Partial Filling of a Fractal Structure by a Wetting Fluid

Abstract

Certain rocks have been found to be fractal in a sizeable range of length scales ${\ell _2} >\ell>{\ell _1}$. We consider such a rock in equilibrium with a wetting fluid, under an adverse pressure p = pgh ( = bulk fluid density; g = gravitational acceleration). Two lengths are relevant: a) the diameter r(h) of a capillary inside which the fluid climbs up by an amount h; b) the thickness e(h) of the liquid film induced at higher levels h by long range forces (Van der Waals, or other), which is systematically smaller than r. We analyse two families of fractal structures: “iterative pits” and “iterative floes”, and find similar conclusions for both. The most interesting regime corresponds to ${\ell _2} >r>{\ell _1}$. Here we expect that macroscopic pockets of liquid will dominate over contributions from the film, and give a fluid fraction ${{\phi }_{L}} \cong \phi {{({{\ell }_{2}}/r)}^{{3 - D}}}$, where D is the fractal dimension of the surface and J the total porosity. This is to be contrasted with the non fractal case $({\ell _2} = {\ell _1} = \ell )$ where we find, ${{\phi }_{L}} \cong \phi {{(\ell /r)}^{2}}(at r < \ell )$ for two distinct models. We also discuss various transport coefficients (conductance E, permeability $\overline K )$, for these structures, and their qualitative effect on the macroscopic concentration profiles during imbibition.