Low-order finite elements for parameter identification in groundwater flow

Improved evaluation and characterization of the geologic subsurface are necessary precursors to enhancing our ability to recover desirable resources and guard against undesirable contaminants. Mathematical models are increasingly used to assess the effects of these activities when the fidelity of material parameters describing the system control the reliability of the resulting prediction. Inverse models provide a formal means of space, or history, matching observed data to determine the unknown spatial distribution of the required parameters. The finite element method is one of the most popular numerical methods and is used in this study for transmissivity identification in steady groundwater flow. To optimize parameter identification, a comparison of different elements is carried out. It is well known that high-order elements usually result in improved accuracy in forward solution of engineering problems. However, this fact might not be true in the inverse solution. The study in this paper uses different element orders for Taylor's series analysis and in coding a finite element program. The analysis indicates that strong variation of transmissivity results in a larger residual error for high-order elements than for low-order elements. An example with an analytical solution is used for numerical comparison. The computed results show that low-order elements, rather than high-order elements, yield better results in parameter estimation. When the variation of unknown parameters is large, the error within the high-order elements is large. It is recommended that low-order finite elements or constant elements be used for an inverse solution. The comparison also illustrates that any minimization procedure may be used to minimize the residual error and to limit numerical difficulties in inverse solution. A two-dimensional example shows that the relative errors are very small when a constant element is used but that errors increase as the measured head distribution flattens.