# V Macroscopic sample scale L

[4.1.2.1] Consider again the stationary limit t→∞
of fluid flow in P.
[4.1.2.2] On the scale of a macroscopic sample,
the viscous forces are dominated by wall friction
and quantified by Darcy’s law for single phase flow

where vD is the magnitude of the (superficial) Darcy velocity,
v=v is the phase velocity of the (interstitial) fluid,
and dP is the magnitude of the viscous pressure drop
across a region of length L.
[4.1.2.3] Here μ is the fluid viscosity, k is the (absolute)
permeability and ϕ the porosity of the porous medium.

[4.1.3.1] The generalization of Darcy’s law to two immiscible fluids
requires some assumptions.
[4.1.3.2] Let Wt0, Ot0 denote the fluid configuration
inside the porous medium at some initial time t0
and consider flooding at constant injection rates
QW≠0, QO=0 or QW=0, QO≠0 with water or oil.
[4.1.3.3] It is observed experimentally and then assumed theoretically that
for long times t→∞ the total volume fractions

| limt→∞WtP=limt→∞SWt=SW=S | | (21a) |

| limt→∞OtP=limt→∞SOt=SO=1-S | | (21b) |

approach constant limiting values.
[4.2.0.1] The values of SW, SO will depend on the phase pressures PW, PO.
[4.2.0.2] Because the injection rates are constant the homogenized
phase velocities vWt, vOt
approach constant values.
[4.2.0.3] Specifically,

| limt→∞vWt=vW, | | | | limt→∞vOt=0 | | (22a) |

| limt→∞vWt=0, | | | | limt→∞vOt=vO | | (22b) |

because QW≠0, QO=0 for water flooding
and QW=0, QO≠0 for oil flooding.
[4.2.0.4] Under these assumptions Darcy’s law is generalized from
single phase flow to two-phase flow as
discussed in [33, 34, 35, 36, 37] to

vD=ϕvi=kkirSμidPiL | | (23) |

where i=W,O indicates the two phases and the
relative permeability functions
kWrS, kOrS quantify the change
in permeability for phase i due to presence of the second phase.
[4.2.0.5] Note that the asymptotic water configuration
W∞ must be path connected and percolating from
inlet and outlet2 in case (22a) and the same holds
for O∞ in case (22b).

2: Limitations and previously unnoticed
implicit assumptions for the validity of
(23) were first addressed in
[15] and
then formulated mathematically and explicitly as
the residual decoupling approximation in
[18, 19, 20].

[4.2.1.1] The asymptotic pressure difference POt-PWtfor t→∞ reflects microscopic capillarity
on macroscales.
[4.2.1.2] The difference is assumed to depend only on saturation

limt→∞POt-PWt=PcS=PbPc^S | | (24) |

but not on the phase velocities
although dynamic capillary
effects have been observed [38, 39].
[4.2.1.3] The function PcS is called capillary pressure
and Pc^S is its dimensionless form.
[4.2.1.4] The functions PcS and kirS are defined
on the interval SWi,1-SOr.
[4.2.1.5] The parameters SWi, SOr, defined as solutions of the equations

| kWrSWi=0 | | (25a) |

| kOr1-SOr=0, | | (25b) |

are the irreducible water saturation SWi and the
residual oil saturation SOr.
[4.2.1.6] Both parameters SWi and SOr are
assumed to be small but nonvanishing, i.e. 0<SWi≪1
and 0<SOr≪1.
[4.2.1.7] With SWi and SOr the parameter Pb in eq. (24)
is defined as

[page 5, §0]
[5.1.0.1] If there is hysteresis, so that the values
Pbim≠Pbdr differ, then
Pb=Pbim+Pbdr/2 will be used.
[5.1.0.2] The dimensionless capillary pressure function
Pc^S can be positive and negative.
[5.1.0.3] The relative permeabilities are positive and monotone functions.

[5.1.1.1] The force balance between the viscous and capillary forces
of macroscale two-phase flow
can now be expressed either as a function of S, vi and L

FiS,vi,L=(viscous pressure drop in phase i)(capillary pressure) | =dPiPO-PW | | | | (27a) |

| =μiϕviLkkirSPcS | | | | (27b) |

| =CaikirSPc^S | | | | (27c) |

| =fiS,Cai | | i=W,O | | (27d) |

or as a function of S and the macroscopic
capillary number Cai in the last two equalities.
[5.1.1.2] The macroscopic capillary number Cai
is defined as in [9] by

Cai=μiϕviLkPbi=W,O | | (28) |

and it depends not only on fluid properties,
but also on properties of the porous medium
such as porosity and permeablity.
[5.1.1.3] Equation (27) seems to be a
new result that has been overlooked so far.
[5.1.1.4] Note that Ca does not explicitly depend on the
interfacial tension σWO between the
two phases and that
the quantity
Ai=kPb/μiϕ
with i=W,O
is dimensionally not a velocity, but a specifc action,
i.e. action per unit mass.
[5.1.1.5] For a derivation of Cai from the traditional
macroscale equations of motion see [32, 9].