A Simple Analytical Model of the Growth of Viscous Fingers in Heterogeneous Porous Media

Fingering of unstable immiscible displacements in moderately heteroge neous reservoirs has been modelled analytically. The displacements are assumed to take place in the horizontal plane. Stabilising mechanisms such as capillary smear out or lateral dispersion are not incorporated. An accurate expression for the viscous growth of an initially small disturbance (finger) in a displacement front is derived first. The porous medium is assumed homogeneous and the saturation of the invading liquid in the fingers constant. For a constant finger width, the derived expression shows exponential growth for small fingers and linear growth for large fingers. Next, this expression is merged with an expression which describes the development of a marginally stable displacement (mobility ratio M=1) in a heterogeneous porous medium. The results of this merging show that the more wide-spread and the more profound the permeability variations are, the faster the fingers initiate and the faster they show linear growth. The results of the analytical model are compared with the results of a revised version of the numerical technique, diffusion-limited aggregation (DLA). The applied version of such a DLA simulator is summarised and its application justified. The results of the analytical model match the results of the numerical simulator, if in the latter only interface movements in the main flow direction (longitudinal) are included.