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Estimating permeability based on pore dimension
This page discusses single phase permeability models that are specifically based on the pore dimensions of the reservoir. Pore dimensions are a critical factor in determining crucial characteristics of the reservoir; including porosity, permeability, and capillary pressure.
Contents
Capillary pressure and pore size
The dimension of interconnected pores plays a major role in determining permeability. Most methods of estimating permeability are indirect methods. A viable direct method requires both adequate theoretical underpinnings relating pore throat dimension to permeability and experimental determination of the critical pore dimension parameters. Many authors have made use of the capillary pressure curve, obtained experimentally by injecting mercury into a dried sample. As mercury pressure is increased, more mercury is forced into progressively smaller pores in the rock, and the resident pore fluid (air) is expelled. A length r, usually referred to as the pore throat radius, is related to the injection pressure by the Washburn equation,
where:
σ is the interfacial tension θ is the wetting angle
The injection process can be visualized by examining the idealized capillary pressure curve of Fig. 1. A finite pressure is required to inject mercury into a 100% water-saturated sample (right side of Fig. 1). At the first inflection point (entry pressure), mercury occupies only a small fraction of the pore volume containing the largest pores. Next, much of the pore space becomes filled with mercury with a comparatively slight increase in pressure (progressing from the circle labeled Katz and Thompson to the circle labeled Swanson in Fig. 1). Finally, large pressure increases are required to force more mercury into the smallest pores (steep curve to left of Swanson circle).
Many authors have linked capillary pressure curves to permeability. Purcell^{[1]} derived an expression relating k to an integral of P_{c}^{-2} over the entire saturation span, achieving a good match with core data. The relationships established by Timur^{[2]} and Granberry and Keelan,^{[3]}, are represented at low water saturation in Fig. 1. Contributions by Swanson,^{[4]} Winland,^{[5]} and Katz and Thompson,^{[6]} symbolized by the circles in Fig. 1, are reviewed below.
Swanson’s equation
Swanson^{[4]} provides a method of determining air and brine permeabilities from a single point on the capillary pressure curve. His regression relationships are based on permeability and capillary pressure data on 203 sandstone samples from 41 formations and 116 carbonates from 33 formations. His method picks the maximum ratio of mercury saturation to pressure, (S_{b}/P_{c})_{max}, from the capillary pressure curve, arguing that at this point all the connected space is filled with mercury and "this capillary pressure corresponds to pore sizes effectively interconnecting the total major pore system and, thus, those that dominate fluid flow." From linear regression, Swanson obtains simple equations of the form,
where the constants a and c depend on the following:
- Rock type (carbonate vs. sandstone)
- Fluid type (air or brine)
For carbonates and sandstones combined, c=2.005. Because S_{b} is defined as the mercury saturation as percent of bulk volume, it must be proportional to Φ(1-S_{w}); through Eq. 1, P_{c} can be linked with a pore throat radius r apex . Thus, Swanson’s result shows that k is proportional to [Φ(1-S_{wi})r_{apex}]^{2}, again demonstrating the dependence of k on the square of a pore throat size.
Winland’s equation and Pittman’s results
An empirical equation relating permeability, porosity, and a capillary pressure parameter is referred to as Winland’s equation.^{[5]}^{[7]} Based on laboratory measurements on 312 samples, Winland’s regression equation is
where:
- r_{35} is the pore throat radius at 35% mercury saturation
- k is air permeability
- Φ is porosity in percent
A log(k)-Φ plot based on Eq. 4 and showing five characteristic lines for pore throat radius is shown in Fig. 2. Note that at a given porosity, permeability increases roughly as the square of the pore throat radius. And for a given throat size, the dependence of permeability on porosity is slightly less than Φ^{2}. Kolodzie^{[5]} states that a pore throat size of 0.5 μm was used as a cutoff for reserves determinations, in preference to the use of k or Φ. Hartmann and Coalson^{[8]} also present Winland’s equation in the same format as Fig. 2. They state that r_{35} is a function of both entry size and pore throat sorting and is a good measure of the largest connected pore throats in a rock with intergranular porosity.
Martin et al.^{[9]} used the r_{35} parameter, along with other petrophysical, geological, and engineering data, to identify flow units in five carbonate reservoirs. With Eq. 4, r_{35} can be computed from permeability and porosity measurements on core samples. Flow units are grouped by the size of pore throats using the designations, as shown in Fig. 2, of:
- Megaport
- Macroport
- Mesoport
- Microport
A completion analysis for the different r_{35} size ranges in a reservoir of medium thickness and medium gravity oil yielded the following:
- Megaport, tens of thousands of barrels of oil per day
- Macroport, thousands
- Mesoport, hundreds
- Microport, nonreservoir
After flow units are identified, well logs and sequence stratigraphy are used to identify zones with similar properties where no core data exist. The method works well in carbonates where flow is controlled by intergranular, intercrystalline, or interparticle pore space but not so well if fractures or vugs are present.
Pittman^{[7]} sheds additional light on Winland’s equation, linking it to Swanson’s results. Pittman used a set of 202 sandstone samples from 14 formations on which k, Φ, and mercury injection data had been obtained. Using Eq. 2, he associated a pore size r_{apex} with the capillary pressure, P_{c}, determined by Swanson’s method and found that the mean value of r_{apex} has a mercury saturation of 36%. That is, on a statistical basis, the points denoted by circles labeled Swanson and Winland in Fig. 1 are practically identical, and the two methods are sampling the same fraction of the pore space.
Pittman^{[7]} also established regression equations for pore aperture sizes ranging from 10% to 75% mercury saturation. His expressions have been rearranged and displayed in Table 1 to show the exponents of r and Φ required to predict k. (Because r was used as the dependent variable in Pittman’s regressions, the coefficients in Table 1 differ somewhat from what would be obtained if k were the dependent variable; however the changes would not invalidate the point of this discussion.) Note that, with increasing mercury saturation:
- r exponent decreases
- Φ exponent increases
That is, the porosity term contributes relatively less to k than does r for mercury saturation values <35%. In fact, Pittman noted that the porosity term was statistically insignificant for r_{10} through r_{35}.
Katz and Thompson’s equation
Another investigation on the influence of pore structure on flow properties comes from Katz and Thompson^{[6]} and Thompson et al.^{[10]} They use percolation theory to derive a deceptively simple relationship,
where:
- k is absolute permeability (same units as )
- σ is electrical conductivity of the rock
- σ_{o} is the conductivity of the saturant
The value of the constant, given as 1/226, is dependent on the geometry assumed for the pore space. They substantiate Eq. 5a with experimental data on 60 sandstone and carbonate samples with permeabilities ranging from <1 md to 5 darcies.
The parameter l_{c} in Eq. 5a represents a dimension of a very particular subset of pores: "The arguments suggest that permeability can be estimated by assuming that the effective pore size is the smallest pore on the connected path of pores containing the largest pores. We call that effective pore size l_{c}." To obtain l_{c}, the pressure at the inflection point on a capillary pressure curve is converted to a diameter. The authors argue that the inflection point marks the pressure at which a sample is first filled continuously end to end with mercury and that the large pores first filled are those that control permeability.
The Katz and Thompson^{[6]} equation and its characteristic curves are given in Fig. 3. To plot curves on log(k)-Φ plots, we assumed the simplest relation between formation factor and porosity (cementation exponent of 2.0), σ/σ_{o}=Φ^{2}. Some data points from Katz and Thompson’s experiments are posted in Fig. 3 to indicate how well their measured l_{c} match the curves (This is not really a test of their model because they used formation factor in their correlations, not Φ^{2}). Their result is similar to that of Swanson’s and Winland’s equations: Permeability is closely proportional to the square of rΦ.
To obtain compatibility with other author’s expressions, we define a critical radius r_{c}=l_{c}/2, keeping both permeability and r_{c}^{2} in units of μm^{2}:
Eq. 5b is identical in form to the Kozeny-Carman equation with tortuosity eliminated, but the percolation concepts used to derive Eq. 5b are quite different from the geometrical arguments used to derive the Kozeny-Carman expression. The Kozeny-Carman coefficient, which is ≈0.4, is considerably greater than that (0.0177) in Eq. 4b. Consequently, the characteristic radius r_{c} is ≈4.7 times greater than the hydraulic radius, r_{h}. Although r_{h} is defined as the ratio of pore volume to pore surface area, it can be determined in a variety of ways, including the use of mercury injection. Conceptually, then, the Kozeny-Carman equation could also be represented by an extended horizontal line across Fig. 1; i.e., as a method that samples a broad spectrum of pore sizes.
It is interesting to compare the Katz and Thompson model (Fig. 3) with Winland’s empirical equation (Fig. 2). The shapes of the curves are comparable; i.e., the models agree on the approximate Φ^{2} dependence. The pore radii given by the Winland equation are smaller than comparable radii in the Katz and Thompson model. This is expected because the Winland equation requires a saturation of 35%, a criterion of greater injection pressure than that of Katz and Thompson. What is noteworthy is the general agreement between the two models regarding the form of the log(k)-Φ relationship. They demonstrate that in the models invoking higher powers of Φ, which we have shown in previous graphs are not well grounded physically, the higher powers of Φ are required to compensate for lack of knowledge regarding the critical pore dimension. It does seem, however, that the empirical data that often show a "straight-line" log(k)-Φ relationship contain some fundamental information regarding how the critical pore dimension relates to porosity.
Flow zone indicator
Amaefule and Altunbay^{[11]} rearranged the version of the Kozeny-Carman equation with specific surface area as a ratio of pore surface to grain volume to obtain a parameter group named the flow zone indicator (I),
where the factor 0.0314 allows k to be expressed in millidarcies. As can be seen from Eq. 5, I has the units of pore size, in micrometers, and can be computed from core measurements of k and Φ, even though it is defined in terms of f, τ, and Σ_{g}, which are not easily measured. The choice of the form used over other forms of the Kozeny-Carman equation that use alternative definition of specific surface area seems a bit arbitrary and results in the particular combination of porosity terms used in Eq. 6.
Amaefule and Altunbay^{[11]} use I to define zones called "hydraulic flow units" on a doubly logarithmic plot incorporating the terms in Eq. 6. For compatibility with other plots in this chapter, a plot in log(k)-Φ coordinates is shown in Fig. 4. Each data point on a log(k)-Φ plot has an I value that associates it with a nearby curve of constant I value. The difficult step is deciding where the boundaries between adjacent I bands should be positioned and how to compute a value of I from well logs in uncored wells. Options for doing so are described in Estimating permeability from well log data.
Nomenclature
f | = | shape factor |
I | = | flow zone indicator |
k | = | permeability |
l_{c} | = | pore-space dimension |
p | = | pressure |
P_{c} | = | capillary pressure |
r_{h} | = | hydraulic radius |
r_{35} | = | pore throat radius at 35% mercury saturation |
R | = | pore throat dimension |
S_{b} | = | mercury saturation |
θ | = | wetting angle |
σ | = | electrical conductivity of rock |
σ_{o} | = | electrical conductivity of saturant |
σ | = | interfacial tension |
Σ_{p} | = | ratio of pore surface area to pore volume |
Σ_{r} | = | ratio of pore surface area to rock volume |
Σ_{g} | = | ratio of pore surface area to grain volume |
Σ | = | specific surface area |
τ | = | tortuosity |
Φ | = | porosity |
References
- ↑ Purcell, W.R. 1949. Capillary Pressures: Their Measurement Using Mercury and the Calculation of Permeability Therefrom. Trans., AIME 1 (2): 39-48. http://dx.doi.org/10.2118/949039-G
- ↑ Timur, A. 1968. An Investigation Of Permeability, Porosity, & Residual Water Saturation Relationships For Sandstone Reservoirs. The Log Analyst IX (4). SPWLA-1968-vIXn4a2.
- ↑ Granberry, R.J., and Keelan, D.K. 1977. Critical Water Estimates for Gulf Coast Sands. Trans., Gulf Coast Association of Geological Societies 27: 41-43.
- ↑ ^{4.0} ^{4.1} 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
- ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} Kolodzie Jr., S. 1980. Analysis of Pore Throat Size And Use of the Waxman-Smits Equation To Determine OOIP in Spindle Field, Colorado. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, 21-24 September. SPE 9382. http://dx.doi.org/10.2118/9382-MS
- ↑ ^{6.0} ^{6.1} ^{6.2} ^{6.3} Katz, A.J. and Thompson, A.H. 1986. Quantitative Prediction of Permeability in Porous Rock. Physical Review B 34 (11): 8179.
- ↑ ^{7.0} ^{7.1} ^{7.2} Pittman, E.D. 1992. Relationship of Porosity and Permeability to Various parameters Derived From Mercury Injection—Capillary Pressure Curves for Sandstone. American Association of Petroleum Geologists Bull. 76 (2): 191-198.
- ↑ Hartmann, D.J. and Coalson, E.B. 1990. Evaluation of the Morrow Sandstone in Sorrento Field, Cheyenne County, Colorado. Morrow Sandstones of Southeast Colorado and Adjacent Areas, 91, eds. S.A. Sonnenberg et al. Rocky Mountain Association of Geologists.
- ↑ ^{9.0} ^{9.1} Martin, A.J., Solomon, S.T., and Hartmann, D.J. 1997. Characterization of Petrophysical Flow Units in Five Carbonate Reservoirs. American Association of Petroleum Geologists Bull. 81 (5): 734.
- ↑ Thompson, A.H., Katz, A.J., and Krohn, C.E. 1987. The Microgeometry and Transport Properties of Sedimentary Rock. Advances in Physics 36 (5): 625. http://dx.doi.org/ 10.1080/00018738700101062
- ↑ ^{11.0} ^{11.1} ^{11.2} Amaefule, J.O., Altunbay, M., Tiab, D. et al. 1993. Enhanced Reservoir Description: Using Core and Log Data to Identify Hydraulic (Flow) Units and Predict Permeability in Uncored Intervals/Wells. Presented at the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October. SPE 26436. http://dx.doi.org/10.2118/26436-MS
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See also
Relative permeability and capillary pressure