A Computational Model of Groundwater Mound Evolution Using the Complex Variable Boundary Element Method and Generalized Fourier Series
B.D. Wilkins1 N. Flowerday1 A. Kratch1 J. Greenberg1 B. Redmond1
T.V. Hromadka II2, AS-0020 K. Hood3 R. Boucher4 H.D. McInvale5
A. Baily1 Abstract Overview
An emerging issue in topics of computational engineering mathematics is the general use of groundwater computational models for solving problems in groundwater flow. Although computational groundwater models are useful for understand- ing and visualizing groundwater flow, computational errors can often result in significant design errors. In this work, an important problem in groundwater flow planning and design is considered, namely, the modeling of unsteady groundwater flow in a groundwater mound located below a groundwater recharge basin.
The actual problem setting is recast into two prototype problems that are suitable for computational assessment. The two test problems are mathematical formulations of unsteady groundwater flow and are designed to assess (1) the ability of computational groundwater models to develop descriptions of the potential surface (groundwater surface), and (2) the abil- ity to develop the associated streamline vector trajectories. Testing the accuracy of computational groundwater models may lead to increased confidence in computational results and may possibly facilitate identifying computational modeling issues before the modeling outcomes move forward towards design and planning actions.
In addition to providing and explaining the two proposed test problems, we also propose a numerical solution technique for this problem. The numerical scheme uses the standard pro- cedure of resolving the global initial-boundary value problem into a steady-state component and a transient component. The steady-state component is modeled by application of the Complex Variable Boundary Element Method (CVBEM),
1. 2. 3. 4. 5.
Cadet, United States Military Academy, West Point, New York
Professor of Mathematics, Department of Mathematical Sciences, United States Military Academy, West Point, New York Instructor, Department of Mathematical Sciences, United States Military Academy, West Point, New York Assistant Professor, Department of Mathematical Sciences, United States Military Academy, West Point, New York
Academy Professor and Associate Professor, Department of Mathematical Sciences, United States Military Academy, West Point, New York.
www.aipg.org
and the transient component is modeled by application of an approximation function that is a linear combination of basis functions that are the product of a two-dimensional Fourier sine series and an exponential function. The accuracy of this coupled procedure is proposed as a benchmark standard for comparison with other computational models.
Test Problem A: Potential Surface Modeling
The computation of a changing water table due to the cre- ation of a groundwater mound is important since such mounds can rise sufficiently such that the mound reaches the base of the recharge basin and interferes with the recharge process. Another possible complication (among others) is the mobiliza- tion of pollutants that are stored in subsurface soils, and the transport of these pollutants to other locations.
In Test Problem A, the steady-state flow situation is modeled as flow around a 90-degree bend. The initial condition is speci- fied as the superposition of the background flow (steady-state flow) and a two-dimensional single-peaked mound geometry. This geometry is picked for its simplicity, however, it is noted that a wide variety of geometric shapes can be specified. For model times t>0, the mound is continuously reducing in spatial coverage due to downward and lateral drainage of the stored groundwater in the mound. This flow regime is associated with difficult-to-solve spatial distributions for both the potential and streamline functions, and hence provides a possibly inter- esting case where the analysis must predict the dissipation of the flow regimes corresponding to the specified background flow superimposed with the groundwater mound.
The full length article is available on the AIPG National website http://
www.aipg.org/TPGPublic.
STUDENT ARTICLE
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