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The extended Theis solution in 2D
We provide an extended theis solution, that incorporates the effectes of a heterogeneous transmissivity field on a pumping test.
In the following this extended solution is compared to the standard theis solution for well flow. You can nicely see, that the extended solution represents a transition between the theis solutions for the geometric- and harmonic-mean transmissivity.
Reference: Zech et. al. 2016
import numpy as np from matplotlib import pyplot as plt from anaflow import ext_theis_2d, theis
We use three time steps: 10s, 10min, 10h
time_labels = ["10 s", "10 min", "10 h"] time = [10, 600, 36000] # 10s, 10min, 10h
Radius from the pumping well should be in [0, 4].
rad = np.geomspace(0.05, 4)
Parameters of heterogeneity, storage and pumping rate.
var = 0.5 # variance of the log-transmissivity len_scale = 10.0 # correlation length of the log-transmissivity TG = 1e-4 # the geometric mean of the transmissivity TH = TG * np.exp(-var / 2.0) # the harmonic mean of the transmissivity S = 1e-4 # storativity rate = -1e-4 # pumping rate
Now let’s compare the extended Theis solution to the classical solutions for the near and far field values of transmissivity.
head_TG = theis(time, rad, S, TG, rate) head_TH = theis(time, rad, S, TH, rate) head_ef = ext_theis_2d(time, rad, S, TG, var, len_scale, rate) time_ticks =  for i, step in enumerate(time): label_TG = "Theis($T_G$)" if i == 0 else None label_TH = "Theis($T_H$)" if i == 0 else None label_ef = "extended Theis" if i == 0 else None plt.plot( rad, head_TG[i], label=label_TG, color="C" + str(i), linestyle="--" ) plt.plot( rad, head_TH[i], label=label_TH, color="C" + str(i), linestyle=":" ) plt.plot(rad, head_ef[i], label=label_ef, color="C" + str(i)) time_ticks.append(head_ef[i][-1]) plt.xlabel("r in [m]") plt.ylabel("h in [m]") plt.legend() ylim = plt.gca().get_ylim() plt.gca().set_xlim([0, rad[-1]]) ax2 = plt.gca().twinx() ax2.set_yticks(time_ticks) ax2.set_yticklabels(time_labels) ax2.set_ylim(ylim) plt.tight_layout() plt.show()
Total running time of the script: ( 0 minutes 0.507 seconds)