Distribution of Rock Properties - New Mexico Institute of ...infohost.nmt.edu/~petro/faculty/Engler524/PET524-2c-permeability.pdf · Figure 3.16 is a schematic representation of the

Chapter 3 – Permeability

3.17

3.3 Porosity-Permeability Relationships

To this point we have independently developed the fundamental properties of

porosity and permeability. Environmental and depositional factors influencing porosity also

influence permeability, and often there is a relationship between the two. The relationship

varies with formation and rock type, and reflects the variety of pore geometry present.

Typically, increased permeability is accompanied by increased porosity. Figure 3.14

illustrates the various trends for different rock types. For example, a permeability of 10 md

can have a porosity range from 6 to 31%, depending on the rock type and its pore geometry.

Constant permeability accompanied by increased porosity indicates the presence of more

numerous but smaller pores.

Figure 3.14 Permeability and porosity trends for various rock types [CoreLab,1983]

For clastic rocks, the k- trend is influenced by the grain size as shown in Figure 3.15. Post

depositional processes in sands including compaction and cementation will result in a shift to

the left of the permeability-porosity trend line. Dolomitization of limestones tends to shift the

permeability- porosity trend lines to the right.


3.18

Figure 3.15 Influence of grain size on the relationship between porosity and permeability [Tiab & Donaldson,

1996]

The inter-relationship of rock properties has lead to numerous correlations to estimate

permeability. Several of the more notable are as follows. Darcy’s Law (1856) uses

empirical observations to obtain permeability as previously shown. Slichter in 1899

performed theoretical analysis of fluid flow through packed spheres of uniform size and

introduced packing as a factor influencing permeability.

sk

dk

22.10

(3.11)

where d is the sphere diameter (cm) and ks is a packing constant and function of porosity ( =

26% & hexagonal packing ks = 84.4; = 45% & cubic packing ks = 13.7).

One of the more well-known correlations was developed by Kozeny (1927) and later

modified by Carmen(1939). It is based on fundamental flow principles by considering the

porous media as a bundle of capillary tubes with the spaces between filled with a non-porous

cementing material. Figure 3.16 is a schematic representation of the capillary tube model.


3.19

Figure 3.16 Capillary tube model

We can define the porosity for the model shown in Figure 3.16 as,

2rnt (3.12)

where r is the radius of the capillary tube and nt is the number of tubes per unit area (A).

Also, the permeability can be derived from combining Poiseuille’s Equation for flow through

a conduit with Darcy’s Law for flow in porous media,

8

4rnk t (3.13)

Combining Eqs (3.12) and (3.13) leads to an expression relating k and .

8

2rk (3.14)

Example 3.3

For the cubic packing arrangement shown in the diagram below, determine the porosity and

permeability.

r


3.20

Solution

The number of tubes per unit area is: 2

)4/(4 rtubes . Substituting into Eq. (3.12) results in an

estimate for porosity.

4

2*

24

1 r

r

The permeability from Eq. (3.14) is r2/32.

To relate the capillary radius, r, to the porous media, we must first define Spv, the

specific surface area per unit pore volume. In the case of cylindrical pore shape, Spv = 2/r.

Similar expressions can be derived for Sbv, specific surface area per unit bulk volume and

Sgv, specific surface area per unit grain volume.

pvS

gvS

pvS

bvS

1

*

(3.15)

Substitution of Spv for pore radius in Eq. (3.14) results in the Carmen-Kozeny equation for

porous media.

2

pvS

zk

k

(3.16)

The Kozeny constant, kz, is a shape factor to account for variability in cross-sectional shape

and length. It can be separated into two components, kz = ko* , where is known as the

tortuosity and describes the variability in length between the capillary tube, La, and unit

length, L.

2

L

aL

(3.17)

ko is a shape factor to account for various cross-sectional shapes; e.g., ko = 2 for circular, =

1.78 for square. If we go back to our example of circular tubes and substitute ko = 2, = 1

(tubes and unit length are equal and parallel), and Spv = 2/r into Eq. (3.16), we obtain Eq.

(3.14).


3.21

Example 3.4

Measurements from capillary pressure, adsorption and statistical techniques are available to

obtain the specific surface area per unit pore volume. A given sample measurement resulted

in a reading of 182 mm-1

. Tortuosity is measured from electrical resistivity and was

determined to be 3.6. Porosity of the sample is 27.7% Assuming circular, capillary tubes to

represent the porous media what is the permeability of the sample?

Solution

For a circular cross-sectional area of a capillary tube, the pore radius is

1

011.0182

22 mm

pvS

r

Substituting into the Carmen-Kozeny equation (Eq. 3.16) results in,

26

10164.1)4)(6.3(2

2)011.0)(277(.

mmxk

Converting to darcies,

Darcys

cmx

darcy

mm

cmmmxk 18.1

2810987.0

1*

2

10

1*

2610164.1

If porosity and pore radius are substituted into Eq. (3.12) we can solve for n,

27292)011.0(

277.

2 mm

rtn

that is 729 tubes per unit area.

Generalized Capillary tube model

The above derivation illustrates the simple capillary tube model. We can modify this

model to be more complex by considering the tube length to be greater than the unit length of

the sample. The results are expressions for porosity and permeability which include

tortuosity.

2rtn (3.18)

8

2rk (3.19)


3.22

Furthermore, let’s introduce the concept of hydraulic radius, the ratio of the volume open to

flow to the wetted surface area or,

pvSgvShr

11

1

(3.20)

thus for a bundle of circular capillary tubes, Eq (3.19) reduces to,

ok

hr

k

2

(3.21)

where ko = 2 for circular tubes. As another example, consider spherical particles with

diameter, dp. In this case, Sgv = 6/dp, substitute into Eq. (3.20) and then the result into Eq.

(3.21) provides,

2172

23

pdk (3.22)

In summary, Kozeny found that variables, such as specific surface area of the pore

system, tortuosity and shape factor, were missing from standard permeability and porosity

relationships. Using a simple model, a bundle of capillary tubes, Kozeny described this new

relationship and extended it to fit porous media. His equation introduced specific surface area

and a constant (generally assumed to be 2).

Carmen elaborated on the specific surface area and Kozeny constant. Carmen

showed that textural parameters, such as the size, sorting, shape and spatial distribution of

grains, drastically affect permeability. Considering Kozeny constant as 5 the Kozeny-Carmen

equation has been adopted for homogeneous rocks with a dominance of nearly-spherical

grains. Since nature provides a random geometry, as encountered in reservoir rocks, this

assumption of homogeneity in the Kozeny- Carmen equation has made its application

questionable, and impossible to transfer from one zone to another. This deficiency has also

curtailed the development of a strong dynamic link between the microscopic and

macroscopic properties measured on reservoir rock. Nevertheless, this model does illustrate

the basic principles of porosity and permeability.

The objective of understanding the Carmen – Kozeny equation is related to the

development of porosity – permeability correlations. The general form of the Carmen –

Kozeny equation demonstrates how permeability depends on the pore geometry. The success


3.23

of this model in explaining and to some extent predicting permeability is surprising because

the model neglects converging – diverging flow, pore entrance effects, dead-end pores, and

local pore arrangements.

No petrophysical tool is more frequently applied than permeability – porosity

correlations and the Carmen – Kozeny model provides the fundamental basis for this

relationship. The three primary methods of estimating permeability are well log correlations,

core analysis, and welltest analysis. Conventional well logs cannot directly measure

permeability; however, they can provide a reasonably accurate value for porosity.

Numerous correlations have been developed to use the abundance of log porosity data to

determine permeability. Unfortunately, the results are only an order of magnitude in

accuracy. However, recent advances in specialized logging methods and specifically the

Nuclear Magnetic Resonance (NMR) log have resulted in directly measuring or inferring

permeability. Core analysis provides accurate permeability values, but is limited by the

small scale of the measurement volume to the reservoir volume. On the other hand,

welltesting investigates the largest portion of the reservoir; however, this permeability is

averaged throughout this volume. Subsequently, the fine details of individual zones or facies

is masked within the averaging process.

A novel approach in enhancing reservoir description is the development of Hydraulic

Flow Units (HFU). This concept is based on the Carmen – Kozeny equation and will be

elaborated on later in this chapter.


3.24

3.4 Distribution of Rock Properties

In most petroleum engineering applications reservoir properties are assumed to be

constant over a spatial direction; i.e., homogeneous. Unfortunately, it is well recognized that

most reservoirs are heterogeneous; i.e., rock properties vary with a spatial direction. Both

vertical and areal variations are possible and usually coexist. It therefore is the objective of

this section to discuss methods of quantifying the reservoir heterogeneity.

A key concept in describing variation in properties is scale. For example, consider

the difference in investigative volumes between core, log, and pressure transient testing. All

three methods are accurate, can determine key properties such as permeability or porosity,

but will likely provide significantly different results. That is because a core is averaging a

property over a 3 ½’’ diameter, a log approximately 12” diameter and well testing from 10s

to 100s of feet. It is suffice to say that not only do rock properties vary spatially but also our

methods of analysis are subject to variations due to averaging different volumes.

Two properties we will focus on are porosity and permeability. Porosities typically

exhibit a normal distribution (Figure 3.17); i.e., a distribution symmetric about the mean.

Figure 3.29 is a histogram of porosity vs. the frequency of samples within a given range of

porosity and the cumulative frequency. Table 3.2 lists the data for this example.

Figure 3.17 Typical porosity histogram


3.25

POROSITY RANGE,% NO. OF SAMPLES FREQUENCY F, % CUMULATIVE F, %

< 10 161 3.78 3.78

10 – 12 257 6.04 9.82

12 – 14 398 9.35 19.17

14 – 16 493 11.58 30.75

16 – 18 608 14.28 45.03

18 – 20 636 14.94 59.97

20 – 22 623 14.63 74.60

22 – 24 447 10.50 85.10

24 – 26 340 7.99 93.09

26 – 28 176 4.13 97.23

>28 117 2.75 100.00

Table 3.2 Classification of Porosity data for distribution analysis [Amyx,et at., 1960]

The mean is a thickness weighted average for n number of beds in parallel.

Th

n

1ii

hi

(3.23)

Common applications of this method are to describe the vertical variability in data from logs

and cores. A drawback is the increase in error due to outliers; i.e., data points which are

considerably different than the others. It is common practice to ignore these points as

measurement errors.

Example 3.4

Core was retrieved from the NBU Well No. 42W-29 in the North Burbank Field of northeast

Oklahoma. The conventional core analysis is shown in Table 3.4. A porosity histogram and

cumulative frequency curve are shown in Figure 3.18 with the accompanying tabulated data

in Table 3.3

The results illustrate a bimodal distribution with an overall mean of 17.5% and median of

15.5%. The dominant mode is centered at 15% porosity while the secondary mode occurs in

the higher porosity range, approximately 25 to 27%.


3.26

Figure3.18. Example porosity histogram and cumulative frequency curve

Table 3.3 Tabulated data for histogram in Figure 3.18

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 14 16 18 20 22 24 26 28

Porosity , %

Fre

qu

en

cy

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Cu

mu

lati

ve

Fre

qu

en

cy

Porosity Frequency Cumulative

interval Frequency

4 1 0.025

5 0 0.025

6 0 0.025

7 0 0.025

8 1 0.050

9 0 0.050

10 1 0.075

11 2 0.125

12 1 0.150

13 4 0.250

14 3 0.325

15 9 0.550

16 3 0.625

17 2 0.675

18 1 0.700

19 0 0.700

20 1 0.725

21 1 0.750

22 1 0.775

23 2 0.825

24 0 0.825

25 2 0.875

26 1 0.900

27 2 0.950

28 1 0.975

29 1 1.000


3.27

The method described above for porosity has been applied also to permeability.

However, the abrupt changes in permeability and lack of representative samples introduce

uncertainty into the interpretation of the data. An alternative relies on an empirical

correlation of permeability data. It has been shown that permeabilities in most reservoirs

exhibit a log normal distribution. That is, geologic processes that create permeability in

reservoir rocks appear to generate distributions about the geometric mean.

nn

ii

kg

k

/1

1

(3.24)

Figure 3.19 illustrates the skewed normal and log normal distributions of typical data.

Figure 3.19 Skewed normal and log normal histograms for permeability [Craig,1971]

To evaluate the heterogeneity of the sample set, plot permeability vs. cumulative

frequency distribution on log-normal probability coordinates. Cumulative frequency

distribution is the fraction of the samples with permeabilities greater than the particular

sample. If a straight line develops then the data exhibits the log normal distribution. The

straight line is a measure of the dispersion or the heterogeneity of the reservoir rock. Dykstra


3.28

and Parsons (1950) recognized this important feature and introduced the permeability

variation, V as,

50k

1.84k

50k

V

(3.25)

where k50 is the mean permeability and k 84.1 is the permeability of the mean + one standard

deviation. The Dykstra-Parsons coefficient ranges from a minimum of 0 (pure

homogeneous) to a maximum of 1.0 (heterogeneous), with most reservoirs falling within V =

0.5 to 0.9. Figure 3.20 is an illustration of the log-normal probability plot and the range of

the coefficient of variation seen in most reservoirs.

Figure 3.20 Characterization of reservoir heterogeneity by permeability variation [Willhite, 1986]

Example 3.5

Given a distribution of permeability data from core samples determine the Dykstra – Parsons

permeability coefficient.

Solution

a. Arrange the permeability data in descending order.

b. Compute the percent of total number of k-values exceeding each tabulated, permeability.

c. Plot the log of permeability vs. the cumulative frequency distribution. (Figure 3.21).

d. From the figure, the mean value = 475 md and the k 84.1 = 324 md, respectively.


3.29

e. Using Eq. 3.25, the Dykstra-Parsons coefficient is V = 0.318. The reservoir has a

reasonably low degree of heterogeneity.

Figure 3.21 Example of log normal permeability distribution [Willhite, 1986]

Example 3.6

The previous example was an ideal case of a single flow unit, which by virtue of the straight-

line relationship follows a log-normal distribution. In comparison, examining the data from

the NBU Well No. 42W-29 shows more variability. The resulting Dykstra-Parsons

coefficient of 0.795 confirms the high degree of variability. Also in this example, four units

are identified by HFU analysis and are illustrated on Figure 3.22 with different symbols.


3.30

Figure 3.22 Example of permeability distribution for the Burbank Sandstone

A second method to describe reservoir heterogeneity is to relate the cumulative flow

capacity to the cumulative storage capacity of the reservoir. A curved relationship will

develop as shown in Figure 3.23. The greater the deviation of this curve from the 45

Figure 3.23 Flow capacity vs storage capacity distribution [Craig, 1971]


3.31

degree line the greater the heterogeneity of the system. To construct this plot, arrange

permeability and porosity in descending order. Determine each intervals flow capacity (kh)

and storage capacity (h), and then sum these values to obtain cumulative curves. Dividing

by the maximum results in the fraction or percent of flow or storage capacity.

The Lorenz Coefficient, Lk, was introduced to characterize permeability distributions

within a formation using the above information. Referring to Figure 3.23, it is defined as,

ADCAArea

ABCAArea

kL (3.26)

The Lorenz coefficient varies from 0 to 1, where uniform permeability is 0. Several

permeability distributions can result for the same value for Lk, therefore the solution is not

unique. However, comparison of Lk for various wells will provide a relative magnitude of

heterogeneity between wells.

Example 3.7

Core analysis for 40 samples from the Burbank Sandstone is given in Table 3.4. Determine

the Lorenz coefficient for this data.

Solution

Table 3.5 presents the sorted data and cumulative capacities for permeability and porosity.

Figure 3.24 is the plot of the fraction of total flow capacity vs. the fraction of total storage

capacity for this formation. Using Eq. (3.26) the Lorenz coefficient is 0.643, suggesting this

well is relatively heterogeneous. (The area of ADCA is a triangle = ½ bh = ½. The Area

under the curve can be integrated using the equation on the figure = 0.822. The area ABCA

= 0.822-0.5 = 0.322.)


3.32

Table 3.4 Input data for Example 3.7

Depth,ft Sample k, md r RQI, m

2905 1 0.224 0.1200 0.136 0.043

2906 2 0.337 0.1015 0.113 0.057

2907 3 0.187 0.1165 0.132 0.040

2908 4 0.653 0.1304 0.150 0.070

2909 5 1.040 0.1153 0.130 0.094

2910 6 0.450 0.0853 0.093 0.072

2911 7 434.000 0.2519 0.337 1.303

2912 8 196.000 0.2159 0.275 0.946

2913.1 9 0.007 0.0467 0.049 0.012

2913.5 10 1156.000 0.2949 0.418 1.966

2915 11 531.000 0.2679 0.366 1.398

2916 12 1059.000 0.2874 0.403 1.906

2917 13 822.000 0.2765 0.382 1.712

2918 14 1014.000 0.2769 0.383 1.900

2934 15 109.000 0.2269 0.293 0.688

2935 16 138.000 0.2330 0.304 0.764

2936 17 166.000 0.2381 0.313 0.829

2937 18 362.000 0.2554 0.343 1.182

2938 19 77.900 0.2009 0.251 0.618

2939 20 64.900 0.1863 0.229 0.586

2940 21 51.100 0.1685 0.203 0.547

2941 22 89.900 0.1555 0.184 0.755

2942 23 84.100 0.1636 0.196 0.712

2943 24 21.200 0.1537 0.182 0.369

2944 25 23.700 0.1676 0.201 0.373

2945 26 39.600 0.1728 0.209 0.475

2946 27 44.400 0.1770 0.215 0.497

2947 28 20.800 0.1578 0.187 0.361

2948 29 13.900 0.1510 0.178 0.301

2949 30 20.800 0.1543 0.182 0.365

2950 31 6.390 0.1365 0.158 0.215

2951 32 10.000 0.1449 0.169 0.261

2952 33 15.300 0.1492 0.175 0.318

2953 34 11.400 0.1447 0.169 0.279

2954 35 22.800 0.1518 0.179 0.385

2955 36 37.200 0.1537 0.182 0.488

2956 37 29.100 0.1537 0.182 0.432

2957 38 5.840 0.1364 0.158 0.205

2958 39 13.900 0.1529 0.180 0.299

2959 40 16.400 0.1387 0.161 0.341

Arith mean 167.76 md

Geo mean 23.28 md

Har mean 0.25 md


3.33

Table 3.5 Calculations for Lorenz Coefficient

Descending Permeability for Lorenz Coefficient

Fraction of Fraction of

cumulative total flow cumulative total

k, md k,md capacity porosity porosity volume

1156.000 1156 0.172 0.2949 0.2949 0.042

1059.000 2215 0.330 0.2874 0.5823 0.083

1014.000 3229 0.481 0.2769 0.8592 0.123

822.000 4051 0.604 0.2765 1.1357 0.162

531.000 4582 0.683 0.2679 1.4036 0.200

434.000 5016 0.747 0.2519 1.6555 0.236

362.000 5378 0.801 0.2554 1.9109 0.273

196.000 5574 0.831 0.2159 2.1268 0.303

166.000 5740 0.855 0.2381 2.3649 0.337

138.000 5878 0.876 0.2330 2.5979 0.370

109.000 5987 0.892 0.2269 2.8248 0.403

89.900 6077 0.906 0.1555 2.9803 0.425

84.100 6161 0.918 0.1636 3.1439 0.448

77.900 6239 0.930 0.2009 3.3448 0.477

64.900 6304 0.939 0.1863 3.5311 0.504

51.100 6355 0.947 0.1685 3.6996 0.528

44.400 6399 0.954 0.1770 3.8766 0.553

39.600 6439 0.960 0.1728 4.0494 0.577

37.200 6476 0.965 0.1537 4.2031 0.599

29.100 6505 0.969 0.1537 4.3568 0.621

23.700 6529 0.973 0.1676 4.5244 0.645

22.800 6552 0.976 0.1518 4.6762 0.667

21.200 6573 0.979 0.1537 4.8299 0.689

20.800 6594 0.983 0.1578 4.9877 0.711

20.800 6615 0.986 0.1543 5.1420 0.733

16.400 6631 0.988 0.1387 5.2807 0.753

15.300 6646 0.990 0.1492 5.4299 0.774

13.900 6660 0.992 0.1510 5.5809 0.796

13.900 6674 0.995 0.1529 5.7338 0.818

11.400 6685 0.996 0.1447 5.8785 0.838

10.000 6695 0.998 0.1449 6.0234 0.859

6.390 6702 0.999 0.1365 6.1599 0.878

5.840 6708 1.000 0.1364 6.2963 0.898

1.040 6709 1.000 0.1153 6.4116 0.914

0.653 6709 1.000 0.1304 6.5420 0.933

0.450 6710 1.000 0.0853 6.6273 0.945

0.337 6710 1.000 0.1015 6.7288 0.960

0.224 6710 1.000 0.1200 6.8488 0.977

0.187 6711 1.000 0.1165 6.9653 0.993

0.007 6711 1.000 0.0467 7.0120 1.000


3.34

Figure 3.24 Fraction of flow vs. storage capacity for determination of Lk.

The previous statistical approaches of estimating reservoir heterogeneity fail to capture an

accurate description of the reservoir for several reasons. First, the data is arranged in a

sequential order, while a true reservoir is not in any ordered sequence (Figure 3.24). [Lake,

1998, Chopra, et al.,1989]

Flow Capacity Distribution

y = -3.8012x4 + 10.572x3 - 11.01x2 + 5.2476x - 0.0146

R2 = 0.9991

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fraction of total Volume

Fra

cti

on

of

tota

l Flo

w C

ap

acit

y


3.35

dep

th

arranged un-arranged

Figure 3.24. Schematic of statistical approach of arranging data in comparison to true reservoir data, which is

not ordered.

In multiphase flow the displacement response is different for both arrangements in

Figure 3.24, unless gravity is neglected (kv 0). Subsequently, the ordering and position of

layers is critical when crossflow occurs. The problem reveals the need for spatial correlation

which is the topic of the next section.

A second drawback of the statistical approaches is the reliance on permeability

variations for estimating flow in layers. It can be shown from mass balance concepts that

the speed fluid travels through a layer is dependent on the phase mobility, pressure gradient,

and the k/ratio. In the hydraulic flow unit method the Reservoir Quality Index is defined as

this ratio to account for the effect of porosity with permeability. Furthermore, variations for

irreducible water saturation are not accounted for in the statistical approaches.

Hydraulic Flow Unit

Introduction

The concept behind hydraulic flow units is to provide a method of identifying and

characterizing zones with similar hydraulic characteristics. Figure 1 illustrates the separation

of a formation into hydraulic flow units.


3.36

HFU1 HFU2

HFU3

HFU4

Figure 3.25 Schematic illustrating the concept of flow units.

The approach is to integrate microscopic and macroscopic measurements into

meaningful relationships and to develop a dynamic link that will allow the prediction of fluid

flow characteristics.

It is desirable to obtain accurate permeability from well logs. Currently, empirical

methods are an order of magnitude in accuracy. However, the preponderance of log data to

core data is a driving motivation in accomplishing this task. Amaefule et al.,1993 showed a

successful application in predicting permeability in uncored sections or wells. However, Lee,

et al., 1999, using principal component analysis, have shown difficulty in identification of

hydraulic flow units in uncored wells.

What is a hydraulic flow unit? It can best be described as unique units with similar

petrophysical properties that affect flow. Hydraulic quality of a rock is controlled by pore

geometry; which is dependent upon mineralogy (type, abundance, morphology) and textural

parameters (grain size, shape, sorting and packing). It is the distinction of rock units with

similar pore attributes, which leads to the separation of units into similar hydraulic units. A

hydraulic unit is the same as a flow unit; however they are not equivalent to a geologic unit.

The definition of geologic units or facies are not necessarily the same as the definition of a

flow unit.

Method

The formulation begins with a generalized form of the Carmen-Kozeny equation,


3.37

2

1

21

3

gvS

ok

k

(3.27)

where is the tortuosity and ko is a shape factor. Previous sections have shown the Sgv can

be measured using imaging or gas adsorption techniques, and from electrical resistivity

measurement, the tortuosity. In this method, the objective is to avoid measuring these

microscopic properties by lumping these parameters into a single variable called the Flow

Zone Indicator (FZI).

Rearranging Eq. (3.27) we obtain,

gvS

ok

k

1

1 (3.28)

We can define the Reservoir Quality Index (RQI), as the ratio of permeability to porosity.

}{0314.0}{

mdkmRQI (3.29)

This term is similar to Leverett’s mean hydraulic radius and is an approximation of the mean

pore throat size. Furthermore, we can define the FZI as a function of specific surface and

tortuosity.

zk

gvS

FZI1

(3.30)

This parameter indicates samples with similar pore throat characteristics and; therefore,

constitute a hydraulic unit. For example, two samples with different grain sizes will result in

different FZIs. The larger grain size will have the greater FZI and also the greatest

permeability. A final definition is the pore-to-grain volume ratio expressed as,

1r (3.31)

Substitution of Eqs (3.29-31) into Eq. (3.28), and taking the logarithm of both sides results

in,

)log()log()log( FZIr

RQI (3.32)


3.38

Subsequently, a log-log plot of RQI vs r will result in a unit slope with a y-intercept equal to

FZI. Samples that lie on the same straight line have similar pore characteristics and are

therefore considered a flow unit. Samples with different FZI values will lie on different but

parallel lines. Figure 3.26 is an example from a sandstone reservoir in Amaefule, et al.

showing six distinct flow units. Figure 3.27 from Lee et al., is from a carbonate reservoir.

Figure 3.26 Plot of RQI vs r for East Texas Well [Amaefule, et al.,1993]

Figure 3.27 Plot of RQI vs r for a carbonate well in Permian Basin [Lee, et al.,1999]


3.39

Determination of the number of hydraulic units

As seen in Figures 3.26 and 3.27, a decision is required to the determination of the

specific number of flow units. This number is constrained by the random errors in

measurement of the porosity and permeability data. The magnitude of the random errors can

be estimated by the root-mean-square technique on FZI.

5.022

1

3*

2

5.0

k

k

FZI

FZI

(3.33)

Any sample with the coefficient of variance (FZI/FZI > 0.5) are considered unreliable and

consequently omitted from the process. In Eq. (3.33), the tolerance for permeability was

20% and for porosity was 0.5%.

Furthermore, the selection of hydraulic flow units must be consistent with core

geologic descriptions. This results in a resolution of numerous hydraulic units on the core

scale; however, the desire is to be able to measure flow units on the log scale. It has been

suggested [Johnson, 1994] that only four hydraulic zones are discernable with well logs:

macropores (r > 1.5 m), mesopores (0.5m < r < 1.5 m), micropores (r < 0.5 m) and no-

flow layers. Amaefule, et al. state the size distribution of pore throat radii are excellent

delineators of hydraulic units; but do not adhere to the idea of only four zones identifiable on

well logs. Unfortunately, the values of FZI to differentiate zones are somewhat subjective,

based on experience as to which pore radii provides the correct flow zones.

Permeability prediction in uncored sections/wells

Mentioned in the introduction was the desire to obtain permeability from well logs.

A flowchart of the procedure taken from Amaefule is shown in Figure 3.28.


3.40

Figure 3.28 Flowchart for estimation of permeability from well logs [Amaefule, et al,1993]

Step 1: Compute the FZI, RQI and z from the core data as described in the previous section

and zone the data into hydraulic units.

Step 2: For each hydraulic unit, develop regression models for FZI based on logging

attributes in cored intervals. A non-parametric regression analysis was used to rank the

logging tool responses to FZI. Then transformation equations were developed after

normalizing the data distribution with logarithmic filtering. In general terms, weighting

coefficients, c1, c2, ….cn are calculated for each log response such that for each flow unit,

....*)*3

(*)*2

(*)*1

(ild

Rcb

ccFZI (3.34)

where is gamma ray, b is bulk density and Rild is deep induction resistivity.

Step 3: Predict the hydraulic unit profiles in uncored sections or wells using probabilistic

methods constrained with deterministic hydraulic unit variables. Probabilities are computed

from estimated distributions of each hydraulic unit by application of Bayes Theorem. The

control wells provide the reference set for the uncored wells to determine the probability of


3.41

having the same hydraulic unit in a given prediction window; thus providing the basis of

qualitatively predicting the hydraulic units in uncored sections/wells.

Step 4: Compute FZI in uncored wells/sections using the regression models based on the

logging attributes. Calculate permeability from the following expression,

2)1(

32

)(1014

FZIk (3.35)

An example from Amaefule, et al., is illustrated in the next sequence of figures. In figure

3.29 is the predicted permeability from a classical multilinear correlation technique without

hydraulic zonation. Note the deviations from the actual measured values are significant.

Figure 3.29. Predicted permeability versus actual core permeability without zonation

[Amaefule, et al, 1993]

In contrast, permeability predicted from the same logging responses after zonation exhibit

excellent correlation with the actual core values (Figure 3.30). Figure 3.31 illustrates the

relationship between porosity and permeability for the two methods. Again, note the

excellent predictive capabilities of the proposed method.


3.42

Figure 3.30. Predicted permeability versus actual core permeability with zonation

[Amaefule, et al, 1993]

Figure 3.31 Relationship for permeability and porosity with and without zonation [Amaefule,

etal.,1993]

An alternative description for predicting permeability in uncored sections is given by

Johnson, 1994. In this work, a predictive permeability database is constructed with unique

log responses for each core permeability. In essence a scaling up procedure was

accomplished between log and core data. The assumption is that the log data provides a


3.43

unique fingerprint for the entire range of permeability. If a given set of log data has multiple

permeability solutions, then the average permeability value will be assigned to those input

log values. Kriging is used on log values which do not precisely match the input database.

Flow Unit A

Sample Cored well log data

1 k1 x1,x2,…

2 k2 x1,x2,…

. . .

. . .

y1,y2,…

Uncored

log data

Figure 3.32 Schematic of permeability database for estimating permeability from uncored wells/sections.

A more conventional approach was presented by Ti, et al., 1995. The determination

of flow units in cored wells was achieved by applying cluster analysis. Cluster analysis is a

statistical method of displaying the similarities and dissimilarities between objects. These

objects form groups or clusters where objects within the group tend to have similar traits

while those in different groups do not. Figure 3.33 is the example presented in Ti, et al. for

the Endicott Field in Alaska. The parameters transmissibility and storativity are normalized

so that each contribute equally to the objective classification scheme.

The next step was to establish a statistical relationship between the core and log data

and then extend these relationships for flow unit determination in the uncored wells. Three

regression relationships were developed;

5)log(*

4)log(

3*

2)log(

log*

1

ch

kcv

k

ccore

ch

k

ccore

(3.36)

The estimate of permeability is related to the cluster analysis to ascertain the flow unit.


3.44

Figure 3.33 Results of cluster analysis,[Ti,et al.,1995]

Example 3.8

Using the same data as in Example 3.7, plot RQI vs r and identify the flow units.

Solution

Table 3.4 contains the input data and calculations for this example. Figure 3.34 is the plot of

the results. Three flow units are identified on the plot.

Figure 3.34 Flow unit identification for Example 3.8.


3.45

3.5 Measurement of absolute permeability

Absolute permeability can be obtained from pressure transient tests (buildup or

drawdowns), from correlations with well log data, or from lab measurements on physical

samples. This section will emphasize core measurement techniques.There are three types of

core analysis techniques: (1) conventional or plug analysis, (2) whole-core analysis and (3)

sidewall core analysis. The technique used depends on the coring method, the type of rock to

be analyzed, and the type of data to be obtained.

Conventional or Plug Analysis. The plug analysis method is used most frequently. In

this method, a small plug sample (3/4” dia, 1 to 1 ½” length), which is easy to work with in

the laboratory, is cut at selected intervals from the whole core. The data obtained from the

small plugs are then assumed to represent the reservoir rock properties of the sampled

interval. The validity of this approach is increased as the rock type becomes more uniform.

It is also necessary to make a decision on the number of samples required for analysis. It is a

generally accepted practice to determine the basic rock properties such as porosity and

permeability on a frequency of one sample per foot with fluid content possibly being

determined less frequently, for example, one for every two to five feet of core. In many

instances, this may turn out to be more data than needed; the indications are that in many

reservoirs the average permeability and porosity determined from one sample for every two

feet did not differ significantly from those determined from one sample per foot.

An additional factor to consider is the sampling procedure since it is most important to avoid

any bias in selecting the samples to be analyzed. A number of techniques can be used that

will eliminate the possibility of selecting the best looking samples.

Whole-Core Analysis. The whole-core analysis method is used when the plug analysis

method becomes invalid because of the presence of heterogeneities such as fractures or vugs.

This method uses the whole core (3 to 3 ½” dia and 6 to 12” length) for rock property

measurement in as long a length as possible. The technique requires larger equipment in the

laboratory, and not all commercial laboratories are equipped to perform this type of analysis.

Sidewall Core Analysis. Considering the process under which these cores are obtained and

the sample size of the core (same size or smaller than plugs), the measured data will have

limited value. Of course, in some areas and in some situations, this rock sample is all that is

available. It is, therefore, desirable to look at the relative value of rock properties as


3.46

determined from sidewall samples and those obtained from conventional cores. Several

studies have been conducted [Helander, 1983] with the results briefly summarized here.

These studies indicate in general that:

1. percussion sample porosities in softer, looser sands are only slightly higher than those

of conventional cores,

2. sidewall sample permeabilities are decreased in higher permeability formations, and

3. water saturations from the sidewall cores are lower and oil saturations slightly higher

than conventional core data.

Based on studies to date, different limiting values and standards of interpretation have been

shown to be necessary when using sidewall sample analysis rather than core analysis. In most

areas where sidewall coring is widely used, however, it appears that suitable relations could

be developed to permit reliable qualitative and possibly even quantitative reservoir

evaluations to be obtained. Certainly the sidewall core data and well log data would be

mutually complementary in the evaluation process.

Permeability is determined by injecting fluid of known density and viscosity through the

core sample and measuring the flow rate and pressure drop. Air is typically selected as the

injectant because of its non-reactive nature, lower cost, and less time consuming through low

permeability samples. Water is also widely used, except in formations with a significant clay

content.


3.47

The apparatus for measuring permeability is shown in Figure 3.35. As shown in the figure,

Figure 3.35 Absolute permeability measuring apparatus [CoreLab, 1983]

both vertical and horizontal permeabilities can be measured. Horizontal permeability is

routinely measured on all sizes of cores. In whole core analysis, two horizontal permeability

values are reported, one in the direction of maximum flow, labeled Kmax, and the other at 90

to the direction of maximum flow, labeled K90. Measurements of permeability in the vertical

orientation are made upon request. Whenever the vertical permeability characteristics are not

well documented, some measurements should be made.

Whenever an insufficient amount of sidewall sample is available for a complete analysis,

it is considered desirable to use the available material for the porosity and saturation

determinations, and to obtain a permeability value from a correlation of permeability with


3.48

porosity (measured) and other textural characteristics available from a careful visual

inspection of the sample.

Comparison of permeability measurements

Whole Core Versus Conventional Samples. Whole core pemeabilities reflect the presence

of vugs and/or fractures that are normally excluded in plug analysis and therefore they are

sometimes higher. Mud solid invasion or buildup of a layer of mud and powdered rock on the

core surface may result in reduced permeability and require sandblasting of the full diameter

sample prior to permeability measurement.

Sidewall Versus Conventional Core Permeability Data. Air permeabilities measured on

percussion sidewall samples are generally too high in hard, low permeability samples less

than about twenty millidarcies. This is due to fracturing and shattering of the samples upon

bullet impact. Percussion sidewall samples from friable and unconsolidated sands with

permeabilities greater than about 20 millidarcies usually yield measured permeabilities that

are too low. This permeability reduction is attributed to partial blocking of pore flow paths by

mud solids and to core compression by the bullet. Figure 3.36 illustrates the differences in

sidewall and conventional core permeabilities for gulf coast samples.

Figure 3.36 Comparison of sidewall and conventional core permeability [Corelab, 1983]

Documents

Distribution of Rock Properties - New Mexico Institute of ...infohost.nmt.edu/~petro/faculty/Engler524/PET524-2c-permeability.pdf · Figure 3.16 is a schematic representation of the