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3 Formulation of the Model

3.1 General Balance Laws

[] The approach presented here is based on the traditional theory presented in Section 1 combined with the distinction between percolating and nonpercolating phases introduced in Refs. [14, 16, 15] and discussed in Section 2. [] The volume fractions of the subsets \mathbb{F}_{i}\subset\mathbb{S},i=1,2,3,4 and \mathbb{M}\subset\mathbb{S} are denoted as \phi _{i}(\mathbf{x},t). [] Let \phi denote the porosity (volume fraction of \mathbb{P}). [] Volume conservation implies the relations

\displaystyle\phi _{1}+\phi _{2}+\phi _{3}+\phi _{4}+\phi _{5} \displaystyle=1 (16a)
\displaystyle S_{1}+S_{2}+S_{3}+S_{4} \displaystyle=1 (16b)
\displaystyle 1-\phi \displaystyle=\phi _{5} (16c)

where \phi _{i}=\phi S_{i} (i=1,2,3,4) are volume fractions, and S_{i} are saturations. [] The water saturation is defined as {S_{{\mathbb{W}}}}=S_{1}+S_{2} and the oil saturation as {S_{{\mathbb{O}}}}=S_{3}+S_{4}.

[] The general law of mass balance is again given by eq. (1) with i=1,2,3,4. [] It provides now four equations instead of two. [] The general law of momentum balance is given by eq. (2), also now with i=1,2,3,4 instead of i=\mathbb{W},\mathbb{O} as before.

3.2 General Constitutive Assumptions

[] As in the traditional theory the porous medium is again assumed to be macroscopically homogeneous so that eq. (4) holds. [] The flows are slow, and hence also eq. (7) continues to hold without change. [] For incompressible fluids one has now

\displaystyle\varrho _{1}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{W}}} (17a)
\displaystyle\varrho _{2}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{W}}} (17b)
\displaystyle\varrho _{3}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{O}}} (17c)
\displaystyle\varrho _{4}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{O}}} (17d)

analogous to eq. (5a).

3.3 Viscous drag

[] The momentum transfer into phase i from all the other phases is assumed to be a simple viscous drag,

\mathbf{m}_{i}=\sum _{{j=1}}^{{5}}R_{{ij}}({\mathbf{v}}_{j}-{\mathbf{v}}_{i}) (18)

where the resistance coefficient R_{{ij}} quantifies the viscous coupling between phase i and j. [] For the rigid rock matrix [page 4, §0]   {\mathbf{v}}_{5}=0. [] Hence -R_{{i5}}{\mathbf{v}}_{i} is the momentum transfer from the wall into phase i. [] Then

\displaystyle\mathbf{m}_{1} \displaystyle=R_{{13}}({\mathbf{v}}_{3}-{\mathbf{v}}_{1})+R_{{14}}({\mathbf{v}}_{4}-{\mathbf{v}}_{1})-R_{{15}}{\mathbf{v}}_{1} (19a)
\displaystyle\mathbf{m}_{2} \displaystyle=R_{{23}}({\mathbf{v}}_{3}-{\mathbf{v}}_{2})+R_{{24}}({\mathbf{v}}_{4}-{\mathbf{v}}_{2})-R_{{25}}{\mathbf{v}}_{2} (19b)
\displaystyle\mathbf{m}_{3} \displaystyle=R_{{31}}({\mathbf{v}}_{1}-{\mathbf{v}}_{3})+R_{{32}}({\mathbf{v}}_{2}-{\mathbf{v}}_{3})-R_{{35}}{\mathbf{v}}_{3} (19c)
\displaystyle\mathbf{m}_{4} \displaystyle=R_{{41}}({\mathbf{v}}_{1}-{\mathbf{v}}_{4})+R_{{42}}({\mathbf{v}}_{2}-{\mathbf{v}}_{4})-R_{{45}}{\mathbf{v}}_{4} (19d)

where R_{{12}}=0 and R_{{34}}=0 was used because there is no common interface and hence no direct viscous interaction between these phase pairs. [] Each R_{{ij}} is a 3\times 3-matrix.

3.4 Capillarity

[] In the present approach the body forces are given by gravity plus capillary forces

\displaystyle\mathbf{F}_{1} \displaystyle=\varrho _{1}\mathbf{g} (20a)
\displaystyle\mathbf{F}_{2} \displaystyle=\varrho _{2}\mathbf{g}+\mathbf{F}_{{c\mathbb{W}}} (20b)
\displaystyle\mathbf{F}_{3} \displaystyle=\varrho _{3}\mathbf{g} (20c)
\displaystyle\mathbf{F}_{4} \displaystyle=\varrho _{4}\mathbf{g}+\mathbf{F}_{{c\mathbb{O}}} (20d)

contrary to eqs. (8). [] The capillary body forces \mathbf{F}_{{c\mathbb{W}}},\mathbf{F}_{{c\mathbb{O}}} are responsible for keeping the trapped fluids inside the medium. [] They are obtained as gradients of capillary potentials

\displaystyle\mathbf{F}_{{c\mathbb{W}}} \displaystyle=-\mathbf{\nabla}{\Pi _{c}}_{{\mathbb{W}}} (21a)
\displaystyle\mathbf{F}_{{c\mathbb{O}}} \displaystyle=-\mathbf{\nabla}{\Pi _{c}}_{{\mathbb{O}}} (21b)

where the capillary potentials {\Pi _{c}}_{{\mathbb{W}}},{\Pi _{c}}_{{\mathbb{O}}} are defined as

\displaystyle{\Pi _{c}}_{{\mathbb{W}}} \displaystyle=\Pi^{*}_{{\rm a}}-\Pi _{{\rm a}}S_{1}^{{-\alpha}} (22a)
\displaystyle{\Pi _{c}}_{{\mathbb{O}}} \displaystyle=\Pi^{*}_{{\rm b}}-\Pi _{{\rm b}}S_{3}^{{-\beta}} (22b)

with constants \Pi^{*}_{{\rm a}},\Pi^{*}_{{\rm b}},\Pi _{{\rm a}},\Pi _{{\rm b}} and exponents \alpha,\beta>0.

[] Next the stress tensor for percolating phases can be specified in analogy with eq. (6) as

\displaystyle\Sigma _{1} \displaystyle=-P_{1}\mathbf{1} (23a)
\displaystyle\Sigma _{3} \displaystyle=-P_{3}\mathbf{1} (23b)

where P_{1} and P_{3} are the fluid pressures. [] The stress tensor \Sigma _{2},\Sigma _{4} for the nonpercolating phases cannot be specified in the same way because the forces cannot propagate in nonpercolating phases. [] Here it is assumed that these stresses are given by the pressure in the surrounding percolating phase modified by the energy density stored in the common interface with the surrounding percolating phases. [] This suggests an Ansatz [4]

\displaystyle\Sigma _{2} \displaystyle=-P_{3}\mathbf{1}+\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi}\frac{\partial A_{{32}}}{\partial S_{2}}\mathbf{1} (24a)
\displaystyle\Sigma _{4} \displaystyle=-P_{1}\mathbf{1}+\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi}\frac{\partial A_{{41}}}{\partial S_{4}}\mathbf{1} (24b)

where \sigma _{{\mathbb{W}\mathbb{O}}} is the oil-water interfacial tension, and the unknowns A_{{32}}(\mathbf{x},t),A_{{41}}(\mathbf{x},t) are the interfacial areas per unit volume of porous medium between phases 3 and 2, resp. 4 and 1. [] (The factor \phi arises from the definition of A_{{ij}} per unit volume of porous medium.) [] To avoid equations of motion for the unknowns A_{{32}} and A_{{41}} it is assumed that geometrical relations of the form

\displaystyle A_{{32}} \displaystyle=A^{*}_{2}\; S_{2}^{\gamma} (25a)
\displaystyle A_{{41}} \displaystyle=A^{*}_{4}\; S_{4}^{\delta} (25b)

hold, where A^{*}_{2},A^{*}_{4} are prefactors assumed to be constant. [] Thus for the nonpercolating phases

\displaystyle\Sigma _{2} \displaystyle=(-P_{3}+\gamma P^{*}_{2}S_{2}^{{\gamma-1}})\mathbf{1} (26a)
\displaystyle\Sigma _{4} \displaystyle=(-P_{1}+\delta P^{*}_{4}S_{4}^{{\delta-1}})\mathbf{1} (26b)

will be used below. [] Here

\displaystyle P^{*}_{2} \displaystyle=A^{*}_{2}\;\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi} (27a)
\displaystyle P^{*}_{4} \displaystyle=A^{*}_{4}\;\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi} (27b)

are constants.

[] The mass transfer rates must depend on rates of saturation change. [] They are here assumed to be

\displaystyle M_{1} \displaystyle=-M_{2}={\eta _{{2}}}\phi\varrho _{{\mathbb{W}}}\left(\frac{S_{2}-{{S_{2}}^{*}}}{{{S_{{\mathbb{W}}}}^{*}}-{S_{{\mathbb{W}}}}}\right)\frac{\partial{S_{{\mathbb{W}}}}}{\partial t} (28a)
\displaystyle M_{3} \displaystyle=-M_{4}={\eta _{{4}}}\phi\varrho _{{\mathbb{O}}}\left(\frac{S_{4}-{{S_{4}}^{*}}}{{{S_{{\mathbb{O}}}}^{*}}-{S_{{\mathbb{O}}}}}\right)\frac{\partial{S_{{\mathbb{O}}}}}{\partial t} (28b)

where {\eta _{{2}}},{\eta _{{4}}} are constants. [] The parameters {{S_{{\mathbb{W}}}}^{*}},{{S_{{\mathbb{O}}}}^{*}}, {{S_{2}}^{*}},{{S_{4}}^{*}} are defined by

\displaystyle{{S_{{\mathbb{W}}}}^{*}} \displaystyle=\frac{1-S_{{\mathbb{O}\,\rm im}}}{2}\left[1+\tanh\left({\tau _{{\mathbb{W}}}}\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}\right)\right] (29a)
\displaystyle+\frac{S_{{\mathbb{W}\,\rm dr}}}{2}\left[1-\tanh\left({\tau _{{\mathbb{W}}}}\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}\right)\right]
\displaystyle{{S_{{\mathbb{O}}}}^{*}} \displaystyle=\frac{1-S_{{\mathbb{W}\,\rm dr}}}{2}\left[1+\tanh\left({\tau _{{\mathbb{O}}}}\frac{\partial{S_{{\mathbb{O}}}}}{\partial t}\right)\right] (29b)
\displaystyle+\frac{S_{{\mathbb{O}\,\rm im}}}{2}\left[1-\tanh\left({\tau _{{\mathbb{O}}}}\frac{\partial{S_{{\mathbb{O}}}}}{\partial t}\right)\right]
\displaystyle{{S_{2}}^{*}} \displaystyle=\frac{S_{{\mathbb{W}\,\rm dr}}}{2}\left[1-\tanh\left({\tau _{{2}}}\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}\right)\right] (29c)
\displaystyle{{S_{4}}^{*}} \displaystyle=\frac{S_{{\mathbb{O}\,\rm im}}}{2}\left[1-\tanh\left({\tau _{{4}}}\frac{\partial{S_{{\mathbb{O}}}}}{\partial t}\right)\right] (29d)

[page 5, §0]   where S_{{\mathbb{W}\,\rm dr}},S_{{\mathbb{O}\,\rm im}} are limiting saturations for S_{2},S_{4} and {\tau _{{\mathbb{W}}}},{\tau _{{\mathbb{O}}}},{\tau _{{2}}},{\tau _{{4}}} are equilibration time scales for reaching capillary equilibrium. [] For simplicity

\tau={\tau _{{\mathbb{W}}}}={\tau _{{\mathbb{O}}}}={\tau _{{2}}}={\tau _{{4}}} (30)

will be assumed below.