An integral equation for immiscible fluid isplacement in a two-dimensional porous medium or Hele-Shaw cell

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Abstract
An integral equation for the normal velocity of the interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell (one fluid displaces the other) is derived in terms of the physical parameters (including interfacial tension), a Green's function and the given interface. When the displacement is unstable, ‘fingering’ of the interface occurs. The Saffman-Taylor interface solutions for the steady advance of a single parallel-sided finger in the absence of interfacial tension are seen to satisfy the integral equation, and the error incurred in that equation by the corresponding Pitts approximating profile, when interfacial tension is included, is shown. In addition, the numerical solution of the integral equation is illustrated for a sinusoidal and a semicircular interface and, in each case, the amplitude behaviour inferred from the velocity distribution is consistent with conclusions based on the stability of an initially flat interface.

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