Criterion for instability of steady-state unsaturated flows

Kapoor, Vivek

doi:10.1007/bf00140986

Cited by 19 publications

(18 citation statements)

References 32 publications

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“…For instance, Kapoor (1996) writes instead of (RE) 1 , after redefining our pressure head to his suction head Ψ := −Ψ,…”

Section: Remarkmentioning

confidence: 99%

“…We do note here that since E ψ (t) is bounded by B ψ (t), the growth of the norm is only a transient phenomenon because B ψ (t) → 0 as t → ∞. Kapoor (1996) derived stability criteria for the various types of steady vertical upward and downward flows in homogeneous, unsaturated soils. These criteria are summarized in Figure 4.…”

Section: Different Norms and Transient Growthmentioning

confidence: 99%

“…Based on experimental evidence that observed fingers often are long and narrow, he assumed that the vertical length scale of the perturbations is large compared to the horizontal length scale and on that basis he simplified the linearized equation for the perturbation of the suction head. Allowing for the sign changes in going from the suction head to the pressure head, Kapoor (1996) in effect ignored in the right hand side of equation (43) Linear stability analysis concerns the process of initiation of the fingers and in that stage the vertical length scale of the perturbations is still small. Therefore we reconsidered the stability of steady vertical flows, without ignoring the vertical gradients.…”

Section: Different Norms and Transient Growthmentioning

confidence: 99%

“…This critically depends on the relaxation parameter involved. Kapoor (1996) derived stability criteria for the various types of steady, vertical upward and downward flows in homogeneous, unsaturated porous media. Using the energy method, he showed that purely gravitational flows are stable.…”

Section: Introductionmentioning

confidence: 99%

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Steady Flows in Unsaturated Soils Are Stable

Duijn¹,

Pieters²,

Raats³

2004

Transport in Porous Media

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Abstract. The stability of steady, vertically upward and downward flow of water in a homogeneous layer of soil is analyzed. Three equivalent dimensionless forms of the Richards equation are introduced, namely the pressure head, saturation, and matric flux potential forms. To illustrate general results and derive special results, use is made of several representative classes of soils. For all classes of soils with a Lipschitz continuous relationship between the hydraulic conductivity and the matric flux potential, steady flows are shown to be unique. In addition, linear stability of these steady flows is proved. To this end, use is made of the energy method, in which one considers (weighted) L 2 -norms of the perturbations of the steady flows. This gives a general restriction of the dependence of the hydraulic conductivity upon the matric flux potential, yielding linear stability and exponential decay with time of a specific weighted L 2 -norm. It is shown that for other norms the ultimate decay towards the steadysolution is preceded by transient growth. An extension of the Richards equation to take into account dynamic memory effects is also considered. It is shown that the stability condition for the standard Richards equation implies linear stability of the steady solution of the extended model.

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“…For instance, Kapoor (1996) writes instead of (RE) 1 , after redefining our pressure head to his suction head Ψ := −Ψ,…”

Section: Remarkmentioning

confidence: 99%

Section: Different Norms and Transient Growthmentioning

confidence: 99%

Section: Different Norms and Transient Growthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Steady Flows in Unsaturated Soils Are Stable

Duijn¹,

Pieters²,

Raats³

2004

Transport in Porous Media

View full text Add to dashboard Cite

show abstract

“…It accounts for gravity, capillarity, and the fact that the permeability to water is reduced because the porous medium is only partially saturated with water. The inability of the Richards equation to explain fingered flow is well documented [35,33,51,78,36,38,37,79,61].…”

Section: Introductionmentioning

confidence: 99%

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

Cueto‐Felgueroso

Juanes

2009

Journal of Computational Physics

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This paper presents the application of adaptive rational spectral methods to the linear stability analysis of nonlinear fourth-order problems. Our model equation is a phase-field model of infiltration, but the proposed discretization can be directly extended to similar equations arising in thin film flows. The sharpness and structure of the wetting front preclude the use of the standard Chebyshev pseudo-spectral method, due to its slow convergence in problems where the solution has steep internal layers. We discuss the effectiveness and conditioning of the proposed discretization, and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation/film precursor, several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows, using just a few hundred grid points.

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Stability of gravity‐driven multiphase flow in porous media: 40 Years of advancements

DiCarlo

2013

Water Resources Research

102

112

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Gravity‐driven multiphase flow in porous media is ubiquitous in the geophysical world; the classic case in hydrology is vertical infiltration of precipitation into a soil. For homogenous porous media, infiltrations are sometimes observed to be stable and laterally uniform, but other times are observed to be unstable and produce preferential flow paths. Since Saffman and Taylor (1958), researchers have attempted to define criteria that determine instability. Saffman and Taylor's analysis consisted of two regions of single phase flow, while Parlange and Hill (1976) integrated this analysis with the multiphase flow equations to provide testable predictions. In the subsequent 40 years, great advances have been made determining the complex interactions between multiphase flow and instability. Theoretically, the stability of the standard multiphase flow equations has been verified, showing the necessity of extensions to the multiphase flow equations to describe the observed unstable flow. Experimentally, it has been shown that the instability is related to a phenomena in 1‐D infiltrations called saturation or pressure overshoot. In this review, the connection between overshoot and instability is detailed, and it is described how models of overshoot can simplify the analysis of current and future models of instability and multiphase flow.

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Criterion for instability of steady-state unsaturated flows

Cited by 19 publications

References 32 publications

Steady Flows in Unsaturated Soils Are Stable

Steady Flows in Unsaturated Soils Are Stable

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

Stability of gravity‐driven multiphase flow in porous media: 40 Years of advancements

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