Two-dimensional Groundwater Flow Problems Research Articles

Review: Optimization methods for groundwater modeling and management

Optimization methods have been used in groundwater modeling as well as for the planning and management of groundwater systems. This paper reviews and evaluates the various optimization methods that have been used for solving the inverse problem of parameter identification (estimation), experimental design, and groundwater planning and management. Various model selection criteria are discussed, as well as criteria used for model discrimination. The inverse problem of parameter identification concerns the optimal determination of model parameters using water-level observations. In general, the optimal experimental design seeks to find sampling strategies for the purpose of estimating the unknown model parameters. A typical objective of optimal conjunctive-use planning of surface water and groundwater is to minimize the operational costs of meeting water demand. The optimization methods include mathematical programming techniques such as linear programming, quadratic programming, dynamic programming, stochastic programming, nonlinear programming, and the global search algorithms such as genetic algorithms, simulated annealing, and tabu search. Emphasis is placed on groundwater flow problems as opposed to contaminant transport problems. A typical two-dimensional groundwater flow problem is used to explain the basic formulations and algorithms that have been used to solve the formulated optimization problems.

An edge-based finite volume scheme for saturated–unsaturated groundwater flow

The application of a computationally efficient finite volume method for modelling groundwater flow in saturated–unsaturated soils is presented. The flow equation is solved by applying a vertex-centred edge-based finite volume scheme, and the discretisation of the flow domain is achieved using structured and unstructured grids. An acceleration algorithm is employed which utilises an explicit non-linear loop for the primary variable update. The scheme is employed to solve a two-dimensional groundwater flow problem and the results obtained are compared against published solutions. It is shown that the edge-based finite volume scheme captures the saturation front more efficiently than the published finite element solution [Comput. Geosci. 27 (2) (2001) 179]. Moreover, an improvement in the solution is shown by the application of a compact stencil scheme to calculate second-derivative terms as compared to the standard edge-based finite volume scheme.

An efficient tool for accelerating the numerical solution of the stochastic subsurface flow problem using neural networks

An efficient numerical solution for the two-dimensional groundwater flow problem using artificial neural networks (ANNs) is presented. Under stationary velocity conditions with unidirectional mean flow, the conductivity realizations and the head gradients, obtained by a traditional finite difference solution to the flow equation, are given as input-output pairs to train a neural network. The ANN is trained successfully and a certain level of recognition of the relationship between input conductivity patterns and output head gradients is achieved. The trained network produced velocity realizations that are physically plausible without solving the flow equation for each of the conductivity realizations. This is achieved in a small fraction of the time necessary for solving the flow equations. The prediction accuracy of the ANN reaches 97.5% for the longitudinal head gradient and 94.7% for the transverse gradient. Head-gradient and velocity statistics in terms of the first two moments are obtained with a very high accuracy. The cross covariances between head gradients and the fluctuating log-conductivity (log-K) and between velocity and log-K obtained with the ANN approach match very closely those obtained by a traditional numerical solution. The same is true for the velocity components auto-covariances. The results are also extended to transport simulations with very good accuracy. Spatial moments (up to the fourth) of mean-concentration plumes obtained using ANNs are in very good agreement with the traditional Monte Carlo simulations. Furthermore, the concentration second moment (concentration variance) is very close between the two approaches. Considering the fact that higher moments of concentration need more computational effort in numerical simulations, the advantage of the presented approach in saving long computational times is evident. Another advantage of the ANNs approach is the ability to generalize a trained network to conductivity distributions different from those used in training. However, the accuracy of the approach in cases with higher conductivity variances is being investigated.

Exact solutions for two-dimensional groundwater flow in a semiconfined aquifer

Two-dimensional groundwater flow problems involving a semipervious boundary can be solved using an analytical function ( V) of the complex potential (ω) and its derivative with respect to the complex coordinate ( z). So far boundary conditions however have been restricted to a semipervious layer of constant resistance, vertical equipotential lines and vertical stream lines. This paper gives solutions for problems in a semiconfined aquifer of finite depth, so involving a semipervious boundary as well as a horizontal stream line boundary. Exact solutions are given for flow with an arbitrary number of polder levels above the semipervious boundary; also sinks and sources may be present.

On a Nonlinear Elliptic-Parabolic Partial Differential Equation System in a Two-Dimensional Groundwater Flow Problem

In this paper a nonlinear elliptic-parabolic system which arises in a two-dimensional groundwater flow problem is studied. Abstract results on evolution equations are employed to obtain existence and uniqueness results. Regularity and stability properties of the solution are also considered.

Mixed Finite Element Simulations in Two-Dimensional Groundwater Flow Problems

A computer code of groundwater flow in two-dimensional porous media based on the mixed finite element method was developed for accurate approximations of Darcy velocities in safety evaluation of radioactive waste disposal. The mixed finite element procedure solves for both the Darcy velocities and pressure heads simultaneously in the Darcy equation and continuity equation. Numerical results of a single well pumping at a constant rate in a uniform flow field showed that the mixed finite element method gives more accurate Darcy velocities nearly 50% on average error than standard finite element method

Mixed finite element simulations in two-dimensional groundwater flow problems.

A computer code of groundwater flow in two-dimensional porous media based on the mixed finite element method was developed for accurate approximations of Darcy velocities in safety evaluation of radioactive waste disposal. The mixed finite element procedure solves for both the Darcy velocities and pressure heads simultaneously in the Darcy equation and continuity equation. Numerical results of a single well pumping at a constant rate in a uniform flow field showed that the mixed finite element method gives more accurate Darcy velocities nearly 50% on average error than standard finite element method

Exact solutions for two-dimensional groundwater flow problems involving a semi-pervious boundary

An analytical function ( V) of the complex potential (Ω) and its derivative with respect to the complex coordinate ( z) is defined that can be used to find exact solutions for two-dimensional groundwater flow problems involving a semi-pervious boundary. Boundary conditions may involve a semi-pervious layer of constant resistance, vertical equipotential lines and vertical stream lines. Solutions are found by conformal mapping of the V-plane on the z-plane and solving the first-order differential equation that is formed by the relationship between V and z. Some examples of the use of the function V are given.