gstools.field.upscaling.var_coarse_graining

gstools.field.upscaling.var_coarse_graining(model, point_volumes=0.0)

Coarse Graning procedure to upscale the variance for uniform flow.

Parameters:

• model (CovModel) – Covariance Model used for the field.

• point_volumes (float or numpy.ndarray) – Volumes of the elements at the given points. Default: 0

Returns:

scaled_var – The upscaled variance

Return type:

float or numpy.ndarray

Notes

This procedure was presented in [Attinger03]. It applies the upscaling procedure ‘Coarse Graining’ to the Groundwater flow equation under uniform flow on a lognormal distributed conductivity field following a gaussian covariance function. A filter over a cube with a given edge-length \(\lambda\) is applied and an upscaled conductivity field is obtained. The upscaled field is again following a gaussian covariance function with scale dependent variance and length-scale:

\[\begin{split}\lambda &= V^{\frac{1}{d}} \\ \sigma^2\left(\lambda\right) &= \sigma^2\cdot\left( \frac{\ell^2}{\ell^2+\left(\frac{\lambda}{2}\right)^2} \right)^{\frac{d}{2}} \\ \ell\left(\lambda\right) &= \left(\ell^2+\left(\frac{\lambda}{2}\right)^2\right)^{\frac{1}{2}}\end{split}\]

Therby \(\lambda\) will be calculated from the given point_volumes \(V\) by assuming a cube with the given volume.

The upscaled length scale will be ignored by this routine.

References

[Attinger03]

Attinger, S. 2003, ‘’Generalized coarse graining procedures for flow in porous media’’, Computational Geosciences, 7(4), 253–273.