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• 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 exact solution for a nonlinear diffusion equation by considering the radially symmetric ν -dimensional case in inhomogeneous medium. This exact solution describes the evolution in space and time of an initial distribution of a diffusing substance. The diffusion coefficient is assumed to be dependent on both, positional coordinate (radial distance) and concentration, with power law nonlinearity. The exact solution is novel and it was obtained thanks to general similarity transformation. A constant time is involved in the similarity variable which allows the definition of the initial amount of the diffusant in the system. We provide an exact expression of the front propagation of the diffusing substance. The proposed exact solution is useful for verification of numerical simulation tools as well as for comparison with laboratory experiments. [ABSTRACT FROM AUTHOR]

Highlights: [•] We develop a complete algorithm for the parameterization in 2D fractured dual media. [•] The parameterization algorithm is based on the refinement indicators. [•] We use adjoint state method for the calculation of the local gradients. [•] We study robustness of the developed algorithm using several numerical experiments. [•] Algorithm provides a good parameterization even in critical cases. [ABSTRACT FROM AUTHOR]

Abstract: Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection–diffusion–reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection–diffusion–reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena. [Copyright &y& Elsevier]

Abstract: The extended -expansion method is introduced to construct more general exact traveling wave solutions to the KdV and Burgers equations with any order nonlinear terms. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters take up special values, new solitary, periodic and complex solutions are also derived from the traveling wave solutions. [ABSTRACT FROM AUTHOR]

Abstract: We present an upscaled model for the vertical migration of a CO2 plume through a vertical column filled with a periodic layered porous medium. This model may describe the vertical migration of a CO2 plume in a perfectly layered horizontal aquifer. Capillarity and buoyancy are taken into account and semi-explicit upscaled flux functions are proposed in the two following cases: (i) capillarity is the main driving force and (ii) buoyancy is the only driving force. In both cases, we show that the upscaled buoyant flux is a bell-shaped function of the saturation, as in the case of a homogeneous porous medium. In the capillary-dominant case, we show that the upscaled buoyant flux is the harmonic mean of the buoyant fluxes in each layer. The upscaled saturation is governed by the continuity of the capillary pressure at the interface between layers. In the capillary-free case, the upscaled buoyant flux and upscaled saturation are determined by the flux continuity condition at the interface. As the flux is not continuous over the entire range of saturation, the upscaled saturation is only defined where continuity is verified, i.e. in two saturation domains. As a consequence, the upscaled buoyant flux is described by a piecewise continuous function. Two analytical approximations of this flux are proposed and this capillary-free upscaled model is validated for two cases of heterogeneity. Upscaled and cell averaged saturations are in good agreement. Furthermore, the proposed analytical upscaled fluxes provide satisfactory approximations as long as the saturation set at the inlet of the column is in a range where analytical and numerical upscaled fluxes are close. [ABSTRACT FROM AUTHOR]

Abstract: The vertical stratification of carbon dioxide (CO2) injected into a deep layered aquifer made up of high-permeability and low-permeability layers, such as Utsira aquifer at Sleipner site in Norway, is investigated with a Buckley–Leverett equation including gravity effects. In a first step, we study both by theory and simulation the application of this equation to the vertical migration of a light phase (CO2), in a denser phase (water), in 1D vertical columns filled with different types of porous media: homogeneous, piecewise homogeneous, layered periodic and finally heterogeneous. For each case, we solve the associated Riemann problems and propose semi-analytical solutions describing the spatial and temporal evolution of the light phase saturation. These solutions agree well with simulation results. We show that the flux continuity condition at interfaces between high-permeability and low-permeability layers leads to CO2 saturation discontinuities at these interfaces and, in particular, to a saturation increase beneath low-permeability layers. In a second step, we analyze the vertical migration of a CO2 plume injected into a 2D layered aquifer. We show that the CO2 vertical stratification under each low-permeability layer is induced, as in 1D columns, by the flux continuity condition at interfaces. As the injection takes place at the bottom of the aquifer the velocity and the flux function decrease with elevation and this phenomenon is proposed to explain the stratification under each mudstone layer as observed at Sleipner site. [Copyright &y& Elsevier]

Abstract: In the analysis of the unsaturated zone, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the porous media. Adaptative multi-scale parameterization consists in solving the problem through successive approximations by refining the parameter at the next finer scale all over the domain and stopping the process when the refinement does not induce significant decrease of the objective function any more. In this context, the refinement indicators algorithm provides an adaptive parameterization technique that opens the degrees of freedom in an iterative way driven at first order by the model to locate the discontinuities of the sought parameters. We present a refinement indicators algorithm for adaptive multi-scale parameterization that is applicable to the estimation of multi-dimensional hydraulic parameters in unsaturated soil water flow. Numerical examples are presented which show the efficiency of the algorithm in case of noisy data and missing data. [Copyright &y& Elsevier]