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# 3 Formulation of the Model

## 3.1 General Balance Laws

[3.1.1.1] 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. [3.1.1.2] The volume fractions of the subsets FiS,i=1,2,3,4 and MS are denoted as ϕix,t. [3.1.1.3] Let ϕ denote the porosity (volume fraction of P). [3.1.1.4] Volume conservation implies the relations

 ϕ1+ϕ2+ϕ3+ϕ4+ϕ5 =1 (16a) S1+S2+S3+S4 =1 (16b) 1-ϕ =ϕ5 (16c)

where ϕi=ϕSi i=1,2,3,4 are volume fractions, and Si are saturations. [3.1.1.5] The water saturation is defined as SW=S1+S2 and the oil saturation as SO=S3+S4.

[3.1.2.1] The general law of mass balance is again given by eq. (1) with i=1,2,3,4. [3.1.2.2] It provides now four equations instead of two. [3.1.2.3] The general law of momentum balance is given by eq. (2), also now with i=1,2,3,4 instead of i=W,O as before.

## 3.2 General Constitutive Assumptions

[3.1.3.1] As in the traditional theory the porous medium is again assumed to be macroscopically homogeneous so that eq. (4) holds. [3.1.3.2] The flows are slow, and hence also eq. (7) continues to hold without change. [3.1.3.3] For incompressible fluids one has now

 ϱ1⁢x,t =ϱW (17a) ϱ2⁢x,t =ϱW (17b) ϱ3⁢x,t =ϱO (17c) ϱ4⁢x,t =ϱO (17d)

analogous to eq. (5a).

## 3.3 Viscous drag

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

 mi=∑j=15Ri⁢j⁢vj-vi (18)

where the resistance coefficient Rij quantifies the viscous coupling between phase i and j. [3.1.4.2] For the rigid rock matrix [page 4, §0]   v5=0. [4.0.0.1] Hence -Ri5vi is the momentum transfer from the wall into phase i. [4.0.0.2] Then

 m1 =R13⁢v3-v1+R14⁢v4-v1-R15⁢v1 (19a) m2 =R23⁢v3-v2+R24⁢v4-v2-R25⁢v2 (19b) m3 =R31⁢v1-v3+R32⁢v2-v3-R35⁢v3 (19c) m4 =R41⁢v1-v4+R42⁢v2-v4-R45⁢v4 (19d)

where R12=0 and R34=0 was used because there is no common interface and hence no direct viscous interaction between these phase pairs. [4.0.0.3] Each Rij is a 3×3-matrix.

## 3.4 Capillarity

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

 F1 =ϱ1⁢g (20a) F2 =ϱ2⁢g+Fc⁢W (20b) F3 =ϱ3⁢g (20c) F4 =ϱ4⁢g+Fc⁢O (20d)

contrary to eqs. (8). [4.0.1.2] The capillary body forces FcW,FcO are responsible for keeping the trapped fluids inside the medium. [4.0.1.3] They are obtained as gradients of capillary potentials

 Fc⁢W =-∇⁡ΠcW (21a) Fc⁢O =-∇⁡ΠcO (21b)

where the capillary potentials ΠcW,ΠcO are defined as

 ΠcW =Πa*-Πa⁢S1-α (22a) ΠcO =Πb*-Πb⁢S3-β (22b)

with constants Πa*,Πb*,Πa,Πb and exponents α,β>0.

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

 Σ1 =-P1⁢1 (23a) Σ3 =-P3⁢1 (23b)

where P1 and P3 are the fluid pressures. [4.0.2.2] The stress tensor Σ2,Σ4 for the nonpercolating phases cannot be specified in the same way because the forces cannot propagate in nonpercolating phases. [4.0.2.3] 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. [4.0.2.4] This suggests an Ansatz [4]

 Σ2 =-P3⁢1+σW⁢Oϕ⁢∂⁡A32∂⁡S2⁢1 (24a) Σ4 =-P1⁢1+σW⁢Oϕ⁢∂⁡A41∂⁡S4⁢1 (24b)

where σWO is the oil-water interfacial tension, and the unknowns A32x,t,A41x,t are the interfacial areas per unit volume of porous medium between phases 3 and 2, resp. 4 and 1. [4.1.0.1] (The factor ϕ arises from the definition of Aij per unit volume of porous medium.) [4.1.0.2] To avoid equations of motion for the unknowns A32 and A41 it is assumed that geometrical relations of the form

 A32 =A2*⁢S2γ (25a) A41 =A4*⁢S4δ (25b)

hold, where A2*,A4* are prefactors assumed to be constant. [4.1.0.3] Thus for the nonpercolating phases

 Σ2 =-P3+γ⁢P2*⁢S2γ-1⁢1 (26a) Σ4 =-P1+δ⁢P4*⁢S4δ-1⁢1 (26b)

will be used below. [4.1.0.4] Here

 P2* =A2*⁢σW⁢Oϕ (27a) P4* =A4*⁢σW⁢Oϕ (27b)

are constants.

[4.1.1.1] The mass transfer rates must depend on rates of saturation change. [4.1.1.2] They are here assumed to be

 M1 =-M2=η2ϕϱW(S2-S2*SW*-SW)∂⁡SW∂⁡t (28a) M3 =-M4=η4ϕϱO(S4-S4*SO*-SO)∂⁡SO∂⁡t (28b)

where η2,η4 are constants. [4.1.1.3] The parameters SW*,SO*, S2*,S4* are defined by

 SW* =1-SO⁢im2⁢1+tanh⁡τW⁢∂⁡SW∂⁡t (29a) +SW⁢dr2⁢1-tanh⁡τW⁢∂⁡SW∂⁡t SO* =1-SW⁢dr2⁢1+tanh⁡τO⁢∂⁡SO∂⁡t (29b) +SO⁢im2⁢1-tanh⁡τO⁢∂⁡SO∂⁡t S2* =SW⁢dr2⁢1-tanh⁡τ2⁢∂⁡SW∂⁡t (29c) S4* =SO⁢im2⁢1-tanh⁡τ4⁢∂⁡SO∂⁡t (29d)

[page 5, §0]   where SWdr,SOim are limiting saturations for S2,S4 and τW,τO,τ2,τ4 are equilibration time scales for reaching capillary equilibrium. [5.0.0.1] For simplicity

 τ=τW=τO=τ2=τ4 (30)

will be assumed below.