VISCOUS FINGERING IN PERIODICALLY HETEROGENEOUS POROUS-MEDIA .1. FORMULATION AND LINEAR INSTABILITY

We are generally interested in viscously driven instabilities in heter ogeneous porous media for a variety of applications, including chromat ographic separations and the passage of chemical fronts through porous materials. Heterogeneity produces new physical phenomena associated w ith the interaction of the flow with the heterogeneity on the one hand , and the coupling between the flow, the concentration of a passive sc alar, and the physical properties (here the viscosity) on the other. W e pose and solve a model in which the permeability heterogeneity is ta ken to be periodic in space, thus allowing the interactions of the dif ferent physical mechanisms to be carefully studied as functions of the relevant length and time scales of the physical phenomena involved. I n this paper: Paper I of a two-part study, we develop the basic equati ons and the parameters governing the solutions. We then focus on ident ifying resonant interactions between the heterogeneity and the intrins ic viscous fingering instability. We make analytical progress by limit ing our attention to the case of small heterogeneity, in which case th e base state flow is only slightly disturbed from a uniform Row, and t o linear instability theory, in which the departures from the base sta te how are taken to be small. It is found that a variety of resonances are possible. Analytic solutions are developed for short times and fo r the case of subharmonic resonance between the heterogeneities and th e intrinsic instability modes. A. parametric study shows this resonanc e to increase monotonically with the viscosity ratio i.e., with the st rength of the intrinsic instability, and to be most pronounced for the case of one-dimensional heterogeneities layered horizontally in the f low direction, as expected on simple physical grounds. When axial vari ation of the permeability field is also considered, a damping of the m agnitude of the response generally occurs, although we find some evide nce of local resonances in the case when the axial forcing is commensu rate with a characteristic dispersive time. The response exhibits a hi gh frequency roll-off as expected. These concepts of resonant interact ion 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, 96 19 (1997)]. (C) 1997 American Institute of Physics.