Multi-phase multi-component flow processes are fundamental to engineering applications in hydrocarbon and geothermal reservoirs but also to many classical geological processes. This review will highlight recent developments in the mathematical modelling and numerical simulations of the underlying physical processes from the pore- to the reservoir scale. Many modern approaches now rely on integrating numerical and analytical methods and incorporate results across the different length scales. This provides new insights into the fundamental properties of multi-phase multi-component flow and helps to mitigate some of the inherent difficulties in quantifying them in subsurface reservoirs.

We present a finite element - finite volume simulation method for modelling fluid flow and solute transport accompanied by chemical reactions in experimentally obtained 3D pore geometries. The advantage of the proposed methodology with respect to other pore-scale modelling approaches is that no simplifications regarding the geometry of the porous space are required and no approximations to the flow equations are introduced. We apply this method in a proof-of-concept study of a digitised Fontainebleau sandstone sample. We use the calculated velocity profile with the finite volume procedure to simulate pore-scale transport and diffusion of the adsorbing solute. We also demonstrate how analysis of the pore geometry can be used to identify the locations of oil during the two-phase flow and couple this with the reactive transport modeling to show how this procedure can be used to estimate the potential of the enhanced oil recovery techniques.

Upscaling pore-scale processes into macroscopic quantities such as hydrodynamic dispersion is still not a straightforward matter for porous media with complex pore space geometries. Recently it has become possible to obtain very realistic 3D geometries for the pore system of real rocks using either numerical reconstruction or micro-CT measurements. In this work, we present a finite element-finite volume simulation method for modeling single-phase fluid flow and solute transport in experimentally obtained 3D pore geometries. Algebraic multigrid techniques and parallelization allow us to solve the Stokes and advection-diffusion equations on large meshes with several millions of elements. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity to simulate pore-scale solute transport and diffusion. From this, we are able to calculate the a priori emergent macroscopic hydrodynamic dispersion coefficient of the porous medium for a given molecular diffusion D_m of the solute species. By performing this calculation at a range of flow rates, we can correctly predict all of the observed flow regimes from diffusion dominated to convection dominated.