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Estimating permeability based on surface area and water saturation
Estimating permeability has been approached using a variety of models considering different rock characteristics. This page discusses methods for estimating permeability considering surface area and water saturation.
Starting from Kozeny-Carman equations
Two ideas inherent in Kozeny-Carman are important for later developments: the dependence of k on a power of porosity and on the inverse square of surface area. The various forms of Eq. 1 have been used as a starting point for predicting permeability from well log data by assuming that residual water saturation is proportional to specific surface area, Σ.
Basic equation: ....................(1a)
Specific surface as ratio of pore surface area to rock volume: ....................(1b)
Specific surface area as ratio of pore surface to grain volume: ....................(1c)
With tortuosity eliminated: ....................(1d)
Granberry and Keelan’s chart
Granberry and Keelan^{[1]} published a set of curves relating permeability, porosity, and "critical water" saturation (S_{ciw}) for Gulf Coast Tertiary sands that frequently are poorly consolidated. Their chart, originally presented with S_{ciw} as a function of permeability with porosity as a parameter, is transposed into log(k)-Φ coordinates in Fig. 1. The S_{ciw} parameter is taken from the "knee" of a capillary pressure curve and is greater than irreducible water saturation, S_{wi}. It is said that if the water saturation in the formation is less than this critical value, the well will produce water free. Because S_{ciw} is taken from the capillary pressure curve, it is a function of the size of interconnected pores. Fig. 1 cannot be used to estimate permeability from porosity and water saturation as determined from well logs because it reflects only the critical water saturation. It was determined from reservoirs in which oil viscosity was approximately twice that of water and requires adjustment for low- or high-gravity oils.
Timur’s model
Timur^{[2]} used a database of 155 sandstone samples from three oil fields (Fig. 2) . The three sandstones exhibit varying degrees of sorting, consolidation, and ranges of porosity. Timur measured irreducible water saturation (S_{wi}) using a centrifuge and then held k proportional to S_{wi}^{−2} in the general power-law relationship,
Coefficients a and b were determined statistically. Timur’s statistical results show that the exponent b can range between 3 and 5 and still give reasonable results. Results for b=4.4 produced a fit somewhat better than other values; it was obtained by taking the logarithm of both sides of Eq. 3 and testing the correlation coefficient with respect to Φ^{b}/S_{wi}^{2}. For b=4.4, the value of a is 0.136 if Φ and S_{wi} are in percent and 8,581 if Φ and S_{wi} are fractional values. There is no theoretical basis for the substitution of S_{wi} for specific surface area Σ, so although the form of Eq. 2 is similar to that of Eq. 1, it is strictly an empirical relationship. The effectiveness of Eq. 2 as a predictor of permeability is shown in Fig. 3, and its form on a log_{10}(k)-Φ plot is shown in Fig. 4.
It is not easy to apply Eq. 2, which is based totally on core data, to an oil reservoir.^{[3]} The S_{wi} core data used to establish Eq. 2 were obtained for a fixed value of capillary pressure (P_{c}). In a reservoir, P_{c} varies with height, and because S_{wi} varies with P_{c}, it is necessary to assume a functional dependence of S_{wi} on P_{c}. There are also some practical difficulties in establishing the coefficients a and b in a reservoir in which the oil/water contact cuts across lithologies because of regional dip or structure. In particular, within the transition zone, only part of the water is irreducible (S_{wi}); the remainder is movable. Thus, a log-based estimate of saturation immediately above the oil/water contact will overestimate S_{wi}.
Dual water model
An algorithm discussed by Ahmed et al.^{[4]} is attributed to Coates. An extension of Eq. 2 and Fig. 4, it assumes that permeability declines to zero as S wi increases to fill the entire pore space:
A further refinement incorporates the presence of clay minerals and is based on the dual water model. It requires log-based estimates of the total porosity (Φ_{t}) and either effective porosity (Φ_{e}) or bound water saturation (S_{bw}). Effective porosity is defined as Φ_{e}=Φ_{t}(1-S_{bw}). The fractional volume of bound water, V_{bw}=S_{bw}Φ_{t}, is computed, and an estimate of a parameter V_{bi}=S_{wi}Φ_{t} called the (fractional) bulk volume irreducible water in clean wet rock must also be provided. Then, computed as a function of depth is the total immovable water,
and the permeability,
The algorithm of Eqs. 5 and 6 uses a pair of parameters, V_{bi} and V_{bw}, which in effect sweep out a broad region of the log(k)-Φ crossplot (Fig. 5). For the solid curves, V_{bw} has been set to 0.0 as if the rock were entirely clay free. As irreducible water V_{bi} increases, the curves shift downward and to the right, into the regime populated by fine-grained rock. The dashed curve is drawn for V_{bw} and V_{bi} each equal to 0.05, thereby representing one of a second family of curves for a fine-grained dirty sandstone. Note how S_{bw} increases with decreasing Φ.
This algorithm produces reasonable results in sandstones if V_{bi} is chosen judiciously. One difficulty is choosing a value for V_{bi} in coarse-grained and gravel-bearing sandstones.
"Tight" sandstones
Predicting permeability becomes much more difficult in formations with small grain size and an abundance of clay minerals. Such rocks are called "tight gas sands" or "submillidarcy reservoirs" (see example in Fig. 6). Kukal and Simons^{[5]} show that the Timur equation produces k values much too high in such formations and establish some predictive equations that decrease the porosity by multiplying Φ by 1-V_{cl}, where V_{cl} is the clay fraction. They show that the water saturation term S_{wi} is not so important in these high-clay rocks. Although their predictive equation is a welcome improvement, the scatter shows the difficulty in dealing with such low-porosity systems.
Nuclear magnetic resonance
Eq. 1a indicates that other measures of specific surface area could be correlated with permeability. A study by Sen et al.^{[6]} provides laboratory data on 100 sandstone samples on exchange cation molarity (Q_{v}), nuclear magnetic resonance (NMR) longitudinal decay time (t_{1}), and pore-surface-area-to-pore-volume ratio (Σ_{p}) from the gas adsorption method. Borgia et al.^{[7]} provide data on Σp and t_{1} on 32 samples. Both studies include measurements of k, Φ, and formation factor (F) on their samples. Both sample suites are made up of samples from different formations, so the log(k)-Φ plots exhibit scatter, as shown by Fig. 7.
Both groups of experimenters found that k correlated best with measures of specific surface when it formed a product with Φ^{m} or Φ^{2}. For example, Sen et al.^{[6]} found k to be strongly correlated (R around 0.9) with (Φ^{m}/Σ_{p})^{2.08}, with (Φ^{m}t_{1})^{2.15}, and with (Φ^{m}/Q_{v})^{2.11}. Two of these correlations are shown as insets in Fig. 7. Borgia et al.^{[7]} did not incorporate m into their regression equations but found k to be best correlated with (Φ^{4}/Σ_{p}^{2})^{0.76} and with (Φ^{4}t_{12})^{0.72}. As an example of these statistical fits, the expression from Sen et al.,^{[6]}
where k is in millidarcies, t_{1} is in milliseconds, and Φ is fractional porosity, is plotted in Fig. 8. Because the porosity exponent is very close to that established by Timur (see Estimating permeability considering mineralogy), the curves in Fig. 8 are quite similar to those in Fig. 4.
Later work showed that the transverse decay time t_{2}, which is a more practical parameter to detect with a logging tool than t_{1}, could also be used to estimate permeability (consult the literature^{[8]}^{[9]}^{[10]} for further details on NMR):
where:
- k is in millidarcies
- t_{2gm} is the geometric mean of t_{2} in milliseconds
- c=4.5 in sandstones and 0.1 in carbonates.^{[10]}
The value of k obtained from Eq. 1 is referred to as k_{SDR}. Better results are obtained if a cutoff can be selected for t_{2L} so that only the pores contributing to permeability are included. Kenyon^{[10]} notes that the NMR measurement is inherently responsive to pore size, whereas permeability depends on pore throat size. He suggests that the experimentally determined Φ^{4} dependence somehow accounts for the way in which the ratio (pore throat size to pore size) varies with porosity.
The Coates equation for estimating permeability is
where:
- k is in millidarcies
- V_{bvi} is the bulk volume irreducible fluid fraction
- V_{ffi} is the free fluid fraction and is equal to Φ-V_{bvi}
- Porosity Φ is taken from the NMR tool.^{[9]}^{[11]}
Eq. 9 closely resembles Eq. 4, which is written in terms of irreducible water saturation; V_{bvi} is computed from the portion of the t_{2} spectrum with the smallest times. Except for the porosity term Φ^{4}, there is little obvious resemblance between Eqs. 8 and 9. However, Sigal^{[12]} argues that a t_{2} cutoff time is implicit in Eq. 9 and that its value is incorporated in the constant c. Even so, the two equations are not equivalent because the two choices of t_{2} result in different weightings of the pore size distribution spectrum. Sigal relates the two choices of t_{2}, one explicit in Eq. 8 and the other implicit in Eq. 9, for different distributions of t_{2} and for several experimental data sets.
As Sigal^{[12]} points out, the problem of selecting a value of t_{2} from NMR data is analogous to the problem of selecting a value of R from capillary pressure data (See Estimating permeability based on pore dimension) : One must capture the length scale appropriate to the estimation of permeability.
Summary
Timur’s equation and its corresponding chart offer a viable method of permeability estimation in which porosity and irreducible water saturation can be estimated. Difficulties arise if there is uncertainty in S_{wi}, as there is within an extensive transition zone. The dual water predictor is an interesting embellishment that can include a clay content parameter. Laboratory data show that, when combined with Φ^{m}, the following all correlate well with k:
- Specific surface area
- Cation exchange molarity
- NMR decay time
Because it responds to the pore size spectrum, NMR is a particularly effective tool in obtaining log-based estimates of permeability.
Nomenclature
f | = | shape factor |
F | = | formation factor |
g | = | gravitational acceleration |
k | = | permeability |
m | = | Archie cementation exponent |
p | = | pressure |
r_{h} | = | hydraulic radius |
S_{bw} | = | bound water saturation |
S_{wi} | = | irreducible water saturation |
t_{1} | = | NMR longitudinal decay time |
t_{2} | = | NMR transverse decay time |
V_{ffi} | = | free fluid fraction |
V_{bw} | = | volume of bound water, fraction |
V_{bi} | = | bulk volume irreducible water, fraction |
Σ_{p} | = | ratio of pore surface area to pore volume |
Σ | = | specific surface area |
τ | = | tortuosity |
Φ | = | porosity |
Φ_{t} | = | total porosity |
Φ_{e} | = | effective porosity |
Subscripts
e | = | effective |
l | = | liquid |
o | = | oil |
t | = | total |
w | = | water |
References
- ↑ ^{1.0} ^{1.1} 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.
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} Timur, A. 1968. An Investigation Of Permeability, Porosity, & Residual Water Saturation Relationships For Sandstone Reservoirs. The Log Analyst IX (4). SPWLA-1968-vIXn4a2.
- ↑ Raymer, L.L. 1981. Elevation And Hydrocarbon Density Correction For Log-derived Permeability Relationships. The Log Analyst XXII (3): 3. SPWLA-1981-vXXIIn3a1.
- ↑ Ahmed, U., Crary, S.F., and Coates, G.R. 1991. Permeability Estimation: The Various Sources and Their Interrelationships. J Pet Technol 43 (5): 578-587. SPE-19604-PA. http://dx.doi.org/10.2118/19604-PA
- ↑ Kukal, G.C. and Simons, K.E. 1986. Log Analysis Techniques for Quantifying the Permeability of Submillidarcy Sandstone Reservoirs. SPE Form Eval 1 (6): 609-622. SPE-13880-PA. http://dx.doi.org/10.2118/13880-PA
- ↑ ^{6.0} ^{6.1} ^{6.2} ^{6.3} ^{6.4} Sen, P.N. et al. 1990. Surface-To-Volume Ratio, Charge Density, Nuclear Magnetic Relaxation, and Permeability in Clay-Bearing Sandstones. Geophysics 55 (1): 61-69. http://dx.doi.org/10.1190/1.1442772.
- ↑ ^{7.0} ^{7.1} Borgia, G.C., Brighenti, G., Fantazzini, P. et al. 1992. Specific Surface and Fluid Transport in Sandstones Through NMR Studies. SPE Form Eval 7 (3): 206-210. SPE-20921-PA. http://dx.doi.org/10.2118/20921-PA
- ↑ Hearst, J.R., Nelson, P.H., and Paillet, F.L. 2000. Well Logging for Physical Properties. New York City: John Wiley & Sons.
- ↑ ^{9.0} ^{9.1} Coates, G.R., Xiao, L., and Prammer, M.G. 1999. NMR Logging, Principles and Applications. Houston, Texas: Halliburton Energy Services.
- ↑ ^{10.0} ^{10.1} ^{10.2} Kenyon, W.E. 1997. Petrophysical Principles of Applications of NMR Logging. Log Analyst 38 (2): 21.
- ↑ Coates, G.R., Peveraro, R.C.A., Hardwick, A. et al. 1991. The Magnetic Resonance Imaging Log Characterized by Comparison With Petrophysical Properties and Laboratory Core Data. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 6–9 October. SPE-22723-MS. http://dx.doi.org/10.2118/22723-MS
- ↑ ^{12.0} ^{12.1} Sigal, R. 2002. Coates and SDR Permeability: Two Variations on the Same Theme. Petrophysics 43 (1): 38.
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See also
Permeability estimation in tight gas reservoirs