Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two‐Dimensional Unsaturated Flow

Bakker, Mark; Nieber, John L.

doi:10.2136/vzj2004.1038

Cited by 21 publications

(10 citation statements)

References 48 publications

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“…These values, although detailed error analysis is not reported here, were found to be more than sufficient for this case. Similar investigations were reported previously (e.g., Bakker and Nieber, 2004) for circular inclusions.…”

Section: Boundary Conditionssupporting

confidence: 92%

“…Alternatively one can use a form of the Gardner model that includes air entry pressure {i.e., use exp [α( h − h a )]}, and use the air entry value h a as a matching parameter. Changes in the AEM algorithm are minor with this addition (e.g., Bakker and Nieber, 2004).…”

Section: Examples and Discussionmentioning

confidence: 99%

“…For contrasting α's, this is a nonlinear equation and the solution must be iterative. We linearized the equation following Bakker and Nieber (2004) in the form

α_{1} ϕ^{-} = {(α_{0} ϕ^{+})}^{\frac{α_{1}}{α_{0}}} = α_{0} ϕ^{+} {(α_{0} {\overset{ϕ}{true}}^{+})}^{\frac{α_{1}}{α_{0}} - 1}

where φ̂ + is the value of φ + evaluated using an estimate of coefficients from a previous iteration.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…Examples include solutions for large number of inhomogeneities (e.g., Barnes and Janković, 1999; Janković and Barnes, 1999). Recently, the method was extended to solutions of the Helmholtz equation (rather than the Laplace equation) hence allowing development of solutions for multi‐aquifers (e.g., Bakker and Strack, 2003), transient flow (through the use of Laplace transform; Furman and Neuman, 2003), and for unsaturated flow through inclusions (Warrick and Knight, 2002, 2004; Bakker and Nieber, 2004).…”

mentioning

confidence: 99%

“…They considered circular and spherical inclusions with a uniform distribution of the Gardner coefficient α. Bakker and Nieber (2004) extended the solution by considering circular inclusions (and circular drains) and allowing contrasts in α. This is a valuable contribution by adding flexibility and realism.…”

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confidence: 99%

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Unsaturated Flow Through Spherical Inclusions with Contrasting Sorptive Numbers

Furman

Warrick

2005

Vadose Zone Journal

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and for unsaturated flow through inclusions (Warrick and Knight, 2002, 2004;Bakker and Nieber, 2004). The analytic element method (AEM) is used to model unsaturated All the AEM solutions for unsaturated flow made flow through a spherical inclusion of contrasting hydraulic properties. use of the Gardner (1958) function (also known as the The steady state Richards' equation is combined with the Gardner model Philip model) to describe the relations between the hyfor unsaturated hydraulic conductivity to form the Helmholtz equation. The later is solved by means of the AEM. The background and inclu-draulic conductivity and the pressure head. This allows sion materials are assumed to have different saturated hydraulic conthe transformation between Richards' equation to the ductivities and different sorptive numbers; hence, the conditions are Helmholtz equation (using the matric flux potential, or more general than treatments of spherical inclusions. Continuity of the the so-called Kirchhoff potential). Warrick and Knight interfacial head boundary condition leads to a nonlinear system of (2002, 2004) were the first to introduce the AEM to deequations, whose solution requires an iterative solution. Analysis inscribe unsaturated flow. They considered circular and cludes the effect of the hydraulic properties and of the background flux spherical inclusions with a uniform distribution of the and evaluation of computational efficiency for contrasting hydraulic Gardner coefficient ␣. Bakker and Nieber (2004) extended properties. To examine water contents, a methodology is presented the solution by considering circular inclusions (and cirfor matching Gardner and van Genuchten parameters. The new solu-cular drains) and allowing contrasts in ␣. This is a valution is more realistic than the previous solution for a spherical inclusion with a sorptive number the same as the background but is computa-

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Section: Boundary Conditionssupporting

confidence: 92%

Section: Examples and Discussionmentioning

confidence: 99%

“…For contrasting α's, this is a nonlinear equation and the solution must be iterative. We linearized the equation following Bakker and Nieber (2004) in the form

α_{1} ϕ^{-} = {(α_{0} ϕ^{+})}^{\frac{α_{1}}{α_{0}}} = α_{0} ϕ^{+} {(α_{0} {\overset{ϕ}{true}}^{+})}^{\frac{α_{1}}{α_{0}} - 1}

where φ̂ + is the value of φ + evaluated using an estimate of coefficients from a previous iteration.…”

Section: Boundary Conditionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Unsaturated Flow Through Spherical Inclusions with Contrasting Sorptive Numbers

Furman

Warrick

2005

Vadose Zone Journal

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Analysis of vadose zone inhomogeneity toward distinguishing recharge rates: Solving the nonlinear interface problem with Newton method

Steward

2016

Water Resources Research

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Recharge from surface to groundwater is an important component of the hydrological cycle, yet its rate is difficult to quantify. Percolation through two‐dimensional circular inhomogeneities in the vadose zone is studied where one soil type is embedded within a uniform background, and nonlinear interface conditions in the quasilinear formulation are solved using Newton's method with the Analytic Element Method. This numerical laboratory identifies detectable variations in pathline and pressure head distributions that manifest due to a shift in recharge rate through in a heterogeneous media. Pathlines either diverge about or converge through coarser and finer grained materials with inverse patterns forming across lower and upper elevations; however, pathline geometry is not significantly altered by recharge. Analysis of pressure head in lower regions near groundwater identifies a new phenomenon: its distribution is not significantly impacted by an inhomogeneity soil type, nor by its placement nor by recharge rate. Another revelation is that pressure head for coarser grained inhomogeneities in upper regions is completely controlled by geometry and conductivity contrasts; a shift in recharge generates a difference Δp that becomes an additive constant with the same value throughout this region. In contrast, shifts in recharge for finer grained inhomogeneities reveal patterns with abrupt variations across their interfaces. Consequently, measurements aimed at detecting shifts in recharge in a heterogeneous vadose zone by deciphering the corresponding patterns of change in pressure head should focus on finer grained inclusions well above a groundwater table.

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Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

Gordeliy¹,

Mogilevskaya²,

Crouch³

2009

Numerical Meth Engineering

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SUMMARYA two-dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non-perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so-called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two-and threedimensional problems demonstrates the effect of the dimensionality.

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Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two‐Dimensional Unsaturated Flow

Cited by 21 publications

References 48 publications

Unsaturated Flow Through Spherical Inclusions with Contrasting Sorptive Numbers

Unsaturated Flow Through Spherical Inclusions with Contrasting Sorptive Numbers

Analysis of vadose zone inhomogeneity toward distinguishing recharge rates: Solving the nonlinear interface problem with Newton method

Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

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