ILLINOIS GEOLOGICAL 
 SURVEY LIBRARY 
 
 STATE OF ILLINOIS 
 
 WILLIAM G. STRATTON, Governor 
 
 DEPARTMENT OF REGISTRATION AND EDUCATION 
 
 VERA M. BINKS, Director 
 
 JOHN a u„ 
 
 FLUID FLOW IN 
 PETROLEUM RESERVOIRS 
 
 III.- Effect of Fluid-Fluid Interfacial 
 Boundary Condition 
 
 Walter Rose 
 
 DIVISION OF THE 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 JOHN C. FRYE, Chief URBANA 
 
 CIRCULAR 291 
 
 1960 
 
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 a. 
 
FLUID FLOW IN PETROLEUM RESERVOIRS 
 
 III. — Effect of Fluid-Fluid Interfacial Boundary Condition 
 Walter Rose 
 
 ABSTRACT 
 
 Relative permeabilities greater than unity are predicted 
 (for a reason different from that developed in Part I of this se- 
 ries), and it is shown that Darcy's Law does not accurately 
 describe mixture flow phenomena in view of the nonzero ve- 
 locity boundary condition that sometimes exists at the fluid- 
 fluid interfacial boundaries. These indications are not to be 
 gathered from examining available experimental data on rela- 
 tive permeability, but explanations are given to reconcile the 
 discrepancy. 
 
 The analyses are based on mathematical models of res- 
 ervoir rock porous media system. To the extent that the mod- 
 els and the reservoir prototype correspond, we can expect 
 valid inferences that are important to an understanding of the 
 mechanics of oil recovery processes. It is suggested that a 
 flowing fluid has an increased mobility when surrounded (in 
 part or entirely) by another immiscible fluid, whether the phe- 
 nomenon is one of mixture flow in porous media or the trans- 
 port of oil in a pipeline containing some annularly located 
 water. The reported lubrication, moreover, is enhanced if the 
 viscosity ratio of surrounding fluid to the flowing fluid has a 
 low value. 
 
 INTRODUCTION 
 
 Previous papers in this series dealt with the fact that Kozeny-Carman 
 theory leads to some unexpected predictions about the microscopies of mixture 
 flow phenomena (Rose, 1957a, b) . Itwas suggested that the relative oil permeability 
 of reservoir rock, containing small amounts of connate water in pendular satura- 
 tion configuration, might be greater than unity, in a way not anticipated from 
 considerations of Darcy's Law. It also was suggested that some slightly consol- 
 idated sandstones might have a specific permeability greater than that of the un- 
 consolidated sand from which they were derived. 
 
 A recent paper by Russell and Charles (1959) considers how a water layer 
 on the internal surface of a pipe might serve as a "lubricant, " thus lessening the 
 power required to move oil through a pipe. The analyses given are limited to in- 
 compressible Newtonian fluids in laminar motion, and it is the phenomenon of 
 transfer of viscous forces across fluid-fluid interfaces which gives rise to the 
 reported effect. 
 
 [1] 
 
2 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 In earlier work, Yuster (1951) discussed how the flow of fluid mixtures 
 through porous media also might depend on the fact that a finite velocity must be 
 taken as the boundary condition to be assigned at the fluid-fluid interfaces. By 
 considering Poiseuille-type concentric flow in a circular tube to be a model of a 
 porous medium saturated with two immiscible fluids, he concluded that the viscosity 
 ratio of the two fluids would influence the volumetric throughput in a way indicating 
 that Darcy's Law of flow could not be applied. His conclusion was contrary to 
 experimental evidence, as Muskat's summary (1949) indicates; hence, it has 
 become urgent to re-examine the theory descriptive of the flow of immiscible fluids 
 through porous media. 
 
 A single liquid flowing without turbulence in a capillary tube (Poiseuille's 
 Law) or in the interconnected and geometrically complicated interstices of porous 
 media (Darcy's Law) has zero velocity at the solid phase boundaries (pore walls), 
 with an ever-increasing velocity out towards the center of the pore space, because 
 of viscous drag . 
 
 In mixture flow, however, the general condition is that each of the immis- 
 cible fluids is bounded by fluid-fluid interfaces, at least in part, and elsewhere 
 is bounded by the pore walls. Yuster (1951) noted that there will usually be a 
 nonzero velocity at the fluid-fluid interfaces; hence, Darcy's Law does not rigor- 
 ously define the conductivity afforded by porous media to the flow of a given fluid 
 which only partially saturates the pore spaces. Milne-Thomson (1955, p. 541) 
 describes the boundary condition at an interface separating two fluids as one where 
 the normal pressure and the viscous stress are continuous, provided surface ten- 
 sion is ignored. Similar recognition had been given by Hall (1956) who attempted 
 to derive a Darcy Law formula applicable to multiphase saturated porous media. 
 
 As an attempt to reconcile Yuster' s theoretical prediction (that mixture flow 
 in porous media should be dependent upon viscosity ratio) with published laboratory 
 data (which fail to show dependence on viscosity ratio), Scott and Rose (1953) 
 chose to assume that Yuster's capillary tube model over-emphasized the magnitude 
 of the ratio of fluid-fluid interfacial areas to fluid-solid surface areas. By re- 
 placing some of the fluid-fluid interface with a thin-walled tube, they were able 
 to demonstrate that there would be a corresponding decrease in the dependence of 
 fluid mixture conductivities upon the viscosity ratio. 
 
 The matter is by no means settled. Therefore, one purpose of this paper 
 is to extend the analytical work of Russell and Charles (1959) and to apply the 
 influence of fluid-fluid boundaries on mixture flow phenomena in other models of 
 porous media prototypes. Another purpose of this paper is to provide analytical 
 results useful for the solution of problems such as those already posed by Russell 
 and Charles, and by Russell, Hodgson, and Govier (1959). Specifically, the 
 "crack" model depicted in figure 1 is discussed for various boundary conditions 
 of stratified flow between wide parallel plates. 
 
 In these connections, it should be noted that Yuster's concentric flow tube 
 model is not one that can be evaluated easily by laboratory experiment. This is 
 because both gravity and surface energy tend to cause the central filament of non- 
 wetting fluid to be broken by the coalescence of the concentrically encompassing 
 wetting fluid. Experimental verifications with the crack model are not handicapped 
 by these factors, however (Ellis and Gay, 1959). 
 
 In another respect, the crack model is unique in having the flow of both 
 fluids depend upon the viscosity ratio, as Russell and Charles (1959) show. By 
 contrast, only the central fluid in the Yuster model flows in a way directly 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 
 
 Fig. 1. The crack model having a total thickness of t and a boundary at Y^ separating fluid 
 phases A and B. Li denotes the extent of the fluid-fluid boundary, and L is the 
 over-all crack length. Flow is in the X-direction. 
 
 dependent upon viscosity ratio. Thus, the concentrically located surrounding fluid 
 flows in a manner proportional to the factor (ro 2 -r2)2 j where ro is the internal ra- 
 dius of the tube and r is the radial distance marking the location of the fluid-fluid 
 interface. For comparable systems, this factor is always greater than that given 
 by Lamb (1932) as: 
 
 (ro' 
 
 (ro 2 - ri 2 ) 2 / In 
 
 frcA)] 
 
 for annular flow in an annulus of external radius, r , and internal radius, q; show- 
 ing that lubrication of the outside liquid occurs in the Yuster model, albeit in a way 
 not dependent upon the viscosity ratio of the inner to the outer fluid. 
 
 STRATIFIED LIQUID FLOW BETWEEN WIDE HORIZONTAL PARALLEL PLATES 
 
 Following the analytical methods of Russell and Charles (1959), equivalent 
 but simplified expressions for the flow of the lower (liquid A) and upper (liquid B) 
 fluids can be given as: 
 
 ,2 _ 
 
 v A = - (dp/dx) / (2u. A ) I y* - yp 
 v B = - (dp/dx) / (2u B ) [y 2 - yp + tp - t 2J 
 Q A = - (W dp/dx) / (12 UA ) [ 3pYi 2 - 2Y! 3 ] 
 Q B = - (W dp/dx)/(l2u. B ) [4t 3 - 6t 2 Y! + 2Y 1 3 - 3p (t - Y x ) 2 ] 
 
 (1) 
 (2) 
 (3) 
 (4) 
 
 where: 
 v = velocities; 
 Q = volumetric flow rate; 
 
 dp/dx = the pressure gradient acting in the direction of flow; 
 u. = viscosity; 
 
 y = the distance parameter in the vertical direction, having a value of 
 zero at the bottom surface of the crack; 
 
4 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 Yi = the value of "y" which marks the location of the fluid-fluid interface; 
 
 t = the value of "y" measuring the thickness of the crack; 
 
 W = the measure of the crack width in the horizontal direction normal to 
 the direction of flow. 
 
 In these equations, the model of figure 1 is being considered, where vis- 
 cous flow in the horizontal direction occurs because of terminal boundary conditions 
 of pressure that develop the same pressure gradient in the upper and lower fluids. 
 Also, in equations (1) through (4), the parameter, p, is defined by: 
 
 P= [(1-Tri Y! 2 -t2]/ [(1 -pr) Y 1 -t] 
 
 (5) 
 
 where-p: is the ratio of the viscosity of the upper to the lower fluid, \±%/\±a . 
 Noting the limits of (3 as the viscosity ratio, -pr, varies from zero through unity to 
 infinity as: 
 
 Limit p = (Y x + t) = t 
 TT-0 
 
 Limit p = t 
 
 Limit (3=Y 1 =t 
 "pr - * oo 
 
 it follows that the parameter, p, serves to measure the effective thickness of the 
 gap. 
 
 One advantage of equations (3) and (4) over the expressions given by 
 Russell and Charles is that their algebraic form leads to the following cubic ex- 
 pression, from which the position of the interface for minimum power requirement 
 can be derived as: 
 
 3<j> (dp/d4>) + 6p - 64, = 
 
 where: 
 
 (dp/d<j>) = [(1 -tt) d> 2 - t 2 l (1 -it) - [(1 -Tr)4>-t1 (1 -a) (24O 
 
 [(1 -pr) <j,-t]2 
 
 * 
 
 Limit 
 
 "~ (Qa + Qb) ~- minimum 
 
 (6) 
 
 Figure 2 shows the velocity profiles predicted by equations (1) and (2) as 
 obtained for values of -pr equal to 0.1, 1, and 10, respectively, when (Y^/t) is 0.5. 
 
 For comparison, figure 3 shows the corresponding velocity profiles for the 
 special case where no pressure energy is imparted to the lower (A) fluid directly so 
 that (dp/dx)^ is zero and (dp/dx)g is finite. In order to avoid a tilt of the fluid- 
 fluid interface due to the unequal pressure gradients in fluids A and B, the fluids 
 should have equal density; otherwise, the crack will have to be inclined, and 
 gravity forces proportional to the density difference will have to be considered in 
 the analysis. 
 
 Thus, by saying: 
 
 
 d 
 
 M-B 
 
 dy 
 
 
 d 
 
 HA 
 
 dy 
 
 d vb 
 
 L dy 
 d va 
 
 ~3y~ 
 
 = (dp/dx) B 
 = 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 
 
 .8 
 .6 
 
 
 
 
 -££l 
 
 ).l cp 
 
 
 
 
 
 1=10 cp 
 
 V 1.0 c 
 
 p 
 
 B ► 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .4 
 
 
 
 
 
 
 
 
 
 
 1/ 
 
 
 
 A ► 
 
 
 
 
 
 
 n 
 
 V 
 
 
 
 
 
 
 
 
 
 .1 
 
 .4 .5 
 
 v cm/sec. 
 
 .7 .8 
 
 Fig. ?.. Velocity profiles when (dp/dx)^ = (dp/dx) B = finite. 
 
 1.0 
 
 .2 
 
 
 \=I.O 
 
 cp 
 
 
 /V^A 
 
 = 0.1 cp 
 
 
 
 
 \=I0 c 
 
 p 
 
 
 
 B — ► 
 
 
 
 ""■"-■-^ 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A ► 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .3 
 
 .4 .5 
 
 v cm/sec. 
 
 .6 
 
 Fig. 3. Velocity profiles when (dp/dx) A = and there is no return of the A phase. 
 
6 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 and imposing the boundary conditions which led to equations (1) and (2), the curves 
 of figure 3 are given by: 
 
 v A = - L(dp/dx) B / n A ] [y (Y x - Y )] (7) 
 
 v B = -[(dp/dx) B / m-bJ ^ 2 " 2 ^ + 2t Y " t 2 ] (8) 
 
 where: 
 
 [(1 - 2pr) Yi 2 - t 2 ] 
 
 Y = 
 
 2 [(1 -tt) Yj - t] 
 
 Equation (7), of course, is a simple straight-line function, since movement 
 of the lower fluid is by viscous drag, originating with the finite velocity of the upper 
 liquid at the fluid-fluid interface and terminating with a zero velocity at the bottom 
 boundary. Equation (8) closely corresponds, term by term, to equation (2), yielding 
 a parabolic velocity distribution; hence, the parameter, Y , must have a significance 
 corresponding to that assigned above to the parameter, (3. 
 
 This modification, where the driving force is applied only to the upper fluid, 
 means that there is an energy loss through the motion imparted by viscous drag to 
 the lower fluid. This is shown, for example, by comparing figures 2 and 3 and noting 
 that, for a given viscosity ratio condition, the velocities indicated in figure 3 are 
 always less than those of figure 2. Also, deriving the volumetric throughputs cor- 
 responding to the conditions of equations (7) and (8) as: 
 
 Q A 
 
 = _ W (d P /dx) B [Yi 2 (Yi _ y)] (g) 
 
 Va 
 
 Qb= _ W(dp/dx) B [(2t3+ Yi 3 ) _3 Y(t _ Yi) 2 . 3t 2 Yi] (10) 
 
 6 M-B 
 
 it is again seen that Qb by equation (10) is always less Qg by equation (4), for all 
 values of (dp/dx)g. 
 
 In any case, it is worth while to realize that capillary systems can be found 
 in nature and are to be encountered in industrial processes that contain two immis- 
 cible fluids, one moving in response to the action of some imposed force field and 
 the other moving only because of viscous drag. For example, a hot vapor being 
 pushed through the center of a tube will impart a forward motion to the liquid con- 
 densate that has formed on the cool tube walls. Similarly, in recovering oil from 
 sedimentary rock, it sometimes happens that gravity acts to move one immiscible 
 fluid through a porous region saturated with another immiscible fluid, which itself 
 would be otherwise stationary because it is not directly acted upon by any other 
 driving force. 
 
 Another case worth considering is one in which the upper fluid is given 
 pressure energy to cause its movement, and movement in the lower fluid is due to 
 the drag exerted across the fluid-fluid interface. In other words, (dp/dx)g is 
 again finite and (dp/dx) A is zero but instead of having the lower fluid always mov- 
 ing forward with the upper fluid, provision is made for circulation of the lower fluid 
 in a manner that can be taken as a model of an eddy swirl. Figure 4 shows the geom- 
 etry of the system now being considered. 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 
 
 B Phase 
 
 Phase 
 
 Fig. 4. The crack model with an eddy pocket containing the A phase circulating in a 
 vortex-like motion. The B phase motion is described by a parabolic velocity 
 profile except where separated from the A phase by a fluid-fluid interface, in 
 which case the velocity profile of figure 5 applies. 
 
 One example of a system that can be recognized as the prototype of the 
 model of figure 4 is the oil recovery system in which oil is flowing in the central 
 pore spaces and here and there is moving over discontinuous elements of a wetting 
 aqueous fluid held by capillary forces as pendular rings at points of particle con- 
 tact (or as the filling of cul-de-sac pores). The wetting liquid in each disconnected 
 pore space necessarily rotates when the continuous nonwetting fluid flows past the 
 fluid-fluid boundary. 
 
 In the approximation of what happens when a flowing fluid imparts an eddy 
 swirl to the adjacent immiscible fluid, it is assumed that the swirl is comprised of 
 four principle directions of motion, as depicted in figure 4. Thus, in the upper 
 center portion, phase A will be flowing in the same direction as phase B, and in 
 the lower central portion of the eddy pocket, the motion is in the opposite direction. 
 This latter provides for a return of the swirling fluid. Elsewhere phase A will be 
 moving either downward (at the downstream end of the pocket) or upward (at the 
 upstream end), such that in these regions there will be no horizontal component 
 of velocity. In any case, a material balance holds everywhere. 
 
 Considering the transfer of fluid A, it is evident that the volume of fluid 
 moving in the upper central portion in the direction of fluid B, must be equal to 
 the volume being returned via the lower portion; that is, there will be a level in 
 phase A (between y = and y = Y^) where the velocity of phase A is zero. As no 
 discontinuity in the velocity profile should be assumed at this level, the velocity 
 of fluid A will be seen to pass smoothly from negative (through zero) to positive 
 values as y increases. It may further be assumed that the velocity profile of the 
 negative velocities in the lower portion (where phase A is swirling back towards 
 the high pressure end of the crack) will be represented by the isosceles triangle 
 shown in figure 5. Thus, from geometry, the boundary condition is imposed that 
 
 = at y = 0.586 Y 1: 
 
 as well as the other boundary conditions that: 
 va = at y = 
 v B = v A at y = Y 1 
 vq = at y = t 
 
 This case is discussed by Ellis and Gay (1959) 
 
8 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 1.0 
 
 .4 
 
 
 
 \\ 
 
 
 
 
 ^b'-° a 
 
 cp 
 
 
 
 = 10 C| 
 
 \ \ 
 
 _J 
 
 = 1.0 cp 
 
 
 B — - 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 // 
 
 
 
 
 
 
 
 A — 
 
 'A 
 
 Y 
 
 
 
 
 
 
 
 
 i 
 
 
 
 - — A 
 
 
 
 
 
 -.2 -.1 .1 .2 .3 .4 .5 .6 .7 
 
 v cm/sec. 
 
 Fig. 5. Velocity profiles when (dp/dx)^ = and there is a return of the A phase, 
 
 Now, introducing the notation: 
 
 t - 0.586 Y 
 
 X = 0.414 Y x 
 
 X= y - 0.586 Y x 
 _ [(l - 2jt) \ } 2 -r 2 } 
 2 [(l --pr) \ 1 -r] 
 
 the velocity in fluid B, by analogy to equation (8), is given by: 
 
 [\ 2 - 2\ + 2 rp -r 2 J 
 
 'B 
 
 (dp/dx)R 
 2 MLB 
 
 (11) 
 
 which is the function plotted for several viscosity ratios in figure 5. The positive 
 and negative segments of the velocity profile for fluid A, of course, have a form as 
 predicted from equations analogous to equation (7). Also for this case: 
 
 Qb= - 
 
 W (dp/dx) 
 
 B 
 
 [(2T 3 + X 1 3 ) - 3p(T-\ 1 ) 2 - (37%)] 
 
 M-B 
 
 (12) 
 
 Comparison of equations (12), (10), and (4) show that for given conditions 
 of-rr, u.g, and (Y /t), a given energy input (as measured by dp/dx) gives less 
 throughput of fluid B when an eddy swirl developes (fig. 5) than when fluid A is 
 simply dragged along (fig. 3), and the latter, in turn, gives less throughput 
 than when dp/dx is acting directly on both fluids (fig. 2). 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 
 
 DISCUSSION 
 
 In order to examine the magnitude of the lubrication effect which gives fluid 
 saturants of porous media (and other capillary systems such as tubes and cracks) an 
 apparent increased mobility because of the fluid-fluid interfacial boundaries, the 
 following notation is introduced: 
 
 Sa = Yj/t = the fractional saturation of the A fluid. 
 
 Sg = (t - Y-J/t = the fractional saturation of the B fluid. 
 
 L-^/L = length of the fluid-fluid interface/total path length. 
 
 A; = LiW = the area of the fluid-fluid interfacial boundary. 
 
 A s = (2L - Li)W = the area of the solid (pore walls) boundaries for 
 values of S less than unity. 
 
 Am = 2LW = the total solid area when S = 1. 
 
 Ai/A T = Li/2L 
 
 Ag/Am = (2L - Li)/(2L) 
 
 Qa 
 
 "RA 
 
 ^RB 
 
 Lim 
 
 
 
 S A = 
 
 = 1 
 
 '_?B_ 
 
 
 Lim 
 
 
 
 s B = 
 
 1 
 
 fractional conductivity of fluid A. 
 
 fractional conductivity of fluid B , 
 
 The S terms are the familiar saturation terms, used in porous media theory, 
 which measure the fraction of the pore space filled by the pore saturant. The K^ 
 terms are the relative permeability parameters which measure the ratio of conduc- 
 tivity afforded by porous media to the flow-transfer of a pore saturant at some 
 value of S (equal to or less than unity) to the conductivity afforded to the flow- 
 transfer of the same saturant when its saturation is unity. 
 
 Heretofore (Muskat, 1949), theoretical deduction and experimental data 
 commonly have been taken to indicate that relative permeabilities are explicit 
 functions of saturation alone and are implicit functions of saturation distribution 
 (as follows from the possibility of saturation hysteresis); but Yuster (1951) has 
 indicated that the viscosity ratio, -pr, may also be a determining factor. This 
 possibility is now examined. 
 
 Equations (3) and (4) yield: 
 
 K 
 
 3S 
 
 RA 
 
 "RB 
 
 A 
 
 [S, z (1 - -pr) - 1J 
 
 2S, 
 
 LS a (l --pr) - lj 
 
 S A (1 -ir) - 1 
 
 S B Z (1 - -pr) + 
 
 2S B - 
 
 S R (1 --pr) - 1 
 
 (13) 
 
 (14) 
 
10 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 for the case where (dp/dx)^ = (dp/dx) B > 0. Figure 6 shows that the conductivity of 
 both the A and B fluids indeed are increased as the viscosity of the contiguous phase 
 decreases (the upper curves) over the values which would be expected if Darcy's 
 Law was applicable in describing mixture flow phenomena (the lowest curve of fig- 
 ure 6) . 
 
 In fact, it is shown in figure 6 that whenever the viscosity ratio differs from 
 unity, there will be a range of saturation values where the more viscous fluid flows 
 more readily (K^ is greater than unity) than if this fluid completely filled the pore 
 spaces (its saturation were unity). For the model described by equations (13) and 
 (14), a maximum of a four-fold increase in conductivity of the more viscous phase 
 is approached as the viscosity ratio approaches zero. 
 
 Comparison of the relative permeability curves of figure 6 with the experi- 
 mental data cited by Muskat (1949) suggests the need for introducing a correction 
 factor if better correspondence between model and prototype is to be achieved. The 
 approach of Scott and Rose (1953) can be followed for this purpose, for it is clear 
 that the models of figures 1 and 4 (as well as the capillary tube model discussed by 
 Yuster [1951] and Russell and Charles [1959]) are defective in suggesting that the 
 term, Aj/Ay., is unity for all values of saturation. 
 
 Thus, equation (14) may be revised as: 
 
 K RR = S B 3 [S B 2 (1 - a) + 2S B - 3] (15) 
 
 ( f") S B tV 1 " G) - « + (4^) [S B 2 ft " «> + 2S B - 3] 
 
 RB 
 
 where a = (pr) 1 = dj. A /|j. B ) 
 
 and figure 7 shows a significantly diminished dependence of the relative permeability 
 function on viscosity ratio, as compared to figure 6, upon making some reasonable 
 assumptions about the surface area functions, as: 
 
 (Ai/A T ) W = 0.715 (1 - S w ) 
 
 (16) 
 and (A S /A T ) W = . 715 S w + . 285 
 
 where the subscript W identifies the B fluid as the wetting phase 
 (cf. discussion in connection with equation [22] below). 
 
 In fact, no values of Kn greater than unity exist for this case, even when the vis- 
 cosity ratio is most unfavorable (approaching zero), a result to be anticipated from 
 an examination of extant experimental data . 
 
 Corresponding to equation (14), equation (10) yields: 
 
 K RB " S B 
 
 3 
 
 S B (4 - G ) - 4 
 Sr (1 - a) - 1 
 
 (17) 
 
 for the model where (dp/dx) A is zero, and fluid A is dragged forward by the motion 
 of fluid B being acted upon by a finite (dp/dx)g. Figure 8, a plot of equation (17), 
 upon comparison with the corresponding curves of figure 6, shows a similar, albeit 
 diminished dependence on viscosity ratio. 
 
 Of more immediate interest, however, are the functions to be derived from 
 equation (12), for these describe the characteristics of the model in which a con- 
 tinuous flowing fluid (for example fluid B) imparts an eddy swirl to contiguous but 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 
 
 11 
 
 4.0 
 
 3.8 
 
 3.6 
 
 3.4 
 
 3.2 
 
 3.0 
 
 2.8 
 
 2.6 
 
 2.4 
 
 2.2 
 K RA 
 or 2.0 
 K 
 
 V RB 
 
 1.8 
 
 1.6 
 
 1.4 
 
 1.2 
 
 1.0 
 
 .8 
 
 .6 
 
 .4 
 
 .2 
 
 
 
 Fig. 6, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 A 
 
 
 
 
 
 
 
 
 
 // 
 
 r 
 
 
 
 
 
 
 
 
 a = 
 
 7/ 
 
 
 
 
 
 
 
 
 
 
 L 
 
 
 
 
 For K RA , o--^ 
 
 B_ 
 
 
 
 
 
 
 
 Ck 
 
 
 
 
 
 
 
 For K RB , a = -jj 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 / C 
 
 ).i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 = 0.5/^ 
 
 
 
 
 
 
 
 
 
 / = 1.0^ 
 
 
 
 
 
 
 
 
 
 
 
 10/^ 
 
 
 
 
 
 
 
 
 
 
 /^OO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .2 .3 .4 .5 
 
 S A or S B 
 
 .6 
 
 .8 .9 
 
 Relative permeability curves for phases A and B us- 
 ing the model of figure 1 with Lj = L at all satura- 
 tions and (dp/dx)^ = (dp/dx) B . 
 
12 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 i.O 
 
 .9 
 
 .7 
 
 'RW 
 
 .2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o/\X\ 
 
 /=I.O / 
 =10// 
 
 = 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .3 
 
 .4 
 
 .7 
 
 >W 
 
 Fig. 7. Relative permeability curves for the wetting phase for the model 
 of figure 1 in a special case, showing a greater dependence on 
 viscosity ratio and proving the necessity for assuming the ex- 
 istence of eddy pockets. 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 13 
 
 2.4 
 
 2.2 
 
 2.0 
 
 1.8 
 
 1.6 
 
 1.4 
 
 >RB 
 
 1.2 
 
 1.0 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 // 
 
 
 
 
 
 
 
 
 
 a = 0j 
 
 / 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 /=0.l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y=o.5 
 
 
 
 
 
 
 
 
 
 
 /^I.O 
 
 
 
 
 
 
 
 
 
 / =ic 
 
 
 /=oo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .4 
 
 .5 
 
 .6 
 
 .8 
 
 .9 
 
 1.0 
 
 Fig. 8. Relative permeability curves for phase B using the model of figure 1 where 
 Li = L for all saturations for the special case where (dp/dx)^ = and 
 there is no return of the A phase. 
 
14 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 disconnected elements of an immobile pore saturant . As noted above, prototypes 
 of this model occur widely and are of special importance in oil recovery systems. 
 Equation (12) gave the flow rate of the continuous phase as it exists over 
 the central portion of the eddy pocket (fig. 4). Assuming that two-thirds of the 
 fluid-fluid interface represents regions where there is no horizontal component of 
 velocity in the A phase, and that in the remaining one-third central region where 
 the A phase is being dragged in the direction of the motion of B (in the upper portion) 
 or returning in the opposite direction (in the lower portion of the eddy pocket), it 
 follows that for L^ = L for all saturations, then: 
 
 K RB _ 2S B 
 
 S B (0.5c - 0.828) + 0.828 
 S B (a - 0.414) + 0.414 
 
 (18) 
 
 It might be felt that the assumptions underlying the derivation of equation 
 (18) provide a poor description of the hydromechanics of an eddy swirl, but it can 
 be shown that in some important cases this defect is of second-order importance 
 (fig. 9). Also, the data of Ellis and Gay (1959) support use of equation (18). 
 
 Equation (18) yields: 
 
 K RB 
 
 2S B 3 [S B (0.5c - 0.82) + 0.82] 
 
 [(S B |a-0.4i+ 0.41)+4(S B |.5a-.82J + . 82) J + f T^J [2Sg (. 5c- . 82) + 1 . 64] 
 
 (19) 
 
 when Li/L values less than unity are to be assumed, since the resistance to flow 
 of phase B through that portion of the crack where the fluid-fluid interface exists 
 is: 
 
 Li w = (AP)LiW (20) 
 
 3t3M t 3 N 12Q bM l b 
 
 and the resistance to flow of phase B through that portion of the crack where no 
 fluid-fluid interface exists is: 
 
 (L-Li) = W AP (L _ u) 
 T^T 12Q^ B 
 
 In equations (20) and (21), AP is the pressure drop across Li or (L - Li) as the case 
 might be, M is the argument of equation (18), and N = S B (the argument of equa- 
 tion [12] when (j. is zero). Thus, equation (19) is derived as equation (15) was 
 derived from equation (14), in the manner first suggested by Scott and Rose (1953). 
 Using the functions of equation (16), and setting: 
 
 (Ai/A^ = 1.67 (1 - S N ) 
 
 (22) 
 
 - n fi7 
 N N 
 
 and (AgAj,)., 1.67 S„ - 0.67 
 
 where the subscript, N, identifies the B fluid to be 
 considered as the nonwetting phase. 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 15 
 
 Figure 9 can be derived from equation (19) to show the apparent second-order de- 
 pendence of relative permeability curves on viscosity ratio. 
 
 Although the conditions which have been imposed (equations [16] and [22]) 
 may be arbitrary, they are qualitatively reasonable. Thus, equation (16) says that, 
 when the wetting phase is decreasing in saturation, (Ai/Ap)^ increases from the 
 proper limit of zero to some maximum value, and (A S /A T ) A decreases from the proper 
 limit of unity and approaches zero. In a like manner, equation (22) gives the proper 
 limits and proper trend for the nonwetting phase surface area functions as the wet- 
 ting phase saturation increases. 
 
 In these latter connections, both figures 6 and 8 show that the greatest de- 
 partures from Darcy's Law occur at the higher values for saturation ( S > 0.7), 
 which are in fact the ones plotted in figure 9. It is an arbitrary consequence of 
 the geometry of the crack model, of course, that the Ai/Aj. ratio varies from zero 
 to 0.5, and the Ag/Ap ratio varies from unity to 0.5, as the Li/L ratio varies from 
 zero to unity. This prevents consideration of the functions of equations (16) and 
 (22) over the entire saturation range (from zero to unity). Inspection shows, how- 
 ever, that if the model could be treated to handle the lower saturation ranges (be- 
 low Sw = 0.3 and S N = 0.7) , the dependence of relative permeability on viscosity 
 ratio would be even less noticeable than the limits depicted in figure 9. 
 
 CONCLUSIONS 
 
 In this paper, use has been made of the self-evident, but sometimes over- 
 looked, consideration that one fluid passing over another contiguous immiscible 
 fluid will always impart a motion to the second fluid. This gives, in general, a 
 non-zero velocity boundary condition to be imposed at the fluid-fluid interface, 
 which must be used in the solution of the appropriate differential equations of 
 motion. When streamline flow is being considered, the equations of motion are 
 derived from the Stokes-Navier Law of Force, the Continuity Equation, and the 
 thermodynamic Equation of State, and their solution may be taken as properly 
 describing the flow transfer process. 
 
 These methods have been applied herein to one-dimensional, two-phase 
 flow of stratified liquids in a capillary space having the geometry of a wide crack 
 with parallel surfaces. Figure 1 shows this model, with the oppositely located 
 source and sink, and the parallel and impermeable edge boundaries in the direction 
 of flow. For flow in the viscous regime, this model may be taken to represent a 
 porous medium prototype such as the petroleum reservoir sedimentary rock system 
 through which oil and water are flowing. By introducing the parameter, (Li/L), a 
 better representation is obtained, as shown by figures 1 and 4, since the model 
 can then be scaled to show a particular dependence of the amount of interfacial 
 (fluid-fluid) surface area on the fluid saturation parameters, S^ and Sg. 
 
 Evidently the models of figures 1 and 4 can be taken to represent other 
 prototypes of mixture flow systems as encountered in other branches of engineer- 
 ing (for example, pipeline transport of oil lubricated by the presence of an aqueous 
 phase, two-phase flow in catalytic reactors, transport of condensate films by mov- 
 ing vapor, etc.) . 
 
 This paper includes results for the cases where an equal driving force acts 
 directly on each of the stratified fluids, where the driving force acts only on one 
 fluid and the other is simply dragged along^and where the driving force acts only on 
 one fluid and the other is caused to rotate in an eddy swirl. 
 
16 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 l.O 
 
 .0 
 
 .8 
 
 .6 
 
 .3 
 
 .9 
 
 .7 
 
 .6 
 
 .5 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 / 
 
 
 
 K RN 
 
 
 
 
 K RW / 
 
 / 
 
 
 co\ \=( 
 
 3 
 
 
 
 N I 1 
 
 ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / < 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .7 
 
 .9 
 
 1.0 
 
 'w 
 
 Fig. 9. Relative permeability curves for wetting and nonwetting phases 
 for the model of figure 4 in a special case, showing the second- 
 order dependence on viscosity ratio. 
 
FLUID-FLUID INTERFACIAL BOUNDARY CONDITION 17 
 
 Other cases of importance, (not treated herein, but derivable by the ana- 
 lytic methods which are presented) are those in which unequal driving forces act 
 on each of the stratified fluids, and those in which equal (or unequal) but oppositely 
 directed driving forces act on each fluid. 
 
 Figures 2,3, and 5 show the magnitude of the lubrication effect as a function 
 of the viscosity ratio, and it is seen that the maximum velocity increase occurs for 
 a particular fluid when the viscosity of the contiguous fluid is the least. It is also 
 apparent from these curves that there is an energy loss (an increase in the power 
 requirement) whenever the driving force acts on only one (instead of both) of the 
 moving fluids, and that this effect is greatest when the rotational swirl is assumed 
 to develop. 
 
 Figures 6 and 8 also express the magnitude of the lubrication effect, by re- 
 lating the relative permeability parameter to saturation and to viscosity ratio param- 
 eters. From these it may be concluded that the volumetric throughput of a particular 
 fluid that only partially fills the capillary pore space is sometimes greater than its 
 throughput when it completely fills the pore space. This condition of increased 
 conductivity is met when the saturation of the fluid approaches unity, as long as 
 the contiguous fluid has a lower viscosity. For example, figure 6 indicates that 
 the more viscous fluid need fill no more than 60 percent of the pore volume in order 
 to have a conductivity greater than when it fills 100 percent of the pore volume, as 
 long as the viscosity ratio (least viscous to the most viscous fluid) is of the order 
 of 0.1, and as long as the fluid-fluid interface extends along the entire flow path. 
 
 Of special importance to the matter of oil recovery are the indications of the 
 curves of figures 7 and 9. These suggest that the relative permeability versus sat- 
 uration functions do have a dependence upon the viscosity ratio as long as finite 
 fluid-fluid interfacial boundaries characterize the prototype reservoir rock system. 
 Even though this dependence may prove to be a second-order effect in some cases, 
 thus explaining why previously obtained experimental data have not shown the de- 
 pendence, it must be concluded that Darcy Law interpretations have limited appli- 
 cation to mixture flow phenomena. This is because an implicit assumption made 
 in the statement of Darcy 1 s Law is that flowing fluids have zero velocities at all 
 fluid-fluid and fluid- solid boundaries, in unnecessary contradiction to the elemen- 
 tary premise of the analyses of this paper, which states that viscous forces extend 
 across all fluid-fluid boundaries. 
 
18 ILLINOIS STATE GEOLOGICAL SURVEY CIRCULAR 291 
 
 REFERENCES 
 
 Ellis, S. R. M., and Gay, B., 1959, The parallel flow of two fluid streams: inter- 
 facial shear and fluid-fluid interaction: Trans. Inst. Chem. Engs., v. 37, 
 p. 206-213. 
 
 Hall, Warren A., 1956, An analytical derivation of Darcy 's equation: Trans. Am. 
 Geophysical Union, v. 37, p. 185-188. 
 
 Lamb, Horace, 1932, Hydrodynamics, 6th ed., Cambridge Univ. Press, England. 
 
 Milne-Thompson, L. M., 1955, Theoretical hydrodynamics , 3rd edition, Macmil- 
 lan Company, New York. 
 
 Muskat, Morris, 1949, Physical principles of oil production: McGraw-Hill Book 
 Company, Inc., New York. 
 
 Rose, Walter, 1957a, Fluid flow in petroleum reservoirs. I. The Kozeny paradox: 
 111. Geol. Survey Circ. 236. 
 
 Rose, Walter, 195 7b, Fluid flow in petroleum reservoirs. II. Predicted effects of 
 sand consolidation: 111. Geol. Survey Circ. 242. 
 
 Russell, T. W. F., and Charles, M. E., 1959, Effect of the less viscous liquid in 
 the laminar flow of two immiscible liquids: Canadian Jour. Chem. Eng., 
 v. 37, p. 18-24, Feb. 
 
 Russell, T. W. F., Hodgson, G. W., and Govier, G. W., 1959, Horizontal pipe- 
 line flow of mixtures of oil and water: Canadian Jour. Chem. Eng., v. 37, 
 p. 9-17, Feb. 
 
 Scott, P. H., and Rose, Walter, 1953, An explanation of the Yuster effect: Jour. 
 Petroleum Technology, v. 5, sec. 1, p. 19-20, Nov. 
 
 Yuster, S. T., 1951, Theoretical considerations of multiphase flow in idealized 
 capillary systems: Proc . Third World Petroleum Congress, v. II, p. 437- 
 445, The Hague. 
 
 Illinois State Geological Survey Circular 291 
 18 p., 9 figs., 1960 
 
nnEnni 
 
 CIRCULAR 291 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 URBAN A <m±*