Time-fractional particle deposition in porous media

In the percolation process where fluids carry small solid particles, particle deposition causes a real-time permeability change of the medium as the swarm of particles propagates along the medium. Then the permeability change influences percolation and deposition behaviors as a feedback. This fact triggers memory effect in the deposition dynamics, which means the particulate transport and deposition behaviors become history-dependent. In this paper, we conduct the time-fractional generalization of the classical phenomenological model of particle deposition in porous media to incorporate the memory effect. We tested and compared the effects of employing different types of fractional operators, i.e. the Riemann–Liouville type, the Hadamard type and the Prabhakar type. Numerical simulation results show that the system behaviors vary according to the change of distinct memory kernels in an expected way. We then discuss the physical meaning of the time-fractional generalization. It is shown that different types of fractional operators unanimously ground themselves on the local-Newtonian time transformation in a complex system, which is equivalent to a class of history integrals. By the introduction of various memory kernels, it enables the model to more powerfully fit and approximate observed data. Further, the fundamental meaning of this work is not to show which fractional operator is 'better', but to argue collectively the legitimacy and practicality of a non-Markovian particle deposition dynamics in porous media, and in fact it is admissible to a bunch of memory kernels which differ greatly from each other in functional forms. Hopefully the presented generalized mass conservation formalism offers a broader framework to investigate transport problems in porous media. (paper)