anaflow.tools.coarse_graining

Anaflow subpackage providing helper functions related to coarse graining.

The following functions are provided

T_CG(rad, trans_gmean, var, len_scale[, …]) The coarse-graining Transmissivity.
T_CG_inverse(T, trans_gmean, var, len_scale) The inverse coarse-graining Transmissivity.
T_CG_error(err, trans_gmean, var, len_scale) Calculating the radial-point for given error.
K_CG(rad, cond_gmean, var, len_scale, anis) The coarse-graining conductivity.
K_CG_inverse(K, cond_gmean, var, len_scale, anis) The inverse coarse-graining conductivity.
K_CG_error(err, cond_gmean, var, len_scale, anis) Calculating the radial-point for given error.
TPL_CG(rad, cond_gmean, len_scale, hurst[, …]) The gaussian truncated power-law coarse-graining conductivity.
TPL_CG_error(err, cond_gmean, len_scale, hurst) Calculating the radial-point for given error.
T_CG(rad, trans_gmean, var, len_scale, T_well=None, prop=1.6)[source]

The coarse-graining Transmissivity.

This solution was presented in ‘’Schneider & Attinger 2008’‘[R3].

This functions gives an effective transmissivity for 2D pumpingtests in heterogenous aquifers, where the transmissivity is following a log-normal distribution and a gaussian correlation function.

Parameters:
  • rad (numpy.ndarray) – Array with all radii where the function should be evaluated
  • trans_gmean (float) – Geometric-mean transmissivity.
  • var (float) – Variance of log-transmissivity.
  • len_scale (float) – Correlation-length of log-transmissivity.
  • T_well (float, optional) – Explicit transmissivity value at the well. Harmonic mean by default.
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

T_CG – Array containing the effective transmissivity values.

Return type:

numpy.ndarray

References

[R3]Schneider, C. and Attinger, S., ‘’Beyond thiem: A new method for interpreting large scale pumping tests in heterogeneous aquifers.’’ Water resources research, 44(4), 2008

Examples

>>> T_CG([1,2,3], 0.001, 1, 10, 2)
array([0.00061831, 0.00064984, 0.00069236])
T_CG_inverse(T, trans_gmean, var, len_scale, T_well=None, prop=1.6)[source]

The inverse coarse-graining Transmissivity.

See: T_CG()

Parameters:
  • T (numpy.ndarray) – Array with all transmissivity values where the function should be evaluated
  • trans_gmean (float) – Geometric-mean transmissivity.
  • var (float) – Variance of log-transmissivity.
  • len_scale (float) – Correlation-length of log-transmissivity.
  • T_well (float, optional) – Explicit transmissivity value at the well. Harmonic mean by default.
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

rad – Array containing the radii belonging to the given transmissivity values

Return type:

numpy.ndarray

Examples

>>> T_CG_inverse([7e-4,8e-4,9e-4], 0.001, 1, 10, 2)
array([3.16952925, 5.56935826, 9.67679026])
T_CG_error(err, trans_gmean, var, len_scale, T_well=None, prop=1.6)[source]

Calculating the radial-point for given error.

Calculating the radial-point where the relative error of the farfield value is less than the given tollerance. See: T_CG()

Parameters:
  • err (float) – Given relative error for the farfield transmissivity
  • trans_gmean (float) – Geometric-mean transmissivity.
  • var (float) – Variance of log-transmissivity.
  • len_scale (float) – Correlation-length of log-transmissivity.
  • T_well (float, optional) – Explicit transmissivity value at the well. Harmonic mean by default.
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

rad – Radial point, where the relative error is less than the given one.

Return type:

float

Examples

>>> T_CG_error(0.01, 0.001, 1, 10, 2)
34.91045016779039
K_CG(rad, cond_gmean, var, len_scale, anis, K_well='KH', prop=1.6)[source]

The coarse-graining conductivity.

This solution was presented in ‘’Zech 2013’‘[R8].

This functions gives an effective conductivity for 3D pumpingtests in heterogenous aquifers, where the conductivity is following a log-normal distribution and a gaussian correlation function and taking vertical anisotropy into account.

Parameters:
  • rad (numpy.ndarray) – Array with all radii where the function should be evaluated
  • cond_gmean (float) – Geometric-mean conductivity.
  • var (float) – Variance of the log-conductivity.
  • len_scale (float) – Corralation-length of log-conductivity.
  • anis (float) – Anisotropy-ratio of the vertical and horizontal corralation-lengths.
  • K_well (string/float, optional) – Explicit conductivity value at the well. One can choose between the harmonic mean ("KH"), the arithmetic mean ("KA") or an arbitrary float value. Default: "KH"
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

K_CG – Array containing the effective conductivity values.

Return type:

numpy.ndarray

References

[R8]Zech, A. ‘’Impact of Aqifer Heterogeneity on Subsurface Flow and Salt Transport at Different Scales: from a method determine parameters of heterogeneous permeability at local scale to a large-scale model for the sedimentary basin of Thuringia.’’ PhD thesis, Friedrich-Schiller-Universität Jena, 2013

Examples

>>> K_CG([1,2,3], 0.001, 1, 10, 1, 2)
array([0.00063008, 0.00069285, 0.00077595])
K_CG_inverse(K, cond_gmean, var, len_scale, anis, K_well='KH', prop=1.6)[source]

The inverse coarse-graining conductivity.

See: K_CG()

Parameters:
  • K (numpy.ndarray) – Array with all conductivity values where the function should be evaluated
  • cond_gmean (float) – Geometric-mean conductivity.
  • var (float) – Variance of the log-conductivity.
  • len_scale (float) – Corralation-length of log-conductivity.
  • anis (float) – Anisotropy-ratio of the vertical and horizontal corralation-lengths.
  • K_well (string/float, optional) – Explicit conductivity value at the well. One can choose between the harmonic mean ("KH"), the arithmetic mean ("KA") or an arbitrary float value. Default: "KH"
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

rad – Array containing the radii belonging to the given conductivity values

Return type:

numpy.ndarray

Examples

>>> K_CG_inverse([7e-4,8e-4,9e-4], 0.001, 1, 10, 1, 2)
array([2.09236867, 3.27914996, 4.52143956])
K_CG_error(err, cond_gmean, var, len_scale, anis, K_well='KH', prop=1.6)[source]

Calculating the radial-point for given error.

Calculating the radial-point where the relative error of the farfield value is less than the given tollerance. See: K_CG()

Parameters:
  • err (float) – Given relative error for the farfield conductivity
  • cond_gmean (float) – Geometric-mean conductivity.
  • var (float) – Variance of the log-conductivity.
  • len_scale (float) – Corralation-length of log-conductivity.
  • anis (float) – Anisotropy-ratio of the vertical and horizontal corralation-lengths.
  • K_well (string/float, optional) – Explicit conductivity value at the well. One can choose between the harmonic mean ("KH"), the arithmetic mean ("KA") or an arbitrary float value. Default: "KH"
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

rad – Radial point, where the relative error is less than the given one.

Return type:

float

Examples

>>> K_CG_error(0.01, 0.001, 1, 10, 1, 2)
19.612796453639845
TPL_CG(rad, cond_gmean, len_scale, hurst, var=None, c=1.0, anis=1, dim=2.0, K_well='KH', prop=1.6)[source]

The gaussian truncated power-law coarse-graining conductivity.

Parameters:
  • rad (numpy.ndarray) – Array with all radii where the function should be evaluated
  • cond_gmean (float) – Geometric-mean conductivity
  • len_scale (float) – upper bound of the corralation-length of conductivity-distribution
  • hurst (float) – Hurst coefficient of the TPL model. Should be in (0, 1).
  • var (float or None, optional) – Variance of log-conductivity If given, c will be calculated accordingly. Default: None
  • c (float, optional) – Intensity of variation in the TPL model. Is overwritten if var is given. Default: 1.0
  • anis (float, optional) – Anisotropy-ratio of the vertical and horizontal corralation-lengths. This is only applied in 3 dimensions. Default: 1.0
  • dim (float, optional) – Dimension of space. Default: 2.0
  • K_well (str or float, optional) – Explicit conductivity value at the well. One can choose between the harmonic mean ("KH"), the arithmetic mean ("KA") or an arbitrary float value. Default: "KH"
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

TPL_CG – Array containing the effective conductivity values.

Return type:

numpy.ndarray

TPL_CG_error(err, cond_gmean, len_scale, hurst, var=None, c=1.0, anis=1, dim=2.0, K_well='KH', prop=1.6)[source]

Calculating the radial-point for given error.

Calculating the radial-point where the relative error of the farfield value is less than the given tollerance. See: TPL_CG()

Parameters:
  • err (float) – Given relative error for the farfield conductivity
  • cond_gmean (float) – Geometric-mean conductivity
  • len_scale (float) – upper bound of the corralation-length of conductivity-distribution
  • hurst (float) – Hurst coefficient of the TPL model. Should be in (0, 1).
  • var (float or None, optional) – Variance of log-conductivity If given, c will be calculated accordingly. Default: None
  • c (float, optional) – Intensity of variation in the TPL model. Is overwritten if var is given. Default: 1.0
  • anis (float, optional) – Anisotropy-ratio of the vertical and horizontal corralation-lengths. This is only applied in 3 dimensions. Default: 1.0
  • dim (float, optional) – Dimension of space. Default: 2.0
  • K_well (str or float, optional) – Explicit conductivity value at the well. One can choose between the harmonic mean ("KH"), the arithmetic mean ("KA") or an arbitrary float value. Default: "KH"
  • prop (float, optional) – Proportionality factor used within the upscaling procedure. Default: 1.6
Returns:

rad – Radial point, where the relative error is less than the given one.

Return type:

float