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VI Mesoscopic cluster scale L_cl

[] 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. [] 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). [] Nevertheless numerous authors have mingled pore and sample scale in an attempt to discuss nonpercolating fluid parts, mesoscopic clusters or trapped ganglia. [] 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


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


a pore-scale capillary “number” that is now a function of saturation S. [] They then interpret this expression as a saturation dependent “critical” capillary number for mobilization of trapped oil ganglia with linear extent Lcl. [] Solving for Lcl gives Lcl=LclS,Ca~i. [] Such approaches were critically examined in [9, 32, 40]. [] 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. [] This derivation shows that the influence of PcS on LclS is lost. [] More importantly, it is clear from eqs. (22a) and (22b) that the generalized Darcy law requires pathconnected and percolating phases. [] Its application to disconnected trapped phases is questionable at least as long as cross terms are not included into the analysis [42, 43]. [] This casts some doubt on the interpretation of LLcl as a length scale of clusters.

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


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


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. [] The quantities k** and Pc** are obtained from pore scale imaging of P, W and O by computations based on digital image analysis. [] Their values depend on numerous numerical and computational parameters summarized as Π. [] Examples are segementation thresholds, lattice constants or density functional parametrizations used by [5] to replace more conventional computational fluid dynamics approaches. [] Within the limits of applicability of the macroscopic constitutive laws (23) and (24) such computational approaches are expected to yield


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


provided LclLiS,vi holds true. [] 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.

[] Note also, that the length scale LiS,vi may often fall in between and L, but it can also exceed beyond these limits. [] In fact 0Li< in general. [] 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]    [] Mesoscopic laws for Lcl defined as the saturation weighted average of the distribution of cluster sizes, are also lacking at present. [] 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×2.5mm (632×607 pixel, 16 bit gray values) at 4.1μm resolution from a 3-dimensional computertomographic image of a VitraPOR P2 Robu® sintered glass specimen. Matrix M is shown in grey, pores P are black.