THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DISPERSION COEFFICIENTS FOR GASES FLOWING IN CONSOLIDATED POROUS MEDIA Max Wilhelm Legatski A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1966 February, 1966 IP-729

ACKNOWLEDGEMENTS The author wishes to express his appreciation to the many people who have contributed to this work. Their advice, interest, patience, and moral support have been instrumental in the completion of this manuscript. Professor Donald L. Katz, as chairman of the committee, has been generous with his advice and assistance. His astute guidance has enhanced this manuscript. Associate Professor Robert H. Kadlec has contributed unselfishly of his time and ability. His guidance during critical stages of the experimental work was an important factor in the development of the experimental and mathematical techniques ultimately employed. Professor M. Rasin Tek has contributed materially to this dissertation by his donation of both time and moral support. Associate Professor Herman Merte, Jr., and Assistant Professor James O. Wilkes have offered helpful criticisms and suggestions. Messrs. Cleatis Bolen, John Wurster, Peter Severn, Robert Reed, Douglas Connell, and Elmer Darling of the Chemical and Metallurgical Engineering Department provided valuable assistance in the construction of experimental apparatus. The Michigan Gas Association has made this manuscript possible by providing a generous fellowship during four years of graduate study. 111

Mr. William H. McDougald provided valuable assistance in constructing an instrumentation and control system for the experimental apparatus. Continental Oil Company and the United States Bureau of Mines provided core samples used in the experimental work. Professor G. Brymer Williams, Mr. L. Karl Legatski, Mr. L. Kent Thomas, and Mr. Finis E. Carleton, III have provided valued advice and moral support. The Industry Program of the College of Engineering has done an excellant job of producing this manuscript in its final form. My parents have made this work possible by their unselfish financial and moral support during the past 26 years. My wife, Pat, and my son, Scott, to whom this manuscript is dedicated, are due particular gratitude. Their inspiration, confidence, moral support, and patience have made this dissertation not only possible, but worthwhile. V'

TABLE OF CONTENTS ACKNOWLEDGEMENTS............... ii ABSTRACT.................... iv LIST OF FIGURES................. viii LIST OF TABLES........... x NOMENCLATURE,..................... xi I. INTRODUCTION.................. 1 II. DISPERSION COEFFICIENTS; THEIR USE AND LIMITATIONS O o 3 A. The Continuity Equations.......... 3 B. Longitudinal and Transverse Dispersion... 4 C. The Capacitance Model............ 5 D. Mixing Cell Model and Dispersion Model. o... 8 E. Linearity................ e. 8 F, Stability............ 9 III. THEORY; PREVIOUS RESEARCH............... 13 A. Dispersion Mechanisms........ 5 B. Parameters Affecting Experimental Results. o2 IV. EXPERIMENTAL PROGRAM.............. 29 Ao Measurement of Physical Properties.. o..o. 32 B. Equipment................. 35 C. Calibration Procedures.......... 5 D. Operating Procedures.......... E. Calculation Procedures............ V. EXPERIMENTAL RESULTS................. A. Short, Dry Samples... O........ B. Short Cores with Immobile Phase a..... a.. 76 C. Long, Dry Samples........,..... 8. VI. SUMMARY AND CONCLUSIONS............... 86 BIBLIOGRAPHY...................... o. 94 APPENDICES e...................... 99 I. SAMPLE CALCULATIONS............... 99 II.o SUPPLIEMENTARY CALCULATIONS............... 112 III. FIELD APPLICATIONS...........0....... 115 vii

LIST OF FIGURES Figure Page 1 The Capacitance Model.................. 6 2 Development of Unstable Fingers when Water Displaces Oil from a Porous Medium...... 10 3 Typical Dispersion Characteristics for Porous Media o o 16 4 Peclet Numbers for an Aqueous System.......... 17 5 Peclet Numbers for a Gaseous System....... 18 6 Comparison of Calculated and Experimental Data..... 19 7 Peclet Numbers for Longitudinal Dispersion in Turbulent Flow. o. o o o. o... o 28 8 Dispersion Model.................. 0 9 Experimental Apparatus................. 36 10 High Pressure Manometer......... 3~ 11 Analog Computer..... 41 12 Schematic Analog Computer Flow Diagram..........2 13 Analog Computer Program................ 43 14 Analog Computer Symbols................. 41 15 Typical Voltage Profiles o.............. -7 16 Thermal Conductivity of Helium-Nitrogen Mixtures.... 49 17 Correction for Helium Concentration Profile.... 51 18 Experimental Apparatus.............. 52 19 Evaluation of Mixing Within Porous Medium....... 60 20 (a-i) Dispersion Characteristics as a Function of Flow Rate............ o o o o 64 viii

LIST OF FIGURES CONT'D Figure Page 21 (a,b) Dispersion Characteristics as a Function of Flow Rate, Immobile Phase in Interstices......... 78 22 (a,b) Dispersion Characteristics as a Function of Flow Rate, Immobile Phase in Interstices....... 80 23 Effect of Core Length upon Computed Dispersion Characteristics......... 84 24 Characteristic Length vso Permeability.......... 88 25 Characteristic Length vs. Resistivity Factor....... 26 Exponent vso Permeability.............. 90 27 Exponent vso Resistivity Factor............. 91 LIST OF FIGURES - APPENDICES I-1 Turbulence Factor Plot.............. 102 I-2 Variance (sec2) vs. Flow Rate (cfm), Core BA-2 = 235.4%, AL = 3077", Bandera Sandstone......... 109 I-3 Inlet Pressure Vs. Flow Rate, Core BA-2 + BA-2A..... 110 II-1 Rotameter Calibration for Nitrogen at 70~F, 14o7 psia,.o 114 ix

LIST OF TABLES Table Page I. Specimen Types................... 74 II. Specimen Properties...................75 III. Effect of Immobile Phase upon Dispersion Characteristics.. 82

NOMENCLATURE Symbol Definition -- ~cross sectional area (L2) 0% particle radius (L) C - capacitance (microfarads) or concentration (vol. fr.) iCA - concentration of component A d: particle diameter (L) DDo - longitudinal or transverse dispersion coefficient (L2T-1) ~DoDgor molecular diffusion coefficient (L2T-1) D!~ Der longitudinal dispersion coefficient (L T-1) at D-= transverse dispersion coefficient (L2T-1) $ = fraction of void volume inaccessible to bulk flow F - electrical resistivity factor K' permeability (L2) L= specimen length (L) kt o ck) Mlk:- Dio ~t/d k(T%-) MK = (scale factor). M I( ( k) IA = molecular weight - power dependence of dispersion upon Peclet number p pressure (FL-2) - Peclet Number, dpU/ 0 or tdP6/~o - injection rate (L3T-1) gas constant or resistance (megohm)

Symbol Definition 1K Reynolds Number'C = radius (L) temperature t = time (T) t - t es-/t. ( T) t mean residence time (T) Ak = interstitial velocity (LT-1 ) V/ ~vector velocity (LT-1) V - void volume (L3)? _- linear (superficial) velocity (LT-1) 9 ~ volumetric flow rate (L3T1) mixing cell volume (L3) VI = mass flow rate -- compressibility factor ".~) ~- rectangular coordinates cylindrical coordinates Qr\4,: spherical coordinates ~ _ turbulence factor (L ) = _ void fraction, or porosity - density (ML- )' viscosity (ML1T ) T/: mean residence time (T) packing, or inhomogeneity, factor - mass transfer coefficient 52: variance of f(t) about mean,/ (TL) a- relative variance change, A (IL;) particle sphericity xii

Io INTRODUCTION Mixing of miscible fluids flowing through porous media has been studied extensively by the petroleum industry and by the chemical industry. Investigators in these fields have been concerned with the behavior of reservoir fluids during miscible secondary recovery operations and with the behavior of fluids flowing through packed columns. While the petroleum industry has studied dispersion in liquid systems flowing through both consolidated and unconsolidated porous media, the chemical industry has investigated both liquid and gaseous systems in unconsolidated porous media. The gas storage industry, however, is interested in the behavior of gaseous systems flowing through consolidated porous media, a process which has not been studied extensively to date. In one case of interest, the United States Bureau of Mines is storing vast quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15% of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure, and rate of movement. The natural gas companies of the midwest also have an important stake in the understanding of gas mixing. Natural gas from the southwest is commonly stored during the summer in depleted reservoirs in Michigan or aquifers in Illinois for later use during winter peak loads. A portion of the gas stored in such reservoirs will be entrapped and lost when the reservoir is finally abandoned. If it were -N1

-2possible to use a cheaper gas (e.g. flue gas) as a cushion, the effective capital investment for the operation would be reduced considerablyo There are also many occasions when it would be helpful to be able to predict gas mixing in the reservoir, such as in evaluating the stripping of liquids when recycling gas through the reservoir. It was the purpose of this research to develop an understanding of gaseous mixing for flow through consolidated porous media, using both the existing literature and experimental measurements. Emphasis was placed upon an experimental technique for conveniently measuring the dispersion coefficient for a variety of gas systems and porous media and a range of flow rates.

II. DISPERSION COEFFICIENTS; THEIR USE AND LIMITATIONS It is pertinent, before discussing the experimental evaluation of dispersion coefficients, to survey the significance of these coefficients and the limitations implied by their use. It is the purpose if this section to make such a survey, and to briefly describe the ultimate use of dispersion coefficients. A. The Continuity Equations For mass transfer due to bulk flow and diffusion, with no generation of a component by chemical reaction, the general equations for the conservation of mass in various coordinate systems may be written as follows: 1o General Accumulation = bulk flow terms + diffusion terms (1) 2. Rectangular Coordinates 3. Cylindrical Coordinates'Y':_(,p (-;- a-~ I-~ "'a" 4. Spherical Coordinates 50 Vector Notation -3

where Cl ( = concentration of component A, lb/cu. ft. = rectangular coordinate directions, fto DP, = diffusivity of component I in solution, ft2/hr. 9C ) 5 = spherical coordinates, ft. C(91 = cylindrical coordinates, ft. &t -- time, hours d~ < = bulk flow velocity component, fto/hro = vector velocity (ftj./2r ), ftj/hr. = vector operator In the usual context of molecular diffusion only, the symbol Ad represents the molecular diffusivity of component 4. For fluids flowing through a porous medium, the symbol DA may be used to represent a dispersion coefficient; the term "dispersion," as opposed to "diffusion," implies flow conditions, Further, the subscript "'" may be neglected if the identity of the diffusing gas is understoodo Bo Longitudinal and Transverse Dispersion It is also necessary to distinguish between a "transverse dispersion coefficient" and a "longitudinal dispersion coefficient" for flow through porous mediao it has been found that the coefficient characterizing dispersion in the direction of bulk flow (longitudinal) can be an order of magnitude greater than the coefficient characterizing mixing perpendicular to the direction of flow (transverse)(lO,45) We shall use the subscripts'I" and "t" to denote longitudinal and transverse dispersion, respectively; this work is concerned with the determination of longitudinal dispersion coefficientso

-5To summarize, the nomenclature used throughout this report is as follows: = molecular diffusion coefficient in the absence of a porous matrix, Dt = transverse dispersion coefficient, and 1D, = longitudinal dispersion coefficient. The symbols g' and < are also found in the literature representing dispersion coefficients. For mass transport in a single direction in a porous medium, with bulk flow, Equation 2 for rectangular coordinates reduces to C o The Capacitance Model C. The Capacitance Model Equations 2-6 imply an "ideal" displacement; ie. the diffusion term in each equation represents mass transport obeying Fick's Law of Diffusion. Fick's Law assumes that there is no gross bypassing of one fluid by another, and that there are no stagnant pockets of gas in the system under consideration. These assumptions are not always valid for flow through porous media, and it is important to recognize the limitations upon Equations 2-6~ Figure 1, "Capacitance Model," schematically illustrates a capillary tube having a volume which is inaccessible to bulk flowo Flow through a capillary system of this sort would not be expected to obey Fick's Law, although under certain conditions Fick's Law may hold

-6f = FRACTION OF VOID --— BULK FLOW INACCESSABLE TO BULK FLOW 0fI= MASS TRANSFER COEFFICIENT Figure 1. Capacitance Model.

-7as a valid approximation. The capacitance model has been studied by Coats and Smith (18) Dispersion in a dead-end capillary system would necessarily be characterized by three parameters. In addition to the dispersion coefficient, l)D, which characterized mixing in a simple capillary model, the dead-end capillary model would assume that a certain fraction, t, of the total void volume of the system was inaccessible to bulk flow, and that a mass transfer coefficient, 6-', existed to describe molecular transport between the inaccessible portion and the mainstream of flow. Thus the capacitance model would be expected to describe mixing better than a simple linear (Fick's Law) model by virtue of the additional parameters characterizing the system. We can make two observations which indicate that Fick's Law is a valid approximation for most of the systems studied in this research program. First, we would expect the void fraction inaccessible to bulk flow to be small for sandstones. For limestones and dolomites, however, considerable error might be encountered by neglecting the inaccessible void fraction,,. Second, for gaseous systems we would expect the mass transfer coefficient, r I, characterizing molecular transport between the inaccessible volume and the mainstream flow, to be large. Thus, except for liquid systems or gaseous systems at high pressures, we may expect the capacitance model to contribute little to our description of dispersion in a porous medium. For the limiting case of 4-( and' C-'. the capacitance model would reduce to a Fick's Law model.

-8D. Mixing Cell Model and Dispersion Model A number of authors(6'40'43) have pursued the mixing problem not in terms of the so called "dispersion model" described by Equations 1-6, but in terms of a "mixing cell model." This model supposes that a porous medium is constructed of a large number of small mixing chambers, and that the concentration of the diffusing component within each mixing chamber is uniform. For a series of -I perfect mixing cells, Aris and Amundson(2) have shown that the concentration of diffusing component in the mixing chamber is given by where, injection rate, - time, and'5I = volume of each cell. This model is discussed further in Section III of this manuscript. Eo Linearity A second assumption of Equations 2-6 is that the dispersion coefficient, Ak', is independent of the concentration of A, i.e. that the mixing equation is "linear." This has been shown by several authors (l721) to be an oversimplification, but the concentration dependence of 1r, is not large enough to contribute significant error to either the present research or the application of Equations 2-6 to field calculations(

-9Fo Stability The most serious limitation upon the assumption of Fick's Law mixing in a porous medium is due to instabilities inherent in some systems because of density and viscosity differences within the fluids and/or inhomogeneities within the porous medium. A stable displacement is one in which the concentration of a component is uniform in a plane perpendicular to the direction of flow; an unstable displacement is one in which "fingers" of one component advance into the other component. Viscous fingering of immiscible fluids in porous media, such as is illustrated in Figure 2, "Development of Unstable Fingers when Water Displaces Oil from a Porous Medium," is a common phenomenon with which the gas storage industry is acquainted. Miscible components may finger in a like manner, although the finger development is not apt to be as extreme, and there is no readily discernable interface between the fluids. The development of such fingers constitutes "gross bypassing" of one fluid by another, as previously mentioned, and hence is in violation of the assumption of Fick's Law model. The formation of fingers within a miscible system is known as a conditional instability, since these fingers must eventually merge due to transverse dispersion. That is, if two fingers are traveling in a nearly parallel direction, transverse or "sideways" mixing will eventually bring the two fingers together, at which time the displacement will again be stable. We thus consider transverse dispersion to be advantageous and longitudinal dispersion to be detrimental in a gaseous displacement in which a minimum of mixing is desired.

-10 - N W.- 2=. 3%o N 13%; W. = 23% N W- 6. 0% N - 3%; W 34-i% N W 9. 5N 34%; W 18o0% P 1 P i p 1 p i Figure 2. Development of Unstable Fingers when Water Displaces Oil from a Porous Medium (from van Meurs, Trans. AIME, (Petr. ), 2153 103, (1958). Np = oil production in pore volumes Wi = cumulative water injection in pore volumes. )

-11There are several factors to be considered in determining whether a gaseous displacement will be stable. The most obvious criterion is reservoir homogeneity; a formation having streaks or layers of varying permeability is very likely to initiate and propagate fingering during displacement. Both density and viscosity differences are factors affecting stability which can be either favorable or unfavorable. If helium displaces a more dense gas such as nitrogen in a downward direction, for example, a stable displacement is apt to result because buoyancy effects will work to damp out fingers as they are formed. On the other hand, if helium displaces nitrogen in an upward direction, or if nitrogen displaces helium in a downward direction, buoyancy effects will tend to increase fingering. Similarly, a less viscous fluid will tend to finger into a more viscous fluid during a displacement in any direction. For gases, a decrease in density corresponds to an ~ncreasein viscosity. For the Rxample of helium displacing nitrogen in a downward direction in a homogeneous medium, we are insured of stability since both viscous effects and density effects will act to promote this stability. For nitrogen displacing helium in an upward direction, a favorable density difference will encourage stability, but an unfavorable viscosity difference wili promote fingering. In this case, we must look more closely at the criteria for stability in order to determine the displacement behavior.

-12Dumore(24), Schowalter(50), Perrine(45) and Perkins (44) have studied the stability problem for liquids. For liquids, a mathematical analysis may be slightly different than for gases, since a low liquid density corresponds to a low viscosity. Schowalter(50) uses a perturbation analysis to show that, for liquids, stability can be predicted from several dimensionless parameters, in agreement with Perrine(45) These authors assume an exponential relationship between viscosity and composition, which is a valid approximation for liquids only. Stability criteria are mentioned further in Section III of this report. A detailed analysis of stability problems for gaseous systems is, however, beyond the scope of this thesis work; we shall endeavor only to mention qualitatively the variables which must be considered.

III. THEORY; PREVIOUS RESEARCH A comprehensive review of diffusion and dispersion in porous (43) media has been presented by Perkins and Johnston. These authors have prepared an extensive list of references on the subject, only. part of which is included in the present bibliography. These authors point out the difficulty in attempting to treat the dispersion and diffusion coefficients as functions of concentration; they also point out that it is usually possible to treat these coefficients as constants by selecting an "effective average," e.g. by using the diffusion coefficient for a 50-50 per cent mixture. Crank (21) has considered the coefficients as functions of concentration. Perkins and Johnston analyze the capillary model, among others. In this model, a porous medium is represented by a bundle of capillary tubes. This is not a very good representation of a porous rock, so that it does not seem worthwhile to expound upon their results in this manuscript. de Jong(23) and Saffman(49 ) have also studied a random network of capillary tubes, but their results still do not agree with those for porous rock. The simplest analysis which provides useful information for the prediction of dispersion coefficients is the electrical resistivity analog, proposed by Brigham, et al. (12), van der Poel(5), and Grane.Recognizing that there is an anal and Gardner(30). Recognizing that there is an analogy between electrical conductivity and diffusion in porous media, it follows that D / at ~F~~~~~~~~~ $ (8) -135

-14for either cemented rocks or unconsolidated packs for very low flow rates, since the law governing electrical conductivity is completely analogous to Equation 6. Equation 8 has been substantiated by Kravik and Bissey(34) for gas flow through Berea sandstone, using a variety of gas systems with mobility ratios ranging from 0.5 to 2.0. In the above equation, = longitudinal or transverse dispersion coefficient within the porous medium (these are identical for extremely low flow rates), or "apparent diffusion coefficient," _D~ = molecular diffusion coefficient in the absence of a porous matrix, F = electrical resistivity factor, and a = fractional porosity of the medium. Equation 8 implies that mixing is defined in terms of the area open to diffusion; if defined in terms of the total cross-sectional area of the sample, then we would write / (8a) For laminar flow it is common practice to correlate the ratio of the effective dispersion coefficient to the molecular diffusion coefficient, D/Do, as a function of a Peclet number, U (characteristic length)/Do. Perkins and Johnston(45) suggest the relationship

-15which is shown in Figure 5, "Typical Dispersion Characteristics for Porous Media." In Equation 9, TG represents a packing or inhomogeneity factor. Other authors prefer a more general expression of the type Do K ) ~t > Io (9a) in which { = a characteristic length such as pore diameter or grain diameter, = a proportionality constant, and.'~ = the power dependence of dispersion upon velocity. Brigham, et.al. (12) states that the proportionality constant, iX, of Equation 9a is a function of the mobility ratio of the flowing fluids, as well as the physical properties of the porous medium. The practice followed in this manuscript is to assume dispersion characteristics of the type __ _ t( D (9b) Determination of the parameters deC~ and f then completely characterizes the mixing properties of a porous medium for a given binary system. A. Dispersion Mechanisms We refer in particular. to Figures 4 and 5 in attempting to analyze the regions of different flow behavior. As the flow rate is increased for any two miscible fluids flowing in a given porous medium,

100 /' /,/, / / Di~~~~~~~~~~~~~~~~~~~~~. De/ // / C`~~~~~ / I I0 - - ~~~~~ / Qz // /j~~~~~ 0.1~~~~~~~~~~~~~~~~~~~~~, ~~~~~/ 4 ~ II 10 0.1. // /10 4 / /1/1/ E C? ////// 0 -/ / FO /~~,4$: // - ~~~,// // _________- ~ ~ ___ /$/,/ EXPERWAENTAL -- _____________ ~ ~ ~ ~ ~ ~ ~ RO ____ RESEARCH // / 0.1 0.001 0.01 0.1 1.0 10 100 Uap or udpo Do Do (lo,43) Figure 3. Typical Dispersion Characteristics for Porous Media.

100 STREAM SPLITTING WITH MASS TRANSFER BY DIFFUSION | nL - aO-. 10 /7 RANSVERSE PECLET |_ TRANSITION FROM -BTURBULENCE Y | |/ NUMBER -| LAMINAR TO TURBULENT -CONTROLS b. FLOW DFFUSION / MIXING CELL z I CONTROLS THEORY (INHOMO) 5L -DIFFUSION DOES -5 2 0- 0.1 LO NGITUDINAL — w-NOT EQUALIZE dp 0.2 cm (I0 MESH) Do =I xlO CM/SEC PECLET NUMBER CON I GM/CC I/FT = 0.7 = I CP a3.5 0.01 I I I I I I 10-5 10-4 10-3 102 I I 10 102 103 10 105 dpUp= REYNOLDS NUMBER Figure 4. Peclet Numbers for an Aqueous System (after Perkins and Johnston).

100 ASSUMPTIONS: dp =0.2 cm. (10 MESH) a 10 P =1.2 xIcf3 gm/cc cr I x I~~~~~~~i-4 -— a —%,-~~~~~~~TURI3ULE?C Lr L=I1.82 x i6- POISE oy TRANSITION FROM "o"CONTROLS Do = 0.15 cm /sec LAMINAR TO TURBULENT DIF ~. FLOW z I I/F#=07 A TN 1~_ a-3.5 -TtjCjL F-CL Lrw L0%GDNP o ~~~~DIFFUSION CONTROLS j I-1 W Q 0.01 1-5 lT4 10-3 I 10 10i 10I dp Up =REYNOLDS NUMBER Figure 5. Peclet Numbers for a Gaseous System (air) (after Perkins and Johnston).

100 ASSUMPTIONS: dp =0.2 cm. u'= 1.82 x 10 POISE D = 0.15 I/F# = 0.7'r - 3.5 10 ac W I,CALCULATED 3= oo -_.o c30 0 ~~z X ~~~~o 8 oO 0 ATA OF McHENRY(40) o w I F (LONGITUDINAL, GASEOUS SYSTEM) _J L, 0.1I I I 1 10 10o 103 104 I05 dpUp = REYNOLDS NUMBER /I' Figure 6. Comparison of Calculated and Experimental Data (longitudinal disperson: gaseous system)(after Perkins and Johnston).

-20the flow undergoes a transition from laminar to turbulent, and there is a concomitant increase in the effect of convective mixing. In the following paragraphs we shall assume that the flow rate is gradually increased from a zero value to a high value and attempt to describe the various mixing regimes encountered. 1. Zero Velocity; Pure Molecular Diffusion If, at time ~- O0, a sharp interface exists between two miscible fluids at rest ( IRe O ), then molecular diffusion alone will cause the two fluids to mix. The concentration, C, of either fluid at time t will be given by,'= 2f, [/ erg ( ), (10) where X - distance measured from the initial position of the interface, and er~ = error function. For unconsolidated porous media, it may be stated as an approx(16) imation that the apparent diffusion coefficient is given by D / (11) Equation 11 is based on the approximation that a randomly chosen fluid particle moving through a porous bed is statistically probable to be moving at 450 wi+th respect to the direction of bulk flow. 2. Low Velocities; Molecular Diffusion Dominating If, as before, a sharp interface exists between the two fluids at time 0- a,but there is bulk flow of the fluids at a low velocity

-21in a direction normal to the interface, then molecular diffusion will still control the mixing. Until a Reynolds number of about 4 1 is reached for liquids, or e /o for gases, mixing will still occur by molecular diffusion alone(43) As long as convective effects do not begin to appear, the longitudinal and transverse dispersion coefficients will be identical, as indicated in Figures 4 and 5. The Peclet number will be proportional to the Reynolds number up to this point, since the velocity term is expressed in the numerator of each dimensionless variable. As the above Reynolds numbers ( O for liquids, or e 1l) 10 for gases) are slightly exceeded, mixing is best described by a combination of diffusion and mixing cell theory. When diffusion alone controlled the mixing, we could assume that the concentration of diffusing material was uniform within each pore space; as the velocity increases, there will be insufficient residence time within each pore space for the concentration to become uniform. This will at first (fkiPcxv for liquids or e(3 for gases) not affect the transverse dispersion, since there is still very little flow perpendicular to the direction of bulk flowo As these Reynolds numbers are surpassed, however, "stream splitting" will occur; ioe, there will be significant mixing effects in the transverse direction. 3. Transition Zone; Molecular and Convective Mixing Above Reynolds numbers of about 4, and up to Reynolds numbers of about 4,000, the flow will undergo a transition from the laminar

-22to the turbulent regime. Inertial effects will begin to appear, and turbulent eddies will occur within the larger pore spaces, resulting in complete mixing. Laminar flow may still exist within the smaller pore spaces. As the flow rate is increased to the upper limit of the transition zone, the Peclet number will approach a limiting value for turbulent flow. 4, Turbulent Flow; Convective Dispersion Above a Reynolds number of 1,000-4,000, the flow may be considered completely turbulent. The limiting value of the Peclet number will be about 2 for longitudinal dispersion, and about 11 for transverse dispersion. The molecular diffusion will play no part in the mixing; inertial effects, or convection, will control the mixing. B. Parameters Affecting Experimental Results Perkins and Johnston(43) identify the following factors as relevant in obtaining consistent measurements of dispersion coefficients in the laboratory: (1) edge effects in packed tubes (2) particle size distribution (35) particle shape (4) packing or permeability heterogeneities (5) viscosity ratios (6) gravity forces (7) amount of turbulence (8) effect of an immobile phase

-23Although several of these factors apply only to unconsolidated packs, it seems appropriate to review their implications upon our current research. 1. Edge Effects (53) For random packs of spherical particles Schwartz and Smith show that there is a zone of high porosity extending two to three particle diameters from the cylinder wallso Data of Singer and Wilhelm(51) (26) (3) Fahien and Smith, and Latinen indicate that error due to wall effects is less than 10% for a particle diameter to tube diameter ratio of less than 0,04; for a particle diameter to tube diameter ratio of 0.1, the error may be as high as 50%. In this research, particle diameter to core diameter ratios of less than 0.01 were encountered. Since all samples were mounted in a rubber sleeve which conformed to the sample boundaries, we feel safe in neglecting edge effectso 2. Particle Size Distribution In a solid pack of particles having a wide distribution of sizes, small particles may be contained in the interstices between the larger particles. The median particle size, in this case, does not adequately characterize the size of the average pore space. Perkins and Johnston used the data of Raimondi, eto al, to conclude that the "effective particle size" for a packed bed (i.e. the particle size used to correlate dispersion data) should be that size corresponding to a 10% cumulative fraction, assuming that the particle size distribution of naturally occuring sands follows a log-normal distribution,

-24Raimondi, et. al. (48) and Orlob and Radhakrishna(42) show that increased dispersion will result from a wide particle size distribution, 3. Particle Shape (6,14,25) Various investigators have studied mixing phenomena in packs of cubes, spheres, rings, saddles, etc., and have found that packs of non-spherical particles produce greater dispersion than packs of spherical particles. The sphericity of a particle is defined as surface area of a sphere having the same _ volume as the particle (12) surface area of the particle No good correlation is evident for the dependence of dispersion on sphericity. 4. Packing and Permeability Heterogeneities Heterogeneities may be present in either random packs or in cemented formations. Increases in heterogeneity will result in increased dispersion. Very little can be said to clarify the effect of heterogeneity upon dispersion, since there is no obvious method of characterizing heterogeneity or homogeneity. Perkins and Johnston(43) have compiled data from the literature in an attempt to characterize a medium by an inhomogeneity factor9, 5, but these data show a wide scatter over a 10-fold range of values. It can be said roughly that packings of smaller particles will have a higher inhomogeneity factor and show a corresponding increase in dispersion. The packing, or inhomogeneity factor, is discussed further in section V of this manuscript.

-25The effect of viscosity ratio upon dispersion was mentioned earlier. For ratios greater than unity (i.e. when the viscosity of the displaced fluid is greater than the viscosity of the displacing fluid), unstable fingering may result. The treatment of these instabilities is beyond the scope of this thesis, although we do consider it essential that the criteria for stability are understood. 6. Gravity Forces Gravity forces, or density differences between displaced and displacing fluids, also fall into the category of instabilities which is beyond the scope of this work. We recognize only that if a dense fluid is above a less dense fluid in a vertical displacement, then gravity may cause redistribution quite separate and distinct from redistribution by dispersion. 7. Turbulence From the standpoint of an engineer designing a packed reactor column for maximum efficiency, turbulence may be desirable. Laminar flow is more often desirable in a petroleum or gas storage reservoir, since turbulence increases dispersion or mixing. We define the laminar flow region as that region in which Darcy's Law is valid, i.e. for Reynolds numbers ( dp ZC/ I ) of less than about 10. Fully developed turbulent flow occurs at Reynolds numbers greater than about 1000.

-26Dispersion coefficients in the turbulent region are normally reported in terms of the Peclet number defined by Equation 13. 11e -_ —-(13) where ~ = particle diameter a( = interstitial velocity = dispersion coefficient, longitudinal or transverse. (When D<-, then; when Q' 4, then e J- ctG.) The Peclet number is commonly expressed as a function of the Reynolds number, as indicated in Figures 4 and 5. There is some ambiguity in the literature in that some authors(l0) have defined the Peclet number in terms of particle radius or bed length, and the symbol K or t is often used to represent the convective dispersion coefficient. The Peclet number may also use the molecular diffusion coefficient, A2, instead of ID A, in other applications, e.g. in the laminar flow range. The amount of turbulence in a porous medium is dependent not only upon the Reynolds number, but also upon the particle shape and packing. These properties of the porous material may be correlated by defining the turbulence factor, A, which modifies Darcy's Law as shown in Equation 14. KdL~~~~~~~ & Ad (14) We consider that the turbulence factor, like the resistivity factor, should provide a measure of the "mixing properties" of a rock.

-27Prausnitz(47), Aris and Amundson(2) and Carberry(l4'15) use the mixing-cell to theoretically predict a longitudinal Peclet number of about 2.0 for fully developed turbulence. Figure 7, "Peclet Numbers for Longitudinal Dispersion in Turbulent Flow," shows literature data verifying the theoretical value for longitudinal mixing. Data for transverse mixing tend to give values slightly lower then the theoretically predicted value of 11 for gaseous systems. By combining theories of laminar and turbulent mixing, we can estimate Peclet numbers for any Reynolds number. Figure 4, IPeclet Numbers for an Aqueous System," and Figure 5, "Peclet Numbers for a Gaseous System," illustrate Peclet numbers for a wide range of Reynolds numbers. Figure 6, "Comparison of Calculated and Experimental Data," includes the data of McHenry(40) for longitudinal dispersion in a gaseous system. Figures 4 and 5 for liquid and gaseous systems, respectively, show the regions where the various mixing theories seem to apply. 8. Effect of an Immobile Phase Orlob and Radhakrishna42) studied the effect of an immobile gas phase upon dispersion in a liquid displacement, and found that an increased "capacitance effect" resulted; i.e. several per cent of the total pore volume was inaccessible to flow because of blocked passages in the interstices of the medium. They also found that dispersion was decreased by gas entrapment. A significant amount of gas entrapment (more than 5% of the total pore volume) is necessary to affect the dispersion, however.

10 DATA FOR 6; GASEOUS SYSTEM z ILuJ THEORETICAL VALUE FOR a_ FULLY DEVELOPED TURBULENCE:2I~ 3!0 DATA FOR LIQUID SYSTEMS z 0 0 0 xo X 0 X 0(9 Mc HENRY (38) (13 ~~d(x xX..O x 0 ox 00 o CARBERRY (15) 0 0 0 — Iwo x EBACH (25) 0~~ TURBULENCE 0.1 I I I I III I I I1I1I 10 102 IO3 104 IO5 dUp = REYNOLDS NUMBER II, Figure 7. Pecilet Numbers for Longitudinal DJ spersion in Turbulent Fl'ow (after Perkins and Johnston).

IVo EXPERIMENTAL PROGRAM A Fick's Law model was chosen to represent mixing for flow through a porous medium, since this model permitted a convenient analytical technique for determining dispersion coefficients for a variety of short core samples. We must recognize that a Fick's Law model is not the most sophisticated model that can be used to describe such mixing, and that it is valid only for a stable displacement, without capacitance effects, and for a dispersion coefficient which is independent of concentrationo Figure 8, "Dispersion Model," illustrates Fick's Law mixing in a porous bed. Dispersion coefficients for this system may be determined by the stimulus-response technique described by Levenspiel(9'36'37'57~) Prior to describing Levenspiel's method, however, it will be well to define the following symbols and terms, Equations 15-21; V' = volumetric flow rate Lk interstitial velocity = bed volume (void) L core or bed length =t / =r -- Q = mean residence time (15) 5K = k-~ rmoment integral of -(t) about the mean./ - 0-7;(t) S~t 5(16) If we define the integral A/ j rf (1-l7)

LI l - Ai 2 B 2 2 tl1 at2 2 a2 v 2' I2(U)2 0 2DD uL or XL 2 t 2 t C(t)dt f tC(t)dt\2 uL (U) A 0 2 - o 2 L c)d Figure 8. Dispersion Model.

-31then the following relationships may be expressed conveniently in terms of the above definitions. d~-t -tf) i,, g 4X(18) =mean residence time (19) 1K':: i t2Ar)t7- (20) q Alt = variance of -6) about mean,/l - aft,/Ct) cLt AI St fit) i t -// /,L ~i/o (21) We can now state the results of Levenspiel s analysis: If;(t) represents the concentration profile at a point in a core or packed bed (see Figure 8, "Dispersion Model"), then the dispersion coefficient, P which characterizes Fick's Law mixing between any two points, 1 and 2, in the bed may be found from Equation 22: _ _ (22)IL,.l- ~ ~ ~ ~ ~ 12

-2 - The dimensionless group AU/DQ is known as the axial Peclet number, and the reciprocal De/L is often called the dispersion number. We must note that the points 1 and 2 indicated in Figure 8 are defined to exclude entrance and exit zones, so that the mixing zone under consideration is "open-ended." If concentration profiles were to be determined at points A and B, so that end effects were included in the mixing zone under consideration, then higher order terms would appear in Equation 22. Analysis of such a model was attempted in this research program, and found to be unsatisfactory for short core samples. A. Measurement of Physical Properties From the above analysis we see that in order to calculate a dispersion coefficient to characterize mixing for flow through a porous medilml, we must measure or calculate: (1) the interstitial velocity, A., of gas through the medium, (2) the specimen length, L, and (3) the inlet and outlet concentration profiles, C( ), and their corresponding moment integrals, ~ M,', and M Porosity. permeability, electrical resistivity factors, and turbulence factors were also determined for each core specimen in order to completely characterize the sample. Core lengths were easily determined, and the porosities of all consolidated samples used were furnished by the suppliers or measured experimentally. Permeabilities, turbulence factors, and electrical

-335resistivity factors of all samples were determined experimentally by (20,52) the method of Katz and Cornell The turbulence factor is defined in terms of a quadratic modification of Darcy's Law, - tB< Yt v (14) where ~ = pressure, atm., L = length, cm., = viscosity, cp., K permeability, Darcys,? = density, gm/cc, and = turbulence factor, atm-sec2/gmo or fto-l For gases, it is convenient to express Equation 14 in terms of mass velocity J/A- Ar since the mass velocity is constant while the linear velocity may vary. et~( dL)~ N.\ - Sr~ @B tZ ~g Lc (23) Using eRT (24) and integrating between points 1 and 2, -tt SI - 1g~z 4 # (a) 2 i, dL (25) where = mass flow rate, gm/sec, A = cross sectional area, sqo cmo, -= molecular weight, = compressibility factor, g = gas constant, and

-54absolute temperature, one obtains M(?J)~+ A (26) 2 \ R-t' L W I a When one plots the parameter t vs. the resulting graphs have straight lines with the turbulence factor, 9, as their slope and 1/K, reciprocal permeability, as intercept. Turbulence factors and permeabilities were obtained by flowing nitrogen through each specimen while it was mounted in the Hassler high pressure sleeve. Upstream pressures and downstream flow rates were recorded for a number of different flow rates and the results were correlated with varying degrees of success, Experimental values were found to be slightly higher than predicted by the correlation of Cornell and Katz (32) Sample calculations are included in Appendix I, Porosities were determined by gravimetric measurements. Dry weights and bulk volumes were first determined for each sample. The core was then evacuated in a bell jar, flooded with water, and returned to atmospheric pressure. The weight of the wet sample then indicated the mass, and therefore the volume, of water contained in the void fraction of the sample. Electrical resistivity factors were determined using a 01ol normal solution of potassium chloride. Each core was first evacuated and saturated with the KC1 solution by the same method used for porosity determinations. The electrical resistance, and thus the electrical resistivity, were measured by mounting the sample between

-55two platinum foil electrodes and then balancing against an AC bridge. The electrical resistivity factor is the ratio of the resistivity of the saturated core sample to the resistivity of pure 0.1 N KC1 solution at the given temperature. Sample calculations of porosity and electrical resistivity factors are also included in Appendix Io B. Equipment Figure 9, "Experimental Apparatus," shows schematically the equipment used to determine dispersion coefficients for the nitrogenargon system. The system includes components for pressure and flow rate regulation, a core holder, a thermal conductivity cell, an amplifierrecorder, an analog computer, and an instrument panel for switching and data monitoring.,o Pressure and Flow Rate Regulation of Gas Supply Since pulses are injected into the system using solenoid valves, sensitive pressure and flow controls are required. In order to minimize pressure pulses upon switching, a system using two pressure regulators, four needle valves, a mano-stat rotameter, and a high pressure manometer was constructed. For a given downstream pressure, usually atmospheric, all of the needle valves must be simultaneously adjusted to obtain the desired flow rate, as read on the downstream rotameter Upstream flowing pressures are dependent upon the permeability and length of the sample being studied, i.e, the pressure drop through the sample, and are therefore determined by the flow rate.

ANALOG COMPUTER FLOWMETER Mo MI c~t2 SAMPLE VENT M2 REFERENCE VENT AMP. a RECORDER I I THERMAL CONDUCTIVITY CELL REFERENCE N2 SLEEVE PRESSURE CORE VENT,AND VACUUM S S SOURCE _ SOURCE NITROGEN _ ARGON/HELIUM N = NEEDLE VALVE H.P MANOMETER S = SOLENOID VALVE Figure 9. Experimental Apparatus.

-37These pressures ranged from essentially zero to 60 psig for the range of permeabilities studied. Needle valves, which are used to control the flow rates during each run, are mounted immediately downstream of the pressure regulators on the gas cylinders. Each pressure regulator and needle valve is adjusted so that sonic flow is maintained through the valve. In this manner, disturbances downstream of the needle valves, such as switching of the solenoid valves, cannot affect the delivery pressure of the regulators. The valves are arranged so that both pressure regulators are: always delivering at equal pressure, no matter which stream is flowing through the core sample. That is, when flow through the core by, say, nitrogen is terminated by closing a solenoid valve, another solenoid valve is simultaneously opened to vent the nitrogen through another needle valve; the needle valve is set to provide the same resistance to flow as the core sample. Although it is somewhat tedious to adjust the four needle valves simultaneously, accurate pressure balancing is obtained by this method. For permeable samples and/or low flow rates, the upstream pressure is essentially atmospheric, so that both vent valves can be left fully open, thereby simplifying the adjustments considerably. We may also note that the blow-over pots of the high pressure manometer (see Figure 10) act as surge tanks, which helps to damp out any pressure pulses which result from switching the solenoid valves.

SOLENOID SOLENOID SOURCE GAS A SOURCE GAS 16, N.O. TO CORE NO. II I' 1 1 MERCURY MERCURY BLOW - OVER BLOW -OVER POT POT N. C. N. C. N.O. NORMALLY OPEN 1/4" POLYPROPYLENE N.C. NORMALLY CLOSE9- TUBING DRAIN PLUG Figure 10. High Pressure Manometer.

-392. Core Holder From the pressure balancing apparatus a line runs to the Hassler sleeve which holds the porous sample. The Hassler sleeve is designed for pressures to 1,000 psi, but auxilliary lines are designed for a limiting pressure of 600 psi. The upstream pressure may be measured at the inflow face of the porous medium mounted in the holder. The core holder is of our own construction, and will hold a | inch diameter core of lengths from 3/4 inch to 6 inches. 3. Thermal Conductivity Cell After leaving the sample at atmospheric pressure, the gas flows to a Gow-Mac Model TR2-B thermal conductivity cello This cell features a fast response time, but is sensitive to flow rate fluctuations, so that it is necessary to balance the bridge of the cell and/or adjust the reference gas (nitrogen) flow rate for each data point taken. 4. Amplifier and Recorder Since output signals from the thermal conductivity cell are only a few millivolts, it is necessary to amplify this signal for further processing on the analog computer. The output signal from the cell was fed to a Beckman-Offner Type RS Dynograph recorder. The amplified output from the recorder (normally used to operate the pen arm), which ranged to 10 volts, was used as input to the analog computer, Use of the recorder amplifier allowed variable amplification and permitted a strip-chart record of the thelrrmal conductivity cell output when desired.

-4052 Analog Computations The advantage of obtaining the concentration profile as a voltage output is to facilitate calculation of the moment integrals ( I0) |I, j using an analog computer. An Applied Dynamics Model 2401 computer (shown in Figure 11) is used for this purpose. Figure 12, "Schematic Computer Flow Diagram," illustrates the basic calculations performed by the computer. Essentially two multiplications and four integrating amplifiers are required for the calculation, but sign changes and gain reductions increase the complexity of the circuitry somewhat. Figure 13, "Analog Computer Program," presents the detailed circuitry which is shown schematically in Figure 12. Figure 14, "Analog Computer Symbols," clarifies this flow diagram. Since the computer performs all calculations in real time, and the equipment does not perform linearly at voltages above 140 volts, it is necessary to introduce several "scale factors" to prevent saturation (overloading) of the integrating amplifiers. Although we first attempted to accomplish this scaling by the use of external resistors and capacitors, it later proved advantageous to use internal components only. Complete flexibility was maintained, since input voltage ranges could be changed by varying the gain of the recorder-amplifier. The circuit illustrated was found to operate satisfactorily as long as the signal was adjusted to have a "peak" value of 2 to 10 volts. The multipliers operate in a fashion to produce 100 volts as output when voltage inputs of 100 volts are multiplied. Thus, a scale factor of 100 is introduced by each multiplication operation. Since

-41Figure 11. Analog Computer.

C(t) Mo- SF x/C(t)dt C (t)(t-/)xSF X t- - e I,= SF xx (t)(t-/L)dt c(t)(t-/L)x SF | { SF(t-l 2,X M-so x S F x jC(t )( -)dt > = INTEGRATOR = MULTIPLIER S.F. = SCALE FACTOR Variance of C(t) about mean, a = 0M2-(M)2 a2 Lt = D 2 x interstitial velocity x length (Equation 22) = longitudinal dispersion coefficient. Figure 12. Schematic Analog Computer Flow Diagram.

1.0 C f(t) 1.0-M Mo = Mo f(t)dt 0.1 0.1 0.1 1.0 MULTI-~ 1.0 I r~~~~~~~~~~~M M=,o PLIER -------- Ml: Ml I o(t)(t-a)dt 10 0 01 I0.1 1.0 0.1 1.0 100Fv 13. Aao Computer 0.Pr t.'= (t L - 0.1 CE, _ ~ ~ M, M I~~~~~~~~~UT-10 I -PLIER 100cE (,u =. I)or 1.. _ _f~t)- t P 0.gure 13, Analog C1(CE Figure~~~I 13 nlo.opue rorm

-44I) INVERTER 0.1:: -e, =-eo eo 0.1 e, 2) AMPLIFIER-MULTIPLIER Rf --- -f e, =e = —-e R eR 3) INTEGRATOR C e- fo R e0dt eo(t) R e, IC = INITIAL CONDITION 4) MULTIPLIER (IF IC NOT SHOWN,THEN IC=O) 0.1 +x ~~~~~~~-x~~ e, =-x.y +y e, - Y 5) POTENTIOMETER e, =Keo * ( a O<K< 1.0 eo eo Figure 14. Analog Computer Symbols.

-45the multipliers are guaranteed only within 70 millivolts, a 7% error can arise from the multiplication of two numbers of 1 volt order of magnitude. For this reason, the input voltage is multiplied by 10 (using an amplifier with gain ig/-iO/0) prior to the multiplication operations. The capacitor C, used in computing the second moment integral, I4, is allowed avalue of either o.f or /.o f to increase flexibility. The value used must of course be considered in the analysis of results. One other modification is necessary for special cases, ioe. for runs in which the response signal was detectable for a period greater than about 80 seconds. Since the computer program described here generates a signal 0.It, a limitation is placed on the duration of the run; if 0e1. reaches 140 volts, then the amplifier which produces this signal will overload. It may be observed from the flow diagram, however, that!,1t is generated by the integrating t, which in turn is obtained by integrating a one volt constant signal. If, instead, 1/2 volt or 1/3 volt is integrated to produce, then an additional scale factor of 1/2 or 1/3 is introduced into the calculated values of /I and dZ. This manipulation will allow the analysis of runs at lower flow rates. Since these various scale factors are introduced into the computation, the moment integrals produced by the computer are not identical to the integrals /o, St, M/z previously defined (Equation 17). The following relationships may be deduced from the flow diagram shown in

-46Figure i.o M0,s t14 and 14Z represent the integrals produced by the computer. M01 0 (2'7) / - / 0ox H (28) fr1' /O= XC xM/ (29) Figure 15, "Typical Voltage Profiles," illustrates parameters as they might be computed for a typical experimental run. The parameters C and 1e are estimated prior to the run to correspond as closely as possible to the time from the initial appearance of the response signal to the mean of the response signal, and are set (by potentiometers) as initial conditions on the integrating amplifiers which generate the t and t / signals. The computer is held in the "reset" mode until the response signal is first detected, and is in "operate" mode only while the response signal is actually being monitored. When the output signal (from the thermal conductivity cell) returns to zero, the computer is switched to "hold" mode, and the results are read on a "Digitek" digital voltmeter (DVM) having an accuracy of better than 0o1%. It will be observed that, if the computation had been started at the beginning of the pulse injection, it would be unnecessary to estimate the initial conditions C and j, also, one integrating amplifier could be eliminated, since the quantity x Clt) could be generated by multiplying t times t x CI, instead of t times

COMPUTER IN COMPUTER IN COMPUTER IN RESET — ~ < OPERATE-' ~- HOLD 60 40 30 20 20 -30 la 206 — o 0 - O PULSE 16t0 t TIME, SECONDS - COMPUTER IN COMPUTER IN COMPUTER IN -| —RESET- - OPERATE -HOLD 100 80 60 T40..... 0 -0 30 No 20 7 10 t-' 0 =t-30 I 0t 4 PULSE TIME, SECONDS Figure 15. Typical Voltage Profiles.

-48tLt)j iri- s method was attempted and found unsatisfactory for several reasons. First, the variance, t, was obtained as the result of subtracting one large number from another, yielding very poor accuracy. Second, it is possible for the thermal conductivity cell output, and the output of the computer amplifiers, to drift several millivolts during the course of the run. When multiplied by t and integrated over the duration of the run, considerable error could result. This error is not eliminated, but is minimized, by avoiding computation when the response signal is not being detected. it is shown in Appendix II that Equation 22 is still valid when the moment integrals are computed in this manner, even if the esti mated values of to and to are not accurate. The computation scheme outlined in this section has sufficient flexibility to compute g, 1, and /'/, and hence a dispersion coefficient, for all of the nitrogei-argon runs attempted. Analysis is simplified by the fact that the thermal conductivity of nitrogenargon mixtures, relative to pure nitrogen, is linearly proportional to argon concentration; thus, the voltage profile produced by the thermal conductivity cell is the same as the concentration profile of the gas passing through the cell. This is not true for helium-nitrogen mixtures, as evidenced by Figuire 16, "Thermal Conductivity of Helium-Nitrogen Mixtures." For these runs, a diode circuit was programmed into the computer to correct the voltage profile to a concentration profile. If the cell is adjusted so that pure heli~um produces a signal which is amplified by the

100 80 60 L. 0 4o 20 0 0 20 40 60 80 100 % He IN N2 * INSTRUCTIONS, GOW-MAC THERMAL CONDUCTIVITY CELL o EXPERIMENTAL Figure 16. Thermal Conductivity of HeliumNitrogen Mixtures.

-50recorder-amplifier to 10o0 volts, and if we denote the amplified cell output by V(t) and the concentration of helium by Ct), then examination of Figure 16 reveals that the concentration profile may be closely approximated by three straight line segments as follows. (26 04 Vlt)?t,44- v- (50) t4o) -06 Vek ) - o. f V&) -.2,44J 2,44 4 V-t), -,87 v. (5oa) C It) S.V) O l.4 Vt)-2.4415 o0,2 f/,t -~7 o7r] gta- 4.D (\30b) The diode circuit shown in Figure 17, "Correction for Helium Concentration Profile, was designed to accomplish this correction,.. 6. Instrumentation The instrument panel constructed for the operation of all electrical equipment is shown in Figure 18, "Experimental Apparatus." An automatic timer was installed for the accurate control of short pulse injections; a semi-manual switch is also available to simultaneously switch the solenoid valves controlling both flowing and diffusing gas streams. Manual switches are provided for completely autonomous control of the solenoid valves, so that they (can be simultaneously on or off when desired. All of the computer outputs are connected to audio-connectors on the instrument panel via a jack panel, so that they rcan be

0.1 0.1 _C(t) + V(t) I0.1 0.1 0.1 - V(t) =v- V(t) g 0. r + 10 V o 0.24 1= D IlODE + IOv Figure 17. Correatior for Helium Concentration Profile.

-52Figure 18* -Mcnerimental Appaxatus ~~IFOIguE L Exprmna Eparts

-53rea d on an ammeter or connected conveniently to chart recorders or the digital voltmeter. C. Calibration Procedures The pressure gauges recording the sleeve and core pressure within the Hassler holder were calibrated using a dead-weight tester. All pressures read from the gauges were corrected using these calibration curves. The Mano-stat rotameter indicating the flow rate through the core holder was calibrated (see Appendix II) using a wet test meter built by Precision Instruments Company. The Hassler sleeve and other pieces of pressure equipment were hydrostatically tested to 900 psi. The analog computer program was analyzed by processing a square wave voltage input of 2,00 volts amplitude. Computer results were found to agree with hand calculated values within 1%. Thermal conductivity as a function of composition was determined experimentally for the helium-nitrogen system by passing samples of known concentration through the thermal conductivity cell. Experimental results were in agreement with published values, as shown in Figure 16. Published values were used for the thermal conductivity of argon-nitrogen mixtures. The diode circuit used to correct voltage output signals from the thermal conductivity cell for helium-nitrogen mixtures was checked by recording the response to a "ramp" input signal generated by the computer. To test thermal conductivity cell response time, step concentrationsof argon were injected into a nitrogen stream close to the entrance of the cell. At flow rates up to about 0.01 cubic feet per minute, a

half-second delay time between zero output and full scale output was considered insignificant. An approximate method was used to check each system for instability, i.e. miscible "fingering." Since stability is controlled by the density and viscosity differences between gases, we reason that if, say, the downward displacement of nitrogen by argon is unstable, then the downward displacement of argon by nitrogen is apt to be stable at the same flow rate. By recording the concentration profile responses to step inputs of nitrogen on argon and argon on nitrogen, we are able to determine visually whether the two displacements are similar. When the two responses are identical, we can conclude that both displacements are stable, but if one response profile is more "smeared" than the others we can conclude that that displacement is unstable. This method indicated stable displacements for the argon-nitrogen system at all flow rates studied, but the displacement of helium by nitrogen (in the downward direction) appeared to be unstable even at low flow rates for the short cores studied. We should observe that some minimum core length to diameter ratio is required in order to observe these instabilities, since a finger will in effect be obliterated by transverse dispersion if it travels a sufficient distance, in which case the response would appear to be stable. D. Operating Procedures From 50 to 150 raw data points were taken to constitute of set of data. A set of data is that which is required to describe the dependence of the dispersion coefficient upon flow rate for a range of flow rates,

-55Flow rates ranged from approximately 0,002 to 0.015 cubic feet per minute (measured at 14.7 psia, 70~F.). An upper limitation is set by the response time of the thermal conductivity cell used, and the lower limitation is effectively set by drift characteristics of the thermal conductivity cell and the analog computer. It is first necessary to obtain a measure of the mixing which occurs in the peripheral plumbing and ends of the flow system, for the entire range of velocities under consideration. This is done by inserting one or two small (1 to 2 inches in total length) core plugs in the holder and measuring responses (i.e. variances, ~ ) to a square wave input, normally of five second duration. Each data point is taken three times in order to obtain a check of the effect of thermal conductivity cell drift, etco The sample being studied is then inserted in the holder, in addition to the core plug(s) used in determining the end effects, If two end plugs were used, the sample is positioned between them; in this case, the "end plugs" may or may not be of the same material as the sample being studied. If only one small plug is used in obtaining the end effects, then it must be of the same material as the sample, Responses (variances) are then determined for the new system, using the same square wave input, for the same range of flow rates. The resulting set of data points expresses variance as a function of flow rate for (1) the core plus ends and plumbing, and (2) the ends and plumbing. By subtracting the second from the first we obtain the change in the cha nge in variance due to mixing in the core itself, which may be used with Equation (22) to determine a dispersion coefficient for a given velocity.

-56The subtraction of variances in this manner is dependent upon the linearity of Equation 6. The superposition principle permits the subtraction of variances if it is assumed that Equation 6 may be quasilinearized by evaluating the velocity and dispersion coefficient at an average flowing pressure. Analysis of the raw data is discussed further in Part E of this section, and sample calculations are included in Appendix I. With the understanding that a number of data points are necessary to characterize the mixing properties of a porous medium, this section will attempt to describe the operational procedures involved in obtaining each data point. (1) The analog computer and thermal conductivity cell are turned on in advance to allow sufficient warm-up time. After warm-up (30 minutes or so) all amplifiers are balanced. The Dynograph amplifierrecorder and the digital voltmeter are turned on and balanced, and the switch is thrown to allow power to the solenoid valves and timer. (2) The core and/or "end" plug is mounted in the Hassler core holder and the sleeve is pressurized to at least 50 psi in excess of the maximum anticipated flowing pressure for the run. All electrical connections must be checked between runs, since others also use the computer and occasionally find it necessary to break connections associated with this apparatus. (3) The gas cylinders are opened, and nitrogen is allowed to bleed slowly through the reference side of the thermal conductivity cell. Argon (or helium) and nitrogen flow rates are adjusted to yield equal pressure drops through the core sample. Vent needle valves must also be adjusted so that the switching of solenoid valves does not

-57cause a pressure perterbation at the core inlet. The inlet pressure and the outlet flow rate are then notedo Ambient temperatures need not be recorded, since the temperature is regulated at 70~F for all runs. (4) The time, in seconds, between the initial appearance of the response signal and the mean, /', of the response signal, must be estimated. The parameters to and to are set as initial conditions on the computer's integrating amplifiers which generate the X and / signals. This is accomplished by setting two potentiometers. An accurate estimate of to is not necessary, but will result in better accuracy. (5) A pulse time is selected and set on the timer switch. A five second pulse is used for all runs with short cores, since calculations are simplified by using a standard pulse lengtho All electrical connections are checked, and the digital voltmeter is connected to read the computer input signal, ioe, the amplified thermal conductivity cell output. (6) With nitrogen flowing through the system ("ends," or "ends plus core sample"), the timer switch is tripped to allow a square concentration wave of argon (or helium) to enter the system. That is, argon flow is started and nitrogen flow is simultaneously stopped, and then five seconds later the solenoid valves are reversed to their original position, with. nitrogen again flowingo (7) As soon as argon begins to appear at the system exit, as monitored by the. thermal conductivity cell, the computer is switched from "reset" to "operate" mode. When the response signal has "peaked"

-58and returned to zero, the computer is switched to "hold" mode. The parameters rlo, MI, and M/Z are then read on the digital voltmeter. (8) Steps 6 and 7 are repeated twice in order to check the reproducibility of the results. The initial conditions tV and t may also be re-estimated if necessary. A new flow rate is then chosen, and steps 3, 4, and 5 are repeated. E. Calculation Procedures Having obtained the computer outputs Io, l,, and Akz for both the "ends" and "ends plus core," over a range of velocities, with inlet pressures, core dimensions and core properties known, a dispersion coefficient may be calculated for each velocity using Equation 22. It is the purpose of this section to summarily describe the calculation procedures used, although more detail may be found in the sample calculations in Appendix I. The variance of the response signal is first determined for each run using Equations 22 and 31. &6o' f (31) These variances are then plotted as a function of flow rate. Two "bands" of data points result; one describes the change of variance due to mixing in the core plus ends and lines, and the other characterizes the change of variance due to mixing in the ends and lines. Each set of points is "banded" by two curves, rather than attempting a fit by a single curve. In this manner both a maximum and a minimum

-59value of 467Y is obtained for each velocity. Figure 19, "Evaluation of Mixing Within a Porous Medium, schematically represents the model by which the change in variance due to mixing within the core sample is determined. A modification to this method is required when a significant pressure drop exists across the core sample, ie,. for high flow rates and/or low permeabilities. The "end effects" or mixing in ends and plumbing, were found to depend upon volumetric flow rate, rather than mass flow rate; hence, if the inlet pressure is significantly above atmospheric, then the "end effects" determined in the absence of a core sample will not apply in the presence of the sample. In order to modify the calculation procedure for such cases, it is necessary to make the approximation that the volume of the plumbing upstream of the core is equal to the volume of the plumbing downstream of the core, and then estimate an "average" volumetric flow rate in terms of the measured mass flow rate (i.eo the measured volumetric flow rate at ambient conditions); the "end effects" curve ( O(-T VSo flow rate) is then modified to describe i\ as a function of this average volumetric flow rate, as follows. Let /K = upstream residence time r = downstream residence time -4 CL

-60oI SYSTEM = "ENDS" POROUS INLET ENDPOROUS THERMAL COND. J-' (IN HOLDER) VALVE CELL 2 Mixing Occurs in Lines, End Plugs, Valve, and Thermal Conductivity Cell. II SYSTEM = ENDS" PLUS CORE ALL IN CORE HOLDER POROUS CORE POROUS THERMAL COND. INLET END SAMPLE END VALVE CELL -i-r 2 2' Mixing Occurs in Lines, Porous Ends, Core Sample, Valve, and Thermal Conductivity Cell. 6a2,i2 + 6+2 2 _ (602 + 62 ) t222' t12 t2-2 t23 1-2 t2-3 2 2 = 2 2 = t tI t at3I II I ~3II 3 = change in variance due to mixing in core. Figure 19. Evaluation of Mixing Within Porous Medium.

-61/4= total residence time Then the ratio of the average volumetric flow rates is equal to the reciprocal of the ratio of residence times, or corrected= measured 32) where j)= upstream pressure when the core is not in place, atm. abs., = upstream pressure when the core sample is in place, atmo abso, zr- m esurd = the flow rate (cubic feet per minute) as measured measured on the rotameter, when the core is not in place, and Fcorrected - the flow rate against which (jD for the end effects should properly be correlated. Having obtained maximum and minimum values of It as a function of flow rate for a sample, it is necessary to convert this value to a "relative change in variance," 4 Q, by dividing by the square of the residence time, (L/) The relative change of variance simply de-dimensionalizes the variance change, since aR has units of 2 time In order to make this conversion, it is necessary to determine the average interstitial velocity, aL, ioe. the interstitial velocity at the average flowing pressureo The average flowing pressure may be found in terms of the inlet pressure (assuming atmospheric pressure at the outlet) by the application of Darcy's Law for gas flow

(see Appendix II). 362a~ pu: Ie -(33) In Equation 33, pressures are expressed in atmospheres absolute. The relative variance change is then used to calculate the dispersion coefficient by Equation 22: _ __ (avg. interstitial velocity)(core length) (22) Appropriate units for the dispersion coefficient are cm2/sec. Following the practice established in the literature, data are correlated by describing the ratio of the dispersion coefficient to the molecular diffusion coefficient, J /D, as a function of a Peclet number, Ad,3/70. Here, ([ is an "inhomogeneity factor" which is conveniently combined with the particle diameter, /, to form a parameter ( dO) which may be used to characterize the mixing properties of the porous medium. Since the interstitial velocity, tZ, and the molecular diffusion coefficient, Do, are both inversely proportional to the flowing pressure, it follows that k/D0 may be plotted vs. the volumetric flow rate measured at atmospheric pressure, and the parameter ( dpo- ) may be determined as that number which makes the data fit the model of Equation 9b. Th mlcar dfi (9b) The molecular diffusion coefficient, 2, may be computed using the Chapman-Enskog or Lennard-Jones collision integral theory(7'8'17) For a given binary system at constant temperature, it is shown in Appendix I that 0 depends only upon pressure.

Vo EXPERIMENTAL RESULTS The results of this experimental program may be considered conveniently in three parts. First, dispersion coefficients were obtained for the nitrogen-argon system using samples 2 to 5 inches in length; unsatisfactory data were obtained for the helium-nitrogen data using these short cores. Second, two of these short cores were treated with a pentane-paraffin solution to provide various degrees of blocking by an immobile phase, and dispersion coefficients were again determined. Third, a longer sample was mounted in epoxy, and dispersion coefficients were obtained in order to check the effect of core length upon results. Dispersion characteristics were also determined for the heliummnitrogen system using the long core, but the interpretation of the results is open to some question. A. Short, Dry Samples Experimental results obtained using short, dry core samples are shown in Figures 20 a-io Approximate error limits are indicated in these illustrations. Experimental accuracy is considered good, since conservative means were used to obtain the indicated error limits. Better data were obtained, as expected, for longer cores, since the analysis of end effects was not as critical in this case. The physical properties of the core specimens are shown in Tables I and IIo All data in Figures 20 ani were obtained for the nitrogen-argon system. Attempts to obtain similar data for the helium-nitrogen system -635

-64IC / SLOPE=1.13 1. - ~/ ~, 0.I, I,, I I I 0.001 0.01 0.1 GAS FLOW RATE, cfm at 14.7 psia,700F Figure 20a. Dispersion Characteristics as a Function of Flow Rate. Core BA- 2, Bandera Sandstone.

SLO:I2 //| 0.:1,,,., 0.001 0.01 0.1 GAS FLOW RATE, cfm at 14.7 psia, 70F Figure 2Cb. Dispersion Characteristics as a Function of Flow Rate. Core BA-3, Bandera Sandstone.

-66I0 / SLOPE=1.27 /0, / 11 0.1: L' I' I I 0.001 0.01 0.1 GAS FLOW RATE, cfm at 14.7 psia, 70 0F Figure 20c. Dispersion Characteristics as a Function of Flow Rate. Core BC-1, Boise Sandstone.

-6710 -- - - ENDS-"=CORE 80-I, L=1.74" "ENDS" =CORE BO-2, L= 2.20 SLOPE =1.0 O ~~~I II 0.1i I 0.001 0.01 0.1 GAS FLOW RATE, cfm at 14.7 psia, 70 F Figure 20d. Dispersion Characteristics as a Function of Flow Rate. Core BO-2, Boise Sandstone.

-6810 SLOPE = 1.0 DI 1.0 - I I I I I I I i I I I I I i Do 0.001 0.01 0.1 GAS FLOW RATE, cfm at 14.7 psia, 70 0F Figure 20e. Dispersion Characteristics as a Function of Flow Rate. Core BO-3, Boise Sandstone.

-6910 _ / 0.1 0.001 0.0o 0.10 GAS FLOW RATE, cfm at 70 F, 14.7 psia Figure 20f. Dispersion Characteristics as a Function of Flow Rate. Core FYX-A, Nodosaria Sandstone.

-7010 SLOPE = 1.2 1.0 -- 0.1 0.1 I I I I I i i i I I II 0.001 0.01 0. 0 GAS FLOW RATE, cfm at 700F, 14.7 psia Figure 20g. Dispersion Characteristics as a Function of Flow Rate. Core FYX-B, Nodosaria Sandstone.

-71100 / SLOPE=1.28 _:_0 - / Do 0. 001 0.01 0.1 GAS FLOW RATE, cfm at 70 F, 14.7 psia Figure 20h. Dispersion Characteristics as a Function of Flow Rate. Core 3501, Dolomite.

-72100 SLOPE = 1.4 DI 1.0 1.0 I I I I I I I I 0.001 0.01 0. I GAS FLOW RATE, cfm at 700F, 14.7 psia Figure 20i. Dispersion Characteristics as a Function of Flow Rate. Core 3514, Dolomite.

-73were unsuccessful, since the mixing in these short cores was dominated by instabilities. That is, experimental results were not reproducible within a tolerable range of error; the variance-responses to square wave stimuli showed particular scatter at higher flow rates. The "characteristic length," Jp-, and power dependence, or slope, y, were determined from experimental data for each specimen (see sample calculations in Appendix I) in order to fit the results to the equation ~ - 1-t 1,- a_- (9b) The calculated values of these parameters are included in Table II, "Specimen Properties." The characteristic lengths, ( (/p-) vary from 0.23 cm to 1.87 cm, in fair agreement with the average value of 0.36 cm (43) reported by Perkins and Johnston The sandstones used here have an average characteristic length of o.I.C _ 0.423 cm, but the dolomite samples show abnormally high values. Comparison of Equation 14 with a friction factor equation of the type (32) L~e - 639,dp('~~1 at" tS3$, dp2 | (34) shows that the particle diameter for a porous medium should be inversely related to the turbulence factor. If a constant value of the packing factor, 6, is assumed for each rock type, then this observation is verified by the data of Table II, in further support of the validity of these experimental results.

-74TABLE I SPECIMEN TYPES (See also Table II) Core Length, Label Inche s Type Source (1) BA-2 4.73 Bandera sandstone Continental BA-2A O0.96 Oil Company BA-2B 1.14 (2) BA-3 4.97 BA-3A 0.83 BA-3B o.096 (3) BO-1 4.88 Boise sandstone BO-1A. 86 BO-1B 0.88 (4) BO-2 3.92 " BO-2A 2.20 (5) BO-3 3.20 i BO-3A 2.02 BO-3B 1.00 (6) FYX-A 2.32 Nodosaria sandstone (7) FYX-B 2.61 it " (8) BE-1 4.20 Berea sandstone Cleveland BE-2 5.22 Quarries BE-3 23.90 (9) 3501 2.66 Dolomite U. S. Bureau of Mines (10) 3514 3.44 I All diameters = 1.50 inches

TABLE II SPECIMEN PROPERTIES (See also Table I) Core Porosity, Perm., Turbulence Res'i Livity idpt Exponent Label Percent Md. Factor,Ft. Factor, F 6,w m, (Eq9O ) (1) BA-2 23.4 40o 1.25 x 109 13o3 0.475 1o13 (2) BA-3 21.8 41o6 6.4 x 108 14,2 0.75 1.20 (3) BO-1 31.0 1030 4,6 x 106 13.9 0.67 1.27 (4) BO-2 32.0 258 1.4 x 107 9o5 o.346 (5) BO-3 32.0 1450 1.72 x 108 8.3 0.25 (6) FYX-A 21.8 1008 not determined 13o6 0.41 1,36 (7) FYX-B 25.2 227 13o3 o,366 1,20 (8) BE-1 19.0 300 152 0o.316 1,24 BE-2 it 267 " ". i BE-3 " 228 i 0,232 1.24 (9) 3501 12.5 61o4 5.2 x 108 27,6 1.87 1,28 (10) 3514 12.1 4o45 3.2 x 10 1 41.6 1.44 1010 ( 1 12.1 445 32 x 10416 144 1,40

-76A study of a Berea sandstone is also discussed in Part C of this section; the results are shown in Table II. The power dependence, P1, of the dispersion coefficient upon velocity, is seen to range from approximately 1.0 to 1.40, which (12) is in good agreement with data obtained by Brigham, et. al. and Blackwell, et. al. (11); these authors obtained values of A r 1.17 to 1.24 for Berea sandstone cores using liquid systems. Brigham, et. al. also found that M 1.2 for unconsolidated bead packs. Figures 20 a-i also indicate the values of F which the data should approach at low flow rates. Reasonable results were obtained. (34) Kravik and Bissey obtained data for gaseous systems at low flow rates, and found that for mobility ratios between 0.5 and 2.0 Equation 8 is valid, i.e. that there is no effect of instability at very low flow rates. B. Short Cores, Immobile Phase in Interstices Two cores (BA-2 and BO-2) were treated with a paraffin and pentane solution to provide varying degrees of pore "blocking" ranging from 4% to 20% of the pore volume. Water was not used as the immobile phase, since thermal conductivity measurements would have become too complicated. Paraffin wax was firstdissolved in commercial grade n-pentane, and the specimens were thensoaked in the solution for several hours. After air drying, the cores were subjected to a vacuum for 30 minutes to an hour to evaporate the pentane, leaving the paraffin deposited

-77in the interstices. The volume of the paraffin deposited was determined by gravimetric measurements, which agreed within 5% with volumes determined by soaking a dry core in a pentane-paraffin solution of precisely measured composition. Dispersion characteristics for the nitrogen-argon system were determined for each saturation by the established procedure. Results are shown in Figures 21 and 22. Permeabilities were estimated for each saturation using the inlet pressure data and Darcy's Law for gas flow. After obtaining data at the highest paraffin saturations (18.7% and 21.5%), the samples were heated to 140OF and blown down with warm nitrogen in an attempt to redistribute the paraffin in the interstices without substantially altering the fraction of the pore space which was blocked. Electrical resistivity factors were determined both before and after this operation, but, as an oversight, were not measured for intermediate runs. It was concluded from this series of runs that the dispersion characteristics of the porous medium were affected by the distribution (microscopic and/or macroscopic) of the immobile phase as well as the amount of the immobile phase. For the Boise sandstone sample, which had a rather large grain size (as determined by mere visual observation), dispersion was in all cases increased by the presence of the immobile phase. The power dependence of dispersion upon flow rate is seen to vary from approximately 1l0 to Lo5, but not as a monotonic function of the volume of immobile phase present.

-78100 3 _4 0 2 1.0 0.1 I I I I I I I I I I I I I I 0.001 0.01 0.1 1.0 FLOW RATE, cfm at 70 F, 14.7 psia Figure 21a. Dispersion Characteristics as a Function of Flow Rate, Imnobile Phase in Interstices (Boise Sandstone).

-79I00 Di Do 1.0 0.001 0.01 0.1 I I I,,1.0 0.001 0.01 0.10 1.0 FLOW RATE, cfm at 700F, 14.7 psia Figure 21b. Dispersion Characteristics as a Function of Flow Rate, Imnobile Phase in Interstices (Boise Sandstone).

-80100 2 I0 D: I.0 0.001 0.01 0.10 1.0 FLOW RATE, cfm at 70 F, 14.7 psia Figure 22a. Dispersion Characteristics as a Function of Flow Rate, Immobile Phase in Interstices (Bandera Sandstone).

100 5 I0 D0 Do 1.0 0.1 I I I I I I I 1 1 0.001 0.01 0.10 1.0 FLOW RATE, cfm at 700F, 14.7 psia Figure 22b. Dispersion Characteristics as a Function of Flow Ratc, Immobile Phase in Interstices (Bandera Sandstone).

-82TABLE III EFFECT OF AN IMMOBILE PHASE UPON DISPERSION CHARACTERISTICS A. Core BA-2, Figure 21. Percent of Porosity, Permeability, Resistivity Power CharacterInterstices, ( K, Factor, Dependence, istic length) Blocked Per Cent Millidarcies Y,,C 1. 0 23.4 40 13.3 1.13.475 2, 4.6 22.3 34.9 - 1.13.365 3. 7.86 21.6 32.2 1.0.272 4. 12.9 20.4 19.1 - 1.25 495 5. 18.7 19.0 26. 23.7 1.50.685 6. 18.2 19.1 20.9 13.2 1.45.605 B. Core BO-2, Figure 22. 1. 0 32.0 258 9.5 1.0.346 2. 4.45 30o.6 263 1.13.475 3. 6.7 29.8 171 1.38.675 4. 10.4 28.7 222 1.13.62 5. 10.45 28.7 230 -- 1.07.535 6. 14.2 27.4 230 1.50.675 7. 21.5 25.2 135 17.9 1.58 1.55 8. 20.4 25.5 338 10.1 1.31.65

We were forced to conclude that the method of depositing the immobile phase in the interstices did not provide uniform deposition. We can nevertheless conclude that the effect of an immobile interstitial phase should be considered in any practical approach to reservoir problems. Co Long, Dry Cores In order to ascertain that experimental results were valid for the short cores used, the experimental technique described earlier was also used to determine dispersion characteristics for a Berea sandstone 23~9 inches in length by 1 1/2 inches in diameter. Three segments of the sandstone were used in the investigation. First, a 4~20 inch specimen was cut, and dispersion characteristics were determined as a function of flow rate with the core mounted in the Hassler sleeve as before. Samples 5~22 inches and 23590 inches in length were mounted in epoxy, and detachable end plates were made from lucite plastic. Response characteristics (ioe. variances of the output concentration profiles) were obtained for each segment using a 10 second pulse injection of argon or helium; the short segment (5~22 inches) represented the "ends" of the core, and the longer (23~9 inches) segment represented the "ends plus sample)" The difference between the two responses was a measure of the mixing in the core itself (18o68 inches), as before. The results of this study are shown in Figure 23, "Effect of Core Length upon Computed Dispersion Characteristics)" Physical

-8410 10 BE-I (AL = 4.2") -i~~~~ / /~~~~~BE - 2,3 (aL = 18.7 ") SLOPE = 1.24 ARGON- NITROGEN HELIUM- NITROGEN 1.0IDi Dg 0 0.10 0.01 I I I I I I i l l I I I 0.001 0.01 0.10 1.0 GAS FLOW RATE, cfm at 700F, 14.7 psia Figure 23. Dispersion Characteristics as a Function of Flow Rate - Effect of Core Length, Berea Sandstone.

-85properties of the core (BE) are included in Table II. The characteristic length, dp6, is seen to be between 0O23 cm and 032 cm, and the power dependence is approximately 1.24, in fair agreement with Brigham, et. a (12), who determined values of:i- 0 39 cm and m = 1.2 for a Berea core of similar porosity and permeability. It will be noted from Table II that the longer cores seem to have a lower permeability than the short (4.2 inch) core. This may be due to a slight amount of epoxy imbibed into the core, although precautions were taken to minimize this possibility. We were able to conclude from this phase of the experimental investigation that reasonable results had been obtained using short core samples. Data obtained for the helium-nitrogen system, using the long Berea core, are also shown in Figure 23. These data are seen to be displaced by a factor of about three from the results anticipated from the argon-nitrogen data. Our only explanation for this anomaly is to hypothesize that the computed value of the molecular diffusion coefficient, Do, was too large. The Chapman-Enskog equation predicted a diffusion coefficient of Do: 0.684 cm2/sec at one atmospheres a value of tk - 0.2 cm /sec would be required to yield the expected dispersion characteristics. In the absence of published experimental values for the diffusion coefficient, we are forced to conclude that our investigation of the helium-nitrogen system was inconclusive.

VI. SUMMARY AND CONCLUSIONS The objectives of this thesis were to review the state of our knowledge of longitudinal mixing for flow through porous media, to develop a new experimental technique for the rapid determination of dispersion coefficients in core samples of convenient size, to ascertain the validity of this technique, to observe the effects of an immobile interstitial phase upon gaseous dispersion, and to correlate the mixing properties of naturally occurring porous media with other physical properties of the media. All of these goals were attained, except that anomalous results were obtained for gases of extreme density difference, and correlations of questionable value were determined for dispersion coefficients as a function of rock properties. The longitudinal mixing properties of a porous medium may be characterized by an equation of the form f + go,( g 0 udp6) j(9b) If the parameters dpa and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two -86

-87dolomite samples. The method features "on-line" gas analysis by thermal conductivity and "on-line" data reduction by analog computation, providing a rapid means of determining longitudinal dispersion properties for core samples only 3 or 4 inches in length by 1-1/2 inches in diameter. By injecting pulses of argon into nitrogen flowing through core samples of various lengths, the relative change in variance of the argon concentration profile may be found for a segment of core sample, exclusive of end effects. The variance, which is determined by processing the output signal of a thermal conductivity cell on an analog computer, is related to the longitudinal dispersion coefficient by the equation t - _ b ( LA ( L (22) It was found that the dispersion ratio:, D2/Do, has a power dependence of greater than 1.0 upon flow rate. The power dependence, ie.o the exponent "m" of Equation (9b), can vary between 1.0 and lo5, and should be considered a property of a porous medium. Poor correlations were found between the dispersion characteristics and rock properties beyond the low velocity range where dispersion consists of molecular effects only, i.e. beyond the range where Equation (8) is applicable. The parameters dpc and m (Equation (9b)) are plotted as functions of permeability and the electrical resistivity factor in Figures 24 to 27. The other correlations attempted were of similar questionable value.

A 1.75 1.50 A DOLOMITE, DRY 0 SANDSTONE, DRY 1.25 0 SANDSTONE, IMMOBILE PHASE I 1.00 z F) I I CO Z0.750 o I- 0 O I[]s Ed 0 z O; o o 0.250 0 0 0O~0 1.0 10 100 1000 10000 PERMEABILITY, K (md) -- Figure 24. Characteristic Length, dpc (cm) vs. Permeability, K (md.)

-892.0 1.9 1.8- DOLOMITE O SANDSTONE, DRY I.7 O SANDSTONE, IMMOBILE PHASE 1.61.5 1.4 E 1.3 b 1.2 I 1.1 1.0J ) 0.9 0.8 O 8 0.7 - 0.6- E] 0.5- 0.4 - 0 0.3- 0 0 0.2 0.1 C L I I I I I I l l 0 5 10 15 20 25 30 35 40 45 ELECTRICAL RESISTIVITY FACTOR, F Figure 25. Characteristics Length, dpa (cm) vs. Electrical Resistivity Factor.

1.6 0 O SANDSTONE, DRY O SANDSTONE, IMMOBILE PHASE 1.5 - 0 0A DOLOMITE, DRY 0 1.4 O O I 1.3 O E | w O 0 -J (nI.2 0 0 I.' 1 L I i 0 10 100 1000 10000 PERMEABILITY, K (md ) Figure 26. Exponent, m, vs. Permeability, K (md.)

-9150 45 0 SANDSTONE, DRY 0 SANDSTONE, IMMOBILE PHASE A DOLOMITE, DRY A 403530 Lh 25w 20 cr w - LU 0 0 Ow 0 5SI I I.0 I~.i 1.2 1.3 1.4 1.5 1.6 1.7 SLOPE, m ~Figure 27. Exponent, m, vs. Resistivity Factor, F

-92Longitudinal dispersion characteristics were determined to be substantially the same for Berea sandstone cores 4 inches in length and 18 inches in length, indicating that the technique yielded reliable results for the shorter samples. Two of the samples studied were saturated with a pentaneparaffin solution of varying proportions and allowed to dry, leaving the paraffin deposited as an immobile phase in the interstices of the rocks. Paraffin saturations ranged from 5 per cent to 20 per cent of the pore volume. It was shown that the longitudinal dispersion characteristics were significantly affected by the presence of the immobile phase, but several anomalous data sets led to the conclusion that the method used to deposit the interstitial paraffin was not reliable, i.e. did not result in uniform depositiono This prevented a positive conclusion which would relate the dispersion characteristics uniquely to the amount of the immobile phase present in the interstices, but we were able to conclude, as a generalization, that the effect of the immobile phase is to increase the exponent, m, and the characteristic length, dpa, of Equation (9b). The presence of an immobile phase should, therefore, be considered in the analysis of any real reservoir problem. Experimental results for the helium-nitrogen system were inconclusive. Poor results were obtained for the short samples due to instability problems. For the longer Berea core, the lack of agreement between experimental and predicted results could only be explained by hypothesizing that an incorrect value of the molecular diffusion coefficient was computed by the Chapman-Enskog and Gilliland equations.

-93No attempt was made to analyze stability problems for gaseous systems. Further research in this area is essential in obtaining the ultimate goal of predicting reservoir behavior. Further research should also include attempts to extend the velocity range which can be studied by the proposed method. A logical approach to this problem would be to verify that similar dispersion characteristics are obtained for gaseous and liquid systems, and then to study liquid dispersion, substituting an electrical conductivity cell for the thermal conductivity cell used in this research.

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-9514. Carberry, James J., "Axial Dispersion and Void-Cell Mixing Efficiency in Fluid Flow in Fixed Beds," Comm. to the Editor, AIChE Jour, 4, 1 (1958) 15. Carberry, J. J., and Bretton, R. H., "Axial Dispersion of Mass in Flow Through Fixed Beds," AIChE Jour 4, 3 (1958) 16. Carman, P. C., "Permeability of Saturated Sands, Soils, and Clays," Jour. Agri. Sci. 29, 262 (1939) 17. Chapman and Cowling, Mathematical Theory of Non-Uniform Gases, Second Edition, Cambridge Univ. Press, (1951) 18. Coats, K. H., and Smith, B. D., "Dead-End Pore Volume and Dispersion in Porous Media," SPE Preprint 647, New Orleans, La., Oct 6-9, 1963 19. Cornell, D., "Flow of Gases Through Consolidated Porous Media," Ph.D. Thesis, Univ. Michigan (1952) 20. Cornell, D., and Katz, D. L., "Flow of Gases Through Consolidated Porous Media," Ind. Eng. Chem. 45, 2145 (1953) 21. Crank, J., The Mathematics of Diffusion, Oxford-Clarendon Press, New York (1957) 22. Davis, B. R., and Scott, D. S., "Measurement of Effective Diffusivities in Porous Particles,t" Preprint 48d, 58th Meeting AIChE, Philadelphia, Pa., Dec. 9, 1965 23. de Jong, G. de Josselin, "Longitudinal and Transverse Diffusion in Granular Deposits," Trans. AGU, 39, 67 (1958) 24. Dumore, J. M., "Stability Considerations in Downward Miscible Displacement," SPE Preprint 961, Houston, Oct. 11-14, 1964 25. Ebach, E. A., and White, R. R., "Mixing of Fluids Flowing Through Beds of Packed Solids," AIChE Jour, 4, 2, (1958) 26. Fahien, R. W., and Smith, J. M., "Mass Transfer in Packed Beds," AIChE Jour, 1, 28, (1955) 27. Gorring, R. L., "Multiphase Flow of Immiscible Fluids in Porous Media," Ph.D. Thesis, Univ. Michigan, 1962 28. Goddard, R. R., "Fluid Dispersion and Distribution in Porous Media Using the Frequency Response Method with a Radioactive Tracer," SPE Preprint 1228, Denver, Colo., Oct. 3-6, 1965

-9629. Gow-Mac Instrument Co., Instructions for Operation of Thermal Conductivity Cell 50. Grane, F. E., and Gardner, G. H. F., "Measurements of Transverse Dispersion in Granular Media," Jour Chem. Eng. Data 6, 2853, (1961) 51. Helander, D. P., "The Effect of Pore Configuration, Pressure, and Temperature on Rock Resistivity," Ph.D. Thesis, Univ. Oklahoma (1965) 52. Katz, et. al., Handbook of Natural Gas Engineering, McGraw-Hill, New York 1959 55. Klinkenberg, R. R., "Residence Time Distributions and Axial Spreading in Flow Systems (With Their Application in Chemical Engineering and Other Fields)," Trans. Instn. Chem. Engrs., 45, T141 (1965) 54. Kravik, G. D., and Bissey, L. T., "A Study of Mixing During Gaseous Displacement at Low Flow Rates in a Consolidated Porous Medium," Proceedings 24th Tech. Conference of Petroleum Production, Oct. 2325, Penn State University, University Park Pa. 55. Latinen, G. A., "Mechanism of Fluid Phase Mixing of Fixed and Fluidized Beds of Uniformly Sized Particles," Ph.D. Thesis Princeton Univ., 1951 56. Levenspiel, O., Chemical Reaction Engineering, Wiley and Sons, New York 1962 (ch. 9) 57. Levenspiel, 0., and Smith, W. K., "Notes on the Diffusion-Type Model for the Longitudinal Mixing of Fluids in Flow," Chem. Eng. Sci. 6, 4 and 5 (1956) 38. Longwell, P. A., and Sage, B. H., "Some Molecular Transport Characteristics in Binary Homogeneous Systems," AIChE Jour, 5, 1 (1965) 59. Mathur, G. P., and Thodos, G., "The Thermal Conductivity and Diffusivity of Gases for Temperatures to 10,000~K," AIChE Jour, 11, 1, 164 (1965) 40. McHenry, K. W., and Wilhelm, R. H., "Axial Mixing of Binary Gas Mixtures Flowing a Random Bed of Packed Spheres," AIChE Jour, 3, 83 (1957) 41. Munnerlyn, R. D. (Division of Helium Resources, U. S. Bureau of Mines), Personal Communication, Oct. 12, 1964

-974'. Orlob, G. T., and Radhakrishna, G. N., "The Effects of Entrapped Gases on the Hydraulic Characteristics of Porous Media," Trans. AGU, 59, 648 (1958) 45. Perkins, T. K., and Johnston, 0. C., "A Review of Diffusion and Dispersion in Porous Media," SPE Jour, March, 1963 44. Perkins, T. K., and Johnston, O. C., "Mechanics of Viscous Fingering in Mixcible Systems," SPE Preprint 1229, Denver, Colo., Oct. 5-6 1965 45. Perrine, R. L., "The Development of Stability Theory for Miscible Liquid-Liquid Displacement," Trans. AIME 222, II-17 (1961) 46. Peterson, E. E., "Diffusion in a Pore of Varying Cross Section," AIChE Jour, 4, 5, (1958) 47. Prausnitz, J. M., "Longitudinal Dispersion in a Packed Bed," Comm. to the Editor, AIChE Jour, 4, 1 (1958) 48. Raimondi, P., Gardner, G. H. F., and Petrick, C. B., "Effect of Pore Structure and Molecular Diffusion on the Mixing of Miscible Liquids Flowing in Porous Media," Preprint 43, AIChE-SPE Joint Symposium on Fundamental Concepts of Miscible Fluid Displacement, Part II, San Francisco, Dec. 6-9, 1959 49. Saffman, P. G., "Dispersion in Flow Through a Network of Capillaries," Chem. Eng. Sci. 11, 125 (1959) 50. Schowalter, W. R., "Stability Criteria for Miscible Displacement of Fluids from a Porous Medium," AIChE Jour, 11, 1 (1959) 51. Singer, E., and Wilhelm, R. H., "Heat Transfer in Packed Beds; Analytical Solution and Design Method; Fluid Flow, Solids Flow, and Chemical Reaction," Chem. Eng. Prog. 46, 343 (1959) 52. Slattery, J. C., and Bird, R. B., "Calculations of the Diffusion Coefficient of Dilute Gases and of the Self-Diffusion Coefficient for Dense Gases," AIChE Jour, 4, 2 (1958) 53. Schwartz, C. E., and Smith, J. M., "Flow Distribution in Packed Beds," Ind. Eng. Chem. 45, 1209 (1953) 54. Skinner, J. L., Personal Communications 55. Toor, H. L., "Diffusion in Three Component Gas Mixtures," AIChE Jour, 3, 2, 198 (1957)

-9850. van Deemter, J. J., Bradler, and Lawrence, "Fluid Displacement in Capillaries," Chem. Eng. Sci. 5, 271 (1956) 57. van der Laan, E. Th., "Notes on the Diffusion-Type Model for Longitudinal Mixing in Flow," Comm. to the Editor, Chem. Eng. Sci. 7, j (1957) 58. van der Poel, C., "Effect of Lateral Diffusivity on Miscible Displacement in Horizontal Reservoirs, SPE Jour, Dec. 1962 59. Wilhelm, R. H., "Rate Processes in Chemical Reactors," Chem. Eng. Prog. 49, 3, 150 (1953)

APPENDIX I SAMPLE CALCUlATIONS 1o Determination of Porosity Core: BE-1, Berea sandstone Dimensions: lo5 in. D x 4,203 in. L Dry weight: 255 15 gmo Bulk Bolume: (j)1 f. 4.?Z3'(2.,5):"Z~ C Weight, saturated with water (lo00 gm/cc): 278.15 gm. Weight of water: 278,15 255.15 23.0 gmo or 23,0 cco Porosity, or fraction of bulk volume occupied by water: 2 _3, ) 2. Determination of Electrical Resistivity Factor: Core: BE-1, Berea sandstone Dimensions: 1t5 ino D x 4.203 in. L Porosity: 19% Measured electrical resistance when saturated with 0.1 N KC1 at 22.00 C: 1180 ohms Conductivity of 0.1 N KC1 @ 220'C: 0.01215 mho/cm (from Chem, Rubber Handbook) -99

-100Resistance of 0.1 N KC1, "equivalent dimensions": R = length _ 4.203 x 2.54 = 77.5 ohms conductivity x c.s. area 11.4 x 0.01215 Resistivity factor: F = 1180 = 15.2 77.5 3. Determination of Permeability Core: BE-1, Berea sandstone Dimensions: 1.5 in. D x 4.203 in. L Gas: Nitrogen Darcy's Law for Gas Flow: P = atm. abs. L = cm. i-'= viscosity- 0.0176 cp. for Nitrogen Q = cc/sec Pb = 1 atm. A = cm K = permeability, Darcys Flow rate = Q = 0.0098 t MI -4 2 -84oocr _ 4 CI- 6~,, 3 se CA = ~ = (,,S s i 54 11.4 c:" 4 Inlet pressure P1 = 3.4 psig = 1.23 atm. abs. Outlet pressure P2 = 0 psig = 1.0 atm. abs. K 2L 9S 2 (2______________________(_____)_____ pL, -W') (_114) c&$ (\,23- Io~ ) 4 4M =0,3 \ - 300 ra.

-1014o Determination of the Turbulence Factor Core: BO-3, Boise sandstone Dimensions: 1o5 ino D x 3,20 in, L Gas: Nitrogen Integrated Flow Equation: IaRTI'L w A' L where M mol, wto = 28 P pressure, atmo abso z compressibility factor 1 o0 R - gas constant T absolute temperature: -- viscosity = 0,0176 cpo L length = 3,20 in. A - c, s, area - 11.4 cm2 W - mass flow rate K permeability, Darcy turbulence factor, atm-sec2/gm or ft.-! The quantity M(p- w/) fP is plotted vso. in Figure I-1o 23g'L T, MrL Intercept - 1/K - 0o7 Permeability - 1/0o7 = 1o43 Darcy (may be checked by method of sample calculation No, 3) Slope - 5,23 atm-sec2/gm ~ 1.72 x 108 ft- 1 5. Calculation of the Molecular Diffusion Coefficient at Laboratory Conditions: Nitrogen-argon system at 70~F, 14o7 psiao Chapman-Enskog (rigid sphere) theory for ideal gases: aD45 - ooo\8f83 ll,,,,'. -

98 t CORE BO-3 K =1.43 Darcy 7 / =1.72 x 10 FT-' x 65- x x 0 ~1N 3 0- 0.2 0.4 0.6 0.8 1.01.2 0 0.2 0.4 0.6 0.8 1.0 1.2 A,L Figure I-1. Turbulence Factor Plot.

-103MA = molo wt. argon = 39~94 MB = mol. wto nitrogen = 28.02 T = 295~K. P = 14.7 psia = 1.0 atm, abs. Lennard-Jones Parameters (8): C'~ = 2 - (3,418 +3,~81) - 3,s05' K raOn Hence QD -: (- T f(3.24) = 0929 (tabulated function, ref. 8) Do = 0.195 cm2/sec. For other pressures at 700F, multiply by 14-7/P to find: Do = 2.84/P (P = psia) 6, Determination of the "Characteristic Length" (dpG): Core: BA-2, Bandera Sandstone Porosity: 235.4% See Figure 20 GL, When ULpIo, thenO,5( d~ ) (the equation of the line fitting the data) is 0.5, and the corresponding value of the flow rate is determined to be 0.00135 cfm at 70~F, 14.7 psiao g: /.,/. 1J6 n.1y -9 30, (,i c - "'i 22 " 84-',~ o'" = o,4 70, 2 3 ~ GlpC= ow47s c

4047. Determination of Effective Porosity for a Core Containing Wax in the Interstices: Core: B3A-2, Bandera sandstone Dimensions: 1.5 in. D x 4.73 in. L. Porosity (dry): 23.4% Void volume: LL 7ft / % -(o.234)(4 73) r)(/ J)/4 - / 96 Dry Weight: 294.225 gmo After soaking in pentane-paraffin solution (prepared using 4.7 vol. % paraffin) and allowing the pentane to evaporate, weight: 295.490 gm. Weight of paraffin in core: 1.265 gm. Measured density of paraffin; 14.1 gm/in.3 Volume of paraffin in core: 1.265/14.1 = 0.09 in.3 Percent of pore space occupied: 100 x 0.09 = 4.6% 1.96 Effective porosity: (I.00 - o.o46) x 23.4 = 22.3% 8. Evaluation of Dispersion Characteristics for a Typical Sample: Core: BA-2) Bandera sandstone A. Determine the (variance) response to a 5.0 second square concentration wave input (argon on nitrogen) for cores BA-2A plus BA-2 B, length 2.10 inches. See Data Sheet No. 1 recall M;__ -( e' ) -_ _ _ _ (31) B. Determine the ~ response to a 5~0 second square wave input for cores BA-2 plus BA-2A, length 5.87 inches. See Data Sheet No. 2.

-105= DATA SHEET NO. 1, 2A-2A *t-2 B, L-= 2Z/,, 2 RAW DATA COMPUTED DATA Rotameter M M 1 M2 Pt, reading psig (suc.) gosr 70"F 7o5 69.97 -11,34 26~57 1o9 CE - 10 3554 0o.oo00165 71.53 - 9.27 27~59 36,8 126,6 -12,46 47091 36.8 9.0 45.96 1.50 74~.32 2,6 CE = 0ol 16.1ol 0.0025 48.46 0,32 77.70 16,1 47.7 3501 75~7 1505 lO4 29.95 2,76 28.34 309 8.60 0,00335 25.54 1.86 22.72 8.27 30~89 2037 26,65 8.05 11o 6 35022 2. 04 2291 4,o0 6,22 oo0041 34.96 1,76 22,99 6.33 35554 5o17 53112 6.64 35004 4,23 29,04 6.84 14.2 43512 -,31 16,96 5.9 3.92 0.0059 44o76.50 18,81 4.19 22,30 -1.68 9.232 3.56 17.2 33~o4 -4,31 18.69 7.35 5395 0o0080 32,58 -3.64 17,26 4,05 29,84 -4,45 17094 3078 20 25084 -4,74 15054 9ol1 265 0o0097 25 44 -4.96 16,41 2.65 25,28 -4753 16013 2,88 26.06 -12,94 69,16 1.o90 18 30.81 4.68 16.50 11o2 5305 0,013 (steel 30,64 1o326 9.594 2.92 float) 30.80 -1o73 10o19 3o00 30.24 -4.56 14.63 2056

DATA SHEET NO. 2a, B'- + BA-lA ) L — A 6 RAW DATA COMPUTED DATA Rotameter Mo M1 M2 P1 Gt RCe reading psig (gec,) l-7;k 70F 7.4 132.8 8.8 167.4 5.0 CE =1.0 126 0.0016 66.56 3.24 84.42 128 19.86 -11.41 29.58 116 40.28 -10o.0 53o54 127 9.7 26.0 - 5.49 11.74 8.0 39.0 0.00285 51.1 -20.34 28.29 39.6 48.36 1.36 21.12 43.6 54.06 9.48 25.92 44.8 11.0 43.22 19.12 22.2 9.8 30.1 0.0037 51.32 1.00 16.14 28.8 51.28 - 1.66 15.50 30.1 14.6 65.10 6.12 11.94 14.5 17.4 0.0061 66.10 - 0.97 13.27 20.0 63.98 - 0.74 11.30 1767 66.12 - 1.39 12.48 18.4 16.0 62.65 2.54 10.62 16.5 16.79 0.0070 57.90 4.50 9.01 15.0 64.11 4.55 10o.45 15.8 18.2 56.38 - 5o90 5.714 18.5 9.05 0.0085 62.36 0.13 71.06 CE = 0.1 11.38 57.92 3.49 64. O0 10o.69 61.08 1.57 52.46 11d74 20.4 61.50 6.11 77.82 21 11.66 0.010 58.33 6.24 72.98 9.56 58.93 4.27 65.66 10o,65 l9ss 57.59 0.12 47.28 26.5 8.30 0.014 (s4ee 57.54 - 0o83 45.70 7.92 - \aet) 59.15 - 0.58 48.94 8.24

-0loC. Determine correction factors for pressure (end effects data, L = 2.10 inches, Data Sheet No, 1): = Psig @ inlet, L —2o10 in, = Psig @ inlet, L=5.87 in. Vt =flow rate, cfm ~ 14,7 psia, 70~F.!tl= the flow rate which would have corresponded to A7 (L = 2.10 in) if the inlet pressure had been 12_ instead of?d, lye 4 j (32) 0,00165 1o 9 44 0O.00173 0,0025 2,6 6,4 0.0028 0o 00335 5 9 8.5 0,o00375 oo0041 4,0o 10.2 Cooo0048 0,0059 509 13.8 0.0072 0o0080 7o3 17.6 0,0102 0.0097 9.1 20.6 0,0126 000130 11.2 25.4 0.0174 Note: F may be found from Figure I-2, "Inlet Pressure vs. Flow Rate, which is obtained from the data of Part B, Data Sheet No. 2. Do Plot variance ( Cit' ) vso flow rate (v~) for L = 5.87 inches (Data Sheet No. 2) and L = 2o10 inches (Data Sheet No. 1, 6.j vs V=); See Figure I-3, "Variance vso Flow Rate." Then "band" each set of points with the lines describing the maximum and minimum variances which could reasonably be ascribed to a given flow rate. (Note that At cannot be less than 2.1, the variance of a 5.0 second square %V wave. This gives a minimum value of ~ which each curve must assymtotically approach for high flow rates ) Thus the maximum and minimum values of A64 may be found for any given flow rate

in the range covered; these values characterize the mixing which occurs in a core of length AL= ~7-02/O3-' iceinches o E.g. for 1) 0,0 cft at 14.7 psia, 70~F, the maximum value of aG is approximately Il z- 2 I'-1 sec2, and the minimum value is approximately 4 6 4, 1 4 $05 sec2. E. The average flow pressure is determined as a function of flow rate and inlet pressure, P2, by an integrated form of Darcy's Law (see Appendix II.) FPRowin 9 2-o4E (33) These results are shown in Figure 1-2. F. For selected increments of flow rate, the dispersion parameter Pe/Ao is computed; the results are shown in Figure 20a. The example calculation shown here is for a flow rate of 0.002 cfm measured at 700F, 14.7 psia. 1. For a flow rate of 0.002 cfm, the average flowing pressure is 17.7 psia (see Figure I-2) 2. The mean interstitial velocity is -E - - - -oo - \. 1 -0 _6_ 40 60,oI 17 tZ34 scC 3. Core length AL:- 3,1 inches 0.314 4-, 4. 1/res. time uIL - oOo6./3l4 - C303 0 S, 5. Max. value of Al\C 677 M value of 4 Figure I-3 Min. value of 40

-109100 x xx L 5.87" \x x < L=:2.10" 10c 0 | I I I I I III I I I I I I! I 0.001 0.01 0.1 GAS FLOW RATE, cfm at 70~F, 14.7 psia Figure I-2. Variance (sec ) vs. Flow Rate (cfm), Core BA-2, 0 = 23.4%, AL = 3.77", Bandera Sandstone.

36 34 32 30- 3 2 P2 28 - 3Pvg- p22_1 26Psig at Inlet 24 22 cm'V 20 I-:: CA 0. 14 0.002 CFM 12- - AVG. FLOWING PRESSURE, Psig I0 8 6 4 2 0 6 8 10 12 14 16 18 20 22 24 26 28 30 ROTAMETER READING Figure I-3. Inlet Pressure vs. Flow Rate, Core BA-2 + BA-2A.

-111Max, LC- A t/L) - 9.4 1) X,4, 67 -- o 0 7 Min, LVS A (Q - 9L,1 ADc 4. o 037 6. -)Q- U 0,08S COtDO56z,4+0 ot314 O ctq?o ^r 2 2 -1,4 F A 0X1

APPENDIX II SUPPLEMENTARY CALCULATIONS 1! The Average Flowing Pressure of a Gas Through a Porous Medium, With Downstream Pressure of One Atmosphere, is Given By: where _=inlet pressure (atm. abs.). Darcy's Law for Gas Flow: or, in integrated form, with X-?%(x)) X-at inlet::/ (P) -p ), )where 2,ca -3L.P-(4 1 2. The Relationship 11 T Is Valid Even If Mo, M1, and M2 are Computed Beginning at t Lt' ED, and if Estimated Values of Ho are not Accurate -112

-113 where t = real time t = 0 at pulse injection to = start of injection tCN ~t) t C t / t~~:(t) d Now, the integrals calculated by the computer program illustrated in Figure -re. Mo &Ord3 CCtL t - OO C tt) dt M1 -=o() a X)!r 00 M - L5: CCtL — t ) L dt —,) at Ml ~C 0 t, si -2, Therefore IA=,So C &t)+~dt-tL3Q / { - (c t. 4)~D''i SD 2 c. c L-t ~ / co ~, tXCt ) t)9* +.

10 z w SAPPHIRE cr 10- ~~~~~~~~~FLOAT w STEEL FLOAT w 2 0 0.0001 0.001 0.01 0.1 FLOW RATE, CUBIC FEET PER MINUTE Figure Il-i. Rotameter Calibration for Nitrogen at 700F, 14.7 psia.

APPENDIX III FIELD APPLICATIONS The Analysis of a Simple Reservoir Problem Assume that methane is to be displaced by ethane in a Boise sandstone formation at 2000 psia, 104Fo. The sandstone contains interstitial water, which occupies 10% of the pore volume. Assume that there is no effect of overburden pressure or temperature on the electrical resistivity factor (an e,0versimplification according to Helander, refo 31). An observed value of the molecular diffusion coefficient has been reported to be 8.4 x 104 c2/sec. The Boise sandstone (porosity 32%,.permeability 260 md, electrical resistivity factor 9.5) is assumed to have the mixing properties of the sample labeled BO2 in this manuscript, and it is desired to estimate the effective longitudinal dispersion coefficient at superficial flow rates of 3 x 10-3 cm/sec and 3 x 10-2 cm/seco From the tabulated results in Tables II and III, V_ lo.1 and d -'~ O,b6cm in the equation _ - ~tB\'s ~e-~ (9b) (NOTE: Experimental values of the molecular diffusion coefficient should always be sought for systems of more than two components, since mathematical approximations are not only cumbersome but subject to considerable error ) Peclet numbers are first computed, converting the given superficial velocities to interstitial velocities: For = XO10 3, 3x - i 0 y _ _ _ c c.3z ( ^o \) seam -115

and go 8,4...O-+o For c - 3LO0 7/. Ie 7 F 1 95X&32x< -9 o366 (approximate, since the value of the resistivity factor obtained for the dry core will be too low when an immobile phase is present.) Hence _D- 06,+ For VU- 3~' Cq dM/~eC: 36 0(,7) - 0773 For 1> Mil lU -.-.o(,4~,, ) = 4z,,o c 7-, _For?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Y,~~~~~~~~~~~~~~~~~~ c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3o6-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, 1Q~~~~~~~~~~~~~~j~es