AN EXPERIMENTAL STUDY OF THE EFFECTS OF PARTIALSATURATION ON ELASTIC WAVE VELOCITIES IN POROUS ROCKSByDavid GoertzB. A. Sc. (Engineering Physics) University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESGEOPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1994© David Goertz, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(SignDepartment ofThe University of British ColumbiaVancouver, CanadaDate H11ABSTRACTElastic wave velocities in porous rocks containing air and water are sensitive to not• only the relative levels of fluid saturation but also to the distribution of the fluids within the—- pore space. Three factors that have significant control over-the relative distributions of fluidsin multiphase saturated porous media are pore space microgeometry, saturation history andwettability. In this thesis, the effect of these factors on the form of dependence of velocitiesupon water saturation level in rocks is investigated experimentally.Ultrasonic elastic wave velocity and drying rate measurements were made as afunction of water saturation in a limestone, a dolomite and two sandstone samples assaturation was reduced through evaporative drying. During the later stages of drying there isa reduction in drying rate that is associated with the transition from capillary transport todiffusive transport due to a loss of hydraulic connectivity of the liquid phase. For the rocksused in this study, this suggests that velocity variations below this point can be associatedwith the removal of disconnected water held in surface roughness and in crack-like porosity.Using these interpretations and simplified models of the pore spaces derived from thinsection analysis, fluid distribution scenarios are proposed for the drying process in theserocks. A numerical modeling routine is then used to predict the form of the velocitysaturation relationships for the rocks. The models were found to be in good agreement withthe form of the experimental results.The effect of wettability on the relationship between velocities and saturation historywas investigated in the sandstone samples by conducting imbibition and drainageexperiments before and after treatment with a chemical that altered their surfaces from beingstrongly water-wet to being oil-wet. In the water-wet sandstones, the results indicate thatgrain contact regions are the last to drain of water and the first to fill with water. At highmsaturation levels, hysteresis is evident and is attributed to differences in the pore scaledistribution of fluids that evolves in pore bodies during the imbibition and drainageprocesses. The results for the oil-wet samples during evaporative drying were found to besimilar to those for the water-wet rocks: water was replaced by air first in the pore bodies andthen in the grain contacts and cracks. In contrast, imbibition produced results that are—consistent with water entering the pore bodies first and being excluded from the cracks andgrain contacts until high saturation levels.ivTABLE OF CONTENTSAbstract.iiTable of Contents ivList of Tables viList of Figures viiAcknowledgments xChapter 1: Introduction 1Chapter 2 : Elastic Wave Velocities During Evaporative Drying 42.1 Introduction 42.2 The Effects of Saturation on Velocities 52.3 Capillary Theory 142.4 The Drying Process 192.5 Sample Descriptions and Experimental Procedures 272.6 Results and Discussion 342.6.1 Drying Data 342.6.2 Comparison of Models with Data 482.6.3 Adsorption Experiments 602.7 Summary 66VChapter 3 : The Effects of Wettabiity on Velocities 683.1 Introduction 683.2 The Effect of Saturation HistoryOlI Fluid Distribution 693.3 Experimental Procedures 753.4 Experimental Results and Discussion 773.4.1 Hydrophilic Samples 783.4.2 Hydrophobic Samples 833.5 Velocity Modeling 993.6Summary 113Chapter 4: Conclusions 114References 116LIST OF TABLES2.1 Summary of rock porosities and permeabilities.273.1 Pore spectrum for Berea 300 100viviiLIST OF FIGURES2. la,b Bulk and shear moduli increases as a function of aspect ratio 102. ic Ratio of bulk and shear moduli as a function oLaspect ratio 112.2 Bulk and shear moduli increases as a functionof saturation 122.3a Contact angle of a liquid droplet on a solid surface 162.3b Force diagram for surface tensions 162.4a Liquid held in a capillary tube 182.4b Liquid held as a pendular ring 182.5 A typical capillary pressure curve 202.6 Drying in a connected two pore system 222.7 A typical drying rate curve 242.8 Fluid distributions during the falling rate period 252.9 Thm section photograph for limestone sample 292.10 Thin section photograph for dolomite sample 292.11 Thin section photograph for Berea 100 sample 302.12 Thin section photograph for Berea 300 sample 302.13 Block diagram of velocity measuring apparatus 322.14 Drying rate curve for limestone sample 352.15 Drying rate curve for dolomite sample 362.16 Drying rate curve for Berea 100 sample 372.17 Drying rate curve for Berea 300 sample 382. 18a Fluid disthbution model for limestone sample 422.1 8b Moduli model for limestone sample 432.19a Fluid distribution model for dolomite sample 45viii2. 19b Moduli model for dolomite sample .462.20a Fluid distribution model for Berea samples 472.20b Moduli model for Berea samples 492.21 Vp and Vs of limestone during drying 502.22 Bulk and shear moduli of limestone during drying 512.23 Vp and Vs of dolomite during drying 532.24 Bulk and shear moduli of dolomite during drying 542.25 Vp and Vs of Berea 100 during drying 552.26 Bulk and shear moduli of Berea 100 during drying 562.27 Vp and Vs of Berea 300 during drying 572.28 Bulk and shear moduli of Berea 300 during drying 582.29 Bulk and shear moduli of Berea 100 during adsorption and drying 622.30 Bulk and shear moduli of Berea 300 during adsorption and drying 632.31 Saturation heterogenieties during the later stages of drying 663.1 Intrinsic, receding and advancing contact angles 703.2 Effects of pore geometry during imbibition and drainage 723.3 Capillary pressure curve during imbibition and drainage 743.4 Vp and Vs of Berea 100 during imbibition and drainage 793.5 Bulk and shear moduli of Berea 100 during imbibition and drainage 803.6 Vp and Vs of Berea 300 during imbibition and drainage 813.7 Bulk and shear moduli of Berea 300 during imbibition and drainage 823.8 Drying rate curves for hydrophilic and hydrophobic Berea 100 samples 843.9 Drying rate curves for hydrophilic and hydrophobic Berea 300 samples 853.10 Sw versus time during imbibition for Berea 100 sample 873.11 Sw versus time during imbibition for Berea 300 sample 883.12 Vp and Vs of Berea 100 during drying, before and after treatment 90ix3.13 Vp and Vs of Berea 300 during drying, before and after treatment 913.14 Vp and Vs in hydrophobic Berea 100 during imbibition and drainage 933.15 Bulk and shear moduli in hydrophobic Berea 100 during imbibition and drainage 943.16 Vp and Vs in hydrophobic Berea 300 during imbibition and drainage 953.17 Bulk and shear moduli in hydrophobic Berea 300 during imbibition.and drainage 963.18 Separating Sw into surface and bulk phases in Berea 300 1023.19 Fluid distributions for model during hydrophilic samples 1033.20 Fluid distributions for model during imbibition in hydrophilic samples 1043.21 Comparison of model and data for bulk modulus in hydrophiic Berea 300 1063.22 Comparison of model and data for shear modulus in hydrophilic Berea 300 1073.23 Separating Sw into surface and bulk phases in hydrophobic Berea 300 1083.24 Fluid distributions for model during imbibition in hydrophobic samples 1093.25 Comparison of model and data for bulk modulus in hydrophiic Berea 300 1103.26 Comparison of model and data for shear modulus in hydrophilic Berea 300 111xACKNOWLEDGMENTSFirst and foremost I would like to express my gratitude to my supervisor RosemaryKnight, a seemingly boundless source of encouragement and enthusiasm. My thanks also goout to other members of the Rock Physics group, particularly Paullette-Tercier and Ana Abad.and Ken Wilks, for their assistance at various stages of this thesis.This research was supported by funding from Imperial Oil, Petro-Canada, ShellCanada and an Industrial Oriented Research Grant from the Natural Sciences andEngineering Research Council of Canada. The author was also supported in part by aUniversity Graduate Fellowship.11 INTRODUCTIONSeismic and sonic well logging surveys are two of the most widely used geophysicalmethods for investigating the earth’s subsurface. In order for these techniques to be useful inproviding information about geologic formations it is necessary to have an understanding theseismic properties of earth materials. Laboratory experiments measuring elastic wavevelocities in rocks provide important constraints in interpreting the results of such surveys.One factor that has been found, both experimentally and theoretically, to have a significantimpact upon elastic wave velocities in porous rocks is nature of the fluids contained withinthe pore spaces. Two pore fluids that are of significant interest when considering applicationof seismic methods to hydrogeology as well as to the detection and monitoring of natural gasreservoirs are gas and water.Laboratory experiments conducted at ultrasonic frequencies have shown there to be aconsiderable degree of variation in the form of dependence of velocities on water saturationlevel (Wyllie et aL, 1956; Gregory, 1976; Murphy, 1982). Much of this complexity arisesfrom ultrasonic velocities being sensitive to not only the level of water saturation but also thegeometric distribution of water and air within the pore space (Endres and Knight, 1989;Knight and Nolen-Hoeksema, 1990). The distribution of fluids is in turn controlled by thecomplex interaction between pore space microgeometry, saturation history and thewettability of the solid surface. The focus of this thesis is to examine experimentally theinfluence of these three factors on the relationship between velocities and saturation level.The thesis is divided into two main chapters. In Chapter Two, the specific nature ofthe distribution of fluids during evaporative drying is investigated in samples possessingthree distinctly different types of pore space microgeometries. Gregory (1976) usedexperimental data collected during evaporative drying to infer general relationships between2porosity and the form of the velocity-saturation relationship. It appears, however, that therelationship between velocities and water saturation level is not simply related to the absoluteporosity value (e.g., Cadoret, 1993). Pore geometry and saturation method are of primaryimportance in determining the distribution of fluids in a porous medium. Further, the natureof the pore space microgeometry also affects the extent to which rocks are sensitive tosaturation state: crack-like pores are much more compressible than equidimensional pores(Walsh, 1965). The objective of this chapter is to explore how the interaction between thedrying process and pore space microgeometry influences the form of the relationship betweenvelocities and saturation level. This is accomplished by first using drying rate versussaturation level curves to resolve the transition from water being held in a funicular state tobeing held in a pendular state. In a funicular state, fully saturated pores will coexist withpartially saturated pores and the liquid phase will be hydraulically connected. In a pendularstate water will tend to be hydraulically disconnected and is held through capillary pressurein cracks, surface roughness and as an adsorbed layer. This transition has been associatedwith the onset of the irreducible water saturation within drying samples of porous media(e.g., Whitaker, 1985). Based on this information and simplified pore space models derivedfrom thin section analysis, fluid distribution scenarios are proposed for the drying process. Inparticular, velocity variations are associated with either the drainage of funicular waterin the-early stages of drying or with pendular water during the later stages of drying. Numericalmodeling is used to model the form of the velocity saturation relationship that would arisefrom these scenarios. The models are then compared with experimental results.In Chapter Three the effect of wettability on the relationship between velocities andsaturation history is investigated. Saturation history has been shown to produce hysteresis inthe relationship between velocities and saturation in rocks (Knight and Nolen-Hoeksema,1990; Cadoret et al., 1992a,b). This can be attributed to differences in the distribution offluids in pore bodies that evolve between imbibition and drainage cycles. The distribution of3fluids during imbibition and drainage cycles in a rock is intimately related to the pore spacemicrogeometry and the wettability of the rock surfaces. In general, capillary pressure willcause a wetting fluid to preferentially occupy smaller pores and cracks. Altering thewettability of the system can therefore influence the relative location of fluids within the porespace, which may in turn affect-elastic wave velocities. To investigate these effects,- imbibition and drainage experiments- were first conducted on strongly water-wet sandstones.The wettability of the pore surfaces were then chemically altered to an oil-wet state and theexperiments were repeated. Pore geometry and pore fluids were therefore kept constant andthe effects of wettability were effectively isolated.The issues addressed in this thesis are of significance in two main respects. The firstissue is to gain a better understanding of how pore space microgeometry, saturation historyand wettability interact to control the form of the relationship between velocities andsaturation level. The second is the potential applicability of the results of this study to fieldsituations. Both the transition of fluids to a disconnected state, and the wettability of the solidphase in mukiphase saturated rocks have important implications to the transport properties ofrocks (see Anderson, 1987a, for a review). The transport of fluids in multiphase saturatedrocks is of considerable interest in hydrogeology as well as in the oil and gas industries. Theability to seismically resolve these properties in the subsurface would therefore be ofsignificant utility. An important and necessary step towards understanding how, or if, thesefactors may impact the relationship between velocities and saturation is to conduct controlledlaboratory experiments.42 ELASTIC WAVE VELOCITIES DURING EVAPORATIVE DRYING2.1 IntroductionLaboratory experiments measuring ultrasonic elastic wave velocities have shownthere to be considerable variation inihe form of dependence of elastic wave velocities on thelevel of water saturation. This variation in functional form can be attributed, in part, todifferences in pore space microgeometry, saturation method, and their resulting control onthe pore scale distribution of fluids. One commonly used method of varying water saturationlevels in rocks during laboratory experiments is evaporative drying. Despite its widespreaduse, there has been little discussion in the geophysical literature about the nature of fluiddistribution that is produced in rocks during the drying process. Of particular interest in thischapter is the specific nature of fluid distribution that is produced within rocks duringevaporative drying and its resulting effect on the form of the relationship between elasticwave velocities and water saturation level.In fields such as chemical engineering and soil science evaporative drying has beenthe subject of a considerable amount of study (see, for example, Scherer, 1990; Whitaker,1977; Keey, 1972 for overviews of the drying process). It has been found that the way inwhich the drying rate of a sample of material varies with saturation level reflects thedominant mass transport processes during the drying process. In the early stages of drying acapillary porous material, the rate is constant as the liquid is in a funicular state and capillarytransport is dominant. During the later stages of drying there is a decrease in rate as theconnectivity of the liquid phase is reduced and there is a transition to the liquid being held ina pendular state (Ceagiske and Hougen, 1937). It has also been observed that sample scalesaturation heterogeneities exist during the later stages of drying. Cadoret et al. (1992a)suggested that such heterogeneities would affect the accurate measurement of velocities.5In this chapter, the application of drying rates to the interpretation of velocity datacollected during evaporative drying is investigated. In particular it is proposed that, for therocks used in this study, velocity changes during the early stages of drying are associatedwith the drainage of pore bodies while towards the later stages of drying they are associatedwith the removal of pendular water. Experiments were conducted on threetypes of rock, eachpossessing distinctly different types of pore space microgeomeiries. Based on theinterpretation of drying rate curves, and a simplified concept of the pore spaces derived fromthin section analysis, fluid distribution scenarios are proposed. A numerical modeling routinedeveloped by Endres and Knight (1989) is used to predict the form of the velocity saturationcurves that would result from such disthbutions, and these are compared with experimentalresults. Finally, velocity data collected as water Saturation was increased through adsorptionis used to explore the extent to which sample scale heterogeneities affect the velocitiesmeasured in the lower ranges of saturation. A better understanding of the way in which thedrying process and pore space microgeometries interact is of significant importance wheninterpreting the form of velocity-saturation relationships produced during the drying process.2.2 The Effects of Saturation on VelocitiesElastic wave velocities in a homogeneous, isotropic medium can be expressed interms of the material properties of bulk modulus (k), shear modulus (p) and density (p) asfollows:Vp=V (2.1)V = (2.2)where VP is the compressional wave velocity and V is the shear wave velocity. The bulk6modulus is a measure of the incompressibility of a solid and the shear modulus is a measureof its shear rigidity.The influence of varying the relative saturation levels of air and water on velocitiescan be attributed to two primary factors. The first is a reduction in moduli that results fromthe addition of a small amount of water to a dry rock (Born and Owen, 1935; Wyllie et al.,1962; Pandit and King, 1979; Clarke et al., 1980; Murphy, 1982).. Murphy et al. (1984)postulated that the mechanism responsible for these effects is a reduction in frame modulithat occurs when the adsorption of a fluid lowers the surface free energy of the solid phase.This occurs because the contact adhesion between grains is proportional to the interfacial freeenergy of the solid surface (Israelachvili, 1985). In sandstones, Knight and Dvorkin (1992)related this to the presence of three to four monolayers of adsorbed water.Velocities are also influenced by the bulk properties of pore fluids, in particular, bythe fluid densities, viscosities and compressibilities. The presence of a dense fluid in the porespace will increase the overall composite density and, as can be seen in equations 2.1 and 2.2,this will have the effect of decreasing both Vp and Vs. A more complex and generally moresubstantial way in which the bulk properties of pore fluids can affect velocities is throughchanges produced in the bulk and shear moduli of the composite system. The extent to whichthe bulk properties of fluids can affect the moduli depends upon the nature of the porosityand the frequency of the elastic waves.Gassmann (1951) and low frequency Biot (1956) theory provides an expression forthe bulk modulus of a fully saturated rock in terms of the porosity and the bulk moduli of thedry rock, the matrix material and the pore fluid. This formulation assumes that there is ahomogeneous and isotropic pore space. It is also implicit that frequencies are low enough sothat inertial effects can be neglected and that pore fluid pressures induced by the passage ofan elastic waves will be equilibrated on a scale that is significantly larger than the averagepore size. Dominico (1976) extended this work to account for partial saturations by assuming7the bulk modulus of the composite pore fluid could be approximated by a simple mixing law.Biot-Gassmann-Dominico (BGD) theory predicts that there will be a gradual decrease in bothP and S wave velocities with increasing saturation due to density effects throughout most ofthe lower range of saturation. At very high saturation levels, the increase in the rigidity of thepore fluid mixture results in an increase in P wave velocity and not shear wave velocity.Murphy (1982) noted that the measured dependence of velocity on saturation could beexplained reasonably well by BGD theory at frequencies less than approximately 1 kHz.Cadoret et al. (1993), however, found that there was departure from the form predicted byBGD theory during drying; this was attributed a lack of homogeneity in the saturationdistribution at a pore scale, which violates a fundamental assumption of BGD theory.Due to heterogeneities in pore shape, orientation and saturation, the passage of anelastic wave can induce pore scale pressure gradients and flow. This is referred to as localflow (O’Connell and Budiansky, 1974, 1977; Mavko and Nur, 1979) which is not explicitlytaken into account in BGD theory. At low enough frequencies local flow is permitted tooccur and such pore scale pressure gradients can be dissipated. At high enough frequencies,the time scale of the wave can be sufficiently small that viscous and inertial effects within thefluid will become significant enough to inhibit local fluid flow (Mavko and Jizba, 1991). Thepore fluid will then become ‘unrelaxed’ and will in turn exert a pressure on the pore wall thatwill act to stiffen the frame moduli in a way that is not accounted for in BGD theory. As aconsequence, ultrasonic velocities tend to be higher (Winkler, 1985) and can be sensitive tothe saturation state of individual pores (Endres and Knight, 1989; Knight and NolenHoeksema, 1990).The stiffening effect of pore fluids in individual pores is dependent upon poregeometry. In equidimensional pores the induced pressure will be relatively low since the8compressional deformation due to the elastic wave will tend to be compensated for, to someextent, by extensional deformation in an orthogonal direction. In compliant regions, such ascracks, which are not equidimensional, the compressionally induced deformation in onedirection will tend to not be balanced by extension in orthogonal direction (Mavko and Jizba,.1991). Consequently, moduli are more sensitive to the saturation state of thin compliantporosity than of equidimensional pores (Walsh, 1965). -The effects of pore geometry can be illustrated using a model developed by Kusterand Toksoz (1974) and expanded upon by Toksoz et al. (1976). It was proposed that themechanical behavior of a porous rock could be modeled by considering that the pore space iscomposed of oblate spheriodal inclusions in a homogeneous background mineral matrix.Different types of pores are accounted for by varying the aspect ratio of the inclusions, wherethe aspect ratio, cx, is defined as the ratio of the lengths of the semi-minor, a, to semi-major,b, axis lengths e.g.,(2.3).Crack-like pores or grain contacts are therefore described by very low aspect ratio inclusionswhereas rounded pores will be represented by higher aspect ratio inclusions. For a sphericalpore cx = 1.Consider a quartz background medium containing a concentration of inclusions of aspecific aspect ratio. Using the numerical approach given by Kuster and Toksoz (1974), thedifferences in bulk and shear moduli between when the pores were air filled, or dry, andwhen they are water filled, or saturated, were calculated as a function of the aspect ratio ofthe inclusions.9The results, shown in Figures 2.la and b, are expressed in terms of the relativeincreases in moduli as defined by the following equations:=(2.4)=t3jy (2.5)where K and are the moduli of the medium with air filled pores and and tat arethe moduli of the medium when the pores are filled with water. From these figures it can beseen that the moduli of a medium containing low aspect ratio pores is very sensitive to thecompressibility of the pore fluids. The moduli are also affected by the relative concentrationlevels of different aspect ratio pores.It is also interesting to note that the relative increase in shear modulus is always lowerthan that of the bulk modulus. This is made more evident in Figure 2.lc, which shows theratio of relative increases in bulk and shear moduli plotted as a function of aspect ratio. Fromthis plot it can be seen that the shear modulus is not affected at all by the saturation state ofspherical pores. As the aspect ratio is decreased, the ratio approaches a constant value, withthe relative increase in shear modulus being about 20 percent that of the bulk modulus.Endres and Knight (1989) adapted this model for partially saturated rocks byconsidering that the rigidity contribution of the pore fluids could be described with a simplemixing law. Using this model the effects of water saturation level, Sw, on the moduli of aporous medium is shown in Figure 2.2. Sw is defined as the volume fraction of the porespace that is occupied by water. The gas is assumed to be distributed evenly at a pore scaleand the aspect ratio is assumed to be small enough that both the bulk and shear moduli are10(a) 0.08006 Shear ModulusX Bulk ModulusC)0.04xC).—-..xx0.00• . ..001 .01 .1Aspect Ratio(b) 0.0003• Shear ModulusX Bulk Modulus.E 0.0002C)C)0.0001x.—C)x•0.0000.1 .Aspect RatioFigure 2.la,b. Relative increases in bulk and shear modulu from unsaturated to saturatedstates as a function of aspect ratio. a) Full range of aspect ratios.b) High end of aspect ratios.110.30.2 B B BB0.10.0 — .i •.001 .01 .1Aspect RatioFigure 2.lc. Ratio of relative increases in bulk and shear modulu from unsaturated tosaturated states as a function of aspect ratio.12•0K‘It-0 1SwFigure 2.2. Bulk and shear moduli as a function of water saturation when the water isdistributed homogeneously at a pore scale.13affected. It can be seen from this figure that the introduction of even a small amount ofcompressible gas into the pore space will reduce the moduli sharply. The subsequent drainageof the remainder of the water has little effect on the moduli.By approximating the pore space-as being composed of a range of different aspectratio pores, and controlling the saturation state of pores with specific aspect ratios, the modelof Endres and Knight (1989) permits a wide range of velocity versus saturation relationshipsto be explored. This model makes two significant approximations. The first is that rock pores,which are often very geometrically complex, can be described by idealized ellipsiodalinclusions. As a first order approximation however, much of the mechanical behavior of arock can be investigated with this type of model (Endres and Knight, 1989). It is alsoassumed that it is the saturation state of individual pores that will dominate the elasticbehavior of the rock and that communication of fluids between pores during the passage ofelastic waves can be neglected. At ultrasonic frequencies, where the time scale of the elasticwave is short enough for viscous and inertial effects to become significant, this was found toproduce reasonable results in fully saturated rocks (Toksoz et al., 1976). It was also used tosuccessfully model the velocity-saturation relationship in a partially saturated rock by Endresand Knight (1991). The numerical modeling routine of Endres and Knight (1989) will beused in this study to explore the effects of different fluid distributions on elastic wavevelocities. In using this model it is recognized that the effects of fluid communication andoversimplification of the pore space microgeometry may contribute to inaccuracies in theresults. However, the main purpose of using this model in this study was to examine howvarying the saturation state of pores with different compliance levels can affect the form ofthe velocity-saturation relationship. Of particular interest are the effects of differences in thesaturation states of stiff equidimensional pore bodies and very compliant crack-like porosity.It is therefore felt that the usefulness of the model in this respect outweighs any of theaforementioned limitations.14Mavko and Nolen-Hoeksema (1994) developed a model that is conceptually similarto that of Endres and Knight (1989), but avoids the use of idealized pore geometries. In thismodel the filling of compliant, or ‘soft’, porosity with water is lilcened to the closure of cracksunder confining pressure. In this portion of the porosity, pore fluid is-assumed to beunrelaxed and produces similar relative increases in bulk and shear moduli as predicted bythe Kuster and Toksoz (1974) model. The remainder of the porosity is assumedzto obey EGDtheory.As a final point, it should be noted that theories developed to account for variations invelocities as a function of saturation generally assume that an effective medium is beingconsidered. When the wavelength of the propagating wave is small with respect to the scaleof heterogeneities in a medium, the medium appears to be homogeneous to the wave and it iscan be treated as an effective medium. If the inhomogeneities approach the scale of thewavelength then wave path dispersion may result and effective medium theory will becomeinvalid.2.3 Canillary TheoryIn the previous section, it was made evident that velocities in partially saturated rocksare sensitive to the distribution of fluids in the pore space. The relative distributions of airand water in a partially saturated rock are controlled by capillary forces. The basic principalsof capillary theory, which are discussed in detail in numerous texts (e.g., Dullien, 1979), arereviewed in the following section.When two immiscible phases are in contact, interfacial tension will exist at theirboundary. It is this interfacial tension that gives rise to capillary phenomena. In a systemcontaining a solid phase, a liquid phase, and a gas phase there are three possible types of15interface, each with a different interfacial tension. This is illustrated in Figure 2.3a whichdepicts a liquid drop placed on a solid surface in the presence of another fluid. At a point ofcontact between the three phases, the three different interfacia-i tensions-must balance formechanical equilibrium to exist. The relationship between the forces, shown in Figure 2.3b,satisfies Young’s equation and can written in the form-a*cosO= cSGGSL (2.6)where e is the contact angle between the liquid/gas and solid/liquid interfaces, Gjj is theliquid/gas interfacial tension, GSG is the solid/gas interfacial tension and GSL is thesolid/liquid interfacial tension.The contact angle is a measure of the affinity of one fluid to spread on a solid surfacein the presence of another fluid. It is frequently used to define the wettability of a solid. If afluid preferentially spreads on a solid in the presence of another fluid it is referred to as thewetting fluid. This will tend to happen if the contact angle is less than 9O. The concept ofwettability and its influence on fluid distributions will be discussed in more detail in ChapterThree.In a multiphase saturated porous medium the relative distributions of the differentfluids within the pore space will be controlled by the geometry of the pore space, thewettability of the solid and the saturation history. For all but the simplest of geometries, it isnot possible to obtain an analytical solution for the shapes of the interfaces between porefluids in a porous medium. Two geometries that are well understood and illustrative toexamine are the capillary tube and two adjacent spheres.16(b)SG aSLFigure 2.3 a) The contact angle of a droplet of liquid on a solid surface.b) Surface tension force diagram.17The shape of the interface between two immiscible fluids present in a capillary tubeis controlled by the radius of the tube, r, the contact angle and the interfacial tension (seeFigure 2.4a). The tension and curvature in the interface gives rise to a pressure differenceacross the boundary, the capillary pressure, P , which is described by - -2acosOrwhere P1 and P2 are the pressures in the non-wetting and wetting fluids, respectively.Capillary pressure is often referred to as a ‘suction’ potential since fluids will tend toflow from regions of low capillary pressure to regions of high capillary pressure. A pore witha smaller radius will give rise to a meniscus with a smaller radius and will therefore have alarger capillary pressure within it. In general, this will cause wetting fluids to preferentiallyoccupy smaller pores.A second geometry, which is of interest in studying partially saturated granular rocks,is that of a pendular ring of wetting fluid held at the region of contact between two spheres,shown in Figure 2.4b. The capillary pressure across the air/gas interface can be described byD _) rl 1— a cost, +where r and r are the two principle radii of curvature that describe the shape of themeniscus. As ri and r decrease in magnitude, reflecting a lower saturation level, the capillarypressure will increase and the liquid will become more tightly held.In a more realistic porous medium containing two fluid phases, capillary pressure ismeasured at a sample scale as a function of wetting phase saturation. The resulting pressureversus saturation plot is referred to as a capillary pressure curve. Capillary pressure curvesmay be either imbibition or drainage curves: drainage occurs when a wetting phase is18(a)(b)P-II )1Figure 2.4 (a) Liquid held in a capillary tube. (b) Pendular ring of liquid held between two spheres.19displaced by a non-wetting phase; imbibition when a non-wetting phase is displaced by awetting phase. A capillary pressure drainage curve is acquired by beginning with a samplethat is fully saturated with a wetting fluid and reducing its saturation level by displacing thewetting fluid with a non-wetting fluid that is under pressure.- A typical capillary pressure curve will have the form showii in Figure 2.5; Initially,only a small amount of pressure is required to reduce the wetting—phase saturation levelsubstantially. In this part of the curve, the wetting phase will tend to be displaced from largerpores. As saturation is lowered, higher pressure levels are required in the nonwetting phasefor drainage to proceed and successively smaller pores will drain. Eventually, the pressurecurve becomes vertical, and the wetting fluid can no longer be displaced through increases inhydraulic pressure. This occurs when the wetting phase loses its hydraulic connectivity. Atthis stage, the wetting fluid will tend to be held in pendular rings, crack-like pores, surfaceroughness and as an adsorbed surface layer. This is referred to as the irreducible watersaturation level.2.4 The Drvin ProcessIn this study, water saturation levels are reduced through evaporative drying, acommonly used method of varying saturation level in rocks during laboratory experiments.Drying is a specific form of drainage by which liquid water is replaced by air in a porousmedium through evaporation. The fluid distributions produced by this process are notnecessarily the same as those produced by those in• immiscible displacement. The twoprimary mechanisms responsible for mass transport in a drying porous medium are capillarytransport in the liquid phase and diffusion in the vapor phase. Capillary transport results froma hydraulic pressure gradient within the sample. Diffusion, a much slower process, is drivenprimarily by a gradient in the concentration of water vapor.20+a)U)U)a)I-.000IrreducibleFigure 2.5 Capillary pressure curve.21The interplay between these two mechanisms during the drying process can beillustrated using a simple model consisting of two adjacent pores connected by a pathway(Oliver and Clarke, 1971). In this model, the ‘pores’ are cylindrical with different radii (rs andrL) and fluid may flow freely between them through the pathway. It is assumed that bothpores are filled with a perfectly wetting liquid (0 = 00).In the initial stages of drying, liquid is removed from the tubes through evaporation arthe air/liquid interface and menisci, of radii rM 1 and rM2, will form (Figure 2.6a). Asevaporation continues, the radii of the menisci will decrease until = r the minimumpossible radius the large pore can accommodate. When this occurs the entry pressure of thelarge pore will have been reached and further evaporation will cause the meniscus to recedeinto the larger pore (Figure 2.6b).In the small pore the meniscus will remain at the surface and rM2 will continue tobecome smaller, thereby reducing the capillary pressure in the smaller pore below thepressure in the large pore. This will in turn result in fluid flow from the larger pore to thesmall pore (Figure 2.6c). Mass is therefore removed from the system by two mechanisms;firstly, the evaporation of liquid from the larger pore and its subsequent diffusion in vaporphase to the surface and, secondly, capillary transport of fluid from the large pore into thesmall pore and its evaporation at the surface of the sample. Since diffusion is a slow processrelative to capillary transport, the dominant mass transport mechanism during this stage iscapillary transport of fluid to the surface of the small pore. Eventually, rM2 will equal rs inthe small pore and the meniscus will recede into the small pore (Figure 2.6d). After thispoint, the dominant mechanism for removal of mass from the system, or the rate limitingstep, will be diffusion of vapor.In a more realistic porous medium, consisting of a large number of interconnectedpores with a wide disthbution of pore sizes, the drying process is considerably morecomplex. A commonly used method of gaining insight into the transport processes occurring22P1=P2P1rS(a)rMl>rL rMl=r>rSP1=P2(c)rMl=rL rMl>r=rSP1 0.95), Cadoret et al. (1992a) noted that fluid distributions were influenced by samplegeometry and that the velocity saturation relationship may therefore be distorted by wavepath dispersion in this region of Sw.During the constant rate period a capillary porous material will be in a funicular state.That is, the liquid phase is continuous and fully saturated pores will tend to be present alongwith partially saturated pores. In the rock samples used in this study the most relevant andconcrete interpretation that can be made is that during the constant rate period continuity of350.0040.003ID0.002I“—‘ 0.001000.0N- • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.14. Drying rate curve for limestone sample.360.00250.0020 El0.00150.00100.0005ElF0.0000 • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.15. Drying rate curve for dolomite sample.370.0040.003CC?Gc .EE 0.002C0•Ec00.001 E0.000.0.2 0:4 0:6 0.8 1.0Figure 2.16. Drying rate curve for Berea 100 sample.380.002e‘I1.0Figure 2.17. Drying rate curve for Berea 300 sample.39the liquid phase will tend to ensure that, since water in the rock will be above a pendularstate, grain contacts and cracks will tend to remain saturated. Any velocity variations cantherefore be atthbuted to the removal of water from pore bodies. The work of Shaw (1987)indicates that saturation levels in partially drained pores are high enough to maintain fluidflow for a considerable time after-the initial partial drainage occurs. Cadoret (1993) notedthat partial saturation, homogeneous at a millimetric scale, was generally evident in rocksduring drying by the middle ranges of saturation. As a simple model, it istherefore proposedthat during the constant rate period there will be a range of saturation over which all poreswill become partially drained but maintain a high degree of hydraulic connectivity. This isfollowed by a period over which the saturation level in the partially drained pore bodies isreduced further but the liquid remains hydraulically connected. Grain contacts and cracks areassumed to remain saturated throughout the entire constant rate period.Another point that warrants discussion is the scale of saturation heterogenieties thatmay exist during this period. Cadoret et al. (1992a,b) used X-ray tomography to image thesaturation distribution within limestone samples during the drying process. Betweenapproximately Sw = 0.15-0.95, it was found that drying produced a relatively homogeneoussaturation distribution, provided the scale of heterogeneity of the porous network was smallwith respect to the wavelength. This form of drainage appears to be present in the samplesabove Sw = 0.20 to 0.30. The pore scale in the Berea and dolomite samples appears to bemuch smaller than the wavelength (—4-5mm). In the limestone sample, millimetric scalepores are present, which may act to distort the measured velocity-saturation relationship.During the falling rate period hydraulic connectivity is lost as the fluids make atransition to the pendular state. In a granular material, the pendular state corresponds to fluidsbeing located at grain boundaries, cracks, in surface roughness and as adsorbed water. It is40therefore interpreted that velocity changes that are associated with the removal of water fromthese areas will occur below the CMC. Guillot et al. (1989) and Cadoret et al. (1992a,b) bothobserved that sample scale saturation heterogeneities begin to develop at low saturationlevels in rocks (Sw < 0.10-0.15). Due to the existence of such sample scale saturationheterogenieites, it should be notedihat the critical moisture content only provides an uppersaturation bound for when the connectivity of the liquid phase begins to be significantlyreduced. Cadoret et al. (1992a) suggested that such large scale saturation heterogeneitieswould distort the measured velocities through wave path dispersion. The effect of theseheterogenieites in the sandstone samples was assessed using velocity data that was collectedduring adsorption, which will be discussed in the final section of this chapter.Based on these interpretations of the drying rate data and simplified pore spacemodels derived from thin section analysis, fluid distribution scenarios will now be proposedfor the drying process. Numerical modeling is then used to explore the form of the velocity -saturation relationships that will result from these distributions. The main interpretation thatis extracted from the drying data is that, in the rocks being considered, crack-like (compliant)porosity will tend to remain saturated until below the CMC; above the CMC, only the porebodies, which will be stiffer, will drain of water. Since the CMC represents the onset of areduction of connectivity of the liquid phase, it is used only to establish an upper Sw boundfor when the crack-like porosity will begin to drain. It should also be noted that since onlythe general characteristics of the pore space were considered in making up the pore spacemodels, it is the general form of the relationships that are considered to be relevant. For thepurposes of modeling the qualitative form of the velocity-saturation relationships, samplescale saturation heterogenieties will not be considered.The pore space model for the limestone sample is composed of high aspect ratioinclusions representing pore bodies and very low aspect ratio pores representing fractures.41The proposed model for fluid distributions during drying, shown in Figure 2.1 8a, is dividedinto three main stages. In the first stage, which occurs during the early part of the constantrate period, all pore bodies will be sequentially and partially drained. A substantial amount ofwater is left in the partially drained pores, which is intended to indicate that the saturationlevel is high enough for hydraulic connectivity to be maintained. In the second stage, whichlasts until just below the CMC, the saturation level in the partially saturated pore bodies isreduced. This represents the drainage of the connected water in the pore bodies until only asmall amount of disconnected water remains, present as water trapped in surface roughnessand as an adsorbed layer. In the final stage, lasting from just below the CMC until Sw = 0,the hydraulically disconnected surface water in pore bodies and water contained in the crackswill drain. The form of the relationship between the moduli and water saturation level that isproduced by such a model is shown in Figure 2.18b. In the first stage, which lasts from Sw =1 until the middle ranges of saturation, a gradual decrease in the bulk modulus occurs as thelarger pores are drained to a partial saturation state. The shear modulus is essentially flat inthis region, since it is relatively insensitive to the saturation state of high aspect ratio pores.In the second stage, both the bulk and shear moduli are constant as the saturation level ofpartially saturated pore bodies is reduced. The moduli are flat during this period because themoduli of a water filled pore is reduced substantially with the addition of a small amount ofcompressible air and the subsequent addition of more air has little effect. This principle wasillustrated in Figure 2.2. In the final stage both the bulk and shear moduli exhibit sharp dropsas water is removed from the cracks and surface roughness of the pore bodies. These changesin moduli are due entirely to the removal of water from the compressible cracks. In using thismodel it is important to note again that it is assumed that frequencies are high enough for thefluids contained in the crack-like porosity to be unrelaxed and not communicate through fluidflow with the partially drained pore bodies.42rDjoa..csdTO______; 0 0__________>4____,-..s.Figure 2.18a Pore saturation scenario during drainage of limestone. A to D: Sequentialdrainage of large and intermediate aspect ratio pores; D to E: Drainage ofconnected water in partially drained pores; E to F: Drainage of cracks andhydraulically disconnected surface water. Black=water, White= air.A B CD E F43K• —0SwFigure 2.18b. Moduli model for limestone sample. Upper line is bulk modulus,lower is shear modulus.44The pore space model for the dolomite sample is composed of intermediate and highaspect ratio pores. For the purposes of modeling the effects of saturation on moduli, the fluiddistribution scenario during drying, shown in Figure 2. 19a, can be divided into two stages. Inthe first stage, which corresponds to the early stages of the constant rate period, all pores willbe sequentially and partially drained. A significant amount of water will be left in the drainedpores, indicating that the saturation level is high enough to maintain hydraulic connectivity.In the second stage the saturation level in the partially drained pores will be then be reducedtowards zero. Since no cracks are present, pendular water will consist primarily of water heldin surface roughness and as an adsorbed layer, which will have no effect on the moduli. Theform of the moduli-saturation relationship that is produced by this model is shown in Figure2.19b. In the first stage, which lasts from Sw = 1 to the middle saturation ranges, the bulkmodulus decreases steadily and the shear modulus shows a modest decrease. These decreasesare due to the introduction of air into all the pore bodies. In the second stage, the subsequentand complete drainage of pore bodies produces no change in the bulk and shear moduli.Berea sandstone porosity is represented by a range of aspect ratios. Pore bodies andthroats are represented by high and intermediate aspect ratios, as in the case of Bakerdolomite. The crack-like porosity evident at grain contacts is represented by very low aspectratio pores. As with the limestones, the fluid distribution scenario during drying, shown inFigure 2.20a, is divided into three stages. In the first stage, which corresponds to the earlystages of the constant rate period, pore bodies, represented by high and intermediate aspectratio pores, sequentially and partially drain. This is then followed by a period, lasting untilbelow the CMC, during which the saturation level of these pores is reduced to a low level.This low level is intended to represent the point at which liquid within the pores becomesdisconnected and is present as water in the surface roughness and as an adsorbed layer.Finally, during the third stage, the remainder of the water contained in the pore bodies as well45D ELJLL)FFigure 2.19a Pore saturation scenario during drainage of dolomite. A to D: Sequentialdrainage of large and intermediate aspect ratio pores; D to E: Drainage ofconnected water in partially drained pores; E to F: drainage of disconnectedsurface water. Black=water, White= air.A B C460 1SwFigure 2.19b. Moduli model for dolomite sample. Upper line is bulk modulus,lower is shear modulus.47Figure 2.20a Pore saturation scenario during drainage. A to D: Sequential drainage oflarge and intermediate aspect ratio pores; D to B: Drainage of connectedwater in partially drained pores; E to F: drainage of cracks and disconnectedsurface water. Black=water, White= air.A B C(JO.*:O:.0.(DC C)___FD E48as water contained in the lowest aspect ratio pores, which represent the grain contactporosity, is drained. Figure 2.20b shows the form of the moduli- saturation relationship that isproduced by this model. In the first stage the bulk modulus decreases steadily and the shearmodulus decreases gradually as air is introduced into the pore bodies. The moduli then leveloff until below the (DMC as the saturation level in the partially drained pore bodies is reducedfurther. Finally, inthe last stages of drying, both the bulk and shear moduli decrease sharplydue to the removal of water from the grain contact regions.2.6.2 Comparison of Models With Experimental DataThe fluid distribution scenarios suggested by the drying rate data and thin sectionanalyses represent a wide range in the form of dependence of velocities upon saturation level.These models will now be compared to the experimental results.For the limestone sample, Vp and Vs are shown as a function of Sw in Figure 2.21;the bulk and shear moduli are shown in Figure 2.22. As Sw is reduced from the point ofmaximum saturation there is a gradual decrease in bulk modulus until Sw 0.15. When Swis decreased from this point towards zero, the bulk modulus decreases rapidly. The shearmodulus remains relatively constant as Sw decreases from its maximum to approximately0.15. After this point it drops sharply.These data are qualitatively similar in form to the proposed relationships in Figure2.1 8b. In the upper ranges of saturation it was interpreted that the larger pore bodies weredraining, which produces only a small effect on the bulk modulus. The decrease in bulk andshear modulus that was associated with the drainage of cracks began at Sw = 0.15,significantly below the CMC (Sw = 0.3) indicated that hydraulic connectivity was becomingsignificantly reduced. That is, the data indicate that there is a distinct and pronounced regionof decrease in moduli at lower saturations that is associated with the removal of water that isheld in the pendular state.49Figure 2.20b. Moduli model for Berea samples. Upper line is bulk modulus,lower is shear modulus.505.0z00 0ElEl4.5 ElC.? C.E 40• El3.5 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw2.62.4‘—S 00 El 00C.? 00 00 000002.2C.—2.01.8 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.21. Velocities in limestone during evaporative drying. Top: Vp; Bottom: Vs.5150•El El El40 El ElElUUUEl30 EFEl.20’10 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw16151413U UD El DUEl DEl12El11.’Cl) 109.8 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.22. Moduli in limestone during evaporative drying. Top: Bulk modulus;Bottom: Shear modulus.52For the dolomite sample, Vp and Vs are plotted as a function of Sw in Figure 2.23; thebulk and shear moduli are shown in Figure 2.24. The Vs data above Sw = 0.85 was of poorquality, therefore both the bulk and shear moduli are shown only below Sw = 0.85. FromSw =- 0.85 to 0.40 the bulk modulus decreases steadily. Below this point it can be seen to—- -level off. The shear modulus is relatively flat throughout most of the saturation range. At low-—-saturations, below approximately Sw = 0.10, it increases gradually as Sw decreases. Theincrease in shear modulus at low saturations is attributed to the effects of adsorbed waterthough the mechanism for this increase in carbonates is not understood (Clarke et al, 1980).Since the modeling routine can only take into account the bulk effects of fluids, onlydata for saturation levels above Sw = 0.10 will be considered. These data are distinctlydifferent from the limestone data and are similar in form to the saturation scenario proposedin Figure 2.l9b. In this case, pores are sequentially and partially drained over a range ofsaturation. This is then followed by the drainage of the remainder of the water from the porebodies.The data for both the Berea 100 and 300 samples are similar in form and will bediscussed together. Vp and Vs for Berea 100 are shown as a function of Sw in Figure 2.25;the bulk and shear moduli are shown in Figure 2.26. Vp and Vs for Berea 300 during dryingare shown in Figure 2.27; the bulk and shear moduli are shown in Figure 2.28.Four distinct regions can be resolved in the bulk moduli. The first region, at highsaturation levels, there is a decrease in bulk modulus with Sw. This occurs from the point ofmaximum saturation until approximately Sw = 0.35 in Berea 100 and Sw = 0.50 in Berea300. This is followed by a flattening until Sw = 0.18 in Berea 100 and Sw = 0.10 in Berea300. In the third region there is a sharp decrease in bulk moduli. As Sw is decreased towardszero, the bulk modulus levels off in Berea 100 and increases in Berea 300. The variations inshear moduli also may be divided into four regions, though the relative magnitude of changes533.121.B2.9B0C.)2.8 BB2.70B2.6 002.5 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw1.501.451.40C? GBB0 B1.35 0 n0• 1.30 U B B1.251.20 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.23. Velocities in dolomite during evaporative drying. Top: Vp; Bottom: Vs.541816BU14B12BBin. B00 DOLU86 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw5.04.5.0000004.0 035. I • • • I0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.24. Moduli in dolomite during evaporative drying. Top: Bulk modulus;Bottom: Shear modulus.553.8D3.6El000ODD-‘ 3•4ElEl— ElC.—ElEl -3002.8 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw2.1cI2.0wDOD0 00.E —El 0000000 00El1.9 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.25. Velocities in Berea 100 during evaporative drying. Top: Vp; Bottom: Vs.562018El00El°16 00014El12ElEl—\ ElI_u ElR6 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw10.510.0:El00000U9.0 Bl 0&1t1°El8.5 08.0 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.26. Moduli in Berea 100 during evaporative drying. Top: Bulk modulus;Bottom: Shear modulus.573.2DEI°Do2.6 g2.4 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw1.71.6C.)I3XU1.51.4 • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.27. Velocities in Berea 300 during evaporative drying. Top: Vp; Bottom: Vs.581614 .tICE1°mcC12 2—010aDa a8‘ci64 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw6.05.5..’22225.04.0 • • • a0.0 0.2 0.4 0.6 0.8 1.0SwFigure 2.28. Moduli in Berea 300 during evaporative drying. Top: Bulk modulus;Bottom: Shear modulus.59are different than those in the bulk moduli. As with the bulk moduli, the shear modulidecrease with Sw in the upper ranges of saturation, and this is followed by a flattening. AsSw is reduced below 0.20 in Berea 100 and 0.10 in Berea 300 there is a small decreasefollowed by a sharp increase.The increases in bulk and shear moduli -at k)w saturation levels are attributed to theincreases in surface free energy of the solid resulting in increase contact adhesion as adsorbedwater is removed. The increase in velocities as Sw decreases towards zero in Berea sandstonehas been previously observed by Wyllie et al. (1962), Pandit and King (1979), and DeVilbiss(1980). These effects are consistent with the interpretation that the removal of adsorbed waterwill occur below the CMC. For the purposes of modeling the effects of bulk fluid properties,the three regions above this saturation level will be compared to the proposed model.The general form of the data in the upper three regions of moduli variations resemblesthat of Figure 2.20b. Pore bodies are partially drained at high saturations, this is thenfollowed by a period over which they are drained more completely. Finally, disconnectedsurface water and water contained in cracks drains below the CMC. The decreases in moduliat lower saturations did not manifest themselves until well below the CMC. This suggeststhat there is a significant drop in velocities that is associated with the removal of water that isheld below the irreducible water saturation level. Another point to note is the lack of decreasethat is evident in the shear modulus in the third region. Core scale saturation heterogenietiesmay be affecting the results in this region in that the surface free energy effect may becompeting with the drainage of cracks. This will be discussed in more detail in the followingsection when the adsorption data are presented.Another feature in the sandstone data that is more pronounced in Berea 300 than inBerea 100, is the decrease in moduli with Sw at high saturations followed by a leveling off.60In terms of the proposed models, this could correspond to a tendency in Berea 100 for themore complete initial drainage of pore bodies. Other factors may also conthbute to thesedifferences in character of the data. Cadoret et a!. (1993) observed variations in the range ofsaturation over which Vp decreased in the upper ranges of saturation in limestone samples. Itwas found that samples with pore sizes that approached the acoustic wavelength tended toexhibit decreases in Vp over a larger range- of saturation. This was attributed to the drainageof individual large pores causing saturation hetrogenieties on the order of the acousticwavelength that in turn resulted in path dispersion. In the Berea samples the Vp wavelengthis approximately 5 mm and the Vs wavelength is approximately 4 mm. Thin section analysisrevealed that the scale of the pores is much smaller than the acoustic wavelength for bothBerea 100 and Berea 300. It does not seem reasonable therefore to interpret that the observeddifferences could be attributed to pore scale induced saturation heterogenieties on the orderof the acoustic wavelength. It is possible, however, that variations in the nature of theporosity on a scale that approaches the wavelength of the elastic waves may be present.Imaging studies of the porosity of the rock were not available to assess this possibility,though upon visual inspection the Berea samples appear to be quite homogeneous incharacter.2.6.3 Adsorption ExperimentsIt is well established that during the falling rate period of drying sample scalesaturation heterogenieties will evolve. The extent to which these effects distort the measuredvelocity-saturation relationship was explored in the sandstones by examining the results ofexperiments that measured velocities as water was adsorbed to dry rocks. Adsorption hasbeen used as a method to increase Sw in the low end of saturation in velocity experiments61(Wyllie et al., 1962; Pandit and King, 1979; Clarke et al., 1980; Knight and NolenHoeksema, 1990) but it appears that the results have not been compared with those obtainedduring drying in the lower ranges of saturation. Adsorption is a much slower process,occurring over the course of six to eight weeks for the samples used in this study, and it istherefore interpreted that a more homogeneous distribution of moisture will evolve duringthis process.The bulk and shear moduli during adsorption and drying are shown in Figure 2.29 forBerea 100 and in Figure 2.30 for Berea 300. As Sw is increased from the dry state in Berea100, both the bulk and shear moduli decrease substantially, to 80 percent of their dry values.This is followed by an increase in bulk and shear moduli until the point of maximumadsorption at Sw = 0.16. Similarly, in Berea 300 the bulk and shear moduli decrease to 69and 73 percent of their dry values. The moduli then increase until the point of maximumadsorption at Sw = 0.15 in Berea 100 and 0.07 in Berea 300.As water is adsorbed to the surfaces of the dry sample there is a reduction in bothbulk and shear moduli. These decreases in moduli can be attributed to a reduction in surfacefree energy of the quartz grain contact surfaces that occurs as water is adsorbed to the dryrock. After an initial drop, both the bulk and shear moduli increase towards the later stages ofadsorption. The interpretation of the fluid distribution in the rock during the later stages ofadsorption follows that by Foster (1932), which was subsequently applied to a study of thedielectric constant in partially saturated rocks by Knight and Nur (1987). As adsorptionproceeds, water layers build up on the pore walls until eventually closure between the layersoccurs and menisci form. In Berea, a granular sandstone, it seems reasonable to assume thatmenisci will form first at the grain contacts rather than in the larger pore bodies.After menisci have been established, condensation can then occur at the menisci62—J)cI—U Drying• Adsorption0.05 0.10 0.15Sw0.2015-00BornB00ElBBBU10- CC0 •I________________.5-0.0010-.BC.0- BBCEl BOBBC0CB B B UB COB B* B. .o.,. .7.0.00 0.05Figure 2.29. Moduli in Berea 100 during adsorption and evaporative drying at lowsaturation levels. Top: Bulk modulus; Bottom: Shear modulus.El DryingAdsorption0.10 0.15Sw0.206310B Drying• Adsorption0.B BCCBB— 6 B B CD DB...4 • • •0.00 0.02 0.04 0.06 0.08 0.10Sw6.00Drying• AdsorptionEl__________ElIi— • ElB CEl EIB5.0I-i.E El El•-45.4.0 • • •0.00 0.02 0.04 0.06 0.08 0.10SwFigure 2.30. Moduli in Berea 300 during adsorption and evaporative drying at lowsaturation levels. Top: Bulk modulus; Bottom: Shear modulus.64surfaces. To gain insight into how the condensation will proceed it is illustrative to examineKelvin’s equation for the vapor pressure, P. across a meniscus held in a cylindrical capillary:P = P0 exp (4) (2.9)where P0 is the ambient vapor pressure, a is the surface tension in the meniscus, v is themolecular volume, r is the radius of curvature of the meniscus, R is the gas constant and T isthe temperature. From the above relationship it can be seen that menisci with smaller radii ofcurvatures will have a lower vapor pressure. Since the rate of condensation is proportional tothe difference between the vapor pressure across the meniscus and the ambient vaporpressure, pores with a smaller radius will tend to fill up before those with a larger radius.Based on this argument, it is interpreted that the increase in saturation towards the later stagesof adsorption is due to the filling of grain contacts, cracks and clays with water. This issupported by the observation that there is a sharp increase in both bulk and shear moduli inthis range of saturation: the saturation state of compliant porosity, such as that at graincontacts, will affect both the bulk and shear moduli.It is interesting to note that the point at which adsorption stopped corresponds to thepoint at which the sharp increase in both the bulk and shear moduli leveled off. In a constanthumidity environment, capillary condensation will cease when the radii of the meniscibecome large enough so that the vapor pressure across the interface is in equilibrium with theambient vapor pressure. In the Berea samples, it appears that adsorption stops when asubstantial amount of the grain contact porosity is saturated.In comparing the adsorption data with the data collected during the final stages ofdrying it is evident that there is a considerable amount of hysteresis. Experimental andtheoretical evidence supports the existence of sample scale saturation heterogeneities during65the later stages of drying. In contrast, adsorption occurs very slowly, over the course ofapproximately eight weeks, and it is interpreted that this will tend to produce a morehomogeneous saturation distribution. The following scenario, shown in Figure 2.31, istherefore proposed to explain the results during the later stages of drying. First, grain contactstowards the outside of the sample are drained of bulk water, which would result in a decreasein moduli, while towards the inside of-the sample they remain saturated. As drying proceeds,adsorbed surface water begins to desorb from the contacts towards the outside of the samplewhile contacts towards the inside of the sample are just beginning to drain of bulk water.That is, the bulk and shear moduli will be increasing towards the outside of the sample anddecreasing towards the inside of the sample. The net result appears to be that these twocompeting effects result in the measured velocities being significantly distorted. Duringadsorption, it appears that a more homogeneous saturation distribution allows these twoseparate effects to be resolved.2.7 SummaryElastic wave velocities and drying rates were measured in three different types ofrock sample types as a function of saturation during evaporative drying. Drying rate curveswere used to establish the transition from water being in a funicular state to water being in apendular state, a transition that has been associated with the onset of the irreducible watersaturation level within drying samples. Velocity variations below this point were therebyassociated with the removal of disconnected pendular water. The implications of thisSAMPLEUnbound Water Draining fromGrain Contacts, CracksSurface Water RemovalSOLID66LIQUIDGASFigure 2.31 Sample scale saturation heterogenieties during the later stages of drying.67information to the form of dependence of velocities upon saturation was found to bedependent upon the nature of the pore space microgeometry.In the dolomite sample, there was a decrease in bulk modulus that was attributed tothe introduction of air into pore bodies. The subsequent drainage of pendular water resultedin no further velocity variations, which was attributed to the absence of any significantamount of crack-like porosity. In contrast, the results for the limestone and Berea sandstonesamples indicate that two distinct regions of velocity variation are present: one that isassociated with the drainage of pore bodies, and another that is associated with the drainageof disconnected, pendular water, held in crack-like porosity. These results may be ofparticular interest since this suggests that velocity variations may sometimes be specificallyassociated with the removal of water that is held below the irreducible water saturation level.The close agreement in the form of the models and data suggests that the combined use ofdrying rates and even simple models of pore space microgeometries can be of considerableuse in interpreting the velocity-saturation relationship.Finally, the differences in measured velocity data between adsorption and dryingindicate that sample scale saturation heterogeneities that exist during the final stages ofdrying can significantly affect velocity measurements.683 THE EFFECTS OF WETTABILITY ON VELOCITIES3.1 IntroductionThere have been a number of recent laboratory and theoretical studies examining theeffect of pore scale fluid distributions on the elastic and electrical properties of rocks. Two ofthe most important parameters in determining the distribution of fluids in partially saturatedrocks are saturation history and wettability. In order to characterize the relationship betweena measured property and fluid distributions in a multiphase saturated porous medium, thecoupled role of these two parameters needs to be understood.Saturation history in porous rocks saturated with air and water has been shown toproduce hysteresis in the relationship between saturation level and a number of geophysicalparameters including elastic wave velocities (Knight and Nolen-Hoeksema, 1990; Cadoret etal., 1992a,b), elastic wave attenuation (Bourbie and Zinszner, 1984), electrical resistivity(Knight, 1992) and dielectric constant (Knight and Nur, 1987). These effects can beattributed to differences in the distribution of fluids that occur due to changes in thesaturation process.Altering the wettability of rocks has been shown in numerous studies to affect therelationship between saturation history and electrical properties in porous rocks containingoil and water (see Anderson, 1986, for a review). However, there appear to have been nopublished studies that examine the effects of altering wettability on the relationship betweensaturation history and velocities in rocks. Wyllie et al. (1958) touched upon this topic bymaking measurements of Vp in synthetic cores using wetting and nonwetting liquidsdisplacing air. Velocities measured when the samples were partially saturated with non-wetting fluids were lower then for air saturated samples. It was interpreted that poor coupling69between the non-wetting fluids and the solid led to the attenuation of signals and,consequently erroneous arrival time selections. Unfortunately, measurements were only madeat a single, unspecified saturation level so that density effects could not be taken intoconsideration, and the effects on the bulk and shear moduli could not be determined.In the following chapter the effects-of altering wettability on the relationship betweenelastic wave velocities and saturation level is investigated. This was accomplished byconducting imbibition and drainage experiments on samples that were initially strongly waterwet and then repeating the experiments after the samples had been treated to make themoil-wet. For the purposes of this study the untreated, strongly water-wet rocks will be referredto as hydrophilic and the treated, oil-wet samples will be called hydrophobic. The proposedfluid distribution scenarios are assessed using numerical modeling.3.2 The Effect of Saturation History On Fluid DistributionSaturation history can affect the distribution of fluids in a porous media through twoprimary mechanisms: contact angle hysteresis and pore geometry effects. The principlesbehind these two effects can be illustrated-in simple pore systems containing a wetting liquidand a non-wetting gas.On a smooth, homogeneous surface contact angle hysteresis will tend to be negligibleand the measured angle on such a surface is referred to as the intrinsic contact angle. If thesurface is roughened, hysteresis will exist between the contact angle when a fluid isadvancing with respect to another fluid (9A) and the contact angle when a fluid is receding(OR). This is illustrated in Figures 3. la and b. In general, OA>91>OR; it has been observed thatthe difference between°A and 0R can be as much as 60° (Johnson and Dettre, 1969).(a)70SMOOTH SURFACE(b)0ROUGH SURFACEFigure 3.1 (a) Intrinsic contact angle (b) Advancing, intrinsic and receeding contact angles71Figure 3.2a illustrates how a difference in contact angles between imbibition anddrainage can result in variations of capillary rise in a cylindrical tube placed in contact with aliquid reservoir. In this type of experiment, the pressure drop that exists across the interfacecauses a liquid column to rise up the tube. The height of the rise, h, is related to the contactangle through the relationshippgh=2SO (3.1)where g is the gravitational constant. During drainage 0R is small, thereby creating ameniscus with a small radius of curvature. Consequently, the resulting capillary pressure (seeequation 2.5) is high, allowing the capillary tube to support a large column of liquid. Incontrast, the lower contact angle present during imbibition creates a lower capillary pressureand the capillary rise is therefore reduced.The effects of pore geometry in the absence of contact angle hysteresis can beillustrated using a tube with a large pore body in the middle (Figure 3.2b). During drainage ofa wetting fluid the radius of the meniscus in the upper part of the pore is small, therebycreating a large enough capillary pressure to support the height of the water column and toretain water in the pore body. During imbibition, the water column rises until it reaches thepore body. At this point the radius of curvature in the meniscus becomes larger causing thecapillary pressure to drop and the meniscus cannot rise any further.Due to the interaction of contact angle hysteresis and the complex geometries of porespaces that exist in rocks, characterizing the behavior of wetting and non-wetting fluidsduring imbibition and drainage cycles becomes difficult. At a sample scale, it has been foundthat these effects result in hysteresis in capillary pressure curves between imbibition anddrainage cycles. A typical imbibition and drainage cycle capillary pressure curve has the72(a)(b)DR 2a COSORrDR 2c cosOr1‘IMB 2a COSOAr‘31MB 2c cosOFigure 3.2 (a) The effects of receding and advancing contact angles on imbibition and d±ainage.Left: drainage; Right: imbibition (b) The effects of pore geometry on imbibition anddrainage. Left: drainage; Right: imbibition.73form shown in Figure 3.3. During drainage, increases of pressure, relative to ambientconditions, in the non-wetting phase result in a reduction in saturation of the wetting phase.As the wetting phase saturation is reduced, higher levels of pressure are required to displacethe wetting fluid until, at the irreducible water saturation level, the wetting phase becomesdisconnected and the saturation cannot be reduced further. The wetting phase will tend to-- exist as pendular rings at grain contacts and in cracks. During imbibition, the—non-wettingphase saturation level is reduced as pressure is dropped. As negative pressure is applied thenon-wetting phase saturation level continues to decrease until it becomes hydraulicallydisconnected and the residual nonwetting phase saturation level is reached. At this point thenon-wetting phase will tend to exist as isolated globules in larger pore bodies.For intermediate wettabilities, the capillary pressure curves can become morecomplex. Morrow and Mungan (1971) and Morrow (1976) studied the effects of the intrinsiccontact angle on imbibition and drainage capillary pressure curves. In these experiments asynthetic porous medium was saturated with air and a liquid. The liquids were selected toprovide a range of intrinsic contact angles from 0° to 108°. During drainage, when the liquidswere displaced by air, it was found that the form of capillary pressure curves were relativelyinsensitive to liquids with intrinsic contact angles below approximately 50°. Duringimbibition however, when liquids were displacing air, the form of the capillary pressurecurve was found to be very sensitive to the intrinsic contact angle above approximately 20°. Itwas interpreted that these differences in the level of sensitivity of capillary pressure curves towettability were due to the receding contact angle being sensitive to intrinsic contact anglesabove approximately 50° and the advancing contact angle to intrinsic contact angles aboveapproximately 20°. These experiments indicate that, for the range of wettabilities consideredin that study, the distribution of fluids is more sensitive to changes in wettability duringimbibition than during drainage.74aIIIIa, IU) I 4%U)‘%a,N0IIIza- I0 Sw 01Sw Irreducible ResidualNon-wettingPhaseFigure 3.3 Capillary pressure curve during imbibiton and drainage cycles.753.3 ExDerimental ProceduresIn this chapter, experiments examining the effects of wettability on the relationshipbetween velocities and saturation wereconducted on the Berea sandstone samples. Bereasandstone was selected because it is well studied and strongly water wet in the untreatedstate.As in the experiments presented in Chapter Two, saturation levels were reduced usingevaporative drying. Sw was increased in the samples in three stages. During the first stagewater was adsorbed to the dry samples by placing them on a stand in a sealed chamber withits bottom filled with deionized water. The samples were thereby exposed to air with relativehumidity that was close to 100 percent (Knight and Nur, 1987). After the samples ceasedadsorbing water by this method, Sw was increased by imbibition through immersion indeionized water (Knight and Nur, 1987; Knight and Nolen-Hoeksema, 1990). In thehydrophobic samples, the saturation level was increased beyond this point by using thedepressurization method. This procedure consists of first placing the samples in a chamberthat is almost completely filled with water, and lowering the pressure in the chamber bypulling a vacuum on the air at the top of the container. This leads to expansion of the air inthe samples, air leaves the samples and when the chamber pressure is returned to atmosphericlevels the samples imbibe water. This is not, strictly speaking, a true imbibition process buthas been shown to produce very uniform saturation distributions (Cadoret et al., 1992a,b).The wettability of the samples were altered by treatment with Quilon-C, acommercially available chromium based complex that attaches to the negatively chargedquartz surfaces and extends fatty acid chains into the pore space. The treatment is amonolayer thick and as such it offers the advantage of altering the wettability of the rock76surfaces without significantly affecting the microgeometry of the pore space. Porosities werealtered by only 0.003 for Berea 100 and 0.002 for Berea 300. Lewis et al. (1988) found thisprocedure altered Berea sandstone samples from being strongly water-wet to being stronglyoil-wet.The treatment procedure was a modified version of that used by Lewis et al. (1988).- -A sample that had been oven driedat 600 C for one day was evacuated in a vacuum chamberfor several hours. A mixture of 20 percent Quilon-C, 80 percent isopropanol (by volume) wasintroduced into the chamber and the sample was soaked in this solution for two days atambient laboratory conditions. The samples were then dried at 60° C for one day and thetreatment process was repeated. Following the second treatment, the samples were evacuated,saturated with distilled water, and soaked for several days. This was repeated until theconductivity of the distilled water reached an equilibrium value, which indicated that theamount of unbound Quilon-C chemical coming into solution was low and therefore that thesample was then ‘clean’.Lewis et al. (1988) used oil and water as pore fluids; in this study, air and water arethe pore fluids. The wettability of a samples in this study was assessed by conductingspontaneous imbibition tests before and after treatment. Spontaneous imbibition rates areproportional to the capillary pressures in the pore network which are directly related to thewettability of the system. Other factors, such as pore size, liquid viscosities and the size andshape of the samples have also been noted to have a significant impact upon imbibition rates(Jadhunandan and Morrow, 1991). In comparing treated and untreated samples of the sametype, we keep these variables essentially constant while isolating wettability as the mainfactor influencing changes in imbibition rates.Imbibition data were acquired by immersing an air saturated sample in water andmeasuring the weight of the sample as a function of time (Denekas et al., 1959). The77experiments were conducted by suspending the samples, with vertical orientation, by a thinfilament into a 600 ml beaker filled with 500 ml of water. The cores were suspended from abottom loading scale, which was interfaced with a computer. Weight measurements were— then made as a function of time for 24 hours. The weights were then converted to- saturationlevel.It should also be noted that in using water to displace air, the density contrast betweenthese fluids results in a hydrostatic gradient across the samples, which is a driving force forincreasing water saturation level. To assess the extent to which this has influenced the results,a second imbibition test was conducted, that of capillary rise. The core samples were placed,vertically oriented, in contact with an interface of water. The base of samples wereapproximately one to two millimeters beneath the surface of the water. After ten days therewere no significant weight changes in the samples and the saturation level at this point wasrecorded. In this case the effect of gravity on hydrostatic pressure is opposite to the previoustest: gravity will tend to resist the capillary rise.It should be emphasized that these are only qualitative wettability tests. However, it isconsidered to be sufficient for the purposes of this study to establish that the wettability hasbeen significantly altered by the treatment process.-3.4 Experimental Results and DiscussionIn the following section the experimental results will be presented and discussed. Dueto the similarity in form of the results for Berea 100 and 300, the results for both samples willbe discussed simultaneously. The hydrophilic case will be presented first, followed by theresults for the hydrophobic case.783.4.1 Hydrophilic SamplesVp and Vs in Berea 100 during imbibition and drainage cycles are shown in Figure3.4; the bulk and shear moduli are shown in Figure 3.5. For Berea 300, Vp andVs data canbe found in Figure 3.6; the bulk and shear moduli are in Figure 3.7.-The drainage results for the Berea 100and 300 samples-are discussed in detail inChapter Two. In both samples there was a strong decrease in bulk modulus with Sw in theupper ranges of saturation. This is followed by a period over which the modulus wasinsensitive to changes in saturation level until, at lower saturation levels the modulus beganto decrease sharply with decreasing Sw. As Sw was reduced further the bulk modulusincreased sharply in Berea 300 and leveled off in Berea 100. The shear modulus followed asimilar pattern to the bulk modulus, with the variations being less pronounced in the upperranges of saturation.As Sw was increased through adsorption, the bulk modulus decreased sharply andthen increased, leaving a substantial amount of hysteresis between the adsorption and dryingresults. Subsequent imbibition through immersion produced only a modest increase in bulkmodulus for Berea 100 while for Berea 300 it was essential flat throughout most of thesaturation range. At high saturations, the bulk modulus increased sharply in Berea 300. Thelack of a sharp increase in bulk modulus in Berea 100 towards higher saturation levels isatthbuted to an inability to increase the saturation to a sufficiently high level by means ofimmersion imbibition. The shear modulus follows a similar pattern though the magnitude ofthe variations is less pronounced than in the bulk modulus.During drying it was interpreted that pore bodies drained in the upper ranges ofsaturation and the removal of bulk and adsorbed water from grain contacts was responsiblefor velocity variations in the lower ranges of saturation. During adsorption, the results79.—04 0.6Sw0.8 1.03.8 -3.6’ DQOD,aElD1D 0::‘I-fl.o,• : • Adsorption2.8’ ‘I Immersion.2.6’0.0 0.22.1rn2.0i i!rnEli UBElDElIUG El El El El El El U••• B DDDElUDU’I U• =Dê flU• •In.. .________________L. •• ...••=1.8 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.4. Velocities in Berea 100 during imbibition and drainage.• B Drying• Adsorption‘I ImmersionTop: Vp; Bottom: Vs.pceC?rJ)Figure 3.5. Moduli in Berea 100 during imbibition and drainage.Top: Bulk modulus; Bottom: Shear modulus.802220181614121086L4— ‘-_El0 EPCmElBQCLDCDrying• AdsorptionImmersion4 • I I I •0.0 0.2 0.4 0.6 0.8 1.0Sw10 —9j ciCC0dIrjIl ao•7.0.0W Drying• AdsorptionImmersionI I0.2 0.4I — I0.6 0.8Sw1.081C?r;jC.—C.?C?C.—3.2DCC:a2.6 .______________2.442.2 —0.0DryingAdsorptiona Immersion0.2 0.4 0.6Sw0.8 1.01.71.6C Drying• AdsorptionImmersion•6I1.5.CDDCCS..S1.4 — - . - - .-0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.6. Velocities in Berea 300 during imbibition and drainage.Top: Vp; Bottom: Vs.821816141210.E 8a Drying6• Adsorption‘ Immersion2 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw6.56.0.5.— b_________1; DryingCl)• Adsorption4.0 Immersion3.5 •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.7. Moduli in Berea 300 during imbibition and drainage.Top: Bulk modulus; Bottom: Shear modulus.83indicate that grain contacts are filling with water. Variations in velocity as water saturationlevel is increased above the point of maximum adsorption is therefore attributed to the fillingof the remainder of the porosity, such as pore bodies and throats. It appears then that crack-like porosity at grain contacts is the last to drain during drying and the first to fill duringimbibition. In the upper ranges of saturation, variations in velocity are attributed to changesin the saturation state of pore bodies and throats.Hysteresis is evident in the bulk and shear moduli in both the upper and lowersaturation ranges. The hysteresis at low saturations was explained the previous chapter interms of sample scale saturation heterogenieties that evolve during the later stages of drying.At higher saturations it is proposed that the hysteresis can be explained by differences in thedistribution of fluids in the pore bodies that evolve between the imbibition and drainageprocesses. During drainage it was interpreted that pore bodies drained sequentially leavingbehind partially drained pores with a connected liquid phase. This was followed by a periodover which the partially drained pores drained further until the liquid phase becamehydraulically disconnected. During imbibition it is interpreted that partial saturation ismaintained at a pore scale until a higher saturation levels. This is followed by a period overwhich the remainder of gas in pore bodies is displaced by water. These scenarios areconsistent with the interpretations of Knight and Nolen-Hoeksema (1990) and Cadoret et al.(1992a,b).3.4.2 Hydrophobic SamplesAs with the hydrophilic sample, an examination of the drying rates can provideimportant insights into the distribution of fluids within the samples during the drying process.Drying rate data for the treated and untreated samples are shown in Figure 3.8 for Berea 100and in Figure 3.9 for Berea 300. The data are remarkably similar for both the treated and840.0040.003CE 0.002.—CC‘.—, 0.001 C0.000’0.2 0:4Sw0:6 0.8 1.00.003CU0.0020.001 mc0.000/0:4 0:6 0:8 1.0Figure 3.8. Comparison of drying rates in hydrophilic (top) and hydrophobic (bottom)Berea 100 samples.852e-3 -Cle-3Oe+0•0:2 0:4 06 0.8 1.05e-5 -4e-5 C:::le-5Oe÷0 • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.9. Comparison of drying rates in hydrophilic (top) andhydrophobic (bottom) Berea 300 samples.86untreated cases. Initially, there is a sharp decrease in rate that is followed by a period ofconstant rate. The constant rate period ensues until Sw = 0.33 in Berea 100 and Sw = 0.24 inBerea 300. The rates then decrease as Sw is reduced towards zero. It should be noted that nosignificance is attached to the absolute value of the drying rate during the constant rate periodsince it can-be influenced by environmental factors such as temperature and humidity.The effect of wettability on drying rates appears not to have been explicitly studied. Itfollows from the background theory presented in section 2.3 that for capillary transport toexist when water is being displaced by air during the evaporative drying process, water mustbehave as a wetting fluid. That is, the receding contact angle must be at least less than 90° forcapillary pressure to exist such that it favors the removal of water through capillarymechanisms. The presence of the constant rate period during the drying of the hydrophobicsamples is therefore interpreted to indicate that capillary transport is the dominant transportmechanism, which in turn implies that when air is displacing water in the evaporative dryingprocess, water behaves like a wetting fluid. As with the water-wet samples, this suggests thatgrain contacts and cracks will tend to remain saturated until below the CMC.The effects of wettability on the imbibition process can be qualitatively assessed byexamining the relative changes in the imbibition versus time data. Figures 3.10 and 3.11show plots of Sw versus time before and after treatment for Berea 100 and 300 respectively.For both Berea 100 and 300, the untreated samples show a sharp increase in saturation uponinitial immersion, indicating high imbibition rates. The curves then begin to level off. For thetreated samples, the saturation level increases very slowly throughout the course ofmeasurement. These results suggest that the wettability of the samples with respect to waterin an air-water system has been substantially reduced when water is displacing air.It was also noted that the contrast in density between the air and water could give rise871.0C Hydrophilic0.8 • Hydrophobic06_____________________CC::•••0.0• I I10 100 1000 10000 100000Time(in seconds)Figure 3.10. Sw versus time for hydrophobic and hydrophilic Berea 100 samplesduring spontaneous imbibition.881.0Hydrophilic0.8 • Hydrophobic10 100 1000 10000 100000Time(in seconds)Figure 3.11. Sw versus time for hydrophobic and hydrophilic Berea 300 samplesduring spontaneous imbibition.89to a hydraulic gradient along the length of the sample that would effectively enhance theimbibition process. Using the capillary rise experiments it was found that the hydrophilicBerea 100 sample reached a saturation level of 0.71 while hydrophobic Berea 100 samplereached a saturation of Sw = 0.08. Similarly hydrophilic Berea 300 sample reached Sw =0.74 while hydrophobic Berea 300 reached Sw = 0.06. These experiments not only suggestthat the wettability has been substantially reduced by the treatment process but also that thehydrostatic gradient was probably a significant factor in contributing to increasing thesaturation level in the hydrophobic samples during immersion imbibition. That is, water wifitend to be forced into the pore spaces during the immersion process.In contrast to the similarity of the hydrophilic and hydrophobic drying rate curves, thetreatment process has produced significant changes in the imbibition test results. Thissuggests that the imbibition process was more significantly affected by the change inwettability than was the drainage process. Given this information about the influence ofwettability on the imbibition and drainage processes, it would now be appropriate to examinethe velocity data.Vp and Vs for Berea 100 during drying, before and after treatment, are show inFigure 3.12. The Berea 300 velocity data are shown in Figure 3.13. One difference betweenthe treated and untreated samples that is immediately apparent is that, at any given saturationlevel, the velocities in the treated samples are higher than those in the untreated samples.Since the treatment process has chemically altered the surfaces of the rock, it seemsreasonable to assume that it has affected the chemical state of the solid/solid interface at thegrain contacts. It appears then that the chemical treatment has resulted in a higher degree ofadhesion between the grain contacts.904.03.8 000003.6.id“ IB -3.2s___________S B Hydrophobic• Hydrophilic3.0 •2.8 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw2.O •••.E 1.90 Hydrophobic1.8 • Hydrophilic1.7— • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.12. Comparison of velocities in hydrophilic and hydrophobic Berea 100 sample.Top: Vp; Bottom: Vs.911.61.51.41.3Hydrophobic• Hydrophilic1.2 I I I I0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.13. Comparison of velocities in hydrophilic and hydrophobic Berea 300 sample.rj3.4.iii°3.23.02.8.•••I’2.62.4C HydrophobicHydrophilic2.2 —0.0 0.2 0.4 0.6 0.8Sw1.01.91.81.7•••. •.••... .• .•••... ••.••Top: Vp; Bottom: Vs.92For the treated Berea 100 sample, Vp and Vs during increasing and decreasingsaturation cycles are shown in Figure 3.14; the bulk and shear moduli are shown in Figure3.15. Similarly for the treated Berea 300 sample, Vp and-Vs-during increasing-and decreasing--saturation cycles are shown in Figure 3.16; the buLk and shear moduli are shown in Figure3.17. In both of the treated Berea samples, the bulk modulus decreases steadily withsaturation until Sw = 0.20 in Berea 100 and Sw = 0.40 in Berea 300. After this point theslope begins to level off in Berea 300. Below Sw = 0.20 in Berea 100 and Sw = 0.10 in Berea300, the bulk moduli drops off steeply. At very low saturations the Berea 100 bulk modulusthen levels off while the Berea 300 bulk modulus increases slightly. The shear modulus canbe seen to decrease slightly with water saturation until Sw = 0.30 in Berea 100 and 0.18 inBerea 300. The moduli then decrease steeply until Sw = 0.04 in Berea 100 and Sw = 0.02 inBerea 300. As Sw is reduced towards zero, the shear modulus then increases steadily.During adsorption, the bulk modulus shows a modest decrease until Sw = 0.03 inBerea 100 and Sw = 0.02 in Berea 300. As the saturation level is increased further, the bulkmodulus increases gradually until Sw = 0.75 in Berea 100 and Berea 300. This is followed byan increase in modulus towards the point of maximum saturation. As with the bulk modulus,the adsorption of water to the dry rock also produces a decrease in the shear modulus untilthe maximum point of adsorption. This is followed by a gradual increase in modulus until thepoint of maximum saturation. The increase that was observed at high saturations in bulkmodulus did not occur in the shear modulus.The form of the relationship between velocities and saturation during evaporativedrying of the hydrophobic rocks closely resembles that of the hydrophilic rock in the upperranges of saturation. In both the treated and untreated cases there is a region of steep decreasein bulk modulus followed by a leveling off. The decreases in bulk moduli are accompanied934.03.8,,.0DOC ODDEl 00003.6•.• DC4 -U U U -a. 3.2’ 0DiyingAdsoiption30U Immersiona Depressurization2.8 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw2.3DOD El El El El D C El2.2 =1= 00000D DODD2.1C? -UU U U U UI2.0Drying• Adsorption19U Immersiona Depressurization1.8 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.14. Velocities in hydrophobic Berea 100 during imbibition and drainage.Top: Vp; Bottom: Vs.94209W918 a9016 001412_________.— 0Drvin10.0 I -AdsorptionIa Immersion8 Depressurization6 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw1200000911__L10-•aI___ ______ ___ ___ __________ ___0Drying• AdsorptionCl) 9• ImmersionDepressurization8 • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.15. Moduli in hydrophobic Berea 100 during imbibition and drainage.Top: Bulk modulus; Bottom: Shear modulus.953.4,El El3.2 ØEIElccEl OW3.0’El a a a a2.8•Sa °DryingAdsorption2.6• ImmersionDepressurization2.4 • • •0.0 0.2 0.4 0.6 0.8 1.0Sw1.85’DOOCOCCOOD 000001.751.65a a a a a0Drying-Adsorption• ImmersionDepressurization1.45- • • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.16. Velocities in hydrophobic Berea 300 during imbibition and drainage.Top: Vp; Bottom: Vs.9616140E112’ El—aU UUI I1,4 11UDrymgI D• Adsorption.6 U ImmersionDepressurization4- • • •0.0 0.2 0.4 0.6 0.8 1.0Sw8.0’7.5’7.0’6.5’U U UU° 6.0’‘.E 5.5___________Drying5.0 • AdsorptionImmersion45’ Depressurization4.0’ • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.17. Moduli in hydrophobic Berea 300 during imbibition and drainage.Top: Bulk modulus; Bottom: Shear modulus.97by gradual decreases in shear modulus over the upper ranges of saturation, which isconsistent with the interpretation that pore bodies are draining. As with the hydrophilic case,it is interpreted that grain contacts remain saturated until the lower ranges of saturation.The earlier onset of decrease in shear modulus relative to the bulk modulus can beexplained in terms of the way in which velocities are measured, keeping in mind that Vs is-affected by the shear modulus and Vp is affected by both the bulk and shear modulus. Thetransducers generate P waves that effectively sample the entire radius of the rock. In contrast,the amplitude of the torsional shear waves must be zero at the center of the sample, and theywill therefore tend to preferentially ‘sample’ towards the outside of the sample. Since theoutside of the sample is likely to have a lower saturation than the inside of the sample, itseems reasonable to assume that the earlier onset of the decrease in shear modulus can beattributed to these saturation heterogeneities.During imbibition, the results for the hydrophobic samples are substantially differentfrom those for the hydrophilic samples. The decrease in moduli that occured with theadsorption of water is attributed to the reduction in surface free energy of the solid surfacesleading to a decrease in contact adhesion. The fact that the magnitude of the drop is smallerin the hydrophobic samples is attributed to a chemical alteration of the grain contact regions.ifl the hydrophilic rocks there was a sharp rise in moduli towards the later stages ofadsorption that was attributed to capillary condensation leading to the filling of grain contactswith water. This was followed by a period over which moduli were relatively constant, whichis consistent with the hypothesis that the saturation level in pore bodies was increased,though partial air saturation was maintained at a pore scale. In contrast, the moduli of thehydrophobic rock rise gradually over a large range of saturation. This clearly indicates theexistence of fundamental differences in the distribution of fluids between the hydrophilic andhydrophobic rocks during the imbibition process.98There are two possible scenarios for the distribution of fluids in the hydrophobicsample during imbibition. The first is based on the assumption that the increases in moduliduring the period of gradual increase are due to the filling of grain contacts. In this case graincontacts fill with water over a large range of saturation while pore bodies maintain partial--saturation until high saturation levels. The second scenario is based on the assumptionthatthe increases in moduli during the middle ranges of-saturation are due to the filling oiporebodies with water. In this case pore bodies fill sequentially and fully while grain contactsremain unsaturated until high saturation levels.In comparing the relative changes in shear modulus with Sw at both high and lowsaturation levels during drying, it can be seen that, at high saturations, there is a modestchange in moduli associated with changing saturation state of pore bodies while at lowsaturations, there is a more substantial change associated with altering the saturation state ofgrain contacts. During imbibition, there is a modest increase in shear modulus throughoutmost of the range of saturation until, at the point of maximum saturation it remains asubstantial amount below its value at Sw = 1. These data support the second hypothesis, thatthe increases in moduli are due to the sequential and complete filling of pore bodies whilegrain contacts remain air filled until high saturation levels. A similar analysis of the bulkmodulus produces ambiguous results, since the relative changes in bulk modulus that areassociated with the removal of water from pore bodies are similar in magnitude to thoseassociated with the removal of water from grain contacts. It is also notable that one of thecontrolling factors of the behavior of menisci in porous media is their tendency to remain‘caught’ on sharp edges and inhibited from moving further (e.g., Anderson, 1987b).Qualitatively, it seems reasonable that this phenomenon might make the entry of menisci intograin contact porosity more difficult, which would also support that latter hypothesis.In either case, it is evident that the fluid distribution occurring during the imbibitionprocess in the hydrophobic rock differs substantially from that in the hydrophiic case. Based99on the above arguments, the scenario in which grain contacts remain air filled until highsaturation levels is preferred and will be adopted for the purposes of modeling.It is also interesting to compare these findings with those of Morrow and Mungan(1974) and Morrow (1976). It was found that capillary-pressure drainage curves were lesssensitive to variations in the wettability of the system than were imbibition capillary pressurecurves. In this study, it was found that during evaporative drying, both the drying rate curvesand the form of the relationship between velocity and saturation were not very sensitive tothe wettability being altered. In contrast, during imbibition it was found that the imbibitionrate tests and the form of the velocity-saturation relationship were substantially affected byaltering the wettability of the system.3.5 Velocity ModelingTo assess the validity of the proposed fluid distribution scenarios, the modelingroutine of Endres and Knight (1989) was used to model the form of the relationship in Berea300. This sample is used since the data set during imbibition is more complete. In the firstchapter the intention was to model general trends in the form of the velocity-saturationrelationship, based on qualitative estimates of the pore spectrums. In this chapter a moreaccurate estimate of the pore spectrum for Berea sandstone is used, based on the model ofCheng and Toksoz (1979). To calculate this pore spectrum, Cheng and Toksoz (1979) usedvelocity versus pressure data to calculate moduli versus pressure relationships. It was thenassumed that pressure applied to the rock will result in closure of pores. Compliant porosity,represented by low aspect ratio pores will close first at low pressures; higher aspect ratiopores close at higher pressures. By inverting the pressure versus velocity data and assumingthat different aspect ratio ellipsoidal pores close at different pressures, a pore spectrum wascalculated. It should be emphasized that when interpreting the results of this model it must betaken into consideration that the complex pore space of Berea sandstone has beenapproximated by assuming a specific idealized geometry.100The pore spectrum used to approximate the porosity of the Berea 300 sample isshown in table 3.1 and was based on one calculated by Cheng and Toksoz (1979) for a 16.3%porosity Berea sample. The aspect ratio concentrations ‘were linearly scaled to account for thehigher porosity of the sample used in this study. A small reduction (0.008% of absolute- porosity) in the concentration of 0.0003 aspect ratio cracks was also made. -This is justifiedby the thin section analysis, which qualitatively indicates a lower concentration of graincontact porosity in Berea 300 than in Berea 100.Aspect Ratio Concentration1.0 19.75%0.1 3.014%0.01 0.0522%0.05 0.01997%0.0017 0.01977%0.0014 0.02259%0.0006 0.01836%0.0003 0.007%Table 3.1 Pore Spectrum used to model Berea 300.As was noted in Chapter Two, the model of Endres and Knight (1989) approximatesthe effects of the bulk properties of pore fluids on the moduli. Effects due to variations insurface free energy that result when fluids are adsorbed to the solid matrix cannot beaccounted for by this model. The water contained within the pore space is therefore dividedinto two types: surface water and bulk water (Endres and Knight, 1991). Surface water will101be considered to be associated with decreases in the bulk and shear moduli that occur duringthe early stages of adsorption. Bulk water will be considered to comprise the remainder of thewater. The saturations used in this section will refer to the bulk water saturation, SWB, whichis related to the surface water saturation, Sw, and the overall saturation through the—following relationship:Sw = (3.2)The division of Sw into surface water and bulk water for Berea 300 is shown inFigure 3.18. The maximum surface water saturation level was selected to be Sw = 0.07,which corresponds to the point at which the bulk and shear modulus begin to increase duringthe adsorption stage.Figure 3.19 shows the proposed fluid distribution scenario during imbibition anddrainage for the hydrophilic rocks. During drainage, intermediate and high aspect ratio pores,representing pore bodies, are sequentially and partially drained over a range of saturation.This is followed by a period over which the saturation level in these pores is reduced further.At low saturation levels, the remainder of the water is drained from the intermediate and highaspect ratio pores, which represents the removal of disconnected bulk water from the porebodies, as well as from the lowest aspect ratio pores, which represent the grain contactporosity. This scenario is depicted in Figure 3.20. It was found that classifying pores withaspect ratios less than or equal to 0.001 as cracks produced the best agreement between themodel and data. It is also notable that the pore spectrum calculated by Cheng and Toksoz(1979) was based on a vacuum dry sample. To take into account the reductions in moduli thatare associated with the adsorption of water to the dry rock, the moduli the values for the solidmatrix bulk and shear moduli were reduced by 31 and 44 percent respectively, a procedureused by Toksoz et a!. (1976) and Endres and Knight (1991). These values are comparable tothe reductions in bulk and shear moduli observed as water was adsorbed to the dry rock.S10216lB414ci 121Uo 10•1 UlIU•8IDng• I • Adsorption I6 fr i . i4• t • • •0.0 0.2 0.4 0.6 0.8 1.0SwS6.5lB46.0••15.0 CCCU U Dj C U‘I.Cl) IBDiying I4.5I I I• Adsorption Ifr immersion4.01•0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.18 Defining the wetted frame for Berea 300. ‘S’ indicates the regionof Sw that is associated with surface water. ‘B’ indicates the regionof Sw that is associated with Bulk water effects.103FFigure 3.19 Pore saturation scenario during drainage. A to D: Sequential drainage oflarge and intermediate aspect ratio pores; D to E: Drainage of connectedwater in partially drained pores; E to F: drainage of cracks and disconnectedsurface water. Black=water, White= air.A B CD E103[cLE::;:II....I.FFigure 3.19 Pore saturation scenario during drainage. A to D: Sequential drainage oflarge and intermediate aspect ratio pores; D to E: Drainage of connectedwater in partially drained pores; E to F: drainage of cracks and disconnectedsurface water. Black=water, White= air.A B CD E104AFFigure 3.20 Pore saturation scenario during imbibition. A to B: Grain contacts fill andsurface moisture accumulates; B to C: Pore bodies fill to a partially saturatedstate; D to F: The remainder of gas in pore bodies is replace sequentiallywith water. Black=water, White= air.B CD E105The model and data are compared for imbibition and drainage cycles in Figures 3.21and 3.22 for the bulk and shear moduli respectively. The model and data are very similar inform: both the hysteresis at high saturation levels and the drop in moduli at low saturations—-are reproduced in the model. The most-notable difference is in the absolute magnitude of-the-shear modulus, which is higher in the model than it is in the data. Such a difference is notsurprising since the rock used to derive the pore spectrum had a much lower porosity, which.suggests that differences in the mechanical behavior of the pore structure might also exist.The interpretations of the fluid distributions in the hydrophobic samples will now beevaluated using numerical modeling. The division of Sw into surface and bulk water isshown in Figure 3.23. The proposed scenario for drainage is the same as that for thehydrophilic case. Large and intermediate aspect ratio pores, representing pore bodies, aredrained in the upper ranges of saturation. At low saturations the lowest aspect ratio poresdrain. In contrast, the model for imbibition, shown in Figure 3.24, begins by sequentially andcompletely filling intermediate and high aspect ratio pores. The lowest aspect ratio pores,representing grain contacts, are the last to fill with water. Since the treatment process alsoaffected magnitude of moduli decreases associated with the adsorption of water to the dryrocks, the solid matrix values were also changed. The bulk and shear solid matrix valueswere reduced by 31 and 27 percent respectively, which are close to the observed reductionsin moduli as water was adsorbed to the dry rock.The model and data are compared in Figure 3.25 for the bulk modulus and 3.26 forthe shear modulus. As with the water-wet rocks, the form of the data is reproduced by themodel. The most notable difference is that the flat portion of the drainage curve in the model,between Sw = 0.20 and 0.50, is a downward slope in the data. Also, the absolute magnitudesare higher in the model, particularly of the shear modulus. Overall however, the agreement inform, which was the primary objective of the modeling process, is quite good.106181614Decreasing Sa12g Saturation4.2 • • • •0.0 0.2 0.4 0.6 0.8 1.0Sw181614ooo12 x10 000xx8OO00°)jDQ X X6’ I 0 Decreasing Saturation4 I X Increasing Saturation2 • • • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.21. Comparison of bulk moduli model and data for hydrophilic Berea 300.Top: model; Bottom: data.107pcrj)0 Decreasing SaturationX Increasing Saturation6.56.0’ Decreasing Saturation[Zgsaation5.0’4.5’ . I0.0 0.2 0.4 0.6 0.8Sw1.05.55.04.5,4.0’)00000008 “Cx xCl)xx3.5, . I0.0 0.2I I — I0.4 0.6 0.8 1.0SwFigure 3.22. Comparison of shear moduli model and data for hydrophiic Berea 300.Top: model; Bottom: data.Cl)S4j108161412108‘B1 000000I 00I 000I 00xxxxl2ptcct xxxx640 Deciasing Saturationx Increasing Saturation8.. I • I • I •0.0 0.2 0.4 0.6 0.8 1.0SwSI4.BI .•••••••7.nB B B Bji cB. Decreasing SaturationI Increasing SaturationS - I I - I0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.23 Defining the wetted frame for Berea 300. ‘S’ indicates the regionof Sw that is associated with surface water. ‘B’ indicates the regionof Sw that is associated with bulk water effects.109Figure 3.24 Pore saturation scenario in hydrophobic sample during imbibition.A to D: Filling pore bodies with water, sequentially and fully;D to E: Grain contacts filling with water. Black=water, White= air.A B CD E11016 -—— %__hicreasing Saturation4.I • I • I •0.0 0.2 0.4 0.6 0.8 1.0Sw16000000014 0o012 000000000—xxxxio.>D0000:o) x8 X—0 Decreasing Saturation I6 X Increasing Saturation• • •0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.25. Comparison of bulk moduli model and data for hydrophobic Berea 300.Top: model; Bottom: data.111Cl)8.58.07.57.06.56.05.5.—0.07.60.2 0.4Sw0.6 0.80 Decreasing SaturationX Increasing Saturation1.05. I • I • I • I0.0 0.2 0.4 0.6 0.8 1.0SwFigure 3.26. Comparison of shear moduli model and data for hydrophobic Berea 300.Top: model; Bottom: data.Decreasing SaturationIncreasing Saturation00000000000000000000000000X xX’Scx x xx x.112The agreement between the form of the data and models for both the hydrophilic andhydrophobic samples indicate that the proposed fluid distribution scenarios are plausible.When interpreting the results of any model it is important to recognize its limitations.In this model, the two primary limitatiausare-the assumption that the effects of fluidcommunication will be negligible and the potential oversimplification of pore spacemicrogeometry. While it is not the focus of this thesis to evaluate the general validity of theseassumptions, it would be appropriate to examine how these factors may affect theinterpretation of the results. Two scales of fluid communication are possible: macroscopicflow, occurring at a scale much larger than the average pore size, and local flow, occurring atthe pore scale due to pore-scale pressure gradients. The recent work of Akbar et al. (1993)and Gist (1994) suggests that macroscopic flow would not occur at ultrasonic frequencies.The impact of local flow is more difficult to evaluate, since it is dependent upon the specificgeomethes of the pore space.The drop in moduli at lower saturation levels was attributed to the drainage of crack-like porosity at grain contacts. This was supported by the drying data, the adsorption data andby the fact that both the bulk and shear moduli were substantially altered. The fluid in theseregions therefore appears to have remained isolated from the rest of the pore space and theassumption of no fluid communication appears to be valid. Additionally, approximating thesecompliant regions by very low aspect ratio pores appears to be reasonable.The remainder of the porosity is comprised of complex pore geometries, frequentlyconsisting of rounded pore bodies that taper towards the grain boundaries. Mavko and Nur(1978) noted that very low aspect ratio ellipsoids are required to approximate the mechanicalbehavior of thin pores that are tapered at their ends. Mendoza (1987), using numericalmodeling, found that pores composed of larger, high aspect ratio bodies that were taperedtowards the edges were highly compressible. It seems plausible then that the inversion113process has approximated the mechanical behavior of these pores in terms of a combinationof intermediate and high aspect ratio ellipsiodal inclusions. Keeping these points in mind, theuse of the model to support the general interpretations that the drops in the lower ranges ofsaturation are due to the drainage of grain contacts and those in the upper ranges of saturationare due to the rest of the porosity seems justified.3.6 SummaryThe differences that were found to exist between the results for hydrophilic andhydrophobic samples indicate that wettability can have a substantial impact upon therelationship between velocities and saturation. From these results, it is apparent thatfundamental differences in the nature of fluid distributions can exist between the hydrophilicand hydrophobic samples at partial saturation levels. In the hydrophilic rocks it wasinterpreted that the cracks are the first to fill with water and the last to drain of water. As Swwas decreased, the form of the results were similar for both the hydrophilic and hydrophobicsamples. However, as Sw was increased in the hydrophobic samples, the data indicated thatwater did not fill the cracks until higher saturation levels. These data suggest that wettabilityis another factor that may have to be taken into consideration when interpreting measuredrelationships between elastic wave velocities and saturation level.1144 CONCLUSIONSThe objective of this thesis was to examine the influence of pore spacemicrogeometry, saturation history and wettability on the relationship between velocities and- water saturation level. Two specific areas of these problems were explored. In Chapter-Twa,the evaporative drying process, pore-space microgeometries and their interrelated influenceon the form of dependence of velocities on saturation level was investigatecL In ChapterThree, the effect of wettability on the relationship between velocities and saturation level wasexplored.Elastic wave velocity and drying rate measurements were made on three differentrock types as saturation level was reduced through evaporative drying. The samplespossessed three distinctly different pore-space microgeometries and exhibited three differentforms of velocity-saturation relationship. Using drying rate curves and simplified versions ofthe pores spaces based on thin section analysis, fluid distribution scenarios were proposed tomodel the form of the relationships and these were found to be in good agreement with thedata. This suggests that the application of these methods can be of considerable use ininterpreting the relationship between ultrasonic velocities measured in non-hygroscopiccapillary porous rocks during evaporative drying. The results for the limestone and Bereasandstone samples also suggest that two regions of velocity decreases were evident: one withthe drainage of pore bodies and the second associated with the removal of pendular waterwhich was held in cracks until low saturation levels.Wettability was also shown to have a significant impact on the relationship betweenvelocities and saturation history. In particular, the main difference found was that in thehydrophobic rocks, water tended to be excluded from grain contacts until higher saturationlevels during imbibition. These results suggest that wettability could be a significant factor115that may have to be taken into consideration when interpreting the results of velocities thatare measured as a function of saturation in rocks.As was noted in the introduction, the state of liquid connectivity and rock surfacewettability can be of primary importance in determining the transport properties of rocks. The - -ability to seismically monitor these properties would therefore be of considerable use.- Theresults of this study indicate that both pendular water and wettability can have a significantinfluence on measured ultrasonic elastic wave velocities. When considering the application ofthese results to the interpretation of seismic and sonic well logging surveys of partiallysaturated geologic formations a key issue is frequency dispersion. 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