Application of the Method of Multiple Scales to the Problem of Occurrence of Solute Convection in Rectangular Region of Porous Medium

Concentration convection has a significant effect on the transport of impurity in porous media. Such transport processes can occur in a variety of geophysical systems. Concentration convection is often considered by analogy with thermogravitational convection in porous media without taking into account the fact that the impurity can be adsorbed on the skeleton of the porous medium. One of the approaches allowing to take this fact into account is the MIM (mobile-immobile media) approach. The paper studies the occurrence of concentration convection in an elongated rectangular region filled with porous medium in the field of gravity at a constant pressure and concentration drop. The Darcy-Boussinesq law is used as a filtration model. The problem is solved analytically by the method of many scales in the approximation of weak buoyancy force. Analytical expressions of concentration and pressure fields for the ground state (the case of no gravity) and the first order of smallness are obtained. The results of numerical calculation are compared with the analytical solution. It is shown that the convective-free case is unstable at any small concentration drop, and the resulting flow is a solitary convective cell. The position of the region of the largest concentration inhomogeneity depends on the ratio of the Peclet and Rayleigh-Darcy numbers.