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Baden, Switzerland

This work develops a simple exact and explicit solution of the one-dimensional transient and nonlinear Richards' equation for soils in a special case of exponential water retention curve and power law hydraulic conductivity. The exact solution is obtained as traveling wave based on the approach proposed by Philip (1957, 1967) and adopted by Zlotnik et al. (2007). The obtained solution is novel, and it expresses explicitly the water content as function of the depth and time. It can be useful to model infiltration into semi-infinite soils with time-dependent boundary conditions and infiltration with constant boundary condition but space-dependent initial condition. A complete analytical inverse procedure based on the proposed analytical solution is presented which allows the estimation of hydraulic parameters. The proposed exact solution is also important for the verification of numerical schemes as well as for checking the implementation of time-dependent boundary conditions. © 2016 Elsevier B.V. Source


Hayek M.,AF Consult Switzerland Ltd
Journal of Hydrology | Year: 2015

An analytical solution for one-dimensional steady vertical flux through unsaturated homogeneous soils is presented. The model assumes power law hydraulic conductivity and diffusivity functions. The soil domain is a finite-depth flow medium overlying a water table. A steady constant flux is applied at the top boundary while a constant saturation value is specified at the bottom boundary. The general form of the analytical solution expresses implicitly the depth as function of the liquid water saturation. It can be used to model both infiltration through the soil surface and evaporation from the bottom, depending on the sign of the flux boundary value. The analytical solution takes into account the prediction of a drying front in the case of evaporation from deep water table. Algebraic expressions of practical and theoretical importance are derived in terms of soil water parameters. These expressions include the stored mass in the system at steady state as well as the drying front when it exists. The general form solution can be inverted back to obtain exact explicit solutions when the power law parameters are related. Numerical results show the effects of soil type, surface flux, capillarity, and gravity on the saturation distribution in the soil. The analytical solution is used for comparing between models, validating of numerical solutions, as well as for estimating the hydraulic parameters. © 2015 Elsevier B.V. Source


Hayek M.,AF Consult Switzerland Ltd
Journal of Hydrologic Engineering | Year: 2016

A general analytical model for one-dimensional (1D) contaminant transport in infinite domain with time-dependent transport parameters is presented in this paper. The model is based on the advection-dispersion equation with a time-dependent flow velocity, a time-dependent dispersion coefficient, and a time-dependent distribution coefficient due to sorption. It takes into account a first-order irreversible reaction (decay), an arbitrary initial distribution of the contaminant, and an arbitrary space- and time-dependent sink/source term. The model can handle any time-dependent transport parameter. Analytical solutions are provided for both the advection dominant and the advection-dispersion transport equations. The proposed analytical solutions are general and they can be used for problems where one or more parameters are time-dependent. It is shown that the presented solutions can be reduced to other existent solutions where one transport parameter is assumed to be time-dependent. The general analytical solutions are obtained by using the Fourier transform, and they are presented in integral forms. Several closed-form solutions can be derived from the general integral form. Such closed-form solutions are of great interest since they allow the dependence of the solutions on the underlying physical parameters to be studied in an analytical manner. In particular, the author presents some closed-form solutions for the case of an initial step function and discusses through the numerical examples some insights about the errors that can be made by assuming constant parameters instead of time-dependent ones. These assumptions may lead to overestimated or underestimated concentrations. The proposed analytical solutions are useful for benchmarking numerical solutions to problems in hydrogeology and chemical engineering. They are also of great importance to the investigation of quantitative accuracy assessment. One of the presented closed-form solutions is used to compare with numerical solutions. © 2016 American Society of Civil Engineers. Source


Hayek M.,Paul Scherrer Institute | Hayek M.,AF Consult Switzerland Ltd | Kosakowski G.,Paul Scherrer Institute | Jakob A.,Paul Scherrer Institute | Churakov S.V.,Paul Scherrer Institute
Water Resources Research | Year: 2012

One of the challenging problems in mathematical geosciences is the determination of analytical solutions of nonlinear partial differential equations describing transport processes in porous media. We are interested in diffusive transport coupled with precipitation-dissolution reactions. Several numerical computer codes that simulate such systems have been developed. Analytical solutions, if they exist, represent an important tool for verification of numerical solutions. We present a methodology for deriving such analytical solutions that are exact and explicit in space and time variables. They describe transport of several aqueous species coupled to precipitation and dissolution of a single mineral in one, two, and three dimensions. As an application, we consider explicit analytical solutions for systems containing one or two solute species that describe the evolution of solutes and solid concentrations as well as porosity. We use one of the proposed analytical solutions to test numerical solutions obtained from two conceptually different reactive transport codes. Both numerical implementations could be verified with the help of the analytical solutions and show good agreement in terms of spatial and temporal evolution of concentrations and porosities. Copyright 2012 by the American Geophysical Union. Source


Pinzer B.R.,WSL Institute for Snow and Avalanche Research SLF | Pinzer B.R.,Paul Scherrer Institute | Schneebeli M.,WSL Institute for Snow and Avalanche Research SLF | Kaempfer T.U.,WSL Institute for Snow and Avalanche Research SLF | Kaempfer T.U.,AF Consult Switzerland Ltd
Cryosphere | Year: 2012

Dry snow metamorphism under an external temperature gradient is the most common type of recrystallization of snow on the ground. The changes in snow microstructure modify the physical properties of snow, and therefore an understanding of this process is essential for many disciplines, from modeling the effects of snow on climate to assessing avalanche risk. We directly imaged the microstructural changes in snow during temperature gradient metamorphism (TGM) under a constant gradient of 50 K mĝ̂'1, using in situ time-lapse X-ray micro-tomography. This novel and non-destructive technique directly reveals the amount of ice that sublimates and is deposited during metamorphism, in addition to the exact locations of these phase changes. We calculated the average time that an ice volume stayed in place before it sublimated and found a characteristic residence time of 2-3 days. This means that most of the ice changes its phase from solid to vapor and back many times in a seasonal snowpack where similar temperature conditions can be found. Consistent with such a short timescale, we observed a mass turnover of up to 60% of the total ice mass per day. The concept of hand-to-hand transport for the water vapor flux describes the observed changes very well. However, we did not find evidence for a macroscopic vapor diffusion enhancement. The picture of {temperature gradient metamorphism} produced by directly observing the changing microstructure sheds light on the micro-physical processes and could help to improve models that predict the physical properties of snow. © Author(s) 2012. Source

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