We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.