DETERMINATION OF RESERVOIR PERMEABILITY FROM INFLOWS INTO A WELL UNDER STEADY-STATE CONDITIONS

L. A. Pukhlyakov

(Presented by Professor A. V. Aksarin)

Usually, determination of reservoir permeability is carried out either from rock samples taken from wells, or from inflow into a well under an unsteady regime (from pressure-build-up curves). A drawback of the first of these methods is that it determines not the permeability that the reservoir has in the bottom-hole zone, but that possessed by a rock sample cleaned of its clay crust. The permeability of rocks in the bottom-hole zone of the well is lower than this, primarily because of clayization of the reservoir. The method of perforation also affects this parameter to a certain extent.

The second of the above-mentioned methods for determining permeability gives precisely that value which production engineers need to know. However, this method requires very complex investigations, which only specialized organizations are capable of carrying out. Such investigations are not conducted by the personnel of production fields or oil-prospecting enterprises. On their own, these enterprises investigate wells only under steady-state regimes. However, determination of reservoir permeability from the results of these investigations is usually not performed.

The point is that, up to the present time, all hydrodynamic calculations have been based on Dupuit’s inflow formula, which can be written in the following form:

\[ P_{\text{pl}} - P_{\text{zab}} = \frac{Q\mu}{2\pi kh}\ln\frac{R}{r}, \tag{1} \]

where \(P_{\text{pl}}\) and \(P_{\text{zab}}\) are the reservoir and bottom-hole pressures in atmospheres, \(Q\) is the inflow into the well in \(\text{cm}^3/\text{sec}\), \(\mu\) is the viscosity of the liquid in centipoises, \(k\) is the permeability of the rocks in darcies, \(h\) is the reservoir thickness in centimeters, and, finally, \(R\) and \(r\) are the radius of influence of the well and the well radius in the same units. Meanwhile, this formula is suitable for calculating inflow only into hydrodynamically perfect wells, that is, wells that are not cased with casing strings and that penetrate the reservoir over its full thickness. But all modern oil wells are hydrodynamically imperfect: they are all cased with casing strings, which are perforated with a certain density. Thus, if Dupuit’s formula is used to process data for these wells, knowingly erroneous results will be obtained.

Known attempts have been made to improve Dupuit’s formula by introducing into it the so-called coefficients of V. I. Shchurov, whose magnitudes depend on the number of openings, their diameters, and other parameters [1, pp. 165–168]. However, the introduction of these coefficients does not make Dupuit’s formula sufficiently accurate, since even after they are introduced it will follow from this formula that over the entire path of the oil from some remote point to the well bottom its motion will obey a single law. In reality, in the zone of influence of the openings the fluid will move as if along the radii of spheres described around the individual openings in the casing, while outside this zone it will move along the radii of a circle lying in the plane of the reservoir. Thus, two laws of fluid motion are present here.

Taking the above into account, a new formula was derived for the inflow into a hydrodynamically imperfect well, which may be written in the following form:

\[ P_{\text{pl}} - P_{\text{zab}} = \frac{Q'\mu}{2\pi K} \left( \frac{s-\lambda}{s\lambda} - \frac{0.25}{r}\ln\frac{s}{\lambda} + 0.0625\frac{s-\lambda}{r^2} - \right. \]
\[ \left. -0.0117187\frac{s^2-\lambda^2}{r^3} + 0.0026042\frac{s^3-\lambda^3}{r^4} - 0.00079346\frac{s^4-\lambda^4}{r^5} + \right. \]
\[ \left. +0.0002075\frac{s^5-\lambda^5}{r^6} - 0.0000610\frac{s^6-\lambda^6}{r^7} +\ldots+ \frac{1}{h'}\ln\frac{R}{r+s} \right), \tag{2} \]

where \(P_{\text{pl}}\) and \(P_{\text{zab}}\) are the reservoir and bottom-hole pressures in atmospheres, \(Q'\) is the inflow into the well per one opening in cm\(^3\)/sec, \(\mu\) is the oil viscosity in centipoises, \(\lambda\) is the mean radius of the openings in cm, \(s\) is the radius of influence of the openings (one-half of the mean distance between adjacent openings) in cm, \(h'\) is the reservoir thickness per one opening in cm, \(R\) is the radius of influence of the well (it may be taken as equal to one-half the distance between neighboring wells) in cm, and \(r\) is the radius of the well before running the casing, in cm.

Using this formula, one can obtain the permeability of the reservoir in the near-bottom-hole zone from the inflow under steady-state conditions. As an example, let us consider the determination of permeability for the productive bed of well 24 of the Klyuchevoe field of the Khadyzhneft Oil Production Administration, which here is the Maikop series.

The initial data for the calculation are as follows. The depth of the filter is 2091–2106 m; hence the filter length is 1500 cm (15 m). The bit diameter (the diameter of the well before running the casing) is 20 cm (8"). The perforation density is 7 openings per meter. The perforation is bullet perforation; hence the design diameter of the openings \((\lambda)\) is taken as 0.6 cm. The density of the separated oil \((\gamma)\) is 0.829 g/cm\(^3\). The formation volume factor of the oil \((b)\) is 1.48. The viscosity of the reservoir oil \((\mu)\) is 0.52 centipoise. The average distance between wells is 420 m; hence the radius of influence of the well \((R)\) may be taken as 21000 cm (210 m). The well was tested for inflow under four regimes. First regime: bottom-hole pressure 214 atm, inflow 93 t/day. Second regime: bottom-hole pressure 205 atm, inflow 117 t/day. Third regime: bottom-hole pressure 197 atm, inflow 129 t/day. Fourth regime: bottom-hole pressure 190 atm, inflow 198 t/day.

A very important role in determining permeability is played by the correct determination of reservoir pressure. The data given above make it possible to determine this quantity by the productivity-coefficient method, i.e., by the formula

\[ P_{\text{pl}}=\frac{P_1q_2-P_2q_1}{q_2-q_1}, \tag{3} \]

where \(P_1\) and \(P_2\) are bottom-hole pressures under different operating conditions, and \(q_1\) and \(q_2\) are the corresponding inflows.

In the present case, when comparing the first and second operating conditions, the reservoir pressure was found to be 248.9 atm; when comparing the first and third conditions, 257.9 atm; when comparing the first and fourth conditions, 253.3 atm; when comparing the second and third, 283.0 atm; when comparing the second and fourth, 226.7 atm; and, finally, when comparing the third and fourth, 210 atm. The arithmetic mean of these values proved to be 246.6 atm, or, rounded, 247 atm. We shall take this value as the reservoir pressure in the case under consideration.

The subsequent processing of the data is carried out in the following order.

1. The total number of openings in the filter, \(n\), is determined.

\[ n = h \cdot 7 = 15 \cdot 7 = 105 \text{ openings.} \]

1. The reservoir thickness per one opening, \(h'\), is determined.

\[ h' = \frac{100}{7} = 14.286 \text{ cm.} \]

1. The average distance between openings, \(l\), is determined. With a high perforation density, this value may be determined, as described earlier [2], through the area per one opening, \(F\), which is expressed by the relation

\[ F = \pi \cdot D \cdot h' = \pi \cdot 20 \cdot 14.286 = 897.62 \text{ cm}^2. \]

In such cases the desired value is determined as the square root of this area:

\[ l = \sqrt{F} = \sqrt{897.62} = 29.96 \text{ cm.} \]

1. The radius of influence of the openings, \(s\), equal to one half of the distance between openings, is determined; in the present case

\[ s = \frac{1}{2}l = \frac{1}{2} \cdot 29.96 = 14.98 \text{ cm,} \]

or approximately 15 cm.

1. The inflow to one opening under reservoir conditions, \(Q'\), is determined. The calculation is made by the formula

\[ Q' = q \frac{b \cdot 1000000}{n\gamma \cdot 86400}, \tag{4} \]

substituting into it \(q = 93\) t/day, \(n = 105\) openings, \(b = 1.48\), and \(\gamma = 0.829\) g/cm\(^3\), we obtain

\[ Q'_1 = \frac{93}{5.0815} = 18.3 \text{ cm}^3/\text{s}. \]

1. The obtained values are substituted into formula (2); as a result, for the first operating condition we obtain

\[ 247 - 214 = \frac{18.30 \cdot 0.52}{2\pi k} \left( \frac{15.0 - 0.6}{15.0 \cdot 0.6} - \frac{0.25}{10}\ln\frac{15}{0.6} + 0.0628\frac{15 - 0.6}{100} \right. \]

\[ \left. - 0.0117187\frac{15^2 - 0.6^2}{10^3} + 0.0026042\frac{15^3 - 0.6^3}{10^4} - 0.0007935\frac{15^4 - 0.6^4}{10^5} + \right. \]

\[ \left. + 0.0002075\frac{15^5 - 0.6^5}{10^6} - 0.000061\frac{15^6 - 0.6^6}{10^7} + \frac{1}{14.286}\ln\frac{21000}{15 + 10} \right) \]

or, after the corresponding transformations,

\[ 33=\frac{18.3}{12.083\,k}\cdot\left(1.5264+\frac{6.7335}{14.286}\right), \]

whence

\[ k=\frac{18.3}{12.083\cdot 33}\cdot 1.9967=0.0916\ \text{darcy} \]

or 91.6 millidarcies.

Continuing the calculation by an analogous method, we find that for the second regime the permeability is 90.6, for the third 82.8, for the fourth 113, and the average over the whole reservoir is 94.8 millidarcies.

Note. Formula (2) remains valid provided that the radius of influence of the opening \(s\) does not exceed the diameter of the well \((2r)\). If this condition is violated, that is, if the distances between openings exceed the well diameter by more than a factor of two, then for calculating the inflow characteristics one should use the formula

\[ P_{\text{res}}-P_{\text{bh}} = \frac{Q''\mu}{2\pi k} \left( \frac{1}{\lambda} - 0.14362\,\frac{1}{r} - \frac{0.25}{r}\ln\frac{2r}{\lambda} - \frac{0.5}{s'} - 0.125\,\frac{r^2}{s'^3} - 0.005625\,\frac{r^4}{s^5} + \frac{1}{h'}\ln\frac{R}{r+s'} \right). \tag{5} \]

REFERENCES

1. I. M. Murav’ev, R. S. Andriasov, Sh. K. Gimatudinov, G. L. Govorova, V. T. Polozkov. Development and Operation of Oil and Gas Fields. Nedra Publishing House, 1965.

2. L. A. Pukhlyakov, M. V. Samoilova. On the question of oil inflow into a hydrodynamically imperfect well. Izvestiya TPI, vol. 196, 1969.