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IV Microscopic pore scale l

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

\text{(viscous pressure gradient)}=\mu|\widetilde{\nabla}\widetilde{\mathbf{v}}|\approx\mu\frac{|\widetilde{\mathbf{v}}(\widetilde{\mathbf{x}})-\widetilde{\mathbf{v}}(\widetilde{\mathbf{x}}_{{\mathrm{wall}}})|}{|\widetilde{\mathbf{x}}-\widetilde{\mathbf{x}}_{{\mathrm{wall}}}|} (16)

where \mu is the fluid viscosity, \widetilde{\nabla}\widetilde{\mathbf{v}} is the phase velocity gradient {}^{1} , and \widetilde{\mathbf{v}}(\widetilde{\mathbf{x}},t)=\widetilde{\mathbf{v}}(\widetilde{\mathbf{x}}) is the phase velocity for stationary flow.
1: In general \widetilde{\nabla}\widetilde{\mathbf{v}}_{i} is a tensor of rank 2 and \mu 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

\displaystyle\text{(capillary pressure)}=\sigma _{{\mathbb{W}\mathbb{O}}}\kappa (17)

where \sigma _{{\mathbb{W}\mathbb{O}}} is the interfacial tension and \kappa the interfacial mean curvature in thermodynamic equilibrium between the two phases. [4.1.0.2] Using the same scale in both laws

|\widetilde{\mathbf{x}}-\widetilde{\mathbf{x}}_{{\mathrm{wall}}}|\approx\kappa^{{-1}}\approx\ell=\text{(pore diameter)}, (18a)

approximating \widetilde{\mathbf{v}}(\widetilde{\mathbf{x}}) by its spatial average \widetilde{\mathbf{v}} as

\widetilde{\mathbf{v}}_{i}(\widetilde{\mathbf{x}})\approx\widetilde{\mathbf{v}}_{i}=\frac{1}{|\mathbb{P}\cap\mathbb{S}|}\int\limits _{\mathbb{S}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}(\widetilde{\mathbf{y}})\widetilde{\mathbf{v}}_{i}(\widetilde{\mathbf{y}})\,\mathrm{d}^{3}\widetilde{\mathbf{y}} (18b)

and using

\widetilde{\mathbf{v}}_{i}(\widetilde{\mathbf{x}}_{{\mathrm{wall}}})=0 (18c)

for both phases i=\mathbb{W},\mathbb{O} one arrives at the microscopic capillary number

\displaystyle\widetilde{\mathrm{Ca}}_{i}=\frac{\mu _{i}|\widetilde{\mathbf{v}}_{i}|}{\sigma _{{\mathbb{W}\mathbb{O}}}} (19)

for phase i=\mathbb{W},\mathbb{O}. [4.1.0.3] Note that \sigma _{{\mathbb{W}\mathbb{O}}}/\mu _{i} is a characteristic flow velocity that depends only on the fluid properties. [4.1.0.4] As a consequence the microscopic capillary number \widetilde{\mathrm{Ca}} depends only on fluid properties, but is independent of the pore space properties.

[4.1.1.1] For a derivation of \widetilde{\mathrm{Ca}}_{i} from the pore scale equations of motion, see [32, 9].