Dake L.P. Fundamentals of reservoir engineering

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IMMISCIBLE DISPLACEMENT 371

the description of two dimensional, segregated flow using one dimensional equations.

Therefore oil recovery calculations, for either stable or unstable, segregated flow, can

be performed using the linear relative permeabilities in conjunction with the Buckley-

Leverett displacement theory. This is because the theory was based simply on the

conservation of water mass, in one dimension, equ. (10.13). Therefore, whether the

water is uniformly distributed with respect to thickness or segregated from the oil does

not matter provided the resulting displacement can be described using one

dimensional mathematics; the same basic principle of mass conservation still applies.

The fractional flow equation can be plotted using the linear relative permeability

functions and the Welge graphical technique applied as illustrated in exercise 10.2. In

this case, the fractional flow curve will have no inflexion point, as shown in fig. 10.22,

since there is no shock front for segregated flow. All points on the fractional flow curve

are used in the recovery calculations after breakthrough.

f = 1

1 - S

Fig. 10.22 Typical fractional flow curve for oil displacement under segregated

conditions

Because the thickness averaged relative permeabilities are linear for segregated flow,

it is also possible to derive a simple analytical expression for the oil recovery as a

function of the cumulative water injected. As mentioned al ready, this is unnecessary

for stable displacement but for unstable displacement it provides a rapid means of

predicting recovery.

The following argument will, for simplicity, be developed for the unstable displacement

of oil by water in a horizontal reservoir. As described in Chapter 4, sec. 9 and

illustrated in exercise 10.1, this will occur if M > 1. An analytical expression for the

fractional flow of water will be derived and used in the oil recovery formula of Welge,

equ. (10.32). The one dimensional equations for the separate flow of oil and water,

under segregated conditions in a horizontal reservoir, are

IMMISCIBLE DISPLACEMENT 372

ro o

(1 b)kk A p

′

−∂

=−

∂

(10.46)

and

rw w

bkk A p

′

∂

=−

∂

(10.47)

in which A is the area of cross-section and

and

are the oil and water phase

pressures referred to the centre line of the reservoir, as shown in fig. 10.23.

Centre line

oil - water interface

}

(

-y)

Fig. 10.23 Referring oil and water phase pressures at the interface to the centre line in

the reservoir. (Unstable segregated displacement in a horizontal,

homogeneous reservoir)

From this diagram it can be seen that

pp y atm

2 1.0133 10

æö

=−−

ç÷

èø

and

pp y atm

2 1.0133 10

æö

=−−

ç÷

èø

where y is the actual thickness of the water; i.e. y = bh. Since the pressures at the

interface, p

and p

, are equal for segregated flow then the phase pressure gradient,

resulting from the differentiation and subtraction of the above equations, is

pgdy

xx dx

1.0133 10

∂

∂∆

−=−

∂∂

For unstable, horizontal displacement the approximation is usually made that the angle

of inclination of the interface, dy/dx, is small and therefore the gradient of the phase

pressure difference can be neglected. In this case, using equs. (10.46) and (10.47),

and following the argument used in sec. 10.3, in deriving the fractional flow equation,

results in

ro w

orw

ro w

k1b

′

⋅

′

−

+⋅

′

IMMISCIBLE DISPLACEMENT 373

which can be simplified as

1(M1)b

+−

in which M is the end point mobility ratio. Until the moment of breakthrough the oil

recovery is simply equal to the cumulative water injected. After breakthrough, let b

the fractional thickness of the water at the producing end of the reservoir block. Then,

for a fully penetrating well, the fractional flow of water into the well is

1(M1)b

+−

(10.48)

Applying equ. (10.27) at the producing end of the block

and using equ. (10.43) for the thickness averaged water saturation, S

we we e we

we w

id e e or wc

df df db df

Wdbdb(1SS)

dS dS

==⋅=⋅

−−

and therefore

we or wc

eidiD

df (1 S S )

db W W

−−

in which W

is the cumulative water injection expressed in movable oil volumes where

1 MOV = PV(1 − S

− S

)

Differentiating equ. (10.48) with respect to b

gives

eiD e

db W

(1 (M 1)b )

+−

from which it can be determined that

()

eiD

bWM1

=−

−

(10.49)

and substituting for b

in equ. (10.48) gives

()

we iD

f1WM

=−

−

(10.50)

The oil recovery equation, (10.32), can be expressed in MOV's as

pD we iD

or wc

N(1f)W

1S S

−

=−

−−

IMMISCIBLE DISPLACEMENT 374

pD e we iD

Nb(1f)W=+−

and substituting in this equation for b

and f

, using equs. (10.49) and (10.50), results

in the simple recovery formula

pD iD iD

N(2WMW1)

=−−

−

(10.51)

in which all volumes are expressed in MOV's. It should again be stressed that

equ. (10.51) is only applicable for horizontal displacement under segregated conditions

and for unstable flow (M > 1).

At the time of water breakthrough N

= W

and solving equ. (10.51) for this condition

gives

(10.52)

which shows that in the limiting case of M = 1, stable, piston-like displacement occurs

for which

N1= . Similarly, when the total amount of oil has been recovered, N

= 1

(MOV), and substituting this condition in equ. (10.51) gives

max

WM= (10.53)

Equations (10.52) and (10.53) clearly demonstrate the significance of the mobility ratio

in characterising oil recovery under segregated flow conditions.

For the more general case of unstable displacement in a dipping reservoir (G<M−1),

the fractional flow equation, equivalent to equ. (10.48), is

ee e

Mb b (1 b )G

1(M1)b

−−

+−

and repeating the steps leading to equ. (10.51 ) will, in this more complex case, yield

the recovery formula

pD iD iD

WG1G (M1)

N2WM11W1G1

M1 M1 M1 (M1)

æö

æöæö

=−−−−−

ç÷

ç÷ç ÷

ç÷

−−−−

èø

(10.54)

which if G = 0 (horizontal reservoir) reduces to the form of equ. (10.51).

Equation (10.54) can readily be solved for the breakthrough condition (N

= W

) to

give

−

(10.55)

while for the maximum recovery (N

= 1)

IMMISCIBLE DISPLACEMENT 375

max

(10.56)

EXERCISE 10.3 DISPLACEMENT UNDER SEGREGATED FLOW CONDITIONS

1) Re-work Exercise 10.2, for the same water injection rate and using precisely the

same reservoir and fluid property data but assuming that displacement takes

place under segregated flow conditions. Compare the results with those obtained

in exercise 10.2.

2) If the same reservoir had a dip angle of 25°, what would be the critical rate for

water displacing oil updip? Compare the breakthrough times and the oil recovery

at breakthrough when injecting at 1000 rb/d and also at 90% of the critical rate.

Additional information

k = 2.0 Darcy

= 1.04 specific gravity in the reservoir

=.81 """"

EXERCISE 10.3 SOLUTION

1) Using the relative permeability relations given in exercise 10.1, table 10.1, and for

= 5 cp,

= 0.5 cp (Case 2 of exercise 10.1), the end point mobility ratio can

be evaluated as

rw w

ro o

.3 .8

M3.750

k/ .5 5

′

===

′

Therefore, using equs. (10.52) and (10.53), breakthrough will occur when

bt bt

iD pD

W N .267 (MOV) .160 (PV)

=== =

since 1 MOV = PV(1 − S

− S

) = PV(1−0.2−0.2) = 0.6 PV; and the maximum oil

recovery when

= M = 3.75 (MOV) = 2.25 (PV)

Between breakthrough and maximum recovery, the oil recovery as a function of W

can be calculated using equ. (10.51), with W

as the independent variable. The results

are listed in table 10.6.

IMMISCIBLE DISPLACEMENT 376

t = 4.39 W

(yrs)

(MOV) (MOV) (PV) (PV) (equ. (10.35) )

.267 (bt) .267 .160 .160 .702

.300 .299 .180 .179 .790

.500 .450 .300 .270 1.317

1.000 .681 .600 .409 2.634

1.500 .816 .900 .489 3.951

2.000 .901 1.200 .540 5.268

3.000 .985 1.800 .591 7.902

3.750 1.000 2.250 .600 9.878

TABLE 10.6

In fig. 10.24 plots of the oil recovery, assuming the segregated flow condition, are

compared with the results from exercise 10.2, assuming diffuse flow. The comparison

shows that, although the breakthrough occurs much earlier for segregated flow, the

ultimate recovery is obtained sooner and for a much smaller throughput of water.

N (PV)

0 5 10 15 20 25 30

W (PV)

time

)

q = 1000 rb / d

DIFFUSE FLOW ( EXERCISE 10.2)

SEGREGATED FLOW (EXERCISE 10.3)

Fig. 10.24 Comparison of the oil recoveries obtained in exercises 10.2 and 10.3 for

assumed diffuse and segregated flow, respectively

2) For the data given in exercise 10.2, the critical rate for by-passing can be

calculated, in field units, as

crit

4.9 10 kk A sin

(M 1)

−

′

×∆

−

(10.42)

IMMISCIBLE DISPLACEMENT 377

4 o

crit

4.9 10 2000 .3 625 40 (1.04 0.81) sin25

q (rb / d)

.5(3.75 1)

520 rb.water / d

−

×× ×××× − ×

−

While for injection at this critical rate

G = M − 1 = 2.75

Comparison of equs. (10.39) and (10.42) indicates that

crit

(M − 1) = q

and therefore, at an injection rate of q

= q

= 1000 rb/d

520

G 2.75 1.430

1000

=×=

Substituting this value of G and the value of M = 3.75 in equ. (10.54), for this unstable

displacement (G < M−1), reduces the latter to

pD iD iD iD

N 0.976 W (1 0.520W ) 0.535 W 0.364=−+− (10.57)

At the time of water breakthrough N

= W

and equ. (10.55) can be applied to

determine the breakthrough recovery as

N 0.431 (MOV) 0.259 (PV)

M G 3.75 1.43

== = =

−−

which, when injecting at 1000 rb/d, will occur after 4.39

W = 1.137 years,

equ. (10.35). Similarly, the maximum cumulative water injection to recover the one

movable oil volume can be determined using equ. (10.56) as

max

M3.75

W 1.543 (MOV) 0.926 (PV)

G1 2.43

=== =

Between breakthrough and total recovery equ. (10.57) can be used to calculate the oil

recovery, as listed in table 10.7.

(MOV)

(PV)

t = 4.39 W

(yrs)

(equ. (10.35))

.431 (bt) .431 .259 .259 1.137

.500 .497 .300 .298 1.317

.750 .697 .450 .418 1.976

1.000 .847 .600 .508 2.634

1.250 .950 .750 .570 3.293

1.543 1.000 .926 .600 4.064

TABLE 10.7

When compared to the water drive performance, at an injection rate of 1000 rb/d in a

horizontal reservoir (table 10.6), it can be seen that, even though the displacement is

IMMISCIBLE DISPLACEMENT 378

unstable in both cases, the gravity force in a reservoir with a 25° dip has a very

favourable effect on the water drive performance, the maximum recovery being

obtained in less than half the time required in a horizontal reservoir.

For stable displacement, at 90% of the critical rate (468 rb/d), the gravity number is

crit

G (M 1) 2.75 3.056

q0.9

=×−=× =

The angle of inclination of the oil water interface to the direction of flow can be

determined using equ. (10.40) as

dy M 1 G 3.750 1 3.056

tan tan 0.4663 0.0467

dx G 3.056

βθ

−− −−

=− = = × =−

and therefore

= 2.673°. The situation after breakthrough, when the water has risen a

height y

at the level of the producing well, is shown in fig. 10.25.

OIL

WATER

Fig. 10.25 The stable, segregated displacement of oil by water at 90% of the critical rate

(exercise 10.3)

If the width, thickness and length of the reservoir are denoted by w, h and L,

respectively, then the total movable volume of oil is whL

(1−S

−S

). For the situation

depicted in fig.10.25, the volume of oil uncontacted by the water is

or wc

(h y) (h y)

w(1 S S )

2tan

−−

×−−

and therefore the oil recovery at this stage, expressed in MOV's, is

(h y )

2hL tan

−

=−

(10.58)

The amount of water injected at this stage can be estimated by ignoring the presence

of the producing well. The volume of water by-passing the well, as shown in fig. 10.25

IMMISCIBLE DISPLACEMENT 379

or wc

w(1 S S )

2tan

⋅−−

and therefore the number of MOV's of water injected is

iD pD

2hL tan

(10.59)

In particular, at breakthrough, y

= 0 and

bt bt

pD iD

NW1

2L tan

==−

(10.60)

For the data relevant to this exercise, the recovery at breakthrough can be calculated

using equ. (10.60) as

N 1 0.786 MOV .472 PV

2 2000 .0467

=− = =

××

At an injection rate of 468 rb/d the relation between water injected and time is now

(refer equ. (10.35)

id id

1000

t 4.39 W 9.38 W (years)

468

=× =

and therefore breakthrough occurs after 9.38 × .472 = 4.43 years. Thereafter the oil

recovery and cumulative water injection, as functions of time, are listed in table 10.8.

These have been calculated using equs. (10.58) and (10.59) for increasing values

of y

(MOV)

(PV)

t = 9.38W

years

0 (bt) .786 .786 .472 .472 4.427

10 .880 .893 .528 .536 5.028

20 .946 1.000 .568 .600 5.628

30 .987 1.107 .592 .664 6.228

40 1.000 1.214 .600 .728 6.829

TABLE 10.8

When compared to the results presented in table 10.7, which were obtained at an

injection rate of 1000 rb/d, the effect of displacing the oil at a rate which is below critical

(468 rb/d) is quite naturally to increase both the breakthrough time (by nearly 300% )

and the total recovery time (by almost 70% ). The main advantage in displacement at

the lower rate is that the total water requirements are reduced from 1.543 to

1.214 (MOV). If there are problems concerning either the availability of injection water

or disposal of produced water it may be considered expedient to inject at a rate which

is just below critical.

IMMISCIBLE DISPLACEMENT 380

The same analytical methods described in this section can be applied to the case of

gas displacing oil downdip, at constant pressure.

As shown in fig.10.26,for unstable displacement the gas will tend to override the oil

causing premature gas breakthrough in the downdip producing wells. For stable

displacement the angle of inclination of the gas-oil interface remains constant.

GAS

OIL

(a)

GAS

OIL

(b)

Fig. 10.26 Segregated downdip displacement of oil by gas at constant pressure;

(a) unstable, (b) stable

Applying the method described in this section for determining the critical rate (in this

case it is algebraically convenient to keep the same sign convention, x-positive in the

updip direction, and reverse the signs of the flow velocities) will again give the

condition for stable displacement as

G > M−1

This leads to the critical rate formula

crit

kk A g sin

1.0133 10 (M 1)

′

∆

×−

in which, for gas displacing oil

ρρρ

∆= −

crit

tgt

kk A g sin

G(M1)

q1.0133 10 q

′

∆

==−

(10.61)

and

rg g

ro o

′

Since

is very small compared to

, the mobility ratio for this displacement is large

and the requirement for unconditional stability (M ≤ 1) is never satisfied. Stability then

depends on the magnitude of G and hence on the dip angle.

An interesting extension of the methods described in this section has been presented

by van Daalen and van Domselaar

who considered segregated displacement in

reservoirs in which there is a defined (absolute) permeability distribution in the dip

normal direction. In addition, Richardson and Blackwell

have analysed some quite