Measurement of capillary pressure and relative permeability
The reliability of measurements of relative permeabilities and capillary pressures is an important issue for reservoir engineering. Although there are many factors that influence reliability, the following three topics are emphasized here:
- Sampling of rocks for measurements
- Measurement methods
- Treatment of data from the measurements
Reliability of measurements
Proper sampling of rocks for measurements is most important for ensuring the reliability of relative permeability and capillary pressure data. If samples are obtained improperly, costly and reliable methods for measuring rock/fluid properties may no longer be necessary or suitable. The goal of sampling should be to avoid or minimize mechanical and chemical damage to the rock. Mechanical and chemical damage can occur during any of four steps in the sampling process:
- Coring and core retrieval
- Shipping and storing
- Cutting samples from the core
- Cleaning and preparing the sample
With all of these opportunities, some damage is inevitable.
It should be obvious that reliable data require good measurement methods and correct treatment of the data obtained. But defining "good" and "correct" requires some not-so-obvious understanding—a goal of this discussion is to touch on the important issues. Measurement of relative permeability and capillary pressure relationships is complicated, particularly because of the intertwining nature of these rock/fluid properties. Indeed, it is the deciphering of these complications that largely defines what is meant by "good" and "correct."
Sample handling and preparation
From cutting a core from a formation to final preparations of a specimen for testing, improper practices will alter the porous structure and wettability of rock samples, which in turn will alter the quality of measurements. The most appropriate procedures depend on the ultimate objective of sample analysis. If the sample is to be used only for porosity and permeability measurement, wettability alterations during handling are not important. But for capillary pressure and relative permeability measurements with native-state wetting conditions, one must work to avoid contamination of the sample. Next, a distillation of industry experience with handling and preparation of samples is presented.
Cutting a core from a formation
Drilling fluids often contain:
- Other particulates in oil- or water-based slurries
To minimize contamination with drilling fluids, cores are sometimes cut with:
- Fresh formation oil
- Synthetic formation brines
- Drilling fluids formulated to minimize invasion
Sometimes, the extent of invasion can be assessed later in the laboratory. Coring also can produce fractures in the rock, either by:
- Mechanical stress of the drilling process
- Relief of the in-situ mechanical and thermal stresses on the core as it travels to the surface
- Retrieval (often with a big hammer) from the core barrel at the surface
A variety of methods are used to ship core. Core has been shipped in containers filled with oil or water from the producing formation, it has been shipped in sealed polymeric bags with metallic barriers to oxygen permeation, and it has been shipped after wrapping with a metallic foil and a wax-coated fabric. Some core has been frozen immediately with dry ice or liquid nitrogen upon arrival at the surface, then shipped in a frozen condition to prevent oxidative and bacterial actions and to minimize evaporation of the water or oil.
Selection of the locations in the core for taking samples is a critical step. For porosity and permeability testing, samples are usually collected at regular intervals (e.g., every foot). Sometimes, the most homogeneous samples are selected. Such a bias likely imposes an undesirable bias on the results. Selecting locations for sampling should follow a logical or a statistical thought process. For example, criteria for sample selection can include:
- Pore structure
- Homogeneity or heterogeneity
Computerized tomography (CT) scanning can help the selection of samples without internal pebbles or fractures that might drastically affect the outcome of measurements. In general, the characteristics of a collected sample should represent a significant portion of the reservoir. After locations are chosen, collection can begin. Collecting a rock sample from the core generally entails a cutting action with diamond-coated saws and boring tools. The coolant for these cutting processes is a source of contamination. To reduce contamination, liquid nitrogen is sometimes used as a coolant, but most often kerosene or water is used—these coolants should be free of contaminants. For preserving water saturations in the sample, kerosene is a common choice for cooling. Excessive exposure of the core to air during sample collection can foster a shift of wettability from its native condition.
Sample cleaning and preparation
After collection from the core, rock samples are thoroughly cleaned in preparation for porosity, permeability, and other measurements. Cleaning by flushing the sample alternately with toluene, chloroform, acetone, and methanol (and mixtures of these) is common and often satisfactory. To preserve delicate clay structures in some samples, a cleaning process with miscible fluids and supercritical drying is sometimes used. Cleaning a sample with boiling solvent (such as toluene) in a reflux-extraction unit should be avoided. During such cleaning, water vaporizes, and any surface coated with water can become coated with high-molecular-weight hydrocarbons that are less soluble at elevated temperatures—this can impart a nearly permanent change in wettability.
Final preparations for testing
After suitable cleaning, the samples can be used in the clean condition for porosity, absolute permeability, and mercury capillary pressure measurements; alternatively, the condition of the samples can be returned to near reservoir condition by processes of flooding with brine and oil from the formation and aging in formation oil. Such restored-state samples can be suitable for capillary pressure and relative permeability measurements with reservoir fluids. Common practice includes centrifugation of brine-saturated samples in air to reduce the brine saturation to near reservoir levels. Some rock samples are not cleaned after collection from the core but are maintained instead in a preserved state by storing them in uncontaminated oil from the reservoir. Such samples are flooded with formation oil in preparation for measurements.
Measuring capillary pressure relationships
Most methods for measuring capillary pressure may be grouped under three headings:
- Mercury methods
- Porous-plate methods
- Centrifuge methods
Each of these is discussed below. Vapor-pressure and gravity-equilibrium methods also have been used, but they are not discussed here. The literature offers few examples of capillary pressure measurements with overburden stress matching that in the reservoir. The magnitude of error caused by this omission will vary, of course, from reservoir to reservoir, but the meaning and significance of capillary pressure data without the appropriate overburden stress are questionable. Some commercial laboratories offer capillary pressure measurements with overburden stress.
In the mercury method, a sample of rock is evacuated, and the volume of mercury that enters the sample at increasing pressures is measured, as shown in Fig. 1. Mercury methods are especially suited for samples of irregular shape, such as those found in drill cuttings. Mercury methods are useful for investigating the porous structure of the sample. Complete mercury capillary pressure curves can be determined within an hour or so, depending on the permeability of the sample. To apply the proper overburden stress, a cylindrical sample could be mounted in a confining sleeve in a variation of the apparatus of Fig. 1. Some commercial laboratories offer mercury measurements with overburden stress.
Use of the mercury method was first documented in the petroleum literature by Purcell in 1949. Purcell showed that mercury capillary pressure and air/water capillary pressure relationships could be correlated. He outlined a method for estimating permeability from mercury measurements. Rose and Bruce proposed a relationship between a rock’s threshold capillary properties and its permeability. Thomeer proposed a correlation for estimating permeability from three parameters of the mercury capillary pressure curve, as mentioned previously. Swanson proposed a correlation between permeability and the intersection of a 45° line with the mercury capillary pressure relationship plotted on a log-log scale.
The porous-plate method can yield very accurate capillary pressure relationships. The following example describes the method. Consider a cylindrical rock sample that is first saturated with water. A flat face of the sample is then pressed against a flat porous plate (or membrane) in a chamber filled with gas, as shown in Fig. 2. The porous plate is also saturated with water. Often, a moist tissue is placed between the sample and the plate to produce good capillary contact.
Then, pressure in the gas phase above the porous plate is increased by a small step, forcing gas to displace the water from the sample through the plate. When the displacement ceases, the difference in pressure between the gas surrounding the sample and the water on the lower side of the plate is the capillary pressure corresponding to the saturation of water remaining in the sample. After a measurement is completed, the pressure in the gas is increased again, forcing more gas into the sample. This process is repeated, increasing the capillary pressure in a series of steps, yielding the capillary pressure relationship for decreasing water saturation.
If the pressure in the gas is raised too far, the gas will penetrate through the porous plate, terminating the test. The highest capillary pressure that can be achieved with the porous-plate method equals the threshold pressure of the plate—that pressure at which gas can penetrate through the plate. The plate threshold pressure depends on the size of pores in the plates and the interfacial properties of the two fluids separated by the plate. The process of gas injection is often reversed before the capillary pressure reaches the threshold pressure of the plate. By decreasing the pressure in the gas in small steps, the capillary pressure relationship for increasing water saturation can be determined.
Porous-plate methods have been applied to gas/water systems, gas/oil systems, and oil/water systems using porous plates with suitable wettability to prevent penetration through the plate of the phase surrounding the rock sample. Considerable time may be needed to reach equilibrium with porous-plate methods—often a week or more at each displacement step for oil/water systems. Christoffersen and Whitson provide an example of the porous-plate technology with automated control of gas/oil displacements. To reduce equilibration time, their system features a thin membrane in place of a porous plate. They could complete the measurements on a 5-md chalk sample in 14 days.
As suggested earlier, capillary pressure relationships should be measured with an applied overburden stress that approximates the stress in the reservoir. The apparatus of Fig. 2 does not provide specifically for overburden stress, but by confining a cylindrical sample in a rubber (or similarly flexible) sleeve, one can apply approximately the needed stress. The downside of applying stress with the sleeve is that much of the surface area of the sample is blocked, which slows the pace of the test.
Centrifuge methods are increasingly favored for measuring capillary pressures. Although not as quick as mercury measurements, centrifuge measurements are much faster than porous-plate methods. To measure a gas/oil capillary pressure relationship with the centrifuge method, a cylindrical sample is first saturated with oil; next, it is mounted in a centrifuge, as shown in Fig. 3, and is spun in steps of increasing spin rate. The centrifugal forces throw oil from the sample while pulling surrounding gas into the sample. The duration of each spin step must be sufficient for production of oil to cease.
The average saturation of oil in the sample at each spin rate may be calculated from the volume of oil that is produced to the collector relative to the porous volume of the sample. Because 4 to 24 hours are needed to reach equilibrium at each spin rate, most centrifuge data sets consist of eight or fewer spin rates. Ruth and Chen recommend at least 15 spin rates for accurate evaluation of capillary pressure.
The capillary pressure distribution Pc(r) at each spin-rate step depends on:
- The gas/oil density difference Δρ
- The spin rate ω
- The dimensions of the sample relative to the axis of rotation:
Here, ro is the radius from the axis of rotation to the outside face of the sample (see Fig. 3), and r is the radial distance to any point in the sample. At each spin rate, the capillary pressure at the inside face of the sample (at ri) is
The process for converting a set of average saturations and Pci from centrifuge measurements to capillary pressure relationships can involve differentiation of the centrifuge data, as first described by Hassler and Brunner. This differentiation process compounds any error in the original data. So, although the centrifuge method is faster than the porous-plate method, it is not as accurate.
Many variations of the centrifuge method are found in the literature and in industry practice. For example, to measure water/oil capillary pressure for increasing water saturation, the bucket in Fig. 3 is reversed so that the rock is on the outside, surrounded by water; the oil that is displaced by the water segregates toward the axis of rotation. Means for applying overburden stress also have been included in centrifuge designs. Baldwin and Spinler and Spinler et al. used magnetic resonance imaging to obtain capillary pressure relationships by direct measurement of fluid saturations in a rock sample taken from a centrifuge test.
Measurement of relative permeabilities
There are many variations in methods for measuring relative permeabilities; only their general features will be discussed here. Usually, a cylindrical porous sample is mounted in a holder similar to that shown in Fig. 4. The cylindrical surfaces of the sample are sealed to prevent flow. The seal is accomplished in Fig. 4 with a rubber sleeve that also allows for application of a radial confining stress. Fluids are injected and produced from the sample through ports at each end. Often, additional ports are added for pressure measurement. In addition to the sample holder, other apparatus are needed to:
- Inject and collect fluids
- Measure pressures
- Apply confining pressure
- Measure saturations, and so forth
Some of these external features are shown in Fig. 5. Phase saturations can be estimated from:
- The change in mass of the rock sample
- The change of electrical conductivity
- The change of absorption of X-rays or other radiation
Acoustic methods and CT scanning are also used. To measure the change in mass, the rock sample is quickly removed from the assembly, weighed, then returned to the assembly. Because this procedure could cause the saturation to change, in-situ techniques such as electrical conductivity or X-ray absorption have an advantage. Steady-state and unsteady-state methods for measuring relative permeabilities are discussed further in the subsections below.
In steady-state methods, both phases (oil and water, gas and oil, or gas and water) are injected simultaneously at constant rates. Injection continues until a steady state is reached, as indicated by constant pressure drop and constant saturations. Four subcategories of the steady-state methods are introduced below.
In the multiple-core method, frequently called the Penn State method, a rock sample is sandwiched between two other rock samples to build a lengthened sample. The upstream and downstream rock samples distribute flow of the multiple phases over the cross section of the rock and reduce the influence of capillary end effects on the central rock sample. Pressure drop is measured across the central sample while two fluid phases are pumped through the sample at constant flow rates. In some early applications of this method, the saturations in the central rock sample were measured by quickly removing the sample and measuring its mass.
In the high-rate method, two fluid phases are injected into a rock sample at high and constant flow rate. The actual magnitude of rate that is required for this method will depend on the length of the rock sample as well as its capillary pressure properties. The injection rate must be sufficient so that capillary end effects are negligible. Of the steady-state methods, the high-rate subcategory is used most frequently.
In the stationary-liquid method, relative permeability of one highly mobile phase is measured in the presence of an essentially immobile second phase. Typically, the immobile phase is a liquid phase, while the mobile phase is usually gas. Because of the high mobility of gas, the liquid phase can be essentially immobile as long as the pressure gradient is small.
In the uniform-capillary-pressure method, often called the Hassler method, the capillary pressure between two flowing phases is kept uniform throughout a rock sample by keeping the pressure gradients in both phases equal. This is accomplished by incorporating porous plates or membranes at the entrance and exit faces of the porous sample (not shown in Fig. 4). The membranes allow passage of just one of the injected fluids, so the pressure drop in each flowing phase can be measured separately. Although the Hassler method is rarely used, measurement methods with selective membranes are frequently encountered in the literature.
In unsteady-state methods, just one phase is injected at either a constant flow rate or a constant pressure drop. Throughout the injection, the pressure drop and production of phases are measured. Three subcategories are described next.
For measurements with high-rate unsteady-state methods, the injection rate must be sufficient so that capillary spreading effects and capillary end effects can be eliminated. The injection and production data and the differential pressure data must be differentiated to obtain the relative permeabilities. These high-rate methods are used most frequently in the oil and gas industry. They provide results for the least cost and with the least delay in time. The quality of the results has been questioned, but there is evidence in the literature that the methods can give results equivalent to those obtained with other methods.
The enormous increase in computing power and its availability in the last 20 years has facilitated measurement of relative permeabilities in low-rate unsteady-state tests. These tests are preferred to high-rate tests for samples that have fines that become mobile at high rates. The test equipment is identical to that used for high-rate methods, but numerical models are used for interpreting the production and pressure-drop data. The low-rate methods are not widely used.
Relative permeabilities can be measured in centrifuge tests using the same apparatus as that described for measurements of capillary pressure (Fig. 3). Standard practice provides for measurement of the relative permeability of the lowest-mobility phase. To obtain this relative permeability, the production of one phase as a function of drainage time must be measured. Then, differentiation of the data per the algorithm devised by Hagoort gives the relative permeability.
Measurement of endpoint saturations
Endpoint saturations are often valued more highly than capillary pressures and relative permeabilities for several reasons. First, the residual oil saturation for a waterflood defines the maximum amount of oil that can be recovered, so it is very useful for economics calculations. Irreducible water saturation is very useful for assessing the volume of oil in place in a reservoir. Furthermore, the endpoints can be measured more accurately than capillary pressure and relative permeability relationships. As such, some discussion of methods for measuring endpoint saturation is included here.
To measure residual oil saturations after a waterflood or gasflood, the apparatus of Figs. 4 and 5 can be used, but pressure-drop data are not needed—just the oil saturation at the end of the flood is required. As a result, residual oil saturations are much less costly to measure than relative permeabilities and capillary pressures. Still, care is needed to ensure proper wetting conditions for the measurements.
In principle, irreducible water saturation should be measured by oilflooding a rock sample that is initially saturated with water. (Of course, the wettability of the sample will influence the results, so care in handling and preparing the sample is important.) However, irreducible water saturations from such oilflooding are often greater than water saturations measured in retrieved cores from reservoirs. Therefore, alternative approaches have been explored. Satisfying estimates of S wi have been obtained from mercury-injection tests, with the mercury representing the oil phase and the vapor phase in the test representing the displaced water phase.
|capillary pressure, m/Lt2, psi
|capillary pressure at inside face of sample in centrifuge test, m/Lt 2, psi
|density difference for fluids in centrifuge tests, m/L3
|radius to a point in sample in centrifuge test, L
|radius to inside face of sample in centrifuge test, L
|radius to outside face of sample in centrifuge test, L
|spin rate for centrifuge tests, 1/t (ω=2π×RPM/60)
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- Rose, W. and Bruce, W.A. 1949. Evaluation Of Capillary Character In Petroleum Reservoir Rock. J Pet Technol 1 (5): 127-142. SPE-949127-G. http://dx.doi.org/10.2118/949127-G.
- Thomeer, J.H.M. 1960. Introduction of a Pore Geometrical Factor Defined by the Capillary Pressure Curve. J Pet Technol 12 (3): 73-77. SPE-1324-G. http://dx.doi.org/10.2118/1324-G
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- Christiansen, R.L. 2001. Two-Phase Flow through Porous Media, Littleton, Colorado: KNQ Engineering.
- Baldwin, B.A. and Spinler, E.A. 1998. A direct method for simultaneously determining positive and negative capillary pressure curves in reservoir rock. J. Pet. Sci. Eng. 20 (3–4): 161-165. http://dx.doi.org/http://dx.doi.org/10.1016/S0920-4105(98)00016-3
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- DiCarlo, D.A., Sahni, A., and Blunt, M.J. 2000. Three-Phase Relative Permeability of Water-Wet, Oil-Wet, and Mixed-Wet Sandpacks. SPE J. 5 (1): 82-91. SPE-60767-PA. http://dx.doi.org/10.2118/60767-PA
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- Hassler, G.L. 1942. Method and Apparatus for Permeability Measurements. US Patent No. 2,345,935. Also see Brownscombe, E.R., Slobod, R.L., and Caudle, B.H. 1990. Laboratory Determination of Relative Permeability: Parts 1 and 2. Oil & Gas J. (9 February): 68 and (16 February): 98.
- Hagoort, J. 1980. Oil Recovery by Gravity Drainage. SPE J. 20 (3): 139–150. SPE-7424-PA. http://dx.doi.org/10.2118/7424-PA
- Rathmell, J., Atkins, L.K., and Kralik, J.G. 1999. Application of Low Invasion Coring and Outcrop Studies to Reservoir Development Planning for the Villano Field. Presented at the Latin American and Caribbean Petroleum Engineering Conference, Caracas, Venezuela, 21-23 April 1999. SPE-53718-MS. http://dx.doi.org/10.2118/53718-MS
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