An alternative continuous form of Arya and Paris model to predict the soil water retention curve of a soil

Highlights

• A new model, ACAP, has been developed, based based on Arya and Paris, 1981, but overcoming some previous detected problems.

• A simple physic-empirical continuous expression is reached to estimate the water retention curve of a soil.

• The resulting expression use of one only parameter β, for a selected scaling parameter D0.

• The inputs for the new expression are basic soil properties: the grain size distribution and porosity.

Abstract

Arya and Paris developed a model in 1981 to predict the water retention curve of a soil from the grain size distribution and the porosity. This model was very convenient due to the simplicity of the input parameters, and because experimental measures are expensive and with no lack of uncertainties. The model was based on very simple equations but the need to discretize the grain size distribution restricted its usability. A new model is presented, preserving the physical meaning but solving this dependency and other issues and allowing a continuous expression that was not possible with the original model. The new model is very convenient and practical as expressions are easy to implement, and its predictions depend on only one parameter for which some recommended values are also included. Limitations and uncertainties of the model are also discussed.

Introduction

Water content has a major influence on the response of soils; many soil-related construction and infrastructure problems are directly or indirectly related to water presence so that understanding the factors that link water to soil behaviour is vital. In general, near-surface soils are in a ‘variable saturated state’ which means that sometimes all the pores are filled by the wetting fluid (water), the ‘saturated state’, or partially filled, the ‘unsaturated state’. In the latter case, due to capillary forces, there is a connection between the suction (the difference between water pressure with air pressure) and the water content, that is defined by the soil-water retention curve (hereinafter referred to as WRC), that may be considered as a constituent property of a soil.

Obtaining the WRC in the laboratory is difficult, expensive and time consuming (Chapuis et al., 2015). Also, it usually leads to different kinds of uncertainties common to the experimental process (Zapata et al., 2000); for that reason, several approaches have been proposed to infer the WRC from other inexpensive properties of the soil. The methods to estimate WRC can be divided into three categories (Buczko and Gerke, 2005; Fredlund et al., 2012): (1) direct stochastic estimation of water contents at various soil suction points, (2) stochastic or empirical estimation of soil parameters using some algebraic functions for the WRC of the soil and (3) physic-empirical models. Some authors group the first two categories into a more general “empirically-based” class, and in general there is not a pure “empirically-based” or a pure “physically-based” category but some mixture of both (Haverkamp et al., 2002).

Within any of those categories can be included the large set of Pedotransfer functions (PTFs) with which to infer desired water retention curves (estimands) from more simple basic soil properties (predictors) (Van Looy et al., 2017). This PTF term was first defined as “translating data that we have into data we need but do not have” by Bouma (1989) and in most cases, the PTFs are simple empirical expressions fitted to the contents of specialized databases like UNSODA (Leij et al., 1996), GRIZZLY (Haverkamp et al., 1997), HYPRES (Wösten et al., 1999) or EU-HYDI (Weynants et al., 2013) which include the required estimands (Van Looy et al., 2017). The number of purely empirical PTFs to infer the WRC from basic properties is substantial and several reviews are available (Patil and Singh, 2016; Van Looy et al., 2017; Vereecken et al., 2010; Wösten et al., 2001). Empirically based methods vary and can extend the original concept of PTFs: methods based on simple look-up tables, regression techniques, neural networks and other machine learning techniques, decision and regression trees, random forests, etc.

Empirical methods are data-driven models which usually elicit a large number of parameters, with no physical meaning, and which are fitted to specific data. Their applicability is usually limited to the dataset to avoid extrapolation outside the data domain. On the other hand, physically based models usually include fewer parameters and are derived from physical concepts. The number of parameters is reduced as the water retention curve expression is usually derived from expected physical behaviour. Sometimes some parameters need to be empirically defined to include some unknowns and assumptions (i.e. in physico-empirical models).

The first approach to obtain the WRC based on physical principles is due to Childs (1940) who applied the capillary concept in which the pores are simplified as a bundle of interconnected capillary tubes, so that when wetting the smaller tubes fill first due to higher suctions. Other more complex physical models consider different packings of spheres (Assouline and Rouault, 1997; Chan and Govindaraju, 2004) or probabilistic expressions for pore throats based on packing of spheres (Jaafar and Likos, 2014; Zhai et al., 2020), fractal models (Chan and Govindaraju, 2004; Perrier et al., 1996), percolation methods (Chatzis and Dullien, 1977; Diaz et al., 1987; Dullien et al., 1976), pore networks defined by prismatic throats and spherical bodies (Ahrenholz et al., 2008; Blunt et al., 2002; Mahabadi et al., 2016; Vogel, 2000; Xiong et al., 2016), etc. These methods relays on pure geometrical interpretations, therefore it seems evident that the water retention curve definition is intimately linked to soil geometry and pore structure. Thus, the GSD and the porosity index can be expected to heavily influence the inferred WRC. This expectation is strengthened in that almost all pedo-transfer functions depend on the GSD and optionally on the porosity index (or on the equivalent void ratio or density).

Physically based models can be accurate but difficult to implement or simpler and less accurate but more practical (Arya et al., 2008, 1999; Arya and Paris, 1981; Haverkamp and Parlange, 1986; Mishra et al., 1989; Mohammadi and Vanclooster, 2011; Nimmo et al., 2007). One of these simple models, that can be considered as the first model to obtain the WRC from the GSD and porosity index based on physical concepts, and with only one empirical parameter, is the model created by Arya and Paris (1981) (hereinafter referred to as the AP₈₁ model) who made good use of the observed similarities between the shape of the PSD and the GSD. Other models also considered this ‘similarity concept’ (Arya et al., 2003; Gupta and Ewing, 1989; Gupta et al., 2004; Haverkamp and Parlange, 1986; Mishra et al., 1989; Mohammadi and Vanclooster, 2011; Nimmo et al., 2007).

The main advantage of the AP₈₁ model was its simplicity, with its use of GSD and void ratio as inputs and with only one parameter, the ‘scaling parameter, α’. For this reason, AP₈₁ became the most applied method that relates pore-scale to grain size based on physical concepts (Buczko and Gerke, 2005), despite that shortly after the publication of the model, some inconveniences were detected by Haverkamp and Parlange (1982) and later by Haverkamp et al. (2002) and Nimmo et al. (2007). Other issues are also detected in the present paper, these problems are listed later in Section 2 (Table 1).

Further insights into the scaling parameter, α, were included in Arya et al. (1999) (hereinafter referred as AP₉₉ model) and a modification of the model was included in Arya et al. (2008) (hereinafter referred to as the AP₀₈ model), however most of the problems remained. Arya and Dierolf (1992) modified the original model, in this case the pore length was assumed to be equal to a parameter α′ for every grain size fraction.

Nevertheless, because of their theoretical basis, AP models became of real interest in WRC prediction and are included in a large number of publications.

Arya et al. (1982) applied the AP₈₁ model to a large dataset of 181 New Jersey soil horizons, Gupta and Ewing (1989) mixed the Arya and Paris approach with a packing model to differentiate between inter and intra-aggregate pore space, Mishra et al. (1989) applied a modified version of the Arya and Paris model to estimate uncertainties of the Van Genuchten parameters, Basile and D'Urso (1997) performed a calibration of the AP model to increase the performance on clay-loamy soils, Fredlund et al. (1997) presented a method to estimate the WRC and although it did not apply the AP model, the conceptual idea was very similar, dividing the GSD to combine specific WRC to each division. Arya et al. (2003) applied the estimation of the PSD by Arya and Paris approach to assess the hydraulic conductivity of soils, Gupta et al. (2004) applied the AP model to estimate the water content on roadbed materials, Buczko and Gerke (2005) applied the AP model to coarse mine soil material and they concluded the model mispredicted water content at high suctions and that the closest fits for the α parameter was a variable expression depending on particle size, Vaz et al. (2005) evaluated the AP model for Brazilian soil types and propose a much smaller α parameter than in the original AP81 model and a variable α parameter depending on the water content, Nimmo et al. (2007) compared several physical models including the Arya and Paris approach, concluding that most of them shares similar concepts and a general framework, they included two continuous modified versions of the Arya and Paris model and Mohammadi and Vanclooster (2011) perform a nearly similar approach to the AP model considering different grain assemblages which result in a very simple equation.

Prevedello and Loyola (2002) developed the SPLINTEX model. This model applies the AP model to a measured grain size distribution with a scaling factor α defined by the AP₈₁ and AP₉₉ models that is corrected to match a measured point in the WRC or to an estimated water content at a specific point of the WRC by using an empirical expression. The output of the model are the parameters of the Van-Genuchten expression fitted to the resulting WRC with the AP model. There are two versions of the model, in the first one the saturated water content can be estimated equal to the porosity or estimated empirically from bulk and solid densities, and no measured point of the WRC is given (SPLINTEX-PTF1), and in the second version the saturated water content and a point of the WRC are inputs to the model (SPLINTEX-PTF2). This model has been evaluated in further papers with satisfactory results (Da Silva et al., 2017; Huf Dos Reis et al., 2018), and the software implemented in C++ resulting in the version 2.0 (Da Silva et al., 2020). The SPLINTEX model however is still based on the original AP₈₁ and AP₉₉ models and therefore shares their limitations.

AP models depend on a particular discretization of the GSD, but a continuous form would be more practical and permit a more convenient implementation of the AP concept. Nimmo et al. (2007) tried to obtain a direct continuous form from the discrete AP₈₁ model by transforming the discrete intervals into differential intervals and integrating, but the expression degenerate to 0 or ∞ when the scaling parameter ,α, is diferent to 1. To avoid this issue, they modified some terms of the expression and built the CNEAP model (Continuous Near-Equivalent of the AP model). Nevertheless, several problems of the AP model remained unresolved.

The purpose of the present paper is to build a closed continuous expression of the WRC or PSD as a function of the GSD and the porosity by performing slight modifications to the original AP₈₁ model so as to overcome the difficulties identified. This new modified Arya and Paris model depends on only one parameter β (parameter β is chosen so as to distinguish it from the original ‘parameter α’). This new parameter will have much less variation across the full suction range than in the original AP₈₁ model. The new model results in a simple expression that relates the grain sizes in the GSD to the pore sizes in the PSD without the need of discretization, and some values or empirical expressions for parameter β will be recommended. Also, expressions to relate previous models with the new parameter β will be given. The new model slightly increases the accuracy of original AP₈₁ model for most simple definitions of parameter β, but substantially increases the accuracy when β is defined by an empirical expression.

Unlike other models that consider capillary forces and adhesion (Aubertin et al., 2003; Kovács, 1981; Qiao et al., 2021), AP based models rely on the capillarity concept only, therefore their predictions are limited to capillary-based water retention, and the water retention due to adsorption is not considered. Adsorption has more influence the smaller the grains (Aubertin et al., 2003; Chan and Govindaraju, 2004; Chang et al., 2019; Lu, 2016; Mohammadi and Meskini-Vishkaee, 2012). This model does not consider either the water retention in the form of pendular rings or the aggregation of clay particles due to chemical interactions. Adsorption and pendular rings could be the cause of the uncertainties detected on the samples with clay content and for high suctions (Chang et al., 2019; Meskini-Vishkaee et al., 2014; Mohammadi and Meskini-Vishkaee, 2012). This is partially solved by using empirical expressions of the β parameter depending on the grain size or the pore size. Therefore, the ACAP model is especially suitable for soils with small clay content and for the mid-range of the water retention curve.