permeability relations in rocks (2)

8/20/2019 Permeability Relations in Rocks (2)

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144 Chapter 1-2 • Fluid Recovery and Modelling: Dynamic Properties

1-2.1.4 Porosity/Permeability relations in rocks

The porosity/permeability relations in rocks are extremely useful in practice, mainly since

permeability is a “no-logging parameter”: despite all attempts (and the corresponding

publications), there is no reliable log analysis method to continuously measure the

permeability of the reservoir rocks in the borehole.

Discrete hydraulic methods are obviously available (e.g. mini-test), but they are costly.

Reservoir geologists therefore would like to find a relation between porosity (easilymeasured using log analysis techniques) and permeability, so that it can be easily deduced.

Most of the time, this is a risky operation.

The complexity of the φ-K relation is related to the complexity of the porous space itself.

We will therefore start by describing the simple case of the “ideal” intergranular porous

medium as encountered in the Fontainebleau sandstones. We will then study the carbonate

rocks and the common sandstones.

A) Simple porous networks: Example of Fontainebleau Sandstone

a) Fontainebleau sandstones (Fig. 1-2.13)

The Fontainebleau sandstones (Paris region, France) are a rare example of simple natural

porous media (intergranular porosity) exhibiting large porosity variations (from about 0.02

to 0.28) with no major change of grain granulometry. This is an ideal example on which to

study the porosity/permeability relation


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1-2.1 • Intrinsic Permeability 145

500 µm


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between 40 mm and 80 mm [Bourbié and Zinszner, 1985] indicates a double trend for the

K – φ relation. For high porosities (between 0.08 and 0.25), all the experimental points lie

fairly well on a curve of type K = f(φ3). Note that the power 3 corresponds to that of the

Carman-Kozeny equation. For low porosities (φ < 0.08), we may observe large exponents

suggesting a percolation threshold (§ 1-2.1.3D).

Example of the Milly la Forêt “normal” sandstones

In this book, we give results of a much larger sampling (but of the same type as the 1985

samples and including them). To simplify the analysis, it is best to identify the origins of thevarious groups of samples studied. The results shown on Figure 1-2.14 concern about 340

samples from a very restricted geographical area (Milly la Forêt). Due to their geological

“unity”, the quality of the porosity/permeability relation is quite exceptional. The

subdivisions (MZ2, etc.) correspond to different blocks measuring several decimetres in

size, obtained from various points in a limited number of quarries. Note that the

permeabilities of block MZ10 are slightly above the average: the granulometry is probably

slightly coarser.Air permeability values are measured in “room condition” (falling head permeameter,

§ 1-2.1.2A, p. 131). The experimental permeabilities below 100 mD have been corrected for

the Klinkenberg effect according to the semiempirical formula (§ 1-2.1.2A)

On Figures 1-2.14 the permeability axis uses a logarithmic scale (corresponding to the

log-normal distribution of permeabilities). We describe both types of axis used for porosity.

On the left hand figure, we use a linear axis (normal distribution of porosities) and on the

right hand figure a logarithmic axis to show the power laws


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The value of the exponent is similar to that observed in the earlier studies. For low porosity

samples (0.04 to 0.06), the Klinkenberg correction increases the value of this exponent even

more, reaching 10. Such high exponents can only be explained by a percolation threshold

(about 0.05 porosity).

To profit from the exceptional quality of the results on the Milly la Forêt sampling, we

calculate the polynomial regression best fitting the experimental values, in order to obtain a

basic datum for estimation of the φ-K laws of intergranular porosity media.

A polynomial regression of order 3, on logarithmic values of φ and K, of type

logK = a(log φ) 3 + b(logφ) 2 + c(logφ) + d,

with the above values (corresponding to the case of porosities expressed as percentages)

gives excellent results for porosities between 4% and 25%.

“Microcrack” facies

Some poorly porous samples have “microcracks”. They consist of strongly pronounced

grain joints that can be observed on thin section but even more clearly on epoxy pore cast

(Figure 1-2.16). These “microcrack” facies have been identified due to the very strong

acoustic anomalies generated by these cracks [Bourbié and Zinszner, 1985]. These facies

must be considered separately when studying the φ-K relation. Figure 1-2.15 shows some

sixty values corresponding to this type of sample (4 different series) As previously the

a b c d

11.17 –40.29 51.6 –20.22


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th t f l l (φ < 0 04) th d i bilit i l i ifi t

Figure 1-2.16 Fontainebleau sandstone with “microcrack” facies (φ = 0.06). Photograph of thin section(red epoxy injected, § 2-2.1.1, p. 325) on the left and of epoxy pore cast, on the right (§ 2-2.1.3, p. 331).

“Microcracks”, which correspond to grain contacts, can be clearly seen on the epoxy pore cast

500 µm


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The percolation threshold at about 0.05 porosity is also clearly determined; it will prove

extremely useful when discussing φ-K relations in double-porosity limestones.

B) Porosity/Permeability relations in carbonate rocks

The situation with carbonates is strikingly different from that observed in Fontainebleau

sandstones. Figure 1-2.17 shows the φ-K relation (air permeability) for a set of about 1 500

limestones and dolomites samples (diameter 4 cm) corresponding to a large variety of

petrogaphic texture. Note that in line with standard practice, and in spite of the fact that it is poorly adapted to the power laws, we have adopted the semilogarithmic representation

which makes the graphs much easier to read on the porosity axis. The permeability

dispersion is very high since, on the porosity interval most frequently encountered in

reservoir rocks (0.1 to 0.3), the values extend over nearly four orders of magnitude. Put so

bluntly, it is clear that there is no φ-K relation! Considering the microtexture of the rocks,

some general trends may nevertheless be observed.

a) Dolomites

On Figures 1-2.17, the points corresponding to dolomites and dolomitic limestones are

separated from the limestones. The dolomite/dolomitic limestone/limestone separation was

made using the criterion of matrix density (§ 1-1.1.5, p. 26) choosing 2 770 kg/m3 as the

lower limit of the dolomite and 2 710 kg/m3 as the upper limit of the limestones. We will

only consider the case of the dolomites. Far fewer points are available (about 50) than for the

limestones We can nevertheless make a few important comments Although dolomites are


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case of a very good producing well (over 10 000 barrels per day) The production log

Figure 1-2.17 Porosity-Permeability relation in carbonate rocks(air permeability, Klinkenberg effect not corrected, about 1500 samples of diameter 4 cm)

0.010.1 0.2

Porosity (%)

Permeability(mD)

0.3 0.4 0.50

0.1

1

10

100

1 000

10 000

100 000

Limestone

Dolomitic Lmst

Dolomite

Fontainebleau

Power 3

Power 5

Power 7


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5 mm

a

b


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Example of oolitic limestones

Oolitic limestones clearly illustrate the variability of the φ K relations These rocks whose

Figure 1-2.20 Porosity-Permeability relation in limestone rocks of permeability greater than 0.1 mD.The values are extracted from the database of Figure 1-2.17

The “Power 3, 5, 7” labels correspond to the relations K = f(φn) with powers 3, 5, 7.

0.010 0.05 0.1 0.15 0.2 0.25

Porosity (Fractional)

Perme

ability(mD)

0.3 0.35 0.4 0.45 0.5

0.1

1

10

100

1 000

10 000

Mudstone Pell-Wacke

Oolite Biocl. Gr/Pk

Crinoidal Grst Fontainebleau

Power 3 Power 5

Power 7


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a b c

1 mm

0 1

1

10

100

1 000

10 000

Permeability(mD)

Oolite

Fontainebleau

Power 3


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Micrites

Micrites (Fig. 1-2.22) are rocks mainly formed from microcrystalline calcite. The φ-K

relation, however, is quite different from those of the oolitic limestones. In the φ-K space,

the micrites are grouped along a line corresponding approximately to a power 3 law going

through φ = 0.1; K = 0.1 mD. This relative simplicity of the φ-K relation in micrites is

explained by the fact that they only have a single type of porosity. The porosimetry spectra

(Fig. 1-2.23) are clearly unimodal and contrast with the other limestones.

10 µm

a


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– Limits of the φ-K space.

The various φ-K relations described in this section respect the “matrix value” criterion

(§ 2-1.3, p. 311). They come from plugs and the minimum homogenisation volume is

millimetric. We can see on Figure 1-2.20 that these values define a φ-K space whose

limits are quite well defined

• By the Fontainebleau sandstone line towards the high permeabilities

• By the mudstone line (power 3 law) towards the low permeabilities

Although these limits are very broad, they are nevertheless practical. When studying

reservoirs, special attention must be paid to values outside this area. They generallyindicate measurement errors or faulty samples (e.g. fissured plug). In the other cases,

however, a special study could prove well worthwhile.

– Main trends

Some main trends in the φ-K relation may also be observed according to the

petrographic texture of the limestones (Fig. 1-2.20).

• The very poorly permeable limestones (K < 0.1 mD) have not been shown on

Figure 1-2.20. The porosity of limestones whose permeability is greater than thislow value is generally more than 10%, apart from the important exception of crinoi-

dal limestones which have very little microporosity, hence the high permeabilities.

Some oolitic grainstones lie within the same area of the φ-K space, for the same rea-

son: proportionally very low microporosity.

• For the other types of limestone, we observe a point of convergence at about

φ = 0.1; K = 0.1 mD. If power 3, 5, 7 law graphs are plotted from this point, we

observe that the mudstones are grouped on the power 3 law (see above), the

k t l t d th 5 li d th bi l ti i k t


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The porosimetric diversity of carbonates is represented on Figure 1-2.23 where the

porosimetry/permeability relation can be checked qualitatively (Purcell model, § 1-2.1.3B,

p. 139). Although a basic point, we must reiterate the fact that the most important value

required to understand φ-K relations is the equivalent pore access radius. The dimension of

the pore itself has virtually no impact on permeability.

0.01

0.1

1

10

100

1000

10000

0 0.1 0.2 0.3 0.4 0.5

Porosity (fractional)

Permeability

(mD)

limestone

Fontainebleau

Power 3

Power 5

Power 7

0

1

2

Porosity(%)

5-Bioclst.

0.01 0.1 1

Equivalent pore access radius ( µm, log)

10 100

0

1

2

Porosity(%)

8-Oolite

0.01 0.1 1


10 100

1

2

sity(%) 7-Oolite

2

4-Pellets

0

1

2

Poro

sity(%)

10-Oolite

0.01 0.1 1


10 1000

1

2

Porosity(%)

6-Bioclst.

0.01 0.1 1


10 100

3-Micrite

0.01 0.1 1


10 100

0

10

Porosity(%)

0

1

2

Poro

sity(%)

9-Oolite

0.01 0.1 1


10 100

2

%) 1-Micrite

5

Porosity(%)

2-Micrite


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Figure 1-2.24 Example of dolomitic limestone (ρma = 2750 kg/m3) with large,

poorly-connected moldic pores. The mean sample characteristics (φ = 0.18; K = 7 mD)are not necessarily representative of the area photographed (heterogeneity). Despite saturation under

vacuum, the red epoxy resin did not have time to invade all the vugs due to the low permeability

5 mm

permeability relations in rocks (2)

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