Use of fractal models to define the scaling behavior of the aquifers' parameters at the mesoscale

AbstractWe present an experimental study aiming at the identification of the hydraulic conductivity in an aquifer which was packed according to four different configurations. The conductivity was estimated by means of slug tests, whereas the other parameters were determined by the grain size analysis. Prior to the fractal we considered the dependence of the conductivity upon the porosity through a power (scaling) law which was found in a very good agreement within the range from the laboratory to the meso-scale. The dependence of the conductivity through the porosity was investigated by identifying the proper fractal model. Results obtained provide valuable indications about the behavior, among the others, of the tortuosity, a parameter playing a crucial role in the dispersion phenomena taking place in the aquifers

A Semi-Automatic Numerical Algorithm for Turing Patterns Formation in a Reaction-Diffusion Model

The Turing pattern formation is modeled by reaction - diffusion (RD) type partial differential equations , and it plays a crucial role in ecological studies. Big data analytics and suitable frameworks to manage and predict structures and configurations are mandatory. The processing and resolution procedures of mathematical models relies upon numerical schemes, and concurrently upon the related automated algorithms. Starting from a RD model for vegetation patterns, we propose a semi-automatic algorithm based on a smart numerical criterion for observing ecological reliable results. Numerical experiments are carried out in the case of spot's formations

Scaling behaviour of braided active channels: a Taylor’s power law approach

none9At a channel (reach) scale, braided channels are fluvial, geomorphological, complex systems that are characterized by a shift of bars during flood events. In such events water flows are channeled in multiple and mobile channels across a gravel floodplain that remain in unmodified conditions. From a geometrical point of view, braided patterns of the active hydraulic channels are characterized by multicursal nature with structures that are spatially developed by either simple- and multi-scaling behavior. Since current studies do not take into account a general procedure concerning scale measurements, the latter behavior is still not well understood. The aim of our investigation is to analyze directly, through a general procedure, the scaling behavior of hydraulically active channels per transect and per reach analyzed. Our generalized stochastic approach is based on Taylor’s law, and the theory of exponential dispersion distributions. In particular, we make use of a power law, based on the variance and mean of the active channel fluctuations. In this way we demonstrate that the number of such fluctuations with respect to the unicursal behavior of the braided rivers, follows a jump-process of Poisson and compound Poisson–Gamma distributions. Furthermore, a correlation is also provided between the scaling fractal exponents obtained by Taylor’s law and the Hurst exponents.Samuele De Bartolo, Stefano Rizzello, Ennio Ferrari, Ferdinando Frega, Gaetano Napoli, Raffaele Vitolo, Michele Scaraggi, Carmine Fallico, Gerardo SeverinoDE BARTOLO, Samuele; Rizzello, Stefano; Ferrari, Ennio; Frega, Ferdinando; Napoli, Gaetano; Vitolo, Raffaele; Scaraggi, Michele; Fallico, Carmine; Severino, Gerard

An indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity

Solute transport takes place in heterogeneous porous formations, with the log conductivity, Y = ln K, modeled as a stationary random space function of given univariate normal probability density function (pdf) with mean Y, variance σY2, and integral scale IY. For weak heterogeneity, the above mentioned quantities completely define the first-order approximation of the longitudinal macrodispersivity αL = σY2IY. However, in highly heterogeneous formations, nonlinear effects which depend on the multipoint joint pdf of Y, impact αL. Most of the past numerical simulations assumed a multivariate normal distribution (MVN) of Y values. The main aim of this study is to investigate the impact of deviations from the MVN structure upon αL. This is achieved by using the concept of spatial correlations of different Y classes, the latter being defined as the space domain where Y falls in the generic interval [Y,Y + ΔY]. The latter is characterized by a length scale λ(Y), reflecting the degree of connectivity of the domain (the concept is similar to the indicator variograms). We consider both “symmetrical” and “non-symmetrical” structures, for which λ(Y′) = λ(−Y′) (similar to the MVN), and λ(Y′) ≠ λ(−Y′), respectively, where Y′ = Y − Y. For example, large Y zones may have high spatial correlation, while low Y zones are poorly correlated, or vice versa. The impact of λ(Y) on αL is investigated by adopting a structure model which has been used in the past in order to investigate flow and transport in highly heterogeneous media. It is found that the increased correlation in the low conductive zones with respect to the high ones generally leads to a significant increase in αL, for the same global IY. The finding is explained by the solute retention occurring in low Y zones, which has a larger effect on solute spreading than high Y zones. Conversely, αL decreases when the high conductivity zones are more correlated than the low Y ones. Dispersivity is less affected by the shape of λ(Y) for symmetrical distributions. It is found that the range of validity of the first-order dispersivity, i.e., αL = IYσY2, narrows down for non-symmetrical structures

Stochastic Analysis of Well-Type Flows in Randomly Heterogeneous Porous Formations

Well-type flow takes place in a heterogeneous porous formation where the transmissivity is modelled as a stationary Random Space Function. General expressions for the covariances of the head and flux are obtained and analyzed. In view of the formation identification problem, the equivalent and apparent transmissivity are computed. Finally, it is shown how the general results can be used for practical applications

Macrodispersion by Point-Like Source Flows in Randomly Heterogeneous Porous Media

A pulse of a passive tracer is injected in a porous medium via a point-like source. The hydraulic conductivity K is regarded as a stationary isotropic random space function, and we model macrodispersion in the resulting migrating plume by means of the second-order radial spatial moment Xrr . Unlike previous results, here Xrr is analytically computed in a fairly general manner. It is shown that close to the source macrodispersion is enhanced by the large local velocities, whereas in the far field it drastically reduces since flow there behaves like a mean uniform one. In particular, it is demonstrated that Xrr is bounded between X∞ corresponding to the short-range (far field), and X0 pertaining to the long-range (near-field) correlation in the conductivity field. Although our analytical results rely on the assumption of isotropic medium, they enable one to grasp in a simple manner the main features of macro-dispersion mechanism, therefore providing explicit physical insights. Finally, the proposed model has potential toward the characterization of the spatial variability of K as well as testing more general numerical codes

Uncertainty quantification of unsteady source flows in heterogeneous porous media

Unsteady flow generated by a point-like source takes place into a -dimensional porous formation where the spatial variability of the hydraulic conductivity is modelled within a stochastic framework that regards as a stationary, normally distributed random space function (rsf). As a consequence, the hydraulic head becomes also stochastic, and we aim at quantifying its uncertainty. Towards this aim, we have derived the head covariance by means of a perturbation expansion which regards the variance of the zero mean rsf (hereafter being the ensemble average operator) as a small parameter. The analytical results are expressed in terms of multiple quadratures which are markedly reduced after adopting specific autocorrelation for . This enables one to obtain simple results providing straightforward physical insight into the spatial distribution of as a consequence of the heterogeneity of . In view of those applications (pumping tests) aiming at the identification of the hydraulic properties of geological formations, we have focused on a flow generated by a source of instantaneous and constant strength. The attainment of the large time (steady-state) regime is studied in detail