A Well in a ‘Target’ Stratum of a Two-Layered Formation: The Muskat–Riesenkampf Solution Revisited

Kasimova, Rouzalia; Kacimov, A. R.

doi:10.1007/s11242-010-9693-6

Cited by 7 publications

(3 citation statements)

References 28 publications

(44 reference statements)

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“…The following mechanisms control the slug motion: gravity (drives the slug down), Darcian resistance (impedes flow), volumedistributed water suction by the roots, capillarity (through two air entrance constants on the two fronts) and accretion/evaporation on the drainage front. These mechanisms are incorporated into the model in a mathematical way, matching the EinsteinKalashnikov epigraph-dictum, and the Guenon concept of "unifying complexity" and limiting contours (Obnosov et al 2011), which in our case are the Green-Ampt fronts. A single-front Green-Ampt model involves hydraulic conductivity, air entrance pressure and effective porosity, while in our case SPF are tackled through five parameters (different porositiespressures at the two fronts).…”

Section: Resultsmentioning

confidence: 99%

Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant–root continuum: the Green-Ampt-Youngs model revisited

Kacimov

2013

Hydrological Sciences Journal

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Editor D. KoutsoyiannisCitation Kacimov, A. and Obnosov, U., 2013. Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant-root continuum: the Green-Ampt-Youngs model revisited. Hydrological Sciences Journal, 58 (1), 237-248.Abstract A tension-saturated water slug descends through a homogenous soil after a rainfall (irrigation) event and shrinks due to transpiration by a distributed root-sink and evaporation. The upper (drainage) and lower (imbibition) sharp fronts of the slug separate it from the superjacent and subjacent vadose zones, where water is immobile. In the slug, the hydraulic conductivity is constant according to the Green-Ampt model. The capillary pressures as well as effective porosities on the fronts are given (generally, different) constants that can be viewed as a kind of hysteresis. A volumetric sink models mild (no desaturation of the slug) soil water withdrawal by the plant roots. The sink intensity varies with the depth from the soil surface and with time. Mathematically, the hydraulic head is immediately expressed by double integration of a governing 1-D flow equation. The pressure and kinematic conditions on the fronts result in a Cauchy problem for a system of two ODEs, which is solved by computer algebra routines.Key words infiltration; two-front Green-Ampt approximation; evapotranspiration; drainage-imbibition; root water uptake; ecohydrology Masses d'eau du sol à double front et pseudo-hystérésis alimentant le continuum plante-racines: retour sur le modèle de Green-Ampt-YoungsRésumé Une bulle saturée en eau descend à travers un sol homogène après un épisode de pluie ou d'irrigation et s'amincit en raison de la transpiration du prélèvement racinaire et de l'évaporation. Les fronts abrupts supérieur (drainage) et inférieur (imbibition) de la bulle la séparent des zones non-saturées sus-jacentes et sous-jacentes, où l'eau est immobile. Dans la bulle, la conductivité hydraulique est constante selon le modèle de Green-Ampt. Les pressions capillaires ainsi que des porosités efficaces sur les fronts sont des constantes données (en général différentes) ce qui peut être vu comme une sorte d'hystérésis. Un prélèvement volumétrique modélise le faible prélèvement d'eau du sol (pas de désaturation de la bulle) par les racines des plantes. L'intensité du prélèvement varie avec la profondeur depuis la surface du sol et au cours du temps. Mathématiquement, la charge hydraulique s'exprime immédiatement par la double intégration de l'équation régissant l'écoulement à une dimension. La pression et les conditions cinématiques sur les fronts permettent de poser un problème de Cauchy pour un système de deux équations différentielles ordinaires, qui peut être résolu par des routines de calcul formel.

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Section: Resultsmentioning

confidence: 99%

Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant–root continuum: the Green-Ampt-Youngs model revisited

Kacimov

2013

Hydrological Sciences Journal

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“…Consequently, we get an infinite series, similarly to Obnosov et al . () for the Laplace equation:

\begin{matrix} lefttrue \\ θ = θ_{1 D} + \frac{Q exp Z}{8 π} [\frac{exp (- r)}{r} - \frac{exp (- r_{1})}{r_{1}} - \frac{exp (- r_{2})}{r_{2}} + \frac{exp (- r_{3})}{r_{3}} + \frac{exp (- r_{4})}{r_{4}} - \dots], \\ r_{2} = \sqrt[]{ρ^{2} + {((), Z + 2 t_{s})}^{2}}, r_{3} = \sqrt[]{ρ^{2} + {((), Z + 2 a + 2 t_{s})}^{2}}, r_{4} = \sqrt[]{ρ^{2} + {((), Z - 2 a - 2 t_{s})}^{2}}, \dots \end{matrix}

…”

Section: Philip's Point Sourcementioning

confidence: 99%

Steady Darcian Flow in Subsurface Irrigation of Topsoil Impeded by a Substratum: Kornev–Riesenkampf–Philip Legacies Revisited

Kacimov

2018

Irrigation and Drainage

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Flows in homogeneous topsoils with a subjacent substratum or horizontal groundwater table generated by line-point emitters are studied and tracked back to the Kornev method of subsurface irrigation. Laplace's equation governs flow in a saturated or tension-saturated hat-shaped zone subtended by the substratum, provided pressure in a porous pipe or mole hole is positive. For low capillarity a free surface (phreatic line or capillary fringe) and a layer-substratum interface of a constant vertical component of velocity bound the flow domain. The free surface is found for various values of source strengths, emitter elevation above the substratum and the ratio of hydraulic conductivities of the topsoil and substratum. Subcritical and supercritical regimes are distinguished. In the limit of an impermeable substratum, the Riesenkampf solution for a line source is analysed. In soils of high capillarity, the J.R. Philip model of a point source and 'exponential mirror principle' give a series solution for a vertical array of alternating sources and sinks. Four topological situations emerge, depending on the layer thicknesses, topsoil potential, source depths strengths, saturated conductivity and sorptive number. The point source, groundwater table and soil surface are hydrologically intertwined, with formation of dividing surfaces (separatrices) and critical lines. Copyright RÉSUMÉ Les flux dans des sols homogènes avec un substrat sous-jacent ou une nappe d'eau horizontale générée par des émetteurs en ligne sont étudiés et suivis par la méthode de Kornev d'irrigation souterraine. L'équation de Laplace régit l'écoulement dans une zone en forme de chapeau saturée ou saturée par la tension sous-tendue par le substrat, à condition de trouver une pression positive dans un tuyau poreux ou un trou de taupe. Pour une faible capillarité, une surface libre (ligne phréatique ou frange capillaire) et une interface couche-substrat d'une composante verticale constante de la vitesse bornent le domaine d'écoulement. La surface libre est trouvée pour différentes valeurs des forces de la source, la hauteur de l'émetteur audessus du substrat et le rapport des conductivités hydrauliques entre la couche superficielle et le substrat. Les régimes souscritiques et supercritiques sont distingués. Dans la limite d'un substrat imperméable, la solution Riesenkampf pour une ligne d'émetteurs est analysée. Dans les sols à forte capillarité, le modèle J.R. Philip d'une source ponctuelle et d'un « principe de miroir exponentiel » donne une solution en série pour un réseau vertical de sources et de puits en alternance. Quatre situations topologiques apparaissent, en fonction de l'épaisseur des couches, du potentiel du sol, de la profondeur des sources, des forces, de la conductivité saturée et du nombre sorbtif. La source ponctuelle, la nappe phréatique et la surface du sol sont entrelacées hydrologiquement, avec la formation de surfaces de séparation (séparatrices) et de lignes critiques. † Ecoulement darcien en régime permanent dans l'irrigation souterr...

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“…Analytical solutions to potential field problems, where the intricate topology of 2D flow nets is controlled by internal heterogeneities of the domain, but the boundary conditions are uniform or follow from the symmetry principle in cases of internal heat sources/sinks, have recently been obtained [6,[9][10][11][12]. In these cases, an elementary cell, where the potential problems were solved, represents a flow tube, which consists of two isotherms and two adiabatic lines.…”

Section: Introductionmentioning

confidence: 99%

Topology of Steady Heat Conduction in a Solid Slab Subject to a Nonuniform Boundary Condition: The Carslaw–jaeger Solution revisited

Kasimova

2012

ANZIAM J.

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Temperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of two-dimensional heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and a heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped onto an auxiliary half-plane. The flow net (orthogonal isotherms and heat lines) of heat conduction is compared with the known Carslaw-Jaeger solution and shows a puzzling topology of three regimes of energy fluxes for temperature boundary conditions common in passive thermal insulation. The simplest regime is realized if cooling of a shaded zone is mild and heat flows in a slightly distorted "resistor model" flow tube. The second regime emerges when cooling is stronger and two disconnected separatrices demarcate the back-flow of heat from a relatively hot segment of the slab surface to the atmosphere through relatively cold parts of this surface. The third topological regime is characterized by a single separatrix with a critical point inside the slab, where the thermal gradient is nil. In this regime the back-suction of heat into the atmosphere is most intensive. The closed-form solutions obtained can be used in assessment of efficiency of thermal protection of buildings.2010 Mathematics subject classification: primary 35J25; secondary 65E04, 80A05.

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A Well in a ‘Target’ Stratum of a Two-Layered Formation: The Muskat–Riesenkampf Solution Revisited

Cited by 7 publications

References 28 publications

Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant–root continuum: the Green-Ampt-Youngs model revisited

Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant–root continuum: the Green-Ampt-Youngs model revisited

Steady Darcian Flow in Subsurface Irrigation of Topsoil Impeded by a Substratum: Kornev–Riesenkampf–Philip Legacies Revisited

Topology of Steady Heat Conduction in a Solid Slab Subject to a Nonuniform Boundary Condition: The Carslaw–jaeger Solution revisited

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