Similarity and iterative solution of the Boussinesq equation

Sidiropoulos, Epaminondas; Tolikas, P.

doi:10.1016/0022-1694(84)90138-0

Cited by 9 publications

(6 citation statements)

References 4 publications

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“…The hypothetical case defined by Sidiropoulos and Tolikas (1984) is solved to examine the accuracy of the approach presented in this work for modeling the stochastic unconfined groundwater flow (Fig. 1).…”

Section: Application and Discussionmentioning

confidence: 99%

“…In this study, an example problem, introduced by Sidiropoulos and Tolikas (1984), is studied (Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

“…Unless the spatial gradient of the state variable is eliminated from the FPE, it will not be possible to obtain a probabilistic solution to the unconfined flow equation process that is governed by a probability distribution function (PDF). One way of eliminating the spatial gradients is by means of the Boltzmann transformation as described by Sidiropoulos and Tolikas (1984). Equation 3 is modified with the introduction of the new dimensionless variable g under the Boltzmann transformation (Sidiropoulos and Tolikas 1984) as:…”

Section: Introductionmentioning

confidence: 99%

“…One way of eliminating the spatial gradients is by means of the Boltzmann transformation as described by Sidiropoulos and Tolikas (1984). Equation 3 is modified with the introduction of the new dimensionless variable g under the Boltzmann transformation (Sidiropoulos and Tolikas 1984) as:…”

Section: Introductionmentioning

confidence: 99%

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Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: an exact second-order model

Cayar

Kavvas

2008

Stoch Environ Res Risk Assess

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A new stochastic model for unconfined groundwater flow is proposed. The developed evolution equation for the probabilistic behavior of unconfined groundwater flow results from random variations in hydraulic conductivity, and the probabilistic description for the state variable of the nonlinear stochastic unconfined flow process becomes a mixed Eulerian-Lagrangian-Fokker-Planck equation (FPE). Furthermore, the FPE is a deterministic, linear partial differential equation (PDE) and has the advantage of providing the probabilistic solution in the form of evolutionary probability density functions. Subsequently, the Boussinesq equation for one-dimensional unconfined groundwater flow is converted into a nonlinear ordinary differential equation (ODE) and a twopoint boundary value problem through the Boltzmann transformation. The resulting nonlinear ODE is converted to the FPE by means of ensemble average conservation equations. The numerical solutions of the FPE are validated with Monte Carlo simulations under varying stochastic hydraulic conductivity fields. Results from the model application to groundwater flow in heterogeneous unconfined aquifers illustrate that the time-space behavior of the mean and variance of the hydraulic head are in good agreement for both the stochastic model and the Monte Carlo solutions. This indicates that the derived FPE, as a stochastic model of the ensemble behavior of unconfined groundwater flow, can express the spatial variability of the unconfined groundwater flow process in heterogeneous aquifers adequately. Modeling of the hydraulic head variance, as shown here, will provide a measure of confidence around the ensemble mean behavior of the hydraulic head.

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Section: Application and Discussionmentioning

confidence: 99%

“…In this study, an example problem, introduced by Sidiropoulos and Tolikas (1984), is studied (Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: an exact second-order model

Cayar

Kavvas

2008

Stoch Environ Res Risk Assess

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“…A numerical solution of the problem discussed in this issue is proposed by Sidiropoulos and Tolikas (1984) and Tolikas et al (1984), who stated that it is advantageous to use the transformation:…”

Section: Available Numerical Solutionsmentioning

confidence: 99%

The Analytical Solution of the Boussinesq Equation for Flow Induced by a Step Change of the Water Table Elevation Revisited

Μουτσόπουλος

2010

Transp Porous Med

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The classical problem of flow induced by a sudden change of the piezometic head in a semi-infinite aquifer is re-examined. A new analytical solution is derived, by combining an expression describing the water table elevation upstream, obtained by the Adomian's decomposition approach, to an existing polynomial expression (Tolikas et al. in Water Resour Res 20:24-28, 1984), adequate for the downstream region; the parameters of both approximations are computed by matching the two solutions at the inflection point of the water table. Although several analytical solutions are available in the literature, we demonstrate that the expression we have developed in this issue is the most accurate for strong or moderate non-linear flows, where the degree of non-linearity is defined as the ratio of the piezometric head elevation at the origin to the initial water table elevation. For this type of flows the perturbation-series solution of Polubarinova-Kochina, characterized by previous studies as the best available analytical solution provides physically unacceptable results, while the analytical solution of Lockington (J Irrig Drain Eng 123:24-27, 1997), used to check the accuracy of numerical schemes, underestimates the penetration distance of the recharging front.

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