Geological uncertainty assessment is most often addressed by considering several geostatistical realizations on some given reservoir grid, and/or by perturbing the geometry of that grid under some hypotheses. These grid-focused approaches are practical but make it difficult to address two major types of geological uncertainties: conceptual uncertainty which can be critical in green field studies, and topological uncertainty which can be important even for mature fields. I argue that these high-level types of geological uncertainties can be better addressed by working directly with geological parameters. Several challenges exist in this type of approach, both educational (changing the way interpreters work) and technical (formalizing geological concepts and defining robust gridding and discretization methods). However, the main benefit of this vision is to provide descriptions of static geological uncertainties which are explicitly compatible with interpretation concepts and data. This strategy is most useful for generating a large set of models which would be considered as discrete scenarios with classical methods and it clears the path for joint inversion and assimilation of reservoir monitoring data.