Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks

Abstract

We study a fractional time derivative generalization of a previous Natale-Salusti model about nonlinear temperature and pressure waves, propagating in fluid-saturated porous rocks. Their analytic solutions, i.e., solitary shock waves characterized by a sharp front, are here generalized, introducing a formalism that allows memory mechanisms. In realistic wave propagation in porous media we must take into account spatial or temporal variability of permeability, diffusivity, and other coefficients due to the system "history." Such a rock fracturing or fine particulate migration could affect the rock and its pores. We therefore take into account these phenomena by introducing a fractional time derivative to simulate a memory-conserving formalism. We also discuss this generalized model in relation to the theory of dynamic permeability and tortuosity in fluid-saturated porous media. In such a realistic model we obtain exact solutions of Burgers' equation with time fractional derivatives in the inviscid case.