Anomalous effects during thermal displacement in porous media under non-local thermal equilibrium

Robust and accurate mathematical models describing fluid and heat transport in naturally occurring geological media can be challenging to formulate due to the spatial heterogeneities occurring at many different scales. Thus, a majority of the widely presented continuum-based mathematical models may not be completely adequate for predicting fluid or/and heat transport in such systems. In this work, two nonlocal temporal constitutive flux relationships are employed to present a novel mathematical model describing the fluid flow and heat transport through a porous medium. Subsequently, existing numerical schemes and well-established numerical discretization methods are applied to solve the resulting set of fractional equations. Parameter sensitivity analysis is presented to illustrate the effect of introduced phenomenological parameters on the fluid and heat transport behavior in the porous medium. Results show that the order(s) of fractional derivative plays a significant role in the pressure and rock temperature evolution with a minimal effect observed in the fluid temperature evolution. Furthermore, the magnitude of the heat transfer coefficient between the fluid and rock phase determines how fast the rock temperature approaches the fluid temperature. The presented mathematical model would find widespread applications in geothermal reservoirs and analyzing temperature profiles in fractured reservoir rocks.