Characteristics of how a soil retains water are represented by a relation between soil water content θ [m³∙m⁻³] and soil water potential ψ [J∙kg⁻¹]. And a curve drawn by a series of ψ-θ plots obtained through drying or wetting the soil is called a soil water retention curve (SWRC). A SWRC can be approximated as a mathematical function θ(ψ) and is used to diagnose soil moisture conditions or to analyze water flow in soils. However, a field-observed ψ-θ relation is not necessarily expressed by a single-valued function of θ(ψ) but may be described as a multiple-valued function, depending on the direction of change in ψ [J∙kg⁻¹]. This is called soil water hysteresis.

In soil water hysteresis, a value of θ [m³∙m⁻³] in a wetting process of a soil becomes larger than that in a drying process for a given ψ [J∙kg⁻¹] so that the two processes draw two different paths on a ψ-θ plane. The path of ψ-θ relation that is drawn when a soil with its possible maximum water content θ_max [m³∙m⁻³] is incrementally drained toward its possible driest condition with the water content θ_min [m³∙m⁻³] is called the main drying SWRC θ_md(ψ), and the path for its reverse process is called the main wetting SWRC θ_mw(ψ). Since a soil in a field is generally exposed to intermediate moisture conditions between the two extremities through drying-wetting cycles, paths of ψ-θ relations observed in the field should be drawn between the two main curves on the ψ-θ plane and are called scanning drying SWRCs θ_scd(ψ) or scanning wetting SWRCs θ_scw(ψ).

Possible causes of soil water hysteresis have been listed like the “bottle-neck effect”, the “contact-angle effect”, the entrapment of air in a wetting soil, and any process through which the pore geometry of a soil may vary as time passes with cycles of wetting and drying ([6]; [2]). And although these phenomenological interpretations are quite rational, they are also too microscopic or too detailed to be practically modelled by mathematical expressions. As a result, more macroscopic approaches have been proposed and tested for modelling soil water hysteresis.

The independent domain theory ([13]; [12]; [8]; [9]) and the dependent domain theory ([10]; [14]) are pioneering works about modelling soil water hysteresis. These theories have evolved mechanistically and have been confirmed to reproduce and predict well hysteretic behaviors of ψ-θ relations. However, it is often cumbersome to effectively incorporate them into the algorithm to describe the trajectory of ψ-θ plots for many repetitions of drying-wetting cycles. Thus, instead, the empirical model proposed by [15] has been widely adopted. The conciseness of this model owes mainly to the feature that it can draw a series of θ_scd(ψ) and θ_scw(ψ) by citing only a pair of θ_md(ψ) and θ_mw(ψ) that are expressed with the same type of mathematical function. Application of the empirical model can certainly improve the accuracy of numerically predicting water flow in soils ([5]) with such a positive feature that the model can be easily calibrated with a limited number of parameters ([11]; [7]).

However, there have been a relatively small number of studies that dealt with features of in-situ ψ-θ relations when modelling and simulating hysteretic behaviors in SWRC, presumably because there had been relatively few ψ-θ relations obtained from long-term monitoring in fields in the era of model development about soil water hysteresis. On the other hand, recent studies tend to observe in-situ ψ-θ relations for longer periods, increasingly clarifying the features of hysteretic behaviors of in-situ ψ-θ relations ([1]; [16]; [4]). Therefore, it seems worth developing a model of soil water hysteresis by referring to features of in-situ ψ-θ relations.

This study aimed at modelling soil water hysteresis based on features of field-observed ψ-θ relations. The model was developed so that it can be available if the two main curves θ_md(ψ) and θ_mw(ψ) share the driest and wettest extremities θ_min [m³∙m⁻³] and θ_max [m³∙m⁻³] on the ψ-θ plain while it can work even when θ_md(ψ) and θ_mw(ψ) are expressed in different types of mathematical functions from each other.

Figure 2 shows the main and scanning curves by the model plotted with the measured ψ-θ relations. The parameter values obtained for Equations (10) and (11) were γ_d = 0.006 [kg∙J⁻¹], γ_w = 39.15 [kg∙J⁻¹], θ_min = 0.353 [m³∙m⁻³], θ_max = 0.636 [m³∙m⁻³], μ_d = 1.146 [kg∙J⁻¹], and μ_w = 0.647 [kg∙J⁻¹], and enabled the resultant two main curves to enclose all the measured ψ-θ plots and to satisfy all the conditions (c1) through (c4).

The trajectory of the modeled ψ-θ relations reproduced the general trend of the measured ψ-θ relations throughout the study period (Figure 2(i)). Particularly, the model successfully explained the measured fact that the ψ-θ plots had consistently resided in the region of θ ≥ 0.50 [m³∙m⁻³] for the first 4 years (Figures 2(ii)-(v)).

Among the subgraphs for the first 4 years, the model showed a sudden unrealistic upward shift in θ [m³∙m⁻³] from 0.52 to 0.60 [m³∙m⁻³] with ψ ≒ −1.3 [J∙kg⁻¹] (Figure 2(v)). However, this shift was apparent because there was a no measurement term of more than 1 week between the dates of these two measurements, suggesting that the soil had become almost saturated and, then, dried again during the no measurement period and, as a result, the model failed to recognize the actual rising and lowering in ψ [J∙kg⁻¹]. Other cases in which no measurement terms deteriorated the model performance were found for the data sets measured in 2013 through 2015. On Figure 2(ii), Figure 2(iii), and Figure 2(iv), the underestimations of θ [m³∙m⁻³] took place several times in the region of 0.50 [m³∙m⁻³] ≤ θ ≤ 0.55 [m³∙m⁻³] with the domain of −100 [J∙kg⁻¹] ≤ ψ ≤ −10 [J∙kg⁻¹]. In this domain, the more times reversals of drying-wetting direction occur, the more θ [m³∙m⁻³] lowered in the study site. The model reproduced well this observed feature and, at the same time, can erroneously predict hysteretic behaviors if the chances of reversals in drying-wetting direction are wrongly input into the model among a series of ψ [J∙kg⁻¹].

Figure 2. The main curves and scanning curves obtained by fitting the model of soil water hysteresis to the time-series of the measured ψ-θ relations. The dashed and dotted lines in each subgraphs indicate the main drying curve θ_md(ψ) and the main wetting curve θ_mw(ψ), respectively, while the solid line denotes the modeled ψ-θ relations. The measured ψ-θ plots are expressed by the plus symbols “+”. The subgraph (i) displays the plots for the entire study period, and each of the subgraphs (ii) through (viii) shows the plots for each year.

The measured θ [m³∙m⁻³] lowered below 0.50 [m³∙m⁻³] only in the last 3 years of the study period (Figures 2(vi)-(viii)), and the measured ψ [J∙kg⁻¹] had never gone above −0.1 [J∙kg⁻¹] except for those in the last year (Figure 2(viii)). The model followed well the long continual drying processes found in 2017 and 2018 (Figure 2(vi) and Figure 2(vii)), while it also reproduced the loop-like behavior of the measured ψ-θ relations in the region of 0.40 [m³∙m⁻³] ≤ θ ≤ 0.45 [m³∙m⁻³] with the domain of −100 [J∙kg⁻¹] ≤ ψ ≤ −0.1 [J∙kg⁻¹] (Figure 2(vii)), suggesting that if the model parameters had been identified without considering the data set measured in the last 3 years, the resultant model would have failed to explain the entire hysteretic behavior observed in the 82 months.

The measured ψ-θ relations traveled from the region of θ ≤ 0.45 [m³∙m⁻³] to the region of θ ≥ 0.55 [m³∙m⁻³] in 2019 (Figure 2(viii)). This continual wetting process took more than half a year and can be separated roughly into two processes. In the first wetting process, the SWRC shifted up into the region of 0.45 [m³∙m⁻³] ≤ θ ≤ 0.50 [m³∙m⁻³] until May in 2019. In the second wetting process, θ [m³∙m⁻³] became more than 0.55 [m³∙m⁻³] by October in 2019. The first wetting process was reproduced by the model relatively well. In the second process, on the other hand, the model partially failed to follow the trajectory of the measured ψ-θ relations so that the modeled curve dropped by the region of 0.50 [m³∙m⁻³] ≤ θ ≤ 0.55 [m³∙m⁻³] while the measured ψ-θ relations skipped this region and directly reached the region of θ ≥ 0.55 [m³∙m⁻³].

The failure of the model to reproduce the wetting process in October in 2019 indicated that it was difficult for the model to explain the process in which water is entering into such large pores as filled with water of potential higher than −1 [J∙kg⁻¹]. This difficulty implied that the model performance can be improved by distinguishing a coarse pore system from the entire pore system of the soil when developing the model of soil water hysteresis.

This study assumed a-priori that the pore size distribution of the soil is of single modal, because single-peaked functions as shown in Equations (10) and (11) were applied to the description of the two main curves. On the other hand, the soil in this study could retain relatively large amount of water in coarse pores and, thus, its in-situ SWRC tended to have roughly 10% smaller θ [m³∙m⁻³] than the corresponding laboratory-measured SWRC under near saturated conditions ([3]). In addition, θ_max determined in this study was 0.636 [m³∙m⁻³], obviously lower than the saturated water content of the soil (0.815 [m³∙m⁻³]), meaning that the coarse pore system had rarely been filled with water for the study period of 82 months and the resultant measured ψ-θ relations must have prevented the modeling process from correctly considering the drying-wetting cycles of the coarse pore system.

In the same manner as the difference between the determined θ_max [m³∙m⁻³] and the actual saturated water content, θ_min determined in this study was 0.353 [m³∙m⁻³], higher than the air-dried water content of the soil (0.050 [m³∙m⁻³]). The gap between these two numerals implied that very fine pores in the soil had had almost no experience of drying-wetting cycles, and their soil water characteristics had no need to be, and were not sufficiently, considered in the model of this study. These features also suggested that it is promising to develop a model that can simulate hysteretic behaviors found in multi-modal pore systems like volcanic ash soils.

This study showed the derivation and algorithm of the model of soil water hysteresis that can reproduce the time series of the relations between soil water potential ψ [J∙kg⁻¹] and soil water content θ [m³∙m⁻³] along in-situ drying-wetting cycles. The assumption of the model derivation was based on the observed features of the in-situ ψ-θ relations. The model requires a pair of the main drying and wetting curves θ_md(ψ) and θ_mw(ψ), both of which can be described by any type of mathematical expression only if they share the driest and wettest extremities. The model can calculate an unknown water content θ₁ [m³∙m⁻³] for a given water potential ψ₁ [J∙kg⁻¹] and any latest known moisture condition (ψ₀, θ₀), when drawing the series of scanning drying and wetting curves θ_scd(ψ) and θ_scw(ψ).

The model reproduced well the general trend of the measured ψ-θ relations throughout the study period. Particularly, the model imitated the observed processes of ψ-θ relations through which the repetitions of reversals in drying-wetting direction in the domain of ψ ≤ −50 [J∙kg⁻¹] caused gradual lowering of θ [m³∙m⁻³]. However, the model underestimated θ [m³∙m⁻³] in this domain when the chances of reversals in drying-wetting direction were wrongly input into the model in the series of ψ [J∙kg⁻¹].

The model also made errors in a long continual wetting process in the domain of ψ ≥ −1 [J∙kg⁻¹] in which water entered into coarse pores. This error implied that the model can be improved by separating the whole pore system of the soil into several pore systems including a coarse pore system, as water can hardly enter in and can easily drain from these coarse pores in soils. Therefore, further studies are necessary to modify the model of soil water hysteresis so that it can deal with multi-modal pore systems like volcanic ash soils or other types of structured soils rich in soil aggregates.

The author thanks Mr. T. Shiozawa, Mr. E. Saito, Mr. K Seino, and Mr. N. Yamaguchi at the Utsunomiya University Farm who have managed the study field. The author also thanks Mr. T. Hirai of the former student in the Graduate school of Agriculture, Utsunomiya University, and Mr. H. Ajito, Ms. M. Anzai, Ms. K. Arakawa, Ms. M. Cho, Ms. N. Goka, Mr. T. Hoshino, Mr. D. Iimura, Mr. J. Kaneko, Mr. T. Kurata, Mr. K. Ogasawara, Mr. Y. Sakanishi, Mr. H. Shoji, Mr. H. Suto, Mr. S. Takaki, Mr. K. Tezuka, Mr. N. Totsuka, Mr. K. Tsukagoshi, and Mr. Y. Usui of the former students in the Faculty of Agriculture, Utsunomiya University, for their works in the field and in the laboratory.