A hyperbolic mathematical modeling for describing the transition saturated/unsaturated in a rigid porous medium

A New Constitutive Relation for Describing the Flow through Porous Media with Unsaturated–Saturated Transition

“…Essentially, the present work proposes a very convenient constitutive relation to describe the filling up of a porous matrix by a fluid, considering a Mixture Theory approach. This relation not only assures that the hyperbolicity of the system is maintained even when saturation is reached, like in Martins-Costa et al [3], but also it consists of a continuous and differentiable function with an increasing function as its first derivative. This important feature allows controlling the (very low) quantity of fluid fraction that may surpass the porosity.…”

“…The new constitutive relation proposed in this work to describe the filling up of a porous matrix by a fluid is a continuous and differentiable function and its first derivative is an increasing function. Besides keeping the good feature of assuring the problem hyperbolicity regardless the fluid fraction value, this relation is more realistic than the previous one [3]. Its advantage is that it allows increasing the porous medium resistance to more fluid inlet when the fluid fraction reaches the porosity, preventing the fluid fraction from being significantly greater than the porosity.…”

“…(Actually, the work [1] is a generalization of [2].) A subsequent work [3] proposes a constitutive relation allowing a small supersaturation. Its advantage over the previous work [1] is that the hyperbolicity of the problem is never lost, even when saturation is reached.…”

“…Although these two mentioned previous works adequately describe the transition from unsaturated flow to saturated one and the latter assures that the hyperbolicity feature is always maintained, in Martins-Costa el al. [3] the first derivative of the constitutive relation is a discontinuous function. Its second part is actually a straight line, so that it is not possible to increase the porous matrix resistance to the incoming of more fluid when the fluid fraction reaches the porosity.…”

Global asymptotics of filtration in porous media

“…Depending on the structure and properties of the porous medium and suspended particles, diffusion, electrostatic, gravitational and hydrodynamic forces, etc., affect the particles retention. [4][5][6]. If the distributions of particle sizes and pores overlap, the size-exclusion mechanism of particle retention plays the main role: suspended particles freely pass through large pores and get stuck in small pores that are smaller than the particle diameter.…”