Mathematical Models of Flow in Porous Media

In this chapter a general model for the two-phase fluid flow in porous media is presented, together with its simplified form, known as the Richards equation, which is applicable (under specific assumptions) to describe water flow in the vadose zone. In each case the governing equations are formulated at the Darcy scale, using the capillary pressure–saturation relationship and an empirical extension of the Darcy equation for the multiphase flow. The validity of these concepts, and the models based on them, is a subject of ongoing scientific debate, due to unclear connections between the pore-scale and Darcy-scale physical quantities, e.g. [25, 29, 32, 56, 66, 88]. Nevertheless, the models described here can be used to simulate many practical cases of multiphase flow in the subsurface with sufficient accuracy, e.g. [17, 31]. Therefore, they have been assumed as the starting point for the analysis presented in this work. The two-phase flowmodel considered here is based on the following assumptions: