Permeability Measurement
Measurement of Permeability
The ability of a formation to transmit fluids is termed permeability; its unit is the darcy (µm2). This unit has been subdivided into 1000 smaller units, called millidarcies, and these units are used in reporting core analysis measured values. A darcy has been defined as that permeability which permits a fluid of one centipoise viscosity to flow at a rate of one cubic centimeter per second through a porous medium with a cross-sectional area of one square centimeter under a pressure gradient of one atmosphere per centimeter. Its value is determined in the laboratory.*
Dry gas has been selected as the standard fluid for use in permeability determinations because it minimizes fluid-rock reaction and is easy to use. In the laboratory, air is caused to flow through a clean and dry sample of measured length and diameter (API-RP-27). The pressure differential and flow rates are measured and the permeability is calculated from the Darcy equation. In the linear flow system, the Darcy equation for noncompressible fluids can be used to calculate permeability to a compressible gas of the rate of flow in cm3/s of the gas is expressed as the mean pressure existing within the core sample. This conversion of a volume at atmospheric pressure to the volume existing at mean pressure is accomplished by using the following equation:
*In SI units a permeability of one meter squared will permit a flow of 1m3/s of fluid of 1 Pa.s viscosity through an area of 1m2 under a pressure gradient of 1 Pa/m.
Noncompressible Fluid Flow (Liquid)
(1)
Thus:
Compressible Fluid Flow (Gas)
(2)
Where:
q = liquid flow rate, cm3/s
qa = gas flow rate at atmospheric pressure, cm2/s
µ = viscosity of fluid flowing, centipoise
L = sample length, cm
A = sample cross-sectional area, cm2
p1 = upstream pressure, atmospheres absolute
p2 = downstream pressure, atmospheres absolute
Pa
= atmospheric pressure, atmospheres absolute
k = permeability, md
The permeability value resulting from use of the compressible fluid equation is known as an air permeability, or "ka."
Data are valid when no reaction between the rock and flowing fluid occurs (normally the case when air is employed), as long as laminar flow exists (where the flow rate remains proportional to the pressure gradient), and as long as one fluid completely saturates the core. These conditions exist during conventional core analysis measurements that yield specific (absolute) permeability.
Permeability is related to rock textural properties. Fine-grained sandstones and intercrystalline limestones have small pores and low permeability. Coarse grained sandstones, fractured lime-stones, and oolitic limestones have large pore channels and high permeability. Combinations of low matrix permeability with high fracture permeability can yield prolific reservoirs, such as those found in Iran, Iraq, and the North Sea.
Samples selected for permeability determinations are drilled parallel to bedding planes. This is easily accomplished in most formations, but others, such as eolian deposits, often show cross-bedding in the plugs selected. If vertical permeabilities are desired, a second plug is drilled perpendicular to the bedding plane. Measurement of this vertically drilled core will indicate the degree of permeability reduction caused by impermeable horizontal layers or grain orientation.
Permeameter
PLUGS: A schematic of the principle involved in permeability measurement is shown in Figure 1 ( Schematic flow diagram of permeameter). A clean, dry sample is placed in a holder; it must fit snugly and allow no air to bypass along the sides of the sample. Upstream and downstream pressures are measured to determine the pressure differential across the core. As Figure 1 shows, the calibrated orifice allows the flow rate in cm3/s to be measured at atmospheric pressure.
Interest in low permeability gas formations has resulted in the development of nonsteady-state permeameters that yield information within a short time span (Freeman and Bush 1983.) This technique allows determination of permeabilities to be made in a matter of minutes rather than taking all or a portion of a day as is required with the steady-state methods. The mathematics allow computation of the air permeability to be made as fluid is flowing through the core. This avoids the need for steady-state equilibrium conditions required in older systems. Hence, low-permeability samples no longer require long-term testing.
FULL DIAMETER CORES:
Full diameter core measurements are normally restricted to carbonates and formations containing vugs and fractures. These measurements can be made on sandstones, but care must be taken to ensure that the invasion of drilling fluid solids does not reduce horizontal permeabilities. This type of reduction can happen in sandstones as well as in chalky limestones. In the latter, ground limestone powder that has been produced by the temperature and by pressure conditions encountered while coring is sometimes plastered on the core, causing permeability reduction. This powder must be removed prior to full diameter horizontal permeability measurements, as it forms a "skin "that will result in erroneously low permeability values. This material can sometimes be cut away easily when the core is wet, otherwise sand blasting or rasping of the dry core may be necessary to remove the plaster coat.
Figure 2 (Permeability comparison of undamaged plug with damaged full-diameter core) offers a comparison of actual plug permeabilities with plastered whole core values. The plugs in this case were cut from the center of the full diameter core and were not subjected to damage. The data indicate that permeability reduction was greatest for the high permeability section, but all full diameter values were reduced.
Vertical permeability on a full diameter sample is easily measured using a Hassler-type core holder, as illustrated in Figure 3 (Full diameter horizontal and vertical permeability measurement apparatus). The sample is placed in the apparatus and the rubber tubing is then collapsed around the core. With the introduction of high pressure air to the holder, the tubing provides a seal along the sides of the sample. Low pressure air is introduced on the upstream end of the core and the measurements needed to compute permeability are easily made.
Obtaining full diameter horizontal permeability measurements is more complex. Screens are placed on opposite sides of the full diameter core ( Figure 3 ) and the sample is then moved into the core holder. Nonpermeable rubber disks are placed on each end of the sample and the rubber tubing is again collapsed around the core. Low pressure air, introduced into the center of the holder, passes through the rubber boot, intersects with the screen, and flows vertically in the screen. The air then flows through the full diameter sample along its full height and emerges on the opposite side, where the screen again allows free flow of the air to the exit port. In this test the flow length is actually a function of the core diameter. The cross-sectional area of flow is a function of the length of the core sample and of the core diameter. The screens are selected to cover designated outer segments of the full diameter sample. In most cases the circumference of the core is divided into four equal quadrants and the screens occupy two opposite quadrants. In this measurement, a geometric factor, G, is substituted for the shape factor, L/A, in the Darcy equation. Mathematics relating to this have been presented by Collins (1952).
It is common to furnish two horizontal permeability measurements on all full diameter samples. The second measurement is made at right angles to the first. When fractures are visible in the core, the first measurement is made parallel to these fractures. The maximum horizontal permeability is recorded as kmax and the other at 90º to the direction of this flow is labeled k90.
SIDEWALL CORES:
When sidewall samples have been mounted in thin-walled jackets, permeability is determined using techniques similar to those employed for normally drilled plugs. When sidewall samples are sufficiently large, a portion of the sidewall core is cut from the sample and mounted in a plastic or wax holding material. This mounted sample can then be tested in normal plug equipment.
In some cases the sidewall core sample is so small that all of the recovered samples are consumed in the porosity and residual fluid determinations. Where samples are large enough for measurements to be made, experience has shown that sidewall permeability measured in soft rock is often equal to one-third or less of the value that would be determined on a conventional core from the same interval. This is believed to be caused by the invasion of the core by mud solids and rearrangement of sand grains at the time the sidewall core is taken. Because of these detrimental effects, as well as the need to use all of the sample for other measurements, it is common to estimate permeability values from previously developed correlations.
The correlation used usually relates permeability with porosity, grain size and distribution, and with shaliness of the core. The correlation is developed from conventional cores on which permeability, porosity, and grain size distribution have been measured. With the value for these parameters available from a sidewall core, an estimation of permeability of the sidewall core is possible. When the permeability value is an estimated one, a note on the core analysis report should state that fact.
The estimation of grain size and distribution, as well as of shaliness, has been subjective. Estimates of permeability can be improved if objective means are available for estimating average grain size and distribution, and shaliness of the core. An apparatus for making these determinations is now available and is being used in many high volume sidewall core analysis laboratories. This apparatus allows determination of the presence of silt-size and smaller particles, including clays, with experimentation under way for enabling differentiation between clay and silt.
Measured permeability values for sidewall cores are erroneously lower than those for conventional cores when both are taken from soft sediment. Measured values are erroneously high in sidewall cores from hard, well-cemented rock, which is shattered by the impact of the bullet used to retrieve sidewall cores.
Factors Affecting Measured Values
Air permeabilities measured in a routine core analysis laboratory on rock samples from nonfractured reservoirs will give higher values than the actual reservoir permeability. This difference is dependent upon the magnitude of permeability as well as the pore geometry. The higher laboratory values are thought to be caused by gas slippage (the Klinkenberg effect), relative permeability, reactive fluids, and overburden pressure effects.
THE KLlNKENBERG GAS SLIPPAGE EFFECT:
The flow of gas through porous media was investigated by Klinkenberg (1941). He found that the permeability of a core sample was not constant, but varied with the gas used to make the measurement, as well as the mean (average) pressure existing in the core at the time of measurement. His investigations indicated that at low mean pressures-for example, at atmospheric pressure-the gas molecules are so far apart that they slip through the pore spaces with little friction loss, and yield a higher value of permeability. At higher mean pressures — for example, 1000 psi (6895 kPa) or greater — the gas molecules are closer together and experience a friction drag at the side of the pore walls. This increases as higher mean pressure increases, with the gas acting more and more like a liquid. This means that the measured value of permeability decreases as reservoir or laboratory mean pressure increases.
Experiments show that a plot of gas permeability versus the reciprocal mean pressure existing at the time of the gas permeability measurement forms a straight line that can be extrapolated to infinite mean pressure. This extrapolated value of permeability, referred to as the Klinkenberg permeability or equivalent liquid permeability, is lower than the measured gas permeabilities and is comparable to the permeability that would be obtained if the core were saturated with a nonreactive liquid such as oil. Figure 1 (Klinkenberg permeability determination) shows an example of this relationship for a low permeability sandstone. Steady-state permeability measurements were made at each of the four points shown at each given net overburden pressure. The Klinkenberg value (kL) can be correlated with the value of permeability determined with air at the mean pressure normally used in the laboratory measurements. Table 1, below, offers examples of the relationship between the air permeability and Klinkenberg-corrected values for some sandstones. The correction, on a percentage basis, is greater in low permeability sands and becomes progressively smaller as permeability value increases.
Noncorrected permeability (md) | Klinkenberg corrected permeability* (md) |
1.0 | 0.7 |
10.0 | 7.8 |
100.0 | 88.0 |
1000.0 | 950.0 |
*Air permeability that has been corrected for gas slippage. The Klinkenberg value is the equivalent liquid permeability assuming no reaction between the rock and the fluid.
Table 1. A comparison of noncorrected and Klinkenberg-corrected air permeability for some sandstones
The table represents the results of laboratory measurements on a suite of core samples that covered a wide range of air permeabilities.
In early core analysis reports, measured air permeability was corrected to the Klinkenberg permeability value, using correlations such as those presented in Table 1., above. These corrections have been found to be good approximations for sandstones and some intergranular Ii me-stones. The correction is usually applied to plug-sized samples but should not be applied to full diameter permeability measurements because the heterogeneity of most full diameter cores renders the correlations unreliable. If the need exists, Klinkenberg permeability values should be determined on individual samples.
The nonsteady-state permeameter allows the calculation of Klinkenberg permeability values on plug-sized samples and provides a second factor related to the pore geometry and expressed as "b." Equation 6.9 relates air permeability to Klinkenberg permeability; knowledge of the Klinkenberg value and the "b" factor from the nonsteady-state tests allows calculation of the air permeability at any mean pressure desired.
Because reservoirs exist at relatively high mean pressures, the Klinkenberg-corrected air permeability (equivalent liquid permeability) value is more representative of the reservoir value than the laboratory-measured ka value.
RELATIVE PERMEABILITY AND REACTIVE FLUIDS EFFECTS:
When a second fluid phase is present in a reservoir, the permeability to each phase is referred to as the effective permeability In an oil or gas reservoir, the second phase present in the pore space is interstitial water. The effective permeability to the hydrocarbon phase in a reservoir with interstitial water present will be less than the measured permeability to air after correction for Klinkenberg effects (absolute permeability). In high permeability rock with relatively low values of interstitial water, the presence of water reduces the gas or oil effective permeability only a minor amount below the absolute permeability. As the permeability of the rock decreases (a condition that is normally associated with a reduction in pore size), the presence of water has an increasingly important and detrimental effect on the effective permeability to oil or gas.
Clays line the pore spaces of many formations and sometimes react with coring, drilling, or other fluids injected into the reservoir. This reaction commonly results in a reduction in permeability that can vary from minor to catastrophic. Two types of damage mechanisms can result. One is a dispersion of clay particles and/or a physical tearing loose of clay platelets that subsequently block pore throats and reduce permeability. A second type of reaction involves clay swelling, which causes an increase in clay volume and subsequent reduction in cross-sectional area open to flow within the pore throats. Because of the difference in the mechanisms, each demands a different treatment for correction, but the effect of both is to cause reservoir permeability to be lower than that measured in the laboratory.
Rock type | Permeability at net overburden pressure: % of original | ||||
| 200 psi | 1000 psi | 3000 psi | 5000 psi | 7000 psi |
Sandstone (well indurated) | 100% | 96% | 91% | 86% | 82% |
Sandstone (unconsolidated) | 100% | 72% | 44% | 35% | 30% |
Chalk | 100% | 97% | 93% | 60% | 30% |
Table 2. Permeability reduction with increasing overburden pressure
OVERBURDEN PRESSURE EFFECTS: Samples tested when making routine core analysis measurements usually have no net overburden pressures applied to them. When simulating reservoir conditions in the laboratory, it has been found that the application of overburden pressure reduces permeability. Reduction in permeability of as little as 7% to as much as 100% has been reported for overburden pressures up to 5000 psi (34475 kPa). (McLatchie, Hem-stock, and Young 1958; Vairogs et a. 1970.) Table 2 presents some permeability reduction data for samples that range from well cemented to unconsolidated in nature. As was noted in porosity measurements, the unconsolidated materials show the greatest percentage reduction in value with increasing overburden pressure. Generally, the reduction in permeability with the application of net overburden pressure is greater than the reduction seen in porosity.
An experiment with flowthrough capillary tubes, as described by the physicist Poiseuille (1797-1869), indicates that the quantity of fluid flowing through a capillary varies with the fourth power of the radius. Because the value of air permeability is directly proportional to flow rate, small changes in pore radii caused by the application of overburden pressure result in correspondingly large reductions in permeability.
In any given reservoir, the effects of gas slippage, reactive fluids, and overburden pressure may be great or small, depending on pore geometry. Laboratory results indicate that these factors become more important as absolute permeability of a rock decreases. In some low permeability gas reservoirs, a high interstitial water level causes a significant reduction in effective permeability, with further reductions caused by an increase in net overburden pressure and the gas slippage effect. Figure 1 illustrates these effects. It represents data on a sample with an initial air permeability of .28 millidarcies. When a nonmobile, interstitial water saturation of 53% of pore space was introduced, the original air permeability determined on the dry core was reduced to an effective permeability of .051 millidarcies. These measurements were made with a minimum net overburden pressure of 200 psi (1379 kpa), to prevent air bypassing the core and flowing down the sample's sides.
The Klinkenberg permeability value, obtained from the extrapolation of four measured air permeabilities at different mean pressures, showed a further reduction in effective permeability, to .035 millidarcies. An increase in overburden pressure from 200 to 6000 psi (1380 to 4137 kPa) resulted in a final effective permeability of .003 millidarcies. This is approximately a 100-fold decrease in permeability. Similar magnitudes of reduction have been observed by Jones and Owens (1979) for tight gas reservoirs.
Accuracy of Measurement
The accuracy of the permeability value measured in the laboratory is approximately 5% of its true value when the permeability is in the range of 10=500 md. As permeability decreases to the <1.0 md, accuracy may decrease to 20% of true value. Above 500 md, the accuracy of measurements is typically 10% of true value.
Horizontal permeability values determined by core analysis yield the vertical distribution of horizontal permeability at the wellbore. They provide a measure of permeability variation, which may be used in subsequent reservoir engineering calculations. Other reservoir evaluation tools, such as buildup pressure tests, provide interwell permeability values, but supply only an average value for the formation permeability and not the distribution of permeability. Both types of data have a place in formation evaluation.
Slippage, relative permeability, reactive fluid, and overburden pressure effects can all be measured in the laboratory. They require additional testing time and are usually not measured at the time that a conventional core analysis is run. The conventional data are adequate to describe permeability distribution profiles, to indicate formation productivity, and to serve as a guide for selecting samples to be used in more complex special core tests.
Exercise 1.
Given:
µg = 0.018 cp
A = 5.06 cm2
I = 2.54 cm
pa = atmospheric pressure = 1.0 atmosphere
p1 = upstream pressure, atmospheres absolute
p2 = downstream pressure, atmospheres absolute
qa = volumetric gas flow rate at atmospheric pressure, cc/sec
qa | p1 | p2 |
3.75 | 1.5 | 1 |
5.56 | 2.25 | 1.75 |
10.48 | 4.25 | 3.75 |
25.20 | 10.25 | 9.75 |
And Darcy's equation for flow of gas in a porous medium:
= millidarcies
Calculate the air permeability and the liquid permeability (Klinkenberg) values.
Solution 1:
qa | p1 | p2 | p | pm | ka | 1/pm |
|
|
| (p1 - p2) | (p1 + p2)/2 |
| |
3.75 | 1.5 | 1 | .5 | 1.25 | 54.2 | 0.8 |
5.56 | 2.25 | 1.75 | .5 | 2.00 | 50.2 | 0.5 |
10.48 | 4.25 | 3.75 | .5 | 4.00 | 47.4 | 0.25 |
25.20 | 10.25 | 9.75 | .5 | 10.00 | 45.5 | 0.10 |
To do:
1. Compute ka (millidarcies) at each of four flow rates.
2. Plot ka versus 1/pm and extrapolate to find equivalent liquid permeability kL where 1/pm = 0 ( Figure 1 ).
3. kL from plot equals 44.2 millidarcies.
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