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: 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])
- rad (
-
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: Examples
>>> T_CG_inverse([7e-4,8e-4,9e-4], 0.001, 1, 10, 2) array([3.16952925, 5.56935826, 9.67679026])
- T (
-
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: Examples
>>> T_CG_error(0.01, 0.001, 1, 10, 2) 34.91045016779039
- err (
-
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: 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])
- rad (
-
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: Examples
>>> K_CG_inverse([7e-4,8e-4,9e-4], 0.001, 1, 10, 1, 2) array([2.09236867, 3.27914996, 4.52143956])
- K (
-
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: Examples
>>> K_CG_error(0.01, 0.001, 1, 10, 1, 2) 19.612796453639845
- err (
-
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
orNone
, 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
orfloat
, 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: - rad (
-
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
orNone
, 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
orfloat
, 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: - err (