MODEL OF GROUNDWATER MOUND EVOLUTION
Test Problem B: Streamline Development
Another situation often encountered in groundwater flow analysis is the identification of sources of groundwater con- tamination. Computational models are typically developed and applied to the flow situation to estimate flow streamlines from which possible locations for sources of groundwater contamina- tion are identified. The analysis procedure presented in Test Problem B uses the flow field streamline function trajectories to work upstream along streamlines toward possible locations for contaminant sources. Here, the initial condition is specified as the superposition of the background flow (which is planar in this case) and a two-dimensional single-peaked mound. It is assumed that a groundwater recharge mound is present, however the steady-state flow situation is a plane rather than fluid flow around a 90-degree bend.
Assumptions
The problem domain is assumed to be homogeneous and isotropic, thus reducing the need for parameter specification in the problem formulation. An initial condition for the ground- water surface is defined that is approximately the situation encountered, and includes a groundwater mound that is drain- ing into the groundwater regime located below the mound. The flow situation involves unsteady flow of the groundwater mound resolving itself towards the steady-state conditions of the problem. Therefore, the conceptual problem can be decomposed into two components: (1) an unsteady flow com- ponent of the groundwater mound reducing in vertical extent over time, with flow moving into the underlying groundwater regime modeled by a two-dimensional Fourier sine series, and (2) a steady-state component representing the considered test situation after the groundwater mound has fully drained into the groundwater regime, which is modeled by the Complex Variable Boundary Element Method procedure using complex variable monomials as basis functions (see Wilkins et al, 2016).
Evaluation Procedure for Common Numerical Methods
Typical groundwater computational models involve thou- sands, or even more, of finite elements or finite difference grid nodes. Here, the procedure used to test the veracity of such large models is to apply test problems designed to represent realistic and important problems in groundwater flow model- ing. Generally, a portion of the target computational model is isolated and the test situation is applied and examined. It is assumed that if the computational model performs adequately for the test situations, that the computational model will per- form similarly well for the actual problems at hand.
A Numerical Scheme Using the CVBEM and Two-Dimensional Fourier Sine Series
The numerical scheme proposed in this work involves decomposing the global initial-boundary value problem into two sub-problems, namely, a steady-state component and a transient component. The steady-state component is gov- erned by the Laplace partial differential equation (PDE) and the transient component is governed by the diffusion partial differential equation. The steady-state component is solved by application of the CVBEM and the transient component
is solved by application of a linear combination of basis func- tions that are the product of a two-dimensional Fourier sine series and an exponential function. It can be shown that the global solution, which is the sum of the modeling outcomes from the steady-state and transient sub-problems, satisfies the governing PDE.
The boundary conditions of the global BVP are satisfied by the CVBEM approximation of the steady-state solution. Thus, the unsteady component is specified with homogeneous bound- ary conditions. The initial condition of the transient problem is specified as the difference between the global initial condition and the CVBEM approximation of the steady-state situation. Consequently, when the two outcomes are summed, the global initial condition is appropriately modeled.
Numerical Test Modeling Results
The spatial domain is identical for both test situations considered in this work. For the planar steady-state of Test Problem B, the specified groundwater mound is superim- posed upon a steady-state condition of a geometric plane. Such a geometry may be appropriate for many problems even outside of the testing situation because within a relatively small distance of the mound, the groundwater regime may be significantly linear. For Test Problem A, the background groundwater flow is assumed to be similar to a flow field in a 90-degree bend. This more difficult flow regime is solved by the complex variable quadratic polynomial function and the poten- tial is readily determined to be the real part, or x^2-y^2 in the first quadrant of the two-dimensional coordinate axis plane.
Conclusion
This work shows that test problems can be applied to large- scale groundwater models to check the modeling veracity of the groundwater model utilized. Such tests are a necessary but not sufficient condition in order to develop an accurate model of the problem being assessed.
References
Asano, T., Artificial Recharge of Groundwater. Butterworth Publisher, 1985.
Hromadka II, T.V., Whitley, R., Foundations of the Complex Variable Boundary Element Method. Springer International Publishing, 2014.
Wilkins, B.D., Kratch, A., Flowerday, N., Hromadka II, T.V., Whitley, R., Johnson, A., Outing, D., McInvale, D., Horton, S., Phillips, M., Assessment of Complex Variable Basis Functions in Approximation of Ideal Fluid Flow Problems. Submitted and currently under review with the Journal, Engineering Analysis with Boundary Elements, 2016.
LTC Randy Boucher is an Assistant Professor and Senior Analyst at the United States Military Academy at West Point where he serves as the Program Director for the Advanced Mathematics Program in the Department of Mathematical Sciences. He holds an M.S. from the University of Washington and a Ph.D. from the Naval Postgraduate School. His research involves optimal control, computational mathematics, and dynamical systems.
Karoline M. Hood is a Captain in the Army and an Instructor in the Department of Mathematical Sciences at the United States Military Academy. She has served in opera-
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