The displacement of foam within a heterogeneous reservoir during foam improved oil recovery is described with the pressure-driven growth model. The pressure-driven growth model has previously been used to study foam motion for homogeneous cases. Here the foam model is modified in such a way that it includes terms for variable permeability. This model gives the evolution of the foam motion over time and the shape of the foam front, a wet foam zone between liquid-filled and gas-filled zones. The foam front shape for a heterogeneous or stratified reservoir develops concave and convex regions. For shapes such as these, the numerical solution of pressure-driven growth requires special numerical techniques, particularly in the case where concavities arise. We also present some analysis of the level of heterogeneity and how it affects the displacement, the shape of the front developing a set of concave corners. In addition to this we consider a heterogeneous and isotropic reservoir, in which case the foam front can sustain concavities, without these concavities having the same tendency to develop into corners. (C) 2016 Elsevier B.V. All rights reserved.