Closure Conditions for Two-Fluid Flow in Porous Media

Transcription

1 Tranport in Porou Media 47: 29 65, Kluwer Academic Publiher. Printed in the Netherland. 29 Cloure Condition for Two-Fluid Flow in Porou Media WILLIAM G. GRAY 1, ANDREW F. B. TOMPSON 2 and WENDY E. SOLL 3 1 Department of Civil Engineering and Geological Science, Univerity of Notre Dame, Notre Dame, IN , U.S.A. 2 GET Diviion, L-204 Lawrence Livermore National Laboratory, PO Box 808, Livermore, CA EES-5 Diviion, F-665, Lo Alamo National Laboratory, Lo Alamo, NM (Received: 8 Augut 2000; in final form: 31 May 2001) Abtract. Modeling of multiphae flow in porou media require that the phyic of the phae preent be well decribed. Additionally, the behavior of interface between thoe phae and of the common line where the interface come together mut be accounted for. One factor complicating thi decription i the fact that geometric variable uch a the volume fraction, interfacial area per volume, and common line length per volume enter the conervation equation formulated at the macrocale or core cale. Thee geometric denitie, although important phyical quantitie, are reponible for a deficit in the number of dynamic equation needed to model the ytem. Thu, to obtain cloure of the multiphae flow equation, one mut upplement the conervation equation with additional evolutionary equation that account for the interaction among thee geometric variable. Here, the econd law of thermodynamic, the contraint that the energy of the ytem mut be at a minimum at equilibrium, i ued to motivate and generate linearized evolutionary equation for thee geometric variable and interaction. The contitutive form, along with the analyi of the ma, momentum, and energy conervation equation, provide a neceary complete et of equation for multiphae flow modeling in the uburface. Key word: multiphae flow, porou media, cloure, unaturated flow, interfacial area, entropy inequality, thermodynamic. Nomenclature Latin A meaure of diequilibrium between change in olid urface area and poroity. a interfacial area per unit volume. B meaure of diequilibrium between change in wn interfacial area and other geometric propertie. b entropy ource term per unit ma. C meaure of diequilibrium between change in common line length and other geometric propertie. C wn common line wn within an averaging volume. D i / material time derivative following the motion of the i component, / t v i. d rate of train tenor, ( v v T )/2. Ê i internal energy of component i per unit volume.

2 30 WILLIAM G. GRAY ET AL. E Lagrangian train tenor of the olid phae. êj i rate of ma exchange from component j to component i. F location of macrocale olid that wa initially at X. G αβ macrocale orientation tenor for αβ interface. G wn macrocale orientation tenor for wn common line. h external upply of energy per unit ma. I unit tenor. J macrocopic curvature of interface baed on unit normal outward from phae. Jαβ α macrocopic curvature of αβ interface baed on unit normal outward from α phae. K θ heat conduction tenor. Kαβ i ma tranfer exchange coefficient between phae i and interface αβ. Kwn ma tranfer exchange coefficient between interface αβ and common line wn. L coefficient in dynamic capillary preure equation. L w coefficient in dynamic preading preure equation. L ɛ coefficient in dynamic total preure equation. l wn length of common line wn per unit volume. M i ma conervation equation for phae, interface, or common line i. M i ma of component i. n unit vector normal to the urface of and pointing outward from the phae. n α αβ unit vector normal to the αβ interface pointing outward from the α phae. P grand canonical potential (GCP) of olid phae per unit volume of olid phae. p preure. ˆQ i j rate of energy exchange from component j to component i. q non-advective heat flux vector. R i j flow reitance coefficient between component i and j. S αβ interfacial area between α and β phae. ˆT i j momentum exchange between component i and j. t i tre tenor for component i. V total volume of ytem. V macrocale, repreentative averaging volume. v i macrocale velocity of component i. X initial location of olid phae. x α fraction of the urface that i in contact with the α phae. Greek Ɣ αβ grand canonical potential (GCP) of αβ interface per unit area of αβ interface. Ɣ wn grand canonical potential (GCP) of wn common line per unit length of wn common line. γ interfacial or lineal tenion. δ variation. δ fixed point variation. ɛ poroity. ɛ α volume fraction of phae α. ˆη entropy per unit volume. θ temperature. κg wn macrocale geodeic curvature of the wn common line. κg wn microcale geodeic curvature of the wn common line. κn wn macrocale normal curvature of the wn common line. κn wn microcale normal curvature of the wn common line.

3 TWO-FLUID FLOW IN POROUS MEDIA 31 rate of entropy generation. λ i E Lagrange multiplier for incorporation of energy conervation contraint for component i into the entropy inequality. λ i M Lagrange multiplier for incorporation of ma conervation contraint for component i into entropy inequality. λ i P Lagrange multiplier for incorporation of momentum conervation equation for component i into the entropy inequality. λ wn microcale unit vector tangent to the wn common line. µ chemical potential. ν αβ wn unit vector normal to the wn common line, tangent to the αβ interface, poitive outward from the αβ interface. ξ microcale patial coordinate. ρ denity, ma per unit volume. ummation over all phae (w, n,and). α β =α ummation over all phae except α-phae. αβ ummation over all interface (wn, w, andn). σ olid phae tre. τ i dynamic momentum exchange term for component i. macrocale effective contact angle between the wetting phae and the olid. i gravitational potential for component i. ϕ microcale contact angle between the wetting phae and the olid. ϕ non-advective entropy flux vector. ˆ grand canonical potential (GCP) per unit volume. Supercript/Subcript i generic reference to either the w, n, or phae; the n, wn,orwn interface; or the wn common line. n relating to the non-wetting phae. n relating to the interface between the non-wetting and olid phae. relating to the olid phae. w relating to the wetting phae. wn relating to the interface between the wetting and non-wetting phae. wn relating to the common line. w relating to the interface between the wetting and olid phae., a relative quantity (e.g. θ w,wn = θ w θ wn ). 1. Introduction Although multiphae flow in porou media ha been germane to the field of petroleum engineering and hydrology for decade, it proper phyical and mathematical repreentation i till a ubject of coniderable tudy. The traditional conceptualization of a two-fluid ytem, for example, begin at the microcale at which the porou medium i compoed of a olid phae and a connected void pace in which the fluid may move and interact (Figure 1). The phae are conidered to be immicible and to have ditinct thermodynamic propertie. They are eparated by very thin tranition region which are typically modeled a two-dimenional interfacial urface. In principle, an interface between a particular pair of phae ha it own thermodynamic propertie that are ditinct from thoe of the phae and

4 32 WILLIAM G. GRAY ET AL. Figure 1. Microcale perpective of a three phae (wn) ytem (below) howing the particular phae, interface, and common line employed in the analyi and the averaging pathway to a macrocale point (top) uing a repreentative volume approach. from other interface. In a three phae ytem, tranition region at the junction of all three phae may alo exit. Thee may be aigned thermodynamic propertie of their own and are typically repreented a one-dimenional common line. The phae are bounded by interface and the interface are bounded by common line. Thu, the microcopic picture of a porou medium ytem i a pace occupied by phae, interface, and common line that exit in mutually excluive domain. Although microcale conervation equation may be formulated for thi ytem, their olution i complicated by the fact that the evolving geometry of each phae throughout the domain mut be known o that boundary condition can be pecified. Interface and common line are moving boundarie. For natural ytem,

5 TWO-FLUID FLOW IN POROUS MEDIA 33 uch geometric detail i unavailable, even for an immobile olid phae, except for very mall ample, and the olution of practical problem over realitic length cale i infeaible. Therefore the equation decribing the phyic of the flow are typically formulated at a larger cale encompaing ten to thouand of pore within a repreentative volume centered on a macrocopic point in the ytem (Figure 1). At uch a point, the phae are conidered to coexit yet are undertood to occupy only a fraction of the region in the repreentative volume itelf. Thu, at the macrocale the phae are overlapping continua. The ma, momentum, and energy conervation equation for interface and common line can alo be integrated from the microcale to the macrocale. Although thee equation are neglected in traditional analye (e.g. Whitaker, 1967, 1969; Bachmat, 1972; Slattery, 1972; Haanizadeh and Gray, 1979), their incluion i needed to realitically and ytematically account for the thermodynamic effect of interface and common line in multiphae flow problem. A a byproduct of changing from the microcale to the macrocale, everal new primary geometric variable enter the formulation that have no counterpart at the microcale. Thee include the familiar phae volume fraction (often expreed in term of aturation and poroity) a well a the interfacial area per volume and common line length per volume. Unfortunately, the change of cale alo introduce an equation deficit into the formulation, a evolutionary equation for the geometric variable are not directly available. In addition, contitutive potulate mut be made for quantitie uch a the macrocale tre tenor, the expreion for ma, momentum, and energy exchange between phae, and the heat conduction vector. Sytematic procee for obtaining the needed contitutive relation have been preented elewhere, a decribed ubequently. The problem of formulating a comprehenive contitutive theory ha led to everal different approache. Whitaker (e.g. 1998) ue an approach that map a microcale quantity into a macrocale quantity uch that a differential equation arie whoe olution complete the cloure problem. Thi approach ha been mot often applied for the cae of a rigid olid matrix. Interface dynamic and macrocale thermodynamic are not conidered. Kalaydjian (1987) ha developed a formulation that employ the entropy inequality and alo ha ome equation for interface dynamic. However, the common line i not included and the development make ome ignificant aumption along the way rather than looking at the complete entropy inequality. Allen (1984) ha developed the multiphae flow equation uing a mixture theory approach, but the cloure condition obtained are incomplete becaue the interface have not been incorporated. A paper by Hilfer (1998) indicate further the difficultie in cloing the complex ytem of multiphae flow equation obtained if one ue the mixture theory approach. Hi heuritic aumption are not able to properly account for interfacial area or the energy of the interface and erroneouly lead to an equation that indicate flow can be driven by gradient in volume fraction. Gray and Haanizadeh (1991) preented an unaturated flow theory that included interface dynamic that employed the averaged entropy

6 34 WILLIAM G. GRAY ET AL. inequality to obtain the needed contitutive relation. Bennethum and Giorgi (1997) and Bennethum (1994) have alo ued the entropy inequality in obtaining porou media flow equation. The potulation of the thermodynamic dependence of internal energy wa improved in Gray (1999) by making the potulation in term of extenive variable. Thi change alo clarified the need for cloure condition involving the dynamic of the geometric variable. The cloure condition developed there are baed on approximation to averaging theorem and do not account for the average orientation of the interface within the averaging volume and thu are omewhat limited in their applicability. Preentation of a ytematic proce baed on exploitation of the econd law of thermodynamic for overcoming the equation deficit and obtaining geometric evolutionary equation i the thrut of thi paper. Thee equation are eential to pecification of a complete model. 2. Macrocale Conervation Equation The multiphae ytem to be conidered i compoed of three phae (Figure 1): a wetting phae, w, a non-wetting phae, n, and a olid phae,. Therefore, three different type of interface will exit denoted a: wn between the wetting and non-wetting phae, w between the wetting and olid phae, and n between the non-wetting and olid phae. The only type of common line i formed where the three phae (or more preciely, the three interface) interect. The common line i deignated a wn. Note that interface are adjacent to phae, phae and common line are adjacent to interface, and interface are adjacent to common line. The derivation of general macrocale equation by averaging or localization i well-undertood and will not be repeated here (ee e.g. Gray and Haanizadeh, 1998). However, the equation themelve are neceary precuror to the contitutive approach. Therefore the balance equation are provided here in tabular form. The ma balance equation appear in Table I with the notation given previouly. Oberve that the entrie in the third column have unit of ma per macrocale volume; the entrie in the fourth column account for exchange of ma between a region and the adjacent region. When the interface and common line are treated a male, only the ource/ ink term urvive. Thi implified form of the interface ma conervation equation i equivalent to the tandard jump condition for the exchange of ma between two phae of different denitie with no accumulation of ma within the interface. The component term of the momentum balance per unit macrocale volume are provided in Table II. The ource and ink term for momentum are each compoed of two part, one aociated with ma exchange and a econd due to urface tre effect between adjacent region due to preure, tenion, and the macrocale accounting for microcale vicou effect. The um of the ource term over all region of the macrocale volume i equal to the um of the ink term over all

7 TWO-FLUID FLOW IN POROUS MEDIA 35 Table I. Conervation of ma M i = Di M i M i d i : I M i ink Mi ource = 0 Region i M i M i ink Mi ource = 0 w-phae w ε w ρ w ê w wn êw w n-phae n ε n ρ n ê n wn ên n -phae ε ρ ê w ê n wn-interface wn a wn ρ wn ê w wn ên wn êwn wn w-interface w a w ρ w ê w w ê w êw wn n-interface n a n ρ n ê n n ê n ên wn wn-line wn l wn ρ wn ê wn wn êw wn ên wn region a momentum i not created within the volume. When the interface and common line are male, the urviving term in the interface momentum equation expre the macrocale form of the tandard jump condition for momentum exchange between phae. If microcale interfacial and lineal tenion are zero, it can be hown that the tre tenor J i for the interface and common line will be zero. The energy balance term appear in Table III. The equation provide the balance of total energy (internal plu kinetic plu potential) for each region per macrocale volume. The ource and ink term appearing in the lat column um over all region to zero. Becaue of the poibility of conidering male interface and common line, the internal energy i expreed directly on a per volume bai. If the interface i male, the kinetic and potential energy term will be zero. Alo, for the cae of a male interface, the interface momentum equation reduce to the tandard jump condition between phae. The male common line energy equation expree the jump condition for energy among the interface that meet at the common line. The entropy inequalitie for the phae, interface, and common line region are given in Table IV. The entropy i expreed per unit of ytem volume in each equation. Note that although the ource and ink term, accounting for exchange among the region, cancel when ummed over all phae and interface, the um of the generation term urvive. Thi um mut be non-negative for the entropy inequality to be atified. Thi condition erve to help guide the development of the contitutive form needed to cloe the problem. Table I through 3 provide five macrocale conervation equation for each of the even phae region that may be olved for 35 primary dependent variable: denity, velocity (diplacement for the olid phae), and temperature. Thee variable may be lited a:

8 36 WILLIAM G. GRAY ET AL. Table II. Conervation of momentum P i = Di (M i v i ) M i v i d i : I J i M i i P i ink Pi ource = 0 Region i M i J 1 P i ink Pi ource w-phae w ε w ρ w ε w t w ê w wn vw ˆT w wn ê w w vw ˆT w w n-phae n ε n ρ n ε n t n ê n wn vn ˆT n wn ê n n vn ˆT n n -phae ε ρ ε t ê w v ˆT w ê n v ˆT n wn-interface wn a wn ρ wn a wn t wn ê w wn vw ˆT w wn ên wn vn ˆT n wn ê wn wn vwn ˆT wn wn w-interface w a w ρ w a w t w ê w w vw ˆT w w ê w v ˆT w ê w wn vw ˆT w wn n-interface n a n ρ n a n t n ê n n vn ˆT n n ê n v ˆT n ê n wn vn ˆT n wn wn-line wn l wn ρ wn l wn t wn ê wn wn vwn ˆT wn wn êw wn vw ˆT w wn ê n wn vn ˆT n wn ρ i, v i,θ i,ρ, F,θ where i = w, n, wn, w, n, wn. (1) In addition, the macrocale equation alo contain even primary geometric variable that do not occur in the correponding microcale problem. Thee variable are geometric in nature and include the volume fraction of each phae, the pecific area per unit bulk volume of the interface, and the pecific length per unit bulk volume of the common line. Thee variable are lited a: ɛ w,ɛ n,ɛ,a wn,a w,a n,l wn. (2) They hould be recognized a patially and temporally variable quantitie that evolve with changing ytem condition. To olve the conervation equation, additional information i needed concerning the inter-relation of thee even variable and their relation to change of the primary variable. Ideally, even dynamic geometric evolutionary equation for thee quantitie would be pecified which would

9 TWO-FLUID FLOW IN POROUS MEDIA 37 Table III. Conervation of energy { D i E i = [ Ê i M i (v i ) 2 ]} i 2 [ {Ê i M i (v i ) 2 ]} i d i : I F i M i H i E i 2 ink Ei ource = 0 Region i M i F i H i E i ink Ei ource w-phae w ε w ρ w ε w (t w v w q w ) h w w t (êw wn êw w )Ew T ε w ρ w ( ˆT w wn ˆT w w ) vw ˆQ w wn ˆQ w w n-phae n ε n ρ n ε n (t n v n q n ) h n n t (ên wn ên n )En T ε n ρ n ( ˆT n wn ˆT n n ) vn ˆQ n wn ˆQ n n -phae ε ρ ε (t v q ) h t (ê w ê n )E T ε ρ ( ˆT w ˆT n ) v ˆQ w ˆQ n wn-interface wn a wn ρ wn a wn (t wn v wn q wn ) h wn wn t ê wn w Ew T ε w ρ w ên wn En T ε n ρ n ˆT w wn vw ˆT n wn vn êwn ˆQ w ˆQ n êwn wn Ewn T ˆT wn ˆQ wn wn wn êwn a wn ρ wn wn vwn wn w-interface w a w ρ w a w (t w v w q w ) h w w t ê w w Ew T ε w ρ w ê w E T ε ρ ˆT w w vw ˆT w v êwn ˆQ w ˆQ êwn wn Ew T ˆT w ˆQ w w w êwn a w ρ w wn vw wn

10 38 WILLIAM G. GRAY ET AL. Table III. (continued) Region i M i F i H i E i ink Ei ource n-interface n a n ρ n a n (t n v n q n ) h n n t ê n n En T ε n ρ n ê n E T ε ρ ˆT n n vn ˆT n v ˆQ n n ên ˆQ ên wn En T ˆT n ˆQ n n ên a n ρ n wn vn wn wn-line wn l wn ρ wn l wn (t wn v wn q wn ) h wn wn t ê wn wn Ewn T a wn ρ wn Ew T êw wn a w ρ w ên wn En T a n ρ n ˆT wn wn vwn ˆT w wn vw ˆT n wn vn ˆQ wn wn ˆQ w wn ˆQ n wn [ E T i = Êi M i (v i ) 2 ] i. 2

11 TWO-FLUID FLOW IN POROUS MEDIA 39 Table IV. Entropy inequality Rgeneration i = Di ˆη i ˆη i d i : I B i M i b i η i ink ηi ource 0 Region i M i B i η i ink ηi ource R i generation w-phae w ε w ρ w ε w ϕ w ê w wn ˆηw /M w ˆ w wn ε w w ê w w ˆηw /M w ˆ w w n-phae n ε n ρ n ε n ϕ n ê n wn ˆηn /M n ˆ w wn ε n n ê n n ˆηn /M n ˆ n n -phae ε ρ ε ϕ ê w ˆη /M ˆ w ε ê n ˆη /M ˆ n wn-interface wn a wn ρ wn a wn ϕ wn ê w wn ˆηw /M w ˆ w wn ên wn ˆηn /M n a wn wn ˆ n wn êwn wn ˆηwn M wn ˆ wn wn w-interface w a w ρ w a w ϕ w ê w w ˆηw /M w ˆ w w ê w ˆη /M a w w ˆ w êw wn ˆηw M w ˆ w wn n-interface n a n ρ n a n ϕ n ê n n ˆηn /M n ˆ n ê n ˆη /M a n n ˆ n ên wn ˆηn M n ˆ n wn wn-line wn l wn ρ wn l wn ϕ wn êwn ˆηwn M wn ˆ wn wn êw wn ˆηw M w w wn ên wn ˆηn M n ˆ n wn l wn wn allow the ytem to be cloed. Unfortunately, only one uch equation i available, the requirement that the um of the volume fraction be 1: ɛ w ɛ n ɛ = 1. (3) The ix additional equation needed are much more eluive. Gray (1999) ha propoed ome form baed on an examination of the averaging theorem in conjunction with the entropy inequality. Here, more le retrictive form are obtained by conidering the condition of mechanical equilibrium at the macrocale in conjunction with the entropy inequality. However, it mut be treed that thee ix relationhip are evolutionary approximation and are not conervation law. They are ubject to further improvement baed on inight derived from numerical and experimental tudie in the future.

12 40 WILLIAM G. GRAY ET AL. For application to a particular ytem, the macrocale conervation equation mut be augmented or cloed by a erie of approximate contitutive equation for the 115 additional variable (uch a ma exchange term, tre tenor, internal energy, etc.) that appear in the equation. Thee function mut ideally be expreed in term of the primary dependent and geometric variable and their derivative in uch a way that the form obtained do not violate the econd law of thermodynamic and are conitent with oberved phyical ytem. For example, the tre tenor ha been previouly hown to be ymmetric baed on the form of the macrocale angular momentum equation (Haanizadeh and Gray, 1979) reducing the number of unknown tre component from nine to ix for each phae. Note alo that although, for example, the heat conduction in an interface will be a twodimenional vector at the microcale, the fact that an interface doe not necearily have a ingle orientation within the macrocale region mean that interfacial heat conduction i a patial proce viewed from the macrocale. Indeed, all the tenor and vector that are two-dimenional at the microcale will be three-dimenional at the macrocale. The procedure for obtaining contitutive form baed on the entropy inequality may be found, for example, in Gray and Haanizadeh (1998). To obtain the additional expreion needed for evolution of the geometric variable, it i neceary to firt et up the econd law of thermodynamic for the full ytem. Thi i obtained from the entropy inequalitie for each of the ytem region ubject to the contraint of the conervation equation. The development of thi contrained entropy inequality i not new to thi work, however it exploitation to obtain the cloure condition a a deviation from the thermomechanical equilibrium tate i new. 3. The Second Law of Thermodynamic The econd law of thermodynamic i a powerful tool that can be ued to guide the development of cloure and contitutive relationhip for the macrocopic equation of multiphae flow (Haanizadeh and Gray, 1980; Gray, 1999). It i epecially ueful in complicated multiphae ytem uch a that conidered here. The tarting point i the general tatement of the econd law, which precribe that the net rate of production of entropy of a ytem inide a large ytem volume V mut be non-negative: dv 0. (4) V The macrocale entropy production per unit volume,, mut be non-negative for thi inequality to hold for any volume. In term of the macrocopic quantitie employed in our three phae ytem, thi may be written a: = ɛ w w ɛ n n ɛ a wn wn a w w a n n l wn wn = i Rgeneration i 0. (5)

13 TWO-FLUID FLOW IN POROUS MEDIA 41 Every olution to the generally-poed field equation in Table I III mut be uch that inequality (5) i atified. Hence, thee equation erve a contraint on inequality (5) and need to be impoed or incorporated within it in order to extract ueful ytem inight. Thi may be accomplihed through a rather long ubtitution proce, uch a that purued by Haanizadeh and Gray (1980) and Gray (1999), or via a conceptually impler approach firt uggeted by Liu (1972). In Liu technique, Equation (5) i modified through the addition of a linear combination of the balance equation appearing in Table I III, = i R i generation λi M Mi λ i P Pi λ i E Ei 0, (6) where λ i M, λi P,andλi E are arbitrary parameter. Since all the term added to the entropy inequality are zero, the calculated rate of entropy generation i unchanged by thi modification. However, the new verion can be rearranged in term of like quantitie that appear throughout the balance equation and ued to identify particular non-zero value of λ i M, λi P,andλi E that trategically implify the reulting relationhip. The uniform extenive (or integrated) energy for each phae, interface, and common line lying within a macrocopic volume V i potulated to be a function of the extenive variable of the ytem, namely it integrated entropy, ma, volume, interfacial area, common line length, etc. To allow for gradient in the energy that may exit, energy denity i obtained a a function of the denitie of the independent variable. Additionally, the firt order homogeneou property of the thermodynamic function (Callen, 1985; Bailyn, 1994) i exploited to obtain the explicit form of the energy denity. The energy of each ytem component i written a depending explicitly on, at leat, the entropy of that component, the ma of the component, and the geometric extent (volume, area, or length) of that component a: Ê α ( ˆη α,ɛ α ρ α,ɛ α,...)=ˆη α θ α ɛ α ρ α µ α ɛ α p α..., Ê ( ˆη,ɛ ρ,ɛ 0 E,...)=ˆη θ ɛ ρ µ ɛ σ :E..., α = w, n, (7a) (7b) Ê αβ ( ˆη αβ,a αβ ρ αβ,a αβ,...)=ˆη αβ θ αβ a αβ ρ αβ µ αβ a αβ γ αβ..., αβ = wn, w, n, (7c) Ê wn ( ˆη wn,l wn ρ wn,l wn,...) =ˆη wn θ wn l wn ρ wn µ wn l wn γ wn... (7d) The incluion of interfacial area and common line length a independent variable in thee expreion are critically important for the thermodynamic decription of the macrocale tate (Gray, 1999). A indicated by the ellipe, additional

14 42 WILLIAM G. GRAY ET AL. independent variable could be included at thi tage (a in Gray (1999)) that would allow for other particular effect to be conidered. To treat film flow, for example, thi might entail allowing the interfacial or contact line energie (Ê αβ or Ê wn ) to be function of the neighboring phae volume (ɛ α and ɛ β ). However, in the development that follow, only the explicit dependence hown in Equation (7a) through (7d) will be retained and incorporated. Before proceeding further, it i ueful to recat the internal energy denitie that appear in Equation (6) into grand canonical potential (GCP) function uing the Lagrange tranformation (Callen, 1985). In eence, thi tranformation provide an energy potential (the GCP) that depend on chemical potential and temperature in place of an energy potential (the internal energy) that depend on ma and entropy per volume. Thi change of independent variable i not required for thi analyi but i convenient in anticipation the form of the final et of equation. A Lagrange tranformation on the entropy per unit volume and the ma per unit volume can be made to obtain the GCP function per unit volume for each phae, deignated here a ˆ i. The reult of the tranformation are: ɛ α P α = ˆ α (θ α,µ α,ɛ α,...)= Ê α ˆη α θ α ɛ α ρ α µ α, α = w, n ɛ :E = ɛ P = ˆ ( ˆθ,µ,ɛ 0 E,...)= Ê ˆη θ ɛ ρ µ, (8a) (8b) a αβ Ɣ αβ = ˆ αβ (θ αβ,µ αβ,a αβ,...)= Ê αβ ˆη αβ θ αβ a αβ ρ αβ µ αβ, αβ = wn, w, n (8c) l wn Ɣ wn = ˆ wn (θ wn,µ wn,l wn,...) = Ê wn ˆη wn θ wn l wn ρ wn µ wn. (8d) The GCP have unit of energy per volume (or force per unit area) and allow the variou phae preure variable to be formally introduced. The capitalized term on the left ide of thee equation are ued to repreent the intrinic GCP. For example, P α in Equation (8a) i the GCP of the α phae per unit volume of α phae, while Ɣ αβ in Equation (8c) i the GCP of the αβ interface per unit area of αβ interface. The ue of a capitalized letter for thee term refer to their mot general functional repreentation. They are equivalent to their lower cae counterpart (e.g. p α, γ αβ,orγ wn ), appearing on the right ide of (7a) through (7d) when the energy pecifically, and only, depend on the particular variable lited. The macrocopic balance law hown in Table I III are now ubtituted into the pointwie entropy inequality (6). The GCP function ( ˆ i ) are ued in place of the internal energie (Ê i ). A a horthand notation to help reveal the tructure of the full entropy inequality, the geometric denitie are given a general notation uch that χ α = ɛ α, χ αβ = a αβ and χ wn = l wn. (9)

15 TWO-FLUID FLOW IN POROUS MEDIA 43 We alo utilize the Gibb Duhem equation 0 = χ i dp i ˆη i dθ i χ i ρ i dµ i, (10) for further implification, yielding = {[ Di ˆη i ] ˆη i d i : I i (1 λ i E θ i ) (χ i ϕ i ) χ i ρ i b i ( ˆη ource i ˆηi ink [ ) D i (χ i ρ i ][ ) χ i ρ i d i : I λ i M λi P vi λ i E (µ i (vi ) 2 )] i 2 (Mource i Mi ink )λi M ( D [χ i ρ i i v i ) ] i (χ i t i ) (λ i P λi E vi ) (P i ource Pi ink ) λi P [ D i ˆ i ( ˆ i I χ i t i ) : d i (χ i q i ) χ i ρ i h i θ i,µ i } (Eource i Ei ink ]λ ) i E 0, (11) where the um over the index i i uch that i = w, n,, wn, w, n and wn. The value of λ i M, λi P,andλi E are now choen uch that the factor multiplying the time derivative of entropy, ma denity, and velocity in Equation (11) are zero. In the ene of Liu (1972), thi tep can be motivated by the need for the inequality to be independent of the derivative quantitie appearing in thee particular term. Thu, λ i E = 1 θ i, (12a) λ i P = vi (12b) θ i and λ i M = 1 [ µ i i (vi ) 2 ]. (12c) θ i 2 Thi reult in a modified entropy inequality that i reflective of the balance law contraint. Two additional aumption are now invoked to provide a implification that i conitent with many ytem found in nature, namely The ytem i conidered to be thermodynamically imple (Eringen, 1980) uch that, for correponding upercript, h i = b i θ i and θ i ϕ i = q i ;and

16 44 WILLIAM G. GRAY ET AL. Temperature in phae, interface, and the common line at a point are conidered to be equal and will be deignated a θ = θ i. In addition, it will be convenient to expre all the material derivative with repect to the olid phae velocity, a the reference velocity, uing the equation D i = D vi,, (13) where v i, = v i v. With thee change, and upon multiplication by θ, Equation (11) take the form θ = i { [ D ˆ i ( ˆ i I χ i t i ): d i v i, ˆ i θ,µ i θ,µ i ] χ i q i θ θ (µ i (vi ) 2 ) i (Mource i 2 Mi ink ) vi (P i ource Pi ink ) (E i ource Ei ink ) θ(ˆηi ource ˆηi ink ) } 0, (14) where it i undertood that the um over i repreent a um over the w, n, and phae, the wn, w, andn interface, and the wn contact line. In general, the olid phae GCP require additional, pecial conideration. Thi i becaue the deformation of a olid i decribed differently from the deformation of a fluid, a accounted for by the preence of the mechanical Lagrangian train tenor in the lit of independent variable in Equation (8b). Callen (1985) ha noted, however, that the conventional thermodynamic theory, in which the volume i the ingle mechanical parameter, fully applie to olid. When thi approach i followed, a olid phae preure i obtained. The incluion of elatic train then erve to provide additional information about the mechanic of the olid phae preure. Here, the prime objective i not to obtain information about olid phae mechanic o that, in the interet of implicity, the olid phae preure, deignated a P in Equation (8b) will appear rather than the detail of it dependence on train. Therefore expanion of Equation (14) (where we revert back to the explicit form of the geometric denitie ɛ α, a αβ,andl wn intead of χ i ) provide: θ = D ˆ α ( ˆ α I ɛ α t α ): d α α θ,µ α D ˆ αβ ( ˆ αβ I a αβ t αβ ): d αβ αβ θ,µ αβ

17 TWO-FLUID FLOW IN POROUS MEDIA 45 D ˆ wn ( ˆ wn I l wn t wn ): d wn θ,µ wn ɛ α q α a αβ q αβ lwn q wn θ θ θ θ α αβ (µ αβ µ i )êαβ i (µ wn µ αβ )êwn αβ αβ i=α,β αβ=wn,w,n v α, ˆ α θ,µ ( ) ααβ vα,αβ ˆT α 2 êα αβ α=w,n β =α { ˆ αβ θ,µ ˆT αβ αβ v αβ, αβ i=α,β v wn, ( iαβ vi,αβ ˆT 2 êi αβ { ˆ wn θ,µ wn )} wn vαβ,wn ij =wn,w,n 2 ê αβ wn ( ijwn vij,wn ˆT 2 ê ij wn )} 0, (15) where the ource term from Table I III have been ubtituted. Thi form of the entropy inequality, will be ued to tudy two apect of the problem of multiphae flow. Firt, mechanical equilibrium condition that mut exit among ome of the contitutive variable may be etablihed. The approach to developing thee condition at the macrocale ha been provided in Gray (2000) and the reult will be collected in Appendix A. Second, near equilibrium condition are examined to provide the evolutionary cloure condition ueful for modeling dynamic problem. Thi part of the analyi i new and will be accomplihed following a review of the mechanical equilibrium ituation. 4. Mechanical Equilibrium Condition Continued analyi of the entropy inequality will be aided by inight derived from the condition of mechanical equilibrium for thi ytem. Thee condition repreent how infiniteimal change in geometric variable affect one another when the ytem i otherwie at thermal and chemical equilibrium. A hown in the next ection, they will ultimately be ueful in determining thermodynamic equilibrium and dynamic relation between change in the geometric variable and thermodynamic tate of the ytem. The mechanical equilibrium condition are obtained uing a variational approach outlined in Appendix A imilar to that of Boruvka and Neumann (1977) for microcale contraint and Gray (2000) for macrocale contraint. A principal

18 46 WILLIAM G. GRAY ET AL. reult of thi analyi i the fact that the following three condition mut hold at equilibrium: p w p n J w wn γ wn = 0, (16a) p w x w p n x n P γ w J w xw γ n J n xn lwn (γ wn κ wn a N γ wn in ) = 0, (16b) γ w γ n γ wn co γ wn κg wn = 0, (16c) where x α = a α /a i the fractional olid phae area and δɛ, δɛ w,andδx w are about otherwie fixed macrocale coordi- infiniteimal variation of ɛ, ɛ w,andx w nate. The macrocale curvature of the interface (Jαβ α, J ) and of the common line, κwn N ) a well a the macrocopic meaure contact angle ( ) are defined in Appendix A. Thee variable can be conidered a econdary, or derivative geometric variable ince they are inherently dependent on the geometric configuration of phae and interface that i repreented by the primary geometric variable (ɛ w, ɛ n, a wn, a w, a n,andl wn ). Thu, in general, relation for the econdary geometric variable mut be obtained a tate function of ome or all of the primary geomet- (κ wn G ric variable, a neceary and appropriate (e.g. Jαβ α = J αβ α (ɛα,a αβ,l wn )). Recall that in the traditional multiphae modeling approach, thi ame philoophy i followed when the capillary preure, which i the product of the interfacial tenion multiplied by the interfacial curvature, i potulated to be a function of aturation. 5. Expanion of the Entropy Inequality in term of Independent Variable Since the GCP depend on time only through it dependence on the independent variable, the time derivative of the GCP function in Equation (15) are expanded in term of the independent variable lited in Equation (8a) through (8d). In thi proce, everal of the reultant term are purpoely collected into form that are conitent with the tructure of the equilibrium condition of Equation (16a) through (16c). To complete thi rearrangement, the curvature and contact angle term have to be added in and ubtracted out, a they do not otherwie appear in the expanded verion of Equation (15). Thi proce yield: { D θ = γ wn a wn [ Jwn w D ɛ w x w Jwn w (in )lwn a (γ w x w [ D γ n x n ) a (co )a Dx w x w ] D (Jw J ɛ )(co ) ] J D ɛ }

19 TWO-FLUID FLOW IN POROUS MEDIA 47 γ wn { D l wn [ κ wn G xw [ D ɛ w D ɛ lwn a κ wn G D x w a (J w J ) κ wn N x w l wn a ] D } ɛ D ] ɛ [p w p n γ wn Jwn w ] { p w x w p n x n P γ w Jw xw γ n Jn xn [γ wn in γ wn κ wn N ] } [a D x w ] x w (Jw J ) D ɛ [ γ wn co γ w γ n γ wn κg wn ] ( ˆ α I ɛ α t α ): d α [ ˆ αβ I a αβ t αβ ]: d αβ α αβ [ˆ wn I l wn t wn ]: d wn { ɛ α q α a αβ q αβ lwn q wn θ θ θ α αβ [µ αβ µ i ]êαβ i αβ αβ=wn,w,n α=w,n i=α,β v a, v αβ, αβ i=α,β v wn, { ˆ α θ,µ α β =α { ˆ αβ θ,µ αβ ˆT αβ ( iαβ vi,αβ ˆT 2 êi αβ { ˆ wn θ,µ wn )} } θ [µ wn µ αβ ]ê αβ ( ˆT α αβ vα,αβ 2 êα αβ wn vαβ,wn ij =wn,w,n ( 2 ê αβ wn )} wn )} ˆT ij wn vij,wn êwn ij 0. (17) 2 Ultimately, the idea in thi manipulation i to develop a form appropriate for exploitation. Retriction and contraint on the form and value of the variou contitutive variable, dictated by thi expreion of the econd law, hould become apparent. Since the entropy production rate mut be zero at equilibrium, each of the product term added together in Equation (17) mut be zero at equilibrium. To properly exploit the inequality, it i alo deirable for each of the individual factor

20 48 WILLIAM G. GRAY ET AL. compriing each product to alo be zero at equilibrium. Unfortunately, thi i not the cae with the current form of (17), a the firt factor in each of the firt three product do not meet thi criterion. To obtain a more manageable form, three approximate evolutionary equation, (52), (60), and (62), have been developed in Appendix B to re-expre the econd factor in each of thee product in term of other variable that appear elewhere in the entropy inequality. The evolutionary equation are determined uing averaging theory, an examination of the mechanical equilibrium tate reviewed in Appendix A, and a linearization around thi tate for near equilibrium condition. Thee particular approximation comprie three of the ix dynamic contraint equation needed to cloe the ytem a dicued previouly in Section 2. They decribe change in geometric variable a the phae, interface, and common line deform. When the approximation in Equation (52), (60), and (62) are ubtituted into Equation (17), it become: θ = [ D ɛ w D ɛ lwn a x w D ɛ ] [p w p n γ wn J w wn ] { p w x w p n x n P γ w Jw xw γ n Jn xn [γ wn (in ) γ wn κn } wn ] [a D x w x w (Jw J ) D ɛ ] [ γ wn (co ) γ w γ n γ wn κg wn ] {( ˆ α I ɛ α t α ): d α } α [ˆ wn I γ wn a wn G wn a wn t wn ]: d wn {ˆ w I [γ w x n (γ wn co γ n γ w γ wn κg wn )]aw G w a w t w }: d w {ˆ n I [γ n x w (γ wn co γ n γ w γ wn κg wn )]an G n a n t n }: d n {[ˆ wn I γ wn l wn G wn l wn t wn ]: d wn } { ɛ α q α a αβ q αβ lwn q wn } θ θ θ θ α αβ { } [µ αβ µ i ]êαβ i αβ i=α,β αβ=wn,w,n { v α, ˆ α θ,µ α ( ˆT α αβ vα,αβ 2 êα αβ α=w,n β =α [µ wn µ αβ ]ê αβ wn )}

21 TWO-FLUID FLOW IN POROUS MEDIA 49 { v wn, ˆ wn θ,µ wn γ wn (G wn a wn ) ˆT wn wn vwn,wn êwn 2 ( )} ˆT i wn vi,wn 2 êi wn i=w,n { v w, ˆ w θ,µ w [γ w x n (γ wn co γ n γ w γ wn κg wn )] (aw G w ) ˆT w wn vw,wn êwn w ( )} ˆT i w vi,w 2 êi w 2 i=w, { v n, ˆ n θ,µ n [γ n x w (γ wn co γ n γ w γ wn κ wn G )] (an G n ) ˆT n wn vn,wn ê n wn [ ( )]} ˆT i n vi,n 2 êi n 2 i=n, { v wn, ˆ wn θ,µ wn γ wn [G wn l wn ] [ ij =wn,w,n ( ˆT )]} ijwn vij,wn êwn ij 0. (18) 2 In thi new, approximate form, the individual factor compriing each product term of the inequality are now all zero at equilibrium making it poible to obtain additional contitutive information. The independent term in thi inequality mut be non-negative to enure that the entropy inequality i atified. 6. Contitutive Equation Inight from the Entropy Inequality In thi ection, the form of the inequality in Equation (18) will be exploited to ugget functional form of everal of the contitutive variable introduced in Section 2. By definition, thee form will be conitent with the econd law of thermodynamic and the current choice of independent variable. In particular, attention will be focued on the tre tenor t i and momentum exchange vector, ˆT i j, that appear in the momentum balance equation, the heat conduction vector, q i, that appear in the energy balance equation, and the ma exchange term, êj i, that appear in the ma and momentum balance equation. Conider, firt, the variou tre tenor t i that appear in thi analyi. Since the entropy production rate ha been aumed independent of the rate of train tenor (that i, d i i not an independent variable), the multiplier appearing in front of d i in (18) mut alway be zero, even when the ytem i not at equilibrium. Uing element of Equation (8a) through (8d), thi ugget that the tre tenor take the following form:

22 50 WILLIAM G. GRAY ET AL. ɛ α t α = ˆ α I = ɛ α p α I, α = w, n, (19a) ɛ t = ˆ I = ɛ P I, a wn t wn = ˆ wn I γ wn a wn G wn = γ wn a wn (I G wn ), (19b) (19c) a w t w = ˆ w I [γ w x n co γ n γ w γ wn κg wn G w = γ w a w (I G w ) [x n co γ n γ w γ wn κg wn G w (19d) a n t n = ˆ n I [γ n x w (γ wn co γ n γ w γ wn κg wn )]an G n = γ n a n (I G n ) [x w (γ wn co γ n γ w γ wn κ wn G )]an G n, (19e) l wn t wn = ˆ wn I γ wn l wn G wn = γ wn l wn (I G wn ). (19f) Now conider the multiplier of the relative velocitie that appear in the entropy inequality (18). Thee term pecifically involve the momentum exchange vector, ˆT i j, and ma exchange term, êi j. A with the velocitie, each of thee term i individually zero at equilibrium, but not necearily zero away from equilibrium. For convenience, thee term have been denoted by the vector τ i which can be reexpreed, for implicity, uing element of Equation (8a) through (8d): ( ) τ α = p α ɛ α αwn vα,wn ˆT 2 êα wn ( ) αα vα,α ˆT 2 êα α, α = w, n (20a) ( τ wn = γ wn [(I G wn )a wn ] ˆT wn i=w,n wn vwn,wn ) êwn 2 ( ) iwn vi,wn ˆT 2 êi wn, (20b) ( τ w = γ w [(I G w )a w ] ˆT w ( ) iw vi,w ˆT 2 êi w i=w, x n wn vw,wn 2 ) êwn w (γ wn co γ n γ w γ wn κ wn G ) (aw G w ), (20c)

23 TWO-FLUID FLOW IN POROUS MEDIA 51 τ n = γ n [(I G n )a n ] ( ˆT n ( ) in vi,n ˆT 2 êi n i=n, x w wn vn,wn 2 ê n wn ) (γ wn co γ n γ w γ wn κ wn G ) (an G n ), (20d) τ wn = γ wn [(I G wn )l wn ] ( ijwn vij,wn ˆT 2 ij =wn,w,n ê ij wn ). (20e) In general, thee vector can be conidered function of independent variable that are zero at equilibrium and approximated by a linearization around the equilibrium tate. Thee linearization may be performed in term of all equilibrium variable or over a ubet conidered to be the mot dominant. Such a linearization for τ i = τ i (v i,,...)will be conidered in the next ection. A a more pecific example, conider how the heat conduction vector appearing in inequality (18) can be approximated uing thi linearization approach. At equilibrium, each element of the term q = α ɛ α q α αβ a αβ q αβ l wn q wn θ (21) mut be zero. Thu, if the imple linearization i employed uch that the heat flux q depend only on θ, a imple linearization yield q = K θ θ, (22) where K θ i a poitive definite coefficient tenor and the negative ign i required to maintain a poitive contribution to the entropy production away from equilibrium. A imilar approximation of the ma exchange term aume êαβ i =êi αβ (µαβ µ i ) and êwn αβ =êwn(µ αβ wn µ αβ ) to obtain and ê i αβ = Ki αβ (µαβ µ i ), αβ = wn, w, n, i = α, β (23a) ê αβ wn = Kαβ wn (µwn µ αβ ), αβ = wn, w, n (23b) where Kαβ i and Kαβ wn are poitive coefficient. We note that thee term apply for ytem where there i an exchange of phae ma between pure phae (e.g. melting ice), a oppoed to the tranfer of pecie ma between phae which i not conidered here. Although no mechanim ha been included for following the propertie of individual pecie within a phae, addition of the appropriate

24 52 WILLIAM G. GRAY ET AL. equation to the formulation can be made directly, albeit with increaed algebraic complexity. Interetingly, thi linearization approach can alo be utilized to identify an approximation to the geometric variable term appearing in the firt three term of the inequality (18). Here, the implet linearization approach i ued in which the effect if cro-coupling among the different dependent variable i neglected. Thu, linearization of the firt term in the inequality provide D ɛ w x w D ɛ where L i a poitive coefficient. Similarly, the econd and third term in Equation (18) linearize to and D ɛ a D x w = L [p w p n γ wn Jwn w ], (24) { = L ɛ p w x w p n x n P γ w Jw xw γ n Jn xn } lwn [γ wn in γ wn κ wn a N ] (25) x w (Jw J ) D ɛ = L w [ γ wn co γ w γ n γ wn κg wn ], (26) where L ɛ and L w are poitive coefficient. Equation (24) through (26) provide three approximate cloure equation for the geometric variable that, along with Equation (52), (60), and (62), comprie the ix extra condition needed to fully cloe the ytem, a dicued previouly in Section 2. Although the reult here are imilar to previou cloure relation developed by Gray (1999), they have been developed here in the context of deviation from thermodynamic equilibrium rather than imply a approximation to mathematical averaging theorem. They are more complete and give rie to the improved approximation to be employed in the mometum equation indicated in contitutive expreion (20a) through (20e). Notice that Equation (24) through (26) are dynamic equation that independently reproduce the equilibrium relationhip uggeted by the mechanical equilibrium analyi of Appendix A (and hown a Eq. (16a) (16c)). There are two apparent and important implication of Equation (24): If the product of the mean macrocale curvature Jwn w and the interfacial tenion, γ wn i identified a the capillary preure, Equation (24) provide the tandard equilibrium condition that the capillary preure i equal to the preure difference between the fluid phae. Since, from previou dicuion and Gray (1999), it i reaonable to potulate that Jwn w = J wn w (ɛw,a wn ), then the uual aumption that capillary preure at equilibrium i only dependent on aturation i not necearily complete. The actual dependence will have to be determined or verified experimentally.

25 TWO-FLUID FLOW IN POROUS MEDIA 53 Away from equilibrium, under tranient, flowing condition, the capillary preure i a dynamic function that i not necearily equal to the phae preure difference. Thi ha alo been uggeted by Kalaydjian (1992), who wa able to meaure a coefficient of proportionality in a imple experiment. Thu, even if Jwn w i a function only of aturation, the difference between it equilibrium and non equilibrium value need to be carefully ditinguihed. Similarly, Equation (25) relate to the dynamic relation among the olid preure and the fluid preure and tenion while tranient in Equation (26) are indicative of a diequilibrium in the force balance at the common line, commonly referred to a preading preure. 7. Cloure and Simplification of the Ma and Momentum Equation A an example, let u conider the cloure of the ma and momentum balance equation uing the above contitutive relationhip. For implicity, ma exchange procee due to phae change will be ignored uch that êαβ i =êαβ wn = 0. Thi i a reaonable approximation in many intance where the exchange term are mall and alo do not influence the momentum tranport ignificantly. Therefore, ma conervation equation for the phae, interfacial area, and common line follow directly from Table I with no further implification. The momentum equation for the fluid phae may be obtained from Table II, Equation (19a) for the tre tenor, Equation (20a) for the momentum exchange between phae, and linearization of τ α around the zero-velocity equilibrium tate. The form obtained i D α (ɛ α ρ α v α ) ɛ α ρ α d α : I ɛ α p α ɛ α ρ α α = R α α vα, R α α vα, R α wn vwn,, α = w, n. (27) In contrat to the previou linearization, here ome coupling between the α phae and the bounding interface i allowed through the tenor R α α and Rα wn. Coupling between the α phae velocity and all other ytem component velocitie could be allowed by including reitance tenor multiplying the other relative velocitie on the right ide of Equation (27). For implicity, thi i not done here. The tenor R α α i repreentative of an invere permeability. It hould be noted that the coefficient that arie in the linearization tep (a well a in preceding and following linearization) are function of the independent variable around which linearization are not performed. For example, the reitance tenor in Equation (27), are reaonably aumed to be function of θ, ɛ α,anda αβ. Similar manipulation and aumption can be ued to cloe the momentum equation for the interface when linearization around the equilibrium tate for the momentum exchange term are employed. When interaction of the interface with the adjacent phae and common line are included, the three cloed interface

26 54 WILLIAM G. GRAY ET AL. momentum equation are: D wn (a wn ρ wn v wn ) a wn ρ wn d wn : I a wn (I G wn ) γ wn a wn ρ wn wn = R wn w vw, R wn n v n, R wn wn vwn, R wn wn vwn,, (28a) and D w (a w ρ w v w ) D n (a n ρ n v n ) a w ρ w d w : I a w (I G w ) γ w a w G w [x n (γ wn co γ n γ w γ wn κg wn )]aw ρ w w = R w w vw, R w w vw, R w wn vwn, a n ρ n d n : I a n (I G n ) γ n a n G n [x w (γ wn co γ n γ w γ wn κg wn )]an ρ n n = R n n vn, R n n vn, R n wn vwn,. (28b) (28c) Similar manipulation involving the momentum equation for the common line in Table II yield: D wn (l wn ρ wn ) l wn ρ wn d wn : I l wn (I G wn ) γ wn l wn ρ wn wn = vwn, R wn w R wn wn R wn n v w, v n, R wn wn vwn,. (29) In the momentum equation, the factor I G αβ and I G wn that are dotted with the gradient of the urface tenion account at the macrocale for the particular orientation that the interface and common line have at the microcale. In particular, microcopic gradient in urface tenion will drive the flow only in direction tangent to the urface. The orientation factor G αβ i equal to 1/3I if there i no preferred orientation of an interface within the averaging volume. It account for the fact that flow of interface ma cannot be driven in direction normal to the urface by gradient in the urface tenion. Similarly, the orientation factor G wn i equal to 2/3I if the common line orientation within the averaging volume i random. It account for the fact that gradient in the common line tenion can only drive flow along the common line and not in direction orthogonal to the line.

27 TWO-FLUID FLOW IN POROUS MEDIA 55 For mot problem involving multiphae flow in porou, geologic media, additional aumption may be applied to the momentum equation uch a negligible advective term or negligible coupling of momentum between phae. Under uch aumption, the momentum equation reduce to Darcian form. 8. Concluion A ytematic procedure ha been preented for obtaining and analyzing the equation decribing two-phae flow in a porou medium. The analyi make ue of the conervation equation of ma, momentum, and energy for the phae, interface, and common line, averaged o that they are expreed at the macrocale or core cale. A a reult of the averaging procedure, a deficit of ix equation i created that involve the principal geometric propertie or variable of the macrocopic ytem (i.e. the volume fraction of the phae, area per volume of the interface, and the common line length per volume). To overcome thi deficit, ix new evolutionary equation (or upplementary condition) were developed to relate thee variable and their rate of change to the primary ytem variable. Three of thee Equation, (52), (60), and (62), were obtained by noting the need to eliminate a product from the entropy inequality that contained a non-zero factor and then making ue of averaging equation. Three more Equation, (24), (25), and (26), were obtained, in linearized form, from analyi of the equilibrium relation that mut exit among the independent thermodynamic variable and from analyi of the dynamic entropy inequality. It mut be emphaized that thee ix upplementary condition are cloure approximation ubject to improvement a future inight might allow. In addition, the current analyi ha alo introduced a erie of econdary geometric factor, uch a interfacial curvature and macrocale contact angle, into the equation that mut be parameterized in term of a erie of tate equation to complete the formulation. Thee factor are related to the macrocale repreentation of the microcale curvature and orientation of the interface and common line. In particular, the orientation tenor appearing in the momentum equation enforce the condition at the macrocale that microcale gradient in urface (lineal) tenion can only produce interfacial (lineal) flow in direction tangent to the interface (common line). Thu, at the macrocale where the orientation of individual interface mut be replaced by information about their average orientation within a core cale volume, thi particular geometric factor provide thi information. Within the analyi, it i poible to identify the relative importance of a number of parameter and equentially implify the form and dependence the variou tate equation. Thi reduce the new equation et to the more traditional et that i currently employed in two-phae flow modeling. Thu, at the very leat, the current work provide explicit information about what aumption are being employed in uing the traditional equation of two-phae flow. If thoe aumption are deemed to be overly retrictive, the expanded et of equation provided here