DERIVATION OF THE INFLOW FORMULA FOR A HYDRODYNAMICALLY IMPERFECT WELL

L. A. Pukhlyakov

(Presented by the scientific seminar of the Department of Higher Mathematics)

This article considers inflow into a well cased with a casing string, cemented, and perforated with a specified density. The drainage region of such a well may be divided into two zones: an outer zone and an inner zone, or the zone of influence of the holes. In the first of these zones, the individual streamlines of the flow move, as it were, along the radii of a plane circle toward its center, through which the well axis passes. Various flow characteristics in this zone can be calculated by the Dupuit formula, which may be written in the following form:

\[ P_c - P_1 = \frac{Q_{\text{pl}}\mu}{2\pi kh}\ln\frac{R}{r+s}, \tag{1} \]

where \(P_0\) is the reservoir pressure in atm, \(P_1\) is the pressure at the boundary of the zone of influence of the holes in atm, \(Q_{\text{pl}}\) is the inflow into the well under reservoir conditions in \(\text{cm}^3/\text{sec}\), \(\mu\) is the viscosity of the oil under reservoir conditions in centipoise, \(k\) is the permeability of the formation in darcies, \(h\) is the formation thickness in cm, \(R\) is the radius of influence of the well in cm, \(r\) is the well radius in cm, and \(s\) is the mean radius of influence of the holes, i.e., half the distance between adjacent holes in cm.

In the zone of influence of the holes, the individual streamlines of the flow move, as it were, along the radii of spheres toward their centers. To characterize the various flow parameters, the Dupuit formula is not applicable here either in its pure form or with the corrections of V. I. Shchurov, which I. M. Murav’ev and others recommend using [1, pp. 164–168; 2, pp. 137–140].

The point is that the named corrections, or coefficients, are introduced into the formula in such a way that their influence extends over the entire drainage region, whereas the laws of fluid motion in the different zones of this region are different.

More accurate in this respect is the formula derived by the author jointly with M. V. Samoilova [3]. One part of this formula characterizes the zone of influence of the holes, or the inner zone, and the other the outer zone. However, the part of the named formula that characterized the zone of influence of the holes contained a correction coefficient \(\varphi\), determined approximately without sufficient mathematical justification. An attempt to justify it mathematically and at the same time to determine it with sufficient accuracy led to the derivation of the new inflow formula for a hydrodyn—

hydrodynamically imperfect well. The derivation of the formula was carried out in the following order.

First of all, the filtration area of the flow \(F_{\phi}\) was determined at a certain distance from the opening in the casing. It is expressed by the following relation

\[ F_{\phi}=4\pi \rho^{2}-F_{\mathrm{v}}. \tag{2} \]

where \(4\pi\rho^{2}\) is the surface area of a sphere of radius \(\rho\), and \(F\) is the area of the cut made in this sphere by the casing. To calculate the latter of these quantities, one quarter of it is taken in the upper half of the given cut (Fig. 1-a), and within it a horizontal strip of width \(\rho\,d\alpha\) is isolated; here \(\alpha\) is the angle between the generatrix of the casing and the radius of the sphere drawn from the center of the opening to any point on the sphere (angle \(\angle AOB\) in Fig. 1).

Fig. 1. Elements determining the area of the cut in the sphere described with radius \(\rho\) around an opening in a casing of radius \(r\), by the outer surface of this casing: \(a\)—side view (casing in longitudinal section); \(b\)—top view (casing in transverse section).

To express the length of this strip, let us cut the casing and the sphere we have chosen by a plane perpendicular to the axis of the well (for simplicity of reasoning we shall regard the well as vertical, and the plane-

plane horizontal). In the section we obtain two intersecting circles (Fig. 1-b), part of one of which (arc CE) will correspond to the strip of interest to us. Its length in the section passing through the center of the opening in the casing (section \(ON\) in Fig. 1-a) can be expressed in terms of the radius of the selected sphere \(\rho\) and the arccosine of the angle \(D^1O_1O_2\) (Fig. 1-b), or by the relation

\[ l'=\rho\arccos\frac{\rho}{2r}. \tag{3} \]

If, however, we choose a section passing above the opening in the casing, for example, in the plane of the strip we have chosen, then the length of the arc under consideration will be expressed through the arccosine of the angle \(DO_1O_2\), or by the relation

\[ l=\rho\cdot\sin\alpha\cdot\arccos\frac{\rho\cdot\sin\alpha}{2r}. \tag{4} \]

Thus, the area of the strip we have chosen can be expressed by the relation

\[ \frac{1}{4}dF_{\mathrm{в}}=\rho\cdot\sin\alpha\cdot\arccos\left(\frac{\rho\cdot\sin\alpha}{2r}\right)\cdot\rho\,d\alpha, \tag{5} \]

integrating which within the limits \(0<\alpha<\frac{\pi}{2}\), we obtain

\[ \frac{1}{4}F_{\mathrm{в}}=\rho^2\int_{0}^{\frac{\pi}{2}}\sin\alpha\cdot\arccos\left(\frac{\rho\cdot\sin\alpha}{2r}\right)d\alpha. \tag{6} \]

For convenience of integration, let us expand the arccosine contained in the integrand. As a result, expression (6) takes the form

\[ \frac{1}{4}F_{\mathrm{в}}=\rho^2\int_{0}^{\frac{\pi}{2}}\sin\alpha\left[ \frac{\pi}{2}-\left(\frac{\rho}{2r}\sin\alpha+ \frac{1}{2\cdot3}\cdot\frac{\rho^3}{8r^3}\sin^3\alpha+ \right.\right. \]

\[ \left.\left. +\frac{1\cdot3}{2\cdot4\cdot5}\cdot\frac{\rho^5}{32r^5}\sin^5\alpha+\ldots\right)\right]d\alpha \tag{7} \]

\[ \frac{1}{4}F_{\mathrm{в}}=\rho^2\int_{0}^{\frac{\pi}{2}}\left( \frac{\pi}{2}\sin\alpha-\frac{\rho}{2r}\sin^2\alpha-\frac{\rho^3}{48r^3}\sin^4\alpha- \right. \]

\[ \left. -\frac{3}{1280}\frac{\rho^5}{r^5}\sin^6\alpha-\ldots\right)d\alpha. \tag{8} \]

Integrating expression (8) over the specified limits, we obtain

\[ \frac{1}{4}F_{\mathrm{в}}=\rho^2\left[ \frac{\pi}{2}(-\cos\alpha)-\frac{\rho}{2r}\left(\frac{1}{2}\alpha-\frac{1}{4}\sin2\alpha\right) -\frac{\rho^3}{48r^3}\left(\frac{3}{8}\alpha-\right. \]

\[ \left. -\frac{1}{4}\sin2\alpha+\frac{1}{32}\sin4\alpha\right) -\frac{3}{1280}\frac{\rho^5}{r^5}\left(\frac{\sin^5\alpha\cdot\cos\alpha}{6}+ \right. \]

\[ \left. +\frac{5\cdot3}{6\cdot8}\alpha-\frac{5}{6\cdot4}\sin2\alpha+\frac{5}{6\cdot32}\sin4\alpha\right) \right]_{0}^{\frac{\pi}{2}}. \tag{9} \]

Finally, substituting the specified values into expression (9) and carrying out the corresponding reductions, we obtain

\[ \frac{1}{4}F_{в}=\rho^2\left[\frac{\pi}{2}(-0+1)-\frac{\rho}{2r}\frac{\pi}{2}-\frac{\rho^3}{48r^3}\cdot\frac{3}{8}\cdot\frac{\pi}{2}-\frac{3}{1280}\frac{\rho^5}{r^5}\cdot\frac{5}{6}\frac{\pi}{2}\right]. \tag{10} \]

whence the total area of the cross section of interest to us is expressed by the relation

\[ F_{в}=2\pi\left(\rho^2-\frac{\rho^3}{4r}-\frac{\rho^5}{128r^3}-\frac{3}{4096}\frac{\rho^7}{r^5}-\ldots\right). \tag{11} \]

Using the obtained expression (11), as well as expression (2), one can obtain the filtration area in the zone of influence of an individual opening at any distance from it. It is expressed by the relation

\[ F_{\varphi}=2\pi\left(\rho^2+\frac{\rho^3}{4r}+\frac{\rho^5}{128r^3}+\frac{3}{4096}\frac{\rho^7}{r^5}+\ldots\right). \tag{12} \]

Now we are able to compose the initial expression for determining the pressure drop in the zone of influence of an individual opening. It proves to be the following:

\[ dP=\frac{Q_{\text{pl}}\mu}{nk2\pi\left(\rho^2+\frac{\rho^3}{4r}+\frac{\rho^5}{128r^3}+\frac{3}{4096}\frac{\rho^7}{r^5}+\ldots\right)}\,d\rho . \tag{13} \]

where \(n\) is the number of openings in the string.

This expression is inconvenient for integration, since its denominator contains a polynomial including the variable with respect to which the integration is performed. Reducing it to a form convenient for integration, we obtain

\[ dP=\frac{Q_{\text{pl}}\mu}{n\,2\pi k}\left( \frac{1}{\rho^2}-\frac{1}{4\rho r}+\frac{1}{16r^2} -\frac{3}{128}\frac{\rho}{r^3} +\frac{1}{128}\frac{\rho^2}{r^4} -\frac{13}{4096}\frac{\rho^3}{r^5} +\frac{17}{16384}\frac{\rho^4}{r^6} -\frac{3}{8192}\frac{\rho^5}{r^7} +\ldots \right)d\rho . \tag{14} \]

Integrating expression (14) within the limits \(\lambda<\rho<s\), we obtain

\[ P_1-P_2=\frac{Q_{\text{pl}}\mu}{n\,2\pi k}\left( -\frac{1}{\rho} -\frac{1}{4r}\ln\rho +\frac{\rho}{16r^2} -\frac{3}{128\cdot2}\frac{\rho^2}{r^3} +\frac{1}{128\cdot3}\frac{\rho^3}{r^4} -\frac{13}{4096\cdot4}\frac{\rho^4}{r^5} +\frac{17}{16384\cdot5}\frac{\rho^5}{r^6} -\frac{3}{8192\cdot6}\frac{\rho^6}{r^7} \right)\bigg|_{\lambda}^{s}. \tag{15} \]

Finally, substituting into expression (15), as limits, first \(s\) and then \(\lambda\), and also carrying out the corresponding transformations of the coefficients and adding to the result obtained the right-hand side of expression (1), we find the inflow formula for a hydrodynamically imperfect well

\[ P_0-P_{\text{zab}}=\frac{Q_{\text{pl}}\mu}{n\,2\pi k}\left( \frac{1}{\lambda}-\frac{1}{s} -\frac{0.25}{r}\ln\frac{s}{\lambda} +0.0625\frac{s-\lambda}{r^2} \right. \]

\[ \left. -0.0117187\frac{s^2-\lambda^2}{r^3} +0.0026042\frac{s^3-\lambda^3}{r^4} -0.0007935\frac{s^4-\lambda^4}{r^5} +0.0002075\frac{s^5-\lambda^5}{r^6} -0.0000610\frac{s^6-\lambda^6}{r^7} +\ldots+\frac{n}{h}\ln\frac{R}{r+s} \right). \tag{16} \]

where \(\lambda\) is the radius of an opening in the string, in cm; \(h\) is the formation thickness in cm; \(Q_{\text{pl}}\) is the inflow under reservoir conditions in \(\text{cm}^3/\text{sec}\), determined by the formula

\[ Q_{\text{pl}}=\frac{1000000\,Q\,b}{86400}, \tag{17} \]

where the coefficient 1,000,000 denotes the number of cm\(^3\) in m\(^3\), 86,400 is the number of seconds in a day, \(Q\) is the total inflow into the well under surface conditions in m\(^3\)/day, \(b\) is the volumetric coefficient of oil, i.e., the ratio of its volume under reservoir conditions to its volume under surface conditions, \(n\) is the number of openings in the screen, and \(P_{\text{bot}}\) is the bottom-hole pressure in atm. The remaining letters were explained in the description of formulas (1), (30), and (31).

For convenience in using formula (16), it is recommended that the last term of the part enclosed in brackets be called the geometric characteristic of the zone of radial-plane flow, and the first nine terms—the geometric characteristic of the zone of influence of the opening.

Formula (16) is valid under the condition that the radius of influence of the opening \(s\) does not exceed \(2r\). Otherwise the expression

\[ \arccos \frac{\rho \sin \alpha}{2r}, \]

which is a component part of relation (4), etc., will have no meaning. At the same time the second condition is also violated. In deriving expression (10), as the upper limit of integration we took \(\pi/2\). If, however, the radius of influence of the opening \(s\) exceeds \(2r\), then the lower edge of the cut made by the cylinder in the sphere will be removed from its highest point not by \(90^\circ\), but by a somewhat smaller value determined by the relation

\[ x_{\max}=\arcsin \frac{2r}{\rho}. \tag{18} \]

In this connection, in such cases it is recommended to represent the zone of influence of the openings in the form of two parts: the first is bounded by a radius equal to the diameter of the well \((2r)\), and the second is the part lying beyond \(2r\). The geometric characteristic of the first of these can be approximately expressed by the relation

\[ G_1=\frac{1}{\lambda}-\frac{1}{2r}-\frac{0.25}{r}\ln \frac{2r}{\lambda} +0.0625\frac{2r}{r^2} -0.0117187\frac{4r^2}{r^3} + \]

\[ +0.0026042\frac{8r^3}{r^4} -0.0007935\frac{16r^4}{r^5} +0.0002075\frac{32r^5}{r^6} -0.000061\frac{64r^6}{r^7}. \tag{19} \]

or, after the corresponding transformations,

\[ G_1=\frac{1}{\lambda}-\frac{0.411}{r}-\frac{0.25}{r}\ln\frac{2r}{\lambda}. \tag{20} \]

As for the geometric characteristic of the second part, to determine it one should take into account, first, that the expression for the area of the cut at \(\rho>2r\) is very complicated, and, second, that the magnitude of this area varies within very limited bounds. Namely, at \(\rho=2r\), according to formula (11),

\[ F_v=4\pi r^2\left(1-\frac{11}{64}\right) \]

or

\[ F_v=3.31\pi r^2 \tag{21} \]

and at \(\rho\to\infty\)

\[ F_v=2\pi r^2. \tag{22} \]

Therefore, for practical use, the magnitude of the cut may be taken as constant and expressed by the relation

\[ F_v=3\pi r^2. \tag{23} \]

The maximum value of the error in determining the pressure drop under this assumption does not exceed 0.2%. Thus, the original expression for determining the pressure drop in the second part of the zone of influence of the openings is expressed by the relation

\[ dP=\frac{Q_{\text{pl}}\mu}{k(4\pi\rho^{2}-3\pi r^{2})}\,d\rho, \tag{24} \]

which is reduced to the form

\[ dP=\frac{Q_{\text{pl}}\mu}{n4\pi k\left(\rho^{2}-\frac{3}{4}r^{2}\right)}\,d\rho \]

\[ dP=\frac{Q_{\text{pl}}\mu}{n4\pi k} \left( \frac{1}{\rho^{2}}+\frac{3}{4}\frac{r^{2}}{\rho^{4}}+ \frac{9}{16}\frac{r^{4}}{\rho^{6}} \right)d\rho. \tag{25} \]

Integrating expression (25) within the limits \(2r<\rho<s\), we obtain

\[ P_{1}-P_{2}= \frac{Q_{\text{pl}}\mu}{n4\pi k} \left( -\frac{1}{\rho}-\frac{r^{2}}{4\rho^{3}}-\frac{9}{16\cdot 5}\frac{r^{4}}{\rho^{5}} \right) \bigg|_{2r}^{s}. \]

\[ P_{1}-P_{2}= \frac{Q_{\text{pl}}\mu}{n4\pi k} \left( \frac{1}{2r}-\frac{1}{s} -\frac{r^{2}}{4s^{3}} +\frac{r^{2}}{4\cdot 8\cdot r^{3}} -\frac{9}{16\cdot 5}\frac{r^{4}}{s^{5}} +\frac{9}{16\cdot 5\cdot 32}\frac{r^{4}}{r^{5}} \right) \]

or, introducing into the parentheses the factor \(\frac{1}{2}\) and carrying out the corresponding transformations,

\[ P_{1}-P_{2}= \frac{Q_{\text{pl}}\mu}{n2\pi k} \left( \frac{0.25}{r} -\frac{0.5}{s} -0.125\frac{r^{2}}{s^{3}} +0.015625\frac{1}{r} -0.05625\frac{r^{4}}{s^{5}} +\frac{0.001758}{r} \right). \tag{26} \]

Thus, the geometric characteristic of the part under consideration is expressed by the relation

\[ C_{2}=0.26738\frac{1}{r}-\frac{0.5}{s} -0.125\frac{r^{2}}{s^{3}} -0.05625\frac{r^{4}}{s^{5}}. \tag{27} \]

The geometric characteristic of the outer zone, as follows from what was said above, is expressed by the relation

\[ C_{3}=\text{[[unclear: formula obscured by blot; contains }} \ln \frac{R}{r+s}\text{]]}. \]

As a result, the total pressure drop in a well with a low perforation density (for \(s>2r\)), taking into account expression (20), is found to be equal to

\[ P_{b}-P_{\text{bh}}= \frac{Q_{\text{pl}}\mu}{n2\pi k} \left( \frac{1}{\lambda} -\frac{0.14362}{r} -\frac{0.25}{r}\ln\frac{2r}{\lambda} -\frac{0.5}{s} -0.125\frac{r^{2}}{s^{3}} \right. \]

\[ \left. -0.05625\frac{r^{4}}{s^{5}} +\frac{n}{h}\ln\frac{R}{r+s} \right). \tag{28} \]

CONCLUSION

The main purpose of the derived formulas (16) and (28) is to determine the permeability of formations from inflows into wells under established regimes, primarily from the results of well tests that are carried out in the course of inducing inflow into them. In addition, these formulas can be used to determine the pressure in

at various points of the formation, in order to establish the relationship between perforation density and inflow into the well and for other purposes. In this case, formula (16) may be applied only in cases of high well-perforation density, and formula (28) at low density. It is recommended that this work be carried out in the following order.

1. The formation thickness per one opening is determined. The calculation is performed by the formula

\[ h'=\frac{h}{n}, \tag{29} \]

where \(h\) is the total formation thickness, or the length of the perforation interval in cm, and \(n\) is the number of openings in the given perforation interval.

1. The radius of influence of an opening is determined; as noted above, it is equal to one half of the mean distance between openings. In this case, if the formation thickness per one opening \(h'\) does not exceed 1.27 well diameters, or 2.55 well radii, the calculation is carried out through the area per one opening, that is, by the formula

\[ s=0.5\sqrt{2\pi r\,\frac{h}{n}} \tag{30} \]

and the pressure drop is determined by formula (16).

If, however, the thickness per one opening exceeds 1.27 well diameters, the calculation is carried out according to the root-mean-square principle, that is, by the formula

\[ s=0.5\sqrt{\left(\frac{h}{n}\right)^2+\left(\frac{2\pi r}{3}\right)^2}, \tag{31} \]

and the pressure drop is determined by formula (28).

REFERENCES

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

2. I. M. Murav’ev, R. S. Andriasov, Sh. K. Gimatudinov, G. L. Govorova, V. T. Polyazkov. Development and Operation of Oil and Gas Fields. Nedra Publishing House, 1970.

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