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Corrections to core measurements of permeability
Before selecting a method of determining permeability in a specific reservoir, one must first be assured that the core measurements are appropriate for reservoir conditions. Sample collection, selection, and preparation are important steps in ensuring that the data set represents the geology at in-situ conditions; some precautions are discussed in Relative permeability and capillary pressure. Adjustments may be necessary for the type of test fluid and for pressure effects.
Klinkenberg effect
The permeability of a sample to a gas varies with the molecular weight of the gas and the applied pressure, as a consequence of gas slippage at the pore wall. Klinkenberg^{[1]} determined that liquid permeability (k_{L}) is related to gas permeability (k_{g}) by k_{L} = k_{g}/(1+b/p), where:
- p is the mean flowing pressure
- b is a constant for a particular gas in a given rock type
The correction parameter b is determined by conducting the test at several flowing pressures and extrapolating to infinite pressure. Alternatively, one can use an empirical correlation established by Jones^{[2]} to estimate b. The correlation, with R^{2} of 0.90, is based on measurements on 384 samples (mostly sandstones) with permeabilities ranging from 0.01 to 2500 md.
For helium, b_{helium}=44.6(k/Φ)^{-0.447} For air, b_{air} = 0.35 b_{helium}
The units of b are psi for permeability in units of md and porosity expressed as a fraction. Another empirical correlation was established by Jones and Owens^{[3]} for tight gas sandstones with permeabilities ranging from 0.0001 to 10 md: b=0.86k^{-0.33}, where:
- b is in atm
- k is in md
The Klinkenberg correction is quite important for low-permeability rocks and less important or unimportant for high-permeability rocks. The value of k_{L} obtained after applying the correction represents the permeability to a gas at infinite pressure or to a liquid that does not react with the component minerals of the rock.
Pore fluid sensitivity
The clays or other materials coating grain surfaces can be sensitive to pore fluid. This complicates the problem of describing permeability because flow properties depend not only on the lithology but also on pore fluid chemistry. This kind of reaction can be seen in Fig. 6 of Rock types. Measured permeabilities on this sample as a function of salinity are shown in Fig. 1. Samples were obtained after drying and storage, and as a result, clays had collapsed. This collapsed state did not significantly change when the rock was saturated with very-high-salinity brine. As the pore fluid decreased in salinity, at a point near 30,000 ppm salt content, the clays expanded and effectively plugged the pore space. This result demonstrates the need to take special precautions in preserving and drying the core.
In low-permeability sandstones, permeability to water (k_{w}) is systematically less than the Klinkenberg-corrected permeability (k_{L}). A correction equation^{[3]} based on >100 samples with a permeability range of 0.0001< k_{L} <1 md is k_{w} = k_{L}^{1.32}. The correction is only approximate, as scatter on a graph of k_{w} vs. k_{L} is high at k_{L} <0.01 md. Surprisingly, the sensitivity to brine concentration in low-permeability sandstones is reported to be less than that in high-permeability sandstones.^{[3]}
Pressure dependence
Permeabilities discussed so far were measured at constant effective pressure. However, as increasing pressure closes fractures and compresses the pore space, permeability will decrease. The magnitude of the change depends on the rock fabric. Weak, unconsolidated rocks will collapse easily, and the drop in permeability can be dramatic. As the rock becomes better consolidated, this pressure dependence decreases. On the other hand, even for tight rocks, as fractures are introduced and begin to dominate the fluid flow, this general trend is reversed and pressure dependence increases.
The pressure dependence can often be fit well with an empirical power law and negative exponential relation,
where:
- k(p_{e}) is the permeability measured at effective pressure p_{e}
- k_{o} is the permeability at zero pressure
- b is a parameter adjusted to fit each rock
This general relation for normalized permeability is shown in Fig.2. A larger b factor yields a stronger pressure dependence.
In Fig. 3, sample D1452-281 (same as Fig. 1 in Rock types) is a clean sandstone with a high porosity of 35%. The pressure dependence is strong, as can be seen in Fig. 4. A b factor of 0.13 fits the general trend of the data (when pressure is expressed in MPa). In more compacted or cemented samples, such as in Fig. 5, with a lower porosity of 18%, the permeability decrease can be fit well with b=0.07. In general, as cementation increases, the pressure sensitivity declines, and the value of b approaches zero. On the other hand, in low-porosity, brittle rocks, flow often becomes fracture dominated. Because fractures are compliant and close easily with pressure, the pressure dependence of permeability again becomes high. In Fig. 6, permeability of a very-low-porosity (0.18%) crystalline rock is plotted on a logarithmic scale. Even though the absolute value of permeability is low to begin with, k drops to 1% of its unconfined value at 100 MPa, and the decline curve can be fit with a b factor of 0.6.
A more elaborate equation relating Klinkenberg-corrected permeability k to effective confining pressure p is^{[4]}
where the following are determined from experimental data:
- k_{o}
- Permeability at zero confining pressure
- Slope a
- Pressure coefficient P*
- Coefficient C
However, if constant values of C=3×10^{-6} psi^{-1} and P*=3,000 psi are used, errors are about 5%, and the two remaining coefficients, k_{o} and a, can be determined if k is determined at only two values of p. Jones^{[4]} recommends the use of 1,500 and 5,000 psi, although very poorly consolidated samples may require that the higher value of p be reduced so as not to exceed the yield strength of the sample.
In practice, one may be required to correct values of k measured with air at ambient conditions to k for brine at reservoir conditions. The correction should be based on samples and conditions for the problem at hand. As a guide, Swanson^{[5]} established a correction of k_{brine}= 0.292k_{air}^{1.186} for a collection of 24 sandstone and 32 limestone samples (0.002< k_{brine} <400 md), where k_{brine} was measured at 1,000-psi effective stress. Swanson’s empirical equation appears to incorporate all three factors discussed above:
- Gas slippage effect
- Presence of brine as opposed to an inert fluid
- Effect of stress
Nomenclature
k | = | permeability |
p | = | pressure |
Subscripts
e | = | effective |
l | = | liquid |
o | = | oil |
w | = | water |
References
- ↑ Klinkenberg, L.J. 1941. The Permeability of Porous Media to Liquids and Gases. Drill. & Prod. Prac. 200.
- ↑ Jones, S.C. 1987. Using the Inertial Coefficient, β, To Characterize Heterogeneity in Reservoir Rock. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 27–30 September. SPE-16949-MS. http://dx.doi.org/10.2118/16949-MS
- ↑ ^{3.0} ^{3.1} ^{3.2} Jones, F.O. and Owens, W.W. 1980. A Laboratory Study of Low-Permeability Gas Sands. J Pet Technol 32 (9): 1631–1640. SPE-7551-PA. http://dx.doi.org/10.2118/7551-PA
- ↑ ^{4.0} ^{4.1} Jones, S.C. 1988. Two-Point Determinations of Permeability and PV vs. Net Confining Stress. SPE Form Eval 3 (1): 235-241. SPE-15380-PA. http://dx.doi.org/10.2118/15380-PA
- ↑ Swanson, B.F. 1981. A Simple Correlation Between Permeabilities and Mercury Capillary Pressures. J Pet Technol 33 (12): 2498-2504. SPE-8234-PA. http://dx.doi.org/10.2118/8234-PA
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See also
Petrophysical properties of gas reservoirs