Supporting Material
Part 1: Figs. S31-S37 support Fig. 3 in the paper.
Fig. S31 Contours of (a)kapp = 0.001, (b) qapp = 0.001 (∆x = 10 cm), (c) kapp = 0.01 and (d) qapp = 0.01 (∆x = 10 cm) for the vG function (Glendale clay loam in Table 2)
Fig. S32(a) Contours of kapp = 0.01 and (b) contours of qapp = 0.01 (∆x = 10 cm) for the vG function (Berino loamy fine sand in Table 2)
Fig. S33(a) Contours of kapp = 0.01 and (b) contours of qapp = 0.01 (∆x = 10 cm)for the BC function (Loamy sand in Table 2)
Fig. S34(a) Contours of kapp = 0.01 and (b) contours of qapp = 0.01 (∆x = 10 cm)for the BC function (Silt loam in Table 2)
Fig. S35(a) Contours of kapp = 0.01 and (b) contours of qapp = 0.01 (∆x = 10 cm)for the BC function (Silty clay in Table 2)
Fig. S36(a) Contours of kapp = 0.01 and (b) contours of qapp = 0.01 (∆x = 10 cm)for for the GE function
Fig. S37. Contours of kapp/kH for the Berino loamy fine sand (9801 nodes homogeneously distributed in the Se1-Se2 coordinate ranged in [0.01, 0.99]).
Part 2: Figs. S4 support Fig. 4 in the paper.
Fig. S4 Comparisons of approximate solutions to the precise solution using the Chinese loamy soil. (a) distributions of h in the profile; (b) comparisons of Se within 49.5 ≤x ≤50.5 cm; (c) distributions of relative errors of h; and (d) distributions of absolute errors of Se. (∆t = 1 s, ∆x= 1 cm, l = 300 cm, and t = 12000 s. The analytical solution denoted by “PRE”;the numerical solutionsdenoted by the subscripts of the kapps.)
Part 3: Fig. S5 supports Fig. 5 in the paper.
Fig. S5Comparisons of relative errors for different soils described by the BC function. (a) relative h errors for the loamy sand at t = 200000 s; (b) relative h errors for the silty clay at t = 60000 s; (c) running MER for the loamy sand; and (d) running MER for the silty clay. (For the loamy sand, ∆t = 30 s, ∆x= 0.2 cm, lx= 10 cm,t = 60000 s, h| t = 0,0 ≤x ≤10 = -500 cm, h| t ≥ 0, x = 0 = -20 cm and h| t ≥ 0, x = 10 = -500 cm. For the silty clay, ∆t = 50 s, ∆x= 0.2 cm, lx= 10 cm,t = 200000 s,Se| t = 0,0 ≤x ≤10 = 0.015, Se| t ≥ 0, x = 0= 0.99 and Se| t ≥ 0, x = 10 = 0.015. For the two cases, the numerical solutions obtained from various kapp were compared with thosefrom kH (with 20% of the original∆t and ∆x)).
Part 4: Fig. S6 supports Fig. 6 in the paper.
Fig. 6SComparisons of relative errors for different soils described by the BC function. (a) relative h errors for the Glendale clay loam at t = 200000 s; (b) relative h errors for the Berino loamy fine sand at t = 60000 s; (c) running MERs for the Glendale clay loam; and (d) running MER for the Berino loamy fine sand. (For the Glendale clay loam, ∆t = 5 s, ∆x= 0.2 cm, lx= 30 cm,t = 5000 s,h| t = 0,0 ≤x ≤10 = -500 cm, h| t ≥ 0, x = 0 = -10 cm and h| t ≥ 0, x = 10 = -500 cm. For the Berino loamy fine sand, ∆t = 1 s, ∆x= 0.2 cm, lx= 50 cm,t = 2000 s,h| t = 0,0 ≤x ≤10 = -1000 cm, h| t ≥ 0, x = 0 = -10 cm and h| t ≥ 0, x = 10 = -1000 cm. For the two cases, the numerical solutions from various kapp were compared with thosefrom the kH (with 50% of the original∆t and ∆x)).
Part 5: Tables S41-S47 support Table 4 in the paper.
Table S41Statistic analysis of the ratios of kapp to kH for the Chinese loamy soil (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 2.46 / 1.04 / 1 / 3.46 / 1.14201 / 0.00026 / 0.25908 / 0.067
kGM-k/kH / 0.781 / 0.98 / 0.219 / 1 / 0.93557 / 0.00011 / 0.10643 / 0.011
kHM-k/kH / 0.9862 / 0.92 / 0.0138 / 1 / 0.82328 / 0.00023 / 0.22482 / 0.051
kAM-S/kH / 2.46 / 1.04 / 1 / 3.46 / 1.14201 / 0.00026 / 0.25908 / 0.067
kGM-S/kH / 0.781 / 0.98 / 0.219 / 1 / 0.93557 / 0.00011 / 0.10643 / 0.011
kHM-S/kH / 0.9862 / 0.92 / 0.0138 / 1 / 0.82328 / 0.00023 / 0.22482 / 0.051
kAM-h/kH / 0.781 / 0.98 / 0.219 / 1 / 0.93557 / 0.00011 / 0.10643 / 0.011
kGM-h/kH / 5.526 / 1.07 / 0.844 / 6.37 / 1.16975 / 0.00028 / 0.27990 / 0.078
kHM-h/kH / 5.912 / 1.16 / 0.998 / 6.91 / 1.35836 / 0.00053 / 0.53304 / 0.284
Table S42 Statistic analysis of the ratios of kapp to kH for the Loamy sand described by the BC function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 92.1 / 1.67 / 1 / 93.1 / 3.46276 / 0.00557 / 5.56429 / 30.961
kGM-k/kH / 0.999777 / 0.8 / 0.000223 / 1 / 0.68209 / 0.00031 / 0.31399 / 0.099
kHM-k/kH / 1 / 0.38 / 5.36E-10 / 1 / 0.44103 / 0.00037 / 0.37130 / 0.138
kAM-S/kH / 11.1 / 1.01 / 1 / 12.1 / 1.16304 / 0.00059 / 0.58870 / 0.347
kGM-S/kH / 0.999777 / 0.8 / 0.000223 / 1 / 0.68209 / 0.00031 / 0.31399 / 0.099
kHM-S/kH / 1 / 0.63 / 4.12E-09 / 1 / 0.55927 / 0.00037 / 0.36577 / 0.134
kAM-h/kH / 0.9999999 / 0.71 / 6.33E-08 / 1 / 0.61003 / 0.00035 / 0.35030 / 0.123
kGM-h/kH / 0.999777 / 0.8 / 0.000223 / 1 / 0.68209 / 0.00031 / 0.31399 / 0.099
kHM-h/kH / 0.589 / 0.89 / 0.411 / 1 / 0.82071 / 0.00018 / 0.18316 / 0.034
kLG-k/kH / 5.83 / 1.07 / 1 / 6.83 / 1.22308 / 0.00043 / 0.42657 / 0.182
Table S43 Statistic analysis of the ratios of kapp to kH for the Silt loam described by the BC function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 24.1 / 1.47 / 1 / 25.1 / 2.35382 / 0.00227 / 2.26879 / 5.147
kGM-k/kH / 0.99963 / 0.82 / 0.00037 / 1 / 0.70218 / 0.00030 / 0.30458 / .093
kHM-k/kH / 1 / 0.46 / 5.45E-09 / 1 / 0.47730 / 0.00037 / 0.36999 / .137
kAM-S/kH / 3.69 / 1 / 1 / 4.69 / 1.08823 / 0.00026 / 0.26349 / .069
kGM-S/kH / 0.99963 / 0.82 / 0.00037 / 1 / 0.70218 / 0.00030 / 0.30458 / .093
kHM-S/kH / 1 / 0.67 / 2.91E-08 / 1 / 0.58217 / 0.00036 / 0.35971 / .129
kAM-h/kH / 0.99999 / 0.79 / 6.97E-06 / 1 / 0.66675 / 0.00033 / 0.32602 / .106
kGM-h/kH / 0.99963 / 0.82 / 0.00037 / 1 / 0.70218 / 0.00030 / 0.30458 / .093
kHM-h/kH / 0.9804 / 0.86 / 0.0196 / 1 / 0.74807 / 0.00027 / 0.26916 / .072
kLG-k/kH / 1.12 / 1.03 / 1 / 2.12 / 1.07391 / 0.00012 / 0.11757 / .014
Table S44 Statistic analysis of the ratios of kapp to kH for the Silty clay described by the BC function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 16.3 / 1.41 / 1 / 17.3 / 2.12744 / 0.00177 / 1.76671 / 3.121
kGM-k/kH / 0.999545 / 0.83 / 0.000455 / 1 / 0.70927 / 0.00030 / 0.30098 / .091
kHM-k/kH / 1 / 0.49 / 1.2E-08 / 1 / 0.49021 / 0.00037 / 0.36894 / .136
kAM-S/kH / 2.64 / 1 / 1 / 3.64 / 1.07219 / 0.00021 / 0.20605 / .042
kGM-S/kH / 0.999545 / 0.83 / 0.000455 / 1 / 0.70927 / 0.00030 / 0.30098 / .091
kHM-S/kH / 0.9999999 / 0.69 / 5.71E-08 / 1 / 0.59028 / 0.00036 / 0.35730 / .128
kAM-h/kH / 0.9999582 / 0.81 / 0.0000418 / 1 / 0.68723 / 0.00032 / 0.31518 / .099
kGM-h/kH / 0.999545 / 0.83 / 0.000455 / 1 / 0.70927 / 0.00030 / 0.30098 / .091
kHM-h/kH / 0.99503 / 0.85 / 0.00497 / 1 / 0.73490 / 0.00028 / 0.28195 / .079
kLG-k/kH / 0.54 / 1.02 / 1 / 1.54 / 1.04070 / 0.00006 / 0.06250 / .004
Table S45 Statistic analysis of the ratios of kapp to kH for the Glendale clay loam described by the vG function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 7.59E+07 / 5.51 / 1 / 7.59E+07 / 3.28E+04 / 854.17726 / 8.53E+05 / 7.28E+11
kGM-k/kH / 1.03975 / 0.67 / 0.00025 / 1.04 / 0.60217 / 0.00033 / 0.32787 / 0.107
kHM-k/kH / 1 / 0.08 / 8.45E-16 / 1 / 0.26969 / 0.00034 / 0.33777 / 0.114
kAM-S/kH / 1.83E+05 / 1.01 / 0.417 / 1.83E+05 / 216.28970 / 4.75360 / 4.75E+03 / 2.26E+07
kGM-S/kH / 0.99991 / 0.59 / 0.00009 / 1 / 0.56089 / 0.00033 / 0.33104 / 0.11
kHM-S/kH / 1 / 0.37 / 7.87E-14 / 1 / 0.42989 / 0.00037 / 0.36873 / 0.136
kAM-h/kH / 1 / 0.22 / 3.35E-15 / 1 / 0.36246 / 0.00037 / 0.36957 / 0.137
kGM-h/kH / 10.39936 / 0.85 / 0.00064 / 10.4 / 0.76699 / 0.00052 / 0.52328 / 0.274
kHM-h/kH / 1.29E+08 / 2.92 / 1 / 1.29E+08 / 1.94E+04 / 733.76549 / 7.33E+05 / 5.37E+11
kLG-k/kH / 2.80E+06 / 1.95 / 1 / 2.80E+06 / 1364.72098 / 32.99457 / 3.30E+04 / 1.09E+09
Table S46 Statistic analysis of the ratios of kapp to kH for the New Mexico clay described by the vG function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 1329 / 2.08 / 1 / 1330 / 10.03206 / 0.05018 / 50.12677 / 2513
kGM-k/kH / 0.999766 / 0.73 / 0.000234 / 1 / 0.64056 / 0.00032 / 0.31592 / 0.1
kHM-k/kH / 1 / 0.25 / 4.36E-11 / 1 / 0.37476 / 0.00036 / 0.36053 / 0.13
kAM-S/kH / 78.3 / 0.99 / 0.4 / 78.7 / 1.40947 / 0.00323 / 3.22675 / 10.412
kGM-S/kH / 0.999878 / 0.68 / 0.000122 / 1 / 0.60954 / 0.00032 / 0.32186 / 0.104
kHM-S/kH / 1 / 0.50 / 4.91E-10 / 1 / 0.49554 / 0.00036 / 0.36350 / 0.132
kAM-h/kH / 1 / 0.58 / 4.93E-10 / 1 / 0.52928 / 0.00037 / 0.36871 / 0.136
kGM-h/kH / 4.9486 / 0.93 / 0.0014 / 4.95 / 0.81531 / 0.00037 / 0.36576 / 0.134
kHM-h/kH / 2279 / 1.33 / 1 / 2280 / 4.13887 / 0.02518 / 25.15536 / 632.792
kLG-k/kH / 83.744 / 1.13 / 0.956 / 84.7 / 1.89128 / 0.00364 / 3.63435 / 13.209
Table S47 Statistic analysis of the ratios of kapp to kH for the Berino loamy fine sand described by the vG function (listed in Table 2) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
Range / Median / Minimum / Maximum / Mean / Std. Deviation / VarianceStatistic / Std. Error
kAM-k/kH / 375.003 / 1.83 / 0.997 / 376 / 5.53779 / 0.01736 / 17.34707 / 300.921
kGM-k/kH / 0.999767 / 0.73 / 0.000233 / 1 / 0.64702 / 0.00031 / 0.31326 / 0.098
kHM-k/kH / 1 / 0.29 / 1.8E-10 / 1 / 0.39533 / 0.00036 / 0.36252 / 0.131
kAM-S/kH / 32.207 / 0.99 / 0.393 / 32.6 / 1.19329 / 0.00150 / 1.49575 / 2.237
kGM-S/kH / 0.999867 / 0.69 / 0.000133 / 1 / 0.61947 / 0.00032 / 0.31912 / 0.102
kHM-S/kH / 1 / 0.53 / 1.55E-09 / 1 / 0.50813 / 0.00036 / 0.36176 / 0.131
kAM-h/kH / 1 / 0.64 / 3.07E-09 / 1 / 0.56601 / 0.00036 / 0.36231 / 0.131
kGM-h/kH / 4.0983 / 0.94 / 0.0017 / 4.1 / 0.82165 / 0.00034 / 0.34074 / 0.116
kHM-h/kH / 554.002 / 1.21 / 0.998 / 555 / 2.35908 / 0.00724 / 7.23695 / 52.373
kLG-k/kH / 25.732 / 1.07 / 0.868 / 26.6 / 1.44232 / 0.00140 / 1.39618 / 1.949
Part 6: Tables S61-S67 support Table 6 in the paper.
Table 61Percentage of one kapp in the first column greater than another in the first row for the Glendale clay loamlisted in Table 2 (%) (998001-couple (Se1, Se2) homogeneously distributed in the Se2-Se1 plane)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 99.9 / 91.48 / 100 / 100
kGM-k / 100 / 10.67 / 99.95 / 100 / 100 / 0 / 0 / 0 / 0.23
kHM-k / 0.09 / 0.1 / 0.12 / 0.01 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 98.93 / 72.7 / 0 / 0 / 59.14
kGM-S / 100 / 98.17 / 0 / 0 / 0 / 0
kHM-S / 96.21 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0
kGM-h / 0 / 4.57 / 20.05
kHM-h / 100 / 100
kLG-k / 100
Table S62Percentage of one kapp in the first column greater than another in the first row for the New Mexico clay in Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 97.91 / 86.12 / 100 / 100
kGM-k / 100 / 9.43 / 100 / 100 / 94.56 / 0 / 0 / 0 / 0
kHM-k / 0.23 / 0.29 / 0.38 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 92.34 / 62.01 / 0 / 0.17 / 38.4
kGM-S / 100 / 82.44 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0.06 / 0
kGM-h / 0 / 17.57 / 24.5
kHM-h / 73.81 / 100
kLG-k / 94
Table S63Percentage of one kapp in the first column greater than another in the first row for Berino loamy fine sand in Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 99.95 / 96.81 / 84.97 / 100 / 99.95
kGM-k / 100 / 8.54 / 100 / 100 / 86.42 / 0 / 0 / 0 / 0
kHM-k / 0.25 / 0.32 / 0.44 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 89.42 / 60.11 / 11.09 / 0.64 / 34.5
kGM-S / 100 / 74.48 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 1.27 / 0.04
kGM-h / 0 / 21.71 / 24.89
kHM-h / 70.35 / 99.54
kLG-k / 84.6
Table S64Percentage of one kapp in the first column greater than another in the first row for Loamy sand in Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100
kGM-k / 100 / 0 / 0 / 100 / 100 / 0 / 0 / 0 / 0
kHM-k / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 100 / 100 / 100 / 3.05 / 96.87
kGM-S / 100 / 100 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0
kGM-h / 0 / 0 / 0
kHM-h / 0 / 0
kLG-k / 100
Table S65Percentage of one kapp in the first column greater than another in the first row for Silt loamin Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100
kGM-k / 100 / 0 / 0 / 100 / 100 / 0 / 0 / 0 / 0
kHM-k / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 100 / 100 / 100 / 11.73 / 96.49
kGM-S / 100 / 100 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0
kGM-h / 0 / 0 / 0
kHM-h / 0 / 0
kLG-k / 99.99
Table S66Percentage of one kapp in the first column greater than another in the first row for Silty clay in Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100 / 100
kGM-k / 100 / 0 / 0 / 100 / 100 / 81.55 / 0 / 0 / 0
kHM-k / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 100 / 100 / 100 / 19.58 / 96.38
kGM-S / 100 / 100 / 81.55 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0
kGM-h / 0 / 0 / 0
kHM-h / 0 / 0
kLG-k / 99.97
Table S67Percentage of one kapp in the first column greater than another in the first row for Chinese loamy soilin Table 2 (%)(998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane excluding those at Se1=Se2)
kGM-k / kHM-k / kAM-S / kGM-S / kHM-S / kAM-h / kGM-h / kHM-h / kLG-k / kHkAM-k / 100 / 100 / 0 / 100 / 100 / 100 / 29.46 / 4.35 / 100 / 100
kGM-k / 100 / 0 / 0.01 / 100 / 47.89 / 0 / 0 / 0 / 0
kHM-k / 0 / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 100 / 100 / 29.46 / 4.35 / 100 / 100
kGM-S / 100 / 47.89 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0
kGM-h / 0 / 99.26 / 99.26
kHM-h / 99.999 / 100
kLG-k / 0
Table S68Percentage of thekapp greater than thekH for the seven-type soils listedin Table 2 (%) (998001-couple (Se1, Se2)homogeneously distributed in the Se2-Se1 plane)
The GE function / The BC function / The vG functionLS / SL / SC / GCL / NWC / BLFS
kAM-k / 100 / 100 / 100 / 100 / 100 / 100 / 99.95
kGM-k / 0 / 0 / 0 / 0 / 0.23 / 0 / 0
kHM-k / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-S / 100 / 96.87 / 96.49 / 96.38 / 59.14 / 38.41 / 34.5
kGM-S / 0 / 0 / 0 / 0 / 0 / 0 / 0
kHM-S / 0 / 0 / 0 / 0 / 0 / 0 / 0
kAM-h / 0 / 0 / 0 / 0 / 0 / 0 / 0.04
kGM-h / 99.26 / 0 / 0 / 0 / 20.05 / 24.46 / 24.89
kHM-h / 99.999 / 0 / 0 / 0 / 100 / 100 / 99.54
kLG-k / 0 / 100 / 99.99 / 99.97 / 100 / 93.98 / 84.6
LS: Loamy sand; SL: Silt loam; SC: Silty clay; GE: Chinese loamy soil; GCL:Glendale clay loam; NMC:New Mexico clay; BLFS: Berino loamy fine sand
1