We present a novel mathematical algorithm for the characterization of non-conventional reservoirs which contain regions of low sensitivity. Our algorithm uses a level set representation of shapes describing different lithofacies in the reservoir. These shapes need to be reconstructed from the production data using a two-phase flow model. In order to deal with regions of low sensitivity, topological perturbations are applied successively during the reconstruction in these low sensitivity regions, and the level set technique will evolve the so created shapes following a gradient direction that minimizes the mismatch between the computed and the production data. New shapes created at wrong locations tend to disappear gradually, whereas those created at locations where a lithofacie is present tend to grow until they approximately match the correct boundaries. We will discuss different strategies and present numerical results which demonstrate and compare their performances for two realistic 2D test cases.