This document presents a technique for representing bulk volume water (BVW) on a Pickett crossplot of log resistivity vs log porosity. Lines of equal BVW are constructed on the crossplot based on the relationships between the cementation exponent (m), saturation exponent (n), and BVW. The slopes of the BVW lines depend on whether m equals n, m is less than n, or m is greater than n. This combined crossplot allows evaluating formation saturation and BVW from a single plot, integrating separate interpretive methods.

1. THE G ICAL REPRESENTATION OF
VOLUME WATER ON
THE PICKETT C Q S S P m
GERALD E. GREENGOLD
Tenneco Oil Exploration & Production
Southwestern Division
San Antonio. Texas
ABSTRACT
For many years both the bulk volume water plot of
saturation vs porosity and the Pickett crossplot of log
resistivity vs log porosity have been used independently
by log analysts in formation evaluation. A graphical
technique is presented whereby the interpretive methods
unique to each of these may now be performed from a
single crossplot. Using a Pickett crossplot as a base, the
construction of lines of equal bulk volume water is
described. The slope of the lines are shown to be depen-
dent upon “m” and “n”; the cementation and satura-
tion exponents of the Archie equation; where slope =
(n-m). Bulk volume water lines are constructed for the
three possible relative values of “m” and “n”: “m”
equal “n”, “m” less than “n”, and “m” greater than
“n”.
~
INTRODUCTION
Bulk volume water (BVW) representsthe percent water
BVW = 4 Sw , (1)
where water saturation (Sw) represents the percentage of
pore volume (4)filled with water.
The concept of BVW has been is use for many years
within the oil industry and is routinely depicted on well
logging service company computed log products. As
shownby Morris and Biggs (1967), BVW data are typically
used for the following evaluations:
1. Determination of zones at irreducible water satura-
tion through the analysis of a BVW (Sw vs 4) plot
(Figure 1)
in a given total rock volume and is defined as
2. Estimation of water cuts and producibility
3. Estimation of permeability
4. Estimation of grain size
5. Estimation of pore type
6. Determination of multiple lithologies
t0
(%
THE LOG ANALYST
sw O!O -
Figure 1: The bulk volume water plot.
21

2. REVIEW OF THE PICKETT CR0SSPU)T
The Pickett crossplot, as developed by G.R. Pickett is
a graphical representation of the solution to the Archie
water saturation equation
log Rt = - m log 8 - n log Sw t log (a Rw)
where a = Tortuosity factor (a function of the com-
Rw = Formation water resistivity at formation
$I = Porosity
Rt = Deep, uninvaded formation resistivity(rock
plus fluids)
m = Cementation exponent
n = Saturation exponent
plexity of the flow path)
temperature
As detailed in Appendix A, equation 2 can be rewritten as
log Rt = -mlog4 - nlogSw + log(aRw) (3)
As shown in equation 3 a log-log plot of “Rt”vs ‘‘4”
will give a family of parallel lines of equal saturation with
a slope of “-m” and a logarithmic spacing controlled
by “n” (see Figure 2). Larger “n” values increase and
smaller “n” values decrease the perpendicular distance
between these lines. The Sw=100% line (also called the
“Ro” line) intercepts the 4=100% line at a value of
“aRw”. As evidenced from equation 3, when $I=100%
the “ -mlog+” term becomes zero and on the “Ro” line
the “-nlogSw” term also becomes zero leaving log
Rt =log(aRw).
By plotting values of “4” and “Rt”, one can graphi-
cally estimate the saturation of any point by its position
relativeto a given saturation line. Using both the absolute
position of the points and pattern analysis, one may
estimate productive, marginal, and wet regions of the plot.
Until now this has been the primary use of the Pickett
crossplot.
GRAPHICAL REPRESENTATION OF BULB
VOLUME WATER
Theory
Substituting Sw =BVW/$I from equation 1 into equa-
tion 2 gives the following equation for a bulk volume
water crossplot (see Appendix B).
log Rt = (n-m)log& - nlogBVW + log(aRw) (4)
As shown in equation 4 a log-log plot of “4” vs “Rt”
gives a family of parallel lines of equal BVW with a slope
of “(n-m)” and a logarithmic spacing of “n”. Larger
“n” values will increase and smaller “n” values will
decrease the perpendicular distance between these lines.
Unlike saturation lines, BVW lines may change slope
depending upon the relative values of “m” and “n”.
When “m” equals “n” regardless of the value, BVW lines
have a slope of zero and the lines are horizontal as shown
log
Rt
lines of
equal saturation
in Figure 3. When “m” is less than “n”, the slope is
positive and the lines slope up and to the right as in Figure
4;and when “m” is greater than “n” the slope is negative
as depicted in Figure 5. Note that the parallel lines shown
in Figures 3,4, and 5 are equivalent to the hyperbolic lines
of equal BVW values of Figure 1.
Graphical Construction
Having shown that constant value BVW lines are
straight and parallel on the log-log resistivity-porosity
Pickett crossplot and noting that for saturations greater
than loo%, BVW values are meaningless, we construct
lines of equal BVW by solving the BVW equation at two
endpoints and connecting them with a straight line. End-
points used are the Ro line and the 4=100% line.
The Archie equation may be solved for “Rt” giving,
On the “Ro” line Sw=l.O and 4=BVW. Therefore,
by substitution into equation 5 we have
22 MAY-JUNE, 1986

3. Similarly,on the 4=100%line, 4=1.0 and Sw=BVW.
Again, substitution into equation 5 gives
aRw
BVW“
Rt =-. (7)
For a given BVW Value, the only differencein the “Rt”
value at Sw=100% and the “Rt” value at 4 =100% is
the exponent to which the BVW value is raised; either
“m” or “n”.
When “m” equals “n”, the value of “Rt” at each end-
point is the same and the BVW lines are horizontal. This
is shown in Figure 6. Therefore, since BVW =4 on the
“Ro” line, when “m” equals “n”, simply construct
horizontal linesat the intersection of the Ro line with “4”
values equal to desired BVW values.
When “m” doesn’t equal “n”, equations 6 and 7 are
still used to calculate “Rt” values on the “Ro” and
4=100% lines respectively. However, we may again find
the endpoint on the “Ro” line at it’s intersection with
the ‘‘4’’ value equal to the desired BVW value. All that
remains to be done is to calculate the “Rt” value of the
upper (4=100%) endpoint using equation 6. Figures 7
and 8 show the enhanced Pickett crossplots with both
saturation and bulk volume water lines for “m” less than
“n” and “m” greater than “n” respectively.
As a simplification to the construction of BVW lines
on a Pickett crossplot, note that for all relativevalues “m”
and “n” the BVW=10% line always has as its endpoints
the intersection of the “Ro” line at 4 = 10% and the
intersection of the Sw=10%line at 4 =100%.Theserela-
tions result directly from equation 1. Therefore, after con-
struction of the “Ro” and the Sw=10% lines, we con-
struct the BVW=lO% line as noted above. All other
desired BVW lines are drawnin parallel to the BVW=1OVo
line, at the intersections of the “Ro” line and 4 =BVW
values. This method of construction may be used with
any value BVW line. For example, the BVW =30% line
will intersectthe “Ro” line at 4=30%,a nd the 4 =100%
line at Sw=30%.
The Pickett crossplot is commonly used to estimatethe
value of “m” when points plot along the “Ro” line, in-
dicating a zone at or near Sw=100Vo. Morris and Biggs
(1973) have shown that for zones at irreducible water
saturation (Swi), BVW is constant. Theoretically, using
a pattern analysis technique similar to that used in the
estimation of “my’, one may estimate the value of the
quantity “(n-m)” when points plot along a BVW line,
I
Figure3: The log-log bulk volume water plotfor m=n.
lines of equal BVW
Figure4: The log-logbulkvolume water plotfor m c n.
THE LOG ANALYST 23

4. I
/lines of equal BVW
log Rt :(n-m) log 0 - n log BVW t log(a Rw)
I10
B
100
Figure5: The log-logbulk volumewater plot for rn > n.
implying that the zone in question is at or near irreduci-
ble water saturation. Therefore, if the analyst can locate
the same formation at a constant rock type both at “Swi”
and at Sw=100V0,then he may estimate the value of “m”
and “(n-m)” and therefore the value of “n”.
CONCLUSIQNS
Both the BVW plot and the Pickett crossplot have been
used for many years as aids in formation evaluation.Each
has its own relative merits in the analysis of log data
through the use of documented methods of pattern
analysis. Recognizing that the hyperbolic lines of equal
BVW on the bulk volume water plot of Figure 1 are
equivalent to the straight BVW lines of the Pickett
crossplot in Figures 6, 7, and 8,the log analyst may now
perform a dual interpretation from a single plot.
I 10
b
loo
Figure 6: The enhanced Pickett crossplot for rn =n.
REFERENCES
Pickett, G. R., 1966, ‘Y4 Review of Current Techniques for
Determination of Water Saturation From Logs,” Jour. Pet.
Techn. Nov., 1966 Pgs. 1425-33
Morrisand Biggs, 1967, “Using LogDerivedValues of Water
Saturationand Porosity,” Transactionsof the SPWLA, 1967
Paper X.
Picket, G. R., 1973, “Pattern RecognitionAs a Means of For-
mation Evaluation,” SPWLA 14th Annual Logging Sym-
posium, 1973 Paper A.
ABOUT THE AUTHOR
JERRY GREENGOLD is a geological engineer with
Tenneco Oil Exploration and Production’s Southwestern
Divisionin San Antonio, Texas. He received his B.S. degree
in Geology from Queens College of the City University
of New York in 1979,and in 1981 he received his M.S.
degree in Engineering Geology from Purdue University
in West Lafayette, Indiana. He is a member of the
SPWLA,AAPG and the Association of EngineeringGeo-
logists, and has presented papers in the fields of Engineer-
ing Geology and Log Analysis.
24 MAY-JUNE, 1986

5. I
f
I1
3
-
D
5
?
i
= log
Rt
10
P
I00 I 10
0
100
Figure7: The enhanced Pickett crossplot for rn < n. Figure8: The enhanced Pickett crossplot for rn > n.
APPENDIX A APPENDIX B
DERIVATION OF THE PICKETT CROSSPIMF
Solving the Archie equation for "Rt" gives
aRw
fb"SW"
Rt =-3
or
Rt = (aRw) ~-"SW-"
Taking the logarithm of each side and rearranging gives
log Rt = -mlog& - nlogSw + log(aRw)
DERIVATION OF THE BULK VOLUME
WATER CROSSPLOT
Substituting the BVW equation for "Sw" in the Archie
equation gives
BVW" - aRw
4 4"Rt
---,
Distributing the exponent "n", multiplying through by
"I$~" and solving for Rt gives
Rt = (aRw) 4n4-" BVW-".
Combining like terms we have
Rt = (aRw) # J ( " - ~ ) BVW-".
Taking the logarithm of both sides and rearranging gives
log Rt = (n-m)log4 - nlogBVW + log(aRw).
THE LOG ANALYST 25