The development of numerical algorithms for simulations of flow processes
in large-scale highly heterogeneous porous formations is challenging
because properties of natural geologic porous formations (e.g.,
permeability) display high variability and complex spatial correlation
structures which can span a hierarchy of length scales. It is usually
necessary to resolve a wide range of length and time scales, which can be
prohibitively expensive, in order to obtain accurate predictions of the
flow, mechanical deformation, and transport processes under investigation.
In practice, some types of coarsening (or upscaling) of the detailed model
are usually performed before the model can be used to simulate complex
processes. Many approaches have been developed and applied successfully
when a scale separation adequately describes the spatial variability of the
subsurface properties (e.g., permeability) that have bounded variations.
The quality of these approaches deteriorates for complex heterogeneities
without scale separation and high contrast. In this talk, I will describe
multiscale model reduction techniques that can be used to systematically
reduce the degrees of freedoms of fine-scale simulations and discuss
applications to preconditioners and coupling to global model reduction
tools. Numerical results will be presented that show that one can improve
the accuracy of multiscale methods by systematically adding new coarse
basis functions, obtain contrast-independent preconditioners for complex
heterogeneities, and get reduced order models at low cost.
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