Oil & Gas - Maximising Recovery Programme (formerly SHARP) IOR Views

There is a possibility that the petroleum industry may be missing opportunities for improving the recovery of hydrocarbons. Most analyses and predictions of reservoir performance assume that the physics of flow in porous media are adequately represented by Darcy’s Law. However, that assumption may not always be justified. There is a need for further research, which may reveal improved methods for developing oil and gas fields.

Today, Darcy’s Law is the foundation for most quantitative work on fluid flow in reservoirs. It has become a paradigm – accepted without questions, and seldom checked. However, if we force ourselves to step back and take a look at the supporting data, we are faced with two uncomfortable realisations [1]:

1. There is no published experimental data demonstrating the validity of Darcy’s empirical law at reservoir conditions, including typically very low flow velocities.
2. There is an appreciable body of published experimental data, mainly from non-petroleum disciplines, showing departures from Darcy’s Law at low velocities.

Darcy’s Law, based on the empirical work he published in 1856 on single-phase flow through coarse sand packs, describes a proportional relationship between the potential gradient causing the fluid to move and the resulting fluid velocity. For a linear horizontal system, Darcy’s Law can be expressed as:

The hypothetical velocity implicit in Darcy’s Law is called here the “superficial velocity”, which is the chemical engineering term, since there is no consistent terminology for it in petroleum engineering. Note that the superficial velocity is essentially the hypothetical velocity that would be observed if the porosity were 100%. Its value is much lower than, for example, the real flood front velocity. On a plot of superficial velocity versus gradient, Darcy’s Law can be represented by a straight line passing through the origin, i.e. the fluid is stationary only when no gradient is acting upon it.

It is instructive to compare the superficial velocity range used in Darcy’s original tests with the superficial velocities for hydrocarbons flowing in the drainage areas of typical wells, Figure 1.

Figure 1: Superficial Velocity Range Tested by Darcy Compared to Reservoir Rates

Clearly, superficial velocities around typical wells are orders of magnitude below the velocities for which Darcy’s Law was empirically established. Petroleum engineers understand that Darcy’s Law has its limitations. For example, there is a lot of data demonstrating an upper limit to Darcy proportionality. At higher velocities, inertial effects cause the “less-than-proportional” relationship commonly called Non-Darcy flow in the petroleum industry. However, the question of a possible lower limit to Darcy’s Law has generally been ignored in the petroleum industry, even though most fluid flow in reservoirs occurs at very low velocities indeed.

Outside the petroleum industry, there has long been debate about the applicability of Darcy’s Law at low velocities. There are published experimental data showing a lower limit to its validity.

In the years following the publication of Darcy’s work in 1856, a number of researchers failed to confirm the proportional relationship he had observed between applied gradient and resulting superficial velocity. In 1899, F. H. King [2] of the United States Geological Survey published an extensive investigation into single phase fluid flow through a variety of porous media, in which he found evidence for a greater-than-proportional relationship. In the hundred years since then, King’s work has been supported and extended by other researchers in Europe, Asia, Australia and North America – mainly outside the petroleum production industry.

In 1977, based on a review of work published on the flow of water through soils, an Indian soil scientist (Dr. Basak [3]) proposed classifying fluid velocity/gradient regimes into three main zones - Pre-Darcy, Darcy, and Post-Darcy. Figure 2 is adapted from his publication:

Figure 2: Pre-Darcy, Darcy, and Post-Darcy Flow Regimes

Basak’s classification recognised departures from Darcy’s Law at low gradients and velocities. There was a possible No-Flow zone at very low applied gradients. The applied gradient has to exceed a Threshold Gradient before any flow can occur. As the applied gradient is increased beyond that threshold, the resulting velocity demonstrates a greater-than-proportional response. Once the gradient reaches an adequate level, Darcy proportionality prevails. At still higher gradients, the resulting velocity once again departs from Darcy’s Law, demonstrating the less-than-proportional response known as Non-Darcy flow in the petroleum industry.

The implications of Basak’s classification for the petroleum industry are immense – especially in light of the curious dearth of any published data confirming the validity of Darcy’s Law at reservoir conditions, including low gradients and velocities. What seems to have happened is that early researchers in the petroleum industry adopted Darcy’s Law and then focused on two-phase flow, where any limitations on Darcy proportionality were subsumed into the uncertainties of relative permeability.

If Darcy proportionality does not prevail at the very low velocities typical for fluid flow in reservoirs, the industry could be missing opportunities to increase the economic recovery of hydrocarbons. For example, if threshold gradients occur in reservoirs, then there will be no drainage from areas where the applied gradient is less than this threshold. In the study of groundwater, such areas are called “Stagnation Zones” [4]. In such a case, sweep efficiencies calculated from Darcy’s Law would be too high, and the benefits of infill drilling or increasing applied gradients would be underestimated. Missed opportunities!

The mechanism for Pre-Darcy flow may be related to an interaction between water and the porous medium. Dr. Polubarinova-Kochina [5] suggested in 1952 that Pre-Darcy phenomena might occur when polar water molecules oriented themselves in electric fields around solid surfaces. This could lead to the formation of a quasi-crystalline layer of bound water at surfaces, held in place by weak hydrogen bonds. This bound water layer could reduce the effective diameters of pore throats, or even completely occlude them. At very low applied gradients, the hydrogen bonds would resist water movement. This could result in a threshold gradient observed at the macroscopic level. As the applied gradient increases, the bound water layer would progressively break down, resulting in the greater-than-proportional relationship observed by King and others.

The implication of Dr. Polubarinova-Kochina’s hypothesis is that Pre-Darcy flow is more likely to be an issue for less permeable, shaly reservoirs than for highly permeable, clean reservoirs. As reservoirs of lower quality are increasingly being brought into production, addressing the possible occurrence of Pre-Darcy flow is becoming ever more important.

There is insufficient published data today to reach any definitive conclusions on whether Pre-Darcy flow occurs in reservoirs. There is therefore an urgent need for research to resolve that question. If Pre-Darcy flow is found to occur in certain reservoirs, the petroleum industry will need to develop ways to take advantage of that phenomenon to maximise the economic recovery of hydrocarbons.

References

1. Longmuir, G.: “Pre-Darcy Flow: A Missing Piece of the IOR Puzzle?”, paper SPE 89433 presented at the 2004 SPE/DOE IOR Symposium, Tulsa OK, April 19 – 21.
2. King, F. H.: “Principles and conditions of the movements of ground water”, in Nineteenth Annual Report of the United States Geological Survey, 1897-98, Part II – Papers Chiefly of a Theoretic Nature, Government Printing Office, Washington (1899) pp. 59 – 294.
3. Basak, P.: “Non-Darcy Flow and Its Implications to Seepage Problems”, Journal of the Irrigation and Drainage Division, American Society of Civil Engineers (Dec. 1977) Vol. 103 IR4 pp. 459 – 473.
4. Bernadiner, M. G. and Protopapas, A. L.: “Progress on the Theory of Flow in Geologic Media with Threshold Gradient”, Journal of Environmental Science and Health A (1994) Vol. 29 pp. 249 – 275.
5. Polubarinova-Kochina, P. Ya: Theory of Ground Water Movement, Moscow (1952). Reprinted in English translation by J. M. R. De Wiest, Princeton University Press, Princeton NJ (1962) p. 17.