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# Immiscible gas injection performance

Techniques described in this page are classic methods for describing immiscible displacement assuming equilibrium between injected gas and displaced oil phases while accounting for differing physical characteristics of the fluids, the effects of reservoir heterogeneities, and injection/production well configurations. Included are modifications to typical displacement equations, evaluating sweep efficiency, and calculating performance.

In simple calculations, the reservoir is treated in terms of average properties for volume of rock, and production performance is described on the basis of an average well. Black-oil-type reservoir simulation models use essentially these same techniques but, by means of 1D, 2D, or 3D cell arrays, account for areal and vertical variations in rock and fluid properties, well-to-well gravity effects, and individual well characteristics. More complex compositional models account 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. It is important to comprehend the physics of displacement to understand the simulation results and to identify incorrect results. The fundamentals of the displacement process presented in this page are intended to provide the background needed to produce good-quality predictions of oil recoveries.

## Contents

## Modifications of displacement equations

Applicability of the basic displacement equations to a given reservoir is governed by whether the underlying assumptions are reasonable. Several authors have reported modifications that eliminate the need to make certain 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,^{[1]} Kern,^{[2]} Shreve and Welch,^{[3]} and others. Jacoby and Berry,^{[4]} Attra,^{[5]} and others have presented equations and simple analytical procedures for calculating performance to account for some of the compositional interchange between the displacing gas and the reservoir oil.

These works are mentioned for completeness. If significant deviations from the basic assumptions of the Buckley-Leverett method are a concern, the more practical approach is to use numerical reservoir simulation to account for reservoir heterogeneities and gravity, capillary, and compositional effects.

## Methods for evaluating sweep efficiency

Some techniques for estimating the volumetric, vertical, and areal sweep efficiency of an immiscible gas/oil displacement are discussed below.

### History matching

If there are sufficient data concerning the location of the gas front and oil recovery as a function of time, past reservoir performance can be used to calculate the volumetric sweep efficiency by dividing observed recovery at various times by the theoretical recovery determined from displacement efficiency calculations.

If there are adequate data to reliably describe spatial variations in reservoir rock and fluid characteristics, numerical reservoir simulation is the best way to predict sweep efficiency, particularly after the historical production and pressure data are matched. If data or sufficient economic justification to undertake a full numerical reservoir simulation study is lacking, the following methods are presented as useful for screening studies and in situations when more detailed studies are inappropriate.

### Vertical sweep efficiency

Several authors have presented methods for determining vertical sweep efficiency based on statistical treatments of routine core analysis data. Some of the most frequently used methods are adaptations of the Stiles^{[6]} method for evaluating the effect of permeability variations on waterflood performance. The same assumptions and calculation procedures may be used for immiscible gas/oil displacements. The relative permeability ratio used in such calculations is considered to be a constant equal to the relative permeability to gas at residual oil saturation (*k*_{rg@Sor}) divided by the relative permeability to oil at initial gas saturation (*k*_{ro@Sgi}).

### Areal sweep efficiency

Several investigators have shown that areal sweep efficiency is primarily a function of injection/production well pattern arrangement, mobility ratio, and volume of displacing phase injected. Various studies have confirmed what would be expected intuitively, that areal sweep efficiency increases with the volume injected and with a lower mobility ratio. Data from model studies that show the influence of mobility ratio and displacement volume on areal sweep efficiency in a regular five-spot pattern are illustrated in **Fig. 1**.^{[7]}

Thin models containing miscible fluids of varying viscosity were used to develop these area sweepout curves. These data are considered applicable to either water/oil or gas/oil displacement. These data are presented to aid in the understanding of the effect of some factors on the gas displacement mechanism and may prove useful in preliminary studies of a potential gas injection project to predict volumetric sweep. However, the quantitative applicability of laboratory data is inherently questionable because of uncertainties in model scaling, laboratory techniques, and associated simplifying assumptions regarding no vertical gas override effects or reservoir heterogeneities. The instability of the very unfavorable mobility ratio gas/oil displacement is most difficult to quantify in laboratory experiments. All these effects can cause a smaller sweepout efficiency than presented in **Fig. 1**. Nevertheless, laboratory model studies do offer a convenient means of making quantitative estimates when simulation is not practical or justified and injected gas remains dispersed in the reservoir.

When the laboratory data are used, the common practice is to calculate a mobility ratio using the viscosity and relative permeabilities of the oil ahead of the gas front and of the gas at the average saturation behind the displacing front.

## Calculating immiscible gas injection performance

Numerical simulation represents the best way to predict the performance of immiscible gas injection if there are sufficient data to characterize the reservoir rocks and fluids adequately. Even simple 2D and 3D black-oil models provide insight into the more important aspects of oil recovery for reservoirs in which compositional effects are not a major concern. When adequate data are unavailable or when screening work is being done, simple models may suffice.

The immiscible displacement of oil by gas is described with fractional flow theory. Muskat^{[8]} presented the basics of this theory more than 60 years ago. Since then, additional work has been done to develop various mathematical calculation methods based on fractional flow theory.^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}

Pattern gas injection is seldom used now because waterflooding performance is much better in those types of reservoirs where pattern gas injection has historically been tried. Therefore, the remainder of this page discusses a simple model used for reservoirs in which stabilized gravity drainage controls the gas/oil displacement process and increases the ultimate oil recovery.

Viscous, gravitational, and capillary forces and diffusion are involved in the displacement of oil by gas, complicating technical analysis of a particular reservoir if each of these forces and flow in all three dimensions are important. Fortunately, there are instances in which one force is dominant and only one dimension is involved in the rate-limiting step. In these circumstances, engineering solutions can be direct and simple. One such circumstance is that of thick reservoirs with high permeabilities.

In steeply dipping oil reservoirs containing sands with high vertical permeabilities, gravity drainage of the oil can be more effective than is calculated from the Buckley-Leverett assumptions alone, as illustrated in **Fig. 2**.When sufficient vertical permeability exists, even at lower oil saturations, oil behind the gas front can continue to flow vertically downward through the reservoir. Thus, the displacement process occurs in two steps:

- Gas invades the originally oil-saturated sand as the gas-oil contact (GOC) moves downdip because of oil production farther downdip
- Oil drains vertically downward through the gas-invaded region and forms a thin layer with high oil saturation (with high
*k*_{ro}) that drains along the base of the reservoir interval to the remaining downdip oil column.

### Mathematical model

A simple mathematical model can be used to describe the displacement of oil by gas drive and gravity drainage when the rate is less than one-half the critical rate. The critical rate is given by

where *q*_{T} = total volumetric flow rate through area *A*, ft^{3}/D, and *k* = permeability, darcies.

The Welge equation for the fractional flow of gas at any gas saturation (*S*_{g}) is calculated as follows:

where

## Nomenclature

The first calculation determines the gas saturation just above the GOC by using **Eq. 2**, plotting *F*_{g} vs. *S*_{g}, and finding the tangent to the curve passing through the origin, as shown in **Fig. 3**. For ease of calculation, the GOC is assumed to move at a constant rate. The next calculation determines the quantity of oil that drains from the region invaded by gas in a given time increment. For ease of calculation, this region is divided into arbitrary lengths, and the amount of oil produced by vertical gravity drainage is calculated for the average time since passage of the gas front.

For vertical drainage of oil, the rate is given by Darcy’s law, with the driving force proportional to the density difference between gas and oil. The assumption is made that resistance to flow of gas and capillary effects are negligible:

where *u*_{ov} = vertical oil flow per unit area, res ft^{3}/ft^{2}-D.

From continuity considerations,

where *z* = vertical distance, ft; *t* = time, days; *ϕ* = porosity, fraction; and *S*_{o} = oil saturation, fraction.

The rate of movement of a particular saturation, (d*z*/d*t*)_{So}, can be determined by plotting *u*_{ov} calculated from **Eq. 3** vs. saturation, taking the slope to determine d*u*_{ov}/d*S*_{o} and dividing by porosity, as indicated in **Eq. 4**. The amount of oil drained from each region since passage of the gas front can be calculated by graphical integration of the height vs. saturation plot.

As a first approximation, the time for oil to flow downdip in the thin layer along the base of the reservoir interval can be neglected, as can the volume of oil in this layer. If the displacement rate exceeds one-half the critical rate, oil tends to accumulate rather than flow away along the bottom of the reservoir. More accurate calculations also include consideration of the thickness of the gas/oil transition zone arising from capillary effects above this layer of oil, especially if the transition zone is < 10 ft thick.

Recoveries calculated by this technique are quite sensitive to the values of *k*_{ro} at low oil saturations. Ways to extrapolate measured information are discussed next. First, conventional laboratory data can be extended to low oil saturations by plotting measured values of *k*_{ro} vs. (*S*_{o} − *S*_{org}*)/(1 − *S*_{wi} − *S*_{org}*), in which *S*_{org}* is the irreducible oil saturation in the presence of gas and connate water. *S*_{org}* can be calculated by material balance for areas of a reservoir that have been invaded by gas if good data are available on GOC movement and oil recovery from the area. A second source is the oil saturation in an associated gas cap as determined in cores from that region. If the gas cap was originally filled with oil, drainage of oil over geologic time as gas migrates into the reservoir establishes an endpoint relict oil saturation. For instance, the Prudhoe Bay Sadlerochit reservoir was originally filled with oil. Gas then migrated into the reservoir several million years later, creating the gas cap. Water-based mud cores from the gas cap interval showed an average routine core analysis oil saturation at discovery of 7% pore volume (PV). The dip of the reservoir is 1 to 3°, but vertical permeabilities throughout the gas and oil columns are generally very high. Interestingly, oil saturations above small shale lenses in the gas cap averaged more than 7% PV, indicating that more time may be required to reach irreducible oil saturations when oil drainage is limited by the dip of this reservoir. A third source is drainage capillary pressure vs. saturation measurements. Experience has indicated that *S*_{org}* should be < 10% PV and sometimes approaches zero. Although these endpoint saturations are seldom realized in the depletion time of a reservoir, it is important to have the correct value for predicting flow behavior and ultimate oil recovery. A benefit of even simple, multidimensional simulation models is that the inclusion of capillary effects controls the oil flow rate and conditions under which irreducible saturations are approached.

If a measured value of *S*_{org}* is unavailable, a value is chosen to yield a straight line through the data, so for

the slope of the line n should be ≈4 according to the theory of Corey *et al.* but may be as large as 6 and as small as 2. Recovery data can be correlated by the dimensionless parameter

which is derived by dividing the time required for vertical drainage,

by the time required for flow along the bedding plane,

where *k*_{v} = vertical permeability, darcies, and *h*_{v} = vertical thickness, ft.

### Example gas/oil gravity-drainage problem

The utility of this simple model can be illustrated by predicting recovery by gas drive and gravity drainage for an actual reservoir, in this case the Hawkins field in east Texas.

**Given:** Average Hawkins Woodbine reservoir properties as presented in **Table 1 and Fig. 4**.

#### Solution

With **Eq. 1**, the critical rate for the Hawkins field is calculated to be 0.173 ft^{3}/D-ft^{2}. The average actual rate, *q*_{T}/*A*, of 0.0365 ft^{3}/D-ft^{2} (see **Table 1**) is 21% of the critical rate, and the simplified model should apply. With **Eq. 2**, the average gas saturation just above the GOC is found to be 46% PV by the Welge procedure, as shown in **Fig. 3**. The rate of frontal movement is

Time to gas breakthrough is 3,500/104 = 34 years. Recovery at breakthrough may now be estimated by dividing the reservoir into seven blocks, each 500 ft long and 49 ft thick. The average vertical movement of saturations in each block can be calculated from **Eqs. 3** and **4**. The relative permeability data for oil were extrapolated to low *S*_{o} values using the correlation term (*S*_{o} – *S*_{org}*)/(1 – *S*_{wi} – *S*_{org}*) discussed above. The resulting plots for two *S*_{org}* assumptions are shown in **Fig. 5**.

In this case, *S*_{org}* = 0 gave a slightly better fit. The resulting plot of *k*_{ro} vs. *S*_{o} is shown in **Fig. 6**. The laboratory data and a plot resulting from use of an *S*_{org}* of 2% PV and an n of 4.5 are also shown to indicate the differences in *k*_{ro} at low *S*_{o} that result.

The average time interval since passage of the gas front is calculated from the calculated rate of frontal movement of 104 ft/yr. The residual oil left in each block is determined by graphical integration of a plot of *S*_{o} vs. height, such as the ones shown in **Fig. 7**. Results of these calculations are shown in **Table 2**.

### Comparisons with field data

A recovery of 87% of original oil in place (OOIP) was observed in the Hawkins field for an area affected by an expanding gas cap. The calculated recovery of 88% compares very favorably. A 2D two-phase reservoir simulation using similar relative permeabilities predicted 87% recovery at breakthrough. The recoveries observed in the field and predicted by models that permit flow in two dimensions are > 15% greater than those calculated by conventional 1D techniques that assume flow only along the bedding planes.

### Model summary

This page has shown how a simple gravity drainage model can be readily applied to predict recoveries by gas drive and gravity drainage when flow rates are less than one-half the critical rate and permeabilities in the vertical direction are high. Some applications of the model have been unsuccessful because of lower-than-expected vertical permeabilities. As a practical matter, the simple model should be used to predict reservoir behavior only when it can be shown to match history or when applied to a field analogous to one that the model fits.

## References

- ↑ Welge, H.J. 1952. A Simplified Method for Computing Oil Recovery by Gas or Water Drive. Trans., AIME 195: 91.
- ↑ Kern, L.R. 1952. Displacement Mechanism in Multi-well Systems. Trans., AIME 195.
- ↑ Shreve, D.R. and Welch, L.W. Jr. 1956. Gas Drive and Gravity Drainage Analysis for Pressure Maintenance Operations. Trans., AIME, 207: 136-143.
- ↑ Jacoby, R.H. and Berry, V.J. Jr. 1958. A Method for Predicting Pressure Maintenance Performance for Reservoirs Producing Volatile Crude Oil. Trans., AIME, 213: 59-64.
- ↑ Attra, H.D. 1961. Nonequilibrium Gas Displacement Calculations. SPE J. 1 (3): 130-136. http://dx.doi.org/10.2118/1522-G.
- ↑ Stiles, W.E. 1949. Use of Permeability Distribution in Water Flood Calculations. J Pet Technol 1 (1): 9-13. SPE-949009-G. http://dx.doi.org/10.2118/949009-G
- ↑
^{7.0}^{7.1}Dyes, A.B., Caudle, B.H., and Erickson, R.A. 1954. Oil Production After Breakthrough as Influenced by Mobility Ratio. J Pet Technol 6 (4): 27-32. SPE-309-G. http://dx.doi.org/10.2118/309-G - ↑ Muskat, M. 1949. Physical Principles of Oil Production, 470-502. New York City: McGraw-Hill Book Co. Inc.
- ↑ Johns, R.T. 1992. Analytical Theory of Multicomponent Gas Drives with Two-Phase Mass Transfer. PhD dissertation, Stanford U., Palo Alto, California.
- ↑ Jessen, K., Wang, Y., Ermakov, P. et al. 1999. Fast, Approximate Solutions for 1D Multicomponent Gas Injection Problems. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3–6 October. SPE-56608-MS. http://dx.doi.org/10.2118/56608-MS
- ↑ Falls, A.H. and Schulte, W.M. 1992. Features of Three-Component, Three-Phase Displacement in Porous Media. SPE Res Eng 7 (4): 426–432. SPE-19678-PA. http://dx.doi.org/10.2118/19678-PA
- ↑ Lake, L.W. 1989. Enhanced Oil Recovery, first edition. Englewood Cliffs, NJ: Prentice-Hall Inc.
- ↑ Juanes, R. and Patzek, T.W. 2003. Relative Permeabilities in Co-Current Three-Phase Displacements with Gravity. Presented at the SPE Western Regional/AAPG Pacific Section Joint Meeting, Long Beach, California, 19–24 May. SPE-82445-MS. http://dx.doi.org/10.2118/83445-MS
- ↑ Azevedo, A.V. et al. 1997. Nonuniqueness of Riemann Problems. Zeitschrift fûr angewandte Mathematik und Physik 47 (6): 977.
- ↑ Marchesin, D., Plohr, B., and Schecter, S. 1997. An Organizing Center for Wave Bifurcation in Multiphase Flow Models. SIAM J. Appl. Math. 57 (5): 1189.

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## See also

Immiscible gas injection in oil reservoirs

Displacement efficiency of immiscible gas injection