Controlling the flow of fluids (e.g. water, oil, natural gas or CO₂) in subsurface porous media is a technical process with many mathematical challenges. The underlying physics can be described with coupled nearly-elliptic and nearly-hyperbolic nonlinear partial differential equations, which require the aid of large-scale numerical simulation. The subsurface rock is strongly heterogeneous on scales ranging from pores (microns) to reservoirs (kilometers), and displays very large spatial variations in the coefficients. Moreover, the limited accessibility of the underground leads to very large uncertainties, while data is sparse and also multi-scale, ranging from lab results on rock samples to seismic profiles over an entire reservoir. In this talk I will address related modeling and system-theoretical aspects.