid mobility ratios for a given reservoir closely approximate those that have been studied in the laboratory, the data on this subject reported in the literature may be used as a basis for estimating the areal sweep efficiencies. Data reported by Dyes et al. ” have been found particularly useful since consideration was given to the influence of production after gas breakthrough. Note that the quantitative applicability of laboratory data is inherently questionable because of uncertainties in model scaling, laboratory techniques, and associated simplifying assumptions. Nevertheless, laboratory-model studies still offer the most convenient means of determining quantitative data concerning areal

c DISPLACEMENT VOLUME

Fig. 43.7-Areal sweep efficiency as a function of injection fluid volume for a mobility ratlo of unity.

sweep efficiencies. For this reason, if mathematical model studies are not practical for the particular reservoir under consideration, published data (tempered by experience) must generally be resorted to as a basis for predicting areal sweep efficiencies even though the well arrangements being investigated do not duplicate those reported in the literature.

For application to performance predictions, it is frequently desirable to construct a curve showing the areal sweep efficiency for a given mobility ratio as a function of the fractional gas flow, fK, or the displacement volumes injected. For example, Fig. 43.7 shows a replot of the data presented in Fig. 43.6 for a mobility ratio of unity. If necessary, the trend established from these data may be adjusted up or down depending on the judgment of the engineer as to the applicability of the model to the reservoir under consideration.

As was discussed in a previous section of this chapter, during gas displacement operations there is a significant gradient in mobility ratios behind the gas front. Therefore, an average mobility ratio must be selected to determine areal sweep efficiencies from published data. Probably the most representative, and certainly the most conservative, value for this purpose is the mobility ratio determined at the average gas saturation behind the front according to the methods presented in connection with unit-displacement efficiencies.

Calculation of Gas Pressure-Maintenance Performance Estimates of gas-injection performance are generally based on the simultaneous solution of one or more forms of the conventional material-balance equations and the displacement equations previously discussed. The manner in which these equations are applied will vary depending on the scope of the investigation. the type of reservoir under consideration, and whether dispersed or external injection is to be used for complete or for partial pressure maintenance. Rigorous treatment of all factors influencing production performance and the displacement processes in a given reservoir can result in the development of calculation procedures that are quite


complex. Specific analytical techniques and procedures as applied to various types of reservoirs have been the subject of numerous articles in the technical literature and several reservoir engineering textbooks. A selected bibliography of technical articles dealing with specific analytical techniques and procedures that can be used for estimating reservoir performance under gas-injection operations is included in Appendix B. Note that these references are indexed according to the type of reservoir under consideration, the major influencing factors, and the type of injection-well arrangement. These references can be used as a basis for developing suitable analytical techniques to estimate future pressure-maintenance performance in any given reservoir. However, since each petroleum reservoir is unique, in the final analysis engineers must rely upon imagination and experience to develop techniques, based on fundamental theory, for the particular reservoir under consideration.

Although the equation forms and specific details of estimating reservoir performance will vary somewhat for each reservoir considered, certain general analytical procedures are common to most investigations and can be used as a basis for developing specific calculation techniques. A complete engineering analysis of a reservoir for the purpose of evaluating gas-injection operations will usually consist of four major phases: (1) assembly, preparation, and analysis of basic data; (2) analysis of past performance; (3) projection of future performance of current operations; and (4) estimation of gas pressure- maintenance performance.

Basic Data The need for adequate and comprehensive basic data has been emphasized in other chapters of this book and is ap- parent when it is realized that the validity and therefore the utility of any engineering analysis is determined primarily by the quality and quantity of basic data. The data requirements for analysis of gas-injection operations are, with few exceptions, the same as the requirements for analysis of other types of fluid-injection operations. Appendix C includes an outline of the usual data requirements for engineering analyses as presented by Pat- ton, I5 with certain additions and modifications.

Analysis of Past Performance The methods used to evaluate past reservoir performance will. of course, vary depending on the active reservoir drive mechanisms present, the quantity of suitable basic data available, and the amount of detail or scope of the investigation. Procedures for analyzing past reservoir performance are discussed in detail in other chapters. The results of such analyses will determine to a large extent the methods used for predicting gas-injection pressure-maintenance performance and will provide the current reservoir pressure and saturation distribution conditions for use in such predictions. Further, proper analysis of past performance will aid in supplementing and establishing the reliability of data required for the projection of reservoir performance under injection operations.

Projection of Future Performance of Current Operations Decisions regarding the installation of gas-injection

operations must be made on the basis of the relative benefits to be derived from such operations compared with competitive recovery techniques. Therefore, any complete analysis of gas-injection operations would include the projection of future reservoir performance under the current production operations. Methods of pro- jecting future primary production performance and other types of injection operations are discussed in detail in other chapters.

Estimation of Gas Pressure-Maintenance Performance Generally, projections of partial pressure-maintenance performance, for either external or dispersed-type gas injection, can be made by use of conventional material- and volumetric-balance techniques in combination with recovery efficiency determinations previously discussed. On the other hand, if complete pressure maintenance is being considered, the project performance can be estimated by only the displacement equations and other analytical procedures presented previously in connection with the discussions of unit displacement, conformance, and area1 sweep efficiency.

Procedures for calculating the future performance of both external and dispersed-type gas-injection operations are included in Appendix A. These example calculations include the determination of displacement efficiency and pressure, producing gas/oil ratio, and recovery performance for primary operations and for various degrees of gas-injection pressure maintenance for two idealized reservoirs.

Performance-Time Predictions. Predictions of future gas-injection performance are necessary for making economic comparisons of various types of future operations. Such predictions will usually include estimates of functions of time such as (1) reservoir pressures; (2) oil-, gas-, and water-production rates; (3) gas- and water- injection rates; (4) GOR’s; (5) cumulative oil, gas, and water recovery; (6) cumulative gas and water injected; (7) number of producing, injecting, and shut-in wells; and (8) recoverable plant products, if applicable.

To estimate these quantities, it is necessary to develop relationships between the hydrocarbon distribution of the subject reservoir and the positions of injection and production wells. Once this is done and with a given injection rate, Eq. 2 can be used to calculate the time necessary for the gas front to reach incrementally selected points in the reservoir.

In gas-cap-drive reservoirs and where external injection is being considered for reservoirs having significant structural relief, it is frequently convenient to relate hydrocarbon PV, cross-sectional area, and well completion intervals to subsea depth within the reservoir. If such relationships are used and if the advancing gas front is assumed to conform to structural depth, displacement equations and fluid inventory equations can be used to predict the rate of advance of the gas front, taking into consideration changes in cross-sectional area and reservoir productivity.

Until the gas front reaches the top of the perforations in the structurally highest well, oil and gas production is controlled by the productivities or allowables of the producing wells ahead of the front: and producing GOR’s

43.10 PETROLEUM ENGINEERING HANDBOOK

TABLE 4X1-BASIC RESERVOIR DATA

Oil reservoir having no original gas cap

Initial oil volume, /V. STB Average porosity, 6, % Average rock permeability. k, md Average interstitial water saturation, S w, % Initial bubblepoint pressure,p,, psig

Oil reservoir with original gas cap

Initial oil volume, N, STB Initial gas-cap gas volume, Mcf Area of gas/oil contact A, acres Ratio of gas-cap to oil-zone volume, m,

fraction Average porosity, 6, % Average rock permeability, k, md Average oil-zone waler saturation, S w. , % Average gas-cap water saturation, S wg , % Bubblepoint pressure at gas/oil level pb, psig

30,650,351 29.5

300.0 30.0

1,375

30,650,351 12,716,OOO

842

0.610 29.5

300.0 30.0 25.0

1,375

are controlled by gas-saturation conditions ahead of the front. If it is assumed that each producing well is shut in as gas breakthrough occurs, the producing GOR will re- main a function of oil-zone gas saturation, and the total OilLproducing rate and gas-injection rate will decline as the front reaches each successively lower-producing well. The oil-producing rate at any position of the gas front can be determined from the productivities or allowables of the wells in the uninvaded portions of the reservoir. If it is assumed that each well is produced to an economically limiting GOR prior to being shut in, production from behind the front must be accounted for by use of the modified displacement equations referred to previously. In such cases, a comprehensive fluid inventory is required to account for the portion of the injected gas being produced at any time and the portion that is advancing down structure. If partial-pressure- maintenance operations are being considered, it is necessary to introduce material-balance equations to calculate, by trial-and-error methods, the pressure decline and relative positions of the advancing gas front.

With complete pressure maintenance in reservoirs having low structural relief or where the gas front is likely to advance parallel to the bedding planes of the formation,

The basic reservoir rock and fluid data used throughout are presented in Tables 43.1 and 43.2 and in Figs. 43.8 and 43.9.

II. Unit Displacement A. Horizontal Gas Flow

1. Equation

1 fg =

where

fK = fractional gas flow, k, = relative permeability to oil at S,, k,, = relative permeability to gas at S,,

PO = oil viscosity at p, cp, and

px = gas viscosity at p, cp.

2. Procedure a. Calculate and construct a fractional-flow curve

for selected increments of gas saturation, S,, as indicated in Table 43.3 and Fig. 43.10.

the cumulative hydrocarbon distribution, cross-sectional area, and reservoir productivity can be related to distance from injection to production wells. Where dispersed gas injection is being considered, calculations can be made for a typical pattern element of the reservoir and the results applied to the total number of patterns present. Care should be taken to select a method of reservoir representation that will conform as nearly as possible to the anticipated frontal advance in a given reservoir.

APPENDIX A Example Calculations of Future Performance

I. Basic Data

Pressure

(Psg) pb = 1,375

1.300 1;200 1,100 1,000

900 800 700 600 500

400 300 200 100

0

TABLE 43.2-SUMMARY OF RESERVOIR-FLUID PROPERTIES

Oil-Volume Solution Gas-Volume Oil Gas Oil Gas Factor GOR Factor Viscosity Viscosity Density Density

1.210 430.1 0.00178 0.480 0.0148 0.765 1.200 414.9 0.00194 0.490 0.0146 0.766 1.186 397.0 0.00211 0.508 0.0143 0.767 1.173 379.0 0.00233 0.527 0.0140 0.769 1.160 361 .O 0.00258 0.544 0.0137 0.771

1.147 1.134 1.120 1.106 1.091

342.0 321.7 301 .o 277.9 254.9

0.00290 0.00329 0.003ao 0.00447 0.00540

0.564 0.587 0.609 0.633 0.661

0.0134 0.0132 0.0129 0.0126 0.0124

0.773 0.775 0.779 0.703 0.788

1.076 230.2 0.00677 0.692 0.0121 0.794 1.060 202.1 0.00904 0.729 0.0119 0.801 1.043 167.9 0.01339 0.773 0.0117 0.809 1.024 125.2 0.02545 0.832 0.0116 0.819

(gGl3) 0.084 0.082 0.079 0.076 0.073

0.068 0.062 0.056 0.050 0.043

0.035 0.027 0.018 0.009 0.001 1.001 0.0 0.19802 0.910 0.0114 0.835

GAS-INJECTION PRESSURE MAINTENANCE IN OIL RESERVOIRS 43-l 1

b. Construct the tangent to the& curve from the equilibrium gas saturation, S,(equal to zero in this case), and read the average gas saturation behind the front at breakthrough from the intercept where f8 = 1 .O. This average gas saturation S R corresponds to the oil recovery as a fraction of total pore volume behind the front.

c. Construct other tangents as required to obtain the average gas saturations and oil recoveries at various other values of fR or frontal gas saturations S,.

B. Downdip Gas Flow 1. Equation

fg = 1+0.489[k,b, -p,)sin dq, k,)l

1 +(krJkr~)(PLgh4

where & = fractional gas flow, k,, = effective permeability to oil, darcies,

P&S = gas density at p, g/cm”, P 0 = oil density at p, g/cm 3,

01 = angle of gas flow (-90”). q, = rate of frontal gas movement, B/D-sq ft.

k, = relative permeability to oil at S,, k % = relative permeability to gas at S,, P II = oil viscosity at p, cp, and

fi, = gas viscosity at p, cp.

2. Procedure a. By using a unit flow, calculate and construct a

fractional-flow curve for selected increments of gas saturation, S,, as indicated in Table 43.4 and in Fig. 43.11.

b. Construct the tangent to thef, curve from the equilibrium gas saturation, S,,, (equal to zero in this case), and read the average gas saturation behind the front at breakthrough from the- intercept where & = 1 .O. This average gas saturation S fi corresponds to the oil recovery as a fraction of the total swept pore volume.

c. Construct the tangent to the fs curve where fx becomes asymptotic to 1 .O and read the ultimate or maximum value for S R .

III. Dispersed Gas-Injection Pressure Maintenance

A. Partial Pressure Maintenance 1. Equations

AN,= (1 -N,;) A[(B,IB,)-R,,]-B~,bA(liB,)

[(B,IB,)-R,] +R(I -AG;) ’

I I I I I I I I

100 80 I

-1

-I I- L\ 4

Fig. 43.8-Reservoir volume and area as functions of height.

T

-

i G/

Flg. 43.9-Relative-permeability data

R = instantaneous GOR, scf/STB, R = average GOR, scf/STB,

R, = solution GOR, scf/STB, SL = total liquid saturation, fraction, S,, = interstitial water saturation, fraction, k 1 = relative permeability to gas at S1,, kz = relative permeability to oil at SL,

p”fi = gas viscosity at p, cp, PLO = oil viscosity at p, cp,

B, = oil FVF, RBISTB, B = oil FVF at Pb, RB/STB, ii = gas FVF, bbllscf, and

AC, = incremental gas injection, fraction of prc- duced gas.

- I.1

-0

-0

-0

-0

AN,, = incremental oil production, fraction of OIP. N, = cumulative oil production, fraction of OIP, N,, = cumulative oil produced from previous step,

fraction of OIP,

43-l 2 PETROLEUM ENGINEERING HANDBOOK

TABLE 43.3-FRONTAL-ADVANCE CALCULATION, HORIZONTAL FLOW

s

(13 k,/k,, W~$,/l”o) l+(3) f, =1/(4)

(2) (4) (5) -0.000 Cl 0 0.02 0.004 7.725 a.725 0.1146 0.05 0.025 1.236 2.236 0.4472 0.10 0.088 0.351 1.351 0.7402

0.15 0.265 0.117 1.117 0.8953 0.20 0.770 0.0400 1.0400 0.9614 0.25 2.300 0.0134 1 .0134 0.9868 0.30 7.35 0.00420 1.00420 0.9958

0.35 25.15 0.00122 1.00122 0.9988 0.40 117.0 0.00026 1.00026 0.9997 0.45 755.0 0.00004 1.00004 1 .oooo

2. Procedure a. Select pressure increments such that any one

mcrement I 10% of the initial or bubble-point pressure and obtain fluid properties as indicated in Table 43.2.

b. Perform material-balance calculation as shown in Table 43.5, with no gas injection, by assuming an incremental oil production AhJ, and verifying this value by the trial-and-error procedure indicated. This procedure can be shortened by use of previous calculated ANNp for the second and third trials at each pressure. (Note that subscript i in Cal. 13 refers to previous step.)

c. Repeat this procedure for various values of gas injection as shown in Table 43.6.

d. Construct performance curves as indicated in Fig. 43.12.

Fig. 43.10-Frontal-advance performance, horizontal 9as flow.

1.0 lg=O983 ’ I

lg = 0.961

0.9 - S13=0285 s,3=0255

0.C

-m "'t I/ I " 1

FRACTIONAL GAS SATURATION, Sg FRACTIONAL GAS SATURATION, Sg

Fig. 43.1 l-Frontal-advance performance, gas-cap expansion.

so (1) 0.20 0.25 0.30 0.35 0.40

0.45 0.50 0.55 0.60 0.65

TABLE 43.4-FRONTAL-ADVANCE CALCULATION, GAS-CAP EXPANSION

k,/k,o (k,Jk,,b&,) 1 +(3) l/(4) k C, x(6)’ f, =

I1 - (7)1(5) (2) (3) (4) (5) 4 (7) (4 0.770 0.04000 1.04000 0.9615 0.1320 366.6 -351.5 2.300 0.01340 1.01340 0.9868 0.0680 188.8 - 185.3 7.35 0.00420 1.00420 0.9956 0.0320 88.9 -87.5

25.15 0.00122 1.00122 0.9988 0.0130 36.1 -35.06 117.0 0.00026 1.00026 0.9997 0.0048 13.3 -12.30

755.0 0.00004 1.00004 0.99996 0.0116 4.44 -3.40 - 0 1 .ooooo 1 .oooo 0.0005 1.39 -0.400 - 0 1 .ooooo 1 .oooo 0.00015 0.417 0.583 - 0 1.00000 1.0000 0.00004 0.111 0.889 - 0 1 .ooooo 1.0000 0.00000 0.000 1.000

‘C, =21,306 [k(Pg -~~)sm irl(q,rro)], where k IS absolute permeability


TABLE 43.5--DEPLETION DRIVE CALCULATION, FINITE-DIFFERENCE METHOD WITH NO REINJECTION OF PRODUCED GAS’

AN, (assumed)

(2) P,=l,375 0

1,300 0.0160 1,200 0.0170 1,100 0.0155 1,000 0.0147

900 0.0142 800 0.0132 700 0.0114 600 0.0110 500 0.0097

400 0.0093 300 0.0098 200 0.0112 100 0.0150

0 0.0590

N, =X(2) 13)

0 0.0160 0.0330 0.0485 0.0632

1 -N, t3,1Bob

(4) (5) -1.0 1.0 0.9840 0.992 0.9670 0.980 0.9515 0.969 0.9368 0.959

0.700 0.683 0.663 0.645 0.6289

0.0774 0.9226 0.948 0.6122' 0.0906 0.9094 0.937 0.5965 0.1020 0.8980 0.9256 0.5818 0.1130 0.8870 0.9140 0.5675 0.1227 0.8773 0.9017 0.5537:

0.1320 0.8680 0.8893 0.5403 0.1418 0.8582 0.8760 0.5262 0.1530 0.8470 0.8620 0.5111 0.1680 0.8320 0.8463 0.4929 0.2270 0.7730 0.8264 0.4472

s, = (1 - S,)(4)(5)

(6) , 1.000 22,035 0.983 20,829 0.963 19,951 0.945 18,969 0.9289 17,841

0.9122 16,651 0.8965 15,357 0.8818 13,919 0.8675 12,409 0.8537 10,777

0.8403 9,089 0.8262. 7,184 0.8111 5,149 0.7929 2,882 017432. 403

A

0 0.0001 0.0160 0.0275 0.0431

(8) x (9) (10)

0 2.1

319.2 521.6 768.9

430.1 414.9 397.0 379.0 361.0

0.0653 1.087.3 342.0 0.0960 1,474.3 321.7 0.1350 1,879.l 301.0 0.1890 2,345.3 377.9 0.2575 2,775.1 254.9

0.3420 3,108.4 230.2 0.4700 3,376.5 202.1 0.6560 3,377.7 167.9 0.9600 2,766.7 125.2 2.580 1,039.7 0.0

R= /q= AN, =

(11)+(10) IV, +~,+,W’ (5,/B,)-R, (13)+(14) A[(6,/B,)-/?,I (4i)(l6)* A(118,) /3,,A(l/Bg) (17)-(19) (20/15)

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) 430.1 - 250.0 417.0 423.6 205.0 716.2 566.6 165.0 900.6 808.4 125.5

1.129.9 1.015.2 86.4

1,429.3 1,279.6 53.5 1,333.l - 34.9 -32.7 -42.6 -51.5 18.8 0.0141 1,796.0 1,612.6 23.4 1,636.0 -30.1 -27.8 -40.5 -49.0 21.2 0.0130 2,180.l 1,988.0 -6.1 1,981.g - 29.5 -26.8 : -41.0 -49.6 22.6 0.0115 2,623.2 2,401.6 -30.7 2,370.g - 24.6 -22.1 - 39.8 -48.2 26.1 0.0110 3.030.0 2,826.6 - 52.7 2,773.g - 22.0 -19.5 -38.2 -46.2 26.7 0.0096

3,338.6 3,184.3 -71.3 3,113.o -18.6 -16.3 -37.6 -45.5 29.2 0.0094 3,578.6 3,458.6 - 84.9 3,373.7 -13.6 -11.8' -37.1 -44.9 33.1 0.0098 3,545.6 3,562.1 - 90.0 3,472.l -5.1 -4.4 - 35.9 -43.4 39.0 0.0112 2,891.g 3,218.8 -85.0 3,133.8 + 5.0 +4.2 -35.4 -42.8 47.0 0.0150 1,039.7 1,965.8 - 5.05 1,960.8 +80.0 +66.6 -34.2 -41.4 108.0 0.0551

- - - - 628.6 -45.0 -45.0 -45.5 731.6 -40.0 -39.4 -42.7 933.9 - 39.5 -38.2 -43.8

1.103.6 - 37.1 -35.3 -42.7

-55.1 -51.7 -53.0 -51.7

‘Hand calculated values rounded ofl and will vary slightly from the computer-generated values given here

‘.I = previous slep

OEPLETION DRIVE, AG = 0

DEPLETION DRIVE. AG, = 0 5

DEPLETION DRIVE, AG, = I 0

PRESS. MAIN., AG,=l 581

I I I 0.10 0.20 0.30 0.40 0.58

OIL PRODUCED, N, FRACTION OF ORIGINAI OIL IN PLACE

Fig.43.12-Dispersed gas-injection pressure-maintenance performance.

- 0 10.1 0.0161 12.3 0.0168 14.8 0.0158 16.4 0.0149


JNP P (assumed)

(2) iv -\G, =0.5 pb =1,375

1,300 1.200 1,100 1,000

0 0

0.0230 0.0230 0.0240 0.0470 0.0245 0.0715 0.0215 0.0930

TABLE 43.6-DEPLETION DRIVE CALCULATION, FINITE-DIFFERENCE METHOD WITH DISPERSED GAS INJECTION

900 800 700 600 500

400 0.0110 0.1810 06190 08893 0.510 0610 9,089 0623 5.662.4 230.2 300 0.0114 0.1924 0.8076 08760 0.495 0795 7,184 0860 6.178.2 202 1 200 0.0126 0.2050 0.7950 08620 0.480 0760 5,149 1 180 6.075.6 167.9 100 0.0167 0.2217 0.7783 08463 0.461 0761 2,882 1790 5,156.a 125.2 0 0.0573 0 2790 07210 0.8264 0.417 0717 403 4850 1.954.5 0 0

AG.=l.O p. =1.375

1,300 1.200 1,100 1,000

R= R= (14)+ U@- AN,= (ll)+(lO) i(R,;&;,)/21 (1 -AG,)R (B,/B,)-R, (15) A[(B,B )-I?,] (20) (21H6)

(12) (14) (15) (16) (4 '4;$7 A(lI8,) B,,A(l/E$)

(19) (20) (21) (22) -___

N,=“(2) l-N, BJB, (3) (4) (5)

s, = (1 -S,)(4)(5)

(‘3)

0.700 0.677 0.654 0.629 0.609

s, = f, = s, + s, (~,~~,)(P,~P,)

(7) (8) k r# ro @)~(Ql R,

(9) (10) (11)

10 1.0 0.9770 0 992 0.9530 0.980 0 9285 0.969 0 9070 0.959

1000 22,035 0 0 430.1 0977 20,829 0 009 1875 414.9 0 954 19,951 0022 438.9 397.0 0 929 18,969 0044 834.6 379.0 0 909 17,841 0072 1.284.6 361.0

0.0200 0.1130 0.8870 0 948 0.589 0 889 16,651 0112 1,864.Q 342.0 0.0170 0.1300 0.6700 0.937 0.571 0671 15,357 0167 2,564.6 321 7 0.0150 0.1450 08550 09256 0.554 0854 13.919 0240 X340.6 301.0 0.0135 0.1585 08415 0.9140 0.538 0838 12,409 0345 4,281 1 277.9 0.0115 0.1700 08300 0.9017 0.524 0624 10,777 0465 5,011.3 254.9

0 0 1.0 1.0 0.700 1000 22,035 0.051 0.051 0 949 0.992 0.659 0 959 20,829 0.083 0134 0.866 0.980 0.594 0 894 19,951 0.150 0.284 0.716 0 969 0.486 0786 18,969 0.284 0.568 0.432 0 959 0.290 0 590 17,841

0 0 430.1 0019 395.8 414.9 0100 1.995.1 397.0 1030 19.538.1 379.0

170.0 3,032,970.0 361.0

430.1 602.4 835.9

1,213.6 1.645.6

- 516.2 719.2

1.024.8 1.429.6

258.1 359.6 512.4 7148

250.0 2050 165.0 125 5 884

463.1 -45.0 -45.0 -45.5 524.6 -40.0 -39.1 -427 637.9 -39.5 -37.6 -43.8 803.2 -37.1 -34.4 -427

- -55.5 -51.7 -53.0 -51.7

105 126 154 173

0 0.0227 0.0240 0.0241 0.0215

2,206,9 1,926.2 963.1 53.5 1.016.6 -34.9 -31.7 -42.6 -51.5 19.8 0.0195 2,886.3 2,546 6 1,273.3 23.4 1.296.7 -30.1 -26.7 -40.5 - 49.0 22.3 0.0172 3,641.6 3,264 0 1,632.0 -6.1 1,625.Q -29.5 -25.7 -41.0 -49.6 23.9 0.0147 4,559 0 4,100 3 2,050.2 -30.7 2,019 5 -24.6 -21.0 -39.8 -48.2 27.2 0.0135 5.266.2 4,912.6 2,456.3 -52.7 2.403.6 -22.0 -18.5 -38.2 -46.2 27.7 0.0115

5.892.6 5.579.4 2.789.7 -71.3 2,718.4 -18.6 -15.4 -37.6 -45.5 30.1 0.0110 6,380.3 6,136.4 3068.2 - 84.9 2,983.3 -13.6 -11.1 -37.1 -44.9 33.8 0.0113 6,243.7 6,312 0 3,156.0 -90.0 3,066.O -5.1 -4.1 -35.9 -43.4 39.3 0.0128 5,284.0 5,763 8 2,881.g -85.0 2,796.g +5.0 + 4.0 -35.4 -42.8 46.8 0.0167 1.954.5 3.619.2 1.809.6 -505 1,804.6 +80.0 +62.3 -34.2 -414 1037 0.0575

430 1 810.7

2.392.1 19.917.1

3,033.331.0 1

- - 250.0 - - 6204 0 205.0 205.0 -450

1,601 4 0 165.0 165.0 -40.0 11.154.6 0 125.5 1255 -39.5

,526,624.0 0 88.4 88.4 - 37.1

- - -45.0 -45.5 -36.0 -42.7 - 34.2 -43.8 -26.6 -42.7

-55.5 -51.7 -53.0 -51.7

- 0 10.5 0.051 13.7 0.083 18.8 0.150 25.1 0.284


TABLE 43.7-GAS-CAP EXPANSION CALCULATION, FINITE-DIFFERENCE METHOD WITHOUT COUNTERFLOW

We, - R,)+ ANN, N, = (6) - R(l -AG,)

[from (II)] W) 1-Np A(B,/B -R,) (2) (3) (4) (4

(4;;y) A(l/B,) (I+ W,(7) (from dep. dr.) $2,

(7) (‘3) (10) 01) - - AGi-0

1.375 0 13QQ 0.0894 1,200 0.0829 1,100 0.0546 1mQ 0.0523

900 0.0423 800 0.0350 700 0.0344

AG, =0.5 1,375 0 1.300 0.0941 1,200 0.0904 1,100 0.0848 1,QQQ 0.0718

AG,=l.O 1,375 0 13QQ 0.2127 1,200 0.3133

x.c694 A:Ez 0.1323 0.6677 0.1669 0.6131 0.2392 0.7608

0.2815 0.7185 0.3165 0.6835 0.3509 0.6491

0 l.OoOO 0.1046 0.6954 0.2103 0.7897 0.3124 0.6876 0.4092 0.5906

0 1.0000 0.2127 0.7673 0.5260 0.4740

- -45.0 -40.0 -39.5 -37.1

- 34.9 -30.1 -17.3

- -45.0 -40.0 -39.5 -37.1

- -45.0 -40.0

- - -46.0 -45.5 -37.2 -42.7 - 34.3 -43.8 - 30.2 -42.7

-88.6 &6 -63.2 46.0 -05.3 51.0 -63.2 53.0

- 626.6 731.6 933.9

1,013.6

0 0.0894 0.0829 0.0548 0.0523

- 26.6 -42.6 -83.0 56.4 1,331,l 0.0423 -21.6 - 40.5 - 78.9 57.3 1.636.0 0.0350 -11.6 -41.0 - 79.9 68.1 1,961.9 0.0344

-45.0 -35.8 -31.2 -25.5

- - - - -45.5 -68.6 43.6 463.1 -42.7 -83.2 47.4 524.6 - 43.8 -85.3 54.1 837.9 -42.7 -83.2 57.7 603.2

0 0.0941 0.0904 0.0848 0.0710

- -45.0 -31.5

- - - - 0 -45.5 -66.6 43.6 205.0 0.2127 -42.7 -63.2 51.7 165.0 0.3133

s, s, (from (13)+(16)

A@ ) (18

~fW4,dW) df;'dr.) f, VS. Sg) AN,NR AG, [17i)(l2) (2Wl5) h,

(13) (14) (15) (lo6 bbl)

(16) (,,"~'~~I) (~~)%) (lo3 bbl) (I$$) (P&&O; bbl)

(17) (18) (19) ___,

- - 0.00016 2,034,560 0.00017 2,161.720 0.00022 2,797,520 0.00025 3,179,000

O.ooO32 4.069,120 0.0878 0.00039 4.959,240 0.1035 o.Ocm51 6,465,160 0.1182

- - 0.00016 2,034,560 0.00017 2.181,720 0.00022 2,797,520 0.00025 3,179,wo

- - 0.00016 2,034,560 0.00017 2,161,720

0 0.017 0.037 0.055 0.0711

0.750 0 0.647' 0 0.647 0 0.647 0 0.647 0

0.647 0 0.645" 0 0.645 0

0 0.750 0.017 0.647' 62i.5 0.037 0.647 1,165.O 0.055 0.647 1,603.5 0.0711 0.647 2,120.6

0 0.750 0 0.017 0.647' 3,927.2 0.037 0.647 15.378

‘Calculated a, correspondmg ,a@ and pressu,e of 1.375 pslg (9480 3 kPa) “Calculated at correspond,ng rate and ,xessure of 900 pslg (6205 3 kPa)

0 0 0

0 0 0

0 0 827.5 1,6xX.4

1,992.5 4,204.2 3,596.0 09378.7 5,716.E 14,749.3

0 0 3,927.2 7,616.E 19.305.2 40.734.0

- 0 0 0 0

0 0 0

- 0 0.3 0.9 2.1

0 1.3

- - 2.034.6 3,145 2,161.7 3,341 2,797.5 4,324 3,179.0 4,913

20.0 17.0 15.4 12.7 10.0

4.069.1 6,289 6.9 4.959.2 7,699 3.4 6,485.2 10,055 -1.0

- 3,640.o 6366.2 11,177.l 17,930.4

5,626.o 9839.6 179275.3 27,713.l

20.0 16.3 10.6 3.7

- 5.2

- 9,65X4

42.897.0 14,920.2 66301.4

20.0 12.1 -1.7

e. For cases where the gas saturation, S, , exceeds the critical gas saturation as determined from an fK vs. S, curve at the appropriate pressure, performance from that point to abandonment must be determined by the frontal-advance method illustrated in III-B, which follows. Abandonment recovery to a limiting GOR can be determined directly from thef, relationship.

B. Pressure Maintenance 1. Equation is same as II-A in preceding section. 2. Procedure

a. Construct fs curve and tangents as shown in II-A.

b. Calculate performance as shown in Fig. 43. IO. c. Construct performance curves as indicated in

Fig. 43.12. d. Calculate injection requirements for complete

pressure maintenance at the bubble-point by the equation

*G,=I+ @o/B,)-R.! I

R.7 .

IV. External Gas-Injection Pressure Maintenance

A. Partial Pressure Maintenance 1. Equation

AN,=

(1 -~,,)A[(WB,)W?,] -(I +~P,,~,Nl~B,)

[cB,/B,)-R,]+R(I -AC,) '

where LW,, = incremental oil production, fraction of OIP, N,,; = cumulative oil production from previous

step, fraction of OIP, R = average GOR, scf/STB,

R,Y = solution GOR, scf/STB, B, = oil FVF, RBISTB,

Bob = oil FVF at pb, RBISTB,


; *ci,-” I I AG =05 I

7ooom 5 .

6000% cc

5000 4

z 4ocDg

2 3000~

z 2000 0

2 &

IO00 a

- 1 02 03 04 05 06 07 08 09 IO0

OIL PRODUCED, NpFRACTION OF ORIGINAL OIL IN PLACE

Fig. 43.13-External gas-injection preSSure-maintenance performance.

B, = gas FVF, bbllscf, AG; = incremental gas injection, fraction of

produced gas, and m = ratio of gas-cap to original oil-zone volume,

fraction. (Note that subscript i refers to previous step.)

2. Procedure a. Select pressure increments such that any one

increment I 10% of the initial or bubble-point pressure and obtain fluid properties as indicated in Table 43.2.

b. Perform depletion drive material-balance calculation with AG; =O, as described in III-A.

c. Perform material-balance calculation as shown in Table 43.7, using R as determined from depletion drive calculation in Point b and S g as determined from unit-displacement calculations.

d. Determine positions of gas/oil level and abandonment conditions, using data in Fig. 43.8 and calculations in Table 43.7.

e. Construct performance curves as indicated in Fig. 43.13.

B. Pressure Maintenance 1. Equation is the same as II-B. 2. Procedure

a. Constructf, curve and tangents as in II-B. b. Calculate recovery and construct performance

curves as indicated in Fig. 43.13. c. Calculate injection requirements for complete

pressure maintenance at the bubble-point by the equation

AG,=l+ [W&-R,]

R,

APPENDIX B Selected References Containing Equations, Calculation Procedures, and Example Calculations Related to Gas- Injectlon Performance Predictions External Injection-Complete Pressure Maintenance

Emphasis on Gravity Drainage and Segregation 1. Combs, G.D. and Knezek, R.B.: “Gas Injection for Upstructure

Drainage,” J. Pet. Tech. (March 1971) 361-72. 2. Craig, F.F. Jr. etal.: “A Laboratory Study ofGravity Segregation

in Frontal Drives,” J. Pet. Tech. (Oct. 1957) 275-81: Trans., AIME, 210.

3. Martin, J.C.: “Reservoir Analysis for Pressure Maintenance Operations Based on Complele Segregation of Mobile Fluids,” Trans., AIME (1958) 213. 220-27.

4. McCord, D.R.: “Performance Predictions Incorporating Gravity Drainage and Gas Cap Pressure Maintenance - LL-370 Area, Bolivar Coastal Field, J. Pet. Tech. (Sept. 1953) 231-48; Trans., AIME. 198.

5. Shreve, D.R. and Welch, L.W. Jr.: “Gas Drive and Gravity Drainage Analysis for Pressure Maintenance Operations,” J. Pet. Tech. (June 1956) 136-43; Trams., AIME. 207.

6. Stewart, F.M., Garthwaite, D.L., and Krebill, F.K.: “Pressure Maintenance by Inert Gas Injection in the High Relief Elk Basin Field,” J. Pet. Tech. (March 1955) 49-55; Trans., AIME. 204.

7. van Wingen, N., Balton, W.C. Jr., and Case, C.H.: “Coalinga Nose Pressure Maintenance Projecl,” J. Per. Tech. (Oct. 1973) 1147-52.

General Frontal-Advance Applications 1.

2.

3.

4.

5.

6.

7.

Buckley. S.E. and Leverett, M.C.: “Mechanism of Fluid Displacement in Sands,” Trans., AIME (1942) 146, 107-16. Craft, B.C. and Hawkins, M.F.: Applied Petroleum Reservoir Engineering, Prentice-Hall Inc., Englewood Cliffs, NJ (1959) 361-75. Dardaganian, S.G.: “The Application of the Buckley-Leverett Frontal Advance Theory to Petroleum Recovery,” J. Pet. Tech. (April 1958) 49-52; Trans., AIME (1958) 213, 365-68. Justus, J.B. et al.: “Pressure Maintenance by Gas Injection in the Brookhaven Field, Mississippi,” J. Pet. Tech. (April 1954) 43-53: Trans.. AIME (1954) 201. 97-107. Kirby, J.E. Jr., Stamm, H.E. 111. and Schnitz. L.B.: “Calculation of the Depletion History and Future Peformance of a Gas-Cap- Drive Reservoir,” J. Pet. Tech. (July 1957) 218-26; Trans., AIME, 210. Pirson, S.J.: Oil Reservoir En#neering, McGraw-HIII Book Co. Inc., New York City (1958) 555-605. Snyder, R.W. and Ramey, H.J. Jr.: “Application of Buckley- Levereu Displacement Theory to Noncommunicating Layered Systems,” J. Pet. Tech. (Nov. 1967) 1500-06: Trans.. AIME, 240.

8. Stutzman, L.F. and Thodos, G.: “Frontal Drive Production Mechanisms-A New Method for Calculatmg the Displacing Fluid Saturation at Breakthrough.” J. Pet. Tech. (April 1957) 67-69; Trans., AIME, 210, 36+66.

9. Welge, H.J.: “A Simplified Method for Computing Oil Recovery by Gas or Water Drive.” Trans., AIME (1952) 195, 91-98.

Gas Displacement Above the Bubble-Point and Production From Behind the Front

I. Kern, L.R.: “Displacement Mechanisms in Multi-Well Systems.” Trans., AIME (1952) 195, 39-46.

2. Shreve, D.R. and Welch, L.W. Jr.: “Gas Drive and Gravity Drainage Analysis for Pressure Maintenance Operations,” J. Pet. Tech. (June 1956) 136-43: Trrms., AIME, 207.

Nonequilibrium Gas Displacement I. Attra, H.D.: “Nonequilibrium Gas Displacement Calculations,”

Ser. Pet. Eng. J. (Sept. 1961) 130-36; Trans., AIME, 222. 2. Jacoby. R.H. and Berry, V.J. Jr.: “A Method for Predicting

Pressure Maintenance Performance for Reservoirs Producing Volatile Crude Oil,” J. Pet. Tech. (March 1958) 59-69: Trans., AIME, 213.

Gas injection in Combination Drive Reservoirs Blair, E.A. et al.: “A Reservoir Study of the Friendswood Field,” J. Pet. Tech. (June 1971) 685-94. Cotter, W.H.: “Twenty-Three Years of Gas Injection Into A Highly Undersaturated Crude Reservoir,” J. Per. Tech. (April 1962) 361-65. Wooddy. L.D. Jr. and Moscrip, R. III: “Performance Calcula- tion& for Combination Drive Reservoirs,” J. Pet. Tech. (June 1956) 128-35: Trans., AIME, 207.


Dispersed Gus Injection-Complete uncl Purtiul Pressure Muintenunce

1.

2.

3.

4.

5.

6.

7.

8.

9.

IO.

Il.

12.

13.

14.

Craft, B.C. and Hawkins. M.F.: Ap/)/ied Prrrolrurn Rrvrrwrr Engirirrring. Prentw-Hall Inc.. Englewood Cliffs, NJ (1959) 37s~90. Crttig, F.F. Jr. and Gcffen. T.M.: “The Determination of Panial Pressure Maintenance Performance by Laboratwy Flow Tests,” J. Per. Trrh. (Feb. 1956) 42-49: ~r&s.. AIME; 207. Craig. F.F. Jr., Geffen, T.M.. and Morwz, R.A.: ‘01 Rccovcry Performance of Pattern Gas or Water Injection Operations from Model Test\,” J. Prr. T&I. (Jan. 19.55) 7-14: Trnnc , AIME. 204. Has, R.L.: “Calculated Effect of Pressure Maintenance on Oil Recovety. ~’ Trm.\. , AIME (1948) 174. 121-30. Kelly. P. and Kennedy, S.I..: “Thirty Year5 of Effective Pmawrc Maintenance By Gas Injection III the Htlbig Field.” J. Per. Tdz. (March 1965) 279-X I. Last. G.J.. Craig. F.F. Jr.. and Reader. P.J.: “Significance 01 PAttial Pressure Maintenance hy Fluid Injectton.“ J Per. T~I. (Jan. 1964) 20-24. Lcihrock. R.M.. H&z. R G.. and Huzarcvich. J.E.: “Results 01 Ga Injection in the Cedar Lake Field,” Trrrrzs.. AIME (1951) 192, 357-66. McGraw, J.H. and Lohec. R.E.: “The Pickton Field-Review 01 a Successful Gas Injection Project.” J. PC,/. Qc,/i. (April 1964) 399-404: discussion, 405. Meltrer, B.D.. Hurdle. J.M., and Cassingham. R.W.: “An Elft- cicnt Gas Displacement Project-Raleigh Field, Mi\aissippl.” J. PH. 7id1. (Mav 1965) 509%14. Muskat, M.: P/&cd Priwiples of Oil Proc/uctior~, McGraw-Hill Book Co. Inc., New York City (1949) 437-53. Patton, E.C. Jr.: “Evaluatmn tit’ Prewure Mamtenance by Internal Gas InJectmn tn Volumetrtcally Controlled Reservoirs,” Trtrn.r. AIME (1947) 170, 112-52: Discussion. 154-55. Pirson, S.J .: 011 Re.c~rwir Efr~~nwn,~fi, McGrawHill Book Co Inc.. New Yorh City (1958) 4X4-532. Shehahi, I A.N.: “Effective Displacement ofOil by Gas ln,jection in B Preferentially Oil-Wet, Low-DIP Rewvoir.” J. Pet. Td?. (Dec. 1979~ 1605Sl3. Tracy. G.W.: “Sm~pltficd Form of the Maternal Balance Equa- tion,” J. Pet. Tdt. (Jan. 195.5). X-56: Tiwts.. ACME. 204. ?‘I~-46.

Mathernaticul Models for Reservoir Simulation Coats. K H.: “An Analysis for Stmulating Reservoir Performaxe Under Preawre Maintenance by Gas and/or Water Injectton.” Srx PH. Eq. J (Dec. 1968) 331-40. Cook. R E.. Jacoby. R.H., and Ramesh, A.B.: “A Beta-Type Reservoir Simulator for Approximating Compositional Effects During Ga Injection.” Sot. Pcv. &IX. J. (Oct. 1974) 471-81. McCulloch. R.C.. Langton. J.R.. and Spivak. A.: “Simulation of High Relief Reservoirs. RainboLc Field. Alhcrta. Canada.” J. PC/. TciIr. (Nov. 1969) 1399-1408. McFarlanc. R.C.. Mueller. T.D.. and Miller. F.G.. “Unsteady- State Distnbutwns of Flutd Compositions in TwovPhase Oil Reserwirs Lindergoing Gas In,jection.” SIC,. Per. EQ. J. (March 1967) 61-74: Twrx.. AIME. 240. Price. H.S. and Donohue. D.A.T.: “Isothermal Di\placrment Procese\ With Interphase Mass Transfer.” .S~x. Pet. Eq. J. (June 1967) 20.5-20. Trcln.\. AIME. 240. Strickland. R.F. and Morse. R.A.: “Gas Injection for Upatructurc 011 Dranage.” J. PC,/. 7d1. (Oct. 1979) 1323-3 I. Thomas. L.K.. Lumpkin. W.B.. and Rchcis. G.M.: “Rc~crwir Slmulatton of- Varlahle Bubble-Point Prohlerm.” Sock. PC,!. E/Q. J. (Feb. 1976) 10-16.

APPENDIX C

Data Requirements for Engineering Analysis of Gas-Injection OperationsI

Analytical Data

1. Core analyses from a representative number of wells

a. Porosity b. Permeability c. Water saturation

2. Special core analyses on a sufficient number of samples to cover permeability range of the reservoir

a. Capillary-pressure data (for determining interstitial saturations)

b. Gas/oil relative permeability, k,/k, c. Relative permeability to oil, k,,,

3. Hydrocarbon compositional analysis a. Gas-cap, casing-head, and trap samples b. Reservoir-fluid samples

4. Reservoir-fluid property analyses a. Solubility

(1) Flash (2) Differential

b. Relative oil volume (1) Flash (2) Differential

c. Oil viscosity d. Oil density e. Gas viscosity f. Gas density

Field Data 1. Development history 2. Abandonment history, if any 3. Production history

a. Oil b. Water c. Gas

4. Injection history, if any a. Gas b. Water

5. Pressure history 6. Well productivity data 7. Gas/oil and oil/water contacts (original and

present) 8. Well and test data

a. Drillstem tests b. Production tests c. Sample cuttings d. Core descriptions e. Electrical and radioactivity logs

9. Average reservoir temperature 10. Well completion data

Interpretive Data (prepared from preceding data)

1. Structure maps a. Top of zone b. Base of zone

2. kopachous maps a. Total net sand b. Net gas sand c. Net oil sand

3. Reservoir volume distribution a. Volume vs. subsea depth, and/or b. Volumes by injection/production units

4. Cross-sectional area a. Area vs. subsea depth, and/or b. Area perpendicular to bedding planes for injec-

tion/production units


5. Volume-weighted reservoir datum 6. Average reservoir fluid properties (as functions of

pressure) a. Differential oil formation volume factor b. Flash relative volume factor c. Gas formation volume factor d. Oil viscosity e. Gas viscosity f. Oil gravity g. Gas gravity h. Gas deviation factor i, Differential gas solubility j. Average oil and gas composition k. Oil and water compressibility

7. Volume-weighted average pressures 8. Permeability distribution 9. Average reservoir-rock properties

a. Porosity b. Permeability c. Interstitial water saturation d. Gas/oil relative permeability ratio, kg/k, e. Oil relative permeability, k,/k

10. Well productivities a. As a function of subsea depth b. By injection/production units c. Productivity indices

Nomenclature

A= B,, = B,, =

B;: 1 A($;. =

k,, = k,.,s = k,.,, =

L= nl = M=

N,, = M,] =

P= Pb =

cross-sectional area gas FVF oil FVF oil FVF at P), fractional gas flow incremental gas injection effective permeability to oil relative permeability to gas relative permeability to oil distance ratio of gas-cap to original oil-zone volume mobility ratio cumulative oil production incremental oil production pressure bubblepoint pressure

p,. = oil/gas capillary pressure ( p. -P,~) q, = total flow rate f = instantaneous GOR R = average GOR

R, = solution GOR S,, = gas saturation S,, =. total liquid saturation S,,. = interstitial water saturation

f = time cy = angle of dip, positive downdip

p s = gas viscosity P ,I = oil viscosity

px = gas density PO = oil density

4 = porosity, fraction

Key Equations in SI Metric Units

f# =

I +(8.639x 10-5)[k,,Aiipq,)] -9.795(p,,-p,)s~1 N 1

where .fg = 91 = A=

P,. = L=

PO =

fractional flow of gas, total flow rate, m3/d, cross-sectional area, m2, oil/gas capillary pressure, P,, -pg. kPa, distance, m, oil specific gravity (water= 1) or density.

g/cm3, Pg = gas specific gravity (water= 1) or density,

g/cm’, Cl= angle of dip, positive downdip, degrees.

kc, = effective permeability to oil,, pm’, k, = effective permeability to gas, pm’, PLO = oil viscosity, Pass. and Ph’ = gas viscosity, Pa* s.

I+ cy (:t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

L=q,r 3 ~A ( > , . . as,s (2)

where t = time, days,

C$ = porosity, fraction, S,q = gas saturation, fraction,

and others are as in Eq. 1.

fq = 1+0.848x lo-‘[k,,(p, -0,)) sin oll(y,p,,)]

l+(t) (2) ’

(A.1I.B)

where

a kc, k,

0, PO

CY 9/ CL,, h

= fractional gas flow, = effective permeability to oil, pm’, = effective permeability to gas. pm’. = gas specific gravity at p (water= 1). = oil specific gravity at p (water= I), = angle of gas flow (-90”). = rate of frontal gas movement, m”/d*m2 = oil viscosity at p, Pa*s. and = gas viscosity at p. Pa.s.

Note: All other material-balance, saturation. and GOR equations that follow are correct for standard SI units, where B,, and B,v are volume factors (in m”/m3) and R. R,, and R, are GOR’s (in m’im’).


References

5.

6.

7.

8.

Muskat. M.: Ph~sica/ Prinqles os0il Pwducrion. McGraw-Hill Book Co. Inc.. New York City (19491 709. Bucklev. S.E. and Leverett. M.C.: “Mechanism of Fhxd Displacement in Sands.” Trcrrlr.. AIME (1942) 146. 107-16. Welge. H.J.: “A Simplified Method for Computing Oil Recovery by Gas or Water Drive,” Trclns.. AIME (1952) 195. 91-98. Dardaganian. S-G.: “The Application of the Bucklev-Leverett Frontal Advance Theory to P&leum Recovery.” J. Per. Tech. (April 1958), 49-52: Trms., AIME, 213. 365-68. Kern, L.R.: “Displacement Mechanism in Multi-well Systems,” Truns. ( AIME ( 1952) 195. 39-46. Shreve. D.R. and Welch, L.W. Jr.: “Gas Drive and Gravity Drainage Analysis for Pressure Maintenance Operations,” J. Per. Tech. (June 1956). 136-43: Tram.. AIME. 207. Jacoby. R.H. and Berry. V.J. Jr.: “A Method for Prechcting Pressure Maintenance Performance for Reservoirs Producmg Volatde Cmde Oil,” J. Per. Tech (March 19%). 59-69: Trans., AIME, 213. Attra H.D.: “Nonequilibrium Gas Displacement Calculation,” Sot. PC!. Eng. .I (Sept. 1961) 130-36; Trms., AIME. 222.

9. Craft, B.C. and Hawkins, M.F.: A&rd Petroleum Reserw;r Engineering, Prentice-Hall Inc., Englewood Cliffs. NJ (1959) 370

10. Anders, E.L. Jr.: “Mile Six Pool-An Evaluation of Recovery Ef- ficiency,” J. Per. Tech. (Nov. 1953) 279-86; Truns.. AIME. 198.

1 I. Craig, F.F. Jr. and Geffen, T.M.: “The Determination of Partial Pressure Maintenance Performance by Laboratory Flow Tests.” J. Per. Tech. (Feb. 1956) 42-49; Trcms.. AIME. 207.

I?. Slobod, R.L. and Koch, H.A Jr.: “High Pressure Gas Injec- tion-Mechanism of Recovery Increase,” 0-i//. and Prowl. Pm-. , API (1953) X2.

13. Stiles, W.E.: “Use of Permeability Distribution in Water Flood Calculations,” Trans.. AIME (1949) 186, 9-13.

14. Dyes, A.B.. Caudle, B.H.. and Erickson, R.A.: “Oil ProductIon after Breakthrough as Influenced by Mobility Ratio.” J. Per. T?ch. (April 1954), 27-32; Truns., AIME (1954) 201, 81-86.

15. Patton, E.C. Jr.: “Evaluation of Pressure Maintenance by Internal Gas Injectcon in Volumetrically Controlled Reservoirs.” Trans., AIME (1947) 170, 112-52, Discussion 154-55.

Water-Injection Pressure Maintenance in Oil Reservoirs

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