On the Rational Number of Holes in the Casing of an Oil Exploration Well
Abstract Generated abstract
This paper examines how many perforation holes in the casing of an oil exploration well are rational for obtaining inflow from a one meter oil bearing bed under specified reservoir and well conditions. Using formulas for the geometric characteristics of the holes’ influence zones and plane radial flow, it calculates inflow for increasing perforation densities and compares the marginal gains from adding holes. The results show that inflow initially rises almost proportionally with the number of holes, but the gain declines sharply at high densities, making very dense perforation unnecessary. For the chosen parameters, the rational density is identified near the point where the two geometric characteristics are equal, about 28 holes per meter, with this number increasing for smaller hole radii and decreasing for smaller well radii.
Full Text
ON THE RATIONAL NUMBER OF HOLES IN THE CASING OF AN OIL EXPLORATION WELL
L. A. Pukhlyakov
(Presented by Professor A. V. Aksarin)
To solve the question of the rational number of holes in an oil exploration well, let us choose an oil-bearing bed with a thickness of \(1\ \text{m}\) and a permeability of \(0.012\) darcy, and open it with a well of radius \(9.5\ \text{cm}\) (bit diameter \(190\ \text{mm}\)); then we lower a metal casing into this well, cement it, and perforate it. In the process of perforation we first make one hole in it, then two, then four, eight, etc., and determine, for each new number of holes, the inflow into the well under otherwise equal conditions: oil viscosity \(0.8\ \text{cP}\), drawdown \(20\ \text{atm}\).
For simplicity of reasoning, let us assume that the bed opened by us is sufficiently loose and therefore the channels at the holes are not preserved. In this case the inflow into the well will be determined by the relation
\[ Q=\frac{2\pi k n}{\Delta P\cdot \mu \cdot (G_s+G_R)} \tag{1} \]
where \(\Delta P\) is the drawdown on the bed, atm; \(k\) is the bed permeability in darcy; \(\mu\) is the oil viscosity in cP; \(n\) is the number of holes; and \(G_s\) and \(G_R\) are the geometric characteristics of the zone of influence of the holes and of the zone of plane-radial flow, in \(1/\text{cm}\). The second of these quantities is calculated by the formula
\[ G_R=\frac{n}{H}\ln\frac{R}{r+S}, \tag{2} \]
where \(H\) is the bed thickness in cm, \(r\) is the well radius in cm, \(R\) is the radius of influence of the well in cm, and \(S\) is the radius of influence of the holes, or half the mean distance between the centers of two adjacent holes, in cm.
The first two of the named quantities were given above; the radius of influence of the well, within the limits of the present problem, may be taken as equal to \(10{,}000\ \text{cm}\). As for the radius of influence of the holes, at high perforation density, when the length of filter per one hole \(\frac{h}{n}\) does not exceed \(2.55r\), it should be calculated by the formula
\[ s=0.5\sqrt{2\pi r\frac{h}{n}}, \tag{3} \]
and at low perforation density, when the length of filter per one hole exceeds \(2.55r\), by the formula
\[ s=0.5\sqrt{\frac{h^2}{n^2}+0.96\pi^2 r^2}. \tag{4} \]
The geometric characteristic of the zone of influence of the holes at high perforation density is calculated by the formula
\[ G_s=\frac{1}{\lambda}-\frac{1}{S}-\frac{1}{4r}\ln\frac{s}{\lambda}+\frac{s-\lambda}{16r^2}-0{,}0117\,\frac{s^2-\lambda^2}{r^3}+0{,}0026\,\frac{s^3-\lambda^3}{r^4}, \tag{5} \]
where \(\lambda\) is the radius of the hole in cm, and at low density by the formula
\[ G'_s=\frac{1}{\lambda}-\frac{0{,}14362}{r}-\frac{1}{4r}\ln\frac{2r}{\lambda}-\frac{1}{2S}-\frac{r^2}{8S^3}-0{,}05625\,\frac{r^4}{S^5}. \tag{6} \]
Taking the radius of the hole in the casing to be \(0{,}5\) cm, we find that, with one hole per meter, the inflow into the well will be \(0{,}976\ \mathrm{cm^3/sec}\), and with two holes per meter, \(1{,}901\ \mathrm{cm^3/sec}\), etc. (table). In other words, we see that, when the number of holes in the casing is small,
Table
| Number of holes per 1 m | Radius of influence of holes, cm | Geometric characteristics: zones of influence of holes, 1/cm | Geometric characteristics: zones of plane-radial flow, 1 cm | Inflows into the well, cm³/s | Degree of increase in inflow compared with the preceding one, in fractions of unity |
|---|---|---|---|---|---|
| 1 | 52,10 | 1,8805 | 0,0509 | 0,976 | — |
| 2 | 29,00 | 1,8715 | 0,1112 | 1,901 | 1,948 |
| 4 | 19,24 | 1,8616 | 0,2036 | 3,651 | 1,920 |
| 8 | 13,66 | 1,8477 | 0,4895 | 6,452 | 1,767 |
| 16 | 9,70 | 1,8243 | 1,0009 | 10,68 | 1,655 |
| 32 | 6,83 | 1,7888 | 2,0535 | 15,70 | 1,471 |
| 64 | 4,83 | 1,7360 | 4,1907 | 20,36 | 1,297 |
| 128 | 3,424 | 1,6591 | 8,5136 | 23,72 | 1,165 |
| 256 | 2,414 | 1,5466 | 17,236 | 25,69 | 1,083 |
| Open bottomhole | 0,0 | 0,00 | \(n\cdot 0{,}0695\) | 27,09 | — |
doubling this number leads to an increase in inflow by almost a factor of two. However, subsequently the rate of increase of the inflow decreases. For example, increasing the number of holes from 128 to 256, we obtain an increment of inflow by only a factor of \(1{,}083\).
It follows from what has been said that there is no need to bring the perforation density up to 256 holes per meter. Similarly, it follows from this calculation that, when perforating wells, one should not limit oneself to one hole per meter. The question arises as to what number of holes in the casing should be considered rational.
It is seen from the table that the geometric characteristic of the zone of influence of the holes decreases as the perforation density increases, while the geometric characteristic of the zone of plane-radial flow increases. Obviously, the break point here will be the one where the characteristics under consideration are equal to one another. This point lies between 16 and 32 holes per meter, and additional calculations make it possible to determine it more precisely. The geometric characteristics under consideration will be equal to one another at a perforation density of 28 holes per meter.
As the radii of the holes decrease, their geometric characteristics become equal to one another at a larger number of holes per meter. Thus, the rational number of holes increases as their radii decrease. As the well radius decreases, it decreases.