# Unsaturated deformable porous media flow with phase transition

## TL;DR

This paper introduces a continuum model for fluid flow in deformable porous media that accounts for phase transitions, deriving equations from fundamental principles and proving the existence of solutions under simplifying assumptions.

## Contribution

It presents a novel coupled PDE-ODE-inclusion system modeling phase-changing flow in deformable porous media, with rigorous mathematical existence results.

## Key findings

- Existence of global solutions for the nonlinear system.
- Coupling of deformation, temperature, and phase fraction dynamics.
- Mathematical validation under neglect of inertia and shear stresses.

## Abstract

In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.