[3.2.2.1] Consider stationary flow of a fluid inside the pore space P.
[3.2.2.2] On the pore scale the viscous forces are given
quantitatively by Newton’s law of internal friction

(viscous pressure gradient)=μ∇~v~≈μv~x~-v~x~wallx~-x~wall | | (16) |

where μ
is the fluid viscosity, ∇~v~
is the phase velocity gradient
1 , and
v~x~,t=v~x~ is the phase velocity
for stationary flow.

1: In general ∇~v~i
is a tensor of rank 2 and μ is a tensor of rank 4
yielding the fluid stress tensor of rank 2.

[page 4, §0]
[4.1.0.1] The capillary forces are quantified by the Young Laplace law as

(capillary pressure)=σWOκ | | (17) |

where σWO is the interfacial tension and κ the
interfacial mean curvature in thermodynamic equilibrium
between the two
phases.
[4.1.0.2] Using the same scale in both laws

x~-x~wall≈κ-1≈ℓ=(pore diameter), | | (18) |

approximating v~x~ by its spatial average v~ as

v~ix~≈v~i=1P∩S∫SχPy~v~iy~d3y~ | | (18) |

and using

for both phases i=W,O
one arrives at the microscopic capillary number

for phase i=W,O.
[4.1.0.3] Note that σWO/μi is a characteristic flow velocity
that depends only on the fluid properties.
[4.1.0.4] As a consequence the microscopic capillary number Ca~ depends
only on fluid properties, but is independent of the
pore space properties.