Modeling a reasonable assumption in many other engineering boundary

Modeling of flow in porous
media is a classical problem in both civil and petrolium engineering, and it
plays as a critical role in analysis and design of civil engineering structures
1, 2. Because of its mathematical challenges, it is also interesting in
mathematical points of view. Thus, a considerable amount of literature has been
published on flow in porous media considering the notion of the phenomenon,
numerical modeling and analytical solution 3,4,5,6,7,8. In traditional
studies, Darcy equation (Darcy’s law) is used which represents a linear
relation between gradient of pressure and the velocity vectors 9,10,11,12.
This assumption is valid as long as the  Reynolds number is very low, while it is not a
reasonable assumption in many other engineering boundary value problems, where
the grain sizes of porous media are not fine and the flow velocity is
considerable. A number of cross sectional studies have been made to modify
Darcy’s law 13, 14. The idea of including velocity by the power of two (or
more) in Darcy equation was introduced by Joceph et al. 15 known as Forchimer
equation converting the linear Darcy’s equation to a nonlinear one. Brinkman
suggested another method to modify Darcy equation being more relayable in porous
media with large porosity 16. The logic of the mentioned modification was the
force due to viscosity of fluid (water) to be taken in account in large
porosities. This force has been considered accurately by the Brinkman
additional term. Several attempts have been made to use some combination of
Forchimer and Brinkman terms to form the momentum equation 17, 18, 19, 20, 21,
22. Recently investigators have used Navier-Stockes type equations in modeling
flow in porous media 23, 24. One of the challenges of modelling of flow can be seen in the
problems where the area of concerned consists of both porous and non-porous
media. 23, 24, 25, 26. These two parts are traditionally solved in two
different procedures, and the interaction of two domain are imposed by using
the result of the model in previous time step on each part for setting the
boundary condition of the other part and vice versa. Recently, a few
researcheres have tried to solve the coupled equations for surface water and
subsurface flow to obtain more accurate results 23, 25, 29