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

Authors


Abstract

[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.

1. 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

equation image

Here qα denotes the Darcy velocity vector for phase α, kr,α is relative permeability, μα is viscosity, K is the intrinsic permeability tensor, ρα is mass density, g the gravitational constant, z the vertical coordinate (increasing downward), and [P]α 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).

[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 〈Pα (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 4.1, 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, e.g., 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 (i.e., the average pressure of a fluid in all pore bodies weighted by the pore body volume occupied by that fluid) [see, e.g., 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 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 4.1, 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 ω and an averaging volume Vx centered at a point x, a general definition of the average of ω is

equation image

where ω may be nonzero over the whole or part of volume Vx. A special case of interest is when ω is defined only over the parts of Vx that are occupied by a phase α. This and other definitions are facilitated by the definition of an indicator function (γα), given by

equation image

[7] The traditional phase volume average, which is commonly applied to extensive variables, is defined as

equation image

while the intrinsic phase volume average, which is commonly applied to intensive variables, is defined as

equation image

[8] Here, the fraction of the medium occupied by the α phase is defined by

equation image

[9] We slightly extend this notation, in the sense that we shall assume we can take intrinsic phase averages over functions ω defined only in the phase, even though the right hand side of equation (5) requires ω 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 σ will denote the solid phase, and the remaining phases, usually denoted by α and β, are the fluid phases. In our treatment we will need the averaging theorem of Slattery [1968]:

equation image

from which we can obtain by the generalized divergence theorem [Whitaker, 1967],

equation image

The interface between two phases α and β is denoted as Aα,β, and the summation in the subscript is the natural interpretation Aα,(σ+β) = Aα,σ + Aα,β. Further, the boundary of the integration volume Vx is denoted ∂Vx, and the vector n is the outward unit normal vector along the boundary of Vx, and n is the outward unit normal vector along the boundary of a phase α within Vx.

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 Vx. 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.

3.1. 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, e.g., 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 α are

equation image
equation image

where Pα is pressure, μα is viscosity, vα is fluid velocity, ρα is density of phase α, 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:

equation image
equation image
equation image

Equations (11) and (12) are no-slip condition boundary conditions, while equation (13) relates the normal stress to the interfacial tension σt and surface curvature H [see, e.g., Whitaker, 1986b]. The total stress is defined for a Newtonian fluid as T = −PI + μ(∇v + ∇vT).

[14] When we take the phase average of the Stokes equation over a volume for problems with constant density, we obtain

equation image
equation image

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]:

equation image

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 rrα,β, 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

equation image

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

equation image
equation image

Note that the first term in equation (18) still contains the average of derivatives of microscopic pressure.

3.2. Consideration of the Average Potential Gradient

[16] To obtain a macroscopic equation from equation (18), we need to find an estimate of 〈γα∇(Pραgz)〉. Traditionally, the intrinsic phase average pressure has been chosen as this estimate [e.g., 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 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 coarse-scale function [ω]α directly.

[18] Consider the decomposition ω = [ω]α + equation image with 〈equation imageα = 0. Note that this decomposition will always exist (take, e.g., [ω]α = ω), however it is not, in general, unique. We are considering cases where [ω]α is smooth on the coarse scale, thus we will make the assumption that a decomposition exists such that high-order derivatives in space of [ω]α are small.

[19] Since it is known that 〈ωα is a good macroscopic representation for statistically uniform fluid distributions, we will consider perturbations around this state, and postulate the following functional dependency:

equation image

[20] For polynomial ω, F is a linear function in its arguments;

equation image

where the superscript (k) indicates that Ck(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 [ω]nα 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

equation image
equation image
equation image

[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 system of equations determining the coefficients Ck,n(k) to be invertible.

[22] We note the physical interpretation of [ω]1α: The average coordinate 〈xα 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.

[23] To proceed further, consider the average of the gradient of a function ω as in equation (18). We then have from the averaging theorem of Whitaker

equation image
equation image

where we have used the identity 〈γαω〉 = εαωα. We now insert the expression for [ω]nα (from equation (24)) to obtain the difference between the intrinsic average gradient of ω and the gradient of the macroscopic variable [ω]nα:

equation image

[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 ∇(Pραgz) = 0, and hence (Pραgz) is constant. From equation (27) and Slattery's averaging theorem equation (7), with ω = γα, we then see that at static conditions,

equation image

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 〈∇(Pραgz) 〉α is a suitable measure in the limit of near static conditions, and apply the linear approximation:

equation image

Note that the matrix Dnα will in general be a function of the fluid distribution. Combining equations (18) and (29), we arrive at our macroscopic equations

equation image

[25] For n ≥ 1, we can simplify the potential terms, since the macroscopic function is exact for linear microscopic functions, [ραgz]nα= ραgz, and thus we have the macroscopic equations

equation image
equation image

These equations are still subject to a constitutive relationship relating [P]nα and [P]nβ.

[26] Let us comment on the intrinsic phase average pressure, equivalent to [P]0α. Then the macroscopic formulation consists of equations (30) and (32). This has the marked disadvantage that [z]0α, the vertical center of mass of phase α, 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 (ID0α)−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 qα = 〈γαv〉, porosity ϕ = (1 − εσ) and saturation sα = εα/ϕ. Introducing relative permeabilities as defined in equation (35) we then obtain

equation image
equation image

[28] The permeability is thus defined as K = ϕA−1, while the (tensor) relative permeabilities Krα,β are defined using the inverse resistivity matrix

equation image

which implies Krα,β = sαkrα,β (IDnβ)−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, e.g., 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.

4.1. 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 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 εσ = 0. This implies that the saturation is equal to the phase fraction sα = εα. 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 xI, 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]

equation image

In this equation, qα is the x component of the fluid velocity at any point along the tube, and the inverse resistance to flow is K = r2/8. Let us denote the fluid occupying x < xI as the nonwetting fluid, α = nw, and the other fluid for the wetting α = w. We will use α to denote either fluid when the equations are symmetrical. From the incompressibility of the fluids and mass conservation, we have that Q = qnw(x) + qw(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 qnw(x) = Q for x < xI, and qw(x) = Q for x > xI. It follows that μα−1∂P/∂x is constant for xxI.

[33] This system is essentially one dimensional, so the averages defined in section 2 take the simple form

equation image

and

equation image

where the length of the averaging volume is denoted equation image. Averaging equation (36) over an averaging volume we get

equation image

[34] This follows from the observation that for linearly varying pressure and 0 < snw < 1, we can apply equation (38) to obtain the exact relationship (see Appendix B):

equation image

Note that the phase saturation snw takes a particularly simple form for this system; snw = (xIx)/equation image + 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,

equation image

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 (〈γαq〉 = [q]α), 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]α = 〈Pα = [P]0α), then the relative permeability function must satisfy

equation image

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, e.g., 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]α = [P]1α), we have from the definition in equation (23) that

equation image

Differentiating this equation with respect to x and applying equation (40), we then have that

equation image

Comparison between Darcy's law and the average of equation (36) implies that when using the macroscopic pressure [P]1nw,

equation image

This results holds when using any [P]kα, 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 Pc,stat. Then integrating equation (36) gives for the traditional averaging approach

equation image

Similarly, the following result is obtained when applying the new macroscopic pressures with k ≥ 1,

equation image

[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.

4.2. 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 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 10 × 10 × 50 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 Sw = 1 to Sw of about 0.2 is followed by a rarefaction wave to residual saturation.

Figure 1.

Saturation profiles for the dynamic network model during drainage (from left).

[41] We have chosen to consider an averaging volume of 10 × 10 × 20, 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 curves, where we observe an abrupt loss of permeability for small nonwetting saturations (Sw > 0.9).

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 dash-dotted and dotted lines are obtained using the first-order macroscopic pressures.

[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.

Figure 3.

Pressure difference curves (Pc(S)) for pressures defined as the intrinsic phase average (solid line) and as the first-order correction (dash-dotted line).

[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]nα, with n ≥ 0. The subscript n should be interpreted as the order of the approximation to a smooth macroscale function [P]α. The approximation is exact when [P]α 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 ω, which is such that it can be decomposed into ω = [ω]α + equation image, where [ω]α is a smooth macroscale function, and equation image is a microscale function with zero mean for some size averaging volume. Herein we will show how to construct [ω]α from ω if [ω]α is a low-order polynomial.

[49] Take [ω]α as an arbitrary nth-order polynomial

equation image

We wish to reconstruct [ω]α from averages of ω, which, because of the definition of equation image, have the property 〈ωα = 〈[ω]αα. We then seek a nth-order approximation to [ω]α of the form

equation image

If we require [ω]nα = [ω]α for all polynomials, of order n, equation (49) must be exact for arbitrary Bk,n(k). We can therefore successively set one Bk,n(k) equal to the identity tensor and the remaining equal to zero. This leads to a system of equations for the tensors Ck,n(k):

equation image

This gives Σj=0ndj equations (where d is the physical dimension of the problem) for the same number of unknowns. It thus follows that our definition of macroscale variables [ω]nα 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,

equation image

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,

equation image

Here we have used xk 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 4.1. In particular, we will exploit the relationship snw = (xIx)/equation image + 1/2. Consider the derivative of the definition of the intrinsic phase average, when the interface location xI is within the averaging volume:

equation image

Acknowledgments

[53] 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 4.2. 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-0309607.

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