Space-time mathematical framework for sedimentary geology

Interpolating physical properties in the subsurface is a recurrent problem in geology. In sedimentary geology, the geometry of the layers is generally known with a precision much superior to that which one can reasonably expect for the properties. The geometry of the layers is affected by folding and faulting since the time of deposition, whereas the distribution of properties is, to a certain extent, determined at the time of deposition. As a consequence, it may be wise to model first the geometry of the layers and then, simplify the geologic equation by removing the influence of that geometry. Inspired from the work of H. E. Wheeler on Time-Stratigraphy, we define, mathematically, a new space where all horizons are horizontal planes and where faults, if any, have disappeared. We surmise that this new space, however approximative, is better to model physical properties of the subsurface whatever the subsequent interpolation method used. The proposed mathematical framework also provides solutions to complex problems such as determination of strains resulting from tectonic events and up-scaling of permeabilities on structured and unstructured 3D grids.