"""
Anaflow subpackage providing flow solutions in homogeneous aquifers.
.. currentmodule:: anaflow.flow.homogeneous
The following functions are provided
.. autosummary::
thiem
theis
grf
"""
# pylint: disable=C0103
import numpy as np
from anaflow.flow.ext_grf_model import ext_grf, ext_grf_steady
from anaflow.tools.special import grf_solution, well_solution
__all__ = ["thiem", "theis", "grf"]
###############################################################################
# Thiem-solution
###############################################################################
[docs]def thiem(rad, r_ref, transmissivity, rate=-1e-4, h_ref=0.0):
"""
The Thiem solution.
The Thiem solution for steady-state flow under a pumping condition
in a confined and homogeneous aquifer.
This solution was presented in [Thiem1906]_.
Parameters
----------
rad : :class:`numpy.ndarray`
Array with all radii where the function should be evaluated.
r_ref : :class:`float`
Reference radius with known head (see `h_ref`).
transmissivity : :class:`float`
Transmissivity of the aquifer.
rate : :class:`float`, optional
Pumpingrate at the well. Default: -1e-4
h_ref : :class:`float`, optional
Reference head at the reference-radius `r_ref`. Default: ``0.0``
Returns
-------
head : :class:`numpy.ndarray`
Array with all heads at the given radii.
References
----------
.. [Thiem1906] Thiem, G.,
''Hydrologische Methoden, J.M. Gebhardt'', Leipzig, 1906.
Notes
-----
The parameters ``rad``, ``r_ref`` and ``transmissivity``
will be checked for positivity.
If you want to use cartesian coordiantes, just use the formula
``r = sqrt(x**2 + y**2)``
Examples
--------
>>> thiem([1,2,3], 10, 0.001, -0.001)
array([-0.3664678 , -0.25615 , -0.19161822])
"""
return ext_grf_steady(rad, r_ref, transmissivity, 2, 1, rate, h_ref)
###############################################################################
# Theis-solution
###############################################################################
[docs]def theis(
time,
rad,
storage,
transmissivity,
rate=-1e-4,
r_well=0.0,
r_bound=np.inf,
h_bound=0.0,
struc_grid=True,
lap_kwargs=None,
):
"""
The Theis solution.
The Theis solution for transient flow under a pumping condition
in a confined and homogeneous aquifer.
This solution was presented in [Theis35]_.
Parameters
----------
time : :class:`numpy.ndarray`
Array with all time-points where the function should be evaluated
rad : :class:`numpy.ndarray`
Array with all radii where the function should be evaluated
storage : :class:`float`
Storage coefficient of the aquifer.
conductivity : :class:`float`
Conductivity of the aquifer.
rate : :class:`float`, optional
Pumpingrate at the well. Default: -1e-4
r_well : :class:`float`, optional
Inner radius of the pumping-well. Default: ``0.0``
r_bound : :class:`float`, optional
Radius of the outer boundariy of the aquifer. Default: ``np.inf``
h_bound : :class:`float`, optional
Reference head at the outer boundary, as well as initial condition.
Default: ``0.0``
struc_grid : :class:`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``
lap_kwargs : :class:`dict` or :any:`None` optional
Dictionary for :any:`get_lap_inv` containing `method` and
`method_dict`. The default is equivalent to
``lap_kwargs = {"method": "stehfest", "method_dict": None}``.
Default: :any:`None`
Returns
-------
head : :class:`numpy.ndarray`
Array with all heads at the given radii and time-points.
References
----------
.. [Theis35] Theis, C.,
''The relation between the lowering of the piezometric surface and the
rate and duration of discharge of a well using groundwater storage'',
Trans. Am. Geophys. Union, 16, 519-524, 1935
"""
if np.isclose(r_well, 0) and np.isposinf(r_bound) and lap_kwargs is None:
return well_solution(time, rad, storage, transmissivity, rate)
return ext_grf(
time=time,
rad=rad,
S_part=[storage],
K_part=[transmissivity],
R_part=[r_well, r_bound],
dim=2,
lat_ext=1,
rate=rate,
K_well=None,
h_bound=h_bound,
lap_kwargs=lap_kwargs,
struc_grid=struc_grid,
)
[docs]def grf(
time,
rad,
storage,
conductivity,
dim=2,
lat_ext=1.0,
rate=-1e-4,
r_well=0.0,
r_bound=np.inf,
h_bound=0.0,
struc_grid=True,
lap_kwargs=None,
):
"""
The general radial flow (GRF) model for a pumping test.
This solution was presented in [Barker88]_.
Parameters
----------
time : :class:`numpy.ndarray`
Array with all time-points where the function should be evaluated.
rad : :class:`numpy.ndarray`
Array with all radii where the function should be evaluated.
storage : :class:`float`
Storage coefficient of the aquifer.
conductivity : :class:`float`
Conductivity of the aquifer.
dim : :class:`float`, optional
Fractional dimension of the aquifer. Default: ``2.0``
lat_ext : :class:`float`, optional
Lateral extend of the aquifer. Default: ``1.0``
rate : :class:`float`, optional
Pumpingrate at the well. Default: -1e-4
r_well : :class:`float`, optional
Inner radius of the pumping-well. Default: ``0.0``
r_bound : :class:`float`, optional
Radius of the outer boundary of the aquifer. Default: ``np.inf``
h_bound : :class:`float`, optional
Reference head at the outer boundary, as well as initial condition.
Default: ``0.0``
struc_grid : :class:`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``
lap_kwargs : :class:`dict` or :any:`None` optional
Dictionary for :any:`get_lap_inv` containing `method` and
`method_dict`. The default is equivalent to
``lap_kwargs = {"method": "stehfest", "method_dict": None}``.
Default: :any:`None`
Returns
-------
head : :class:`numpy.ndarray`
Array with all heads at the given radii and time-points.
References
----------
.. [Barker88] Barker, J.
''A generalized radial flow model for hydraulic tests
in fractured rock.'',
Water Resources Research 24.10, 1796-1804, 1988
"""
if np.isclose(r_well, 0) and np.isposinf(r_bound) and lap_kwargs is None:
return grf_solution(
time=time,
rad=rad,
storage=storage,
conductivity=conductivity,
dim=dim,
lat_ext=lat_ext,
rate=rate,
h_bound=h_bound,
struc_grid=struc_grid,
)
return ext_grf(
time=time,
rad=rad,
S_part=[storage],
K_part=[conductivity],
R_part=[r_well, r_bound],
dim=dim,
lat_ext=lat_ext,
rate=rate,
K_well=None,
h_bound=h_bound,
struc_grid=struc_grid,
lap_kwargs=lap_kwargs,
)
if __name__ == "__main__":
import doctest
doctest.testmod()