Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media. (English) Zbl 1355.76066
Summary: In the propagation of nonlinear waves in fluid saturated porous media, a key role is played by the Darcy law. In many cases one should take into account a realistic spatial or temporal variability of rock parameters, as permeability, porosity, diffusivity\(\ldots \). To this purpose we here introduce a nonlocal Modified Darcy Law (MDL) by considering a fractional derivative generalization of the classical case. As an application we here study the effect of a generalized Darcy law on the propagation of nonlinear thermoelastic waves in porous media. In more details, we discuss an application of this MDL to an early Natale and Salusti model about quick and strong pressure and temperature waves in fluid saturated rocks. The original model leads to the Burgers equation, while with the MDL here we obtain a nonlocal formulation of the Burgers equation. We moreover find the analytic solution in the case that diffusion plays a secondary role. With this nonlocal model we obtain a richer analysis of realistic characteristic of such transient phenomena, in particular the spatial delays in wave propagation.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 26A33 | Fractional derivatives and integrals |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
Keywords:
nonlocal Darcy law; nonlinear thermoelastic waves; nonlinear fractional differential equationsReferences:
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