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## Water Retention Theory |

Soil holds water in its matrix by adsorption onto particles and by capillarity in the pores (Marshall et al., 1996). Soil water will contain energy; its potential energy, and more specifically the pressure potential, is most important as it characterizes its physicochemical condition and movement. The pressure potential is considered to be negative as the water pressure is sub-atmospheric. It is often known as matric potential, tension or scution (Hillel, 1982). Soil-water potential is expressed in terms of energy per unit mass or volume.

Water in unsaturated soil is constrained by the capillary and absorptive forces and so energy is required to remove it from the soil. Finer pores exert greater force per unit cross-section area of soil water meniscus than larger pores and so at a given tension soils with smaller pores will retain more water (Klocke and Hergert, 1990). The relationship between soil water content and soil water tension is presented graphically and termed the soil-moisture characteristic curve or water retention curve. In saturated soil at equilibrium with atmospheric pressure the tension is zero. As the water content decreases, the tension used to hold water increases. Soil structure will affect the shape of the curve, as at low tensions ( 0 - 100 kPa) the amount of water retained is a function of capillarity and therefore pore-size distribution. At high tensions, water is retained due to adsorption and so texture is more influential (Hillel, 1982)

Soil water can be classified into three categories: (1) gravitational water, which drains readily by gravitational force, (2) available water, which is retained by capillary forces and is available for extraction by plants, and (3) unavailable water, which is held by adsorptive forces and is unavailable for plant uptake (Klocke and Hergert, 1990). Field capacity refers to the water content at the upper limit of the available water range. This can be defined as the amount of water retained in a soil after it has been saturated and allowed to drain for 24 hours (Klocke and Hergert, 1990) and corresponds to 5 kPa suction under British conditions and in sandy soils (Hall et al., 1977). The water content at 1500 kPa is an approximation of the permanent wilting point; this lower limit of the available water range is the point where plants have extracted all available water and will wilt and die. The available water capacity is a measure of the amount of water held between field capacity and wilting point, and varies with soil texture. Soil water content (θ ) is often expressed as a percentage by mass (θM) or volume (θv).

Unsaturated hydraulic conductivity is the primary measure of the ease of transport of water and dissolved or suspended chemicals through the vadose region. Mualem (1976) has shown that unsaturated hydraulic conductivity can be approximately calculated from a soil's water retention characteristic. Water retention characteristics are difficult and time consuming to measure, but are nevertheless easier to measure than unsaturated hydraulic conductivities over a full range of saturations. Therefore, water retention characteristics tend to be crucial in estimating hydraulic conductivity, and hence water and chemical transport. To provide more global estimates of water retention, the property is often related to the more readily measured properties of texture, density and organic carbon content using multiple regression functions known as pedo-transfer functions. Pedo-transfer functions have been used for a wide range of uncultivated soils, amongst others US soils (Pachepsky et al., 2006), Danish soils (Borgesen and Schapp, 2005) and English and Welsh soils (Mayr and Jarvis, 1999). Pedo-transfer functions are also used more generally to relate a wider range of hydraulic properties, including run-off, infiltration and meterological heat balance due to soil moisture (Pachepsky et al., 2006). Despite their usefulness, pedo-transfer functions have weaknesses which are well known. These include their tendency to be based on laboratory measurements; water retention in the field situation tends to be lower (Pachepsky and Rawls, 2003).

In the context of the standard experimental procedure for measuring water retention, there exist a series of major problems associated with the study of the void structure of soil, and here we need to make a brief description of five of them. The first, (i), arises from the fact that there is an implicit assumption within much of soil physics that all voids within soil are fully accessible to the exterior of the sample with respect to fluid for flow, imbibition or drainage. Such accessibility can be thought of in terms of a bundle of capillary tubes which open to the surface. Each tube is implicitly assumed to be of a constant size, and not connected with others of a different size. Under these circumstances, it is possible to assume that the void size distribution can be directly derived from the first derivative (i.e. slope) of the water retention curve. On this basis, Mualem (1976) calculated unsaturated hydraulic conductivity from water retention, and Dexter (2004) derived the S-factor for measuring soil health. The assumption of complete accessibility is also implicit in fractal approaches, including the pore-solid fractal (Bird and Perrier, 2003) also used by Huang and Zhang (2005).

In practice, however, voids generated by geophysical processes rather than soil macro fauna are often surrounded by smaller connecting 'throats', a phenomenon often referred to as the 'shielding' or 'shadowing' of the voids. In the case of the porous rocks, the extent of this shielding can be discovered by filling a sample with low-melting Woods metal, and dissolving away the rock (Wardlaw et al., 1987). But Wood's metal destroys the structure of soil, and resin or carbowax leaves a structure from which the soil cannot easily be dissolved away. Neither do thin sections give an unambiguous determinant of the extent of shielding, because in two dimensions it is impossible reliably to differentiate between pores and throats. So one has to guess the extent of this phenomenon on the basis of the known water retention characteristics and the overall porosity of the soil. Guesswork is unsatisfactory, but is nevertheless better than disregarding the shielding. Furthermore, the reliability of the guesswork can be estimated by carrying out a series of stochastic realisations of the model. See Matthews et al. 2018 for our latest work on this problem.

Another disadvantage of using the first derivative of the water retention, rather than the shape of the whole curve, is that it induces a lack of experimental rigour. In practice many water retention curves are incomplete, because the sample drains by gravity at the lowest tension or the investigator has not measured water retention at the lowest tension. Often, the highest tension is not enough to remove water from e.g. clays. Another problem is that during water retention measurement, the sample can expand or shrink. If such volume change is not compensated for when calculating the total sample water retention gravimetrically, illogical results may be derived, for example that the amount of water removed from the sample is greater than the amount it can contain.

In practice, complete curves are almost impossible to obtain, so we have to model incomplete curves. We then have to decide whether the modelled porosity should be that picked up from the observed water retention curve, or the total porosity measured gravimetrically from the total water retention capacity. If one uses the observed water retention curve equivalent to a fractional water retention range of 0.81 to 0.20 (v/v), it avoids the danger of modelling immobile water, but opens one to the possibility of not modelling large, gravity drained pores. The range is converted to air-intrusion values but one must make sure that only voids within the appropriate size range are modelled.

Problem (iii) is the intractability of the shapes of the water retention characteristic curves. The curves vary monotonically from maximum water retention at low tensions to maximum at high tensions, and usually, but not always, exhibit a point of inflection and position of maximum slope at an intermediate tension. Such behaviour is not much on which to base a mathematical fitting function. However, the necessity of parameterising water retention curves for input to pedo-transfer functions has spawned a host of fitting functions, such as the van Genuchten function (1980), Brooks-Corey function (Ma et al., 1999), and modified Brooks-Corey function (Mayr and Jarvis, 1999). Although very useful for input to pedo-transfer functions, and convenient for applying to other soil characteristics (Zhu et al., 2004), the straight-jackets of the assumed functionalities of these fitting functions tend to mask subtle effects such as those caused by roots. The arbitrariness of the functions also tends to result in different predictions of water retention by pedo-transfer functions based on different fitting curves. McBratney et al. (2002) suggests that this problem can be overcome by using Monte Carlo methods to choose the results from the pedo-transfer function which gives the least variance.

To avoid the problem of intrusion curve shape, the void network model performs a point by point fit to the experimental water retention curves. This procedure then exposes two further problems, (iv) and (v), which are both theoretical and practical. Problem (iv) is that the standard protocol for measurement of water retention curves is to measure around five points (ISO 11274:1998); although investigators tend to measure a minimum of eight points. Even this can take many weeks. Allowing for the fact that the minimum and maximum tend to be fixed within the fitting procedure, one is left with fitting three variable points using a fitting function with two or three parameters. There are a therefore a minimal or zero number of statistical degrees of freedom. Coupled with this is that the fitting parameters of the van Genuchten and Brooks-Corey functions are not mathematically orthogonal, so a range of fits are possible, which in practice are constrained to some narrower band thought appropriate to the sample.

The PoreXpert void network model, even though it is carrying out a point by point fit, is also short of statistical degrees of freedom. So we recommend you fit all the water retention curves allowing the model's fitting parameters to have unrestrained variation between stochastic realisations. The curves are then re-fitted, constraining the fitting parameters to a common range of variation for each parameter which does not include outliers. This procedure is only partially satisfactory, but is better than the ignoring of the problem when fitting van Genuchten or Brooks-Corey functions.

Problem (v), also uncovered by the use of the point-by-point fit, is the absence of water retention data at low or zero applied tensions. Saturated soil samples mounted on water retention tables drain by gravity initially, and this initial drainage is usually ignored. However, this gravity drainage occurs through the largest voids within the sample, which have the greatest hydraulic conductivity. The functionality of the van Genuchten and Brooks-Corey functions overlooks this absence, by assuming that the gravity drainage can be inferred from the rest of the drainage curve. However, this inference is based on the mathematical functionality of the fitting function, and has no relation to the structure of the soil. A previous attempt has been made to address this problem by use of a 'matching point' for pedo-transfer functions at a tension of 1 kPa (Jarvis et al., 2002). The PoreXpert void network model sees no data in this region and allows itself to vary as much as it wishes to fit the data at the other known points. There are therefore wide variations between stochastic realisations, and hence wide variations in the model's prediction of saturated hydraulic conductivity.