Distribution functions dimensionality of soil pore and particle sizes

The argument that soil volumetric water content is proportional to the volume of the parallel body in fractal geometry has been used in the literature to give an explanation of the power representation of the moisture characteristic, which relates water content and soil water pressure by using Laplace's law, which, in turn, relates soil water pressure to pore size; the power is equal to the codimension, i.e. the difference between the Euclidean dimension of space, in which the soil is imbedded, and its fractal dimension. This assessment is only valid when the absolute value of soil water pressure is high, i.e. when the pore size is small. Using this result to estimate the soil fractal dimension from the experimental moisture characteristic has yielded unacceptable fractal dimension values, which result from large pore size values in the data; the precise boundary between large and small pore size values is difficult to assess. Therefore, an approach is being proposed which estimates the fractal dimension based on the fact that the pore number and pore volume distribution functions accept density functions, establishing a link between them. It is imposed that the microcanonical distributions supplied by fractal formalism, valid for small pore size values, satisfy the relationship established between its densities. The empirical generalization of the pore volume microcanonical distribution over the full pore size domain is called canonical distribution; in the latter the exponent is not necessarily equal to the codimension. By satisfying the theoretical boundaries of the fractal dimension based on the canonical distribution, it is possible to obtain an integral expression to calculate the fractal dimension, called an integral fractal dimension due to the way it is calculated. The integral fractal dimension allows to move through distinct distribution functions, such as Gauss's, Cauchy's, and Brutsaert's symmetrical logistics. The integral fractal dimension is the link between asymmetrical functions, such as the double exponential, zeta, and asymmetrical logistics. Since the van Genuchten distribution is widely used in soil hydrology, it is analyzed using different relationships among its exponents from the 660 soil types found in the GRIZZLY. database; it is shown that the integral dimension function stays constant, although the product of the exponents depends on the accepted relationship among them. This supports the fact that the fractal dimension is a soil property, and should not depend on the accepted theoretical curve that represents the moisture characteristic.