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IV Microscopic pore scale l

[] Consider stationary flow of a fluid inside the pore space P. [] 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]    [] 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. [] Using the same scale in both laws

x~-x~wallκ-1=(pore diameter),(18)

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


and using


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


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

[] For a derivation of Ca~i from the pore scale equations of motion, see [32, 9].