The law of Darcy does not provide detailed information on the pore scale (say submillimeter scale). It relates the gradient of average pressure to the average velocity. This average could be taken over volumes of decimeters. The gradients could be over hundreds of meters (oil field).

Hence the law of Darcy does not provide any information on the pore level.

For a correct answer it is important to recall the definition of the groundwater flow:

Under Ground water in the sense of groundwater hydraulic, means the water that fills the interconnected pore spaces of the soil and moves exclusive under the effect of gravity and / or pressure.

The basic equation of the fluid flow (i.e., Darcy’s-Law) in porous media can be derived from the Basic Equation of hydrodynamics at the pore scale (i.e. momentum balance and mass balance of fluid in interconnected pore spaces). Using for this equation the integral - averaging technique on the so called REW (Representative- Elementary- Volume) we get a generalised form of the Darcy’s-Law at the macroscopic scale (i.e. equation which connect the Darcy velocity and the pressure gradient). Because the pore spaces are interconnected the pressure at the macroscopic scale is identically to averaged pressure at the pore scale in the REW.

The simplified form coincides with the empirical Darcy’s-Law.

An extended mathematical presentation and derivation are given in

David, I.: Grundwasserhydraulik. Strömung und Transportprozesse. Vieweg-Springer, 1997

but unfortunately, in German only.

Dear Dr Ioan David

I completely agree with your point.

It essentially conveys that the pore pressure at both micro-scopic as well às at the macro-scopic scale merges together - when the single-phase groundwater flow is purely concerned with the “ continuous fluid flow through hydraulically inter-connected pore spaces “.

‘Given this background,

‘can we expect the following statements to remain true?

1. The pore pressure at the pore-scale (capillary pressure) and it’s associated averaged pressure at the macro-scopic scale (REV) would merge together - in case of describing multi-phase fluid flow. [capillary pressure is a function of hydraulic pore radius as well - which will be varying along the flow direction].

2. Whether the same concept of assuming pore-scale pore-pressure and it’s associated macro-scopic scale averaged (REV) averaged pressure to merge together - in case of a fractured reservoir.

Thanks for your time and reply.

Dear Dr Govindarajan

Yes, we can expect certainly that also the two mentioned statements remain true.

Best Regards

Dear Dr David,

Whether Integral averaging technique will do justice in up-scaling the pore-scale hydrodynamics to the macroscopic-scale Hydrodynamics?

Whether the integral averaging of pore-scale capillary-pressure would lead to macroscopic-scale averaged reservoir pressure?

What happens to the pore-scale interfacial tension, when we up-scale to a larger macroscopic-based Darcy’s-scale?

How can we relate pore-scale capillary pressure with that of the larger-scale averaged reservoir pressure?

Pls enlighten me.

Thanks.

Dear Dr Govindarajan,

I hope that the following information are usefull.

The integral averaging technique (REV) is general accepted and allows the transition of physical quantities from pore-scale (e.g. microscopic phase pressure, capillary pressure etc.) to its associated averaged quantities at the macroscopic scale.

In this regard, a detailed derivation of the Darcy's law for a single-phase fluid (e.g. water or oil) can see (pg.10-18) in:

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

pg. 10-18.

The obtained result is the well-known Darcy law:

v= -(k/η)∙(∇P+ρgz)

P-pressure; v -Darcy velocity vector; k- absolute intrinsic permeability; η-viscosity; z- vertical coordinate positive direction upward

In the same way can be obtained the extended Darcy law for two (or multiple) phase fluid flow:

vα= -(k∙krα/ηα)∙(∇Pα+ραgz)

α -is the fluid phase index (e.g. α=o for oil phase and α=w for water phase)

Pα- is the pressure in phase α; v – the Darcy velocity vector; k- absolute intrinsic permeability; η-viscosity; z- vertical coordinate positive direction upward and krα- is the relative permeability of phase α;

The relative permeability krα depend strongly from fluid saturation Sα , wettability, pore size distribution, effective porosity etc.

In reservoir modelling of two (multi) phase flow using the extended Darcy’s law, the relative permeability must be predetermined using site-specific samples. The relation between relative permeability and saturation can be determine experimentally and using formulas as functions of phase-saturations like Orey’s, Henderson,s formula.

Capillary pressure PC is the pressure difference between the pressure of nonwetting and wetting phases.

In the attached study is an example to prove the quantitatively transition from pore-scale to macroscopic scale for capillary pressure, fluid saturations, and interfacial area between fluid phases using numerical network model:

https://www.sciencedirect.com/science/article/pii/S0309170816304377

Best regards

Sir,

"Grundwasserhydraulik. Strömungsß und Transportvorgänge" - Could you pls send me this material written in English?

Thanks.

Dear Suresh:

your statement : "The pore pressure at the pore-scale (capillary pressure) and it’s associated averaged pressure at the macro-scopic scale (REV) would merge together -"

It is only valid if both phases are in hydraulic continuity along the length of the porous media. One example when this does not work is in foam flow and slug flow as each bubble or drop is not in hydraulic continuity and preserves the pressure of the moment that become disconnected due to an instability.

The same applies to multi phase Darcy eqn.

Dear Prof Carlos,

I completely agree with you.

However, a small correction.

That was not my statement; and I raised that question (and not a statement) following Dr David's reply.

Thanks.

Dear Carlos,

You're right, but I have made it clear in my answer for three months ago, that it only considers groundwater flows (one phase or multi-phase) in the sense of groundwater hydraulic. That means that the fluids (i.e. water, oil, air …) fill the interconnected pore spaces of the porous media and moves exclusive under the effect of gravity and / or pressure gradient. Darcy’s law is only valid under these conditions for both cases, mono or multi-phase flow in porous media. Therefore, all the discussions that we do, refer exclusive to these types of movements.

As you mentioned, the given examples (e.g slug flow) do not belong to this kind of mathematically representations.

Regards

Dear Sureh,

I will soon come back with the requested translation.

Regards

Hi Ioan: I am coming from the oil industry so we may be saying the same thing with different terminology. For example: we don't use ground water flow or ground water hydraulics... " That means that the fluids fill the interconnected pore spaces of the porous media and moves exclusive under the effect of gravity and / or pressure gradient." The first part defines fully saturated (water +gas + hydrocarbon), the second refers to the driving force. None defines how it flow within the rock and does not help that we cannot see withing the rock. We know that 2 phases go in and 2 come out at steady state with certain pressure drop and the hydraulic continuity is an assumption.

Dear Suresh,

Attached the promised English translation (Darcy law derivation from the basic equation of hydrodynamic for water flow in porous media) Chap 3 from:

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

Farther I try to summarize, according to my opinion, some Aspects of our discussions and my answer:

The main objective of discussions is based on your question:

“How can we relate pore-scale capillary pressure with that of the larger-scale averaged reservoir pressure?”

This regard my answer refer to a light reformulated/completed question:

How can we relate pore-scale physical quantities (i.e. fluid flow velocity, pressures etc. in microstructure of porous media, namely in rock pore space) with that assigned of the large-scale reservoir (flow velocity, pressure etc. in porous media as macrostructure i.e. porous media as continuum)? In other words, how can we obtain the theoretically derivation of Darcy’s law from the basic equations of fluid flow in pore space?

In my statements/answers to this question are considered the assumptions that fluids (one or more phase) fill the interconnected pore spaces of the porous media (hydraulic continuity along the length of the porous media) and the fluids moves under the effect of gravity and / or pressure gradient. In term of oil industry that means, as Carlos noticed as well, that porous media is fully saturated (e.g. water only in case of one phase or water + hydrocarbon in case of two phase and so on) and the driving force are gravity and pressure gradient.

I would mention that different terms (specific for hydraulics or for oil industry) not change the main objective of the question/answer namely the theoretically derivation of Darcy’s law from the basic equations of fluids flow in pore space. By the way, the basic equations of the fluid d\flows and the Darcy's law for fluid flow in porous media (as experimental law) have their origins in hydraulics (water flow) and was extended later to other fluids like oil, gas etc.

I resume my answer as follow:

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 flow equation in porous media as continuum whose simplified form coincides with the empirical Darcy’s-Law.

As I mentioned it in my previous answer can be obtained in the same way the extended Darcy law for two (or multiple) phase fluid flow.