Generalized Coarse Graining Procedures for Flow in Porous Media |
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| Authors: | Sabine Attinger |
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| Institution: | (1) Institute of Hydromechanics and Water Resources Management (IHW), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland |
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| Abstract: | This paper focuses on heterogeneous soil conductivities and on the impact their resolution has on a solution of the piezometric
head equation: owing to spatial variations of the conductivity, the flow properties at larger scales differ from those found
for experiments performed at smaller scales. The method of coarse graining is proposed in order to upscale the piezometric
head equation on arbitrary intermediate scales. At intermediate scales large scale fluctuations of the conductivities are
resolved, whereas small scale fluctuations are smoothed by a partialy spatial filtering procedure. The filtering procedure
is performed in Fourier space with the aid of a low-frequency cut-off function. We derive the partially upscaled head equations.
In these equations, the impact of the small scale variability is modeled by scale dependent effective conductivities which
are determined by additional differential equations. Explicit results for the scale dependent conductivity values are presented
in lowest order perturbation theory. The perturbation theory contributions are summed up with using a renormalisation group
analysis yielding explicit results for the effective conductivity in isotropic media. Therefore, the results are also valid
for highly heterogeneous media. The results are compared with numerical simulations performed by Dykaar and Kitanidis (1992).
The method of coarse graining combined by a renormalisation group analysis offers a tool to derive exact and explicit expressions
for resolution dependent conductivity values. It is, e.g., relevant for the interpretation of measurement data on different
scales and for reduction of grid-block resolution in numerical modeling.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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| Keywords: | coarse graining filtering procedures flow heterogeneous porous media multiscale modelling |
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