The Shape of a Gravity Finger in a Rectangular Channel in Homogeneous Porous Media
Using conformal mapping techniques, we derive analytical expressions for the shape of a propagating finger in a rectangular channel in homogeneous porous media, in the absence of interfacial tension, but in the presence of gravity, acting in a direction transverse to the direction of displacement. The gravity finger propagates either along the top or the bottom boundaries of the channel, depending on the density contrast between displacing and displaced fluids. Thus, the model describes the respective cases of gravity override or gravity underrunning, which occur when a lighter fluid phase displaces a heavier one and vice-versa. The solution is expressed in terms of the finger thickness, which is a free parameter in this model. When gravity is neglected, the solution reduces to the classical solution of the Saffman–Taylor finger. Numerical illustrations are provided to examine the sensitivity of the finger geometry to the various parameters, following the scaling theory of Brener et al. (1991).