Abstract. A numerical multifractal analysis was performed for five river networks extracted from Calabrian natural basins represented on 1:25000 topographic sheets. The spectrum of generalised fractal dimensions, D(q), and the sequence of mass exponents, τ(q), were obtained using an efficient generalised box-counting algorithm. The multi-fractal spectrum, f(α), was deduced with a Legendre transform. Results show that the nature of the river networks analysed is multifractal, with support dimensions, D(0), ranging between 1.76 and 1.89. The importance of the specific number of digitised points is underlined, in order to accurately define, the geometry of river networks through a direct generalised box-counting measure that is not influenced by their topology. The algorithm was also applied to a square portion of the Trionto river network to investigate border effects. Results confirm the multifractal behaviour, but with D(0) = 2. Finally, some open mathematical problems related to the assessment of the box-counting dimension are discussed. Keywords: River networks; measures; multifractal spectrum

[1] Two fixed-size algorithms were used to investigate natural river networks: the boxcounting method and the sandbox method. The first produces good results for nonnegative moment orders, q, but suffers from border effects. The second solves border problems and is particularly adapted to negative moment orders for the reconstruction of the right side of the multifractal spectrum, f(a). In the box-counting applications the influence of the mesh number was investigated in the range 20 to 100, for moment orders from 0 to 10, showing that the error in the assessment of the generalized fractal dimensions is not greater than 3%. The sandbox method was applied to the river networks for the first time, and the theoretical expression of the information entropy was derived. Results were obtained also for negative moment orders, showing that the analyzed river networks are multifractal and non-plane-filling structures. The sandbox method avoids border effects, as verified through an application to a square portion extracted from a river network. Result analysis suggests a dependence of the f(a) spectra from the lithologic characteristics of the source rocks.

Abstract:We study the scale dependence of the saturated hydraulic conductivity K s through the effective porosity n e by means of a newly developed power-law model (PLM) which allows to use simultaneously measurements at different scales. The model is expressed as product between a single PLM (capturing the impact of the dominating scale) and a characteristic function κ ⋆ accounting for the correction because of the other scale(s). The simple (closed form) expression of the κ ⋆ -function enables one to easily identify the scales which are relevant for K s . The proposed model is then applied to a set of real data taken at the experimental site of Montalto Uffugo (Italy), and we show that in this case two (i.e. laboratory and field) scales appear to be the main ones. The implications toward an important application (solute transport) in Hydrology are finally discussed.

[1] Diverging radial flow takes place in a heterogeneous porous medium where the log conductivity Y ¼ ln K is modeled as a stationary random space function (RSF). The flow is steady, and is generated by a fully penetrating well. A linearly sorbing solute is injected through the well envelope, and we aim at computing the average flux concentration (breakthrough curve). A relatively simple solution for this difficult problem is achieved by adopting, similar to Indelman and Dagan (1999), a few simplifying assumptions: (i) a thick aquifer of large horizontal extent, (ii) mildly heterogeneous medium, (iii) strongly anisotropic formation, and (iv) large Peclet number. By introducing an appropriate Lagrangian framework, three-dimensional transport is mapped onto a one-dimensional domain ( , t) where and t represent the fluid travel and current time, respectively. Central for this approach is the probability density function of the RSF that is derived consistently with the adopted assumptions stated above. Based on this, it is shown that the travel time can be regarded as a Gaussian random variable only in the far field. The breakthrough curves are analyzed to assess the impact of the hydraulic as well as reactive parameters. Finally, the travel time approach is tested against a forced-gradient transport experiment and shows good agreement.

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