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    2ATURATION ESTIM TIONINTRODUCTION

    The cornerstone of the saturation interpretation of resistivity measurements isthe evaluation of the Archie relationship, which was presented in Chapter 3.Because of its simplicity, it has many shortcomings, and it is not directlyapplicable to shaly formations. And despite its simplicity, its application topractical interpretation problems is not always straightforward; constantsappropriate to the formation must be determined. In clean formations, this isrelatively easy to do. Two graphical solutions of the Archie relation fordetermining the water resistivity and saturation are examined in this chapter.One of them also permits the determination of the cementation exponentappropriate to the given zone.There are no such clear-cut methods for determining saturation in shalyformations. Dozens of different prescriptions exist. As a background to thisbewildering array, this chapter considers in detail the effect of clay onresistivity measurements, alluded to in earlier chapters. The observationalresults of the influence of clay on rock resistivity is discussed, along withmodels to explain the behavior.Many of the saturation equations for shaly formations are based onempirical observations and are of limited validity, despite good predictivesuccess in certain applications. Two current models which are based on animportant property of clays (the cation exchange capacity) re also discussed.

    471

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    472 Well Logging or Earth ScientistsCLEAN FORMATIONSThe basic interpretation problem, given the corrected resistivity of theuninvaded formation Rt and the porosity @ is in the evaluation of the Archierelation. In its simplest form, it can be written as:

    The first question, in a practical application, concerns the value to use for R,.This may present a problem if there is no proper SP development, as is oftenthe case. In addition, if the matrix values for the formation are not known,there may even be some considerable doubt about the porosity values to beassociated with the resistivity values measured. Finally, there can beuncertainty about the cementation exponent to be used. In the simple caseabove, it was taken to be 2 (i.e., F = - . However, this need notalways be the case.

    There are two graphical methods available for interpreting the watersaturation of a zone when R, is assumed to be constant but unknown. Thebasic measurements necessary are R, , corrected for environmental effects, anda porosity log (usually density or sonic). A further requirement is thepresence of a few water-bearing zones of different porosity in the loggedinterval, and, of course, the formations of interest must be clean (shale-free).

    The first cross plot technique to be considered is the Hingle plot.' In thiscase, assuming that a density or sonic measurement is available, even if thematrix values are unknown, a plot can be constructed which will giveporosity and water saturation directly. This expedience, coupled with theease with which sonic and resistivity logs can be run in a single pass, hascontributed much to the success of the inductiodsonic logging combination.

    To see the logic behind the Hingle plot, note that the simplified saturation

    a [email protected] 4

    1expression of Eq. (1) indicates that will vary as t a fixed value ofwater saturation, assuming, of course, that the water resistivity is constant.This leads to the construction of a plot, shown in Fig, 20-1, of inverse squareroot of resistivity versus porosity. It is obvious that formations of constantwater saturation will lie on straight lines. Since we can rewrite Eq. (1) as:

    it is clear that the lOO%-water-saturated points will fall on a straight line ofmaximum slope. Less-saturated points, at any fixed porosity, must have alarger resistivity and thus fall below this line. Once these points have beenidentified, the line corresponding to S, = 100% can be drawn, as shown inFig. 20-1. It is relatively easy to construct lines of the appropriate slopescorresponding to partial water saturations.

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    Saturation Estimation 473

    At (pseclft)0 10 20 30 40-6

    Figure 20-1. The Hingle plot, which combines resistivity and porosity (in this casethe At measurement) to estimate water saturation. Implicit use ismade of the square-root saturation relation in constructing the chart.

    The value of R, can be determined immediately from inspection of thegraph. In the construction of Fig. 20-1, the uppermost line corresponds toR,, since it is fully water-saturated and satisfies the relationship:% 1

    R w $2F = - = -

    The implication is that at a porosity of 10% the indicated value of R, will be100 times the value of R,. For the example given, the value of R, at 10 PUis 12 Q-m, which indicates that the water resistivity is 0.12 a.m.In the case of unknown porosity values, the horizontal axis may be scaledin the raw log reading: At or Pb. The intersection of the R., line with thehorizontal axis (corresponding to an infinite resistivity) will give the matrixvalue for constructing a porosity scale.

    The second useful graphical technique is the result of work by Pickett. Aknowledge of porosity is required, but the values of m (the appropriatecementation exponent), R,, and S, can be obtained. In this method, thepower law expression for saturation is exploited by using log-log graph paper.

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    Saturation Estimation 475line by shifting a line parallel to the S, = 1 line by a factor of four inresistivity, and the 25%-saturation case by shifting another factor of four, andso forth. A n example of the result of this procedure is shown in Fig. 20-2.As an exercise in applying these two techniques, let us compare them in asimple interpretation. The logs to be used for the evaluation are shown inFigs. 20-3 and 20-4. In the bottom section, four clean zones are indicated.With reference to the Rx,,/Rt and SP overlay in track 1 of Fig. 20-3, the lowerthree zones seem to indicate the presence of hydrocarbon. Zone 4 appears tocontain water. How can this be confirmed?

    deep.. ........... ........... ..... ..... ... ..... .Rx, QW1 10 100 1 DO

    Figure 20-3. A resistivity log with zones for evaluation indicated. Adapted fromHil~hie.~

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    476 Well Logging for Earth Scientists

    Figure 20 4. A companion density log for use in the Pickett plot and Hingle plot.Adapted from H i l ~ h i e . ~

    Since the formation rock type is not given, the porosity scaling for thedensity log (Fig. 20-4) is not obvious. However, the Hingle plot can be usedto determine the saturation of the four zones in question. First, a number ofpoints of resistivity and pb are selected from the upper section of the log inthe clean section, where the RJR, and SP indicate water-bearing formations.These establish the 100%-saturation line. format, withdensity along the linear scale. (Note that the resistivity on this figure may beby any multiplicative factor desired to accommodate the log readings.) The100%-saturation line is quite easy to identify, and with appropriate scaling thesaturations of the four zones of interest can be determined. Zone 4 seems tohave a water saturation of about 40%, or the same as zone 2, which appearsless promising on the resistivity log.

    Fig. 20-5 shows the selected points, in the inverse

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    Saturation Estimation 477

    Figure 20-5. Hingle plot for selected data from the logs of Figs. 20-3 and 20-4.Points selected as water-bearing have been used to define theS = 100% line. Points representing the four suspected hydrocarbonwnes are indicated.

    Using the porosity scale, which can be derived from the Hingle plot, asimilar analysis can be made using the Pickett technique, which is left as anexercise. The points corresponding to the water zones will nearly fall on astraight line, but with a slope nearer 1.4 than the value of 2 which wasimplicit in the graphical construction of the preceding analysis. Constructingthe saturation lines parallel to the line defined by the water zones willindicate that the water saturations of zones 2 nd 4 are very similar and closeto a value of 40%.

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    478 Well Logging for Earth ScientistsSHALY FORMATIONSBefore reviewing the variety of saturation equations used in the analysis ofshaly formations, let us consider why they are necessary. The reason is thatfrequently real rocks do not follow the simple conductivity behavior describedby the Archie equation. Before considering the complication of partialsaturation, it is useful to see how complicated the behavior of fully water-saturated rock samples is.

    The experimental data which confirm that there is an additionalcomplication in the interpretation of the resistivity of clay-bearing or shalyrocks is given in Chapter 7 (Figs. 7-13 and 7-14). Fig. 20-6 shows theconductivity of the fully saturated rock as a function of the saturating waterconductivity. The clean sand response, shown as a dashed line, represents theArchie relation; its slope is the reciprocal of the formation factor F. At largevalues of water conductivity, the response of the shaly formation is seen to besimply displaced with respect to the Archie-type behavior. This additionalconductivity associated with the clay can be put into the Archie relation as:

    X ,c,c =where X, he additional term that results from shale, must decrease to zero asthe clay content vanishes. Above some value of water conductivity, it simplyappears as a linear shift. The slope of the line beyond this region yields thesame formation factor F as would be obtained for the rock without thepresence of clay. However, it is seen that at very low values of water salinitythere is a nonlinear region in which the additional clay conductivity appearsto be a function of C,.

    Non- I Linear

    c o

    Figure 20-6. Schematic behavior of the conductivity of water-saturated shaly rocksshowing the nonlinear behavior at low values of water conductivityand offset at higher values. From Worthington

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    Saturation Estimation 479

    302015-10-8 -6 -

    To further illustrate this problem, Fig. 20-7 shows the results of somemeasurements of the formation factor as a function of porosity for somesandstone core samples. In the top figure, the samples have been saturatedwith very saline water (C, is large). The behavior for these shaly samples isas expected for clean cores. There is a definite porosity relationship for theformation factor. Shown in the lower portion of the figure are results ofmeasurements on the same cores, this time with quite fresh saturating water.In this instance, the apparent formation factors are seemingly quite random.The intrinsic conductivity of the shale portion dominates the conductivity inthis case.

    --

    0

    0

    410 15 20 30Porosity 010)

    l10 15 20Porosity (010)

    Figure 20-7. Data from Worthington which demonstrate the influence of claycontent at low values of water cond~ct iv i ty .~he formation factorsof the cores in the upper figure were determined with highlyconductive saturating water, the lower set with very fresh water. Thespread in the lower set is a reflection of the clay content and cationexchange capacity of the rock.

    From this we can conclude that the abnormal electrical behavior of shalysands is of minor importance when the resistivity of the formation is low.However, it is much more important when the sand is saturated with a dilutebrine or when the saturated sample contains a large fraction of nonconductive

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    480 Well Logging or Earth Scientistshydrocarbon. Because of this last point, a number of empirical techniques tocope with the clay-affected resistivity measurements have been developedover the years.

    Vsh ModelsThe first model for attempting to quantify this shale conduction term, and theonly one which we will consider in any detail, applies the simplified Archierelation:

    Wc, = 4 wFto the shale. When all of the interstitial electrolyte is replaced by a wettedshale which completely fills the porosity, the volume of shale, Vsh will beequal M . It will have, by analogy with the equation above, an additionalconductivity of magnitude Vh2Ch, where Gh is the conductivity of thewetted shale. Thus the completely water-saturated conductivity of a shalyformation can be written as:

    C WFx =- v&, .Wc =The interpretation task is to evaluate the shale conductivity and the volumefraction contained in a given formation in order to correct the resistivityreading for the perturbation due to the shale.

    Table 20-1. Four empirical conductivity relationships for fully water-saturatedrock. The shale content of the rock is described by a single bulkparameter V From WorthingtonTable 20-1 presents four of the general types of vsh models which havebeen developed over the years to cope with local situations. Of interest is the

    third expression, attributed to an unpublished work by H. G. Doll. It seemsto be obtained from the first expression by simply taking the square root ofeach of the terms. If the Doll expression is then squared, the result is:

    c, = v*2c* * v * g f i .

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    Saturation Estimation 481Despite the fact that there seems to be no logic for the procedure, thecross-term provides the ability to match the behavior of the conductivity atlow salinities in the nonlinear region of Fig. 20-6.

    The problem with the four V, models presented here is that C,, theshale conductivity, needs to be altered to fit the linear and nonlinear regions.From this sampling it should be noted that there is no universal V,, model tofit all interpretation needs. Different approaches work in differentcircumstances. Most importantly, these V, models do not take into accountthe mode of distribution of the shale or any other physical attribute of theshale. In fact, the models may be partially driven by the method employed todetermine V,.

    Effect of Clay Minerals on ResistivityAs briefly mentioned in Chapters 7 and 19, one of the most importantproperties of shale in terms of electrical effects is the cation exchangecapacity, or CEC. The CEC of a shale is related to the ability of the shale toadsorb electrolytic cations, such as Na, onto or near the surface. Anotherfactor which must be related to this adsorption is the amount of surfaceaccessible to the electrolyte. This is a combination of the specific surfacearea of the clay minerals and their distribution.

    Figure 20-8. A conceptualization of the distribution of clay in a sand which affectselectrical conductivity. From Winsauer et al?The mechanism for the excess conductivity was first proposed by

    Winsauer et al. in 1953 and is illustrated in Fig. 20-8. In this representation,

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    482 Well Logging for Earth Scientiststhe clay is seen to coat some of the sand grains. The negative surface chargeof the clay platelets causes the attraction of the positive Na cations. Thus inthe regions of fluid close to the clay there will be a much larger concentrationofpositive charge carriers than in the rest of the solution. The conductanceof this "near" fluid will be higher than that of the undisturbed portion of thefluid. This visualization is usually referred to as a double layer model andgives rise naturally to the image of two conduction paths for electricalcurrent: one through the unbound "far" fluid and another through the moreconductive "near", or bound, layer.

    Region Region,

    i Y :i.h.p. 0.h.p.A- :Anion c Cation - Water Dipole

    ErE0.-calsc0-., Distance From Surface-

    Figure 20-9. The distribution of water molecules and electrolytes near the surfaceof a clay crystal with excess negative surface charge. The lowerportion of the figure indicates the distribution of ions as a function ofdistance from the surface. Increased conductivity is expected in thelayer next to the clay as a result of the increased cation density. Thedistance at which equilibrium between cation and anion densities(e nd n+ s reached depends on temperature and water salinity.Adapted from Yariv and Cross: and Berner?

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    Saturation Estimation 483Electrical double layer theory, known to physical chemists long before its

    application to shaly formations, attempts to describe the distribution of ions insolution near the interface of a solid. For our purposes, the solid is a claycrystal with excess negative surface charge. Such a region is sketched in theupper portion of Fig. 20-9. Several distinct regions of the electrolyte havebeen identified. In region A, there are preferentially oriented water moleculesand some unhydrated exchangeable counter ions. The next region, referred toas the outer Helmholtz layer, is the limit of closest approach of fully hydratedcations. Beyond this layer is a diffuse region where the concentration ofcations decreases until reaching equilibrium with the bulk of the electrolyte.The lower portion of Fig. 20-9 schematically shows the ion concentrationnear the surface, indicating an excess of cations which tails off to theequilibrium value. It is this region, whose dimensions are temperature andsalinity dependent, which is considered to provide an additional path ofconduction in shaly formations.

    Double Layer ModelsThe recognition of the importance of the CEC spurred the development ofanother set of models for dealing with clay effects on conductivity which area bit more sophisticated than the V, models. The first of these, proposed byWaxman and Smits of Shell, used the concept of CEC directly to explain theincreased conductivity.* Rather than use CEC, it is more convenient to definea new quantity, Qv, which is the CEC normalized to the pore volume. Thedefinition of Qv is given by:

    CEC P b ( 1 4 )Q v =Since the dimensions of CEC are in meq/g, the dimensions of Q are inmeq/cm3.The Waxman-Smits model then expresses the conductivity relation as:

    Cw BQv+- F 'o=where B is the conductance of the Na cations. This quantity has beenmeasured in the laboratory, and is known as a function of temperature andconcentration of the NaCl solution.In this model, B is not a constant and must change at low values of Cw inorder to fit the nonlinear region of the conductivity data. Waxman andThomas find that it can be related simply to an exponential function of thewater conductivity? The most important shortcoming is that, at present, thereis no direct way to obtain CEC from log measurements aside from inducedpolarization measurements and the geochemical inference discussed in the lastchapter." For the present, CEC values must be painstakenly obtained from

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    484 Well Logging for Earth Scientistsmeasurements on cores or estimated from other logging parameters.

    Another model, the "dual water" model, has been developed to rectifysome of the deficiencies of the Waxman-Smits model." In thisrepresentation, the authors view the clay as attracting not only the Na cations,but also a layer of polar water molecules, as illustrated in Fig. 20-9. In thismanner, the porosity is viewed as having two components, a bound watercomponent which is directly in contact with the clay, and a free componentwhich is not associated with the clay particles.

    This model also holds the shale-bound water to have a certain invariableconductivity (although there is a temperature dependence) which is notdependent on the type of clay. In contrast to the Waxman-Smits approach,the conductivity of the enriched layer of Na ions near the surface is dilutedby the presence of the bound water. A crucial parameter in the dual watermodel is the fractional portion of the porosity which is bound to the clay. Tocompute this volume, one considers the thickness of the cationenriched layerand multiplies by the specific surface area of the clay. Drawing upon thecorrelation between specific surface area and CEC, the fractional volume ofbound water is directly proportional to Q,,. The dual water modelrepresentation of conductivity is given by:

    Cw (Cbw-Cw)vqQvF O F O

    co 9where Cbw is the conductivity of the bound water, and vq is the constant ofproportionality between the fractional pore volume of bound water and Q,.Dewan details the steps for applying the "dual water" model to a logexamp1e.l2

    Saturation EquationsSo far the discussion has been centered on the description of the electricalbehavior of fully water-saturated rock. What will be the effect of changing

    water saturation by introducing nonconductive hydrocarbon? The result mostcertainly will depend on the details of the fluid distribution. If thehydrocarbon is distributed in irregular drops surrounded by water-wet porewalls, it seems difficult to imagine a generalized relationship between theresistivity index /RJ and saturation which works in al l ranges of porosityand pore distribution systems. However, empirical observations seem toindicate that the saturation index is independent of porosity and has a valueof about 2. In the absence of definitive physical models for the understandingof saturation relations in clean rocks, we have but one recourse for clay-bearing rocks: We must resort to the empirical Archie relation:

    c,c = s; ,F

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    Saturation Estimation 485where n is the saturation exponent. If only the clean portion of theconductivity is expected to change by reducing the saturation, we mightexpect the general saturation equation to be given by:

    c,ct= -g xRHowever, experimental evidence indicates that the shale conductivity termcan be considered to vary with saturation and is generally expressed as:

    FCWct = -s: +xx

    where s is the shale saturation exponent. It has also been recognized that adecrease in the water saturation increases the importance of the electricalXdouble layer. The enhanced shale conductivity is often approximated by -Consequently the generalized saturation equation above usually has a shaleconductivity term which is linear in Sw.

    Worthington has made a comprehensive survey of the more than thirtysaturation equations which have been used over the years A summary of thefour basic types of saturation equations is given in Table 20-2. Of the fourtypes listed, only one, that based on the double layer model, is less empiricalthan the others. Why are there so many approaches? Certainly one reason isrelated to the good predictive performance in localized applications. Onewonders, also, if the number of models proliferated through some basicmisunderstanding of the porosity derived from the neutron-densitycombination. We have seen how different clay types can affect the neutronporosity, and thus the cross-plot-derived porosity. Another reason has beenthe drive to put some saturation models on a sound scientific basis. But forthe moment this has not entirely been achieved.

    s w

    G - p + Xc,= S.2 .+xs:G =* aa=-g&f i+m

    Table 20-2. Four basic saturation equations written in terms of the excessconductivity due to the contained shale X. Th e Waxman-Smits and"dual water" model use an equation of the second form.

    It is clear that the CEC is one reasonable way to approach the correctionof shaly resistivity measurements, despite the difficulty in obtainingreasonable values to use for CEC. It is probably through the use of such aparameter that a unified approach can be obtained. As an example of this,

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    486 Well Logging for Earth Scientists

    1 o0.95

    tFa 0.1.-F

    Worthington has contributed the following observation If we rearrange thebasic conductivity equation:Ic o = - + x ,Fwe can obtain the following: co-x c w 1- = - -co F c o

    cCOwhere the term s recognized as the apparent formation factor Fapp.Thus the ratio of Fap+F represents the fraction of total conductivity whichcannot be attributed to shale. Plots of the ratio as a function of the waterconductivity for four sets of data with widely varying clay content (and thuswidely varying Q,) are shown in Fig. 20-10. It is obvious that the fourcurves resemble one another almost as if each were shifted according to theCEC value. Casting the available data into this framework may provide aunifying basis for comparing further work. However, the definitive work onsaturation determination from electrical measurements has not yet appeared.

    - .Q v = O . O O ~ 0.023

    1.470.17I0

    Figure 20-10. An attempt to demonstrate the generally observed trends ofconductivity in clay-bearing rocks. The ratio of apparent formationfactor to F is the fraction of the total conductivity not attributed toshale conductivity. It is plotted as a function of saturating waterconductivity for four core samples of varying clay content, quantifiedby a particular value of Qy. he curves have a similar shape andseem to be offset by a value related to Qv. rom Worthington

    References1. Hingle, A. T., "The Use of Lqy in Exploration Problems," Twenty-ninth Annual International Meeting of SEG, Los Angeles, 1959.2 Pickett, G. R., "Acoustic Character Logs and Their Application," JPT,June 1963.

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    Saturation Estimation 4873.4.5.

    6.7.8.9.

    10.

    11.

    12.

    Hilchie, D. W., Applied Openhole Log Interpretation D. W. Hilchie,Golden Colorado, 1978.Worthington, P. F., The Evolution of Shaly-Sand Concepts inReservoir Evaluation, The Log Analyst Jan.-Feb. 1985.Winsauer, W. O., and McCardell, W. M., Ionic Double-LayerConductivity in Reservoir Rock, Petroleum Transactions, AIME, Vol.198, 1953.Yariv, S., and Cross, H., Geochem istry of Colloid Systems fo r EarthScientists Springer-Verlag, Berlin, 1979.Berner, R. A., Principles of Chemical Sedimentology McGraw-Hill,New York, 1971.Waxman, M. H., and Smits, L. J. M., Electrical Conductivities inOil-Bearing Sands, SPEJ, June 1968.Waxman, M. H., and Thomas, E. C., Electrical Conductivity in ShalySands, I. The Relation Between Hydrocarbon Saturation andResistivity Index, 11. The Temperature Coefficient of ElectricalConductivity, Paper SPE 4094, SPE-AIME Forty-seventh AnnualMeeting, 1972.Vinegar, H. J., Waxman, M. H., Best, M. H., and Reddy, I. K.,Induced Polarization Logging-Borehole Modeling, Tool Design andField Tests, SPWLA Twenty-sixth Annual Logging Symposium,1985.Clavier, C., Coates, G., and Dumanoir, J., The Theoretical andExperimental Basis for the 'Dual Water' Model for the Interpretationof Shaly Sands, SPE Paper 6859, 1977.Dewan, J. T., Essentials of Modern Open-hole Log InterpretationPennWell Publishing Co., Tulsa, 1983.

    Problems1. The Hingle plot of Fig. 20-5 shows the fully water-saturated linecorresponding to sandstone (since the intercept is at 2.65 g/cm3).

    a.b.

    Estimate the value of R,.Two of the data points at low apparent porosity do not lie on thetrend line for S =O Give several plausible reasons to explain thepositions of these two points on the plot.Redraw the Hingle plot assuming that the matrix is limestone.How different are the saturation estimates for zones 1 and 3?c.

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    488 Well Logging for Earth Scientists

    23

    2. Using estimates of porosity from Fig. 20-5, use the Pickett plottechnique of analyzing the resistivity data. Graph ically determine thefollowing:a. water resistivityb. cementation exponen tc.

    3. The table below lists the values of Rt and At observed in a number ofclean zones in a well with zones of hydrocarbons, as well as some waterzones of different po rosities.a. Using the Hingle plot technique on the graph paper of Fig. 20-3,

    determine the value of R, to be used in the analysis of thesuspected hydrocarbon zones.From the plot of part a, what value should be used for the matrixtravel time A h , in order to convert P b o porosity?W hich zones have an oil saturation greater than 50 ?What is the porosity of level 16?

    water saturation in zones 1-4

    b.c.d.

    87 1.9 11 75 5.8107 4.1 12 89 5.9

    8995

    96 1.2 17 82 1065 23 18 102 1.8, I10 I 85 3.2 19 89 I 154. Using your knowledge of porosity from the preceding analysis, use thePickett plot technique of analyzing the resistivity data. Graphically

    determine the following:a. water resistivityb cementation exponentc. water saturation in zones 1-4