On the definition of macroscale pressure for multiphase flow in porous media

Transcription

1 WATER RESOURCES RESEARCH, VOL 44, W06S02, doi:101029/2006wr005715, 2008 On the definition of macroscale pressure for multiphase flow in porous media J M Nordbotten, 1,2 M A Celia, 2 H K Dahle, 1 and S M Hassanizadeh 3 Received 7 November 2006; revised 29 November 2007; accepted 11 December 2007; published 11 June 2008 [1] We consider immiscible two-phase flow in porous media, starting with the Stokes equations Our analysis leads to Darcy s law but with notable differences from the usual interpretation The most immediate difference is the interpretation of macroscale pressure, which, contrary to previous derivations, does not equal the intrinsic phase average pressure We recover the intrinsic average only when systematic subscale heterogeneities, in material properties or fluid distribution, are absent Examples using capillary tube and dynamic pore network models are given These results impact our understanding of multiphase flow and have a direct effect on numerical upscaling efforts, including calculations of continuum-scale flow parameters from pore-scale network models Citation: Nordbotten, J M, M A Celia, H K Dahle, and S M Hassanizadeh (2008), On the definition of macroscale pressure for multiphase flow in porous media, Water Resour Res, 44, W06S02, doi:101029/2006wr Introduction [2] When Darcy introduced his well-known equation 150 years ago, it was solely based on one-dimensional flow of water, as a constant density incompressible fluid, in a homogeneous nondeformable porous medium under isothermal conditions That simple formula has subsequently been extended (but actually keeping its original form) to be valid for simultaneous flow of many (possibly compressible) fluids in a heterogeneous anisotropic deformable porous medium under nonisothermal conditions in three dimensions One major change has been to formulate Darcy s equation in terms of fluid pressure instead of the hydraulic potential This is particularly relevant for multiphase flow where an auxiliary equation for the difference in pressures of the fluids, often simply referred to as the capillary pressure, is introduced [3] The common form of Darcy s law for two fluids reads q a ¼ k r;a Kðr½PŠ a rgrzþ: ð1þ m a Here q a denotes the Darcy velocity vector for phase a, k r,a is relative permeability, m a is viscosity, K is the intrinsic permeability tensor, r a is mass density, g the gravitational constant, z the vertical coordinate (increasing downward), and [P] a is the Darcy-scale fluid pressure The pressure has been placed in square brackets to emphasize that this is a macroscale quantity In this multiphase extension, it is not clear what exactly the macroscale pressure is, and in particular how it relates to the pressure at the microscale (pore scale) 1 Department of Mathematics, University of Bergen, Bergen, Norway 2 Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA 3 Department of Earth Sciences, Utrecht University, Utrecht, Netherlands Copyright 2008 by the American Geophysical Union /08/2006WR W06S02 [4] Derivations of Darcy s law using volume averaging methods have always involved the assumption that the macroscale pressure is equal to the intrinsic phase average pressure hpi a (to be defined later) In all volume-averaging based derivations, many assumptions are made and a number of terms are neglected In a separate paper [Nordbotten et al, 2007], we have shown that these assumptions can lead to unacceptable restrictions on the range of applicability of Darcy s law In particular, when there are gradients in fluid distribution and pressure, use of the intrinsic phase average leads to untraditional gravitational forces This has also been shown in a recent paper by Gray and Miller [2004] In the case of two-phase flow, intrinsic phase average pressure will involve weighting of the microscopic pressure by saturation, as well as porosity So, even in the case of homogeneous media, because we will in general have gradients in saturation and pressure on all scales, use of the intrinsic phase average pressure will lead to additional terms in Darcy s law In fact, as we show in section 41, use of the intrinsic phase average pressure leads to relative permeability being larger than unity in some flow situations [5] These observations lead us to question the definition of macroscale pressure in terms of the microscale pressure While this question has significant theoretical implications, it is also relevant to many upscaling studies that involve computation of the pressure field at a given scale and the associated inference of parameters at a larger scale (see, eg, Barker and Thibeau [1996] for a review pointing out the ambiguity of pressure definition in upscaling) For example, in dynamic pore-scale network models, the macroscale pressure field for a given fluid is almost always defined as the intrinsic phase pressure (ie, the average pressure of a fluid in all pore bodies weighted by the pore body volume occupied by that fluid) [see, eg, Dahle and Celia, 1999; Gielen et al, 2004, 2005; Manthey et al, 2005] This means that results of computations from dynamic pore-scale network models may lead to predictions of unphysical macroscopic behavior This problem does not 1of8

2 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 appear to a similar extent in static upscaling models, where the pressure in connected phases is essentially constant over the sample (see Blunt [2001] for a recent review) We illustrate this in section 41, where we study the displacement of a wetting phase by a nonwetting phase in a capillary tube, which is the simplest pore-scale model one can construct We expand upon the analysis by also analyzing the results obtained using the dynamic pore-scale model of Nordhaug et al [2003] Our objective in this paper is to develop a definition of macroscopic variables from microscopic variables which gives physically consistent results at the macroscale The variable we focus on is pressure, but the concepts apply to other upscaled variables as well 2 Definitions [6] For a microscale function w and an averaging volume V x centered at a point x, a general definition of the average of w is hwi ¼ 1 V x ZV x wdv; ð2þ where w may be nonzero over the whole or part of volume V x A special case of interest is when w is defined only over the parts of V x that are occupied by a phase a This and other definitions are facilitated by the definition of an indicator function (g a ), given by g a ðx; tþ ¼ 8 < 1 if phase a exists at point x at time t; : 0 otherwise: [7] The traditional phase volume average, which is commonly applied to extensive variables, is defined as ð3þ hg a wi¼ 1 V x ZV x g a wdv; ð4þ while the intrinsic phase volume average, which is commonly applied to intensive variables, is defined as hwi a ¼ 1 Z g a V a wdv: ð5þ x V x [8] Here, the fraction of the medium occupied by the a phase is defined by a ¼hg a i [9] We slightly extend this notation, in the sense that we shall assume we can take intrinsic phase averages over functions w defined only in the phase, even though the right hand side of equation (5) requires w to be everywhere defined inside the averaging volume This will have no consequence on the derivation given here [10] We will herein restrict ourselves to the consideration of three phases, where subscript s will denote the solid phase, and the remaining phases, usually denoted by a and b, are the fluid phases In our treatment we will need the averaging theorem of Slattery [1968]: ð6þ rhwi ¼ 1 V x ZV x nwds; ð7þ from which we can obtain by the generalized divergence theorem [Whitaker, 1967], hg a rwi ¼rhg a wiþ 1 V x Z n a wds; ð8þ A a; ðsþbþ The interface between two phases a and b is denoted as A a,b, and the summation in the subscript is the natural interpretation A a,(s+b) = A a,s + A a,b Further, the boundary of the integration volume V x is denoted V x, and the vector n is the outward unit normal vector along the boundary of V x, and n is the outward unit normal vector along the boundary of a phase a within V x 3 Averaging Microscale Equations [11] In a recent paper, we showed that for single-phase flow in porous media, the macroscale pressure in Darcy s law is not simply the intrinsic phase average of the microscale pressure, unless the porosity (the phase occupancy) could be assumed constant [Nordbotten et al, 2007] For two-fluid flow, the saturation acts as an indicator for phase occupancy In general porous media we expect conditions under which it is inappropriate to consider the saturation as constant on the scale of the averaging volume V x In particular, this will be the case when sharp saturation fronts arise In this section we will address how our formulation of Darcy s law applies in these regions of significant saturation changes at the scale of the averaging volume 31 Traditional Derivation [12] We start by recalling the main points of the derivation of the two-phase extension of Darcy s law from the microscale flow equations, referring the interested reader to literature for detailed accounts [see, eg, Gray and O Neill, 1976; de la Cruz and Spanos, 1983; Whitaker, 1986b; Muccino et al, 1998] We will deviate from these expositions at the point where they restrict their analysis to regimes where the saturation gradients on the scale of the averaging volume are negligible [13] The Stokes equations for an incompressible fluid a are rp a m a r 2 v a r a grz ¼ 0 rv a ¼ 0; ð9þ ð10þ where P a is pressure, m a is viscosity, v a is fluid velocity, r a is density of phase a, g is the gravitational constant, and z is the vertical coordinate Inertial terms have been neglected because of the slow flow regimes under consideration As usual, we will only consider boundary conditions between the phases, as it has been established that the outer boundaries of the domain do not significantly affect the results [Whitaker, 1986b] Thus we have the following boundary conditions: v a ¼ 0 on A a;s ð11þ v a ¼ v b on A ð12þ T a T b n ¼ 2s t Hn on A : ð13þ Equations (11) and (12) are no-slip condition boundary conditions, while equation (13) relates the normal stress to 2of8

3 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 the interfacial tension s t and surface curvature H [see, eg, Whitaker, 1986b] The total stress is defined for a Newtonian fluid as T = PI + m(rv + rv T ) [14] When we take the phase average of the Stokes equation over a volume for problems with constant density, we obtain hg a rðp r a gzþim a hg a r 2 vi ¼0 ð14þ hg a rvi ¼0: ð15þ Note that we omit the subscripts on P and v when averaging over phases, since they are only multivalued at the interfaces, which have measure 0 The second term can be approximated by exploiting the no-slip condition at fluid interfaces [de la Cruz and Spanos, 1983; Whitaker, 1986b; Muccino et al, 1998]: hg a r 2 vi ¼A rr a;a hg a viþrr hg b vi ; ð16þ where A is a material property accounting for the medium resistance to flow In accordance with Whitaker [1986b], we have neglected Brinkman-like terms Note that the generalized relative resistivity coefficients r r a,b, appear in this equation (and are functions of saturation), representing the effect of shear forces over the fluid-fluid interfaces [15] Next, we consider equation (15) Using Whitaker s averaging theorem (8) in divergence form we obtain suggested a new macroscopic pressure This macroscopic definition of pressure extends the usual intrinsic phase average by correcting for systematic dependencies on the length scale of the averaging volume [17] For the application to multiphase flow we find that the macroscopic pressure defined by Nordbotten et al [2007] is not appropriate, since while more general subscale variations in fluid distributions are permitted, there are still restrictions that the microscale fluid distribution satisfies an REV condition This is not compatible with strong spatial variations in saturation, which are known to appear on all scales [Lenormand et al, 1988] We will herein take a different approach, by attempting to approximate a coarsescale function [w] a directly [18] Consider the decomposition w =[w] a + ~w with h~wi a = 0 Note that this decomposition will always exist (take, eg, [w] a = w), however it is not, in general, unique We are considering cases where [w] a is smooth on the coarse scale, thus we will make the assumption that a decomposition exists such that high-order derivatives in space of [w] a are small [19] Since it is known that hwi a is a good macroscopic representation for statistically uniform fluid distributions, we will consider perturbations around this state, and postulate the following functional dependency: ½wŠ a ¼ Fðhwi a ; rhwi a ; rrhwi a ; Þ: ð20þ [20] For polynomial w, F is a linear function in its arguments; hg a rvi ¼rhg a viþ 1 V x ZA n a vds: ð17þ ½wŠ a ¼ X1 k¼0 C ðkþ k : k x k hwia ; ð21þ We recognize the product in the integral as the interface velocity, thus the last term is the time derivative of the volume fraction Applying equations (16) and (17) to equations (14) and (15), we obtain hg a rðp r a gz Þiþm a A r a;a hg a viþr hg b vi ¼ 0 ð18þ r a t þrhg avi ¼0: r ð19þ Note that the first term in equation (18) still contains the average of derivatives of microscopic pressure 32 Consideration of the Average Potential Gradient [16] To obtain a macroscopic equation from equation (18), we need to find an estimate of hg a r(p r a gz)i Traditionally, the intrinsic phase average pressure has been chosen as this estimate [eg, Gray and O Neill, 1976; Hassanizadeh and Gray, 1993a, 1993b; Manthey et al, 2005; Muccino et al, 1998; Quintard and Whitaker, 1988; Whitaker, 1986a, 1986b] However, this is only valid under assumptions of statistically uniform distribution of fluids within the averaging volume (as discussed by Gray and Miller [2004], Quintard and Whitaker [1994] and Whitaker [1986a, 1986b]) To circumvent this restriction, Nordbotten et al [2007] analyzed the case of single-phase flow, and under less strict assumptions on the existence of an REV where the superscript (k) indicates that C k (k) is a tensor of order k All tensors will have rank d, the dimension of the system in consideration It is now possible to construct approximations [w] n a of any order n by requiring that (21) is satisfied exactly for polynomial functions of order n, see Appendix A These approximations take the form ½wŠ a 0 ¼hwia ½wŠ a 1 ¼hwia þ ðrhxi a Þ 1 ðx hxi a Þ rhwi a ½wŠ a n ¼hwia þ Xn k¼1 C ðkþ k;n : k x k hwia : ð22þ ð23þ ð24þ [21] It is important to note that no assumptions are made on the existence of an REV (we will show in the first example below a case where we apply equation (23) for a problem where no REV exists) We further require the (k) system of equations determining the coefficients C k,n to be invertible [22] We note the physical interpretation of [w] a 1 : The average coordinate hxi a is the centroid of the phase Therefore the second term on the right-hand side of equation (23) corrects for the distance between the centroid of the phase and the centroid of the averaging volume 3of8

4 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 [23] To proceed further, consider the average of the gradient of a function w as in equation (18) We then have from the averaging theorem of Whitaker hg a rwi ¼rhg a wiþ 1 V x Z ¼ a rhwi a þr a hwi a þ 1 V x Z n a wds ð25þ A a; ðsþbþ n a wds; ð26þ A a; ðsþbþ where we have used the identity hg a wi = a hwi a Wenow insert the expression for [w] n a (from equation (24)) to obtain the difference between the intrinsic average gradient of w and the gradient of the macroscopic variable [w] n a : hrwi a rw ½ Š a n ¼ r a hwi a þ 1 Z n a wds a a V x A a; ðsþbþ r Xn k¼1 C ðkþ k;n : k x k hwia : ð27þ [24] By analogy to the procedure applied by Whitaker [1967] and Gray and O Neill [1976], we consider the case of stagnant fluids, for which equations (14) and (15) imply that r(p r a gz) = 0, and hence (P r a gz) is constant From equation (27) and Slattery s averaging theorem equation (7), with w = g a, we then see that at static conditions, hrðp r a gzþi a ¼rP ½ r a gzš a n ¼ 0: ð28þ Thus we can expand the difference between the two terms on the left hand side as a function of some measure of distance from static conditions We assume that hr(p r a gz) i a is a suitable measure in the limit of near static conditions, and apply the linear approximation: hrðp r a gz Þi a rp ½ r a gzš a n ¼ Da n hr ð P r agzþi a : ð29þ Note that the matrix D n a will in general be a function of the fluid distribution Combining equations (18) and (29), we arrive at our macroscopic equations I D a n 1r ½ P ra gzš a n þ 1 a m aa rr a;a hg a viþrr hg b vi ¼ 0: ð30þ [25] For n 1, we can simplify the potential terms, since the macroscopic function is exact for linear microscopic functions, [r a gz] n a = r a gz, and thus we have the macroscopic equations I D a n 1r ½PŠ a n r agz þ 1 a m aa rr a;a a t þrhg avi ¼0: hg a viþrr hg b vi ¼ 0 ð31þ ð32þ These equations are still subject to a constitutive relationship relating [P] n a and [P] n b [26] Let us comment on the intrinsic phase average pressure, equivalent to [P] 0 a Then the macroscopic formulation consists of equations (30) and (32) This has the marked disadvantage that [z] 0 a, the vertical center of mass of phase a, is a function of volume fraction gradients Therefore, the gravitational body force acting on the system will be dependent on the gradient of saturation in the system (see Nordbotten et al [2007] for a lengthy discussion) Nonconstant gravitational body forces are not consistent with the usual interpretation and understanding of the multiphase extension of Darcy s law The more general definition of macroscale pressure alleviates this problem, since it is exact for linear variations such as coordinates For the special case of n =0,Whitaker [1986b] has derived local closure equations for calculating (I D 0 a ) 1 For the general case of n 1, such closure equations have not yet been obtained [27] To complete this section we rewrite the macroscale equations (31) and (32) in terms of fluxes q a = hg a vi, porosity f =(1 s ) and saturation s a = a /f Introducing relative permeabilities as defined in equation (35) we then obtain q a ¼ K m a r r ½PŠ a n r agz þ K r r ½PŠ b n r bgz K a;a 0 ¼ f sa þrq a : t ð33þ ð34þ [28] The permeability is thus defined as K = fa 1, while the (tensor) relative permeabilities K r a,b are defined using the inverse resistivity matrix kr a;a kr b;a kr kr b;b ¼ 1 ; ð35þ ra;a r rr b;a rr rr b;b which implies K r a,b = s a k r a,b (I D n b ) 1 4 Examples [29] In this section we will consider two examples, to illustrate the implications of the definition of pressures The first example considers flow in a single tube and serves as a cartoon of upscaling From this example, which in many ways mimics the evolution of a sharp (Buckley-Leverett) front, we see that the conventional interpretation of pressure necessarily allows for relative permeabilities that are discontinuous and exceed unity, while the first-order macroscale pressure defined herein leads to no such problems These observations will likely explain some of the high relative permeability values observed in previous investigations [see, eg, Bartley and Ruth, 2001; Hewett et al, 1998] As a second example, we have considered one of the dynamic network simulations presented by Nordhaug et al [2003] Again, we see clear differences between the relative permeability curves derived when applying the intrinsic average pressure in contrast to the phase average pressure 41 Flow in a Single Tube [30] Here we analyze an example of two-fluid flow in a single tube The fluids are separated by a sharp interface (note that the problem violates the assumption on an REV 4of8

5 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 for saturation) We compare the results obtained from both the intrinsic phase average and the macroscale pressure derived in the previous section, equation (24) Note that while a single tube for many applications cannot be considered a valid representation of a porous medium, we can in this example consider it as mathematically equivalent to a sharp saturation front passing through a homogeneous porous material, which has been observed in experiments [Lenormand et al, 1988] A single tube may also be viewed as the limit of a bundle of tube analysis [Bartley and Ruth, 2001; Dahle et al, 2005] [31] Consider a tube of constant radius and sufficiently large length that the ends of the tube do not affect the solution near the front For simplicity of exposition, we will neglect the width of the fluid interface We will also neglect the presence of a solid phase outside of the tube, setting s = 0 This implies that the saturation is equal to the phase fraction s a = a For this example, we will also consider the tube to be smooth enough so that there is no residual wetting fluid behind the front [32] If the flow is horizontal, we can for sufficiently small tubes neglect gravity from our discussion Then on each side of the interface location x I, we have single-phase flow, which for low-enough velocities allows us to approximate the solution of the Stokes equations by the Washburn equation [Dullien, 1992] q a ¼g a K m a P x ; for x 6¼ x I: ð36þ In this equation, q a is the x component of the fluid velocity at any point along the tube, and the inverse resistance to flow is K = r 2 /8 Let us denote the fluid occupying x < x I as the nonwetting fluid, a = nw, and the other fluid for the wetting a = w Wewillusea to denote either fluid when the equations are symmetrical From the incompressibility of the fluids and mass conservation, we have that Q = q nw (x) +q w (x) is constant in space This also implies that the boundary conditions at the endpoint of the tube become immaterial for the discussion Observe then from equation (36) that because only one fluid flows on either side of the interface, we have that q nw (x) =Q for x < x I, and q w (x) =Q for x > x I It follows that m 1 a P/x is constant for x 6¼ x I [33] This system is essentially one dimensional, so the averages defined in section 2 take the simple form and hwi ¼ 1 hwi a ¼ 1 s a Z xþ =2 x =2 Z xþ =2 x =2 wdl g a wdl; ð37þ ð38þ where the length of the averaging volume is denoted Averaging equation (36) over an averaging volume we get 8 K >< m hg nw qi¼ nw x hpinw ; x x I =2; K >: 2s nw m nw x hpinw ; otherwise: ð39þ [34] This follows from the observation that for linearly varying pressure and 0 < s nw < 1, we can apply equation (38) to obtain the exact relationship (see Appendix B): x hpinw ¼ 1 nw 2 x P : ð40þ Note that the phase saturation s nw takes a particularly simple form for this system; s nw =(x I x)/ + 1/2, bounded above and below by 1 and 0, respectively [35] We compare equation (39) to Darcy s law, which we expect to be valid over a collection of tubes, ½qŠ a ¼K r ðs a Þ ½KŠ x ½PŠa : m a ð41þ Here brackets indicate macroscale variables We have omitted viscous coupling terms, since these are derived as proportional to shear forces over the interface For a single tube, all motion is perpendicular to the fluid-fluid interface, and thus there are no shear forces over this interface, and viscous coupling is nonexistent To obtain a mass-conservative flow field, it is natural to associate the average flow velocity in the tubes with the Darcy flux (hg a qi =[q] a ), and define the permeability [K] = K We now observe by comparison of equations (39) and (41) that if we take the macroscopic pressure to equal the intrinsic phase average pressure ([P] a = hpi a =[P] a 0 ), then the relative permeability function must satisfy 8 K r ðs a < 2s a ; s a < 1; Þ ¼ : 1; s a ¼ 1: ð42þ This function is discontinuous, and exceeds unity Both of these observations are in contrast with the expected behavior (and indeed measurements) of relative permeability We note in passing that similar qualitative results have been observed in the field of upscaling [see, eg, Hewett et al, 1998] [36] In contrast to defining macroscopic pressure according to Equation equation (38), if we considering the macroscopic pressure to be equal to the first-order approximation ([P] a = [P] a 1 ), we have from the definition in equation (23) that ½PŠ nw 1 ¼hPinw þ ð1 s nw Þ x hpinw : ð43þ Differentiating this equation with respect to x and applying equation (40), we then have that x ½PŠnw 1 ¼h x Pinw : ð44þ Comparison between Darcy s law and the average of equation (36) implies that when using the macroscopic pressure [P] 1 nw, K r ðs nw Þ ¼ s nw : ð45þ This results holds when using any [P] k a, with k 1 [37] Related observations can be made with respect to the pressure difference Let the microscopic pressure jump over the interface due to interfacial tension be denoted by P c,stat 5of8

6 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 Figure 1 Saturation profiles for the dynamic network model during drainage (from left) Then integrating equation (36) gives for the traditional averaging approach 8 Undefined; s nw ¼ 0; >< hpi nw hpi w ¼ P c;stat þ 2 m nw s nw þ m w s w s nw ; 0 < s nw < 1; 2 K t >: Undefined; s nw ¼ 1: ð46þ Similarly, the following result is obtained when applying the new macroscopic pressures with k 1, ½PŠ nw 1 8 ½PŠw 1 Undefined; s nw ¼ 0; >< ¼ P c;stat þ 2 m nw m w ðs nw 1=2Þ s nw ; 0 < s nw < 1; 2 K t >: Undefined; s nw ¼ 1: ð47þ [38] Equations (46) and (47) imply that for both formulations we introduce a dynamic term (referring to the dependency on the rate of change of saturation) in the pressure difference relationship, the capillary pressure, which scales with the square of the averaging volume This dynamic term has the same length-squared scaling as has been observed in previous pore-scale investigations, as discussed by Dahle et al [2005] Note that both equations (46) and (47) imply that for certain flow regimes, depending on the sign and magnitude of the time derivative of saturation, the capillary pressure can become negative This is interesting, and an understanding of the implication of this phenomena certainly requires further study 42 Dynamic Flow in a Pore Network Model [39] We expand on the above example by investigating the results from a three-dimensional dynamic pore network model We consider the model given by Nordhaug et al [2003], which was designed to investigate the relationships among saturation, interfacial area, and interfacial velocities The model comprises spherical pore bodies connected by cylindrical tubes, and is simplified by allowing only a single fluid to occupy any pore throats at any time Interfacial tension is included to determine 6of8 interface movement between pore bodies and throats, however no local capillary pressure is prescribed in the pore bodies themselves The geometry of the network is that of a regular lattice, and the radii of pore bodies and throats are prescribed randomly according to cutoff lognormal distributions [40] Of the cases presented by Nordhaug et al [2003], we will herein consider a drainage experiment with stable displacement The viscosity ratio is set at 1:10 For this case, it was determined that a network, where the longest direction is parallel to flow, is sufficient to obtain representative results A typical saturation profile is shown in Figure 1 Note the significant amount of trapped fluid, most of which is trapped in pore throats Overall this profile resembles the usual solution one would obtain by solving a Buckley-Leverett problem: A quasi-static front wave with jump from saturation S w =1toS w of about 02 is followed by a rarefaction wave to residual saturation [41] We have chosen to consider an averaging volume of , starting at the 21st layer of pore bodies This allows for 20 layers upstream and 10 layers downstream, which avoids influence of the boundaries Our results are insensitive to small perturbations in averaging volume size and location Since flow is one-dimensional at the macroscopic scale, we have used an assumption of symmetry perpendicular to the flow direction, such that the only nonzero derivatives appear parallel to flow [42] We have calculated relative permeability curves for both the wetting and the nonwetting phase, using both the average pressure and the first-order macroscale pressure The relative permeability curves were calculated from equation (24) under the assumption of a diagonal relative permeability tensor The results are shown in Figure 2 We note that the relative permeability results are in accordance with the results from the single tube example: The relative permeability curves obtained using the intrinsic phase average are nonmonotone and exceed unity Conversely, the curves obtained using the first-order macroscale pressures remain below one, and are essentially monotone A notable effect of the increased complexity of network models can be seen in the nonwetting relative permeability Figure 2 Relative permeability curves for the wetting and nonwetting phases The solid and dashed lines are obtained using the intrinsic phase average pressures, while the dashdotted and dotted lines are obtained using the first-order macroscopic pressures

7 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 Figure 3 Pressure difference curves (P c (S)) for pressures defined as the intrinsic phase average (solid line) and as the first-order correction (dash-dotted line) curves, where we observe an abrupt loss of permeability for small nonwetting saturations (S w > 09) [43] For completeness, we also include the macroscale capillary pressure curves in Figure 3 We see that the results are qualitatively similar to with those predicted by equations (46) and (47): The difference between the intrinsic phase average pressures is consistently positive, while the difference between the first-order macroscale pressures go both above and below zero [44] To conclude these two examples, we observe that the theoretical development leading to a family of macroscopic pressures as defined in equation (24), not only solves the theoretical problem of the gravitational term, but also leads to more consistent results in terms of relative permeabilities in the presence of a sharp front In regards to dynamic capillary pressure, this phenomena appears in both formulations, but takes on different form A complete understanding of dynamic capillary pressure is still elusive, and is not explored further herein 5 Discussion [45] In section 3, we showed the validity of the two-phase extension of Darcy s law for a family of macroscale pressures [P] n a, with n 0 The subscript n should be interpreted as the order of the approximation to a smooth macroscale function [P] a The approximation is exact when [P] a is a polynomial of order n or lower The importance of the new macroscale variables becomes clear when considering the macroscale Darcy s law, given in equation (31) We see that the gravity potential can only be simplified to the usual form, equation (1), when n 1, which implies that the intrinsic phase average, which is equivalent to n =0, does not satisfy the usual form of Darcy s law [46] For flow problems, the difference between the zeroth-order macroscale pressure and higher-order macroscale pressures was highlighted through examples in section 4 In particular, the relative permeability functions needed to reproduce a macroscale flow field exceed 1 when the intrinsic phase average pressure is used They are also discontinuous functions of saturation Conversely, for the higher-order (n 1) macroscale pressure, relative permeabilities do not exhibit these problems We further observe that for a more complex flow system, as in the second example of section 4, a nonmonotonic system capillary pressure versus saturation relationship is observed with the macroscale pressure defined as the intrinsic average [47] These results have the following implication A modeler of porous media flow has a choice in defining macroscale variables Some choices lead to macroscale parameters that have constrained behavior consistent with their traditional functional forms (like relative permeability), while others do not If these are used to guide our choice for macroscale variables, then the new definitions of macroscale pressure proposed herein have clear advantages Consider the examples discussed in section 4 Most modelers of porous media would have chosen, by intuition, to use relative permeability functions bounded between 0 and 1 With this choice, we immediately know that the pressures in the model are not the intrinsic phase average pressures, but must be closer to the higher-order macroscale pressures discussed herein Thus we can argue that the coarse-scale modeler today already uses the macroscale pressures, thus uncovering an inconsistency between the parameters determined by the fine-scale investigation and the parameters used at the coarse scale We believe this new perspective provides valuable insight into the modeling process, and can provide both theoretical and practical guidance in upscaling studies Appendix A [48] We consider a microscopic function w, which is such that it can be decomposed into w =[w] a + ~w, where [w] a is a smooth macroscale function, and ~w is a microscale function with zero mean for some size averaging volume Herein we will show how to construct [w] a from w if [w] a is a loworder polynomial [49] Take [w] a as an arbitrary nth-order polynomial ½wŠ a ¼ Xn k¼0 B ðþ k k;n : xk : ða1þ We wish to reconstruct [w] a from averages of w, which, because of the definition of ~w, have the property hwi a = h[w] a i a We then seek a nth-order approximation to [w] a of the form ½wŠ a n ¼ Xn k¼0 C ðkþ k;n : k x k hwia : ða2þ If we require [w] a n =[w] a for all polynomials, of order n, equation (49) must be exact for arbitrary B (k) k,n We can (k) therefore successively set one B k,n equal to the identity tensor and the remaining equal to zero This leads to a system of equations for the tensors C (k) k,n : C ð0þ 0;n ¼ 1 P n j¼0 C ðþ j j;n : j x hxi a ¼ x j P n j¼0 C ðþ j j;n : j x hxxi a ¼ xx j ða3þ ¼ P n j¼0 C ðþ j j;n : j hx n i a ¼ x n : x j This gives S n j=0 d j equations (where d is the physical dimension of the problem) for the same number of 7of8

8 W06S02 NORDBOTTEN ET AL: MACROSCALE PRESSURE FOR MULTIPHASE FLOW W06S02 unknowns It thus follows that our definition of macroscale variables [w] a n only exists and is unique subject to the solution of equation (50) A1 Special Case n =1 [50] For n = 1, the system (A3) can be solved directly, C ð1þ 1;1 ¼ 1 x hxia ðx hxi a Þ ¼ ðrhxi a Þ 1 ðx hxi a Þ: ða4þ A2 Special Case d =1 [51] For d = 1, all the tensors will have rank 1 and are thus scalars System (A3) then becomes a matrix equation, 0 x hxi a 1 x 2hxia x n hxia C 1;n x hxi x hx 2 i a x 2hx2 i a x nhx2 i a C 2;n x 2 hx 2 i ¼ B CB : C B C A A A x hx n i a x 2hxn i a x nhxn i a x n hx n i C n;n ða5þ Here we have used x k as shorthand for the kth derivative This system can be inverted analytically for small n Appendix B [52] In this appendix we will derive equation (40) Thus we consider flow in a single tube, under the assumptions and simplifications of section 41 In particular, we will exploit the relationship s nw =(x I x)/ + 1/2 Consider the derivative of the definition of the intrinsic phase average, when the interface location x I is within the averaging volume: x hpinw ¼ 1 Z xi nw x Py ð Þdy þhg nw Pi x 1 ¼ 1 nw x x =2 Z xi x =2 nw Py ð Þdy þ hg nw Pi nw ¼ 1 Px ð =2Þþ 1 hpi nw nw nw ¼ 1 Pððx I þ x =2Þ=2ÞPx ð =2Þ 2 ðx =2 þ x I Þ=2 ¼ 1 2 x P ¼ 1 nw 2 x P : [53] Acknowledgments The authors thank R van Dijke and S Manthey for interesting discussions Furthermore, the example of flow in a single tube was motivated from a discussion with I Neuweiler Finally, H F Nordhaug supplied the network model of section 42 This work was supported in part by BP and Ford Motor Company through funding to the Carbon Mitigation Initiative at Princeton University and by the National Science Foundation under grant EAR References Barker, J W, and S Thibeau (1996), A critical review of the use of pseudorelative permeabilities for upscaling, paper presented at European 3-D Reservoir Modelling Conference Soc of Pet Eng, Stavanger, 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