General Solution for Steady Infiltration and Water Uptake in Strip-Shaped, Rectangular, and Cylindrical Domains

Soil Science Soc of Amer J

2012

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We recently derived analytical solutions of the linearized infiltration equation for steady flows from point sources into a cylindrically confined homogeneous soil domain. The same basic concepts were used in this study to obtain the solution for unsteady flows from a surface point source toward a subsurface point sink in a cylindrically confined soil with saturated hydraulic conductivity that varies exponentially with depth. As in the previous study, the infinite and finite Hankel transformations, now coupled with the Laplace transform, were used to obtain general solutions for point sources in a cylindrical domain. The solutions were incorporated into a coupled source-sink model to illustrate the influence of soil vertical heterogeneity on steady, maximum water uptake rates and on the rate of attainment of apparently steady conditions in the root zone. Illustrative simulations include: (i) steady flow patterns that develop in coupled source-sink systems in various soil textures with vertical heterogeneity; and (ii) effects of soil vertical heterogeneity on temporal variations in water uptake. Applications of a steady coupled sourcesink model that assumes only a time-dependent sink resistance, and of an unsteady model that also takes account of a time-dependent water source, for studying the effects of the frequency and the duration of cyclic water applications on water uptake are considered in detail in the companion article.Abbreviations: MFP, matric flux potential; RWUR, relative water uptake rate; SHC, saturated hydraulic conductivity.T o achieve high efficiency of irrigation water use, trickle irrigation systems should be designed, with respect to lateral spacing, spacing between plants and emitters, and emitter discharge rates, so that the delivery of water and fertilizers into the rooting zone matches the requirements for optimal plant growth. Management of irrigation, which together with water delivery also provides proper leaching of accumulating salts from the rooting zone while ensuring minimal groundwater pollution by percolated water, is also important. Quantitative description of water movement from surface sources through the rooting zone is essential for the design and management of trickle irrigation systems.Mathematical description of flows arising from acting trickle irrigation systems may be based on numerical solutions, but accurate analytical solurions are preferable, not only because of their greater ease of use but also because they provide a direct link between input parameters and resulting flow patterns, thus facilitating both design and management. Analytical solutions for multidimensional infiltration from irrigation sources are usually based on two restrictive assumptions: that hydraulic conductivity is an exponential function ofthe pressure head (Gardner, 1958) and that it is a linear function of the water content (Warrick,

supporting

confidence: 62%

Generalized Coupled Source–Sink Model for Evaluating Transient Water Uptake in Trickle Irrigation: I. Model Formulation for Soils with Vertical Heterogeneity

Soil Science Soc of Amer J

2012

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“…The use of this constitutive relationship enables linearization of the Richards equation for steady flow and, with the additional assumption that that hydraulic conductivity varies linearly with soil water content, the equation becomes linear for unsteady flow also. Pullan (1990) presented an extended review of the quasi‐linear theory of infiltration and, more recently, Basha (2000), Warrick (2003), Tracy (2007), and Communar and Friedman (2011, 2013) presented a variety of multidimensional solutions for steady and unsteady infiltration for various simple source and flow domain geometries.…”

mentioning

confidence: 99%

“…Analytical solutions for steady infiltration from shallow circular ponds at the soil surface (Wooding, 1968; Warrick, 1985) and from subsurface and surface point sources (Raats, 1971; Philip, 1971) were found useful for designing the geometry of drip irrigation systems (Bresler, 1978; Amoozegar‐Fard et al, 1984; Revol et al (1997a, 1997b); Communar and Friedman, 2010a, 2010b, 2010c, 2010d, 2011). Different methods, such as the dripper (pond radius) method (Shani et al, 1987; Or, 1996; Revol et al, 1997a, 1997b; Yitayew et al, 1998) and a multiple‐disk approach (Smettem and Clothier, 1989; Lazarovitch et al, 2007) have been used for evaluating the two parameters α and K s of Gardner's relationship from in situ measurements.…”

mentioning

confidence: 99%

Determination of Soil Hydraulic Parameters with CyclicIrrigation Tests

2014

Vadose Zone Journal

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A method for determining soil hydraulic parameters based on periodic point source solutions of the linearized Richards equation is proposed. Closed‐form solutions were derived for buried and surface point sources with a sinusoidally varying flux. These solutions describe the dependence of the matric flux potential (MFP) amplitude and phase shift on the distance from the point sources, frequency of source flux alternations, and soil hydraulic parameters. The Fourier series representation of square waves was used to extend the point source solutions for sinusoidally varying flux to square‐wave cyclic inputs of water with a 50% duty cycle—a preferable operating mode for field experiments. Close to the sources, the amplitudes and shape of MFP waves from a step input were more pronounced than those from a sinusoidal input, but the difference diminished as distance increased. The proposed method involves recording the pressure head variations with a continuously reading tensiometer installed below a pressure‐compensating emitter controlled by an irrigation computer. Among 16 cyclic step‐input irrigation tests, one series involved pressure‐head measurements at a single depth and varied water‐pulse durations; the second involved pressure‐head measurements at various depths with a fixed water‐pulse duration. The soil saturated hydraulic conductivity, the coefficient α of Gardner's conductivity model, and the coefficient k of the linearized unsteady‐flow equation were estimated by matching the periodic solution for cyclic step inputs to the measured pressure‐head data with a two‐step inversion procedure: an amplitude‐shift (AS) estimation procedure followed by the Levenberg–Marquardt (LM) optimization algorithm. The AS procedure was effective, and the LM optimization yielded only minor improvements. Our main findings are: (i) the periodic solution for step inputs yields reasonably good predictions of pressure‐head variations measured at various depths below an emitter for different periods of water application; and (ii) the proposed point‐source method seems to yield consistent and reliable estimates of the three soil hydraulic parameters.

“…In this context, steady‐state solutions to the quasilinear flow equation for a homogeneous soil with a spatially distributed extraction sink term (Raats, 1974; Warrick, 1974; Philip, 1997) appeared useful for the interpretation of laboratory and field results. In practice, other factors such as soil heterogeneity (Philip, 1972; Philip and Forrester, 1975; Srivastava and Yeh, 1991; Shan and Stephens, 1995; Chen and Gallipoli, 2004), lateral confinement of the flow domain (Raats, 1977; Merrill et al, 1978; De Rooij et al, 1996; Communar and Friedman, 2010d, 2011), and evaporation from the soil surface (Lomen and Warrick, 1978; Philip, 1991) may also affect the water flow and uptake patterns in subsurface zones.…”

mentioning

confidence: 99%

“…The only previous analytical study that yielded the wetting patterns from a buried point source in homogeneous root‐containing soil is that of Philip (1997). Raats (1977) and later De Rooij et al (1996) and Communar and Friedman (2010d, 2011) presented solutions for steady flow from point sources in cylindrically confined homogeneous (β = 0) soil domains with no water uptake (λ = 0). De Rooij et al (1996) first constructed the Hankel‐transformed solution for a surface disk source and then, by allowing the disk radius to approach zero, found the point‐source solution, whereas Communar and Friedman (2011) applied the Hankel transform directly to the point source problems.…”

mentioning

confidence: 99%

Steady Infiltration from a Surface Point Source into a Two-Layered Cylindrically Confined and Unconfined Soil Region with Root Water Extraction

2012

Vadose Zone Journal

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Steady infi ltra on from surface point sources into two-layered cylindrically confi ned and unconfi ned soil regions with roots extrac ng water is analyzed with a linearized form of the Richards equa on. In the upper layer, from which plant roots withdraw water with a uniform extrac on coeffi cient (λ), the hydraulic conduc vity at satura on decreases (−1 ≤ β < 0) or increases (β > 0) exponen ally with depth. The lower soil layer is homogeneous, and the soil texture coeffi cient α is the same for both layers. The evapora on loss is taken to be propor onal to the matric fl ux poten al (MFP) at the soil surface, with a propor onality coeffi cient m. It is shown that the point-source solu ons for the modeled systems are physically relevant, i.e., consistent with an exponen al hydraulic conduc vity func on, not only for posi ve but also for nega ve β if λ ≥ 0.5α 2 (1 − β). Also, the frac ons of water lost via evapora on at the soil surface, extrac on by plant roots, and deep percola on are found to be iden cal for both laterally confi ned and unconfi ned regions. Numerical examples illustrate the behavior of the solu ons for various combina ons of the coeffi cients λ, β and m, the root-zone depth, and the lateral extent of the soil domain. The water uptake rate (q up ) increases with increasing λ and root-zone depth and the we ed regions contract as the water uptake increases. Without evapora on, the water uptake rate decreases in the order q up (β < 0) > q up (β = 0) > q up (β > 0). Evapora on losses signifi cantly reduce both the water uptake rate and the extent of we ng. For β < 0 the eff ect of evapora on on q up is larger than for β ≥ 0.Abbrevia ons: ev, evapora on; MFP, matric fl ux poten al; per, deep percola on; up, water uptake.