Mitigating climate change which is driven by carbon dioxide release into the atmosphere requires scientific advancements in the area of carbon dioxide capture, conversion and storage. We report progress in our research of fluid flow scenarios in geological samples that involve carbon dioxide injection at pore scale. Specifically, we generate capillary network representations by extracting the geometric boundaries of the connected pore space from x-ray microscale computer tomography data of porous reservoir rock samples. We present our initial results of capillary network flow simulations obtained with hierarchical, multi-phase flow models which are calibrated through molecular dynamics simulation. The methods developed here are more broadly applicable for studying problems of fluid flow in porous media and their application will benefit the understanding and improvement of carbon dioxide conversion and storage.