A memory model of sedimentation in water reservoirs

We consider a one-dimensional model of water reservoir, where the sediment is diffusing according to the Fourier law modified with the introduction of a derivative of fractional distributed orders as memory formalism. The fractional order is equivalent to a time-varying diffusivity and the distributed orders represent a variety of memory mechanisms to model a sediment with a varied distribution of grain sizes. Using the Laplace transform (LT), we find the solution in the case when the flux is constant at the source and is arbitrarily given at the output. Then, the time-domain solution is obtained by means of a numerical Fourier transform. We apply a one-dimensional simplified model, with the diffusion governed by two parameters, to the Quarto Nuovo (Italy) reservoir, where the flux of sediment at the output is obtained from observed data. It is found that the flux increases when one of the parameters defining the diffusion model, the pseudo-diffusivity, is increasing or when the other parameter defining the diffusion, the order of fractional differentiation, is decreasing. When the latter parameter is nil, one obtains the classic diffusion with maximum flux.