Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.1 An important problem in petroleum engineering is the prediction of oil recovery during displacement by water. Two common examples are a natural water drive and secondary waterflood. The latter is displacement of oil by bottom or edge water, the former is the injection of water to enhance production. In this chapter we will begin with the development of equations of multiphase, immiscible flow, concluding with the frontal advance and Buckley-Leverett equations. Next, we will discuss factors that control displacement efficiency followed by limitations of immiscible displacement solutions. 9.1 Development of equations The development of equations for describing multiphase flow in porous media follows a similar derivation as given previously for single phase, i.e., combination of continuity equation, momentum equation and equation of state. The mass balance of each phase can be written as: increment in time s accumulate that phase of mass increment in time leaving phase of mass increment in time entering phase of mass Shown in Figure 9.1 is the differential element of porous media for oil. u ox │ x u ox │ x+x x y z Figure 9.1 Differential element in Cartesian coordinates. Only x-direction velocity is shown. As an example, the mass of oil entering and leaving the element is given by: Entering: t A u t A u t A u z z oz o y y oy o x x ox o (9.1) Leaving: t A u t A u t A u z z z oz o y y y oy o x x x ox o (9.2) Oil can accumulate by: (1). Change in saturation, (2). Variation of density with temperature and pressure, and (3). Change in porosity due to a change in confining stress. Thus we can write, t o o t t o o V S V S (9.3)
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Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.1
An important problem in petroleum engineering is the prediction of oil recovery
during displacement by water. Two common examples are a natural water drive and
secondary waterflood. The latter is displacement of oil by bottom or edge water, the
former is the injection of water to enhance production. In this chapter we will begin with
the development of equations of multiphase, immiscible flow, concluding with the frontal
advance and Buckley-Leverett equations. Next, we will discuss factors that control
displacement efficiency followed by limitations of immiscible displacement solutions.
9.1 Development of equations
The development of equations for describing multiphase flow in porous media
follows a similar derivation as given previously for single phase, i.e., combination of
continuity equation, momentum equation and equation of state. The mass balance of
each phase can be written as:
increment in time saccumulate
thatphase of mass
increment in time
leaving phase of mass
increment in time
entering phase of mass
Shown in Figure 9.1 is the differential element of porous media for oil.
uox│x uox│x+x
x
y z
Figure 9.1 Differential element in Cartesian coordinates. Only x-direction velocity is
shown.
As an example, the mass of oil entering and leaving the element is given by:
Entering: tAutAutAu zzozoyyoyoxxoxo (9.1)
Leaving: tAutAutAu zzzozoyyyoyoxxxoxo
(9.2)
Oil can accumulate by: (1). Change in saturation, (2). Variation of density with
temperature and pressure, and (3). Change in porosity due to a change in confining stress.
Thus we can write,
toottoo VSVS
(9.3)
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.2
Substitute Eqs. (9.1-9.3) into the conservation of mass expression, rearrange terms, and
take the derivative as t, x, y, z 0, then the phase dependent continuity equations
can be written as;
ooozooyooxo St
uz
uy
ux
(9.4)
wwwzwwywwxw St
uz
uy
ux
(9.5)
The oil and water continuity equations assume no dissolution of oil in the water phase.
That is, no mass transfer occurs between phases and thus flow is immiscible.
The next step is to apply Darcy’s Law to each phase, i. For example in the x-
direction,
x
ku i
i
iixix
(9.6)
where uix is the superficial velocity of phase i in the x-direction, kix, is the effective
permeability to phase i in the x-direction, and is the phase potential. Substitute Eq.
(9.6) into (9.4), apply Leibnitz rule of differentiation, and combine terms, results in,
ooo
o
o
ozoo
o
oyoo
o
oxo St
gz
pk
zy
pk
yx
pk
x
(9.7)
www
w
w
wzww
w
wyww
w
wxw St
gz
pk
zy
pk
yx
pk
x
(9.8)
Even though Eqs. (9.7) and (9.8) are written in Cartesian coordinates, they both can be
solved for a particular geometry. The solution will provide not only pressure and
saturation distributions, but also phase velocities at any point in the porous media.
To combine Eqs. (9.7) and (9.8) requires a relationship between phase pressures
and between phase saturations. The latter is easily understood from the definition of
saturations in Chapter 4, So + Sw = 1.0. The relationship between pressures was
developed in Chapter 5, and is known as capillary pressure.
wP
oPor
wP
nwP
cP (9.9)
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.3
9.2 Steady state, 1D solution
As a simple example, let’s consider the steady state solution to fluid flow in a
linear system as shown in Figure 9.2. This example is of primary interest in lab
experiments to determine relative permeabilities.
qo
qw L
poi
Pwi
poL
PwL
D
Figure 9.2 Steady state core flood of oil and water.
Oil and water are injected simultaneously, rates and pressures are measured, and core
saturation is determined gravimetrically. Permeability is unknown.
The steady state, incompressible fluid diffusivity equations are given by:
0
0
dx
dpk
dx
d
dx
dpk
dx
d
ww
oo
(9.10)
Integrating and combining with Darcy’s equations,
A
qc
dx
dpk
A
qc
dx
dpk
www
ww
ooo
oo
(9.11)
If water saturation is uniform throughout the core, then effective permeability is
independent of x. Therefore, for oil,
L
oo
o
p
p
o dxk
cdp
oL
oi
(9.12)
which upon integrating, becomes,
)( oLoi
ooo
ppA
Lqk
(9.13)
If kbase = ko at Swi is known, then it is possible to calculate relative permeability.
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.4
9.3 Capillary End Effect
During laboratory experiments, capillary equilibrium must be maintained; that is,
Pc = Po – Pw. Unfortunately, under certain conditions capillary end effects occur due to a
thin gap existing between the end of the core and the core holder. As shown in Figure
9.3, capillary pressure in this gap is zero.
gap
Pc=0
Figure 9.3 Schematic of gap between core and holder
The result is a rapid change in capillary pressure from a finite value immediately adjacent
to the outlet to zero in the gap. As a consequence, the saturation of the wetting phase
must increase to a value of Pc = 0. A generic core profile is shown in Figure 9.4 for both
pressures and saturation.
Sw
0 0 L L
Po
Pw
Pc=0+ Swc
Sor
P
Figure 9.4 Pressure and saturation profile through a core of length, L, with capillary end
effect.
Mathematically, we can describe this effect by investigating Darcy’s Law for the non-
wetting phase.
x
S
S
p
x
pAkq w
w
cw
nw
nw
nw
(9.14)
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.5
At the outlet, knw 0, but qnw ≠ 0; therefore,
x
S
Lx
wlim
(9.15)
Two plausible methods have been applied to avoid capillary end effect. The first
is to inject at a sufficiently high rate such that the saturation gradient is driven to a small
region at the end of the core. The second method is to attach a thin, (high porosity and
high permeability) Berea sandstone plug in series with the test core sample. The result is
to confine the saturation gradient in the Berea plug and thus have constant saturation in
the sample of interest.
A consequence of the saturation gradient in the core is that effective permeability
can no longer be considered constant from 0 < x < L. Subsequently, the convenient
steady state method of obtaining relative permeability outlined in Section 9.2 is not valid.
A solution to the saturation gradient can be obtained be combining the definition of
capillary pressure with the steady state, incompressible diffusivity equations. Begin with
defining the boundary conditions. Illustrated in Figure 9.5 is the capillary pressure –
saturation relationship in a core with end effects.
Sw
pc
0
inlet
outlet
Figure 9.5 Schematic representation of capillary pressure – saturation relationship in a
core sample with end effect.
From this figure we can deduce the following conditions,
Pc = 0 for both oil and water phases at x = L.
Sw = Swi at x = 0, thus Pc = Poi – Pwi
Sw = SwL at x = L, thus Pc = 0
From the definition of capillary pressure,
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.6
dx
dp
dx
dp
dx
dp woc (9.16)
Since pc = f(Sw),
dx
dS
S
p
dx
dp w
w
cc (9.17)
Substituting Eq. (9.17) into (9.16) for the capillary pressure gradient term, and Eq. (9.11)
into (9.16) for the oil and water gradient terms and rearranging, results in,
L
x
S
S
o
oo
w
ww
w
w
c
dx
Ak
q
Ak
q
dSS
pwL
w
(9.18)
Equation (9.18) can be solved either graphically or numerically for saturation gradient.
The result will be a calculated saturation profile similar to the one shown in the right-
hand side of Figure 9.4.
9.4 Frontal advance for unsteady 1D displacement
The unsteady-state displacement of oil by water is due to the change in Sw with
time. This can be visualized by looking at the schematics in Figure 9.6. These
schematics represent
Figure 9.6 Progression of water displacing oil for immiscible, 1D
Swi
Sor
Sw
A
Swi
Sor
Sw
C
0 1x/L
Swi
Sor
Sw
B
Swi
Sor
Sw
D
0 1x/L
Swi
Sor
Sw
A
Swi
Sor
Sw
A
Swi
Sor
Sw
C
0 1x/L
Swi
Sor
Sw
B
Swi
Sor
Sw
B
Swi
Sor
Sw
D
0 1x/L
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.7
snapshots in time of the frontal boundary as water is displacing oil. In sequence, A
depicts the initial state of the sample (or reservoir) where saturations are separated into
irreducible water, residual oil and mobile oil components. After a given time of
injection, the front advances to a position as shown in B. Ahead of the front water
saturation is at irreducible, but behind the front water saturation is increased. Continuing
in time, eventually the water will breakthrough the end of the core (reservoir) and both oil
and water will be produced simultaneously, C. Continued injection will increase the
displacing phase saturation in the core (reservoir), D.
Two methods to predict the displacement performance are 1) the analytical
solution by Buckley – Leverett (1941), and 2) applying numerical simulation. Only the
analytical solution will be described in this chapter.
9.4.1 Buckley – Leverett (1941)
The derivation begins from the 1D, multiphase continuity equations.
oooxo St
ux
(9.19)
wwwxw St
ux
(9.20)
In terms of volumetric flow rate,
oooo St
Aqx
(9.21)
wwww St
Aqx
(9.22)
Assume the fluids are incompressible and the porosity is constant. Eqs. (9.21) and (9.22)
simplify to,
t
SA
x
q oo
(9.23)
t
SA
x
q ww
(9.24)
Combining,
0
t
SSA
x
qq owow (9.25)
The result is qT = qo + qw = constant, the total flow rate is constant at each cross-section.
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.8
From the definition of fractional flow,
Two
Tww
qfq
qfq
)1(
(9.26)
Substitute into Darcy’s equation for each phase,
sin)1( gx
pAkqfq o
o
o
oTwo (9.27)
singx
pAkqfq w
w
w
wTww (9.28)
Rearranging Eqs. (9.27) and (9.28), we can substitute into Eq. (9.16) for the pressure
gradient terms. Solving the resulting equation for fractional flow of water, provides the
complete fractional flow equation.
ow
wo
c
To
o
ow
wow
k
k
gx
p
q
Ak
k
kf
1
sin
1
1 (9.29)
In the analytical solution it is difficult to analyze the derivative term (dpc/dx). If we
expand this derivative to,
x
S
S
p
x
p w
w
cc (9.30)
In linear displacement, dpc/dSw 0 at moderate to high water saturations as observed by
the capillary pressure curve such in Figure 9.7. As a result, dpc/dx 0.
Sw
Pc
0
w
c
S
p
Figure 9.7 Capillary pressure curve illustrating flat transition region at moderate to high
water saturations.
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.9
If the derivative term is negligible, and flow is in the horizontal direction such that no
gravity term is present, then the fractional flow equation reduces to,
ow
wow
k
kf
1
1 (9.31)
If we define mobility ratio as,
wo
ow
k
kM
(9.32)
then fw = 1/(1+1/M).
If we return to Eq. (9.24) and substitute for qw, we obtain,
t
S
q
A
x
f w
T
w
(9.33)
To develop a solution, Eq. (9.33) must be reduced to one dependent variable, either Sw or
fw. Observe, Sw = Sw(x,t) or,
dtt
Sdx
x
SdS
x
w
t
ww
(9.34)
Let dSw(x,t)/dt = 0, (Tracing a fixed saturation plane through the core) then
t
w
x
w
S
xS
tS
dt
dx
w
(9.35)
where the left-hand side is the velocity of the saturation front as it moves through the
porous media.
Observe fw = fw(Sw) only, then,
t
w
tw
w
t
w
x
S
S
f
x
f
(9.36)
Substitution of Eqs. (9.35) and (9.36) into Eq. (9.33), results in the frontal advance
equation.
tw
wT
S S
f
A
q
dt
dx
w
(9.37)
Chapter 9 – Simultaneous Flow of Immiscible Fluids
9.10
Equation (9.37) represents the velocity of the saturation front. Basic assumptions in the
derivation are incompressible fluid, fw(Sw) only and immiscible fluids. Furthermore, only
oil is displaced; i.e., the initial water saturation is immobile, and no initial free gas
saturation exists; i.e., not a depleted reservoir.
The location of the front can be determined by integrating the frontal advance
equation,
dtqS
f
Adx T
t
tw
w
x
S
wS
w
00
1
(9.38)
If injection rate is constant and if the dfw/dSw = f(Sw) only, then
w
w
Sw
wT
S S
f
A
tqx
(9.39)
We can evaluate the derivative from the fractional flow equation (Eq. 9.31), either
graphically or analytical. Figure 9.8 illustrates the graphical solution.
Swf Sw Swc
fw
fwf
Swbt
Figure 9.8 Fractional flow curve
The fractional flow of water at the front, fwf, is determined from the tangent line
originating at Swc. The corresponding water saturation at the front is Swf. The average
water saturation behind the front at breakthrough, Swbt, is given by the intersection at fw =
1. The location of the front is determined by Eq. (9.39), with the slope of the tangent to
the fractional curve used for the derivative function.
Chapter 9 – Simultaneous Flow of Immiscible Fluids