Over the past two decades, multicomponent lattice-Boltzmann (LB) modelling has become a popular numerical technique to study the porous medium systems. For this technique to become a mature platform at a production level and to solve realistic problem that can be readily incorporated in the digital core analysis services for the oil and gas industries, there are still some challenges to resolve. This thesis intends to resolve some of issues confronted by the LB community. The first part of the thesis investigates the impact of the fundamental trade-off between image resolution and field of view on LB modelling. This is of practical value since 3D images of geological samples rarely have both sufficient resolution to capture fine structure and sufficient field of view to capture a full representative elementary volume of the medium. To optimise the simulations, it is important to know the minimum number of grid points that LB methods require to deliver physically meaningful results, and allow for the sources of measurement uncertainty to be appropriately balanced. We choose two commonly used multicomponent LB models, Shan-Chen and Rothman-Keller models, and study the behaviour of these two models when the phase interfacial radius of curvature and the feature size of the medium approach the discrete unit size of the computational grid. Both simple, small-scale test geometries and real porous media are considered. Models' behaviour in the extreme discrete limit is classified ranging from gradual loss of accuracy to catastrophic numerical breakdown. Based on this study, we provide guidance for experimental data collection and how to apply the LB methods to accurately resolve physics of interest for two-fluid flow in porous media. Resolution effects are particularly relevant to the study of low-porosity systems, including fractured materials, when the typical pore width may only be a few voxels across. The second part of the thesis explores the two-fluid displacement mechanism, especially the Haines jump dynamics and associated snap-off during drainage, by using a novel flux boundary condition, which is numerically more stable, and can more realistically replicate experiments given a prescribed capillary number. Irreversible events such as Haines jump in multiphase flow is what ultimately determines the hysteric behaviour of the porous medium systems. The high temporal resolution of LB methods makes it a suitable candidate to capture the dynamics of fast events (e.g. Haines jump in millisecond). We study the impacts of both the geometries of porous medium using persistent homology and the dynamic factors of fluids (i.e. viscosity ratio and capillary number) on the occurrence and frequency of snap-off events during drainage
