Adv Petrophysics Fall07 Chapter8

CHAPTER 8
RELATIVE PERMEABILITY
8.1 DEFINITION OF RELATIVE PERMEABILITY

In a petroleum reservoir, it is possible for two or three fluids to flow
simultaneously. Examples are (a) the flow of gas and water in a gas reservoir,
(b) the flow of oil and water in an oil reservoir, (c) the flow of oil and gas in an
oil reservoir and (d) the flow of oil, water and gas in an oil reservoir. In
multiphase flow situations, the absolute permeability of the porous medium is
no longer sufficient to calculate the flow rate of each fluid type or to calculate
the total flow rate of all the fluids.
In order to make quantitative predictions for multiphase flow, we need

to know the permeability to each fluid in the presence of the other fluids in
the rock. The permeability of one fluid in the presence of the other immiscible
fluids is known as the effective permeability to that fluid. To calculate the
flow rate of each fluid in multiphase flow, we extend Darcy’s Law to
multiphase flow. For example, for simultaneous flow of oil, water and gas in
an inclined linear system, Darcy’s Law is applied to each phase as follows:
ko A ⎛ ∂Po ⎞
qo = − ⎜ ∂x + ρo g sin α ⎟ (8.1)
μo ⎝ ⎠
k w A ⎛ ∂Pw ⎞
qw = − ⎜ + ρ w g sin α ⎟ (8.2)
μ w ⎝ ∂x ⎠
8-1
k g A ⎛ ∂Pg ⎞
qg = − ⎜ + ρ g g sin α ⎟ (8.3)
μ g ⎝ ∂x ⎠
where α is the angle of inclination with the horizontal. Eqs.(8.1) to (8.3) show
that using the concept of an effective permeability, Darcy’s Law is applied to
each phase as if the other phases did not exist. Capillary equilibrium between
the phases gives
Po − Pw = Pc / ow ( S w ) (8.4)
Pg − Po = Pc / go ( So ) (8.5)
Pc / ow + Pc / go = Pc / gw (8.6)
It is often more convenient to work with a dimensionless effective

permeability known as the relative permeability obtained by dividing the
effective permeability by a base permeability such as the absolute
permeability of the porous medium. Thus, for the three phase example, using
the absolute permeability of the porous medium as the base permeability, the
relative permeabilities to oil, water and gas are given by
ko
kro = (8.7)
k
kw
krw = (8.8)
k
kg
krg = (8.9)
k
In terms of relative permeabilities, Eqs.(8.1) through (8.3) become
kkro A ⎛ ∂Po ⎞
qo = − ⎜ + ρ o g sin α ⎟ (8.10)
μo ⎝ ∂x ⎠
8-2
kkrw A ⎛ ∂Pw ⎞
qw = − ⎜ + ρ w g sin α ⎟ (8.11)
μ w ⎝ ∂x ⎠
kkrg A ⎛ ∂Pg ⎞
qg = − ⎜ + ρ g g sin α ⎟ (8.12)
μ g ⎝ ∂x ⎠
Sometimes, the effective permeability to the non-wetting phase at the

irreducible wetting phase saturation is used as the base permeability for
defining the relative permeability. In this case, the end point relative
permeability to the non-wetting phase will be 1.0. As the base pressure
appears in Darcy's law as shown in Eqs.(8.10) to (8.12), it is necessary to
ascertain the base permeability used to define a relative permeability curve
before such a curve is used in performance calculations. Failure to do so will
lead to wrong results.
Figure 8.1 shows typical imbibition relative permeability curves for a

two-phase system. The following observations can be made about the key
features of the relative permeability curves.
1. The relative permeability curves are nonlinear functions of fluid

saturation.
2. The sum of the relative permeabilities at each saturation is always less

than 1.0.
3. There is an irreducible wetting phase saturation (Swirr) at which the
relative permeability to the wetting phase is zero and the relative

permeability to the non-wetting phase attains a maximum end point
value (knwr)
4. There is a residual non-wetting phase saturation (Snwr) at which the
relative permeability to the non-wetting phase is zero and the relative
8-3
permeability to the wetting phase attains a maximum end point value
(kwr).
Figure 8.1. Typical imbibition relative permeability curves.
5. The relative permeability curves are not defined in the saturation ranges
given by (1 − S nwr ) < S w < 1 and 0 < S w < S wirr .
6. Two phase flow occurs over the saturation range S wirr < S w < (1 − S nwr ) .
Imbibition relative permeability curves typically are used to perform the

following reservoir performance calculations:
• Waterflood calculations in a water wet reservoir in which water

displaces oil and/or gas.
8-4
• Natural water influx calculations in a water wet reservoir in which water
displaces oil and/or gas.
• Oil displaces gas, which occurs when oil is forced into a gas cap.
Figure 8.2 shows typical drainage relative permeability curves with

features that are very similar to the imbibition curves of Figure 8.1. The
obvious differences between the drainage and imbibition curves are that the
drainage curve for the wetting phase starts at a wetting phase saturation of
1.0 and that of the non-wetting phase is zero at the wetting phase saturation
of 1.0. This is because at the start of the drainage relative permeability
measurements, the porous medium was fully saturated with the wetting
phase. The permeability of the wetting phase is then equal to the absolute
permeability of the porous medium. Of course, at the start of the experiment,
there was no non-wetting phase in the medium. Therefore, the relative
permeability to the non-wetting phase must be zero. This is the only occasion
in which the sum of relative permeabilities is equal to 1.0 because there was
only one phase present. As will be shown later, because of capillary pressure
hysteresis, the drainage and imbibition relative permeability curves will be
different.
Drainage relative permeability curves typically are used to perform the

following reservoir performance calculations:
• Solution gas drive calculations in which gas displaces oil.
• Gravity drainage calculations in which gas replaces drained oil.
• Gas drive calculations in which gas displaces oil and/or water.
• Oil or gas displacing water in tertiary recovery processes.
8-5
Figure 8.2. Typical drainage relative permeability curves.
8.2 LABORATORY MEASUREMENT OF TWO-PHASE RELATIVE

PERMEABILITIES BY THE STEADY STATE METHOD
The most straight-forward laboratory measurement technique for
relative permeabilities is the steady state method. For imbibition relative
permeability measurement, the test starts with the core initially saturated
with an irreducible wetting phase saturation (Swirr) and a non-wetting phase
saturation of (1-Swirr). Then a mixture of the two phases is injected into the
⎛q ⎞
inlet face of the core at a fixed ratio of ⎜ nw ⎟ until steady state is achieved.
⎝ qw ⎠
Steady state is achieved when the pressure drop across the core no longer
changes with time and the ratio of the produced fluids is the same and the
ratio of the injected fluids. The steady state pressure drop across the core and
8-6
the injection rate of each phase are measured. The relative permeabilities are
calculated with the integrated forms of Darcy's law for two phase flow as
shown later. The saturations are usually calculated by material balance.
At steady state, the continuity equations for the wetting and non-
wetting phases for horizontal flow are
∂vw
=0 (8.13)
∂x
∂vnw
=0 (8.14)
∂x
Darcy's law applied to each phase gives
k w ∂Pw
vw = − = a constant (8.15)
μ w ∂x
knw ∂Pnw
vnw = − = a constant (8.16)
μnw ∂x
From capillary equilibrium,
Pnw − Pw = Pc ( S w ) (8.17)
where Pc(Sw) in this case is the imbibition capillary pressure. If the

saturations, Sw and Snw, are uniform throughout the porous medium, then
kw, knw and Pc are independent of x. Then, Eqs.(8.15) and (8.16) can be
integrated to give
μ w vw L μ w qw L
kw = = (8.18)
ΔPw AΔPw
μnwvnw L μnw qnw L

knw = = (8.19)
ΔPnw AΔPnw
Since Pc ( S w ) is uniform, ΔPw and ΔPnw are equal and the pressure drop across
the core can be measured in either phase and used to calculate the effective
8-7
permeabilities with Eqs.(8.18) and (8.19). The steady state saturation
distribution in the core can be calculated with Eq.(7.83), which is reproduced
here for convenience:
⎛ fw ⎞
⎜ − 1⎟
dS w
= ⎝ Fw ⎠
(7.83)
dxD N k dJ
cap rnw
dS w
with a specified inlet boundary condition. Of course, to do so, krw , krnw and
Pc ( S w ) must be known.
If the saturation distribution in the core is not uniform because of

capillary end effect, then Eqs.(8.18) and (8.19) are not valid and cannot be
used to calculate the effective permeabilities. As there are no alternative
equations to use, the relative permeability experiment will be a failure.
Therefore, in the steady state experiment, the total injection rate ( q = qw + qnw )
should be sufficiently high to minimize capillary end effect as outlined in

Chapter 7.
Figure 8.3 shows an apparatus that can be used for the steady state
experiment. A typical sequence of steps for obtaining the imbibition relative
permeability curves might be as follows:
1. Install the clean, dry core sample in the Hassler apparatus as shown in
Figure 8.3. Evacuate the core and saturate with the wetting phase.
Determine the absolute permeability of the core by wetting phase flow.
2. Displace the wetting phase with the non-wetting phase until no more
wetting phase flows from the core. Calculate the irreducible wetting
phase saturation and the initial non-wetting phase saturation.
Measure the steady state pressure drop and the non-wetting phase
8-8
injection rate and calculate the relative permeability to the non-wetting
phase at the irreducible wetting phase saturation by use of Eq.(8.19) as
μnw qnw L
krnw = (8.20)
kAΔPnw
3. Inject a mixture of the wetting and non-wetting phases at rates qnw and
qw such that the ratio, qw/qnw, is very much less than 1 until steady
state is achieved. Steady state is achieved when the injected and

produced qw/qnw ratios are equal and the pressure drop no longer
changes with time.
4. Measure the pressure drop and calculate the wetting phase saturation
by material balance. Calculate the relative permeabilities to the non-
wetting and wetting phases at the latest wetting phase saturation using
Eq.(8.20) and (8.18) as
μ w qw L
krw = (8.21)
kAΔPw
5. Increase the ratio qw/qnw and repeat steps 3 and 4 to calculate the
relative permeabilities at higher and higher wetting phase saturations.
6. Finally, inject only the wetting phase until no more non-wetting phase
flows from the core. Calculate the residual non-wetting phase
saturation. Measure the steady state pressure drop and the wetting
phase injection rate and calculate the relative permeability to the
wetting phase at residual non-wetting phase saturation. This completes
the relative permeability measurements.
In the steady state relative permeability experiment, it is necessary to

minimize capillary end effect. This can be accomplished by injecting at a
sufficiently high total rate or by other means as discussed by Richardson et
al. (1952). Figures 8.4 and 8.5 show the pressure profiles in the gas and oil
8-9
Figure 8.3. Hassler’s apparatus for relative permeability measurement (Osoba
et al., 1951).
phases for a gas-oil steady state relative permeability experiment conducted at

two rates by Richardson et al. (1952). At the lower injection rate (Figure 8.4),
capillary end effect is apparent whereas at the higher rate (Figure 8.5), there
is little or no capillary end effect. Note also, that when the capillary end effect
has been eliminated, the pressure drop in each phase is the same. Therefore,
the pressure drop measured in either phase is sufficient for calculating both
relative permeabilities.
8-10
The various steady state methods such as the Penn State method,
single core dynamic method, dispersed feed method, Hafford method and
Hassler method differ primarily in the techniques used to minimize or
eliminate capillary end effect (Richardson et al., 1952). When capillary end
effect has been eliminated, all the steady state methods give the same results
as shown in Figures 8.6 and 8.7.
Figure 8.4. Steady state oil and gas pressure profiles at a relatively low
injection rate (Richardson et al., 1952).
8-11
Figure 8.5. Steady state oil and gas pressure profiles at a relatively high
injection rate (Richardson et al., 1952).
8-12
Figure 8.6. Relative permeability curves from six steady state methods, short
core section (Richardson et al., 1952).
8-13
Figure 8.7. Relative permeability curves from six steady state methods, long
core section (Richardson et al., 1952).
The major problem with the steady state method for relative
permeability measurements is that it takes too long to complete. It is not
unusual for a steady state experiment to take several weeks to complete. An
alternative and much faster technique is the unsteady state method or the
dynamic displacement method based on immiscible displacement theory.
Because the calculation of relative permeabilities from unsteady state
experiment is based on the solution of two-phase immiscible displacement
equation, we must first solve the two-phase immiscible displacement problem
before we can discuss the unsteady state relative permeability measurements.
8-14
8.3 THEORY OF ONE DIMENSIONAL IMMISCIBLE
DISPLACEMENT IN A POROUS MEDIUM
8.3.1 Mathematical Model of Two-Phase Immiscible Displacement
Consider the displacement of a non-wetting phase by a wetting phase in
a linear inclined core as shown in Figure 8.8. Darcy’s Law applied to each
phase gives
kkrnw A ⎛ ∂Pnw ⎞
qnw = − ⎜ + ρ nw g sin α ⎟ (8.22)
μnw ⎝ ∂x ⎠
kkrw A ⎛ ∂Pw ⎞
qw = − ⎜ + ρ w g sin α ⎟ (8.23)
μ w ⎝ ∂x ⎠
Capillary equilibrium gives
Pnw − Pw = Pc ( S w ) (8.17)
Assuming incompressible fluids, mass conservation requires that
q = qw + qnw (8.24)
The true fractional flows of the wetting and non-wetting phases are defined as
follows:
qw qw
fw = = (8.25)
q qw + qnw
qnw qnw
f nw = = = 1 − fw (8.26)
q qw + qnw
The continuity equation for the wetting phase is
∂S w ∂qw
φA + =0 (8.27)
∂t ∂x
8-15
Figure 8.8. Displacement of a non-wetting phase by a wetting phase in an
inclined core.
The saturation constraint gives
S w + Snw = 1 (8.28)
Subtracting Eq.(8.22) from (8.23) and rearranging gives
qw μ w qnw μnw ∂Pnw ∂Pw

− = − − ( ρ w − ρ nw ) g sin α (8.29)
kkrw A kkrnw A ∂x ∂x
8-16
Substituting Eqs.(8.17) and (8.24) into (8.29) gives upon rearrangement
kkrnw A ⎡ ∂Pc ⎤
1+ ⎢ − ( ρ w − ρ nw ) g sin α ⎥
qw q μnw ⎣ ∂x ⎦
= (8.30)
q k μ
1 + rnw w
krw μnw
Let an approximate fractional flow of the wetting phase be defined as
1
Fw = (8.31)
k μ
1 + rnw w
krw μnw
Substituting Eqs.(8.30) and (8.31) into (8.25) gives the true fractional flow of
the wetting phase as
⎧ kk A ⎡ ∂P ⎤⎫
f w = Fw ⎨1 + rnw ⎢ c − ( ρ w − ρ nw ) g sin α ⎥ ⎬ (8.32)
⎩ q μ nw ⎣ ∂x ⎦⎭
Let the dimensionless distance from the inlet end be defined as
x
xD = (8.33)
L
Let the spontaneous imbibition capillary pressure curve be given in terms of

its Leverett J-function as
σ cos θ
Pc ( S w ) = J ( Sw , Γ ) (8.34)
k /φ
Substituting Eqs.(8.83) and (8.34) into (8.32) gives the true fractional flow of
the wetting phase as
⎡ ⎛ Aσ cos θ kφ ⎞ ∂J ⎛ kA ( ρ w − ρ nw ) g sin α ⎞ ⎤
f w = Fw ⎢1 + ⎜ ⎟ krnw − krnw ⎜ ⎟⎥ (8.35)
⎜ q μnw L ⎟ ∂xD q μnw
⎢⎣ ⎝ ⎠ ⎝ ⎠ ⎥⎦
Eq.(8.35) can be written as
8-17
⎡ ∂J ⎤
f w = Fw ⎢1 + N cap krnw − krnw N g ⎥ (8.36)
⎣ ∂xD ⎦
where N cap is given by
Aσ cos θ kφ
N cap = (7.61)
q μnw L
and N g is given by
kA ( ρ w − ρ nw ) g sin α
Ng = (8.37)
qμnw
N cap is the same dimensionless number we encountered in the analysis of
capillary end effect. It represents the ratio of capillary to viscous forces in the
displacement. N g is a new dimensionless number, which represents the ratio
of gravity to viscous forces in the displacement. The mobility ratio of the

displacement is given by
krw μnw
M ( Sw ) = (8.38)
krnw μ w
The mobility ratio as defined in Eq(8.38) is a function of saturation and will be

different at each point in the porous medium depending on the saturation. A
characteristic mobility ratio for the displacement can be defined in terms of
the end-point relative permeabilities as
kwr μnw
ME = (8.39)
knwr μ w
where kwr and knwr are the end-point relative permeabilities for the wetting and
non-wetting phases. The mobility ratio given in Eq.(8.39) is a characteristic

dimensionless number for the displacement that is independent of saturation.
Eq.(8.36) can also be written as
8-18
∂J
1 + N cap krnw − krnw N g
∂xD
fw = (8.40)
⎛ 1 ⎞
⎜1 + ⎟
⎝ M⎠
where the approximate fractional flow of the wetting phase is given by
1
Fw = (8.41)
⎛ 1 ⎞
⎜1 + ⎟
⎝ M⎠
In order to maximize the displacement efficiency of the non-wetting

phase, we need to minimize the fractional flow of the wetting phase at each
point in the porous medium. Much can be deduced about the immiscible
displacement from the fractional flow equation, Eq.(8.40). Examination of this
equation leads to the following qualitative deductions about immiscible
displacements in porous media:
1. The fractional flow of the wetting phase is a strong function of

saturation.
2. The displacement behavior can be rate-sensitive if the effect of

capillarity or gravity is significant.
3. Capillarity is detrimental to the displacement efficiency as it increases

the fractional flow of the wetting phase at a given saturation.
4. Gravity is beneficial to the displacement efficiency for up-dip

displacement of the lighter non-wetting phase by the heavier wetting
phase as it reduces the fractional flow of the wetting phase at a given
saturation. Conversely, gravity will be detrimental to the displacement
efficiency for down-dip displacement of the lighter non-wetting phase by
the heavier wetting phase.
5. The displacement efficiency can be increased by reducing the mobility

ratio. This can be accomplished in practice by increasing the viscosity
8-19
of the wetting phase (the injected fluid) by use of a polymer. This is the
basis for polymer flooding as an improved oil recovery technique.
6. The effects of gravity and capillarity on the displacement can be reduced

by increasing the injection rate.
There are additional facts about the immiscible displacement that are
not apparent from the fractional flow equation. The fractional flow equation
indicates that the displacement efficiency can be improved by injecting the
wetting phase at a high enough rate to minimize capillary smearing of the
displacement front. This is generally true for a favorable mobility ratio
displacement. If the mobility ratio is unfavorable, an increase in rate can
result in viscous instability which reduces the displacement efficiency. The
fractional flow equation suggests that the effect of gravity will be eliminated if
the porous medium is horizontal. This is misleading because, in practice, if
there is a density contrast between the fluids, the injection rate is sufficiently
low and the core has a vertical dimension (which it does), gravity segregation
will occur even in a horizontal medium. In this case, the one dimensional
displacement model is inadequate to describe the displacement. A
multidimensional model is needed to correctly describe the gravity-dominated
displacement. The only fail proof way to eliminate the effect of gravity is to
eliminate the density contrast between the fluids or perform the displacement
in outer space. One can create a gravity number for displacement in a
horizontal core by replacing sin α in Eq.(8.37) by the aspect ratio (d/L).
The partial differential equation for the wetting phase saturation can be
derived as follows. Let
Ψ ( S w ) = Fw − krnw N g (8.42)
dJ
Ω ( S w ) = krnw (8.43)
dS w
Eq.(8.36) then becomes
8-20
∂S w
f w ( S w ) = Ψ ( S w ) + N cap Ω ( S w ) (8.44)
∂xD
Eq.(8.27) can be written in dimensionless form as
∂S w ∂f w
+ =0 (8.45)
∂t D ∂xD
where t D is given by
qt
tD = (7.63)
Aφ L
Substituting Eq.(8.44) into (8.45) gives the partial differential equation for the
wetting phase saturation as
∂S w d Ψ ∂S w ∂ ⎡ ∂S w ⎤
+ + N cap ⎢Ω ( S w ) ⎥=0 (8.46)
∂t D dS w ∂xD ∂xD ⎣ ∂xD ⎦
We have reduced the immiscible displacement problem to the solution of a

second order, nonlinear, parabolic partial differential equation for the wetting
phase saturation. When supplemented with appropriate initial and boundary
conditions, Eq.(8.46) can be solved, usually numerically, to obtain the wetting
phase saturation in time and space.
8.3.2 Buckley-Leverett Approximate Solution of the Immiscible

Displacement Equation
Eq.(8.46) cannot be solved analytically for the saturation profiles. Here,
we examine the approximate solution obtained by Buckley and Leverett
(1941). The continuity equation, Eq.(8.45), can be written as
∂S w df w ∂S w
+ =0 (8.47)
∂t D dS w ∂xD
where the true fractional flow of the wetting phase for horizontal
displacement is given by
8-21
dJ ∂S w
1 + N cap krnw
dS w ∂xD
fw = (8.48)
⎛ 1 ⎞
⎜1 + ⎟
⎝ M⎠
∂S w
It should be observed that the true fractional flow function contains ,
∂xD
which is unknown. Buckley and Leverett (1941) obtained an approximate
solution to Eq.(8.47) by making a key simplifying assumption. They dropped
the capillary pressure term from Eq.(8.48) and as a result, they approximated
the fractional flow of the wetting phase as
f w Fw (8.49)
Substituting Eq.(8.49) into (8.47) gives the partial differential equation for the
wetting phase saturation as
∂S w dFw ∂S w
+ =0 (8.50)
∂t D dS w ∂xD
Eq.(8.50) is known as the Buckley-Leverett equation in the petroleum

industry. The Buckley-Leverett approximation changes the original second
order parabolic partial differential equation for the wetting phase saturation to
a first order, hyperbolic partial differential equation. This is a radical change
in the structure of the mathematical problem. However, the change allows an
approximate analytical solution to be obtained for the wetting phase
saturation profiles that is adequate for making gross performance calculations
for the immiscible displacement.
Eq.(8.50) is a nonlinear, first order, hyperbolic partial differential

equation that can be solved by the method of characteristics. From calculus,
the total time derivative of S ( xD , t D ) is given by
∂S w ⎛ dxD ⎞ ∂S w dS w
+⎜ ⎟ = (8.51)
∂t D ⎝ dt D ⎠ ∂xD dt D
8-22
Subtracting Eq.(8.50) from (8.51) gives
⎛ dxD dFw ⎞ ∂S w dS w
⎜ − ⎟ = (8.52)
⎝ dt D dS w ⎠ ∂x D dt D
Eq.(8.52) can be decomposed into the following two simultaneous equations:
dxD dFw
− =0 (8.53)
dt D dS w
dS w
=0 (8.54)
dt D
Eq.(8.53) gives the characteristic path for the hyperbolic partial differential
equation given by Eq(8.54). Eq.(8.54) shows that along the characteristic
path given by Eq.(8.53), the saturation is a constant.
Eq.(8.53) can be integrated to determine the distance traveled by a

constant saturation at a given time as
dFw
xDSw ( tD ) − xDSw ( 0) = ( t D − t D 0 ) (8.55)
dS w
If there was no prior injection, t D 0 will be zero and all the saturations from Swi
to (1 - Snwr) will be located at the inlet end of the system, making xDSw ( 0) equal
to zero. In this case, Eq.(8.55) becomes
dFw
xDSw ( tD ) = t D (8.56)
dS w
Eq.(8.56) can be written as
dFw
xD = t D (8.57)
dS w
8-23
where xD is the dimensionless distance traveled by a given saturation at time
t D . Eq.(8.57) can be written in dimensional form as
Qi ( t ) dFw
x= (8.58)
φ A dS w
Eq.(8.58) is usually referred to in the petroleum industry as the Buckley-

Leverett frontal advance equation. It should be emphasized that Eq.(8.57) or
(8.58) applies to a particular wetting phase saturation. To determine the
dimensionless distance traveled by a particular saturation S w1 at time t D , we
use Eq.(8.57) to compute the distance as
⎛ dF ⎞
xD = t D ⎜ w ⎟ (8.59)
⎝ dS w ⎠ Sw1
where the derivative of the approximate fractional flow curve is evaluated at

S w1 . Eq.(8.57) can be used to derive a similarity transformation for an
immiscible displacement. The similarity transformation is given by
xD dFw df w
z= = = (8.60)
t D dS w dS w
If the saturation profiles for an immiscible displacement are plotted as S w
xD
versus , all the saturation profiles will collapse into one curve. If the
tD
saturation profiles in an immiscible displacement are imaged say by CT or by
NMR, then Eq.(8.60) can be used to calculate the true fractional flow curve,
including the effect of capillarity, as
xD
f w ( Sw ) = ∫
Sw
dS w (8.61)
S wirr tD
Given the relative permeability curves and the viscosity ratio, the
approximate fractional flow function and its derivative can be computed and
8-24
plotted as shown in Figure 8.9. In this figure, the S-shaped curve ADBC is
the approximate fractional flow curve (Fw) obtained from the relative
permeability curves and the viscosity ratio. The curve AFE is the derivative of
⎛ dF ⎞
this function ⎜ w ⎟ . Using Eq.(8.59) and this derivative function, the distance
⎝ dS w ⎠
traveled by each wetting phase saturation between Swirr and (1-Snwr) at a
given time t D can be computed. Figure 8.10 shows the saturation profile that
will be obtained before wetting phase breakthrough by use of Eq.(8.59) and

the approximate derivative function. We see that the Buckley-Leverett
approximation gives rise to multiple-valued saturations at various xD which is
physically impossible. This multiple-valued solution is caused by neglecting

the capillary term in the fractional flow equation. It is no accident that the
multi-valued solution occurs in the saturation range S wirr < S w < S wf where the
capillary pressure gradient is high and should not have been neglected.
To eliminate the multiple-valued solution, we appeal to physical reality

as follows. At time t, Qi(t) of wetting phase has been injected and the flood
front has traveled a distance x f into the medium. A volumetric balance of the
injected wetting phase can be used to calculate x f as follows:
Qi ( t ) = ∫ φ A ( S w − S wirr )dx
xf
(8.62)
0
Integration of Eq.(8.62) by parts and substitution of Eq.(8.58) gives
Qi ( t ) = φ Ax f ( S wf − S wirr ) − Qi ∫
S wf dFw
dS w (8.63)
1− Snwr dS w
Upon performing the integration in Eq.(8.63) and rearranging, one obtains
8-25
Figure 8.9. Approximate fractional flow function and its first derivative. Note
the tangent construction.
Figure 8.10. Calculated water saturation distribution based on the Buckley-

Leverett approximation showing the discontinuity in saturation as required by
a material balance.
8-26
Fw ( S wf )
φ Ax f = Qi ( t ) (8.64)
S wf − S wirr
From the Buckley-Leverett frontal advance equation, Eq.(8.58), one can also
obtain
⎛ dFw ⎞
φ Ax f = Qi ( t ) ⎜ ⎟ (8.65)
⎝ dS w ⎠ S wf
A comparison of Eqs.(8.64) and (8.65) gives
⎛ dFw ⎞ Fw ( S wf )
⎜ ⎟ = (8.66)
⎝ dS w ⎠ Swf S wf − S wirr
The saturation distribution in Figure 8.10 will be single valued if all the
saturations between Swirr and the frontal saturation, Swf, are eliminated.
Eq.(8.66) shows that the frontal saturation (Swf) is the saturation at which the
straight line passing through the point Sw = Swirr and Fw = 0 is tangent to the
approximate fractional flow curve, Fw. This line is shown in Figure 8.9 as AB.
This tangent construction was first suggested by Welge (1952). The effect of
the tangent construction is to correct the approximate fractional flow curve Fw
for the capillary term that was neglected to obtain the true fractional flow
curve fw. Such a correction is needed at the front (low wetting phase
saturation) where the capillary pressure gradient is high and should not have
been neglected. With the tangent construction correction in place, the true
fractional flow curve, fw, is now given by the curve ABC (Fig. 8.9) thereby
eliminating the S-shaped lower portion of Fw, which led to the tripple-valued
saturation solution of Figure 8.10. With this correction, the derivative of the
⎛ df ⎞
true fractional flow curve ⎜ w ⎟ used in the solution is given by the curve
⎝ dS w ⎠
EFG (Fig. 8.9). After the tangent construction, the true fractional flow curve
and its derivative are given by
8-27
Figure 8.11. Similarity transformation for an immiscible displacement.
Figure 8.12. Integration of the transformed saturation data to calculate the

true fractional flow curve including capillarity.
8-28
⎧⎛ S − S ⎞ ⎛ dFw ⎞
⎪⎜
⎪
w wirr
⎟ F ( S ) = ( S − S ) ⎜ ⎟ for S wirr ≤ S w ≤ S wf
f w ( S w ) = ⎨⎜⎝ S wf − S wirr ⎟⎠
w wf w wf
⎝ w ⎠ Swf
dS (8.67)
⎪
⎪⎩ Fw ( S w ) for S wf ≤ S w ≤ 1.0
and
⎧⎛ dFw ⎞
⎪⎜ ⎟ = a constant for S wirr ≤ S w ≤ S wf
df w ⎪⎝ w ⎠ Swf
dS
=⎨ (8.68)
dS w ⎪⎛ dF ⎞
⎪⎜ dS ⎟ for S wf ≤ S w ≤ 1.0
w
⎩⎝ w ⎠ Sw
The similarity transformation for the immiscible displacement is given

by the curve EFGH (Fig. 8.9) and is shown in Figure 8.11. Figure 8.12 shows
how the transformed saturation data can be integrated to obtain the true
fractional flow curve that includes the effect of capillarity.
We now show that the intersection of the tangent line with the line Fw =
1 (point J in Fig. 8.9) gives the constant average wetting phase saturation
behind the front before and at wetting phase breakthrough. The slope of the
tangent line can be written as
⎛ dFw ⎞ 1 − Fw ( S wf )
⎜ ⎟ = (8.69)
⎝ dS w ⎠ Swf S wav − S wf
which can be rearranged as
1 − Fw ( S wf )
S wav = S wf + (8.70)
⎛ dFw ⎞
⎜ ⎟
⎝ dS w ⎠ Swf
Before breakthrough, the average wetting phase saturation behind the front is
given by
8-29
1− Snwr
S wav =
∫
0
φ AxdS w
(8.71)
φ Ax f
Substituting Eq.(8.58) into (8.71) and integrating gives the average wetting
phase saturation behind the front as
1 − Fw ( S wf )
S wav = S wf + (8.72)
⎛ dFw ⎞
⎜ ⎟
⎝ dS w ⎠ Swf
Eq.(8.72) is identical to Eq.(8.70), which confirms that the intersection of the

tangent line and the line Fw = 1 gives the wetting phase saturation (Swav)
corresponding to point J in Figure 8.9. Thus, the average wetting phase
saturation at breakthrough can be obtained graphically from the tangent
construction.
The average wetting phase saturation after breakthrough can be

obtained by a tangent construction at the outlet wetting phase saturation
between Swf and 1 - Snwr. The intersection of the tangent line and the line
Fw = 1 gives the average wetting phase saturation in the porous medium
corresponding to the outlet wetting phase saturation. By this tangent

construction, the Buckley-Leverett approximation can be used to predict the
performance of the one-dimensional immiscible displacement after wetting
phase breakthrough. Before breakthrough, the amount of non-wetting phase
recovered is equal to the amount of fluid injected. Thus, the entire
displacement performance can be predicted for a given set of wetting and non-
wetting relative permeability curves and wetting and non-wetting viscosity
ratio.
8-30
8.3.3 Waterflood Performance Calculations from Buckley–Leverett
Theory
We now apply Buckley-Leverett theory to calculate a waterflood
performance from beginning to end. It is assumed that the true fractional flow
curve and its derivative have been computed using the relative permeability
curves, the viscosity ratio and the Welge tangent construction. Therefore, the
equations in this section are written in terms of the true fractional flow curve.
The methodology presented also applies to the calculation of the performance
of a gas flood using gas-oil drainage relative permeability curves.
Oil Recovery at any Time

The oil recovery at any time after the initiation of water injection is
given by
S wav − S wirr
R= (8.73)
1 − S wirr
where R is the oil recovery as a fraction of the initial oil in place, Swav is the
average water saturation in the porous medium at the time of interest and
Swirr is the initial water saturation in the porous medium before water
injection which is assumed to be the irreducible water saturation. Thus, in

principle, the oil recovery can be calculated at any time by first calculating the
average water saturation in the porous medium at that time and applying
Eq.(8.73). However, depending on the stage of water injection, Eq.(8.73) may
not offer the most direct way to calculate the oil recovery. Let us examine the
waterflood performance at various stages of the flood.
Oil Recovery Before Water Breakthrough

Consider a constant rate water injection project. Assuming
incompressible fluids, the amount of oil recovered before water breakthrough
must equal the amount of water injected. Thus, at reservoir conditions,
8-31
qBwt = Qi ( t ) = Qo ( t ) (8.74)
where q is the constant water injection rate, in surface units, Bw is the water
formation volume factor, t is the time of interest before water breakthrough,
Qi is the cumulative water injected at time t in reservoir units and Qo is
the cumulative oil produced at time t in reservoir units. The cumulative oil
produced at surface conditions is
qBwt Qi ( t ) Qo ( t )
Cumulative Oil Produced = = = (8.75)
Bo Bo Bo
where Bo is the current oil formation volume factor. The oil recovery as a
fraction of the initial oil in place is given by
qBwt Qi ( t ) Wi
R= = = (8.76)
φ AL (1 − S wirr ) φ AL (1 − S wirr ) (1 − S wirr )
where Wi is the pore volume of water injected.
Oil Recovery at Water Breakthrough

From the Buckley–Leverett frontal advance equation, Eq.(8.58), the
distance traveled by a given saturation is given by
qBwt ⎛ df w ⎞ Qi ( t ) ⎛ df w ⎞
x= ⎜ ⎟ = ⎜ ⎟ (8.77)
φ A ⎝ dS w ⎠ S φ A ⎝ dS w ⎠ S
w w
Let us apply Eq.(8.77) to the frontal water saturation Swf to get
xf = ⎜ ⎟ = ⎜ ⎟ (8.78)
wf wf
At the moment of water breakthrough, the frontal saturation arrives at the

outlet end of the porous medium and x f equals L. At the moment of water
breakthrough, Eq.(8.78) then becomes
8-32
L= ⎜ ⎟ = ⎜ ⎟ (8.79)
wf wf
Eq.(8.75) can be rearranged as
qBwt Qi ( t ) 1
Wi = = = (8.80)
φ AL φ AL ⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ Swf
where Wi is the pore volume of water injected. The cumulative oil recovery at
water breakthrough is equal to the cumulative water injected in reservoir

volumes. The fractional oil recovery at water breakthrough is obtained from
Eq.(8.80) as
Wi qBwt Qi ( t ) 1
R= = = = (8.81)
1 − S wi φ AL (1 − S wi ) φ AL (1 − S wi ) ⎛ df ⎞
(1 − S wi ) ⎜ w ⎟
⎝ dS w ⎠ S wf
The breakthrough time, tbt, can be obtained from Eq.(8.80) as
φ AL
tbt = (8.82)
⎛ df ⎞
qBw ⎜ w ⎟
⎝ dS w ⎠ Swf
or in dimensionless form as
1
t Dbt = (8.83)
⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ Swf
The average water saturation in the porous medium behind the displacement
front before and at water breakthrough is given by
1− Sor
S wav =
∫ 0
φ AxdS w
(8.71)
φ Ax f
8-33
Figure 8.13 shows a typical water saturation distribution at time t before
breakthrough. From Figure 8.13, we see that the integral (area under the
curve) in Eq.(8.71) can be split into two parts as follows:
1− Sor
φ AS wf x f + ∫ φ AxdS w
S wf
S wav = (8.84)
φ Ax f
Substituting Eq.(8.58) into (8.84) gives the average water saturation as
1− Sor
Qi ( t ) ∫ dFw
S wf
S wav = S wf + (8.85)
φ Ax f
Performing the integration in Eq.(8.85) gives
Qi ( t ) ⎡⎣ f w (1 − Sor ) − f w ( S wf ) ⎤⎦
S wav = S wf + (8.86)
φ Ax f
But Fw at Sw = (1 – Sor) is equal to 1.0. Thus, Eq.(8.81) can rewritten as
Qi ( t ) ⎡⎣1 − f w ( S wf ) ⎤⎦
S wav = S wf + (8.87)
φ Ax f
Substituting Eq.(8.78) into (8.87) gives the average water saturation behind
the front as
⎡1 − f w ( S wf ) ⎤
S wav = S wf + ⎣ ⎦ (8.72)
⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ S wf
It should be observed in Figure 8.9 that the average water saturation behind
the front up until water breakthrough as given in Eq.(8.72) is the same as the
water saturation at which the tangent to the fractional flow curve intersects
the Fw = 1 axis. Thus, the average water saturation in the porous medium at
water breakthrough can easily be determined graphically. The average water
8-34
saturation can then be substituted into Eq.(8.73) to calculate the oil recovery
at water breakthrough. We can easily show that the result obtained by this
approach will be the same as that obtained by Eq.(8.81). Substituting
Eq.(8.72) into Eq.(8.73) gives
Figure 8.13. Typical water saturation profile at time t before water

breakthrough.
8-35
⎡1 − f w ( S wf ) ⎤
S wf + ⎣ ⎦ −S
⎛ df w ⎞
wirr
⎜ ⎟
⎝ dS w ⎠ S
R= wf
(8.88)
1 − S wirr
From the equation of the tangent line in Figure 8.9, we find that
⎛ df w ⎞ f w ( S wf )
⎜ ⎟ = (8.66)
⎝ dS w ⎠ Swf S wf − S wirr
Substituting Eq.(8.66) into (8.88) gives the oil recovery at water breakthrough
as
1
R= (8.89)
⎛ df ⎞
(1 − S wi ) ⎜ w ⎟
⎝ dS w ⎠ S wf
which is identical to Eq.(8.81).
Oil Recovery After Water Breakthrough

After water breakthrough, Eq.(8.77) applied to the outlet end of the
porous medium gives
L= ⎜ ⎟ = ⎜ ⎟ (8.90)
w2 w2
where Sw2 is the water saturation at the outlet end of the porous medium
which now lies between Swf and (1 – Sor). Rearrangement of Eq.(8.90) gives
the pore volumes of water injected as
qBwt Qi ( t ) 1
Wi = = = (8.91)
φ AL φ AL ⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ Sw 2
8-36
where Wi is the pore volumes of water injected since the initiation of water
injection. Eq.(8.91) is analogous to Eq.(8.80) before breakthrough.
A material balance for the water after water breakthrough gives
Qi ( t ) − ∫ φ A ( S w − S wirr )dx = Qw ( t )
L
(8.92)
0
The integral in Eq.(8.92) can be performed using integration by parts.

Performing integration by parts, Eq.(8.92) can be written as
{
Qi ( t ) − ⎡⎣φ A ( S w − S wirr ) x ⎤⎦ 0 − ∫
L
Sw 2
1− Sor
φ AxdS w = Qw ( t ) } (8.93)
Substituting the limits for the first integral gives
{
Qi ( t ) − φ AL ( S w 2 − S wirr ) − ∫
1− Sor
Sw 2 }
φ AxdS w = Qw ( t ) (8.94)
Substituting Eq.(8.77) into (8.94) gives
{
Qi ( t ) − φ AL ( S w 2 − S wirr ) − Qi ( t ) ∫
1− Sor
Sw 2 }
df w = Qw ( t ) (8.95)
{ }
Qi ( t ) − φ AL ( S w 2 − S wirr ) − Qi ( t ) ⎡⎣ f w (1 − Sor ) − f w ( S w 2 ) ⎤⎦ = Qw ( t ) (8.96)
or
{
Qi ( t ) − φ AL ( S w 2 − S wirr ) − Qi ( t ) ⎡⎣1 − f w ( S w 2 ) ⎤⎦ = Qw ( t ) } (8.97)
since fw at Sw = 1–Sor is equal to 1. Eq.(8.97) can be rearranged as
Qi ( t ) − Qw ( t ) Qi ( t )
S w 2 = S wirr + − ⎡1 − f w ( S w 2 ) ⎤⎦ (8.98)
φ AL φ AL ⎣
8-37
S w 2 = S wirr + N pD − Wi ⎡⎣1 − f w ( S w 2 ) ⎤⎦ (8.99)
N pD = S w 2 − S wi + Wi ⎡⎣1 − f w ( S w 2 ) ⎤⎦ (8.100)
where NpD is the oil recovery as a fraction of the total pore volume. We observe
that the sum of the first two terms on the right hand side of Eq.(8.99) is the
average water saturation in the porous medium after water breakthrough.
Thus, Eq.(8.99) can be rewritten as
S w 2 = S wav − Wi ⎡⎣1 − f w ( S w 2 ) ⎤⎦ (8.101)
Substituting Eq.(8.91) into (8.101) and rearranging gives the average water
saturation in the porous medium after water breakthrough as
⎡1 − f w ( S w 2 ) ⎤⎦
S wav = S w 2 + ⎣ (8.102)
⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ S w2
which is analogous to Eq.(8.72) at water breakthrough. Figure 8.14 shows

that the average water saturation after water breakthrough as given by
Eq.(8.102) is equal to the water saturation where the tangent line to the
fractional flow curve at the outlet water saturation intersects the Fw = 1 axis.
The average water saturation in the porous medium after water

breakthrough could also have been derived using Eq.(8.71) and the water
saturation profile shown in Figure 8.15. The average water saturation is then
given by
1− Sor
φ AS w 2 L + ∫ φ AxdS w
S wav = (8.103)
Sw 2
φ AL
Substituting Eq.(8.77) into (8.103) gives the average water saturation as
8-38
1− Sor
Qi ( t ) ∫ df w
S wav = S w 2 + (8.104)
Sw 2
φ AL
Figure 8.14. Average water saturation after water breakthrough.
Qi ( t ) ⎡⎣ f w (1 − Sor ) − f w ( S w 2 ) ⎤⎦
S wav = S w 2 + (8.105)
φ AL
But fw at Sw = 1 – Sor is equal to 1.0. Thus, Eq.(8.105) can be rewritten as
8-39
Figure 8.15. Typical water saturation profile at time t after water
breakthrough.
Qi ( t ) ⎡⎣1 − f w ( S w 2 ) ⎤⎦
S wav = S w 2 + (8.106)
φ AL
Substituting Eq.(8.90) into (8.106) gives the average water saturation after
water breakthrough as
⎡1 − Fw ( S w 2 ) ⎤⎦
S wav = S w 2 + ⎣ (8.107)
⎛ dFw ⎞
⎜ ⎟
⎝ dS w ⎠ S w2
8-40
which is identical to Eq.(8.102).
Water Production
There is no water production before water breakthrough. After water
breakthrough, the water oil ratio is given by
⎛ fw ⎞
⎜ ⎟
qw ⎝ Bw ⎠ Bo ⎛ Fw ⎞
WOR = = = ⎜ ⎟ (8.108)
qo ⎛ f o ⎞ Bw ⎝ 1 − Fw ⎠
⎜ ⎟
⎝ Bo ⎠
The pore volumes of water produced is given by material balance on the water
as
Water produced = Cumulative water injected − Water stored (8.109)
Substituting appropriate symbols into Eq.(8.109) gives
W p = Wi − ( S wav − S wirr ) (8.110)
Substituting Eq.(8.91) into (8.110) gives the pore volumes of water produced
as
1
Wp = − ( S wav − S wi ) (8.111)
⎛ df w ⎞
⎜ ⎟
⎝ dS w ⎠ Sw 2
Example 8.1
A waterflood is to be performed in a linear reservoir. The relative permeability
curves for the reservoir are adequately described by the following analytical
models:
krw = k wr Se3 (8.112)
8-41
krnw = knwr (1 − Se )
2
(8.113)
where Se is defined as
S w − S wirr
Se = (8.114)
1 − S wirr − Snwr
The other pertinent data are as follows:
S wirr = 0.20
S nwr = 0.30
knwr = 0.95
k wr = 0.35
μnw = μo = 10 cp
μ w = 1 cp
Bo = 1.20 RB/STB
Bw = 1.0 RB/STB
1. Calculate and plot graphs of the relative permeability curves.
2. Calculate and plot graphs of the approximate fractional flow curve

⎛ dFw ⎞
( Fw ) and its derivative ⎜ ⎟.
⎝ dS w ⎠
3. Perform the Welge tangent construction and from it determine the

frontal water saturation (S ) ,
wf the average water saturation at water
breakthrough ( Swav ) and the true fractional flow curve ( fw ) and its
⎛ df ⎞
derivative ⎜ w ⎟ .
⎝ dS w ⎠
8-42
4. Plot the graphs of the true fractional flow curve and its derivative.
5. Calculate the end point mobility ratio for the waterflood.
6. Calculate and plot graphs of the water saturation profiles at tD = 0.20,

0.30 and 1.0.
7. Calculate the dimensionless breakthrough time.
8. Calculate the breakthrough oil recovery as a fraction of the initial oil in

place.
9. Calculate and plot the graph of oil recovery versus pore volume of water
injected before and after water breakthrough.
10. Calculate and plot the graph of water oil ratio versus oil recovery.
Solution to Example 8.1

The results of the calculations are summarized in Table 8.1.
1. The relative permeability curves calculated with Eqs.(8.112) and (8.113)

are shown in Figure 8.16.
2. Figure 8.17 shows the approximate fractional flow curve calculated with
Eq.(8.41) and its derivative calculated by differentiating Fw with respect
to Sw analytically.
Table 8.1. Calculated Results for Example 8.1.
tD tD tD
0.20 0.30 1.00
Sw krw krnw Fw dFw fw df w xD xD xD Wi R WOR
dS w dS w
0.200 0.00000 0.950 0.00000 0.000 0.000 2.775 0.555 0.833 2.775 0.000 0.000 0.000
0.210 0.00000 0.912 0.00003 0.009 0.028 2.775 0.555 0.833 2.775 0.008 0.023 0.000
0.220 0.00002 0.876 0.00026 0.039 0.056 2.775 0.555 0.833 2.775 0.016 0.045 0.000
0.230 0.00008 0.839 0.00090 0.094 0.083 2.775 0.555 0.833 2.775 0.025 0.068 0.000
0.240 0.00018 0.804 0.00222 0.176 0.111 2.775 0.555 0.833 2.775 0.033 0.091 0.000
8-43
0.250 0.00035 0.770 0.00453 0.290 0.139 2.775 0.555 0.833 2.775 0.041 0.113 0.000
0.260 0.00060 0.736 0.00815 0.441 0.167 2.775 0.555 0.833 2.775 0.049 0.136 0.000
0.270 0.00096 0.703 0.01348 0.632 0.194 2.775 0.555 0.833 2.775 0.057 0.158 0.000
0.280 0.00143 0.670 0.02094 0.866 0.222 2.775 0.555 0.833 2.775 0.066 0.180 0.000
0.290 0.00204 0.639 0.03097 1.147 0.250 2.775 0.555 0.833 2.775 0.074 0.202 0.000
0.300 0.00280 0.608 0.04403 1.473 0.278 2.775 0.555 0.833 2.775 0.082 0.223 0.000
0.310 0.00373 0.578 0.06057 1.844 0.305 2.775 0.555 0.833 2.775 0.090 0.243 0.000
0.320 0.00484 0.549 0.08103 2.254 0.333 2.775 0.555 0.833 2.775 0.098 0.263 0.000
0.330 0.00615 0.520 0.10575 2.693 0.361 2.775 0.555 0.833 2.775 0.106 0.281 0.000
0.340 0.00768 0.492 0.13496 3.150 0.389 2.775 0.555 0.833 2.775 0.115 0.299 0.000
0.350 0.00945 0.466 0.16875 3.607 0.416 2.775 0.555 0.833 2.775 0.123 0.315 0.000
0.360 0.01147 0.439 0.20703 4.044 0.444 2.775 0.555 0.833 2.775 0.131 0.330 0.000
0.370 0.01376 0.414 0.24949 4.439 0.472 2.775 0.555 0.833 2.775 0.139 0.343 0.000
0.380 0.01633 0.389 0.29560 4.772 0.500 2.775 0.555 0.833 2.775 0.147 0.355 0.000
0.390 0.01921 0.365 0.34465 5.024 0.527 2.775 0.555 0.833 2.775 0.156 0.365 0.000
0.400 0.02240 0.342 0.39576 5.181 0.555 2.775 0.555 0.833 2.775 0.164 0.374 0.000
0.410 0.02593 0.320 0.44794 5.238 0.583 2.775 0.555 0.833 2.775 0.172 0.381 0.000
0.420 0.02981 0.298 0.50019 5.195 0.611 2.775 0.555 0.833 2.775 0.180 0.388 0.000
0.430 0.03407 0.277 0.55153 5.058 0.638 2.775 0.555 0.833 2.775 0.188 0.393 0.000
0.440 0.03871 0.257 0.60109 4.842 0.666 2.775 0.555 0.833 2.775 0.197 0.398 0.000
0.450 0.04375 0.238 0.64815 4.561 0.694 2.775 0.555 0.833 2.775 0.205 0.403 0.000
0.460 0.04921 0.219 0.69216 4.234 0.722 2.775 0.555 0.833 2.775 0.213 0.407 0.000
0.470 0.05511 0.201 0.73274 3.879 0.749 2.775 0.555 0.833 2.775 0.221 0.411 0.000
0.480 0.06147 0.184 0.76969 3.511 0.777 2.775 0.555 0.833 2.775 0.229 0.416 0.000
0.490 0.06829 0.168 0.80296 3.144 0.805 2.775 0.555 0.833 2.775 0.237 0.421 0.000
0.491 0.06900 0.166 0.80608 3.107 0.808 2.775 0.555 0.833 2.775 0.246 0.423 0.000
0.492 0.06971 0.164 0.80917 3.071 0.810 2.775 0.555 0.833 2.775 0.254 0.426 0.000
0.493 0.07043 0.163 0.81222 3.035 0.813 2.775 0.555 0.833 2.775 0.262 0.428 0.000
0.494 0.07115 0.161 0.81524 2.999 0.816 2.775 0.555 0.833 2.775 0.270 0.430 0.000
0.495 0.07188 0.160 0.81822 2.964 0.819 2.775 0.555 0.833 2.775 0.278 0.432 0.000
0.496 0.07262 0.158 0.82117 2.928 0.822 2.775 0.555 0.833 2.775 0.287 0.434 0.000
0.497 0.07335 0.157 0.82408 2.893 0.824 2.775 0.555 0.833 2.775 0.295 0.436 0.000
0.498 0.07410 0.155 0.82695 2.857 0.827 2.775 0.555 0.833 2.775 0.303 0.438 0.000
0.499 0.07485 0.154 0.82979 2.822 0.830 2.775 0.555 0.833 2.775 0.311 0.440 0.000
0.500 0.07560 0.152 0.83260 2.788 0.833 2.775 0.555 0.833 2.775 0.319 0.442 0.000
8-44
0.500 0.07568 0.152 0.83288 2.784 0.833 2.775 0.555 0.833 2.775 0.328 0.444 0.000
0.500 0.07575 0.152 0.83316 2.781 0.833 2.775 0.555 0.833 2.775 0.336 0.445 0.000
0.500 0.07583 0.152 0.83343 2.777 0.833 2.775 0.555 0.833 2.775 0.344 0.447 0.000
0.500 0.07583 0.152 0.83343 2.777 0.833 2.775 0.555 0.833 2.775 0.352 0.449 0.000
0.500 0.07586 0.151 0.83357 2.775 0.834 2.775 0.555 0.833 2.775 0.360 0.450 6.010
0.501 0.07636 0.150 0.83537 2.753 0.835 2.753 0.551 0.826 2.753 0.363 0.451 6.089
0.502 0.07712 0.149 0.83810 2.718 0.838 2.718 0.544 0.816 2.718 0.368 0.452 6.212
0.503 0.07789 0.147 0.84081 2.684 0.841 2.684 0.537 0.805 2.684 0.373 0.453 6.338
0.504 0.07866 0.146 0.84347 2.650 0.843 2.650 0.530 0.795 2.650 0.377 0.454 6.466
0.505 0.07944 0.144 0.84611 2.616 0.846 2.616 0.523 0.785 2.616 0.382 0.455 6.598
0.506 0.08023 0.143 0.84871 2.583 0.849 2.583 0.517 0.775 2.583 0.387 0.456 6.732
0.507 0.08102 0.142 0.85127 2.549 0.851 2.549 0.510 0.765 2.549 0.392 0.457 6.868
0.508 0.08181 0.140 0.85380 2.516 0.854 2.516 0.503 0.755 2.516 0.397 0.458 7.008
0.509 0.08261 0.139 0.85630 2.483 0.856 2.483 0.497 0.745 2.483 0.403 0.459 7.151
0.510 0.08341 0.137 0.85877 2.450 0.859 2.450 0.490 0.735 2.450 0.408 0.460 7.297
0.520 0.09175 0.123 0.88169 2.137 0.882 2.137 0.427 0.641 2.137 0.468 0.469 8.943
0.530 0.10062 0.110 0.90160 1.850 0.902 1.850 0.370 0.555 1.850 0.540 0.479 10.995
0.540 0.11005 0.097 0.91878 1.591 0.919 1.591 0.318 0.477 1.591 0.628 0.489 13.575
0.550 0.12005 0.086 0.93351 1.360 0.934 1.360 0.272 0.408 1.360 0.736 0.499 16.849
0.560 0.13064 0.074 0.94606 1.154 0.946 1.154 0.231 0.346 1.154 0.866 0.508 21.048
0.570 0.14183 0.064 0.95668 0.974 0.957 0.974 0.195 0.292 0.974 1.027 0.518 26.502
0.580 0.15364 0.055 0.96561 0.816 0.966 0.816 0.163 0.245 0.816 1.226 0.528 33.693
0.590 0.16609 0.046 0.97306 0.678 0.973 0.678 0.136 0.203 0.678 1.474 0.537 43.348
0.600 0.17920 0.038 0.97923 0.559 0.979 0.559 0.112 0.168 0.559 1.788 0.546 56.589
0.610 0.19298 0.031 0.98430 0.456 0.984 0.456 0.091 0.137 0.456 2.191 0.555 75.235
0.620 0.20745 0.024 0.98841 0.368 0.988 0.368 0.074 0.110 0.368 2.716 0.564 102.358
0.630 0.22262 0.019 0.99171 0.292 0.992 0.292 0.058 0.088 0.292 3.420 0.573 143.471
0.640 0.23852 0.014 0.99430 0.228 0.994 0.228 0.046 0.068 0.228 4.392 0.581 209.224
0.650 0.25515 0.009 0.99629 0.172 0.996 0.172 0.034 0.052 0.172 5.798 0.589 322.295
0.660 0.27254 0.006 0.99777 0.126 0.998 0.126 0.025 0.038 0.126 7.966 0.597 537.909
0.670 0.29070 0.003 0.99882 0.086 0.999 0.086 0.017 0.026 0.086 11.663 0.605 1020.015
0.680 0.30966 0.002 0.99951 0.052 1.000 0.052 0.010 0.016 0.052 19.193 0.612 2444.665
0.690 0.32942 0.000 0.99988 0.024 1.000 0.024 0.005 0.007 0.024 42.067 0.619 10402.648
0.700 0.35000 0.000 1.00000 0.000 1.000 0.000 0.000 0.000 0.000
8-45
Figure 8.16. Relative permeability curves for Example 8.1.
8-46
Figure 8.17. Approximate fractional flow curve and its derivative for Example
8.1.
3. The Welge tangent construction is shown in Figure 8.17. From the

tangent construction,
S wf = 0.500035
⎛ df w ⎞
⎜ ⎟ = 2.775
⎝ dS w ⎠ Swf
S wav = 0.5603
4. The true fractional flow curve and its derivative obtained from the
tangent construction are shown in Figure 8.18.
8-47
Figure 8.18. True fractional flow curve and its derivative for Example 8.1.
5. The end point mobility ratio for the waterflood is given by
M E = ( krw / μ w ) / ( krnw / μnw ) = ( 0.35 /1) / ( 0.95 /10 ) = 3.68
6. The water saturation profiles calculated with Eq.(8.57) are shown in

Figure 8.19.
8-48
Figure 8.19. Water saturation profiles for Example 8.1.
7. The dimensionless breakthrough time is calculated with Eq.(8.83) as
1 1
t Dbt = = = 0.460 pore volume injected.
⎛ df w ⎞ 2.775
⎜ ⎟
⎝ dS w ⎠ Swf
8. The breakthrough oil recovery as a fraction of the initial oil in place is

calculated with Eq.(8.81) as
1 1
Rbt = = = 0.450
⎛ df w ⎞ (1 − 0.20 )( 2.775 )
(1 − S wirr ) ⎜ ⎟
⎝ dS w ⎠ Swf
9. Before water breakthrough, the oil recovery is a linear function of the

pore volume injected and can be calculated with Eq.(8.86). After water
8-49
breakthrough, the oil recovery is calculated with Eq.(8.100) as
N pD
R= .
1 − S wirr
Figure 8.20 shows the calculated oil recovery curve.
Figure 8.20. Oil recovery curve for Example 8.1.
10. The producing water oil ratio is zero before water breakthrough. After
water breakthrough, the producing water oil water ratio is calculated
with Eq.(8.108). After breakthrough, the producing water oil ratio
increases rapidly as shown in Figure 8.21.
8-50
Figure 8.21. Producing water oil ratio for Example 8.1.
8.4 LABORATORY MEASUREMENT OF TWO-PHASE RELATIVE

PERMEABILITIES BY THE UNSTEADY STATE METHOD
The major problem with the steady state method for relative
permeability measurements is that it is too slow. An alternative and much
faster technique is the unsteady state method or the dynamic displacement
method (Welge, 1952; Johnson et al., 1959; Jones and Roszelle, 1978). In
this method, for an imbibition test, the core is first saturated with the non-
wetting phase at irreducible wetting phase saturation as in the steady state
method. However, only the wetting phase is injected into the core to displace
the non-wetting phase. As the experiment progresses, the wetting phase
breaks through at the outlet end of the core and over time a higher and higher
fraction of the total produced fluid is the wetting phase.
8-51
By measuring the produced fractions of the wetting and non-wetting
phases at the outlet end of the core and the pressure drop across the core
versus time, the relative permeability curves can be calculated from the
production and pressure data using the theory of immiscible displacement in
porous media. This method is much faster than the steady state method,
usually requiring a few hours to complete compared to several weeks for the
steady state method. If adequate precautions are taken, the dynamic
displacement method will give relative permeability curves that are
comparable to those obtained by the steady state method.
Figure 8.22 shows the experimental setup and the measured data.
Because the point of observation is the outlet end of the core, it is necessary
that capillary end effect be minimized otherwise the calculated relative
permeability-saturation relationship will be wrong. It should be noted that
relative permeability curves can only be obtained over the saturation range Swf
to 1-Snwr. Therefore, it is necessary to choose the fluid viscosities that will give
the widest possible saturation window. This is obtained by using performing
and adverse mobility ratio displacement. A favorable mobility ratio
displacement will be unsuitable because for such a displacement, Swf is equal
to (1-Snwr) and there is no saturation window for calculating the relative
permeability curves. The relative permeability to the wetting phase below Swf
can only be obtained by extrapolating the data above Swf.
The technique for calculating relative permeability curves from

unsteady state measurements was developed by Welge (1952) and Johnson,
Bossler and Neumann (JBN, 1959). The fractional flow of the non-wetting
phase at the outlet end of the core is given by
1
f nw 2 = (8.115)
k μ
1 + rw nw
krnw μ w
8-52
Figure 8.22. Unsteady state method for determining two-phase relative
permeability curves; (a) coreflood; (b) measured data.
It should be noted that for saturations above Swf, Eq.(8.115) gives the true
fractional flow of the non-wetting phase because above Swf, the true fractional
flow and the approximate fractional curves are equal. Eq.(8.115) can be
rearranged to calculate the wetting-non-wetting phase relative permeability
ratio as
krw μ ⎛ 1 ⎞
= w ⎜ − 1⎟ (8.116)
krnw μ nw ⎝ f nw 2 ⎠
The fractional flow of the non-wetting phase at the outlet end of the core is
also given by
8-53
qnw dQnw ( t ) dN pD
f nw 2 = = = (8.117)
q dQi ( t ) dWi
where Qnw(t) and Qi(t) are the cumulative non-wetting phase produced and
the cumulative wetting phase injected and N pD and Wi are their dimensionless
counterparts as fractions of the total pore volume. Eqs.(8.116) and (8.117)

were first presented by Welge (1952). It should be noted that these equations
give no useful information before breakthrough because the fractional flow of
the non-wetting phase at the outlet end of the core is 1 and the relative
permeability to wetting phase is zero. This is why the unsteady state relative
permeability method is limited to only post breakthrough wetting phase
saturations between Swf and 1-Snwr.
After wetting phase breakthrough, we need to associate the computed

relative permeability ratio with the wetting phase saturation at the outlet end
of the core, the point of observation. To determine the wetting phase
saturation at the outlet end of the core, we perform a material balance for the
wetting phase after breakthrough to obtain
S w 2 = S wirr + N pD − Wi ⎡⎣1 − Fw ( S w 2 ) ⎤⎦ (8.99)
Eq.(8.99) can be written in terms of the fractional flow of the non-wetting

phase as
S w 2 = S wirr + N pD − Wi f nw 2 (8.118)
krw
Using Eqs.(8.116) and (8.118), versus S w 2 can be computed.
krnw
Johnson, Bossler and Neumann (JBN, 1959) presented equations for

calculating the individual relative permeability curves by the unsteady state
method by incorporating the pressure drop into the computations. The
pressure drop across the porous medium at time t is given by
8-54
L ∂P
ΔP = − ∫ dx (8.119)
0 ∂x
Darcy’s Law for the non-wetting phase gives
kkrnw A ∂P
qnw = − (8.120)
μnw ∂x
Dividing Eq.(8.120) by q and rearranging gives the pressure gradient as
∂P ⎛ q μnw ⎞
= −⎜ ⎟ f nw (8.121)
∂x ⎝ kkrnw A ⎠
⎛ qμ ⎞ L f
ΔP = ⎜ nw ⎟ ∫ nw dx (8.122)
⎝ kA ⎠ 0 krnw
Applying the Buckley-Leverett frontal advance equation, Eq.(8.77), at the

outlet end of the core after breakthrough gives
Qi ( t ) ⎛ df w ⎞
L= ⎜ ⎟ (8.90)
φ A ⎝ dS w ⎠ S
w2
Dividing Eq.(8.77) by (8.90) gives
x f'
= w' (8.123)
L f w2
where f w' and f w' 2 are the derivatives of the fractional flow functions at any
distance and at the core outlet, respectively. Differentiating Eq.(8.123) with

respect to f w' gives
L
dx = '
df w' (8.124)
f w2
8-55
Substituting Eq.(8.124) into (8.122) and rearranging gives
f w' 2 f nw ' ΔPkAf w' 2

∫ 0 krnw
df w =
qμnw L
(8.125)
Let
⎛ q ⎞ kA
⎜ ⎟ = = a constant (8.126)
⎝ ΔP ⎠ s μ nw L
⎛ q ⎞ '
⎜ ⎟ f w2
f w' 2 f nw ⎝ ΔP ⎠ s
∫ df w = (8.127)
'
0 krnw ⎛ q ⎞
⎜ ⎟
⎝ ΔP ⎠
Let a relative injectivity ratio be defined as
⎛ q ⎞
⎜ ⎟
ΔP ⎠
Ir = ⎝ (8.128)
⎛ q ⎞
⎜ ⎟
⎝ ΔP ⎠ s
f w' 2 f nw ' f w' 2

∫ 0 krnw
df w =
Ir
(8.129)
Differentiating Eq.(8.129) with respect to f w' 2 gives
f nw 2 d ⎛ f' ⎞
= ' ⎜ w2 ⎟ (8.130)
krnw df w 2 ⎝ I r ⎠
f nw 2 d ⎛ 1 ⎞
= ⎜ ⎟ (8.131)
krnw ⎛ 1 ⎞ ⎝ Wi I r ⎠
d⎜ ⎟
⎝ Wi ⎠
8-56
Eq.(8.131) can be used to calculate the relative permeability of the non-
wetting phase as
f nw 2
krnw = (8.132)
d ⎛ 1 ⎞
⎜ ⎟
⎛ 1 ⎞ WI
d⎜ ⎟⎝ i r ⎠
⎝ Wi ⎠
Knowing the relative permeability of non-wetting phase, the relative

permeability of the wetting phase can be calculated from Eq.(8.116) as
μw ⎛ 1 ⎞
krw = ⎜ − 1⎟ krnw (8.133)
μnw ⎝ f nw 2 ⎠
The advantage of the unsteady method over the steady state method of
relative permeability measurement is that it is considerably faster. Because
the method is based on the Buckley-Leverett displacement model, the
unsteady state method can only be used to calculate relative permeability
curves between Swf and the wetting phase saturation at the residual
nonwetting phase saturation (1-Snwr) as previously noted. If Swf is high as in
the case of a favorable mobility ratio displacement, then much of the relative
permeability curves cannot be obtained because one is limited to a very small
saturation observation window. To solve this problem, unfavorable mobility
ratio displacements are typically used to determine relative permeability
curves by the unsteady state method. Further, in order to minimize capillary
end effect, high displacement rates are also typically used. The combination
of high rate and adverse mobility ratio can lead to viscous instability that will
make the displacement performance to be rate sensitive. If this happens, the
relative permeability curves obtained by the unsteady state method will be
rate sensitive and can be quite different from the relative permeability curves
of the same porous medium obtained by the steady state method (Peters and
Khataniar, 1987).
8-57
Eqs.(8.117) and (8.132) call for differentiating the measured
experimental data. The challenge in calculating the relative permeability
curves from these equations is to ensure that the curves are smooth. Any type
of finite difference approximation of the derivatives will result in numerical
noise leading to noisy relative permeability curves. The best way to process
the experimental data is by fitting well behaved functions to the experimental
data and then differentiating the functions. Peters and Khataniar (1987) have
suggested the following curve fits, which they have shown to work well.
N pD = A1 + A2 ( ln Wi ) + A3 ( ln Wi )
2
(8.134)
2
⎛ 1 ⎞ ⎛ 1 ⎞ ⎡ ⎛ 1 ⎞⎤
ln ⎜ ⎟ = B1 + B2 ln ⎜ ⎟ + B3 ⎢ln ⎜ ⎟ ⎥ (8.135)
⎝ Wi I r ⎠ ⎝ Wi ⎠ ⎣ ⎝ Wi ⎠ ⎦
Example 8.2
Table 8.2 gives the experimental data for an unsteady state relative
permeability measurement for a sandpack. In the experiment, water was used
to displace a viscous oil at a constant injection rate. The pore volume of water
injected (Wi), the cumulative oil produced (Qo) and the pressure drop across
the sandpack (ΔP) were measured as functions of time.
Table 8.2. Experimental Data for Unsteady State Relative Permeability

Measurements.
Wi Qo ΔP
PV %IOIP psi
0.339 38.28 9.02
0.351 38.95 8.30
0.395 40.10 6.91
0.439 40.91 6.07
0.502 41.92 5.42
0.587 42.95 4.87
8-58
0.670 43.77 4.55
0.840 45.11 4.00
1.137 46.55 3.32
1.604 47.96 2.78
2.029 48.96 2.52
2.624 50.08 2.42
3.225 50.78 2.30
4.346 51.78 2.13
5.719 52.67 1.99
7.092 53.23 1.90
8.464 53.67 1.83
10.516 54.16 1.79
11.203 54.34 1.75
12.578 54.60 1.74
13.271 54.71 1.70
14.644 54.82 1.70
16.016 54.90 1.70
Other data for the experiment are as follows:

Injection rate = 100 cc/hr
Irreducible water saturation = 11.90%
Length of porous medium = 54.7 cm
Diameter of porous medium = 4.8 cm
Average porosity of porous medium = 30.58%
Absolute permeability of porous medium = 3.42 Darcies
Oil viscosity = 108.37 cp
Oil density = 0.959 gm/cm3

Water viscosity = 1.01 cp
Water density = 0.996 gm/cm3

Oil-water interfacial tension = 26.7 dynes/cm
8-59
Effective permeability to oil at irreducible water saturation = 3.16
Darcies
Oil recovery at water breakthrough = 38.28 % IOIP
Final oil recovery at termination of experiment = 54.9 % IOIP
1. Plot graphs of the raw experimental data.

2. Perform the curve fits suggested in Eqs.(8.134) and (8.135) and display
the results graphically.
3. Calculate the oil-water relative permeability curves for the porous

medium using the Johnson-Bossler-Neumann (JBN) method.
4. Plot graphs of the relative permeability curves.
5. Plot the graph of the true fractional flow curve measured in the
experiment.
6. How long was this test?

The results of the calculations are summarized in Table 8.3.
1. Figure 8.23 shows the graphs of the raw experimental data.
2. Figures 8.24 and 8.25 show the curve fits of N pD versus ln Wi and
⎛ 1 ⎞ ⎛ 1 ⎞
ln ⎜ ⎟ versus ln ⎜ ⎟ . The curve fit equations are
⎝ Wi I r ⎠ ⎝ Wi ⎠
N pD = 0.4026 + 0.0474 ln Wi + 0.0066 ( ln Wi )

2
2
⎛ 1 ⎞ ⎛ 1 ⎞ ⎡ ⎛ 1 ⎞⎤
ln ⎜ ⎟ = −2.3600 + 1.5798ln ⎜ ⎟ + 0.1130 ⎢ln ⎜ ⎟ ⎥
⎝ Wi I r ⎠ ⎝ Wi ⎠ ⎣ ⎝ Wi ⎠ ⎦
These equations can be differentiated analytically to obtain
8-60
dN pD 0.0474 − ( 2 )( 0.0066 ) ln Wi
f nw 2 = =
dWi Wi
⎛ 1 ⎞ ⎡ ⎛ 1 ⎞⎤ ⎛
d⎜ ⎟ ⎢ ( 2 )( 0.1130 ) ln ⎜ 2⎞
⎟ ⎥ ⎜⎜ −2.3600+1.5798ln ⎛⎜ W1 ⎞⎟ + 0.1130 ⎡⎢ln ⎛⎜ W1 ⎞⎟⎤⎥ ⎟⎟
= ⎝ i r ⎠=⎢ ⎝ Wi ⎠ ⎥ e⎝
f nw 2 WI 1.5798 ⎝ i⎠ ⎢⎣ ⎝ i ⎠ ⎥⎦
+ ⎠
krnw ⎛ 1 ⎞ ⎢⎛ 1 ⎞ ⎛ 1 ⎞ ⎥
d⎜ ⎟ ⎢ ⎜ ⎟ ⎜ ⎟ ⎥
⎝ Wi ⎠ ⎢⎣ ⎝ Wi ⎠ ⎝ Wi ⎠ ⎦⎥
Table 8.3. Calculated Results for Example 8.2.
Wi NpD ΔP
PV PV psi ln(Wi) ⎛ 1⎞ fnw2 Sw2 Ir 1 ⎛ 1 ⎞ f nw 2 krnw krw
ln ⎜ ⎟ ln ⎜ ⎟
⎝ Wi ⎠ Wi I r ⎝ Wi I r ⎠ k rnw
0.119 0.924 0.000
0.339 0.337 9.02 -1.082 1.082 0.182 0.395 4.335 0.680 -0.385 0.368 0.494 0.021
0.351 0.343 8.30 -1.047 1.047 0.174 0.401 4.711 0.605 -0.503 0.356 0.490 0.022
0.395 0.353 6.91 -0.929 0.929 0.151 0.413 5.659 0.447 -0.804 0.319 0.473 0.025
0.439 0.360 6.07 -0.823 0.823 0.133 0.421 6.442 0.354 -1.040 0.290 0.457 0.028
0.502 0.369 5.42 -0.689 0.689 0.113 0.432 7.214 0.276 -1.287 0.258 0.436 0.032
0.587 0.378 4.87 -0.533 0.533 0.093 0.443 8.029 0.212 -1.550 0.226 0.411 0.037
0.67 0.386 4.55 -0.400 0.400 0.079 0.452 8.594 0.174 -1.751 0.203 0.388 0.042
0.84 0.397 4.00 -0.174 0.174 0.059 0.467 9.775 0.122 -2.106 0.170 0.349 0.052
1.137 0.410 3.32 0.128 -0.128 0.040 0.483 11.778 0.075 -2.595 0.136 0.295 0.066
1.604 0.423 2.78 0.473 -0.473 0.026 0.500 14.065 0.044 -3.116 0.108 0.237 0.084
2.029 0.431 2.52 0.708 -0.708 0.019 0.512 15.517 0.032 -3.449 0.094 0.199 0.097
2.624 0.441 2.42 0.965 -0.965 0.013 0.526 16.158 0.024 -3.747 0.082 0.162 0.113
3.225 0.447 2.30 1.171 -1.171 0.010 0.534 17.001 0.018 -4.004 0.074 0.135 0.125
4.346 0.456 2.13 1.469 -1.469 0.006 0.547 18.358 0.013 -4.379 0.064 0.100 0.144
5.719 0.464 1.99 1.744 -1.744 0.004 0.559 19.649 0.009 -4.722 0.057 0.074 0.162
7.092 0.469 1.90 1.959 -1.959 0.003 0.566 20.580 0.007 -4.983 0.053 0.057 0.175
8.464 0.473 1.83 2.136 -2.136 0.002 0.573 21.367 0.006 -5.198 0.050 0.045 0.185
10.516 0.477 1.79 2.353 -2.353 0.002 0.580 21.844 0.004 -5.437 0.047 0.033 0.197
11.203 0.479 1.75 2.416 -2.416 0.001 0.582 22.344 0.004 -5.523 0.047 0.030 0.200
12.578 0.481 1.74 2.532 -2.532 0.001 0.586 22.472 0.004 -5.644 0.045 0.025 0.206
8-61
13.271 0.482 1.70 2.586 -2.586 0.001 0.588 23.001 0.003 -5.721 0.045 0.022 0.208
14.644 0.483 1.70 2.684 -2.684 0.001 0.590 23.001 0.003 -5.820 0.044 0.019 0.213
16.016 0.484 1.70 2.774 -2.774 0.001 0.592 23.001 0.003 -5.909 0.043 0.016 0.217
Figure 8.23. Raw experimental data for the unsteady state relative
permeability measurements of Example 8.2.
8-62
Figure 8.24. Curve fit of N pD versus ln Wi for Example 8.2.
⎛ 1 ⎞ ⎛ 1 ⎞
Figure 8.25. Curve fit of ln ⎜ ⎟ versus ln ⎜ ⎟ for Example 8.2.
⎝ Wi I r ⎠ ⎝ Wi ⎠
3. The oil-water relative permeability data calculated with Eqs.(8.132) and

(8.133) are presented in Table 8.2.
4. Figure 8.26 shows the oil-water relative permeability curves from the
unsteady state experiment. It should be noted that the relative
permeability curves are obtained over the limited saturation range of
0.395 ≤ S w ≤ 0.592 . The relative permeability curves between S wirr = 0.119
and S wf = 0.395 cannot be obtained from the experiment. They can only
be obtained by extrapolation of the computed data to the conditions at

the irreducible water saturation where krw = 0.000 and knwr = 0.924 . The
8-63
experiment predicts a residual oil saturation of 40% in this
homogeneous high permeability sand.
Figure 8.26. Computed relative permeability curves for Example 8.2.
5. The true fractional flow curve measured in the experiment is shown in

Figure 8.27. It is interesting to note that it is only that portion of the
true fractional flow curve that is equal to the approximate fractional
flow curve that can be measured in the experiment. If the saturation
profiles in the experiment could be imaged, then it would possible to
calculate the true fractional flow curve between S wirr and S w by the
similarity transformation and the integration outlined in Figures 8.11

and 8.12.
6. The unsteady state experiment lasted 48.48 hours compared to several

weeks for the steady state experiment.
8-64
Figure 8.27. True fractional flow curve measured in the unsteady state
experiment of Example 8.2.
8.5 FACTORS AFFECTING RELATIVE PERMEABILITIES

The factors that affect or could affect relative permeability curves
include (1) fluid saturation, (2) fluid saturation history, (3) Wettability, (4)
injection rate, (5) viscosity ratio, (6) interfacial tension, (7) pore structure, (8)
temperature and (9) heterogeneity.
8.5.1 Fluid Saturation

Relative permeabilities are strongly dependent on fluid saturations. The
higher the fluid saturation, the higher the relative permeability to that fluid.
In general, relative permeabilities are nonlinear functions of fluid saturation
as shown in Figures 8.6 and 8.7.
8-65
8.5.2 Saturation History
Like capillary pressure curves, relative permeability curves show
saturation hysteresis. Figure 8.28 shows typical relative permeability curves
for drainage and imbibition. The imbibition non-wetting phase relative
permeability curve is generally lower than the drainage curve at the same
saturations. The imbibition wetting phase relative permeability curve is
slightly greater than the drainage curve. These differences can easily be
explained. During drainage, the non-wetting phase flows through the large
pores displacing the wetting phase along the way. The thin film of wetting
phase that coats the grain surface acts as a lubricant for the flow of the non-
wetting phase. Therefore, the relative permeability to the non-wetting phase
will be high during drainage. That of the wetting phase also will be high
because it starts from 1 and decreases as the non-wetting phase begins to
occupy some of the pores that were previously occupied by the wetting phase.
During imbibition, some of the non-wetting phase will be trapped in the large
pores. This capillary trapping reduces the amount of non-wetting phase
available to flow during imbibition compared to during drainage. It also
reduces the cross-sectional area of the medium occupied by the connected
non-wetting phase. As a result, the imbibition relative permeability to the
non-wetting phase is reduced compared to that during drainage. Because of
capillary trapping of the non-wetting phase during imbibition, the wetting
phase is forced to occupy and flow through pore sizes that are larger than it
would otherwise have flowed if there was no trapping of the non-wetting
phase. This forcing of the wetting phase to flow through larger pores than it
would otherwise have done in the absence of trapping enhances the relative
permeability of the wetting phase on the imbibition cycle compared to the
drainage cycle. These observations are in accord with the experimental results
shown in Figure 8.28.
8-66
Figure 8.28. Relative permeability hysteresis (Osoba et al., 1951).
8.5.3 Wettability
Relative permeability curves are markedly affected by the wettability of
the medium. Jennings (1957) measured steady state oil water relative
permeability curves on a core that was initially strongly water wet. He then
treated the core with a surface active agent (organo chlorosilane) that
rendered the core oil wet and repeated the relative permeability
measurements. The results are shown Figure 8.29. In general, the relative
permeability to oil decreases while the relative permeability to water increases
as the medium changes from a strongly water wet to a strongly oil wet
8-67
medium. It is interesting to replot the relative permeability curves of Figure
8.29 as functions of wetting phase saturation instead of water saturation. The
replotted curves are shown in Figure 8.30. We see that when plotted against
the wetting phase saturation, the relative permeability curves for the oil wet
core and the water wet core are close to each other. They are not identical
because the degree of wettability preference in the two experiments may be
different. However, the relative permeability curves for the wetting phase and
the non-wetting phase from the two experiments are essentially the same.
Based on experimental observations, Craig (1971) gives the following

rules of thumb about the relative permeabilities for water wet and oil wet
media. (1) The irreducible water saturation for a water wet medium is usually
greater than 20% to 25% whereas that of an oil wet medium it is generally
less than 15%, and frequently less than 10%. (2) The water saturation at
which the oil and water relative permeabilities are equal is greater than 50%
for a strongly water wet medium whereas it is less than 50% for a strongly oil
wet medium. (3) The end-point relative permeability to water is generally less
than 30% for a strongly water wet medium and greater than 50% and
approaches the oil end point for a strongly oil wet medium. These
observations are consistent with the effect of wettability on the fluid
distribution and displacement discussed in Section 6.3.4.
It should be emphasized that the above rules of thumb are applicable

only to systems that show a strong preferential wettability to either water or
oil. In general, one cannot infer the wettability of a porous medium based
solely on the relative permeability curves. For example, relative permeability
curves that intersect at a water saturation of 50% does not mean that the
medium is of “neutral” wettability.
8-68
Figure 8.29. Effect of strong preferential wettability on steady state relative
permeability curves (Jennings, 1957).
Figure 8.30. Relative permeability curves from Figure 8.28 replotted as

functions of wetting phase saturation (adapted from Jennings, 1957).
8-69
At a given saturation, the relative permeability for a phase is higher
when that phase is the non-wetting phase than when it is the wetting phase.
This is observation can be seen in Figure 8.29. At any water saturation, the
relative permeability to water is higher when the water was the non-wetting
phase than when it was the wetting phase. Similarly, at any water saturation,
the relative permeability to oil is higher when the oil was the non-wetting
phase than when it was the wetting phase.
Owens and Archer (1971) measured relative permeability curves of

sandstones that were rendered progressively oil wet with a surface active
agent. Figure 8.31 shows their results for contact angles ranging from 0 to
180°. Note the general decrease in the oil relative permeabilities and increase
in the water relative permeabilities as the system was made progressively
more oil wet. Note also, that the strongly preferentially wet systems with
contact angles of 0 and 180° generally obey Craig’s rules of thumb regarding
the end-point water relative permeability and the water saturation at which
the water and oil relative permeabilities are equal. The rules of thumb do not
strictly apply to the intervening degrees of wettability.
8.5.4 Injection Rate

Injection rate usually does not affect relative permeabilities obtained by
the steady state method provided the rate is sufficiently high to minimize
capillary end effect. However, Peters and Khataniar (1987) have shown that
relative permeabilities obtained by the unsteady state displacement method
can show rate sensitivity due to viscous instability. Figures 8.32 and 8.33
show the effects of rate and viscosity ratio on relative permeability curves for
oil wet and water wet sandpacks. Both the curves for the oil wet medium and
the water wet medium shift to lower water saturations as the injection rate
(stability number) is increased. It should be noted that for the water wet
system, the relative permeability curves obtained by the unsteady state
method deviate from the steady state curves as the degree of instability is
8-70
increased. The water curve increases and the oil curve decreases away from
the steady state curves as the degree of instability of the displacement
experiment increases.
Figure 8.31. Relative permeabilities for range of wetting conditions (Owens

and Archer, 1971).
8-71
Figure 8.32. Effect of stability number on unsteady state relative permeability
curves for oil wet sandpacks (Peters and Khataniar, 1987).
8-72
Figure 8.33. Effect of stability number on unsteady state relative permeability
curves for water wet sandpacks (Peters and Khataniar, 1987).
8.5.5 Viscosity Ratio

Viscosity ratio usually does not affect relative permeabilities obtained by
the steady state method since there is no displacement involved. Figure 8.34
shows the relative permeability curves obtained with the steady state method
at various viscosity ratios. Clearly, no viscosity ratio effect is apparent.
However, Peters and Khataniar (1987) have shown that relative permeabilities
obtained by the unsteady state displacement method at adverse viscosity
ratios can show sensitivity to injection rate and viscosity ratio due to viscous
instability (Figures 8.32 and 8.33).
8-73
Figure 8.34. Effect of viscosity ratio on relative permeability curves obtained
by the steady state method (Leverett, 1939).
8.5.6 Interfacial Tension

Relative permeability curves are affected by interfacial tension only at
interfacial tensions lower than 0.1 dyne/cm. Above this value, relative
permeabilities are unaffected by interfacial tension.
Bardon and Longeron (1978) studied the effect of interfacial tension on

gas-oil relative permeabilities using methane and normal heptane
displacement experiments. In their study, interfacial tensions were calculated
using parachors and the gas-oil relative permeabilities were calculated using
relative permeability models and a numerical simulator to history match the
displacement data. Figure 8.35 shows the relative permeability curves that
gave the best fit to the displacement recovery data at the calculated interfacial
tensions. The results show that the relative permeabilities to gas and oil
8-74
increased as the interfacial tension decreased. The residual fluid saturations
decreased as the interfacial tension decreased as expected from the effect of
capillary number on residual fluid saturations. In the limit, at ultra-low
interfacial tensions, the relative permeability curves were approximately
straight lines. These general trends in the effect of interfacial tensions on
relative permeability curves have been confirmed by Amaefule and Handy
(1981).
Figure 8.35. Effect of interfacial tension on gas-oil relative permeability curves

(Bardon and Longeron, 1978).
8.5.7 Pore Structure

Morgan and Gordon (1970) have presented results that show that rocks
with large pores and correspondingly small specific surface areas have low
irreducible water saturations that leave a relatively large amount of pore
8-75
space available for multiphase flow. Therefore, for such rocks, end point
relative permeabilities are high and a large saturation change may occur
during two phase flow. By contrast, rocks with small pores have larger
specific surface areas and larger irreducible water saturations that leave less
room for multiphase flow. As a result, the end point relative permeabilities
are lower and the saturation range for two phase flow is smaller than in rocks
with large pores. Finally, rocks having some relatively large pores connected
by small pores have a large surface area, resulting in high irreducible water
saturation and relative permeability behavior that is similar to rocks with
small pores only. These observations are summarized in Figure 8.36.
8.5.8 Temperature
There are data in the literature that suggest that relative permeability
curves are affected by temperature. Poston et al., (1970) found that
temperature causes residual oil saturation to decrease and irreducible water
saturation to increase, with corresponding increases in relative permeability
curves (Figure 8.37). On the other hand, there are data in the literature that
also show that relative permeabilities are not temperature dependent (Miller
and Ramey, 1985). Apparently, the effect of temperature on relative
permeabilities is still and open question. This situation is understandable
because temperature can affect rock and fluid properties which in turn can
affect relative permeability curves. For example, high temperature can change
the wettability of the rock which affects relative permeabilities. It can also
reduce interfacial tensions, which can affect relative permeabilities and the
irreducible saturations. Because of the effect of temperature on the other
properties of the system that can affect relative permeabilities, it is difficult to
categorically determine the effect of temperature on relative permeabilities.
8-76
Figure 8.36. Effect of pore structure on relative permeability curves; (a)
sandstone with large, well-connected pores with k = 1314 md; (b) sandstone
with small, well-connected pores with k = 20 md; (c) sandstone with a few
large pores connected with small pores with k = 36 md (Morgan and Gordon,
1970).
8-77
Figure 8.37. Effect of temperature on relative permeability curves (Poston et
al., 1970).
8.5.9 Heterogeneity
Relative permeabilities are typically measured on homogeneous core
samples. These curves are then used in numerical simulators to model the
performance of heterogeneous reservoirs. It is often necessary to adjust the
laboratory measured relative permeability curves in order to successfully
history match the performance of the heterogeneous reservoirs. Gharbi and
Peters (1993) simulated the waterflood performance of a heterogeneous
reservoir using a set of input relative permeability curves and then used the
simulated oil recovery versus pore volumes of water injected and the
simulated pressure drop to calculate the equivalent relative permeability
curves for the heterogeneous medium by the JBN method. Figure 8.38
8-78
compares the input relative permeabilities with the computed equivalent
relative permeabilities for the heterogeneous medium. The effect of the
heterogeneity is to shift the oil and water relative permeabilities to low water
saturations thereby increasing the water relative permeability curve and
decreasing the oil relative permeability curve. Thus, the relative
permeabilities for the heterogeneous medium are similar to the relative
permeabilities for a strongly oil wet medium.
Figure 8.38. Effect of heterogeneity on relative permeability curves (Gharbi

and Peters, 1993).
8.6 THREE-PHASE RELATIVE PERMEABILITIES

Three phase relative permeabilities are required to predict the
performance of three phase flow of oil, water and gas. There are considerably
less experimental data in the literature on three phase relative permeabilities
than two phase relative permeabilities. Figure 8.39 shows, on a ternary
8-79
saturation diagram, the approximate regions of single phase flow, two phase
flow and three phase flow in an oil, water and gas system ( Leverett and Lewis,
1941). It can be seen that the three phase flow region is small compared to
single phase and two phase flow regions. Figures 8.40, 8.41 and 8.42 show
the three phase water, oil and gas relative permeabilities measured by
Leverett and Lewis (1941). They found that the relative permeability to water
was only a function of the water saturation. However, the relative
permeabilities to oil and gas were functions of all three fluid saturations.
Figure 8.39. Approximate limits of saturations giving 5 per cent or more of all
components in flow stream for the flow of nitrogen, kerosene and brine.
Arrows point to increasing fraction of respective components in stream
(Leverett and Lewis, 1941).
8-80
Figure 8.40. Three phase relative permeability to water (Leverett and Lewis,
1941).
Figure 8.41. Three phase relative permeability to oil (Leverett and Lewis,
1941).
8-81
Figure 8.42. Three phase relative permeability to gas (Leverett and Lewis,
1941).
Three phase relative permeabilities are not routinely measured in the

laboratory as two phase relative permeabilities. Instead, three phase relative
permeabilities are usually calculated from two phase relative permeability
data using various relative permeability models. Delshad and Pope (1989)
have reviewed the various three phase relative permeability models and found
that some of them do not always agree with the available experimental three
phase relative permeability data.
8.7 CALCULATION OF RELATIVE PERMEABILITIES FROM

DRAINAGE CAPILLARY PRESSURE CURVE
In Section 7.12.2, we derived the following approximate drainage
relative permeability curves for wetting and non-wetting phases from the
drainage capillary pressure curve:
8-82
Sw
dS w
kw ∫ Pc 2
krw ( S w ) = = 0
1
(7.159)
k dS
∫0 Pc 2w
and
1
dS w
knw
∫ Pc 2
krnw ( S w ) = = (7.160)
Sw
1
k dS
∫0 Pc 2w
We found that these models were defective in two respects: (1) they do not
include trapped residual saturations and (2) the sum of the relative
permeabilities is equal to 1, which is contrary to experimental observations.
These deficiencies result from the fact that the models neglect certain facts
about the nature of two phase flow in porous media. First, the cross-sectional
area open to the flow of the wetting phase is not a constant as assumed in the
models but is a function of the wetting phase saturation. Second, the
tortuosity for the flow of the wetting phase, which was neglected in the
models, is also a function of the wetting phase saturation. Burdine (1953)
proposed the following normalized drainage relative permeability models,
which account for these saturation dependencies in the cross-sectional area
and the tortuosity for two phase flow:
S w*
1
* ∫P 2
dS w*
= ( S w* )
kw ( S ) 2
krw ( S w* ) = w 0 c
(8.136)
k w ( S w = 1)
* 1
1
∫0 Pc2 dSw
*
1
1
* ∫P 2
dS w*
= (1 − S w* )
knw ( S ) 2 S w* c
krnw ( S w* ) = w
(8.137)
knw ( S w = 0)
* 1
1
∫0 Pc2 dSw
*
8-83
where S w* is the normalized wetting phase saturation given by
S w − S wirr
S w* = (8.138)
1 − S wirr
In Eqs.(8.136) and (8.137), the ratios of the integrals on the right side
account for the cross-sectional area changes with saturations and the terms
(S )
* 2
w and (1 − S )
* 2
w account for the tortuosity changes with saturations. It
should be noted that the base permeability used in Eq.(8.136) to define the
normalized relative permeability to the wetting phase is equal to the absolute
permeability of the medium, whereas the base permeability used to define the
normalized relative permeability to the non-wetting phase in Eq.(8.137) is
equal to the effective permeability to the non-wetting phase at the irreducible
wetting phase saturation. Thus, the normalized wetting phase relative
permeability given by Eq.(8.136) is also the true relative permeability to the
wetting phase. However, the normalized non-wetting phase relative
permeability given by Eq.(8.137) must be multiplied by the end point relative
permeability to the non-wetting phase (knwr) in order to obtain the true non-
wetting phase relative permeability. Furthermore, the normalized non-wetting

phase relative permeability of Eq.(8.137) starts at a wetting phase saturation
of 1.0 or a non-wetting phase saturation of zero. Normally, a critical non-
wetting phase saturation is required before the non-wetting phase can flow.
Thus, the end-point non-wetting phase relative permeability and a critical
non-wetting phase saturation must be introduced into Eq.(8.137) to obtain
the true relative permeability for the non-wetting phase. Given the drainage
capillary pressure curve, the integrals in Eqs.(8.136) and (8.137) can easily be
calculated numerically to obtain the normalized drainage relative permeability
curves.
An alternative approach to evaluating the integrals in Eqs.(8.136) and

(8.137) is to fit the Brooks-Corey (1966) model to the drainage capillary
8-84
pressure curve and then integrate the resulting linear function. As discussed
in Section 7.13.1, the Brooks-Corey drainage capillary pressure model is given
by
ln S w* = −λ ln Pc + λ ln Pe (7.161)
or
1
ln Pc = − ln S w* + ln Pe (7.162)
λ
and
1
Pc = Pe ( S w* )
−
λ (7.164)
where λ is the pore size distribution index obtained from the straight line
given by Eq.(7.161) or (7.162). Substituting Eq.(7.164) into Eqs.(8.136) and
(8.137) and performing the integrations gives the normalized drainage relative
permeability curves as
2 + 3λ
krw ( S w ) = ( S w* ) λ (8.139)
and
2+λ
2 ⎡ ⎤
krnw ( S w ) = (1 − S w* ) ⎢1 − ( S w* ) λ ⎥ (8.140)
⎣ ⎦
A critical saturation can be introduced into the relative permeability model for
the non-wetting phase as
2
⎛ S − S wirr ⎞ ⎡ 2+ λ
⎤
krnw ( S w ) = ⎜ 1 − w ⎟ ⎢ ( w) λ ⎥
− (8.141)
*
1 S
⎝ Sm − S wirr ⎠ ⎣ ⎦
8-85
where Sm is the wetting phase saturation corresponding to the critical non-
wetting phase saturation. Finally, the true relative permeability curve for the
wetting and non-wetting phases are given by
2 + 3λ
krw ( S w ) = ( S w* ) λ (8.142)
2
⎛ S − S wi ⎞ ⎡ 2+λ
⎤
krnw ( S w ) = knwr ⎜ 1 − w ⎟ ⎢1 − ( S *
) λ
⎥ (8.143)
⎝ Sm − S wi ⎠ ⎣
w
⎦
where knwr is the non-wetting phase relative permeability at the irreducible
wetting phase saturation.
Example 8.3
Use the air-water capillary pressure data of Table 8.4 to calculate the
drainage relative permeability curves by the method of Brooks and Corey for a
core sample.
Table 8.4. Drainage Capillary Pressure Curves for Example 8.3.

Saturation Capillary Pressure
(psi)
1.000 1.973
0.950 2.377
0.900 2.840
0.850 3.377
0.800 4.008
0.750 4.757
0.700 5.663
0.650 6.781
0.600 8.195
0.550 10.039
0.500 12.547
0.450 16.154
0.400 21.787
0.350 31.817
8-86
0.300 54.691
0.278 78.408

*
Figure 8.43 shows the graph of lnPc versus lnS w for Swirr = 0.10. The
equation of the resulting straight line is given by
ln Pc = −2.1443ln S w* + ln 2.2238
Therefore,
1
− = −2.1443
λ
λ = 0.4664
Pe = 2.2238
*
Figure 8.44 shows the graph of lnS w versus lnPc for Swi = 0.10. It also is
linear and could have been used for the subsequent calculations. The Brooks-
Corey drainage capillary pressure equation is given by
1
Pc = Pe ( S )
* −λ
= 2.2238 ( S w* )
−2.1443
w
Figure 8.45 shows the normalized drainage relative permeability curves

calculated with Eqs.(8.139) and (8.140). Figure 8.46 shows the true drainage
relative permeability curves for Sm = 0.95 and knwr = 0.961. The results of the
calculations are summarized in Table 8.5.
8-87
*
Figure 8.43. Log-log graph of Pc versus S w for Example 8.3.
*
Figure 8.44. Log-log graph of S w versus Pc for Example 8.3.
8-88
Figure 8.45. Normalized drainage relative permeability curves for Example
8.3.
Figure 8.46. True drainage relative permeability curves for Example 8.3.
8-89
Table 8.5. Results of Drainage Relative Permeability Calculations for Example
8.3
Brooks-Corey
Original Data Model Drainage Relative Permeability Curves
Pc Pc
Sw psi S *
w psi krw ( S w ) krnw ( S w ) krw krnw
1.000 1.973 1.000 2.224 1.000 0.000 1.000 0.000
0.950 2.377 0.944 2.514 0.659 0.001 0.659 0.000
0.900 2.840 0.889 2.863 0.424 0.006 0.424 0.002
0.850 3.377 0.833 3.288 0.265 0.017 0.265 0.008
0.800 4.008 0.778 3.812 0.160 0.036 0.160 0.022
0.750 4.757 0.722 4.468 0.093 0.063 0.093 0.044
0.700 5.663 0.667 5.305 0.052 0.098 0.052 0.073
0.650 6.781 0.611 6.393 0.028 0.140 0.028 0.111
0.600 8.195 0.556 7.843 0.014 0.189 0.014 0.156
0.550 10.039 0.500 9.831 0.006 0.244 0.006 0.208
0.500 12.547 0.444 12.656 0.003 0.304 0.003 0.266
0.450 16.154 0.389 16.851 0.001 0.371 0.001 0.331
0.400 21.787 0.333 23.452 0.000 0.443 0.000 0.401
0.350 31.817 0.278 34.672 0.000 0.521 0.000 0.479
0.300 54.691 0.222 55.947 0.000 0.605 0.000 0.562
0.278 78.408 0.198 71.829 0.000 0.643 0.000 0.601
0.250 0.167 103.678 0.000 0.694 0.000 0.652
0.200 0.111 247.331 0.000 0.790 0.000 0.749
0.150 0.056 1093.394 0.000 0.892 0.000 0.852
0.100 0.000 0.000 1.000 0.000 0.962
8-90
NOMENCLATURE
A = cross sectional area in the flow direction
Bo = oil formation volume factor
Bw = water formation volume factor
fw = fractional flow of wetting phase
fw = fractional flow of water
fnw = fractional flow of non-wetting phase
fnw2 = fractional flow of non-wetting phase at the outlet end of porous
medium
fo = fractional flow of oil
Fw = approximate fractional flow of wetting phase
g = gravitational acceleration
Ir = relative injectivity
J = Leverett J-function
k = absolute permeability of the medium
ko = effective permeability to oil
kw = effective permeability to water
kwr = end-point relative permeability to wetting phase
kg = effective permeability to gas
kro = relative permeability to oil
krw = relative permeability to water
krg = relative permeability to gas
krw = relative permeability to wetting phase
krnw= relative permeability to non-wetting phase
knwr= end-point relative permeability to non-wetting phase
L = length
M = mobility ratio
ME = end-point mobility ratio
Ncap = dimensionless capillary to viscous force ratio
Ng = gravity number
NpD = dimensionless cumulative production
8-91
Nvcap = capillary number
P = pressure
Pc = capillary pressure
Pe = displacement pressure for Brooks-Corey model
Pg = pressure in the gas phase
Pnw= pressure in the non-wetting phase
Po = pressure in the oil phase
Pc/ow = oil-water capillary pressure curve
Pc/go = gas-oil capillary pressure curve
Pc/gw = gas-water capillary pressure curve
Pw = pressure in the water phase
Pw = pressure in the wetting phase
q = total volumetric injection rate
qo = volumetric flow rate of oil
qg = volumetric flow rate of gas
qnw= volumetric flow rate of non-wetting phase
qw = volumetric flow rate of water
qw = volumetric flow rate of wetting phase
Qi = cumulative injection
Qnw = cumulative non-wetting phase produced
Qo = cumulative oil produced
R = oil recovery as a fraction of initial oil in place
Se = effective wetting phase saturation
Sg = gas saturation
So = oil saturation
Sor = residual oil saturation
Sw = water saturation
Sw = wetting phase saturation
Swirr = irreducible wetting phase saturation
8-92
Swro= wetting phase saturation at which imbibition capillary pressure is
zero
Sw2 = wetting phase saturation at the outlet end of porous medium
Snw= non-wetting phase saturation
Snwr = residual non-wetting phase saturation
Swav = average wetting phase saturation
Swav = average water saturation
Swf = frontal saturation
S w* = normalized wetting phase saturation
t = time
tbt = breakthrough time
tD = dimensionless time
G
v = flux vector, Darcy velocity vector
vw = Darcy velocity for the wetting phase
vnw = Darcy velocity for the non-wetting phase
x = distance in the direction of flow
xf = distance to the displacement front
xD = dimensionless distance
xDf = dimensionless distance to the displacement front
Wi = dimensionless pore volume injected

Wp = cumulative water produced
WOR = water oil ratio

δx = small length in the neighborhood of the outlet end of porous
medium
ρg = density of gas
ρo = density of oil
ρw = density of water
ρw = density of wetting phase
ρ nw = density of non-wetting phase
σ = interfacial tension
8-93
θ = contact angle
λ = pore size distribution index
μ = viscosity
μg = gas viscosity
μο = oil viscosity
μw = water viscosity
μw = wetting phase viscosity
μnw= non-wetting phase viscosity
φ = porosity, fraction
τ = tortuosity
γ = liquid specific gravity
ω = angular velocity of centrifuge
ΔP = pressure drop
ΔPw = pressure drop in the wetting phase
ΔPnw = pressure drop in the non-wetting phase
Γ = pore structure
REFERENCES AND SUGGESTED READINGS

Amaefule, J.O. and Handy, L.L. : “The Effect of Interfacial Tensions on
Relative Oil-Water Permeabilities of Consolidated Porous Media,”
SPE/DOE 9783, presented at the SPE/DOE 2nd Joint Symposium on
Enhanced Oil Recovery of the Society of Petroleum Engineers, Tulsa,
OK, April 5-8, 1981.
Anderson, W.G. : “Wettability Literature Survey - Part 5: The Effects of
Wettability on Relative Permeability,” J. Pet. Tech. (November 1987)
1453-1468.
Bardon, C. and Longeron, D. : “Influence of Low Interfacial Tensions on
Relative Permeability,” SPE 7609, presented at the 53rd Annual Fall
Technical Conference and Exhibition of the Society of Petroleum
Engineers, Houston, Tx, October 1-3, 1978.
Brooks, R.H. and Corey, A.T. : “Properties of Porous Media Affecting Fluid
Flow,” Jour. Irrigation and Drainage Div., Proc. Amer. Soc. Of Civil Engr.
(June, 1966) 61-88.
8-94
Brutsaert, W. : "Some Methods of Calculating Unsaturated Permeability,"
Trans. of the Amer. Soc. of Agricultural Engineers, Vol. 10 (1967) 400-
404.
Buckley, S.E. and Leverett, M.C. : “Mechanism of Fluid Displacement in
Sands,” J. Pet. Tech. (May 1941) 107-116.
Burdine, N.T. : “Relative Permeability Calculations From Pore Size
Distribution Data,” Trans., AIME (1953) 71-78.
Campbell, G.S. : "A Simple Method for Determining Unsaturated Conductivity
from Moisture Retention Data," Soil Sci., Vol. 117, No. 6 (1974) 311-
314.
Charbeneau, R.J. and Daniel, D.E. : "Contaminant Transport in Unsaturated
Flow," Chapter 15, in Handbook of Hydrology, D.R. Maidment (Ed.),
McGraw-Hill Inc., New York, 1993.
Collins, R.E. : Flow of Fluids Through Porous Materials, Research &
Engineering Consultants Inc., 1990.
Corey, A.T. : Mechanics of Heterogeneous Fluids in Porous Media, Water
Resources Publications, Fort Collins, Colorado, 1977.
Craig, F.F., Jr. : The Reservoir Engineering Aspects of Waterflooding, SPE
Monograph Vol. 3, Society of Petroleum Engineers, Richardson, Texas,
1971.
Daniel, D.E. : "Permeability Test for Unconsolidated Soil," Geotechnical
Testing Journal., Vol. 6, No. 2 (1983) 81-86.
Daniel, D.E., Trautwein, S.J., Boyton, S.S. and Foreman, D.E. : "Permeability
Testing with Flexible-Wall Permeabilities for Unconsolidated Soil,"
Geotechnical Testing Journal., Vol. 7, No. 3 (1984) 113-122.
Delshad, M. and Pope, G.A. : “Comparison of the Three-Phase Oil Relative
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CHAPTER 8

8.1 DEFINITION OF RELATIVE PERMEABILITY

In order to make quantitative predictions for multiphase flow, we need

It is often more convenient to work with a dimensionless effective

In terms of relative permeabilities, Eqs.(8.1) through (8.3) become

Sometimes, the effective permeability to the non-wetting phase at the

Figure 8.1 shows typical imbibition relative permeability curves for a

1. The relative permeability curves are nonlinear functions of fluid

2. The sum of the relative permeabilities at each saturation is always less

3. There is an irreducible wetting phase saturation (Swirr) at which the

relative permeability to the wetting phase is zero and the relative

4. There is a residual non-wetting phase saturation (Snwr) at which the

relative permeability to the non-wetting phase is zero and the relative

Figure 8.1. Typical imbibition relative permeability curves.

Imbibition relative permeability curves typically are used to perform the

• Waterflood calculations in a water wet reservoir in which water

Figure 8.2 shows typical drainage relative permeability curves with

Drainage relative permeability curves typically are used to perform the

• Solution gas drive calculations in which gas displaces oil.

• Gravity drainage calculations in which gas replaces drained oil.

• Gas drive calculations in which gas displaces oil and/or water.

• Oil or gas displacing water in tertiary recovery processes.

8.2 LABORATORY MEASUREMENT OF TWO-PHASE RELATIVE

Darcy's law applied to each phase gives

From capillary equilibrium,

where Pc(Sw) in this case is the imbibition capillary pressure. If the

μnwvnw L μnw qnw L

If the saturation distribution in the core is not uniform because of

should be sufficiently high to minimize capillary end effect as outlined in

state is achieved. Steady state is achieved when the injected and

changes with time.

relative permeabilities at higher and higher wetting phase saturations.

In the steady state relative permeability experiment, it is necessary to

phases for a gas-oil steady state relative permeability experiment conducted at

Capillary equilibrium gives

Assuming incompressible fluids, mass conservation requires that

The continuity equation for the wetting phase is

The saturation constraint gives

Subtracting Eq.(8.22) from (8.23) and rearranging gives

qw μ w qnw μnw ∂Pnw ∂Pw

Let an approximate fractional flow of the wetting phase be defined as

Let the dimensionless distance from the inlet end be defined as

Let the spontaneous imbibition capillary pressure curve be given in terms of

Eq.(8.35) can be written as

where N cap is given by

N cap is the same dimensionless number we encountered in the analysis of

of gravity to viscous forces in the displacement. The mobility ratio of the

The mobility ratio as defined in Eq(8.38) is a function of saturation and will be

non-wetting phases. The mobility ratio given in Eq.(8.39) is a characteristic

where the approximate fractional flow of the wetting phase is given by

In order to maximize the displacement efficiency of the non-wetting

1. The fractional flow of the wetting phase is a strong function of

2. The displacement behavior can be rate-sensitive if the effect of

3. Capillarity is detrimental to the displacement efficiency as it increases

4. Gravity is beneficial to the displacement efficiency for up-dip

5. The displacement efficiency can be increased by reducing the mobility

6. The effects of gravity and capillarity on the displacement can be reduced

Eq.(8.36) then becomes