A technique in using integrodifferential equations for model simulation of multiaquifer systems

Abstract

A numerical technique for modeling a multiaquifer system is developed on the basis of a quasi threedimensional flow model. The model consists of many horizontal aquifers interconnected by leakage flux from aquitards. The direct flow across aquitards and yield from aquitard storage are taken into account in leakage flux terms in the form of convolution integrals. Mathematical formulation results in flow functions which diminish with time and can be approximated by few exponential terms. They have been proved to be very convenient in a numerical solution scheme involving the convolution integrals. The approach reduces the dimensionality of the problem and uncouples the equations corresponding to each of the aquifers. The technique is flexible and can be used for any basin geometry with an arbitrary pumping scheme. The validity of the technique is demonstrated by comparing the model results with available analytical solutions. In general, it appears that this technique is more economical than the full three-dimensional model or even the quasi three-dimensional models that use one-dimensional elements to represent the aquitards. Figueroa V. [1969]. Herrera and his co-workers [Herrera, 1970; 1974; Herrera and Rodarte, 1973; Herrera and Yates, 1977] formulated the problem of a multiaquifer system in the form of integrodifferential equations in which the influences of adjacent aquitards and aquifers are expressed by means of 'memory' and 'influence' functions. In the latest paper, Herrera and Yates [1977] suggested that the associated functions can be approximated by a few terms of the exponential series and then they can be readily employed in a step-by-step finite element model. The 'memory function' g vanishes at large values of time, and it can be approximated roughly by a truncated exponential series and a residual in the form of Dirac's delta function. Herrera and Yates mentioned that there are difficulties in finding a satisfactory approximating function for the 'influence function' h, which is zero at the initial val. ue of time and is unity at large values of time. The difficulties remain unsolved. In their work, rigorous integrodifferential equations were elegently derived, and the formulation was presented with clarity. However, practical approximating functions were not given for model application, the formulation has not been verified with analytical formulae (derived by Neuman and Witherspoon [1969]), and it is not yet employed in practical field situations.In this paper the method used to formulate the multiaquifer model that was developed in connection with the Bangkok land subsidence study [Premchitt, , 1979 is described. An integrodifferential equation is used in the formulation of a -PREMCHITT: MULTIAQUIFER SYSTEMS 163 s s 1 (t•) ,(.'••.• s ' (t')

A Block Iterative Finite Element Model for Nonlinear Leaky Aquifer Systems

“…Two theoretical approaches have been developed to build computationally efficient quasi three-dimensional models of groundwater flow in multiaquifer systems. The first approach assumes linearity, that is, hydrogeological parameters independent of the head distribution, and transforms the set of partial differential equations into a set of integrodifferential equations [Herrera and Yates, 1977;de Marsily et al, 1978;Premchitt, 1981;Gambolati et al, 1986]. The second approach relies on the full numerical integration of the original equations which therefore need not be linear [Fujinawa, 1977;Chorley and Frind, 1978;Neuman et al, 1982;Rivera, 1990].…”

Recent advances in modeling of well hydraulics

“…Numerical solutions for flow in the leaky and multilayered aquifer systems have been accomplished by finite element methods [284,285], finite difference methods [285,286], and boundary element methods [269].…”

Green element simulations of multiaquifer flows with a time-dependent Green's function

“…They have either been applied to the coupled aquifer -aquitard equations along the lines of the theory of Neuman and Witherspoon (1969a) or to the integro-differential equations of Herrera (1970). Making use of the former theory are the FE model of Chorley and Frind (1978), the BE model of Zakikhani and Aral (1989) and the GE model of Taigbenu and Onyejekwe (2000), while the latter theory has been incorporated into the FE model of Herrera and Yates (1977), the FD models of Premchitt (1981); Cheng and Ou (1989), and the BE models of Cheng and Morohunfola (1993). The AE method has been largely limited in application to steady groundwater flows (Strack, 1999).…”