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Relative permeability

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Relative permeability behavior

Figs. 1 and 2 show typical behavior for a gas/oil system. The semilog scale of Fig. 2 is convenient for reading the relative permeabilities less than 0.05. Although the curves are labeled "gas" and "oil" in these figures, the phase identity of a curve can be deduced without the labels. For example, the relative permeability that increases in the direction of increasing oil saturation must be the oil relative permeability. The endpoints of the relative permeabilities in Figs. 1 and 2 are defined by the critical gas saturation Sgc and the residual oil saturation Sor. Common names and symbols for some saturation endpoints are listed in Table 1.


Hysteresis

As is the case for capillary pressure, the relative permeabilities depend on the direction of saturation change, as shown schematically in Fig. 3. For this gas/oil system, hysteresis is much greater for the gas relative permeability. Usually, the hysteresis of the wetting phase (oil, in this example) is very small. The trapped-gas saturation Sgt that remains at the end of the imbibition process is a key feature of hysteresis.

Actual observations of hysteresis for water/oil systems are shown in Figs.4[1] through 6. These three figures share some common characteristics. For example, one phase shows large hysteresis, while the other phase shows small hysteresis. Interestingly, the imbibition tracks in Figs. 5[2] and 6 are above the secondary-drainage trends. Jones and Roszelle[3] report large variations in krw and small variations in kro in what they consider to be a water-wet sample.

Wettability

Wettability affects the position of relative permeabilities, as shown in Fig. 7 (from Owens and Archer[4]). The authors measured oil/water relative permeabilities for varying wettabilities with a Torpedo sandstone sample. Wettability was controlled by the concentration of additives in the oil and water. Advancing contact angles were measured on a flat quartz surface.

Fig. 7 shows two important trends. With increasing wetting by the water, the intersection of the oil and water relative permeabilities shifts to the right, and the maximum krw decreases. Similar trends were documented by Morrow et al.[5] and by McCaffery and Bennion.[6] Reservoir engineers use these trends as indicators of wettability.

As mentioned previously, Treiber et al.[7] reported wettabilities for 55 oil-producing reservoirs. A rock was deemed:

  • Water-wet if krw at Sor is less than 15% of kro at Swi
  • Intermediate-wet if krw at Sor is between 15 and 50% of kro at Swi
  • Oil-wet if krw at Sor is greater than 50% of kro at Swi

In addition to the shape of the relative permeability relationships, the authors used the following to supplement their judgment of wettability:

  • Connate water saturations
  • Gas/oil and gas/water relative permeabilities
  • Contact-angle measurements

The judgments of Treiber et al.[7] relied heavily on the results of Schneider and Owens[8] and Owens and Archer.[4] Treiber et al.[7] emphasized that interpretation of wettability from relative permeability behavior is subject to large error because the relative permeabilities depend on connate water saturations and pore-size distribution in addition to wettability. Furthermore, the authors recognized that laminations and other heterogeneities can dramatically alter the relative permeability behavior and, hence, the interpretation of wettability. To prevent such mistaken interpretations, the authors selected rock samples with a high degree of homogeneity.

Interfacial tension

Relative permeabilities change with decreasing interfacial tension (IFT), especially when IFT falls below 0.1 dyne/cm2. The sensitivity of relative permeability to decreasing IFT is of great interest for enhanced oil recovery processes, such as miscible-gas processes and surfactant processes, and for the recovery of fluids from retrograde gas reservoirs.

The change in gas/oil relative permeabilities with decreasing gas/oil IFT as reported by Bardon and Longeron[9] is shown in Fig. 8. At very low IFT, the relative permeabilities approach an "X" shape, with endpoints close to oil saturations of 0 and 1, while at higher IFT, the relative permeabilities display more curvature and have endpoints more distant from the edges of the water-saturation scale. Significant changes in relative permeabilities are not usually observed until the IFT falls below approximately 0.1 dyne/cm2. Another example of the effect of IFT on relative permeabilities as reported by Haniff and Ali[10] is shown in Fig. 9. Asar and Handy[11] also reported on the changes in relative permeabilities for gas/condensate systems as the gas/condensate IFT decreased from approximately 10 to 0.01 dyne/cm2. Amaefule and Handy[12] reported relative permeabilities for low-IFT oil/water displacements.

Endpoint saturation relationships

The most frequently encountered saturation endpoints are:

  • Residual oil saturation
  • Irreducible water saturation
  • Trapped-oil and -gas saturations
  • Critical gas and condensate saturations

Residual oil, irreducible water, and trapped-gas and trapped-oil saturations all refer to the remaining saturation of those phases after extensive displacement by other phases. Critical saturation, whether gas or condensate, refers to the minimum saturation at which a phase becomes mobile.

The endpoint saturation of a phase for a specific displacement process depends on:

  • The structure of the porous material
  • The wettabilities with respect to the various phases
  • The previous saturation history of the phases
  • The extent of the displacement process (the number of pore volumes injected)

The endpoint saturation also can depend on IFTs when they are very low, and on the rate of displacement when it is very high.

Results reported by Chatzis et al.[13] give general insight on the combined effects of wettability and porous structure on residual saturations. In tests with an unconsolidated sand of nonuniform grain size, the wetting phase (oil) was displaced by a nonwetting phase (air) from an initial saturation of 100% to a residual value. The authors observed residual wetting-phase saturations Swr of 7 to 8%. They also found that heterogeneities in the porous medium can lead to Swr greater or less than 7 to 8%, depending on the nature of the heterogeneities. Chatzis et al.[13] also reported residual nonwetting-phase (air) saturations Snwr for displacements by a wetting phase (oil). They reported that Snwr is approximately 14% for an unconsolidated sand of fairly uniform size. In tests on sandpacks of distributed grain size, Snwr rose to an average of 16%. Chatzis et al.[13] also measured Snwr for glass-bead packs consisting of lightly consolidated clusters of glass beads of one grain size distributed in unconsolidated glass beads of another size. They reported that Snwr was 11% for clusters of smaller beads surrounded by larger beads. For larger beads surrounded by smaller beads, Snwr rose to 36%. These results suggest two general conclusions:

  • Residual saturation of a wetting phase is less than the residual saturation of a nonwetting phase
  • Residual saturation of a nonwetting phase is much more sensitive to heterogeneities in the porous structure

General conclusions on the effects of wettability are useful, but the diverse array of wetting alternatives suggests caution, especially in oil/water reservoir systems. This wide range of wetting possibilities is an obstacle to interpreting or predicting the effect of wettability on endpoint saturations. Indeed, conflicting results for different porous media are likely. For example, Jadhunandan and Morrow[14] report that residual oil saturation displays a minimum value for mixed-wet media as wettability shifts from water-wet to oil-wet—counter to the results of Bethel and Calhoun,[15] who reported a maximum for media of uniform wettability.

Critical gas saturation

The critical gas saturation is that saturation at which gas first becomes mobile during a gasflood in a porous material that is initially saturated with oil and/or water. If, for example, the critical gas saturation is 5%, then gas does not flow until its saturation exceeds 5%. Values of Sgc range from zero to 20%.

Critical gas condensate saturation

Interest in the mobility of condensates in retrograde gas reservoirs developed in the 1990s, as it was observed that condensates could hamper gas production severely in some reservoirs, particularly those with low permeability. The trend of increasing critical condensate saturations with decreasing permeability, as summarized by Barnum et al.,[16] is reproduced in Fig. 10.

Trapped, or residual, gas saturation

As shown in Fig. 11, the remaining gas saturation after a waterflood depends on the gas saturation before the waterflood. The relationship of Fig. 11[17] is often called a "trapping relationship." The amount of gas that is trapped in gas reservoirs is of considerable economic significance. For example, in a gas reservoir, encroachment of the aquifer will lead to trapping of some portion of the gas.

Several correlations and summaries for residual gas saturation are found in the literature:

  • Katz and Lee[18] provide a summary of residual gas saturations in a graphical form that is useful for estimates.
  • According to the model presented by Naar and Henderson[19] for multiphase flow through rock, the trapped or residual gas saturation is one-half of its initial saturation; this Naar-Henderson rule is the simplest correlation for residual gas.
  • Agarwal[20] correlated a large collection of residual gas saturations for consolidated and unconsolidated sandstones, for unconsolidated sands, and for limestones.

The ranges of parameters in the correlations are summarized in Table 2. The correlations may be erroneous outside of these ranges. Three of the Agarwal correlations are listed below:

 ....................(1)

 ................(2)

 ....................(3)

In these expressions, residual gas saturation Sgr, initial gas saturation Sgi, and porosity Φ are fractional quantities, not percents. Permeability k is in millidarcies.

Land[21] suggested the following form for estimating trapped-gas saturation Sgr as a function of initial gas saturation Sgi:

 ....................(4)

To calculate C, a limited data set is needed, consisting of the maximum trapped-gas saturation Sgr,max for Sgi=1-Swi . Then,

 ....................(5)

Land[22] reported C = 1.27 for four Berea sandstone samples.

Residual oil relationships

Residual oil saturations after waterflooding or gasflooding are clearly significant for oil recovery. Here, the dependence of residual oil saturation on initial oil saturation and capillary number for a waterflood will be considered.

The relationship between initial and residual oil saturation is termed the oil-trapping relationship. For strongly water-wet rocks, the oil-trapping relationship should be identical to the gas-trapping relationship. Indeed, because of this analogy and because it is easier to measure gas-trapping relationships, few oil-trapping relationships have been measured. A set of oil-trapping relationships reported by Pickell et al.[23] are shown in Fig. 12. Oil-trapping relationships are important for estimating reserves in transition zones. In conventional reservoir engineering, residual oil saturation refers to the remaining oil saturation after a displacement that starts near the maximum initial oil saturation, which generally equals one minus the initial water saturation.

In the remainder of this section, the dependence of residual oil saturation on capillary number is discussed for processes starting with initial oil saturation at a maximum value: So = 1– Swi. This topic has received much more attention in the literature than oil-trapping functions. The capillary number is the ratio of viscous forces to capillary forces. It is represented quantitatively with various expressions, as summarized by Lake.[24] These expressions are derived from the ratio of pressure drop in the water phase to the capillary pressure between the oil and water phases. A popular definition of the capillary number is as follows:

 ....................(6)

with v representing the velocity of the water. The capillary number is small (less than 0.00001) when capillary forces dominate the flow processes. The example below shows just how small capillary numbers can be.

As the capillary number for an oil-displacing process increases, residual oil saturation decreases in the manner sketched in Fig. 13. Above the "critical capillary number," the rate of decrease of Sor is particularly rapid. The critical capillary number is 10–5 to 10–4 for porous media with fairly uniform pore sizes. With increasing distribution of pore sizes, the critical capillary number decreases, the Sor at low Nc increases, and the domain for decreasing S or becomes broader. Extensive discussion of these relationships is available elsewhere.[25] King et al.[26] suggested centrifuge methods for measuring these relationships. Pope et al.[27] correlated residual phase saturation with a modified form of the capillary number, which was termed the "trapping number." Adjusting a parameter in their correlation fits the effects of wetting on residual saturation.

Example 1

Use the following quantities to estimate a capillary number for a waterflood with Eq. 6, where

  • μw = 1 cp = 0.01 g/cm/s
  • v = 1 ft/D = 30.48 cm/(24 × 3,600 s) = 0.00035 cm/s
  • σow = 30 dynes/cm

Therefore, the capillary number is as follows:

 

Capillary forces do indeed dominate flow processes for waterfloods. Even in high-velocity regions, such as the vicinity of a well that is producing oil and water, the capillary number will remain very small.

Residual water saturation

Residual, or irreducible, water saturation Swi is the lowest water saturation that can be achieved by a displacement process, and it varies with the nature of the process—gas displacement or oil displacement. Also, Swi varies with the extent of the displacement, as measured by pore volumes of oil or gas injected or by time allowed for drainage.

To be more specific, the results of Chatzis et al.[13] (discussed above) can be extended to suggest irreducible water saturations of 7 to 9% for displacements in unconsolidated sand and glass beads that are water-wet. Furthermore, Swi should increase slightly with increasing breadth of grain-size distribution. Significant variations in Swi should occur when small clusters of consolidated media of one grain size are surrounded by media of another grain size:

  • If the grains of the clusters are smaller than those of the surrounding media, Swi increases.
  • If the grains of the clusters are larger than those of the surrounding media, Swi decreases.

The saturation of water in an oil or gas reservoir at discovery is called the connate water saturation, or Swc. The connate water saturation and the irreducible water saturation can differ. If the reservoir processes that produced the connate water saturation can be replicated, then the Swi for the replicated processes should be the same as Swc. Swc is significant for its connection to initial oil or gas saturation in a reservoir.

  • For an oil reservoir, So = 1– Swc
  • For a gas reservoir, Sg = 1– Swc

The connate water saturation will also affect initial oil or gas relative permeability and, hence, the economic viability of a reservoir. Bulnes and Fitting[28] concluded that low-permeability limestone reservoirs are more viable than sandstone reservoirs of the same permeability because the connate water saturation is lower in the limestones than in the sandstones; as a result, the relative permeabilities to oil are higher in the limestones than in the sandstones.

Salathiel[29] observed that the connate water saturations in carefully retrieved rock samples from some oil reservoirs are substantially lower than can be achieved when the rock is waterflooded and then oilflooded. He attributed this effect to the mixed-wettability condition. When the reservoir was first invaded by oil, the rock was water-wet, and low water saturations were obtained. However, the wettability of the rock surfaces that were now in contact with oil changed from water-wet to oil-wet as portions of the hydrocarbons adsorbed onto the solid surfaces. So, when such a rock is waterflooded and then oilflooded, the connate water saturation is not obtained because the water in the oil-wet portions of the rock becomes trapped.

Temperature

The effects of temperature on relative permeability have been studied primarily for applications to steamflooding and in-situ combustion. Mechanistically speaking, temperature can affect relative permeability by altering the IFT between flowing phases or by altering the wettability of the porous material. IFT between water and oil should decrease with increasing temperature, but to substantially influence relative permeability, the IFT would need to decrease to 0.1 dyne/cm2 or less, according to the discussions in other pages. Such reductions would be possible only at very high temperatures with light oils. Therefore, temperature-related IFT reductions could influence relative permeabilities for in-situ combustion processes, but they would not be important for typical steamflooding.

The influence of temperature on wettability and, hence, on relative permeability is more likely to be important for most applications. With increasing temperature, the wettability could shift either to more water-wet or more oil-wet conditions, depending on the reservoir fluids and the chemical composition of the porous medium.

Akin et al.[30] reviewed a wide variety of published studies of relative permeabilities for heavy oil and water at different temperatures. Some of the studies concluded that these relative permeabilities were unaffected by temperature changes, while other studies concluded the opposite. In the light of the previous paragraph, these contradictory observations in the literature are not surprising. However, Akin et al.[30] concluded that viscous instability—not wettability change—is the cause of most reported changes in relative permeability with increasing temperature (see note below). With increasing temperature, the viscosity of the heavy oil decreases, and the water/oil displacement process becomes more stable. The changing stability of the displacement (estimated with the expression of Peters and Flock[31]) causes the apparent relative permeabilities to change with temperature. Nevertheless, it is possible that relative permeabilities do change with temperature for some systems. As Akin et al.[30] conclude, further study of this subject is needed.

Note: Viscous instability results from displacement of a viscous (low-mobility) phase by a less-viscous (high-mobility) phase. The high-mobility phase is prone to bypass or "finger" through the low-mobility phase. With "viscous fingering," the displacement must be 2D or 3D rather than 1D. One-dimensional displacements are preferred for measurement of relative permeabilities.

Nomenclature

C = parameter in the Land function
k = permeability, L2, md
Nc = capillary number
Sg = saturation of gas
Sgi = initial saturation of gas
Sgr = residual saturation of gas
So = saturation of oil
Sw = saturation of water
Swc = critical saturation of water
Swi = irreducible or residual saturation of water
μw = viscosity of water, m/Lt, cp
σow = oil/water interfacial tension, m/t 2, dyne/cm
Φ = porosity

References

  1. 1.0 1.1 Geffen, T.M., Owens, W.W., Parrish, D.R. et al. 1951. Experimental Investigation of Factors Affecting Laboratory Relative Permeability Measurements. J Pet Technol 3 (4): 99-110. SPE-951099-G. http://dx.doi.org/10.2118/951099-G
  2. 2.0 2.1 2.2 Braun, E.M. and Holland, R.F. 1995. Relative Permeability Hysteresis: Laboratory Measurements and a Conceptual Model. SPE Res Eng 10 (3): 222–228. SPE-28615-PA. http://dx.doi.org/10.2118/28615-PA
  3. Jones, S.C. and Roszelle, W.O. 1978. Graphical Techniques for Determining Relative Permeability From Displacement Experiments. J Pet Technol 30 (5): 807–817. SPE-6045-PA. http://dx.doi.org/10.2118/6045-PA
  4. 4.0 4.1 4.2 Owens, W.W. and Archer, D.L. 1971. The Effect of Rock Wettability on Oil-Water Relative Permeability Relationships. J Pet Technol 23 (7): 873-878. SPE-3034-PA. http://dx.doi.org/10.2118/3034-PA
  5. Morrow, N.R., Cram, P.J., and McCaffery, F.G. 1973. Displacement Studies in Dolomite with Wettability Control by Octanoic Acid. SPE J. 13 (4): 221–232. SPE-3993-PA. http://dx.doi.org/10.2118/3993-PA
  6. McCaffery, F.G. and Bennion, D.W. 1974. The Effect OfWettability On Two-Phase Relative Penneabilities. J Can Pet Technol 13 (4). PETSOC-74-04-04. http://dx.doi.org/10.2118/74-04-04
  7. 7.0 7.1 7.2 Treiber, L.E. and Owens, W.W. 1972. A Laboratory Evaluation of the Wettability of Fifty Oil-Producing Reservoirs. SPE J. 12 (6): 531–540. SPE-3526-PA. http://dx.doi.org/10.2118/3526-PA
  8. Schneider, F.N. and Owens, W.W. 1970. Sandstone and Carbonate Two- and Three-Phase Relative Permeability Characteristics. SPE J. 10 (1): 75-84. SPE-2445-PA. http://dx.doi.org/10.2118/2445-PA
  9. 9.0 9.1 Bardon, C. and Longeron, D.G. 1980. Influence of Very Low Interfacial Tensions on Relative Permeability. SPE J. 20 (5): 391–401. SPE-7609-PA. http://dx.doi.org/10.2118/7609-PA
  10. 10.0 10.1 Haniff, M.S. and Ali, J.K. 1990. Relative Permeability and Low Tension Fluid Flow in Gas Condensate Systems. Presented at the European Petroleum Conference, The Hague, Netherlands, 21-24 October 1990. SPE-20917-MS. http://dx.doi.org/10.2118/20917-MS
  11. Asar, H. and Handy, L.L. 1988. Influence of Interfacial Tension on Gas/Oil Relative Permeability in a Gas-Condensate System. SPE Res Eng 3 (1): 257-264. SPE-11740-PA. http://dx.doi.org/10.2118/11740-PA
  12. Amaefule, J.O. and Handy, L.L. 1982. The Effect of Interfacial Tensions on Relative Oil/Water Permeabilities of Consolidated Porous Media. SPE J. 22 (3): 371-381. SPE-9783-PA. http://dx.doi.org/10.2118/9783-PA
  13. 13.0 13.1 13.2 13.3 Chatzis, I., Morrow, N.R., and Lim, H.T. 1983. Magnitude and Detailed Structure of Residual Oil Saturation. SPE J. 23 (2): 311–326. SPE-10681-PA. http://dx.doi.org/10.2118/10681-PA
  14. Jadhunandan, P.P. and Morrow, N.R. 1995. Effect of Wettability on Waterflood Recovery for Crude-Oil/Brine/Rock Systems. SPE Form Eval 10 (1): 40–46. SPE-22597-PA. http://dx.doi.org/10.2118/22597-PA
  15. Bethel, F.T. and Calhoun, J.C. 1953. Capillary Desaturation in Unconsolidated Beads. J Pet Technol 5 (8): 197-202. SPE-953197-G. http://dx.doi.org/10.2118/953197-G
  16. 16.0 16.1 Barnum, R.S., Brinkman, F.P., Richardson, T.W. et al. 1995. Gas Condensate Reservoir Behaviour: Productivity and Recovery Reduction Due to Condensation. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 22-25 October 1995. SPE-30767-MS. http://dx.doi.org/10.2118/30767-MS
  17. 17.0 17.1 Keelan, D.K. and Pugh, V.J. 1975. Trapped-Gas Saturations in Carbonate Formations. Society of Petroleum Engineers Journal 15 (2): 149-160. SPE-4535-PA. http://dx.doi.org/10.2118/4535-PA
  18. Katz, D.L. and Lee, R.L. 1990. Natural Gas Engineering, 83-86. New York City: McGraw-Hill Publishing Co.
  19. Naar, J. and Henderson, J.H. 1961. An Imbibition Model—Its Application to Flow Behavior and the Prediction of Oil Recovery. SPE J. 1 (2): 61–70; Trans., AIME, 222. SPE-1550-G. http://dx.doi.org/10.2118/1550-G
  20. Agarwal, R.G. 1967. Unsteady-state performance of water-drive gas reservoirs, 46-59. PhD thesis, Texas A&M University, College Station, Texas (May 1967).
  21. Land, C.S. 1968. Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties. SPE J. 8 (2): 149–156. SPE-1942-PA. http://dx.doi.org/10.2118/1942-PA
  22. Land, C.S. 1971. Comparison of Calculated With Experimental Imbibition Relative Permeability. SPE J. 11 (4): 419–425. SPE-3360-PA. http://dx.doi.org/10.2118/3360-PA
  23. 23.0 23.1 Pickell, J.J., Swanson, B.F., and Hickman, W.B. 1966. Application of Air-Mercury and Oil-Air Capillary Pressure Data In the Study of Pore Structure and Fluid Distribution. SPE J. 6 (1): 55–61. SPE-1227-PA. http://dx.doi.org/10.2118/1227-PA
  24. 24.0 24.1 Lake, L.W. 1989. Enhanced Oil Recovery, 71. Englewood Cliffs, New Jersey: Prentice Hall.
  25. Green, D.W. and Willhite, G.P. 1998. Enhanced Oil Recovery, Vol. 6, 18-27. Richardson, Texas: Textbook Series, SPE.
  26. King, M.J., Falzone, A.J., Cook, W.R. et al. 1986. Simultaneous Determination of Residual Saturation and Capillary Pressure Curves Utilizing the Ultracentrifuge. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 5-8 October 1986. SPE-15595-MS. http://dx.doi.org/10.2118/15595-MS
  27. Pope, G.A., Wu, W., Narayanaswamy, G. et al. 2000. Modeling Relative Permeability Effects in Gas-Condensate Reservoirs With a New Trapping Model. SPE Res Eval & Eng 3 (2): 141–178. SPE-62497-PA. http://dx.doi.org/10.2118/62497-PA
  28. Bulnes, A.C. and R. U. Fitting, J. 1945. An Introductory Discussion of the Reservoir Performance of Limestone Formations. Trans. of AIME 160 (1): 179-201. http://dx.doi.org/10.2118/945179-G
  29. Salathiel, R.A. 1973. Oil Recovery by Surface Film Drainage in Mixed-Wettability Rocks. J Pet Technol 25 (10): 1216–1224. SPE-4104-PA. http://dx.doi.org/10.2118/4104-PA
  30. 30.0 30.1 30.2 Akin, S., Castanier, L.M., and Brigham, W.E. 1998. Effect of Temperature on Heavy-Oil/Water Relative Permeabilities. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 27-30 September 1998. SPE-49021-MS. http://dx.doi.org/10.2118/49021-MS
  31. Peters, E.J. and Flock, D.L. 1981. The Onset of Instability During Two-Phase Immiscible Displacement in Porous Media. SPE J. 21 (2). SPE-8371-PA. http://dx.doi.org/10.2118/8371-PA. See also Peters, E.J. and Khataniar, S. 1987. The Effect of Instability on Relative Permeability Curves Obtained by the Dynamic-Displacement Method. SPE Form Eval 2 (4): 469-474. SPE-14713-PA. http://dx.doi.org/10.2118/14713-PA

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