Permeability is one of the most important and critical petrophysical properties to determine the economic value of a reservoir. Prediction of permeability serves as a platform and prerequisite for any integrated reservoir studies. Single-phase permeability measurements are important to understand fluid flow through porous media. Permeability can be estimated indirectly using log data directly with core sample. Engineers and geologists observed that there is not a specifically defined trend line between porosity and permeability. In addition, their relationship is qualitative and is not direct or indirect in any way because it is possible to be high porosity without permeability as in clay and shale, Qays . In addition, it is true to be high permeability with low porosity as in microfractured carbonates. However, there often can be found a very useful correlation between them, Tiab and Donaldson .
Kozeny  proposed the first empirical equation relating measurable rock properties with permeability using bundle of straight capillary tubes model with constant and uniform surface area. Carman  modified Kozeny equation by describing permeability in packs of uniformly sized spheres. Timur  proposed a generalized equation based on the work of Kozeny .
Amaefule  introduced a new practically and theoretically based technique, which has been developed to identify and characterize units with similar pore throat geometrical attributes (hydraulic units). This theory of hydraulic flow units (HFUs) and reservoir quality index pending the pore-throat, pore and grain arrangement and different macroscopic parameters.
2. FIELD DESCRIPTION
Bai Hassan field (Fig. 1) is located in the north of Iraq to the northwest of Kirkuk governorate and to the north of Kirkuk giant oil field. It is elongated from the north-west to the south-east; the surface structure differs from the subsurface structures because of the faults effect. The field is of a very complex nature with several main opposite faults and mini faults spreading in the field. It is 40 km in length and 3.8 km in width, it was discovered in 1929 and the production started on 1959, Sadeq .
Fig. 1. The location of studied Bai Hassan field, Sadeq and Bhattacharya .
3. DYKSTRA-PARSONS COEFFICIENT VK
Dykstra and Parsons used the log-normal distribution of permeability to define the coefficient of permeability variation, Tiab and Donaldson .
S is the standard deviation for permeability, n is a number of samples, K and Kare the permeability, and mean is the permeability of main direction.
The range of this index is as and can be interpreted as follows:
If =0, the reservoir is ideal homogeneous
If 0<<0.25, then the reservoir is slightly heterogeneous
If 0.25<<0.50, the reservoir is heterogeneous
If 0.50<<0.75, the reservoir is very heterogeneous
If 0.75<<1, the reservoir is extremely heterogeneous, and
If = 1, then the reservoir is perfectly heterogeneous, Tiab and Donaldson .
Hence, for Bai Hassan cretaceous middle carbonate reservoir, V = 11.4/12.59 = 0.9. This value of Dykstra-Parsons coefficient shows that the reservoir is extremely heterogeneous because it is a fractured carbonate reservoir.
4. PERMEABILITY PREDICTION DEPENDING ON ROCK TYPE METHODS
Conventional method for rock typing is based on simple regression evaluating permeability from log-derived porosity. In most cases, a linear relationship between log permeability and porosity is obtained, but in carbonate formations, it does not close to the actual case. Fig. 2 shows the conventional permeability-porosity relationship for the entire reservoir. There is a poor relationship between permeability and porosity (= 0.191).
Fig. 2. Permeability porosity cross-plot for the entire reservoir
In this approach, a mathematical relationship introduced between the petrophysical properties such as porosity, permeability, and capillary pressure to pore-throat radius measured in a mercury injection capillary pressure experiment at mercury saturation of 35%, Gunter. . The Winland equation is:
Where, R is the calculated pore-throat radius at 35% mercury saturation from mercury injection capillary pressure test, k is permeability (md), and Ø is porosity (percentage). The core samples of similar R values represent a single rock type. Petrophysical units can be defined using below classification of R values:-
Megaport: When the value of R35 is >10 µ
Macroport: When the value of R35 is between 2 and 10 µ
Mesoport: When the value of R35 is between 0.5 and 2 µ
Microport: When the value of R35 is between 0.1 and 0.5 µ.
Figs. 3-5 show the conventional permeability-porosity relationship for well A, well B, and well C depending on Winland method. As shown in these figures, there are weak relationships between permeability and porosity (≤ 0.19). Winland method is experimental measurements, which means that it does not work for all circumstances.
Fig. 3. Winland plot between porosity and permeability for well A
Fig. 4. Winland plot between porosity and permeability for well B
Fig. 5. Winland plot between porosity and permeability for well C
The inverse (negative) relationship between the porosity and permeability in this well resulted from the high shale volume (Vsh)5. HFU METHOD (HFUM)
Geologists and engineers specified the definition of units to shape the description of reservoir zones as storage containers and reservoir conduits for fluid flow. Literatures confirmed that the flow units, as the resultant of the depositional environment and diagenetic process. The hydraulic (pore geometrical) unit is the representative elementary volume of the total reservoir rock within the geological and petrophysical properties of the rock volume, Bear .
HFUs consider as map-able portion of the reservoir within which the geological and petrophysical properties that affect the flow of fluid are consistent and predictably different from the properties of other reservoir rock volume, Ebanks Jr. .
Hearn.  defined flow unit as a reservoir zone that is laterally and vertically has similar permeability, porosity, and bedding characteristic. A continuous stratigraphically interval of similar reservoir processes that honor the geologic framework and maintains the characteristic of the rock type, Gunter. . The rock types are classified according to the following equations:
Where, k is the permeability (md), is the effective porosity (fraction), RQI is rock quality index (µm), is the normalize porosity, and FZI is flow zone indicator. On a log-log plot of RQI versus, all samples with similar FZI values will lie on other parallel lines. The values of FZI constant can be determined from the intercept of unit slope straight line at = 1, Haghighi. .
HFUM works perfectly to characterize and predict permeability for uncored wells as shown in Figs. 6, 8, and 14. They indicate that permeability can be accurately predicted from porosity if the rock type is known.
Fig. 6. Log-log plot of RQI versus Øz indicating the presence of four flow units for well A
Fig. 7. Permeability porosity cross-plot for well A depending on FZI
Fig. 8. Log-log plot of RQI versus Øz indicating the presence of five flow units for well B
Fig. 9. Permeability porosity cross-plot for well B depending on FZI
Fig. 10. Comparison between measured permeability and predicted permeability depending on Eq. (7) for well A shows excellent matching between a measured and predicted values, this represents that VK value is very close to zero, the reservoir is ideal homogeneous
Fig. 11. Comparison between measured permeability and predicted permeability depending on k versus phi-dependent FZI for well A
Fig. 12. Comparison between measured permeability and predicted permeability depending on Eq. (7) for well B shows excellent matching between a measured and predicted values, this represents that VK value is very close to zero, the reservoir is ideal homogeneous
Fig. 13. Comparison between measured permeability and predicted permeability depending on k versus phi-dependent FZI for well B
Fig. 14. Log-log plot of RQI versus Øz for the three studied wells
Porosity alone is not sufficient to describe the permeability variations, even if the porosity-permeability data that were used came from same field. This study shows good relationship between porosity and permeability depending on ZI as shown in Figs. 7-15.
Fig. 15. Permeability porosity cross-plot for the three studied wells depending on FZI
The relationship is excellent because ≈ 1 and the presence of good matching between measured permeability and predicted permeability as shown in Figs. 11 and 13. The observations from this study refer to insignificant differences between porosity and permeability equations for each well alone and generalize equations (all three wells data). As shown in Table I, the high FZI values indicate high permeability values within four HFUs for well A and five for well B.
TABLE I Generalized permeability depending on FZI