Ching-Sheng Huang

An analytical solution for tidal fluctuations in unconfined aquifers with a vertical beach

[1] The perturbation technique has been commonly used to develop analytical solutions for simulating the dynamic response of tidal fluctuations in unconfined aquifers. However, the solutions obtained from the perturbation method might result in poor accuracy for the case of the perturbation parameter being not small enough. In this paper, we develop a new analytical model for describing the water table fluctuations in unconfined aquifers, based on Laplace and Fourier transforms. In the new approach, the mean sea level is used as the initial condition and a free surface equation, neglecting the second‐order slope terms, as the upper boundary condition. Numerical results show that the present solution agrees well with the finite different model with the second‐order surface terms. Unlike Teo et al.’s (2003) approximation which restricts on the case of shallow aquifers, the present model can be applied to most of the tidal aquifers except for the very shallow one. In addition, a large‐time solution in terms of sine function is provided and examined graphically with four different tidal periods.

Groundwater flow to a pumping well in a sloping fault zone unconfined aquifer

This study develops a mathematical model for simulating the hydraulic head distribution in response to pumping in a sloping fault zone aquifer under a water table boundary condition. A two-dimensional equation with a sink term representing the pumping is used for describing the head distribution in the aquifer. In addition, a first-order free surface equation is adopted to represent the change in water table at the outcrop. The analytical solution of the model, derived by the Laplace and finite Fourier cosine transforms, is expressed in terms of a double series. A finite difference solution within a deformable grid framework is developed to assess the solution obtained by specifying the free surface equation at the outcrop. Based on the analytical solution, we have found that the models prediction tends to overestimate drawdown in a late pumping period. The temporal head distribution is independent of the aquifer slope if the water table change is small, and exhibits a double-humped shape due to the effect of the free surface. The temporal drawdown predicted from the analytical solution is further compared with those measured from a pumping test conducted in northern Portugal.

Approximate Solution for a Transient Hydraulic Head Distribution Induced by a Constant-Head Test at a Partially Penetrating Well in a Two-Zone Confined Aquifer

AbstractA mathematical model describing the transient hydraulic head distribution induced by constant-head pumping/injection at a partially penetrating well in a radial two-zone confined aquifer is a mixed-type boundary value problem. The analytical solution of the model is in terms of an improper integral with an integrand having a singularity at the origin. The solution should rely on numerical methods to evaluate the integral and handle the problems of convergence and singularity. This study aims at developing a new approximate solution describing the transient hydraulic head distribution for a constant-head test (CHT) at a partially penetrating well in the aquifer. This approximate solution is acquired based on a time-dependent diffusion layer approximation proposed in the field of electrochemistry. The diffusion layer can be analogous to the radius of influence in the area of well hydraulics. The approximate solution is in terms of modified Bessel functions for aquifers with a partially penetrating w...

An analytical approach for the simulation of flow in a heterogeneous confined aquifer with a parameter zonation structure

This study introduces an analytical approach to estimate drawdown induced by well extraction in a heterogeneous confined aquifer with an irregular outer boundary. The aquifer domain is divided into a number of zones according to the zonation method for representing the spatial distribution of a hydraulic parameter field. The lateral boundary of the aquifer can be considered under the Dirichlet, Neumann or Robin condition at different parts of the boundary. Flow across the interface between two zones satisfies the continuities of drawdown and flux. Source points, each of which has an unknown volumetric rate representing the boundary effect on the drawdown, are allocated around the boundary of each zone. The solution of drawdown in each zone is expressed as a series in terms of the Theis equation with unknown volumetric rates from the source points. The rates are then determined based on the aquifer boundary conditions and the continuity requirements. The estimated aquifer drawdown by the present approach agrees well with a finite element solution developed based on the Mathematica function NDSolve. As compared with the existing numerical approaches, the present approach has a merit of directly computing the drawdown at any given location and time and therefore takes much less computing time to obtain the required results in engineering applications. This article is protected by copyright. All rights reserved.

Estimating stream filtration from a meandering stream under the Robin condition: FILTRATION FROM MEANDERING STREAM

This study applies image well theory to estimate the stream depletion rate (SDR) due to pumping near a meandering stream with a clogged streambed treated as the Robin condition. The stream is considered as an irregular boundary represented by discrete nodes. Image wells are arranged along the stream and near those nodes. On the basis of the Theis (1935) solution and the principle of superposition, the solution for the aquifer drawdown subject to the stream can then be expressed as the sum of the Theis solution and a simple series representing the effect of those image wells. The discharge rates of the image wells are determined by solving a system of equations obtained by substituting the drawdown solution into the Robin condition. Quantitative criteria for assessing the applicability of the image well theory are provided. On the basis of the drawdown solution and Darcys law, the analytical solution for SDR can then be obtained. A finite element solution is also developed to verify the SDR solution. Temporal SDR distributions predicted by both the analytical solution and finite element solution agree well over the entire period except at late time when the stream filtration rate approaches the pumping rate (i.e., SDR ≅ 1). It is found that a meandering stream has a significant effect on SDR compared with a rectilinear one and the effect should be taken into account in estimating SDR.

Discussion of “Integral and Closed-Form Analytical Solutions to the Transport Contaminant Equation Considering 3D Advection and Dispersion” by Luan Carlos de S. M. Ozelim and André Luís Brasil Cavalcante

where t 5 time; (x, y, z) 5 variables of the Cartesian coordinate system; c 5 contaminant concentration; R 5 retardation factor; l 5 first-order reaction coefficient; v5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vx 1 vy 2 1 vz p in which vx, vy, and vz are the components of the groundwater velocity v in the x, y, and z directions, respectively; Dd 5 molecular diffusion coefficient of the formation; al and at 5 longitudinal and transverse dispersivities, respectively; and the matrix defines the components of the dispersion coefficient tensor. The authors assumed that al=at 5 1 (i.e., al 5at) and the flow velocity is steady and uniform; therefore, Eq. (1) reduces to