A Method of Averaging Capillary Pressure Curves

7/28/2019 A Method of Averaging Capillary Pressure Curves

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SPWLA FIFTEENTH ANNUAL LOGGING SYMPOSIUM, JUNE 2-5,

A METHOD OF AVERAGING CAPILLARY PRES SURE CURVESby

G. M. HESELDIN - Shell Canada Limited - Calgary, Alberta, Canada

ABSTRACTCapillary pressure data can be useful to calculate reservoir sat-

urations provided a realistic average capillary pressure curve can be derivedfor a particular rock type. Since capillary pressure characteristics are knownto vary with porosity d% permeability, a mathematical procedure is developedwhereby the resulting average curve is a single relationship dependent on thatparameter. This averag& curve is especially useful in evaluating low porositycarbonate reservoirs as well as providing a basis for generating one or morerepresentative capillary pressure curves for reservoir simulation.

Capillary pressure data must be one of the most neglected sourcesof information for determining reservoir saturations. Many so-called watertable anomalies can usually be resolved with the help of capillary pressuredata which more often than not has already been measured and filed away. Thisis not to say that capillary pressures have never been used in a quantitativemanner, as they often have been a factor in determining Unit equity in lowporosity carbonate reservoirs where the characteristic high resistivitiesnecessary for conventional water saturation calculations are difficult tomeasure. Typical examples of fields in Western Canada where capillary pres-sures have been used to calculate reserves and/or equities are:

Oil GasHarmat~ East Jumpin~ound WestHarmattan Elkton Lookout ButteHomeglen-Rimb ey RigelSimonette WatertonSturgeon Lake Wildcat HillsWeyburn

If it is accepted that capillary pressure data can be useful tocalculate saturations, why are these applications not more common? One reasonmust be the difficulty in combining the widely varying shapes of curves toestablish a single representative curve for a particular rock type. This paperpresents a mathematical procedure to derive a single capillary pressure rela-tionship from which a saturation can be calculated for a given rock type at theappropriate subsea depth; the rock type can be expressed as a continuouslyvarying parameter such as porosity or permeability. This relationship can alsobe used as a basis for generating one or more representative capillary pressurecurves for reservoir simulation.

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SPWLA FIFTEENTH ANNUAL LOGGING SYMPOSIUM, JUNE 2-5, 1974

STATING THE PROBLEMIt is an easy task to average the four capillary pressure curves

shown in Figure 1 at high values of capillary pressure (Pc); likewise it isalso easy to derive an average entry pressure at 100 percent air saturation.However, the shape of the average curve between these end points is subject toconsiderable interpretation and therefore will vary according to the personalpreferences of the engineer.

When data varies as much as these four capillary pressure curves, thequestion always should be posed as to whether an average curve would have anymeaning at all. Clearly, the rock depicted in Figure 1 with the highest entrypressure (Curve D) obviously has poorer reservoir potential than the others.Assuming all four curves were measured on essentially the same rock type, thenit would be expected that the porosity and/or permeability of this sample wouldbe less than that for the other three. If this poor quality rock does not, forexample, contribute to reservoir net pay, then it should not be included in theaverage curve which is going to be applied to net pay. In this case it is rel-atively easy to combine the remaining three curves to come up with a compositeaverage.

On the other hand, if all the four curves in Figure 1 are potentialreservoir rock and have the expected trend of generally increasing porosityor permeability from the poorest (Curve D) to best (Curve A) capillary pressurecharacteristics, then the data must be presented as a function of that varyingparameter. Although permeability would be the logical choice of the parameter,the resulting average curve could not be applied to a non-cored interval; there-fore the more practical choice is porosity (since it can be measured by logging)and this will be used as the rock quality parameter for the remainder of thispaper.

DISPLAYING CAPILLARY PRESSURE DATAIncluding porosity as an additional parameter has increased the

dimensions of the analysis by one. Consequently the method of displaying thedata is important and the two presented in Figure 2 are recommended. By ex-pressing the saturations in terms of bulk instead of pore volume, the largescatter of saturation values usually observed at low porosities has a minimaleffect on the overall trend. Of the two displays, the plot using bulk volumeoccupied by the non-wetting phase is preferred for the reason that the trendthrough most reservoir data will have the smallest change of curvature makingit more suitable for curve fitting methods.

Figure 3 shows an example of one of these plots at a specific capillarypressure which is within the range to which the average relationship will beapplied. Note that samples which were interpreted as having no mercury in-jected at the specific capillary pressure are not included in the plot; thoseobserved to be on the zero line actually had small quantities injected. Alsoshown on this plot are two least square fit lines to the data; the displaced

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rectangular hyperbola not only appears to fit the data points better, but alsohas a lower standard error of estimate than the second order polynomial. Howeverthe type of line chosen to fit the data is immaterial, as the proposed averagingprocedure can handle whatever relationship the engineer prefers, albeit a straighline, constant water-filled porosity line or sophisticated curve.

Having decided which line to use, the process is repeated for variousvalues of capillary pressure as shown in Figure 4. Note how these linescontain minor irregularities in shape from one another. For instance, theporosity separation between the Pc = 200 and 400 lines is almost constantthroughout the range of bulk volume occupied by mercury whereas elsewhere theseparation decreases with increasing bulk volume occupied. These inconsistenciesare probably not characteristic of the rock type but are due to non-randomsampling of the capillary pressure plugs. The following averaging processeffectively smooths out these inconsistencies and presents the series of curvesin the form of a single mathematical formula.

THE AYERAGING PROCESSThe equation of a displaced rectangular hyperbola has two constants,

one describing the shape of the curve and the other its position. The sum ofthese two constants happens to be the entry porosity or intercept on theporosity ordinate in Figures 3 and 4. These constants can be plotted againstcapillary pressure as shown in Figure 5. The proposed averaging processsmooths out the inconsistencies in these constants by expressing them as asimple function of capillary pressure. From Figure 5 it will be seen that thetrends of the constants are essentially linear with the logarithm of capillarypressure and two further least square fits minimizing the deviations of theconstants completely define the capillary pressure system.

By using the lines to determine what the two constants should be, adisplaced rectangular hyperbola can be drawn for ~ value of capillary pressure.Thus these two lines represent a composite set of average capillary pressurecurves for the reservoir. Figure 6 shows the average hyperbola correspondingto the same capillary pressures at which the original least square fits wereobtained. Note how there is now a gradual change in shape and position of thecurves from high to low capillary pressure.

SOME COMMENTS ON THE AVERAGE CURVES

The average curves can bepercent saturation versus capillaryshown in Figure 7. Note how better

more conveniently displayed in the form ofpressure for various porosity values asporosity rock can still contain hydro-

carbons when poorer rock is 100 percent water bearing at the same capillarypressure, i.e. same structural elevation. For example, at a capillary pressureof 100 psia rock less than 6.9 percent porosity is statistically 100 percentwater-bearing, whereas rock greater than 7.9 percent could qualify for pay usinga 60 percent water saturation cut-off criterion.

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Of course an appropriate relationship between capillary pressure andheight above the free water table (zero capillary pressure) must be determined.The free water table must also be determined from log, oil base core and/orproduction test data; the free water table is constant in a common aquifer andshould not be confused with the 100 percent water level which will be higher thanthe free water table and vary with porosity.

This analysis was done using two constants to define the shape ofthe curves. If the second order polynomial had been used, then there wouldhave been three constants to plot against capillary pressure. The constantwater-filled porosity line is a 45-degree line through the data and is definedby only one constant. If the constant water-filled porosity concept had beenused, the average curves shown on Figure 8 would be developed using the same data.However, the data in Figure 3 does not appear to fit a 45-degree line which wasconfirmed by a higher standard error of estimate from the least square fit.Accordingly the curves shown on Figure 8 are not as representative of thereservoir under analysis as the displaced rectangular hyperbola curves givenin Figure 7.

For reservoir simulation purposes, a single representative capillarypressure curve can be obtained using the field average porosity, or alternatively,several curves may be derived for different porosity ranges.

A COMPARISON WITH INDUCTION LOG SATUMTIONSThe free water table was chosen in the example reservoir by matching

induction log with average capillary pressure saturations in the transitionzone. The whole reservoir was then re-evaluated using average capillarypressure saturations and an example well evaluation is given in Figure 9.The lowest interval in this well actually produced oil even though it wassome 20 feet below the accepted field water level which was determined fromlogs and drill-stem tests; in this instance the field water level appears tohave been based on the 100 percent water level for rock of lower porosity.Note how the capillary pressures confirm this pay as well as effectively matchthe remaining saturation values within the accuracy of the induction log andthe statistical variation of the average capillary pressure curves. As wouldbe expected the lower porosity and tighter rock have higher water saturationsby both measurements.

CONCLUSIONSA procedure for averaging capillary pressure curves has been developed.

Minor water level anomalies in an example reservoir can be resolved using thederived average capillary pressure curves. The agreement between water sat-urations calculated by two independent methods gives confidence in both theinduction log values as well as those derived from capillary pressures. Theaverage curves can also be used to represent typical capillary pressure character-istics in reservoir simulation.

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A Method of Averaging Capillary Pressure Curves