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.