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.

  • 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


head – Array with all heads at the given radii and time-points.

Return type



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