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• 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]

The development of the adjoint state method for the system of partial differential equations (PDEs) describing radionuclide transport in heterogeneous discrete fractured porous media in steady‐state regime is presented in this study for both continuous and discrete adjoint approaches. The continuous adjoint approach has its own advantage in the sense that the adjoint system can be derived in terms of PDEs which can be solved analytically for specific performance measures. These PDEs can then be used to obtain a physical interpretation of the adjoint state functions. The discrete adjoint approach permits a simpler mathematical treatment based on the use of the same matrix (or its transpose) of the linear system of the discretized transport equations referring to from hereon as the forward problem. Analytical solutions for both the forward problem and its associated adjoint problem have been derived for radionuclide transport in layered formations including a fracture using an analytical inversion of tridiagonal matrices. A set of 14 formulas of sensitivity coefficients with respect to pertinent system parameters has been derived and presented in integral forms. Analytical results have been compared with numerical results and have shown excellent agreements. The discrete adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. The results show that when a fracture is present in the vicinity of waste repository, the state parameter controlling flow through the fracture, referred to herein as fracture transmissivity, is among the important parameters which should be considered in sensitivity analysis studies. Key Points: A full mathematical/numerical adjoint sensitivity model for steady‐state RN transport in heterogeneous discrete fractured media is presentedAnalytical solutions for concentrations and adjoint states have been derived for the case of layered formations including a fractureThe developed adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case [ABSTRACT FROM AUTHOR]

• A new explicit closed-form approximate analytical solution for both recharging and discharging aquifers is derived • The proposed solution is accurate, straightforward, do not require any numerical treatment, and can be implemented in a simple spreadsheet • It is compared against efficient numerical solutions, series expansion solutions, and closed-form solution The classical problem used to model the response of an unconfined aquifer to a sudden change in boundary head is considered in this paper. This problem is usually modeled using the nonlinear groundwater Boussinesq equation. Due to the nonlinearity of this problem, no exact solutions exist. Solutions to the Boussinesq equation are therefore obtained using numerical techniques or approximate analytical methods. In this paper, we present a novel closed-form approximate analytical solution to this problem. The Boussinesq equation is converted to a first-order ordinary differential equation (ODE) by means of the Boltzmann transformation and by introducing a new variable related to the water flow. The first-order ODE is then solved analytically after introducing an intermediate approximation involving two fitting parameters. To avoid any numerical treatment, closed-form polynomial expressions of the fitting parameters are proposed. The final-form solution is simple to use and is obtained in terms of the incomplete gamma function, which is valid for both recharge and discharge. The derived solution is tested and compared to efficient numerical solutions, as well as to two types of analytical solutions: an accurate series expansion solution and an equivalent closed-form solution. The results show excellent agreement between the proposed solution and the numerical and series solutions. The proposed solution offers a key advantage in terms of both accuracy and simplicity; notably, it can be implemented using a simple spreadsheet. [ABSTRACT FROM AUTHOR]

We present an efficient mathematical/numerical adjoint sensitivity analysis model for transient radionuclide transport through heterogeneous porous media. This work extends our previous research on radionuclide transport in a steady‐state regime. Both continuous and discrete adjoint approaches have been developed in this current research. The mathematical equations associated with the adjoint system and required for the continuous approach have been derived. The methodology for deriving the discrete adjoint solution is also presented. For the homogeneous case, the adjoint system for radionuclide decay chain is solved analytically. The four‐member decay chain (238Pu → 234U → 230Th → 226Ra) is considered for validation. The validity of the analytical adjoint sensitivity model has been shown using two illustrative examples associated with two performance measures. The analytical adjoint states are compared with calculated numerical results and show excellent agreements. The sensitivity coefficients are also computed numerically using the perturbation method. The numerical sensitivity coefficients successfully reproduce the analytically derived sensitivities based on adjoint states. The adjoint model has been coupled with a derivative‐based global sensitivity method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. An illustrative case study of spent fuel and high‐level waste (SF/HLW) repository involving 6 radionuclides and 48 parameters has been considered for identifying the most important parameters with respect to the maximum radionuclide dose at the interface geosphere/biosphere. Based on the results of sensitivity analysis, the uncertainty of the maximum radionuclide dose has been performed using Monte Carlo simulations. Key Points: An adjoint sensitivity model for transient sequentially coupled radionuclide transport in heterogeneous media is presentedAnalytical expressions including adjoint state functions and sensitivity coefficients are derived in closed forms for the homogeneous caseA derivative‐based global sensitivity method coupled with the adjoint approach is presented and applied to a real field case [ABSTRACT FROM AUTHOR]

Highlights • An efficient analytical model for horizontal infiltration in soils is presented. • The model covers many existing realistic models in the literature. • It doesn’t make any prior assumptions about the form of the diffusivity function. • It is useful for studying the effects of soil type and checking numerical schemes. • A new method to predict the distribution of WC and estimate the hydraulic parameters. Abstract An efficient method for nonlinear approximate solutions of the one-dimensional horizontal absorption equation is presented. The method pertains to a semi-infinite soil with uniform initial condition and constant soil water content at the beginning of a soil column. It does not make any prior assumptions about the form of the diffusivity function. The nonlinear solution is of simple combination of power and exponential functions and expresses in explicit form the distance as function of time and soil water content. It involves all the physical parameters and four additional fitting parameters which can be obtained by simple fitting procedure. Algebraic expressions that quantify the main infiltration process are derived from the basic equation. These include the sorptivity, the infiltration rate, the cumulative infiltration rate, and the wetting front. A comparison with exact, numerical and other analytical solutions validates the accuracy of the approximation and shows its advantage to other approximate solutions. The analytical expressions are used to predict experimental soil moisture profiles from simple measurements such as sorptivity, wetting front, and initial and final soil water contents. The method is applied to estimate soil hydraulic properties using horizontal water absorption experiments. The results of the parameter estimation procedure are compared to other existing results and show good agreements. The solution method is also tested to various diffusivity functions and its limitation is discussed in detail. [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 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]