Darrell I. Leap

Modeling Three-Dimensional Groundwater Flows by the Body-Fitted Coordinate (BFC) Method: II. Free and Moving Boundary Problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.

Simulation of a free-surface and seepage face using boundary-fitted coordinate system method

Abstract The boundary-fitted coordinate (BFC) system method is applied to simulate steady groundwater seepage with a free-surface and seepage face using the finite-difference method. The BFC system method eliminates the difficulty of fitting finite-difference grids to a changeable free-surface which is not known a priori but will be obtained as part of a solution. Also, grid generation with this approach is simpler than with the finite-element method. At each iterative sweep, the changeable free-surface becomes a part of the boundary-fitted grid lines, making boundary condition implementation easy and accurate. An example problem demonstrating the simulation procedure and numerical results compares very well with the analytical solution.

Modeling three-dimensional groundwater flows by the body-fitted coordinate (BFC) method : I. Stationary boundary problems

Based on the body-fitted coordinate (BFC) method, a three-dimensional finite difference computer code, BFC3DGW, was developed to simulate groundwater flow problems. Methodology and solution procedures of the BFC method for simulating groundwater flows, particularly when the flow domain is stationary as in the case of confined aquifers, are described. The code was verified by comparing numerical results with analytical solutions for well-flow problems in an isosceles right-triangular aquifer. An example simulation is made to demonstrate capability of the code for solving flow problems in anisotropic aquifers where directions of anisotropy change continuously. The method differs from the conventional finite difference method (FDM) in the ability to use a flexible, nonorthogonal, and body-fitted grid. The main advantages of the method are the convenience of grid generation, the simplified implementation of boundary conditions, and the capability to construct a generalized computer code which can be consistently applied to problem domains of any shape.

Application of Boundary-Fitted Coordinate Transformations to groundwater flow modeling

The Boundary-Fitted Coordinate (BFC) Transformation method is a very powerful, efficient and accurate method of modeling heat or fluid flow in two- or three-dimensional domains with complex boundary shapes and abrupt changes in internal properties. Since the late 1970s it has become the modeling method of choice among many aerodynamicists and heat-flow modelers. It is being presented here for the first time as a new approach to modeling groundwater flow, based on successful research results in two dimensions. The BFC transformation method was employed to simulate two hypothetical well-flow scenarios in isotropic and anisotropic domains, and actual groundwater flows in the area of West Lafayette, Indiana. The numerical solutions in those cases were at least as accurate as and/or consistent with those obtained by purely finite difference and finite element methods, but with the added advantage of more accurate representation and implementation of the boundary condition in the region of great sensitivity. The BFC method successfully applied to two-dimensional simulations should be easily extended to simulations of three-dimensional flow and transport and thus, this research is continuing in that direction.

Modeling groundwater flow by the element-free Galerkin (EFG) method

The element-free Galerkin (EFG) method is one of meshless methods, which is a very powerful, efficient and accurate method of modeling problems of fluid or solid mechanics with complex boundary shapes and large changes in boundary conditions. This paper reports the theory and the first-known application of the EFG method to groundwater flow modeling. The EFG method constructs shape functions based on moving least square (MLS) approximations, which do not require any element but only a set of nodes. Thus, the EFG method eliminates time-consuming mesh generation procedure with irregular shaped boundaries. The coupled EFG-FEM technique was used to treat Dirichlet boundary conditions. A computer code EFGGW was developed for the problems of steady-state and transient groundwater flow in homogeneous or heterogeneous aquifers. Solutions by the EFG method were similar in accuracy to that by the FEM. The main advantages of the method are the convenience of node generation and the enforced implementation of boundary conditions.

Modeling groundwater flow with a free and moving boundary using the element-free Galerkin (EFG) method

This paper presents the results of the first-time application of the element-free Galerkin (EFG) method for simulations of groundwater flow with free and moving boundaries. The EFG method does not require any element or grid, therefore it eliminates time-consuming remeshing procedures in modeling of moving boundary problems. The EFG method employs time dependent shape functions based on the moving least square (MLS) approximations. A coupled EFG-FEM technique was used to treat Dirichlet boundary conditions that are difficult in the EFG method. The EFG method was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate applicability of the EFG method dealing with flow problems with moving boundaries, a transient free-surface and seepage problem in an unconfined aquifer was simulated. Difficulties associated with modeling a free and moving irregular boundary have been successfully overcome by employing the EFG method.