upscaling immiscible two-phase dispersed flow in homogeneous porous media: a mechanical equilibrium...

Author's Accepted Manuscript

Upscaling immiscible two-phase dispersedflow in homogeneous porous media: A me-chanical equilibrium approach

O.A. Luévano-Rivas, F.J. Valdés-Parada

PII: S0009-2509(14)00718-0DOI: http://dx.doi.org/10.1016/j.ces.2014.12.004Reference: CES12028

To appear in: Chemical Engineering Science

Received date: 5 September 2014Revised date: 19 November 2014Accepted date: 1 December 2014

Cite this article as: O.A. Luévano-Rivas, F.J. Valdés-Parada, Upscalingimmiscible two-phase dispersed flow in homogeneous porous media: Amechanical equilibrium approach, Chemical Engineering Science, http://dx.doi.org/10.1016/j.ces.2014.12.004

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Upscaling immiscible two-phase dispersed flow in homogeneous

porous media: A mechanical equilibrium approach

O.A. Luevano-Rivasa, F.J. Valdes-Paradaa,∗

aDivision de Ciencias Basicas e Ingenierıa, Universidad Autonoma Metropolitana-Iztapalapa.Av. San Rafael Atlixco 186, col. Vicentina, 09340 Mexico, Mexico

Abstract

In this work, we model immiscible two-phase dispersed flow in homogeneous porous me-dia by upscaling the governing mass and momentum transport equations at the pore scaleusing the method of volume averaging. The model consists of a closed set of macroscopicequations for mass and momentum transport applicable for the dispersed and continuousphases. Furthermore, under the local mechanical equilibrium assumption, only one macro-scopic equation arises for momentum transport, which resembles an extension of Darcy’slaw; whereas for mass transport, the equilibrium model reduces to the continuity equation.These macroscopic models are written in terms of effective medium coefficients that arecomputed from solving the associated closure problems in representative regions of the porescale. After performing a parametric analysis, we observe that the magnitude of the lon-gitudinal component of the permeability-like coefficient increases with the saturation andviscosity of the dispersed phase. We validated the model by comparing the predictions ofthe permeability coefficient with experimental data available in the literature. The resultsexhibit a relative error percent that ranges from 1 % to 15 %.

Keywords: Immiscible two-phase dispersed flow, upscaling, volume averaging,permeability predictions.

∗Corresponding author. E-mail: iqfv@xanum.uam.mx

Preprint submitted to Chemical Engineering Science December 9, 2014

1. Introduction1

Immiscible dispersed flow (or emulsion flow) in porous media is important in several2

areas that include chemical, industrial, environmental science, bioenginnering, CO2 storage,3

soil remediation, enhanced-oil recovery (EOR), among others (Ouyang et al., 1995; Jain and4

Demond, 2002; Durucana et al., 2013). This type of flow in porous media, is important5

in many EOR processes where emulsions can be injected to increase the volumetric sweep6

efficiency (McAuliffe, 1973). Moreover, emulsion transport is a key component in surfactant-7

enhanced aquifer remediation technologies in subsurface zones contaminated with organic8

liquids (Jain and Demond, 2002).9

Design and improvement of processes based on emulsion flow through porous media rely10

on a better understanding of dispersed flow and its displacement mechanisms (Guillen et al.,11

2012b). Several experimental analyses have shown that the magnitude of the permeability12

coefficient may be reduced due to interception and/or deposition of the dispersed-phase13

(emulsion drops) that causes a reduction in the pore throats diameter (McAuliffe, 1973;14

Alvarado and Marsden, 1979; Soo and Radke, 1984). On the one hand, for situations in15

which the drop and pore diameter ratio is close to or larger than unity (i.e., large drops),16

it is reasonable to think interception to be relevant. On the other hand, if the ratio of drop17

and pore diameter is less than 0.2 (i.e., small drops), deposition is to be expected (Soo18

and Radke, 1984; Cobos et al., 2009). In this way, both large and small drops affect the19

permeability value; large drops block the pores, tending to reduce flow through the entire20

porous medium. In addition, several small drops, with diameters smaller than the pore21

diameter, may also be accumulated in the porous medium crevices causing the same effect22

as the large drops. Clearly, the permeability reductions depend on the volume occupied by23

the retained emulsion phase and its capacity to reduce flow (McAuliffe, 1973; Alvarado and24

Marsden, 1979; Soo and Radke, 1984; Hofman and Stein, 1991; Jain and Demond, 2002).25

Modeling single-phase flow in homogeneous porous media is usually carried out by the26

well-known Darcy’s law (Darcy, 1856). In this context, we refer to a homogeneous porous27

medium in the same sense as Quintard and Whitaker (1987), i.e., “A porous medium is28

homogeneous with respect to a given averaging volume and a given process when the effective29

transport coefficients in the volume averaged transport equations are independent of posi-30

tion”. In multiphase flow, an extension of Darcy’s law is normally used, incorporating the31

influence of each phase in a relative permeability coefficient (Dullien, 1992). In particular,32

for dispersed flow, Alvarado and Marsden (1979) presented a modified Darcy’s law model,33

which included the permeability reductions caused by partial plugging, to describe emulsion34

flow in porous media. The so-called “droplet retardation model” was originally proposed35

by McAuliffe (1973) and was mathematically described by Devereux (1974) who modified36

the classical Buckley and Leverett (1942) theory for two-phase flow in porous media that37

includes a retardation factor for the dispersed phase. In the works of Soo and Radke (1986);38

Soo et al. (1986), the filtration theory concept was used to develop a model for dilute flow39

of stable emulsion in porous media. The flow redistribution (i.e., drop capture) and large40

permeability reductions are included in this model through a filter coefficient and the effec-41

tiveness parameter of retained drops in the pore throats. The parameters were determined42

2

based on experimental data (Soo and Radke, 1984), and a redistribution parameter related43

the diversification of the flow due to the drops retained in the pores. The model was modi-44

fied by Jain and Demond (2002) to determine the hydraulic conductivity reduction caused45

by incorporating multilayer covering and viscosity variations. A Darcy-scale model was de-46

veloped by Islam and Ali (1994) for stable emulsions, including in-situ generation. The47

model was formulated in terms of conservation equations for both aqueous and oil phases,48

and coupling with the filtration model of Soo and Radke (1986); Soo et al. (1986), which49

takes into account capture mechanisms. The permeability reduction was compared with the50

experimental data of Soo and Radke (1984) obtaining good agreement. Recently, a capillary51

network model has been proposed (Cobos et al., 2009; Romero et al., 2011; Guillen et al.,52

2012b) to obtain a Darcy-scale response by using experimental pore-level flow descriptions.53

This approach incorporates flow information from a single constriction into a large capillary54

network (pores) and connecting constraints (pore throats), in order to predict the global55

permeability from the flow equation within each pore.56

Despite the progress in the study of dispersed flow in porous media, it appears that57

fundamental understanding of flow is still required in order to develop reliable models for58

this type of transport process. Evidently, the complex nature of immiscible dispersed flow59

operates at different scales that go from the pore-scale (or droplet size scale) to the Darcy-60

scale. This suggests the use of upscaling methodologies in order to derive macroscopic61

models that capture the essential features of the pore-scale by means of an averaging proce-62

dure. Upscaled models are often written in terms of effective-medium coefficients that are63

responsible of saving the essential pore-scale information. Many works have been devoted64

to carry out upscaling of immiscible two-phase flow in porous media by means of different65

approaches, such as homogenization (Auriault et al., 1989), volume averaging (Whitaker,66

1986a, 1994), and recently TCAT (thermodynamically constrained averaging theory) (Gray67

and Miller, 2014). Using the volume averaging method, Whitaker (1986a, 1994) derived68

effective-medium equations for each fluid phase. These equations were similar to Darcy’s69

law but included additional terms representing the influence of the fluid-fluid interfacial70

drag. The permeability and viscous drag coefficients involved in this approach can be com-71

puted from the solution of the associated closure problems (Whitaker, 1994; Lasseux et al.,72

1996). Later on, Lasseux et al. (1996) derived the relations between the permeability and73

viscous drag tensors and showed that both coefficients can be predicted from the same clo-74

sure problems. As indicated by Whitaker (1986a, 1994), this modeling approach involves a75

set of length-scale constraints and assumptions, some of them are summarized as follows:76

1. The characteristic length associated to the averaging domain, r0, must be much smaller77

than the characteristic length associated to the macroscale, L and, at the same time,78

r0 must be much larger than the characteristic lengths of the fluid phases, �γ and �β.79

These characteristic lengths are sketched in Fig. 1.80

2. No significant variations of the interface curvature take place within the averaging81

volume. This assumption is justified when the capillary and Bond numbers are small82

compared to one.83

3. For the closure problem solution, it was assumed that the effect of the moving contact84

3

lines can be neglected, thus restricting the model to quasi-steady flow.85

Certainly, for cases in which the coupling between both fluid phases is negligible, the model86

proposed by Whitaker reduces to Darcy’s law extension for two-phase flow, in which the87

effective permeability of each phase can be computed from the intrinsic permeability and88

the saturation (Bear and Cheng, 2010).89

For dispersed flow, the permeability cannot be determined only from the pore structure90

and the modified emulsion viscosity (Guillen et al., 2012a). Indeed, two approaches can91

be proposed for modeling dispersed flow: 1) a non-equilibrium model consisting of two92

effective-medium equations, one for each fluid phase, as Whitaker (1986a) proposed and 2)93

an equilibrium model where a single effective-medium equation (with a structure similar to94

Darcy’s law) can be used to model transport everywhere (Dullien, 1992). Both alternatives95

are discussed in this work using a volume averaging framework; however, as a first approach,96

the computations are focused on determining the effective medium coefficients associated to97

the equilibrium model.98

The paper is organized as follows: In Section 2, we present the governing equations for99

mass and momentum transport at the microscale. When applying the volume averaging100

method to these equations, we obtain the non-equilibrium model as detailed in Section101

3. To simplify this model, we adopt the local mechanical equilibrium assumption in order102

to obtain a one-equation model for describing macroscopic momentum transport in both103

fluid phases (Section 4). We clearly identify the length-scale constraints and assumptions104

supporting this macroscopic model. In Section 5, we carry out a parametric analysis of the105

permeability coefficient with the main variables of the system and we validate our predictions106

of the transport coefficients by comparison with the experimental data from Soo and Radke107

(1984) and Jain and Demond (2002). Finally, we provide the discussions and conclusions of108

our work. In addition, the details about the derivation and solution of the closure problem109

are presented in Appendix A and the details behind the local mechanical equilibrium model110

are given in Appendix B.111

2. Microscopic model112

Let us consider the immiscible flow of a dispersed phase (the γ-phase) inside a continuous113

fluid phase (the β-phase) through the pores of a homogeneous porous medium (the κ-phase).114

Both fluid phases are assumed incompressible and Newtonian, whereas the solid matrix is115

assumed to be rigid and impermeable. In addition, we constrain the analysis to situations116

in which there is only contact between two phases. In this way, the dispersed phase, which117

is regarded as a stable emulsion, is in contact with the continuous fluid phase, but not with118

the solid phase. Therefore, the governing equations for mass and momentum transport at119

the microscale are120

4

∇ · vα = 0 , in the α-phase α = β, γ (1a)

0 = −∇pα + ραg + μα∇2vα, in the α-phase α = β, γ (1b)

vβ = vγ , at the β-γ interface (1c)

nβγ ·[−pβI+ μβ

(∇vβ +∇vTβ

)]= nβγ ·

[−pγI+ μγ

(∇vγ +∇vTγ

)]+ 2σHnβγ, at the β-γ interface (1d)

vβ = 0, at the β-κ interface (1e)

In the above expressions, ρα and μα represent the density and viscosity of the α-phase121

(α = β, γ), which are assumed to be constant in the upcoming derivations. Notice that, we122

are assuming that no mass exchange is taking place between both fluid phases, hence we can123

impose continuity of the velocity as indicated in Eq. (1c). In addition, we have considered a124

jump in the momentum stress caused by purely normal capillary effects as shown in Eq. (1d).125

The interfacial tension is represented by σ and the mean curvature of the interface byH . The126

total stress tensor has been decomposed according to the Newtonian character of the fluids.127

Finally, we have imposed non-slip conditions at the boundaries between the continuous fluid128

phase and the solid in Eq. (1e). This boundary-value problem differs conceptually from129

previous studies for immiscible two-phase flow reported by Whitaker (1986a, 1994), who130

considered the contact of both fluid phases with the solid.131

3. Upscaling132

In order to obtain the effective medium equations from the microscale equations by133

means of an averaging process, we introduce a volume averaging domain, V , with norm V .134

In terms of this averaging domain, the superficial averaging operator for a given variable135

defined in either fluid phase, ψα (α = β, γ), is expressed as (Whitaker, 1999)136

〈ψα〉 = 1

V

∫Vα

ψαdV, ψ = p,v, α = β, γ (2)

where Vα is the domain occupied by the α-phase included in V . The intrinsic averaging137

operator is defined as138

〈ψα〉α =1

Vα

∫Vα

ψαdV, ψ = p,v, α = β, γ (3)

Both averaging operators are related by 〈ψα〉 = εα 〈ψα〉α, where εα is the volume fraction139

of the α-phase within the averaging domain, εα = Vα/V . The derivation of volume-averaged140

equations requires the use of the averaging theorem (Howes and Whitaker, 1985). For the141

β-phase, we have142

〈∇ψβ〉 = ∇〈ψβ〉+ 1

V

∫Aβκ

nβκψβdA+1

V

∫Aβγ

nβγψβdA (4)

5

In addition, since we have assumed that there is no contact between the dispersed phase143

and the solid, this theorem takes the following form for the γ-phase144

〈∇ψγ〉 = ∇〈ψγ〉+ 1

V

∫Aβγ

nγβψγdA (5)

The first step in the averaging process consists on applying the superficial averaging145

operators to the mass and momentum balance equations for both fluid phases. A detailed146

development of this step is available from Whitaker (1986a), here we simply provide the147

resulting expression from averaging the continuity equation, which is148

∇ · 〈vα〉 = 0, α = β, γ (6)

In order to obtain this result, it was necessary to interchange temporal differentiation with149

spatial integration, which required the use of the general transport theorem. Moreover, since150

we are limiting the analysis to situations in which there is no coalescence or shrinkage of the151

dispersed phase and the solid phase is assumed rigid, we discarded the temporal variations152

of the volume fraction in Eq. (6).153

As mentioned above, in the upscaling process, the size of the averaging domain, r0, is154

usually constrained according to (Whitaker, 1999)155

�α � r0 � L, α = β, γ (7)

In this relation, �α represents the characteristic length of the α-phase at the microscale and156

the system macroscopic length is denoted by L, as sketched in Fig. 1.157

Furthermore, the resulting expression from the application of the superficial averaging158

theorem to the momentum balance equation for the β-phase is (see Whitaker, 1986a, for159

details)160

0 = −∇〈pβ〉β+ρβg+μβ∇2 〈vβ〉β+ 1

Vβ

∫Aβκ

nβκ·[−pβI+ μβ∇vβ] dA+1

Vβ

∫Aβγ

nβγ ·[−pβI+ μβ∇vβ ] dA

(8a)and for the γ-phase is161

0 = −∇〈pγ〉γ + ργg + μγ∇2 〈vγ〉γ + 1

Vγ

∫Aβγ

nγβ · [−pγI + μγ∇vγ] dA (8b)

In writing Eqs. (8), the spatial variations of the physical properties (density and viscosity) of162

the fluid-phases in the averaging domain have been neglected. In addition, we have taken into163

account the non-slip condition in the derivation of these equations, as well as the separation164

of characteristic lengths expressed in (7). Furthermore, since we are directing the analysis165

to the homogeneous regions of a porous medium, we have regarded the volume fractions as166

6

constants. Finally, we have expressed the result in terms of both intrinsic averaged properties167

and their spatial deviations, the latter are defined from the decomposition (Gray, 1975)168

ψα = ψα + 〈ψα〉α (9)

The process of determining the functionality of the spatial deviations in terms of the169

corresponding averaged quantities is known as closure. The development of the formal170

solution of the closure problem is quite extensive and the details are available in Appendix171

A. The formal solutions of the closure problems for the deviations variables can be expressed172

as follows173

vα = Aα · 〈vβ〉β + Aαλ ·(〈vβ〉β − 〈vγ〉γ

)+ dα

(〈pβ〉β − 〈pγ〉γ

)+ fα, α, λ = β, γ, α �= λ

(10a)

pα = μα

[aα · 〈vβ〉β + aαλ ·

(〈vβ〉β − 〈vγ〉γ

)+ sα

(〈pβ〉β − 〈pγ〉γ

)+ ϕα

](10b)

where Aα, Aαλ, aα, aαλ, dα, fα, sα and ϕα (α = β, γ), are the so-called closure variables,174

which arise from the non-homogeneous terms involved in the boundary-value problem for175

the deviations variables. The closure variables solve the boundary-value problem given in176

Eqs. (A.8). The derivation of this problem requires considering the length-scale constraint177

given in (7) (see Appendix A for details).178

Substitution of eqs. (10a) and (10b) into eqs. (8a) and (8b) gives rise to the non-equilibrium (NE) model for the β-phase:

0 = −∇〈pβ〉β + ρβg + μβ∇2 〈vβ〉β + μβεβ(−Kne

ββ

)−1 · 〈vβ〉β + μβKneβγ ·(〈vβ〉β − 〈vγ〉γ

)+

μβuβ

(〈pβ〉β − 〈pγ〉γ

)+ μβhβ (11)

here, the effective-medium coefficients are defined as follows179

−εβ(Kne

ββ

)−1=

1

Vβ

∫Aβκ

nβκ · [−aβI+∇Aβ] dA+1

Vβ

∫Aβγ

nβγ · [−aβI+∇Aβ] dA (12a)

180

Kneβγ =

1

Vβ

∫Aβκ

nβκ · [−aβγI+∇Aβγ ] dA+1

Vβ

∫Aβγ

nβγ · [−aβγI+∇Aβγ ] dA (12b)

181

uβ =1

Vβ

∫Aβκ

nβκ · [−sβI+∇dβ] dA+1

Vβ

∫Aβγ

nβγ · [−sβI+∇dβ] dA (12c)

182

hβ =1

Vβ

∫Aβκ

nβκ · [−ϕβI+∇fβ] dA+1

Vβ

∫Aβγ

nβγ · [−ϕβI+∇fβ] dA (12d)

183

7

The closed form of the upscaled non-equilibrium model for the γ-phase is

0 = −∇〈pγ〉γ + ργg + μγ∇2 〈vγ〉γ + μγεγ(−Kne

γβ

)−1 · 〈vβ〉β + μγKneγγ ·(〈vβ〉β − 〈vγ〉γ

)+ μγuγ

(〈pβ〉β − 〈pγ〉γ

)+ μγhγ (13)

in this case, the effective-medium coefficients are defined as follows184

−εγ(Kne

γβ

)−1=

1

Vγ

∫Aβγ

nγβ · [−aγI+∇Aγ] dA (14a)

185

Kneγγ =

1

Vγ

∫Aβγ

nγβ · [−aγβI+∇Aγβ] dA (14b)

186

uγ =1

Vγ

∫Aβγ

nγβ · [−sγI+∇dγ] dA (14c)

187

hγ =1

Vγ

∫Aβγ

nγβ · [−ϕγI+∇fγ] dA (14d)

188

In this model, we notice that the first three terms in eqs. (11) and (13) are volume-189

averaged versions of the micro-scale Stokes’ equation and the rest are the result of the190

upscaling process. Regarding this upscaled model, the following comments are in order:191

• The main contributions from microscopic momentum transport (represented by the192

closure variables) are captured in the effective-medium coefficients, which are defined193

in terms of integrals (i.e., filters) of the closure variables. These coefficients are: 1) The194

permeability tensors Kneββ and Kne

γβ, which are defined in terms of the closure variables195

aα and Aα (α = β, γ). These closure variables arise from solving a closure problem196

in which the only source is located at the β − κ interface, just like in the case of197

the permeability tensor in Darcy’s law (see eqs. (4.2-27) in Whitaker, 1999). 2) The198

tensors Kneβγ and Kne

γγ , which may be interpreted as interfacial momentum transport199

coefficients, because they are defined in terms of the closure variables aαλ and Aαλ200

(α, λ = β, γ and α �= λ). These closure variables arise from solving a closure problem201

in which the source term is located in the velocity boundary condition at the fluid-202

fluid interface. 3) The vectors uα (α = β, γ), are similar to the previous ones but they203

account for the microscopic differences in stress at the fluid-fluid interface. 4) The204

vectors hα (α = β, γ), are the result of considering the capillary effects of curvature205

and superficial tension in the closure problems.206

• It is important to point out that the validity of eqs. (11) and (13) is bounded by207

the length-scale constraint given in (7). In addition, the application of the model is208

limited to non-coalescing emulsions, for this reason the flow is assumed quasi-static,209

8

and the volume fractions for the β and γ-phases are regarded as given parameters.210

This means that the predictions of the effective-medium coefficients in this model211

(or in the equilibrium model shown below) are only functions of the porosity and212

the saturation. This set of constraints and assumptions can be visualized as scaling213

postulates as suggested by Wood and Valdes-Parada (2013).214

• It should be stressed that the model derived here is different from the model derived215

by Whitaker (1986a) because of the last two terms in eqs. (11) and (13). Indeed, the216

pressure difference in the first term can be related to the average curvature of the217

fluid-fluid interface, according to (Whitaker, 1986a)218

−(〈pβ〉β − 〈pγ〉γ

)= 2σ 〈H〉βγ (15)

where the average curvature is defined as219

〈H〉βγ =1

Aβγ

∫Aβγ

H dA (16)

It should be stressed that Eq. (15) is valid under the capillary restriction, Ca � 1220

(Whitaker, 1994). Here the Capillary number is defined as Ca = μ〈v〉/σ, with μ〈v〉221

representing the magnitude of the largest of μα〈vα〉α (α = β, γ). Furthermore, in order222

for the mean curvature to undergo negligible variations within the averaging volume,223

it is necessary to impose that (Whitaker, 1994)224

Bo =(ρβ − ργ)g�

2

σ� 1 (17)

with Bo denoting the Bond number and � is a characteristic length representing the225

largest of �α (α = β, γ). Under these conditions, the contributions of these two terms226

become negligible as noted by Whitaker (1986a) and the non-equilibrium model can227

be arranged as228

0 = −∇〈pβ〉β+ρβg+μβ∇2 〈vβ〉β+μβ

[εβ(Kne

ββ

)−1+Kne

βγ

]·〈vβ〉β−μβK

neβγ ·〈vγ〉γ (18a)

229

0 = −∇〈pγ〉γ+ργg+μγ∇2 〈vγ〉γ+μγ

[εγ(Kne

γβ

)−1+Kne

γγ

]·〈vβ〉β−μγK

neγγ ·〈vγ〉γ (18b)

In order to recover Whitaker’s result, the following definitions are introduced[εβ(Kne

ββ

)−1+Kβγ

ne]=− εβK

−1β (19a)

−Kneβγ = εγK

−1β ·Kβγ (19b)[

εγ(Kne

γβ

)−1+Kne

γγ

]= εβK

−1γ ·Kγβ (19c)

−Kneγγ = − εγK

−1γ (19d)

9

In this way, after some manipulations, eqs. (18) can be expressed as230

εβ 〈vβ〉β = −Kβ

μβ

(∇〈pβ〉β + ρβg

)+Kβ∇2 〈vβ〉β +Kβγ · εγ 〈vγ〉γ (20a)

231

εγ 〈vγ〉γ = −Kγ

μγ(∇〈pγ〉γ + ργg) +Kγ∇2 〈vγ〉γ +Kγβ · εβ 〈vβ〉β (20b)

232

These equations correspond to the modified version of Darcy’s law with the Brinkman233

correction and the viscous drag indicated originally in previous works (Whitaker,234

1986a, 1994; Lasseux et al., 1996).235

Finally, it is worth remarking that this work is restricted to cases in which the γ-phase is236

dispersed in the β-phase and there is no triple-phase contact. This means that the predictions237

of the transport coefficients provided below are not extensible to the situations studied by238

Whitaker (1986a, 1994); Lasseux et al. (1996). However, to the best of our knowledge,239

no numerical solution to the closure problems stated in these works has been reported in240

the literature. We believe that the computations presented in the following paragraphs241

for dispersed flow in the local mechanical equilibrium model constitute a relevant advance242

towards the solution of the closure problems presented by Lasseux et al. (1996).243

4. Local mechanical equilibrium model244

The derivations presented in the previous section consist in two effective-medium equa-245

tions for each fluid phase in the system (eqs. 11 and 13). However, previous works for246

studying heat and mass transport in multiphase systems have suggested the use of local247

equilibrium models, which rely on a single weighted-averaged temperature or concentration248

to provide a reasonable description of the whole transport process (Whitaker, 1991; Quin-249

tard and Whitaker, 1995; Ochoa-Tapia et al., 1986; Wood and Whitaker, 1998). Certainly,250

this modeling approach is simpler and thus more attractive than the non-equilibrium model.251

However, the use of equilibrium models requires imposing additional assumptions aside from252

those involved in the averaging process. For situations in which these constraints are met,253

the comparison of the predictions from the equilibrium model, with those resulting from254

laboratory or numerical experiments yields good agreement (Quintard and Whitaker, 1993,255

1995). Furthermore, the use of equilibrium models is not restricted to the bulk of porous256

media and has been recently extended to the fluid-porous medium boundary as shown by257

Aguilar-Madera et al. (2011). In that work, the equilibrium model was shown to satis-258

factorily reproduce the predictions from both the non-equilibrium model and from direct259

numerical simulations. For momentum transport, equilibrium models have been proposed260

for the study of heterogeneous porous media as shown by Quintard and Whitaker (1988).261

Motivated from the above, in this work we propose a local mechanical equilibrium (LME)262

approach for modeling dispersed flow in homogeneous porous media. In this model, the263

10

dependent variables are written in terms of the following intrinsic average (hereafter denoted264

as the equilibrium average)265

〈ψ〉 = 1

Vf

∫Vf

ψdV =1

Vf

∫Vβ

ψβdV +1

Vf

∫Vγ

ψγdV, ψ = v, p (21)

here Vf = Vβ +Vγ, is the volume occupied by the fluid phases. At this point, it is opportune266

to define the porosity as ε = Vf/V and the volume fractions for each phase as εα = Vα/V ,267

so that Eq. (21) can be rearranged as268

ε 〈ψ〉 = εβ 〈ψβ〉β + εγ 〈ψγ〉γ = 〈ψβ〉+ 〈ψγ〉 (22)

On the basis of this definition, we may add the upscaled continuity equations for each269

phase [eqs. (6)] and obtain, without any further simplification, the following expression for270

the equilibrium mass transport model271

∇ · ε〈v〉 = 0 (23)

However, for momentum transport, the derivations of the equilibrium model is not as272

straightforward. We will follow a procedure previously used for studying heat transfer in273

porous media (see Whitaker, 1999, Chap. 2), as a first step towards this derivation, it is274

convenient to decompose the intrinsic averages of each phase in terms of the equilibrium275

average, 〈ψ〉, and the macroscopic deviations, ψ, as follows276

〈ψα〉α = 〈ψ〉+ ψα, ψ = v, p, α = β, γ (24)

In this way, we can derive the governing equation for 〈ψ〉 by substituting Eq. (24) into277

the non equilibrium model (eqs. (11) and (13)) and combine the resulting expressions278

accordingly to obtain279

0 = −∇〈p〉+ 〈ρ〉 g + 〈μ〉∇2 〈v〉 − μβ (Ke)−1 · ε 〈v〉+ 〈μ〉he +Φ (25)

here we introduced the definitions280

〈ρ〉 = Sβρβ + Sγργ (26a)

281

〈μ〉 = Sβμβ + Sγμγ (26b)282

μβ (Ke)−1 = Sβμβ

(Kne

ββ

)−1+ Sγμγ

(Kne

γβ

)−1(26c)

283

μβhe = Sβμβhβ + Sγμγhγ (26d)

In the above expressions, we used Sα = εα/ε, α = β, γ to represent the saturation for a284

given porosity and volume fraction of the α-phase. The term Φ in Eq. (25) includes all the285

terms involving the macroscopic deviations, ψα,286

11

Φ = ε−1 (μβ − μγ)SβSγ∇2(〈vβ〉β − 〈vγ〉γ

)+(μβSβK

neβγ + μγSγK

neγγ

) · (〈vβ〉β − 〈vγ〉γ)

(−μβSβ

(Kne

ββ

)−1+ μγSγ

(Kne

γβ

)−1)SβSγ ·

(〈vβ〉β − 〈vγ〉γ

)+

(μβSβuβ + μγSγuγ) ·(〈pβ〉β − 〈pγ〉γ

)(27)

Under the assumption of local mechanical equilibrium, we expect that Φ can be assumed287

negligible with respect to the other terms in Eq. (25). The assumptions and length-scale288

constraints required to determine the validity of the local mechanical equilibrium model are289

addressed in Appendix B. These conditions can be loosely summarized as follows:290

• One fluid phase is in much larger proportion than the other, for example, εγ � εβ.291

• The physical properties of both phases are alike, i.e., ρβ ∼ ργ and μβ ∼ μγ.292

• There is a disparity of characteristic lengths of the type �� L.293

Under the local mechanical equilibrium assumption, we may drop the Φ term in Eq. (25)294

and simplify it to obtain the following expression295

0 = −∇〈p〉+ 〈ρ〉 g + 〈μ〉∇2 〈v〉 − μβ (Ke)−1 · ε 〈v〉+ 〈μ〉he (28)

If the LME assumption is valid, and it is applied to the closure problem derived inAppendix A, the formal solution for the deviations variables given by Eq. (10) reduces to

vα = Aα · 〈v〉+ fα, α, λ = β, γ, α �= λ (29a)

pα = μα [aα · 〈v〉+ ϕα] (29b)

and the associated boundary-value problems are Problems I and IV from Table A.2, where296

the only sources will be 〈v〉 and H , respectively. To close the model of the LME model we297

have to solve these boundary value problems in order to compute the fields of the closure298

variables and thus obtain predictions of the effective tensors and vectors. The expressions299

for Ke and he in terms of closure variables are given in eqs. (A.11) and (A.12). The closure300

problems are solved in the unit cell for dispersed flow sketched in Fig. A.1. However, as301

mentioned above, in order to neglect significant variations of the interfacial curvature, within302

the averaging volume, it is necessary to impose that Ca� 1 and Bo� 1. As consequence,303

we have that304

〈μ〉he � μβ (Ke)−1 · ε 〈v〉 (30)

and thus, Eq. (28) reduces to305

ε 〈v〉 = −Ke

μβ· (∇〈p〉 − 〈ρ〉g) + 〈μ〉

μβKe · ∇2 〈v〉 (31)

12

This result can be further modified if the definition of Ke is related to the intrinsic306

permeability and a correction factor . We can introduce a new definition for the effective307

coefficient in terms of these contributions as308

(Ke)−1 = K−1 · (I+Km) (32)

where K is the intrinsic permeability coefficient defined by Whitaker (see Eq. (4.2-32) in309

Whitaker, 1999) and Km is the correction factor that takes into account the effect of the310

dispersed phase. As explained by Whitaker (1999), in order to compute K, it is necessary311

to solve a Stokes-like problem in a periodic unit cell (see eqs. (4.2-37) in Whitaker, 1999).312

Furthermore, in order to compute the values of Km, it is necessary to solve the closure313

problems defined in Appendix A. This correction factor is only influenced by the viscosity314

ratio and by the volume fraction occupied by each phase in the system. Examples of the315

solution of the closure problems are provided in the following section along with a parametric316

study of the functionality of the effective medium coefficients.317

Substitution of Eq. (32) into Eq. (31), gives rise to the following form of the upscaled318

model that resembles to an extension of the Darcy-Brinkman model319

ε 〈v〉 = − K

μβ· (∇〈p〉 − 〈ρ〉 g) + 〈μ〉

μβK · ∇2 〈v〉 −Km · ε 〈v〉 (33)

In this expression, the first term on the right-hand side represents the Darcy term,320

the second one is the Brinkman correction term and finally the effect of the dispersed γ-321

phase is accounted in the last term. As explained by Ochoa-Tapia and Whitaker (1995),322

the Brinkman correction term is crucial in the study of transport phenomena near the323

boundaries of a porous medium. However, in the porous medium bulk, this term can be324

safely discarded. Under these conditions, Eq. (33) can be reduced to325

ε 〈v〉 = − K

μβ· (∇〈p〉 − 〈ρ〉 g)−Km · ε 〈v〉 (34)

Or, in terms of Ke,326

ε 〈v〉 = −Ke

μβ

· (∇〈p〉 − 〈ρ〉 g) (35)

Notice that, although the structure of the above expression resembles to Darcy’s law, the ef-327

fective medium coefficient, Ke, does not, in general, correspond to the intrinsic permeability,328

K. As a matter of fact, from Eq. (32), it can be deduced that (Ke)−1 ≈ K−1, only for condi-329

tions in which Km � I. These conditions will be discussed in the following paragraphs. As330

mentioned in the introduction, Darcy’s law has been largely used in the literature to model331

multiphase flow in porous media (Alvarado and Marsden, 1979; Soo and Radke, 1984; Hof-332

man and Stein, 1991; Islam and Ali, 1994; Jain and Demond, 2002; Romero et al., 2011;333

Nogueira et al., 2013). The effective-medium coefficient involved in this upscaled model is334

usually predicted from best-fits of experimental data from emulsion flow in porous media335

and, more recently, in microchannels flow (e.g. Cobos et al., 2009). In this work, Ke is336

predicted from the solution of the associated closure problems in a representative unit cell,337

as detailed below.338

13

5. Prediction of the effective coefficients339

5.1. Parametric analysis340

In this section, we analyze the parametric dependence of the effective transport coef-341

ficients involved in the LME model with the main degrees of freedom. These are: the342

saturation (Sα = Vα/Vf = εα/ε), the viscosity ratio ν = μβ/μγ and the void fraction ε.343

The solution of the closure problems and the computations of the effective parameters were344

performed using the commercial finite element solver Comsol MultiphysicsTM (version 4.3a).345

The default element types and solvers were used in all simulations. In addition, adaptive346

mesh refinements algorithms were applied in order to ensure that the results were indepen-347

dent of the computational elements. All closure problems were solved in the 2D unit cells348

illustrated in Fig. A.1. In this model of the microscale geometry, the γ-phase is represented349

by a series of circular particles with five different diameters, rγi, i = 1, · · · , 5 (details are350

presented in Appendix A). Certainly, we are assuming flow conditions in which the droplets351

do not modify their shape, size and position. The latter was conveniently fixed in order352

to satisfy the saturation values. The κ-phase is represented by a circular particle with the353

radius determined by the value of the void fraction; rκ =√(1− ε) /π. The closure prob-354

lem solution was carried out in order to obtain the longitudinal component of the effective355

transport coefficients (i.e., the x-direction). Examples of the fields of longitudinal closure356

variables (Aα)xx , α = β, γ are presented in Fig. 2.357

Once the solution fields of the closure variables are obtained, they are substituted into358

Eq. (A.11) that defines the effective coefficient for LEM model given in Eq. (35). In Fig. 3359

we show the dependence of the longitudinal component of the effective coefficient (Ke = Kexx)360

involved in the LEM model, with the main parameters (ε, ν and Sβ). Analyzing these results,361

the following comments are in order:362

• As expected from previous studies about one-phase flow in porous media, Ke increases363

with the void fraction. As a matter of fact, for the case in which ν = 1, the results364

reproduce those from the intrinsic permeability. In our computations we had to limit365

the porosity values to be larger than 0.5, otherwise the largest drops tend to have a366

diameter that is almost equal to the pore diameter.367

• We observe in Fig. 3 that, if the volume in the pore is occupied by the continuous phase368

in a minor proportion with respect to the dispersed phase (Sβ = 0.35), the predictions369

are more sensitive to changes with ν than when the pore volume is occupied in major370

proportion by the continuous phase (Sβ = 0.8). Regarding the influence of ν, we371

note that Ke increases with this parameter. This is to be expected from the previous372

analysis of the fields in Fig. 2; especially as Sβ decreases, which translates in a larger373

volume fraction of the dispersed phase.374

• In Figs. 3a,b, we observe that, for ν > 1 (μβ > μγ), the values of Ke are larger than375

the ones corresponding to the intrinsic permeability given by ν = 1. Furthermore,376

for ν < 1 (μβ < μγ), the predictions of Ke are smaller than K, and Ke decreases as ν377

decreases (Figs. 3c,d).378

14

It should be noted that the analysis of the results provided above does not show very379

clearly the influence of each phase in the effective medium coefficient Ke. To gain a better380

understanding of the role played by the continuous and the dispersed phases, it is convenient381

to direct the attention to Eq. (32), in which Ke is expressed in function of the intrinsic382

permeability and a correction factor Km. From this expression, we observe on the one hand383

that if Km > 0, it turns out that Ke < K, which means that the dispersed phase hinders384

the overall momentum transport. On the other hand, for situations in which −1 < Km < 0,385

it results in the effective coefficient being larger than the intrinsic permeability, i.e., an386

enhanced momentum transport.387

Using the solution fields of the closure involved in the computation of Ke (eqs. A.8a -388

A.8i) and solving the corresponding boundary value problem to obtain K (see Eq. (4.2-32)389

in Whitaker, 1999) in the same unit cell shown in Fig A.1, we can compute Km = (Km)xx.390

The predictions of Km are plotted in Fig. 4 as a function of ε for different values of ν taking391

Sβ = 0.8. We provide the following comments regarding the results shown in the Fig. 4:392

• For ν > 1, the negative values of Km results in Ke > K. Therefore, as the ν increases,393

the effective coefficient Ke increases and, as consequence, the global dispersed flow394

is enhanced by the presence of the dispersed γ-phase. Although the results are only395

presented for one saturation value, from the results in Fig. 3, it is appealing to think396

that if Sβ increases, the volume of the dispersed phase decreases and the enhanced397

effect is reduced. At this point, it is worth recalling that the assumptions adopted so398

far lead to neglecting the influence of the interfacial tension between the continuous399

and dispersed phases, thus the viscosity ratio and the volume occupied by each phase400

are the main parameters that determine the values of the correction factor.401

• In the case of ν = 1, the effect of the γ-phase over the effective coefficient can be402

neglected because Km � I, which results in values of Ke ≈ K. The presence of the403

both immiscible phases in the homogeneous porous media is similar to one-phase flow.404

• In contrast, the values Km are positive for ν < 1, therefore Ke < K. This results suggest405

the global flow is detracted by the presence of the dispersed fluid where the γ-phase406

is acting like “drag particles”. Therefore, reduction the Sβ increase the volume of this407

drag particles as consequence the drag will be increased.408

To conclude this section, we provide a validation of the effective coefficient predictions409

with experimental data obtained from the literature. This analysis is presented in the fol-410

lowing paragraphs.411

5.2. Comparison with experiments412

In this section, the prediction of the effective coefficient for the LME model, Ke, are413

compared with the experimental data reported by Soo and Radke (1984); Jain and Demond414

(2002). The experimental data reported by Soo and Radke (1984) were obtained in a system415

where a dilute emulsion of known concentration and drop-size distribution is slowly flowed416

to a sandpack of known pore size distribution and permeability under a constant volumetric417

15

flow rate. The permeability reduction is determined by changes in the pressure drop and418

the reduction of drops concentration after the flush. The conditions for several experiments419

reported by Soo and Radke (1984) are presented in Table 1.420

The effective coefficient for the LME model is computed using the solution of the closure421

problem in the unit cell shown in Fig. 5 that attempts to capture the essential geometric422

information obtained by the experimental system described above. In Fig. 5, the sand grain423

of the porous media are represented by circles with radius: rκ =√

(1− ε)/2π and the pore424

diameter is: dp =(√

1/2)�c − 2rκ. For given values of Sγ and ε, the number of circles425

needed to represent the dispersed γ-phase, Nγ , is computed by: Nγ = Sγε/ (πrγ) using the426

data in Table 1. The localization of the circles was chosen to be near the pore throats on427

the basis of previous studies (Soo and Radke, 1984) that concluded that the dispersed phase428

reduces the pore throats diameter by the deposition and interception of the droplets. It429

should be stressed that in the experiments by Soo and Radke (1984), the droplets were not430

monodispersed. However, the droplets sizes did not exhibit large deviations from the average431

drop size as shown in Fig. 3 of Soo and Radke (1984). For this reason, the droplet size, 2rγ,432

used in the unit cell shown in Fig. 5 corresponded to the average droplet size reported by433

Soo and Radke (1984). In addition, note that, in all the experimental data studied here, the434

values of 2rγ were smaller than the pore diameter in the unit cell, dp, as shown in Table 1.435

In Fig. 6 we present the comparison between the experimental data of Soo and Radke436

(1984) and the predicted coefficients for the LME model obtained by the closure problem437

solution illustrated in Fig. 5. The predictions were compared with several emulsified systems,438

which included variations in the viscosity of γ - phase, as well as variations in the pore439

diameter and drop size, as shown in Table 1. To have a more qualitative perspective of the440

predictive capabilites of the LME model, in Fig. 6b we plot the computed relative error441

percent between the theoretical predictions and the experimental data. We observe that,442

except for the last experimental value in experiment 1, all the results exhibit a relative error443

percent that is below 7%, which may be considered acceptable in many situations. It should444

be stressed that the predictions of the experimental data showed a strong dependency with445

the geometric parameters and, for this reason, the largest errors were found for the higher446

values of 1− Sβ.447

The predictions of the effective-medium coefficients involved in the LME model are com-448

pared with the experimental data reported by Jain and Demond (2002) in Fig. 7. The449

experimental methodology is essentially the same described above and the parameters used450

for the simulations are provided in Table 1. In this case, the range of saturation values451

is larger and, not surprisingly, the relative error percent between theory and experiments452

increases up to 15 % for the smallest value of Sβ . This increase of the error percent with453

1−Sβ can be associated with the increases in the number of particles necessary to complete454

the saturation and to the set of assumptions involved in the derivation of the model, among455

which is the assumption that there is no interaction between the dispersed phase and the456

solid. In addition, more elaborated geometries can be used in the unit cell, which may457

provide better agreement with the experimental data. However, we believe that the simple458

geometries used in this work, as a first approach, yield to reasonable agreement with the459

16

experimental data.460

6. Discussion and conclusions461

In this work, we applied the method of volume averaging to obtain an upscaled model462

for modeling immiscible two-phase dispersed flow in homogeneous porous media. We have463

considered the conditions of stable-steady emulsion flow in the microscale model. We ob-464

tained closed macroscopic mass and momentum transport equations that, under the local465

mechanical equilibrium hypothesis can be applied to both continuous and dispersed phases.466

In this model, the momentum transport equation is written in terms of effective-medium467

coefficients that incorporate the microscopic geometry effects and the momentum transport468

between both phases. These coefficients can be predicted from the solution of the associated469

closure problem in representative periodic unit cells that capture the essential features of470

the pores scale. In this way, we found conditions in which the presence of the dispersed471

phase enhances or hinders the macroscopic momentum transport.472

It should be noted that the local mechanical equilibrium, despite being a relatively473

simple approach, contains more restrictions than its non-equilibrium counterpart. The non-474

equilibrium model is in agreement with previous applications of the volume averaging method475

by Whitaker (1986b), where the resulting model is a set of macroscopic equations, one for476

each phase, that resemble Darcy’s law with an extra term that is related to the interaction477

between both phases (see eqs. 20). On the other hand, the equilibrium model, as expressed478

in Eq. (35), can be written in a way that resembles Darcy’s law. However, the coefficient Ke479

is not intrinsic, in general, because it depends of the phases interaction. For this reason, it is480

more convenient to use Eq. (32) and express the equilibrium model in the form given in Eq.481

(34), which contains the permeability tensor K involved in one-phase flow and a correction482

factor accounting for the interaction between the phases. Indeed, more research is needed483

and an exhaustive comparison of the equilibrium and non equilibrium formulations should484

be addressed in a future work.485

The results of our parametric analysis evidence that, for conditions in which μβ > μγ486

(i.e., ν > 1) momentum transport is enhanced and thus Km < 0 and the opposite is true.487

This observation is relevant because in many EOR or soil remediation applications (Ouyang488

et al., 1995; Jain and Demond, 2002; McAuliffe, 1973) it is desirable to enhance global flow.489

In this way, for conditions in which the assumptions supporting the LME model, one may490

predict operational conditions in which the transport can be enhanced and thus look for491

optimal operation conditions.492

The relatively good agreement between the theoretical predictions and experimental493

data encourages us to believe that the LME approach may be reasonable in applications in494

which the saturation Sβ is large. We attribute the largest deviations between theory and495

experiments to the geometrical limitations in the unit cell used here and to the length scale496

constraints and assumptions involved in the derivation of the model. Briefly, we assumed497

that there is no contact between the dispersed phase and the solid phase, that the influences498

of curvature and interfacial tension are negligible and that there should be a separation499

of characteristic scales between the pore scale and the macroscale. In future works we500

17

shall expand the range of applicability of the model derived here by relaxing some of these501

assumptions. Nevertheless, the relative error in most of the results remained below 10%502

with respect to experimental data. We can thus conclude that, in general, the predictions503

exhibit acceptable agreement with the experimental data examined here.504

In future works we will investigate the use of more complicated unit cells, in particular, it505

would be enlightening to use three-dimensional domains with more realistic representations506

of the porous matrix and of the dispersed phase location and size distribution. Another507

topic that should be investigated is the possibility to consider droplets’ coalescence and508

their interaction with the solid phase. These and other extensions of the current research509

will be explored in future work.510

Appendix A. Closure problem for NLME511

The objective in this section is to present the details in relation with the functionality of512

the spatial deviations and the average quantities in order to close the macroscopic model.513

From Eq. (9), we have that the spatial deviations of any quantity are: ψα = ψα − 〈ψα〉α.514

Directing the attention to mass transport, it is convenient to write Eq. (6) in terms of515

intrinsic averages as follows516

∇ · 〈vα〉α = 0, α = β, γ (A.1)

here we have taken into account the assumption that the porous medium is homogeneous517

and, consequently, εα is treated as a constant. Subtracting Eq. (A.1) to Eq. (1a), gives rise518

to519

∇ · vα = 0 in the α-phase, α = β, γ (A.2)

For momentum transport, the governing equations for the deviations result from sub-520

tracting eqs. (8) and (1b) and can be written as521

0 = −∇pβ + μβ∇2vβ − 1

Vβ

∫Aβκ

nβκ · [−pβI+ μβ∇vβ ] dA− 1

Vβ

∫Aβγ

nβγ · [−pβI+ μβ∇vβ ] dA

in the β-phase (A.3a)

522

0 = −∇pγ + μγ∇2vγ − 1

Vγ

∫Aβγ

nγβ · [−pγI+ μγ∇vγ ] dA, in the γ-phase (A.3b)

523

The closure problem is completed by the following interfacial boundary conditions that524

arise from substituting the spatial decompositions for the pressure and velocity into eqs.525

(1c)-(1e)526

18

vβ = vγ −(〈vβ〉β − 〈vγ〉γ

), at the β-γ interface (A.4a)

nβγ ·[−pβI+ μβ

(∇vβ +∇vTβ

)]=nβγ ·

[−pγI+ μγ

(∇vγ +∇vTγ

)]+ nβγ

(〈pβ〉β − 〈pγ〉γ

)+ 2σHnβγ, at the β-γ interface

(A.4b)

vβ = −〈vβ〉β , at the β-κ interface (A.4c)

Notice that, in Eq. (A.4b) we have followed Whitaker (1994) and neglected the contribution527

of the macroscopic viscous terms. This assumption is justified by the length-scale constraint528

�α � L (α = β, γ). In addition, the fields of the deviations are bounded by the following529

average constraint530 ⟨ψα

⟩α= 0, ψ = p,v, α = β, γ (A.5)

Finally, following previous approaches (e.g., Whitaker, 1986a), the closure problem is531

solved in a periodic representative domain of the microscale (i.e., a unit cell). This motivates532

imposing the following boundary conditions at the entrances and exits of the unit cell.533

ψα (r+ li) = ψα (r) , ψ = p,v; α = β, γ; i = 1, 2, 3 (A.6)

From this statement of the closure problem, we identify the sources 〈vβ〉β,(〈vβ〉β − 〈vγ〉γ

),(

〈pβ〉β − 〈pγ〉γ)and 2σH . Given the linear nature of this boundary-value problem, we can

propose a formal solution in terms of the sources as follows

vα = Aα · 〈vβ〉β + Aαλ ·(〈vβ〉β − 〈vγ〉γ

)+ dα

(〈pβ〉β − 〈pγ〉γ

)+ fα, α, λ = β, γ, α �= λ

(A.7a)

pα = μα

[aα · 〈vβ〉β + aαλ ·

(〈vβ〉β − 〈vγ〉γ

)+ sα

(〈pβ〉β − 〈pγ〉γ

)+ ϕα

](A.7b)

534

In equations (A.7a) and (A.7b), the variables Aα, Aαλ, aα, aαλ, dα, fα, sα and ϕα are535

known as the closure variables. Indeed, one may rearrange the first two terms on the right536

hand side of the above expressions to be proportional to the intrinsic averaged velocity in each537

phase; however, we find it more convenient to express the formal closure problem solution538

in its current form because, under equilibrium conditions, the terms involving differences539

of volume averaged quantities are null and eqs. (A.7) reduce to eqs. (29). Each closure540

variable can be conceived as a mapping function of a source onto the deviations fields. In541

this way, for example, dα maps the difference of the intrinsic averaged pressures onto vα;542

whereas fα and ϕα map H and μαH onto vα and pα, respectively.543

Substitution of the formal solution given by eqs. (A.7) into eqs. (A.2)-(A.5) and (A.6),544

gives rise to four closure problems. For the sake of brevity in presentation, these problems545

are compacted into the following one546

19

∇ · Zβ = 0 in the β-phase (A.8a)

−∇ζβ +∇2 Zβ = Δβ in the β-phase (A.8b)

Zβ = Zγ +ΔIβγ , at the β-γ interface (A.8c)

μβ

[−ζβI+(∇Zβ +∇ZT

β

)] · nβγ = μγ

[−ζγI+(∇Zγ +∇ZT

γ

)] · nβγ

+ΔIIβγ , at the β-γ interface (A.8d)

Zβ = Δβκ, at the β-κ interface (A.8e)

∇ · Zγ = 0 in the γ-phase (A.8f)

−∇ζβ +∇2 Zβ = Δγ in the γ-phase (A.8g)

Zα (r+ li) = Zα (r) , i = 1, 2, 3 (A.8h)

〈Zα〉α = 0 (A.8i)

where Zα and ζα represent the closure variables and the terms ΔI,IIβγ and Δβκ represent the547

non-homogeneous terms according to Table A.1. In addition, in eqs. (A.8b) and (A.8g), we548

have denoted the integral terms as follows549

Δβ =1

Vβ

∫Aβκ

nβκ ·[−ζβI+∇Zβ

]dA+

1

Vβ

∫Aβγ

nβγ ·[−ζβI+∇Zβ

]dA (A.9a)

Δγ =1

Vγ

∫Aβγ

nγβ ·[−ζγI+∇Zγ

]dA (A.9b)

The domain of solution of these closure problems are periodic unit cells that reasonably550

represent the complicated pore structure and the distribution of the dispersed phase. An551

example of a 2-D unit cell used in this work is sketched in Figure A.1; here we represent552

the κ-phase as a circle of radius rκ =√

(1− ε) /π. The radii of the circles representing the553

γ-phase, rγ1, · · · , rγ5 are related to each other as: rγi = λirγ1 (i = 2, · · · , 5) and554

rγ1 =√Sγε/

{π(4 + 4 (λ2)

2 + 8 (λ3)2 + 8 (λ4)

2 + 4 (λ5)2]} (A.10)

The values of λi, i = 2, · · · , 5 are provided in Table A.2 for each porosity used in this work.555

The position of these circles was chosen in a way that the assumptions of non-coalescence556

and no contact with the solid phase are met.557

To conclude this section, we analyze the expression of the effective-medium coefficients558

involved in the LEM, Eq. (26c) and Eq-(26d) in terms of the corresponding closure variables559

in order to obtain an equivalent definition that expresses the intrinsic permeability and the560

relative effect of the dispersed flow contributions. From the Table A.1 we can define the561

effective coefficients as follows562

20

εμβ (Ke)−1 = μβ

1

Vf

∫Aβκ

nβκ · [−aβI+∇Aβ] dA+ μβ1

Vf

∫Aβγ

nβγ · [−aβI+∇Aβ] dA

+ μγ1

Vf

∫Aβγ

nγβ · [−aγI+∇Aγ ] dA (A.11)

μβhe = μβ

1

Vf

∫Aβγ

nβγ · [−ϕβI+∇fβ] dA+ μγ1

Vf

∫Aβγ

nγβ · [−ϕγI+∇fγ] dA (A.12)

Using the boundary condition at the β−γ interface given by the Eq. (A.8d) under LME563

conditions, (i.e., ΔIIβγ = 0 and ΔII

βγ = 2Hσnβσ) we have564

μβ

[−aβI+(∇Aβ +∇AT

β

)] · nβγ = μγ

[−aγI+(∇Aγ +∇AT

γ

)] · nβγ

and565

μβ

[−ϕβI+(∇fβ +∇fTβ

)] · nβγ = μγ

[−ϕγI+(∇fγ +∇fTγ

)] · nβγ + 2Hσnβσ

Integrating these boundary conditions along the fluid-fluid interface and taking into566

account the solenoidal nature of the fields of the closure variables Aβ, Aγ, fβ and fγ , leads567

to the following identities568

μβ1

Vf

∫Aβγ

nβγ · [−aβI+∇Aβ] dA = −μγ1

Vf

∫Aβγ

nγβ · [−aγI+∇Aγ ] dA (A.13)

569

μβ1

Vf

∫Aβγ

nβγ · [−ϕβI+∇fβ] dA = −μγ1

Vf

∫Aβγ

nγβ · [−aγI +∇Aγ] dA (A.14)

Under these conditions, we may simplify eqs.(A.11) and (A.12) to570

−ε (Ke)−1 =1

Vf

∫Aβκ

nβκ · [−aβI+∇Aβ ] dA (A.15)

571

he = − 1

Vf

∫Aβγ

2HσnβγdA (A.16)

21

Appendix B. Constraints for the local mechanical equilibrium572

This part of the paper is dedicated to derive the length-scale constraints supporting the573

assumption of local mechanical equilibrium. As a first step towards this goal, let us use the574

macroscopic decomposition given by Eq. (24) and the definition of the intrinsic macroscopic575

properties expressed in Eq. (22) in order to obtain576

εβψβ = (εβ − ε)ψγ =εβ (ε− εβ)

ε

(〈ψβ〉β − 〈ψγ〉γ

)(B.1)

Certainly, for conditions in which(〈ψβ〉β − 〈ψγ〉γ

)= 0, it follows from the above result that577

the macroscopic deviations will be zero. However, this is rarely the case. In practice, the578

main assumption behind the local mechanical equilibrium is the following579

〈ψβ〉β ≈ 〈ψ〉 ≈ 〈ψγ〉γ , local mechanical equilibrium is valid (B.2)

From Eq. (25), we notice that, whenever the following assumptions are reasonable580

ε−1 (μβ − μγ)SβSγ∇2(〈vβ〉β − 〈vγ〉γ

)� 〈μ〉 (Ke)−1 · ε 〈v〉 (B.3a)

581 [(−μβSβKββ−1 + μγSγKγβ

−1)SβSγ + μβSβKβγ + μγSγKγγ

]·(〈vβ〉β − 〈vγ〉γ)� 〈μ〉 (Ke)−1·ε 〈v〉

(B.3b)582

(μβSβuβ + μγSγuγ) ·(〈pβ〉β − 〈pγ〉γ

)� 〈μ〉 (Ke)−1 · ε 〈v〉 (B.3c)

we can discard the Φ term in Eq. (25) and thus obtain an equilibrium model. An order of583

magnitude analysis in (B.3), gives rise to the following inequalities584

ε−1 (μβ − μγ)SβSγ

ε 〈μ〉 (Ke)−1Lv1Lv

(〈vβ〉β − 〈vγ〉γ

〈v〉

)� 1 (B.4a)

585 (−μβSβK−1ββ + μγSγK

−1γβ

)SβSγ + μβSβKβγ + μγSγKγγ

ε 〈μ〉 (Ke)−1

(〈vβ〉β − 〈vγ〉γ

〈v〉

)� 1 (B.4b)

586

(μβSβuβ + μγSγuγ)

ε 〈μ〉 (Ke)−1

(〈pβ〉β − 〈pγ〉γ

〈v〉

)� 1 (B.4c)

in which Kββ, Kγβ , Kβγ and Kγγ ,are the norms of Kββ, Kγβ, Kβγ and Kγγ respectively. In587

(B.4a) we have assumed that the length scales for ∇〈v〉 and 〈v〉 are Lv1 and Lv, respectively.588

From the above inequalities it is clear that it is necessary to have estimates of the differences589

〈vβ〉β − 〈vγ〉γ and 〈pβ〉β − 〈pγ〉γ in order to derive the length-scale constraints supporting590

the local mechanical equilibrium assumption. With this in mind, let us subtract Eq. (13)591

to Eq. (11) to obtain592

22

0 = −∇(〈pβ〉β − 〈pγ〉γ

)+ (ρβ − ργ)g + μβ∇2 〈vβ〉β − μγ∇2 〈vγ〉γ

+(μβεβ (−Kββ)

−1 − μγεγ (−Kγβ)−1) · 〈vβ〉β + (μβKβγ − μγKγγ) ·

(〈vβ〉β − 〈vγ〉γ

)+

(μβuβ − μγuγ)(〈pβ〉β − 〈pγ〉γ

)+ (μβhβ − μγhγ) (B.5)

At this point, it is convenient to notice that a proper combination of eqs. (B.1) and (24)593

leads to the following relation594

〈ψα〉α = 〈ψ〉+ ελε

(〈ψβ〉β − 〈ψγ〉γ

), α, λ = β, γ, α �= λ, ψ = v, p (B.6)

Substituting the above expression into Eq. (B.5) and after some algebraic manipulations595

we can eventually obtain596

∇(〈pβ〉β − 〈pγ〉γ

)μβγ

− eβγμβγ

(〈pβ〉β − 〈pγ〉γ

)−∇2

(〈vβ〉β − 〈vγ〉γ

)−

(SγCβγ +Dβγ) ·(〈vβ〉β − 〈vγ〉γ

)− (ρβ − ργ)

μβγg − hβγ =

(μβ − μγ)

μβγ∇2 〈v〉+ Cβγ · 〈v〉

(B.7)

Here, we introduced the following definitions597

μβγ = μβSγ − μγSβ (B.8a)

598

Cβγ =μβεβ (−Kββ)

−1 − μγεγ (−Kγβ)−1

μβγ(B.8b)

599

Dβγ =μβKβγ − μγKγγ

μβγ(B.8c)

600

eβγ = μβuβ − μγuγ (B.8d)601

hβγ = μβhβ − μγhγ (B.8e)

From Eq. (B.7), it can be demonstrated that, for cases in which the volumetric fraction of602

one phase is much larger than the other (for example, if εγ εβ) or if the properties of603

both fluid phases are equal, the local mechanical equilibrium assumption is automatically604

satisfied. However, we are interested in cases in which the above conditions may not be605

met. For this reason, in the following paragraphs we will perform an analysis of orders of606

magnitude in order to estimate the ratio(〈vβ〉β − 〈vγ〉γ

)/ 〈v〉. We begin our derivations by607

estimating the order of magnitude of each term on the left-hand side of Eq. (B.7)608

23

∇(〈pβ〉β − 〈pγ〉γ

)μβγ

= O

(〈pβ〉β − 〈pγ〉γ

μβγL

)(B.9a)

eβγμβγ

(〈pβ〉β − 〈pγ〉γ

)= O

(〈pβ〉β − 〈pγ〉γ

μβγ�

)(B.9b)

∇2(〈vβ〉β − 〈vγ〉γ

)= O

(〈vβ〉β − 〈vγ〉γ

L2

)(B.9c)

(SγCβγ +Dβγ) ·(〈vβ〉β − 〈vγ〉γ

)=⎧⎨

⎩O

⎡⎣Sγ

(〈pβ〉β − 〈pγ〉γ

)Lμβγ

⎤⎦+O

(1

�2

)⎫⎬⎭(〈vβ〉β − 〈vγ〉γ

)(B.9d)

−(ρβ − ργ) g

μβγ− hβγ = O

(〈pβ〉β − 〈pγ〉γ

μβγL

)(B.9e)

For the sake of simplicity, we have taken L to represent the characteristic length associated to609

the spatial variations of any volume-averaged quantity, whereas � denotes the characteristic610

length related to the changes of properties defined at the microscale. In addition, the order611

of magnitude for the tensor Cβγ was obtained from the definition given in Eq. (B.8b) and612

the NLME model (eqs. (11) and (13)). Assuming that the pressure gradient term and the613

term that includes the effective coefficients (−Kββ)−1 and (−Kγβ)

−1 are of the same order,614

we have615

O(−K−1

ββ

)= O

(∇〈pβ〉βμβεβ 〈vβ〉β

), O

(−K−1γβ

)= O

( ∇〈pγ〉γμβεβ 〈vγ〉γ

)(B.10)

Furthermore, for conditions in which the estimate O(〈vβ〉β

)= O (〈v〉) is reasonable, it616

follows that617

Cβγ = O

(〈pβ〉β − 〈pγ〉γLμβγ 〈v〉

)(B.11)

To advance in our derivations, it is convenient to write the complete expressions for Dβγ,

24

eβγ and hβγ in terms of the corresponding closure variables,

Dβγ =μβε

−1β

μβγ

⎛⎜⎝ 1

V

∫Aβκ

nβκ · [−aβγI+∇Aβγ ] dA+1

V

∫Aβγ

nβγ · [−aβγI+∇Aβγ] dA

⎞⎟⎠−

μγε−1γ

μβγ

⎛⎜⎝ 1

V

∫Aβγ

nγβ · [−aγβI+∇Aγβ] dA

⎞⎟⎠ (B.12a)

eβγ = μβε−1β

⎛⎜⎝ 1

V

∫Aβκ

nβκ · [−sβI+∇dβ] dA+1

V

∫Aβγ

nβγ · [−sβI+∇dβ ] dA

⎞⎟⎠−

μγε−1γ

⎛⎜⎝ 1

V

∫Aβγ

nγβ · [−sγI+∇dγ ] dA

⎞⎟⎠ (B.12b)

hβγ = μβε−1β

⎛⎜⎝ 1

V

∫Aβκ

nβκ · [−ϕβI +∇fβ] dA+1

V

∫Aβγ

nβγ · [−ϕβI+∇fβ ] dA

⎞⎟⎠−

μγε−1γ

⎛⎜⎝ 1

V

∫Aβγ

nγβ · [−ϕγI+∇fγ] dA

⎞⎟⎠ (B.12c)

Therefore, the order of magnitude estimates for Dβγ , eβγ and hβγ can be expressed as618

O (Dβγ) = O

[μβε

−1β

μβγ

1

�2,μγε

−1γ

μβγ

1

�2

](B.13a)

619

O (eβγ) = O

[μβε

−1β

1

μβ�, μγε

−1γ

1

μγ�

](B.13b)

620

O (hβγ) = O

[μβε

−1β

σ

μβ�2, μγε

−1γ

σ

μγ�2

](B.13c)

here, we have taken into account the derivations provided in Appendix A in order to write621

the following estimates622

Aβγ = O(1), Aγβ = O(1), aβγ = O

(1

�

), aγβ = O

(1

�

)(B.14a)

25

623

dβ = O

(�

μβ

), dβ = O

(�γμγ

)sβ = O

(1

μβ

), sγ = O

(1

μγ

)(B.14b)

624

fβ = O

(σ

μβ

), fβ = O

(σγμγ

)ϕβ = O

(σ

μβ�

), ϕγ = O

(σ

μγ�

)(B.14c)

We continue our derivations by examining the right-hand side of Eq. (B.7) and providing625

the estimate626

(μβ − μγ)

μβγ∇2 〈v〉 = O

(〈v〉L2

)(B.15a)

Cβγ · 〈v〉 = O

[(〈pβ〉β − 〈pγ〉γ

Lμβγ

)](B.15b)

On the other hand, we can estimate the pressure difference from eqs. (11) and (13) to obtain627

O(〈pβ〉β − 〈pγ〉γ

)= O

⎡⎣μβγ

(〈vβ〉β − 〈vγ〉γ

)�

⎤⎦ (B.16)

Substitution of the estimates for the left- and right-hand side of Eq. (B.7), gives rise,628

after some manipulation, to the following expression, which is applicable at the first order629

for〈vβ〉β−〈vγ〉γ

〈v〉 ,630

〈vβ〉β − 〈vγ〉γ〈v〉 = O

(�

L

)2{

1

1 +O(�L

)+O

(�L

)2}

(B.17)

here, the terms with the same order of magnitude have been collapsed into one term (eqs.631

B.9a, B.9e and B.15b). For simplicity, we have replaced Lv, Lv1 and Lp with L. Furthermore,632

under the assumption633

〈vβ〉β − 〈vγ〉γ〈v〉 � L

�(B.18)

and using the length scale constraint �� L we have634

〈vβ〉β − 〈vγ〉γ〈v〉 = O

(�

L

)2

� 1 (B.19)

which is the desired constraint for local mechanical equilibrium assumption to be valid.635

26

Nomenclature636

Aβγ continuous-dispersed phases interface, m2

Aβκ continuous-solid phase interface, m2

Aα closure variables tensors that map 〈vβ〉β onto vα

Aαλ closure variables tensors that map 〈vβ〉β − 〈vγ〉γ onto vα

aα closure variables vectors that map μα 〈vβ〉β onto pα, m−1

aαλ closure variables vectors that map μα

(〈vβ〉β − 〈vγ〉γ

)onto pα, m

−1

dα closure variables vectors that maps 〈pβ〉β − 〈pγ〉β onto vα, m s−1Pa−1

dp pore diameter in the unit cell (=(√

1/2)�c − 2rκ)

fα closure variables vectors that maps H onto vα, ms−1

g gravitational acceleration vector, ms−2

H mean curvature, m−1

hα effective vector, Pa m−1

he effective vector, Pa m−1

I unit tensorK Darcy’s permeability tensor, m2

Ke effective tensor, m2

Km effective tensorKne

αα effective tensor, (α = β, γ) m2

Kneαλ effective tensor, (α, λ = β, γ, α �= λ) m−2

�c width of the unit cell, m�α characteristic length for the α-phase, (α = β, γ) m�p small scale representation of the mean pore diameter, mL characteristic length associated with volume averaged quantities, mNγ number of circles needed to represent the dispersed γ-phase in the unit cellnβκ unit normal vector pointing from the β-phase towards the κ-phase.nβγ unit normal vector pointing from the β-phase towards the γ-phase (= nγβ)pα pressure in the α-phase, (α = β, γ), Pa〈pα〉 superficial averaged pressure in the α-phase, (α = β, γ) Pa〈pα〉α intrinsic averaged pressure in the α-phase, (α = β, γ), Papα pressure deviations in the α-phase, Papα non-equilibrium spatial deviations of pressure of the α-phase, Pa〈p〉 average pressure in the equilibrium, Par0 radius of the averaging volume V , mr position vector, m

rκ radius of the κ-phase in the unit cell (=√

(1− ε) /π), mrγ radius of the γ-phase in the unit cell, mSα saturation (= εα/ε, α = β, γ)

sα closure variable that maps μα

(〈pβ〉β − 〈pγ〉β

)onto vα, Pa s−1

t time, st∗ characteristic process time, s

637

27

uα effective vector, (Pa m s )−1

vα velocity vector in the α-phase, ms−1

〈vα〉 superficial averaged velocity in the α-phase, ms−1

〈vα〉α intrinsic average velocity in the α-phase, ms−1

vα velocity deviations in the α-phase, ms−1

vα local non-equilibrium spatial deviation velocity of α-phase, ms−1

〈v〉 average velocity in the equilibrium model, ms−1

Vα volume of the α-phase, contained within the averaging volume, m3

V averaging volume, m3

Vf (= Vβ + Vγ) fluid-phase volume in the averaging domain, m3

V magnitude of the averaging domain, m3

yα position vector that locates points in the α-phase relative to the centroid of V , m

638

Greek letters639

ε porosityεα volume fraction of the α-phaseμα viscosity of the α-phase, Pa · s〈μ〉 average viscosity, Pa · sρα density of α-phase, kg m−3

ν (= μβ/μγ) viscosity ratio〈ρ〉 average density, kg m−3

σ interfacial tension between the β- and γ-phase N/mϕα closure variable that maps μαH onto pαΦ non-equilibrium mechanical source,

640

Sub and superscripts641

β fluid continuous phaseγ fluid dispersed phaseκ solid phasene non-equilibrium modele equilibrium model

642

Acknowledgments643

This work was supported by Fondo Sectorial de Investigacion para la Educacion from644

CONACyT (Project number: 12511908; Arrangement number: 112087). OALR thanks645

CONACyT for the doctoral scholarship 274032. Both authors are grateful to Dr. Didier646

Lasseux, Dr. J. Alberto Ochoa-Tapia and M.in Sc. Raquel de los Santos for their valuable647

comments and suggestions.648

28

Aguilar-Madera, C., Valdes-Parada, F., Goyeau, B., Ochoa-Tapia, J., 2011. One-domain approach for heat649

transfer between a porous medium and a fluid. International Journal of Heat and Mass Transfer 54, 2089650

– 2099.651

Alvarado, D., Marsden, S., 1979. Flow of oil-in-water emulsions through tubes and porous media. Society652

of Petroleoum Engineers Journal 19, 369–377.653

Auriault, J., Lebaigue, D., Bonnet, G., 1989. Dynamics of two immiscible fluids flowing through deformable654

porous media. Transport in Porous Media 4, 105 – 128.655

Bear, J., Cheng, A.-D., 2010. Modeling Groundwater Flow and Contaminant Transport. Springer.656

Buckley, S., Leverett, M., 1942. Mechanism of fluid displacements in sands. Transactions of the AIME 146,657

107–116.658

Cobos, S., Carvalho, M., Alvarado, V., 2009. Flow of oil-water emulsions through a constricted capillary.659

International Journal of Multiphase Flow 35, 507–515.660

Darcy, H., 1856. Les fontaines publiques de la ville de Dijon. Librairie des corps impriaux des ponts et661

chausses et des mines.662

Devereux, O., 1974. Emulsion flow in porous solids-I. A flow model. Chem. Engng. J. 7, 121–128.663

Dullien, F., 1992. Porous media: fluid transport and pore structure. Academic Press Inc.664

Durucana, S., Ahsan, M., Syed, A., Shi, J.-Q., Korre, A., 2013. Two phase relative permeability of gas and665

water in coal for enhanced coalbed methane recovery and CO2 storage. Energy Procedia 37, 6730 6737.666

Gray, W., 1975. A derivation of the equations for multiphase transport. Chem. Engng. Sci. 30, 229–231.667

Gray, W., Miller, C., 2014. Introduction to the Thermodinamically Constrained Averaging Theory. Springer.668

Guillen, V., Carvalho, M., Alvarado, V., 2012a. Pore scale and macroscopic displacement mechanisms in669

emulsion flooding. Transp Porous Media 94, 197206.670

Guillen, V., Romero, M., da Silveira Carvalho, M., Alvarado, V., 2012b. Capillary-driven mobility control671

in macro emulsion flow in porous media. International Journal of Multiphase Flow 43, 62–65.672

Hofman, J., Stein, H., 1991. Permeability reduction of porous media on transport of emulsion through them.673

Colloids and Surfaces 61, 317–329.674

Howes, F., Whitaker, S., 1985. The spatial averaging theorem revisited. Chem. Eng. Sci. 40, 13871392.675

Islam, M. R., Ali, S. M. F., 1994. Numerical simulation of emulsion flow through porous media. Journal of676

Canadian Petroleum Technology 33, 59–63.677

Jain, V., Demond, A., 2002. Conductivity reduction due to emulsification during surfactant enhanced-aquifer678

remediation. 1. Emulsion transport. Environ. Sci. Technol. 36, 5426–5433.679

Lasseux, D., Quintard, M., Whitaker, S., 1996. Determination of permeability tensors for two-phase flow in680

homogeneous porous media: Theory. Transport in Porous Media 24, 107–137.681

McAuliffe, C., 1973. Oil-in-water emulsions and their flow properties in porous media. Journal of Petroleum682

Technology 25, 727–733.683

Nogueira, G., Carvalho, M., Alvarado, V., 2013. Dynamic network model of mobility control in emulsion684

flow through porous media. Transp Porous Med 98, 427–441.685

Ochoa-Tapia, J., Stroeve, P., Whitaker, S., 1986. Diffusion and reaction in cellular media. Chemical Engi-686

neering Science 41 (12), 2999–3013.687

Ochoa-Tapia, J., Whitaker, S., 1995. Momentum transfer at the boundary between a porous medium and688

homogeneous fluid - I. Theorical development. International Journal Heat Mass Transfer 38, 2635 – 2646.689

Ouyang, Y., Mansell, R. S., Rhue, R., 1995. Emulsionmediated transport of nonaqueousphase liquid in690

porous media: A review. Critical Reviews in Environmental Science and Technology 25, 269–290.691

Quintard, M., Whitaker, S., 1987. Ecoulement monophasique en milieu poreux: effet des heterogeneites692

locales. J. Mca. Thorique et Applique 6, 691–726.693

Quintard, M., Whitaker, S., 1988. Two phase flow in heterogeneous porous media: The method of large-scale694

averaging. Transport in Porous Media 3, 357–413.695

Quintard, M., Whitaker, S., 1993. One- and two-equations models for transient diffusion process in two-phase696

system. Adv. Heat Transfer 23 (369-464).697

Quintard, M., Whitaker, S., 1995. Local thermal equilibrium for transient heat conduction: Theory and698

comparison with numerical experiments. Int. J. Heat Mass Transfer 38 (15), 2779–2796.699

29

Romero, M., Carvalho, M., Alvarado, V., 2011. Experiments and network model of flow of oil-water emulsion700

in porous media. Physical Review E. 84, 046305.701

Soo, H., Radke, C., 1984. The flow mechanism of dilute, stable emulsions in porous media. Ind. Eng. Chem.702

Fundam. 342 - 347, 342–347.703

Soo, H., Radke, C., 1986. A filtration model for flow of dilute, estable emulsion in porous media - I. Theory.704

Chemical Engineering Science 41, 263–272.705

Soo, H., Williams, M., Radke, C., 1986. A filtration model for the flow of dilute, stable emulsions in porous706

mediaII. Parameter evaluation and estimation. Chemical Engineering Science 41, 273–281.707

Whitaker, S., 1986a. Flow in porous media I: A theoretical derivation of Darcy’s law. Transport in Porous708

Media 1, 3–25.709

Whitaker, S., 1986b. Flow in porous media II: The governing equations for immiscible, two-phase flow.710

Transport in Porous Media 1, 105–125.711

Whitaker, S., 1991. Improved constraints for the principle of local thermal equilibrium. Ind. Eng. Chem.712

Res. 30, 983–997.713

Whitaker, S., 1994. The closure problem for two-phase flow in homogeneous porous media. Chemical Engi-714

neering Science 49, 5.715

Whitaker, S., 1999. Theory and Applications of Transport in Porous Media: The Method of Volume Aver-716

aging. Kluwer Academic Publishers.717

Wood, B., Valdes-Parada, F., 2013. Volume averaging: Local and nonlocal closure using a Green’s function718

approach. Advances in Water Resources 51, 139–167.719

Wood, B., Whitaker, S., 1998. Diffusion and reaction in biofilms. Chemical Engineering Science 53 (3),720

397–425.721

30

Figures captions722

Fig. 1. Characteristic length scales and averaging volume.723

Fig. 2. Examples of the fields of the closure variables (Aβ)xx and (Aγ)xx under different724

conditions: (a) ε = 0.5, Sβ = 0.8, ν = 1, (b) ε = 0.5, Sβ = 0.8, ν = 0.1, (c) ε = 0.8,725

Sβ = 0.8, ν = 1 and (d) ε = 0.8, Sβ = 0.8, ν = 10726

Fig. 3. Predictions of the longitudinal component of the effective coefficient Ke for the LME727

model as a function of the void fraction ε taking Sβ = 0.35 (a, c) and 0.8 (b, d) for728

several values of the viscosities ratios, ν.729

Fig. 4. Predictions of the correction factor, Km as a function of the void fraction, ε, for730

several values of ν, taking Sβ = 0.8.731

Fig. 5. Fields of the closure variables (Aβ)xx and (Aγ)xx for the experimental parameters732

corresponding to experiment number 3 of Soo and Radke (1984).733

Fig. 6. a) Comparison of the experimental data reported by Soo and Radke (1984) and734

the predictions of the effective coefficient for the LME model, Ke with respect to the735

intrinsic permeability, K. b) Relative error percent between theory and experiments.736

Fig. 7. a) Comparison of the experimental data reported by Jain and Demond (2002) and737

the predictions of the effective coefficient for the LME model, Ke with respect to the738

intrinsic permeability, K. b) Relative error percent between theory and experiments.739

Fig. A.1. Representative unit cell for dispersed flow.740

Tables captions741

Table 1. Parameters used for predicting the experimental data.742

Table A.1. Closure variables and non-homogeneous terms for the associated boundary743

value problems.744

Table A.2. Values for geometry parameter λi745

31

L Average scale (mm to 10cm)

Macroscopic scale (1m to 100m)

r0

Microscopic scale (mm to �m)

�-phase ℓ�ℓβℓκ

�-phase

�-phase

Figure 1: Characteristic length scales and averaging volume.

1

Figure 2: Examples of the fields of the closure variables (Aβ)xx and (Aγ)xx under different conditions: (a)ε = 0.5, Sβ = 0.8, ν = 1, (b) ε = 0.5, Sβ = 0.8, ν = 0.1, (c) ε = 0.8, Sβ = 0.8, ν = 1 and (d) ε = 0.8,Sβ = 0.8, ν = 10

2

Figure 3: Predictions of the longitudinal component of the effective coefficient Ke for the LME model asa function of the void fraction ε taking Sβ = 0.35 (a, c) and 0.8 (b, d) for several values of the viscositiesratios, ν.

Figure 4: Predictions of the mixture coefficient, Km as a function of the void fraction, ε, for several valuesof ν, taking Sβ = 0.8.

3

Figure 5: Fields of the closure variables (Aβ)xx and (Aγ)xx for the experimental parameters correspondingto experiment number 3 of Soo and Radke (1984).

Figure 6: a) Comparison of the experimental data reported by Soo and Radke (1984) and the predictionsof the effective coefficient for the LME model, Ke with respect to the intrinsic permeability, K. b) Relativeerror percent between theory and experiments.

4

Figure 7: a) Comparison of the experimental data reported by Jain and Demond (2002) and the predictionsof the effective coefficient for the LME model, Ke with respect to the intrinsic permeability, K. b) Relativeerror percent between theory and experiments.

r�

�-phase

ℓcr�1r�3

r�4

r��

r�5

�-phase

�-phase

Figure A.1: Representative unit cell for disperse flow.

5

Table 1: Parameters used for predicting the experimental data.

Experiment number ε ν Sβ 2rγ [µm] dp [µm]Soo and Radke (1984)

1 0.34 0.660 0.16-0.1 3.1 29.52 0.34 0.660 0.20-0.7 6.1 29.53 0.34 0.044 0.01-0.06 3.4 29.5

Jain and Demond (2002)1 0.336 1.18 0.02-0.34 1.1 47

Table A.1: Closure variables and non-homogeneous terms for the associated boundary value problems.

Problem I Problem II Problem III Problem IV

Source: 〈vβ〉β 〈vβ〉β − 〈vγ〉γ 〈pβ〉β − 〈pγ〉β HClosure variables:Zα Aβ, Aγ Aβγ, Aγβ dβ, dγ fβ, fγζα aβ, aγ aβγ, aγβ sβ, sγ ϕβ, ϕγHomogeneous terms in α-phase:∆β −εβK−1

β Kβγ δβγ hβ∆γ −εγK−1

γ Kγβ δγβ hγNon-homogeneous terms at the interfaces∆βκ −I 0 0 0∆I

βγ 0 I 0 0∆II

βγ 0 0 +nβγ +2Hσnβγ

Table A.2: Values for geometry parameter λi

ε = 0.5 = 0.6 = 0.7 = 0.8 = 0.9λ2 0.55 0.55 0.55 0.75 1λ3 0.9 0.9 0.8 0.75 1λ4 0.62 0.62 0.65 0.5 0.95λ5 0.51 0.6 0.77 0.65 0.7

An  upscaled  model  was  derived  to  study  two-­‐phase  dispersed  flow  in  porous  media    A  Darcy’s-­‐law  type  model  was  obtained  using  a  local  mechanical  equilibrium  approach    The  effective-­‐medium  coefficient  was  sensitive  to  flow  and  geometrical  parameters    Good  agreement  was  found  with  available  experimental  data