Graphical Techniques for Determining Relative Permeability From Displacement Experiments

S. C. Jones, SPE-AIME, Marathon Oil Co.
W. 0. Roszelle; SPE-AIME, Marathon Oil Co.
Introduction
To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core
usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either
constant or variable) across the entire core and water
injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and
oil in each is measured. Augmented by the absolute
permeability and pore volume of the core and by oil and
water viscosities, these data are sufficient to develop
relative permeability curves.
. The average saturation in the core at any time in the
flood can be found from· an over-all material balance.
However, to calculate relative permeability, the saturation history at some point in the core must be determined,
not the average saturation history. The Welge1 equation
yields saturations at the effluent end of the core when the_
average saturation history is known.
Similarly, to compute relative permeability, the point
. pressure gradient per unit injection rate is needed, not the
average. The equation developed by Johnson et al. 2
converts average relative injectivity to a point value,
accomplishing the required task.
While the equations of Welge and Johnson et al. have
been used successfully for years, they require tedious
computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented
0149-2136/78/0005-6045$00.25
© 1978 Society of Petroleum Engineers of AI ME
in this study are equivalent to these equations, but are
easier to use and can give a more accurate evaluation of
relative permeability.
Lefebvre du Prey 3 - 5 has presented graphical constructions based on curves of volume of oil produced vs time
and pressure drop vs time to develop the required point
functions. These constructions are limited to constant
rate displacements. The constructions presented here are
general and apply to constant rate, constant pressure, or
variable rate-pressure displacements. Constant-rate and
constant-pressure examples are given to help clarify the
methods.
The graphical techniques make it easy to see that
double or triple saturation values, so extensively discussed in the past6 - 9 simply do not result from the fractional flow curve generated by a single displacement,
such as a waterflood or an.oilflood.
Theory
Ignoring gravity effects and capillary pressure, water
and oil relative permeabilities (expressed as functions of
saturation) are
·
and
kro = f.LJo 2 fA 2 -
1
.
(2)
· · · · · · · · ...............
To use these equations, the fractional flow of water or oil
and effective viscosity, A. - 1 , must be determined as functions of saturation. These must be point values, not aver-
This paper presents graphical constructions that simplify the calculation of relative
permeability from displacement data. These constructions convert raw data to relative
penneability in a less tedious, more accurate manner than the usual computations.
Fractional-flow saturation curves·derivedfrom waterflood displacements are always concave
downward and never yield multiple-value saturations.
MAY; 1978
807
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Graphical Techniques for Determining Relative
Permeability From Displacement Experiments
age ones. The most convenient location is 'the outlet end
of the core because fractional flow there is the same as the
produced oil or water cut.
Determining Saturation
Data from a constant-rate waterflood are presented in
Table 1. Injected water volumes are converted into pore
volumes, Qi. Given the starting water saturation, average
water saturation as a function of Qi is calculated from
Sw =Swi
+ Np/VP . ........... : ..........
(3)
Sw 2 =Sw- Qi dSw · .................... (4)
dQi
Careful inspection of the construction reveals that all
terms in Eq. 4 are represented. Several more tangents
may be drawn to generate a complete Sw 2 vs Qi curve
(Fig. 2).
Note that all tangents drawn to the average saturation
curve (Fig. 1) before water breakthrough intersect the
ordinate axis at the exact, beginning water saturation,
fw2 = 1 - fo2· · · · · · · · · · · · · · · · · · · · · · · · · · · (6)
Fig. 1 could be extended to include all data in Table 1
(to 10 PV injection) and the remaining point-saturation
and fractional-flow values could be obtained as above.
However, a plot of average saturation vs the reciprocal of
pore volume injected has two advantages (Fig. 3). First,
long tangent extensions are ayoided; long lever arms tend
to amplify errors. Second, the reciprocal plot provides a
convenient means to extrapolate to (1 - S0 r) at infinite
throughput (1/Qi = 0).
As before, tangents to the curve are extended to the
vertical axis. Intercept values (d{moted S w+) are no longer
equal to the outflow-end saturations. Instead, the latter
are given by
Sw2 = 2Sw - Sw +, . . . . . . . . . . · · . · · . · · · · · · · (7)
which is derived in Appendix A. Fig. 3 shows both the
average saturation and the outflow-face saturation
curves, computed from Eq. 7 using the tangent intercepts, and the extrapolation to infinite throughput. Here,
the average and point saturations become equal .
0.7
TANGENT TO CURVE
c.
>
.........
c.
z
J
l.l
fo2 = (Sw- Sw2)/Qi. · · · · · · · · · · · · · · · · · · · · · (5)
Also,
0.4
o.J o~--::o':-.1---='"o.2=----:Lo.J=---o~.4~----:o:-'-.5--o-'-.6--o"'-.7----'o.a
PV WATER INJECTED=W;/Vp=O;
Fig. 1-Construction for determining point saturation from
average saturation.
Effective Viscosity
Eqs. 1 and 2 require point values of effective viscosity as
a function of saturation for the calculation of water and oil
relative permeabilities. First, compute average effective
viscosity or reciprocal relative mobility of fluids in the
core. This is found by comparing ilp/q measured during
the waterflood with the single- ph~e flow value, ilpb/qb,
measured to find the absolute permeability of the core:
0.7
0.6
, EXTRAPOLATION TO { 1- 5 0 ,) AT INFINITE THRUPUT.
0.5
.68
0.4
.66
Sw2
0.3
.64
0.2
.62
0.1
.60 L------'------L..--....L....--..L..----,..-JL------1,--....J
0
0.2
0.6
0.8
1.0
1.2
1.4
0
0
~
~
~
~
M
M
Q;, P.V.
Fig. 2-Saturation at outlet end of core.
808
~
0.8
1/0i = Vp/Wi
Fig. 3-Construction for determining point saturation at large
throughputs, showing extrapolation to residual saturation.
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Average saturation is plotted in Fig. 1 for the early
portion of the example waterflood. Note that a line
tangent to the curve at 0. 30 PV extends to intersect the
ordinate axis at a saturation of 0. 534. This intercept value
is the saturation at the outlet end of the core when 0. 30 PV
water has been injected. The tangent construction is
equivalent to the Welge equation, d_erived in Appendix A
as
S wi; that is, the saturation at the outlet end of the core is
unchanged during a waterflood until water breakthrough.
Also, the water saturation at this location is always less
than the average saturation until all oil production terminates, at which time the slope of the average saturation
curve becomes zero and the intercept has the same value
as the average saturation.
Because fractional flow of oil is equal to the slope of
the tangent line in Fig. 1 at any given injection, it is
calculated conveniently from the average and intercept
saturations:
TA~LE 1-DATA FOR CALCUl..ATION OF OIL/WATER RELATIVE
PERMEABILITIES FROM A CONSTANT-RATE WATERFLOOD DISPLACEMENT
wi
Np
(ml)
(psi)
Qi
=W/Vp
0.00
3.11
7.00
7.84
8.43
8.93
9.30
9.65
9.96
10.11
10.30
138.6
120.4
97.5
91.9
87.9
83.7
78.5
74.2
70.0
68.1
65.4
0.000
0.100
0.225
0.360
0.523
0.780
1.260
2.001
3.500
5.000
10.000
(ml)
0.00
3.11
i.OO*
11.20
16.28
24.27
39.2
62.3
108.9
155.6
311.3
-ap
A.-1
=apf.Lbqb!(apbq)
Sw
. =Sw 1+Np/Vp
(cp)
0.350
0.450
0.575
0.602
0.621
0.637
0.649
0.660
0.670
0.675
0.681
13.50
11.73
9.50
8.95
8.56
8.15
7.65
7.23
6.82
6.63
6.37
/LI, = 1'-w = 0.970 cp
/1-o = 10.45 cp
su•i
= 0.350
fj,p 1,/q1, = 0,1245 psi/(ml/hr) ... atS,. = 1.000
k = 35.4 md
<P = 0.215
q = 80.0 ml/hr.
A_, = IL•
fl: j 1e:.·j . ............
~
(8)
where /J-b is the viscosity of the fluid used to find absolute
permeability.
Average effective viscosities for the example calculation are listed in Table 1 and plotted in Fig. 4. The
graphical construction for finding point effective viscosity at the outlet end of a core is identical to that for point
saturation. A tangent to the average viscosity curve is
extended to the axis; the intercept is the point value. Fig.
4 shows tangents drawn at cumulative injections of
0.225, 0.300, and 0.600 PV.
The equation that corresponds to the tangent construction is
A2 -1
= A-1 -
Qi dA. -1
dQi
.
. .............•...
(9)
Eq. 9 is derived in Appendix B. Outflow-end effective
viscosities (intercept values) are plotted in Fig. 5.
Comparison of Fig. 2 with Fig. 5 shows that effective
viscosity at the outflow end of the core changes at the
same throughput that saturation changes - that is, ef-
fective viscosity does not change at this location until
water breakthrough. To achieve an unchanging effective
viscosity before water breakthrough (considering the
tangent-intercept construction), the average effective
viscosity curve (Fig. 4) must be linear from the start of the
flood until water breakthrough. This requirement is useful in evaluating .experimental data - a plot of A.plq vs
volume injected must be a straight line until breakthrough. Nonlinearity can result from at least two conditions: (1) the initial saturation throughout the core is
nonuniform ~ insufficient fluid was injected in the pre~
vious flood to achieve uniform s~Huration and (2) the core
is heterogeneous.
In our experience, the first condition is the most frequent cause of nonlinearity and can be che~ked easily. If
oil injection in a completely water-saturated core results
in a linear Ap/q before oil breakthrough, but a subsequent
waterflood does not yield a linear plot before water breakthrough, then the oil saturation just before the waterflood
probably was not uniform. However, if the initial oil
·injection results in a nonlinear curve, then Condition 2
must be suspected. Injection of non-Newtonian polymer
14,---.----.--~~--~--~--~----~--~
13
12
"'1-t.
:l..
11
12
#<J
~,,.
).,-;, cp
11
10
8~--~~~----~--~--~--~~--~--~
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Q;
Fig. 4-Construction for determining outlet-end effective
viscosity from average values.
MAY, 1978
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Q;, P.V.
Fig. 5-Effective viscosity at outlet end of core.
809
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*Point of water breakthrough.
V" = 31.13 ml; L =5.002 in.; 0 = 1.500 in.
TABLE 2-CALCULATED VALUES FOR OIL/WATER RELATIVE PERMEABILITIES
Pore Volume
Injected
0;
0.000
0.100
0.225
0.300
0.600
1.000
2.500
5.000
iO.OQO
00
Av~age
__1_
. 0;
10.000
4.444
3.333
1.667
1.000
0.400
0.200
0.100
0.000
Sw
0.350
0.450
0.575
0.5~3
0.627
0.643
0.664
0.675
0.681
0.687
Intercept
Sw2
0.350
0.350
0.511
0.534
0.580
0.617
0.646
0.664
0.676
0.687
Average
A.-1
(cp)
13.50
11.73
9.50
9.16
8.42
7.88 .
7.05
6.63
6.37
6.07
Intercept
A.2 - 1
(cp)
13.50
13.50
10.66
10.30
9.45
8.95
7.79
7.11
6.69
6.07
A2 - 1 =2A- 1
-
A-1+ ............... : ..... (10)
Average and point values as well as the extrapolation to
infinite throughput are shown in Fig. 6. Fig. 6 is totally
analogous to the saturation treatment in Fig.
3:
Calculating Relative Permeability
Table 2 lists average and intercept saturations and effective viscosities (taken from Figs. 1 through 3 and 4
through 6, respectively) and oil and water fractional flow
values (calculated from ~qs. 5 and 6). Each line represents a particular cumulative water injection, Qi. Water
and oil relative permeabilities then can be calculated from
Eqs. 1 and 2, respectively, using the water and oil vis.
cosities given in Table 1.
The first line of Table 2 corresponds to the start of the
waterflood and the second line to 0.100 PV injection
before water breakthrough. Note that effluent-end saturations, effective viscosities, fractional flows, and relative
1.000
1.000
d.291
0.197
0.078
0.026
0.007
0.002
o~oo1
0.000
k,,..
'"'2
=1-fo2
0.000
0.000
0.709
0.803
0.922
0.974
0.993
0.998
0.999
1.000
k,.
=fwJLwiA.2 -1
=foJL.fA2 - 1
0.000
0.000
0.065
0.076
0.095
0.106
0.124
0.136
0.145
0.160
0.774
0.774
0.285
0.200
0.087
0.030
0.010
0.003
0.002
0.000
permeabilities are identical in these two lines~ In fact,
these values would be identical for all cumulative injections until water breakthrough; For this reason, it is
impossible to obtain relative permeabilities for saturations between the initial and breakthrough values using
the unsteady-state method.
. Fig. 7 shows oil and water relative permeabilities
plotted as functions of water saturation (Sw 2 in Table 2).
Because all the variables ate functions of cumulative
irijectioil; relative permeability is also a function of
cumulative injection. Thus, the curves are directional,
starting with the lowest water saturation (0. 35) and proceeding to the right. The next higher water-saturation
point shown, 0.511, corresponds to water breakthrough
at 0.225 PV injection. The unobtainable relativepermeability curves between this saturation and the initial, or irreducible, water saturations are drawn as dashed
iines. Moving to the right, the points represent increasing
throughput to infinite injection at the last pair of extrapolated points (Sw = 0.687).
·
Constant Pressure Example
Data for a constant pressure-drop. (- D..p = 100 psi) displacement are shown in Table 3. Core and fluid properties equal those in Table 1. Cumulative water injection,
assumed equal to the sum of cumulative water and oil
production, and cumulative oil production are plotted vs
time (Fig. 8).
.\
\
kro
\
X)
cp
\
\
\
\
\
\
\
0.6
0.8
1.4
1/Qi = Vp/Wi
Fig. &-Construction for determining point effective viscosity at
large throughputs.
810
o~~--~--~~~--~--~~~~~~--~
0
0.1
0.2
0.5
1.0
Sw
Fig. 7-Relative permeability curves from example calculation.
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solutions also can cause nonlinearity. Prebreakthrough
linearity is necessary for the valid application of the
Welge and Johnson et al. eqmitions or the corresponding
graphical constructions.
Large-throughput, average effective-viscosity data are
treated similarly to large-throughput, average saturation
data; A- l is plotted vs 1/Qi as in Fig. 6. Tangents are
drawn and outlet-end values are calculated from the intercepts, A-1+, by
'··
=(Sw-Sw~IO;
TABLE 3-DATA FROM CONSTANT-PRESSURE WATERFLOOD DISPLACEMENT
~~.-1
q
Elapsed
Time
(minutes)
W;
Np
(ml)
(ml)
0.00
3.00
6.20*
9.00
12.00
15.00
20.00
26.00
60.00
100.00
150.00
0.00
3.09
7.00
10.90
15.28
19.89
27.9
37.8
99.5
176.8
276;9
0.00
3.09
7.00
'7.80
. 8.33
8.70
9.01
9.32
9.90
10.09
10.31
(from slope)
(ml/hr)
57.7**
66.4
8.2.0
86.4
90.2
93.4
97.5
98.6
113.0
118.7
121.5
0;
=W;IVP
0.000
0.099
0.225
0.350
0.491
0.639'
0.896
1.214
3.196
5.679
8.895
Sw
=Sw;+NP/VP
0.350
0.449
0.575
0.601
0.618
0.629
0.639
0.649
0.668
0.674
0.681
= t.p p.bqbl (t.pbqJ
(cp)
13.50
11.73
9.50
9.02
8.64
8.34
7.99
7.90
6.89
6.56
6.41
Core and fluid properties are the same as those shown in Table 1.
S,.; = 0.350
-!lp = 100.0 psi
q · = variable.
Instantaneous injection rates must be determined to
calculate average effective viscosities. These are found
by drawing lines tangent to the cumulative injection vs
time curve (Fig. 8). The slope of the curve at any point is
equal to the instantaneous rate.
This rate often changes most rapidly between the start
of water injection and water breakthrough. Because of
the short time span involved, accurate measurements are
difficult to obtain, especially at the point of water breakthrough. The requirement of a linear - D..p/q vs Wi relationship until water breakthrough permits an alternative
calculation of rate at breakthrough,
qBT = l/(2fltBrfWiBT - 1f%), ............. (11)
hand side of Eq. 14 to Eq. 9. However, it is important
to recognize. a subtle difference. Water and oil perineabilities presented here are relative to the core's absolute permeability to water. Johnson's permeabilities are
relative to the prewaterflood condition - that· is, oil
flowing at irreducible water saturation. Thus; to convert
our relative permeabilities to Johnson's basis, we must
divide these by kroCSwi) or 0. 774. We prefer the absoiute
basis and recommend its use, ·but the choice is arbitrary.
Fractional Flow Curves
Fig. 9 shows the satunltion/fractiomil-flow relationship
for the example waterflood. The portion of the curve
where D..tBr and WiBr are cumulative time and injected
volume, respectively, from the start of water injection
until water breakthrough. The initial rate, q 0 , is the same
as the steady-state rate (at the same D..p) obtained at the
end of oil saturation and just before water injection.
Once the instantaneous rates have been determined,
values of A- 1 can be calculated fromEq. 8. The values of
Qi, Sw, and A- 1 are plotted exactly as in the constant rate
example, and all subsequent constructions and calculations are identical to this example.
Equivalence of Graphical Technique and Johnson
Equation
Eq. 2 shows that
{:: = ~:
1
..........................
(12)
But, Johnson et al. 2 have shown that
d
(_1___)
fo2 = -~Q+-l_,__r~
kro
d
(Ji)
, , , , , , , , , , , , , , .... ,· .. (13)
where/r is defined as a relative injectivity (see Appendix
B). Therefore, by comparing Eq. 12 with Eq. 13,
................. (14)
TIME, minutes
Appendix B demonstrates the equivalence of the rightMAY, 1978
Fig. a-Cumulative water injected and cumulative oil produced
vs time for constant pressure example.
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*Water breakthrough.
**This rate is calculated from oil injection data at the end of oil saturation.
Sw2 AT 0.5 PV
INJECTED
OL-~----~----~~~----~----~----~~----~----~
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0,9
Sw
Fig. g___;Fractional flow curve for example waterflood.
812
1.0
increases in saturation with continued injection. The initial straight-line portion before breakthrough is followed
by a curve of positive, but continually decreasing slope
with continued injection (because the slope is equal to
1/Qi). In contrast, most fractional flow curves presented
in the waterflooding literature 10 have an "S" shape, or
even are concave upward.
Fractional flow curves that are not concave downward
can be computed from kro and krw data. If relative permeability curves generated from an unsteady-state displacement (Fig. 7) are completed by interpolation between the connate water and breakthrough saturations
(the dashed portions of Fig. 7); then a family of fractional
flow curves can be calculated from
where the viscosity ratio, Mwl p.,0 , is a parameter. The
shape of a computed curve depends on the value of the
viscosity ratio compared with the ratio actually observed
in the experiment to generate the kro and krw data. If the
viscosity ratio used iri Eq. 15 is the same as the experimental ratio and if the relative permeability curves are
interpolated correctly, then the calculated fractional flow
curve will have the shape of the one in Fig. 9 - a
straight-line, prebreakthrough portion followed by a
curve with a posit~ve, but gradually decreasing slope.
This curve is corlcave downward; however, higherthan-experimental viscosity ratios will produce curves
that have an ''S '' shape or that are everywhere concave
upward. On the other hand, lower-than-experimental
.ratios yield fractional flow curves that are concave
downward, brlt that have no prebreakthrough straight-·
line portion.
Steady-state relative permeability experiments also
can yield fractional flow curves that are not concave
downward. In addition to computing fractional flow
curves fromEq. 15 and using steady-state-derivedkro and
ktw curves and higher-than-experimental viscosity ratios,
another possibility exists. A fraCtional flq~ curve derived
from steady-state data does not represetH a single· displacement. Instead, it is the locus of end points from
several displacements, in each of which a different
water.,.oil ratio is injected. Indeed, the actual fractional
flow curve for each displacement is concave downward.
Fig. 10 illustrates a hypothetical, four-point, steady-state
fractional flow curve that really is comprised of four
separate displacements. Thefwi values listed are injected
water fractions.
· The important point is that any fractional flow vs
saturation curve that represents a single experimental
displacement always will be concave downward when
the displacing phase is plotted- fw 2 vs Sw 2 for a waterflood orj02 vs S02 for an oilflood. This is true regardless of
the wettability of the core or the water/oil viscosity ratio
used in the displacement. Other shapes of fractional flow
curves are obtained only from composite experiments or
by. calculation, using relative permeability data and a
water/oil viscosity ratio that is higher than the experimental ratio ..
A saturation profile can be constructed frorri the derivative of the fractional flow curve. The derivative of
'the curve shown in Fig. 9, dfwldSw, is.plotted vs Sw in
Fig. 11. A horizontal line corresponding to the initial
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between the initial water saturation (0.35) and water
breakthrough is a straight-line segment with a slope equal
to the reciprocal of pore volumes injected at water breakthrough. The straight line is a consequence -of rotating
successive tangent lines (Fig. 1) around the discontinuity
at water breakthrough from the initial slope ( = 1. 0)
before water breakthrough to tangency just after breakthrough. Because the intercept of each tangentline is Sw 2
and its slope is ( 1 .;_ f w 2 ), saturation and fractional flow of
water are related linearly when these liries originate from
a single point. If satuation changes were continuous at
water breakthrough, caused by smearing of the waterbank front by capillary presstire forces, then lines that are
truly tangent to the average saturation curve at water
breakthrough could be drawn. Points generated from
these tangents also would form a virtually straight-line
segment of the fractional flow curve. The main point is
that a fractional flow curve that describes a waterflood or
oilflood is essentially a straight line from the initial saturation and fractional flow to the point that describes
saturation and fractional flow of the displacing phase
immediately after breakthrough.
As Welge1 demonstrated, the slope of the fractional
flow curve, dfwldS w, at any point is equal to the reciprocal
of pore volumes of water injected. Furthermore, a line
segment that is drawn tangent to the curve and extended
tofw = l.Owill temiinate at the average water saturation.
TQ illustrate, a line (slope = 2.0) is drawn tangent to
the fractional flow curve (Fig. 9) with a slope indicating a
cumulative injection of 1/2.0 = 0. 5 PV. The extension of
this line to fw = 1.0 shows that the average water saturation in the coreat 0.5 PV injection is 0.616. The saturation and fractional flow of water at the outflow end of the
core (found from the point of tangency) are 0.573 and
0.910, respectively. Thus, Qi, fw 2 , Sw2• and Sw can be
determined from the tangent construction.
Some confusion regarding the shape of fractional flow
curves exists. We have implied that any fractional flow
curve that describes an actual displacement is .concave
downward when the plotted phase (for example, water)
water saturation, Swi' has been added to the right side of
the plot. Fig. 11 is a generalized saturation profile 11 from
which one can compute the saturation profile in the core
at any stage in the waterflood. For example, suppose we
wish to find the profile at 0. 2 PV water injected. The
outflow end of the core (x = 1.0) corresponds to an
abscissa value of 5. 0 .(x/qi = 1. 0/0. 2); the midpoint
corresponds to 2.5, and so forth. Note that the water
bank has not reached the outlet· end of the core at
0.2 PV injection. At 0.5 PV injection, the outlet-end
saturation occurs at 1.0/0.5 = 2.0. The midpoirtt of the
core corresponds to an abscissa value of 1. 0, and so forth.
Construction of a saturation profile from the derivative
of an S-shaped, fractional flow curve can lead to errors. A
notable example is the triple value of saturation front that
has been discussed for decades. The multiple-value saturation problem has been circumvented by use of the well
known secant-tangent construction. 1 Here, the secant is
drawn from initial conditions ifw = 0 andSwi) to the point
of tangency on the composite fractional flow curve and
become.s the straight.;. line portion of the fractional flow
curve observed for a single displacement.
Hysteresis
DISPLACEMENT
END POINT
DISPLACEMENT 3
fwi = 0.75
I
I
I
I
I
DISPLACEMENT 2
fwi = 0.50
I
I
I
I
The imbibition relative-permeability curves obtained
from a waterflood (Fig. 7) are reproduced in Fig. 12. Two
more curves have been added that result from an oilflood
following the waterflood. These are drainage curves for a
water-wet core. Relative permeability to water is considerably reduced in the drainage cycle, but permeability to
oil, is relatively unchanged from imbibition values. This
behavior is typical of our observations of water-wet
cores. Like capillary pressure, relative permeability exhibits hysteresis and depends on history. The wide divergence among krw curves makes it mandatory to choose
the correct one for predicting flooding behavior, particularly for tertiary recovery calculations.
The waterflood and oilflood fractional flow curves
(Fig. 13) also form a hysteresis loop. The oilflood part of
I
I
DISPLACEMENT 1
fwi = 0.25
0~--~--~~----~--~----~~--~--~
0.2
0.3
0.4
0.6
0.5
0.7
0.8
0.9
Sw
Fig.10-Composite steady-state fractional flow curve.
OL_~~~--~-=~====~~~~--L_
0
0.3
0.4
0.6
0.5
0.7
_____J
0.8
Sw
Fig. 11-Generalized saturation profile for waterflood example.
MAY, 1978
Fig. 12-Hysteresis in waterflood vs oilflood relative permeability
curves.
813
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DISPLACEMENT 4
fwi = 1.00
Conclusions
1. Unsteady-state, displacement relative permeability data can be calculated easily and accurately using
graphical constructions that are equivalent to the standard
equations. Permeabilities calculated from the Johnson
et al. equation are relative to the effective permeability
at prewaterflood saturation. Permeabilities calculated by
the techniques presented in this study are relative to
absolute water permeability.
2. True end-point saturation, Sor• and effective viscosity can be estimated by graphical techniques presented
here.
3. A fractional flow curve for a single displacement is
always concave downward when the displacing phase is
plotted.
4. Relative permeability and fractional flow curves
exhibit distinct hysteresis for an oilflood following a
waterflood.
5. Multiple- value saturation profiles never result
from a fractional flow curve developed from a single
displacement.
6. Because krw and kro are history dependent, water-
flood performance should be predicted more accurately
by unsteady-state, waterflood-derived data than by
steady-state data, provided that capillary end effects are
negligible.
Nomenclature
A = area normal to the axis of a linear core
f = fractional flow of water or oil
Ir =relative injectivity
k = absolute permeability
kr = relative permeability
.t = length of a linear core from the injection face to
an arbitrary point within the core
L = total length of a linear core
NP = volume of oil produced
p =pressure
q = volumetric injection rate
Qi =pore volumes injected relative to the entire core
S = saturation
VP =pore volume of an entire core
wi =volume of water injected
x = fraction of the total length of a core
z =pore volumes injected with respect to an interior
location in a core
d = denotes a difference between two quantities
A- l =effective viscosity or reciprocal relative
mobility
J.L = viscosity
Subscripts
b =refers to single-phase flow or fluid in a core on
which the absolute permeability is based
i = initial (prewaterflood) or irreducible
~ =refers to an interior measurement in a linear core
L =refers to an over-all measurement in a linear
core
o =oil
r = refers to residual saturation
w =water
x =refers to an interior location in a core
2 =refers to outlet end location
Superscript
+ = intercept value
Acknowledgment
The authors thank H. J. Ramey, Jr., and W. E. Brigham
for their encouragement to publish this paper.
References
ru
~
0~
OA
0~
Q6
OJ
~
Sw
Fig. 13-Hysteresis in fractional flow curves.
814
M
~
1. Welge, Henry J.: ''A Simplified Method for Computing Oil Recovery by Gas or Water Drive, "Trans., AIME(1952) 195,91-98.
2. Johnson, E. F., Bossler, D.P., and Naumann, V. 0.: "Calculation of Relative Permeability from Displacement Experiments,"
Trans., AIME (1959) 216, 370-372.
3. Buckley, S. E. and Leverett, M. C.: "Mechanism of Fluid Displacement in Sands," Trans., AIME (1942) 146, 107-116.
4. Lefebvre du Prey, E.: ''Deplacement de I 'huile par I 'eau dans un
milieu consolide," Report Ref. 11.456 lnstitut Franc;ais du
Petrole, Paris (Jan. 1965).
5. Lefebvre du Prey, E.: ''Mesure des Permeabilities relatives par Ia
methode de W~lge~" ~eport Ref. 15.120 Institut Franc;ais du
Petrole, Paris (Oct. 1967).
6. ''Mesure des Permeabilities relatives par Ia methode de Welge
(enquete)," Groupe Petrophysique de Ia sous~Commission
Laboratoires d'Exploitation du Comire des Techniciens, Chambre
Syndicale de Ia Recherche et de Ia Production de Petrole et du Gaz
Nature!, Paris (Sept.-Oct. 1973) 28, No.5, 695.
JOURNAL OF PETROLEUM TECHNOLOGY
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the loop is concave upward because the displaced phase,
not the displacing phase, is plotted.
History dependence raises the question of uniqueness.
If one chooses to generate steady-state relative permeabilities using eight steps instead of four, would the
same krolkrw curves be generated? Because two sets of
relative permeability curves are formed when the direction of saturation change is reversed, might not a family
of curves be generated that depends on the saturation
history of the core? These questions can be answered only
with careful, accurate experimental measurements.
Because relative permeability and fractional flow
curves depend on history, we believe that waterflood
performance is described more accurately by unsteadystate, waterflood-derived, relative permeability curves
than by the composite curves obtained from simultaneous
injection of water and oil.
7. Cardwell, W. T., Jr.: ''The Meaning of the Triple Value in NonCapillary Buckley-Leverett Theory," Trans., AIME (195.9) 216,
271-276.
8. Fayers, F. J. and Sheldon, J. W.: "The Effect of Capillary Pressure and Gravity on Two-Phase Fluid Flow in a Porous Medium,"
Trans., AIME (1959) 216, 147-155.
9. Hovanessian, S. A. and Fayers, F. J.: "Linear Water Flood With
Gravity and Capillary Effects," Soc. Pet. Eng. J. (March 1961)
32~36; Trans., AIME, 222.
10. Craig, F. F., Jr.: The Reservoir Engineering Aspects of Waterflooding, Monograph Series, Society of Petroleum Engineers of
AIME, Dallas (1971).
11. Parsons, R. W. and Jones, S. C.: "Linear Scaling in Slug-Type
Processes- Application to Micellar Flooding," Soc. Pet. Eng. J.
(Feb. 1977) 11-26.
Therefore, from the definition of a derivative,
So[Qi,x] =So[Qi,x]
+xaSa~~i,x]
.
.............................. (A-3)
If z is defined as Q1/x, then from the chain rule we get
aS::[Qi,X] - a"S:"[Qi,X] . ~ , ........... (A-4)
ax.
az
ax
but, for single-fluid injection:
a"S:"[Qi,x J= d5;;[Qi,x] , ................. (A-5) ·
az
d(Qi/x)
and,
~ = -Q/x2 •
Calculating Point Saturations in a Linear
Core From Production Data
ax
is odx =
lim
~~o~
x+.U
lim
~~o
(A-6)
Therefore,
For a given porous medium and fluid system, a displacement resulting from the injection of a single fluid is said
to be linearly scalable if saturation at any point is a
function of only the number of pore volumes of fluid
injected with respect to that point. 11 The pore volumes of
fluid injected in a core segment can be increased in two
equivalent ways. First, more fluid can be injected. Second, the volume of rock on which the injected _PV of fluid
- is calculated can be decreased. Regardless of basis size,
however, the average saturation (of oil, for example)
remaining in the rock and the shape of the saturation
profile depend on only the pore volumes of fluid, Q/x,
injected in the rock, or portion of the rock, under
consideration.
For a linearly scalable displacement, the saturation at
any point in a linear core can be obtained from a material
balance around a small segment of the core. All that is
required is a plot of average saturation as a function of the
pore volumes of fluid injected. Although data obtained for
this plot /are the average saturation over the entire core
and the pore volumes injected based on the entire core,
these may be plotted as if applied to only an arbitrary
segment of the core (from 0 to x), based on the above
arguments (see Fig. 14). The shaded area in the ~x
increment is the difference in shaded areas from 0 to x +
~ and from 0 to x. These areas also can be expressed as
an average saturation times the appropriate distance. As
~approaches zero, the average saturation over the increment approaches the point saturation at x. These considerations lead to the following equations.
x+.U
• •••••••••••••••••••
So[Q i•x J = -so[Q·v x J - Qi Ji-SJQi,x
I J . . . . . (A-7)
----,;c...
x
.
..
d(Qi x)·
At the outflow face, x = 1, and Eq :-A-7 becomes
d~r Qi T
(A-8)
- -so [Qi J - nSo2 [Q
. i Jlrt.i --dQi.- · · · · · · · · ·
Because Sw = 1 - S0 , Eq. A-8 can be written as
Sw2[Qi] =Sw[Qi]
-Qid~~fi],_ ........ (A-9)
which is Eq. 4 in the text.
_
Large throughput data, plotted as Sw vs x!Qi, are handled slightly differently. Starting with Eq. A-3, this time
written in terms of water instead of oil saturation:
x
x+Ax
x
Lsodx -
[ Sodx
~
· {from 0 to x)
~
lim
(x+~)Sa(Qi,x+~)-xSa(Qi,x)
dx~o
~x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)
AVERAGE SATURATION FROM 0 to
x +Ax, AT A VOLUME, W;, OF
WATER INJECTED.
AVERAGE OIL SATURATION FROM 0 to x,
AT THE SAME W; .
and
S0 (Qi,x) =
lim
~~o
[s(Q·
+~) +
o vx
xe5:"[Qi,x+~]~so[Qi,x])]
Llx
PV INJECTED = Wi /( Vpx)
·
.............................. (A-2).
MAY, 1978
Fig.14-Relationship between the saturation profile at a given
instant in the injection history and the average saturation vs
injection history.
815
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APPENDIX A
Application of the chain rule results in
S [x}Q.] =S[xiQ·]
w
w
t
A-1 =
+~ dS:[x/Qi]
Qi
l
d(x/Qi)
kA = qb/1-t/..- , . : ...................... (B-2)
-dpb
where - tJ.pb is the pressure drop across the entire core
which, at the outlet face, becomes
[VQ-] =S[l/Q.] +-1 dS:[ +Qi] :
t
w
t
Qi d(l/Qi)
w2
............................. (A-12)
during the single-phase flow of a fluid of viscosity /1-b at
rate qb (for example, during an absolute permeability
determination); also, since
But from the tangent construction (see Fig. 3),
s
w
(B-1)
But,
· · · · · ........................ (A-ll)
s
k:- (- ~~) ....................
R=xL, ............ ·................. (B-3)
then
+[liQ·] =S[+Q·] __
1 dS:[liQi] .
l
w
l
Qi d(l/Qi)
Substituting Eq. A-13 into Eq. A-12 yields
Sw2[11Qi] =2Sw[11Qi] -sw+[l!Qi], ..... (A-14)
Similarly (referring to Fig. 15), the average effective
viscosity from 0 to x is
which is Eq. 7 in the text.
Ax -
APPENDIXB
Derivation of Point Effective Viscosity From
Over-All Pressure/Rate Measurements in a
Linear Core
(~;:) ~(:) ............. ~ ... (B-4)
1
= e:~;:
y~x
. .. ................ (B-5)
Point effective viscosity now is found in a similar
manner to point saturation, using linear scaling principles:
x+Ax
A [Q,,x] = lim LA -tdx
By definition, the effective viscosity (reciprocal relative
mobility) at any point in a linear core is
-1
tJ.x
tJ.x~o
x+Ax
lim
tJ.x~o
x
lim
LA - dx -faA - dx
ax~o
tJ.x
1
1
(x+fJ.x~ [Qi,x+fJ.x] -x0[Qi,x]
11x ·
........................... , .. (B-6)
or
A-1[Qi,x J =A -1[Qi,x] +x
aFtJ_~hx J .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-7)
PV INJECTED WITH RESPECT TO ENTIRE CORE=Wi/Vp= Qi
a.
Because A- 1 is a function of Q/x (Fig. 15), we must apply the chain rule to evaluate the partial derivatives in
Eq. B-7. Ifz = Q/x, thenforsingle-fluidinjection
u
a8[Qi,x] = aA - 1
ax
-----az-
~,..ll
:ta.
X\._ , EVAL~ATED
AT x
'-<
PV INJECTED WITH RESPECT TO SEGMENT OF CORE FROM
wQ·
0 to i = Vp(t/L) =
-f
0
0
-t.pL
-t.p
1 or -t.p.
-t.px+ll.x
.............................
i 1 +t:.L
I
r--t.p ll.x
! i
~
(B-9)
At the outflow face (x = 1), the effective viscosity is
I
I I
~
................ (B-8)
Substituting Eq. B-8 into Eq. B-7, the effective viscosity
at any point in the core is
1
A-1[Q· J =A-1[Q· J _Qi dA- [Qi,x] .
uX
uX
X
d(Qi/x)
AND THE SAME W;
~)(
1-
A,-'[Q,] = A- 1 [Q,]- Q,~~;,] ,
.
I
I
x
x + t.x
Fig. 15-lllustrating the basis for calculating point-pressure
gradients from over-all pressure measurements in a linear core.
816
az = d8[Qi,x]
ax
dz
(~~i)
1.::.
'
•
............................. (B-10)
which is Eq. 9.
FromEq. 2,
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1
A- =
............................. (A-13)
fo2 = A2 - t . . . . . . . . . . . . . . . . . . (B-11)
kro
JLo
But Johnson et al. 2 have shown that
Substituting Eq. B-16 into Eq. B-14, we obtain
JLo
d .
d
............ (B-12)
tQ~r) -_----:-.----.--,------.
p.,(flp/q), d (~J
(Ji)
JLb(Ap/q)i d
(Ji)
............................. (B-17)
But it is apparent from Eqs. B-11 and B-15 that
where the relative irtjectivity, Ir, is
lr
=
(lp) I tlPji' ...... (B-l3)
_k_ =
kro
!o2
kro [swt.]
kro 'kro [Swi ]
=
=
JLJ'o2
A2 - 1
"·o
r
d
d
To demonstrate the equivalence of the two derivations,
we must show that
d ( 1 )
--'--=Q-7--/,__,r:. .:.___· . .. .. .. .. . (B-14)
d(Ji)
e:) l (a:),' ....
(~)
(Ji) -kro [s ·]
Wt
dQi
dQi
d(l!Qi)
............................. (B-20)
d~/Qi) = _ Q.2 d~/Qi)
·d(l/Qi)
dQi
l
' ........ .
(B-21)
- ----=1 - Qi dA-1
-A
- , .......... (B-22)
dQi
.
which is equaJ to our A2 -
1•
Therefore,
(B- I 5)
and the two derivations are equiyalent.
Therefore,
Ir = J.Lb(Aplq)i . . ............ (B-16)
A- 1(Ap/q)b
MAY, 1978
1
But,
.............................. (B-11a)
= !Lb
'
JLb(Ap/q)i
because at irreducible water saturatioh,/02 = 1 and A2 is equal to the average value,· A- 1 .
Therefore, Eq. B-17 becomes
d(l!Qi)
A-•
= JLo(Ap/q)b
............................. (B-18)
d~/Qi) - d~/Qi)
By definition,
1-Lo
A- 1
JPT
Original manuscript received in Society of Petroleum Engineers office July ~Q, 1976.
Paper accepted for publication June 18, 1977. Revised manuscript received Dec. 20,
1977. Paper (SPE 6045) was presented at the SPE-AIME 51st Annual Fall Technical
Conference and Exhibition, held in New Orleans, Oct. 3-6, 1976.
This paper will be included in the 1978 Transactions volume.
817
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where (q/ Ap )i is the injectivity just before the start of
waterflood when oil flows at irreducible water saturation.
Because two different bases were used in Eqs. B-11 and
B-12, that is, our permeabilities are relative to the absolute brine permeability and Johnson's permeabilities are
relative to the prewaterflood effective permeability, the
left side of the equation can be written as
kro [Swi]