Relative Permeability of Petroleum Reservoirs Authors Mehdi Honarpour Associate Professor of Petroleum Engineering Department of Petroleum Engineering Montana College of Mineral Science and Technology Butte, Montana Leonard Koederitz Professor of Petroleum Engineering Department of Petroleum Engineering University of Missouri Rolla. Missouri A. Herbert Harvey Chairman Department of Petroleum Engineering University of Missouri Rolla, Missouri @frc') CRC Press, Inc. Boca Raton, Florida PREFACE In 1856 Henry P. Darcy determined that the rate of flow of water through a sand filter could be described by the equation h,-h. q:KA -L where q represents the rate at which water flows downward through a vertical sand pack with cross-sectional area A and length L; the terms h, and h, represent hydrostatic heads at the inlet and outlet, respectively, of the sand filter, and K is a constant. Darcy's experiments were confined to the flow of water through sand packs which were 1007o saturated with water. Later investigators determined that Darcy's law could be modified to describe the flow of fluids other than water, and that the proportionality constant K could be replaced by k/ p, where k is a property of the porous material (permeability) and p is a property of the fluid (viscosity). With this modification, Darcy's law may be written in a more general form AS k l- dz dPl u':*LPgos-dsl where S v Distance in direction of flow, which is taken as positive Volume of flux across a unit area of the porous medium in unit time along flow path S Vertical coordinate, which is taken as positive downward Density of the fluid Gravitational acceleration Pressure gradient along S at the point to which v. refers The volumetric flux v. may be further defined as q/A, where q is the volumetric flow rate and A is the average cross-sectional area perpendicular to the lines of flow. It can be shown that the permeability term which appears in Darcy's law has units of length squared. A porous material has a permeability of I D when a single-phase fluid with a viscosity of I cP completely saturates the pore space of the medium and will flow through it under viscous flow at the rate of I cm3/sec/cm2 cross-sectional area under a pressure gradient of 1 atm/cm. It is important to note the requirement that the flowing fluid must completely saturate the porous medium. Since this condition is seldom met in a hydrocarbon reservoir, it is evident that further modification of Darcy's law is needed if the law is to be applied to the flow of fluids in an oil or gas reservoir. A more useful form of Darcy's law can be obtained if we assurne that a rock which contains more than one fluid has an effective permeability to each fluid phase and that the effective permeability to each fluid is a function of its percentage saturation. The effective permeability of a rock to a fluid with which it is 1007.o saturated is equal to the absolute permeability of the rock. Effective permeability to each fluid phase is considered to be independent of the other fluid phases and the phases are considered to be immiscible. If we define relative permeability as the ratio of effective permeability to absolute perme- ability, Darcy's law may be restated for a system which contains three fluid phases as tirllows: Z p g D dP dS ,t Ir l5 r ''J.: ntJtCnal i\ :.,'nrhlc cl'lirfl - : F)n\lbilit\ .\ l'lllcn c()n5enl I Vo.:T(0.,*K-*) V*.:*(o-'13-t) Vo,:H(o-r#-k) Dr. lfcL lhc \ltntrna .{r(arrnl hrr r\rfi.Rr{le I tnLlt.rs t> nl rstn : rrrluhng drc h t-;xrlrr Ti lrrya I \lrsr.n.R.i R.{1. [}r }ri (-}rrrrrrr.n r I rcrtr rrltcrj t f- lldrr .rl e Fb t) qrtYln\ll Erjt n (tlr.run DcFtur r where the subscripts o, g, and w represent oil, gas' and water, respectively' Note that k,,,' k.", and k,* are the relative permeabilities to the three fluid phases at the respective saturations of the phases within the rock' Darcy's law is the basis for almost all calculations of fluid flow within a hydrocarbon reservoir. In order to use the law, it is necessary to determine the relative permeability of the reservoir rock to each of the fluid phases; this determination must be made throughout the range of fluid saturations that will be encountered. The problems involved in measuring and predicting relative permeability have been studied by many investigators. A summary of the major results of this research is presented in the following chapters' ltr.' \r,tc thlt k ,. re.} : r'. . .sturations Iri:' " ., hrJrttarbon tt: . - :.o.':-tlrcahilitl of I h\ ' .'.ic throughout !\. . :.: tn lllt'a\uring [r :: -: '\ ruilflrof)' Plc:. THE AUTHORS Dr. Mehdi "Matt" Honarpour is an associate professor of petroleum engineering at the Montana College of Mineral Science and Technology, Butte, Montana. Dr. Honarpour obtained his B.S., M.S., and Ph.D. in petroleum engineering from the University of Mis- souri-Rolla. He has authored many publications in the area of reservoir engineering and core analysis. Dr. Honarpour has worked as reservoir engineer, research engineer, consultant, and teacher for the past 15 years. He is a member of several professional organizations, including the Society of Petroleum Engineers of AIME, the honorary society of Sigma Xi, Pi Epsilon Tau and Phi Kappa Phi. Leonard F. Koederitz is a Professor of Petroleum Engineering at the University of Missouri-Rolla. HereceivedB.S., M.S., andPh.D. degrees fromtheUniversityof Missouri- Rolla. Dr. Koederitz has worked for Atlantic-Richfield and previously served as Department Chairman at Rolla. He has authored or co-authored several technical publications and two texts related to reservoir engineering. A. Herbert Harvey received B.S. and M.S. degrees from Colorado School of Mines and a Ph.D. degree from the University of Oklahoma. He has authored or co-authored numerous technical publications on topics related to the production of petroleum. Dr. Harvey is Chairman of both the Missouri Oil and Gas Council and the Petroleum Engineering Department at the University of Missouri-Rolla. ACKNOWLEDGMENT The authors wish to acknowledge the Society of Petroleum Engineers and the American Petroleum Institute for granting permission to use their publications. Special thanks are due J. Joseph of Flopetrol Johnston and A. Manjnath of Reservoir Inc. for their contributions and reviews throughout the writing of this book. ctf, rh t n m n l \l fslc CLI tr I u I t\ I rl ru rltr tt t u ll* tu trl t I I n I r| n.j thc Anrerican li :::.,nk. are due rr: - 'ntributions TABLE OF CONTENTS Chapter I Measurement of Rock Relative Permeability . I. Introduction. . . il. Steady-State Methods . A. Penn-State Method B. Single-Sample Dynamic Method C. Stationary Fluid Methods D. Hassler Method. E. Hafford Method F. Dispersed Feed Method . I I 1 I 2 4 4 5 5 6 8 9 10 t2 III. IV. V. VI. Unsteady- State Methods Capillary Pressure Methods Centrifuge Methods Calculation from Field Data . References. Chapter 2 Two-Phase Relative Permeability 15 I. Introduction 15 II. Rapoport and Leas ' 15 III. Gates,Lietz,andFulcher 16 IV. Fatt, Dykstra, and Burdine. 16 V. Wyllie, Sprangler, and Gardner. ' . 19 VI. Timmerman, Corey, and Johnson . .20 VII. Wahl, Torcaso, and Wyllie VIII. Brooks and Corey . . . .27 XIIX. Wyllie, Gardner, and Torcaso . . . . .29 X. Land, Wyllie, Rose, Pirson, and Boatman 30 XI. Knopp, Honarpour et al., and Hirasaki . . . . . .37 References 41 Chapter 3 Factors Affecting Two-Phase Relative Permeability 45 I. Introduction 45 il. Two-Phase Relative Permeability Curves 45 n. Effects of Saturation States 49 IV. Effects of Rock Properties 50 V. Definition and Causes of Wettability. 54 VI. DeterminationofWettability 58 A. Contact Angle Method 58 B. ImbibitionMethod. 60 C. Bureau of Mines Method 63 D. Capillarimetric Method 63 E. FractionalSurfaceAreaMethod 64 F. Dye Adsorption Method ' .64 G. Drop Test Method. . . 64 H. Methods of Bobek et al. 64 I. Magnetic Relaxation Method 64 J. Residual Saturation Methods .65 27 K. Permeability Method 65 L. Connate Water-Permeability Method 66 M. Relative Permeability Method 66 N. Relative Permeability Summation Method 61 O. Relative Permeability Ratio Method 67 P. Waterflood Method 68 a. Capillary Pressure Method . 68 R. Resistivity Index Method . 68 VII. Factors Influencing Wettability Evaluation . 68 VIII. Wettability Influence on Multiphase Flow . . .72 IX. Effects of Saturation History '74 X. Effects of Overburden Pressure ' 78 K)(I. Effects of Porosity and Permeability 79 XII. Effects of Temperature. . .82 XIII. Effects of Interfacial Tension and Density . . .82 XIV. Effects of Viscosity . .; . ' ' 83 XV. Effects of Initial Wetting-Phase Saturation 89 XVI. Effects of an Immobile Third Phase . '. 90 XVII. Effects of Other Factors . . .92 References 97 Chapter 4 Three-Phase Relative Permeability f 03 I. Introduction 103 il. DrainageRelativePermeability '.104 A. Leverett and Lewis ' . . 104 B. Corey, Rathjens, Henderson, and Wyllie 105 C. Reid. 107 D. Snell. l0g E. Donaldson and Dean . . I l0 F. Sarem 113 G. Saraf and Fatt I 15 H. WyllieandGardner .'ll5 m. Imbibition Relative Permeability 117 A. Caudle,slobod,andBrownscombe 117 B. Naar and Wygal I 17 C. Land. 120 D. SchneiderandOwens 123 E. Spronsen .' 123 IV. Probability Models . .123 V. ExperimentalConfirmation 126 U\/I. LaboratoryApparatus 127 VII. Practical Considerations for Laboratory Tests ' 132 VIII. ComparisonofModels '133 References""' """'134 Appendix Symbols. 137 Tbc I hr crth r3th\ rrl c{ehlr. \,ilUt-3ll irlurltl thc crr Itrf ft\ thc Ha ln tt thc tc. drrqlg urcfrr| fa nx A.h Tht d'er ad' Frgun nrun alrr P Thc t r alCr Ftrst .r hrs L-Tth rltc\ rlctcn rnU\\ ktt t rrcrgl tlr .i Th than TTE:N a. flt Itr lfi' rnarl ln ci r-all, thYl. 6-i 66 66 6- 6- 6,\ hs h\ 6\ -: l Chapter I MEASUREMENT OF ROCK RELATIVE PERMEABILITY I. INTRODUCTION The relative peffneability of a rock to each fluid phase can be measured in a core sample by either "steady-state" or "unsteady-state" methods. In the steady-state method, a fixed ratio of fluids is forced through the test sample until saturation and pressure equilibria are established. Numerous techniques have been successfully employed to obtain a uniform saturation. The primary concern in designing the experiment is to eliminate or reduce the saturation gradient which is caused by capillary pressure effects at the outflow boundary of the core. Steady-state methods are preferred to unsteady-state methods by some investigators for rocks of intermediate wettability,' although some difficulty has been reported in applying the Hassler steady-state method to this type of rock.2 ln the capillary pressure method, only the nonwetting phase is injected into the core during the test. This fluid displaces the wetting phase and the saturations of both fluids change throughout the test. Unsteady-state techniques are now employed for most laboratory meas- urements of relative permeability.3 Some of the more commonly used laboratory methods for measuring relative perrneability are described below. II. STEADY-STATE METHODS A. Penn-State Method This steady-state method for measuring relative perrneability was designed by Morse et al.a and later modified by Osoba et aI.,5 Henderson and Yuster,6 Caudle et a1.,7 and Geffen et al.8 The version of the apparatus which was described by Geffen et al., is illustrated by Figure l. In order to reduce end effects due to capillary forces, the sample to be tested is mounted between two rock samples which are similar to the test sample. This arrangement also promotes thorough mixing of the two fluid phases before they enter the test sample. The laboratory procedure is begun by saturating the sample with one fluid phase (such as water) and adjusting the flow rate of this phase through the sample until a predetermined pressure gradient is obtained. Injection of a second phase (such as a gas) is then begun at a low rate and flow of the first phase is reduced slightly so that the pressure differential across the system remains constant. After an equilibrium condition is reached, the two flow rates are recorded and the percentage saturation of each phase within the test sample is determined by removing the test sample from the assernbly and weighing it. This procedure introduces a possible source of experimental error, since a small amount of fluid may be lost because of gas expansion and evaporation. One authority recommends that the core be wgighed under oil, eliminating the problem of obtaining the same amount of liquid film on the surface of the core for each weighing.3 The estimation of water saturation by measuring electric resistivity is a faster procedure than weighing the core. However, the accuracy of saturations obtained by a resistivity measurement is questionable, since resistivity can be influenced by fluid distribution as well as fluid saturations. The four-electrode assembly which is illustrated by Figure I was used to investigate water saturation distribution and to determine when flow equilibrium has been attained. Other methods which have been used for in situ determination of fluid saturation in cores include measurement of electric capacitance, nuclear magnetic resonance, neutron scattering, X-ray absorption, gamma-ray absorption, volumetric balance, vacuum distilla- tion, and microwave techniques. .le Relative Permeabilin of Petroleum Reservoirs El-ectrodes Outl-et Differential Pressure Taps Inlet Inlet FIGURE l. Three-section core assembly.8 After fluid saturation in the core has been determined, the Penn-State apparatus is reas- sembled, a new equilibrium condition is established at a higher flow rate for the second phase, and fluid saturations are determined as previously described. This procedure is re- peated sequentially at higher saturations of the second phase until the complete relative permeability curve has been established. The Penn-State method can be used to measure relative permeability at either increasing or decreasing saturations of the wetting phase and it can be applied to both liquid-liquid and gas-liquid systems. The direction of saturation change used in the laboratory should cor- respond to field conditions. Good capillary contact between the test sample and the adjacent downstream core is essential for accurate measurements and temperature must be held constant during the test. The time required for a test to reach an equilibrium condition may be I day or more.3 B. Single-Sample Dynamic Method This technique for steady-state measurement of relative permeability was developed by Richardson et al.,e Josendal et al.,ro and Loomis and Crowell.ttThe apparatus and exper- imental procedure differ from those used with the Penn-State technique primarily in the handling of end effects. Rather than using a test sample mounted between two core samples (as illustrated by Figure 1), the two fluid phases are injected simultaneously through a single core. End effects are minimized by using relatively high flow rates, so the region of high wetting-phase saturation at the outlet face of the core is small. The theory which was presented by Richardson et al. for describing the saturation distribution within the core may be de- veloped as follows. From Darcy's law, the flow of two phases through a horizontal linear system can be described by the equations -dP*, : Q*, F*,dL k*, A tL* tl rEC I rr rrl (l) kir F. rfi cFr g:f rdt tqr ll er G f,F: 5X and ,n Q. Fr" dL -dPn: =i^ Q) where the subscripts wt and n denote the wetting and nonwetting phases, respectively. From the definition of capillary pressure, P", it follows that 1.0 o a 0 lel . ICsr- J ii- *i'trDd CE'.i-:; ir [C- plcir :Jtrtr\r' 3T .:'. :t.t.tIlS id ;:J end I ri,' J r-trf- J li. ; ., .: ' .ric rll nr-' \' hcld tr\. : - mJ\ lc.l. ,i*-J b) ! -::- C\F'r-f- D r:. ' rn thC Cr':; :::lplCr BJ ,,.:l'l!ls' f3h " : nrsh Jil. l-: s'ntcrj ! n-:. re' Jc- iz '. a(rr 5 10 15 20 25 Distance from Outflow Face, cffi FIGURE 2. Comparison of saturation gradients at low flow rate.e dP.:dP dP*, These three equations may be combined to obtain qP. : /Q*, Fr,*, _ 9"U=\ / o dL \ k*, kn // where dP"/dL is the capillary pressure gradient within the core. Since dP. : dP. ds*, dL dS*, dL it is evident that (3) (4) (s) (6) dS*, dL | /Q*, Fr*, Q"p.\ I :A\ k* - L" /op.rus* ,lt Richardson et al. concluded from experimental evidence that the nonwetting phase sat- uration at the discharge end of the core was at the equilibrium value, (i.e., the saturation at which the phase becomes mobile). With this boundary condition, Equation 6 can be integrated graphically to yield the distribution of wetting phase saturation throughout the core. If the flow rate is sufficiently high, the calculation indicates that this saturation is virtually constant from the inlet face to a region a few centimeters from the outlet. Within this region the wetting phase saturation increases to the equilibrium value at the outlet face. Both calculations and experimental evidence show that the region of high wetting-phase saturation at the discharge end of the core is larger at low flow rates than at high rates. Figure 2 illustrates the saturation distribution for a low flow rate and Figure 3 shows the distribution at a higher rate. a r _l Ftt', c.r From \o \.o >{^ / -i- -o- Theoretical saturation gradient f nf low f ace 1> [...]... empirical or laboratory techniques Poor agreementbetween relative permeability determined from production data and from may include laboratory experiments is not uncommon The causesof these discrepancies the following: t2 l 2 3 Relative Permeability of Petroleum Reservoirs The core on which relative permeability is measuredmay not be representative the of reservoir in regard to such factors as fluid distributions,... the calculation permeability of therefrom, Trans AIME, 186, 39 1949 30 Fatt, I and Dyksta, H., ,Relative permeabilitystudies,Trans AIME, 192,41, 1951 31 Burdine, N T., RelativePermeability Calculations from Pore Size DistributionData, Trans AIME, lg8, 7t,1953 l5 Chapter 2 TWO-PHASE RELATIVE PERMEABILITY I INTRODUCTION to Direct experimentalmeasurement determinerelative permeabilityof porous rock has... equation equation.2 parameters rock permeabilitywas the Kozeny-Carmen to measured the expresses permeabilityof a porousmaterialas a function of the productof the effective throughwhich path lengthof the flowing fluid and the meanhydraulicradiusof the channels the fluid flows Purcell3formulated an equation for the permeability of a porous system in terms of the the curve of that systemby simply considering... influenceof capillary number (ratio of viscousto the capillary forces)on two-phaseoil-water relativepermeabilitycurves IV FATT, DYKSTRA,AND BURDINE Fatt and Dykstrarr developedan expression relativepermeabilityfollowing the basic for methodof Purcell for calculatingthe permeabilityof a porousmedium They considered a lithology factor (a correction for deviation of the path length from the length of the... has been recordedin petroleumrelatedliterature.However, empirical methodsfor deterlong mining relative permeabilityare becomingmore widely used, particularlywith the advent of digital reservoirsimulators.The generalshapeof the relative permeabilitycurves may k.* : A(S*)'; k , : B(l - S*)"'; where A, by be approximated the following equations: B n and m are constants Most relative permeability mathematicalmodels... extensiveconsolidationis present Application of Corey's equationpermits oil relative permeabilityto be calculatedfrom are of measurements gas relative permeability. Since k., measurements easily made while are k.o measurements made with difficulty, Corey's equationis quite useful The procedure of involves the measurement gas relative permeability at severalvalues of gas saturationin and then performing the... o.lo FIGURE 7 2 3 4 5 6 Data Data of Vlelge points o.20 0.30 0.40 0.50 0.60 0.70 Sg Example of the use of the Corey equations.rl Preparea tabulation of k., vs So" for values of k,, ranging from 0.001 to 0.99 in stepwisefashion Determinevaluesof So"for eachexperimental valueof k., by usingthe above-described tabulation Plot these values of So againstthe values of S" coffespondingto the k., values on... constantduring the test.2 Relative Permeabilin of Petroleum Reservoirs Laboratory equipment is available for making the unsteady-state measurements under simulated reservoirconditions.2a In addition to the JBN method, several alternative techniquesfor determining relative permeabilityfrom unsteady-state data have been proposed.Saraf and McCaffery2detest veloped a procedurefor obtainingrelative permeabilitycurves... work of Purcell3and Burdiner3into a form that has considerable for utility and is widely accepted its simplicity It requireslimited input data (sinceresidual neededto developa set of relativepermeabilitycurves)and is saturation the only parameter it is fairly accuratefor consolidatedporous media with intergranularporosity Corey's equationsare often used for calculationof relative permeabilityin reservoirssubjectto... media; however, values of S,, were found to be greaterthan unity when there was stratificationperpendicular the direction of flow and to less than unity in the presence stratificationparallel to the direction of flow They also of concludedthat oil relative permeabilities were less sensitiveto stratificationthan the gas relativepermeabilities The gas-oil relative permeabilityequationis often used for testing, . Relative Permeability of Petroleum Reservoirs Authors Mehdi Honarpour Associate Professor of Petroleum Engineering Department of Petroleum Engineering Montana College of Mineral. of these discrepancies may include the following: t2 Relative Permeability of Petroleum Reservoirs l. The core on which relative permeability is measured may not be representative of. Montana Leonard Koederitz Professor of Petroleum Engineering Department of Petroleum Engineering University of Missouri Rolla. Missouri A. Herbert Harvey Chairman Department of Petroleum Engineering University