The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media

Transcription

1 Transport in Porous Media 44: , c 2001 Kluwer Academic Publishers. Printed in the Netherls. 109 The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media RAMON G. BENTSEN School of Mining Petroleum Engineering, Department of Civil Environmental Engineering, University of Alberta. Edmonton, Alberta T6G 2G7, Canada (Received: 23 July 1999; in final form: 23 July 1999) Abstract. Recently developed transport equations for two-phase flow through porous media usually have a second term that has been included to account properly for interfacial coupling between the two flowing phases. The source magnitude of such coupling is not well understood. In this study, a partition concept has been introduced into Kalaydjian s transport equations to construct modified transport equations that enable a better understing of the role of interfacial coupling in two-phase flow through natural porous media. Using these equations, it is demonstrated that, in natural porous media, the physical origin of interfacial coupling is the capillarity of the porous medium, not interfacial momentum transfer, as is usually assumed. The new equations are also used to show that, under conditions of steady-state flow, the magnitude of mobilities measured in a countercurrent flow experiment is the same as that measured in a cocurrent flow experiment, contrary to what has been reported previously. Moreover, the new equations are used to explicate the mechanism by which a saturation front steepens in an unstabilized displacement, to show that the rate at which a wetting fluid is imbibed into a porous medium is controlled by the capillary coupling parameter, α. Finally, it is argued that the capillary coupling parameter, α, is dependent, at least in part, on porosity. Because a clear understing of the role played by interfacial coupling is important to an improved understing of two-phase flow through porous media, the new transport equations should prove to be effective tools for the study of such flow. Key words: capillarity, interfacial coupling, mobility, transport equations, two-phase flow. Nomenclature Roman Letters k i effective permeability for phase i, i = 1, 2. k ij generalized effective permeability for phase i, i, j = 1, 2. p i pressure for phase i, i = 1, 2. P c macroscopic capillary pressure. R 12 function relating the pressure gradient in phase 1 to that in phase 2. S i saturation of phase i, i = 1, 2. v total velocity. v i darcy velocity of phase i, i = 1, 2. x distance in direction of flow. Greek Letters α capillary coupling parameter. α i capillary coupling parameter for phase i, i = 1, 2.

2 110 RAMON G. BENTSEN λ i generalized partition coefficient for phase i, i, j = 1, 2. λ i (k i )/(µ i ) = mobility of phase i, i = 1, 2. λ ij (k ij )/(µ i ) = generalized mobility of phase i, i = 1, 2. µ i viscosity of phase i, i = 1, 2. θ angle between p 1 / x p 2 / x. Superscripts steady-state, cocurrent flow. steady-state, countercurrent flow. 1. Introduction On the basis of experimental results presented in the literature (Lelièvre, 1966; Bourbiaux Kalaydjian, 1990; Bentsen Manai, 1991, 1993), it appears that mobilities determined in a countercurrent flow experiment are less than those determined, for the same s fluid system, in a cocurrent flow experiment. Such a result cannot be explained, if one assumes that the conventional transport equations (Muskat, 1982) describe correctly two-phase flow through porous media. That is, one must resort to more sophisticated transport equations, such as those constructed by Kalaydjian (1987), or others (de la Cruz Spanos, 1983; Whitaker, 1986) to explain such a result. In Muskat s transport equations (Muskat, 1982), the flux is taken to be proportional to one driving force, the pressure gradient acting across the phase. On the other h, in the more sophisticated transport equations, Muskat s equation for a given phase is modified to include a cross, or coupling, term that is proportional to the pressure gradient of the other phase. The need for such a modification is argued usually on the basis of symmetry, or results arising out of irreversible thermodynamics (Katchalsky Curran, 1975). Moreover, it is postulated usually that the coupling effect arises from the interfacial contact between the wetting nonwetting fluids. In constructing the more sophisticated transport equations, it is usual to start at the microscopic scale with the Navier Stokes equation, then use volume averaging, or some equivalent technique, to construct equations that pertain at the macroscopic scale. When the volume averaging approach is used, the mobilities that appear in the transport equations are defined in terms of surface integrals. Because of the complicated nature of the pore space in natural porous media, it is necessary to determine these integrals experimentally. As a consequence, one must be aware that the information lost in passing from the microscopic to the macroscopic scale may not be recaptured easily in the experiment used to determine the mobilities. Thus, when conducting experiments to determine mobility, it is important to keep in mind two realities. First, the definition for mobility is an operational definition. That is, mobility is defined by the equation in which it appears. If the form /or underlying physics of the defining equation are inappropriate, model error

3 THE PHYSICAL ORIGIN OF INTERFACIAL COUPLING 111 will be introduced into the experimentally determined mobilities. Second, in the transport equations, the mobilities pressure gradients appear as products. This can make it difficult to discern whether a particular interfacial coupling effect should be associated with the mobility, or whether it should be associated with the pressure gradient. Hence, if care is not taken when determining mobilities experimentally, interfacial effects that should be associated more properly with the appropriate pressure gradient may be incorporated into the definition for mobility. Again, model error may be incorporated into the experimentally determined mobilities. Whether a particular interfacial effect should be associated with the mobility, or the pressure gradient, depends upon the source of the interfacial coupling. If the effect is due to interfacial momentum transfer, it should be associated with the mobility. If, however, the effect is due to the capillarity of the porous medium, it should be associated with the appropriate pressure gradient. The purpose of this paper is to investigate whether the interfacial coupling that takes place in two-phase flow through natural porous media is truly attributable to interfacial momentum transfer, or whether it should be attributed more properly to the capillarity of the porous medium. This is done by introducing the partitioning concept developed in an earlier paper (Bentsen, 1998b) into Kalydjian s transport equations (Kalaydjian, 1987) to construct modified transport equations that enable a better understing of the role of interfacial coupling in two-phase flow through porous media. 2. Theory The following analysis is confined to the stable, collinear, horizontal flow of two immiscible, incompressible fluids through a water-wet, isotropic homogeneous porous medium where phase 1 is the wetting phase phase 2 is the nonwetting phase BASIC EQUATIONS Kalaydjian (1987) has shown that, consistent with the assumptions made above, the transport equations for the flow of two continuous phases may be written as p 1 ν 1 = λ 11 x λ p 2 12 x, (1) p 1 ν 2 = λ 21 x λ p 2 22 x, (2) where λ ij = k ij /µ i ; ij = 1, 2. Bentsen (1992,1994,1997,1998a) has established, for all types of one-dimensional flow, that P c x = R 12 cos θ p 2 x p 1 x (3)

4 112 RAMON G. BENTSEN where p 1 p 2 cos θ = x i p x 1 x i p 2 x where R 12 is a weak function of normalized saturation that is introduced to account for the fact that, for horizontal, steady-state cocurrent flow, the pressure profile for the wetting phase is not parallel to that for the nonwetting phase (Bentsen Manai, 1991, 1993). Introducing Equation (3) into Equations (1) (2) yields ( ν 1 = λ 11 + λ ) 12 p1 R 12 cos θ x λ 12 P c R 12 cos θ x, (5) ν 2 = (λ 22 + R 12 cos θλ 21 ) p 2 x + λ P c 21 x. (6) Moreover, the conventional transport equations, again consistent with the assumptions made above, may be written as (4) ν 1 = λ 1 p 1 x, (7) p 2 ν 1 = λ 2 x, (8) where λ i = k i /µ i, i = 1, DETERMINATION OF PARTITION COEFFICIENTS Based on the experimental results presented by Bentsen Manai (1991, 1993), it can be inferred that (Bentsen, 1998b) λ ij = α ij λ i, i,j = 1, 2, (9) where the α ij are generalized partition coefficients for phase i, i, j = 1, 2, where the λ i,i = 1, 2, are mobilities determined in a steady-state, cocurrent flow experiment. In one-dimensional, cocurrent flow, the pressure gradients, p 1 / x p 2 / x, act in the same direction. As a consequence, in view of Equation (4), cos θ = 1. Thus, upon introducing Equation (9) into Equations (5) (6), one obtains [( ν 1 = λ 1 α 11 + α ) 12 p1 R 12 x + α ] 12 P c (10) R 12 x

5 THE PHYSICAL ORIGIN OF INTERFACIAL COUPLING 113 [ ν 2 = λ 2 (α 22 + R 12 α 21 ) p ] 2 x α P c 21. (11) x Thus, given the validity of Equation (10), it appears that the total pressure force per unit volume available to act on a volume element of phase 1, p 1 / x, may be partitioned into two components. The first is a phase component, α 11 p 1 / x, the second, a coupling (or capillary) component, α 12 /R 12 p 1 / x, that arises because of the introduction of the capillary pressure equation (Equation (3)) into Equation (1). As the pressure forces per unit volume, p 1 / x p 2 / x, act in the same direction, the phase coupling components of the total pressure force per unit volume also act in the same direction. Similar comments can be made with respect to Equation (11), the equation that determines the flux of phase 2. Moreover, the partition coefficients, α ij, may be viewed (see Equations (20) (21)) as being the fraction of the pressure force per unit volume of phase j that is available to act on a volume element of phase i, i, j = 1, 2. Under conditions of steady-state flow, the capillary pressure gradient, P c / x, is identically equal to zero. Moreover, the flux of a given phase may be defined by either Kalaydjian s transport equations (Kalaydjian, 1987) or by the conventional transport equations. Thus, for steady-state, cocurrent flow, Equations (7) (8) may be combined with Equations (10) (11) to obtain α 11 + α 12 R 12 = 1 (12) α 22 + R 12 α 21 = 1. (13) In countercurrent flow, the pressure gradients for the two flowing phases, p 1 / x p 2 / x, act in opposite directions. Hence, in view of Equation (4), cos θ = 1. Thus, introducing Equations (9) into Equations (5) (6), one obtains, for countercurrent flow, ν 1 = λ 1 ν 2 = λ 2 [( α 11 α 12 R 12 ) p1 x α ] 12 P c R 12 x (14) [ (α 2 R 12 α 21 ) p ] 2 x α P c 21. (15) x Note that, because the pressure forces per unit volume, p 1 / x p 2 / x, act in opposite directions, the phase coupling (or capillary) components of the total pressure force per unit volume, p 1 / x, must also act in opposite directions. Moreover, as shown below, the phase coupling components can sum algebraically to only a fraction of the total pressure force per unit volume, p 1 / x. Similar

6 114 RAMON G. BENTSEN comments can be made with respect to Equation (15), the defining equation for the phase 2 flux. Based on the experimental results presented by Bentsen Manai (1991,1993), it can be inferred that (Bentsen, 1998b) λ i = α i λ i, i = 1, 2. (16) where the λ i,i = 1, 2, are the mobilities measured in a steady-state, countercurrent flow experiment, where the α i,i = 1, 2, are parameters that control the amount of capillary coupling that can take place. By combining Equations (7) (8) with Equations (14) (15), by introducing Equation (16) into the resulting equations, it may be shown that, for steady-state, countercurrent flow, α 11 α 12 R 12 = α 1 (17) α 22 R 12 α 21 = α 2. (18) To ensure that the capillary pressure gradient for countercurrent flow is consistent with that for cocurrent flow, one more condition must be imposed. Because, for countercurrent flow, the pressure gradients of the two flowing phases are opposite in sign, Equation (3) becomes, for countercurrent flow, P c x = R p 2 12 x + p 1 x. (19) Introducing Equation (9) into Equations (1) (2), setting the resulting equations equal to Equations (7) (8), respectively, yields p 1 α 11 x + α p 2 12 x = p 1 x (20) p 1 α 21 x + α p 2 22 x = p 2 x. (21) Multiplying Equation (21) through by R 12, adding the resulting equation to Equation (20), collecting like terms, one obtains, for the negative of the countercurrent capillary pressure gradient, p 2 R 12 x + p 1 x = (α 12 + R 12 α 22 ) p 2 x + (α 11 + R 12 α 21 ) p 1 x. (22) Equating the coefficients of like terms yields α 11 + R 12 α 21 = 1 (23)

7 THE PHYSICAL ORIGIN OF INTERFACIAL COUPLING 115 α 12 + R 12 α 22 = R 12. (24) Note that only one of Equations (23) (24) is needed, that is, use of either of these two equations leads to the same result. Equations (12), (13), (17), (18) either Equation (23) or (24) comprise a set of five equations involving six unknowns: α 11, α 12, α 21, α 22, α 1 α 2. This system of equations may be solved for five of the unknowns in terms of the sixth unknown to obtain α 11 = α 22 = 1 + α, (25) 2 where α 12 = R 12(1 α) 2 α 21 = (26) (1 α) 2R 12, (27) α 1 = α 2 = α. (28) The parameter α controls the amount of capillary coupling that can take place. Mathematically, α can range between 0 1. For physical reasons (see Equation (16)), it seems unlikely that α would ever take on a value of zero. However, by setting α = 0, an upper limit can be placed on the amount of capillary coupling that can take place. Thus, if α = 0, α 12 = 0.5R 12 while α 21 = 0.5/R 12.That is, when α = 0, the phase coupling components of the force per unit volume of a given phase have the same magnitude. Consequently, for countercurrent flow, the pressure gradient for a given phase (see Equation (14) or (15)) is eliminated, leaving only the capillary pressure gradient as a driving force, an unlikely scenario. If α = 1,α 12 = α 21 = 0, that is, no capillary coupling can take place. Note that, because α 1 = α 2 = α, only three sets of experiments are needed to determine the α ij : two sets of steady-state, cocurrent flow experiments, one for each phase, to determine R 12, one set of steady-state, countercurrent flow experiments to determine α. Finally, it should be noted that, even though it is suggested herein that the physical origin of the coupling is different, Equations (25) (28) are consistent with recently presented results for viscous coupling (Bentsen, 1998b; Babchin et al., 1998).

8 116 RAMON G. BENTSEN 2.3. ONE-DIMENSIONAL TRANSPORT EQUATIONS The introduction of Equations (25) (27) into Equations (10) (11) leads, for one-dimensional, cocurrent flow, to ( ν 1 = λ p1 1 x + 1 α ) P c (29) 2 x ( ν 2 = λ p2 2 x 1 α ) P c. (30) 2R 12 x Note that, if α = 1, /or if P c / x = 0, the conventional transport equations for one-dimensional, cocurrent, two-phase flow are obtained, as should be the case. Note also that P c / x p i / x, i = 1, 2, are opposite in sign. Consequently, the unsteady-state fluxes, ν 1 ν 2, are less than or greater than, respectively, the steady-state, cocurrent fluxes, ν1 ν 2. Moreover, in view of Equations (29) (30), it appears that the capillary-pressure-gradient term is the mechanism by which the wetting-phase saturation front steepens in an unstabilized displacement. The introduction of Equations (25) (27) into Equations (14) (15) yields, for one-dimensional, countercurrent flow, ν 1 = λ 1 ( α p 1 x 1 α 2 P c x ) ( ν 2 = λ 2 α p 2 x 1 α ) P c. (32) 2R 12 x Again note that if α = 1 /or P c / x = 0, the conventional equations for one-dimensional, two-phase flow are obtained. Note also that because, in countercurrent flow, the phase 1 phase 2 pressure gradients act in opposite directions, only a fraction (α) of the total pressure force per unit volume of a given phase is available to act on a volume element of that phase. Moreover, in view of Equations (31) (32), it appears that the same mobility may be used for both cocurrent countercurrent flow. (31) 2.4. PURE COUNTERCURRENT IMBIBITION For pure countercurrent imbibition ν = ν 1 + ν 2 = 0. (33) Equations (3), (31), (32) (33) may be combined to show that (Bentsen, 1998a) ν 1 = λ 1 λ 2 R 12 λ 1 + λ 2 α P c x. (34)

9 THE PHYSICAL ORIGIN OF INTERFACIAL COUPLING 117 Note that the rate at which fluid is imbibed depends on the capillary coupling parameter, α. If there is no capillary coupling, α = 1, Equation (34) degenerates to the usual form (Collins, 1990), provided R 12 = 1. If α = 0, no imbibition can take place. Thus, it appears that the magnitude of the driving force causing imbibition, α P c / x, depends upon the amount of capillary coupling that can take place. 3. Discussion 3.1. CAPILLARY COUPLING PARAMETER If one is to underst completely the role of capillary coupling in two-phase flow it is essential to come to a better understing of the physics underlying the capillary coupling parameter, α. Some insight into the nature of α can be gained, provided the channel-flow theory (Rapoport Leas, 1951; Chatenever Calhoun, 1952) for two-phase flow is accepted. In channel-flow theory, it is postulated that, during steady-state flow, each fluid flows through a different set of channels, that the channels are bounded in part by fluid solid surfaces, in part by fluid fluid interfaces. The channel-flow model may be used to consider two limiting cases: one for which α = 1 one for which α = 0. First, let us suppose that the channels are bounded entirely by fluid solid surfaces. This is equivalent to assuming that the porous medium is made up of a bundle of capillary tubes (channels). If such is the case, there can be no capillary coupling between adjacent channels that are flowing wetting nonwetting fluids, respectively, that is, α = 1. Moreover, because there is no capillary coupling, the magnitude of the driving force causing imbibition (see Equation 34)), α P c / x, takes on its maximal value. The second limiting case occurs when one assumes that the channels are bounded completely by fluid fluid interfaces. This is equivalent to assuming that the flow is taking place in free space, rather than within the confines of a porous medium. Such an assumption permits the maximum amount of capillary coupling to take place; that is, α = 0. However, because there is no porous medium, no imbibition can take place. Thus, in view of Equation (34), the magnitude of the imbibition driving force, α P c / x, takes on its minimal value, zero. When a wetting fluid imbibes into a porous medium, high-energy (nonwetting) surface is replaced by lower-energy (wetting) surface, resulting in a decrease in the amount of surface free energy stored in the fluid-solid surfaces in the porous medium. Moreover, the surface free energy that is released, as a consequence of the decreae in stored surface free energy, provides a source of energy to drive the imbibition process. In a homogeneous, isotropic porous medium, the amount of solid surface area, in any given plane, is dependent on the porosity of the porous medium. As a consequence, it is thought that the capillary coupling parameter, α, must also be dependent, at least in part, on porosity. Whether other parameters have a role to play is, as yet, unknown. Based on the limited amount of data currently available (Bentsen Manai, 1991, 1993), it appears that α may be assumed to be a

10 118 RAMON G. BENTSEN constant for a given s fluid system. However, a direct test of the validity of this assumption is yet to be undertaken. In natural porous media, the value of α must fall between 0 1. For the high permeability, homogeneous, unconsolidated porous medium fluids used by Bentsen Manai (1991,1993), α was determined to have a value of 0.7. It is important to keep in mind, however, that the magnitude of α might differ significantly for other s fluid systems GENERALIZED MOBILITIES In several earlier papers (Bentsen, 1997, 1998a, b), it has been suggested that, in order to describe completely the flow characteristics of a porous medium, four generalized mobilities are required. In view of Equations (29) (30), it appears that such is not the case. That is, only the two conventional mobilities, λ 1 λ 2,are required, provided a proper partitioning of the driving forces has been undertaken. Moreover, in the light of Equations (31) (32), it seems that the mobility of a given phase for countercurrent flow is the same as that for cocurrent flow, if the driving forces have been partitioned properly. Thus, there is no need to suppose that the countercurrent mobility for a given phase is less than that for cocurrent flow, as has been done in the past (Bentsen, 1997, 1998a, b). In the earlier experimental studies (Bentsen Manai, 1991, 1993), different mobilities for cocurrent countercurrent flow were determined because of the operational nature of the definition for mobility. That is, without a proper partitioning of the driving forces, the parameter α is incorporated as a part of the definition for mobility, whereas, in view of Equations (31) (32), it is more properly associated with the driving force (pressure gradient) for a given phase SHOCK FORMATION In a one-dimensional displacement in which the conditions upon which Buckley Leverett (Buckley Leverett, 1942) theory is based are not violated, a sharp front (or shock) will eventually establish itself. Displacements in which the saturation front steepens as the front progresses along the core are commonly referred to as being unstabilized (Sarma Bentsen, 1989). The mechanism by which stabilization (steepening) of the saturation profile occurs can be understood by referring to Equations (29) (30). With regard to these two equations, it is important to note that, while the phase pressure gradients are both negative, the capillary pressure gradient is positive. Thus, the effect of the capillary pressure gradient is to decrease the magnitude of the wetting-phase flux, increase that of the nonwetting-phase flux. As the magnitude of the capillary pressure gradient increases as the wetting phase saturation decreases, the magnitude of this decrease (or increase) in flux increases as the wetting phase saturation decreases. This is so because the magnitude of the slope of the capillary pressure curve increases

11 THE PHYSICAL ORIGIN OF INTERFACIAL COUPLING 119 markedly as the wetting-phase saturation decreases. This is particularly the case in the immediate vicinity of the irreducible saturation to the wetting phase. The overall result of a decreasing wetting phase flux an increasing nonwetting phase flux is a significant steepening of the saturation front, particularly in the immediate vicinity of the irreducible saturation to the wetting phase VISCOUS COUPLING VERSUS CAPILLARY COUPLING It has been determined experimentally (Bentsen Manai, 1991,1993) that the effective mobilities determined in a steady-state, countercurrent flow experiment are less than those determined in a steady-state, cocurrent flow experiment. Such a result cannot be explained, if the conventional transport equations (Equations (7) (8)) are assumed to describe correctly two-phase flow through porous media. That is, in order to explain such results, one must resort to more sophisticated transport equations, such as those constructed by Kalaydjian (1987) others (de la Cruz Spanos, 1983; Whitaker, 1986). It has been usual, in such equations, to postulate that the coupling that takes place between the two flowing phases arises because of interfacial momentum transfer. However, in view of Equations (31) (32), it seems preferable to refer to such interfacial coupling as capillary rather than viscous coupling. This position is taken for two reasons. First, as can be seen in Equations (31) (32), the parameter α is associated with the pressure gradients, not with the mobilities. Second, the coupling component of the driving force for a given phase (see Equations (10) (11)) arises because of the capillarity of the porous medium, not because of interfacial momentum transfer. Over the years, simple analogous models of porous media (see, for example, Bacri et al., 1990; Rose, 1990, 1993; Ayub Bentsen, 1999) have been used to gain insight into the role interfacial momentum transfer plays in flow through porous media. In these models, it is possible to specify the conditions that must apply on the boundary separating the two flowing fluids. Moreover, it is practicable to derive not only transport equations that are analogous to those of Kalaydjian (1987), but also to specify the functional forms of the mobilities. However, two problems arise when one tries to extend these results to flow through natural porous media. First, in order to meet the requirement that the velocity must be zero at the wall(s) of the analogous porous medium, it is necessary that the cross mobilities become zero at both limiting values (S = 0S = 1) of saturation. In natural porous media, on the other h, the cross mobilities were found to be zero at only one of the limiting values of saturation, not both (Bentsen Manai, 1991, 1993). Second, in the analogous models of porous media, it is usually found that the cross mobilities are equal, a result that is consistent with ideas arising out of nonequilibrium thermodynamics (Katchalsky Curran, 1975). Again, however, available experimental results do not support the extension of this result to natural porous media (Dullien Dong, 1996; Avraam Payatakes, 1995; Bentsen Manai, 1993).

12 120 RAMON G. BENTSEN In analogous models of porous media, the interfacial coupling that takes place is truly due to interfacial momentum transfer. On the other h, it appears that, in natural porous media, such coupling is due to the capillarity of the porous medium. Thus, because viscous coupling occurs in analogous models of porous media, while capillary coupling occurs in natural porous media, results obtained for the former are not extendable to the latter. Moreover, it should be pointed out that, while the results presented herein provide a plausible explanation as to why, in natural porous media, steady-state, countercurrent mobilities appear to be less than steady-state, cocurrent mobilities, they do not preclude the possibility that some viscous coupling is also taking place. On the basis of recent results presented in the literature (Zarcone Lenormond, 1994; Rakotomalala et al., 1995), however, it can be inferred that, in natural porous media, the viscous coupling effect is likely to be negligibly small. 4. Concluding Remarks In this study, a partitioning concept developed in an earlier paper (Bentsen, 1998b) was introduced into Kalaydjian s transport equations (Kalaydjian, 1987) to construct modified transport equations that enable a better understing of interfacial coupling in two-phase flow through natural porous media. Using these new equations, it is shown that, in natural porous media, interfacial coupling arises not because of interfacial momentum transfer, as is usually assumed, but rather because of the capillarity of the porous medium. Moreover, it is demonstrated that the new equations can be used to show that the mobilities measured in a steadystate, cocurrent flow experiment are the same as those measured in a steady-state, countercurrent flow experiment. The reason, in earlier experiments (Bentsen Manai, 1991, 1993), that the countercurrent permeabilities appeared to be less than the cocurrent mobilities is that a proper partitioning of the pressure gradients in the equations used to determine the mobilities had not been undertaken. In addition, it is demonstrated that the capillary-pressure term appearing in the new equations can be used to explicate the mechanism by which the saturation front steepens in an unstabilized displacement. Also, it is shown that, in view of the new equations, the rate at which fluid imbibes into a water-wet porous medium is dependent upon the capillary coupling parameter, α. Finally, it is argued that the capillary coupling parameter, α, is, at least in part, dependent on the porosity of the porous medium through which the fluids are flowing. References Avraam, D. G. Payatakes, A. C.: 1995, Generalized relative permeability coefficients during steady-state two phase flow in porous media, correlation with flow mechanisms, Transport in Porous Media 30, Ayub, M. Bentsen, R. G.: 1999, Interfacial viscous coupling: a myth or reality, J. of Petr. Sci. Engng 23,

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14 122 RAMON G. BENTSEN Whitaker, S., 1986, Flow in porous media: II. The governing equations for immiscible, two-phase flow, Transport in Porous Media 1, Zarcone, C. Lenorm, R.: 1994, Détermination expérimentale du couplage visqueux dans les écoulements diphasiques en milieu poreux, C.R. Acad. Sci., Paris, Series II, 318,