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Capillary pressure models
This page provides an overview of models for calculating capillary pressure based on the relationship to other reservoir properties.
Leverett j-function
Leverett and coworkers,^{[1]} based on the evaluation of gas/water capillary pressure data for drainage and imbibition in unconsolidated sands, proposed the following definition:
The function j(S_{w}), defined in Eq. 1, is known to many as the "Leverett j-function." The j-function is obtained from experimental data by plotting against S_{w}. The combination is often considered an estimate of the mean hydraulic radius of pore throats. However, the directional dependence of permeability complicates this interpretation: Which permeability should be used? While permeabilities for just one direction are one answer, this choice is often not available.
The j-function has been used for correlating capillary pressure data for rocks with similar pore types and wettability, but with different permeabilities.^{[2]} Applications include allotment of oil reserves during unitization negotiations. However, one should be very careful when correlating data with Eq. 1 to use permeabilities that are measured in the same direction. Perhaps some of the scatter typical of j-function correlations results from inconsistent directions of permeability measurement.
The proportionalities of 'Eq. 1 summarize an intuitive expectation for the relationship between capillary pressure, interfacial tension (IFT), and permeability that is widely used. Occasionally, cos θ is included adjacent to σ_{gw} in the definition of the j-function, although Leverett did not write it that way. While the contact angle should affect capillary pressure, the cosine function may not be the correct function to include in the definition of the j-function. Indeed, the dependence of capillary pressure on the contact angle could be quite complex. Nevertheless, the traditional procedure for adjusting capillary pressures for wettability uses the cosine function:
This procedure of Eq. 2 has been applied to the conversion of mercury capillary pressures to oil/water, gas/oil, and gas/water capillary pressures. The results are mixed: sometimes it works well, and sometimes it does not.
Thomeer model
Thomeer^{[3]} proposed a model of the following form for describing mercury-injection capillary pressure data:
in which S_{Hg} is the saturation of mercury. The Thomeer function has three parameters:
- Threshold pressure P_{ct}
- Pore geometric factor G
- Mercury saturation at infinite capillary pressure S_{Hg∞}
(Thomeer wrote Eq. 3 with bulk mercury saturation instead of mercury saturation. Bulk mercury saturation equals the product of porosity and mercury saturation. Bulk mercury saturation is appropriate for irregularly shaped samples as collected from drill cuttings.) Thomeer related absolute permeability to the three parameters of Eq. 3.
Brooks and Corey
Brooks and Corey,^{[4]} extending the earlier work of Corey,^{[5]} suggested the following relationship for capillary pressure during primary drainage of oil from an oil-saturated porous medium during gas invasion:
S_{or} is the residual oil saturation that remains trapped in the pore at high capillary pressure. P_{ct}, the threshold pressure, corresponds approximately to the pressure at which the gas phase is sufficiently connected to allow flow. Brooks and Corey related the parameter λ to the distribution of pore sizes. For narrow distributions, λ is greater than 2; for wide distributions, λ is less than 2. Eq. 4 should be representative of any primary-drainage process as long as the porous medium is homogeneous and strongly wetted by the drainage phase; that is, the contact angle measured through the wetting phase must be small.
Brooks and Corey provide no suggestions for estimating the residual oil saturation and the threshold pressure in Eq. 4. Typical values of residual oil saturation vary from 8 to 40%. Thomas et al.^{[6]} suggested the following expression for roughly estimating gas/water threshold pressures of low-permeability (less than 1 md), water-saturated sandstones and limestones:
(0.43 is near to the 0.5 in the Leverett function). For this expression, the appropriate units are millidarcies for permeability k and psi for threshold pressure P_{cgwt}. Eq. 5 should be applicable to fluid pairs other than gas and water if it is adjusted for IFT differences.
Bentsen and Anli
Bentsen and Anli^{[7]} proposed the following expression for capillary pressure for a primary-drainage process in which a porous sample initially saturated with water is invaded by an oil phase:
P_{cs} is a parameter with pressure units for controlling the shape of the capillary pressure function. Bentsen and Anli developed Eq. 6 following a qualitative argument. These authors reported a range of parameters for several rock/oil/water systems, but they did not suggest means for estimating those parameters.
Alpak-Lake-Embid model
An interesting model was proposed by Alpak et al.^{[8]} for representing both capillary pressure and relative permeability relationships. For capillary pressure, they suggested an expression that can be obtained by applying basic thermodynamic arguments to capillary pressure concepts. Their model relates capillary pressure to the change of oil/water interfacial area and water/solid interfacial area with water saturation. Alpak et al.^{[8]} applied their model to drainage and imbibition data with fair success. Research in the years to come may show whether this approach to interpreting capillary pressure is useful.
Nomenclature
G | = | pore geometric factor in Thomeer function |
j ( S_{w}) | = | Leverett j-function |
k | = | permeability, L^{2}, md |
P_{c} | = | capillary pressure, m/Lt^{2}, psi |
P_{cgo} | = | capillary pressure between gas and oil phases, m/Lt^{2}, psi |
P_{cow} | = | capillary pressure between oil and water phases, m/Lt^{2}, psi |
P_{cgw} | = | capillary pressure between gas and water phases, m/Lt^{2}, psi |
P_{cgwt} | = | threshold capillary pressure between gas and water phases, m/Lt^{2}, psi |
P_{ct} | = | threshold capillary pressure, m/Lt ^{2}, psi |
P_{cs} | = | shape parameter in Bentsen-Anli function , m/Lt ^{2}, psi |
S_{o} | = | saturation of oil |
S_{or} | = | residual saturation of oil |
S_{orw} | = | residual saturation of oil for a water/oil displacement |
S_{oS} | = | scaled saturation of oil for Stone I model |
S_{w} | = | saturation of water |
S_{wi} | = | irreducible or residual saturation of water |
θ | = | contact angle, degrees |
λ | = | pore-size-distribution parameter in Corey functions |
σ_{gw} | = | gas/water interfacial tension, m/t ^{2}, dyne/cm |
Φ | = | porosity |
References
- ↑ Leverett, M.C. 1941. Capillary Behavior in Porous Solids. Trans. of AIME 142 (1): 152-169. http://dx.doi.org/10.2118/941152-G.; see also Leverett, M.C., Lewis, W.B., and True, M.E. 1942. Dimensional-model Studies of Oil-field Behavior. Trans. of AIME 146 (1): 175-193. SPE-942175-G. http://dx.doi.org/10.2118/942175-G
- ↑ Brown, H.W. 1951. Capillary Pressure Investigations. J Pet Technol 3 (3): 67-74. SPE-951067-G. http://dx.doi.org/10.2118/951067-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
- ↑ Brooks, R.H. and Corey, A.T. 1964. Hydraulic properties of porous media. Hydrology Paper No. 3, Colorado State University, Fort Collins, Colorado, 22–27.
- ↑ Corey, A.T. 1954. The interrelation between gas and oil relative permeabilities. Producers Monthly 19 (November): 38–41.
- ↑ Thomas, L.K., Katz, D.L., and Tek, M.R. 1968. Threshold Pressure Phenomena in Porous Media. SPE J. 8 (2): 174–184. SPE-1816-PA. http://dx.doi.org/10.2118/1816-PA
- ↑ Bentsen, R.G. and Anli, J. 1976. A New Displacement Capillary Pressure Model. J Can Pet Technol 15 (3). PETSOC-76-03-10. http://dx.doi.org/10.2118/76-03-10. For further applications of the model, see Bentsen, R.G. and Anli, J. 1977. Using Parameter Estimation Techniques To Convert Centrifuge Data Into a Capillary-Pressure Curve. SPE J 17 (1): 57-64. SPE-5026-PA. http://dx.doi.org/10.2118/5026-PA
- ↑ ^{8.0} ^{8.1} Alpak, F.O., Lake, L.W., and Embid, S.M. 1999. Validation of a Modified Carman-Kozeny Equation To Model Two-Phase Relative Permeabilities. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October 1999. SPE-56479-MS. http://dx.doi.org/10.2118/56479-MS
Noteworthy papers in OnePetro
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See also
Relative permeability and capillary pressure
Measurement of capillary pressure and relative permeability
Methods to determine pore pressure
PEH:Relative_Permeability_and_Capillary_Pressure