Existence and Uniqueness of Solution to Unsteady Darcy-Brinkman Problem with Korteweg Stress for Modelling Miscible Porous Media Flow

TL;DR

This paper proves the existence and uniqueness of solutions for a complex model combining Darcy-Brinkman flow with Korteweg stress, relevant for modeling various porous media flows with variable permeability and external forces.

Contribution

It establishes well-posedness results for a coupled convection-diffusion and flow system with Korteweg stress, extending previous models to include variable permeability and external forces.

Findings

Well-posedness of the model is proven for a broad class of initial data.

Techniques are applicable to non-reactive and homogeneous porous media flows.

The model captures complex behaviors in porous media such as fractures and industrial materials.

Abstract

The work investigates a model that combines a convection-diffusion-reaction equation for solute concentration with an unsteady Darcy-Brinkman equation for the flow field, including the Kortweg stress. Additionally, the flow field experiences an external body force term while the permeability fluctuates with solute concentration. Such models are used to describe flows in porous mediums such as fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. The system of equations has Neumann boundary conditions for the solute concentration and no-flow conditions for the velocity field, and the well-posedness of the model is discussed for a wide range of initial data. The proofing techniques remain applicable in establishing the well-posedness of non-reactive and homogeneous porous media flows under the specified simplifications.