The problem of the encroachment of water into an oil sand is formulated as a new type of potential problem, namely, that of finding potential distributions in two regions of different ``constants'' (``conductivities'') which are separated by a surface, each point of which has a velocity proportional to the vector gradient of the potential at the point, and such that the area swept out by the moving interface assumes the ``constant'' appropriate to that of the encroaching side of the interface. The cases of strictly linear, radial and spherical systems, in which the shapes of the interfaces are evident from symmetry requirements, are solved in detail and discussed graphically. The zeroth approximation to the general problem which gives the history of a line of fluid particles in a homogeneous system is also treated in detail. Analytical and graphical solutions are presented for (1) systems with elliptical boundaries, (2) an infinite linear source driving fluid into an isolated sink, and (3) the history of a ring of fluid particles travelling from a source to a sink. The relation of the analytical results to the practical problems of the encroachment of water into oil bearing sands is discussed both for the solutions of the general problem and those of the zeroth approximation.
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