anaflow.flow.ext_thiem_int_3d

ext_thiem_int_3d(rad, r_ref, cond_gmean, var, len_scale, roughness=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 the integral variogram 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 an integral correlation function incorporating a roughness parameter.

The roughness parameter controls the field roughness of log-transmissivity. It’s limits are a pure nugget model for r -> 0 and the gaussian model for r -> infinity.

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.

  • roughness (float, optional) – Roughness of the model. Should be positive. Default: 1

  • 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)