Background

Groundwater is an important source of water supply for drinking, agriculture, and industry. It represents 98% of freshwater readily available to humans [Schwartz and Zhang, 2002]. Groundwater is found in aquifers, which have the capability of both storing and transmitting groundwater. Recently, the groundwater problems, such as industrial wastewater injected into groundwater system and seawater intrusion, have been attracted public attention. The hydraulic properties of the aquifer systems have to be determined prior to characterizing or investigating the pollutant and its plume. Groundwater hydrologists often conduct aquifer tests to determine the in situ hydraulic properties of the soil formation, such as hydraulic conductivity and storage coefficient. These parameters are necessary information for quantitative and/or qualitative groundwater studies.

The pumping test is a very reliable method for estimating aquifer parameters. Figures 1 (a) and (b) display the sketch of the pumping tests in leaky and unconfined aquifers, respectively. Typically, a pumping test consists of a pumping well and one or more observation wells. The observation wells are located at varying distances from the pumping

pumping. The term drawdown (s in Figures 1 (a) and (b)) shows the change in water levels through the test. The drawdown curve which describes a conical shape is cone of depression.

In the past, the pumping test data was usually analyzed using a graphical procedure with type curves to estimate the aquifer parameters. In addition, the parameters can also be obtained by computer methods, which usually estimate parameters using the least-square approach by taking the derivative of the sum of square errors between the observed and estimated drawdowns with respect to the parameters. The gradient-type methods are then utilized to solve the nonlinear least-square equations to determine the best-fit parameters.

However, two disadvantages might be occurred when the gradient-type methods were used to solve the nonlinear least-square equations to obtain the parameters. First, those methods may yield divergent results if the initial guesses of parameters are not close enough to the target parameter values. Second, those methods may give poor results if improper increments were made when applying finite difference formula to approximate the derivative terms appeared in the least-square equations.

Recently, the computer-based parameter estimation models (PEM) were developed promptly. The models of aquifer parameter estimation usually combine a suitable solution for describing the pumping test with an optimization approach such as simulated annealing (SA) or a recursive approach, such as extended Kalman filter (EKF). Some commercial

softwares, like AQTESOLV [Duffield, 2002], use nonlinear weighted least-squares approach to fit the time-displacement data obtained from an aquifer test to the type curve.

A pumping test was usually required to perform for a long period of time if a graphical approach is chosen to analyze the measurement data. Otherwise, the estimated result may not be in good accuracy if the data is too short and the data points are too sparse to give a good visual fit to the type curve. However, such a test would spend a lot of time, money, and groundwater resources. These problems were aggravated when analyzing the data from the leaky and unconfined aquifers.

In a leaky aquifer, the semi-pervious bed (also shown as the aquitard in Figure 1 (a)), although of very low permeability, may yield significant amounts of water to the adjacent pumped aquifer. As time increased, leakage across the semi-pervious bed may become appreciable and flow is not restricted to the pumped aquifer alone. The additional water may be derived from storage of the aquitard and adjacent unpumped aquifers. During the pumping, the water is immediately withdrawn from the aquifer and then the head difference between two aquifers induces a flow across the aquitard. Therefore, the parameters of the confining bed (aquitard) may not be accurately estimated if only first few drawdown data points are used. Two approaches have been developed for dealing with leaky aquifers, one considers the aquitard storage while the other does not consider.

Physically, the drawdown in an unconfined aquifer can be divided into three segments

[Charbeneau, 2000]. In the first stage, water is immediately released from storage due to the compaction of the aquifer and the expansion of the water. In the second stage, the vertical gradient near the water table causes drainage of the porous matrix. The vertical hydraulic conductivity Kz begins to contribute to the pumping and the rate of descent in the hydraulic head slows or stops after a period of time. Finally, the flow is horizontal and most of the pumping is supplied by the specific yield, Sy. Therefore, the analysis of Sy requires sufficient long drawdown data fallen at the third section. In some cases, the effect of well bore storage needs to be considered since the diameter of pumping well is large. The water is withdrawn first from the casing at the beginning of pumping. Then the groundwater flow into the well because the head difference between the well and the adjacent formation.

Most physical systems can be viewed as input-output models that relate the output information to the proper input parameters. Unfortunately, the input parameters can not be known perfectly in the real world. Hence, the basic concept of sensitivity analysis is to investigate how the errors in the input parameters influence the outputs, or, in particular, to study if a small perturbation in the input parameters causes a large change in the output.

Now the sensitivity analysis is being wildly applied in all sciences.