ON INFLOW INTO A WELL WITH SEVERAL TYPES OF HYDRODYNAMIC IMPERFECTIONS

L. A. Pukhlyakov

(Presented by Professor A. V. Aksarin)

First of all it is necessary to consider the influence on inflow into a well of channels at the openings in it. Suppose that, in some well, near each opening there is a channel of cylindrical form with a hemisphere at the end, of radius $\lambda$ and with cylindrical-part length $y$ (Fig. 1). To determine the character of the resistances in the zone of influence of such a channel, let us isolate around it a cylinder-like body ending on both sides in hemispheres, one of which is

Fig. 1. Diagram of an opening in the casing and cement ring of a well with a channel of radius $\lambda$ and length of the cylindrical part $y$, and of the thin-walled body isolated around it, needed for deriving the formula for the geometric characteristic of the zone of influence of this channel. A longitudinal section is given at the top, a transverse section below. The dense oblique hatching indicates the casing; the sparse hatching, the cement ring.

continuous, and the second is incomplete (because it is intersected by the outer surface of the cement sheath). Let us further assume that the radius of both hemispheres and of the cylindrical part of this body is equal to \(\rho\), and the thickness of its walls is \(d\rho\).

To express the surface of the given body, one should take as a basis the surface of a sphere cut by a cylinder, whose value was established in the derivation of the formula for inflow into a well with holes without channels [2]. It is only necessary to add to this expression the value of the surface of the cylindrical part of the body under consideration, expressed by the relation

\[ F_1 = 2\pi \rho y . \tag{1} \]

Thus, the surface of the body under consideration will be expressed by the following relation:

\[ F_{\phi}=2\pi\left(\rho y+\rho^2+\frac{\rho^3}{4r}+\frac{\rho^5}{128r^3}+\ldots\right), \tag{2} \]

where \(r\) is the radius of the well before the casing is run (half the bit diameter), in cm.

It is now possible to construct the initial expression for determining the pressure drop in the zone of influence of an individual opening. It will have the form

\[ dP=\frac{Q_{\mathrm{pl}}\mu}{2\pi k n\left(\rho y+\rho^2+\frac{\rho^3}{4r}+\frac{\rho^5}{128r^3}+\ldots\right)}\,d\rho, \tag{3} \]

where \(Q_{\mathrm{pl}}\) is the inflow into the well under reservoir conditions, in \(\text{cm}^3/\text{s}\); \(\mu\) is the viscosity of oil or another fluid under reservoir conditions, in centipoise; \(k\) is the reservoir permeability in darcies; and \(n\) is the number of openings in the well within the reservoir interval under study.

Dropping the constants contained in expression (3), we find the initial expression for determining the geometric characteristic of the zone of influence of an individual opening

\[ dG=\frac{1}{\rho y+\rho^2+\frac{\rho^3}{4r}+\frac{\rho^5}{128r^3}+\ldots}\,d\rho, \tag{4} \]

which, after the corresponding transformations, takes the form

\[ dG=\frac{1}{y}\left(\frac{1}{\rho}-\frac{1}{y}+\frac{\rho}{y^2}-\frac{\rho}{4ry}+\frac{\rho^2}{2ry^2}-\frac{\rho^2}{y^3}-\frac{\rho^3}{128r^3y}+\frac{\rho^3}{16r^2y^2}-\frac{\rho^3}{4ry^3}+\frac{\rho^3}{y^4}+\ldots\right)d\rho . \tag{5} \]

We integrate the left-hand side of the obtained series over the limits \(0<G<G_s\), and the right-hand side over the limits \(\lambda<\rho<s\), where \(s\) is the radius of influence of the openings, or one-half of the mean distance between adjacent openings. At high perforation density, when the reservoir thickness per opening \(h/n\) is less than \(2.55r\), this parameter can be determined by the formula

\[ s=0.5\sqrt{2\pi r\,\frac{h}{n}}, \tag{6} \]

where \(h\) is the length of the well perforation interval. At low perforation density, when the reservoir thickness per opening \(h/n\) exceeds \(2.55r\), this parameter can be determined by the formula

\[ s^*=0.5\sqrt{\frac{h^2}{n^2}+0.96\pi^2r^2}. \tag{7} \]

As a result of integration we obtain

\[ G_{s-1}=\frac{1}{y}\left(\ln \frac{(y+\lambda)s}{(y+s)\lambda} -\frac{0.25}{r+y}\ln \frac{s+y}{\lambda+y} +0.0625\,\frac{s-\lambda}{(r+y)^2}\right). \tag{8} \]

Here it is necessary to note that formula (8) is valid only under the condition

\[ s<2r. \tag{9} \]

If this condition is violated, that is, if

\[ s>2r, \tag{10} \]

then, in order to derive the formula for the geometric characteristic of the zone under consideration, it must be divided into two subzones: an inner one, bounded by the radius \(2r\), and an outer one, bounded by the limits

\[ 2r<\rho<s. \tag{11} \]

The geometric characteristic of the inner subzone can be obtained from formula (8) by substituting the quantity \(2r\) into it in place of \(s\).

To obtain the geometric characteristic of the outer subzone, it is first necessary to determine the filtration area in it. If, in doing so, we neglect the cross-sectional area of the well itself, which here plays a very limited role, then this parameter is expressed by the formula

\[ F_{\phi}=2\pi(\rho y+2\rho^2), \tag{12} \]

whence the initial expression for determining the geometric characteristic of this subzone takes the form

\[ dG=\frac{d\rho}{\rho(y+2\rho)}. \tag{13} \]

Integrating the left-hand side of this expression over the limits \(0<G<G_2\), and the right-hand side over the limits \(2r<\rho<s\), we obtain

\[ G_2=\frac{1}{y}\ln \frac{(y+2\cdot 2r)s}{(y+2s)2r}. \tag{14} \]

Thus, the complete geometric characteristic of the zone of interest to us in the case under consideration is expressed by the formula

\[ G_{s-\mathrm{IV}}=\frac{1}{y}\ln \frac{(y+\lambda)\cdot 2r}{(y+2r)\lambda} -\frac{0.25}{r+y}\ln \frac{2r+y}{\lambda+y} +0.0625\,\frac{2r-\lambda}{(r+y)^2} +\frac{1}{y}\ln \frac{(y+4r)s}{(y+2s)2r}. \tag{15} \]

Beyond the zone of influence of the openings in the well drainage region, two more zones can be distinguished: the zone of flow constriction due to incomplete perforation, and the zone of plane-radial flow, in each of which the laws of fluid motion toward the well are different.

The zone of flow constriction due to incomplete perforation is characteristic only of wells that are commonly called hydrodynamically imperfect with respect to the degree of penetration of the formation. To determine the geometric characteristic of this zone, let us assume that a certain well has penetrated the formation perpendicular to its bottom and top over the full thickness \(H\), but that during the perforation process only some part \(h\) of this formation was opened. In this case, between the upper opening of the screen and the top of the formation, and also between the lower opening of the screen and the bottom of the formation, there remained unperforated intervals \(\delta_2\) and \(\delta_1\), of which \(\delta_2\)

less than \(\delta_1\). Below, these quantities will be called the components of incomplete perforation.

Let us isolate around the perforated part of the wellbore a thin-walled body in the form of a cylinder ending at both ends in hemispheres of radius \(\rho\), with the length of the cylindrical part equal to the length of the perforated part of the wellbore \(h\). Obviously, the surface of this body is expressed by the relation

\[ F=2\pi \rho h+4\pi \rho^2 . \tag{16} \]

The thickness of its walls will be equal to \(d\rho\), and the initial expression for determining the geometric characteristic of the flow-constriction zone will have the form

\[ dG=-\frac{n}{\rho(h+2\rho)}\,d\rho \tag{17} \]

where \(n\) is the number of openings in the filter. This quantity is introduced into the numerator of expression (17) because in formula (3) and others it appears in the denominator.

Integrating the left-hand side of expression (17) over the limits \(0<G<G^*\), and the right-hand side over the limits \(x_0<\rho<\delta_2\), where \(x_0\) is expressed by the relation

\[ x_0=r+y+s, \tag{18} \]

we obtain

\[ G^*=-\frac{n}{h}\ln \frac{h+2\rho}{\rho}\bigg|_{x_0}^{\delta_2} \tag{19} \]

\[ G^*=-\frac{n}{h}\left(\ln \frac{h+2\delta_2}{\delta_2}-\ln \frac{h+2x_0}{x_0}\right) \tag{20} \]

or, after the corresponding transformations,

\[ G^*=\frac{n}{h}\ln \frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0}. \tag{21} \]

The formula obtained characterizes the resistance to the motion of the liquid inside a cylinder-like body ending at both ends in hemispheres, one of which touches either the roof or the bottom of the bed, while the other is separated from the opposite boundary by a distance determined by the difference

\[ \delta_1-\delta_2. \tag{22} \]

To account for the remaining part of the space characterizing the hydrodynamic imperfection of the well according to the degree of penetration of the bed, it is necessary to isolate a second thin-walled cylinder-like body, differing from the first in that it ends in one hemisphere. Its surface is expressed by the relation

\[ F=2\pi \rho(h+\delta_2)+2\pi \rho^2, \tag{23} \]

and the initial expression for determining the geometric characteristic of the considered part of the flow-constriction zone will have the form

\[ dG=-\frac{n}{\rho(h+\delta_2+\rho)}\,d\rho . \tag{24} \]

Integrating the left-hand side of this expression over the limits \(0<G<G^{**}\), and the right-hand side over the limits \(\delta_2<\rho<\delta_1\), we obtain

\[ G^{**}=-\frac{n}{h+\delta_2}\ln \frac{h+\delta_2+\rho}{\rho}\bigg|_{\delta_2}^{\delta_1} \tag{25} \]

or, after the corresponding transformations,

\[ G^{**}=\frac{n}{h+\delta_2}\ln\frac{(h+2\delta_2)\delta_1}{(h+\delta_2+\delta_1)\delta_1}. \tag{26} \]

Finally, adding expression (21) to expression (26), we find the complete geometric characteristic of the flow-constriction zone due to incomplete perforation:

\[ G_{\delta}=\frac{n}{h}\ln\frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0} +\frac{n}{h+\delta_2}\ln\frac{(h+2\delta_2)\delta_1}{(h+\delta_2+\delta_1)\delta_2}. \tag{27} \]

This formula will be suitable for solving problems both in those cases where the unperforated intervals are identical on the side of the roof and the floor, and in those where they are different, and in those where one of them, \(\delta_2\), tends to its minimum value \(x_0\). Only one rule is obligatory: \(\delta_2\) must denote the smaller interval, and \(\delta_1\) the larger one, that is, the relation

\[ \delta_2 \leq \delta_1. \tag{28} \]

There is no need to derive the formula for the geometric characteristic of the plane-radial flow zone, since it is a part of Dupuit’s formula. If one proceeds from the assumption that, in the absence of a flow-constriction zone, both characteristics of incomplete perforation tend to their minimum values, that is, if one assumes the relation

\[ r+y+s \leq \delta_2, \tag{29} \]

to hold, then the geometric characteristic of the plane-radial flow zone in general form will be expressed by the relation

\[ G_R=\frac{n}{H}\ln\frac{R}{\sigma_{\max}}, \tag{30} \]

where \(H\) is the total thickness of the formation in cm and \(R\) is the radius of influence of the well in cm.

Several methods are known for determining the radius of influence of wells. In the investigation of wells during exploration of oil fields, the most convenient appears to be the method based on the compressibility of oil. To determine the compressibility of oil, V. N. Mamuna et al. [1] recommend using the formula

\[ \alpha=\frac{1}{V_0}\cdot\frac{\Delta V}{\Delta P}, \tag{31} \]

where \(V_0\) is the initial volume of the formation-oil sample in \(\text{cm}^3\), \(\Delta V\) is the change in this volume in \(\text{cm}^3\) when the pressure changes by the amount \(\Delta P\), expressed in atmospheres.

To determine the radius of influence of a well, let us isolate in the plane-radial flow zone a thin-walled cylinder whose axis will coincide with the axis of the well, and whose radius will be equal to some variable \(x\). The wall thickness of this cylinder will be equal to \(dx\), the height \(H\) (the formation thickness), and the wall volume will be expressed by the relation

\[ V'=2\pi Hx\,dx. \tag{32} \]

Obviously, the amount of oil that will be released from the walls of this cylinder will be expressed by the following relation:

\[ dV_p=2\pi Hm\alpha xP_s\ln\frac{R}{x}\,dx, \tag{33} \]

where \(m\) is the porosity of the formation, expressed as a fraction of unity, and \(P_s\) is the de-

pressure on the formation at the outer boundary of the zone of influence of the holes in atmospheres. In the absence of a flow-constriction zone, this quantity can be calculated by the formula

\[ \frac{P_s}{P_{\text{pl}}-P_z}=\frac{G_R}{G_s+G_R}, \tag{34} \]

and in the presence of a flow-constriction zone, by the formula

\[ \frac{P_s}{P_{\text{pl}}-P_z}=\frac{G_\delta+G_R}{G_s+G_\delta+G_R}. \tag{35} \]

Expression (33) is the initial one for determining the radius of influence of the well. Integrating its left-hand side over the limits \(0<V<V_p\), and the right-hand side over the limits \(x_0<x<R\), we obtain

\[ V_p=2\pi Hm\alpha P_s\int_{x_0}^{R} x(\ln R-\ln x)\,dx, \tag{36} \]

whence

\[ V_p=2\pi Hm\alpha P_s \left[ \frac{x^2}{2}\ln R - x^2\left(\frac{\ln x}{2}-\frac{1}{4}\right) \right]\Bigg|_{x_0}^{R} \tag{37} \]

\[ V_p=\pi Hm\alpha P_s \left( R^2\ln R-x_0^2\ln R-R^2\ln R+x_0^2\ln x+\frac{R^2}{2}-\frac{x_0^2}{2} \right) \tag{38} \]

or, after the corresponding transformations,

\[ V_p=\pi Hm\alpha P_s \left( \frac{R^2}{2}-\frac{x_0^2}{2}-x_0\ln\frac{R}{x_0} \right). \tag{39} \]

Here it is necessary to note that the value \(x_0\) always differs from the value \(R\) by two or three orders of magnitude, while their squares differ from one another already by four to six orders of magnitude. Therefore, in practical calculations the value \(x_0\) in formula (39) may be neglected. As a result, it takes the simpler form

\[ R^2=\frac{2V_p}{\pi Hm\alpha P_s}. \tag{40} \]

Within the limits of the flow-constriction zone, the dependence between the amount of oil withdrawn and the radius of influence of the well will be still more complex; however, here too expression (40) may be used without great detriment to the accuracy of the solution of the problem posed.

For each regime, the radius of influence of the well is determined separately.

The geometrical characteristics calculated in accordance with the formulas given above are substituted into the general formula for the pressure drop

\[ P_{\text{pl}}-P_z=\frac{Q_{\text{pl}}\mu}{2\pi kh}\left(G_s+G_\delta+G_R\right), \tag{41} \]

where \(P_{\text{pl}}\) and \(P_z\) are the reservoir and bottom-hole pressures in atm, \(\mu\) is the viscosity of the oil or other fluid under reservoir conditions in centipoises, \(k\) is the permeability of the formation in darcies, \(n\) is the number of holes in the casing within the limits of the formation under study, and \(Q_{\text{pl}}\) is the inflow into the well of oil or other fluid under reservoir conditions in \(\text{cm}^3/\text{s}\), determined by the formula

\[ Q_{\text{pl}}=\frac{1\,000\,000}{86\,400}Q_{\text{pv}}\cdot b, \tag{42} \]

where \(Q_{\mathrm{pv}}\) is the inflow into the well, measured under surface conditions in \(\mathrm{m}^3/\mathrm{day}\); 1,000,000 and 86,400 are coefficients expressing the number of \(\mathrm{cm}^3\) in \(\mathrm{m}^3\) and the number of seconds in a day; \(b\) is the formation volume factor of the oil or other fluid, i.e., the ratio of its volume under reservoir conditions to its volume under surface conditions.

References

1. V. N. Mamuna, G. F. Trebin, B. V. Ulyaninskii. Experimental Study of Reservoir Oils. GOSINTI, 1960.

2. L. A. Pukhlyakov. Derivation of the formula for inflow into a hydrodynamically imperfect well. Izvestiya TPI, vol. 201, 1972.