SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

1

PATTERN RECOGNITION IN A DIGITAL AGE: A GAMEBOARD APPROACH TO

DETERMINING PETROPHYSICAL PARAMETERS

Daniel A. Krygowski, Robert M. Cluff: The Discovery Group

Copyright 2015, held jointly by the Society of Petrophysicists and Well Log

Analysts (SPWLA) and the submitting authors.

This paper was prepared for presentation at the SPWLA 56th Annual Logging

Symposium held in Long Beach, California, USA, July 18-22, 2015.

ABSTRACT

Graphical pattern recognition interpretive techniques

have been part of petrophysics since quantitative

interpretation began, as a way to quickly determine

properties of interest with a minimum of calculations.

When calculators and computers were introduced to

petrophysics, the focus of the techniques changed from

determining the quantities themselves to determining

the parameters needed to calculate those quantities. As

an example, Hingle plots (1959) and Pickett plots

(1966, 1973), first used to quickly determine water

saturation for a few points in a reservoir, can now

instead be used to determine the parameters needed in

Archie’s (1942) water saturation equation, so that the

parameters and associated well log data can be used to

calculate water saturation in much more detail and with

more precision than before.

An extension of those graphical techniques is shown

here, where Hingle, Pickett, and Buckles (1965) plots

(Morris and Biggs, 1967) are displayed simultaneously.

In this “gameboard” display in Excel (©Microsoft),

data is displayed on all the plots. The selection and

modification of computational parameters is

immediately reflected in all plots, leading to a more

coherent prediction of those parameters than from the

same plots used independently.

Pickett plots, with bulk volume water lines added as

shown by Greengold (1986), Hingle plots, with bulk

volume water lines added as shown by Krygowski and

Cluff (2012), and Buckles plots (using both linear and

logarithmic scales) can predict in a common

environment the following parameters: Matrix

parameters to derive porosity from bulk density or sonic

slowness, Archie porosity (cementation) exponent (m),

saturation exponent (n), and water resistivity (Rw), and

irreducible bulk volume water (BVWirr).

The display uses those three plots not commonly

displayed simultaneously, and has the plots linked so

that changes made to the parameters determined from

one plot are reflected in the other plots and the

computations derived from those plots.

By being able to change the values of any of the

parameters while seeing how those changes impact the

other parameters and the calculated porosity, water

saturation, and bulk volume water, the user can quickly

try different interpretive scenarios and determine which

results best honor all the data at hand. Other

information, such as from core measurements, can limit

or set the values of some of the parameters while still

allowing the values of other parameters to be

determined in the context of that other data.

INTRODUCTION

From its beginnings, petrophysics has used graphical

(“pattern recognition”) displays to identify zones of

interest, to initially speed the determination of reservoir

properties of interest, and currently to determine

parameters needed in the calculation of those

properties. Such displays range from depth displays of

raw log data (“QuickLook” methods) to x-y plots

(“crossplots”) of acquired data. In the era before

calculators and computers, calculations were tedious

and time-consuming, compounded by urgency when

drilling activities were suspended waiting for

interpreted results. While slide rules and nomographs

helped speed the calculation process, development of

pattern recognition techniques provided the means for

interpreters to quickly assess a well and focus on

intervals of critical interest. With the advent of

machine-assisted calculations, individuals could

concentrate on interpretation instead of arithmetic, and

the focus of pattern recognition techniques turned from

determining reservoir properties of interest (like

porosity and water saturation) to predicting the

parameters needed to calculate those properties in a

more detailed and robust manner.

This paper shows a proof-of-concept in the combination

of three pattern recognition techniques; Buckles,

Hingle, and Pickett plots, which are used to

simultaneously determine three Archie saturation

equation parameters (m, n, Rw), matrix parameters for

SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

2

conversion of bulk density or acoustic slowness to

porosity, and irreducible bulk volume water to help

predict water production from hydrocarbon-bearing

zones. By predicting the parameters simultaneously, the

interpretation process can be shortened through the

immediate iteration process, and the interpretation

parameters are more coherent, having been determined

together. The spreadsheet is referred to as a

“gameboard” because as with a board game, changes

are made directly on the “board” and other parts (like

game pieces) respond immediately.

THE PLOTS: HISTORY AND CURRENT

USAGE

Buckles Plot Buckles (1965), in creating a method to

determine average water saturation, found that in

intervals at irreducible water saturation, the product of

water saturation and porosity would be a constant

related to pore surface area. When porosity was plotted

against water saturation for those points, the resulting

curve was a hyperbola while points in a transition zone

plotted at values above the points at irreducible

saturation. Morris and Biggs (1965) extended the use of

the constant (often referred to as the “Buckles number”)

to not only identify transition zones from zones at

irreducible saturation, but also to estimate permeability.

They also noted that the porosity-water saturation

product was the bulk volume water fraction of the

porosity, or bulk volume water, BVW.

Bulk Volume Water, BVW Sw [1]

Figure 1 shows Buckles plots in two forms; the linear

plot shown by Buckles, and Morris and Biggs, and the

plot with logarithmic scales, as shown by Bateman

(1984) and others. In the linear plot, the equal BVW

values are hyperbolas, while in the full logarithmic plot,

the iso-BVW lines are linear. The displays provide the

same information; the choice of which to use is up to

the interpreter, as the one easiest to personally interpret.

For intervals at irreducible water saturation, Swirr, and

with a range of porosities, the bulk volume water is

irreducible, BVWirr.

Pickett Plot Pickett (1966, 1973) proposed a graphical

solution to Archie’s equation. Equation 2shows the

solution in terms of resistivity, and Equation 3 shows

the equation solved for porosity, adapted to the graph

paper available at the time (and the usual form of the

crossplot).

[2]

[3]

As shown in Figure 2, the plot is full logarithmic; that

is, both scales are logarithmic, with resistivity on the x-

axis and porosity on the y-axis. In use, data points are

plotted by porosity and resistivity. From the location of

the points, a water-bearing line can be drawn at the

“southwest” edge of the data. From that line, and an

assumption of a value for the saturation exponent, n, a

family of lines of decreasing water saturation can be

drawn, which are parallel to the water-bearing line. As

originally designed, the water saturation, Sw, of each

plotted point can be read directly from the plot. The use

of Archie’s equation is bypassed, as is the knowledge of

formation water resistivity, Rw. In addition, two

parameters in Archie’s equation are predicted by the

plot: the porosity exponent, m, from the negative

inverse of the slope of the water-bearing line, and

formation water resistivity, Rw, from the intercept of

the water-bearing line at porosity equal to 1 (100%),

with a user estimate of tortuosity factor, a.

To use Pickett’s plot, the interpreter plotted points on

full logarithmic paper, placed the water-bearing line

based on the location of the data points, and placed the

lines of decreasing water saturation based on the

location of the water-bearing line and an assumed value

for the saturation exponent, n. The saturation value at

each point could then be read directly from the plot.

With the advent of machine-aided computations, the

Pickett plot can still be used to quickly determine from

the pattern of data points if there are any intervals with

promising water saturations, but it is not necessary to

actually read Sw values from the crossplot. The

crossplot does provide an estimate of the porosity

exponent from the slope of the Sw lines, and an

estimate of Rw from the intercept of the Sw = 1 line at

porosity = 1.

Greengold (1986) added bulk volume water lines to the

Pickett plot. Rearranging equation [1]:

[4]

And substituting for Sw, equation [2] becomes:

[5]

As shown in Figure 3, zones at the lowest value of

BVW, on the “eastern” edge of the data, are at

log( ) log( ) log( ) log( )Rt m n Sw a Rw

1 1log( ) log( ) *log( ) log( )

nRt Sw a Rw

m m m

BVWSw

log( ) ( ) log( ) log( ) log( )Rt n m n BVW a Rw

3

irreducible bulk volume water, BVWirr, (and therefore

Swirr), and should produce only hydrocarbons. If there

is a sufficient range of porosity at irreducible water

saturation, the slope of the BVWirr line can be

determined. The slope is (m-n), so knowing m from the

slope of the water-bearing line, the saturation exponent,

n, can be estimated. If n = m, the BVW lines will be

vertical. Those points at the “southwestern” edge of the

data are in the water-bearing zone, and will produce

only water. The points between those two edges are in

the transition zone, where some combination of water

and hydrocarbons will be produced.

Hingle Plot Previous to Pickett’s method, Hingle

(1959) proposed the first widely-used graphical solution

to Archie’s equation. To do so, he solved the equation

for resistivity in this form:

[6]

Hingle’s work, including the form of the equation,

appears to be an extension of the work of Tixier et al,

1958, in determining water saturation from resistivity

and sonic slowness. Tixier’s method required the

knowledge of formation water resistivity, Rw, but by

using Hingle’s method the value of Rw can be

determined from the data. As shown in Figure 4 (right

plot), the x-axis is porosity increasing to the right.

Instead of using a calculated porosity, raw bulk density

(as shown) or acoustic slowness can be used. The y-axis

is a non-linear scale, shown in the figure in both

resistivity and conductivity. In use, data points are

plotted by “porosity” and resistivity. From the location

of the points, a water-bearing line is drawn at the

“northwest” edge of the data. From that line, and the

assumption of a value for saturation exponent, n, a

family of lines of decreasing water saturation can be

drawn, which fan out from the intercept of the water-

bearing line at the x-axis. As originally designed, the

water saturation, Sw, of each plotted point can be read

directly from the plot. The use of Archie’s equation is

bypassed, and the knowledge of formation water

resistivity, Rw, is not needed, nor is a calculated

porosity. In addition, if bulk density or acoustic

slowness is plotted instead of porosity, the x-intercept

of the water-bearing line predicts the matrix value.

From that value, and an estimate of fluid value, the

porosity can be calculated, with the data having

predicted the matrix value, instead of the interpreter

estimating the value.

The value of formation water resistivity, Rw, can be

predicted from the intercept of the water-bearing line at

porosity equal to 1 (100%), with a user estimate of

tortuosity factor, a. This prediction, however, was

rarely done using the chart, as the porosity scale that

was usually used, spread the points in the expected

porosity range over the available space, and the

resistivity scale of the plot was often insufficient to

both plot data points accurately and visually determine

Rw.

The plot at the top left of Figure 4 shows in the y-axis

the calculated value of (1/Rt)^(1/m), and is displayed on

a linear scale. The value is calculated and displayed in

Microsoft © Excel, and its interpretation is the same as

with the right-hand plot, but resistivity values cannot be

read directly from the plot. This plot also has the

porosity scale extended to 100% porosity ( = 1.0). The

intercept of the water-bearing line at 100% porosity is

a*Rw. The plot at the bottom left shows the same data,

but with scales that spread the data over the plot space

in a manner most usually seen in use.

To use Hingle’s plot, the interpreter needed to use

specially-constructed graph paper (usually available in

logging company chartbooks), and had to assume a

value for porosity exponent, m, as the y-axis changes

with that value. The interpreter then placed the water-

bearing line based on the location of the data points,

and placed the lines of decreasing water saturation

based on the location of the water-bearing line and an

assumed value for saturation exponent, n. The

saturation value at each point could then be read

directly from the plot. With the advent of machine-

aided computations, the Hingle plot can still be used to

quickly determine if there are intervals with promising

water saturations, but it is not necessary to actually read

Sw values directly from the crossplot. The crossplot

does provide an estimate of matrix density or matrix

slowness; a parameter needed in determining porosity.

Analogous to the work of Greengold (1986) on Pickett

plots, Krygowski and Cluff (2012) added bulk volume

water lines to Hingle plots. This is shown in Figure 5.

The points at the lowest value of BVW, on the

“southern” edge of the data, are at irreducible bulk

volume water, BVWirr, (and therefore Swirr), and

should produce only hydrocarbons. Those at the

“northwestern” edge of the data are in the water-bearing

zone, and will produce only water. The points between

those two edges are in the transition zone, where some

combination of water and hydrocarbons will be

produced.

11

1 n mm Sw

Rt a Rw

SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

4

BEHAVIOR OF THE GAMEBOARD

Figure 6 (left illustration) shows the concept of the

behavior of the gameboard. This iterative process, from

Gael (1981) and Bassiouni (1994) shows an iteration

between Hingle and Pickett plots to converge on

cementation exponent and matrix density. The

implication, from the figure, is that the iteration takes

place one step at a time, with the interpreter picking a

value for one variable from one plot and using it in

another to determine the value of a second variable. By

using a spreadsheet with a simultaneous display of the

plots, changes made in one plot can be immediately

reflected in other plots, thereby decreasing the time

needed for the iteration.

Figure 6 (right illustration) shows the gameboard

control area in detail. The user can change the Archie

parameter values, matrix and fluid parameter values,

and irreducible bulk volume water values by the slider

bars and arrows. The porosity calculation (input

calculated porosity, density porosity, or sonic porosity)

can be selected, which activates the appropriate matrix

and fluid values. The user can also change the water

saturation and BVW lines that are displayed.

Figure 7 shows the entire spreadsheet with data plotted,

but modification of the parameters not yet begun. The

spreadsheet consists of a control area (upper left), two

Buckles plots (linear and logarithmic), three Hingle

plots (one for calculated porosity, one with bulk

density, and one with sonic traveltime (slowness)), and

a Pickett plot. The Pickett and Hingle plots all have

both water saturation lines and bulk volume water lines.

Porosity, water saturation, and bulk volume water are

calculated each time a parameter value is changed.

The data in Figure 7, shown in log format in Figure 8,

are constructed to have a range of porosity, with a water

zone, a transition zone, and a zone of irreducible bulk

volume water. The value of Swirr varies with porosity

so that BVWirr is a constant.

The gameboard in Figure 7 has parameters defaulted to

common values; m = n = 2.0, a = 1.0, RHOma = 2.71

(limestone), and Rw = 0.01 as an arbitrary value. The

Hingle bulk density plot shows the choice of RHOma to

be incorrect (not aligning with the plotted points), as

does the Pickett plot, where the “southwestern” points

show a curved line instead of a straight line (Pickett,

1966). The misalignment of the suspected water-

bearing points on the Pickett and Hingle plots is similar

to points at the east side of the Buckles plots as well.

The user could start the iteration of the data by

changing any of the variables, but probably the

variables having the most effect would be RHOma and

Rw. As the Sw lines and data points begin to converge,

the user would see from the Pickett plot that a change in

m would be of benefit. Once the data was aligned with

the Sw lines, n and BVWirr could be modified to bring

the BVW lines in alignment with the appropriate “sides”

of the data cloud. The results of the modification of the

parameters are shown in Figure 9, where the points in

all the plots show an alignment with the Sw and BVW

lines, and with the intercepts which specify RHOma

and Rw.

Again, the order in which the parameters are changed is

up to the individual interpreter, and interpreters may

find one sequence of change to be especially efficient.

When actual data is used, and depending on the range

of porosities and saturations, there may be some

ambiguity in the results, with different sets of

parameters delivering the same fit of the lines and

intercepts to the data. As always, any other available

appropriate data should be used to arrive at a solution

which honors all the data at hand.

CONCLUSIONS

By using a number of pattern recognition techniques;

namely Pickett, Hingle, and Buckles plots, one can

determine parameters for the calculation of porosity and

water saturation in an interactive mode, which makes

that determination faster and more coherent than using

the same techniques individually.

The methods, used individually or in concert, provide a

method to quickly identify zones of interest in what are

often long intervals with no production potential. The

identification of zones can be quantitative, through the

determination of specific values for calculation

parameters, or can be qualitative, through observation

of the pattern of points on the plots with respect to

water saturation and bulk volume water lines.

While Microsoft Excel was used to implement this

method as a proof of concept, the methodology would

be better suited for implementation in an existing

petrophysical or geological software program, either as

a user program (given the appropriate software

functionality), or as an enhancement to existing

functionality.

5

DEFINITIONS

a Archie equation tortuosity factor

m Archie equation porosity (cementation)

exponent

n Archie equation saturation exponent

Rw Formation connate water resistivity

(ohmm)

Rt “True” or undisturbed formation resistivity

(ohmm)

total porosity (decimal)

Sw formation water saturation (decimal)

Swirr irreducible water saturation (decimal)

BVW bulk volume water (decimal)

BVWirr irreducible bulk volume water (decimal)

RHOma formation matrix or grain density (g/c3)

REFERENCES

Archie, G.E., 1942, The electrical resistivity log as an

aid to determining some reservoir characteristics:

Petroleum Technology, January; SPE of AIME.

Bassiouni, Zaki, 1994, Theory, Measurement, and

Interpretation of Well Logs, SPE Textbook Series

Volume 4, Society of Petroleum Engineers,

Richardson Texas.

Bateman, R.M, 1984, Watercut prediction from logs

run in feldspathic sandstone with fresh formation

waters; SPWLA 25th Annual Logging Symposium,

New Orleans, Paper EE.

Buckles, R.S., 1965, Correlating and averaging connate

water saturation data; Journal of Canadian

Petroleum Technology, 9(1), pp.42-52.

Gael, T.B.,1981, Estimation of petrophysical

parameters by crossplot analysis of well log data;

MS Thesis; Louisiana State University, Baton

Rouge, Louisiana, May.

Greengold, Gerald E., 1986, The graphical

representation of bulk volume water on the Pickett

crossplot: The Log Analyst, 27(3); Society of

Petrophysicists and Well Log Analysts, Houston,

Texas.

Hingle, A.T., 1959, The use of logs in exploration

problems: paper presented at the SEG 29th

International Annual Meeting, Los Angeles,

November.

Krygowski, Daniel A., Robert M. Cluff, 2012, Pattern

recognition in a digital age: a gameboard approach

to determining petrophysical parameters: Poster

Session Theme 11, P98, AAPG Annual Conference

and Exhibition, Long Beach, April.

Morris, R.L, and W.P. Biggs, 1967, Using log-derived

values of water saturation and porosity; Society of

Professional Well Log Analysts, 8th Annual

Symposium Transactions, Paper X.

Pickett, G.R., 1966, Review of current techniques for

determination of water saturation from logs;

Journal of Petroleum Technology: Society of

Petroleum Engineers, November, pp.1425-1433.

Pickett, G.R., 1973, Pattern recognition as a means of

formation evaluation; SPWLA Fourteenth Annual

Logging Symposium, Paper A.

Schlumberger, 2005, Log Interpretation Charts, 2005

Edition, Appendix A, p.264: Schlumberger, Sugar

Land, Texas.

Tixier, M.P., R.P. Alger, C.A. Doh, 1958, Sonic

logging: presented at 33rd Fall Meeting of the

Society of Petroleum Engineers, paper T.P. 8063.

SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

6

ABOUT THE AUTHORS

Daniel A. (Dan) Krygowski

Senior Petrophysical Advisor, The Discovery Group

Dan has over 35 years of

experience in the art and

science of petrophysics, and

in the design and

development of petro-

physical software. Dan

earned a B.A. in Physics

from the State University of

New York at Geneseo. After

earning M.S. and Ph.D.

degrees in geophysics (with

a focus on petrophysics)

from the Colorado School of Mines, he joined Cities

Service Company, and worked in Denver and Tulsa.

After Citco, he joined Atlantic Richfield Company

(ARCO). In both companies, he gained experience in a

variety of geologic and geographic areas in both

technical and management positions in petrophysics.

After ARCO, he joined Landmark Graphics, and was a

member of the PetroWorks development team,

providing petrophysical expertise and was also involved

in interface design, and development of documentation

and training materials. When Landmark closed its

Austin, Texas office, Dan joined Chevron, working in

deep Gulf of Mexico and Chad, Africa projects. He also

supported internal petrophysical training efforts.

Dan joined The Discovery Group in late 2006.

Since the late Cretaceous, Dan has taught the AAPG

Basic Well Log Analysis course annually with Dr.

George Asquith of Texas Tech University. Dan also

teaches Basic Openhole Log Interpretation, a similar,

but shorter course, and Petrophysics Elements, a long-

term, low-intensity course.

In 2004, the AAPG published George and Dan's book,

Basic Well Log Analysis, a second edition of George's

1983 similarly-named book which was one of the

AAPG's all-time best selling publications.

Dan is a member of the Society of Petrophysicists and

Well Log Analysts (SPWLA), American Association of

Petroleum Geologists (AAPG), Society of Petroleum

Engineers (SPE), Society of Exploration Geophysicists

(SEG), the Rocky Mountain Association of Geologists

(RMAG), and the Denver Well Logging Society

(DWLS).

Dan is a Texas Registered Professional Geoscientist

(#5014).

Robert M (Bob) Cluff

President, The Discovery Group

Bob Cluff is a geologist and

petrophysicist with over 35

years experience in oil and

gas exploration, develop-

ment, and research. His

principal areas of expertise

are petrophysics, petroleum

geology of carbonate and

clastic reservoirs, and the

integration of petrophysical

data with geological data in

detailed reservoir studies. He has worked and published

extensively in the fields of non-conventional gas from

both tight sandstones and shales, petrophysics, source

rock analysis and basin modeling. He has conducted

and supervised projects in most sedimentary basins of

North America as well as South America, Europe,

Southeast Asia, and Australia.

Bob received his BS degree in Geology from the

University of California at Riverside (high honors), an

MS in Geology from the University of Wisconsin at

Madison, and a BA in Mathematics from the

Metropolitan State College of Denver. From 1976 to

1981 he was a geologist with the Coal and Oil and Gas

Sections of the Illinois State Geological Survey, worked

as an independent consulting geologist from 1982-

1986, founded The Discovery Group in 1987.

Bob is active in several professional societies including

the American Association of Petroleum Geologists,

Society of Petroleum Engineers, the Rocky Mountain

Association of Geologists (past-President 2006), the

Denver Well Logging Society (past-President), and the

Society of Petrophysicists and Well Log Analysts (Vice

President Technology; Vice President Membership,

Regional Director North America).

He is a registered geologist in the states of Texas

(1873), Illinois (196-000177), and Wyoming (313), and

is a DPA Certified Petroleum Geologist (3168).

7

FIGURES

Figure 1: Buckles plots with linear scales (left) and logarithmic scales (right).

Figure 2: Pickett plot with saturation lines placed based on the location of data points.

BVWirr

BVWirr

0.01

0.1

1

0.01 0.1 1 10 100 1000

po

rosit

y

Resistivity

Pickett Plot

Sw = 1

Sw = 0.8

Sw = 0.6

Sw = 0.4

Sw = 0.2

Data

a*Rw

Slope = -1/cementation exponent, m

Sw = 1.0

SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

8

Figure 3: Pickett plot with Bulk Volume Water lines added.

Figure 4: Hingle plots: In the original format (right) with porosity from zero to 100%, and with a calculated y-axis

(left) with two different ranges of porosity and calculated Hingle value [ (1/Rt)^(1/m) ].

a*Rw

Slope = -1/cementation exponent, m

Sw = 1.0

Increasing BVW

Slope = (m-n)

BVWirr

Increasing porosity

Incr

easi

ng c

ondu

ctiv

ity,

mS

1.00.750.25 0.500.0

Incr

easi

ng

resi

stiv

ity,

oh

mm

Porosity

Sw = 1.0

a*Rw @ Phi = 100%

Schlumberger, 2005

0

0.2

0.4

0.6

0.8

1

2.002.102.202.302.402.502.602.702.80

(1/R

t)^

(1/m

)

Bulk density

Hingle (RHOB)

0

0.5

1

1.5

2

2.5

3

1.001.201.401.601.802.002.202.402.602.80

(1/R

t)^

(1/m

)

Bulk density

Hingle (RHOB)

Resistivity scaling reflects m = 2

Calculation from Rt and m

9

Figure 5: Hingle plot with Bulk Volume Water lines added.

Figure 6: Gameboard concept and gameboard controls.

Sw = 1.0

Incre

asing

BV

W

RHOma

BVWirr

assume m

RHOma

RHOma

m Iterate until convergence m

Gael, 1981, from Bassiouni, 1994

SPWLA 56th

Annual Logging Symposium, July 18-22, 2015

10

Figure 7: Pattern recognition gameboard with ideal data and initial (default) parameter values.

Figure 8: Ideal log data: a range of porosities in wet, transition, and irreducible BVW zones. ir

red

uci

ble

tr

ansi

tio

nw

et

11

Figure 9: Pattern recognition gameboard with ideal data and final parameter values.