The creation of a 3D pore-scale model of a porous medium is often an essential step in quantitatively characterising the medium and predicting its transport properties. Here we describe a new stochastic pore space reconstruction approach that uses thin section images as its main input. The approach involves using a third-order Markov mesh where we introduce a new algorithm that creates the reconstruction in a single scan, thus overcoming the computational issues normally associated with Markov chain methods. The technique is capable of generating realistic pore architecture models (PAMs), and examples are presented for a range of fairly homogenous rock samples as well as for one heterogeneous soil sample. We then apply a Lattice–Boltzmann (LB) scheme to calculate the permeabilities of the PAMs, which in all cases closely match the measured values of the original samples. We also develop a set of software methods – referred to as pore analysis tools (PATs) – to quantitatively analyse the reconstructed pore systems. These tools reveal the pore connectivity and pore size distribution, from which we can simulate the mercury injection process, which in turn reproduces the measured curves very closely. Analysis of the topological descriptors reveals that a connectivity function based on the specific Euler number may serve as a simple predictor of the threshold pressure for geo-materials.