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Replacing Wax With > in eq. [17] yields an equation for calculating the wetting scanning curve starting at a suction, Ww, on the boundary drying curve. The Mualem (1984a) independent model has the same as- sumptions as the Mualem (1977) model. Better results can be obtained with the aid of an additional wetting scanning curve. To predict scanning curves, the boundary wetting curve must first be calculated using eq. [15]. The scanning curves can then be calculated using the equations presented for the Mualem (1974) model-II.

Figure 17 Replacing Wax With > in eq. [17] yields an equation for calculating the wetting scanning curve starting at a suction, Ww, on the boundary drying curve. The Mualem (1984a) independent model has the same as- sumptions as the Mualem (1977) model. Better results can be obtained with the aid of an additional wetting scanning curve. To predict scanning curves, the boundary wetting curve must first be calculated using eq. [15]. The scanning curves can then be calculated using the equations presented for the Mualem (1974) model-II.

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Fig. 1. Commonly used definitions for hysteretic soil—water hys- teresis curves. predict the boundary wetting curve and scanning curves; and (ii) development of a simple method for estimating the hysteretic curves. These objectives are addressed through a review of the historical development of hysteresis models; a comparison of selected models using datasets collected from the literature; and a statistical analysis of the relationships among the primary hysteretic curves.
Fig. 1. Commonly used definitions for hysteretic soil—water hys- teresis curves. predict the boundary wetting curve and scanning curves; and (ii) development of a simple method for estimating the hysteretic curves. These objectives are addressed through a review of the historical development of hysteresis models; a comparison of selected models using datasets collected from the literature; and a statistical analysis of the relationships among the primary hysteretic curves.
Fig. 3. Boundary drying and boundary wetting processes in a soil and the corresponding Néel (1942, 1943) diagram. (a) Boundary drying and wetting curves. (b) Néel (1942, 1943) diagram for boundary drying process. (c) Néel (1942, 1943) diagram for boundary wetting process. Fig. 2. Schematic illustration of volumetric water distribution in the soil. (a) Boundary drying and wetting curves. (b) Perspective view of water content distribution as a function of the wy, W, plane for the Néel (1942, 1943) diagram.
Fig. 3. Boundary drying and boundary wetting processes in a soil and the corresponding Néel (1942, 1943) diagram. (a) Boundary drying and wetting curves. (b) Néel (1942, 1943) diagram for boundary drying process. (c) Néel (1942, 1943) diagram for boundary wetting process. Fig. 2. Schematic illustration of volumetric water distribution in the soil. (a) Boundary drying and wetting curves. (b) Perspective view of water content distribution as a function of the wy, W, plane for the Néel (1942, 1943) diagram.
(Fig. 3a). When the soil is drying, the vertical line passing through a suction of y moves from left to right. Similarly, when the soil is wetting, a horizontal line passing through a soil suction of y is moving downward. The filled areas in Fig. 3a and 3b describe groups of pores that are filled with water. The water content along the boundary drying curve at any specified soil suction, y, can be calculated as follows (Fig. 3b):
(Fig. 3a). When the soil is drying, the vertical line passing through a suction of y moves from left to right. Similarly, when the soil is wetting, a horizontal line passing through a soil suction of y is moving downward. The filled areas in Fig. 3a and 3b describe groups of pores that are filled with water. The water content along the boundary drying curve at any specified soil suction, y, can be calculated as follows (Fig. 3b):
Fig. 4. Boundary drying and boundary wetting processes in a soil and the corresponding Mualem (1974) diagram. (a) Boundary drying and wetting curves. (b) Mualem (1974) diagram for boundary drying process. (c) Mualem (1974) diagram for boundary wetting process. The water content along the boundary wetting curve at any specified soil suction, y, can likewise be calculated as follows (Fig. 3c): Mualem (1974) proposed a diagram that is similar to the Néel (1942, 1943) diagram. Two variables; namely, normal- ized neck pore diameter, 7, and normalized body pore diam- eter, P, are used to define points corresponding to the drying soil suction variable, yy, and the wetting soil suction vari- able, y,,, on the Néel (1942, 1943) diagram shown in Fig. 4. The normalized pore diameter R can be defined as follows:
Fig. 4. Boundary drying and boundary wetting processes in a soil and the corresponding Mualem (1974) diagram. (a) Boundary drying and wetting curves. (b) Mualem (1974) diagram for boundary drying process. (c) Mualem (1974) diagram for boundary wetting process. The water content along the boundary wetting curve at any specified soil suction, y, can likewise be calculated as follows (Fig. 3c): Mualem (1974) proposed a diagram that is similar to the Néel (1942, 1943) diagram. Two variables; namely, normal- ized neck pore diameter, 7, and normalized body pore diam- eter, P, are used to define points corresponding to the drying soil suction variable, yy, and the wetting soil suction vari- able, y,,, on the Néel (1942, 1943) diagram shown in Fig. 4. The normalized pore diameter R can be defined as follows:
Fig. 5. Poulovassilis’s solution for the Néel (1942, 1943) diagram. hypothesis; and (4) dependent models without the similarity hypothesis. A flow chart describing the historical develop- ment of 17 domain physically based hysteresis models for SWCCs is shown in Fig. 6.
Fig. 5. Poulovassilis’s solution for the Néel (1942, 1943) diagram. hypothesis; and (4) dependent models without the similarity hypothesis. A flow chart describing the historical develop- ment of 17 domain physically based hysteresis models for SWCCs is shown in Fig. 6.
Fig. 7. Summary of the empirical hysteresis models. The shaded boxes indicate the models that are compared in this paper.
Fig. 7. Summary of the empirical hysteresis models. The shaded boxes indicate the models that are compared in this paper.
Table 1. Summary of hystersis models.
Table 1. Summary of hystersis models.
where w,, is the predicted water content at a suction, and Wms 1S the measured water content at a suction. curve into 99 equally spaced suction values distributed along an arithmetic scale (Fig. 8). The two criteria were calculated as follows:
where w,, is the predicted water content at a suction, and Wms 1S the measured water content at a suction. curve into 99 equally spaced suction values distributed along an arithmetic scale (Fig. 8). The two criteria were calculated as follows:
Fig. 8. Schematic illustration of dividing a hysteresis curve ac- cording to the soil suction range.
Fig. 8. Schematic illustration of dividing a hysteresis curve ac- cording to the soil suction range.
“T, initial drying curve; D, boundary drying curve; W, boundary wetting curve; n, number of scanning curves. ’Some soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20, 32, 35, and 50 °C, respectively. Table 2. List of the collected soils.
“T, initial drying curve; D, boundary drying curve; W, boundary wetting curve; n, number of scanning curves. ’Some soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20, 32, 35, and 50 °C, respectively. Table 2. List of the collected soils.
The two parameters b,, and d,, can then be calculated us- ing eqs. [21] and [22] The simplified version of the Feng and Fredlund (1999) model assumes that the boundary wetting curve and the boundary drying curve are parallel when soil suction is plot- ted on a logarithmic scale. Consequently, only one point on the boundary wetting curve is required to calibrate the model. where w,, is the water content on the boundary drying curve at zero soil suction; b, c, and d are curve-fitting parameters of the boundary drying curve; y, is the soil suction at the first additional point; and w, is the water content at the first additional point.
The two parameters b,, and d,, can then be calculated us- ing eqs. [21] and [22] The simplified version of the Feng and Fredlund (1999) model assumes that the boundary wetting curve and the boundary drying curve are parallel when soil suction is plot- ted on a logarithmic scale. Consequently, only one point on the boundary wetting curve is required to calibrate the model. where w,, is the water content on the boundary drying curve at zero soil suction; b, c, and d are curve-fitting parameters of the boundary drying curve; y, is the soil suction at the first additional point; and w, is the water content at the first additional point.
Fig. 9. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Mualem (1977) model. Predictions of the boundary wetting curve using five selected models
Fig. 9. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Mualem (1977) model. Predictions of the boundary wetting curve using five selected models
Fig. 10. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 1971b) applying the Mualem independ- ent (1984a) model. Fig. 11. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Hogarth et al. (1988) model.
Fig. 10. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 1971b) applying the Mualem independ- ent (1984a) model. Fig. 11. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Hogarth et al. (1988) model.
Fig. 12. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Feng and Fredlund (1999) improved by Pham et al. (2003) model.
Fig. 12. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the Feng and Fredlund (1999) improved by Pham et al. (2003) model.
tion. In general, it may be necessary to test a series of soil suction decrements in the laboratory to determine the water entry value as required for the Hogarth et al. (1988) model.
tion. In general, it may be necessary to test a series of soil suction decrements in the laboratory to determine the water entry value as required for the Hogarth et al. (1988) model.
Table 4. Ranking of the five soil—water hysteresis models for predicting the boundary wetting curve based on R squared. (3) Hogarth et al. (1988) model. The selection of the five models was based on the criteria that the models must be physically based and easy to use, requiring relatively simple data for calibration. Of the five models, only the Hogarth et al. (1988) model uses the Néel (1942, 1943) diagram. The remainder utilized the Mualem (1974) diagram. The five hysteresis models apply the “similarity hypothesis” to re- duce the amount of measured data required for calibration. A brief description of the theories associated with the mod- els is given in the following sections.
Table 4. Ranking of the five soil—water hysteresis models for predicting the boundary wetting curve based on R squared. (3) Hogarth et al. (1988) model. The selection of the five models was based on the criteria that the models must be physically based and easy to use, requiring relatively simple data for calibration. Of the five models, only the Hogarth et al. (1988) model uses the Néel (1942, 1943) diagram. The remainder utilized the Mualem (1974) diagram. The five hysteresis models apply the “similarity hypothesis” to re- duce the amount of measured data required for calibration. A brief description of the theories associated with the mod- els is given in the following sections.
Table 3. Ranking of the five soil—water hysteresis models for predicting the boundary wetting curve based on APD.
Table 3. Ranking of the five soil—water hysteresis models for predicting the boundary wetting curve based on APD.
Fig. 13. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the simplified Feng and Fredlund (1999) model (presented in this paper). (1974); (2) Mualem (1977); (3) Hogarth et al. (1988); (4) Mualem (19845); (5) Hanks et al. (1969); and (6) Scott et al. (1983) using the datasets of seven soils. The authors found that the Mualem (1974) model-II and the Mualem (1984b) dependent model gave comparable results and pro- vided the best prediction of scanning curves.
Fig. 13. Predicted and measured boundary wetting curves for the Caribou silt loam (Topp 19715) applying the simplified Feng and Fredlund (1999) model (presented in this paper). (1974); (2) Mualem (1977); (3) Hogarth et al. (1988); (4) Mualem (19845); (5) Hanks et al. (1969); and (6) Scott et al. (1983) using the datasets of seven soils. The authors found that the Mualem (1974) model-II and the Mualem (1984b) dependent model gave comparable results and pro- vided the best prediction of scanning curves.
Table 5. Ranking of the five hysteresis models for predicting the scanning curve based on the APD
Table 5. Ranking of the five hysteresis models for predicting the scanning curve based on the APD
Table 6. Ranking of the five hysteresis model for predicting the scanning curve based on R squared. best-fitted to the measured data to obtain better input param- eters for the model (ie., Wye, A, and Wy). The wetting scan- ning curve starting at the highest soil suction that is available for each soil was used as the input wetting scan- ning for the Mualem (1984a) independent model. The dry- ing scanning curve starting at a water content approximately 50% of the effective degree of saturation of each soil was used as the input drying scanning curve for the Mualem and Miller (1979) model-II],,,). only for the drying processes. Therefore, predictions of the boundary wetting curve for the Mualem (1974) model-I, Mualem and Miller (1979) model-III,,,;, and Mualem (1984b) dependent model are similar. The differences among the calculated wetting curves for these models are caused by the calculation procedures and the input data used in the models. The Mualem and Miller (1979) model-III,,.,1 predicts the drying scanning curve well with the aid of an additional drying scanning curve. Some mathematical diffi- culties were encountered for some soils when using the Mualem and Miller (1979) model-III,,,), and the pore block- age function (Pj) cannot be determined. When an additional wetting scanning curve, starting at high suction, was used for calibration, the Mualem (1984a) independent model ac- curately predicted the wetting scanning curve starting at a ower suction. The Hogarth et al. (1988) model was not able to predict scanning curves well since the equations proposed for the boundary drying and boundary wetting curves did not fit the measured data well. The model appears to predict better for poor-graded soils.
Table 6. Ranking of the five hysteresis model for predicting the scanning curve based on R squared. best-fitted to the measured data to obtain better input param- eters for the model (ie., Wye, A, and Wy). The wetting scan- ning curve starting at the highest soil suction that is available for each soil was used as the input wetting scan- ning for the Mualem (1984a) independent model. The dry- ing scanning curve starting at a water content approximately 50% of the effective degree of saturation of each soil was used as the input drying scanning curve for the Mualem and Miller (1979) model-II],,,). only for the drying processes. Therefore, predictions of the boundary wetting curve for the Mualem (1974) model-I, Mualem and Miller (1979) model-III,,,;, and Mualem (1984b) dependent model are similar. The differences among the calculated wetting curves for these models are caused by the calculation procedures and the input data used in the models. The Mualem and Miller (1979) model-III,,.,1 predicts the drying scanning curve well with the aid of an additional drying scanning curve. Some mathematical diffi- culties were encountered for some soils when using the Mualem and Miller (1979) model-III,,,), and the pore block- age function (Pj) cannot be determined. When an additional wetting scanning curve, starting at high suction, was used for calibration, the Mualem (1984a) independent model ac- curately predicted the wetting scanning curve starting at a ower suction. The Hogarth et al. (1988) model was not able to predict scanning curves well since the equations proposed for the boundary drying and boundary wetting curves did not fit the measured data well. The model appears to predict better for poor-graded soils.
Fig. 16. Predicted and measured scanning curves for Caribou silt loam soil (Topp 1971b) applying the Mualem’s independent model (1984a) with the aid of a wetting scanning curve starting at a relatively low water content.
Fig. 16. Predicted and measured scanning curves for Caribou silt loam soil (Topp 1971b) applying the Mualem’s independent model (1984a) with the aid of a wetting scanning curve starting at a relatively low water content.
Fig. 15. Predicted and measured scanning curves for Caribou silt loam soil (Topp 1971b) applying the Mualem and Miller’s inde- pendent model (1979) with the aid of a drying scanning curve starting at a relatively high water content (i.e., w = 35%).
Fig. 15. Predicted and measured scanning curves for Caribou silt loam soil (Topp 1971b) applying the Mualem and Miller’s inde- pendent model (1979) with the aid of a drying scanning curve starting at a relatively high water content (i.e., w = 35%).
Fig. 14. Predicted and measured scanning curves for Caribou silt loam soil (Topp 19715) applying the Mualem model-II (1974).
Fig. 14. Predicted and measured scanning curves for Caribou silt loam soil (Topp 19715) applying the Mualem model-II (1974).
Fig. 17. Predicted and measured scanning curves for Caribou silt loam soil (Topp 19715) applying the Mualem’s dependent model (1984D).
Fig. 17. Predicted and measured scanning curves for Caribou silt loam soil (Topp 19715) applying the Mualem’s dependent model (1984D).
Fig. 18. Predicted and measured scanning curves for Caribou sil loam soil (Topp 19715) applying the Hogarth et al. (1988) model.
Fig. 18. Predicted and measured scanning curves for Caribou sil loam soil (Topp 19715) applying the Hogarth et al. (1988) model.
Fig. 19. Schematic illustration of the slope and distance between the two boundary hysteretic SWCCs. SWCCs were studied; namely, (7) the distance between the boundary drying and the boundary wetting curves, (ii) the relationship between the slopes of the boundary drying and the boundary wetting curves, and (iii) the air entry value of the initial drying and boundary drying curves. The distance between the boundary drying curve and the boundary wet- ting curve is defined as the horizontal distance between the inflection point of the boundary drying curve and the point having the same water content on the boundary wetting curve (Fig. 19).
Fig. 19. Schematic illustration of the slope and distance between the two boundary hysteretic SWCCs. SWCCs were studied; namely, (7) the distance between the boundary drying and the boundary wetting curves, (ii) the relationship between the slopes of the boundary drying and the boundary wetting curves, and (iii) the air entry value of the initial drying and boundary drying curves. The distance between the boundary drying curve and the boundary wet- ting curve is defined as the horizontal distance between the inflection point of the boundary drying curve and the point having the same water content on the boundary wetting curve (Fig. 19).
Table 7. Distance between the two boundary hysteresis curves and the slopes of the boundary hysteresis curves on the arithmetic soi suction coordinate system. “Some soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20, 32, 35, and 50 °C, respectively.
Table 7. Distance between the two boundary hysteresis curves and the slopes of the boundary hysteresis curves on the arithmetic soi suction coordinate system. “Some soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20, 32, 35, and 50 °C, respectively.
Table 8. Suggested value, along with percentage deviation, of the slope ratio and the distance between the two boundary curves for different soil types.
Table 8. Suggested value, along with percentage deviation, of the slope ratio and the distance between the two boundary curves for different soil types.
Table 9. Chosen values for the slope ratio and the distance among the four boundary curves for chalks and silty sands and the calcu- lated results.
Table 9. Chosen values for the slope ratio and the distance among the four boundary curves for chalks and silty sands and the calcu- lated results.
Fig. 20. Estimated boundary drying curve for Aiken clay loam (Richards and Fireman 1943) obtained by scaling from the initial drying curve at 85%, 90%, and 95% saturated water content. where w,, is the water content on the estimated curve; w,,, 1S the water content on the measured (actual) curve; and n is the number of measured points.
Fig. 20. Estimated boundary drying curve for Aiken clay loam (Richards and Fireman 1943) obtained by scaling from the initial drying curve at 85%, 90%, and 95% saturated water content. where w,, is the water content on the estimated curve; w,,, 1S the water content on the measured (actual) curve; and n is the number of measured points.
Fig. 22. Estimated boundary wetting curves for the loose and dense silty sands (Croney and Coleman 1954). hysteretic effects can be drawn from the results of the study reported in this paper:
Fig. 22. Estimated boundary wetting curves for the loose and dense silty sands (Croney and Coleman 1954). hysteretic effects can be drawn from the results of the study reported in this paper:
Fig. 21. Estimated boundary wetting curves for the soft and hard chalks (Croney and Coleman 1954).
Fig. 21. Estimated boundary wetting curves for the soft and hard chalks (Croney and Coleman 1954).
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Environmental EngineeringCivil EngineeringUnsaturated soilPhysics based Modeling
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