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Published in Soil Sci Soc Am J 23:91-96 (1959)
© 1959 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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A Unifying Numerical Solution for Two-Dimensional Steady Flow Problems in Porous Media With an Electrical Resistance Network1

Herman Bouwer and W. C. Little2

ABSTRACT

Numerical solutions of two-dimensional steady flow systems that are partially or completely unsaturated involve satisfying the tension vs. conductivity relations of the media in addition to some approximative expression of the Laplace equation for nonhomogeneous conductivity. Where tensions and conductivities are initially unknown, assumed conductivity values are used in the first relaxation procedure after which the correct values are determined by a process of repeated relaxation analyses with conductivity to tension adjustments after each relaxation. Relatively rapid solutions can be obtained by use of an electrical resistance network for relaxing the system.

A numerical procedure is developed for expressing the equivalent conductivity of the medium between any two adjacent points of the network in terms of the conductivities or tensions at the two points. This procedure is independent of the nature of the relation between tension and conductivity and utilizes experimentally determined values in the form of graphs or tables.

The equivalent conductivity between adjacent points of the network differs from the arithmetic mean of the conductivities at the points, especially when the conductivity values are relatively far apart. The arithmetic mean appears to be an accurate measure of the equivalent conductivity only in event of horizontal flow with a linear relation between tension and conductivity.

Examples of completed analyses include a solution of flow in tile-drained soil with the saturated and unsaturated zone united into one system and a solution of unsaturated subirrigation flow from perforated underground tubing in the presence of a deep water table. The shape of the water table in the drainage example was immaterial to the solution of the problem, but it could be evaluated after the analysis as the atmospheric-pressure contour. The streamlines above the water table deviated from the vertical direction causing the flow rate across the water table to be nonuniform.

NOTES

1 Paper presented before Div. I, Soil Science Society of America, Aug. 6, 1958, Lafayette, Ind.

2 Associate Agricultural Engineer, Alabama Polytechnic Institute and Agricultural Engineer, ARS, USDA, Auburn, Alabama, respectively.

Received for publication July 14, 1958. Accepted for publication October 7, 1958.






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