anaflow.tools.coarse_graining¶
Anaflow subpackage providing helper functions related to coarse graining.
The following functions are provided
|
The coarse-graining Transmissivity. |
|
The inverse coarse-graining Transmissivity. |
|
Calculating the radial-point for given error. |
|
The coarse-graining conductivity. |
|
The inverse coarse-graining conductivity. |
|
Calculating the radial-point for given error. |
|
The gaussian truncated power-law coarse-graining conductivity. |
|
Calculating the radial-point for given error. |
- 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 evaluatedcond_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])
- 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 conductivitycond_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
- 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 evaluatedcond_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])
- 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 evaluatedcond_gmean (
float
) – Geometric-mean conductivitylen_scale (
float
) – upper bound of the corralation-length of conductivity-distributionhurst (
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: 1.0dim (
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
- 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 conductivitycond_gmean (
float
) – Geometric-mean conductivitylen_scale (
float
) – upper bound of the corralation-length of conductivity-distributionhurst (
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: 1.0dim (
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
- 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 evaluatedtrans_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])
- 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 transmissivitytrans_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
- 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 evaluatedtrans_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])