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Abstract
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Since a nonlinear partial differential equation was developed by Boussinesq based on Darcy’s law and the Dupuit-Forchheimer assumption, it has played an essential role in the development of simulation models for the solution of various flow problems in porous media. To solve Boussinesq’s equation, both analytical and numerical solutions, such as finite difference (FD) or finite element (FE), have been sought. The differential quadrature method (DQM) is a new numerical method frequently used by researchers to solve partial differential equations. In this paper, DQMs have been coupled with explicit, implicit, and Crank-Nicholson FD and applied to solve Boussinesq’s equation for a drainage problem. This case has been solved previously both analytically and numerically, including by DQM, by other researchers. Those who had employed DQM had linearized Boussinesq’s equation first and solved the linearized form implicitly. In this work, the nonlinear form of Boussinesq’s equation has been solved using three models based on DQM for solving the dimensionless form and on one approach for solving the dimensional form of Boussinesq’s equation. The obtained results and their degree of accuracy are compared with available experimental and numerical data found in the literature, and on that basis, it is concluded that DQM generates accurate results, is very easy to formulate and operate, does not need large mesh size, and is very time efficient.
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