Inversion of 3-D structural geometry using geological least-squares criteria

Oil exploration requires quantitative determination of structural geometry in sedimentary basins. This leads to back-and-forth use of geological methods, e.g. cross-section balancing and geophysical techniques, such as tomography, and the synthesis becomes tedious, especially in three dimensions. This suggests that they should be as much as possible quantitatively integrated into a single consistent framework. For this integration, we propose using inversion techniques, i.e. multicriteria optimization. We locally model a geological structure as a ( geometric) foliation , the leaves of which represent deposition isochrons. We consider a geological structure as a set of foliations joined along faults and unconformities. We propose five kinds of geological data to constrain structural geometry quantitatively: dip measurements that may be available along wells, developability and smoothness of deposition isochrons, the directions of fold axes, and layer parallelism. Using concepts of differential geometry, we formulate these data in terms of least-squares criteria. To solve the canonical non-uniqueness problem raised by the inversion of parametric representations of geometrical objects such as foliations (many parametrizations describe the same object), we introduce the additional criterion method which consists of adding an unphysical objective function to the physical objective function, so as to make the solution unique. Assuming well trajectories and borehole correlations to be known, we optimize, with respect to these criteria, several simple structures comprising one foliation, including a field example.