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Electrical resistivity of partially saturated sandstones Tercier, Paulette E. 1992

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ELECTRICAL RESISTIVITY OF PARTIALLY SATURATED SANDSTONESbyPAULETTE E. TERCIERB.Sc. (Honours) Geology, University of Alberta, 1982A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF GEOLOGICAL SCIENCESWe accept this thesis as conformingto the required standardsTHE UNIVERSITY OF BRITISH COLUMBIAJanuary 1992© Paulette Tercier, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of Geolo,c_c,\ Sc \ ev-> ce: The University of British ColumbiaVancouver, CanadaDate man. ZZ /9 2DE-6 (2/88)11ABSTRACTElectrical resistivity measurements are used to evaluate the in situ nature of porefluids. These measurements can provide information on the level of water saturation, Sw, andthe pore-scale fluid distribution. The objective of this thesis was to study the effect of fluiddistribution on electrical resistivity measurements. In this study two methods were used tochange the fluid/air distribution within sandstone and teflon cores. The first method involvedchanging saturation technique, imbibition or drainage; the second method involved varyingwetting conditions.Electrical resistivity measurements were made on partially saturated sandstone andteflon cores. The resistivity measurements were made at various levels of water saturation onthree sandstone cores during the imbibition and drainage of distilled water. The relationshipbetween electrical resistivity and Sw was shown to be non-unique. Measurements madeduring imbibition differed from those measurements made during drainage; this is a reflectionof varying fluid geometries within the pore space. Electrical resistivity measurements werealso made at various levels of saturation on a teflon core during drainage of nonwetting(distilled water) and a wetting (methanol) conducting fluids.Results from experiments on the sandstone and teflon cores were analyzed in terms ofArchie's equation. The results from the sandstones indicate Archie's equation can beextended into the low Sw region for both imbibition and drainage. The saturation exponentswere found to vary between imbibition and drainage indicating different pore-scale fluiddistributions for the two saturation processes. It is suggested that during imbibition the waterforms thick surface layers which are separated by a continuous thin cental air phase. Duringdrainage this continuous air phase is not recreated and therefore the fluid/air distribution williiibe different for drainage than imbibition. The saturation exponent for the nonwetting system,water/teflon, was found to be higher than the saturation exponent for the wetting system,methanol/teflon. The higher saturation exponent found for the nonwetting system indicates adifferent pore-scale fluid geometry. Wetting conducting fluids will form a connected pathalong the surface of the pores at very low levels of saturation. This surface layer is notformed when the conducting fluid is nonwetting and therefore higher levels of saturation arenecessary for conduction.The drainage data were assessed in light of percolation theory. It is suggested thatArchie's saturation exponent from the water saturated sandstones and the methanol saturatedteflon can be compared to the critical exponent for conductivity in two-dimensions. Archie'sexponent for the water saturated teflon can be compared to the critical conductivity exponentin three-dimensions. For the samples studied, it appears that if the conducting fluid wets theinsulating matrix the system will be dominated by a two-dimensional conducting mechanism.This is seen in the drainage data for both the water-wet rocks and the methanol-wet teflon. Ifthe conducting fluid no longer has an affinity for the insulating matrix, conduction is througha three-dimensional system. This is seen in the drainage data for the water saturated teflonsample.ivTABLE OF CONTENTSTITLE PAGE ^  iABSTRACT  iiTABLE OF CONTENTS ^  ivLIST OF TABLES  viLIST OF FIGURES ^  viiACKNOWLEDGEMENTS  xCHAPTER 1: INTRODUCTION ^  1CHAPTER 2: SAMPLE CHARACTERISATION ^  15Introduction  ^15Surface Area Measurements  ^16Measurement Procedure ^  18Results  ^19Porosity Measurements^  20Measurement Procedure ^  20Results  ^23Mineralogy ^  24X-Ray Diffraction ^  24Measurement Procedure ^  25Results  ^26Scanning Electron Microscopy ^  32Measurement Procedure  32Results  ^33Summary of Results ^  41CHAPTER 3: ELECTRICAL RESISTIVITY: MEASUREMENT PROCEDURE ANDRESULTS ^  47Introduction  ^47Measurement Procedure ^  47Results  ^50vPrimary Drainage Data: Natural Core Samples ^  50Primary Drainage Data: Teflon Core ^  62Imbibition-Drainage Data: Natural Core Samples  ^66CHAPTER 4: DISCUSSION^  85Variations in n and n * Due to Saturation Technique: Natural Core Samples^85Primary Drainage Data ^  85Imbibition-Drainage Data  ^87Variations in n and n* Due to Wetting Conditions: Teflon Core  ^89Application of Percolation Theory  ^91CHAPTER 5: CONCLUSION^  97REFERENCES ^  100APPENDIX 1: SURFACE AREA PROGRAM ^  104APPENDIX 2: ADSORPTION ISOTHERMS  110APPENDIX 3: POROSITY PROGRAM^  115viLIST OF TABLESTable1.1. Summary of Saturation Exponents in the Literature ^  32.1. Surface Areas  ^192.2. Porosities  ^232.3. XRD Whole Rock Semi-Quantitative Analyses ^  272.4. XRD Clay Semi-Quantitative Analyses  272.5. Summary of Sample Characterisation ^  464.1. Saturation Exponents for Drainage Data  ^87viiLIST OF FIGURESFigures1.1 Schematic illustration of the matrix/air/water distribution ^ 81.2 Frequency distribution of reported values of conductivity exponent (t) ^ 122.1. Sketch of pycnometer used in porosity measurements. ^ 212.2. X-ray diffraction pattern for 22-11-12-10069. ^ 282.3. X-ray diffraction pattern for API236 No.1433. 292.4. X-ray diffraction pattern for API263 No.9021. ^ 302.5. X-ray diffraction pattern for Berea100A(5). 312.6. SEM Micrograph - 22-11-12-10069 - (Magnification — 45X) ^ 342.7. SEM Micrograph - 22-11-12-10069 - (Magnification — 700X) 352.8. SEM Micrograph - API 236 No.1433 - (Magnification — 25X) ^ 362.9. SEM Micrograph - API 236 No.1433 - (Magnification — 2250X) 372.10. SEM Micrograph - API 263 No.9021 - (Magnification — 45X) ^ 382.11. SEM Micrograph - API 263 No.9021 - (Magnification — 110X) 392.12. SEM Micrograph - API 263 No.9021 - (Magnification — 2250X) ^ 402.13. SEM Micrograph - Berea100A(5) - (Magnification — 55X) 422.14. SEM Micrograph - Berea100A(5) - (Magnification — 125X) ^ 432.15. SEM Micrograph - Teflon5 - (Magnification — 250X) 442.16. SEM Micrograph - Teflon5 - (Magnification — 1250X) ^ 453.1. Log resistivity versus Sw for 22-11-12 #10069 during primary drainage. ^ 513.2. Log resistivity versus Sw for API263 No.9021 during primary drainage. ^ 523.3. Log resistivity versus Sw for Berea100A(5) during primary drainage. ^ 533.4. Log resistivity versus Sw for API236 No.1433 during primary drainage. ^ 543.5. Log resistivity versus number of monolayers of water for 22-11-12 #10069during primary drainage. ^ 573.6. Log resistivity versus number of monolayers of water for API263 No.9021during primary drainage. ^ 583.7. Log resistivity versus number of monolayers of water for Berea100A(5) duringprimary drainage^ 59viii3.8. Log resistivity versus number of monolayers of water for API236 No.1433during primary drainage.  ^603.9. Log I versus log Sw for 22-11-12 #10069 during primarydrainage.  ^613.10. Log I versus log Sw for API263 No.9021 during primarydrainage.  ^633.11. Log I versus log Sw for Berea100A(5) during primarydrainage.  ^643.12. Log resistivity versus Sw for Teflon5 during primary drainage. Water saturatedsample^  653.13. Log resistivity versus Sm for Teflon5 during primary drainage. Methanolsaturated sample.  ^673.14. Log I versus log Sw for Teflon5 during primary drainage. Water saturatedsample^  683.15. Log I versus log Sm for Teflon5 during primary drainage. Methanol saturatedsample^  693.16. Log resistivity versus Sw for 22-11-12 #10069 during imbibition (increasingSw) and drainage (decreasing Sw).  ^703.17. Log resistivity versus Sw for API263 No.9021 during imbibition (increasingSw) and drainage (decreasing Sw).  ^723.18. Log resistivity versus Sw for Berea100A(5) during imbibition (increasing Sw)and drainage (decreasing Sw) ^733.19. Log resistivity versus number of monolayers of water for 22-11-12 #10069during imbibition (increasing Sw) and drainage (decreasing Sw).  ^743.20. Log resistivity versus number of monolayers of water for API263 No.9021during imbibition (increasing Sw) and drainage (decreasing Sw).  ^763.21. Log resistivity versus number of monolayers for water for Berea100A(5) duringimbibition (increasing Sw) and drainage (decreasing Sw).  ^773.22. Log I versus log Sw for 22-11-12 #10069 duringimbibition.  ^78ix3.23. Log I versus log Sw for 22-11-12 #10069 duringdrainage. ^ 793.24. Log I versus log Sw for AP1263 No.9021 duringimbibition. ^ 813.25. Log I versus log Sw for API263 No.9021 during drainage^ 823.26. Log I versus log Sw for Berea100A(5) during imbibition. 833.27. Log I versus log Sw for Berea100A(5) during drainage^ 844.1. Log I versus log (Sm - Sm,„) for Teflon5 during methanol drainage. ^ 944.2. Log I versus log (Sw - Sw„„) for Teflon5 during water drainage. 95ACKNOWLEDGMENTSI am grateful to my supervisor, Dr. Rosemary Knight, for her endless enthusiasm,guidance and friendship through the duration of this project. Thanks to the Rock PhysicsGroup, especially Ana Abad for her help in the laboratory. My supervisory committee Dr. L.Groat, Dr. M. Barnes and Dr. W. Barnes provided numerous helpful suggestions. Specialthanks to Doug for his continuous encouragement and support.Funding for this project was supplied through a Natural Sciences and EngineeringResearch Council of Canada postgraduate scholarship. This research was funded by MobilExploration and Production Services Inc., and by a consortium of oil companies: ShellCanada Limited, Esso Resources Canada Limited, and Petro-Canada Resources.xCHAPTER 1INTRODUCTIONThe electrical conductivity, a, of porous geological materials is primarily determinedby the magnitude of the conduction through the fluids contained in the pore space. Themeasurement of cr, or its inverse the resistivity of a material (Rm), can thus provideinformation about the volume and nature of pore fluids. Due to this link between resistivityand pore fluids, the measurement of electrical resistivity is used extensively in the petroleumindustry and in hydrogeological applications to obtain information about the in situ nature ofthe fluids.A simple model of the dependence of resistivity on the fluids in a porous material isgiven by the Archie equation (Archie, 1942). If the pore space is only partially saturated withconducting fluid, the level of saturation of the conducting fluid, Sw, is related to the electricalresistivity through Archie's equation:Rm_ - Sw - nRowhere Rm is the measured resistivity, Ro is resistivity of the rock when it is fully saturatedwith the conducting fluid and n is the saturation exponent. This equation can also be writtenas:12I^Sw - "^ (1.2)where 1 is the electrical resistivity index; 1 = Rm/Ro. Archie's equation works well when thefollowing assumptions are satisfied:1. The conducting fluid has a high salinity.2. Conduction is only through the conducting fluid occupying the pore space; therock matrix is insulating.3.^The level of water saturation, Sw, is high enough that the connectivity of theconducting phase is maintained.The saturation exponent, n, depends on the geometry of the fluid phases. Of recentinterest is the variation in n with changing pore scale fluid distributions. The dependence ofn on pore scale fluid distribution has been most extensively studied by changing thewettability of the porous medium. Table 1.1 contains a summary of some of these studies.Wettability can be defined as the preference a solid surface will have for one fluidover another fluid. Figure 1.1 is a schematic illustration of the effect wettability has on thevolumetric fluid distribution. If the system is water-wet (Figure the water will coat thesurface of the grains forming a continuous network and the other component in the system,e.g. air or oil, will fill the centre of the pores. If the system is nonwater-wet (Figure relative positions of the water and air/oil will switch; the water now forming discretedroplets in a continuous air/oil background phase. These changes in volumetric fluiddistribution will be reflected in Archie's saturation exponent. Many studies; Keller (1953),Rust (1957), Sweeney and Jennings (1960), Morgan and Pirson (1964), Swanson (1980),Donaldson and Bizerra (1985), Lewis et al. (1988), and Donaldson and Siddiqui (1989) haveused chemical or oil treatments to change the wettability of rock samples. Mungan andTable 1.1. Summary of Saturation Exponents in the LiteratureReference Saturationrange (%)Saturationexponent (n)Material Measurement CommentsArchie, 1942 > 15 2 clean consolidatedand unconsolidatedsands?Guyod, 1948 10 - 100 1.7 - 4.3 unconsolidatedsands? summary ofRussian dataDunlap et al., 1949 10 - 100 1.08 - 2.24 sandstones 4 electrode,15 - 100 1.75 - 3.37 unconsolidatedsandscapillary pressurefor desaturationRust, 1952 33.8 - 100 2.31 - 2.40 Woodbine sand 2 & 4 electrode,capillary pressurefor desaturation,showed both 2and 4 electrodemethods workwellWhiting et al, 1953 15 -100 1.32 - 2.66 Edwards limestone 4 electrode,1.84 - 2.41 BenedumEllenberger andcapillary pressurefor desaturationShafter LakeKeller, 1953 8 -30 4.4 - 5.3 Bradford third sand, 2 electrode,30 - 100 1.5 - 2.3 water-wet evaporation for8 - 60 2.4 -3.1 Bradford third sand,desaturation60 - 100 8.9 -^11.7 nonwater-wetTable 1.1. Summary of Saturation Exponents in the LiteratureReference Saturationrange (%)Saturationexponent (n)Material Measurement CommentsRust, 1957 9 - 90 1.68 -1.77 U-1 sands (?sst)water-wet4 electrode,capillary pressurefor desaturationair or oil asdisplacing media20 -87 3.4 - 3.5 U-1 sands (?sst)nonwater-wetair displacement60 - 87 13.5 oil displacementSweeney and JenningsJr., 1960—17 - 100 1.61 water-wetcarbonates2 electrode,capillary pressurefor desaturation—17 - 100 1.92 neutral carbonates—25 - 35—40 - 8012.278.09oil-wet carbonatesMorgan and Pirson,1964—20 - 100 2.48 100% water-wetunconsolidated glassbeads2 electrode,capillary pressurefor desaturation—80 -100 9.28 50% water-wet +50% oil-wetunconsolidated glassbeads—90 - 100 25.17 - 26.43 100% oil-wetunconsolidated glassbeads4=,Table 1.1. Summary of Saturation Exponents in the LiteratureReference Saturationrange (%)Saturationexponent (n)Material Measurement CommentsWalther, 1968 38.2 2.21 Frio sandstone 2 & 4 electrode,capillary pressurefor desaturationusing oil81.6 1.69Mungan and Moore,1968—15 - 90 1.91 teflon cores 4 electrode,capillary pressurefor desaturationusing air or oilmethanol/airsystem, wheremethanol wasboth wetting andconducting28.4 - 33.939.5 - 66.28.9 - 4.062.11 - 1.97brine/air system,where brine isnonwetting andconducting31.0 - 34.336.8 - 64.19.0 - 4.02.81 - 2.35brine/oil systemwhere brine isnonwetting andconductingCoates and Dumanoir,1974? 1.28 - 2.5 sandstones ?1.15 - 3.8 limestonesUiTable 1.1. Summary of Saturation Exponents in the LiteratureReference Saturationrange (%)Saturationexponent (n)Material Measurement CommentsSwanson, 1980 15 - 74 1.49 - 2.07 Berea sandstone 2 electrode,capillary pressurelittle change in"n" despitesoaking cores inoil for 6 to 30days14 - 67 1.51 - 4.67 Austin limestoneDiederix, 1982 < 57> 571.28 -1.462.09 - 2.20sandstone - roughgrains2 electrode,capillary pressure"n" valuescalculated usingWaxman & Smitscorrection forclays? 1.45 - 1.91 sandstone -smoothgrainsDonaldson andBizerra, 198515.5 - 66.218.4 - 66.63.824.90Berea sandstone -water-wet(w = 0.498 - 0.269)2 electrode,centrifugecapillary pressureshowed linearrelationshipbetween "n" andDonaldson'swettability index(w)31.3 - 89.736.0 - 95.27.478.45Berea sandstone -oil-wetw = -0.130 - 0.150Garrouch, 1987 — 10 - 100 1.93 - 2.41 water-wet glassbeads4 electrode,capillary pressureusing naptha4.13 - 5.40 oil-wet glass beadsLewis et al., 1988 ? 1.2 - 5.18 water-wet and oil-wet Berea sandstone2 & 4 electrodecapillary pressureimbibition anddrainage "n's" forvariouswettabilitiesONTable 1.1. Summary of Saturation Exponents in the LiteratureReference.Saturationrange (%)Saturationexponent (n)Material Measurement Comments.Longeron et al.,1986,198928 - 6619 - 511.62 - 2.661.30 - 1.96sandstonecarbonate2 electrode,capillary pressure"n's" collectedduring imbibitionand drainageZemanek, 1989 ? 1.72 - 1.92 Pleistocene sands,Gulf of Mexicolog data -inductionresistivityDonaldson and ? 1.54 - 8.45 Berea sandstone 4 electrode, linear relationshipSiddiqui, 1989 1.45 - 4.60 Elgin sandstone centrifugecapillary pressurebetween "n" andDonaldson'swettability index(w)8Figure 1.1^Schematic illustration of the matrix/air/water distribution in(a) water-wet rock; continuous phase is water (b) nonwater-wet teflon; continuousphase is air.9Moore (1968) used teflon cores combined with wetting and nonwetting conducting fluids tostudy changes in resistivity due to changes in wettability. In all these studies Archie'ssaturation exponent was found to increase for the nonwetting systems. This increase in thesaturation exponent is the result of poor connectivity of the conducting phase. Theconducting phase does not form a continuous path along the surface of the grains thereforehigher saturations are needed before a good conducting path will form.Similar variation in n is found by changing the geometry of the fluid phases throughsaturation technique. Specifically it has been found that changing saturation technique,imbibition or drainage, causes variation in n (Dunlap et al., 1949, Whiting et al., 1953, Lewiset al., 1988, Longeron et al., 1989, Knight, 1991). Imbibition can be defined as thespontaneous uptake of water by a water-wet sample (increasing Sw) while drainagecorresponds to the expulsion of water (decreasing Sw). Knight (1991) presents a geometricalmodel to explain the hysteresis found in electrical resistivity measurements collected duringimbibition and drainage. In this model the fluid/air distributions for imbibition and drainageare the same at very low levels of water saturation. In this region the adsorption of waterduring imbibition and the desorption of water during drainage produce the same air/watergeometries. At higher saturations, in the hysteretic region, the air/water geometry duringimbibition is thought to be different than during drainage. In this region, during imbibitionwater continues to coat the pore surfaces forming thick layers of water separated by a thin,continuous air phase. During drainage this thin, continuous air phase and these thick surfacewater layers are not recreated; instead the air phase occurs as isolated bubbles. Thus, theprimary difference between imbibition and drainage is the geometry of the air phase. As aresult of the varying pore-scale fluid distribution we see a change in the Archie saturationexponent.10In the literature Archie saturation exponents have been compared, often assuming themethods for determining n are the same. Frequently, this is not the case. Using equation 1.2,a plot of log I versus log Sw should yield a straight line with slope equal to Archie'ssaturation exponent, n. Keller (1953), Whiting et al. (1953), Sweeney and Jennings (1960),Diederix (1982), and Longeron et al. (1986, 1989) use the slope of the best fit line to theirdata to determine "n"; thus a modification of Archie's equation is required as given below:I - K(Sw) -" .^ (1.3)where n* is defined as the saturation exponent in this modified expression and K is a constant.If equation 1.3 is written in log form it can be seen that log K is the y-intercept. If K isequal to 1 the modified equation will reduce to Archie's equation. Mungan and Moore(1968), Swanson (1980), Garrouch (1987) and Lewis et al. (1988) appear to calculate ndirectly from Archie's equation for each measured Sw and thus present a series of n valuescorresponding to discrete Sw's. Dunlap et al. (1949), Rust (1952, 1957), Morgan and Pirson(1964), Donaldson and Bizerra (1985), and Donaldson and Siddiqui (1989) present their dataas log I versus log Sw plots. Using these log-log plots lines are found which obey Archie'sequation (equation 1.2). These three interpretations of Archie's equation will produce slightlydifferent values for n.Archie's equation is one specific example of a more general relationship between theconductivity of a multicomponent system and the volume fraction of the conducting phase.Percolation theory, originally developed by Broadbent and Hammersley (1957), shows thata o a (P - Pc)`^ (1.4)11where ao is the normalized conductivity, P is the fraction of conducting bonds, Pc is thepercolation threshold and t is the critical exponent for conductivity. Using a square grid as anexample the percolation threshold, Pc, would be the fraction of conducting squares needed toform a continuous path from one side of the grid to the opposite side. The formation of thiscontinuous path establishes the connectivity necessary for conduction.Computer simulations and analog experiments measuring the electrical conductivity ofrandom resistor networks have been studied using percolation theory (Last and Thouless,1971, Adler et al., 1973, Kirkpatrick, 1973). Various values for t have been determined fromsuch studies. Sahimi (1984) provides an overview of the published values for the criticalconductivity exponent, t. The frequency of distribution of the reported values for t in two andthree dimensions is given in Figure 1.2. It can be seen that in two-dimensions t varies from1.0 to 1.5; in three-dimensions t varies from 1.5 to 2.4. Currently accepted values for t are1.3 for two-dimensions and 2.0 for three-dimensions (Sahimi, 1984, Stauffer, 1985). Thecritical exponent, t, is thought to be dependent only on dimension (Stauffer, 1985).The correspondence between percolation theory and Archie's equation is apparent ifwe let o = 111, P = Sw, Pc = SWcrit and t = n. Sw„„ is defined as the saturation below whichconduction in the system is lost, presumably due to lack of connectivity of the water. If weallow K to be the constant of proportionality then the modified Archie's equation agrees wellwith the percolation equation for conductivity. Writing the modified Archie equation in termsof the expression for percolation yields:I^K(Sw - Sw crit) -4.^(1.5)3D(a) 0.40.3 122DU-0.10.0(b)1 . 0^1 . 1^1 . 2^1 . 3^1 . 4^1 5Percolation conductivity critical exponent, t0.30.20.01 5^1 . 7^1 . 9^2 . 1^2 . 3Percolation conductivity critical exponent, tFigure 1.2^Frequency distribution of reported values of conductivity exponent (t).(a) Two-dimensional values; sample size is equal to 27 and currently accepted value isequal to 1.3). (b) Three-dimensional values; sample size is equal to 20 and currentlyaccepted value is equal to 2.0). (Sahimi, 1984)13It is of interest that Archie (1942) found the saturation exponent to be equal to 2 for cleanunconsolidated sands. This is the same value as the currently accepted conductivity exponentfor three dimensions. Can percolation theory be applied to the study of conduction in highporosity, water-wet sandstones during the drainage and/or imbibition of water?The objective of this thesis is to study the effect of fluid distribution on electricalresistivity measurements. If Archie's equation is valid, these geometrical effects should bereflected in the saturation exponent, n. From previous studies (Table 1.1), the geometricaldistribution of the fluids appears to affect the Archie relationship. Of particular interest is thechange in the electrical resistivity measured at low levels of water saturation. Is the Archieequation still applicable at low levels of water saturation? In the literature, only two studies(Keller, 1953, Rust, 1957) were found where electrical resistivity had been measured belowSw equal to 0.10. Studies by Knight and Nur (1987b) and Knight and Endres (1990) haveshown that there is a dramatic change in electrical properties at low levels of water saturation.It has been suggested (Knight and Endres, 1990) that this change is due to geometrical andelectrochemical effects. To test whether the change in resistivity at low levels of watersaturation is an electrochemical effect, a geometrical effect or a combination of both, a highdensity of data needs to be collected in the low Sw region.In this study electrical resistivity measurements are made on partially saturated naturaland synthetic cores. All data are assessed using the modified Archie equation as this is themost useful for comparison with existing studies. In some cases it will be shown that K isequal to 1 and thus the modified Archie equation can be reduced to Archie's equation.Archie's saturation exponent, n or n*, will be shown to vary with fluid distributions.Electrical resistivity measurements are made on three natural core samples during14imbibition and drainage of distilled water. Data are collected over the entire saturation range;with a high density of points in the low Sw region. In this study Sw is defined as the volumefraction of the pore space filled with water, the remaining space being filled with air. Thesedata will show that the variation in the saturation exponent, n or n', is due to varying fluiddistributions which are caused by saturation technique.Electrical resistivity measurements are also made on a porous teflon disc duringdrainage using both nonwetting (distilled water) and wetting (methanol) conducting fluids.These data will show that the variation in the saturation exponent, n or n* , is due to varyingfluid distributions which are caused by different wetting properties of the two conductingfluids. These data, collected on both the natural and synthetic cores are also assessed interms of percolation theory. It will be shown that n, n* or t are primarily determined by thegeometry of the fluids within the pore space. It is suggested that the electrical resistivitymeasured during the drainage process on the natural core samples and the synthetic teflonsample in this study can be modelled using two-dimensional and three-dimensionalpercolation.CHAPTER 2SAMPLE CHARACTERISATIONIntroduction Four natural core samples and one synthetic sample were chosen for electricalresistivity studies. Three of the natural core samples are sandstone reservoir samples suppliedby Mobil Exploration and Production Services, Dallas, and are referred to as 22-11-12-10069,API236 No.1433 and API263 No.9021. The fourth natural core sample is from a quarriedsandstone, Berea100, and is referred to as Berea100A(5). The synthetic sample is pressedteflon powder and is referred to as Teflon5. To characterise each sample, surface area andporosity were measured and the mineralogy described using x-ray diffraction and scanningelectron microscopy.Each of the natural core samples was cut into a number of thin discs, from which onedisc of each core was chosen for electrical and porosity measurements. Qualitatively thenatural cores appeared homogeneous. In preparation for electrical measurements the thindiscs, ranging from 0.513 to 1.254 cm in thickness were surface ground to a smoothness of±30 p.m. Smoothness is the variation in the height of the disc measured perpendicular to theface of the disc. Gold electrodes, 100 to 140 nm in thickness were sputtered onto each sideof the discs. The remaining discs were used in the surface area analysis, x-ray diffraction andscanning electron microscopy work.1516Teflon powder (#7A) was obtained from Dupont of Canada, and using a hydraulicpress combined with a 1 inch cylindrical die, thin discs, 1.471 cm in thickness, were made.Two discs with similar porosities were made; one disc to be used in electrical measurementsand the second disc to be used in surface area and scanning electron microscopy work. Goldelectrodes, 100 nm in thickness, were sputtered onto the ends of the disc chosen for electricalstudy.Surface Area Measurements The surface area of a rock is the total area formed by the walls of the pores; in awater saturated sample, this becomes the size of the rock/water interface. In general, whilethe grain size of a rock affects the surface area, the dominant factor is the percentage of highsurface area material (such as clay) that is present.The physical adsorption of a gas onto a solid can be used to determine the surfacearea of the solid. Gregg and Sing (1982) provide a discussion of the adsorption methods usedto determine the surface area of porous solids. Relevant points from their discussion aresummarized below.The volume of gas taken up by a solid is proportional to the mass of the solid (W e). Italso depends on the temperature, the pressure of the gas and the nature of both the solid andthe gas. Experimentally the volume of gas adsorbed (V a) is measured at various pressures (p)and a plot Va/14/, versus plpo, where p o is the saturation vapour pressure of the gas, yields anadsorption isotherm. Brunauer, Deming, Deming and Teller (1940) classified adsorptionisotherms into five types determined by the nature of both the adsorbate and adsorbent. Theadsorption isotherm generally obtained from sandstones is a Type II adsorption isotherm(Knight and Nur, 1987a).17The BET equation (Brunauer et al., 1938) describes the linear portion of a Type IIadsorption isotherm where V m is the monolayer capacity and c is a constant related to heats ofadsorption. The monolayer capacity is the quantity of adsorbate needed to completely coverthe surface of a solid with a single layer of molecules, and is referred to as the monolayer.When the BET equation is written in linear form,p^1 + (c-1) pV a(po -P)^V mc^V mc Po(2.1)it can be seen that a plot of plVa(po-p) against plpo will yield a straight line with an interceptof PV„,c and a slope of (c-/)/V„,c. It follows then that the monolayer capacity V„, can then bedetermined from the plot, according toV - ^1^ (2.2)slope + interceptThe range of validity for the BET equation is usually 0.05 < plpo < 0.30 (Gregg and Sing,1982).The surface area (SA) is directly proportional to the monolayer capacity, therelationship between the two being given by the simple equationaSA ^N V (2.3)Mgwhere am is the cross-sectional area of the adsorbate molecule, N is Avogadro's constant andMg is the molar gas volume. Thus knowing the volume of gas needed for monolayer18coverage of the surface and the physical dimensions of the gas molecule one can readilycalculate the surface area of the solid.Measurement ProcedureThe surface area of the samples was measured using a Micromeritics Model 2100DOrr Surface-Area Pore-Volume Analyzer with nitrogen as the adsorbate. Nitrogen and heliumwere supplied by Linde Union Carbide and were of ultra high purity (a guaranteed minimumpurity rating of 99.999%). The description and operating instructions for the instrument canbe found in the Micromeritics Manual supplied with the instrument. A brief summary of themeasurement procedure and surface area calculation follows.Samples were prepared for surface area analysis by breaking slices of the samples intopieces 2 to 3 mm in diameter. For the adsorption measurements the specimen must have asurface area of at least 10 m2 . It was assumed, based on petrographic work, that all thesandstones had a surface area of approximately 1 m 2/g; therefore approximately 10 grams ofeach sample were used.After crushing, the specimens were placed into the sample flasks, weighed, andattached to the Micromeritics surface analyzer. To free the samples of all physisorbedvapours and gases, the samples were heated to 125°C and evacuated overnight under highvacuum (— 10' torr). The adsorption measurements were made at low temperature (— 77 K)by immersing the sample flasks in liquid nitrogen. During the first step of the adsorptionmeasurements nitrogen gas was introduced into the manifold, which has a known volume,was held at a constant temperature, and the equilibrium pressure was recorded. The valvebetween the manifold and the sample was then opened and the gas allowed to expand into theavailable space. The new equilibrium pressure was recorded. This process was repeated for19each incremental increase in pressure thus obtaining a series of V a values for each pressurestep. A plot of MEVa/Ws(Po-P)1 versus pip, yields the BET equation isotherm from which themonolayer volume (V„,) can be determined. Using equation 2.3 the surface area can becalculated. A computer program supplied with the Micromeritics instrument for calculationof surface area was modified for use on the Macintosh and is included in Appendix 1.ResultsAn alumina standard (catalogue No. 2007) from Quantachrome Corporation was run tocheck the performance of the analyzer. The surface area given with the standard was 2.04 ±.008 m2/g; we measured an average surface area of 2.04 m 2/g.The surface areas of the natural and synthetic core samples are given in Table 2.1.Table 2.1: Surface AreasSAMPLE NAME SURFACE AREA:RUN #1in2/gSURFACE AREA:RUN #2m2igQuantachrome 2.05 2.03Standard No.2007API236 No.1433 0.964 0.960API263 No.9021 1.06 1.0622-11-12 #10069 0.864 0.873Berea 100A(5) 1.234* 1.22Teflon5 2.49 not doneKnight and Nur (1987a)Duplicate analyses of the surface area of samples 22-11-12 #10069, API236 No.1433, API263No.9021 and Berea100A(5) were completed, while a single analysis of Teflon5 was made.20The average surface area calculated from the duplicate analyses on each of the samples was:0.869 m2/g for 22-11-12 #10069, 0.962 m2/g for API236 No.1433, 1.06 rii 2/g for API263No.9021 and 1.23 m2/g for Berea100A(5). Teflon5 had a surface area of 2.49 m 2/g. Thelinear portions of the nitrogen adsorption isotherms for each analysis can be found inAppendix 2. All data showed an excellent agreement with the BET equation.Porosity Measurements The porosity of a rock is the ratio of the void volume to the total external rockvolume. Monicard (1980) provides a discussion on the various methods used to measureporosity. In this study the measurement of the total external rock volume, using calipers, andthe grain volume, using gas expansion, was used to calculate porosity.Measurement ProcedureBefore porosity measurements were made the physical dimensions of the discs weremeasured to determine the external volume. Water vapour was then removed by placing thesamples in a desiccator over calcium sulphate for three to four days. The porosity of eachsample was measured using a Micromeritics multivolume pycnometer #1305. A detaileddescription and the operating instructions of the pycnometer can be found in the manualsupplied with the instrument. A brief summary of the measurement procedure is describedbelow.A simplified diagram of the pycnometer is shown in Figure 2.1. Both the samplechamber (Kell) and the expansion (Vey) volumes are known. The core sample is added to thesample chamber and with the valve between Ka and Vey open, the pressure transducer iszeroed (Pa). The valve is then closed, helium gas is introduced into the sample cell, thepressuretransducer0valve0Vcell^Vexp21Figure 2.1. Sketch of pycnometer used in porosity measurements.22pressure allowed to equilibrate and then recorded as P1. The equation for the system is:P I (Vcell — Vsamp + P aVex,'^n celiRT + nexpRT^(2.4)where Vsa„,p is the volume occupied by the sample, T is the temperature, cell is the number ofmoles of gas in sample cell, nev is the number of moles of gas in expansion volume and R isthe gas constant. When the valve is opened the pressure will fall to P2 and the equation forthe system becomes:P 2(V cell — V camp + V exp) = n eeuRT + n,RT^ (2.5)Solving for Vsamp yields:V s^= V - ^1 Pa)V sxpDeft (2.6)(P2 — a)The difference between the total external volume (VE) and the volume measured by thepycnometer (Vsa,np) yields the void volume. The porosity (0) is then simply the ratio of thevoid volume (Vvoid) to the total external volume.V.E^sampVE^VEV - V (2.7)23A computer program written by A. Abad (UBC Rock Physics Group) was used to calculatethe porosities and is included in Appendix 3.The porosity of each of the samples was measured on two different occasions tomonitor the effects of the various saturation experiments. The first porosity measurementswere made before any saturation experiments were started. The second measurements weremade following all saturation experiments on the samples. Close monitoring of the porositywas important as a change in the pore volume will change the pore geometry as well as thewater saturation calculation.ResultsPorosities of both the natural and synthetic core samples can be found in Table 2.2.The four natural core samples, API236 No.1433, API263 No.9021, 22-11-12 #10069, andBerea100A(5) have porosities of 0.332, 0.301, 0.281 and 0.21 respectively. The syntheticsample, Teflon5 has a porosity of 0.248. None of the cores showed a significant change inporosity due to the saturation experiments.Table 2.2: PorositiesSAMPLE NAME POROSITYRUN #1 RUN #2API236 No.1433 0.332 0.333*API263 No.9021 0.301 0.30022-11-12 #10069 0.281 0.281Berea100A(5) 0.21— 0.215Teflon5 0.248 0.249!A.this measurement obtained on the largest piece of a broken sample.this measurement obtained July 1985 at Stanford University by R. Knight24Mineralogy The petrographic microscope is commonly used to determine the mineralogy of elasticsediments. Although thin sections were examined, detailed petrographic descriptions were notcompleted for two reasons: many of the grains had been disturbed during sample preparationand the clays were too fine grained for identification. Instead, powder x-ray diffraction wasused to determine the mineralogy, and scanning electron microscopy was used to gain insightinto the pore structure and grain relationships.X-Ray DiffractionPowder x-ray diffraction is used to identify minerals and in particular is very usefulfor the identification of clay minerals which, because of their cryptocrystalline nature, arevery hard to identify using a petrographic microscope. The principles of x-ray diffraction arediscussed fully in Cullity (1959). A brief overview of the principles and instrumentation arediscussed below.Minerals are made up of ordered arrays of atoms which produce characteristic crystalstructures. When an x-ray beam hits an ordered array of atoms scattering occurs, andalthough many of the reflected rays interfere with one another, some of the scattered wavesare in phase which then combine to form a diffracted beam. In a crystalline material x-raydiffraction can be thought of as a reflection of the incident beam along crystal planes. Thisreflection obeys the Bragg equation:jX - 2dsine^ (2.8)25where j is an integer, X is the wavelength of the x-ray, 8 is the reflection angle where0° < 8 < 180°, and d is the interplanar spacing in the crystal. Thus knowing X. and 8, the dspacings can be readily calculated and matched to standard mineral patterns.The x-ray diffractometer is designed to detect and record reflections through a rangeof operator chosen 28 values. A closer look at the instrumentation reveals: a pivoting stageto which the sample is attached, a source for monochromatic x-rays, an x-ray detector, and ameans of recording the reflections (in this case a computer). As the sample rotates through 0x-rays are reflected which correspond to the d spacings of the minerals present. The detector,which is mounted on a separate arm, moves through 28 collecting these reflections. The 28values are then converted to d spacing using equation 2.8 and matched to the mineralstandards in the Joint Committee on Powder Diffraction Standards (JCPDS) database.Measurement Procedure Each of the four natural core samples was prepared for qualitative x-ray diffraction bygrinding to a fine powder in water and then placing a thin layer of the paste on a flat glassslide. Two slides of each sample were prepared, one representative of the whole rock and thesecond representative of the clay fraction. The fine fraction was obtained by dispersing thepowder in a beaker of water, waiting a few minutes (allowing time for the heavier fraction tosettle) and then pipetting 1-2 ml of the supernatant liquid containing the clay fraction onto aslide. This was then allowed to dry. Preparing the samples in this manner (oriented) willproduce strong basal reflections from the clays. Heat treatment, at 550°C overnight, was alsoused to help further in the identification of the clays. Wilson (1987) provides a gooddiscussion on the preparation and identification of clay minerals.For the semi-quantitative XRD new samples were prepared for whole rock analysis. A26random orientation of the mineral grains is needed for the whole rock semi-quantitativeanalysis; therefore dry powdered samples were packed into a sample holder.Measurements were made on a computer controlled Siemens D5000 x-raydiffractometer. Three runs (whole rock, untreated clay and heat treated clay) for each of thefour samples (22-11-12-10069, API 236 No.1433, API 263 No.9021, Berea100A(5)) werecarried out. The diffractometer was set to sweep from 20° to 60 ° (20) for the whole rockanalyses and from 5° to 30 ° (20) for the clay analyses. The heat treated samples were runfrom either 5° to 30 °(20) or 11° to 13°(20). Standard x-ray diffraction patterns from theJCPDS computer database were used to identify the patterns produced by the samples.Semi-quantitative XRD was completed on whole rock and clay separates. For semi-quantitative analysis the highest intensity peaks for each mineral present were identified andthe area under these peaks measured. This area is proportional to the amount (in weightpercent) of the mineral present (Wilson, 1987).Results Results from the whole rock semi-quantitative analysis are given in Table 2.3. Due tosample preparation and differences in x-ray scattering from different mineral structures theerror in these analyses may be as high as ±2.5% (L. Groat, per. comm.) The x-ray diffractionpatterns obtained from the whole rock analysis for samples 22-11-12 #10069, API236No.1433, API263 No.9021 and Berea100A(5) are seen in Figures 2.2a, 2.3a, 2.4a, and 2.5arespectively. All four samples contained quartz and kaolinite; 22-11-12-10069, API 236No.1433 and Berea100A(5) contained illite; Berea100A(5) contained K-feldspar; API 263No.9021 contained muscovite.^Figures 2.2b, 2.3b, 2.4b, 2.5b contain the x-ray diffractionpatterns for the clay analyses. Identification of clay minerals by x-ray diffraction is27complicated because the strong peak at approximately 12.5° (20) could belong to chlorite,kaolinite or montmorillonite. The heat treated slide was used to distinguish between kaoliniteand chlorite. When kaolinite is heated to 550°C the structure collapses and thus the peak at12.5° (20) will disappear (Wilson, 1987). This is seen in Figures 2.2c, 2.3c, 2.4c and 2.5c.Table 2.3: XRD Whole Rock Semi-Quantitative AnalysesSAMPLENAMEWHOLE ROCK MINERALOGY (approximate weight %)quartz illite kaolinite K-feldspar muscoviteAPI236 No.1433API263 No.902122-11-12 #10069Berea100A(5)99999997trace---tracetrace1111---------2---trace------The ratio of the clay minerals to each other was also calculated in a similar manner,API236 No.1433 contained 82% kaolinite and 18% illite; API263 No.9021 contained 92%kaolinite and 8% muscovite; 22-11-12 #10069 contained 94% kaolinite and 6% illite; andBerea100A(5) contained 77% kaolinite and 23% illite. Results from the clay separatesanalysis are given in Table 2.4.Table 2.4 : XRD Clay Semi-Quantitative AnalysesSAMPLE NAMECLAY MINERALOGY (approximate weight %)kaolinite illite muscoviteAPI236 No.1433 82 18API263 No.9021 92 822-11-12 #10069 94 6Berea 100A(5) 77 23a)11. 088 x^2theta y : 38 Linear 12.990)1 28. WAb)x : 2theta y : 826. Linear. 59.986)x^2thota 9^436. Linear 29.990)c)28Figure 2.2. X-ray diffraction pattern for 22-11-12-10069. Standard Peaks: Q -quartz, K - kaolinite, I - illite a) whole rock; b) clay separate; c) heated clay separate -kaolinite structure has collapsed therefore shows only background.0C) K OK11 J 59.980)K K CI^a,^ . ki,(^.x : 2theta y :^874. Linear( 20.00029a)b)c)<^5.000^x : 2theta y :^323. Linear^29.990)11.000 x 2theta y : 30. Linear 12.990)Figure 2.3. X-ray diffraction pattern for API236 No.1433. Standard Peaks: Q - quartz, K -kaolinite, I - illite a) whole rock; b) clay separate; c) heated clay separate - kaolinite structurehas collapsed therefore shows only background.59.909)< 29.999 x : 2theta y : 1651. Linear29.9961)5.099^ x : 2theta 9 :^647. Linear5.999 29.999)329. Linearx : 2th.ta y :b)Note: kaolinite peaks are missingK 0aMc)30a)Figure 2.4. X-ray diffraction pattern for API263 No.9021. Standard Peaks: Q - quartz, K -kaolinite, M - muscovite a) whole rock; b) clay separate; c) heated clay separate - kaolinitestructure has collapsed therefore shows only quartz and muscovite peaks.a)x : 2theta y :^967. Linear 29.990)<^5.000c)< 11.008 x : 2theta y 40. Linear 12.990)b)Figure 2.5. X-ray diffraction pattern for Berea100A(5). Standard Peaks: Q - quartz, K -kaolinite, F - K-feldspar, I - illite a) whole rock; b) clay separate; c) heated clay separate -kaolinite structure has collapsed therefore shows only background.I440,04•446.0410011*.wwewawel31aKaI^II\Aroa.onwo,e0.1.~0.rojeterrf4 4Nol.`111'""°94"032Scanning Electron MicroscopyThe scanning electron microscope (SEM) and energy dispersive x-ray system (EDX)can provide detailed information on grain size and shape, pore structures as well as mineralidentification through both crystal morphology and characteristic x-rays. Gabriel (1985)provides a complete discussion on theory and operation of the SEM as well as in depthdiscussion on sample preparation.The SEM can be thought of as an extension of the light microscope. The SEM useselectrons for image formation rather than light photons, thus enhancing resolution,magnification, and depth of field. The combination of these three features makes the SEM avery useful tool in the study of surfaces.The scanning electron microscope produces a finely focussed beam of electrons whichinteracts with a sample to produce various forms of radiation such as secondary electrons,characteristic x-rays, and backscattered electrons. In this study only secondary electronimages and characteristic x-ray patterns were used. Secondary electron imaging was used toreveal the three-dimensional topographic image. Characteristic x-rays were used to identifythe minerals present.Measurement ProcedureBoth polished sections and rough specimens of the rock samples were gold sputteredfor scanning electron microscopy (SEM) work. It was quickly realized that the polishedsections were not of much use as the preparation process had plucked and disturbed many ofthe grains; all clays appeared to have been lost. All work then concentrated on the roughspecimens. Secondary electron images, revealing crystal morphology, along with energydispersive x-rays (EDX) were used to identify the minerals present. The SEM Petrology33Atlas by Welton (1984) was used extensively to aid in mineral identification. A roughspecimen of Teflon5 was prepared in a similar manner. Secondary electron images were usedto examine the size and shape of the pores in Teflon5.Results In all four core specimens the pores were found to be relatively clean with only minordevelopment of bridging clays. Often pore walls were formed by well developed quartzovergrowths which also appears to be the main cementing material. Representative SEMmicrographs for each of the specimens may be found in Figures 2.6 through 2.16.Figures 2.6 and 2.7 contain SEM micrographs of 22-11-12 #10069. Figure 2.6reveals well developed quartz overgrowths on many of the detrital grains. These overgrowthsappear to be the main cementing material. The average grain size measured is 286 p.m; poresizes ranged from 2 to 100 p.m. Figure 2.7 shows clay; illite mats and kaolinite books liningthe pore. These clays are probably the main contributing factor to the surface area measured.Representative micrographs of API236 No.1433 are found in Figures 2.8 and 2.9. InFigure 2.8 the detrital grains are well rounded although many show the development ofauthigenic quartz overgrowths. The average grain size is 400 p.m; with pores ranging from 1to 210 p.m. Most of the larger pores appear open and free of infilling clays. Figure 2.9reveals well developed kaolinite books oriented perpendicular to the quartz overgrowth.Figures 2.10 through 2.12 contain SEM micrographs of API263 No.9021. Figure 2.10shows quartz overgrowths developed on most of the detrital grains. The average grain size is169 ilm; with pores ranging from 1 to 175 pm. Figure 2.11 reveals well developed quartzovergrowths with kaolinite infilling some of the pores. Figure 2.12 shows euhedral kaoliniteinfilling a pore.Figure 2.6. SEM Micrograph - 22-11-12-10069 - (Magnification — 45X)SEM micrograph showing well developed quartz overgrowths. The overgrowthsappear to be the main cementing material.34Figure 2.7. SEM Micrograph - 22-11-12-10069 - (Magnification — 700X)SEM micrograph shows illite mats (a,b) and euhedral kaolinite books, — 6 pm (c)lining pore. EDX analysis at (a,b) revealed Al, Si, K, Fe, and Au (due to samplecoating) peaks. Positive identification of the illite was confirmed by XRD analysis.At (d) is probably a cube of pyrite.35Figure 2.8. SEM Micrograph - API 236 No.1433 - (Magnification — 25X)Detrital quartz grains are well rounded (a). Authigenic quartz overgrowths arebeginning to develop on many of the grains (b,c). Pores appear to be relatively clean.The pores with infilling contain both illite and kaolinite. (d) marks the location ofFigure 2.9.36Figure 2.9. SEM Micrograph - API 236 No.1433 - (Magnification — 2250X)At (a) euhedral kaolinite books (— 6 grn across) are oriented perpendicular to a welldeveloped quartz overgrowth (b). See (d) in Figure 2.8 for location of thismicrograph.37Figure 2.10. SEM Micrograph - API 263 No.9021 - (Magnification — 45X)Most detrital grains show well developed quartz overgrowths (a). Clays coat and infillpores (b). (c) marks the location of Figure 2.11.38Figure 2.11. SEM Micrograph - API 263 No.9021 - (Magnification — 110X)Shows well developed quartz overgrowths (a) and kaolinite (b) infilling the pores.Some of the detrital grains (c) do not have well developed quartz overgrowths and areinstead coated with a mixture of kaolinite and quartz which was confirmed using EDXanalysis. (b) marks the location of Figure 2.12. See (c) in Figure 2.10 for thelocation of this micrograph.39Figure 2.12. SEM Micrograph - API 263 No.9021 - (Magnification — 2250X)Shows euhedral kaolinite (— 2 8 p.m) infilling pore. See (b) in Figure 2.11 for thelocation of this micrograph.4041Figures 2.13 and 2.14 contain SEM micrographs of Berea100A(5). Figure 2.13 showsthe subrounded detrital grains with only minor development of quartz overgrowths. Theaverage grain size is 144 .t.m; pores range in size from 1 to 70 1.1m. In Figure 2.14 illite matsand kaolinite coat the quartz overgrowths.SEM micrographs of Teflon5 are found in Figures 2.15 and 2.16. Figure 2.15 revealsthe wide variation of both, particle and pore, shape and size. The average particle size was 7wri; with pores ranging in size from 1 to 35^Outlined in Figure 2.15 is the location ofFigure 2.16 which further reveals the variation in pore shape and size.Summary of ResultsThe four natural core samples can be described as high porosity fine to mediumgrained quartzose sandstones which all contain a small percentage of kaolinite. Berea100A(5)contains a small percentage of K-feldspar; trace illite was found in 22-11-12 #10069, API236No.1433, and Berea100A(5), and trace muscovite was found in API263 No.9021. A summaryof the results can be found in Table 2.5.Figure 2.13. SEM Micrograph - BerealOOA(5) - (Magnification — 55X)SEM micrograph reveals subrounded detrital grains with an average grain size of 144pm. Most of the grains are quartz and the finer grained material kaolinite.42Figure 2.14. SEM Micrograph - BerealOOA(5) - (Magnification — 125X)SEM micrograph reveals authigenic quartz overgrowths (a,b) developing on many ofthe detrital grains. Later growth of kaolinite (c,d) overtop of quartz overgrowths.Detrital K-feldspar grain at (e). flute mat coats detrital grain at (f). All mineralidentification in this micrograph was confirmed by EDX analysis.43Figure 2.15. SEM Micrograph - Teflon5 - (Magnification — 250X)SEM micrograph reveals teflon particles varying in shape and size, from — 5 - 40 1-1.M.The largest pores are — 20 p.m varying in size and shape down to 1 (a) marks thelocation of Figure 2.16.44Figure 2.16. SEM Micrograph - Teflon5 - (Magnification — 1250X)Enlargement of (a) in Figure 2.15 reveals the irregular shape and size of the pores.45Table 2.5: Summary of Sample CharacterisationSAMPLENAMEAPPROXIMATE COMPOSITION (weight %) AverageSurfacearea(m2/0Porosity Averagegrainsize(1.1m)Mineralogyquartz kaolinite K-feldspar muscovite illite PTFE*API236No.143399 1 --- --- trace --- 0.962 0.332 400API263No.902199 1 --- trace 1.06 0.301 16922-11-12#1006999 1 --- trace --- 0.869 0.281 286Berea 1 00A(5) 97 1 2 --- trace --- 1.23 0.21 144Teflon5 --- --- 100  2.49 0.248 7*PTFE TEFLON PTFE Fluorocarbon Resin Grade 7A obtained from Dupont of CanadaCHAPTER 3ELECTRICAL RESISTIVITY: MEASUREMENT PROCEDURE AND RESULTSIntroduction Electrical resistivity measurements can be used to determine the volume of conductingfluid within the pore space of a rock. These measurements can also provide information onpore-scale fluid geometries. In this study two-electrode measurements are made on partiallysaturated sandstone and synthetic samples. Data are analyzed in terms of Archie's equationwhich relates the level of saturation of the conducting fluid to the measured electricalresistivity.Measurement Procedure Each of the thin discs, previously prepared for electrical measurements, was cleaned ofresidual salts by soaking in distilled water for a period of three weeks. During this timeresistivity was monitored. Upon completion of each resistivity measurement the distilledwater was changed. When the changes in resistivity became insignificant (less than 500 Q m)the disc was assumed clean. The disc was then dried in a vacuum oven, stored in adesiccator over calcium sulphate, and weighed to obtain a dry mass.Electrical measurements were made using a Hewlett-Packard Model 4192A lowfrequency impedance analyzer controlled by an HP computer. The impedance analyzer wasset to measure complex admittance Y,4748Y' - Gp + iBp^ (3.1)where Gp is the conductance and Bp is the susceptance. Data were collected in the frequencyrange of 10 Hz to 10 MHz by a logarithmic sweep. The oscillation level was set to 1 volt.To correct for the impedance of the sample holder and cables a "zero offset" on the4192A can be set. A disc of a standard material with a known dielectric constant is then runto ensure the proper zero offset. Details of this calibration procedure may be found in Knight(1985) and Knight and Nur (1987a).Knowing the physical dimensions of the sample area, A, and thickness, Th, and theconductance value, Gp, the resistivity, Rm, can be calculated:Rm IMI A (3.2) Gp ThOne of the principal objectives of this study was to determine the effects of watersaturation on measured resistivity. Sw was varied by considering both imbibition anddrainage.The primary drainage experiment was designed to allow collection of data at variouswater saturations from a fully saturated sample as it dried through evaporation. A highpressure saturating set-up was used to saturate samples with distilled water. The saturatingsetup was evacuated to 10 -2 torr, a desiccator dried sample was loaded, evacuated to 10 -2 torrand then distilled water was introduced and pressurized to 13.79 MPa. The sample was leftunder pressure in the distilled water overnight. In the morning the sample was removed fromthe saturating setup and weighed immediately to get the maximum saturation. As the sample49dried, resistivity measurements were made every 0.05 change in Sw, thus producing adrainage curve.The imbibition-drainage experiment was designed to collect resistivity values duringboth imbibition and drainage. Imbibition data were obtained by starting with a vacuum drysample. The first few measurements were made as water was adsorbed from the atmosphere,then from a humidifier, and finally by soaking the sample in a beaker of distilled water.Once the sample had imbibed as much water as possible it was allowed to dry andmeasurements were again made every 0.05 change in Sw to complete the drainage portion ofthe experiment. This experiment produced two curves, one corresponding to the imbibition ofdistilled water and the second to the drainage.The procedure followed for each measurement is given below:1. Weigh sample.2. Load sample into sample holder; connect to impedance analyzer (this places thesample in an electric field) and allow to equilibrate for 10 minutes.3. Collect data.4. Unload sample and weigh.5.^Continue saturation or desaturation for next measurement, then return to 1.Using the mass of the sample determined after the measurement, the water saturationfor each electrical measurement can be calculated using the following equation:Swm - and•;1)(3.3)• A • Th • p 50where m is the mass of the sample after measurement, and is the dry mass of the sample, P isthe porosity of the sample and p is the density of water.Results Primary Drainage Data: Natural Core SamplesAll data presented are for a single frequency, 100 kHz. Data from the drainageexperiment completed on 22-11-12 #10069 are shown in Figure 3.1 as a plot of log Rmversus Sw. The maximum Sw reached was 0.88; this was the starting point for the drainageexperiment. As the water evaporated there was a slow but steady increase in log resistivity;for the region defined by 0.10 < Sw < 0.88 resistivity increases by approximately one order ofmagnitude. For water saturations below 0.10 resistivity increases dramatically (three orders ofmagnitude) producing the very steep region seen at low Sw. Data for the other three rocks,API263 No.9021, Berea100A(5), and API236 No.1433 are qualitatively very similar and areshown in Figures 3.2 through 3.4.Low Saturation Region Of particular interest to this study is the change in the electrical resistivity measured atlow levels of water saturation. At low levels of water saturation it is reasonable to assumethat the water is present in the rock as a thin layer coating the surfaces of the pore space. Itis useful to think of water saturation in terms of monolayers of water; so Sw is converted tomonolayers of water.For this calculation a spherical shape and a hexagonal close packing of the waterC)022-11-12 #100690 primary drainage0000 0 00 0 0 0 00 o 0 0 0 0 0 0 0 000 0000 0^0 . 2^0 . 4^0 . 6^0 . 8^1 0SwFigure 3.1.^Log resistivity versus Sw for 22-11-12 #10069 during primarydrainage.512cm07o^ API263 No.9021o primary drainage5526-O 00 0 0 0 o o 0 0 o 0 o 00 0 0 0 00 0 0 0 0 ap432 I^I^I0 0^0 . 2 0 . 4 0 . 6S wFigure 3.2.drainage.Log resistivity versus Sw for API263 No.9021 during primary0 .8 1 00)07536Berea100A(5)o primary drainage I54 ik00000 0 000 0 0 0000 0 000 0 0000030 : 82 ^1 00 . 0 0 .2^0 . 4^0 . 6SwFigure 3.3.^Log resistivity versus Sw for Berea100A(5) during primary drainage.0........ECV'6 -o0754API236 No.14330 primary drainage4^0 0C) 0O00 00 0 e 00 00O 0 00  0 00I^T^I^•^I^ I0 0^0 . 2 0 . 4 0 . 6 0 . 8^1 0S wFigure 3.4.drainage.Log resistivity versus Sw for API236 No.1433 during primary355molecules is assumed. The number of water molecules, W„„ needed to cover the surface areaof the solid (SA determined from nitrogen adsorption) is calculated:SW -Am A wwhere A,,, is the area occupied by one water molecule; A w = 10.5 x 10' m2 (McClellan andHarnsberger, 1967). The conversion of molecules of water to mass of water gives themonolayer capacity:WMmonolayer capacity - mNwhere M is the molecular mass of water, M = 18.01528 g/mole and N is Avogadro's constant.To calculate the amount of water needed for monolayer coverage of the solid in question,W Mwater needed for monolayer coverage -  Nm ^• and (3.6)The number of monolayers is therefore given by:number of monolayers -m - andwater needed for monolayer coverage- Sw(A •0 - Th •p •A w -Nand ' SA • M\^ /(3.4)(3.5)i56- Sw A - ep - Thand ' SA 13542 (3.7) Figure 3.5 shows the resistivity data plotted as log Rm versus number of monolayersfor sample 22-11-12 #10069. Here we can see that the addition of a very small quantity ofwater, in this case 4 monolayers, causes a dramatic drop (11/2 orders of magnitude) in theresistivity measured. Similar results, that is, this dramatic change at 3 to 4 monolayers, werealso found for samples API263 No.9021 and Berea100A(5) and can be seen in Figures 3.6and 3.7. API236 No.1433 shows this dramatic change at 2 monolayers (Figure 3.8). Furtherdata analysis of API236 No.1433 are not presented as a crack developed in the sample whilein the high pressure saturating set-up.Application of Archie's Equation The drainage data for each sample, 22-11-12 #10069, API263 No.9021 andBerea100A(5) are plotted as log I versus log Sw in Figures 3.9 through 3.11. Since I =Rm/Ro, data from all samples were extrapolated to determine Ro (the resistivity of the core at100% water saturation). The method of least squares is used to determine the best fit lines tothese data. The coefficient of determination, R2, indicates how well the line fits the data.Figure 3.9 contains the resistivity data from sample 22-11-12 #10069 plotted as log Iversus log Sw. These data are best described by two lines. The data collected over most ofthe saturation range, 0.004 < Sw < 0.88, fit Archie's equation very well. These data define aline with slope, n, equal to 1.07 (R2 = 0.995). For the data in the region Sw < 0.004 the0.007S w0.013^0.020.^1.026t57J022-11-12 #10069o primary drainage6005- 00000 00 040 4 8 12 16 20number of monolayers of waterFigure 3.5.^Log resistivity versus number of monolayers of water for 22-11-12#10069 during primary drainage.7S w0.008^0.015^0.023.^I^,^I^.0.031158API263 No.90210o primary drainage6 -00O00050 0 0 0^04 -^1^0^4^8^12^16^20number of monolayers of water^Figure 3.6.^Log resistivity versus number of monolayers of water for API263No.9021 during primary drainage.59S w0.014^0.027^0.0411Berea100A(5)o primary drainage005.5 -006.5 -4.5 -0 000 00 0 0 0 003.54 8 12 16number of monolayers of waterFigure 3.7.^Log resistivity versus number of monolayers of water forBerea100A(5) during primary drainage.SW0.015^0.030^0.045^0.060API236 No.14331^o primary drainage6-000 000 -0 0 0 o 0 0004I^.^r^•^I^0^4^8 12 16^20number of monolayers of water^Figure 3.8.^Log resistivity versus number of monolayers of water for API236No.1433 during primary drainage.6053OM=C)061.0 0 1^. 0 1^1SwFigure 3.9.^Log I versus log Sw for 22-11-12 #10069 during primary drainage.62modified Archie's equation is used to obtain n* equal to 4.05 and K equal to 1.27x10 -7 (R2 =1.000, used only two data points). The modified Archie equation is used when K does notequal 1.Figure 3.10 contains the resistivity data from API263 No.9021 plotted as log I versuslog Sw. These data are best represented by a single line over the entire saturation rangemeasured, which, in this case was 0.001 < Sw < 0.91. These data fit the modified Archieequation very well with n . is equal to 1.19, and K is equal to 1.70 (R2 = 0.988).Figure 3.11 contains the resistivity data from Berea100A(5) plotted as log I versus logSw. Here it can be seen that the data are best described by three lines; the first correspondingto the saturation range defined by 0.010 < Sw < 0.92 where n is equal to 0.77 (R 2 = 0.987),the second corresponding to the saturation range defined by: 0.002 < Sw < 0.007 where n * isequal to 2.10 and K is equal to 2.9x10" 3 (R2 = 0.988) and the third corresponding to thesaturation range defined by Sw < 0.002 where n. is equal to 0.17 and K is equal to 508 (R2 =1.000, used only two data points).Primary Drainage Data: Teflon CoreFigure 3.12 contains the drainage data plotted as log Rm versus Sw for the teflon-water-air system. The maximum Sw reached was 0.85. In the region defined by 0.70 < Sw <0.85 resistivity decreased slightly with decreasing saturation. This may be a problem causedby the pressure differential felt by the sample when it is first removed from the saturatingsetup. For the region defined by 0.60 < Sw < 0.70 the log Rm versus Sw curve is very flatindicating little change in measured resistivity. In the region defined by 0.013 < Sw < 0.60resistivity increases 3.5 orders of magnitude. At Sw = 0.019 the resistance of the teflon has1^v. 0001^. 001 . 0 1^. 1^1SwFigure 3.10. Log I versus log Sw for API263 No.9021 during primary drainage.63.0001^. 001^. 01^1SW64Figure 3.11. Log 1 versus log Sw for Berea100A(5) during primary drainage.0)O 438650Teflon50 primary drainage: water......^7EC:S....•000O o 0 00 o0I^ I^•^I^.^I^0 0^0 . 2 0.4 0.6 0.8^1 0Sw^Figure 3.12.^Log resistivity versus Sw for Teflon5 during primary drainage.6-O5- 0000066increased beyond the measuring capabilities of the impedance analyzer.Figure 3.13 contains drainage data plotted as log Rm versus Sm for the methanolsaturated teflon sample. Sm is defined as the volume fraction of the pore space filled withmethanol, the remaining space being filled with air. In the region defined by 0.90 < Sm <0.60 little change is seen in the measured resistivity. The region defined by 0.10 < Sm < 0.60resistivity increases by 11/2 orders of magnitude. For Sm < 0.10 resistivity increases by 2orders of magnitude producing the very steep region for Sm < 0.10.Application of Archie's Equation The drainage data for the water saturated Teflon5 sample are plotted as log / versuslog Sw in Figure 3.14. These data can be represented by a single line over the saturationrange defined by 0.03 < Sw < 0.66. These data fit the Archie equation very well with nequal to 2.96 (R2 = 0.996). Data in the region defined by 0.70 < Sw < 0.85 are not analyzedbecause there may have been a problem caused by pressure differences when the sample wasfirst removed from the saturating setup.Figure 3.15 contains the drainage data from the methanol saturated Teflon5 sampleplotted as log I versus log Sm. These data can be represented by a single line over thesaturation range defined by 0.02 < Sm < 0.73. These data fit the Archie equation very wellwith n equal to 1.98 (R 2 = 0.983).Imbibition-Drainage Data: Natural Core SamplesData from the imbibition-drainage experiment completed on 22-11-12 #10069 areshown in Figure 3.16 as a plot of log Rm versus Sw. There are two distinct curves over most 677 0Teflon50 0 primary drainage: methanol60O054O000 o o oo O o 0 0 0a)0 O 0 0 0 000 O3 ^^0 . 0 0 . 2^0 . 4^0 .6^0 . 8^1 0Sm^Figure 3.13.^Log resistivity versus Sm for Teflon5 during primary drainage.Methanol saturated sample.0)068Figure 3.14. Log I versus log Sw for Teflon5 during primary drainage."^r " ' •. 0 1^ . 1SmFigure 3.15. Log / versus log Sm for Teflon5 during primary drainage. Methanolsaturated sample.69V 1-17007 •22-11-12 #100696-a o imbibition• drainage•54 -3-Qir ••• •o •• • • • • g • Do• • • • • •▪ `n1:13• • • 011) a30 a2 ^ • 1^0 0 0 . 2^0 . 4^0 . 6^0 . 8^1 . 0Sw^Figure 3.16.^Log resistivity versus Sw for 22-11-12 #10069 during imbibition(increasing Sw) and drainage (decreasing Sw).71of the saturation range, the lower curve corresponding to imbibition (increasing Sw) and theupper curve corresponding to drainage (decreasing Sw). This hysteresis, dependent onsaturation history, has also been noted by Longeron et al. (1989) and Knight (1991) inlaboratory measurements of electrical resistivity. In Figure 3.16, at very low saturations, Sw< 0.007 the drainage and imbibition curves are indistinguishable. In the region defined by0.45 < Sw < 0.53 log Rm increases with increasing Sw. For Sw > 0.55 log Rm decreases withincreasing saturation for both imbibition and drainage.Figure 3.17 contains the data collected during an imbibition-drainage cycle on API263No.9021 plotted as log Rm versus Sw. This sample also shows hysteresis between theimbibition and drainage curves in the saturation range defined by 0.005 < Sw < 0.60. For Sw< 0.005 and Sw > 0.60 the drainage and imbibition curves are indistinguishable. There is asudden drop in resistivity at Sw = 0.06. This corresponds to the point at which the samplewas wrapped in Saran WrapTM and stored overnight in a humidity chamber. This drop mayreflect a redistribution of the water within the pores.In Figure 3.18 is plotted log Rm versus Sw for Berea100A(5). Similar to the previoustwo imbibition-drainage plots, this too shows a hysteretic region defined by 0.007 < Sw <0.25. Above and below this hysteretic region the imbibition and drainage curves combine toform one curve.Low Saturation Region The imbibition-drainage data for 22-11-12 #10069 are plotted in Figure 3.19 as log Rmversus number of monolayers of water. Included in this plot are the data from the primarydrainage experiment discussed previously. The drainage curves from the two differentCD06 ZI••5 kw•cl II •4 - cl^•oaRSO 0O7723-API263 No.9021o imbibition• drainage• •• • • •S. • _X 00CI 1:3 ■ • • ,.a CI El 13 0 D O rliwi VI%2 I^r^1^,^I^0 0^0 . 2 0 . 4 0 . 6^0 . 8^1 0S w^Figure 3.17.^Log resistivity versus Sw for API263 No.9021 during imbibition(increasing Sw) and drainage (decreasing Sw).730)07Berea100A(5)6B1:1^imbibition• drainage54-fi•3-Se21^• • • WOO 0 ri41 °0*ID ibla•:1••11•9cl2 ^ 1^0 . 0 0 . 2^0 . 4^0 . 6^0 .8^1 0SW^Figure 3.18.^Log resistivity versus Sw for Berea100A(5) during imbibition(increasing Sw) and drainage (decreasing Sw).4-3 I^,^r0•B etB B16 201280 422-11-12 #10069o imbibition• drainageo primary drainage• 0• •• 0•B• •o • 0 a•00• 8o•0•o07650.007Sw0.013^0.020 0.02674number of monolayers of waterFigure 3.19.^Log resistivity versus number of monolayers of water for 22-11-12#10069 during imbibition (increasing Sw) and drainage (decreasing Sw). Also includedare the data for primary drainage.75experiments correspond very well. Again, as in the drainage data discussed previously; wecan see that the addition of a very small quantity of water, in this case 4 monolayers, causes adramatic drop (11/2 orders of magnitude) in the resistivity measured. The point at whichhysteresis starts also corresponds to 4 monolayers.Figures 3.20 and 3.21 contain the imbibition-drainage data plotted as log Rm versusnumber of monolayers of water for samples API263 No.9021 and Berea100A(5) respectively.Qualitatively these data are very similar to the data shown in Figure 3.19 for 22-11-12#10069.Application of Archie's Equation The imbibition data for 22-11-12 #10069 are plotted in Figure 3.22 as log I versus logSw. These data are best represented by four lines, all defined by the modified Archieequation as outlined below:Sw < 0.0090.014 < Sw < 0.46n. = 4.22. = 1.05nK = 2.36 x 10 -7K = 0.32R2 = 1.000R2 = 0.9950.48 < Sw < 0.51 n* = -16.05 K= 1.77 x 105 = 0.9990.54 < Sw < 0.77 n` =2.7 8 K = 0.73 R2 = 0.985The drainage data for 22-11-12 #10069 are plotted in Figure 3.23 as log I versus log Sw.These data are best described by two lines. The data collected over most of the saturationrange, 0.006 < Sw < 0.79, fit the Archie equation very well. These data define a line withslope, n, equal to 1.13 (R2 = 0.998). For the data in the region Sw < 0.003 the modifiedArchie equation is used to obtain n* equal to 6.06 and K equal to 2.27x1042 (R2 = 1.000, usedonly data points).Irci imbibition• drainageo primary drainage7 S w760.010 0.019 0.0290 AP1263 No.9021650• 013^o• •130• • o• fl) o• so 9 0^o■ • • • • o•a•4aaao320T15 201^ 15 10number of monolayers of waterFigure 3.20. Log resistivity versus number of monolayers of water for AP1263No.9021 during imbibition (increasing Sw) and drainage (decreasing Sw). Alsoincluded are the data for primary drainage.77Sw0.014^0.027^0.041^0.05570Berea100A(5)a imbibition• drainageo primary drainage•o% % 70 •0 •a• 0 • • •a a6- a o0 5-4-32 1 i^112 16 200^4^8number of monolayers of waterFigure 3.21.^Log resistivity versus number of monolayers for water forBerea100A(5) during imbibition (increasing Sw) and drainage (decreasing Sw). Alsoincluded are the data for primary drainage.0)078. 0 0 1^. 0 1^. 1^1SWFigure 3.22.^Log 1 versus log Sw for 22-11-12 #10069 during imbibition.C)079. 0 01^. 0 1^1SwFigure 3.23.^Log 1 versus log Sw for 22-11-12 #10069 during drainage.80The imbibition data for API263 No.9021 are plotted as log I versus log Sw in Figure3.24. These data are best represented by three lines as outlined below:Sw < 0.54 . = 1.02n K = 0.65 R2 = 0.9380.54 < Sw < 0.65 n* = -2.96 K = 8.40 R2 = 0.7220.65 < Sw < 0.71 . = 4.84n K = 0.30 R2 = 0.999Figure 3.25 contains the drainage data for API263 No.9021 plotted as log I versus log Sw.Data collected over the entire saturation range, 0.004 < Sw < 0.70, can be fitted to themodified Archie equation with n` equal to 1.04 and K equal to 1.52 (R2 = 0.979).Figure 3.26 contains the imbibition data for Berea100A(5) plotted as log I versus logSw.^These data are best represented by four lines as given below:. = 0.17Sw < 0.003^n^K = 348^R2 = 1.0000.007 < Sw < 0.11 n. = 1.81 K = 0.02 R2 = 0.9830.14 < Sw < 0.35 . = -0.69n K = 5.73 R2 = 0.9430.39 < Sw < 0.77 n = 1.09. R2 = 0.871The drainage data for Berea100A(5) are plotted as log I versus log Sw in Figure 3.27. Thesedata can be represented by a single line over the entire saturation range, 0.001 < Sw < 0.76,with n equal to 0.94 (R2 = 0.983).n* = 4.848100 0 1^.001^.01SwFigure 3.24. Log 1 versus log Sw for AP1263 No.9021 during imbibition.SwFigure 3.25. Log I versus log Sw for API263 No.9021 during drainage.82.0001^.001^.01^. 1^1SWFigure 3.26. Log 1 versus log Sw for Berea100A(5) during imbibition.838432.01•MIC)O10.001^.01^ 1SWFigure 3.27.^Log I versus log Sw for Berea100A(5) during drainage.CHAPTER 4DISCUSSIONIn this study two methods were used to change the fluid/air distribution withinsandstone and teflon cores. The first method involves changing saturation technique, thesecond method involves varying wetting conditions. The drainage data in this study can beassessed in light of percolation theory. Percolation theory will not be applied to theimbibition data because of the variability in saturation exponents found.Variations in n and n * Due to Saturation Technique: Natural Core SamplesPrimary Drainage DataThe Archie saturation exponents, n and n`, calculated for 22-11-12 #10069, API263No.9021, and Berea100A(5) during primary drainage (Figures 3.9 through 3.11) are lowerthan most values found in the literature (Table 1.1). In this study, the values for thesaturation exponent during primary drainage in the saturation region defined by Sw > 0.01are: n equal to 0.77 for Berea100A(5), n equal to 1.07 for 22-11-12 #10069 and n* equal to1.19 for A_PI263 No.9021. In the literature saturation exponents range from 1.08 (Dunlap etal., 1949) to 26.43 (Morgan and Pirson, 1964). Since the saturation exponents in this studyrepresent water-wet sandstones a more appropriate comparison would be to compare with8586studies also on water-wet sandstones. Thus, published values of n for water-wet sandstonesrange from 1.2 (Lewis et al., 1988) to 2.60 (Longeron et al., 1989). Even after appropriatecomparisons are made, the values for the saturation exponents in this study are still low.Of specific interest to this study is the validity of Archie's equation in the lowsaturation region, Sw less than 0.10. The drainage data collected on the natural core samplesshow a rapid change in resistivity at lower saturations. It has been suggested (Knight andEndres, 1990) that this dramatic change in resistivity at low levels of water saturation is dueto geometrical and electrochemical effects. To evaluate the geometrical and/orelectrochemical effects at the solid/liquid interface a high density of data was collected in thelow Sw region. If the conduction mechanism is different in this low Sw region we wouldexpect to see a change in the Archie saturation exponent.The primary drainage data for 22-11-12 #10069 are plotted as log I versus log Sw inFigure 3.9. A single saturation exponent, n equal to 1.07 was found to extend from Sw equalto 0.88 down to Sw equal to 0.004. This shows Archie's equation can be extended down intothe low saturation region. There is no change in n at low saturations; thus, it appears thegeometry of the fluid phase is the controlling factor for n. Electrochemical interactions arenot needed to explain the rapid change in resistivity seen at low levels of water saturation.Although n * equal to 4.05 has been calculated for Sw less than 0.004, there are insufficientdata at the very low levels of water saturation to evaluate the usefulness of Archie's equationin this region and therefore no significance is placed on the saturation exponent found here.The two other natural core samples, API263 No.9021 and Berea100A(5), showed similarresults; above a critical saturation a single Archie saturation exponent, n or n', can be used torepresent the rock during primary drainage.87Imbibition-Drainage DataIn this study, for both primary drainage and the drainage portion of the imbibition-drainage experiments, one saturation exponent, n or n*, was defined over most of thesaturation range. These saturation exponents, n and ?I% determined for the drainage portion ofthe imbibition-drainage experiments (Figures 3.23, 3.25 and 3.27), can be compared to thesaturation exponents determined during primary drainage (Figures 3.9 through 3.11). Thesimilarity of these saturation exponents for the two different drainage experiments can be seenin Table 4.1.Table 4.1. Saturation Exponents for Drainage DataSample Primary DrainageExperiment: drainageImbibition-DrainageExperiment: drainagesaturationrangesaturationexponentsaturationrangesaturationexponent22-11-12 #10069API263 No.9021Berea100A(5)> 0.004> 0.001> 0.0101.071.19*0.77> 0.004> 0.004> 0.0011.131.04*0.94* These are n* values calculated using the modified Archie equation.Knight (1991) has suggested that the reproducibility of electrical resistivitymeasurements made during drainage is a reflection of the stability of the fluid geometry. Inthis study, the similarity of these saturation exponents for the two different drainageexperiments is a reflection of the reproducibility of the fluid geometries during drainage.Changing saturation techniques, imbibition or drainage, will produce variation in88pore-scale fluid distributions. The distribution of the air and water within the pore space willdiffer depending upon whether the saturation level was achieved by adding water or byremoving water. The hysteresis seen in the log R m versus Sw plots for the imbibition-drainage experiments (Figures 3.16 through 3.18) is attributed to varying fluid distributionscaused by changing saturation technique. Knight and Nur (1987b) presented a simple modelof pore-scale fluid geometries to describe dielectric hysteresis. This model has recently beenextended to hysteresis found in electrical resistivity data collected during imbibition anddrainage (Knight, 1991). The variation in pore-scale fluid distribution will be reflected in theArchie saturation exponents as well.This study shows there is variation in the saturation exponents between imbibition anddrainage for each of the natural core samples. This has been found by other investigators(Dunlap et al., 1949, Whiting et al., 1953, Lewis et al., 1988, Longeron et al., 1989) and hasbeen attributed to variations in fluid distribution caused by saturation technique. Longeron etal. (1989) also found, not only did n vary between imbibition and drainage, but in addition,during imbibition n was found to vary with the level of water saturation.In this study the saturation exponents for imbibition varied with level of watersaturation (Figures 3.22, 3.24 and 3.26). Using the modified Archie equation, up to fourdifferent saturation exponents were found for each core over the entire saturation rangemeasured. For example, the Archie saturation exponents during imbibition for 22-11-12#10069 were; rt . equal to 4.22, 1.05, -16.05, and 2.78. As with the drainage data, nosignificance is placed on the saturation exponent calculated in the very low Sw region, n*equal to 4.22. The value of n. equal to -16.05 corresponds to the point where resistivitysuddenly increases. In the model proposed by Knight (1991) this sudden increase in89resistivity corresponds to the rearrangement of the fluid/air geometry within the pore space.A brief description of the model follows. During imbibition, thick metastable surface layersare formed which are separated by a thin central air phase. This water/air interface creates alarge surface area along which there is surface conduction. At some critical point thisgeometrical arrangement of the water and air may become unstable. The water and air willrearrange to a more stable geometry which no longer contains this continuous air phase. Theloss of this large water/air interface results in an increase in resistivity which leads to thenegative saturation exponents found in this study. No other references to negative saturationexponents were found in the literature. This negative value for a saturation exponent mayindicate a drastic rearrangement of fluid-air geometry within the pore space. It appearsArchie's equation is not applicable in this region. In general, the variation in n withsaturation level during imbibition may be a reflection of a lower stability of the water/airgeometry.Variations in n and n* Due to Wetting Conditions: Teflon Core The effect of pore-scale fluid distribution on Archie's saturation exponent hasfrequently been studied by changing the surface properties of the porous material, forexample, in rocks from water-wet to nonwater-wet (Table 1.1). In this study the drainagedata for both the methanol and water saturated teflon are analyzed in terms of Archie'sequation. The differences in the Archie saturation exponents calculated for the methanol andwater saturated teflon can be attributed to variations in pore-scale fluid distribution caused bythe varying wetting properties of the conducting fluid used.The value for n, equal to 1.98, as calculated from the drainage data for the methanolsaturated Teflon5 compares well with that found by Mungan and Moore (1968), n equal to901.91, also for a methanol/teflon system. For the water saturated Teflon5, a single value of nequal to 2.96 was found, whereas Mungan and Moore (1968) showed n to vary with level ofsaturation. See Table 1.1, Chapter 1 of this study for the n values found by Mungan andMoore (1968).In this study, the higher value of n found for the nonwetting conducting fluid is asexpected. This agrees with many studies (Table 1.1) indicating increasing values for n as thesystem moves from water-wet to nonwater-wet. The higher n value found for the teflon/watersystem has in the past (Mungan and Moore, 1968) and is in this study attributed to thefluid/air distribution within the pore space. Higher water saturations are necessary to formthe connected path needed for conduction. This increase in n reflects the increasing tortuosityof the conducting path.Figure 1.1 is a schematic illustration of the water/air distributions envisioned forwater-wet and nonwater-wet systems. Figure 1.1a could also represent the geometryenvisioned for the methanol/teflon/air system, where instead of water, methanol would be thewetting conducting fluid. In the methanol-wet teflon, methanol is the continuous phase andremains the continuous phase down to low levels of saturation. This continuous conductingphase along the pore surfaces provides an efficient path for conduction. The lower value forn, equal to 1.98, reflects this less tortuous path. In the teflon/water system, at low levels ofwater saturation, the air is the continuous phase with the water forming discrete droplets. Forthe water to form the continuous phase we must move to much higher saturations. Thisresults in higher n' s. The log Rm versus Sw curve shows this very well.91Application of Percolation Theory Archie's equation is one expression describing the variation in conduction in a systemwith variation in the volume fraction of the conductor. The drainage data collected in thisstudy can also be assessed in light of percolation theory, a more generalized treatment of therole of the volume fraction of conducting phase in the conduction of a system. As reviewedin the Introduction, the values of the critical conductivity exponent, t, depend on dimension;1.0 < t < 1.5 for two-dimensions and 1.5 < t < 2.4 for three-dimensions. The balancebetween three-dimensional conduction through the fluid occupying the pore space and two-dimensional conduction along the surface of the pore space can be clarified by comparison ofresults from Archie's equation with the results from percolation theory.Archie's equation assumes conduction through the volume of the pore fluid; a three-dimensional conduction process. As the water saturation is decreased it is expected the roleof the rock/water interface in the conduction process will become more important. Thus, inmoving from three-dimensional conduction through the volume of the pore fluid to two-dimensional conduction along the pore surfaces one would expect to see a change in thesaturation exponent. The drainage data for these water-wet sandstones did not show this. Asshown in Table 4.1, during drainage, one saturation exponent for each rock was found to berepresentative over most of the saturation range measured. These values obtained for thesaturation exponents are low compared with those found in the literature (Table 1.1). Theselower saturation exponents may be due to the conducting fluid used. In this study distilledwater was used; in other studies brines were used as the conducting fluid. It appears that byusing distilled water, surface conduction becomes more important than conduction through thebulk liquid. Parkhomenko (1967) recognized that rocks with large specific surface areascould exhibit appreciable amounts of surface conduction. She noted that the surface92conduction is further enhanced when deionised water is used as a conducting fluid. For therocks in this study it appears distilled water has enhanced the surface conduction.The modified Archie equation (equation 1.5) is used to analyze data in terms ofpercolation theory. To calculate n`, Sw„„ must be determined. For the natural cores, Sw c,„was of the order of 10 -4 , and was therefore considered to be zero. On a practical level thismeans that once the water-wet core is exposed to the atmosphere, water is adsorbed, aconnected surface phase is formed, and thus there will be conduction along the surface of thepores. The values found for n and n* summarized in Table 4.1 can be compared to thecritical exponent for conductivity, t (Figure 1.2). The values of n and n`, for two of the threenatural core samples, falls into the range given for t in two-dimensions (Figure 1.2a). Themonolayer plots (Figures 3.19 through 3.21) indicate the importance this envisioned thinsurface layer has on conduction. This connected adsorbed water phase is present down tovery low levels of water saturation and accounts for the main conduction path. The ability ofthese rocks to maintain this connected surface layer of water down to very low levels ofwater saturation results in a two-dimensional surface along which the current will flow. Ittherefore does not seem unreasonable that the water-wet natural core samples can be modelledusing two-dimensional percolation.These values, n equal to 1.07 and 1.13 for 22-11-12 #10069 and n * equal to 1.19 and1.04 for API263 No.9021 are lower than the currently accepted value for t in two-dimensions,equal to 1.3. Berea100A(5) is much lower with n equal to 0.77 and 0.94. The SEMmicrographs for the natural core samples revealed variation in the mineralogy spatially withinthe pore space. Some pores were lined with high surface area clays while other pores wereclean. The lower values for the saturation exponents may be a reflection of this anisotropy93on the pore-scale. It should be remembered that the value of t, equal to 1.3, is for randomtwo-dimensional percolation. Redner (1983) completed a study on random resistor-diodenetworks which would allow for directed percolation. He found the directed conductivityexponent, t+ , to be smaller than the conductivity exponent, t, for a random resistor network inthe same dimension. In two-dimensions he found a - 0.6 whereas the currently acceptedvalue for t is 1.3. The biased nature of Redner's resistor-diode network may be analogous tothe anisotropy found in these rocks at the pore-scale.Unlike the natural core samples where the percolation threshold, Sw,„ was taken aszero both the methanol and water saturated teflon samples exhibited a measurable percolationthreshold. For the water saturated Teflon5 Sw„„ is equal to 0.020. For the methanolsaturated Teflon5 Sm c ,„ is equal to 0.029. Plots of log I versus log (Sm - Sm„„) for themethanol and log I versus log (Sw - Swc,„) for the water saturated samples are seen in Figures4.1 and 4.2. In Figure 4.1 n* is equal to 1.44 (R 2 = 0.984) which is lower compared to the ncalculated earlier from Archie's equation. Similar results are seen in Figure 4.2 where n . isequal to 2.34 (R 2 = 0.991), again, lower than the earlier calculated value. Comparing these n*values to the values for t given earlier, one can see that the methanol saturated sample fitsinto the range of t's given for two-dimensional percolation while the water saturated teflonfits into the range of t's given for three-dimensional percolation (Figure 1.2). This three-dimensional percolation for the water saturated teflon sample is expected, since the water inthe teflon sample does not wet the teflon and therefore conduction is through the bulk fluid.There should be no surface conduction at the teflon/water interface; therefore conduction isthrough a three-dimensional system.For the samples studied, it appears that if the conducting fluid wets the insulatingrn094Sm-SmcritFigure 4.1.^Log I versus log (Sm - Sm„„) for Teflon5 during methanol drainage.a)095. 0 1^ 1Sw-SwcritFigure 4.2.^Log I versus log (Sw - Swcni.) for Teflon5 during water drainage.96matrix, the system will be dominated by a two-dimensional conducting mechanism. The useof high resistivity conducting fluids, distilled water and methanol, enhances the surfaceconduction. This is seen in the drainage data for both the water-wet rocks and the methanol-wet teflon. If the conducting fluid no longer has an affinity for the insulating matrix,conduction is through a three-dimensional system. This is seen in the drainage data for thewater saturated teflon sample.CHAPTER 5CONCLUSIONThe results of this study clearly show that the measurement of electrical resistivity notonly provides information on the volume of the conducting fluid present, but in addition,provides insight into pore-scale fluid distribution. Archie's equation can be used to relatemeasured resistivity to level of water saturation. The comparison of Archie's saturationexponents with results from percolation provide insight into the importance of three-dimensional conduction through the bulk of the pore fluid and/or two-dimensional conductionalong the surface of the pores.The differences found in Archie's saturation exponent between imbibition and drainagehas been attributed to changes in pore-scale fluid distribution. These imbibition-drainageexperiments completed on the three natural core samples; 22-11-12 #10069, API263 No.9021,and Berea100A(5), showed hysteresis in the electrical resistivity measurements made. At agiven Sw, the geometry of the air/water phase differed depending upon whether the saturationlevel was achieved by the addition of water or the removal of water. While a singlesaturation exponent can be used to describe the "drainage" geometry, the saturation exponentsfound for imbibition vary with level of water saturation. The differences in the saturationexponents are a reflection of the varying fluid/air geometries during the different saturationprocesses.Varying the wetting properties of the system also resulted in variations in fluid9798geometry. These variations in fluid geometry are reflected in the changing Archie saturationexponents. The primary drainage experiments completed on the water-wet natural coresamples, the methanol-wet teflon and the nonwater-wet teflon clearly showed the increase inthe saturation exponent with the decreasing wetting ability of the conducting fluid. Thesaturation exponent for the three natural core samples; 22-11-12 #10069, API263 No.9021,and Berea100A(5), ranged from 0.77 to 1.19. The saturation exponents for the methanol andwater saturated Teflon5 were 1.98 and 2.96 respectively. These varying saturation exponentsreflect the different fluid distributions caused by changing the wetting properties.In this study Archie's equation was shown to be valid in the low Sw region. Thelower saturation exponents found for the natural core samples indicate a water-wet system,which, by first coating the surfaces of all pores with water, tends to provide an efficient pathfor conduction. In the water-wet rocks, the covering of the pore surfaces with a layer ofwater is present down to very low levels of water saturation. The ability of these rocks tomaintain water connectivity through an adsorbed water phase results in low values forArchie's saturation exponent. Higher saturation exponents for nonwetting systems indicatesincreasing tortuosity. The surface connected path does not exist and therefore a goodconducting path is not established until discrete water droplets in the centre of the porescombine to form a continuous network.The saturation exponents obtained for the water-wet rocks and the methanol-wet teflonare within the range given for the percolation conductivity critical exponent in twodimensions. In a wetted system, one can assume that during drainage, the adsorbedconducting phase maintains a connected path along the surface of the pores down to very lowsaturation levels. This connected path can be represented as a two dimensional surface. Then and n* calculated from the drainage data for the water-wet rocks and methanol-wet teflon99indicates these systems can be modelled using two-dimensional percolation.The saturation exponent obtained for the water saturated teflon sample is within therange given for the percolation conductivity exponent in three dimensions. Teflon is ahydrophobic material, and water will not adsorb to the surface; thus at lower levels of watersaturation the distribution of the water within the pores will be as discrete droplets. In theabsence of this surface conducting layer of water, the conduction path must be developedthrough the bulk water phase. The conduction path is formed through the clustering of thewater droplets and therefore the conduction is through a three dimensional network. Thedrainage data for the water saturated Teflon5 indicates the nonwetting system in this studycan be modelled using three-dimensional percolation.All these data indicate the importance of fluid distributions on measured resistivity.Although this complicates the calculation of in situ water saturations, in the laboratory wheresaturations can be controlled, insight can be gained into fluid distributions and saturationhistory. An understanding of how fluid distributions affect measured resistivity must first bedetermined before in situ water saturations can be calculated.100REFERENCESAdler, D., L.P. Flora and S.D. Senturia^1973:^Electrical conductivity in disordered systems; Solid State Communications, v.12, p. 9-12.Archie, G.E.1942:^The Electrical Resistivity Log as an Aid in Determining Some ReservoirCharacteristics; Petroleum Development and Technology, AIME, v. 146, p. 54-62.Broadbent, S .R. and J.M. Hammersley1957:^Percolation Processes; Proc. Camb. Phil. Soc., v.53, p. 629.Brunauer, S., Emmett, P.H. and E. Teller1938:^Adsorption of Gases in Multimolecular Layers; American Chemical SocietyJournal, v. 60, p. 309-319.Brunauer, S., Deming, L.S., Deming, W.E. and E. Teller1940:^On a Theory of the van der Waals Adsorption of Gases; American ChemicalSociety Journal, v. 62, p. 1723-1732.Coates, G.R. and J.L. Dumanoir1974:^A new approach to improved log-derived permeability; The Log Analyst, v. 5,p. 17-31.Cullity, B.D.1959:^Elements of X-ray Diffraction; Addison-Wesley Publishing Company, Inc.,Reading, Massachusetts, U.S.A.,514 p.Diederix, K.M.1982:^Anomalous Relationships Between Resistivity Index and Water Saturations inthe Rotliegend Sandstone (The Netherlands); SPWLA Twenty-Third AnnualLogging Symposium, July 6-9, 1982, Issue 10, p. 1-16.Donaldson, E.C. and T.K. Siddiqui1989:^Relationship Between the Archie Saturation Exponent and Wettability; Societyof Petroleum Engineers Formation Evaluation, p. 359-362.Donaldson, E.C. and M.J. Bizerra1985:^Relationship of Wettability to the Saturation Exponent; Third InternationalConference on Heavy Crude and Tar Sands, July 22-31, UNITARJUNDPInformation Centre for Heavy Crude and Tar Sands, p.664 -684.101Dunlap, H.F., Bilhartz, H.L., Shuler, E. and C.R. Bailey^1949:^The Relation Between Electrical Resistivity and Brine Saturation in ReservoirRocks; American Institute of Mining and Metallurgical Engineers, Transactions,Petroleum Branch, v. 186, p. 259-264.Gabriel, B.L.1985:^SEM: A Users Manual for Material Science; American Society for Metals, 198P.Garrouch, A.1987:^The Effect of Wettability, Stress, and Temperature on the Saturation andCementation Exponent of the Archie Equation; M.Sc.Eng., University of Texasat Austin, Austin, Texas, 129 p.Gregg, S.J. and K.S.W. Sing1982:^Adsorption, Surface Area and Porosity; Second edition, Academic Press,London, 303 p.Guyod, H.1948:^Electrical Logging Developments in the U.S.S.R.; Part 6, World Oil, v. 128,no. 4, p. 110-120.Keller, G.V.1953:^Effect of Wettability on the Electrical Resistivity of Sand; Oil and Gas Journal,v. 51, no. 34, p. 62-65.Kirkpatrick, S.1973:^Percolation and Conduction; Reviews of Modern Physics, v. 45, p. 574-588.Knight, R.J.1985:^The Dielectric Constant of Sandstones, 5 Hz to 13 MHz: Ph.D. dissertation,Stanford University, Stanford, California, U.S.A.1991:^Hysteresis in the electrical resistivity of partially saturated sandstones;Geophysics, v. 56, p. 2139-2147.Knight, R.J.1990:Knight, R.J.1987a: •and A. EndresA new concept in modeling the dielectric response of sandstones: Defining awetted rock and bulk water system; Geophysics, v. 55, p. 586-594.and A. NurThe dielectric constant of sandstones; 60 kHz to 4 MHz, Geophysics, v. 52, p644-654.1987b:^Geometrical Effects in the Dielectric Response of Partially SaturatedSandstones; The Log Analyst, v. 28, p. 513-519.102Last, B.J. and D.J. Thouless^1971:^Percolation Theory and Electrical Conductivity; Physical Review Letters, v. 27p. 1719-1721.Lewis, M.G., M.M. Sharma and H.F. Dunlap1988:^Wettability and Stress Effects on Saturation and Cementation Exponents;Transactions SPWLA Twenty-Ninth Annual Logging Symposium, v. 1, PaperK.Longeron, D.G., Argaud, M.J. and J.P. Feraud1986:^Effect of Overburden Pressure, Nature, and Microscopic Distribution of theFluids on Electrical Properties of Rock Samples; Society of PetroleumEngineers, Paper 15383, 12 p.1989:^Effect of Overburden Pressure and the Nature and Microscopic Distribution ofFluids on Electrical Properties of Rock Samples; Society of PetroleumEngineers Formation Evaluation, v. 4, p. 194 -202.McClellan, A.L. and H.F. Harnsberger1967:^Cross-sectional Areas of Molecules Adsorbed on Solid Surfaces; Journal ofColloid and Interface Science, v. 23, p. 577-599.Micromeritics Instrument Corporation1974:^Instruction Manual; Model 2100D, Orr Surface-Area Pore-Volume Analyzer,p.I-VIII.Monicard, R.P.1980:^Properties of Reservoir Rocks: Core Analysis; Gulf Publishing Company, Paris,165 p.Morgan, W.B. and S.J. Pirson1964:^The Effect of Fractional Wettability on the Saturation Archie Exponent;Transactions SPWLA Fifth Annual Symposium, Sec. B, p. 1-13.Mungan, N. and E.J. Moore1968:^Certain Wettability Effects on Electrical Resistivity in Porous Media; Journal ofCanadian Petroleum Technology, v. 7, p. 20-25.Parkhomenko, E.I.1967:^Electrical Properties of Rocks; Plenum Press, New York, 314 p.Redner, S.1983:^Percolation and Conduction in Random Resistor-Diode Networks: inPercolation Structures and Processes; edited by G. Deutscher, R. Zallen and J.Adler, Annals of the Israel Physical Society, v.5, p. 447-476.103Rust, C.F.^1952:^Electrical Resistivity Measurements on Reservoir Rock Samples By The Two-Electrode and Four-Electrode Methods; American Institute of Mining andMetallurgical Engineers, Transactions, Petroleum Branch, v. 195, p. 217-224.1957:^A Laboratory Study of Wettability Effects on Basic Core Parameters; Societyof Petroleum Engineers Paper 986-G, p. 1-16.Sahimi, M.1984:^On the relationship between the critical exponents of percolation conductivityand static exponents of percolation; Journal of Physics A: Mathematical andGeneral, v. 17, p. L601-L607.Stauffer, D.1985:^Introduction to Percolation Theory; Taylor and Francis Ltd., London, 124 p.Swanson, B.F.1980:^Rationalizing the Influence of Crude Wetting on Reservoir Fluid Flow WithElectrical Resistivity Behavior; Journal of Petroleum Technology, p. 1459-1464.Sweeney, S .A. and H.Y. Jennings, Jr.1960:^The Electrical Resistivity of Preferentially Water-Wet And Preferentially Oil-Wet Carbonate Rock, Producers Monthly, v. 24, no. 7, p. 29-32.Walther, H. C.1968:^Saturation From Logs - Laboratory Measurements of Logging Parameters;Journal of Petroleum Technology, p. 251-258.Welton, J.E1984:•SEM Petrology Atlas; American Association of Petroleum Geologists, ed. R.Steinmetz, 237 p.Whiting, R. L., Guerrero, E.T. and R.M. Young1953:^Electrical Properties of Limestone Cores; Oil and Gas Journal, vol. 52, no. 12,p. 309-315.Wilson, M. J.1987:^X-ray powder diffraction methods; in A Handbook of Determinative Methodsin Clay Mineralogy, edited by M.J. Wilson, Blackie, Glasgow and London, 308p.Zemanek, J.1989:^Low-Resistivity Hydrocarbon-Bearing Sand Reservoirs; SPE FormationEvaluation, p. 515-521.APPENDIX 1: SURFACE AREA PROGRAM*******************************************************************MICROMERITICS SURFACE AREA PROGRAM : JULY 1990.Language: FortranCompiler: MacFortran/020* Version 1.0: This program was originally transferred onto the Mac by A. Abad* from N. Rose's IBM version used at Stanford. A.A's version was edited by M.* Knoll and lastly by P. Tercier. This program calculates surface area (SA) using* the BET equation and finds the standard error of the least squares line, the* correlation coefficient (r), and r squared. An input file using edit on the MAC* must first be created for use in the program. An example of the input file can* be found in the surface area binder or in the file SAUBCIN. Constants in the* program (X1,X2,VD) correspond to the Micromeritics surface area analyser in* UBC's rock physics lab. If the Stanford surface analyser was used, the program* SASTANFORD.apl contains the constants for that machine. Results of the* program are stored in an output file created during execution of the program. An* example of the output file can also be found in the surface area binder.** Version: Dec 6, 1990. Changed by AA to have a better output********************************************************************C CONSTANTS GIVEN WITH MICROMERITICS MACHINECC X1 is the volume (130.9 ml) of the small extra volume bulbC X2 is the volume (526.0 ml) of the large extra volume bulbC VD is the volume (29.16 ml) of the manifoldC TD is the temperature (307.2 K) of the manifoldC VI is the volume (3.65 ml) of the interconnecting tubingC S is the area occupied by one molecule of the adsorbing gasC ALPHA is the gas correction factor from Table I in the Micromeritics ManualC STEMP is temperature at standard conditions (273 K)C SPRESS is pressure at standard conditions (760 mm Hg)CC VARIABLES MEASUREDCC W1 is the weight of the sample flask, sample and stopperC W2 is the weight of the sample flask and stopperC TS is the temperature of the sample (Kelvin)C H1 is the initial pressure when helium is let into the manifoldC H2 is the final pressure when helium is let into the sampleC P1 is the initial pressue when gas is let into the manifoldC P2 is the final pressure (equilibrium) when the gas is let into the sampleC PE is the equilibrium pressure (P2) from the previous stepCCC VARIABLES CALCULATEDCC WS is the weight of the sample (g)C TI is the temperature of the interconnecting tubing (Kelvin)C VS is the "deadspace" volume: volume (ml) of the free space about the sample104C Vd is volume (ml) of manifold + extra volumeC Va is the volume of adsorbed gas at each pressure step measuredC STDIVSP is temperature divided by pressure at standard conditionsCC**********************************************************************INTEGER SETS,WHICHX(10)REAL INT,INTER1,INTER2,CORREL,RSQUAREDREAL X(10),Y(10),ERR(10),A(10),P1(10),P2(10),V(10)CHARACTER*20 CODE,INFNAME,OUTFNAME*WRITE (9,*) "*** SURFACE AREA PROGRAM ***"WRITE (9,*)WRITE (9,*) "Number of sets of data?"READ (9,*) SETSWRITE (9,*) "Name of input file?"READ (9,*) INFNAMEWRITE (9,*) "Name of output file?"READ (9,*) OUTFNAMEOPEN (UNIT=10,FILE=INFNAME,ACTION='READ)OPEN (UNIT=11,FILE=OUTFNAME,STATUS='NEW')ITER = 1200 CONTINUEREAD (10,*) CODE,W1,W2,H1,H2,TS ,PS ,S ,ALPHA,NX1 = 130.9X2 = 526.0VD = 29.16TD = 307.2VI = 3.65STEMP = 273.0SPRESS = 760.0CC*****C Calculation of sample weightC*****CWS = Wl-W2CC*****C Set initial volume of adsorbed gas = 0C*****CV(1) = 0.0CC*****C Set initial in sample chamber pressure = 0C*****CP2(1) = 0.0CC*****C Calculate the temperature of the interconnecting tubingC*****105CTI = (TD+TS)/2.0CC*****C Calculate volume of the deadspaceC*****CVS = (TS/H2)*((VD*(H1-H2)/TD)-(VI*H2/11))I = 2NX = N+164 CONTINUECCC*****C WHICHX is a flag; 0 = no extra volume was usedC^1 = small extra volume used (X1)C 2 = large extra volume used (X2)C The following IF statements set the extra volume to the proper setting forC each P1/P2 set of readings.C*****CREAD(10,*) WHICHX(I),P1(I),P2(I)IF(WHICHX(I).EQ.0) THENXX = 0.0ELSEIF (WHICHX(I).EQ. I) THENXX = X1ELSEXX = X2END IFEND IF8 CONTINUECC*****C Equation 5 (in Micromeritics manual p. 1-3) has been split up into a numberC of parts for ease of calculation. Parts A, B, and C group together the volume,C temperature and weight of the sample at STP while parts D, E, and F are theC pressure differences used in EQ. 5. A, B, and C are then multiplied by D, E,C and F respectively to produce G, H and DLJ. Finally DLK represents Va/WS(volumeC of gas adsorbed divided by weight of the sample) for that step. For furtherC clarification see the theory section in the Micromeritics manual.C*****CC*****C A is the first term in EQ 5 : 273/(760*Td)*(Vd/WS) where Vd = VD+XX.C*****CSTDIVSP = STEMP/SPRESSA(I)=((STDIVSP/TD)*(VD+XX))/WSCC*****C B is the second term of EQ 5C*****106CB.((STDIVSP*VS)/(TS*WS))+((STDIVSP*VI)/(TI*WS))CC*****C C is the last term in EQ 5: (VS*ALPHA/TS)(273/760*WS)C*****CC=(STDIVSP*VS*ALPHA)/(WS*TS)Cc*****C PE is the equilibrium pressure which is always the final pressure P2C from the previous step. I was set to 2 initially.C*****CPE=P2(I-1)CC*****C D, E, and F are the pressure differences from the first second and thirdC terms in EQ 5.C*****CD=PI(I)-P2(I)E=P2(I)-PEF=P2(I)**2-PE**2CC*****C G, H, and DLJ complete the multiplications needed for the first, second andC and third parts of EQ 5.C*****CG=A(I)*DH=B*EDLJ=C*FCC*****C DLK is the volume of adsorbed gas divided by the weight of the sampleC (Va/WS) for each pressure step done.C*****CDLK=G-H-DLJCc*****C V is summation of Va/WS over all pressure steps done.C*****CV(I)=V(I-1)+DLKCC*****C X values ( P2/PS ) and Y values ( (P2/PS)/summation of Va/WS) for BET plot.C*****CX(I)=P2(I)/PSY(I)=X(I)/(V (I)* ( 1.0-X(I)))1F(N-I+1) 10,10,91079 I=I+1GO TO 64*Cc*****C Start of least squares line. MM is a counter.C*****C10 CONTINUESUMX = 0.0SUMY = 0.0SUMXY = 0.0SUMXSQ = 0.0SUMYSQ = 0.0NX = N+1MMJDO 11 I = 2,NXCC*****C This IF statement limits the P1-P2 values used in the BET plot to only thoseC within the linear range for nitrogen (ie. 0.05<P2/PS<0.35).C*****CIF (X(I).LT.0.05).OR.(X(I).GT.0.35) GOTO 11MM=MM+1SUMX = SUMX+X(I)SUMY = SUMY+Y(I)SUMXY = SUMXY+X(I)*Y(I)SUMXSQ = SUMXSQ+X(I)**2SUMYSQ = SUMYSQ+Y(I)**211 CONTINUECC*****C Calculation of the y intercept (INT), SLOPE, and finally the surface area (SA)C*****CINT = ((SUMY*SUMXSQ)-(SUMX*SUMXY))/((MM*SUMXSQ)-(SUMX**2))SLOPE = ((MM*SUMXY)-(SUMX*SUMY))/((MM*SUMXSQ)-(SUMX**2))SA = (0.2687*S)/(SLOPE+INT)CC*****C Calculation of the correlation coefficient-r (CORREL) and RSQUARED-r*r.C*****CINTER1 = (MM*SUMXY)-(SUMX*SUMY)INTER2 = SQRT(((MM*SUMXSQ)-(SUMX**2))*((MM*SUMYSQ)-(SUMY**2)))CORREL = INTERVINTER2RSQUARED = CORREL**2CC*****C Formats and outputs the data to the output file.C*****108CWRITE (11,101) CODE,W1,H1,X1,PS,W2,H2,X2,ALPHA,WS ,TS,VD,S101 FORMAT(T26,'Surface Area Measurement',///,1 'Sample Identification Code: ',A20,///,2 'W1 = ',F8.4,' gm',4X,'H1 = ',F6.2,' mmHg',3X,'X1 = ',F6.2,3 ' ml ',3X,'PS = ',F7.3,' mmHg', / ,'W2  = ',F8.4,' gm',4X,'H2 = ',4 F6.2,' mmHg',3X,'X2 = ',F6.2,' ml ','ALPHA = ',E7.2,/,'WS = ',5 F8.4,' gm',4X,'TS = ',F6.2,' K',6X,'VD = ',F6.2,' ml ',4X,6 'S = ',F6.1)WRITE (11,102)102 FORMAT(//,1X,'Iteration',4X,'WHICH X',7X,'P1 ',7X,'P2',2 8X,'V',7X,'X',7X,'Y',/)*DO 106 I = 2,NXJ = I-1WRITE(11,103) J,WHICHX(I),P1(I),P2(I),V(I),X(I),Y(I)103 FORMAT(I7,I11,F14.2,F9.2,3F8.3)106 CONTINUEWRITE(11,104) SA,CORREL,RSQUARED,SLOPE,INT104 FORMAT(///,T20,'Surface Area = ',F9.3,' m2/g'//,3^T20,'Correlation Coefficient = ',F6.3,//,4^T20,'R squared = ',F6.3,//,5^T20,'Slope = ',F6.3,//6^T20,'Intercept = ',F6.3,5/)CC*****C This repeats the program for each set of data in the input file.C* ****CIF (SETS-ITER) 400,400,300300 CONTINUEITER = ITER+1GO TO 200400 CONTINUEWRITE(9,*)WRITE(9,1020) OUTFNAME1020^FORMAT('program done --- results in ',A20,/,2^'press return')CLOSE(UNIT=10)CLOSE(UNIT=11)PAUSESTOPEND109APPENDIX 2: ADSORPTION ISOTHERMSAdsorption isotherm for Quantachrome Standard: Run 10.5y^1.8e-2 + 2.104x RA2 = 1.0000.4 -0.3 -0.2 -0.1 -0.0^ •^I0.04^0.08^0.12^0.16^0.20P2/PsAdsorption isotherm for Quantachrome Standard: Run 20.5y = 1.3-2 + 2.132x RA2 = .9990.4 -0.2 -0.1 -0.0 ,^0.04^0.08^0.12^0.16^0.20P2/Ps1100.250.4^0.00^0.05^0.10^0.15^0.20P2/PsAdsorption isotherm for 22-11-12 #10069: Runt0.00^0.05^0.10^0.15P2/Ps•0.20Adsorption isotherm for 22 - 11 - 12 #10069: Run 2111y = 5.7e-2 + 4.038x RA2 = 1.0000.90.8 -• 0.7> 0.6 -0.4 -0.3 ^ T^ 1^ I1.0ay = 5.9e-2 + 4.038x RA2 = 0.999a0.8 -•0.2 ^0.00 0.05t0.10 0.15• •0.20Adsorption isotherm for API263 No.9021: Run 10.00^0.05^0.10^0.15^0.20P2/PsAdsorption isotherm for API263 No.9021: Run 2P2/Ps112Adsorption isotherm for API236 No.1433: Run 1P2/PsAdsorption isotherm for API236 No.1433: Run21.2-^y = 6.8e-2 + 4.468x R A2 = 0.9991.0 - rg 0.6-0.4 -0.20.00^0.10^0.20P2/Ps113Adsorption isotherm for Berea100A(5)0.7y = 5.1e-2 + 3.53x RA2 = 1.0000.6 -0.5 -0.4 -114EU)0.3 -0.2 ^0.00^0.05^0.10^0.15P2/PsAdsorption isotherm for Teflon50.00^0.05^0.10^0.15^0.20^0.25^0.30P2/PsAPPENDIX 3 : POROSITY PROGRAMLanguage: FortranCompiler: MacFortran/020**** poros(AA)* Written by A. Abad* Program to calculate the density, specific volume and porosity* of a rock sample from porosimeter data.** Version: May 25, 1989* Revised: June 2, 1989 to include spacers with volume calculations***character*20 namecharacter*20 fnamecharacter*2 ansreal*4 pl(12),p2(12),p1s(12),p2s(12)** Prompt for required data and echo data into a file*write(9,*)"Program to calculate the density, specific"write(9,*)" volume, and porosity of a rock sample"write(9,*)"^ i,write(9,*)write(9,*) "Enter sample name:"read(9,*) namewrite(9,*) "Enter diameter and length of sample:"read(9,*) diameter,heightwrite(9,*) "Enter gross weight and cup weight:"read(9,*) grosswt,cupwtwrite(9,*) "Enter cell and expansion volumes:"read(9,*) vcell,vexpwrite(9,*) "Enter the name of a file for the results:"read(9,*) fnameopen(unit=10,file=fname,status="new")write(10,*) "^Porosity Data"write(10,*)write(10,*) "Sample Name: ",namewrite(10,*)sampwt = grosswt-cupwtwrite(10,1020) " Gross wt. (gm) =",grosswtwrite(10,1020) " Cup wt. (gm) =",cupwtwrite(10,1020) " Sample wt. (gm) =",sampwtwrite(10,*)write(10,1030) " Diameter (cm) =",diameterwrite(10,1030) " length (cm) =",heightwrite(10,*)write(10,1040) " Cell Volume (cc)^=",vcellwrite(10,1040) " Expansion Volume (cc) =",vexpwrite(9,*)"Enter number of runs for this sample (max. 12):"read(9,*) numwrite(9,*) "Are you using spacers with this sample? (y/n):"read(9,*) ans115if (ans.eq.'y') thenwrite(10,*)write(9,*)"Enter charge (P1) and expansion (P2) pressures for"write(9,*)"sample + spacers."write(10,*)write(10,*) "^Charge^Expansion"write(10,*) "Run # Press. (psi) Press. (psi) Volume (cc)"write(10,*) "  ^ IIwrite(10,*)call getpres(num,p 1 s,p2s)sum = 0do 10 i = 1,numvl = vcell - vexp/(pls(i)/p2s(i)-1)sum = sum+v 1write(10,1010) i,p 1 s(i),p2s(i),v 110^continuevmatl = sum/numendifwrite(9,*)"Enter charge (P1) and expansion (P2) pressures"write(9,*)"for spacers only."if (ans.='y') thenwrite(10,*)write(10,*) "For spacers:"endifwrite(10,*)write(10,*) "^Charge^Expansion"vvrite(10,*) "Run # Press. (psi) Press. (psi) Volume (cc)"write(10,*) "write(10,*)call getpres(num,p1,p2)sum = 0do 20 i = 1,numv2 = vcell - vexp/(pl(i)/p2(i)-1)sum = sum+v2write(10,1010) i,p1(i),p2(i),v220^continuevmat2=sum/num** Do calculations and output results*if (ans.='y') thenvmatrix = vmatl-vmat2elsevmatrix = vmat2endifdensity = sampwt/vmatrixusamp = vmatrix/sampwtvtotal = 3.14159/4*diameter**2*heightphi = (vtotal-vmatrix)/vtotal*100*write(10,*)write(10,1020) " Density (gm/cc) =",densitywrite(10,*)write(10,1050) " Spec. Volume (cc/gm) =",usamp116w-rite(10,*)write(10,1040) " Matrix Volume (cc) =",vmatrixwrite(10,1040) " Total Rock Volume (cc) =",vtotalwrite(10,*)write(10,1060) " Porosity =",phi," %"close(unit=10)write(9,*) "Porosity = ",phi," %"write(9,*)write(9,*) "porosity results for ",name," in file ",fnamewrite(9,*) "program done, press return to exit ---."pause*1010^format(i3,f12.3,f15.3,f14.3)1020^format(a18,f10.3)1030^format(a16,f10.4)1040^format(a26,f10.3)1050^format(a23,e14.6)1060^format(all,f10.2,a2)*stopend***subroutine getpres(num,p1,p2)*real*4 pl(num),p2(num)*do 10 i=l,numwrite(9,*)"Enter next (P1,P2):"read(9,*) p1(i),p2(1)10^continuewrite(9,*)write(9,*) "run #^P1^P2"write(9,*) " _Ifwrite(9,*)do 20 i=1,numwrite(9,1000)i,p1(i),p2(i)20^continuewrite(9,*)write(9,*) "If you wish to change any (P1,P2) data"write(9,*) "1. enter the run # or 0 when done, press return."write(9,*) "2. enter the correct (P1,P2) data press return."write(9,*) "3. repeat 1 and 2 for each incorrect (P1,P2) data."30^continueread(9,*) nif (n<>0) thenread(9,*) pl(n),p2(n)goto 30endif*1000^format(i3,2f10.3)returnend117


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