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Pressure Inversion and Material Balance Calculations

ABSTRACT

In a reservoir when gas comes out of solution and rises, additional pres-sure is created because of the change in position of this gas in the bounded volume. If this pressure effect is not taken into account as a pseudo-influx in material balance calculations on reservoirs in which there is evolving gas, an error is introduced that is directly proportional to Ihe length of oil column and amount of gas evolved and inversely proportional to the res-ervoir pressure. In reservoirs with small oil column this pseudo-influx is of little importance. The larger the oil column the more important the effect becomes. The most important application, however, will be for res-ervoirs with declining pressure in which increasing water entry has been indicated by material balance calculations. It could be that this in-dication is the result of the cited pseudo-influx and not water entry at all. This paper attempts to explain I his phenomenon in relation to ma-terial balance calculations and pre-sents an approximate method of de-termining the pseudo-influx.

INTRODUCTION

Only recently has the pressure-vol-ume relationship of gas rising in a liquid appeared in engineering lit-erature, although its effect has been discussed for several years-espe-cially among engineers engaged in drilling operations. The writer pub-lished a paper in 1957 in which this phenomenon was examined in rela-tion to blowouts in drilling wells.' Also, in 1957 Stegemeier and

Original manuscript received in Society of Petroleum Engineers office Oct. 10, 1958. Re-vised manuscript received March 20, 1959.

'References given at end of paper.

MAY, 1959 SPE 1185-G

HOMER N. MEAD MEMBER AIME

MARTIN, SYKES & ASSOCIATES, c. A. CARACAS, VENEZUELA

Matthews' authored a paper in which the relationship was presented with respect to pressure build-up tests when shutting in producing wells. In this paper the effect will be called "pressure inversion" which is de-scriptive of the phenomenon.

Fundamentally, the effect just men-tioned is the result of changing the position of a quantity of energy con-tained within the gas itself (as a re-sult of its compression) within the bounded volume in which the gas ex-ists. If the gas were allowed to ex-pand "beyond all bounds" there could be no pressure inversion effect. This is because all of the energy con-tained within the gas as a result of compression would have been re-leased.

PRESSURE INVERSION

A phase that has not been dis-cussed in the literature is the effect of pressure inversion upon material balance calculations in reservoirs in which the reservoir pressure drops with reservoir withdrawal and is be-low the bubble point. This action releases solution gas which, after the critical gas saturation has been reached, will tend to rise and collect at the top of the reservoir. The pseu-doinflux caused by this effect could be great enough in reservoirs of large oil column and high permeability to render invalid any analyzer studies made on them - if pressure inver-sion has not been taken into account. Most suspect would be those reser-voirs with large oil column in which increasing water entry has been pre-dicted but the bottom wells have not yet produced water.

EXAMPLE 1

Simply stated, this phenomenon exists in oil reservoirs because of the

difference in density and expansibil-ity between gas and oil. For exam-ple, if a bubble of gas exists at the bottom of a 1,000-ft column of oil having a gradient of 0.30 psi/ft with the pressure at the top of the column zero, the bubble of gas will be com-pressed to a pressure of 300 psi. If temperature and the combined vol-ume of the oil column and the gas bubble remain constant, as the gas bubble rises in the oil column it will have the same volume and conse-quently the same pressure at any point.

Under these conditions when the bubble reaches the top of the oil col-umn, the pressure will be approxi-mately 300 psi at the surface and ap-proximately 600 psi at the bottom of the oil column. If it is assumed that the oil is incompressible the pressures will be exactly 300 and 600 psi, re-spectively.

Vgc 100 Units

P -GAS /v gc 10 Units

10 Mois CAP '"gc-

OIL

Gas Bubble

Vb - 1 Unit

Pb = 100 Units

j nb 1 Mol

FIG. I-PRESSURE INVERSION WITH A GAS CAP.

EXAMPLE 2 Let us take an example with a gas

cap. An ideal example is shown in Fig. 1. The gas-cap volume (Vgc ) is 100 units and the gas-cap pressure (Puc) is 10 units. A bubble of gas exists at the bottom of the oil column which bubble has a volume (Vb) of one unit and a pressure (Pb ) of 100 units. The volume of the oil column including the gas bubble is 1,001 units. Reservoir temperature is as-sumed to be constant throughout; the gas will not re-dissolve in the oil; the bubble contains 1 mol (N b ); and the gas cap contains 10 mols (Ngc ). Pressure-volume relationships are assumed to follow the perfect gas law.

Therefore,

Pgc X Vue = Pb X Vb -= 100. Nuc Nb

The ,otal volume the gas will oc-

cupy when the bubble reaches the gas cap is 101 units and there will be 11 mols of gas. Assume p. is the pressure in the gas cap after the bub-ble of gas has risen to the top. So,

p. X 101 = 100 11 '

where p. = 10.89 units. It can be seen from the aforemen-

tioned that the gas bubble rising through the oil column to the gas cap increased the gas-cap pressure by 0.89 units. Therefore, the pres-sure at any point in the reservoir will have been increa&ed by the same amount. Since material balance cal-culations equate change in pressure against volume withdrawal, it fol-lows that this increase of 0.89 units of pressure indicates a pseudo-influx that must be accounted for in ma-terial balance calculations.

It is recognized that gas does not rise through the porous media in the form of "bubbles". As the reservoir pressure drops below bubble point, gas is evolved in the form of bub-bles. These bubbles grow in size until by diffusion they coalesce forming continuous gas passages. At this time the gas begins to move, passing up-wards through the reservoir in the gas channels in a relatively continu-ous phase. Actually, the pressure re-lationships discussed herein are the same no matter how the evolved gas rises. Therefore, because it is easier to explain the phenomenon using bubbles, it will be assumed that the gas is evolved in the form of bub-bles and moved in some manner to the gas cap.

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RECOMMENDED ApPROXIMATE

FORMULA

The following is an approximate formula that can be used to calcu-late the pseudo-influx.

XVb,(l)

where Gp , is pseudo-influx in barrels during a given period of production, t; P b is average pressure at the level in the oil column where it can be as-sumed that all of the gas has evolved and moved up-dip to the gas cap dur-ing the same production period, t; Puc is average gas-cap pressure dur-ing the same period of production, t; and Vb is volume in reservoir bar-rels of the gas evolved which reaches the gas cap during the same produc-tion period, t, under the conditions of Pb and reservoir temperature.

ApPROXIMATE SOLUTION FOR

DETERMINING PSEUDO-INFLUX

In a reservoir in which the pres-sure is dropping and is below the bubble point throughout, the addi-tional pressure created by the pres-sure inversion will be the sum of a relationship involving the mass and volume of each bubble evolved which will migrate to the gas cap and the difference in pressure for each bub-ble between the point at which it is evolved and the gas-cap pressure. Since this is a transient condition and since it would be impossible to tell how much of the evolved gas is pro-duced from up-dip wells and how much of the gas has re-dissolved, there is no exact solution. It would appear that approximate methods must be employed. The following is one approach to the solution of this problem.

Examination of solubility vs pres-sure curves for oil and its dissolved gas will show that as the pressure is decreased from the bubble point, gas will be evolved at nearly a con-stant rate per pound drop until the pressure has decreased to a few hun-dred psi. Even at these pressures the change is gradual so that an approxi-mate straight line could be drawn through the curve for a pressure in-crement of as much as 300 psi. Since it is the rare reservoir that has an oil column of over 1,000 ft and an original pressure of less than 500 psi, it will be assumed that when the pressure of the entire oil column is below the bubble point, the same amount of gas will be evolved from each barrel of reservoir oil per pound pressure drop regardless of its posi-tion in the reservoir.

The determination of the point in

FIG. 2-EFFECT OF POSITION OF THE GAS IS CONSIDERED WHEN (A) OIL COLUMN ASSUMED OF CONSTANT VOLUME FROM THE OILWATER CONTACT TO THE GAS-OIL CONTACT. (B) GRAPHICAL REPRESENTA TION OF WHAT EACH EVOLVED BUBBLE WILL CONTRIBUTE TO PRESSURE INVER-

SION EFFECT WITH RESPECT TO ITS

POSITION OF EVOLUTION.

the reservoir which can be used as that from which all gas has been evolved is explained in two steps. In Fig. 2 (A) it is assumed that the oil column is of constant volume from the oil-water contact to the gas-oil contact. The effect of position of the gas will thereby be considered. In Fig. 3 the quantity of gas with relation to depth is considered. The final approximation is the product of these two effects.

In Fig. 2 (AY it is assumed that the entire oil column is below bub-ble-point pressure, the pressure is dropping and gas is evolving over the entire oil column. Fig. 2 (B) is a graphical representation of what each evolved bubble will contribute to the pressure inversion effect with respect to its position of evolution. A bubble of gas evolved at the oil-gas contact will contribute no additional pressure due to its change in position in ris-ing to the gas cap. On the other hand, a bubble of gas evolved at the oil-water contact will contribute an additional pressure of P b less (what the gas-cap pressure is when this quantity of gas reaches it) by its rise --or, since pressure and volume are inversely related in the gas law, if the pressure drops by the aforemen-tioned increment, it will contribute an increase in volume due to its change in position.

In Fig. 2 (B), therefore, this pres-sure differential is related in an ap-proximate straight line to depth and the average effect of pressure inver-sion may be considered at the line where A, = A,. This line is 0.293 of the length of the oil column from the oil-water contact to the gas cap.

If the reservoir is not uniform,

JOUR"'AL OF PETROLEt:M TECHNOLOGY

-Q-G Contact

_~~=~_---J-O-W ContClet

FIG. 3-QUANTITY OF GAS CONSIDERED WITH RELATION TO DEPTH IN WHICH THE CROSSSECTIONAL AREA OF THE OIL COL-

UMN IS PLOTTED AGAINST DEPTH.

then a graph similar to Fig. 3 should be prepared in which the cross-sec-tional area of the oil column is plotted against depth. The distance, (b), from the oil-water contact where the planimetered area, A, = A" should be determined. The level at which all gas can be considered to have been evolved is 0.586b above the oil-water contact.

ApPLICATION

If it is assumed that an equal quan-tity of gas is evolved with each pound pressure drop, it follows that the pseudo-influx effect will get lar&er as reservoir pressure decreases. This is

MAY, 1959

because this effect is inversely pro-portional to a measure of reservoir pressure. It is also necessary to re-member that the effect will increase at increasing rate with decreasing res-ervoir pressure. It is this fact, that magnitude of error will increase at increasing rate, that is of the greatest importance, for it will influence res-ervoir interpretation even in reser-voirs of decreasing pressure where the oil column is relatively small. Material balance calculations on these reservoirs may show an in-creasing influx that is interpreted as water when as a matter of fact there might be no influx at alI.

CONCLUSIONS

To use this method it is necessary to know: (1) the size and configura-tion of the reservoir, (2) the oil-gas and oil-water contacts, (3) complete PVT analysis, and (4) pressure-pro-duction history.

The basic steps for the analysis would be as follows. .

1. Determine the depth from which it is assumed that all the gas has evolved. Use method just described.

2. Calculate the quantity of gas that has evolved from the oil during the production period.

3. From experience determine a "lag" factor to apply if the produc-

tion rate varies greatly between con-secutive time periods.

4_ Determine the reservoir pres-sure (at depth calculated in Step 1) and the gas-cap pressure for the mid-period and with the quantity of gas calculated in Step 2, substitute in Eq. 1. This will give a quantity that represents the pseudo-influx for the period unless the production rate has varied greatly from the previous in-terval of time.

5. If the production rate has varied greatly, multiply the result obtained in Step 4 by the lag factor obtained in Step 3. This will give an estimated pseudo-influx for the period.

The quantity obtained in either Step 4 or 5 is used in material bal-ance calculation as gas influx into the gas cap.

ACKNOWLEDGMENT

The author wishes to express his appreciation to Martin, Sykes and Associates, C. A., for permission to publish this paper.

REFERENCES

1. Mead, Homer N.: "Blowout Control Begins Before They Occur", Pet. Engr. (Sept., 1957) 29, No. 10.

2. Stegemeier, G. L. and Matthews, C. S.: "A Study of Anomalous Pressure Build-Up Behavior", Trans. AIME (1958) 213, 44. ***

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