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Bibliography HYDRAULIC GRADIENT HYDRAULIC HEAD Bibliography 368 HYDRAULIC GRADIENT Bibliography studying or predicting site-specific water flow and solute Introduction to Environmental Soil Physics (First Edition) 2003 transport processes in the subsurface. This includes using Elsevier Inc. Daniel Hillel (ed.) http://www.sciencedirect.com/ models as tools for designing, testing, or implementing science/book/9780123486554 soil, water, and crop management practices that optimize water use efficiency and minimize soil and water pollution by agricultural and other contaminants. Models are equally needed for designing or remediating industrial HYDRAULIC GRADIENT waste disposal sites and landfills, or assessing the for long-term stewardship of nuclear waste repositories. The slope of the hydraulic grade line which indicates the Predictive models for flow in variably saturated soils are change in pressure head per unit of distance. generally based on the Richards equation, which combines the Darcy–Buckingham equation for the fluid flux with a mass conservation equation to give (Richards, 1931): HYDRAULIC HEAD @yðhÞ @ @h ¼ KðhÞ À KðhÞ (1) @t @z @z The sum of the pressure head (hydrostatic pressure rela- tive to atmospheric pressure) and the gravitational head in which y is the volumetric water content (L3 LÀ3), h is (elevation relative to a reference level). The gradient of the pressure head (L), t is time (T), z is soil depth (positive the hydraulic head is the driving force for water flow in down), and K is the hydraulic conductivity (L TÀ1). porous media. Equation 1 holds for one-dimensional vertical flow; similar equations can be formulated for multidimensional flow Bibliography problems. The Richards equation contains two constitutive Introduction to Environmental Soil Physics (First Edition) 2003 relationships, the soil water retention curve, y(h), and the Elsevier Inc. Daniel Hillel (ed.) http://www.sciencedirect.com/ unsaturated soil hydraulic conductivity function, K(h). science/book/9780123486554 These hydraulic functions are both strongly nonlinear func- tions of h. They are discussed in detail below. HYDRAULIC PROPERTIES OF UNSATURATED SOILS Water retention function The soil water retention curve, y(h), describes the relation- Martinus Th. van Genuchten1, Yakov A. Pachepsky2 ship between the water content, y, and the energy status of 1Department of Mechanical Engineering, COPPE/LTTC, water at a given location in the soil. Many other names may be found in the literature, including soil moisture Federal University of Rio de Janeiro, UFRJ, Rio de – Janeiro, Brazil characteristic curve, the capillary pressure saturation 2Environmental Microbial and Food Safety Laboratory, relationship, and the pF curve. The retention curve histor- Animal and Natural Resource Institute, Beltsville ically was often given in terms of pF, which is defined as Agricultural Research Center, USDA-ARS, Beltsville, the negative logarithm (base 10) of the absolute value of MD, USA the pressure head measured in centimeters. In the unsatu- rated zone, water is subject to both capillary forces in soil pores and adsorption onto solid phase surfaces. This leads Definition to negative values of the pressure head (or matric head) Hydraulic Properties of Unsaturated Soils. Properties relative to free water, or a positive suction or tension. reflecting the ability of a soil to retain or transmit water As opposed to unsaturated soils, the pressure head h is and its dissolved constituents. positive in a saturated system. More formally, the pressure head is defined as the difference between the pressures of Introduction the air phase and the liquid phase. Capillary forces are the Many agrophysical applications require knowledge of the result of a complex set of interactions between the solid hydraulic properties of unsaturated soils. These properties and liquid phases involving the surface tension of the liq- reflect the ability of a soil to retain or transmit water and uid phase, the contact angle between the solid and liquid its dissolved constituents. For example, they affect the phases, and the diameter of pores. partitioning of rainfall and irrigation water into infiltration Knowledge of y(h) is essential for the hydraulic charac- and runoff at the soil surface, the rate and amount of redis- terization of a soil, since it relates an energy density tribution of water in a soil profile, available water in the (potential) to a capacity (water content). Rather than using soil root zone, and recharge to or capillary rise from the pressure head (energy per unit weight of water), many the groundwater table, among many other processes in agrophysical applications use the pressure or matric the unsaturated or vadose zone between the soil surface potential (energy per unit volume of water, usually mea- and the groundwater table. The hydraulic properties are sured in Pascal, Pa), cm = rwgh, where rw is the density also critical components of mathematical models for of water (MLÀ3) and g the acceleration of gravity (L TÀ2). HYDRAULIC PROPERTIES OF UNSATURATED SOILS 369 Figure 1 shows typical soil water retention curves for initially very dry sample to produce the main wetting relatively coarse-textured (e.g., sand and loamy sand), curve, which is generally displaced by a factor of medium-textured (e.g., loam and sandy loam), and fine- 1.5–2.0 toward higher pressure heads closer to saturation. textured (e.g., clay loam, silty loam, and clay) soils. The This phenomenon of having different wetting and drying curves in Figure 1 may be interpreted as showing the equi- curves, including primary and secondary scanning curves librium water content distribution above a relatively deep is referred to hysteresis. Hysteresis is caused by the fact water table where the pressure head is zero and the soil that drainage is determined mostly by the smaller pore in fully saturated. The plots in Figure 1 show that coarse- a certain pore sequence, and wetting by the larger pores textured soils lose their water relatively quickly (at small (this effect is often referred to as the ink bottle effect). negative pressure heads) and abruptly above the water Other factors contributing to hysteresis are the presence table, while fine-textured soils lose their water much more of different liquid–solid contact angles for advancing gradually. This reflects the particle or pore-size distribu- and receding water menisci, air entrapment during wet- tion of the medium involved. While the majority of pores ting, and possible shrink–swell phenomena of some soils. in coarse-textured soils have larger diameters and thus drain at relatively small negative pressures, the majority Hydraulic conductivity function of pores in fine-textured soils do not drain until very large tensions (negative pressures) are applied. The hydraulic conductivity characterizes the ability of As indicated by the plots in Figure 1, the water content a soil to transmit water. Its value depends on many factors varies between some maximum value, the saturated water such as the pore-size distribution of the medium, and the content, y , and some small value, often referred to as tortuosity, shape, roughness, and degree of interconnec- s tedness of the pores. The hydraulic conductivity decreases the residual (or irreducible) water content, yr. As a first approximation and on intuitive ground, the saturated considerably as soil becomes unsaturated since less pore water content is equal to the porosity, and y equal to zero. space is filled with water, the flow paths become increas- r ingly tortuous, and drag forces between the fluid and the In reality, however, the saturated water content, ys, of soils is generally smaller than the porosity because of entrapped solid phases increase. The unsaturated hydraulic conductivity function gives and dissolved air. The residual water content yr is likely to be larger than zero, especially for fine-textured soils with the dependency of the hydraulic conductivity on the water their large surface areas, because of the presence of content, K(y), or pressure head, K(h). Figure 2 presents examples of typical K(y) and K(h) functions for relatively adsorbed water. Most often ys and especially yr are treated as fitting parameters without much physical significance. coarse-, medium-, and fine-textured soils. Notice that the Soil water retention curves such as shown in Figure 1 hydraulic conductivity at saturation is significantly larger are not unique but depend on the history of wetting and for coarse-textured soils than fine-textured soils. This differ- drying. Most often, the soil water retention curve is deter- ence is often several orders of magnitude. Also notice mined by gradually desaturating an initially saturated soil that the hydraulic conductivity decreases very significantly by applying increasingly higher suctions, thus producing as the soil becomes unsaturated. This decrease, when a main drying curve. One could similarly slowly wet an expressed as a function of the pressure head (Figure 2; right), is much more dramatic for the coarse-textured soils. The decrease for coarse-textured soils is so large that at 100,000 a certain pressure head the hydraulic conductivity becomes smaller than the conductivity of the fine-textured soil. The 10,000 water content where the conductivity asymptotically becomes zero (Figure 2; left) is often used as an alternative working definition for the residual water content, y . 1,000 r Soil water diffusivity 100 Another hydraulic function often used in theoretical and management application of unsaturated flow theories is À Pressure head, |h| (cm) 10 the soil water diffusivity, D(y), (L2 T 1), which is defined as dh 1 DðyÞ¼KðyÞ : (2) 0.0 0.1 0.2 0.3 0.4 0.5 dy Volumetric water content [–] This function appears when Equation 1 is transformed into a water-content-based equation in which y is now Hydraulic Properties of Unsaturated Soils, Figure 1 Typical the dependent variable: soil water retention curves for relatively coarse- (solid line), medium- (dashed line), and fine-textured (dotted line) soils. The @ @ @ y ¼ ð Þ h À ð Þ : curves were obtained using Equation 6a assuming hydraulic @ @ D y @ K y (3) parameter values as listed in Table 1.
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