For example, Akbari and Namin 24

used this approach in an SPH numerical method in wave interaction with

structures. Their developed model solves porous and pure fluid (water) flows

simultaneously by means of an equation that is equivalent to the unsteady 2D

Navier–Stokes (N-S) equations for the flows outside the porous media, while the

extended Forchheimer equation has been used for the flows inside the porous

media. In their model, the interface boundary between pure fluid (water) and

porous media is effectively taken into account by the SPH integration technique24.

One of the main challenges free surface modeling is the tracking of free

surface evolutions. Hydrostatic assumption has been used by many authors in

previous researches 30, 31. However, the solutions based on the simple

hydrostatic assumption are not as accurate in most of engineering problems

where the vertical velocities and accelelerations are not ignorable. For this

reason, detailed investigation based on the hydrodynamic treatment of the

phenomenon is required where the vertical plane of the model area has to be

devided into a finite number of elements. Two different approaches have been

followed based on hydrodynamic assumption, which are updated mesh and fixed

mesh approaches. Studies with updated mesh mostly use an Arbitrary Lagrangian

Eulerian (ALE) technique 32, 33. When using the ALE

technique in simulations the mesh for the domain has to be regenerated and the

nodes on the boundaries of the domain have to move along with the materials to

preciesely track the boundaries, accordingly. In fixed mesh (Eulerian) approach, on the other hand, level set method usually is used by which

the free surface is represented by a signed distance function that takes the

value 0 on the free surface, takes negative values in the fluid domain and

positive values outside the fluid domain. 29, 34, 35. In most of the applications the above two

mentioned methods can track surface evolution accurately, however, the main

disadvantages of these methods are their high cost and complexities. The

improved accuracies due to the mentioned methods do not justify to suffer the

high computation expensis and model complexities in the most of the real

applications.