Optimisation technique for evaluation of aquifer parameters

Abstract

This thesis presents results of an extensive computational study on the evaluation of parameters in different types of aquifers based on the optimisation of an objective function using the least squares approach. The objective function is defined as the sum of the weighted squares of the difference between the observed and computed drawdown values. The aquifer systems considered for the analysis are:

Isotropic confined aquifer,

Isotropic semiconfined aquifer,

Anisotropic confined and semiconfined aquifers,

Confined and semiconfined aquifers with partially penetrating wells,

Confined and semiconfined aquifers in the presence of a barrier boundary,

Isotropic semiconfined aquifer with storage release from the aquitard, and

Unconfined aquifer.

Besides avoiding the subjectivity inherent in the traditional graphical method, the optimisation techniques enable the evaluation of aquifer parameters without any approximation, even in aquifer systems in which there are more than two independent dimensionless parameters in the drawdown solution.

The methods used for the solution of the optimisation problems are:

The sensitivity analysis technique,

The Marquardt algorithm, and

The Newton-Raphson technique.

In the sensitivity analysis technique, the expression for the drawdown is linearised by the Taylor series expansion in terms of the aquifer parameters. The objective function is minimised with respect to the perturbations in the parameters. The sensitivity analysis technique requires only the first derivatives of the objective function with respect to the parameters for the solution of the minimisation problem. The Marquardt algorithm is a modification of the sensitivity analysis technique where a convergence factor is added to the diagonal terms of the sensitivity matrix. In the Newton-Raphson technique, the second derivatives of the objective function for the solution of the optimisation problem are required.

A comparison of the three methods for isotropic confined and semiconfined aquifers shows that the sensitivity analysis technique has the advantages of simplicity and good convergence characteristics. In view of this, only the sensitivity analysis technique is used for all the aquifer systems referred to earlier.

The sensitivity analysis technique has been used successfully for all these aquifer systems, as demonstrated by applications to field and hypothetical problems. Convergence to the global minimum has been obtained with any type of initial assumptions, by imposing the following constraints on the relative changes in the parameters in each iteration:

0.2< P<0.5 0.2< P<0.5

where P P refers to the parameters in the systems considered. For example, in a semiconfined aquifer, P P refers to the parameters T T (the transmissivity), S S (the storage coefficient), and r/B r/B (the leakage parameter). These constraints prevent the parameters from assuming physically unrealistic values. Convergence is achieved even when the initial values of the parameters differ from the true values by four orders of magnitude.

Convergence to the global minimum is obtained without imposing constraints such as those stated above on the corrections to the parameters, by choosing the initial estimates in regions of the parametric space where the objective function exhibits a uniform convex nature. The iterative algorithm reaches the global minimum even when the initial estimates of the parameters differ from the true values by six orders of magnitude.

For the semiconfined aquifer with storage release from the aquitard and the unconfined aquifer, for which computational times are significant, initial assumptions of the parameters differing from the true values within one order of magnitude only were used.

The objective function exhibits a convex nature when the eigenvalues of the Hessian matrix are positive. The convex nature of the objective function is established through the computation of the eigenvalues for isotropic confined and semiconfined aquifers, with the wells fully and partially penetrating the aquifer, as well as for a confined aquifer in the presence of a barrier boundary. For a confined aquifer, a uniform convex nature of the objective function is found in the T T- S S plane in the quadrant in which T T and S S are underestimated. For a semiconfined aquifer, the octant in which the parameters T T, S S, and r/B r/B are underestimated, the response surface exhibits a convex nature. The eigenvalues are found to be uniformly positive for the case of a confined aquifer with partially penetrating wells in the region where the parameters T T, S S, and r/B r/B are underestimated. For the case of a confined aquifer with a barrier boundary, such a feature is observed in the regions where the parameters T T and S S are underestimated and the parameter Rj2 R j 2

(square of the distance between the observation well and the image well) is over- or underestimated.

The presence of local minima is observed in the case of isotropic confined and semiconfined aquifers, anisotropic confined aquifer, confined aquifer with partially penetrating wells, and in the presence of a barrier boundary, in regions of the parametric space where some of the parameters are overestimated and others underestimated. In these regions, the objective function may improve in the wrong direction in the iterative process.

For the evaluation of parameters in a semiconfined aquifer with storage release from the aquitard and an unconfined aquifer, the Neuman and Witherspoon and Neuman solutions, respectively, are used in view of their validity for all time intervals. The computational details for the evaluation of drawdown for the former case are presented in view of the complex nature of the solution. The convergence characteristics of the sensitivity analysis technique are good even for these cases. But the computational time required for each iteration is significant. In view of this, it may be desirable to provide fair initial assumptions based on an approximate graphical analysis or a knowledge of the local hydrogeology.