s 14.GS: CIR263 c. 1 STATE OF ILLINOIS WILLIAM G. STRATTON, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M.BINKS, Director ELECTROKINETICS I. — Electroviscosity and the Flow of Reservoir Fluids Norman Street DIVISION OF THE ILLINOIS STATE GEOLOGICAL SURVEY JOHN C. FRYE, Chief URBANA CIRCULAR 263 1959 Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://archive.org/details/electrokinetics263stre ELECTROKINETICS I. — Electroviscosity and the Flow of Reservoir Fluids Norman Street ABSTRACT Liquids flowing through narrow capillaries frequently exhibit a viscosity that is higher than normal; usually it is high for low-conduct- ivity liquids but is low for very conductive liquids or concentrated so- lutions. This effect may have some influence on the flow of oil and water through petroleum reservoirs, although little note has yet been taken of the possibility. The theory underlying this electroviscous effect is simply devel- oped and some calculations as to its order of magnitude are presented. An equation is given for the electroviscous effect in two-phase flow through a "Yuster" model. INTRODUCTION Experimental observations of an increase in the viscosity of a liquid flowing through a narrow capillary have been reported by a number of workers (Reekie and Aird, 1945; Terzaghi, 1931; Macauly, 1936; Henniker, 1952), and a recent worker has published a theoretical analysis of the origin of such increase in viscosity (Elton, 1948). The electrical origin of the effect was early recognized. In fact, Smoluchow- ski (1916) had given an equation for the "electroviscous" effect in suspension flow; Bull (1932) gave a simple derivation of the effect for flow through capillaries; and White, Monaghan, and Urban (1935) discussed the effect of electrical charges on flow through cellophane membranes. More recently, Lorenz (1952), in a sophis- ticated approach to electrokinetic phenomena, dealt with electroviscosity of liquid flow through both capillaries and porous plugs. The effect is apparent only for flow through very small capillaries when it is controlled by the charge at the solid-liquid interface (perhaps more correctly the zeta potential), the conductivity of the flowing liquid, and its dielectric constant. We shall see that high interfacial potential and low liquid conductivity are the factors that contribute most to high apparent viscosity. If an aqueous solution is relatively concentrated its conductivity is high, and even though the solid surface has a high charge, the interfacial or zeta poten- tial nevertheless will be low because the double layer (vide infra) is compressed and there will be no discernable electroviscous effect. If the aqueous solution is dilute, however, conductivity is much lower, zeta potentials generally are high, and a real electroviscous effect may exist. For example, Reekie and Aird (1945) report five to ten times the normal bulk viscosity for water flowing through dia- phragms of rouge, French chalk, and carborundum. In non-aqueous systems we can expect that the conductivity normally will be very low, but, on the other hand, the zeta potentials may also be low although [1] 2 ILLINOIS STATE GEOLOGICAL SURVEY there is some evidence that this is not always true (van der Minne and Hermanie, 1953). If the zeta potentials have some reasonable value coupled with very low conductivity, the electroviscous effect may be large. Of course the flow of reservoir fluids is not generally a matter of simple single-phase flow. But even if oil alone were flowing the effect would be compli- cated because the mineral surfaces are usually water wet and therefore, the oil would not be in direct contact with the solid surface. An immobile water film may exist, thick enough and conductive enough to provide a path for the back flow of any current generated, without such current flow affecting the fluid flow of the hydrocarbon. Or it is possible for a water film to be much thicker and to be flowing along with the oil. In either case the problem is further complicated by the fact that there will be set up not only a solid-water potential but also an oil-water potential which must be taken into account in con- sidering the final effect. In the following discussion some attempt is made to investigate these pos- sibilities and to indicate the order of magnitude of the effects themselves. As the concepts of zeta potential, streaming potential, and electro-osmosis may be some- what unfamiliar, a small space is first devoted to a discussion of the origin of surface charge and potential and to simple derivations of the equations for stream- ing potential and electro-osmosis. BASIC PRINCIPLES Surface Charge Mineral surfaces in contact with aqueous solutions are almost invariably charged. The most important charging mechanisms are dissociation of ionogenic groups and the preferential adsorption of one ion from the solution (Mukherjee, 1920). The electrical potential resulting from these charges is called the zeta potential . Inasmuch as Coulomb attraction exists between the charged surface and any oppositely charged ions in the solution, it may seem surprising that the surface remains charged rather than being immediately neutralized by combination with op- positely charged ions from the solution. In order to understand this, it is neces- sary to consider the forces existing between simple dissolved ions. Consider the case of two oppositely charged univalent ions, for example, Na + and Cl~ in aqueous solution. The attractive force between them is (1) Dx 2 where e = electronic charge (4.8 x 10" 10 e.s.u.) D = dielectric constant (80 for water) x = distance separating the centers of the ions However, as is well known, such ions do not coalesce by the operation of such a force because although the Coulomb attraction between unlike ions tends to draw them together, thermal (Brownian) motions tend to distribute them throughout the solution. Let us compare the thermal energy of simple ions with the energy necessary to separate them. The radii of hydrated Na + and CI" ions are 2.5 and 2.0 A, FLOW OF RESERVOIR FLUIDS respectively, so that at their closest distance of approach the charges are sepa- rated by 4.5 A. So by equation (1) (4.8 x lCT 10 ) 2 , . ... fi . = -* r— = 1.4x10 D dyne 80(4.5 x 10" 8 ) 2 To obtain a value for the work necessary to separate these ions we must in- tegrate force times distance from x = r to x = oo, that is f e+e" e + e~ (4.8 x 10" 10 ) 2 _ . lf1 -i4 = J 7 dx = ~n7~ = ft " x { r Dx z ur 80(4.5 x 10"°) The thermal energy of a molecule or ion is given by kinetic theory and is Kinetic energy = 3/2 kT where k = Boltzmann's constant (1.37 x 10 -1 erg/molec./deg.) T = absolute temperature Thus at 20°C Kinetic energy = 6.10 x 10 erg So we see that in water, the thermal energy of the separated ions and the energy necessary to separate them are very nearly the same. In solvents with dielectric constants smaller than it is in water, the force attracting the ions and the energy necessary to separate them will be much greater, and such ions are not dissociated in solution. Similar considerations apply to the ions adsorbed on the mineral surfaces and the oppositely charged ions in the solution surrounding the surface. Electric Potential The work necessary to bring together from infinity two ions of opposite sign has the same magnitude as that necessary to separate them to infinity from their distance of closest approach. It is convenient to define the electric potential as the work required to bring unit charge from infinity to a charged point of like sign, or alternatively, as the work released when a unit charge of unlike sign is brought to this point from infinity. The potential function is a property of the space surrounding electric charges, every point in space has a potential due to the presence of the ion, and if there are other ions in the space, the total potential at any point is given by the alge- braic sum of the individual potentials at that point due to each ion. The work ne- cessary to bring unit charge from infinity to a distance r from the center of an ion is equal to e/Dr, and this is the potential at a distance r. Zeta Potential and Double Layer Thickness The charged particle surface attracts water dipoles and is covered by a layer of strongly bound water molecules that become part of the kinetic unit. Trapped among the water molecules are commonly some positive charges that also become part of the kinetic unit and by their presence reduce the net charge on the particle (fig. 1). 4 ILLINOIS STATE GEOLOGICAL SURVEY When there is relative movement between the particle and the liquid, the plane of shear is at the outermost edge of the solvated layer, and so it is the net charge that is important in electrokinetic phenomena . The zeta potential then is determined by the work necessary to bring unit charge from infinity to the surface of shear. Surrounding the particle, but relatively distant from it, is an atmosphere of ions in constant thermal movement. The number of positive ions (assuming the particle surface to be negative) in this atmosphere is greater than the number of negative ions, and there are enough positive ions, on a time average, to bal- ance out the net negative charge on the particle. The ions of the ionic atmosphere form the "dif- fuse double layer. " They are not immobilized by the Coulomb attraction of the particle but constantly move in and out between the double layer and the main body of the liquid. It is convenient to consider that the excess positive charges are on a concentric shell at a fixed distance from the particle, the shell is the electri- cal "center of gravity" of the ion cloud, and the distance from the surface of shear to the shell is the thickness of the double layer. If a suspended charged particle is subjected to an electric field it moves to one or the other of the electrodes, at the same time the oppositely charged ionic atmosphere tends to move in the opposite direction and consequently to retard the motion of the particle. The distance from the surface of shear to the hypothetical concentric shell of oppositely charged ions is chosen so that if the ions were ac- tually on this shell they would have the same retarding effect as the ion atmosphere. Thus, although we assume that the opposite charges are present only on the sur- face of the shell, nevertheless we can feel confident that their effect is the same as when they are scattered through the atmosphere. With this model, a large, non-conducting particle, together with its double layer, constitutes a parallel plate condenser with its plates separated by a dis- tance X, the "thickness" of the double layer. In the next two sections we shall examine (a) the effect on zeta of varying the distance of separation of two such plates (at a fixed surface charge density), and (b) the effect of concentration and type of electrolyte in solution, on the double layer thickness. The two taken to- gether show the effect of concentration on zeta at constant surface charge density. Fig. 1. - A charged spherical particle and its bound water molecules. Effect of Double Layer Thickness on the Zeta Potential Consider a particle of radius r and charge Q surrounded by a concentric shell of radius (r + X ) and charge -Q (fig. 2). FLOW OF RESERVOIR FLUIDS The potential at the surface of the sphere is Q/Dr, this being the work ne- cessary to bring unit charge of like sign from infinity to a distance r from the center of the sphere. The resultant po- tential of a condenser consisting of two such concentric spheres is the algebraic sum of the potentials due to the inner sphere at its surface and the outer sphere at the surface of the inner sphere. The potential on the surface of the inner sphere in the absence of the outer sphere would be Q/Dr; the potential due to the outer sphere at any point inside it is -Q/D(r + X ); therefore, C- Dr Dr X r + X D(r + X) and since X</kT_ e + e^/kT) ne = -2ne sinh e^/kT and if the potential is small so that e 400 E p = -2ne 2 3 = !/>!+ I// 5 2 and because 1//0J = tt(R 2 - r 2 )K 1 /L; l/p 2 = irr 2 K 2 /L; I//O3 = ttR 2 K 3 /L therefore K 3 = r 2 /R 2 • K 2 + (R 2 - r 2 )/R 2 • Kj Now water saturation, S w = tt(R 2 - r 2 )L/VR 2 L = and so K 3 = (1 - S W )K 2 + S w Kj In terms of thickness of the water layer K 3 - K 2 - 28/R • (K 2 - K x ) (R< : 2 )/R 2 and if K„<< K, or K 3 - 28KJ/R K, S w K l Figure 6 is a plot of 77 a /77 against S w showing the rapid decrease of7^ a as the wetting phase saturation increases. The tube considered has a radius of 0.5 x 10"" 4 cm and exhibits a zeta potential of 30 mv when filled with a hydro- carbon of D = 2, K 2 = 10 -12 , K x - 10 -6 . TWO- PHASE FLOW In considering flow of two phases through a capillary tube we will take the model used by Yuster (1951) of "a single capillary with the non-wetting phase flowing in a cylindrical portion of the capillary and concentric with it. The wet- ting phase will flow in the annulus between the capillary wall and the non-wetting phase" . FLOW OF RESERVOIR FLUIDS 13 Assuming that both the capillary surface and the oil surface are negatively charged, the charges at both surfaces are balanced by double layer charges car- ried in the solution in a Helnholtz double layer at a distance X from the surfaces. As the ionic concentration in the aqueous phase in contact with both solid and oil is identical, X is the same for both surfaces. By first deriving an equation for streaming potential, then for electro-osmosis, and combining them as was done above, (Street, to be published) it is possible to obtain expressions for the apparent viscosities in both the oil and the water phases. These expressions are R^v 'aw 1 f /w i \ D 2 (£i + S 2 £ 2 ) (£j - S [1 + 4X77 S 2]£ 2 ) = 1 + a - y — ^^^_ (16) ' W 1 S w Z = (1 + S 2) (1 + s - -3) in S "2 -^ = 1- - ^5o_ 8ir 2 RXT7 w K S w where £ ■, and £~ are tne zeta potentials at the mineral and oil surfaces, respec- tively; K is the conductivity of the aqueous phase; rj is the viscosity of the aqueous phase; and 77 that of the oil phase. There are several points to be considered in the use of these expressions. First it is assumed that X is small in relation to R-r, that is, the two double layers in the water must not overlap. In fact, the double layers will affect each other even at quite large distances of separation. In the simple case of two equal po- tentials, ty , separated by a distance 2h, Elton and Hirschler (1949) show that m cosh h/X where \j/ m is the potential at a distance h from either surface. The oil phase will have a dielectric constant much less than the aqueous phase, most likely of the order of 2 rather than 80, and this affects the development of potential, the thickness of X , and the concentration of dissolved substance in the oil phase (Verwey and Overbeek, 1948). It is perhaps easier to see the effect on X using the approach of Klinkenberg and van der Minne (1958) who show that K where E Q = absolute dielectric constant of vacuum A m = coefficient of molecular diffusion Since A is approximately the same value for both water and a hydrocarbon, then V^w = V8^ • 80 Ko 14 ILLINOIS STATE GEOLOGICAL SURVEY Thus the double layer thickness developed in the hydrocarbon phase is much greater than that developed in the aqueous phase. The electroviscous effect is apparent only when the capillary radius is small (1 x 10"^ cm or less) so normal- ly the double layer thickness in the oil phase would be greater than its radius. Under these conditions we can assume a homogeneous distribution of charge through- out this phase and use Rutgers, de Smet, and de Moyer's expression (1957) in calculating the contribution of this phase to the total streaming potential. These considerations have been borne in mind in the development of equa- tions (16) and (17), and these expressions should not be used when R-r approaches X and certainly not if R-r< X . In terms of water saturation we can say that the definite lower limit of applicability is S w = 2X/R Because the movement of the two liquids through a capillary moves positive charges in the water phase and negative charges in the oil phase, the magnitude and sign of the streaming potential set up depends on the relative sizes of these charges, their sign, the viscosity ratios of the liquids, and their saturation in the tube, that is, on £,, £ ?' ^ ' ^ o' anc ^ ^w* Since both positive and negative charges are transported it is possible for the streaming potential to be either positive or negative downstream, and the effect on liquid viscosity may be either to increase it or to decrease it. It is perhaps appropriate to point out that £ 2 is the zeta potential measured in the water phase against the oil-water interface. DISCUSSION It has been shown in the foregoing that viscosity will be increased when an interfacial charge exists, but the increase will be unimportant unless the charge is high, the liquid conductivity low, and the flow channel narrow. Although it could be expected that hydrocarbons would show a large effect because of their low conductivity, this need not necessarily follow because thin conducting films in narrow pores will increase the apparent conductivity so as to considerably reduce the back pressure. When both water and oil flow through a capillary the interaction of the various factors may cause the apparent viscosity of either phase to be less than the bulk viscosity instead of greater. Normally the high ionic concentration of an oil-field brine will reduce X so as to give such low zeta potentials at both interfaces that the effect will be negligible. It is possible, however, that the effect could be significant in a fresh-water flood or, more important, that electrolytes could be added to the flood water to alter £ ■, and £~ so as to g ive maximum recovery. In the laboratory, core experiments are frequently conducted with both fresh water and brine in order to determine their relative effects; any low permeability to fresh water is ascribed to the presence of swelling clays. While admitting that this is a potent factor in permeability reduction, it is also possible, especially in relative permeability experiments, that the electroviscous effect is also oper- ating. The presence of clays will in itself tend to increase the zeta at the water- mineral interface (Street and Buchanan, 1956) and this may increase the viscosity even though the clays are of the non-swelling variety. If the predictions can be borne out experimentally, then we would expect to find a dependence of relative permeability on ionic concentration in the aqueous phase during the flow of oil and water through low permeability cores. FLOW OF RESERVOIR FLUIDS 15 At constant surface charge density, increase of ionic concentration decreases the zeta potential so that very little charge transport occurs in the water phase; however, the balancing negative charges in the oil phase will still be carried with the stream, and the tendency at higher concentrations should be for the oil phase viscosity to increase while that of the water phase stays relatively constant. How- ever, if the ionic concentration or S w increases beyond a certain point, back flow of current through the solution annulus will cause the effect on the oil phase to become negligible also. It is hoped to initiate laboratory tests in the near future with the object of correlating measured zeta potentials at oil-water and mineral-water surfaces with apparent viscosities of the flowing oil and water phases. APPENDIX The identity of \/k with the double layer thickness is by no means obvious and although an understanding of it is not essential to the solution of the problems involved here, nevertheless it would seem more complete to include it. The fol- lowing is modeled closely on the treatment given by Abramson, Moyer, and Gorin (1942). Let equation (5) be written as y2^ =< 2^ ( 5a ) Because for a flat surface, or one of large radius of curvature, the potential ^ de- pends only on the distance x from the surface, equation (5a) becomes and a general solution of equation (5b) is ^r= A e-* x + Be + * x (5c) and because y= when x -► coand ty = £ when x = 0, B = and A = £ , so = ■£■ +*/' 1 (5e) Then Dr £4 _Q_ d±_ W " ' Dr2 + dr x=r 16 ILLINOIS STATE GEOLOGICAL SURVEY and because anywhere inside the sphere and at its surfaced' is constant, thus dii/ — , = and so dr and because Q = 4ttt , $L\ dx/ x=r " Dr 2 t^ which for a particle of any shape has the general form - Ancf/D (5f) ■ dx /x=o Differentiation of equation (5d) gives dx ^ and substitution into equation (5f) at x=o gives dx therefore £ = 4tto/Dk (5g) Comparison of equations (2) and (5g) shows that X = \/k. It should also be noted that at any distance x from the surface, the potential \ff is given by FLOW OF RESERVOIR FLUIDS 17 REFERENCES Abramson, H.A., Moyer, L. S., and Gorin, M. H., 1952, Electrophoresis of proteins: Reinhold Pub. Corp., New York. Bull, H. B., 1932, Die Bedeutung der Kapillarenweite fur das Stromungspotential: Kolloid Zeit., v. 60, p. 130. Butler, J. A. V., 1940, Electrocapillarity: Methuen, London, p. 93. Elton, G. A. H., 1948a, Electroviscosity I: The flow of fluids between surfaces in close proximity: Royal Soc. Proc, v. A194, p. 259. Elton, G. A. H., 1948b, Electroviscosity II: Experimental demonstration of the electroviscous effect: Royal Soc. Proc, v. A194, p. 275. Elton, G. A. H., and Hirschler, F. G., 1949, Electroviscosity IV: Some exten- sions of the theory of flow of liquids in narrow channels: Royal Soc. Proc, v. A198, p. 581. Henniker, J. C, 1952, Retardation of flow in narrow capillaries: Jour. Colloid Sci., v. 7, p. 443. Klinkenberg, A., and van der Minne, J. L., 1958, Electrostatics in the petroleum industry: Elsevier Pub. Co., Amsterdam. Lorenz, P. 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Smoluchowski, M., 1916, Theoretische Bemerkungen uber die Viskositat der Kolloide: Kolloid Zeit ., v. 18, p. 190. Street, N., and Buchanan, A. S., 1956, The zeta potential of kaolinite particles: Australian Jour. Chem., v. 9, p. 450. Terzaghi, C, 1931, The static rigidity of plastic clays: Jour. Rheology, v. 2, p. 253. 18 ILLINOIS STATE GEOLOGICAL SURVEY van der Minne, J. L., and Hermanie, P. H., 1953, Electrophoresis measurements in benzene - correlation with stability. II. - Results of electrophoresis, stability and adsorption: Jour. Colloid Sci., v. 8, p. 38. Verwey, E. J. W., and Overbeek J. Th . G., 1948, Theory of the stability of lyo- phobic colloids: Elsevier Pub. Co., Amsterdam. White, H. L., Monaghan, B., and Urban, F., 1935, Electrical factors influencing the rate of filtration of aqueous electrolyte solutions through cellophane membranes: Jour. General Physiol., v. 18, p. 515. Yuster, S. T., 1951, Theoretical considerations of multiphase flow in idealised capillary systems: Third World Petroleum Congress Proc, Sec. II, p. 437. Illinois State Geological Survey Circular 263 18 p., 6 figs., 1 table, 1959 CIRCULAR 263 ILLINOIS STATE GEOLOGICAL SURVEY URBANA