Comparative study of gravity-driven discharge from reservoirs with translationally invariant and fractal pore networks

Abstract

We perform a comparative study of water discharge from Euclidean and fractal reservoirs. The Euclidean reservoir is modeled by a filter with a translationally invariant cubic network of straight square channels. As a model of fractal reservoir with the scale invariant pore network we use a pre-fractal inverse Menger sponge. The gravity-driven discharge experiments were carried out with two different setups. In the first setup, a porous filter is fixed in container above a lateral exit duct. In the second setup, the exit duct is connected to one of lateral exits of the pore network. In both cases, we found that the squared outflow velocity is a linear function of water table elevation above exit orifice. Hence, the water discharge is governed by the Bernoulli's principle rather than by the Darcy’s law. Furthermore, we found that filters placed above the exit duct have no effect on the discharge rate, because absolute permeabilities of both filters are large enough to supply water into the volume between filter’s bottom and exit orifice. Conversely, when the exit duct is connected to the lateral exit of the pore network, the water discharge from the inverse Menger sponge is faster than from the Euclidean reservoir. Experimental data fittings with a modified Torricelli’s equation suggest that a loss of mechanical energy of flowing water in the fractal pore network is less than in the translationally invariant network of channels. This finding reveals that an effective wetted area along a flow path of least resistance in the self-similar pore network is smaller less than in the periodic network of straight channels. So, the fractal features of the pore network can have a strong impact on hydrological processes in natural reservoirs and karst aquifers which commonly possess statistical scale invariance.