The use of "stratigraphic grids" allows the construction of geostatistical models controlled by stratigraphic information. A stratigraphic grid is a 3D grid with boundaries defined by stratigraphic surfaces and cells which respect the stratigraphic organisation (proportional, onlap, toplap,…) The GOCAD Research Group recently developed object-based stochastic simulations using curvilinear sedimentary bodies and generalized the concept of stratigraphic grids within each simulated object. We propose a technique for simulating petrophysical parameters within an object-based reservoir model, using this framework provided by stratigraphic grids within each object. The complete stochastic simulation approach is then defined by the following sequence of operations: - Define a 3D stratigraphic grid respecting the geometry of the sedimentary limits of the reservoir. - Next, in the layers where it is required, stochastically simulate the distribution of 3D curvilinear sedimentary bodies (e.g. representing sand bodies). - Once this object simulation is performed, create a local 3D curvilinear stratigraphic grid contained within each sedimentary body. - In each of these local 3D curvilinear stratigraphic grids, use geostatistical simulation tools to simulate the distribution of petrophysical properties. The local curvilinear stratigraphic grids so defined honours the sand bodies stratigraphy and allow to generate more realistic simulations than classical global rectilinear grid. The ultimate goal is to merge all the local 3D curvilinear stratigraphic grids into one global 3D curvilinear reservoir stratigraphic grid filling the whole reservoir. The petrophysical properties of the reservoir can then be transferred from the local stratigraphic grids to the global reservoir stratigraphic grid. Erosion rules between sand bodies can also be accounted for. With such an approach, based on local stratigraphic grids within sedimentary bodies, we can take into account both the geometry and the topology of the geological bodies.