anaflow.flow.heterogeneous¶

Anaflow subpackage providing flow solutions in heterogeneous aquifers.

The following functions are provided

`ext_thiem_2d`(rad, r_ref, trans_gmean, var, …)	The extended Thiem solution in 2D.
`ext_thiem_3d`(rad, r_ref, cond_gmean, var, …)	The extended Thiem solution in 3D.
`ext_thiem_tpl`(rad, r_ref, cond_gmean, …[, …])	The extended Thiem solution for truncated power-law fields.
`ext_thiem_tpl_3d`(rad, r_ref, cond_gmean, …)	The extended Theis solution for truncated power-law fields in 3D.
`ext_theis_2d`(time, rad, storage, …[, …])	The extended Theis solution in 2D.
`ext_theis_3d`(time, rad, storage, cond_gmean, …)	The extended Theis solution in 3D.
`ext_theis_tpl`(time, rad, storage, …[, …])	The extended Theis solution for truncated power-law fields.
`ext_theis_tpl_3d`(time, rad, storage, …[, …])	The extended Theis solution for truncated power-law fields in 3D.
`neuman2004`(time, rad, storage, trans_gmean, …)	The transient solution for the apparent transmissivity from [Neuman2004].
`neuman2004_steady`(rad, r_ref, trans_gmean, …)	The steady solution for the apparent transmissivity from [Neuman2004].

ext_thiem_2d(rad, r_ref, trans_gmean, var, len_scale, rate=-0.0001, h_ref=0.0, T_well=None, prop=1.6)[source]¶

The extended Thiem solution in 2D.

The extended Thiem solution for steady-state flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a 2D statistical framework, where the transmissivity distribution is following a log-normal distribution with a gaussian correlation function.

Parameters:	rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated r_ref (`float`) – Radius of the reference head. trans_gmean (`float`) – Geometric-mean transmissivity. var (`float`) – Variance of log-transmissivity. len_scale (`float`) – Correlation-length of log-transmissivity. rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 h_ref (`float`, optional) – Reference head at the reference-radius r_ref. Default: `0.0` T_well (`float`, optional) – Explicit transmissivity value at the well. Default: `None` prop (`float`, optional) – Proportionality factor used within the upscaling procedure. Default: `1.6`
Returns:	head – Array with all heads at the given radii.
Return type:	`numpy.ndarray`

References

[Zech2013]

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

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

Examples

>>> ext_thiem_2d([1,2,3], 10, 0.001, 1, 10, -0.001)
array([-0.53084596, -0.35363029, -0.25419375])

ext_thiem_3d(rad, r_ref, cond_gmean, var, len_scale, anis=1.0, lat_ext=1.0, rate=-0.0001, h_ref=0.0, K_well='KH', prop=1.6)[source]¶

The extended Thiem solution in 3D.

The extended Thiem solution for steady-state flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a 3D statistical framework, where the conductivity distribution is following a log-normal distribution with a gaussian correlation function and taking vertical anisotropy into account.

Parameters:	rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated r_ref (`float`) – Reference radius with known head (see h_ref) cond_gmean (`float`) – Geometric-mean conductivity. var (`float`) – Variance of the log-conductivity. len_scale (`float`) – Corralation-length of log-conductivity. anis (`float`, optional) – Anisotropy-ratio of the vertical and horizontal corralation-lengths. Default: 1.0 lat_ext (`float`, optional) – Lateral extend of the aquifer (thickness). Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 h_ref (`float`, optional) – Reference head at the reference-radius r_ref. Default: `0.0` 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:	head – Array with all heads at the given radii.
Return type:	`numpy.ndarray`

References

[Zech2013]

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

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

Examples

>>> ext_thiem_3d([1,2,3], 10, 0.001, 1, 10, 1, 1, -0.001)
array([-0.48828026, -0.31472059, -0.22043022])

ext_thiem_tpl(rad, r_ref, cond_gmean, len_scale, hurst, var=None, c=1.0, dim=2.0, lat_ext=1.0, rate=-0.0001, h_ref=0.0, K_well='KH', prop=1.6)[source]¶

The extended Thiem solution for truncated power-law fields.

The extended Theis solution for steady flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a d-dimensional statistical framework, where the conductivity distribution is following a log-normal distribution with a truncated power-law correlation function build on superposition of gaussian modes.

Parameters:	rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated r_ref (`float`) – Reference radius with known head (see h_ref) cond_gmean (`float`) – Geometric-mean conductivity. You can also treat this as transmissivity by leaving ‘lat_ext=1’. len_scale (`float`) – Corralation-length of log-conductivity. hurst (`float`) – Hurst coefficient of the TPL model. Should be in (0, 1). var (`float`) – Variance of the log-conductivity. If var is 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` dim (`float`, optional) – Dimension of space. Default: `2.0` lat_ext (`float`, optional) – Lateral extend of the aquifer: sqare-root of cross-section in 1D thickness in 2D meaningless in 3D Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 h_ref (`float`, optional) – Reference head at the reference-radius r_ref. Default: `0.0` K_well (`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:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

ext_thiem_tpl_3d(rad, r_ref, cond_gmean, len_scale, hurst, var=None, c=1.0, anis=1, lat_ext=1.0, rate=-0.0001, h_ref=0.0, K_well='KH', prop=1.6)[source]¶

The extended Theis solution for truncated power-law fields in 3D.

The extended Theis solution for transient flow under a pumping condition in a confined aquifer with anisotropy in 3D. The type curve is describing the effective drawdown in a 3-dimensional statistical framework, where the conductivity distribution is following a log-normal distribution with a truncated power-law correlation function build on superposition of gaussian modes.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated storage (`float`) – Storage of the aquifer. cond_gmean (`float`) – Geometric-mean conductivity. len_scale (`float`) – Corralation-length of log-conductivity. hurst (`float`) – Hurst coefficient of the TPL model. Should be in (0, 1). var (`float`) – Variance of the log-conductivity. If var is 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. Default: 1.0 lat_ext (`float`, optional) – Lateral extend of the aquifer (thickness). Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` K_well (`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` far_err (`float`, optional) – Relative error for the farfield transmissivity for calculating the cutoff-point of the solution. Default: `0.01` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

ext_theis_2d(time, rad, storage, trans_gmean, var, len_scale, rate=-0.0001, r_well=0.0, r_bound=inf, h_bound=0.0, T_well=None, prop=1.6, struc_grid=True, far_err=0.01, parts=30, lap_kwargs=None)[source]¶

The extended Theis solution in 2D.

The extended Theis solution for transient flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a 2D statistical framework, where the transmissivity distribution is following a log-normal distribution with a gaussian correlation function.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated storage (`float`) – Storage of the aquifer. trans_gmean (`float`) – Geometric-mean transmissivity. var (`float`) – Variance of log-transmissivity. len_scale (`float`) – Correlation-length of log-transmissivity. rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` 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` far_err (`float`, optional) – Relative error for the farfield transmissivity for calculating the cutoff-point of the solution. Default: `0.01` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

Examples

>>> ext_theis_2d([10,100], [1,2,3], 0.001, 0.001, 1, 10, -0.001)
array([[-0.33737576, -0.17400123, -0.09489812],
       [-0.58443489, -0.40847176, -0.31095166]])

ext_theis_3d(time, rad, storage, cond_gmean, var, len_scale, anis=1.0, lat_ext=1.0, rate=-0.0001, r_well=0.0, r_bound=inf, h_bound=0.0, K_well='KH', prop=1.6, far_err=0.01, struc_grid=True, parts=30, lap_kwargs=None)[source]¶

The extended Theis solution in 3D.

The extended Theis solution for transient flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a 3D statistical framework, where the transmissivity distribution is following a log-normal distribution with a gaussian correlation function and taking vertical anisotropy into account.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated storage (`float`) – Storage of the aquifer. cond_gmean (`float`) – Geometric-mean conductivity. var (`float`) – Variance of the log-conductivity. len_scale (`float`) – Corralation-length of log-conductivity. anis (`float`, optional) – Anisotropy-ratio of the vertical and horizontal corralation-lengths. Default: 1.0 lat_ext (`float`, optional) – Lateral extend of the aquifer (thickness). Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` K_well (`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` far_err (`float`, optional) – Relative error for the farfield transmissivity for calculating the cutoff-point of the solution. Default: `0.01` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

Examples

>>> ext_theis_3d([10,100], [1,2,3], 0.001, 0.001, 1, 10, 1, 1, -0.001)
array([[-0.32756786, -0.16717569, -0.09141211],
       [-0.5416396 , -0.36982684, -0.27798614]])

ext_theis_tpl(time, rad, storage, cond_gmean, len_scale, hurst, var=None, c=1.0, dim=2.0, lat_ext=1.0, rate=-0.0001, r_well=0.0, r_bound=inf, h_bound=0.0, K_well='KH', prop=1.6, far_err=0.01, struc_grid=True, parts=30, lap_kwargs=None)[source]¶

The extended Theis solution for truncated power-law fields.

The extended Theis solution for transient flow under a pumping condition in a confined aquifer. The type curve is describing the effective drawdown in a d-dimensional statistical framework, where the conductivity distribution is following a log-normal distribution with a truncated power-law correlation function build on superposition of gaussian modes.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated storage (`float`) – Storage of the aquifer. cond_gmean (`float`) – Geometric-mean conductivity. You can also treat this as transmissivity by leaving ‘lat_ext=1’. len_scale (`float`) – Corralation-length of log-conductivity. hurst (`float`) – Hurst coefficient of the TPL model. Should be in (0, 1). var (`float`) – Variance of the log-conductivity. If var is 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` dim (`float`, optional) – Dimension of space. Default: `2.0` lat_ext (`float`, optional) – Lateral extend of the aquifer: sqare-root of cross-section in 1D thickness in 2D meaningless in 3D Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` K_well (`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` far_err (`float`, optional) – Relative error for the farfield transmissivity for calculating the cutoff-point of the solution. Default: `0.01` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

ext_theis_tpl_3d(time, rad, storage, cond_gmean, len_scale, hurst, var=None, c=1.0, anis=1, lat_ext=1.0, rate=-0.0001, r_well=0.0, r_bound=inf, h_bound=0.0, K_well='KH', prop=1.6, far_err=0.01, struc_grid=True, parts=30, lap_kwargs=None)[source]¶

The extended Theis solution for truncated power-law fields in 3D.

The extended Theis solution for transient flow under a pumping condition in a confined aquifer with anisotropy in 3D. The type curve is describing the effective drawdown in a 3-dimensional statistical framework, where the conductivity distribution is following a log-normal distribution with a truncated power-law correlation function build on superposition of gaussian modes.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated storage (`float`) – Storage of the aquifer. cond_gmean (`float`) – Geometric-mean conductivity. len_scale (`float`) – Corralation-length of log-conductivity. hurst (`float`) – Hurst coefficient of the TPL model. Should be in (0, 1). var (`float`) – Variance of the log-conductivity. If var is 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. Default: 1.0 lat_ext (`float`, optional) – Lateral extend of the aquifer (thickness). Default: `1.0` rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` K_well (`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` far_err (`float`, optional) – Relative error for the farfield transmissivity for calculating the cutoff-point of the solution. Default: `0.01` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

Notes

If you want to use cartesian coordiantes, just use the formula r = sqrt(x**2 + y**2)

neuman2004(time, rad, storage, trans_gmean, var, len_scale, rate=-0.0001, r_well=0.0, r_bound=inf, h_bound=0.0, struc_grid=True, parts=30, lap_kwargs=None)[source]¶

The transient solution for the apparent transmissivity from [Neuman2004].

This solution is build on the apparent transmissivity from Neuman 2004, which represents a mean drawdown in an ensemble of pumping tests in heterogeneous transmissivity fields following an exponential covariance.

Parameters:	time (`numpy.ndarray`) – Array with all time-points where the function should be evaluated. rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated. storage (`float`) – Storage of the aquifer. trans_gmean (`float`) – Geometric-mean transmissivity. var (`float`) – Variance of log-transmissivity. len_scale (`float`) – Correlation-length of log-transmissivity. rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 r_well (`float`, optional) – Radius of the pumping-well. Default: `0.0` r_bound (`float`, optional) – Radius of the outer boundary of the aquifer. Default: `np.inf` h_bound (`float`, optional) – Reference head at the outer boundary as well as initial condition. Default: `0.0` struc_grid (`bool`, optional) – If this is set to `False`, the rad and time array will be merged and interpreted as single, r-t points. In this case they need to have the same shapes. Otherwise a structured r-t grid is created. Default: `True` parts (`int`, optional) – Since the solution is calculated by setting the transmissivity to local constant values, one needs to specify the number of partitions of the transmissivity. Default: `30` lap_kwargs (`dict` or `None` optional) – Dictionary for `get_lap_inv` containing method and method_dict. The default is equivalent to `lap_kwargs = {"method": "stehfest", "method_dict": None}`. Default: `None`
Returns:	head – Array with all heads at the given radii and time-points.
Return type:	`numpy.ndarray`

References

[Neuman2004]

Neuman, Shlomo P., Alberto Guadagnini, and Monica Riva. ‘’Type-curve estimation of statistical heterogeneity.’’ Water resources research 40.4, 2004

neuman2004_steady(rad, r_ref, trans_gmean, var, len_scale, rate=-0.0001, h_ref=0.0)[source]¶

The steady solution for the apparent transmissivity from [Neuman2004].

This solution is build on the apparent transmissivity from Neuman 1994, which represents a mean drawdown in an ensemble of pumping tests in heterogeneous transmissivity fields following an exponential covariance.

Parameters:	rad (`numpy.ndarray`) – Array with all radii where the function should be evaluated r_ref (`float`) – Radius of the reference head. trans_gmean (`float`) – Geometric-mean transmissivity. var (`float`) – Variance of log-transmissivity. len_scale (`float`) – Correlation-length of log-transmissivity. rate (`float`, optional) – Pumpingrate at the well. Default: -1e-4 h_ref (`float`, optional) – Reference head at the reference-radius r_ref. Default: `0.0`
Returns:	head – Array with all heads at the given radii.
Return type:	`numpy.ndarray`

References

[Neuman2004]

Neuman, Shlomo P., Alberto Guadagnini, and Monica Riva. ‘’Type-curve estimation of statistical heterogeneity.’’ Water resources research 40.4, 2004