The quadratic velocity term which is frequently added to Darcy's law, and is commonly called the Forchheimer term. Essentially Darcy's law arises when the flow on the microscopic scale is Stokes flow, which is governed by the Navier-Stokes equations without the nonlinear advective terms. Thus the microscopic Reynolds is negligible. On the other hand, once the microscopic Reynolds number becomes sufficiently large, then steady recirculating regions form downstream of particles, and even before this happens, the flow past, say, a regular array of cylinders or spheres, become nonsymmetric due to the increasing influence of the advective terms in the Navier-Stokes equations. This is still well within the laminar regime and the Reynolds is still single digit or perhaps a little larger.

So while turbulent microscopic flow is definitely non-Darcy, non-Darcy flow is not necessarily turbulent and there remains quite a large range of values of Re for which the microscopic flow remains laminar.

Darcy's law is only valid for laminar flow through porous medium. A turbulent flow doesn't follow Darcy's law. So for flow through porous medium, turbulent flows are always non-Darcy flow.

Thanks a lot, Mohammad

Do you believe that all Non-Darcy flow originates from turbulent regime?

Yes.

The quadratic velocity term which is frequently added to Darcy's law, and is commonly called the Forchheimer term. Essentially Darcy's law arises when the flow on the microscopic scale is Stokes flow, which is governed by the Navier-Stokes equations without the nonlinear advective terms. Thus the microscopic Reynolds is negligible. On the other hand, once the microscopic Reynolds number becomes sufficiently large, then steady recirculating regions form downstream of particles, and even before this happens, the flow past, say, a regular array of cylinders or spheres, become nonsymmetric due to the increasing influence of the advective terms in the Navier-Stokes equations. This is still well within the laminar regime and the Reynolds is still single digit or perhaps a little larger.

So while turbulent microscopic flow is definitely non-Darcy, non-Darcy flow is not necessarily turbulent and there remains quite a large range of values of Re for which the microscopic flow remains laminar.

Thanks a lot for your clear answer, Andrew.

Could you please provide some numerical criteria 

Hi Reza,

My arguments before, based on the asymmetry of the flow past a single cylinder or sphere, were only to show that Stokes flow (i.e. when the flow is symmetric and when the advective inertia terms are absent) does not arise at moderately small values of the Reynolds number when the flow is still a long way from being turbulent. The nonlinear coefficient, the Forchheimer coefficient, will depend strongly on the geometry of the microstructure of the porous medium.

In order to try to answer your past question I have taken a quick google search. Here is one nice experiment: http://www.uwm.edu.pl/wnt/technicalsc/tech_17_4/b02.pdf with some interesting references. But there is a huge amount of information in the following thesis:

https://spiral.imperial.ac.uk/bitstream/10044/1/24389/2/Takhanov-D-2011-Pet-Eng-MSc-thesis.pdf

I hope that you find the information which you are looking for.

Indeed all turbulent flows are non-Darcy, but deviation from Darcy's law occurs long before (at lower Reynolds numbers based on the pore sizes) turbulence develops. Darcy law corresponds to a Stokes regime, when inertia can be neglected.

Noted with appreciation

Another issue that arises is the Reynolds number at onset of turbulence in porous media. Is it same as that observed in classical fluid mechanics?

The onset of turbulence in classical fluid mechanics depends on the situation. I don't think that there is a deifinitive definition of how such onset is defined, but free jets become turbulent at much lower Reynolds numbers than does flow in a pipe. In additionm and formally, flow in a circular pipe is linearly stable, but can admit localised turbulent bursts if the Reynolds number is high enough and a disturbance is large enough. Other flows have sequences of successive destabilisations. So in classical fluid mechanics, there is no generic sequence of behaviours as Re increases.

I do not study turbulence in porous media, but some others do. Marcelo de Lemos has written many papers and a book on the topic: https://www.elsevier.com/books/turbulence-in-porous-media/de-lemos/978-0-08-098241-0

The previous answer is very good. Typically the Reynolds number U D/nu (velocity  U, Length scale D, kinematic viscosity nu) for the onset of turbulence depends on the details of the geometry, but also on the specific choice of the length scale D and velocity U. For example when considering a long pipe with circular cross section of diameter D one knows that for Re1000. However the exact limit will strongly depend on details of geometry and the is certainly no universal threshold Re for turbulence.

Avraham and Andrew: appreciate for detailed answer

one more point is when and where the turbulent flow may occur; that is, shall we expect turbulent flow in natural oil flow through porous reservoir? What about the water flooding and/or gas flooding? How will be the case for flow through fractures? Under what conditions Shall we expect turbulent flow in these cases?

Dear Reza,

I am afraid that you will need an oil and gas expert to answer your latest questions - I apologise.

Dear Avraham

in your answer, you mention that " A fair guess for porous media is that when Re is based on the average pore diameter D turbulence can occur for Re>1000" 

May I request for a published reference to address this?

Not necessary non-Darcy's flow is regarded as turbulent flow.

Basically, one does only expect non-Darcy flow in the region very close to a well bore or in a fracture (in tight gas reservoir). In the bulk of a reservoir when water drive is considered, typical flow velocities are a "foot/day" (3E-6 m/s). For pore sizes of 1E-4 m (very low resistance) and a dynamic water viscosity of 1E-6 m^2/s, we have in the bulk of the reservoir a Reynolds number of the order of Re=3E-4. This is definitely a Darcy law condition. As crude oil has usually a much high viscosity this also olds for oil. For gasses at high pressure the kinematic viscosity is typically low compared to oil. Near a well bore non-dairy conditions are reached because of high velocities. In a fracture of a tight gas reservoir, gas flow can reach non-Darcy condition both because both the velocity is high and the pore size is very large. My answer is: Non-Darcy flow is only a local phenomenon in reservoir engineering.

First of all, Darcian flow is characterized by the flow rate being directly proportional to the applied head (straight line relationship). Therefore in non_Darcian flow, the flow rate is not directly proportional to the applied head. In fully turbulent flow the square of the flow rate is proportional to the applied head.

non-Darcian flow in porous media (both unconsolidated sediments and fractured rock) is fundamentally different than turbulent flow in pipes. The cross sectional flow area in pipes is completely open and therefore flow regime changes from laminar to turbulent flow is predominantly a function of flow velocity. as the velocity increases, the friction factor decreases and flow begins to occur perpendicular to the main direction of flow causing mixing and creating eddies. The roughness of the pipe surface will cause the friction factor to level off at a larger value than would be seen in smooth pipes.

In porous media the geometry of the flow path physically causes the water to change direction, consider the tortuous path of a pore channel in unconsolidated sediments, or a fracture in rock where the aperture, roughness, contact area, obstructions, and deadwater volumes change with distance. This 'geometry effect' causes the flow to become non-Darcian at much lower flow rates and the transition zone between laminar and turbulent flow to be much larger than in pipe flow. Flow can become non-Darcian at Re numbers much less than 2100, typically on the order of 10-20 for fractures. In other words because of the geometry of the flow paths flow can occur perpendicular to the main flow direction and eddies can form at much lower flow rates than observed in pipe flow. Because of this many researchers refer to non-Darcian flow in porous media as nonlinear laminar flow.

Darcy law is laminar flow in porous media. Darcy flow usually occurs in porous media where there is high friction losses, irregular flow paths and small velocities.

Interesting answers.