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The hydraulic conductivities of each sand sample determined from constant-head measurements kD and the values calculated based on the mathematical model using numerical optimization techniques kW are presented in Tables 2–7. The values of absolute and percent errors between kD and kW are given in the above-mentioned tables. Moreover, examples of rate-of-rise curves are depicted in Figures 4–9 for every experimental series. In each figure three curves are presented: (1) the experimentally recorded rate-of-rise curve for an experiment with two sand types fully filling the chambers of the apparatus; (2) the corresponding curve calculated using the mathematical model for hydraulic conductivities derived from numerical optimization (kW values); and (3) the corresponding curve calculated using the mathematical model for hydraulic conductivities obtained from constant-head tests in the apparatus (kD values). A logarithmic timescale was used for each figure. Every set of curves h = h(t) was normalized using the following equation (Equation (11)):
formula
11
Furthermore, a breakdown of hydraulic conductivity identification results (kD and kW values) for all experimental series is shown in Figure 10.
Table 2

Results of hydraulic conductivity identification for experimental series I_II

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.31 × 10−3 2.87 × 10−3 4.40 × 10−4 13.3 
I50 I 3.31 × 10−3 2.96 × 10−3 3.50 × 10−4 10.6 
II25 II 2.88 × 10−4 3.00 × 10−4 1.20 × 10−5 4.2 
II50 II 2.88 × 10−4 2.33 × 10−4 5.50 × 10−5 19.1 
I25_II25 I 3.31 × 10−3 3.42 × 10−3 1.10 × 10−4 3.3 
II 2.88 × 10−4 2.77 × 10−4 1.10 × 10−5 3.8 
I25_II50 I 3.31 × 10−3 3.60 × 10−3 2.90 × 10−4 8.8 
II 2.88 × 10−4 2.43 × 10−4 4.50 × 10−5 15.6 
I50_II25 I 3.31 × 10−3 3.42 × 10−3 1.10 × 10−4 3.3 
II 2.88 × 10−4 3.03 × 10−4 1.50 × 10−5 5.2 
I50_II50 I 3.31 × 10−3 3.52 × 10−3 2.10 × 10−4 6.3 
II 2.88 × 10−4 2.52 × 10−4 3.60 × 10−5 12.5 
      average 1.40 × 10−4 8.8 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.31 × 10−3 2.87 × 10−3 4.40 × 10−4 13.3 
I50 I 3.31 × 10−3 2.96 × 10−3 3.50 × 10−4 10.6 
II25 II 2.88 × 10−4 3.00 × 10−4 1.20 × 10−5 4.2 
II50 II 2.88 × 10−4 2.33 × 10−4 5.50 × 10−5 19.1 
I25_II25 I 3.31 × 10−3 3.42 × 10−3 1.10 × 10−4 3.3 
II 2.88 × 10−4 2.77 × 10−4 1.10 × 10−5 3.8 
I25_II50 I 3.31 × 10−3 3.60 × 10−3 2.90 × 10−4 8.8 
II 2.88 × 10−4 2.43 × 10−4 4.50 × 10−5 15.6 
I50_II25 I 3.31 × 10−3 3.42 × 10−3 1.10 × 10−4 3.3 
II 2.88 × 10−4 3.03 × 10−4 1.50 × 10−5 5.2 
I50_II50 I 3.31 × 10−3 3.52 × 10−3 2.10 × 10−4 6.3 
II 2.88 × 10−4 2.52 × 10−4 3.60 × 10−5 12.5 
      average 1.40 × 10−4 8.8 
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Table 3

Results of hydraulic conductivity identification for experimental series I_III

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.48 × 10−3 3.11 × 10−3 3.70 × 10−4 10.6 
I50 I 3.48 × 10−3 3.46 × 10−3 2.00 × 10−5 0.6 
III25 III 7.71 × 10−5 1.11 × 10−4 3.39 × 10−5 44.0 
III50 III 7.71 × 10−5 8.60 × 10−5 8.90 × 10−6 11.5 
I25_III25 I 3.48 × 10−3 3.59 × 10−3 1.10 × 10−4 3.2 
III 7.71 × 10−5 1.12 × 10−4 3.49 × 10−5 45.3 
I25_III50 I 3.48 × 10−3 3.74 × 10−3 2.60 × 10−4 7.5 
III 7.71 × 10−5 7.91 × 10−5 2.00 × 10−6 2.6 
I50_III25 I 3.48 × 10−3 3.85 × 10−3 3.70 × 10−4 10.6 
III 7.71 × 10−5 1.15 × 10−4 3.79 × 10−5 49.2 
I50_III50 I 3.48 × 10−3 3.97 × 10−3 4.90 × 10−4 14.1 
III 7.71 × 10−5 8.63 × 10−5 9.20 × 10−6 11.9 
      average 1.46 × 10−4 17.6 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.48 × 10−3 3.11 × 10−3 3.70 × 10−4 10.6 
I50 I 3.48 × 10−3 3.46 × 10−3 2.00 × 10−5 0.6 
III25 III 7.71 × 10−5 1.11 × 10−4 3.39 × 10−5 44.0 
III50 III 7.71 × 10−5 8.60 × 10−5 8.90 × 10−6 11.5 
I25_III25 I 3.48 × 10−3 3.59 × 10−3 1.10 × 10−4 3.2 
III 7.71 × 10−5 1.12 × 10−4 3.49 × 10−5 45.3 
I25_III50 I 3.48 × 10−3 3.74 × 10−3 2.60 × 10−4 7.5 
III 7.71 × 10−5 7.91 × 10−5 2.00 × 10−6 2.6 
I50_III25 I 3.48 × 10−3 3.85 × 10−3 3.70 × 10−4 10.6 
III 7.71 × 10−5 1.15 × 10−4 3.79 × 10−5 49.2 
I50_III50 I 3.48 × 10−3 3.97 × 10−3 4.90 × 10−4 14.1 
III 7.71 × 10−5 8.63 × 10−5 9.20 × 10−6 11.9 
      average 1.46 × 10−4 17.6 
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Table 4

Results of hydraulic conductivity identification for experimental series I_IV

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.30 × 10−3 3.46 × 10−3 1.60 × 10−4 4.8 
I50 I 3.30 × 10−3 3.45 × 10−3 1.5 × 10−4 4.5 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
I25_IV25 I 3.30 × 10−3 4.75 × 10−3 1.45 × 10−3 43.9 
IV 1.90 × 10−5 2.11 × 10−5 2.10 × 10−6 11.1 
I25_IV50 I 3.30 × 10−3 4.52 × 10−3 1.22 × 10−3 37.0 
IV 1.90 × 10−5 2.29 × 10−5 3.90 × 10−6 20.5 
I50_IV25 I 3.30 × 10−3 4.00 × 10−3 7.00 × 10−4 21.2 
IV 1.90 × 10−5 2.00 × 10−5 1.00 × 10−6 5.3 
I50_IV50 I 3.30 × 10−3 4.53 × 10−3 1.23 × 10−3 37.3 
IV 1.90 × 10−5 2.04 × 10−5 1.40 × 10−6 7.4 
      average 4.11 × 10−4 18.1 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
I25 I 3.30 × 10−3 3.46 × 10−3 1.60 × 10−4 4.8 
I50 I 3.30 × 10−3 3.45 × 10−3 1.5 × 10−4 4.5 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
I25_IV25 I 3.30 × 10−3 4.75 × 10−3 1.45 × 10−3 43.9 
IV 1.90 × 10−5 2.11 × 10−5 2.10 × 10−6 11.1 
I25_IV50 I 3.30 × 10−3 4.52 × 10−3 1.22 × 10−3 37.0 
IV 1.90 × 10−5 2.29 × 10−5 3.90 × 10−6 20.5 
I50_IV25 I 3.30 × 10−3 4.00 × 10−3 7.00 × 10−4 21.2 
IV 1.90 × 10−5 2.00 × 10−5 1.00 × 10−6 5.3 
I50_IV50 I 3.30 × 10−3 4.53 × 10−3 1.23 × 10−3 37.3 
IV 1.90 × 10−5 2.04 × 10−5 1.40 × 10−6 7.4 
      average 4.11 × 10−4 18.1 
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Table 5

Results of hydraulic conductivity identification for experimental series II_III

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
II25 II 2.12 × 10−4 2.16 × 10−4 4.00 × 10−6 1.9 
II50 II 2.12 × 10−4 2.03 × 10−4 9.00 × 10−6 4.2 
III25 III 6.21 × 10−5 1.11 × 10−4 4.89 × 10−5 78.7 
III50 III 6.21 × 10−5 7.97 × 10−5 1.76 × 10−5 28.3 
II25_III25 II 2.12 × 10−4 2.77 × 10−4 6.50 × 10−5 30.7 
III 6.21 × 10−5 1.15 × 10−4 5.29 × 10−5 85.2 
II25_III50 II 2.12 × 10−4 2.28 × 10−4 1.60 × 10−5 7.5 
III 6.21 × 10−5 7.97 × 10−5 1.76 × 10−5 28.3 
II50_III25 II 2.12 × 10−4 1.59 × 10−4 5.30 × 10−5 25.0 
III 6.21 × 10−5 1.37 × 10−4 7.49 × 10−5 120.6 
II50_III50 II 2.12 × 10−4 2.02 × 10−4 1.00 × 10−5 4.7 
III 6.21 × 10−5 8.17 × 10−5 1.96 × 10−5 31.6 
      average 3.24 × 10−5 37.2 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
II25 II 2.12 × 10−4 2.16 × 10−4 4.00 × 10−6 1.9 
II50 II 2.12 × 10−4 2.03 × 10−4 9.00 × 10−6 4.2 
III25 III 6.21 × 10−5 1.11 × 10−4 4.89 × 10−5 78.7 
III50 III 6.21 × 10−5 7.97 × 10−5 1.76 × 10−5 28.3 
II25_III25 II 2.12 × 10−4 2.77 × 10−4 6.50 × 10−5 30.7 
III 6.21 × 10−5 1.15 × 10−4 5.29 × 10−5 85.2 
II25_III50 II 2.12 × 10−4 2.28 × 10−4 1.60 × 10−5 7.5 
III 6.21 × 10−5 7.97 × 10−5 1.76 × 10−5 28.3 
II50_III25 II 2.12 × 10−4 1.59 × 10−4 5.30 × 10−5 25.0 
III 6.21 × 10−5 1.37 × 10−4 7.49 × 10−5 120.6 
II50_III50 II 2.12 × 10−4 2.02 × 10−4 1.00 × 10−5 4.7 
III 6.21 × 10−5 8.17 × 10−5 1.96 × 10−5 31.6 
      average 3.24 × 10−5 37.2 
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Table 6

Results of hydraulic conductivity identification for experimental series II_IV

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
II25 II 2.38 × 10−4 1.95 × 10−4 4.30 × 10−5 18.1 
II50 II 2.38 × 10−4 1.91 × 10−4 4.70 × 10−5 19.7 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
II25_IV25 II 2.38 × 10−4 2.22 × 10−4 1.60 × 10−5 6.7 
IV 1.90 × 10−5 1.80 × 10−5 1.00 × 10−6 5.3 
II25_IV50 II 2.38 × 10−4 2.48 × 10−4 1.00 × 10−5 4.2 
IV 1.90 × 10−5 2.37 × 10−5 4.70 × 10−6 24.7 
II50_IV25 II 2.38 × 10−4 2.09 × 10−4 2.90 × 10−5 12.2 
IV 1.90 × 10−5 1.69 × 10−5 2.10 × 10−6 11.1 
II50_IV50 II 2.38 × 10−4 2.14 × 10−4 2.40 × 10−5 10.1 
IV 1.90 × 10−5 2.52 × 10−5 6.20 × 10−6 32.6 
      average 1.56 × 10−5 14.0 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
II25 II 2.38 × 10−4 1.95 × 10−4 4.30 × 10−5 18.1 
II50 II 2.38 × 10−4 1.91 × 10−4 4.70 × 10−5 19.7 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
II25_IV25 II 2.38 × 10−4 2.22 × 10−4 1.60 × 10−5 6.7 
IV 1.90 × 10−5 1.80 × 10−5 1.00 × 10−6 5.3 
II25_IV50 II 2.38 × 10−4 2.48 × 10−4 1.00 × 10−5 4.2 
IV 1.90 × 10−5 2.37 × 10−5 4.70 × 10−6 24.7 
II50_IV25 II 2.38 × 10−4 2.09 × 10−4 2.90 × 10−5 12.2 
IV 1.90 × 10−5 1.69 × 10−5 2.10 × 10−6 11.1 
II50_IV50 II 2.38 × 10−4 2.14 × 10−4 2.40 × 10−5 10.1 
IV 1.90 × 10−5 2.52 × 10−5 6.20 × 10−6 32.6 
      average 1.56 × 10−5 14.0 
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Table 7

Results of hydraulic conductivity identification for experimental series III_IV

ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
III25 III 8.09 × 10−5 1.05 × 10−4 2.41 × 10−5 29.8 
III50 III 8.09 × 10−5 8.41 × 10−5 3.20 × 10−6 4.0 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
III25_IV25 III 8.09 × 10−5 1.14 × 10−4 3.31 × 10−5 40.9 
IV 1.90 × 10−5 1.78 × 10−5 1.20 × 10−6 6.3 
III25_IV50 III 8.09 × 10−5 8.66 × 10−5 5.70 × 10−6 7.0 
IV 1.90 × 10−5 2.42 × 10−5 5.20 × 10−6 27.4 
III50_IV25 III 8.09 × 10−5 4.41 × 10−5 3.68 × 10−5 45.5 
IV 1.90 × 10−5 3.69 × 10−5 1.79 × 10−5 94.2 
III50_IV50 III 8.09 × 10−5 9.00 × 10−5 9.10 × 10−6 11.2 
IV 1.90 × 10−5 2.74 × 10−5 8.40 × 10−6 44.2 
      average 1.24 × 10−5 27.9 
ExperimentSand sampleDarcy test kD [m/s]W-tube test kW [m/s]Absolute error Δk [m/s]Percent error δk% [%]
III25 III 8.09 × 10−5 1.05 × 10−4 2.41 × 10−5 29.8 
III50 III 8.09 × 10−5 8.41 × 10−5 3.20 × 10−6 4.0 
IV25 IV 1.90 × 10−5 1.50 × 10−5 4.00 × 10−6 21.1 
IV50 IV 1.90 × 10−5 1.85 × 10−5 5.00 × 10−7 2.6 
III25_IV25 III 8.09 × 10−5 1.14 × 10−4 3.31 × 10−5 40.9 
IV 1.90 × 10−5 1.78 × 10−5 1.20 × 10−6 6.3 
III25_IV50 III 8.09 × 10−5 8.66 × 10−5 5.70 × 10−6 7.0 
IV 1.90 × 10−5 2.42 × 10−5 5.20 × 10−6 27.4 
III50_IV25 III 8.09 × 10−5 4.41 × 10−5 3.68 × 10−5 45.5 
IV 1.90 × 10−5 3.69 × 10−5 1.79 × 10−5 94.2 
III50_IV50 III 8.09 × 10−5 9.00 × 10−5 9.10 × 10−6 11.2 
IV 1.90 × 10−5 2.74 × 10−5 8.40 × 10−6 44.2 
      average 1.24 × 10−5 27.9 
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Figure 4
Experimentally recorded rate-of-rise curve for I50_II50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_II50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 4
Experimentally recorded rate-of-rise curve for I50_II50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_II50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 5
Experimentally recorded rate-of-rise curve for I50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 5
Experimentally recorded rate-of-rise curve for I50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 6
Experimentally recorded rate-of-rise curve for I50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 6
Experimentally recorded rate-of-rise curve for I50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for I50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 7
Experimentally recorded rate-of-rise curve for II50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for II50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 7
Experimentally recorded rate-of-rise curve for II50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for II50_III50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 8
Experimentally recorded rate-of-rise curve for II50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for II50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 8
Experimentally recorded rate-of-rise curve for II50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for II50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 9
Experimentally recorded rate-of-rise curve for III50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for III50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 9
Experimentally recorded rate-of-rise curve for III50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).
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Experimentally recorded rate-of-rise curve for III50_IV50 test (bold line) compared to the results of simulations with kD values from constant-head tests (dotted line) and kW values from numerical optimization (dashed line).

Figure 10
Breakdown of hydraulic conductivity identification results for all experimental series: (a) experiments conducted in the U-tube; (b) experiments conducted in the W-tube.
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Breakdown of hydraulic conductivity identification results for all experimental series: (a) experiments conducted in the U-tube; (b) experiments conducted in the W-tube.

Figure 10
Breakdown of hydraulic conductivity identification results for all experimental series: (a) experiments conducted in the U-tube; (b) experiments conducted in the W-tube.
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Breakdown of hydraulic conductivity identification results for all experimental series: (a) experiments conducted in the U-tube; (b) experiments conducted in the W-tube.

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