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4 Constitutive Assumptions

[page 145, §1]   
[145.1.1] The porous medium is assumed to be macroscopically homogeneous

\phi(\mathbf{x})=\phi={\rm const} (12)

although this assumption rarely holds in practice [26]. [145.1.2] Let us further assume that the fluids are incompressible so that

\displaystyle\varrho _{1}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{W}}} (13a)
\displaystyle\varrho _{2}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{W}}} (13b)
\displaystyle\varrho _{3}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{O}}} (13c)
\displaystyle\varrho _{4}(\mathbf{x},t) \displaystyle=\varrho _{{\mathbb{O}}} (13d)

where the constants \varrho _{{\mathbb{W}}},\varrho _{{\mathbb{O}}} are independent of \mathbf{x} and t.

[145.2.1] Flows through porous media often have low Reynolds numbers. [145.2.2] Thus accelerations and the inertial term

\frac{{\rm D}^{i}}{{\rm D}t}{\mathbf{v}}_{i}=0 (14)

can be neglected in the momentum balance equation (10).

[145.3.1] The momentum transfer into phase i from all the other phases is assumed to arise from viscous drag,

\mathbf{m}_{i}=\sum _{{j=1}}^{{5}}R_{{ij}}({\mathbf{v}}_{j}-{\mathbf{v}}_{i}) (15)

with resistance coefficients R_{{ij}} quantifying the loss due to viscous friction between phase i and j. [145.3.2] The matrix is assumed to be rigid so that {\mathbf{v}}_{5}=0. [145.3.3] Hence -R_{{i5}}{\mathbf{v}}_{i} is the momentum transfer from the wall into phase i. [145.3.4] 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} (16a)
\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} (16b)
\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} (16c)
\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} (16d)

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. [145.3.5] The viscous resistance coefficients R_{{ij}} may be rewritten in term of dimensionless coefficients r_{{ij}} as

\displaystyle R_{{ij}} \displaystyle=\mu _{{\mathbb{W}}}k^{{-1}}r_{{ij}},\qquad i=1,2 (17a)
\displaystyle R_{{ij}} \displaystyle=\mu _{{\mathbb{O}}}k^{{-1}}r_{{ij}},\qquad i=3,4 (17b)

where \mu _{{\mathbb{W}}},\mu _{{\mathbb{O}}} are the viscosities of water and oil, k is the absolute permeability tensor of the medium, and r_{{ij}} are dimensionless viscous drag coefficients. [145.3.6] Each R_{{ij}} is a 3\times 3-matrix. [145.3.7] In practice viscous coupling terms between the two fluid phases are often neglected.

[page 146, §1]    [146.1.1] The stress tensor is written as a pressure term plus a capillary correction term. [146.1.2] The reference pressure for the nonpercolating phases is the pressure of the surrounding percolating phase [24]. [146.1.3] Thus

\displaystyle\Sigma _{1} \displaystyle=-{P_{{\mathbb{W}}}}\mathbf{1}+{\Sigma _{c}}_{1} (18a)
\displaystyle\Sigma _{2} \displaystyle=-{P_{{\mathbb{O}}}}\mathbf{1}+{\Sigma _{c}}_{2} (18b)
\displaystyle\Sigma _{3} \displaystyle=-{P_{{\mathbb{O}}}}\mathbf{1}+{\Sigma _{c}}_{3} (18c)
\displaystyle\Sigma _{4} \displaystyle=-{P_{{\mathbb{W}}}}\mathbf{1}+{\Sigma _{c}}_{4}, (18d)

where {\Sigma _{c}}_{i} are capillary stresses resulting from the presence of fluid-fluid and fluid-matrix interfaces.

[146.2.1] Similarly, the body forces are augmented with capillary body forces as

\mathbf{F}_{i}=\varrho _{i}\mathbf{g}+{\mathbf{F}_{c}}_{i} (19)

with i=1,2,3,4. [146.2.2] The capillary body forces {\mathbf{F}_{c}}_{i} are responsible for keeping the trapped fluids inside the medium. [146.2.3] They are assumed to be potential forces

{\mathbf{F}_{c}}_{i}=-\mathbf{\nabla}{\Pi _{c}}_{i} (20)

where {\Pi _{c}}_{i} are the capillary potentials.

[146.3.1] One has considerable freedom to specify the capillary stresses {\Sigma _{c}}_{i} and potentials {\Pi _{c}}_{i}. [146.3.2] General thermodynamic considerations suggest ideas to restrict this freedom. [146.3.3] Let F denote the total Helmholtz free energy of the system with and oil-water interface, and let F_{\mathbb{W}} and F_{\mathbb{O}} denote the individual Helmholtz free energies of bulk water and bulk oil. [146.3.4] Then [27, 28]

{\rm d}F=-{P_{{\mathbb{W}}}}\;{\rm d}V_{\mathbb{W}}-{P_{{\mathbb{O}}}}\;{\rm d}V_{\mathbb{O}}+\sigma _{{\mathbb{W}\mathbb{O}}}\;{\rm d}\widetilde{A}_{{\mathbb{W}\mathbb{O}}}+\sigma _{{\mathbb{W}\mathbb{M}}}{\rm d}\widetilde{A}_{{\mathbb{W}\mathbb{M}}}+\sigma _{{\mathbb{O}\mathbb{M}}}{\rm d}\widetilde{A}_{{\mathbb{O}\mathbb{M}}} (21)

where {P_{{\mathbb{W}}}},{P_{{\mathbb{O}}}} are the oil and water pressure, V_{\mathbb{W}},V_{\mathbb{O}} are the volumes of oil and water, and \widetilde{A}_{{\mathbb{W}\mathbb{O}}},\widetilde{A}_{{\mathbb{W}\mathbb{M}}},\widetilde{A}_{{\mathbb{O}\mathbb{M}}} are the total interfacial areas between oil and water, water and matrix, resp. oil and matrix. [146.3.5] The oil-water surface tension \sigma _{{\mathbb{W}\mathbb{O}}} and the fluid-matrix interfacial tensions \sigma _{{\mathbb{W}\mathbb{M}}},\sigma _{{\mathbb{O}\mathbb{M}}} are related by Youngs equation

\sigma _{{\mathbb{O}\mathbb{M}}}=\sigma _{{\mathbb{W}\mathbb{M}}}+\sigma _{{\mathbb{W}\mathbb{O}}}\cos(\vartheta) (22)

where \vartheta is the contact angle of water. [146.3.6] The interfacial areas obey

\displaystyle\widetilde{A}_{{\mathbb{W}\mathbb{O}}} \displaystyle=\widetilde{A}_{{31}}+\widetilde{A}_{{32}}+\widetilde{A}_{{41}}+\widetilde{A}_{{42}} (23)
\displaystyle\widetilde{A}_{{\mathbb{P}\mathbb{M}}} \displaystyle=\widetilde{A}_{{\mathbb{W}\mathbb{M}}}+\widetilde{A}_{{\mathbb{O}\mathbb{M}}} (24)
\displaystyle\widetilde{A}_{{\mathbb{W}\mathbb{M}}} \displaystyle=\widetilde{A}_{{15}}+\widetilde{A}_{{25}} (25)
\displaystyle\widetilde{A}_{{\mathbb{O}\mathbb{M}}} \displaystyle=\widetilde{A}_{{35}}+\widetilde{A}_{{45}} (26)

where \widetilde{A}_{{ij}} is the total interfacial area between phase i and j, and the volumes are related by eqs. (5b) and (5c).

[page 147, §1]    [147.1.1] In equilibrium {\rm d}F=0 holds. [147.1.2] Also, sample volume and internal surface are constant because the porous medium is rigid. [147.1.3] This implies {\rm d}V=0 and {\rm d}\widetilde{A}_{{\mathbb{P}\mathbb{M}}}=0. [147.1.4] Using eq. (5c) one arrives at

0={P_{{\mathbb{O}}}}-{P_{{\mathbb{W}}}}+\sigma _{{\mathbb{W}\mathbb{O}}}\left(\frac{\partial\widetilde{A}_{{\mathbb{W}\mathbb{O}}}}{\partial V_{\mathbb{W}}}-\cos\vartheta\frac{\partial\widetilde{A}_{{\mathbb{W}\mathbb{M}}}}{\partial V_{\mathbb{W}}}\right) (27)

where Youngs equation (22) was also used.

[147.2.1] These considerations suggest one particular way to specify the capillary stresses and potentials. [147.2.2] Following earlier ideas [24] the capillary stresses are specified as

\displaystyle{\Sigma _{c}}_{1} \displaystyle=0 (28a)
\displaystyle{\Sigma _{c}}_{2} \displaystyle=-\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi}\frac{\partial A_{{\mathbb{W}\mathbb{O}}}}{\partial{S_{{\mathbb{W}}}}} (28b)
\displaystyle{\Sigma _{c}}_{3} \displaystyle=0 (28c)
\displaystyle{\Sigma _{c}}_{4} \displaystyle=-{\Sigma _{c}}_{2} (28d)

where local equilibrium was assumed and intensive quantities (per unit volume of porous medium) were introduced. [147.2.3] The capillary potentials may be associated with the last term in eq. (27). [147.2.4] They are specified as

\displaystyle{\Pi _{c}}_{1} \displaystyle=0 (29a)
\displaystyle{\Pi _{c}}_{2} \displaystyle=\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{\phi}\cos\vartheta\frac{\partial A_{{\mathbb{W}\mathbb{M}}}}{\partial{S_{{\mathbb{W}}}}} (29b)
\displaystyle{\Pi _{c}}_{3} \displaystyle=0 (29c)
\displaystyle{\Pi _{c}}_{4} \displaystyle=-{\Pi _{c}}_{2} (29d)

in analogy with [23, 24].

[147.3.1] The mass transfer rates are M_{i}=\sum _{{j=1}}^{5}M_{{ij}} where M_{{ij}} is the mass transfer rate from phase j into phase i. [147.3.2] Neglecting chemical reactions one assumes M_{{ij}}=0 for all pairs (i,j) except the pairs (1,2),(2,1),(3,4),(4,3). [147.3.3] These remaining transfer rates are assumed to be given as

\displaystyle M_{{12}}=-M_{{21}} \displaystyle=\phi\varrho _{{\mathbb{W}}}\left\{\left[b_{1}(1-S_{2})-b_{2}S_{2}\right]\Theta^{*}_{\mathbb{W}}+{\eta _{{2}}}\left(\frac{S_{2}-{{S_{2}}^{*}}}{{{S_{{\mathbb{W}}}}^{*}}-{S_{{\mathbb{W}}}}}\right)\frac{\partial{S_{{\mathbb{W}}}}}{\partial t}\right\} (30a)
\displaystyle M_{{34}}=-M_{{43}} \displaystyle=\phi\varrho _{{\mathbb{O}}}\left\{\left[b_{3}(1-S_{4})-b_{4}S_{4}\right]\Theta^{*}_{\mathbb{O}}+{\eta _{{4}}}\left(\frac{S_{4}-{{S_{4}}^{*}}}{{{S_{{\mathbb{O}}}}^{*}}-{S_{{\mathbb{O}}}}}\right)\frac{\partial{S_{{\mathbb{O}}}}}{\partial t}\right\} (30b)

[page 148, §0]    with

\displaystyle{{S_{{\mathbb{W}}}}^{*}} \displaystyle=\Theta\left(\frac{{\tau _{{\mathbb{W}}}}\partial{S_{{\mathbb{W}}}}}{\partial t}\right)+{{S_{2}}^{*}}-{{S_{4}}^{*}} (31a)
\displaystyle{{S_{{\mathbb{O}}}}^{*}} \displaystyle=1-{{S_{{\mathbb{W}}}}^{*}} (31b)
\displaystyle{{S_{2}}^{*}} \displaystyle=\min({S_{{\mathbb{W}}}},S_{{\mathbb{W}\,\rm i}})\left[1-\Theta\left(\frac{{\tau _{{\mathbb{W}}}}\partial{S_{{\mathbb{W}}}}}{\partial t}\right)\right] (31c)
\displaystyle{{S_{4}}^{*}} \displaystyle=\min({S_{{\mathbb{O}}}},S_{{\mathbb{O}\,\rm r}})\Theta\left(\frac{{\tau _{{\mathbb{W}}}}\partial{S_{{\mathbb{W}}}}}{\partial t}\right) (31d)

as in [22, 23, 24]. [148.0.1] The limiting saturations for S_{2},S_{4}, called irreducible water resp. residual oil saturation,

\displaystyle S_{{\mathbb{W}\,\rm i}} \displaystyle=S_{{\mathbb{W}\,\rm i}}({\mathbf{v}}_{1},{\mathbf{v}}_{3})=\frac{b_{1}({\mathbf{v}}_{1},{\mathbf{v}}_{3})}{b_{1}({\mathbf{v}}_{1},{\mathbf{v}}_{3})+b_{2}({\mathbf{v}}_{1},{\mathbf{v}}_{3})} (32a)
\displaystyle S_{{\mathbb{O}\,\rm r}} \displaystyle=S_{{\mathbb{O}\,\rm r}}({\mathbf{v}}_{1},{\mathbf{v}}_{3})=\frac{b_{3}({\mathbf{v}}_{1},{\mathbf{v}}_{3})}{b_{3}({\mathbf{v}}_{1},{\mathbf{v}}_{3})+b_{4}({\mathbf{v}}_{1},{\mathbf{v}}_{3})} (32b)

are velocity dependent, because they depend on the velocity dependent “reaction rates” b_{i}({\mathbf{v}}_{1},{\mathbf{v}}_{3}),i=1,2,3,4. [148.0.2] The relation between residual oil saturation S_{{\mathbb{O}\,\rm r}} and flow velocity is also known as capillary correlation or capillary desaturation curve [29, 30, 31]. [148.0.3] The factors \Theta^{*}_{\mathbb{W}},\Theta^{*}_{\mathbb{O}} are defined as

\displaystyle\Theta^{*}_{\mathbb{W}} \displaystyle=\left\{\Theta(\cos\vartheta)+[1-\Theta(\cos\vartheta)]\Theta(S_{2}-S_{{\mathbb{W}\,\rm i}}^{0})\right\} (33a)
\displaystyle\Theta^{*}_{\mathbb{O}} \displaystyle=\left\{\Theta(\cos\vartheta)\Theta(S_{4}-S_{{\mathbb{O}\,\rm r}}^{0})+[1-\Theta(\cos\vartheta)]\right\} (33b)

where \vartheta denotes the contact angle of water,

\displaystyle S_{{\mathbb{W}\,\rm i}}^{0} \displaystyle=\lim _{{{\mathbf{v}}\to 0}}S_{{\mathbb{W}\,\rm i}}({\mathbf{v}},{\mathbf{v}}) (34a)
\displaystyle S_{{\mathbb{O}\,\rm r}}^{0} \displaystyle=\lim _{{{\mathbf{v}}\to 0}}S_{{\mathbb{O}\,\rm r}}({\mathbf{v}},{\mathbf{v}}). (34b)

are the low velocity limits of S_{{\mathbb{W}\,\rm i}},S_{{\mathbb{O}\,\rm r}}, and

\Theta(x)=\begin{cases}1,&x>0\\ 0,&x\leq 0\end{cases} (35)

denotes the Heaviside unit step function. [148.0.4] The velocity dependent “reaction rates” b_{i} are chosen such that they vanish for vanishing velocities. [148.0.5] In this paper it will be assumed that [page 149, §0]   

\displaystyle b_{1} \displaystyle=b_{1}({\mathbf{v}}_{1},{\mathbf{v}}_{3})={\tau _{{\mathbb{W}}}}^{3}A^{4}{\mathbf{v}}_{1}^{2}{\mathbf{v}}_{3}^{2} (36a)
\displaystyle b_{2} \displaystyle=b_{2}({\mathbf{v}}_{1},{\mathbf{v}}_{3})={\tau _{{\mathbb{W}}}}^{3}A^{4}{\mathbf{v}}_{3}^{4}\left(\frac{1-S_{{\mathbb{W}\,\rm i}}^{0}}{S_{{\mathbb{W}\,\rm i}}^{0}}\right) (36b)
\displaystyle b_{3} \displaystyle=b_{3}({\mathbf{v}}_{1},{\mathbf{v}}_{3})={\tau _{{\mathbb{O}}}}^{3}A^{4}{\mathbf{v}}_{1}^{2}{\mathbf{v}}_{3}^{2} (36c)
\displaystyle b_{4} \displaystyle=b_{4}({\mathbf{v}}_{1},{\mathbf{v}}_{3})={\tau _{{\mathbb{O}}}}^{3}A^{4}{\mathbf{v}}_{1}^{4}\left(\frac{1-S_{{\mathbb{O}\,\rm r}}^{0}}{S_{{\mathbb{O}\,\rm r}}^{0}}\right) (36d)

consistent with eq. (34). [149.0.1] The parameters {\tau _{{\mathbb{W}}}},{\tau _{{\mathbb{O}}}} are time scales, and {\eta _{{2}}},{\eta _{{4}}},b_{{ij}} are dimensionless constants.

[149.1.1] The first terms in the curly brackets of (30) model an equilibrium reaction between nonpercolating and percolating fluids. [149.1.2] The reaction, i.e. breakup and coalescence, takes only place when both percolating phases move, i.e. have nonvanishing velocity. [149.1.3] The prefactors \Theta^{*}_{\mathbb{W}},\Theta^{*}_{\mathbb{O}} reproduce the experimental observation that nonpercolating nonwetting fluid phases show little breakup or coalescence below the low velocity limit of the residual nonwetting saturation. [149.1.4] The prefactors also ensure that sign and dimensions are correct.

[149.2.1] The specific internal surfaces A_{{\mathbb{W}\mathbb{O}}},A_{{\mathbb{W}\mathbb{M}}} depend on saturation. [149.2.2] Here it is assumed that

A_{{\mathbb{W}\mathbb{M}}}=A_{{\mathbb{P}\mathbb{M}}}{S_{{\mathbb{W}}}}. (37)

[149.2.3] The dependence of A_{{\mathbb{W}\mathbb{O}}}(S_{1},S_{2},S_{3},S_{4}) is suggested by inverting the classic hydraulic radius theory

k=C_{1}\frac{S_{1}^{3}}{A_{1}^{2}}=C_{3}\frac{S_{3}^{3}}{A_{3}^{2}} (38)

where A_{1}=A_{{31}}+A_{{41}}+A_{{51}} and A_{3}=A_{{31}}+A_{{32}}+A_{{35}}.

[149.3.1] Finally, the system is closed selfconsistently using the condition

\displaystyle 0 \displaystyle=\left(\frac{R_{{13}}}{\phi _{1}}+\frac{R_{{14}}}{\phi _{1}}+\frac{R_{{15}}}{\phi _{1}}+\frac{R_{{31}}}{\phi _{3}}-\frac{R_{{41}}}{\phi _{4}}+\frac{M_{1}}{\phi _{1}}\right){\mathbf{v}}_{1}+\varrho _{1}\frac{{\rm D}^{1}}{{\rm D}t}{\mathbf{v}}_{1}
\displaystyle+\left(-\frac{R_{{23}}}{\phi _{2}}-\frac{R_{{24}}}{\phi _{2}}-\frac{R_{{25}}}{\phi _{2}}+\frac{R_{{32}}}{\phi _{3}}-\frac{R_{{42}}}{\phi _{4}}+\frac{M_{1}}{\phi _{2}}\right){\mathbf{v}}_{2}-\varrho _{2}\frac{{\rm D}^{2}}{{\rm D}t}{\mathbf{v}}_{2}
\displaystyle+\left(-\frac{R_{{13}}}{\phi _{1}}+\frac{R_{{23}}}{\phi _{2}}-\frac{R_{{31}}}{\phi _{3}}-\frac{R_{{32}}}{\phi _{3}}-\frac{R_{{35}}}{\phi _{3}}-\frac{M_{3}}{\phi _{3}}\right){\mathbf{v}}_{3}-\varrho _{3}\frac{{\rm D}^{3}}{{\rm D}t}{\mathbf{v}}_{3}
\displaystyle+\left(-\frac{R_{{14}}}{\phi _{1}}+\frac{R_{{24}}}{\phi _{2}}+\frac{R_{{41}}}{\phi _{4}}+\frac{R_{{42}}}{\phi _{4}}+\frac{R_{{45}}}{\phi _{4}}-\frac{M_{3}}{\phi _{4}}\right){\mathbf{v}}_{4}+\varrho _{4}\frac{{\rm D}^{4}}{{\rm D}t}{\mathbf{v}}_{4} (39)

written here in its most general form. [149.3.2] It is obtained by demanding that the closure condition should be consistent with the capillary pressure saturation relation obtained in the residual decoupling limit (see [22, 23, 24]).