In this presentation, we will delve into the fundamental aspects of utilizing Virtual Element Methods (VE methods) to simulate real-world problems. VE methods are attracting attention due to their adaptability in mesh generation for intricate geometries. We will initiate our exploration with a novel approach that integrates mesh generation and refinement based on a fundamental geometric description. Our particular emphasis will be on refining polygonal meshes to tackle challenges related to preserving and improving quality in complex domains like the ones displayed in the flow simulations in Discrete Fracture Networks. A distinctive and highly optimized refinement technique designed for convex cells will be introduced. This technique incorporates properties that contribute to addressing concerns related to convergence and optimality in adaptive methods. Key elements in refining general convex polygons involve a cell refinement strategy dependent solely on marked cells at each step, a partial enhancement of mesh quality, or, at the very least, the maintenance of non-degenerate mesh quality throughout refinement iterations. Additionally, there will be a constraint on the number of unknowns in the discrete problem relative to the number of cells in the mesh.