Experimental and Theoretical Memory Diffusion of Water in Sand

The basic equations used to study the fluid diffusion in porous media have been set by Fick and Darcy in the mid of the XIXth century but some data on the flow of fluids in rocks exhibit properties which may not be interpreted with the classical theory of propagation of pressure and fluids in porous media. Concerning the fluids and the flow, some fluids carry solid particles which may obstruct some of the pores diminishing their size or even closing them, some others may chemically and physically react with the medium enlarging the pores; so permeability changes during time and the flow occurs as if the medium had a memory, intending that at any instant the process of diffusion is also affected by the previous local value of pressure and flow of the fluid. This phenomenon should be taken into account when writing equations for diffusion of fluids in porous media. The scope of our work is to show, with experimental data, that during the diffusion the permeability of sand layers may decrease due to reassessment of the grains and consequent compaction. We also provide a memory model for diffusion of fluids in porous media, obtained by introducing a derivative of fractional order n in the classical equations, which fits well the flux rate observed in our laboratory experiments of diffusion of water in sand. The classic theory, in the case of constant diffusivity, with constant boundary and initial conditions, would give a constant flux contrary to the results of our laboratory experiments and so one would have to introduce in the equations a time variable diffusivity which is a priory unknown and would have to be determined monitoring the permeability changes caused by the flux in the sand. We have also seen that, with the boundary and initial conditions used, the relaxation time of the flux, that is the time to reach stability, is about 10 hours which in turn implies that the compaction of the sand in the sample has the same relaxation time. However in terms of the memory model the flux and the associated relaxation time are now defined by two parameters, and not only one as in the classic theory; the parameters are the order of fractional derivative n and the pseudodiffusivity. Finally we show that the flux rate variations observed during the experiments are compatible with the compaction of sand, due to the amount of fluid which went through the grains locally, and therefore with the reduction of porosity, in fact in all experiments we observed that flux decreases in time to about 70 percent of initial value and that the volume of sand reduces of about 3 percent; moreover, using empirical Fair and Hatch law for permeability, the sand volume and flux reductions seem compatible; which proves that mechanical compaction occurring during diffusion is caused by the permeability changes which in turn cause the flux variations.