Area sinks in the analytic element method for transient groundwater flow

Abstract

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 as a well or the leakage through an aquitard. The analytic element method was originally developed for regional steady groundwater flow. In this paper the method is extended to transient flow. Several transient analytic elements and a method of determining a transient solution are presented. The governing differential equation that is used is the heat equation in two spatial dimensions with a sink term. Solutions for a transient well and a transient line sink are available in the literature. Both have a discharge that is equal to zero before the starting time and has a constant value after the starting time. A solution for a transient area 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 constant in space. A validation of the approach presented here is obtained by comparison with an exact solution for a case of one‐dimensional transient groundwater flow. 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 element method is illustrated by an example model with analytic elements both for steady and for transient flow. The former elements represent the initial steady state. The transient elements simulate variations of the groundwater flow due to seasonal variations in recharge and pumping.

Transition from transient Theis wells to steady Thiem wells

“…The discussion in this paper will be limited to Thiem and Theis wells. Many other solutions to the differential equations (see Strack, 1989;Zaadnoordijk & Strack, 1993) show a similar behaviour so that the conclusions can easily be extended.…”

“…This function is constructed as the combination of the transient uniform recharge and a transient area sink, which provides a uniform evaporation inside a rectangle (Zaadnoordijk & Strack, 1993).…”

“…This behaviour could be changed if the boundary condition at infinity is replaced by a boundary condition at a finite distance. This can be done by adding the potential of a so-called ring well <f> R (Zaadnoordijk, 1988):…”

“…This difference can be eliminated by adding a function to the potential (equation (15)) that changes the potential uniformly. Such a function is the transient uniform recharge for a limited period of time (see Zaadnoordijk, 1988):…”

Laplace-transform analytic-element method for transient porous-media flow

“…The corresponding space discretization offset the mesh-free benefit normally associated with AEM. Another early approach [15] combined steady and transient elements, using line and area sources to model transient effects; transient storage effects were assumed piecewise-constant in time and the method required zero net withdrawal of water from the aquifer. While not an AEM solution, Butler and Liu [16] developed a solution for transient flow to a well in the presence of a single circular inhomogeneity, using an approach similar to that taken here.…”