The Application of the Buckley-Leverett Frontal Advance Theory to Petroleum Recovery

I N T R O D U C T I O N

STEPHEN G. DARDAGANIAN* JUNIOR MEMBER AlME

Basically, the Buckley-Leverett theory involves two systems which are similar in nature but are differen- tiated by time. These systems may be described by the fractional flow and frontal advance equations which es- sentially characterize the mechanics of oil movement while being expelled from the reservo'ir.

A & M COLLEGE O F TEXAS COLLEGE STATION, TEX.

The fractional flow equation orig- inally developed by Leverettl may be expressed in the more usable form

[I221 - .433 AY sin e I The development of this equationz is based on Darcy's law describing fluid flow through porous media and ap- plies to the flow at only one point (it is a point function). For simpli- fication, the fractional flow equation is written in the above form because the capillary forces always increase the fractional flow of the displacing phase regardless of the direction of flow or the displacing phase.

If, for simplicity, the effects due to gravity and capillary pressure dif- ferences are neglected, the fraction of the displacing fluid, f,, at any point in the flowing stream is related

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Original manuscript received in Society of Petroleum Engineers ofice on March 11. 1957. Revised manuscript received Feb. 2 6 . 1958. Paper presented at Joint University of Texas- A & M College of Texas Student Chapters of AIME Regional Meeting in Austin, Tex., Feb. 14-15, 1957.

*Presently completing 'The Atlantic Refin- ing Co. training program, Dallas. Tex.

'References given at em1 of paper.

to the properties of the system by

0 k* Thus, it is seen that in the absence of capillary and gravitational effects, f d for a given sand and fluids varies only with saturation and pressure. The magnitude of the viscosity ratio,

Pa -, has an effective range (range of Po about 30) in the system where gas is displacing oil and a much smaller range for water displacing oil.

In order to make the fractional flow equation more versatile, it is necessary to connect the fractional flow at a given point and saturation with time. This problem was ap- proached by Buckley and Leverett" who developed the frontal advance equation,

( ) Af AS,, S., . . ( 3 )

Eq. 3 states that the rate of advance of a plane that has a fixed saturation, Sd, is proportional to the change in composition of the flow stream caused by a small change in the saturation of the displacing fluid. I t is, essential- ly, a transformation of a material balance equation representing the net rate of accumulation of the displac- ing phase within a homogeneous sand block. This accumulation is propor- tional to the difference between the rate at which the displacing fluid enters the sand and that at which it leaves.

Eq. 3 describes the velocity with which a plane of constant displacing phase saturation advances through a porous system. Buckley and Leverett,"

Babson,' Kern," Welge ,hnd others have adequately discussed the basic mechanism and application of the fractional flow and frontal advance equations.

APPLICATIONS OF THE FRONTAL ADVANCE THEORY TO PETROLEUM RECOVERY

Two general applications of the Buckley-Leverett frontal advance the- ory involve the system in which the oil is being displaced by an expand- ing gas cap overlying the oil zone and that in which the oil is being displaced by water.

A system in which gas is the dis- placing phase may be thought of as having two forces effecting the dis- placement process. These forces are the gravitational force and that force exerted by the displacing gas. The gravitational effects control the dis- placing efficiency of the gas. The gra- vitational effect will be less at higher rates of flow, thereby reducing the effectiveness of the displacement of the oil by the gas. The more efficient displacements occur at flow rates which are less than the gravity free fall rate. Capillary forces can be neg- lected without materially changing the magnitude of the gas saturation.

The Mile Six pool is used herein to illustrate the calculating proce- dures in evaluating gas drive-gravity drainage field perfomlance. These calculations represent the determina- tion of the gas-oil contact when the distribution of the hydrocarbon pore volume is considered. Two methods

V O L . 2 1 3 , 1 9 5 8 365

Jre ava~lable lor making such calcu- lations: ;I lincar block method and a volumetric invasion method.

The lincur hlock method involves the treatment of the reservoir as a u n i f ~ ) r n ~ block. The displacement of oil by thc advancing gas cap is evaluated by

!!,

k A . 1 4- .001127--2--( - ,433 a; sin 6 ;

/J& 0, .. - ..~ .. -

I + -5,;- 1 , k,l

. . . . . . . . . . (4) which is the same as Eq. 1 except that the capillary forces are neglec- ted. It should be noted that the cross- sectional area normal to flow must be determined before the fractional flow can be evaluated. The re a re two methods presented by Amys, Bass. and Whiting which can be ~ ~ s e d for the determination of this area. A res- ervoir voidage rate of 9,177 BID and o ther propert ies reported by Welge and Anders' were used in the calculations shown in Fig. 1.

In order to determine the position of the gas-oil contact at any time, it is necessary to solve the following equations for various values of time or gas-oil contact position

These are essentially forms of Eq. 3. Eq. 5 may be restated in terms of hydrocarbon pore space provided the fractional flow equation is expressed with gas saturation as a fraction of hydrocarbon pore volume (as shown in Fig. I ) . The following is noted:

f'=f(l - S , ) .

5 ,:x 4 ( I - S l l

F r c . ~ - (FROAI Ittr. 7 ) FRACTIONAI. E'LO\V OF A GAS A S A F ~ J N C T I ~ N OF GAS SATURA- TION EXPRISSSCD AS 4 FK\ ( :TI~N O F H Y D R O -

CAR1:OT ~)( : ( 'uT ' I I ' I ) 1'0~1: ~ l ' r ~ ~ ~ ; .

rlf , 'I lld --

1 (If - - - - - -. -

d S , - ( I S , ) (1s' Using the above definitions, Eq. 5 may hc rewritten as

5.61 L Af' Q,T = , . . ( 6 )

where Q,T is the cumulative reser- voir voidage.

The volumetric invasion method involves the use of the hydrocarbon pore distribution. The invaded hy- drocarbon pore volume appears in Eq. 6 as: hydrocarbon pore volume = L A f (1 - S , , ) , cu f t such that its value can be read directly from Fig. 2 which shows the distribution of the hydrocarbon pore volume of the oil zone with depth in the Mile Six pool. The distance along the bed- ding planes (L ) is converted to ver- tical heights by h = L sin 0. This conversion is performed because the vertical displacement of the gas-oil contact is of basic importance in res- ervoir study.

Using the modified form of Eq. 6 such that

Q T sin 0 ( d!. ) Ah I= -:-- - 5.61 A f' ( 1 - S, ) dS', '

. . . . . . . . . ( 7 ) the saturation distribution with height may be calculated. As the position of the front is specified in each case, the cumulative reservoir voidage term may be replaced by Eq. 6 such that Eq. 7 becomes

Ah -= L sin OF$/ (2:) f] . . . . . . . . . . ( 8 )

Thus, for any given position of the gas-oil contact denoted by L, the sat- uraticn dis tr ibut ion can be deter- mined.

Sample calculations for the deter- mination of the gas-oil contact when the distribution of the hydrocarbon pore volume is taken into considera- tion are presented in 'Table 1. The data calculated in Table 1 are plotted in Fig. 3, which shows a comparison of the computed position of the gas- oil contact to that observed from field data. Only small deviations are noticed at low values of cumulative production with the deviation increas- ing with increased production.

The time and cumulative produc- tion are readily obtained as it was as- sumed that thc rate of reservoir voi- dage was a constant and that no pro-

FIG. 2-(FROM REF. 7 ) DISTRIBUTION OF HYD~IOCARDON PORE V O L U M E I N T H E OIL

ZONE WITH DEPTH, MILE SIX POOL.

duction occurred behind the front. 'Thus, @,T = N,B, , where Q, and B, are known, and Q,T is expressed and calculated as a function of the gas-oil contact position by Eq. 6.

The fractional flow and frontal ad- vance calculations for a system in which water is the displacing phase are basically the same as the calcu- lations for a system in which gas is the displacing phase. However, the latter generally occurs in a system in which a third immobile phase (water) is usually present, whereas, in a water-drive system, the third phase (gas) may or may not be present and may exist either in a mobile or immobile state.

Pressure is the controlling factor which defines the state of the third phase (gas) in a water-drive system. If the reservoir exists at a pressure above the bubble-point pressure of the oil, no free gas phase will exist, and the system may be evaluated as a two-phase system. Assuming the reservoir exists at a pressure slightly below the bubble-point pressure of the oil, the gas that has been evolved from the oil may be thought of as being immobile. This system may be evaluated as a three-phase system with an immobile third phase. If, however, the reservoir exists at a pressure sufficiently below the bub- ble point such that the evolved gas has mobility, the problem will then become one involving three mobile phases. The treatment of such a prob- lem will now be discussed.

Consider a reservoir in which a gas phase has been created by pro- duction and pressure depletion. The saturation distribution is such that three mobile phases exist. The frac- tional flow-frontal advance theory breaks down when applied to this system as a whole. However, this problem may be simplified remem- bering that an oil bank develops ahead of a waterflood f ron t when flooding in the presence of a mobile

P E T R O L E U M T R A N S A C T I O N S , A I M E

TABLE I-CALCULATION OF GAS-CAP POSITION AS A FUNCTION OF CUMULATIVE PRODUCTION (Voidage Rate-9,177 B / D , B o = 1.196)

, Q, = L A f I - S ( 0 5.61 (Sgalr

. * Np =TQI- B"

-I From Fig. 2

-

gas saturation. This complex system of three mobile phases may now be thought of as two separate systems in which only two mobile phases exist. One system is that in which the oil bank displaces the gas to an i n ~ m o - bile saturation. In the second system, the advancing water bank displaces the oil from the porous rock in the presence of an immobile gas phase. By combining the evaluations of these systems, it is then possible to arrive at an evaluation of the system as a whole. The methods for such evalua- tions are developed in the following paragraphs.

The residual gas sa tura t ion de- veloped during a flood is a function of the rock and fluid properties and can be determined from flow tests on the rock. During a water flood, the oil bank that builds ahead of the water front tends to displace any free gas from the system. Since the viscosity of the displacing oil is con- siderably greater than that of the displaced gas, the oil bank essen- tially displaces the gas with a pis- ton-like effect. The gas is reduced to a "trapped" saturation assuming the pressure is such that it does not go into solution. Fig. 4 presents data adapted from flow tests performed by Kruger%n the amount of gas trapped by wate r flooding a gas-saturated core. This shows that the magnitude of the residual gas saturation ob- tained is a function of the amount of gas initially present.

The frontal advance theory may be used to evaluate 1:he displacement of oil by water in the presence of

the immobile gas. It is concluded that the residual oil saturation by water displacement will be a function of the r e s i d ~ ~ a l gas within the oil bank. The effect of the residual gas satura- tion on the residual oil may be com- puted by the fractional flow-frontal advance theory if appropriate relative permeability data are available. From such data and frontal advance con- cepts the residual oil saturation at breakthrough of water may be calcu- lated as a function of the viscosity ratio, pU/p,. For a given reservoir, the viscosity of water at reservoir conditions is essentially independent of the pressure. Therefore, the only variable effecting the residual oil saturation that varies with pressur: is the oil viscosity. Using the oil and water viscosity data applying to a particular reservoir, the variation of the viscosity ratio with pressure may be calculated. Fig, 5 illustrates a family of curves defining the residual oil saturation as a function of the pressure and pas saturation.

The initial free gas saturation may be determined from material balance calculations, and by utilizing da ta such as that shown in Fig. 4, the residual gas saturation existing in the oil bank may be estiinated. The resid- ual oil saturation then may be de- termined from the residual gas satu- ration and the reservoir by the utilization of such data as pre- sented in Fig. 5.

INIT 'AL GAS SATURATION, PERCENT

FIG. &-(FROM REF. 9 ) ESTIMATED GAS ENTRAPMENT AFTER OIL FLOOD.

In the development of this evalu- ating process, the rcsidual gas satu- ration cleveloped by the advancing oil bank is considered to be the same as the residual gas saturation hehind the waterflood front such that the water flood has n o effcct on the gas sntura- tion. Tn addition, the residual oil saturation is thought of as essentially being equal to the average oil satura- tion behind the front. Thus, the re- sidual oil may be defined by (1 - S,, - 5 ,,,, ) . The second phase of re- covery (displacement of oil a f te r breakthrollgh of the flood front) is not considered since the economic value of such recovery is sometimes relatively low.

By evaluating the performance of solution gas-drive reservoirs, the free gas saturation and recovery as a func- tion of pressurc decline may be cal- culated. The effect of initiating and maintaining a water flood at a given pressure may be determined as shown in Table 2. The results of applying this procedure to other systems are shown in Fig. 6. Curves A, B, C, and D are based on the calculated per- formance of a solution gas-drive given by Muskat'". The waterflood re- coveries were estimated from the data in Fig. 4 and curves corresponding to Fig. 5 prepared for each API oil. The curves in Fig. 6 consider 100 per cent pattern efficiency. Note that Curves A and B show maximums in total oil recovery a t pressures below the bubble-point pressure (3.000 psi for all of the oils).

The maximums in total oil recov- ery are attributed to the effect o f the free gas saturation in reducing the residual oil following a water flood. Maximums do not occur in Curves C and D. This is attributed to the ef- fect of the greater shrinkage of the higher gravity oils. Curve E illus- trates the effect of pattern efficiency on the optimum flooding pressure.

F L ~ J t i P R t 5 5 1 ~ E c o

FIG. 5-(FROM RLF. 8) EFFECT or PATS- SLRE ON RESIDI~AL OIL S A T U R ~ T I O N BY WATER 1 7 ~ o o o ~ u c FOR A 30' API CRUDE OIL.