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 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.

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  Fig. 1 – Empirical chart relating permeability to porosity with critical interstitial water S_ciw as a parameter, after Granberry and Keelan.

Timur’s model

Timur 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,

....................(2)

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_wi². 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 log₁₀(k)-Φ plot is shown in Fig. 4.

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  Fig. 2 – Permeability/porosity data from three US oil fields by Timur.

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  Fig. 3 – Permeability data from three US oil fields as a function of ϕ^4.4/S_wi², after Timur. Two bounding lines represent the standard error band that includes 68% of the sample points.

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  Fig. 4 – Permeability/porosity relationship with irreducible water saturation as a parameter, after Timur.

It is not easy to apply Eq. 2, which is based totally on core data, to an oil reservoir. 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. 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:

....................(4)

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_bwΦ_t, is computed, and an estimate of a parameter V_bi=S_wiΦ_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,

....................(5)

and the permeability,

....................(6)

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.

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  Fig. 5 – Permeability/porosity relationship based on the dual-water model incorporating irreducible (V_bi ) and bound water (V_bw) as parameters. Four solid curves show effect of increasing V_bi with V_bw = 0.0. Two dashed curves show effect of increasing V_bw with V_bi = 0.05.

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 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.

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  Fig. 6 – Permeability/porosity plot showing newly deposited beach sands and three sample suites from wells in oil and gas fields. Permeability scale ranges from 1,000 darcies to 0.1 μd.

Nuclear magnetic resonance

Eq. 1a indicates that other measures of specific surface area could be correlated with permeability. A study by Sen et al. provides laboratory data on 100 sandstone samples on exchange cation molarity (Q_v), nuclear magnetic resonance (NMR) longitudinal decay time (t₁), and pore-surface-area-to-pore-volume ratio (Σ_p) from the gas adsorption method. Borgia et al. provide data on Σp and t₁ 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.

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  Fig. 7 – Permeability/porosity data from Sen et al. Inserts show log (k) plotted agains log (ϕ^m/Σ_p) and against log(ϕ^mt₁).

Both groups of experimenters found that k correlated best with measures of specific surface when it formed a product with Φ^m or Φ². For example, Sen et al. found k to be strongly correlated (R around 0.9) with (Φ^m/Σ_p)^2.08, with (Φ^mt₁)^2.15, and with (Φ^m/Q_v)^2.11. Two of these correlations are shown as insets in Fig. 7. Borgia et al. did not incorporate m into their regression equations but found k to be best correlated with (Φ⁴/Σ_p²)^0.76 and with (Φ⁴t₁₂)^0.72. As an example of these statistical fits, the expression from Sen et al.,

....................(7)

where k is in millidarcies, t₁ 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.

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  Fig. 8 – Empirical relationship among permeability, porosity, and t₁ from NMR measurement by Sen et al.

Later work showed that the transverse decay time t₂, which is a more practical parameter to detect with a logging tool than t₁, could also be used to estimate permeability (consult the literature for further details on NMR):

....................(8)

where:

• k is in millidarcies
• t_2gm is the geometric mean of t₂ in milliseconds
• c=4.5 in sandstones and 0.1 in carbonates.

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 notes that the NMR measurement is inherently responsive to pore size, whereas permeability depends on pore throat size. He suggests that the experimentally determined Φ⁴ 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

....................(9)

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.

Eq. 9 closely resembles Eq. 4, which is written in terms of irreducible water saturation; V_bvi is computed from the portion of the t₂ spectrum with the smallest times. Except for the porosity term Φ⁴, there is little obvious resemblance between Eqs. 8 and 9. However, Sigal argues that a t₂ 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 t₂ result in different weightings of the pore size distribution spectrum. Sigal relates the two choices of t₂, one explicit in Eq. 8 and the other implicit in Eq. 9, for different distributions of t₂ and for several experimental data sets.

As Sigal points out, the problem of selecting a value of t₂ 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
rh	=	hydraulic radius
Sbw	=	bound water saturation
Swi	=	irreducible water saturation
t1	=	NMR longitudinal decay time
t2	=	NMR transverse decay time
Vffi	=	free fluid fraction
Vbw	=	volume of bound water, fraction
Vbi	=	bulk volume irreducible water, fraction
Σp	=	ratio of pore surface area to pore volume
Σ	=	specific surface area
τ	=	tortuosity
Φ	=	porosity
Φt	=	total porosity
Φe	=	effective porosity

Subscripts

e	=	effective
l	=	liquid
o	=	oil
t	=	total
w	=	water

Noteworthy papers in OnePetro

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External links

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See also

Single phase permeability

Permeability estimation in tight gas reservoirs

Permeability estimation with NMR logging

Permeability determination

PEH:Single-Phase_Permeability