The physical problem dealt with in this article is the unsteady ground-water flow in an aquifer of limited horizontal extent and arbitrary boundary shape.

The unsteady ground-water level is described by a differential equation, the solution of which includes the solution of an eigen-value problem by a numerical method.

Boundary conditions are provided by nature in the form of watertight rocks, rivers, lakes or any kind of constant water level in hydraulic contact with the aquifer.

The run-off is found to behave as a sum of flows through infinitely many linear reservoirs, corresponding to the eigen-functions.

The resulting equations for ground-water level and run-off discharge are sums of convolution integrals of the infiltration, an easy process to handle in a digital computer.

Finally, the mathematical model derived is used to analyse the inflow into a water reservoir in Iceland from a nearby lava field, and the runoff is compared with flow data.

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