Viscous fingering in periodically heterogeneous porous media .1. Formulation and linear instability

We are generally interested in viscously driven instabilities in heterogeneous porous media for a variety of applications, including chromatographic separations and the passage of chemical fronts through porous materials. Heterogeneity produces new physical phenomena associated with the interaction of the flow with the heterogeneity on the one hand, and the coupling between the flow, the concentration of a passive scalar, and the physical properties (here the viscosity) on the other. We pose and solve a model in which the permeability heterogeneity is taken to be periodic in space, thus allowing the interactions of the different physical mechanisms to be carefully studied as functions of the relevant length and time scales of the physical phenomena involved. In this paper: Paper I of a two-part study, we develop the basic equations and the parameters governing the solutions. We then focus on identifying resonant interactions between the heterogeneity and the intrinsic viscous fingering instability. We make analytical progress by limiting our attention to the case of small heterogeneity, in which case the base state flow is only slightly disturbed from a uniform Row, and to linear instability theory, in which the departures from the base state how are taken to be small. It is found that a variety of resonances are possible. Analytic solutions are developed for short times and for the case of subharmonic resonance between the heterogeneities and the intrinsic instability modes. A. parametric study shows this resonance to increase monotonically with the viscosity ratio i.e., with the strength of the intrinsic instability, and to be most pronounced for the case of one-dimensional heterogeneities layered horizontally in the flow direction, as expected on simple physical grounds. When axial variation of the permeability field is also considered, a damping of the magnitude of the response generally occurs, although we find some evidence of local resonances in the case when the axial forcing is commensurate with a characteristic dispersive time. The response exhibits a high frequency roll-off as expected. These concepts of resonant interaction are found to be useful and to carry over to the strongly nonlinear cases treated by numerical methods in Paper II [J. Chem Phys. 107, 9619 (1997)]. (C) 1997 American Institute of Physics.