Example

Zhou’s paper

As illustrated below, curves similar to those in Figure 6 of Zhou (2013) can be reproduced using this Python code, with the same parameter set applied in each figure. This demonstrates that the hystfit tool effectively implements the algorithm proposed by Zhou (click to enlarge).

As illustrated in these figures, a larger advancing contact angle (γ_A)—depicted in the upper figures—results in a greater deviation between the main wetting and drying curves. Conversely, a larger value of ‘b’ (shown in the right figures) causes a slower transition from the main drying curve to the main wetting curve, and vice versa.

The sample code utilizes matplotlib for rendering these figures. For further details, refer to the matplotlib documentation. The smooth_theta method in the hystfit class is used to draw smooth curves. For instance, executing the following code:

theta = f.smooth_theta((0.3, 0.7, 0.55, 0.9, 0.8))

results in a numpy array named ‘theta’, which begins at 0.3, gradually increases to 0.7, then decreases to 0.55, and continues similarly. The code:

h = f.h(p, theta)

calculates ‘h’ from this theta value.

UNSODA dataset

This program utilized the UNSODA dataset and produced 15 figures, with full data links available through the UNSODA viewer. The model exhibits an excellent fit for sand in these figures, particularly for disturbed samples; see samples 1410, 2310, 4890, 4940 and 4941.

Method: The Python code used to generate the figures is provided. Laboratory drying curves were fitted using either the VG or FX model, depending on which had the lower corrected AIC. θ_s is fixed when measured and optimized otherwise. θ_r is set to 0. Laboratory wetting curves were fitted using the Zhou model. The saturated point may be omitted when fitting the wetting curve. In some cases, θ_s was altered in the wetting curve, as documented. Data points where h = 0 cm are displayed as h = 1 cm in the figures.

UNSODA 1270 Tuffaceous Rock (undisturbed)

• Full data
• Rasmussen et al., Unsaturated Fractured Rock Characterization Methods and Data Sets at the Apache Leap Tuff Site, NUREG/CR-5596.
• FX: a = 3.22e+02 m = 0.463 n = 1.516 R² = 0.9950
• Zhou: cosγ_A = 0.749 b = 0.50 R² = 0.887

  UNSODA 1410 sand (disturbed)

• Full data
• Shen and Jaynes, J. Hydr. Eng. (China) 10:11-20.
• FX: a = 22.1 m = 0.841 n = 12.259 R² = 0.9985
• Zhou: cosγ_A = 0.541 b = 1.00 R² = 0.865

  UNSODA 2020 silty clay loam (undisturbed)

• Full data
• Ahuja and El-Swaify, 1975. Hydrologic Characteristics of Benchmark Soils of Hawaii’s Forest Watersheds. Univ. of Hawaii.
• FX: a = 10.9 m = 0.127 n = 1.701 R² = 0.9966
• Zhou: cosγ_A = 0.328 b = 0.40 R² = 0.963
• θ_s = 0.684 for drying and θ_s = 0.711 for wetting

  UNSODA 2022 silty clay loam (undisturbed)

• Full data
• Ahuja and El-Swaify, 1975. Hydrologic Characteristics of Benchmark Soils of Hawaii’s Forest Watersheds. Univ. of Hawaii.
• FX: a = 11.5 m = 0.080 n = 2.888 R² = 0.9969
• Zhou: cosγ_A = 0.029 b = 0.50 R² = 0.906
• θ_s = 0.671 for drying and θ_s = 0.711 for wetting

  UNSODA 2310 sand (disturbed)

• Full data
• Stauffer and Dracos, 1986. J. Hydrol. 84:9-34.
• FX: a = 37.4 m = 0.889 n = 16.851 R² = 0.9987
• Zhou: cosγ_A = 0.145 b = 1.00 R² = 0.965
• θ_s = 0.348 for drying and θ_s = 0.296 for wetting

  UNSODA 3340 sand (undisturbed)

• Full data
• FX: θ_s = 0.311 a = 20.3 m = 0.762 n = 4.790 R² = 0.9925
• Zhou: cosγ_A = 0.359 b = 1.00 R² = 0.718

  UNSODA 4700 silty clay loam (disturbed)

• Full data
• Moore, R.E., Hilgardia 12(6):383-427, 1939
• FX: θ_s = 0.322 a = 54.8 m = 0.199 n = 7.935 R² = 0.9980
• Zhou: cosγ_A = 0.276 b = 0.87 R² = 0.64
• θ_s = 0.322 for drying and θ_s = 0.396 for wetting

  UNSODA 4710 loam (disturbed)

• Full data
• Moore, R.E., Hilgardia 12(6):383-427, 1939
• FX: θ_s = 0.364 a = 38.5 m = 0.710 n = 2.772 R² = 0.9973
• Zhou: cosγ_A = 0.503 b = 1.00 R² = 0.891
• θ_s = 0.364 for drying and θ_s = 0.406 for wetting

  UNSODA 4890 sand (disturbed)

• Full data
• Bull. IASH XI No.1:69-110. 1966
• FX: a = 1.17e+02 m = 0.858 n = 6.542 R² = 0.9962
• Zhou: cosγ_A = 0.393 b = 1.00 R² = 0.951
• θ_s = 0.298 for drying and θ_s = 0.301 for wetting

  UNSODA 4910 loam (disturbed)

• Full data
• Elrick and Bowman, 1964. Soil Sci. Soc. Am. Proc. 28:450-452
• FX: θ_s = 0.520 a = 59.7 m = 0.416 n = 2.244 R² = 0.9956
• Zhou: cosγ_A = 0.000 b = 0.79 R² = 0.998

  UNSODA 4920 silt loam (undisturbed)

• Full data
• Green, et al., 1964. Soil Sci. Soc. Am. Proc. 28:15-19
• FX: θ_s = 0.539 a = 1.3e+02 m = 0.672 n = 1.433 R² = 0.9995
• Zhou: cosγ_A = 0.000 b = 0.98 R² = 0.814

  UNSODA 4922 silt loam (undisturbed)

• Full data
• Green, et al., 1964. Soil Sci. Soc. Am. Proc. 28:15-19
• FX: θ_s = 0.521 a = 77.3 m = 0.678 n = 1.550 R² = 0.9995
• Zhou: cosγ_A = 0.104 b = 1.00 R² = 0.781

  UNSODA 4923 silt loam (undisturbed)

• Full data
• Green, et al., 1964. Soil Sci. Soc. Am. Proc. 28:15-19
• FX: θ_s = 0.521 a = 77.3 m = 0.678 n = 1.550 R² = 0.9995
• Zhou: cosγ_A = 0.104 b = 1.00 R² = 0.781

  UNSODA 4940 sand (disturbed)

• Full data
• Poulovassilis, 1970. Soil Science 109(1):5-12
• FX: θ_s = 0.305 a = 6.23e+02 m = 2678.129 n = 2.345 R² = 0.9952
• Zhou: cosγ_A = 0.586 b = 1.00 R² = 0.992

  UNSODA 4941 sand (disturbed)

• Full data
• Poulovassilis, 1970. Soil Science 109(1):5-12
• VG: θ_s = 0.275 α = 0.0339 n = 4.84 R² = 0.9963
• Zhou: cosγ_A = 0.438 b = 1.00 R² = 0.957