PET524-2c-permeability.pdf

Chapter 3 – Permeability 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.

3.17

Chapter 3 – Permeability

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.

10.2d k k s

2 (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.

3.18

Chapter 3 – Permeability

Figure 3.16 Capillary tube model

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

  ntr 2 

(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,

nt r 4 k 8 

(3.13)

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

k

 r2 8

(3.14)

Example 3.3 For the cubic packing arrangement shown in the diagram below, determine the porosity and permeability.

r

3.19

Chapter 3 – Permeability Solution

2 The number of tubes per unit area is: 4tubes /( 4r ) . Substituting into Eq. (3.12) results in an estimate for porosity.



2  1 * r  2 4 4r

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.

S bv   * S pv     S S gv   pv 1    

(3.15)

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

k

 2 k z S pv

(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.

L   a  L 

   

2 (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 k o = 2,  = 1 (tubes and unit length are equal and parallel), and Spv = 2/r into Eq. (3.16), we obtain Eq. (3.14).

3.20

Chapter 3 – Permeability 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

r

2 S pv



1 2  0.011 mm 182

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

(.277)(0.011) k 2(3.6)(4)

2  1.164 x10

6

mm

2

Converting to darcies,

k  1.164 x10

6

 2  1cm  2  1 darcy   1.18 Darcys  * mm *     8 2  10 mm   0.987 x10 cm 

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

nt 

 r 2



.277

 (0.011)2

 729 mm2

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.

  ntr 2 

(3.18)

 r2 8

(3.19)

k

3.21

Chapter 3 – Permeability Furthermore, let’s introduce the concept of hydraulic radius, the ratio of the volume open to flow to the wetted surface area or,

   1 1  rh     1    S gv S pv

(3.20)

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

rh2 ko

(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,

k

 3d 2p 721   2

(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

3.22

Chapter 3 – Permeability 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.

3.23

Chapter 3 – Permeability 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

3.24

Chapter 3 – Permeability 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.

n  i h i   i 1 h T

(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%.

3.25

Chapter 3 – Permeability

10

1.2

9

Frequency

7

0.8

6 5

0.6

4 0.4

3 2

Cum ulative Frequency

1.0

8

0.2

1 0

0.0 4

6

8

10

12

14

16

18

20

22

24

26

28

Porosity , %

Figure3.18. Example porosity histogram and cumulative frequency curve Porosity interval 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Frequency

Cumulative Frequency

1 0 0 0 1 0 1 2 1 4 3 9 3 2 1 0 1 1 1 2 0 2 1 2 1 1

0.025 0.025 0.025 0.025 0.050 0.050 0.075 0.125 0.150 0.250 0.325 0.550 0.625 0.675 0.700 0.700 0.725 0.750 0.775 0.825 0.825 0.875 0.900 0.950 0.975 1.000

Table 3.3 Tabulated data for histogram in Figure 3.18

3.26

Chapter 3 – Permeability 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.

1/ n  n    k g   ki    i 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

3.27

Chapter 3 – Permeability and Parsons (1950) recognized this important feature and introduced the permeability variation, V as,

k  k 84.1 V  50 k 50

(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.

3.28

Chapter 3 – Permeability 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 straightline 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.

3.29

Chapter 3 – Permeability

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]

3.30

Chapter 3 – Permeability 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,

Lk 

Area ABCA Area ADCA

(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.)

3.31

Chapter 3 – Permeability

Depth,ft 2905 2906 2907 2908 2909 2910 2911 2912 2913.1 2913.5 2915 2916 2917 2918 2934 2935 2936 2937 2938 2939 2940 2941 2942 2943 2944 2945 2946 2947 2948 2949 2950 2951 2952 2953 2954 2955 2956 2957 2958 2959

Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Arith mean Geo mean Har mean

k, md 0.224 0.337 0.187 0.653 1.040 0.450 434.000 196.000 0.007 1156.000 531.000 1059.000 822.000 1014.000 109.000 138.000 166.000 362.000 77.900 64.900 51.100 89.900 84.100 21.200 23.700 39.600 44.400 20.800 13.900 20.800 6.390 10.000 15.300 11.400 22.800 37.200 29.100 5.840 13.900 16.400

 0.1200 0.1015 0.1165 0.1304 0.1153 0.0853 0.2519 0.2159 0.0467 0.2949 0.2679 0.2874 0.2765 0.2769 0.2269 0.2330 0.2381 0.2554 0.2009 0.1863 0.1685 0.1555 0.1636 0.1537 0.1676 0.1728 0.1770 0.1578 0.1510 0.1543 0.1365 0.1449 0.1492 0.1447 0.1518 0.1537 0.1537 0.1364 0.1529 0.1387

RQI, m r 0.136 0.043 0.113 0.057 0.132 0.040 0.150 0.070 0.130 0.094 0.093 0.072 0.337 1.303 0.275 0.946 0.049 0.012 0.418 1.966 0.366 1.398 0.403 1.906 0.382 1.712 0.383 1.900 0.293 0.688 0.304 0.764 0.313 0.829 0.343 1.182 0.251 0.618 0.229 0.586 0.203 0.547 0.184 0.755 0.196 0.712 0.182 0.369 0.201 0.373 0.209 0.475 0.215 0.497 0.187 0.361 0.178 0.301 0.182 0.365 0.158 0.215 0.169 0.261 0.175 0.318 0.169 0.279 0.179 0.385 0.182 0.488 0.182 0.432 0.158 0.205 0.180 0.299 0.161 0.341

167.76 md 23.28 md 0.25 md Table 3.4 Input data for Example 3.7

3.32

Chapter 3 – Permeability

Descending Permeability for Lorenz Coefficient

k, md 1156.000 1059.000 1014.000 822.000 531.000 434.000 362.000 196.000 166.000 138.000 109.000 89.900 84.100 77.900 64.900 51.100 44.400 39.600 37.200 29.100 23.700 22.800 21.200 20.800 20.800 16.400 15.300 13.900 13.900 11.400 10.000 6.390 5.840 1.040 0.653 0.450 0.337 0.224 0.187 0.007

Fraction of Fraction of cumulative total flow cumulative total k,md capacity porosity porosity volume 1156 0.172 0.2949 0.2949 0.042 2215 0.330 0.2874 0.5823 0.083 3229 0.481 0.2769 0.8592 0.123 4051 0.604 0.2765 1.1357 0.162 4582 0.683 0.2679 1.4036 0.200 5016 0.747 0.2519 1.6555 0.236 5378 0.801 0.2554 1.9109 0.273 5574 0.831 0.2159 2.1268 0.303 5740 0.855 0.2381 2.3649 0.337 5878 0.876 0.2330 2.5979 0.370 5987 0.892 0.2269 2.8248 0.403 6077 0.906 0.1555 2.9803 0.425 6161 0.918 0.1636 3.1439 0.448 6239 0.930 0.2009 3.3448 0.477 6304 0.939 0.1863 3.5311 0.504 6355 0.947 0.1685 3.6996 0.528 6399 0.954 0.1770 3.8766 0.553 6439 0.960 0.1728 4.0494 0.577 6476 0.965 0.1537 4.2031 0.599 6505 0.969 0.1537 4.3568 0.621 6529 0.973 0.1676 4.5244 0.645 6552 0.976 0.1518 4.6762 0.667 6573 0.979 0.1537 4.8299 0.689 6594 0.983 0.1578 4.9877 0.711 6615 0.986 0.1543 5.1420 0.733 6631 0.988 0.1387 5.2807 0.753 6646 0.990 0.1492 5.4299 0.774 6660 0.992 0.1510 5.5809 0.796 6674 0.995 0.1529 5.7338 0.818 6685 0.996 0.1447 5.8785 0.838 6695 0.998 0.1449 6.0234 0.859 6702 0.999 0.1365 6.1599 0.878 6708 1.000 0.1364 6.2963 0.898 6709 1.000 0.1153 6.4116 0.914 6709 1.000 0.1304 6.5420 0.933 6710 1.000 0.0853 6.6273 0.945 6710 1.000 0.1015 6.7288 0.960 6710 1.000 0.1200 6.8488 0.977 6711 1.000 0.1165 6.9653 0.993 6711 1.000 0.0467 7.0120 1.000

Table 3.5 Calculations for Lorenz Coefficient

3.33

Chapter 3 – Permeability

Flow Capacity Distribution 1 0.9

Fraction of total Flow Capacity

0.8 0.7 0.6 0.5 0.4 0.3 y = -3.8012x 4 + 10.572x 3 - 11.01x 2 + 5.2476x - 0.0146 R2 = 0.9991

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Fraction of total Volum e

Figure 3.24 Fraction of flow vs. storage capacity for determination of L k.

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]

3.34

depth

Chapter 3 – Permeability

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.

3.35

Chapter 3 – Permeability

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,

3.36

Chapter 3 – Permeability

   1    k 1   2  k S 2   o gv 



3

(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,      1        1     k  S   o gv  k

(3.28)

We can define the Reservoir Quality Index (RQI), as the ratio of permeability to porosity. RQI {m}  0.0314

k{md}



(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.

FZI 

1 S gv k z

(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,

r 

 1

(3.31)

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

log( RQI )  log(  r )  log( FZI )

3.37

(3.32)

Chapter 3 – Permeability 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]

3.38

Chapter 3 – Permeability 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.

2 2  2 FZI    3     k     *     0.5      1    FZI  k    

0.5 (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 (0.5m < r < 1.5 m), micropores (r < 0.5 m) and noflow 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.

3.39

Chapter 3 – Permeability

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,

FZI  (c1 *  ) * (c2 * b ) * (c3 * Rild ) * ....

(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 3.40

Chapter 3 – Permeability 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, k  1014( FZI )



2

3

(1   )

2

(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. excellent predictive capabilities of the proposed method.

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Again, note the

Chapter 3 – Permeability

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

3.42

Chapter 3 – Permeability 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,… . . . . . .

Uncored log data

y1,y2,…

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;

core  c1 * log log( k )  c *  c h 2 core 3

(3.36)

log( k )  c * log( k )  c v 4 h 5 The estimate of permeability is related to the cluster analysis to ascertain the flow unit.

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Chapter 3 – Permeability

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.

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Chapter 3 – Permeability 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

3.45

Chapter 3 – Permeability 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.

3.46

Chapter 3 – Permeability 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

3.47

Chapter 3 – Permeability 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]

3.48