Determining the Rational Number of Perforations in the Casing of an Oil Exploration Well

L. A. Pukhlyakov

(Presented by Professor A. V. Aksarin)

Many authors have dealt with the problem of fluid inflow into hydrodynamically imperfect wells, above all L. S. Leibenzon [1, 2], G. B. Pykhachev [7, 8], M. N. Tikhov [9, 10], V. I. Shchurov, I. G. Akhmatov [11], and also the author of the present article [3, 4, 5, 6], who, for determining various inflow parameters, recommends using the following formula:

\[ P_{\text{pl}} - P_{z} = \frac{Q_{\text{pl}}\mu}{2\pi k n}\,(G_s + G_{\delta} + G_R), \tag{1} \]

where \(P_{\text{pl}}\) and \(P_z\) are the reservoir and bottom-hole pressures in atm, \(\mu\) is the viscosity of the oil under reservoir conditions in centipoise, \(k\) is the average permeability of the formation in darcies, \(n\) is the number of openings in the casing, and \(Q_{\text{pl}}\) is the inflow of oil into the well under reservoir conditions in \(\text{cm}^3/\text{s}\), calculated by the formula

\[ Q_{\text{pl}} = \frac{1000000}{86400}\,Q_{\text{pv}} b, \tag{2} \]

where \(Q_{\text{pv}}\) is the oil inflow into the well, measured under surface conditions in \(\text{m}^3/\text{day}\), \(b\) is the oil volume factor, and 1000000 and 86400 are the coefficients for converting \(\text{m}^3\) to \(\text{cm}^3\) and days to seconds. \(G_s\), \(G_{\delta}\), and \(G_R\) are the geometric characteristics of the zone of influence of the openings, the zone of stream constriction due to incomplete perforation, and the zone of plane-radial flow. To determine them, the following quantities are needed: \(r\) is the radius of the well before lowering the casing, in cm; \(R\) is the radius of influence of the well, in cm; \(h\) is the length of the perforation interval, in cm; \(H\) is the total thickness of the formation, in cm; \(\lambda\) is the radius of the opening and of the associated channel, in cm; \(s\) is the radius of influence of the openings, in cm, understood as half the average distance between the centers of the openings; \(y\) is the depth (length) of the channels at the openings in the casing, in cm, measured from the outer surface of the cement sheath to the bottom of the channel; \(x_0\) is the radius of influence of the filter, determined by the formula

\[ x_0 = r + y + s, \tag{3} \]

as well as the components of incomplete perforation.

The components of incompleteness are understood to be the unperforated sections when they adjoin either the roof or the bottom of the formation, or halves of these sections if on both sides of them there are located

perforated. In doing so, only those sections or halves of them that exceed the radius of influence of the filter are taken into account.

The first component is denoted by \(\delta_1\). By this is meant the component that exceeds all the others, if there are several of them. The second component of incompleteness is taken to be a component that either is paired or can be placed within the limits of the first. In the latter case it is shorter than the first. In the former case, that is, in the case of equality of the two largest components, both are regarded as second components. The second component is denoted by \(\delta_2\).

The third component is considered to be one that can be laid off within the limits of the other three times. It is denoted by \(\delta_3\). The remaining components are named analogously.

The definition of the radius of influence of the openings requires some clarification. At high perforation density, when the formation thickness per one opening, \(\frac{h}{n}\), does not reach \(2.55\,r\), this quantity is calculated by the formula

\[ s = 0.5\sqrt{2\pi r\frac{h}{n}}, \tag{4} \]

and if the length of the filter per one opening, \(\frac{h}{n}\), exceeds \(2.55\,r\), then by the formula

\[ s^* = 0.5\sqrt{\frac{h^2}{n^2} + 0.96\pi^2 r^2}. \tag{5} \]

Here it is necessary to note that, in determining the rational number of openings in the casing of an oil exploration well, we shall have to use exclusively the formulas for high perforation density.

In the absence of channels at the openings and with high perforation density, the geometric characteristic of the zone of influence of the openings is calculated by the formula

\[ G_s = \frac{1}{\lambda} - \frac{1}{s} - \frac{1}{4r}\ln\frac{s}{\lambda} + \frac{1}{16}\frac{s-\lambda}{r^2} - 0.0117\frac{s^2-\lambda^2}{r^3}. \tag{6} \]

Under the same conditions, but in the case where channels are present at the openings, the formula for this characteristic has a somewhat different form:

\[ G_s^* = \frac{1}{y}\ln\frac{(y+\lambda)s}{(y+s)\lambda} - \frac{0.25}{r+y}\ln\frac{s+y}{\lambda+y} + 0.0625\frac{s-\lambda}{(r+y)^2}. \tag{7} \]

The geometric characteristic of the flow-constriction zone for one-component perforation incompleteness is calculated by the formula

\[ G_{\delta-1} = \frac{1}{h}\ln\frac{(h+x_0)\delta_1}{(h+\delta_1)x_0}. \tag{8} \]

When there is a larger number of components of incompleteness, other formulas are used; however, the rational number of openings in the casing is determined on the basis of their uniform distribution along the entire length of the borehole. This means that below we shall consider the case in which the flow-constriction zone in the drainage region of the well will be absent and, consequently, none of these formulas will be used.

The geometric characteristic of the plane-radial-flow zone in this case can be calculated by the formula

\[ G_R = \frac{n}{H}\ln\frac{R}{r+y+s}. \tag{9} \]

In the presence of a flow-constriction zone, it is calculated by the formula

\[ G_R^*=\frac{h}{H}\ln\frac{R}{\delta_{\max}}. \tag{10} \]

Let us consider an unconsolidated formation, that is, a case in which there are no channels at the openings in the casing, and in which, in order to determine the geometric characteristic of the zone of influence of the openings, it is necessary to use formula (6). To determine the geometric characteristic of the zone of plane-radial flow, it is obvious that formula (9) should be used.

Analysis shows that, with an increase in the number of openings \(n\) and a decrease in the radius of influence of the openings \(s\), the geometric characteristic of the zone of plane-radial flow will increase, while the geometric characteristic of the zone of influence of the openings will decrease. Therefore, as a criterion for determining the rational number of openings, it is recommended to take such a number of them at which the geometric characteristics under consideration will be equal to one another.

Suppose that, in a certain case, the radius of an opening \(\lambda\) is equal to \(0.5\) cm, the radius of the well before the casing string is run is \(r=9.5\) cm, the formation thickness is \(H=100\) cm, and the radius of influence of the well is \(R=10000\) cm (\(10\) m). With 28 openings per meter, the radius of influence of the openings, according to formula (4), proves to be equal to \(7.3\) cm; the geometric characteristic of the zone of influence of the openings, according to formula (6), is 1.80, and the geometric characteristic of the zone of plane-radial flow, according to formula (9), is also 1.80. Thus, under these conditions, this number of openings per meter of filter will be rational.

If other radii of the openings are taken as initial data, then other numbers of them per linear meter of filter will be obtained (see the table).

Table

Rational numbers of openings per linear meter of well filter for different radii of openings and wells

It is interesting to note that by doubling the number of holes relative to the rational number (indicated in the table), other conditions being equal, we obtain an increase in inflow relative to the initial value by a factor of 1.3; and by again doubling the number of holes, we obtain an additional increase in inflow by only a factor of 1.18. If, however, we reduce the number of holes relative to the rational number (given in the table) by a factor of two, then, other conditions being equal, we obtain a decrease in inflow by a factor of 1.5; and if we reduce it again by a factor of two, the inflow will decrease by a further factor of 1.76.

Changing the well diameter has little effect on the rational number of holes. However, the presence of channels at the holes substantially reduces their number. Thus, if it is assumed that the length of the channels at the holes reaches 2 cm and their average radius is 0.2 cm, then, for a well radius of 9.5 cm, only 20 holes per meter prove to be rational.

REFERENCES

1. L. S. Leibenzon. Handbook of Oilfield Mechanics, Part 1. Moscow, State Scientific and Technical Publishing House, 1931.

2. L. S. Leibenzon. Oilfield Mechanics, Part II. Moscow, Gorneonefteizdat, 1934.

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

4. L. A. Pukhlyakov. Determination of reservoir permeability from inflows into a well under steady-state conditions. Izvestiya TPI, vol. 177, TSU Publishing House, 1971.

5. L. A. Pukhlyakov. Derivation of an inflow formula for a hydrodynamically imperfect well. Izvestiya TPI, vol. 201, TSU Publishing House, 1972.

6. L. A. Pukhlyakov. On inflow into a well possessing several types of hydrodynamic imperfection. Izvestiya TPI, vol. 264, TSU Publishing House, 1976.

7. G. B. Pykhachev. On the production rate of a well in a heterogeneously permeable reservoir. Grozny, Grozny Book Publishing House, 1944.

8. G. B. Pykhachev. Underground Hydraulics. Moscow, Gostoptekhizdat, 1961.

9. M. N. Tikhov. The influence of perforations in casing pipes on well productivity. Neftyanoe khozyaistvo, No. 5, 1947.

10. M. N. Tikhov. Mathematical Theory of the Motion of Liquid and Gas toward a Centrally Imperfect Well. Kharkov, Kharkov University Press, 1964.

11. V. I. Shchurov, I. S. Akhmatov. Certain questions in the study of oil wells under steady withdrawals. In: Physics and Hydrodynamics of an Oil Reservoir. Proceedings of MINKh and GP, Issue 57. Moscow, Nedra, 1966.