Derivation of finite-element equations and comparisons with analytical solutions

MODFE, a modular finite-element model for simulating steadyor unsteady-state, area1 or axisymmetric flow of ground water in a heterogeneous anisotropic aquifer is documented in a three-part series of reports. In this report, part 2, the finite-element equations are derived by minimizing a functional of the difference between the true and approximate hydraulic head, which produces equations that are equivalent to those obtained by either classical variational or Galerkin techniques. Spatial finite elements are triangular with linear basis functions, and temporal finite elements are one dimensional with linear basis functions. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining units; (3) specified recharge or discharge at points, along lines, or areally; (4) flow across specified-flow, specified-head, or head-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) headdependent fluxes from springs, drainage wells, leakage across riverbeds or confining units combined with aquifer dewatering, and evapotranspiration. The matrix equations produced by the finite-element method are solved by the direct symmetric-Doolittle method or the iterative modified incomplete-Cholesky conjugate-gradient method. The direct method can be efficient for smallto medium-sized problems (less than about 500 nodes), and the iterative method is generally more efficient for larger-sized problems. Comparison of finite-element solutions with analytical solutions for five example problems demonstrates that the finite-element model can yield accurate solutions to ground-water flow problems. INTRODUCTION This report is the second part of a three-part series of reports (parts 1 and 3 are by Torak, 1992a and 1992b) that document the computer program MODFE (modular finite-element model), which simulates steadyor unsteadystate, area1 or axisymmetric flow of ground water in a heterogeneous, anisotropic aquifer. The model incorporates a variety of physical processes necessary to simulate ground-water flow in the complicated settings that often characterize actual field problems. Flow may be confined, unconfined (using the Dupuit assumption), or a combination of both; known recharge and discharge may be distributed areally, along lines such as specified-flow boundaries, or at point sources and sinks such as pumping wells; and headdependent leakage may be distributed areally, such as through confining units or wide riverbeds, or along lines such as narrow riverbeds. Confining units may be rigid or may have elastic storage capacity. Special nonlinear, head-dependent source and sink functions allow simulation of springs, drainage wells, rivers or confining units combined with aquifer dewatering, and evapotranspiration. 1 The material in the three reports has evolved over the past 10 years from material presented by the authors in the courses entitled "FiniteElement Modeling of Ground-Water Flow" held at the U.S. Geological Survey National Training Center in Denver, Colorado. These reports formalize the course material and incorporate valuable suggestions and comments from attendees of the courses. Features that appear to be new, at least to published finite-element programs for ground-water flow, include (1) the method of deriving the finite-element equations from a functional of the difference between the true and approximate solutions, (2) the method of approximating the variability of transmissivity over an element so that the coefficient matrix does not have to be reassembled element by element each time the saturated thickness changes, (3) the method of treating decreases of aquifer thickness to zero under conditions of extreme water-table decline and increases of aquifer thickness from zero as the water-table rises, (4) the finite-element in time method for deriving (a) the finite-element equations for unconfined flow and (b) the functions for nonlinear, head-dependent sources and sinks, and (5) the method for incorporating transient leakage from confining units.