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Analysis and Interpretation of Water-Oil Ratio in Waterﬂoods
Article
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SPE Journal · December 1999
DOI: 10.2118/59477-PA
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Analysis and Interpretation of Water/OilRatio in Waterﬂoods
Y.C. Yortsos,
SPE,
Youngmin Choi,
*
and
Zhengming Yang,
**
SPE, U. of Southern Californiaand
P.C. Shah,
†
SPE, Dowell
Summary
An important problem in water control is the identiﬁcation of thedominant reservoir or production mechanisms. Recently, Chan
Chan, K.S.: ‘‘Water Control Diagnostic Plots,’’ paper SPE 30775presented at the 1995 SPE Annual Technical Conference and Ex-hibition, Dallas, 22–25 October
postulated that a log-log plot of the water/oil ratio
WOR
produced vs. production time may beused to diagnose these factors. In this paper we provide analyticaland numerical results for a variety of waterﬂood conditions toexplore this possibility.We show analytically that the late-time slope of the log-log plotcan be related to the well pattern and the relative permeabilitycharacteristics, speciﬁcally the power-law exponent in the
k
ro
S
w
relationship or the reservoir heterogeneity. Analytical re-sults are provided for the behavior immediately following waterbreakthrough. In certain simple cases
mobility ratio equal to 1,layered systems
, analytical type curves can be derived. In thegeneral case, we use numerical simulation to provide numericaltype curves.The results are summarized in terms of the various power-lawscalings in different time regimes and by a catalog of numericaltype curves. They conﬁrm the potential of WOR-time plots asdiagnostic tools for reservoir analysis and characterization.
Introduction
An important problem in water control is the identiﬁcation of dominant reservoir or production factors
for example, channelingor water coning
from produced water/oil data. Water productioncan be the result of a natural waterdrive, edge or bottom water-drive or a waterﬂood. Despite its relevance, however, the problemhas received rather scant attention in the literature so far. Aspectsof waterﬂooding under various conditions have been extensivelydescribed in the literature
see, e.g., the SPE monographs of Craig
1
and Willhite
2
. Limited, however, are studies on the depen-dence of the water/oil ratio
WOR
on production time. Based onthe solution of the one-dimensional
1D
Buckley–Leverett equa-tion and under a certain assumption on the functional relation of the relative permeability ratio, Ershaghi and Omoregie
3
see alsoErshaghi and Abdassah
4
proposed the so-called X plot to inter-pret and extrapolate water/oil production. In this approach, a com-bination of the fractional ﬂow function was found to vary linearlywith the cumulative oil production in an appropriate semilog plot.The X-plot method successfully matched various ﬁeld data.Lo
et al.
5
explored the applicability of the X plot further byconducting numerical simulations in two-dimensional
2D
andthree-dimensional
3D
systems and by investigating various ef-fects. They concluded that a linear relationship between theln
WOR
and the cumulative oil production adequately ﬁt manyof their results. Like Ershaghi and Omoregie,
3
they interpretedtheir ﬁndings using the 1D Buckley–Leverett equation using thepreviously assumed dependence on the relative permeability ratio.In a more recent study, Chan
6
used numerical simulation toexamine the sensitivity of the curve of WOR produced vs. pro-duction time on various reservoir and production factors. He con- jectured that a log-log plot of this curve can be used to diagnosethe srcin of the water production, and speciﬁcally to determinewhether it is due to channeling
heterogeneity
or to coning. Anessential part of his conjecture is that a log-log plot of WOR vs.production time contains linear segments, the slopes of which aredifferent in the cases of channeling or coning, hence they can beused for diagnostic purposes. No expressions for these slopeswere postulated or derived, however. Chan’s conjecture is pre-sumed to have strong support from numerical simulations andﬁeld data.Motivated by Chan’s work we provide in this paper a funda-mental investigation by conducting analytical and numerical stud-ies of waterﬂooding under a variety of conditions. The key objec-tive is to analyze the behavior of the WOR vs. time curve invarious time domains
for example, following breakthrough or atlate times
and to develop a methodology for interpreting the be-havior observed. There are two important differences from theprevious works: We extend the 1D analysis by considering themore realistic power-law dependence of the relative permeabilityratio at high water saturations, and we also analyze the behaviorof multidimensional patterns. In certain cases, analytical expres-sions are derived under simplifying assumptions, which are sub-sequently supported by numerical studies. In this paper, the mainfocus is on waterﬂooding.In order of increasing complexity the analysis proceeds by in-vestigating the following:
one-dimensional
single-layer or homogeneous
displace-ment;
two-dimensional homogeneous displacement in various pat-terns;
two-dimensional heterogeneous displacement
channeling
;
three-dimensional displacement
including layering
.We show that the X-plot approach is a special case of the 1Ddisplacement at intermediate times. At later times, the log-log plotof the WOR vs. time is asymptotically a straight line, with a slopethat reﬂects relative permeability and fractional ﬂow effects in the1D case and the particular pattern geometries in the 2D and 3Dcases. Ultimately, and at sufﬁciently large times, all patterns re-ﬂect 1D displacement behavior. Various properties of log-logplots of the WOR vs. time are also established for heterogeneoussystems represented as a permeability streak and layers, respec-tively. In the latter case, effects of cross ﬂow and communicationbetween layers are also discussed. Finally we provide an interpre-tation methodology for the analysis of the WOR vs. time re-sponse. Numerical simulations conﬁrm these ﬁndings in the ap-propriate geometries.Before we proceed we note that the analysis below is in termsof the dimensionless time
fraction of pore volumes injected
, inorder to compare our results with Chan.
6
Extending the results toWOR vs. cumulative oil recovery plots as in Lo
et al.
5
is straight-forward and discussed in Appendix A. Reference to these plotswill be made where appropriate.
Analysis
The dependence of the WOR on time
or oil produced
resultsfrom the interaction of two effects: the ﬂow rate partition in dif-ferent streamtubes, due to the ﬂow pattern, the viscosity ratio,layering and heterogeneity, and the displacement in a given
*
Now with Jason Natural Products.
**
Now with MSC Software Corp.
†
Now with Landmark Graphics Corp.Copyright © 1999 Society of Petroleum EngineersThis paper (SPE 59477) was revised for publication from paper SPE 38869, presented atthe 1997 SPE Annual Technical Conference and Exhibition held in San Antonio, Texas,5
–
8 October. Original manuscript received for review 18 December 1997. Revised manu-script received 3 December 1998. Manuscript peer approved 13 September 1999.
SPE Journal
4
4
, December 1999 1086-055X/99/4
4
/413/12/$3.50
0.15 413
streamtube, which is affected by the viscosity ratio
M
and frac-tional ﬂow. For the case of a passive tracer
M
1, no fractionalﬂow effects
, the streamlines are ﬁxed, dictated by the well patternand the system heterogeneity, and the displacement within astreamtube is one dimensional. For a waterﬂood with variable
M
and fractional ﬂow effects, these assumptions will fail, in prin-ciple. However, when the mobility ratio is close to unity, theresulting error is expected to be small, and an approach based ondecoupling areal heterogeneity and fractional ﬂow effects appearsto be reasonable, at least as a ﬁrst approximation. The problemthen reduces to
i
understanding the features of 1D displace-ments, and
ii
obtaining the probability distribution,
p
(
t
), of breakthrough
arrival
times of various multidimensional prob-lems. This simplifying approach is adopted for analytical purposesin this paper, where the above two topics are discussed separately.In the ﬁrst, we extend the Buckley–Leverett analysis beyond therange considered by Ershaghi and Omoregie.
3
The second topic,involving multidimensional effects, has not been addressed ana-lytically in the literature so far. Throughout this paper an
immo-bile
initial water saturation is assumed, so that before water break-through the WOR is negligibly small.
1D Displacements.
Consider a 1D immiscible displacement
lin-ear or radial
, which in the absence of capillarity is described bythe traditional Buckley–Leverett equation. Although relativelysimple, the behavior of 1D displacements is important, since itemphasizes petrophysical
relative permeability and fractionalﬂow
effects. As shown below, it also controls the late-time be-havior of more general problems
for example, that of a layeredsystem after water breaks through in all layers
. In appropriatedimensionless variables, with
x
expressing the fractional distancebetween the injector and the producer and
t
the number of porevolumes injected, the standard conservation equation applies,
S
w
t
f
w
x
0,
1
where
f
w
is the fractional ﬂow function,
f
w
11
S
w
M
,
2
expressed in terms of the viscosity ratio,
M
o
/
w
, and therelative permeability ratio,
k
ro
/
k
rw
. Note that Eq. 1 appliesalso for variable injection rates. The solution of Eq. 1 is wellknown,
xt
f
w
S
w
.
3
Then the saturation,
S
w
*
, at the producer (
x
1) is implicitlygiven as a function of time from the solution of
S
w
*
1
S
w
*
M
2
M t
,
4
obtained by substituting Eq. 2 in Eq. 3. Denoting by
W
the WORat the producing well, it is then easily shown that
W
M
S
w
*
.
5
Hence, to obtain the relation between
W
and
t
, one needs to ﬁrstsolve Eq. 4 for
S
w
*
and subsequently substitute the result in Eq. 5.Clearly, this relation will be directly inﬂuenced by the assumeddependence of the ratio
on the saturation.An inspection of a typical plot of
vs.
S
w
Fig. 1
shows thefollowing two different regimes.
At intermediate values of
S
w
*
,
can be roughly approximatedby the exponential
A
exp
BS
w
,
6
where
A
,
B
0 are appropriate constants. This is the essence in theErshaghi and Omoregie
3
X-plot method. For the hypothetical casein Fig. 1
a
, this regime applies up to
S
w
*
0.5. Use of Eq. 6 inEqs. 4 and 5 then leads to
1
W
2
W
Bt
.
7
Note that at large
W
and under the condition that the exponentialbehavior is still obeyed, Eq. 7 suggests a linear log
W
-log
t
plotwith slope 1.
On the other hand, for values of
S
w
*
relatively close to 1
S
or
,
can be approximated by a power law,
a
1
S
w
S
or
b
,
8
where
a
0 is a constant, and
b
1 is the exponent of the relativepermeability to oil. In deriving Eq. 8 it was assumed that
k
ro
varies in this region as
k
ro
1
S
w
S
or
b
,
9
as can be veriﬁed by percolation theory
for example, see Ref. 7
.For the hypothetical case of Fig. 1
b
, where
b
2.1, this regimeapplies after
S
w
*
0.55. Use of Eq. 8 in Eqs. 4 and 5 leads to
W
1
2
W
1
1/
b
ba
1/
b
M
1/
b
t
,
10
which at large
W
suggests a linear log
W
-log
t
plot, but now withslope
b
/(
b
1).We will use these two different limiting regimes to analyze theproperties of 1D displacements. We will focus on two different
Fig. 1–Relative permeability ratio
„
Sw
…
for two different expo-nents
b
plotted vs.
„
a
…
the X-plot coordinates or
„
b
…
the power-law coordinates
„
here
S
or
0.25
…
.
414 Yortsos
et al.
: Water/Oil Ratio in Waterﬂoods SPE Journal, Vol. 4, No. 4, December 1999
time regimes, one following breakthrough and a late-time regime.Equivalent results for the relationship between
W
and the dimen-sionless cumulative oil production,
Q
o
, are given in Appendix A.
Behavior Following Breakthrough.
Of most interest in practi-cal applications is the early part of the WOR curve that immedi-ately follows breakthrough. For unfavorable mobility displace-ments, where
S
w
*
is relatively small, hence where X-plotconditions prevail, we can use the Buckley–Leverett theory aboveto obtain the following equation after breakthrough:1
t
1
t
B
B
W
W
1
2
W
B
W
B
1
2
;
W
B
1.
11
Here
t
B
and
W
B
are, respectively
dimensionless
breakthroughtime and the WOR at breakthrough
which in the Buckley–Leverett theory is nonzero
. In an appropriate log-log plot, there-fore, this equation represents a straight line of slope 1. On theother hand, for more favorable mobility displacements, where thebreakthrough saturation is relatively high and where the power-law regime is obeyed, the equation near breakthrough is1
t
1
t
B
ba
1/
b
M
1/
b
W
b
1/
b
W
1
2
W
Bb
1/
b
W
B
1
2
;
W
B
b
1
b
1 .
12
This equation is also a straight line with a unit slope in the appro-priate log-log plot.The validity of Eqs. 11 and 12 was tested numerically using acommercial ﬁnite-difference simulator.
Fig. 2
shows that, forrather large
M
and at early times following breakthrough, Eq. 11is well obeyed.
Fig. 3
shows the corresponding test for Eq. 12. Asexpected, Eq. 12 is satisﬁed better under conditions of more fa-vorable mobility.Either Eq. 11 or 12 can be linearized near
t
B
to yield the scal-ing,
W
W
B
t
t
B
,
13
which indicates a straight line of slope 1 in a log-log plot of increments of the WOR vs. increments of time. The numericalresults of Figs. 2 and 3 conﬁrm this scaling at early times. Inprinciple, Eqs. 11 through 13 can be used for diagnostic purposes.However, such tests will be sensitive to the particular choice of
W
B
, the value of which needs to be estimated by other means.
Late-Time Behavior.
Following breakthrough, the saturation atthe producing end is relatively high. Again, the WOR behaviordepends on the particular regime of the relative permeability de-pendence. For sufﬁciently large
M
, such that the water saturationat breakthrough probes the X-plot region
where Eq. 6 is appro-priate
, Eq. 7 can be rearranged to yield
1
W
2
Wt
B
.
14
Under such conditions, the ratio
(1
W
)
2
/
Wt
is independent of production time. Furthermore, at large
W
assuming that exponen-tial behavior is still obeyed
, Eq. 14 can be approximated bylog
W
log
t
C
,
15
where
C
is a constant, and which shows that a log
W
-log
t
plot islinear with slope 1.For sufﬁciently low
M
, on the other hand, or at sufﬁcientlylarge times and for any
M
, the power-law regime, Eq. 8, applies.In contrast to the X-plot prediction, Eq. 14, the ratio
(1
W
)
2
/
Wt
is not constant but varies with time following Eq. 10,which can be rearranged to
1
W
2
Wt
W
1/
b
.
16
As
W
increases, this can be further simpliﬁed to
W
1
1/
b
ba
1/
b
M
1/
b
t
17
orlog
W
bb
1 log
t
H
,
18
where
H
is a constant. Under these conditions, therefore, a plot of log
W
vs. log
t
is asymptotically a straight line, but now with slope
b
/(
b
1). This slope provides information on the
exponent
of thepower-law dependence of the oil’s relative permeability on satu-ration. As will be shown below using numerical simulations, thisasymptotic behavior is eventually common to all systems tested.Using the numerical simulator we proceeded to test numeri-cally the validity of these two regimes for practically relevantvalues of the WOR.
Figs. 4 and 5
show log-log plots of the WOR
Fig. 2–Behavior following breakthrough using X-plot coordi-nates
†
X
1/
t
B
1/
t
,
Y
W
B
/
„
1
W
B
…
2
W
/
„
1
W
…
2
‡
, for
M
0.1, 1, 2, 5, and 10.Fig. 3–Behavior following breakthrough using power-law coor-dinates
†
X
1/
t
B
1/
t
,
Y
W
B b
1/
b
/
„
1
W
B
…
2
W
b
1/
b
/
„
1
W
…
2
‡
,for
M
0.1, 1, 2, 5, and 10.Fig. 4–Log-log plot of the WOR vs. time for a 1D displacementand
M
0.5, 1, 2, 5, and 10. The late-time slope is equal to 1.5.
Yortsos
et al.
: Water/Oil Ratio in Waterﬂoods SPE Journal, Vol. 4, No. 4, December 1999 415

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