If we take one-dimensional, steady, incompressible assumptions into N-S equation, we can get Dacy law. Thus, the darcy law is a simplified equation to dearl with flow inside a porous medium.

Minghou Liu Thank you very much!

Hi Shekufe,

I think so. As I have used the Darcy Law and NS in a 2D coupled well-reservoir problem and solved them using finite element method. You can find more details here:

https://www.researchgate.net/publication/270468723_A_coupled_wellbore-reservoir_flowmodel_for_numerical_pressure_transient_analysis_in_vertically_heterogeneous_reservoirs

Thank you Pascale Royer .

On the basic of CFD definition like following:

"Computational fluid dynamics (CFD) is a science that, with the help of digital computers, produces quantitative predictions of fluid-flow phenomena based on the conservation laws (conservation of mass, momentum, and energy) governing fluid motion". (From: Fluid Mechanics, Fifth Edition, 2012)

we can consider that the modelling fluid flows in porous media like those in petroleum engineering is fully integrated in CFD.

In relation to this it is to mention that the Darcy’s-Law for fluid flow in porous media obtained empirical for about 150 years ago can be derived theoretically from the Basic Equation of the fluid flow at the pore scale (i.e. momentum balance and mass balance of fluid in interconnected pore spaces of the rock saturated with fluids). The derivation is based on the integral – averaging of the basic equations of the hydrodynamics in pore space on the so-called REV (Representative- Elementary- Volume) of the Porous media.

The obtained result is a generalised form of fluid flow equations in porous media as continuum whose simplified form coincides with the very known empirical Darcy’s-Law.

As example attached (PDF) you find the English translation of the chap. 3 of my book:

https://www.researchgate.net/publication/270214318_Grundwasserhydraulik_Stromungsss_und_Transportvorgange

where the Darcy’s-Law derivation is presented.

Additionally remarks:

Through replace the Darcy equation in continuity equation on obtain the governing equations of the fluid flow in porous media which are generally partially differential equations. These equations, with additionally boundary and initial conditions corresponding to a real groundwater or petroleum exploitation flow system lead mathematically to Boundary Value Problems (BVP) or Initial-Boundary Value Problems (IBVP). To solve such problems, we use mathematical methods of CFD (e.g. FEM, FDM, BEM, coupled BEM-FEM and so on).

Best regards

The computational fluid dynamics (CFD) is: using finite difference (FD) or Finite volume (FV) for 3-D problems to solve fluids problems that are governed by governing equation (NS). So any simplified version of NS is still can be solved using CFD.