A three-dimensional finite element model is developed for the simulation of saltwater intrusion in single and multiple coastal aquifer systems with either a confined or phreatic top aquifer. The model formulation is based on two governing equations, one for fluid flow and the other for salt transport. Density coupling of these equations is accounted for and handled using a Picard sequential solution algorithm with special provisions to enhance convergence of the iterative solution. Flexibility in the formulation allows for either three-dimensional simulations or quasi three-dimensional simulations, where flow and transport in aquitards are treated using one-dimensional analytical and/or numerical approximations. Spatial discretization of three-dimensional regions is performed using a vertical slicing approach designed to accommodate complex geometry with irregular boundaries, layering, and/or lateral discontinuity. This approach is effectively combined with the use of simple linear elements such as rectangular and triangular prisms, and composite hexahedra and pentahedra made up of tetrahedra. For these elements, computation of element matrices can be performed efficiently using influence coefficient formulas that avoid numerical integration. New transport influence coefficient formulas are presented for rectangular and triangular prism elements. Matrix assembly is performed slice by slice, and the matrix solution is achieved using a slice successive relaxation scheme. This permits a fairly large number of nodal unknowns (of the order of five to ten thousand) to be handled conveniently on small or medium-size minicomputers. Flexibility of the formulation and matrix handling procedures also allows two-dimensional and axisymmetric problems to be solved efficiently using single slice representations. Four examples are presented to demonstrate the model verification and utility. These problems represent a fair range of physical conditions. Where possible, simulation results are compared with previously published solutions.

Saltwater intrusion in aquifers: Development and testing of a three-dimensional finite element model

This study focuses on the determination of the Forchheimer equation coefficients a and b for non-Darcian flow in porous media. Original theoretical equations are evaluated and empirical relations are proposed based on an investigation of available data in the literature. The validity of these equations is checked using existing experimental data, and their accuracy versus existing approaches is studied. On the basis of this analysis, some insight into the physical background of the phenomenon is also provided. The dependence of the coefficients a and b on the Reynolds number is also detected, and potential future research areas, e.g. investigation of inertial effects for consolidated porous media, are pointed out. Copyright © 2006 John Wiley & Sons, Ltd.

Determination of Forchheimer equation coefficients a and b

Approximate analytical solutions of the Forchheimer equation

Two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied. They are the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into four schemes: moving mesh, pseudo-Lagrangian (FEM1); stationary mesh, pseudo-Lagrangian (FEM2); pseudo saturated-unsaturated, Eulerian (FEM3); and non-stationary element, Eulerian (FEM4).

Experiments on a one-dimensional flow case are performed to illustrate the schemes and to determine the effect of discretization on accuracy. In two-dimensional flow the above methods are compared with experimental results from a sand box model. Results indicate that for a similar degree of accuracy, the FEM requires less computational effort than the MOC. Among the four FEM schemes, FEM4 appears to be most attractive as it is the most efficient and most convenient to apply.

Numerical solutions for solute transport in unconfined aquifers

The hydrodynamics of a rubble-mound weir are theoretically and experimentally examined. This type of weir is considered to be environmentally friendly, since its permeability allows substances and aquatic life to pass through longitudinally. By performing a one-dimensional analysis on a steady nonuniform flow through the weir, discharge is described as a function of related parameters, such as flow depths on the up- and downstream sides of the weir, porosity, and grain diameter of the rubble mound, weir length, etc. A laboratory experiment is carried out to determine the empirical coefficients included in the analytical model. The theoretical solution of the discharge is compared with the experimental data to verify the analysis. It is confirmed that agreement between theory and experiment is satisfactory for a wide range of flow conditions. The present study makes it possible to apply the rubble mound weir for practical use as a discharge control system.

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Discharge through a Permeable Rubble Mound Weir

Numerical and experimental studies of non-Darcy flow in porous media are examined. Laboratory experiments using a screened gravel include radial flow to a simulated well using a sector and two-dimensional flow through a bank with vertical sides in a flume. Permeameter tests on the gravel were used to estimate coefficients in the nonlinear relation between head loss and velocity. Results are presented of analyses performed using a finite difference solution of the appropriate partial differential equation boundary value problem. The flow nets and discharges obtained are compared with the experimental results and the corresponding solutions for Darcy flow.

Solutions for unconfined non-darcy seepage

The weighted residual technique in conjunction with the finite element method is employed to formulate numerical models to solve flow and transport problems in a nonstationary saturated ground-water zone. Through the formulation presented, elements are allowed to move to accommodate the nonstationary saturated domain without the necessity for interpolation when proceeding from one time step to the next in the solution of the mass transport equation. The models can simulate the flow of water and the movement of a pollutant in the saturated zone of a homogeneous isotropic aquifer. The moisture influx from the unsaturated zone can exert some influence over mass distribution in the neighborhood of the free surface.

Raymond E. Volker

Papers

Solutions for unconfined non-darcy seepage

Simulating Mass Transport in Unconfined Aquifers