Groundwater flow theory for perforations in well casings and soil drain tubes
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A theoretical solution and numerical results of the effect of size and disposition of perforations in a well casing have been obtained. The solution is for steady state flow in a confined horizontal aquifer from an external boundary of constant potential head to a series of equally spaced rectangular perforations (three-dimensional space) at equal constant potential on a well surface. The solution includes flow into continuous vertical slots (two-dimensional) and into ring openings (three-dimensional space, two coordinates). For simplicity, it was assumed that Darcy's law is valid and the flow medium is homogeneous and isotropic. The exact analytical solution of the potential flow problem that satisfies Laplace's equation in polar coordinates and the mixed boundary conditions was developed by converting the Kirkham and Powers (1972) modified Gram-Schmidt process for two-dimensional space to three-dimensional space. This method of solution is very lengthy because of the three-dimensional nature of the problem. A simpler and sufficiently accurate solution was developed for homogeneous boundary conditions by use of Fourier series. The numerical result using the approximate method was found to give good results in comparison with the exact mixed boundary solution for certain parameters. The results of the approximate solution also showed that the flow into rectangular perforations (three-dimensional problem) can be approximated, using the flow analysis into continuous vertical slots, for a number of perforation geometries used in practice, provided that the percentage of open to unopen space of comparison is the same;The solution of the perforation problem includes an exact analytical solution, with numerical results, of the problem of continuous vertical well casing slots. The numerical results of this solution are tabulated for a wide range of parameters. These results showed that the flow into perforations can be found with a great degree of accuracy at locations less than 1 cm distance from the well surface and thus in the region where the largest hydraulic gradients occur;The basic theory and the numerical results also apply to flow into perforations in drain tubes used in land underdrainage. The numerical results in this study are presented as dimensionless parameters so that the flow for any particular system or units can readily be found.