Modeling Three-Dimensional Groundwater Flow and Solute Transport by the Finite Element Method With Parameter Estimation.

Abstract

A general, systematic and improved numerical formulation for the three-dimensional (3-D) groundwater flow and solute transport problems was derived, using the finite element method. In the numerical formulation, a hidden mistake, which has appeared in many textbooks and journal papers dealing with the advective term of the general 3-D solute transport equation, was found and corrected. Some simpler and practical expressions for the leaky boundary condition, surface flux condition, and sources and sinks were proposed. To improve the conventional formulation for the time derivatives, a combination of the Galerkin method and the collocation method was developed. A more accurate scheme was derived to solve the resulting system of ordinary differential equations using the finite integration. Based on the numerical formulations, three computer models were developed for modeling (1) 3-D steady groundwater flow, (2) 3-D unsteady groundwater flow, and (3) 3-D solute transport. These models are rather general in terms of initial conditions, boundary conditions, and fluid and aquifer properties. The three models were tested using a variety of one-dimensional and two-dimensional analytical solutions and numerical models. Computed results showed that all these models are relatively simple, stable and accurate, and have little chance of experiencing any numerical problems. Component and parameter sensitivity analyses were also made. In general, these models are relatively insensitive to time step size, and the new solute transport model yields reasonable results even if the Peclet number reaches 50. In order to estimate model parameters, a fast, reliable, and derivative-free subroutine was developed by combining the quadratic interpolation search, the Golden section search, and the side-search algorithms, for finding the minimum of any user-defined 1-D function. With this 1-D subroutine, a general conjugate gradient search program was then developed to find the optimal set of parameters for any type of model. The program was tested using a variety of analytical functions and found to be accurate, fast and efficient. The program was applied to estimate parameters of the 3-D solute transport model, using two types of sampling methods and four types of data sets.