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 Fi⊂S,i=1,2,3,4
and M⊂S 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) 
1ϕ  =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.
3.3 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,
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  =R13v3v1+R14v4v1R15v1 
 (19a) 
m2  =R23v3v2+R24v4v2R25v2 
 (19b) 
m3  =R31v1v3+R32v2v3R35v3 
 (19c) 
m4  =R41v1v4+R42v2v4R45v4 
 (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 3×3matrix.
3.4 Capillarity
[4.0.1.1] In the present approach
the body forces are given by gravity
plus capillary forces
F1  =ϱ1g 
 (20a) 
F2  =ϱ2g+FcW 
 (20b) 
F3  =ϱ3g 
 (20c) 
F4  =ϱ4g+FcO 
 (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
FcW  =∇ΠcW 
 (21a) 
FcO  =∇ΠcO 
 (21b) 
where the capillary potentials ΠcW,ΠcO are
defined as
ΠcW  =Πa*ΠaS1α 
 (22a) 
ΠcO  =Πb*ΠbS3β 
 (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  =P11 
 (23a) 
Σ3  =P31 
 (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  =P31+σWOϕ∂A32∂S21 
 (24a) 
Σ4  =P11+σWOϕ∂A41∂S41 
 (24b) 
where σWO is the oilwater 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+γP2*S2γ11 
 (26a) 
Σ4  =P1+δP4*S4δ11 
 (26b) 
will be used below.
[4.1.0.4] Here
P2*  =A2*σWOϕ 
 (27a) 
P4*  =A4*σWOϕ 
 (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(S2S2*SW*SW)∂SW∂t 
 (28a) 
M3  =M4=η4ϕϱO(S4S4*SO*SO)∂SO∂t 
 (28b) 
where η2,η4 are constants.
[4.1.1.3] The parameters SW*,SO*, S2*,S4*
are defined by
SW*  =1SOim21+tanhτW∂SW∂t 
 (29a) 
 +SWdr21tanhτW∂SW∂t 

SO*  =1SWdr21+tanhτO∂SO∂t 
 (29b) 
 +SOim21tanhτO∂SO∂t 

S2*  =SWdr21tanhτ2∂SW∂t 
 (29c) 
S4*  =SOim21tanhτ4∂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
will be assumed below.