# Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

## TL;DR

This paper derives a nonlocal pressure form of the fractional porous medium equation to model superdiffusive moisture movement in porous media, emphasizing the use of the fractional gradient for better representation.

## Contribution

It provides a deterministic derivation of the nonlocal pressure form, highlighting the fractional gradient's role in modeling superdiffusive phenomena.

## Key findings

- The nonlocal pressure form naturally models superdiffusive jump phenomena.
- The fractional gradient operator is more suitable than the fractional Laplacian in this context.
- The derivation incorporates nonlinear effects in moisture evolution modeling.

## Abstract

In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.