On boundedness and growth of unsteady solutions under the double porosity/permeability model

TL;DR

This paper investigates the stability and growth behavior of unsteady solutions in the double porosity/permeability model, providing theoretical insights and verification tools for numerical solutions in complex porous media flows.

Contribution

It proves Lyapunov stability and linear growth bounds for unsteady solutions under the DPP model, aiding numerical solution verification.

Findings

Unsteady solutions are Lyapunov stable.

Solutions grow at most linearly with time.

Results assist in verifying numerical solutions.

Abstract

There is a recent surge in research activities on modeling the flow of fluids in porous media with complex pore-networks. A prominent mathematical model, which describes the flow of incompressible fluids in porous media with two dominant pore-networks allowing mass transfer across them, is the double porosity/permeability (DPP) model. However, we currently do not have a complete understanding of unsteady solutions under the DPP model. Also, because of the complex nature of the mathematical model, it is not possible to find analytical solutions, and one has to resort to numerical solutions. It is therefore desirable to have a procedure that can serve as a measure to assess the veracity of numerical solutions. In this paper, we establish that unsteady solutions under the transient DPP model are stable in the sense of Lyapunov. We also show that the unsteady solutions grow at most linear with time. These results not only have a theoretical value but also serve as valuable a posteriori measures to verify numerical solutions in the transient setting and under anisotropic medium properties, as analytical solutions are scarce for these scenarios under the DPP model.