# VI Mesoscopic cluster scale L_cl

[5.1.2.1] As of today there does not seem to exist a rigorous
connection between the microscopic Newton and
Laplace law and the macroscopic generalized Darcy law.
[5.1.2.2] This fact was discussed at length in
[9, 32, 40, 16, 17, 20] and it
is the reason why the microscopic interfacial tension
σWO does not appear explicitly
in the macroscopic capillary number in eq. (28).
[5.1.2.3] Nevertheless numerous authors have mingled
pore and sample scale in an attempt to discuss
nonpercolating fluid
parts, mesoscopic clusters or trapped ganglia.
[5.1.2.4] A classic example, that led to some confusion, is
given in [41]
where Darcy’s law for single phase flow
(20) is inserted into
eq. (19) to write

Ca~iOPC=kdPiσWOϕLi=W,O | | (29) |

replacing velocity and viscosity by permeability,
porosity and pressure gradient.
[5.1.2.5] Subsequently [4, eq. (9)] used
the generalized Darcy law eq. (23)
in eq. (19) to obtain

Ca~iMB=kkirSdPiσWOϕLcli=W,O | | (30) |

a pore-scale capillary “number” that is now a function of
saturation S.
[5.1.2.6] They then interpret this expression
as a saturation dependent “critical”
capillary number for mobilization of
trapped oil ganglia with linear extent Lcl.
[5.2.0.1] Solving for Lcl gives Lcl=LclS,Ca~i.
[5.2.0.2] Such approaches were critically examined in
[9, 32, 40].
[5.2.0.3] The problem with eq. (30) emerges by noting that
the same relation (30) can be obtained from the
equality between the expression in (27a)
and expression (27b) by using
eq. (24) and multiplying with
1/σWO.
[5.2.0.4] This derivation shows that the
influence of PcS on LclS
is lost.
[5.2.0.5] More importantly, it is clear from eqs.
(22a) and (22b)
that the generalized Darcy law requires
pathconnected and percolating phases.
[5.2.0.6] Its application to disconnected trapped
phases is questionable at least as long
as cross terms are not included into the analysis
[42, 43].
[5.2.0.7] This casts some doubt on the interpretation
of L≈Lcl as a length scale of clusters.

[5.2.1.1] More recently this cluster length Lcl was
discussed using Ca instead of Ca~
in [5] following [40].
[5.2.1.2] The idea is to assume that mesoscopic (nonpercolating)
clusters or trapped ganglia are roughly of size

Lcl≈LiS,vi=kkirSPcSμiϕvi,i=W,O | | (31) |

where the length scale LiS,vi
is obtained from the macroscopic force balance by setting
FiS,vi,L=1 in eq. (27).
[5.2.1.3] The capillary correlation from [5, eq. (7)]
is defined following eq. (27b) as (i=W,O)

CaiAGOKB=μiϕviLclS,v,Πk**S,v,ΠPc**S,v,Π | | (32) |

by replacing the macroscopic length L with
the mesoscopic Lcl, the effective permeability
kkirS with a computed permeability
k**=k**S,v,Π and the macroscopic
capillary pressure PcS with a computed
pore scale capillary pressure Pc**=Pc**S,v,Π.
[5.2.1.4] The quantities k** and Pc**
are obtained from pore scale imaging of P, W and O
by computations based on digital image analysis.
[5.2.1.5] Their values depend on numerous numerical and computational
parameters summarized as Π.
[5.2.1.6] Examples are segementation thresholds, lattice constants
or density functional parametrizations used by [5]
to replace more conventional computational fluid dynamics approaches.
[5.2.1.7] Within the limits of applicability of the
macroscopic constitutive laws (23) and
(24) such computational approaches are
expected to yield

k**S,v,Π | ≈kkirS | | (33a) |

Pc**S,v,Π | ≈PcS | | (33b) |

independent of v and Π.
[5.2.1.8] If this holds true, then
inserting eq. (31) is expected to give

provided Lcl≈LiS,vi holds true.
[5.2.1.9] Measuring the cluster length Lcl
from fast X-ray computed microtomography
[5, Fig. 1] finds values of

from saturation weighted averaging of the
cluster size distribution.

[5.2.2.1] Note also, that the length scale LiS,vi may often fall
in between ℓ and L, but it can also exceed
beyond these limits.
[5.2.2.2] In fact 0≤Li<∞ in general.
[5.2.2.3] The length scale Li cannot be considered
a new mesoscopic length scale, because it is not
derived from a new mesoscopic constitutive law.
[page 6, §0]
[6.1.0.1] Mesoscopic laws for Lcl defined as the
saturation weighted average of the distribution
of cluster sizes, are also lacking at present.
[6.1.0.2] Although constitutive laws for disconnected fluids have been
proposed in [19] and discussed in
[21] the present article will stay
within the confines of the traditional
Darcy based constitutive theory.