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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 \sigma _{{\mathbb{W}\mathbb{O}}} 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

\displaystyle\widetilde{\mathrm{Ca}}_{i}^{{\mathrm{OPC}}}=\frac{k\,\mathrm{d}\! P_{i}}{\sigma _{{\mathbb{W}\mathbb{O}}}\phi\, L}\qquad i=\mathbb{W},\mathbb{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

\displaystyle\widetilde{\mathrm{Ca}}_{i}^{{\mathrm{MB}}}=\frac{k\, k^{r}_{i}(S)\,\mathrm{d}\! P_{i}}{\sigma _{{\mathbb{W}\mathbb{O}}}\,\phi\, L_{{\mathrm{cl}}}}\qquad i=\mathbb{W},\mathbb{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 L_{{\mathrm{cl}}}. [5.2.0.1] Solving for L_{{\mathrm{cl}}} gives L_{{\mathrm{cl}}}=L_{{\mathrm{cl}}}(S,\widetilde{\mathrm{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/\sigma _{{\mathbb{W}\mathbb{O}}}. [5.2.0.4] This derivation shows that the influence of {P_{\mathrm{c}}}(S) on L_{{\mathrm{cl}}}(S) 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\approx L_{{\mathrm{cl}}} as a length scale of clusters.

[5.2.1.1] More recently this cluster length L_{{\mathrm{cl}}} was discussed using \mathrm{Ca} instead of \widetilde{\mathrm{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

\displaystyle L_{{\mathrm{cl}}}\approx L_{i}(S,v_{i})=\frac{k\, k^{r}_{i}(S)|{P_{\mathrm{c}}}(S)|}{\mu _{i}\,\phi\, v_{i}},\quad i=\mathbb{W},\mathbb{O} (31)

where the length scale L_{i}(S,v_{i}) is obtained from the macroscopic force balance by setting F_{i}(S,v_{i},L)=1 in eq. (27). [5.2.1.3] The capillary correlation from [5, eq. (7)] is defined following eq. (27b) as (i=\mathbb{W},\mathbb{O})

\displaystyle\mathrm{Ca}_{i}^{{\mathrm{AGOKB}}}=\frac{\mu _{i}\,\phi\, v_{i}\, L_{{\mathrm{cl}}}(S,v,\mathbf{\Pi})}{k^{{**}}(S,v,\mathbf{\Pi})\,{P_{\mathrm{c}}}^{{**}}(S,v,\mathbf{\Pi})} (32)

by replacing the macroscopic length L with the mesoscopic L_{{\mathrm{cl}}}, the effective permeability k\, k^{r}_{i}(S) with a computed permeability k^{{**}}=k^{{**}}(S,v,\mathbf{\Pi}) and the macroscopic capillary pressure |{P_{\mathrm{c}}}(S)| with a computed pore scale capillary pressure {P_{\mathrm{c}}}^{{**}}={P_{\mathrm{c}}}^{{**}}(S,v,\mathbf{\Pi}). [5.2.1.4] The quantities k^{{**}} and {P_{\mathrm{c}}}^{{**}} are obtained from pore scale imaging of \mathbb{P}, \mathbb{W} and \mathbb{O} by computations based on digital image analysis. [5.2.1.5] Their values depend on numerous numerical and computational parameters summarized as \mathbf{\Pi}. [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

\displaystyle k^{{**}}(S,v,\mathbf{\Pi}) \displaystyle\approx k\, k^{r}_{i}(S) (33a)
\displaystyle{P_{\mathrm{c}}}^{{**}}(S,v,\mathbf{\Pi}) \displaystyle\approx{P_{\mathrm{c}}}(S) (33b)

independent of v and \mathbf{\Pi}. [5.2.1.8] If this holds true, then inserting eq. (31) is expected to give

\displaystyle\mathrm{Ca}_{i}^{{\mathrm{AGOKB}}}\approx 1 (34)

provided L_{{\mathrm{cl}}}\approx L_{i}(S,v_{i}) holds true. [5.2.1.9] Measuring the cluster length L_{{\mathrm{cl}}} from fast X-ray computed microtomography [5, Fig. 1] finds values of

\displaystyle\mathrm{Ca}_{i}^{{\mathrm{AGOKB}}}\approx 10 (35)

from saturation weighted averaging of the cluster size distribution.

[5.2.2.1] Note also, that the length scale L_{i}(S,v_{i}) may often fall in between \ell and L, but it can also exceed beyond these limits. [5.2.2.2] In fact 0\leq L_{i}<\infty in general. [5.2.2.3] The length scale L_{i} 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 L_{{\mathrm{cl}}} 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.

Figure 1: Cross section of 2.6mm\times 2.5mm (632\times 607 pixel, 16 bit gray values) at 4.1\mum resolution from a 3-dimensional computertomographic image of a VitraPOR P2 Robu® sintered glass specimen. Matrix \mathbb{M} is shown in grey, pores \mathbb{P} are black.