An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards’ Equation

Fahs, Marwan; Younès, Anis; Lehmann, François

doi:10.1016/j.envsoft.2009.02.010

Cited by 57 publications

(51 citation statements)

References 27 publications

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“…The MOL has been successfully used to solve RE in many studies (e.g., Farthing et al, 2003;Miller et al, 2006;Li et al, 2007;Fahs et al, 2009). Details on the use of the MOL for solving RE are described in Fahs et al (2009).…”

Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

“…Error checking, robustness, order selection and adaptive time step features, available in sophisticated solvers, are applied to the time integration of partial differential equations (Tocci et al, 1997). The MOL has been successfully used to solve RE in many studies (e.g., Farthing et al, 2003;Miller et al, 2006;Li et al, 2007;Fahs et al, 2009). Details on the use of the MOL for solving RE are described in Fahs et al (2009).…”

Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

“…A uniform mesh of 600 cells is employed. Temporal discretization is performed with the highorder method of lines (MOL) (e.g., Miller et al, 1998;Tocci et al, 1997;Younes et al, 2009;Fahs et al, 2011). Error checking, robustness, order selection and adaptive time step features, available in sophisticated solvers, are applied to the time integration of partial differential equations (Tocci et al, 1997).…”

Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

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Hydraulic and transport parameter assessment using column infiltration experiments

Younès¹,

Mara²,

Fahs³

et al. 2017

Hydrol. Earth Syst. Sci.

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Abstract. The quality of statistical calibration of hydraulic and transport soil properties is studied for infiltration experiments in which, over a given period, tracer-contaminated water is injected into an hypothetical column filled with a homogeneous soil. The saturated hydraulic conductivity, the saturated and residual water contents, the Mualem-van Genuchten shape parameters and the longitudinal dispersivity are estimated in a Bayesian framework using the Markov chain Monte Carlo (MCMC) sampler. The impact of the kind of measurement sets (water content, pressure inside the column, cumulative outflow and outlet solute concentration) and that of the solute injection duration is investigated by analyzing the calibrated model parameters and their confidence intervals for different scenarios. The results show that the injection period has a significant effect on the quality of the estimation, in particular, on the posterior uncertainty range of the parameters. All hydraulic and transport parameters of the investigated soil can be well estimated from the experiment using only the outlet concentration and cumulative outflow, which are measured non-intrusively. An improvement of the identifiability of the hydraulic parameters is observed when the pressure data from measurements taken inside the column are also considered in the inversion.

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Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

Section: Unsaturated Flow-transport Modelmentioning

confidence: 99%

See 1 more Smart Citation

Hydraulic and transport parameter assessment using column infiltration experiments

Younès¹,

Mara²,

Fahs³

et al. 2017

Hydrol. Earth Syst. Sci.

Self Cite

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“…4 and 5), analytical solution of RE does not exist except in special cases (Celia et al, 1990;van Dam and Feddes, 2000). Therefore, numerical methods such as finite difference (Feddes et al, 1988;Romano et al, 1998;van Dam and Feddes, 2000), finite element (Gottardi and Venutelli, 2001), and mixed finite element (Bause and Knabner, 2004;Bergamaschi and Putti, 1999;Fahs et al, 2009;Farthing et al, 2003) are used to solve RE.…”

Section: Introductionmentioning

confidence: 99%

Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Maina

Ackerer

2017

Hydrol. Earth Syst. Sci.

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Abstract. The solution of the mathematical model for flow in variably saturated porous media described by the Richards equation (RE) is subject to heavy numerical difficulties due to its highly nonlinear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the timestep magnitude within the iteration procedure while the nonlinear parameters are computed with the state variable at the previous time. The Ross method is an example of this type of scheme, and we show that it is equivalent to the NewtonRaphson method with a time-adaptive algorithm.Both algorithms are coupled to different time-stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change in water saturation. Three different test cases are used to evaluate the efficiency of these algorithms.The numerical results highlight the necessity of implementing an estimate of the time truncation errors.

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“…These techniques include Finite-Di erence Method (FDM), Finite-Element Method (FEM), Finite Volume Method (FVM), hp-FEM and time splitting method [2][3][4][5][6][7][8][9][10][11]. Analytical solutions, on the other hand, are mainly o ered for one-dimensional ow of water through the soil and for restrictive boundary and initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

2017

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Abstract. Fluid in ltration into unsaturated soil is of vital signi cance from many perspectives. Mathematically, such transient in ltrations are described by Richards' equation: a nonlinear parabolic Partial Di erential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1D horizontal and vertical water in ltration are derived and compared to numerical nite-di erence method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water in ltration are derived for constant, no-ow, and sinusoidal boundary and initial conditions. The presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use di erent computational techniques.

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An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards’ Equation

Cited by 57 publications

References 27 publications

Hydraulic and transport parameter assessment using column infiltration experiments

Hydraulic and transport parameter assessment using column infiltration experiments

Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

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