Generalized Hooke’s Law in a Curvilinear Space Application to Upscaling

@inproceedings{RoyerRM2007,
 abstract = { Up-scaling on unstructured grid has become an important step in mechanical problem applied to geologically complex heterogeneous formations, including restoration, fracture prediction, and reservoir subsidence studies. This comes from the fact that fine levels of details for mechanical rock properties cannot be handled directly in mechanical reservoir studies because they lead to unrealistical computing time. It is hence necessary to up-scale rock properties defined or simulated on fine gridded mesh onto coarse cells defined on the large-scale reservoirs. This paper aims at providing a theoretical background for the generalized Hooke equation in a local parametric space and to use it for up-scaling the compliance and the stiffness tensors. Formulas to write mechanical equations such as the energy conservation equation or the momentum equation in a curvilinear space are given using results derived from the tensorial calculus. They are then applied to estimate the up-scaled stiffness or/and equivalently the compliance tensor on structured or unstructured grids. In particular, the classical up-scaling methods including analytical, simple averaging and power averaging procedures, are transposed in a curvilinear space. The resulting formulas lead to an exact computation of the up-scaled stiffness tensor of a volume of a reservoir in the framework of the Geochron model. },
 author = { Royer, Jean-Jacques AND Titeux, Marc-Olivier },
 booktitle = { 27th gOcad Meeting },
 month = { "june" },
 publisher = { ASGA },
 title = { Generalized Hooke’s Law in a Curvilinear Space Application to Upscaling },
 year = { 2007 }
}