Analysis and parameter estimation models for on aquifer-water table aquitard system

Abstract

The present study is concerned with developing analysis and parameterestimation models for groundwater flow to a well in a twolayer leaky system comprising an aquifer-watertable aquitard. Previous studies analyzing such a system did not consider the compressibility of the aquitard, and the boundary condition at the water table was treated approximately, restricting their use to small pumping times. Also, no parameterestimation studies have been reported for such a system. In this study, the direct problem is solved using a finitedifference model, and the inverse problem is solved using the weighted leastsquares method. The model, which generalizes the leakyaquifer concept, is also applicable in hardrock regions, as the system closely resembles the doubleporosity models developed specifically for fracturedrock aquifers. The aquifer-watertable aquitard system consists of a watertable aquitard overlying an aquifer. The aquifer is anisotropic, with storage coefficient S and transmissivities T and T along the principal Cartesian directions x and y. The aquitard has a vertical hydraulic conductivity K, specific storage S, and specific yield S. Flow in the aquifer is lateral, while flow in the aquitard is vertical. The aquifer and aquitard equations form a coupled system of partial differential equations due to the presence of the leakage term in the aquifer equation and the continuity requirements at the aquifer-aquitard interface. The model is quasithreedimensional, with drawdown in the aquifer being a function of (r, t), while the aquitard drawdown is a function of (r, z, t), where r is the radial distance in the equivalent isotropic domain. The analysis accounts for compressibility and gravity drainage in the aquitard, and treats the water table as an unknown boundary. The coupled aquifer-aquitard equations are solved numerically by an iterative procedure in which the watertable elevation and flowtransfer term are updated at each iteration. The system is governed by the nondimensional parameters P (= Q / T h), P (= T / T), A (= T / S), and P (= S / S), where Q is the pumping discharge, h is the prepumping head, T is the equivalent transmissivity, and S is the aquitard storage coefficient. Type curves for aquifer and watertable drawdowns were generated over a wide range of parameter values. These curves reveal the effects of progressively increasing leakage, followed at longer times by declining watertable elevation and associated reduction in leakage. At sufficiently long times, watertable drawdown approaches aquifer drawdown. The effects of parameters T, P and P on aquifer and watertable drawdowns, vertical distribution of aquitard drawdown, and vertical gradients at the aquifer-aquitard interface are examined. The type curves are used to illustrate the model’s application to a 7day field pumping test for estimating parameters by graphical matching. The analysis model is coupled with an optimization algorithm for parameter estimation. A weighted leastsquares approach is used, and the optimization problem is solved by the sensitivityanalysis technique. Such a simulation-optimization algorithm is important because graphical matching is highly subjective and extremely difficult given the number of parameters involved. Although the analysis assumes an equivalent isotropic domain, the simulation-optimization algorithm can estimate anisotropic transmissivities by treating the direction of principal axes () as an unknown. The parameters estimated are T, T, S, A / h, S, and . Sensitivity coefficients are obtained using a modified parameterperturbation technique based on the nondimensionalized governing equations. Computational workload analysis indicates that this method performs comparably to the adjointstate method for this problem. A suitable choice of weights for the weighted leastsquares objective function is proposed. The inverse model is applied to six test problems, including three field pumping tests with observations from multiple wells over pumping periods of 3-7 days. For two field problems, the principalaxis directions are assumed known; for the third, this parameter is estimated. Constraints on parameter corrections are found necessary to ensure convergence to the global minimum for any initial guess in parameter space. A firstorder approximation of the parametercovariance matrix-reflecting the uncertainty in parameter estimates-is applied to a test problem with: (i) errorfree data and correct model, (ii) noisy data and correct model, (iii) errorfree data and incorrect model, and (iv) noisy data and incorrect model. The method is also applied to field data. The uncertainty estimates provide a useful qualitative measure of the effect of data noise and, to a lesser extent, the relative merits of alternative models. The study is extended to flow near a dugcumbore well in an anisotropic aquifer-aquitard system, accounting for well storage and well loss. The dug well partially penetrates the watertable aquitard. This introduces an additional parameter for well loss (S). The parameters estimated include T, S, K / h, S, S, D, , and S. The analysis treats the wellwater level as an additional unknown boundary, updated each iteration using mass balance and wellloss equations. Variations in relative contributions of aquifer and well storage with time are illustrated for cases with and without well loss. The parameterestimation procedure is applied to two test problems involving aquifer and watertable drawdowns over 7day pumping periods.