On the Problem of Oil Inflow into a Hydrodynamically Imperfect Well

L. A. Pukhlyakov, M. V. Samoilova

(Presented by Prof. A. V. Aksarin)

When determining the inflow of oil into a well, one usually uses Dupuit’s formula

\[ Q=\frac{2\pi kh\left(P_{0}-P_{3}\right)}{\mu \ln \dfrac{R}{r}}, \tag{1} \]

where \(Q\) is the inflow in tons/day, \(P_{0}\) and \(P_{3}\) are the reservoir and bottom-hole pressures in atm, \(\mu\) is the viscosity of the reservoir fluid in poises, \(k\) is the permeability of the formation in darcies, \(h\) is the thickness of the formation in meters, \(R\) is the radius of influence of the well in meters, and \(r\) is the radius of the well in meters. This formula is valid for hydrodynamically perfect wells, i.e., wells that have penetrated the formation from roof to floor and that are not cased by columns in the formation zone. Meanwhile, it is known that wells are usually hydrodynamically imperfect, and their imperfection consists primarily in the fact that they are cased with metal columns, so that oil inflow into them occurs only through specially made openings, sometimes spaced at considerable distances from one another.

To determine the inflow into a well of this kind, let us assume that a certain well of radius \(\rho_{0}\) has entered a formation of thickness \(h\) and stopped at its very roof (Fig. 1, \(a\)). Let us further assume that the formation is sufficiently strong and therefore oil production from it is carried out with an open bottom. Around the bottom of this well, let us isolate a hemisphere of radius \(\rho_{1}\), and inside it a spheroidal layer of radius \(\rho\) and thickness \(d\rho\), described about the same center (Fig. 1, \(a\)). Obviously, with a constant inflow into this well, the pressure drop at the boundaries of this layer will be expressed by the relation

\[ dP=\frac{Q\mu}{2\pi \rho^{2}k}\,d\rho, \tag{2} \]

whence the pressure drop at the boundaries of the entire hemisphere will be expressed through the integral of this expression

\[ \int_{P_{2}}^{P_{1}} dP=\frac{Q\mu}{2\pi k}\int_{\rho_{0}}^{\rho_{1}}\frac{d\rho}{\rho^{2}}, \]

\[ P_1-P_2=\frac{Q\mu}{2\pi k}\cdot \frac{1}{-\rho}\bigg|_{\rho_0}^{\rho_1}, \]

\[ P_1-P_2=\frac{Q\mu}{2\pi k}\left(\frac{1}{\rho_0}-\frac{1}{\rho_1}\right) =\frac{Q\mu}{2\pi k}\cdot\frac{\rho_1-\rho_0}{\rho_0\cdot \rho_1}. \tag{3} \]

If, for the investigation, we were to choose the same kind of well, but one drilled to the middle of the formation, and at its bottom-hole we created a small cavity of approximately spherical shape with radius \(\rho_0\) (Fig. 1, б), then,

Fig. 1. Various types of hydrodynamically imperfect wells: \(a\)—a well has penetrated the top of the formation; б—a well has reached a cavity located in the middle of the formation; в—a well exploits the formation through a single opening located in its middle. Explanation of the drawing: wavy lines indicate clays; dots—sandstones; \(\rho_0\)—radius of the opening through which oil enters the well; \(\rho_1\)—radius of influence of this opening, i.e., the radius of the space in which the pressure drop can be expressed by formula (5).

carrying out the calculation in an analogous manner, we would obtain the following expression for the pressure drop in the sphere

\[ P_1-P_2=\frac{Q\mu}{4\pi k}\cdot\frac{\rho_1-\rho_0}{\rho_0\cdot \rho_1}. \tag{4} \]

It is not difficult to see that expression (3) differs from expression (4) only in that the denominator of the first contains the coefficient “2,” while the denominator of the second contains the coefficient “4.” If, for the investigation, we take a well cased with a column in which only one opening has been made, lying exactly at the middle of the formation, then, isolating a sphere around it (Fig. 1, в), we obtain on it a pressure drop that will be expressed by the relation

\[ P_1-P_2=\frac{Q\mu}{2\varphi\pi k}\cdot\frac{\rho_1-\rho_0}{\rho_0\cdot \rho_1}, \tag{5} \]

where \(\rho_0\) is the radius of the opening, \(\rho_1\) is the radius of influence of the opening, and \(\varphi\) is a coefficient whose value lies within the limits

\[ 1<\varphi<2. \]

Approximately, this coefficient may be determined from the formula

\[ \varphi = 1 + \frac{\rho_1}{\rho_1 + 2r}, \tag{6} \]

where \(r\) is the radius of the well.

Beyond the radius of influence of the opening, the pressure drop must be expressed by Dupuit’s formula, which in the present case takes the form

\[ P_0 - P_1 = \frac{Q\mu}{2\pi k h}\ln \frac{R}{\rho_1 + r}, \tag{7} \]

where \(P_0\) is the pressure beyond the radius of influence of the well, and \(R\) is the radius of influence of the well. As a result, the total magnitude of the pressure drop is expressed by the relation

\[ P_0 - P_2 = \frac{Q\mu}{2\pi k} \left( \frac{\rho_1-\rho_0}{\varphi \rho_0 \rho_1} + \frac{1}{h}\ln \frac{R}{\rho_1 + r} \right). \tag{8} \]

Let us analyze the expression obtained for large values of the radius of influence of the openings. Suppose that a certain well has penetrated a bed of thickness \(4\ \text{m}\), has been cased with a 6-inch casing string (i.e., has a radius of \(7.5\ \text{cm}\)), and has been perforated with a density of 3 openings per meter. Thus, the number of openings in it is 12, and the thickness of the bed corresponding to one opening is \(h' = 33.3\ \text{cm}\). Suppose further that the radius of the openings is \(0.5\ \text{cm}\), and that, with a pressure drop of 30 atmospheres, an inflow of \(6\ \text{t/day}\) is obtained. It is required to determine what the pressure drop will be at the same inflow if the number of openings in the casing string is doubled and if the radius of influence of the well in this case was and remains equal to \(100\ \text{m}\) (\(10000\ \text{cm}\)).

Obviously, the amount of fluid entering the well under consideration through each opening must be equal to \(0.5\ \text{t/day}\). Further, the surface area of one meter of casing string is determined by the relation

\[ F = \pi \cdot 15.0 \cdot 100 = 4712\ \text{cm}^2, \]

whence the area corresponding to each opening will be equal to \(1571\ \text{cm}^2\), the average distance between openings is \(39.64\ \text{cm}\), and the radius of influence of each opening, equal to half this distance, is \(19.8\ \text{cm}\). Substituting the obtained value of \(\rho_1\), as well as the value of the casing radius \(r = 7.5\ \text{cm}\), into formula (6), we find that the coefficient \(\varphi\) in the case under consideration will be equal to 1.569. Finally, substituting all these quantities into formula (8), we obtain

\[ P_0 - P_2 = \frac{0.5\mu}{2\pi k} \left( \frac{19.8-0.5}{1.569 \cdot 0.5 \cdot 19.8} + \frac{1}{33.3}\ln \frac{10000}{19.8+7.5} \right), \]

\[ 30 = \frac{0.5\mu}{2\pi k}(1.2425 + 0.1771), \tag{9} \]

whence we find that the constants contained in expression (9) give

\[ \frac{\mu}{2\pi k} = 42.265. \]

If the number of openings in the casing string under consideration is doubled, then the average inflow per opening (with a constant inflow into the well as a whole) will decrease to \(0.25\ \text{t/day}\), the average distance between openings to \(28\ \text{cm}\), the radius of influence of an opening to \(14\ \text{cm}\), the thickness

of the formation per one opening—to 16.67 cm, and the value of \(\varphi\) must reach 1.4827. Substituting these data into formula (8), we obtain

\[ P_0 - P_2 = 0.25 \cdot 42.265\,(1.3007 + 0.3685), \tag{10} \]

whence we find that the pressure drop in this case must reach 17.637 atm.

Thus, by increasing the perforation density from 3 to 6 openings per meter, we can obtain the same inflow at a pressure drop 1.7 times smaller than the initial one. If, however, the pressure drop is thereby brought up to its former value, we obtain an increase in inflow 1.7 times greater than the initial one. Here, however, it should be noted that with a further increase in perforation density, the effect of reducing the pressure drop, or of increasing the inflow at the same pressure drop, will already be less significant. Thus, if the well under consideration is perforated with a density of 50 openings per meter, then we obtain

\[ P_0 - P_2 = 0.03 \cdot 42.265\,(1.4416 + 3.3482) = 6.073\ \text{atm}, \tag{11} \]

and if it is perforated with a density of 100 openings per meter, then we obtain

\[ P_0 - P_2 = 0.015 \cdot 42.265\,(1.4212 + 6.8188) = 5.224\ \text{atm}. \tag{12} \]

Thus, in this case the pressure drop decreases only by a factor of 1.16. If the number of openings is increased by another factor of two and brought up to 200 openings per meter, then we obtain

\[ P_0 - P_2 = 0.0075 \cdot 42.265\,(1.393 + 13.831) = 4.826\ \text{atm}, \tag{13} \]

which corresponds to a reduction of the pressure drop by only a factor of 1.09. Finally, if this well is regarded as hydrodynamically perfect, i.e., if the resistance of the casing string is neglected, then the first term in the parentheses of expression (8) becomes zero and the entire expression takes the form

\[ P_0 - P_2 = 6.0 \cdot 42.265\left(0 + \frac{1}{400}\ln \frac{10\,000}{7.5}\right) = 4.526\ \text{atm}. \tag{14} \]

From what has been said there follows one important practical conclusion. Namely, when the distances between openings in the casing string are considerable, the inflow into the well can be increased by additional perforation, i.e., by increasing the number of openings in the string; however, this number should not be made too large, since at a high perforation density an increase in the number of openings has little effect on the inflow, while the strength of the string is reduced to a significant extent. Apparently, the number of openings should not be brought to more than 50 per meter.