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• Analytical results to flow in unconfined aquifer induced by a decaying conductivity. • Exact solutions for drainage into penetrating channels at the aquifer boundaries. • Analytical transit time for calculating particle travel from water table to boundary. • Analytical inverse model to estimate decay parameter by solving a nonlinear algebraic equation. This paper presents some analytical results related to groundwater flow through an unconfined horizontal aquifer induced by areal recharge on the top and a decaying hydraulic conductivity. Under steady-state conditions, and in the context of Boussinesq equation, exact analytical solutions describing drainage into one or two penetrating channels at the aquifer boundaries have been derived in closed forms. The analytical expressions have been used to investigate the existence of water divide inside the unconfined aquifer. Based on the derived analytical solutions, a closed-form analytical formula is developed for calculating groundwater transit time within Dupuit-type flow system. The analytical transit time expression is simply expressed in terms of common mathematical functions. Analytical results are verified against a numerical model and show excellent agreements. The analytical transit time is validated against a numerical one obtained from a two-dimensional model representing a rectangular unconfined aquifer. Based on the explicit closed-form solution, a fully analytical inverse problem to estimate the decay parameter is proposed using a least-square objective function. The solution of the inverse problem simply corresponds to the solution of a nonlinear algebraic equation obtained from the definition of the objective function and the form of the analytical solution. [ABSTRACT FROM AUTHOR]

• Van Genuchten-Mualem soil's steady flux analyzed; analytical solutions developed. • First solution for flow equation without prior assumptions on pressure head's form. • Solution derived using double Maclaurin series and multinomial theorem. • Analytically identify hydraulic parameters, avoiding numerical inverse procedures. Various analytical solutions for steady-state vertical flux through unsaturated soils exist in the literature. These solutions often assume specific hydraulic conductivity forms such as exponential form, rational power form, or power-law form such as the Brooks-Corey model. The van Genuchten-Mualem model is widely used in practical applications of flow through unsaturated soils. It is among the most complicated hydraulic conductivity forms for which no analytical solutions are available. This paper deals with the development of an analytical solution for steady-state vertical flux using van Genuchten-Mualem model. The analytical solution is obtained using Maclaurin series expansions and the multinomial theorem. The general form expresses the depth as a function of pressure head. The analytical solution can be used to model both infiltration (positive flux rate) and evaporation (negative flux rate). The derived analytical solution is compared with numerical results and shows excellent agreement. It can also be used to identify the hydraulic parameters analytically, eliminating the need of a numerical inverse procedure. As far as the author's knowledge, the proposed solution is the first analytical solution dealing with van Genuchten-Mualem model, which is directly derived from the flow equation without any predefined mathematical form of the pressure head. [ABSTRACT FROM AUTHOR]

• Analytical solutions for steady vertical flux with generalized hydraulic conductivity. • The solutions generalize existing solutions based on Brooks-Corey and rational models. • The proposed model provides better estimate of drying fronts for evaporation problems. • The analytical solutions allow a better fitting of experimental data. In this paper, analytical solutions for one-dimensional nonlinear steady-state vertical flux through unsaturated soils are presented. The analytical solutions are applicable for homogeneous soils for which the hydraulic conductivity function is a generalized form of the well-known Brooks-Corey and rational models. The adopted generalized soil–water retention model involves two additional parameters which reflect the curvature of hydraulic functions near saturation. The main advantage of the used generalized hydraulic conductivity function is that it fits better experimental data. The soil domain is a finite-depth medium overlying a fixed groundwater level. The derived analytical solutions are obtained using the Maclaurin series expansion technique. Analytical solutions for both infiltration and evaporation are developed. For evaporation from a fixed groundwater level, the derived solutions can be used to predict the existence of a vaporization plane below the soil surface. For this case, analytical expression of the position of drying front separating liquid and gas regions is derived. The analytical results are compared to numerical ones for various infiltration and evaporation examples and excellent agreements are obtained. The generalized model is also used to fit experimental data obtained for two soils. The results show that the use of the standard Brooks-Corey model may lead to an overestimation of the position of the drying front. [ABSTRACT FROM AUTHOR]

A general analytical model for one-dimensional transient vertical infiltration is presented. The model is based on a combination of the Brooks and Corey soil water retention function and a generalized hydraulic conductivity function. This leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index. Accordingly, the proposed analytical solution covers many existing realistic models in the literature. The general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time. It can be used to model infiltration through semi-infinite dry soils with prescribed water content or flux boundary conditions. Some mathematical expressions of practical importance are also derived. The general form solution is useful for comparison between models, validation of numerical solutions and for better understanding the effect of some hydraulic parameters. Based on the analytical expression, a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented. [ABSTRACT FROM AUTHOR]

Summary 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. [ABSTRACT FROM AUTHOR]

Summary 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. [ABSTRACT FROM AUTHOR]