Macroscopic description of quasi-static and quasi-stationary processes in unsaturated porous materials

Abstract

Abstract The main aim of this work is to derive equations describing quasi-static and quasi-stationary processes of capillary transport of liquid and gas in unsaturated porous materials. Considerations are based on the new macroscopic model of the capillary transport in porous media, proposed in the paper (Cieszko, 2016). This model has been formulated in the spirit of the theory of interacting continua and shows that the capillary processes of transport in porous materials take place in five-dimensional pressure-time-space. This means that the equations for quasi-static and quasi-stationary processes are not a simple consequence of reduction of the general system of equations. An extended definition of quasi-static processes is proposed, allowing such processes to be analysed as a special case of the general description. A nonlinear first-order evolution equation is obtained for pore saturation with mobile liquid in quasi-static processes; it simultaneously describes the non-stationary processes of saturation evolution taking place in the pressure-space continuum. A new class of processes of capillary transport in unsaturated porous materials is proposed, called quasi-stationary, and a system of strongly coupled nonlinear equations describing these processes is derived. It is shown that even the geometrically very simple boundary value problem of non-wetting liquid flow through an unsaturated porous layer is described by a system of three nonlinear coupled equations for the saturation, pressure and velocity of the liquid.