CHAPTER ELEVEN: CONSTRUCTION AND USE OF CROSSPLOTS

Table of Contents
11.00 Introduction to This Chapter
11.01 Types of Crossplots
11.02 The Common Crossplots
11.03 Examples and Uses of Crossplots
1. Shaly Sand
2. Carbonate
11.04 Statistical Analysis
11.05 Lithodensity Crossplots
11.06 Porosity Resistivity Crossplot (Hingle)
11.07 Resistivity Porosity Crossplot (Pickett)
11.08 Cumulative (Holgate) Plots
11.09 In Conclusion
11.10 Exercises For Chapter Eleven
11.11 Bibliography For Chapter Eleven

Continue to Chapter Twelve.

Publication History: This Chapter formed Chapter Eleven of The Log Analysis Handbook published by Pennwell in 1986. Updated for this electronic version Aug 2002.

The illustrations in this Chapter were originally created in colour, but the originals were lost so the black and white published versions are shown. All figure numbers have "13" as a prefix instead of "11". This will be fixed when time permits.

CHAPTER ELEVEN: CONSTRUCTION AND USE OF CROSSPLOTS

11.00 Introduction to This Chapter
Crossplots assist in selection of interpretation parameters, identification of trends and problems, and compress large amounts of data to a few pages. Several hundred thousand different crossplots can be made on the same zone, but only a few are helpful. Some of these are described in detail in this Chapter.

Most crossplots described here have already been used or discussed in previous Chapters. However, this Chapter consolidates that information and offers more detail of constructing some of the more exotic forms.

Case histories showing the use of crossplots in a number of real situations are shown in Chapter Twelve.

11.01 Types of Crossplots
We first must distinguish between a nomograph, a chart, and a crossplot. A nomograph is a mechanism for solving an equation. It consists of ruled lines, scaled and placed to allow solution of the equation with a pencil and straightedge. An example is the resistivity salinity temperature nomograph in Figure 11.01.

FIGURE 11.01: Nomograph	FIGURE 11.02: Chart

A chart can be used for the same purpose. If a chart is properly constructed, no straight edge is necessary and results can be read directly from the graph. The chart in Figure 11.02 performs the same mathematical function as the nomograph described above.

When data points are plotted on a chart, they may form recognizable patterns, which may be useful in evaluating the data. A chart or graph on which such points are plotted is called a crossplot.

FIGURE 11.03: Crossplot	FIGURE 11.04: 4-D Scatter plot

In some cases, one or more of the axes of a graph can be transformed into other measurements to enhance the appearance of the patterns. In Figure 11.03, the temperature axis of the salinity graph has been transformed into depth units so that salinity variations with depth for various wells can be analyzed.

A crossplot is a graph of two parameters on X-Y coordinate graph paper. A third parameter may be plotted by assigning values to different symbols plotted at the X-Y position, thus creating a Z-plot or 3-D plot.

Figure 11.04A: META/LOG 4-D Crossplot

In our operation, we have devised a 4-D plot, which provides information about a fourth parameter, coded by the color of the plotted point. Thus, two Z-plots can be combined, allowing more information on one page. The fourth dimension is often shale volume or frequency of occurrence of the X, Y pairs and is called the W axis.

Crossplots can be drawn by hand on graph paper, or any computer and presented on a printer, plotter, or CRT monitor.

If data is presented on a plotter or CRT, the X-Y points fall at their exact position on the paper or screen. This is called a scatter plot, since the points are scattered all over the page. The symbols used to represent the Z and W axes may overlap and be hard to read. Such a plot is shown in Figure 11.04.

FIGURE 11.05: Grouped crossplot	FIGURE 11.06: Printer plot

To reduce confusion, data can be summed into cells with the points plotted falling at the centroid of a cell. If the cell size is chosen correctly for the character size used in the plots, no overlap will occur. These plots are called grouped plots as shown in Figure 11.05. The Z and W axis values plotted are the average of all points that fall anywhere in the cell.

If plots are to be output on a printer, data must be assembled into cells proportional to the printer's horizontal and vertical spacing. A scatter plot is thus impossible on a computer printer, unless the graph is first plotted to a CRT and then dumped dot for dot to a dot matrix printer. Such plots are called graphic dumps, and can be done with any kind of crossplot. Because the dots on a printer are unequally spaced (horizontally and vertically), the plot may change shape from that seen on the CRT. See Figure 11.06.

A composite plot is one which comprises data from two or more zones or wells. On a scatter plot, the Z or W axis may distinguish which well or zone the data is from. This cannot usually be done to a grouped plot due to the summation process required to place data into the cells.

A histogram is a crossplot in which one of the X or Y axes is frequency of occurrence. These graphs may use absolute frequency, percent frequency or fractional frequency on the axis. Histogram examples are shown in Figure 11.07 to 11.11.

FIGURE 11.07: Histogram of GR	FIGURE 11.08: Histogram of DELT

FIGURE 11.09: Histogram of PHID	FIGURE 11.10: Histogram of PHIN

In a cumulative plot, the X or Y axis is the cumulative value of some core, log, or derived data over an interval. It is possible to have a composite plot of cumulative data, which can reduce statistical error.

FIGURE 11.11 Histogram of RESD

When two cumulative plots are compared over the same rock interval, (such as cumulative core porosity and cumulative sonic log readings), a calibration curve between the two measurements can be derived. This is often called a Holgate plot, after the man who first publicized the method. The Holgate plot is accurate even if the two sets of data are off depth from one another, although data should encompass the same physical interval of rock. Examples are shown in Figures 11.12 to 11.19. These plots are explained more fully in Section 11.09.

FIGURES 11.12 and 11.13: Holgate plot and regression of PHIe and COR-P

FIGURES 11.14 and 11.15: Holgate plot and regression of PHID and COR-P

FIGURES 11.16 and 11.17: Holgate plot and regression of DELT and COR-P

FIGURES 11.18 and 11.19: Holgate plot and regression of PHIN and COR-P

Other terms define drawing methods, as opposed to data used for the plot. In line charts, plotted points are joined by short line segments. Several sets of data may be plotted on a line chart using different symbols, colors, or line types to distinguish the data. Line charts can only be drawn for single valued functions and cannot be used to replace a scatter plot for example, but may outline a histogram. The normal depth plot of log curve data is a line chart.

Bar charts can be used in place of line charts, for single valued functions. The color or shading of the bars can vary so that several variables can be plotted on one graph. Examples of combined bar and line charts are given in Figures 11.20 and 11.21.

FIGURES 11.20 and 11.21: Line and Bar charts

FIGURE 11.22: Pie chart

Pie charts can also plot histograms. It is often shaded or colored so that the pie segments can be seen easily. (See Figure 11.22)

11.02 The Common Crossplots
A very common crossplot is the resistivity-porosity plot. It is often called a Hingle plot, when porosity is on a linear scale, or a Pickett plot, when both porosity and resistivity are on logarithmic scales. See Figures 11.23 and 11.24. These plots are described in more detail in Section 11.07 and 11.08.

FIGURES 11.23 and 11.24: Hingle plot and Pickett plot

Because lithology is easily interpreted from them, density neutron and sonic neutron crossplots are very common. Examples are shown in Figure 11.25 (sand-shale) and 11.26 (carbonates).

FIGURES 11.25 and 11.26: 4-D Crossplots of density-neutron data

Plots of MLITH versus NLITH also illustrate lithology patterns (Figure 11.27), as do plots of matrix travel time versus matrix density or matrix density versus photo electric cross section (not illustrated). These plots are interpreted by observing the location of the data points relative to the pure mineral points.

FIGURE 11.27: MLITH vs NLITH 4-D plot FIGURE 11.28 PHIe vs SW 4-D plot

Porosity versus water saturation crossplots are also very common. They can be used to identify pore geometry or rock types, and predict potential water-cut problems in new wells, such as the data in Figure 11.28. Horizontal patterns of points indicate potential water cut, whereas hyperbolic patterns indicate clean oil or gas production. Changes in porosity type may also be recognized on this plot when data follows more than one hyperbolic trend line.

Core porosity versus core permeability crossplots are helpful in choosing cutoff values (Figure 11.29 and 11.30) and for deriving permeability from porosity.

FIGURES 11.29 and 11.30: Core permeability versus core porosity and regression results

Core porosity versus log porosity will help confirm the validity of an analysis (Figure 11.31 and 11.32), and demonstrate the possible error in the porosity calculations. They will also show whether a calibration shift is necessary.

FIGURES 11.31 and 11.32: Log porosity versus core porosity and regression results

Many other crossplots may portray log analysis problems; one merely has to choose suitable axes and scales.

11.03 Examples and Uses of Crossplots
Case histories showing the use of crossplots in a number of real situations are shown in Chapter Twelve. The crossplots for the Shaly sand Case History are repeated here, with an explanation of their use.

1. Shaly Sand Example
The following material illustrates a number of plots from Coriband data - taken from "Log Interpretation, Volume II - Applications" by Schlumberger c.1972. While not all plots are always useful, this set illustrates some of the more complex functions which may aid the analyst.

This example is a shaly sand - shale - clean sand sequence. The clean sand has oil over water and the shaly sand appears to also be oil bearing. The following crossplots were made:

1. Porosity vs Resistivity - shows water saturation lines (shale data falls below 100% Sw line).

2. Porosity vs Saturation - shows constant water volume lines. Data streaming above and to the right indicate transition and water zones. Shale data falls to the bottom of the graph.

3. Density vs Neutron - shows all data below limestone line, indicating either no perfectly clean sand or mixed lithology sand (GR suggests clean sand). Shale data falls towards bottom and right.

4. Core porosity vs core permeability - shows a data cluster which cannot be used to derive a regression line mathematically. A line drawn thru the lower left corner will work fine.

FIGURE 11.33A: Basic crossplots for Shaly Sand Example - Part 1

5. Matrix density vs matrix cross section - confirms that sand is not pure quartz, but the plot does not tell us which minerals to expect. Sample description suggests quartz, calcite, and glauconite (plots past anhydrite at top right).

6. Apparent water resistivity vs density - shows RW@FT and RWSH points relative to spread of data for both shale and hydrocarbon zones.

7. Apparent water resistivity vs effective porosity - similar to above but uses effective porosity. Shale plots near origin, water zone at top left, oil at right.

8. Apparent water resistivity vs gamma ray - shows where to pick GR0 and GR100 (also can be picked from raw logs). Best oil zone is off scale to the right.

FIGURE 11.33B: Basic crossplots for Shaly Sand Example - Part 2

Just to illustrate that you don't need a $5000 to $75000 log analysis package to do good work, all the calculations and crossplots shown here were made with a Lotus 1-2-3 spreadsheet program, called META/LOG, written by the author, and available for a mere $50. The depth plot shown below was made with a $50.00 shareware plot utility called LAS/PLOT. This presentation is the bare minimum that would be given; more complete plots are shown in the next two case histories. Most log analysis packages can make similar or more elaborate plots.

FIGURE 11.33C: Basic depth plot for Shaly Sand Example

2. Carbonate Example
The basic set of Coriband plots may, of course, be hand generated or created by nearly any spreadsheet package or computer aided log analysis system. All these plots in this example are printer plots and are thus grouped plots. The example plots and commentary are taken from “Log Interpretation Applications” by Schlumberger.

Figures 11.33 through 11.53 are for an interval indicated by sample description to be silica (quartz) and limestone with some shaliness and secondary porosity.

Figure 11.33 shows an MLITH versus NLITH frequency crossplot. Typical pure-mineral points for hard-rock interpretation are silica, limestone, dolomite and anhydrite. These points are quickly located by overlaying the crossplot with a permanent plastic sheet on which the points are inscribed.

Figures 11.33, 11.34, and 11.35

The distributions of frequency data are studied to see if the concentrations of various known lithologies fall on the pure mineral points. Consistent shifts of the concentration of points from the pure mineral points, may indicate the need for an adjustment in calibration (normalization) of one or more of the logs. Such indications are looked for on all lithology crossplots.

In Figure 11.34, the frequency values indicate concentrations of levels between limestone and dolomite, and between limestone and silica. There is some indication of secondary porosity in the limestone-dolomite mixture. Note the points plotted above the line joining limestone and dolomite, with a maximum upward displacement near 75 percent dolomite.

The points above the silica-limestone line may indicate secondary porosity in a silica-limestone mixture, or gas if the lithology is lime. Gas produces a displacement to the northeast from the mineral in which the gas saturation occurs. Unconsolidated sand plots below the silica point.

No anhydrite exists in this example. The pattern of points between the silica and shale points are in the shaly trend, and can be seen by referring to the gamma ray Z-plot of Figure 11.34. Clean points at the upper right of the distribution, show low gamma ray Z-values. The highest gamma ray Z-values are at the shale end at lower left. This pattern serves to indicate that the gamma ray will be a useful shale indicator at this interval and that the shale point can be picked at the lower left-hand corner of the distribution (assuming some pure shales are present).

If there are no pure shales, beds with some shaliness may show a trend towards the 100 percent shale point which will enable the point to be approximated. Shales generally plot below and adjacent to anhydrite. The location of the shale area on the crossplot becomes rather well known after some experience in a particular geological province.

The value of MLITHSH from the MLITH vs NLITH plot is used with other crossplots to aid in the selection of other shale parameters.

Other lithological trends sometimes observed on the MLITH vs NLITH plot may be: a displacement of points from anhydrite to the northeast toward Halite (salt) at MLITH = 1.23, NLITH = 1.01; a displacement from anhydrite to the northwest toward gypsum; and a displacement from silica to the southwest toward 50/50 pyrite/siderite at about MLITH = 0.4, NLITH = 0.3.

Any minerals with known matrix coefficients, such as volcanic tuff, granite wash, etc., may be plotted on the MLITH versus NLITH plot to aid interpretation in unusual conditions.

In using the data from the MLITH versus NLITH frequency plot, due attention is also given to the Caliper Z-Plot of Figure 11.35. Caliper Z-Plots indicate washouts by high Z (high caliper) values. Note that the l's above the lime-silica line are due to enlarged-hole. The grouping of high caliper values shown by Figure 11.35 in the mid-range of the shale trend indicates that these shaly formations are washed out more than those closer to the "shale" point. This may be due to the latter containing more clay and less friable material. It is important to notice on the caliper Z-Plot, any occurrence of caved hole in the non-shaly regions, since it indicates that precautions must be exercised in the interpretation of these regions. Note that these two Z-plots can be combined into one 4-D plot by choosing the correct Z and W axis.

MLITH vs DENS plots (Figures 11.36 and 11.37) are useful for determining DENSSH. They are examined first for the distribution of plotted points, relative to the pure mineral points, as located by the corresponding acetate overlay. This is also another check on log calibration. The zero-porosity, pure mineral points are shown at the right edge of the distribution. Since porosity increases to the left from any mineral point, the circles at the left ends of the horizontal lines indicate the 25-percent-porosity points for the given mineral.

The numbers at the top and right edges of the frequency plot (Figure 11.36) show the sums of the points occurring vertically or horizontally on the plot; they may be thought of as histograms of frequency versus, in this case, MLITH or DENS. They aid in giving a clearer picture of which minerals predominate in the zone.

If varying percentages of shale are filling the pore spaces, two trend lines will develop toward the shale point near the bottom of the plot. One trend originates from the maximum porosity, and the other trend from the minimum porosity of the mineral. As a guide for pinpointing the shale point, the MLITH value for shale (MLITHSH), as determined from the MLITH vs NLITH plots, may be entered to intersect with these two trends at the DENSSH value. The location of the shale point is confirmed on the gamma ray Z-Plot (Figure 11.37) by the higher Z values.

Also, it may be possible to tell from which basic minerals the shale trends extend. In the present case, the main trend to shale appears to be from silica.

On the MLITH vs DENS plots, secondary porosity will usually produce points displaced vertically upward from a line drawn between limestone and dolomite. Gas, if present, will develop towards the northwest from a mineral point.

Figures 11.38 through 11.41

Figures 11.38 through 11.41 show MLITH vs PHIN and MLITH vs DELT plots. They can be used in the same manner as the MLITH vs DENS plots to check log calibration, to give a better idea of predominate minerals in the zone, and to obtain values of PHINSH and DELTSH respectively.

The MLITH vs gamma ray plot of Figure 11.42 checks for the validity of the gamma ray log as a shale indicator. The concentration of clean points at the right are grouped in the range of MLITH values expected for the lithologies present. It is easy to pick GRO at approximately the right edge of the main concentration using the histograms at the top of the plot.

A shale trend is seen downward to the left intersecting with MLITH at about GR100 = 100. From this shale point another small trend is seen upward and to the left. This may result from shales containing radioactive dolomite.

Figures 11.43 and 11.44 are MLITH vs COND plots. The points furthest to the right, represent the highest resistivities in this interval. Displacements of plotted points to the left indicate the development of water-filled porosity. The shale trend is shown by a pattern of points displaced downward and to the left, with gamma ray increasing as the shale point is approached, as shown on the Z-Plot. RSH, needed for the computation of saturation in shaly formations, is obtained from the intersection of this trend with the MLITHSH line.

The DENS vs DELT plots of Figures 11.45 and 11.46 have good definition for shale indication and evaporite determinations (anhydrite, salt, polyhalite, etc.). An acetate overlay is again used for locating the typical sandstone, limestone, and dolomite porosity lines. The frequency plot (Figure 11.45) is perused for predominant minerals and major trends. The shale point is verified, and, if necessary, rechecked against the MLITH vs DELT plots.

The DENS vs PHIN plots of Figures 11.47 and 11.48 are the most important plots in CORIBAND and in most computer aided analysis. They are used for shale and hydrocarbon corrections, and for lithology (i.e., matrix density) and porosity determination. Many other computer aided log analysis systems present PHID vs PHIN crossplots, which are often easier to use with compatibly scaled logs.

In analysis of the DENS vs PHIN plots, the shale point is plotted from previously determined values of DENSSH and PHINSH. Using the appropriate plastic overlay, the shale point and mineral trends are verified. To use the DENS vs PHIN plot as a shale indicator, the clean line is defined as for the DENS vs DELT plot. The Z-Plot, Figure 11.48 is useful here.

In certain areas, gypsum may plug dolomite porosity. This can be recognized if a trend is developed from the dolomite porosity toward the gypsum point. The MLITH vs NLITH Plot also aids in detecting such trends.

The PHIN vs DELT plots, Figures 11.49 and 11.50, are important in predominantly dolomite zones for determining shale content. If radioactive materials other than shale are present in dolomite, these plots may be the best shale indicator available for dolomite.

To use the PHIN vs DELT crossplot as a shale indicator, the shale point and clean line are found as described for the DENS vs DELT and DENS vs PHIN shale indicators.

Secondary porosity, with its smaller apparent DELT, produces a PHIN vs DELT plot pattern with a slightly greater slope than those of the mineral lines. This increase in slope can be decreased by gas effect, which reduces PHIN more than it increases DELT. In clean, water-bearing formations, resolution is good for distinguishing between lithologies and for porosity determination.

Figures 11.51, 11.52 and 11.53 are crossplots of PHIxdn vs DELT, where PHIxdn is the porosity derived from the density neutron crossplot. The primary-porosity lines in clean, water-bearing limestone and dolomite are shown. Displacement of a plotted point from the corresponding primary-porosity line results from the following reasons:

- shale effects produce a shift toward the northeast (see shaly trend on gamma ray Z-Plot, Figure 11.52).

- lack of compaction produces shifts to the right

- secondary porosity produces shifts to the left (see points to left of dolomite line)

- rugosity and washouts produce shifts upward, (see Caliper Z-Plot, Figure 11.53) unless density data has been eliminated by caliper and density correction limits.

Note the washed-out hole indicated by the Caliper Z-Plot in the mid-range on the shaly trend.

From the PHIxdn vs DELT plots, the value of PHIMAX may be selected. This is done by looking for the maximum porosity of the clean-formation points. At the same time, the Caliper Z-Plot (Figure 11.53) is checked to make sure these porosity values are not being taken from levels with enlarged hole.

The value of PHIMAX is used in some programs to limit spuriously large values of porosity which may be computed in enlarged hole or in shales with erratic composition. The maximum porosity value in a zone is limited as described in Chapter Seven.

Another limit derived from the PHIxdn vs DELT plots is PHISlim, useful for intervals of low-porosity formations which implode under stress into the borehole. A plot is made of apparent total porosity versus DELT from adjacent sections where smooth borehole conditions exist. A limit line is found which gives for each value of DELT a maximum value of PHIxdn. This value is PHISlim.

In Figure 11.51 the limit line is to the left of the dolomite line. This may be due to secondary porosity, which produces shifts to the left, but could be due to rugosity and washouts which produce shifts upward (although the Caliper Z-Plot seems to rule this out). The maximum porosity in the low-porosity caved zone is limited to PHIsc = PHISlim (for the given value of DELT).

Other crossplots aid in determining fluid parameters. Figure 11.54 illustrates use of these plots in a sandstone reservoir. For determination of RW@FT, a line is drawn from the zero-porosity point (PHIN=0, 1/RESD ^ 0.5 = 0) through the cleanest water-bearing (upper right-hand) points. The line intersects a given value of RESD ^ 0.5 at a given value of PHIN (in the example RESD ^ 0.5 = 1 and PHIN = 0.315). RW@FT is computed from these values using the Archie saturation formula with Sw = 1. (See insert at upper right of figure).

This value of RW@FT is checked against values found in a similar manner from the DENS vs 1/RESD ^ 0.5 and DELT vs 1/RESD ^ 0.5 crossplots. Values of RMF@FT are found in a similar manner from the crossplots made versus 1/RXO ^ 0.5.

A shale trend is seen at the lower right of Figure 11.54. With the help of the PHINSH value previously determined, a value of RSH can be picked. As before, this value is checked against RSH values from other crossplots. RSHS values are determined in a similar manner from the crossplots made versus 1/RXO ^ 0.5.

On these plots, oil-bearing points would be shifted to the left from their corresponding water-bearing position (due to decreased 1/RESD ^ 0.5). Gas-bearing points would be shifted to the left and also upward (due to decreased PHIN).

Some crossplots can be contoured, if the data is well behaved. This is especially useful in comparing units within a single zone. Three such plots are shown in Figure 11.55 through 11.57, in which three units (upper, middle and lower) have been plotted on PHID vs PHIN crossplots. Note that the scales vary on the X and Y axis. These plots show how the choice of shale points (PHINSH and PHIDSH) can be influenced by zoning and other lithology. As well, the shape of the contours suggest that the GR is not a good indicator for shale in the upper and lower units.

11.04 Statistical Analysis of Crossplots
The data groupings and patterns on crossplots can be analyzed statistically to aid interpretation and quantify differences or similarities between zones.

Crossplot programs should provide the ability to plot a least-squares "best" fit line to the crossplot data. These are called regression lines. Three lines are plotted. One uses the X axis as the "independent" axis (that is; Y depends on X). The second using the Y axis as the "independent" axis and the third is halfway between and is called the reduced major axis line. These three lines intersect at the mean of the X and Y data. The mean is the weighted average of the data.

The equations used are as follows:

Slope of Best Fit Line
1: A1 = (Sum (XiYi) - Sum (Xi) * Sum (Yi) / Ns) / (Sum (Xi ^ 2) - Sum (Xi) ^ 2) / Ns)
2: A2 = (Sum (XiYi) - Sum (Yi) * Sum (Xi) / Ns) / (Sum (Yi ^ 2) - Sum (Yi) ^ 2) / Ns)

Intercept on Y Axis
3: B1 = (Sum (Yi) - Al * Sum (Xi)) / Ns
4: B2 = (Sum (Xi) - A2 * Sum (Yi)) / Ns

Equation of Best Fit Lines
5: Y = Al * X + B1 (X dependent axis)
6: X = A2 * Y + B2 (Y dependent axis)

The Reduced major axis regression line is the regression line that usually represents the most useful relationship between the X and Y axes. It assumes that both axes are equally error prone. An approximation to this line is halfway between the two independent regression lines. Solve equation 6 for Y:
7: Y = (1/A2) * X + B2 / A2

Average slope and intercept of equations 5 and 7:
8: A3 = (A1 + 1/A2) / 2
9: B3 = (B1 + B2 / A2) / 2
10: Y = A3 * X + B3 (reduced major axis)

Coefficient of Determination
11: Cd = (Bl * Sum (iY) + Al * Sum (Xi * Yi) - (Sum (Yi) ^ 2) / Ns) /
(Sum (Xi ^ 2) - (Sum (Xi) ^ 2) / Ns)

The coefficient of determination is a measure of "best fit" and is capable of being calculated as data is entered and processed (e.g.: as in a hand calculator). Other measures of fit require two passes through the data - the first to find the average X and average Y values, then a second pass to find the differences between each individual X and the average X, and the differences between the individual Y and the average Y values.

An alternate form of the above equation is:
12: Cd = (Sum (XiYi) - Sum (Xi) * Sum (Yi) / Ns) / (((Sum (Xi ^ 2) - Sum (Xi) ^ 2) / Ns) *
(Sum (Yi ^ 2) - Sum (Yi) ^ 2) / Ns)) ^ 0.5

Both equations give the same answer.

These data are used in the following statistical measures.

Arithmetic Mean
13: Xbar = Sum (Xi) / Ns
14: Ybar = Sum (Yi) / Ns

Variance
15: Vx = Sum ((Xi - Xbar) ^ 2) / (Ns - 1)
16: Vy = Sum ((Yi - Ybar) ^ 2) / (Ns - 1)

Standard Deviation
17: Sx = Vx ^ 0.5
18: Sy = Vy ^ 0.5

Correlation Coefficient
19: Rxy = A1 * Sx / Sy

T Ratio
20: Txy = Rxy * ((Ns - 2) / (1 - (Rxy ^ 2))) ^ 0.5

Skew
21: Ux = (Sum ((Xi - Xbar) ^ 3) / Ns) / ((Sum ((Xi - Xbar) ^ 2) / Ns) ^ 1.5)
22: Uy = (Sum ((Yi - Ybar) ^ 3) / Ns) / ((Sum ((Yi - Ybar) ^ 2) / Ns) ^ 1.5)

Kurtosis
23: Kx = (Sum ((Xi - Xbar) ^ 4) / Ns) / ((Sum ((Xi - Xbar) ^ 2) / Ns) ^ 2)
24: Ky = (Sum ((Yi -Ybar) ^ 4) / Ns) / ((Sum ((Yi - Ybar) ^ 2) / Ns) ^ 2)

Geometric Mean
25: Gx = (PROD (Xi)) ^ (1 / Ns)
26: Gy = (PROD (Yi)) ^ (1 / Ns)

Harmonic Mean
27: Hx = Ns / (Sum (1 / Xi))
28: Hy = Ns / (Sum (1 / Yi))

WHERE:
A1 = slope of best fit line (x dependent)
A2 = slope of best fit line (y dependent)
A3 = slope of best fit line (reduced major axis)
B1 = intercept of best fit line (x dependent)
B2 = intercept of best fit line (y dependent)
B3 = intercept of best fit line (reduced major axis)
Cd = coefficient of determinations
Gx = geometric mean of X values
Gy = geometric mean of Y values
Hx = harmonic mean of X values
Hy = harmonic mean of Y values
Kx = kurtosis of X values
Ky = kurtosis of Y values
Ns = number of X - Y pairs or number of samples
Rxy = correlation coefficient
Sx = standard deviation of X values
Sy = standard deviation of Y values
Txy = T ratio
Ux = skew of X values
Uy = skew of Y values
Vx = variance of X values
Vy = variance of Y values
Xi = individual X data values
Xbar = arithmetic mean of X values
XiYi = product of individual X - Y pairs
Yi = individual Y data values
Ybar = arithmetic mean of Y values

11.05 Lithodensity Crossplots
Lithodensity logs may be analyzed by the simultaneous use of two crossplots. One crossplot is the standard density neutron crossplot with its pure mineral lines in place. The second plot is a density (or density porosity) versus photo-electric effect (PE), with its pure mineral lines also plotted. Data points are plotted on both graphs, with four possible outcomes, as shown in Figure 11.58, and listed below:

PHIN/DENS PE/DENS Must Be
Case 1 QL or QD LD or QL QL
Case 2 QL or QD QL or QD QL or QD
Case 3 QD or LD LD or LQ LD
Case 4 QD or LD QL or QD QD

WHERE:
D = dolomite region
L = limestone region
Q = quartz region

Only Case 2 is ambiguous and a density sonic or sonic neutron plot may help.

See Chapter Nine (Lithology from Lithodensity Log) for other restraints.

A more useful crossplot using lithodensity data is a plot pf matrix density (DENSma) versus matrix capture cross section (Uma). This will require preprocessing of density, neutron, and PE curve data to obtain the "porosity-free" parameters DENSma and Uma:

1: Uma = (PE * DENS - Vsh * USH) / (1 - PHIe)
2: DENSma = (DENS - PHIe * DENSW - Vsh * DENSSH) / (1 - PHIe - Vsh)

Where:
DENSma = calculated matrix density (gm/cc or Kg/m3)
DENSSH = density log reading in 100% shale (gm/cc or Kg/m3)
DENSW = density log reading in 100% water (gm/cc or Kg/m3)
PHIe = effective porosity (fractional)
Uma = computed matrix photoelectric absorption cross section (barns/cm3)
Vsh = shale volume from any method (fractional)

FIGURE 11.58B Lithodensity crossplot DENSma vs Uma

This plot is especially powerful with GR on the Z axis and frequency of occurrence on the W axis.

11.06 Porosity Resistivity Crossplot (Hingle)
The porosity resistivity crossplot is a venerable tool, still used in many areas. In the most common version, porosity is plotted on a linear scale, and resistivity on a scale such that straight lines on the graph represent constant water saturation as determined by the Archie formulae:
1: Sw = (F * RW@FT / RES) ^ (1 / N)
2: F = A / (PHIe ^ M)

WHERE:
A = tortuosity exponent (fractional)
F = formation factor (fractional)
M = cementation exponent (fractional)
N = saturation exponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)
Sw = water saturation (fractional)

This plot is often called the Hingle plot after the man who first publicized the method. The graph requires a special grid, since the Y axis is linear in the function RESD ^ (-1 / M) but not linear in RESD. RESD or COND lines are used to plot and read data points, so these are plotted to fall non-linearly on the graph paper.

On such graph paper (Figure 11.59) the saturation lines fan out from the zero porosity, infinite resistivity point. The 100% water saturation line can be placed by calculating RESD for any positive value of porosity from the Archie formula. Similarly other saturation lines can be placed on the graph. By rearranging the Archie equation we get:
3: RESD = A * RW@FT / (PHIe ^ M) * (Sw ^ N)

WHERE:
A = tortuosity exponent (fractional)
F = formation factor (fractional)
M = cementation exponent (fractional)
N = saturation exponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)
Sw = water saturation (fractional)

If we take A = 1.0, M = N = 2.0, PHIe = 0.1 and RW@FT = 0.25, then:
Sw = 1.0, RESD = 0.25 / (0.1 ^ 2) / (1 ^ 2) = 25
Sw = 0.7, RESD = 0.25 / (0.1 ^ 2) / (0.7 ^ 2) = 50
Sw = 0.5, RESD = 0.25 / (0.1 ^ 2) / (0.5 ^ 2) = 100
Sw = 0.2, RESD = 0.25 / (0.1 ^ 2) / (0.25 ^ 2) = 625

Therefore, for this example we would draw a line from the PHIe = 0, RESD = infinity point to a point defined by PHIe = 0.1 and RESD = 25, to obtain the 100% water saturation line. The 50% water saturation line joins the origin with the point PHIe = 0.1 and RESD = 100 and so on, as shown in Figure 11.59.

If RW@FT is unknown, a line can be drawn slightly above the most northwesterly points on the graph to intersect at the origin and RW@FT back calculated from any point on the line by using:
4: RW@FT = RESD * (PHIe ^ M) / A

WHERE:
A = tortuosity exponent (fractional)
M = cementation exponent (fractional)
PHIe = effective porosity (fractional)
RESD = deep resistivity (ohm-m)
RW@FT = water resistivity (ohm-m)

If sufficient porosity range exists in the water zone, the northwesterly line can be drawn without knowledge of the porosity origin, thus helping to find the matrix point. In Figure 11.59, the data suggests a matrix density of 2.7 gm/cc, so the porosity scale origin is set at this point. If data was in porosity units to begin with, this technique would define the matrix offset to correct the porosity log to the actual matrix rock present.

Any of the three porosity logs, (sonic, density, neutron) or any derived porosity, such as density neutron crossplot porosity, can be used for the porosity axis. Any deep resistivity or conductivity reading can be used on the Y axis.

If shallow resistivity data are available, the parameter RESS*RW/RMF can be plotted below the RESD points. The distance between the RESD and normalized RESS points represents the moveable hydrocarbon - the larger the better.

The manual construction of this crossplot can be summarized as follows:
1. Select proper crossplot paper (Figure 11.60 or Figure 11.61).
2. Scale the X-axis in linear fashion for raw logging parameters (DELT, DENS, PHIN or PHID) and establish porosity scale. Porosity will be zero at the matrix point and increases to the right.
3. Plot resistivity (RESD) vs log data (DELT, DENS, PHIN or PHID). The resistivity scale can be changed by any order of magnitude to fit the log data. This can be done without changing the validity of the graph paper grid.
4. The straight line drawn through the most north westerly points defines Sw = 1.0. Extrapolate this to the intersection with X-axis (PHIe = 0, RESD = infinity).
5. At the intersection, determine the matrix value (DELTMA or DENSMA) for a proper porosity scaling of the X-axis. If logs are in porosity units, this line will determine the matrix offset.
6. Calculate RW@FT from any corresponding pair of PHIe and RESD data along the water line.
7. Determine lines of constant Sw values based on the Archie equation (for any given PHIe value). Keep in mind that all these lines must converge at the matrix point.
8. Read and evaluate Sw values for all points plotted on the crossplot. Make sure points are numbered to avoid confusion, particularly if very long sections are analyzed.
9. As an extension of this method, in case RESS data are also available, the moveable hydrocarbon can be determined by plotting RESD * RW / RMF below each RESD point.

The grid for a Hingle plot is difficult to draw by hand as the resistivity axis is non-linear. Blank forms are available in most service company chart books, as well as here:

FIGURE 11.60: Hingle plot M = 2.00

FIGURE 11.61: Hingle plot M = 2.15

11.07 Resistivity Porosity Crossplot (Pickett)
Since the non-linear graph paper of the Hingle plot is difficult to construct, another style of porosity resistivity plot is popular. It is called a Pickett plot, and both resistivity and porosity are plotted on logarithmic scales.

Again, by rearranging the Archie equation we get:
1: log RESD = -M * log PHIe + log (A * RW@FT) - N * log Sw

When Sw = 1.0, then:
2: log RESD = -M * log PHIe + log (A * RW@FT)
3: M = (log(A*RW@FT) - log(RESD)) / log(PHIe)

This is the equation of a straight line on log - log paper. The line has a slope of (-M) and the intercept when PHIe = 1 is the value of A * RW@FT.

If resistivity increases upward and porosity increases to the right, a line drawn slightly below the south westerly data points should represent the 100% water saturation line (as long as a water zone exists in the interval). If A * RW@FT is known, the line should pass through this point at PHIe = 1.0.

If the cementation exponent M is known, the line can be drawn with this slope to find A * RW@FT. Remember that M is seldom less than 1.7 or more than 2.8 in non-fractured reservoirs. See Chapter Twenty-Nine for details on using this plot in fractured reservoirs.

The slope is determined manually by measuring a distance on the RESD axis (in cm. or inches) and dividing it by the corresponding distance on the porosity axis, or by using equation 3. The result will always be negative.

To construct the other water saturation lines, first draw a line upward from the point where the 100% water saturation line meets the line RESD = 1.0. Then mark points on the vertical line at RESD values of 2.0, 4.0 and 25.0. Draw a line through each of these marks parallel to the 100% water saturation line. These lines are 70%, 50% and 20% water saturation lines respectively.

An example is shown in Figure 11.62, using the same data as in Figure 11.59. Because A and M and the matrix values for rocks are seldom the world wide averages commonly assumed, the porosity resistivity crossplot is often used to find reasonable values prior to or in lieu of special core studies.

If RW@FT varies, this may be noticed by parallel groupings of data belonging to several distinct water zones. The sequence should be zoned to create a separate plot for each different water resistivity value. Comparisons of these plots between wells are often useful. A shift of data in the porosity direction may indicate a mis-calibrated porosity log.

A shift in the resistivity direction may indicate a mis-calibrated resistivity log, differences in invasion, a change in pore geometry, or a change in A * RW@FT. In the example above, the W axis (colour) is coded red for PE near 3.0 and blue for PE near 5.0, thus segregating dolomite from limestone. Note that the porosity distribution and the slope of the line through the red data is different than that through the blue data. This demonstrates that the pore geometry for the dolomite interval is different than that for the limestone. The M value for the dolomite is less than 2.0 for the dolomite and considerably higher than 2.0 for this limestone. (RESD is on the X axis in this plot).

If RW@FT is known from water samples, it may help define the value for A, which varies primarily with grain size and sorting. This is a function of position in the basin and distance from source rock.

Again, as for the Hingle plot, values of RESS * RW / RMF can be plotted to estimate moveable hydrocarbon. If no water zone exists in the interval, plotting RESS vs PHIe may find the slope M, since RESS sees mostly a water filled zone.

Figure 11.63 Example of Computer-drawn Pickett Plot

The Pickett plot in Figure 11.63 shows that cementation exponent (M) varies with lithology. The slope of the line through the dolomite data (red) is less than that through the limestone (blue). For a good estimate of water saturation in both zones, the appropriate value of M must be used in each zone. An average line through this data set will make the high porosity dolomite look too wet. The low porosity limestone would appear not wet enough.

11.08 Cumulative (Holgate) Plots
A Holgate plot is a special crossplot constructed in order to calibrate one log response to another, or to calibrate a log response or computed result with core data. The usual form is a sonic log versus core porosity plot, but any two correlatable properties may be compared. However, the construction is quite a bit more complicated than merely plotting X-Y data as in previous plots. A Holgate plot requires cumulative data over an interval of the formation. For example, assume a series of log or core values such as:

The data is sorted into ascending (or descending) values and placed into cells with discrete ranges:

The crossplot is created by plotting the lower row of numbers (the accumulated number of samples) on the Y axis versus the centroid of the range of data values represented on the X axis. Usually these points are connected by a series of straight lines. If the range of values in each cell is very small, a smooth cumulative curve can be created. This is normally done on a computer.

If two such curves are made, one for a log value, and the other for a core property such as porosity, a calibration curve can be constructed. Assume our previous data reflected core porosity data and the sonic data had the following values:

The resulting calibration would relate the centroid of each range to its corresponding value in the other table. Thus:

A best fit regression analysis on this paired data would generate the equation of the line which calibrates sonic log readings to porosity. The relationship need not be linear.

The data for the two sets of values must come from the same interval of rock, but the two sets do not need to be "on depth" with each other since no actual depth values are used. In fact, an upside-down core will still produce the same log calibration as a right-side-up core.

Although the Y-axis accumulations were a number of samples in this example, the accumulation can be any one of:
- frequency of occurrence (same as number of samples)
- actual thickness
- percent or fractional frequency
- percent or fractional thickness

A compact form of this plot comprises three separate plots on one page, with axes appropriately labeled. The three plots are:
1. Number of samples versus data-type-one accumulated in ascending order
2. Number of samples versus data-type-two accumulated in descending order
3. Values of data-type-one versus data-type-two picked from the accumulated curves at equal intervals

The first two curves will create two "S" shaped curves facing in opposite directions and crossing at their median values. The third curve, when fitted with a regression line, will provide the calibration equation. Many examples are shown in Section 11.02 in this Chapter.

11.09 In Conclusion
Crossplots are part of modern quantitative log analysis methods. They are easy to generate on computers, and are therefore often over-produced, resulting in data overload or confusion. Choose the crossplots you need carefully and generate only those which will materially assist in parameter selection or lithology identification.

I personally use very few crossplots since many parameters can be found by observation of depth plots. Often, a few crossplots are generated to show clients important relationships in their data set.

The most useful crossplots are core permeability vs core porosity, capillary pressure water saturation vs porosity (along with log analysis water saturation vs log analysis porosity), and matrix density vs matrix capture cross section. Other lithology crossplots shown in Chapter Nine are also useful in special circumstances. The Pickett plot is necessary in some jobs if water zones are present. A resistivity vs gamma ray plot may be helpful but you can find everything on this plot from a depth plot.

If you do regression analysis, use the reduced major axis method and a Holgate plot to generate the data set for the regression. RMA gives a more realistic relationship on clustered data, and Holgate eliminates depth shift problems, as long as the same rock interval is covered.

Use the Z and W axes to improve the data quality and information content of your crossplots. Gamma ray, caliper, frequency of occurrence, PE, or Uma are commonly used on these axes to indicate rock type or data quality.

11.10 Exercises For Chapter Eleven
1. Define the following types of plots. (10 marks)
- nomograph
- chart
- crossplot.
- Z-plot or 3-D plot.
- 4-D plot
- scatter plot
- grouped plots
- printer plots
- plotter plots

- graphic dumps
- composite plot
- histogram
- cumulative plot
- Holgate plot
- line charts
- depth plot
- bar charts
- pie charts

2. What are the common statistical measures used to analyze crossplotted data? Describe how to derive the reduced major axis regression line from the X and Y dependent lines. (20 marks)

3. Describe how to construct and use a porosity resistivity crossplot (Hingle Type). (20 marks)

4. What are the differences between a Hingle and Pickett plot? (10 marks)

5. What steps are taken to create a cumulative plot (Holgate Type)? (20 marks)

6. Using the META/LOG spreadsheet and the data for the mixed lithology example in generate crossplots of:
- density vs neutron
- porosity vs water saturation
- core porosity vs core permeability
- core porosity vs log porosity
- log porosity vs resistivity
- Holgate plot of log porosity vs core porosity

Choose appropriate X, Y, Z, and W axes, scales and annotate pertinent comments on each plot. (20 marks)

11.11 Bibliography For Chapter Eleven
1. The Use of Logs in Exploration Problems A.T. Hingle SEG, 1959

2. Crossplotting - A Neglected Technique in Log Analysis T.B. McFadzean CWLS, 1972

3. Pattern Recognition as a Means of Formation Evaluation G.R. Pickett SPWLA, 1972

4. New Developments in Induction and Sonic Logging AIME, 1960

5. A Review of Current Techniques for Determination of Water Saturation from Logs G.G. Pickett JPT, 1966

6. Use of Differential Sonic Resistivity Plots to Find Moveable Oil JPT, 1961

7. Using Log Derived Values of Water Saturation and Porosity SPWLA, 1967

8. The Litho-Porosity Crossplot J.A. Burke, R.L. Campbell, A. W. Schmidt SPWLA, 1969

9. Mixed-Lithology Analysis Using MN Product A. Heslop CWLS, 1971

10. The Fluid Identification Plot R.M. Bateman SPWLA, 1977

11. Well Logging and Interpretation Techniques Home Study Course, Dresser Atlas,1982

12. Log Interpretation Volume 1 - Principles Schlumberger, 1972

13. Log Interpretation Volume II - Applications Schlumberger, 1974 .PA

ABOUT THE AUTHOR

E. R. (Ross) Crain, P.Eng. is a Consulting Petrophysicist and a Professional Engineer with over 35 years of experience in reservoir description, petrophysical analysis, and management. He has been a specialist in the integration of well log analysis and petrophysics with geophysical, geological, engineering, and simulation phases of oil and gas exploration and exploitation, with widespread Canadian and Overseas experience.

His textbook, "Crain's Petrophysical Handbook on CD-ROM" is widely used as a reference to practical log analysis. Mr. Crain is an Honourary Member and Past President of the Canadian Well Logging Society (CWLS), a Member of Society of Petrophysicists and Well Log Analysts (SPWLA), and a Registered Professional Engineer with Alberta Professional Engineers, Geologists and Geophysicists (APEGGA)