Fundamentals of the Pickett Plot

8/11/2019 Fundamentals of the Pickett Plot

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Fundamentals of the Pickett Plot

When Dick Pickett introduced his plot he emphasized that it should not merely be used as agraphical substitute for calculations that could be done with a slide rule. Instead, it was to

be a means for pattern recognition, so that trends and discriminations within the clouds of

crossplotted points could be related to pay zone evaluation and reservoir structure. Eventoday, the same plots are displayed on computer monitor screens as a reconnaissance deviceto recognize reservoir patterns and to validate the values of parameters used in log analysisequations. Newer concepts of textural controls on productivity can be evaluatedimmediately on these plots and used in the distinction of potential pay and assessment oflikely water cut. Contour lines of bulk volume water may be plotted directly as cut-off

boundaries for immediate decisions concerning testing or abandonment. Someinterpretations of permeability are also possible in favorable situations.

These more perceptive uses of crossplots reflect a clearer understanding of the role of poresize and geometry in the determination of reservoir characteristics. This knowledge has

come from geological studies of the relation between rock textures and the depositional anddiagenetic history of the rock, as well as engineering studies of productivity as related to porosity, water saturation, and rock type. New insights have gone a long way to theexplanation as to why some zones that appear to be wet have significant production, whileother, supposedly good pay zones, produce only water.

The principles of the Pickett plot will be illustrated with the hypothetical data set ofresistivity and porosity values logged in the Rottweiler Sandstone described earlier, withArchie rock propertie s and a simple reservoir profile (Figure 1). The example consists ofa pay section at irreducible water saturation (zones A-E), a transition zone (Zone F), abovea water leg (zones G-J). The sandstone has been logged in the hypothetical wildcat , Toto

Rottweiler #1. We will suppose that the log analyst has no idea whatsoever concerningeither the formation water resistivity or any of the other parameter values that he or shewould need to solve the Archie equations. However, the log analyst is very familiar withthe Pickett plot and sets to work to evaluate the section.


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Figure 1: Gamma ray, porosity and resistivity logs of a section of Rottweiler Formation(Triassic), a hypothetical oil-productive sandstone

The Pickett plot is made on conventional double logarithmic (to base ten) scale graph paper. An examination of a crossplot of the porosity and resistivity values of the ten zones(Figure 4) shows a pattern that is easy to interpret. Zones A to E have much higherresistivities than zones H to J, even though they have similar porosities, and suggests thatthese zones may have appreciable hydrocarbon content. Zone F is intermediate betweenthese two clusters, and its depth relationship to them indicates that it may be in a transitionzone. The zone H-J trend of a systematic decline in resistivity with porosity also favorstheir interpretation as possible water zones.


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These qualitative generalizations can be made considerably more specific, using themathematical properties of the graph form and its ability to represent many useful

petrophysical relationships. The basic method for the plot that bears his name wasdescribed by Pickett (1966). In a later paper, Pickett (1973) described in detail the patternrecognition properties of the plot which made it a particularly powerful method for log

interpretation. The mathematics of the Pickett plot are simpler than those of the Hingle plot(Hingle, 1959), and is another transformation of the Archie equation:

Rearranging the Archie equation and substituting the resistivity index, I, gives :

Taking logarithms, the equation becomes: whichdescribes a family of parallel lines for different resistivity index values whose slope is thenegative of the cementation exponent (-m). When the resistivity index, I is unity, the line isthe water line with an intercept equal to a*Rw. Other resistivity lines are displaced to thenortheast and are drawn easily as lines parallel to the water line and with resistivities whichare the water line resistivities multiplied by the index at common values of porosity.

In common with most visual methods, these concepts are more obvious when sketched outgraphically, as in figures 5 to 7. The water line can be established by eye, or numerically,

by stipulating the formation water resistivity, Rw, and the Archie constants of a and m

(Figure 5). Many log analysts prefer to work with the value of a held at a value of unity.Using this convention, the intercept is equated directly with Rw, and the slope of m becomes an average estimate of cementation factor within the reservoir. If a trend of(relatively) low resistivity points is suspected to sketch out a water line, then the exact

position can be established through the selection of water resistivity and Archie constantsthat are most compatible with preexisting knowledge while simultaneously honoring thegraphed data points in a convincing manner. (The use of an interactive graphics computer

program is most effective for this type of operation.)

In our hypothetical example, the intercept of the water line predicts a formation waterresistivity of 0.10 ohm-m and a cementation factor, m, of 1.8 (assuming the a constant to beone). In a real analysis, these numbers would have local implications that could be assessedfor credibility. The cementati on factor of the Rottweiler sandstone implies that thesandstone is moderately cemented; the formation water resistivity would be compared withavailable measurements of Rottweiler water resistivities in the immediate area.


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Figure 4: Plotting zones on a Pickett Plot

Figure 5: Location of the "water line"


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Figure 6: Location of the 50% saturation water line

Figure 7: Location of the 20% water saturation line

The location of other water saturation lines will be on the higher resistivity side of the waterline (which is the 100% water saturation line). They all have the same slope of -m , and soare parallel. The resistivity index, I, for any given water saturation determines the value of


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Rt on the associated line at any porosity as a multiple of Ro on the water line at the same porosity. This concept is explained graphically on figures 6 and 7.

Crossplotted points that lie above the water line have water saturations of less than 100%and complementary hydrocarbon saturations. However, their location on the plot does not

immediately answer the question concerning the fluids the zones will produce when eithertested or perforated. Water free hydrocarbons, water-cut hydrocarbons or water alone areall possibilities. The product of porosity and water saturation is the bulk volume of water(BVW) which can give important clues to producibility when related to pore character andreservoir type.

As an additional feature, each pair of contiguous zones on Pickett plots generated byPfEFFER are linked by a straight line segment. Taken collectively, the lines sketch out atrace that is the reservoir "trajectory" in the depth dimension of the covariation of resistivityand porosity. Trends, deviations, cut-backs and other features of the trajectory giveimportant clues regarding hydrocarbon column structure, reservoir heterogeneity, cyclicrepetition, and changes in pore size.

The Hough Transform Crossplot Method

Although the Pickett plot has many useful properties for pattern recognition, there is stillroom for improvement. When fitting either the water line or a line of irreducible saturation,the resulting values of cementation and saturation exponents are not immediately obvious,

but must be calculated from the slopes. The normal range of porosities also means thatthere is often a fair degree of uncertainty in the estimate of water resistivity whenextrapolating to the intercept at 100% porosity. An alternative approach to this problem isthrough the use of the Hough transform, by plotting the data directly in parameter space.

The Hough transform was introduced by Hough (1962) as a means to detect patterns of points in binary image data. The basic concept is very simple, which is an immediaterecommendation of its power. When searching for trends such as straight lines in a plot ofX Y coordinate data, the original points can be considered to be represented in "imagespace," although the purpose of the analysis is to extract the parameters of any systematicfeatures. If parameters are the goal, then the problem can be simplified by replotting thedata in "parameter space." The basic idea can be clarified by referring to Figure 8. Here thetask is to locate any linear trend through the set of points plotted in X Y Cartesian space. Aline is specified by the equation: Y = a +b X

where is the intercept and is the slope. The objective of the Hough transform then becomesto replot the data in parameter space, rather than X Y image space. Any single X Ycoordinate pair can be "back-projected" as a line on an plot which gives the slopes andintercepts of all possible lines that could pass through this point. The equation for this lineis found easily by rearranging the Cartesian space equation as: a = Y- b X

where X and Y are now fixed and the intercept a and the slope b are variables. Each pointin Cartesian space is now a line in parameter space and the intersections of lines mark


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slopes and intercept values of lines that pass through two or more points in Cartesian space.Therefore, a linear trend through a subset of data on a Cartesian plot will be marked by a

bundle of lines whose intersections are clustered in the area of a representative commonslope and intercept.

Now that the basic concept of the Hough transform has been explained, we will follow itsapplication to the Rottweiler Sandstone data. The parameters of the water line are the waterresistivity intercept (assuming the constant " " to be unity) and the cementation exponentslope. We can combine the concepts of the reconnaissance water resistivity (Rwa) and thereconnaissance cementation exponent in a single equation: log(Rt) = log(Rwa) - ma*log(

This equation describes every possible line that could be drawn through the resistivity- porosity coordinates of either a water zone or a hydrocarbon zone. When mapped into the parameter space of water resistivity and cementation exponent, each zone that plots as a point on the Pickett plot becomes a line on the Hough transform. Water zones should berecognized as a bundle of sub-parallel lines with lower Rw and m values than other zones.

Ideally, the water zone lines would have a mutual intersection at the coordinates that locatethe true formation water resistivity and cementation exponent. In reality, the ragged trendthat characterizes most water lines results in a range of possibilities which is an accuratereflection of the spread of alternative lines that could be drawn as possible water line on theoriginal Pickett plot. The advantage of the Hough transform is that it immediately gives thevalues of all possible combinations of Rw and m and, since both of these have

petrophysical meaning, this aids the analyst in his/her selection of the most likely pair. Thisdiscussion can be understood by thoughtful examination of the Hough transform of theRottweiler Sandstone data shown in Figure 9.

Although the Hough transform is a pattern recognition method in its own right, itsapplication to the Pickett plot as described here can equally be considered as a "universalRwa plot" in which all possible values of cementation exponent within a petrophysicallyreasonable range are applied successively in the computation of corresponding Rwa valuesfrom zone resistivity-porosity log values. This is the spirit in which the Hough transformoption is implemented in PfEFFER in which the cells of a grid of Rw (rows) and m(columns) are filled with frequencies of occurrence of values generated when mappinglogged zones into the parameter space. As a result, the Hough transform lines are sampleddiscretely and individual cells will tend to have higher counts at parameter valuecombinations where several lines converge or intersect. However, as stated already, theinterpretation of these plots should be made with reference to what is known and/or what isreasonable concerning the reservoir formation's cementation exponent and water resistivity.


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Figure 8: Simple example of a Hough transform applied to the projection of points in Cartesian "image space parameter space of slope and intercept.


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Figure 9: Hough transform of the Rottweiler Sandstone resistivity-porositydata into the parameter space of formation water resistivity (Rw) and Archieequation cementation exponent, m.

Fundamentals of the Pickett Plot

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