Lower Bound to Darcy's Law for Water flow

Darcy's Law (1856) describes a linear relationship between flowing pressure gradient and the fluid velocity for viscous flow in a porous media under certain conditions.

An upper velocity bound, associated with gas flow, was established by Forcheimmer in 1901. At high gas velocities inertial forces can increase to the point at which the gas flow becomes turbulent. Forchheimers equation describes a non-linear relationship beween the flowing pressure gradient.

Two lower velocity bounds have also been established;

A lower velocity bound, associated with low pressure gas flow, this bound was demonstrated Klinkenberg in 1941. At low pressures gas slippage occurs (a layer of gas on the pore wall is stationary) which results in a deviation from Darcy's Law.

 A lower velocity bound, associated with water flow, was pointed out by King 1898 (Ref 1).

The lower and upper velocity bounds to Darcy's Law can be described graphically;

It can be seen that the pressure gradient for low velocity water flow is greater than expected using Darcy's Law

Following King's statement that;

The evidence which has been presented regarding the flow of water through rock and sands demonstrates beyond question that it does, under certain conditions, increase faster than the pressure.

numerous experimental investgations into the lower velocity bound of water have been performed by hydrology, civil and chemical engineering practitioners. The understanding of the mechanisms causing the deviation from Darcy resides in the literature of those discplines. Experimental investigation has been prone to experimental artefacts. Many mechanisms have been cited as the cause of the phenomenon and a widely accepted understanding is elusive.

A common denominator is the recognition that at low velocity electro-chemical forces at the grain-water interface become significant relative to the prevaling viscous forces.

The exsistence of electro-chemical forces at the grain-water interface has long been recognised in the field of petrophysics. There is a gendre of water saturation models, using electric log data, which are adopted if those electro-chemical forces are significant.

A single SPE Paper 89433  leverages knowledge of a pre-Darcy flow phenomena in the context of reservoir performance. The paper provides a comprehensive discussion of the phenomena. However, the absence of data  demonstrating pre-Darcy oil flow is not clearly reported. There is no public domain experimental data to support the suggestion that 'staggant oil zones' may exsist due to a lower bound to Darcy's Law for oil. However, the implications for oil and gas production associated with water injectivity and aquifer influx are well made.

The appetite of reservoir engineers to explore the implications of the phenomena may well be limited by;

• its association with the movement of water not oil
• the limited experimental data acquired on porous samples relevant to the industry
• the absence of quantitative evidence of the influence of the phenomena on reservoir performance predictions
• the absence of observed reservoir performance demonstrating the influence of the phenomena

However, this may change in light of two developments in the field of increased or enhanced oil recovery;

• De-pressurisation - where the impact of aquifer performance will be greater than under oil recovery by waterflood
• Field application of a  low salinity waterflood strategy by BP, called LoSal^TM. - The prevailing electro-chemical forces will be influenced by the modification in the injected water salinity.

References

1. Principles and Conditions of the Movement of Ground Water

King FH , US Geol. Survey 19th Ann. Rpt Part 2 pp59 - 294