We consider two-phase fluid flow through heterogeneous cores with given injection rates. In these problems, measured pressure drop over the core is an important information source for recovering permeability. This gives rise to the inverse problem of inverting the pressure data, p(t), into knowledge of the permeability distribution, K(x), in the core. Given a global pressure model, a description of the resolution power of pressure data with respect to permeability identification is possible. We examine how the flow pattern - width and front structure of the two-phase zone - affects the resolution power. We further briefly discuss the value of additional data, e.g., lateral saturation data, for the ability to recover permeability.