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

1 Capillary pressure and pore size

2 Swanson’s equation

3 Winland’s equation and Pittman’s results

4 Katz and Thompson’s equation

5 Flow zone indicator

6 Nomenclature

7 References

8 Noteworthy papers in OnePetro

9 External links

10 See also

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,

....................(1)

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).

Fig. 1 – Capillary pressure curve (idealized) showing measures used by different

authors for determination of characteristic pore dimension.

Many authors have linked capillary pressure curves to permeability. Purcell[1]

derived an

expression relating k to an integral of Pc-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, (Sb/Pc)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,

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

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 Sb is defined as the mercury

saturation as percent of bulk volume, it must be proportional to Φ(1-Sw); through Eq. 1, Pc can

be linked with a pore throat radius r apex . Thus, Swanson’s result shows that k is proportional to

[Φ(1-Swi)rapex]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

....................(3)

where:

r35 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 Φ2. 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 r35 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.

Fig. 2 – Empirical model based on regression attributed to Winland, from Kolodzie. [5]

Labels for four ranges of r35 are taken from Martin et al.[9]

Martin et al.[9]

used the r35 parameter, along with other petrophysical, geological, and

engineering data, to identify flow units in five carbonate reservoirs. With Eq. 4, r35 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 r35 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 rapex with the capillary pressure,

Pc, determined by Swanson’s method and found that the mean value of rapex 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 r10 through

r35.

Table 1

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,

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

where:

k is absolute permeability (same units as )

σ is electrical conductivity of the rock

σo 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 lc 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 lc." To obtain lc, 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/σo=Φ2. Some data points from Katz and Thompson’s

experiments are posted in Fig. 3 to indicate how well their measured lc match the curves (This is

not really a test of their model because they used formation factor in their correlations, not Φ2).

Their result is similar to that of Swanson’s and Winland’s equations: Permeability is closely

proportional to the square of rΦ.

Fig. 3 – Permeability equation with critical pore-size radius (Rc) as a parameter,

from Katz and Thompson.[6]

Values of rc posted next to data points are from

mercury injection tests.

To obtain compatibility with other author’s expressions, we define a critical radius rc=lc/2,

keeping both permeability and rc2 in units of μm

2:

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

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 rc is

≈4.7 times greater than the hydraulic radius, rh. Although rh 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 Φ2 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 ratio of pore surface to grain volume to obtain a parameter group named the flow

zone indicator (I),

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

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 Σg, 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.

Fig. 4 – Zonation of permeability and porosity data based on a parameter called the

flow zone indicator (I) in μm. Data from southeast Asia and algorithm taken from

Amaefule and Altunbay.[11]

Nomenclature

f = shape factor

I = flow zone indicator

k = permeability

lc = pore-space dimension

p = pressure

Pc = capillary pressure

rh = hydraulic radius

r35 = pore throat radius at 35% mercury saturation

R = pore throat dimension

Sb = mercury saturation

θ = wetting angle

σ = electrical conductivity of rock

σo = electrical conductivity of saturant

σ = interfacial tension

Σp = ratio of pore surface area to pore volume

Σr = ratio of pore surface area to rock volume

Σg = ratio of pore surface area to grain volume

Σ = specific surface area

τ = tortuosity

Φ = porosity

References

1. ↑ 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

2. ↑ Timur, A. 1968. An Investigation Of Permeability, Porosity, & Residual Water

Saturation Relationships For Sandstone Reservoirs. The Log Analyst IX (4). SPWLA-

1968-vIXn4a2.

3. ↑ 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. ↑ 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. ↑ 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. ↑ 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. ↑ 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.

8. ↑ 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. ↑ 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.

10. ↑ 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. ↑ 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

Noteworthy papers in OnePetro

Use this section to list papers in OnePetro that a reader who wants to learn more should

definitely read

External links

Use this section to provide links to relevant material on websites other than PetroWiki and

OnePetro

See also

Single phase permeability

Relative permeability and capillary pressure

Corrections to core measurements of permeability

Permeability determination

PEH:Single-Phase Permeability

Estimating permeability from well log data

Many approaches to estimating permeability exist. Recognizing the importance of rock type,

various petrophysical (grain size, surface area, and pore size) models have been developed. (See

links to these in Single phase permeability). This page explores techniques for applying well logs

and other data to the problem of predicting permeability [k or log(k)] in uncored wells.

If the rock formation of interest has a fairly uniform grain composition and a common diagenetic

history, then log(k)-Φ patterns are simple, straightforward statistical prediction techniques can be

used, and reservoir zonation is not required. However, if a field encompasses several lithologies,

perhaps with varying diagenetic imprints resulting from varying mineral composition and fluid

flow histories, then the log(k)-Φ patterns are scattered, and reservoir zonation is required before

predictive techniques can be applied.

Contents

1 Multiple linear regression o 1.1 Predictors with one or two input variables o 1.2 Predictors with several input variables o 1.3 Predictors using computed parameters

2 Database approach 3 Fuzzy clustering techniques 4 Artificial neural networks 5 References 6 Noteworthy papers in OnePetro 7 External links 8 See also

Multiple linear regression

A widely used statistical approach is multiple linear regression.[1][2]

Linear regression techniques

are popular for establishing predictors of geological variables because the methods are effective

at predicting mean values, are fast computationally, are available in statistical software packages,

and provide a means of assessing errors.

Predictors with one or two input variables

When a straight-line relationship between log(k) and Φ exists, as it does in Figs. 1 and 2, the

computation of a predictor for log(k) by Eq. 1 is straightforward and merits little discussion.

Curvature in the log(k)-Φ relationship is treated by adopting a polynomial in Φ. Increased

accuracy is also afforded by dividing the field by area or vertically and computing regression

coefficients for each area. In one area, curvature in the statistical predictor may be rather

pronounced; in another, curvature may be absent.

....................(1)

Fig. 1– Permeability/porosity data from the Lower Cretaceous Hosston Sandstone from Thomson.[3]

Fig. 2 – Permeability/porosity data from Oligocene and Miocene sandstones from Bloch.[4]

Predictors with several input variables

The quality of the predictor can often be enhanced by adding a variable such as gamma ray

response or depth normalized to top of formation. As variables are added to Eq. 1, families of

curves are required to present graphically the effect of combinations of variables. When one or

two parameters are varied, the curves sweep out a large area on the log(k)-Φ plot. Predictive

power can be increased by adding other parameters. Predictive accuracy does not increase

indefinitely as parameters are added but instead usually reaches a limit after several (anywhere

from two to six) parameters are included in the regression (see Fig. 17 of Wendt et al.[2]

for an

example).

Predictors using computed parameters

Computed logs such as shale volume and differences between porosities from different logs can

be included as independent variables. In this way, petrological information can also be

incorporated into the predictive relationships. A petrological parameter (cement or gravel) is first

"predicted" from well logs using core observations as "ground truth." The predicted petrological

parameters can then be included in a relationship to estimate permeability.

As the complexity of the log(k)-Φ plot increases (i.e., as the data deviate from a linear trend),

more variables must be incorporated into the predictive model to maintain predictive accuracy,

although instability can result from having too many variables. The better the understanding is of

petrological controls on permeability, the more effective the predictor and its application will be.

Other complications with regression methods are mentioned by various authors.[1][2][5]

. These

complications include:

Underestimation of high-permeability zones Overestimation of low-permeability zones

At some point, it becomes necessary to adopt a method of zoning the reservoir.

Database approach

A database approach equivalent to an n -dimensional lookup table can also be used for predicting

permeability within a field or common geology.[5]

In this approach, the user must first select the

logs or log-derived variables that offer sufficient discriminating power for permeability. One

must also choose a suitable bin size for each variable on the basis of its resolution. Then, a

database is constructed from the core permeability values and associated log values. Each n-

dimensional bin or volume is bounded by incremental log values and contains mean and standard

deviation values of permeability plus the number of samples. In application, permeability

estimates are extracted from a bin addressed by the log values. An interpolation scheme is used

to extract an estimate from an empty bin. Like the regression method, the database approach can

be used only when adequate core data are available to build the model, and results generally

cannot be transferred to other areas.

Fuzzy clustering techniques

Fuzzy clustering techniques provide a means of determining the number of clusters (bins in the

preceding paragraph) and their domains.[6]

The term "fuzzy" indicates that a given input/output

pair can belong (partially) to more than one cluster. Finol and Jing[6]

applied the technique to a

shaly sandstone reservoir in which permeability ranged from 0.05 to 2,500 md. Six clusters were

defined. In each cluster, permeability is determined by

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

where:

Φ is porosity Qv is the cation exchange capacity per unit pore volume

The final determination of log(k) is a weighted sum of the six log(ki), with weights determined

by the degree of membership of Φ and log(Qv) in their respective clusters. An average

correlation coefficient of 0.95 was obtained on test sets.[6]

Implementation of Eq. 2 in uncored

wells requires that Qv be determined from a porosity log and requires an estimate of grain density

and shale fraction (Vsh).

Artificial neural networks

Artificial neural networks are a third method of establishing a predictor specific to an area of

interest. A back-propagation neural network is optimized on a training set in which the desired

output (permeability at a given depth) is furnished to the network, along with a set of inputs

chosen by the user. Rogers et al.[7]

established a predictor for a Jurassic carbonate field using

only porosity and geographic coordinates as inputs. For each value of permeability to be

predicted, porosity values spanning the depth of the desired permeability value were provided as

inputs, rather than a single porosity value at a single depth. Permeability values predicted by the

neural network in test wells were generally closer to the core measurements than were the values

predicted by linear regression.

References

1. ↑ 1.0 1.1 Allen, J.R. 1979. Prediction of Permeability From Logs by Multiple Regression. Trans., Society of Professional Well Log Analysts.

2. ↑ 2.0 2.1 2.2 Wendt, W.A., Sakurai, S., and Nelson, P.H. 1986. Permeability Prediction From Well Logs Using Multiple Regression. Reservoir Characterization, 181-222. Eds. L.W. Lake and H.B. Carroll, Jr. New York City: Academic Press, Inc.

3. ↑ Thomson, A. 1978. Petrography and Diagenesis of the Hosston Sandstone Reservoirs at Bassfield, Jefferson Davis County, Mississippi. Trans., Gulf Coast Association of Geological Societies 28: 651-664.

4. ↑ Bloch, S. 1991. Empirical Prediction of Porosity and Permeability in Sandstones. American Association of Petroleum Geologists Bull. 75 (7): 1145-1160.

5. ↑ 5.0 5.1 Nicolaysen, R. and Svendsen, T. 1991. Estimating the Permeability for the Troll Field Using Statistical Methods Querying a Fieldwide Database. Trans., Society of Professional Well Log Analysts, paper QQ.

6. ↑ 6.0 6.1 6.2 Finol, J. and Jing, X-D.D. 2002. Permeability Prediction in Shaly Formations: The Fuzzy Modeling Approach. Geophysics 67 (3): 817-829. http://dx.doi.org/ 10.1190/1.1484526

7. ↑ Rogers, S.J. et al. 1995. Predicting Permeability From Porosity Using Artificial Neural Networks. American Association of Petroleum Geologists Bull. 79 (12): 1786-1797.

Noteworthy papers in OnePetro

Use this section to list papers in OnePetro that a reader who wants to learn more should

definitely read

External links

Use this section to provide links to relevant material on websites other than PetroWiki and

OnePetro

See also

Single phase permeability

Permeability determination

Permeability estimation with NMR logging

Permeability estimation with Stoneley waves

Lithology and rock type determination

PEH:Single-Phase Permeability