Diffuse-interface Modeling of Two-phase Flow for a One-component Fluid in a Porous Medium

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1 Transport in Porous Media (2006) 65: Springer 2006 DOI /s Diffuse-interface Modeing of Two-phase Fow for a One-component Fuid in a Porous Medium P. PAPATZACOS and S. M. SKJÆVELAND University of Stavanger, Stavanger 4036, Norway (Received: 26 January 2005; accepted in fina form: 7 December 2005) Abstract. The diffuse-interface (DI) mode for the two-phase fow of a one-component fuid in a porous medium has been presented by Papatzacos [2002, Transport Porous Media 49, ] and by Papatzacos and Skjæveand [2004, SPE J. (March 2004), 47 56]. Its main characteristics are: (i) a unified treatment of two phases as manifestations of one fuid with a van der Waas type equation of state, (ii) the incusion of wetting, and (iii) the absence of reative permeabiities. The present paper competes the presentation by incuding the impementation of wetting in the genera case of a mixed-wet rock. As a resut of this impementation, some statements are made about capiary pressure, confirming simiar statements by Hassanizadeh and Gray [1993, Water Resour. Res. 29, ]. As an appication of the mode, we show that reative permeabiities depend on the spatia derivatives of the saturation. Key words: diffuse-interface, two-phase fow, wetting. 1. Introduction A new mode for two-phase fow in porous media has recenty been presented in two papers, referred to beow as P1 (Papatzacos, 2002) and P2 (Papatzacos and Skjæveand, 2004). It is based, at the pore eve, on the diffuse-interface (DI) mode of a one-chemica-component fuid (see Anderson et a., 1998; Papatzacos, 2000) characterized by (i) the two phases are manifestations of one and the same fuid, (ii) the transition from one phase to the other is taken care of by an additiona term in the Navier Stokes equation and by an equation of state of the van der Waas type, and (iii) the wetting properties are described by a boundary condition invoving the norma gradient of the fuid density (the Cahn theory). The upscaing from the pore eve to the Darcy eve is performed in P1, where the assumption of a constant and uniform temperature is aso introduced. (We point out that that the assumption of constant temperature was This paper buids on resuts presented as SPE84546 at the 2003 SPE Annua Technica Conference and Exhibition, hed October 5 8 in Denver. Author for correspondence: e-mai: pau.papatzacos@uis.no

2 214 P. PAPATZACOS AND S. M. SKJÆVELAND introduced as a simpification the energy-baance equation drops out and is not a imitation inherent to the DI mode.) A genera method for the incusion of wetting at the Darcy eve is given in P1, but it is reativey compicated and its expicit impementation is not attempted in that paper. It is then shown in P2 that a considerabe simpification can be obtained, for the impementation of wetting at the Darcy eve, if one assumes that the wetting anges at the pore eve are neither 0 nor 180, but anywhere in between. The expression incompete wetting approximation is used in P2 for this simpification. The expicit impementation of wetting is iustrated in P2, in the restricted case of a vapor wet rock. It is one of the purposes of this paper to generaize the impementation to a mixed wet rock. One then obtains a mode which we sha ca the DI-mode in the seque. This mode is presented in detai in Section 2 beow. Its main characteristics are as foows. It concerns a one-chemica-component fuid capabe of existing in two phases, which are convenienty referred to as the iquid and vapor phases. It has just one fow equation, in contradistinction to the traditiona mode of two-phase fow where each phase has its fow equation. This fow equation is a partia differentia equation of the Cahn Hiiard type. The traditiona constants of permeabiity and porosity are input parameters, in addition to a constant which is directy reated to the thickness of the transition region. It is to be noted that the mode does not use reative permeabiities. An equation of state of the van der Waas type is needed, and the wetting properties are deduced from the information contained in the traditiona capiary pressure function. The DI-mode is thus ideay suited to the description of steam-water systems or simiar, if the temperature gradients are negigibe. But it can aso be appied to oi gas fows in situations where oi and gas can be described, thermodynamicay and to an acceptabe approximation, as the two phases of one and the same fuid. Since the DI-mode does not use reative permeabiities it shoud be possibe to use it as a too to test various hypotheses about these quantities. It must be pointed out, however, that such a use of the mode is not straightforward, because its dependent variabes (density, veocity,... ) characterize the fuid and there is no obvious method for finding phase quantities such as, say, vapor and iquid veocities. We give one possibe method in P2, which satisfies some obvious physica criteria (conservation of momentum is one exampe); we then proceed to cacuate reative permeabiities through their definitions in terms of phase veocities and pressure gradients, after having carried out numerica experiments. The cacuations in P2 are restricted to a stabiized fow type (i.e., a fow type that can be described as a constant veocity traveing wave) and it is found there that the formuas defining reative permeabiities can be turned into anaytic expressions. These give the reative permeabiities in terms of the equation of state, the

3 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 215 capiary pressure, and some parameters. See Section 3 beow, where the expressions are generaized to the case of a mixed wet rock. (A two-phase saturated reservoir rock is often cassified as having mixed wettabiity if each fuid wets part of the interna rock surface.) Section 2 beow presents the DI-mode. Anaytic expressions for the reative permeabiities of stabiized fows in a mixed wet porous medium are given in Section 3. We return to the reative permeabiity concept in Section 4. Even if the concept is absent from the DI-mode, it is certainy of interest to traditiona two-phase fow, and therefore an important subject. We see it here as an interesting numerica appication of the DI-mode. In Section 4 we consider a one-dimensiona mixed-wet porous medium, and use the DI-mode to perform numerica experiments describing non-stabiized fow types, thus generaizing the investigation carried out in P2. We cacuate experimenta reative permeabiities and draw some concusions. 2. Description of the DI-mode The DI-mode is a Darcy eve mode of fow in a porous medium, of a fuid consisting of one chemica component. The rock matrix is assumed to be rigid, and the temperature is assumed uniform, constant, and subcritica. The upscaing from the pore eve to the Darcy eve, is done in P1 by using the averaging technique of Mare (1982). It is shown in P2 that the assumption of incompete wetting at the pore eve resuts in just one fow equation at the Darcy eve, the mass baance equation, invoving one dependent variabe, the upscaed fuid density. This we denote by R instead of the more traditiona ρ, according to the notation introduced in P1 (and ater used in P2), and foowing Mare s recipe of using capita etters for upscaed quantities. The fow equation and the boundary conditions invoving R are described in Section 2.1. The thermodynamics of the fuid, comprising the equation of state and the interaction energy due to wetting, is described in Section fow equation, initia and boundary conditions The centra equation of the DI-mode is the Darcy eve fow equation, ( R KR 2 ( )) d t = φη dr 2 R + G, (1) where K is the absoute permeabiity, φ the porosity, and a constant reated to the thickness of the transition region between the vapor and iquid phases (see P1); η is the fuid viscosity, is tota Hemhotz free

4 216 P. PAPATZACOS AND S. M. SKJÆVELAND energy per unit voume of buk fuid (buk being defined at the beginning of Section 2.3), and G is the gravitationa potentia causing an acceeration equa to G. Functions and η depend on R aone, and the ways to determine them are given in Sections 2.3 and 2.2. Note that Equation (1) is a mass baance equation, with a fuid veocity defined as V = KR ( ) d φη dr 2 R + G, (2) (It is shown in P1 that Equation (2) is the Darcy eve momentum baance equation of the fuid, and that the traditiona veocity resuts from setting =0.) Note aso that Equation (1) is (when G is identicay zero) a Cahn Hiiard equation (see, for exampe, Novick Cohen and Sege, 1984). The boundary conditions of the mode are usuay written in terms of two quantities, u 1 and u 2, defined as foows: u 1 = R, u 2 = d dr 2 R + G. (3a) (3b) A we-posed probem for Equation (1) is obtained by suppying an initia condition R(x, 0)=F(x) (a given function of x), and a boundary condition having one of the forms (a) to (e) beow. (a) H = 0, { u1 = α (b) 1, u 2 = α 2 { n u2 = G (c) 2 u 2 = α 2, (4a) (d) { u1 = α 1 n u 1 = G 1, (e) { n u1 = G 1 n u 2 = G 2. (4b) In these equations, n is the unit norma to the boundary, pointing out; H is a function of u 1 and u 2 ; α 1 and α 2 are two constants; and G 1 and G 2 are functions of x, and of u 1 and u 2 and their derivatives. A boundary condition on u 1 in the present mode is equivaent, because of the equation of state, to a condition on the pressure in the traditiona mode. A boundary condition on n u 2 is a condition on the veocity. The boundary conditions invoving n u 1 are new to this mode and have been discussed in P1: they specify the ange between the isodensity ines and the reservoir surface. The boundary conditions which are reevant to reservoir studies are thus the ones abeed (d) and (e) above. For the specia case of one-dimensiona studies, as the ones presented in the present paper, it is natura to use G 1 = 0 (see P1).

5 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL the fuid viscosity Fuid viscosity must be known as a function of R in order to sove the fow equation (1). The formua for η(r) that has been used in P1 and P2 is a modified form of a formua proposed by Arrhenius, η = η S η S v v, (5) where η and η v are the viscosities of the pure iquid and vapor phases, and S and S v are the iquid and vapor saturations. In the present mode, these are interpreted as foows: S = R R v R R v, S v = R R R R v. (6) 2.3. thermodynamics We now define the thermodynamics of the Darcy eve fuid, and thereby determine the function (R). We remind the genera assumptions. The temperature is uniform and constant. The fuid consists of one chemica component, and is capabe of undergoing a phase transition so that two coexisting phases are possibe. The phases are caed iquid and vapor. There is a transition region of definite thickness between the phases, where the density gradients are arge. We define buk fuid to be the fuid which is far enough from the transition region that the density gradients are negigibe and thus do not contribute to its free energy. The buk fuid has an equation of state of the van der Waas type. It is shown in P2 that the tota Hemhotz free energy per unit voume of buk fuid consists of two parts and is written (R)= b (R) + I(R). (7) The first part, b, is the free energy per unit voume of buk fuid. The equation of state can be expressed through it as foows: P b (R) = R 2 d ( b ), (8) dr R where P b is the pressure of buk fuid. The determination of b is given in Section The second part, I, is introduced in P1 for the description of wetting. An exampe of how I can be determined is given in P2. The genera method is given in Section beow.

6 218 P. PAPATZACOS AND S. M. SKJÆVELAND The Equation of State and b Let us consider a fuid of one chemica component, capabe of existing in two phases with densities R v and R, and where the chemica potentia and the pressure at equiibrium are two constants denoted M and P respectivey. Then it can be shown (see Appendix A in P1, and references given there) that the intrinsic Hemhotz free energy per unit voume of buk fuid has the form b (R) = W(R)+ MR P, (9) where W(R) has the shape of a fourth order poynomia, with two minima of vaue zero occurring at two distinct vaues. (We sha in the seque refer to such functions as being W-ike.) The minima of W(R) are at R = R v and R = R. It is shown in P1 that, when assuming uniform temperature as we do here, the vaue of M is irreevant and one can set M = 0. (10) If quantitative resuts are expected from soving the we-posed probem outined in Section 2.1, then it is important to obtain R v,r and W(R),at the specified temperature, from the reevant equation of state. Densities R v and R are usuay found by the Maxwe equa area rue. With an equation of state where buk pressure P b is given as a function of density R, this is R ( 1 ). (11) P b (R ) dr R v R = P b (R 2 v ) 1 R v R The W(R)-function is found by integrating Equation (8). As shown in P2, this eads to W(R)= P b (R v ) (1 RRv ) + R R R v P b (R ) dr R 2. (12) Using the van der Waas equation of state as an exampe, i.e., PvdW b NkT R (R) = 1 BR AR2 (13) (N is the number of moecues, k is Botzmann s constant, T is the temperature, and A and B are two positive constants) we obtain for the van der Waas W-function, ) W vdw = PvdW b (R v) (1 RRv + AR(R v R)+ NkT R n R(1 BR v) R v (1 BR). It can be checked that W vdw is W-ike; its two minima of vaue zero are at R = R v and R = R. There are no anaytica expressions for R v and R and

7 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 219 their actua cacuation by the use of Equation (11) is somewhat tedious and is not given here. More reaistic equations of state have of course the same probem. If one aims at quaitative resuts, as we do in this paper, then the numerica awkwardness just pointed out can easiy be avoided by postuating a W -function, and naturay choosing the simpest possibe form, i.e., W(R)= (P c /R 4 c )(R R v) 2 (R R ) 2 (14) (see P1 and P2 where the fuid having such a W -function is caed pseudo-van-der-waas ). Here P c and R c are the pressure and density at the critica point. For this simpe mode, R c = (R v + R )/2, (15) whie P c is arbitrary positive. Numerica vaues for P c and R c are not needed because a the equations of the mode can be written in dimensioness form and P c and R c are absorbed in dimensioness groups. Dimensioness variabes are indicated with a tide overstrike: R = R/R c, R v = R v /R c, R = R /R c, W = W/P c. (16) Equation (14) becomes W( R)= ( R R v ) 2 ( R R ) 2. (17) The constants R v < 1 and R > 1 are now parameters that are given as input, instead of being the resut of a numerica cacuation. Using this simpe W -function in Equation (9), then using (8) we obtain an equation of state. The constant M is given by Equation (10) and P can be obtained by the condition that the pressure must vanish at zero density. The dimensioness b is then b (R) = b /P c = ( R R v ) 2 ( R R ) 2 R v 2 R 2. (18) We add for competeness that this free energy function is vaid for just one isotherm. It is possibe to generaize to arbitrary dimensioness temperatures T (temperature divided by critica temperature) by using R v = 1 1 T, R = T (19) (see Equations (90) and (91) in P1; here we have put β = 1). Substitution of these expressions in Equation (18) eads, after some obvious agebra, to an expression that is vaid for a temperatures, even those arger than the critica. There resuts an equation of state with temperature as a parameter, which has a the characteristics of the van der Waas equation of state (see Appendix A in P1).

8 220 P. PAPATZACOS AND S. M. SKJÆVELAND We set down here for future reference the expression of fuid viscosity in terms of density that resuts from using the pseudo van der Waas fuid. Firsty, we easiy find the viscosity η c at critica density by using Equation (15) in Equations (5) and (6): η c = η v η. (20) The expression for the dimensioness viscosity, is then η = η ( ) r ηv =, r= R + R v 2R η c η 2(R R v ). (21) To summarize: Equation (18) is a simpe formua for the Hemhotz free energy per unit voume for buk fuid, we adapted to quaitative cacuations at constant subcritica temperatures. We sha use this formua in this paper, together with formua (21) for viscosity. For quantitative resuts, the b must be obtained through the equation of state, using Equations (9), (10), (11), and (12) Wetting and the I-function The I-function is a Hemhotz free energy per unit voume introduced in P1 for the description of wetting. It is in genera a function of two variabes and its determination is not given in P1. However, it is shown in P2 that, if wetting is incompete at the pore eve (i.e., the wetting ange is between 0 and 180 ), then I is, to a good approximation, a function of R aone that can be determined in its essentia characteristics by the traditiona capiary pressure function. It is easy to show, using Equation (2), that a pressure that can be identified with a capiary pressure exists in the DI-mode, and that it is reated to the I-function. This can be done by using Equation (7) to write R (d /dr) = R (d b /dr) + R (di/dr). (22) The first term on the right-hand side can be transformed as foows: R d b dr = R d2 b dpb R = dr2 dr R = P b, where the second equaity comes from taking the derivative of Equation (8), obtaining: dp b dr = R d2 b dr. (23) 2 The second term on the right-hand side of Equation (22) can be transformed in the same way, i.e., R di dr = P c,

9 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 221 by defining a pressure P c Equation (23), i.e., due to wetting by an equation anaogous to dp c dr = R d2 I dr. 2 The veocity, Equation (2) can then be written (24) V = K ( (P b + P c ) + R G R 2 R ). (25) φη Note that it is essentia for the definition of P c by the thermodynamica reation (24) that I be a function of R aone. We sha ca P c the capiary pressure function. We sha now use Equation (24) backwards: we assume that P c is empiricay known and we use the equation to cacuate I(R). We assume that P c (R) is identica with the traditiona capiary pressure function p c (S), provided a conversion from saturation S to density R is used in p c (S). As we sha see, the identification can ony be vaid for a imited interva of the density R, impying that I can be constructed from empirica data ony inside such a imited interva. Some sort of extrapoation must be done to extend the construction outside the interva. We show beow that I must obey the requirement that W + I must be W -ike. See Figure 1a. It foows that its cacuation from p c is the first experimenta chaenge met by the DI-mode. The argument eading to the requirement on I is as foows. Assume a very ong, one-dimensiona porous medium in dynamica equiibrium: its veocity is zero everywhere and the variation of density with the space coordinate x is found by soving the differentia equation obtained by equating to zero the expression inside the parentheses in Equation (2). Let us assume that I = 0 (neutra wetting). The first effect of gravitation, represented by G, is to separate the fuid in a iquid phase (downwards) and (a) (b) (c) (d) W Jv W W Rv R* v W+I R* R s s W+I -C Rv R* v R* R J C J C J J - v v -Cv J - s s v Rv R* R* v R Rv R* v R* R Figure 1. In (a) The W-ike functions W(R) and W(R) + I(R). In (b) Functions J (R) and J v (R) defined by Equations (32b) and (33b). In (c) Functions W(R), C v J v (R), C J (R), and W(R)+ I(R) where I = C v J v + C J. In (d) The smoothing of I(R) producing I s (R) = C s J s (R) + Cs v J v s (R). In (b) and (c), it is assumed that a,a v < 1.

10 222 P. PAPATZACOS AND S. M. SKJÆVELAND a vapor phase (upwards). Gravitation has aso the effect of distorting the density profie due to a compression effect downwards. We assume that G is so sma that this distortion can be negected: the density profie at equiibrium is then determined by W and by the boundary conditions specifying that we have iquid at the ower boundary and vapor at the upper boundary. It is then easy to see (the cacuation can be done by hand if the simpe W of Equation (14) is used) that R(x) is a smooth curve, with one pateau where R is equa to R v, another where R is equa to R, and a sigmoid-ike curve in between. The smoothness of the curve agrees with observations. Let us now assume mixed wetting so that I 0. The equiibrium is now determined by W + I and the ony difference with the I = 0 case, shoud be a shift of the two pateaus: the one at R v is raised at a vaue Rv, the one at R is owered at R.(WehaveR v <Rv because of the presence of residua iquid in the pores. Simiary, R >R because of the presence of residua vapor.) The sigmoid must fit smoothy to the pateaus to agree with observations; it can have a different shape from the neutrawetting sigmoid, but that is of secondary importance. Thus W + I must be W-ike. The traditiona p c (S) is an experimentay measured function. We sha use an anaytica expression, which depends on four parameters and which covers a arge range of measurements (Skjæveand et a., 2000). It is given here with our notation, i.e., with iquid and vapor as the names of the two phases: ( ) 1 a ( ) Sr 1 av Svr p c (S ) = C + C v. (26) S S r 1 S vr S S is the iquid saturation. The residua iquid saturation S r, and the residua vapor saturation S vr are considered as input parameters. The constants C and C v are usuay referred to as entry pressures and 1/a and 1/a v as pore size distribution indices. They are here adjustabe parameters. The transation from saturation to density is done through Equations (6). We begin by defining Rv and R in terms of input parameters: S r = R v R v Rv R R = R v + (R R v )S r, v (27a) S vr = R R R = R (R R v )S vr. R R v (27b) We now identify P c (R) with p c (S ), where S is given by Equation (6) (eft). For ater convenience, we write P c = P c + P c v, (28)

11 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 223 where P c = C ( R R v R R v ) a, P c v = C v ( R ) R av v R (Rv R <R<R ). (29) Note that P c is ony known in the open interva (Rv,R ). The boundaries are excuded because the DI-mode does not accept infinite pressures. The integration of Equation (24), using Equation (28), gives I(R)= I (R) + I v (R), with P c Ia(R) = R a R dr α 2 a + β a R (a=, v), (31) where α a and β a are integration constants. We now refer to the second paragraph after the paragraph containing Equation (25): the bounds and constants of integration in Equation (31) must be determined, together with constants C and C v, in such a way that W + I is a W -ike function of R, with two minima at R = Rv, and R = R, of vaue zero. It is shown beow that this is possibe, impying that the mode is in agreement with experimenta resuts on capiary pressure. There is a certain reservation to the ast statement, as we sha see, reating to the possibe infinities in p c (S ). We first obtain I and I v for Rv <R<R. I is that part of the I- function which is due to the iquid-wet part of the capiary pressure function. We impose the condition that W +I shoud be as much as possibe equa to W forr in the neighborhood of R : (30) I (R ) = 0, I (R ) = 0 (where the prime denotes differentiation). Simiary, W + I v shoud be as much as possibe equa to W forr in the neighborhood of Rv : I v (R v ) = 0, I v (R v ) = 0. Using these four conditions to determine the constants in Equation (31) with a = and with a = v, one easiy gets I (R) = C J (R), ( R Rv J (R) = R Rv and I v (R) = C v J v (R), ( R J v (R) = R v R Rv ) a ( 1 R ) R R R R ) av ( 1 R ) + R Rv R R v ( R R v R R v ) a dr R 2, ( R ) R av v dr R R R. 2 (32a) (32b) (33a) (33b)

12 224 P. PAPATZACOS AND S. M. SKJÆVELAND It turns out that J and J v are monotonicay increasing functions for Rv < R<R (see Figure 1b). There are difficuties connected to the infinity of J as R Rv with a 1, and to the infinity of J v as R R with a v 1. We sha temporariy, and for the sake of iustration, assume that the a s are ess than 1. Figure 1b shows J v and J for a v = a = 0.5. It is now easy to determine the constants C and C v in such a way that W + I = 0atRv and R : it suffices to put C = W(R v ) J (Rv ), C v = W(R ) J v (R ). (34) (It is cear, incidentay, that C > 0 and that C v < 0.) The resuting W + I- function is shown in Figure 1c, together with W, C v J v, and C J. To obtain I(R) outside of the interva [Rv,R ] we must now resort to extrapoations. However, W + I must reach the horizonta axis with a horizonta tangent for the extrapoated W + I to be W-ike, and it is easy to see from expressions (32b) and (33b) that I, and thus necessariy W + I, have vertica tangents at R = Rv and R = R. This is due to the infinities of p c at S = S r and S = 1 S vr. The situation is worse when a or a v or both are arger than 1, because then we can not even satisfy W + I = 0atRv and R Ẇe make W +I a W -ike function by using the fact that the behavior of p c (S ), when S approaches S r from above or 1 S vr from beow, is not we estabished (Hassanizadeh and Gray, 1993). Without contradicting experimenta evidence we then smooth out the I-function, or rather its two components I and I v, as described beow. There are probaby many ways to perform the smoothing. We have chosen to start at the eve of the J and J v functions, and to shift their possibe infinities away from Rv and R : choosing a sma number ɛ, we shift the singuarities at Rv + ɛ and R ɛ, as shown on Figure 1d. We then define a function J s as foows: { J s j (R), R = v R<R v + 2ɛ, J R v +ɛ Rv + 2ɛ R R, (35) where J R v +ɛ is J with Rv repaced by R v + ɛ. A typica vaue for ɛ, or rather for its dimensioness counterpart ɛ = ɛ/r c is (For the sake of readabiity, the vaue of ɛ in Figure 1d is greaty exaggerated.) We choose a second degree poynomia for j (R) such that J s and its derivative are continuous at R = Rv + 2ɛ: it can then be seen that j has just one remaining degree of freedom. A function Jv s is defined in a simiar manner, with a poynomia j v (R). We now define a smoothed I-function by I s (R) = C s J s (R) + Cs v J v s (R), (36)

13 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 225 and determine the constants C s,cs v, and the two degrees of freedom in j and j v, with the four conditions W(Ra ) + I s (Ra ) = 0 W (Ra ) + I s (Ra ) = 0 (a =, v) (the primes denote differentiation). The cacuations are eementary but somewhat ong and are not given here. It is to be noted that the C s appearing in the capiary pressure correation are not free parameters, according to the DI-mode. We now ook at the probem of extending the I s -function, to the eft of Rv and to the right of R. We denote this function by I se. The extension is done as foows. To the eft of Rv we take W +Ise to be equa to W, transated by the amount Rv R v: this is the vapor phase region and we expect it to behave approximatey as a pure vapor. To the right of R we want the iquid to behave as a neary incompressibe fuid and we impose therefore that W(R)+ I se (R) = P c (R R )2, Rc 2 4 ɛ where ɛ is the sma parameter used previousy. The function W(R)+I se (R) is shown in Figure 1 as the curve abeed W + I. To summarize: Function I se (R) is cacuated from the empirica capiary pressure function and the equation of state on an interva [Rv + 2ɛ,R 2ɛ] which excudes the possibe infinities of the capiary pressure function. Outside this interva it is extrapoated as simpy as possibe, keeping in mind that W + I must be W-ike the mode in summary The mode consists of: (i) the fow equation (1), (ii) an initia condition on the fuid density R, and (iii) boundary conditions of type (d) or (e) (see Equations (4b)). There are two input functions: the fuid viscosity η(r) (see Section 2.2) and the tota Hemhotz free energy per unit voume of buk fuid (R) (see Section 2.3). The input parameters are as foows. Firsty the permeabiity K, the porosity φ, a constant characterizing gravity (usuay the acceeration g), and. The ast mentioned is reated to the thickness of the transition region as shown in P1 and P2. Secondy, the constants inherent to the functions η and. The former has a minimum of two parameters, namey the viscosities η v and η of the pure phases. The atter has a minimum of six parameters: the densities R v and R of the pure phases, the residua saturations S vr and S r, and the two a s of the capiary pressure correation (Equation (26)).

14 226 P. PAPATZACOS AND S. M. SKJÆVELAND 3. Anaytica Reative Permeabiities for Stabiized Fows The formuas defining reative permeabiities can be turned into anaytic expressions if the fow can be characterized as a constant veocity traveing wave. See P2 where these formuas were restricted to the case of a vapor wet rock. We sha here generaize them to the case of a mixed wet rock. We need the derivative of the capiary pressure with respect to R. The smoothing and continuation of the I-function performed in Section impies that we must use the derivative of a function P cs (a smoothed and continued P c ). Using the earier spitting up into a iquid and a vapor wetting part (Equation (28)), we introduce dp cs dr = dpcs dr + dpcs v dr, (37) with dp cs dr = R d2 dr 2 Cs J s (R), dpv cs dr = R d2 dr 2 Cs v J v s (R) (38) (see Equations (24) and (36)). The reative permeabiities (k r and k rv for the iquid and vapor phases) can now be given the same form as in P2, namey S k r = 1 + (Rv /R )γ (39a) (S ), k rv = 1 S 1 + γ(1 S ), (39b) where S = R R v. R R v (40) The generaization to the mixed wet case proceeds from the foowing straightforward generaization of the function γ given by Equation (61) of P2: γ(s ) = 1 dp v cs g R v d R 1 d P cs g R v d R 2 [ W( R)+ I s ( R)] 2 [ W( R)+ I s ( R)] R= R ( R R v )S R= R v +( R R v )S, (41)

15 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 227 where the quantities with tides are dimensioness. Densities ( R, R v ) are referred to R c, pressures ( P v cs, P cs ) and potentias ( W, I s ) are referred to P c, and g = (R c L/P c )g, = (R 2 c /(P cl 2 )), (42) where L is the ength of the porous medium aong which the wave is traveing. It can be shown that, if a v and a in Equation (26) are both ess that 1, then the smoothing of the I-function is not necessary for the cacuation of the reative permeabiities. The superscript s can be dropped in Equation (41): Equation (29) can be used to cacuate the derivatives of P c and P v c, and one can use the I-function defined by Equations (30), (32a), (33a), and (34). Thus knowedge of the equation of state, of the capiary pressure, of the parameters beonging to these functions, and of the constants g and, ead to the cacuation of the reative permeabiities for stabiized fows. 4. A Numerica Appication In this section we appy the DI-mode presented above to a numerica investigation of reative permeabiities for non-stabiized fows. We force the fows to be nonstabiized by the use of start and boundary conditions. We begin with an outine of a method for cacuating reative permeabiities numericay experimenta reative permeabiities We assume a one-dimensiona porous medium of ength L, mixed wetting, and we begin with the definition of reative permeabiities: Ṽ = k ( r(exp) η( R ) ) p x + g R, Ṽv = k ( rv(exp) η( R v ) ) p v x + g R v. (43) The dimensioness quantities not aready defined are as foows: x = x/l, and x is the coordinate which increases in the direction of the force of gravity; p and p v are pressures, referred to P c as earier in the paper. The veocities on the eft-hand sides are made dimensioness by the foowing genera formua, vaid for a other veocities used ater in the paper: Ṽ = (φη c L/(KP c ))V. (44) The DI-mode does not provide phase veocities, nor any other phase quantities. To obtain them we generaize the method used in P2. We begin with

16 228 P. PAPATZACOS AND S. M. SKJÆVELAND the dimensioness veocity Ṽ and momentum Ɣ, defined as foows: Ṽ = R ( ) d η( R) x d R g x 2 R, Ɣ = RṼ. (45) x 2 We now introduce the foowing notation for the veocities and momenta at, respectivey, S = 0 and S = 1: Ṽ 0 = Ṽ S =0, Ṽ 1 = Ṽ S =1, Ɣ 0 = Ɣ S =0, Ɣ 1 = Ɣ S =1. (46) Foowing P2 we then postuate that Ṽ = Ɣ 0 Ɣ Ṽ 1, Ɣ 0 Ɣ 1 Ṽ v = Ɣ Ɣ 1 Ɣ 0 Ɣ 1 Ṽ 0. (47) As pointed out in P2, these equations ensure that the oca momenta carried by each phase add to the tota fuid momentum. We can, with Equation (47), obtain the phase veocities during the cacuation of a given process, since a quantities on the right-hand sides are cacuabe. Turning to the pressure gradients in the two phases, the centra argument eading to their cacuation is given in P2 as foows. If P b denotes the buk pressure in the fuid, then the gradient of P b is zero at equiibrium and when wetting is absent since, foowing the Maxwe construction, the van der Waas oops in the equation of state are repaced by an horizonta ine. With wetting, where the W -function is repaced by W + I s,it is the gradient of P b + P cs that must vanish at equiibrium. It foows that [ ] P b x eq = P cs x = d P cs R d R x. (48) For the simpe case of a vapor wetting medium considered in P2, one identifies the pressure gradient in each phase with the eft-hand side of Equation (48), taken at the proper vaue of the saturation. Generaizing to mixed wetting, we put p x = g R v γ exp(s ), where γ exp is γ exp (S ) = [ 1 g R v dp v cs d R p v x = g R v γ exp(1 S ). (49) ] R x 1 S [ 1 g R v dp cs d R ] R x S. (50) In this expression, the derivatives of P cs and P v cs are given by Equation (38), whie the partia derivative of R is obtained by using the soution of

17 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 229 the partia differentia equations. (In stabiized fow, an anaytica expression can be found for this partia derivative, eading to the expressions of Section 3.) One can thus cacuate k r(exp) = Ṽ η( R ) 1 g R 1 + (Rv /R )γ exp(s ), (51a) k rv(exp) = Ṽ v R v ) 1 g R v 1 + γ exp (1 S ), (51b) at any time step and at vaues of x where the density between R v and R. R is intermediate 4.2. numerica experiments We have done three numerica experiments, where one can be characterized as a drainage, and two as imbibitions. We have abeed them Drainage, Imbibition 1, and Imbibition 2: see Appendix A for the parameter specifications, and for the initia and boundary conditions. In the Drainage and Imbibition 1 cases the boundary conditions have been determined in such a way that fow veocities decrease and static fina states are approached, with both phases ceary present. In the Imbibition 2 case the boundary conditions do not aow a static fina state to occur before most of the medium is fied with iquid. The resuts of the drainage cacuation are shown in Figure 2. The soution R( x, t) of the fow Equation (1) is shown in (a). Note the quaitative agreement with the experimenta resuts of Terwiiger et a. (1951) (see in particuar their Figure 2). Parts (b) and (c) are pots of the iquid and vapor veocities (Equation (47)), versus the normaized iquid saturation (Equation (40)). The essentia information contained in Figures (b) and (c) is that we have counter-current fow: iquid veocity is positive so that the iquid phase moves in the direction of the gravitationa force, whie the vapor phase moves in the opposite direction. In other words, the iquid fas, eaving a rising vapor behind it. Simiar figures for the imbibition cacuations are not shown here. The main features are as foows. For Imbibition 1 we again have counter-current fow, with the iquid phase being drawn upwards, against the gravitationa force, whie the vapor phase fas into the rising iquid and iquefies. The inter-phase region (i.e., the diffuse interface) rises and iquid is drawn in the medium from beow thus justifying the name imbibition. The ony non-conventiona feature is the gas being drawn in the medium from above and iquefying.

18 230 P. PAPATZACOS AND S. M. SKJÆVELAND (a) (b) (c) t = t = 0.02 t = t = t = Figure 2. Drainage simuation. In (a) Fuid density versus x, given by the soution of Equation (1) with the specifications given in Appendix A, for t-vaues of 0, 0.02, 0.08, 0.16, 0.32, 0.40 (increasing from the broken ine and to the right). In (b) Dimensioness iquid veocity versus normaized iquid saturation S, for t-vaues of 0.02, 0.08, 0.16, 0.32, 0.40, increasing from top to bottom as indicated. In (c) Dimensioness vapor veocity versus normaized iquid saturation S, for the same t-vaues as in (b), increasing from bottom to top as indicated. The physics is somewhat more compex for Imbibition 2: fow is now co-current, both phases moving with positive veocities, i.e., in the direction of the gravitationa force. But the vapor veocity is about ten times arger than the iquid veocity so that we have the same overa effect as in Imbibition 1, i.e., vapor faing into iquid and iquefying. The inter-phase region rises, but iquid is expeed from beow and vapor is drawn in from above. Thus we do not have imbibition in the traditiona sense, but we have kept the name since the overa iquid saturation in the medium increases and the interface rises discussion of the resuts of the numerica experiments Experimenta reative permeabiities are cacuated for each of the three processes presented in Section 4.2, and for a number of t-vaues between 0 and 0.4. The resuts of these cacuations are shown on Figure 3, top row. We first take up the fact, apparent from the figure, that the mode predicts reative permeabiities arger than one. The expanation is found in the expression for fuid veocity, Equation (2). In one spatia dimension and using the thermodynamica reations of Section 2.3, one finds V = K ( P b ) φη x gr KR φη ( d 2 I R dr 2 x R 3 x 3 ). (52) We sha refer to the first and second terms on the right-hand side as the Darcy and DI-non-Darcy (not to be confused with the inertia non-darcy veocity) veocities. Obviousy, the cacuations of reative permeabiities are a two step process where the first step is to partition the fuid veocity in its iquid and vapor parts. The second step consist essentiay in taking away the DI-non-Darcy term and accounting for its contribution by a factor (the reative permeabiity) mutipying the Darcy term.

19 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL 231 Drainage Imbibition 1 Imbibition t = t = t = 0.02 t = 0.02 t = t = Figure 3. Liquid (ascending curves) and vapor (descending curves) reative permeabiities versus normaized iquid saturation S, for the three processes specified in Appendix A and for t-vaues of 0.02, 0.04, then up to 0.40 by steps of Top row: cacuated, see Equation (51). Bottom row: drawn in gray are the curves of the top row, with ordinates divided by their (argest) end-point vaues; drawn in back are the theoretica reative permeabiities of Section One important difference between stabiized fow types considered in P2 and the non-stabiized ones considered here is iustrated in Figure 4. For stabiized fow types (eft) the arge gradients in the density are confined to the transition region so that the DI-non-Darcy term ony contributes to the veocity inside that region. This contribution graduay vanishes as one approaches the 100% iquid or vapor regions, which eads to a reative permeabiity that behaves as expected, being in particuar ess than 1. For non-stabiized fow types (right) the arge gradients show a tendency to spi over in the 100% iquid and vapor regions, so that the DI-non-Darcy term remains important in the pure-phase regions, giving a veocity that is arger than what is expected from the Darcy expression. Accounting for the non- Darcy term by a mutipicative factor eads in this case to the factor being arger than 1. The question that immediatey arises is whether permeabiities arger than 1 are experimentay observabe. To answer this question we note that the very arge reative permeabiities occur for sma times, and for the phase for which the dimensioness veocities are roughy arger than 1: see Figures 2 and 3. The reference veocity (see Equation (44)) is KP c /(φη c L).

20 232 P. PAPATZACOS AND S. M. SKJÆVELAND Density Stabiized fow (paper P2) Non-stabiized fow (this paper) Density Position Position 100% vapor 100% iquid 100% vapor 100% iquid Figure 4. In stabiized fow types the arge gradients are confined to the transition region. Using the foowing aboratory vaues K = 100 md φ = 0.3 L = 20 cm P c = N/m 2, η c = cp (where critica pressure P c and critica viscosity η c beong to the water steam system) we get a reference veocity of the order of 1 m/s. This is at east 10,000 times arger than the usuay occurring veocities, which expains that reative permeabiities arger than one are not observed. The other obvious feature of the top row of Figure 3 is the absence of a unique reative permeabiity curve. The present mode thus confirms the we-accepted fact that reative permeabiities are not functions of saturation aone. In fact, the foregoing discussion has shown that the reative permeabiities are inked to the second term on the right-hand side of Equation (52), i.e., to the density and its derivatives. In view of our simpe reation (Equations (6)) between density and saturation, a reasonabe assumption woud be that k ra = k ra (S, S; P) (a=, v). Here we have used S is a shorthand notation for the dependence of k ra on the partia derivatives of S of (in principe) a orders; P stands for a set of parameters which usuay do not vary with time and space, ike pure phase densities, viscosities, and other parameters incuded in the definition of the equation of state, and the capiary pressure. The bottom row of the Figure 3 is a redrawing of the curves of the top row, after a renormaization consisting in dividing the ordinates of each curve by the curve s end-point ordinate (eft end point for the vapor, right end point for the iquid reative permeabiities). The anaytica curves of Section 3 are aso drawn. A gathering of the renormaized curves, ike the one happening for the Drainage case, woud have indicated that reative permeabiities are functions of the type f(s; P)g( S; P). Obviousy, the DI-mode ceary indicates that such simpified functiona dependence does not exist in genera.

21 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL Concusions We have presented the DI-mode of two-phase fow in porous media. This paper carifies questions eft open in two previous papers (referred to above as P1 and P2), concerning the description of wetting. In particuar, it eaborates on the assumption of incompete wetting at the pore eve, introduced in P2. The main imitation of the mode is in its assumption of one chemica component, which restricts its appicabiity to iquid vapor systems, or to systems whose thermodynamica description can be approximated by that of a iquid vapor system. The advantage of the mode is a theoretica description which incorporates the separation of the phases as an integra part of the soution of the fow equation. Thus numerica dispersion, with spurious breakthroughs of a mobie phase, does not occur. The absence of reative permeabiities is an additiona advantage. Concerning capiary pressure: a function P c (R) has been defined, not as the difference P n P w between the pressures in the non-wetting and the wetting phases, but as a quantity with the forma properties of a pressure. The P c -function is reated to that part of the Hemhotz free energy per unit voume (the I-function) which pertains to the description of wetting, in the same forma manner as the buk pressure P b is reated to b : see Equations (23) and (24). It is then natura to identify P c (R) with the traditiona capiary pressure function p c (S), with the purpose of cacuating the I-function, provided a definition of saturation as a function of density is given, and provided one keeps in mind that this identification does not yied I(R) for a R. (See Section ) Some statements about p c (S) resuting from this identification are as foows: (i) p c is not, in genera, a function of state, in the thermodynamic sense (see P1); (ii) p c is, to a good approximation, a function of state if one assumes incompete wetting, i.e., wetting anges stricty between 0 and 180, at the pore eve, or if one assumes permeabiities of the order of one Darcy and above (see P2); (iii) p c can not become infinite. It is to be noted that simiar statements have been made in Hassanizadeh and Gray (1993): see their comments in connection with their Equations (14), (15), and (44). Concerning reative permeabiities: the DI-mode does not use them but it can, assuming its experimenta correctness, be used to perform simuations eading to their evauation. The concusions from such simuations are as foows. Anaytica expressions exist (see Section 3) and can be used for constant veocity traveing wave fow types. For other fow types, the mode confirms the beief that reative permeabiities, as functions of saturation aone, cannot capture the fu compexity of two-phase fow. In fact, the mode indicates a ikey dependence on the spatia derivatives of the

22 234 P. PAPATZACOS AND S. M. SKJÆVELAND saturation. In addition, the mode does not confirm the hypothesis of a separation of the functiona dependence on saturation from the functiona dependence on derivatives of saturation. Concerning experimenta verification: quaitative verification resuts first from the fact that I can be cacuated from p c (after adjusting its commony assumed infinities) in such a way that W + I is W -ike. We woud aso ike to point out the resembance of the drainage curves of Figure 2 with the cassica resuts of Terwiiger et a. (1951). Quantitative verification with pubished resuts is, for the time being, not feasibe since the information on thermodynamica properties, which is necessary for the simuations with the DI-mode is usuay not provided. Appendix. Specifications for the Numerica Experiments of Section 4.2 The fow equation in one dimension, with the space axis pointing in the direction of the gravitationa force (downwards), is [ ( )] R t = R 2 d x η( R) x d R 2 R g x, x 2 using dimensioness quantities defined in the main text and with t = (KP c /(φη c L 2 ))t. We have chosen = 0.01, g = 0.5, η v /η = 0.1. We have used the pseudo-van-der-waas fuid, i.e., the fuid having the b given by Equation (18). The parameters describing this function and the I- function of Section are chosen as foows: R = 1.6, R v = 0.4, S r = 0.4, S vr = 0.1, a = 0.50, a v = 1.15, for drainage, (A.1) a = 1.00, a v = 1.40, for imbibition. The numerica vaues on the first ine impy that the minima of the W- function are at R v = 0.88 (density of phase consisting of pure vapor and residua iquid) and R = 1.48 (density of phase consisting of pure iquid and residua vapor). Note the different choices for a and a v, depending on whether one simuates drainage or imbibition. The two resuting capiary pressure functions, given by Equation (26), are shown in Figure A.1 (see Section for the cacuation of C and C v or rather of C s and Cv s.). The initia conditions are as foows: R( x,0) is first given by a step function, where R = R v for x ess than the position of the discontinuity, and

23 TWO-PHASE ONE COMPONENT FLOW IN THE DI-MODEL S -1 Figure A.1. The capiary pressure functions versus iquid saturation S, resuting from constants given in (A.1), see Equation (26). (The eft vertica asymptote is at S =S r, the right one at S =1 S vr.) The upper curve is used for drainage, the ower for imbibition simuations. R = R otherwise. The position of the discontinuity is at about 0.1 for drainage, at about 0.9 for imbibition. The step function is then smoothed (see the broken ine in Figure 2a). The boundary conditions are of the type (d) (see Equation (4b)), i.e., R( x, t) and R x ( x, t) (the partia derivative of R with respect to x) are given for a t at x = 0 and x = 1: R(0, t)= R 0, R x (0, t)= 0, (A.2) R(1, t)= R 1, R x (0, t)= 0. The vaues of R 0 and R 1 depend on the simuated process and are as foows: Drainage: R 0 = R 1 = Imbibition 1: R 0 = R 1 = Imbibition 2: R 0 = R 1 = (A.3) References Anderson, D. M., McFadden, G. B. and Wheeer, A. A.: 1998, Diffuse-interface methods in fuid mechanics, Ann. Rev. Fuid. Mech. 30, Hassanizadeh, S. M. and Gray, W. G.: 1993, Thermodynamic basis of capiary pressure in porous media, Water Resour. Res. 29, Mare, C. M.: 1982, On macroscopic equations governing mutiphase fow with diffusion and chemica reactions in porous media, Int. J. Eng. Sci. 20, Novick-Cohen, A. and Sege, L. A.: 1984, Non-inear aspects of the Cahn Hiiard equation, Physica D 10, Papatzacos, P.: 2000, Diffuse-interface modes for two-phase fow, Phys. Scripta 61, Papatzacos, P.: 2002, Macroscopic two-phase fow in porous media assuming the diffuseinterface mode at pore eve, Transport Porous Media 49, Papatzacos, P. and Skjæveand, S. M.: 2004, Reative permeabiity from thermodynamics, SPE J. (March 2004),

24 236 P. PAPATZACOS AND S. M. SKJÆVELAND Skjæveand, S. M., Siqveand, L. M., Hammervod, W. L. and Virnovsky, G. A.: 2000, Capiary pressure correation for mixed-wet reservoirs. SPEREE (February 2000), Terwiiger, P. L., Wisey, L. E., Ha, H. N., Bridges, P. M. and Morse, R. A.: 1951, An experimenta and theoretica investigation of gravity drainage performance, Trans. AIME 192,