STATE OF ILLINOIS 
 
 WILLIAM G. STRATTON, Governor 
 
 DEPARTMENT OF REGISTRATION AND EDUCATION 
 
 VERA M. BINKS, Director 
 
 ELECTROKINETICS 
 
 Development of Equations for 
 Two-Phase Flow Through 
 Single Capillaries 
 
 Norman Street 
 
 DIVISION OF THE 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 JOHN C. FRYE, Chief URBANA 
 
 CIRCULAR 276 
 
 1959 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://archive.org/details/electrokineticsi276stre 
 
ELECTROKINETICS 
 
 I.— Development of Equations for Two-Phase 
 
 Flow Through Single Capillaries 
 
 Norman Street 
 
 ABSTRACT 
 
 This report is the second part of a study on the electro- 
 kinetics of the flow of oil and water through petroleum reservoirs. 
 
 The development of electric charges at the oil-water inter- 
 face is discussed and equations are presented for the streaming 
 potential, electro-osmotic pressure, and electroviscous effect dis- 
 played by a single capillary filled with anannulus of wetting phase 
 (water) and having a central core of immiscible nonwetting phase 
 (oil). 
 
 Although it is unlikely that oil and water would flow through 
 a porous medium following the theoretical pattern set forth here, 
 it seems certain that some fraction of the flow is of this type and 
 would modify considerably the measured electrokinetic effects. 
 
 INTRODUCTION 
 
 Although the effect of electrokinetics in retarding the flow of a single phase 
 through capillaries and porous plugs has been treated theoretically and observed 
 experimentally, no comparable work has yet been done on the simultaneous flow of 
 two immiscible fluids. There seems little doubt that during such flow both fluids, 
 under suitable conditions, undergo an electroviscous retardation; but it is also 
 possible that, under other conditions, one of the phases may be accelerated so 
 that the measured viscosity is less rather than greater than the bulk viscosity. 
 Such decreased viscosity actually has been observed by Templeton (1954) and 
 Templeton and Rushing (1956) during capillary flow of water and oil through micro- 
 scopic capillaries. 
 
 Before it is possible to derive an equation for the electroviscous effect re- 
 sulting from two-phase flow through a porous medium, it is necessary to develop a 
 model for that flow. Generally such a model is built up from some combination of 
 capillary tubes. Thus the simplest theoretical approach is to equate the porous 
 medium to a single capillary of suitable length and radius for the purpose of 
 describing its hydrodynamic behavior. The logical extension is to a combination 
 of tubes of varying radii, arranged in parallel and series. Such a series model 
 was used by Scott and Rose (1953) to modify the Yuster model and thus help explain 
 the nondependence of relative permeability on viscosity ratio. 
 
 [1] 
 
2 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Network models (interconnecting networks of parallel and series elements) 
 have been analyzed by Fatt (1956) who found them useful in predicting flow behav- 
 ior; this work has been extended by Rose (1957). The use of network models for 
 analyzing electrokinetics in porous media has been treated by Overbeek and Wijga 
 (1946), Mazur and Overbeek (1951),and by Lorenz (1953a). 
 
 Wyllie and Gardner (1958) suggested that one might start with a bundle of 
 capillary tubes, cut them into thin sections, and rearrange these sections at ran- 
 dom laterally to produce a "pseudo-model. " 
 
 As pointed out by Tempi eton, the first concern is to gain knowledge of the 
 mechanism of the flow through individual capillaries. 
 
 Using a "bundle of capillary tubes" model, Brownell and Katz (1947) sug- 
 gested that both phases flow through all the tubes, whereas Wyllie and Gardner 
 suggested that each phase follows a separate path. 
 
 It is possible to combine these views (as I plan to do in another article) 
 and to divide the flow into three regions: a) tubes flowing both wetting and non- 
 wetting phase; b) tubes flowing only wetting phase; and c) the remainder of the 
 tubes, wetted by an immobile film of wetting fluid and flowing the nonwetting 
 phase. 
 
 The percentage of the total flow in each region can be determined by analy- 
 sis of relative permeability curves carried out when the electrokinetic effect is 
 negligible. From the contribution of each flow type to the total streaming potential, 
 or volume flowed during electro-osmosis, one can then calculate the total streaming 
 potential or electro-osmotic pressure to be expected, and thus also predict the 
 electroviscous retardation to flow. 
 
 CHARGED OIL-WATER INTERFACE 
 
 There is much experimental evidence that charged interfaces develop be- 
 tween oil and water, but the distribution of charges at this interface is open to 
 discussion. Thus Booth (1951), in deriving equations for the calculation of zeta 
 potentials from the electrophoretic mobility of oil droplets dispersed in an aqueous 
 medium, considers three possible distributions of charge: a) the charge is concen- 
 trated in a thin layer at the interface; b) the charge is distributed in uniform density 
 throughout the droplet; c) the charge is distributed in an electric double layer de- 
 veloped in the oil phase. 
 
 The equations developed for distribution c reduce to those for distribution b 
 when the double-layer thickness becomes greater than the radius of the droplet. 
 One may consider that distribution c is the intermediate one and that the double 
 
 layer can expand into uniform density, b, 
 or compress into a thin interfacial layer, 
 a. Thus if we picture the general charge 
 distribution at the oil-water interface to 
 be that of two diffuse double layers, we 
 can illustrate it schematically as in 
 figure 1 . 
 
 Here, since the double diffuse double 
 layer as a whole must be electrically 
 neutral 
 
 /°0 /° CO 
 
 f yOjMdx + J p 2 (x)dx = (1) 
 
 "CO ^° 
 
 Aqueous phase 
 
 Oil phase 
 
 Fig. 1. - Schematic representation of the 
 
 distribution of space charges in the 
 
 oil and water phases 
 
TWO-PHASE FLOW THROUGH SINGLE CAPILLARIES 
 
 which will also describe the charge condition in the event of reduction to either 
 a or b. p i, A, = charge density in phase 1 and phase 2, respectively. 
 
 The double-layer thickness in either phase is generally given by the 
 expression 
 
 where 
 
 X- /^ Do 2 kT 2 (2) 
 
 2nz z e z 
 
 k = Boltzmann's constant 
 
 T = absolute temperature 
 
 n = number of dissociated molecules per m^ 
 
 z = valency of ions 
 
 e = charge of an electron 
 
 D = relative dielectric constant 
 
 Do = absolute dielectric constant of a vacuum 
 
 However, for purposes of comparison between aqueous solutions and hydro- 
 carbons the expression given by Klinkenberg and van der Minne (1958) is much to 
 be preferred. This expression may be arrived at from (2) by a series of substitu- 
 tions, thus 
 
 kT 
 
 where 
 
 and 
 
 m 61777a 
 
 A m - coefficient of molecular diffusion 
 
 a = radius of an ion 
 
 77 = viscosity of medium 
 
 where 
 
 6TT77a 
 
 v = ionic velocity under the influence of an electric force, F 
 F = zeE (E is electric field strength) 
 
 It is also true that the current, I, flowing under the influence of a potential 
 difference, E, in a liquid of conductivity, K, is 
 
 I = KE = 2nvze 
 
 Hence it follows that in (2) kT may be replaced by 
 
 * c j o ? ? „ Eze Eze 6-m?a v , c 
 
 A m 6tT77a and 2nz^e^ = K — — = K — y~ L =K6-n-r;a 
 
 and so 
 
 ^ = DD A m 
 
 K 
 
 Klinkenberg and van der Minne point out that A m may be assumed to have 
 about the same value in hydrocarbon and water, because in the hydrocarbon the 
 somewhat larger size of the organic ions will be compensated for by the lower 
 viscosity of the medium. 
 
 Early experiments on the mobility of oil droplets dispersed in water were 
 reported by Ellis (1912), Mooney (1924), and Limberg (1926). Some authors have 
 reported their results in terms of mobilities (microns per second per volt per centi- 
 
_L 
 
 SOLID-WATER INTERFACE 
 
 I 
 
 OIL PHASE 
 
 WATER PHASE 
 
 Fig. 2. - The flow model showing the dis- 
 tribution of the phases and of their 
 associated double layers 
 
 4 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 meter), and others as zeta potential 
 (millivolts). Although for large spherical 
 solid particles dispersed in water at 25°C 
 the zeta potential is numerically equal to 
 the mobility multiplied by 14.3, such a 
 simple relationship does not apply be- 
 tween the mobility and zeta potential of 
 oil droplets. 
 
 First there is the problem of the 
 charge distribution, which may be one of 
 the three possible distributions listed by 
 Booth, and, second, there is the fact that 
 the droplets are liquid, which causes them 
 to have a velocity different from that of 
 
 an equivalent solid particle. Booth suggests that the mobility of the equivalent 
 solid particle will be greater, but Jordan and Taylor (1952) suggest it will be less; 
 Jordan and Taylor's hypothesis is supported to some extent by experimental evi- 
 dence. 
 
 It would appear that the zeta potentials of paraffin oil droplets may approach 
 values of -86 mV in water and -151 in 0.0036M sodium oleate (Urbain and Jensen, 
 1936). Powney and Wood (1940) report mobilities of 4.35 u./sec/v/cm for Nujol 
 droplets decreasing to about 2 at 0.005N. The mobility was considerably increased 
 when the droplets were dispersed in sodium oleate and sodium dodecyl sulphate 
 solutions. Newton (1930) had reported velocities of 3.9 in water with a conduc- 
 tivity of 4xl0~6 ohm~l cm~l. 
 
 More recently Douglas (1950) found that dissolved sugar reduced the mobil- 
 ity of hydrocarbons in solutions of nonionic substances. He explained this as being 
 due to the preferential adsorption of the sugar, which reduced the ionic adsorption 
 and consequently the surface charge density. 
 
 Many workers have studied hydrocarbons other than paraffin. Taylor and 
 Wood (1957) found zeta potentials of -70 to -100 mV for decalin droplets in NaCl 
 solutions, and Carruthers (1938) attempted to correlate measured mobilities with 
 chemical composition using n-Octadecyl, n-Undecyl, n-Octyl, and A^'^n-Un- 
 decenyl alcohols and n-Octadecane and A-^ ^n-Octadecene a n dispersed in solu- 
 tions 0.01N with respect to sodium ion. Douglas (1943) determined the mobility 
 of droplets of paraffin wax, dodecane, octadecane and A^-' ^octadecene, and of 
 mixtures of hexane-dodecane, cyclohexane-dodecane, benzene-dodecane and 
 decalin-dodecane, getting values of 1 - 1.5 u,/sec/v/cm in solutions kept 0.01N 
 in sodium ion at various pH's. 
 
 DEVELOPMENT OF EQUATIONS 
 
 This paper deals with the electrokinetics to be expected when two phases 
 are present in one capillary tube, and considers the flow through a single capillary 
 under the conditions suggested by Brownell and Katz (1947) and by Yuster (1951), 
 namely that the nonwetting phase (oil) should flow in a cylindrical portion of the 
 wetted (by water) capillary and concentric with it (fig. 2). 
 
 These authors assume that the velocities of the oil and aqueous phases are 
 equal at the interface during such flow, and Yuster gives the following expression 
 for the velocity distribution in the oil phase 
 
TWO-PHASE FLOW THROUGH SINGLE CAPILLARIES 
 
 P 
 
 4L 
 
 ( R 2- r 2) t ( r 2_ a 2) 
 
 (4) 
 
 where u Q = velocity of laminar layer of radius a 
 
 R = radius of capillary 
 r = radius of oil phase 
 7] w = viscosity of water phase 
 7] Q = viscosity of oil phase 
 P = pressure drop across capillary 
 L = length of capillary 
 
 The average velocity of the oil phase is 
 
 P 
 
 8'7 Lr2 
 
 2 ( R 2 r 2_ r 4) 7 7o + r 4 
 
 (5) 
 
 /w — L / VV J 
 
 and the velocity (v w ) of any laminar layer (radius a) in the aqueous phase is 
 
 '» = T^r ' R2 - a2 l < 6 » 
 
 Streaming Potential 
 
 Assume that both the capillary and the oil surfaces are negatively charged 
 so that mobile positive counter ions are located in the aqueous phase. (The 
 analysis will remain valid, with due attention to signs of the zeta potentials, in 
 the event of other charge distributions.) As only one ionic solution is in contact 
 with both the capillary wall and the oil surface, it is apparent that the double 
 layers associated with both surfaces will be of the same thickness (X). Then 
 assuming, for simplicity in the derivation, that all the counter ions are located in 
 a Helmholtz double layer, it is possible to derive an expression for the streaming 
 potential caused by the movement of charges in the aqueous phase (Wood, 1946; 
 Butler, 1940; Perrin, 1904). 
 
 The streaming potential arises partly from the transport of the positive ions 
 of both the double layers in the aqueous solution, namely those associated with 
 the capillary wall charge and those associated with the oil surface, and partly from 
 transport of the negative charges associated with the oil itself. It is of course 
 essential that the thickness of the aqueous phase be such that the double layers 
 do not overlap and that the distribution of charges in the oil phase be known. 
 
 It is possible to suggest that the charges in the oil phase are located in a 
 Helmholtz double layer at a distance (X ) of approximately 
 
 X = DqK . x 
 
 from the oil-water interface (D Q and K are the dielectric constant and the conduc- 
 tivity of the oil, respectively). 
 
 However, the double-layer thickness in the hydrocarbon phase will be great, 
 and, as we consider only capillaries of small diameter, it is more likely that the 
 charge is homogeneously distributed throughout this phase as suggested by 
 Klinkenberg and van der Minne (1958). 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 (1) Oil Phase 
 
 -/: 
 
 If electrokinetic measurements in the aqueous phase indicate a surface 
 
 charge density of 
 
 'CO 
 /^Mdx = -CT2 
 
 'o 
 on the oil phase, then the surface charge per unit length is -2-rrra2; an expression 
 
 that describes the charge per unit length of this phase whatever the charge dis- 
 tribution. 
 
 If the oil should move with an average velocity, v , the current of con- 
 vection is 
 
 I = -2Trrcr 2 v 
 and substituting the value of v from equation 5 gives 
 
 Io = " 
 
 ?!l2l.T 9l o2r2_Wh3o^ 
 
 4 Vo 
 
 -[2(R^-r4)^ + r«] (7) 
 
 1 ' w 
 
 (2) Aqueous Phase 
 
 Assuming that the surface-charge density in the Helmholtz double layer 
 associated with the capillary interface is <r ^ and that associated with the oil- 
 water interface is a 2 , the current caused by the transport of these charges is 
 given by 
 
 I w = 2-n-RaiVj + 2Trro"2V2 
 
 where v^ and v 2 are the velocities in the double layer associated with the cap- 
 illary surface and the oil surface, respectively — that is, v\ is the velocity of 
 radius R-X and V2 is the velocity of the annular layer of radius r+X . Thus 
 
 v l = 4^-lR 2 -(R-A) 2 ], v 2 - 7^-[R 2 -(r+X)2] 
 ^7w HT ]w 
 
 and 
 
 I w = ^T" t(2RX)Rai + (R 2 -r 2 -2rX)rcr 2 ] (8) 
 
 (3) Total Streaming Potential 
 
 Because the conductivity of the oil phase (10~ 10 to 10~ 12 ) is much smaller 
 than that of the aqueous phase (10~3 to 10~6) the current of conduction will be 
 given by 
 
 I s = KTr(R 2 -r2)E 
 
 where 
 
 K = conductivity of the aqueous phase 
 E = streaming potential 
 
 The total current carried in the streaming liquids is I = I w +I , and at 
 equilibrium I = I s , and so it follows when 
 
 R 2 - i-2 r 2 
 
 Sw = 5 — and s o = 7T2' tnat 
 
 R z R 
 
TWO-PHASE FLOW THROUGH SINGLE CAPILLARIES 
 
 PX f r ^w R i. 
 
 (9) 
 
 or putting 
 
 then 
 
 d£i d£ ; 
 
 0"t = i — r , cr = 
 
 1 4ttX ' "2 4^ 
 
 DP f, Rt7 w i 1 
 
 ELECTRO-OSMOTIC PRESSURE 
 
 By considering the development of an electro-osmotic pressure in each phase 
 of this system we can consider conditions that exist when there is no movement at 
 the fluid-fluid boundary. Hence the problem in the aqueous phase is simply that of 
 the electro-osmotic pressure developed in an annulus, and the problem in the oil 
 phase is the electro-osmotic pressure produced in a capillary tube with a homo- 
 geneous charge distribution in the liquid. 
 
 (1) Aqueous Phase 
 
 Following the method of Porter (1920) we have: total charge in unit length 
 of annulus of radii R and r is 
 
 2ttR<t^ + 2tttct2 
 
 and hence with an applied potential difference of E the electric force causing move- 
 ment is 
 
 Fj = 2ir(Ro- 1 + ro- 2 )E 
 
 The frictional forces opposing flow are 
 
 F 2 = -2iTR7^ w dv/da+ 2-TTrT^dv/da 
 
 The velocity within the double layer varies linearly with distance from 
 either wall and is zero at a=r and a=R, and so within the double layer extending 
 from R to R-X, dv/da = -v/X , whereas in the double layer extending from r to 
 r+X, dv/da = v/A . It follows that 
 
 F 2 = 2 T T(R+r) 7?w v/X 
 
 At equilibrium F^ = ?2' anc * so ^ f°H° ws tnat 
 
 _ (Rg- 1 + ro-2)EX 
 
 V= ( R+r ^w 
 
 or 
 
 DE(R£ 1+ r^ 2 ) 
 
 v = — 
 
 4Tr(R+r)77 w 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 or when V is the volume rate of flow 
 
 v = DE(R-r)(R£ 1+ r(; 2 ) 
 
 4 ^w 
 
 (ID 
 
 In order to express this in terms of an electro-osmotic pressure, equation 
 11 should be equated with the equation for flow through an annulus (Lamb, 1932). 
 Thus 
 
 EB»zlMillM . «& (R 2 . r 2, (r2+ r 2 _ ^, 
 
 47 7w 8l 7w ^ n R / r 
 
 and so 
 
 Pw ~~ 
 
 2DE(^ 1+ S 2^ 2 ) 
 
 irR 2 ( 1 + S 2 ) (1 + S -T-T^i 
 
 tnS 
 
 o 2 
 
 (12) 
 
 (2) Oil Phase 
 
 Total charge per unit length is -l-nv^, and with an applied field strength 
 of E the electric force causing motion is 
 
 -2iTro"2E 
 
 The viscous force opposing motion is 
 
 2irr.Pr/ 277.77 
 
 and so 
 
 P = -2Ea 2 /r 
 
 or, expressing -03 in terms of the zeta 
 potential measured in the aqueous phase 
 
 Po = -dr (13) 
 
 The negative sign indicates that the 
 motion in the oil phase will be in a 
 direction opposite to that in the aqueous 
 phase with the assumed charge distri- 
 bution. 
 
 (3) Total Electro-Osmotic Pressure 
 
 The aqueous phase electro-osmotic 
 pressure (P w ) as derived acts over an 
 area of ttR^Sw, and the oil phase pres- 
 sure (P Q ) as derived acts over an area 
 of ttR2So. The total electro-osmotic 
 pressure acting simultaneously over the 
 area ttR^ is therefore 
 
 P = PwSw + PqSq 
 
 ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 
 
 Water saturation 
 Fig. 3. -Water saturation against 
 
 Z = ( 1 + S Q 2 ) ( 1 + S Q - 
 
 In S -2 
 
TWO-PHASE FLOW THROUGH SINGLE CAPILLARIES 
 Substituting from the expressions (12) and (13) gives 
 
 2DE{Sw£l + SoMSw-H)£2} 
 
 P = ~ 7 " (14) 
 
 S w 
 
 Z = (1 + S 2) (1 + S - . _i ) and is plotted in figure 3. 
 "nSo 2 
 
 ELECTROVISCOUS EFFECT 
 
 In considering the electroviscous effect that arises out of this streaming 
 potential and electro-osmosis, one must decide whether it is possible to test the 
 effect microscopically or macroscopically . For example, Lorenz (195 3b) in testing 
 the effect during the flow of acetone through quartz plugs shorted out the ends of 
 the plug and judged the existence of the effect from the differences in shorted and 
 unshorted flow. He therefore, at least implicitly, considered that the effect was 
 macroscopic in that he ignored the possibility that backflow might tend to occur 
 in the liquid during its transport between the electrodes, and assumed that a 
 potential difference must exist for the effect to be seen. 
 
 On the other hand, if certain viscosity increases in suspension flow in a 
 concentric cylinder rotational viscometer are ascribed to the effect of electro- 
 viscosity, one is implicitly supposing that the effect is microscopic and that it 
 affects the particle at the instant of its displacement because in this case there 
 is a closed circuit and no current flow. 
 
 The relevance of these concepts to the two-phase problem is that we could 
 consider that each phase is reacted on independently by the streaming potential or, 
 conversely, that both are reacted on together and to the same extent. In the first 
 case we can sometimes predict an increase and sometimes a decrease for each 
 phase, depending on such factors as saturation, viscosity ratio, and relative 
 values of the surface charges. In the second, the electroviscous effect would 
 always increase viscosity in the same ratio for each phase. 
 
 An expression for the electroviscous effect can be derived by considering 
 the volume of electro-osmotic backflow, resulting from the streaming potential, 
 that must be subtracted from flow caused by the pressure difference imposed on a 
 tube. Or one can consider that the electro-osmotic back pressure reduces the 
 pressure causing flow, a simpler idea to apply in the present case, although it is 
 not to be supposed that these back pressures have any physical significance. 
 
 An expression for the electroviscous effect may be simply derived (Street, 
 1959) by finding a value for the electro-osmotic back pressure (P-.) that arises from 
 the streaming potential and subtracting it from the applied pressure (P) to get the 
 pressure (P2) actually causing flow. Then 
 
 P 2 = P- Pl 
 
 If the liquid is flowing through a capillary tube of radius R and length L 
 with a velocity v, its apparent viscosity (-n a ) has a value of 
 
 PttR 2 
 la 8Lv 
 
 while the true viscosity (77) has a value of 
 
 P 2 ttR 2 
 77 8Lv 
 
10 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 That is 
 
 773/77 = P/P 2 (15) 
 
 For both the aqueous and oil phases, P 2 is given by rearrangement of 
 equation 10 as 
 
 D ^- s oU^s o i, Q 
 
 The streaming potential E gives rise to an electro-osmotic back pressure 
 (P^) that has the value of P w by equation 12 for the aqueous phase and of P Q by 
 equation 13 for the oil phase. 
 
 Combining equation 15 with these values for the electro-osmotic pressure 
 gives 
 
 ? I r R7) w 1 i 
 
 law n D (£i + Vt 2 )Ui-s [i + n^s ^ 2 } 
 
 -1+ — (16) 
 
 ^w 2-n l R l Tj w KZ 
 
 =1- 5 ' (17) 
 
 V 8tt^RX77 w KS w 
 
 where 
 
 77 aw , 77 ao = apparent viscosity of water and oil phases, respectively, 
 If the less likely macroscopic approach is used, then 
 
 2DE{s w £l + Soi(Sw-|f)C2} 
 
 and so 
 
 1 " .R 2 Z 
 
 2D 2 [S w C 2 + S I(S w -§|)^][^-So(l+^So*)^] 
 
 f = 1+ — ~ (18) 
 
 Tr 2 R 2 Z77 w KS w 
 
 DISCUSSION 
 
 I do not believe that two immiscible fluids will actually flow through a 
 porous medium according to this simple (and unstable) pattern. However, it seems 
 certain that some fraction of the flow through porous media is of this type, and that 
 even a small fraction of such flow would modify considerably the measured electro- 
 kinetic effects. 
 
 It is probably impossible to test the equations in capillary flow experiments 
 except at very low wetting-phase saturations, and then of course care must be taken 
 to make sure that the double layers developed in the aqueous phase do not overlap, 
 as has already been discussed (Street, 1959). The best test of the equations would 
 appear to lie in experiments conducted with porous media for which the fraction of 
 the various flow types has been determined. 
 
TWO-PHASE FLOW THROUGH SINGLE CAPILLARIES 11 
 
 The equations for streaming potential, electro-osmotic pressure, and electro- 
 viscosity for porous media supposedly made up of bundles of capillary tubes of 
 equal and varying radii will be presented in a future publication. 
 
 REFERENCES 
 
 Booth, F. J., 1951, The cataphoresis of spherical fluid droplets in electrolytes: 
 Jour. Chem. Physics, v. 19, p. 1331. 
 
 Brownell, L. E., and Katz, D. L., 1947, Flow of fluids through porous media, 
 Part II - 2 . Simultaneous flow of two homogeneous phases: Chem. Eng. 
 Progress, v. 43, p. 601. 
 
 Butler, J. A. V., 1940, Electrocapillarity: Methuen, London. 
 
 Carruthers, J. C, 1938, The electrophoresis of certain hydrocarbons and their 
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 Douglas, H. W., 1943, The electrophoretic behaviour of certain hydrocarbons and 
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 Douglas, H. W., 1950, The influence of sugars on the electrokinetic potential 
 
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 Ellis, R. Z., 1912, Die Eigenschaften der Olemulsionen. I.- Die elektrische 
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 I - Capillary pressure characteristics, p. 144. 
 
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 III - Dynamic properties of networks with tube radius distributions, p. 164. 
 
 Jordan, D. O., and Taylor, A. J., 1952, The electrophoretic mobilities of hydro- 
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 Klinkenberg, A., and van der Minne, J. L., 1958, Electrostatics in the petroleum 
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 Lamb, H., 1932, Hydrodynamics: Cambridge University Press. 
 
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 Lorenz, P. B., 1953a, Electrokinetic processes in parallel and series combinations: 
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 electrokinetic effects: Rec. Travaux Chim. des Pays-Bas, v. 70, p. 83. 
 
12 ILLINOIS STATE GEOLOGICAL SURVEY 
 
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 Newton, D. A., 1930, Investigation of the cataphoresis of small particles in 
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 Overbeek, J. Th. G., and Wijga, P. W. O., 1946, On electro-osmosis and 
 
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 Perrin, Jean, 1904, Mecanisme de L'Electrisation de contact et solutions 
 colloidales: Jour. Chim. Phys., v. 6, p. 601. 
 
 Porter, A. W., 1920, The physics and chemistry of colloids and their bearing on 
 industrial questions. Section VI - Electric endosmosis and cataphoresis: 
 Faraday Soc. Discussion. 
 
 Powney, J., and Wood, L. J., 1940, The properties of detergent solutions. Part 
 IX - The electrophoretic mobility of oil droplets in detergent solutions: 
 Faraday Soc. Trans., v. 36, p. 57. 
 
 Rose, Walter, 1957, Studies of waterflood performance. III.- Use of network 
 models: Illinois Geol. Survey Circ. 237. 
 
 Scott, P. H., and Rose, W., 1953, An explanation of the Yuster effect: Jour. 
 Petroleum Technology, v. 5, Tech. Note 192. 
 
 Street, Norman, 1959, Electrokinetics. I. - Electroviscosity and the flow of 
 reservoir fluids: Illinois Geol. Survey Circ. 263. 
 
 Taylor, A. J., and Wood, F. W., 1957, The electrophoresis of hydrocarbon drop- 
 lets in dilute solutions of electrolytes: Faraday Soc. Trans., v. 53, p. 523, 
 
 Templeton, C. C, 1954, A study of displacements in microscopic capillaries: 
 Jour. Petroleum Technology, v. 6, p. 37. 
 
 Templeton, C. C, and Rushing, S. S., 1956, Oil-water displacements in micro- 
 scopic capillaries: Jour. Petroleum Technology, v. 8, no. 9, p. 211. 
 
 Urbain, W. M., and Jensen, L. B., 1936, Soaps: Electric charge effects and 
 dispersing action: Jour. Physical Chem. , v. 40, p. 821. 
 
 Wood, L. A., 1946, An analysis of the streaming potential method of measuring 
 the potential at the interface between solids and liquids: Jour. Amer. 
 Chem. Soc, v. 68, p. 432. 
 
 Wyllie, M. R. J., and Gardner, G. H. F., 1958, The generalized Kozeny-Carman 
 equation: World Oil, v. 147. 
 Part 1. Review of existing theories, p. 121. 
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 Yuster, S. T., 1951, Theoretical considerations of multiphase flow in idealized 
 
 capillary systems: Third World Petroleum Congress Proc, Sec. II, p. 437. 
 
 Illinois State Geological Survey Circular 276 
 12 p., 3 figs., 1959 
 
ILLINOIS 
 
 Xcutd of£iaco6v 
 
 i 
 
 CIRCULAR 276 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 URBANA