WATER RESOURCES RESEARCH, VOL. 29, NO. 12, PAGES 4121--4129, DECEMBER 1993 Area Sinks in the Analytic Element Method for Transient Groundwater Flow WILLEM J. ZAADNOORDIJK Regional Office West, IWACO, Consultants for Waterand Envitvnment, Rotterdam.Netherlands OTTO D. L. 8TRACK Department of CMI and Mineral Engineering, University qoe Minnesota, Minneapolis, Minnesota In the analytic element method,regional groundwater flow is modeled by superposition of particular solutions to the governing differential equation. The domain of the solutions is the x, y plane with the possible exception of isolated points. The solutions are referred to as analytic elements and represent a feature of flow in the aquifer, such asa well or the leakage through an aquitard. The analytic element methodwas originally developed for regional steady groundwater flow. In this paper the methodis extended to transient flow. Several transient analytic elements anda method of determining a transient solutionare presented. The governing differential equation that is usedis the heat equation in two spatial dimensionswith a sink term. Solutions for a transientwell and a transient line sink are available in the literature. Both have a discharge that is equalto zero beforethe starting time and hasa constant value after the startingtime. A solution for a transientarea sink is presented that also has a constant strength after the starting time. The area sink is a polygon with an extraction inside that is constantin space.A validation of the approach presented here is obtainedby comparison with an exact solution for a case of one-dimensional transientgroundwaterflow. The domain is semi-infinite with a prescribed head at one side. Initially, the water is at rest, and then the head is suddenly raised at the boundary. The use of the transient analytic elementmethod is illustratedby an example model with analytic elementsboth for steadyand for transient flow. The former elementsrepresentthe initial steady state. The transient elements simulate variations of the groundwater flow due to seasonal variations in recharge and pumping. INTRODUCTION This paper concernsthe modeling of transient regional groundwater flow usingthe analytic element method. Strack initially developed the method for steady regionalground- water flow [Strack and Haitjema 1981a, b]. A concise introduction is given by Strack [1987].Strack [ 1989] presents the method in great detail. The analyticelement method is based on the superposition of particular solutions to the governing differential equation. The particular solutions, valid in an infinite domain,repre- sent a variety of aquifer features such as a well. Two of these are of specialinterest in this paper:the well and the area sink, used to simulate infiltration over an area, or leakage from one aquifer to anotherthroughan aquitard. Haitjema and $track [1985] modeled transient groundwa- ter flow by means of steady analytic elements. Changes in the aquifer storage were represented by area sinks. The strengths of the areasinks are determined for each time step; the resulting solution is notcontinuous in time.Strack [1989] mentions that it is possible to addtransient wells to a steady state analytic element model. Thisidea is worked outhereto extendthe analytic element methodto transient regional flow in which the transient responses of a groundwater system can be calculated.The result is a solution that is continuous bothin space and time, except at isolated points. Thegoverning differential equation is theheat equation in two spatialdimensions with a sink term. The following simplifying assumptions are used [Bear and Verru•jt, 1987]: (1) the properties of the groundwater and the matrix of the aquifer are constant;(2) Darcy's law is valid and the hydrau- lic conductivityk is isotropic and constant; (3) the base of the aquifer is horizontal and the saturated thickness H is constant; t,4) the flow is shallow so that the Dupuit- Forchheimerassumption is adopted: and (5) the storage and release of water in the aquifer is linear, occurs instanta- neously and is fully reversible, so that the amount storedper unit area per unit time is equal to SO•/Ot, where S is the storativity and • the piezometric head. The storativity S is constant. The continuity equation can be written as oQx OQy oqo •+ + E + S • = 0 (1) Ox Oy Ot where x and y are horizontal Cartesian coordinates, the extraction E is a sinkterm, and Qx and Qy are the compo- nents of the discharge vector. A discharge potential ß is used as dependent variable [Strack, 1989] with Qx = dp Qy= ß c•y (2) Copyright 1993 by the American Geophysical Union. Paper number 93WR02278. 0043-1397/93/93WR-02278505.00 where dp= kH• •3) 4121