In this study, we explore the two-dimensional optimal system of subalgebras for a ( \(2+1\) )-dimensional unsteady incompressible fluid flow model through a porous medium featuring low porosity. We analyze the exact solutions for each inequivalent subalgebra associated with the Lie symmetries. The unsteady Darcy-Brinkman equations are the governing equations for the considered flow problem. The considered Darcy-Brinkman model captures the interplay of viscous and Darcy resistance effects in low-porosity media. The governing partial differential equations are challenging to solve analytically. Although numerical methods can provide approximate solutions and dynamics of such problems, the exact group-invariant solutions play a fundamental role in uncovering the hidden structures and understanding explicit features of the flow problems. The exact solutions are also essential for benchmarking the numerical schemes. Consequently, we conduct a comprehensive symmetry analysis for the considered model and construct an optimal system of Lie subalgebras aligned with the one-parameter Lie group of transformations that preserves the invariance property for the system of governing partial differential equations. Utilizing the adjoint action of the one-parameter Lie group on the associated Lie algebra, we unveil a set of invariant solutions, which is crucial for understanding the underlying dynamics of the porous media flow. The process of optimal classification plays a pivotal role in our Lie symmetry analysis, enabling us to identify all conceivable inequivalent group-invariant solutions. By extracting various two-dimensional Lie subalgebras from the optimal classifications of the unsteady Darcy-Brinkman equations, we gain insight into diverse similarity solutions for the governing system. Each subalgebra yields some new kinds of special analytical solutions, which have possible applications in groundwater hydrology, oil recovery, biological transport, and filtration technologies.
