RESULTS OF DETERMINING THE PERMEABILITY OF THE Yu-1 RESERVOIR OF THE PERVOMAYSKOYE OIL FIELD FROM INFLOWS UNDER STEADY-STATE CONDITIONS

L. A. PUKHLYAKOV

(Presented by Professor A. V. Aksarin)

The initial material for determining the permeability of the Yu-1 reservoir of the Pervomayskoye oil field from inflows under steady-state conditions consisted of the results of tests of wells 260, 261, 262, and 264 (Table 1), as well as the characteristics of the oils obtained in the course of studying them at UIPGN-2M, from which the oil formation volume factor \(b\), its viscosity under reservoir conditions \(\mu\), and the compressibility coefficient \(\alpha\) were used. Oil samples for analysis were taken from two wells: 260 and 264. The average value of the formation volume factor for both samples proved to be 1.181, the viscosity 0.71 cP, and the compressibility coefficient \(17.72 \cdot 10^{-5}\ 1/\text{atm}\).

As the calculation formula, the formula from paper [3] is taken:

\[ P_{\text{pl}} - P_{3} = \frac{Q_{\text{pl}}\mu}{2\pi K_{\text{ef}} n} \left(G_s + G_{\hat{\delta}} + G_R\right), \tag{1} \]

where \(P_{\text{pl}}\) and \(P_{3}\) are the reservoir and bottom-hole pressures in atm; \(\mu\) is the viscosity of the oil under reservoir conditions in centipoise; \(K_{\text{ef}}\) is the effective permeability of the reservoir in darcies; \(n\) is the number of openings in the string; \(Q_{\text{pl}}\) is the inflow of oil into the well under reservoir conditions in \(\text{cm}^3/\text{s}\), determined, in turn, by the formula

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

where \(Q_{\text{pv}}\) is the inflow of oil into the well, measured under surface conditions in \(\text{m}^3/\text{day}\); \(b\) is the oil formation volume factor, equal in the case considered to 1.181; 1000000 and 86400 are coefficients for converting \(\text{m}^3/\text{day}\) into \(\text{cm}^3/\text{s}\); \(G_s\) is the geometrical characteristic of the zone of influence of the openings in \(1/\text{cm}\); \(G_{\hat{\delta}}\) is the geometrical characteristic of the flow-constriction zone due to incomplete well perforation in \(1/\text{cm}\); and \(G_R\) is the geometrical characteristic of the plane-radial-flow zone, also in \(1/\text{cm}\).

To determine the first of these characteristics, \(G_s\), it is first necessary to determine the filter length per opening. The calculation is carried out by the formula

\[ h' = \frac{h}{n}, \tag{3} \]

where \(h\) is the length of the well perforation interval in cm.

Table 1

Initial data for determining the productivity of the Yu-1 formation in wells 260, 261, 262, and 264 of the Pervomayskoye oil field

In well 260 the perforation interval of the well proved to be 1100 cm, the number of holes 180, and the filter length per hole 6.111 cm; in well 261 the perforation interval was 3400 cm, the number of holes 360, the filter length per hole 9.444 cm; in well 262 the perforation interval was 1500 cm, the number of holes 250, the filter length per hole 6.000 cm; in well 264 the perforation interval was 1500 cm, the number of holes 288, the filter length per hole 5.208 cm.

Next, the filter length per hole is compared with the value \(2.55r\), on the basis of which it is determined whether the perforation density is high or low. The exploratory wells on the Pervomayskaya area, at the time of opening the oil pool, were drilled with bits of diameter 190 mm; hence the value \(2.55r\) proves to be equal to 24.2 cm. It exceeds the filter length per hole in all wells; hence the perforation density in all wells is high. Then the radius of influence of the holes \(s\) is determined. At low perforation density it is calculated by the formula

\[ S=0.5\sqrt{\frac{h^2}{n^2}+0.96\pi^2 r^2}, \tag{4} \]

at high density—by the formula

\[ S=0.5\sqrt{2\pi r\frac{h}{n}}. \tag{5} \]

In the wells under consideration the perforation density is high; formula (5) is used for the calculation. As a result of the calculation we obtain: in well 260 the radius of influence of the holes is 9.55 cm, in well 261—11.87 cm, in well 262—9.46 cm, and in well 264—8.82 cm.

Next, to determine the geometric characteristic of the zone of influence of the holes it is necessary to know the radii of the holes \(\lambda\) and the depth of the channels at them \(y\). In the process of exploration of the Pervomayskoye oil field, the PK-80 cumulative perforator was used for perforating the wells; according to E. M. Visheni [1], it creates holes with channels of depth (length) 5–6 cm, of conical, gradually tapering shape. As for the calculated length of these channels \(y\), i.e., that part of it which is outside the cement ring, it may be taken as equal to 2 cm.

The radius of the channel at the inner cut of the casing, according to E. M. Visheni [1], is equal to 0.5 cm; on the outer side of the cement ring it is close to 3.5 cm and on average may be taken as equal to 0.2 cm.

In the case where channels are present at the holes and the perforation density of the wells is high, the geometric characteristic of the zone of influence of the holes is calculated by the formula

\[ G_s=\frac{1}{y}\ln\frac{(y+\lambda)s}{(y+s)\lambda} -\frac{0.25}{r+y}\ln\frac{s+y}{\lambda+y} +\frac{1}{16}\cdot\frac{s-\lambda}{(r+y)^2}. \tag{6} \]

Under the same conditions and with low perforation density \((s>2r)\), the calculation is made by the formula

\[ G_s^{*}=\frac{1}{y}\ln\frac{(y+\lambda)2r}{(y+2r)\lambda} -\frac{0.25}{r+y}\ln\frac{2r+y}{\lambda+y} +\frac{1}{16}\cdot\frac{2r-\lambda}{(r+y)^2} +\frac{1}{r+y}\ln\frac{(y+4r)s}{(y+2s)2r}. \tag{7} \]

In the case where there are no channels at the holes (unconsolidated formation, \(y\to0\)), formulas (6) and (7) are inconvenient to use, since in this case these formulas give an indeterminacy that is difficult to resolve. In connec-

In this connection, at high perforation density, instead of formula (7) it is recommended to use the following formula:

\[ G_s^{**}=\frac{1}{\lambda}-\frac{1}{s}-\frac{0.25}{r}\ln\frac{s}{\lambda} +0.0625\,\frac{s-\lambda}{r^2} -0.0117\,\frac{s^2-\lambda^2}{r^3}, \tag{8} \]

and at low perforation density \((s<2r)\), the formula

\[ G_s^{***}=\frac{1}{\lambda}-\frac{0.1436}{r} -\frac{0.25}{r}\ln\frac{2r}{\lambda} -\frac{0.5}{s} -0.125\,\frac{r^2}{s^3} -0.05625\,\frac{r^4}{s^5}. \tag{9} \]

In the wells we considered, the reservoir is competent and the perforation density is high. Therefore we shall use formula (6), substituting into it the data pertaining to well 260; we obtain

\[ G_s=\frac{1}{2}\ln\frac{(2+0.2)\cdot 9.55}{(2+9.55)\cdot 0.2} -\frac{0.25}{9.5+2}\ln\frac{9.55+2.0}{0.2+2.0} + \]

\[ +\frac{1}{16}\cdot\frac{9.55-0.2}{(9.5-2.0)^2} =1.0732\ \text{cm}^{-1}. \tag{10} \]

For well 261, the geometrical characteristic of the zone of influence of the openings turned out to be equal to \(1.0891\ \text{cm}^{-1}\), for well 262—\(1.0732\ \text{cm}^{-1}\), and for well 264—\(1.0679\ \text{cm}^{-1}\).

The geometrical characteristic of the flow-constriction zone depends on several parameters, but above all on the nature of the perforation incompleteness. Perforation incompleteness may be one-component, two-component, three-component, four-component, etc.

One-component perforation incompleteness is understood to mean such incompleteness in which, within the reservoir interval opened by the well from top to bottom, one interval adjacent either to the top or to the bottom remains unperforated.

Two-component incompleteness is understood to mean such perforation incompleteness in which, within the reservoir, two intervals remain unperforated, one of which adjoins the top of the reservoir and the other its bottom. Also considered two-component is such perforation incompleteness in which one interval remains unperforated, but this interval is located somewhere in the middle of the reservoir.

Three-component perforation incompleteness is understood to mean such incompleteness in which two intervals remain unperforated near the well, one of which adjoins either the top or the bottom of the reservoir, while the other is located in the middle, i.e., it adjoins neither the bottom nor the top.

Four-component perforation incompleteness is understood to mean such incompleteness in which, within the reservoir, either two intervals remain unperforated, neither of which adjoins either the top or the bottom of the reservoir; or three intervals, two of which adjoin the top and bottom of the reservoir, while the third is located somewhere in the middle.

Five-component perforation incompleteness is understood to mean such incompleteness in which three intervals remain unperforated, one of which adjoins either the top or the bottom of the reservoir, while two are located somewhere in the middle.

Theoretically, perforation incompleteness may have any number of components; however, in practice, incompleteness with a number of components greater than four apparently does not occur. In any case, the probability of such incompleteness appearing is extremely small.

A component of incompleteness is understood to mean either half of an unperforated interval, if this interval adjoins neither the bottom,

to the top, or its full length if it adjoins the top or bottom of the formation. Naturally, only those components that exceed the radius of influence of the holes, \(s\), are taken into account.

The first component of the incompleteness is considered to be the one which, in length, exceeds all the others and occurs only once. It is denoted by \(\sigma_1\). If, in a well with multicomponent incompleteness, the two largest components are equal, then they are considered second components and are denoted by \(\sigma_2\). If, in a well with multicomponent incompleteness, the three largest components are equal, then they are considered third components and are denoted by \(\sigma_3\).

If, in a well with multicomponent incompleteness, all components have different lengths, then the largest is considered the first and is denoted by \(\sigma_1\), the next largest is the second and is denoted by \(\sigma_2\), etc.

The formula for the geometric characteristic of the flow-constriction zone has its simplest form in the case of single-component incompleteness of well perforation. To derive it, let us assume that some well penetrated the formation perpendicular to its top and bottom over the full thickness \(H\), but during perforation only some part \(h\) of it was opened. In this case the upper opening of the screen proved to be in the immediate vicinity of the top of the formation, and the lower one was removed from the bottom by the amount \(\sigma_1\).

Let us single out, around the perforated part of the borehole, a thin-walled body in the form of a cylinder ending at the lower end in a hemisphere of radius \(\rho\), equal to the radius of its cylindrical part. Obviously, the length of the cylindrical part of the body we have singled out will be equal to \(h\). The surface of this body is expressed by the relation

\[ F = 2\pi \rho h + 2\pi \rho^2, \tag{11} \]

the wall thickness will be equal to \(d\rho\), and the initial expression for determining the geometric characteristic will have the form

\[ dG = \frac{n}{\rho(h+\rho)}\, d\rho, \tag{12} \]

where \(n\) is the number of openings in the screen. This quantity has been introduced into the numerator of expression (12) because we calculate the resistance to the flow attributable to one opening.

Integrating the left-hand side of this expression over the limits \(0 < G < G_1\), and the right-hand side over the limits \(x_0 < \rho < \sigma_1\), where

\[ x_0 = r + y + s, \tag{13} \]

we obtain

\[ G_1 = -\frac{n}{h}\ln \frac{h+\rho}{\rho}\bigg|_{x_0}^{\sigma_1} \tag{14} \]

\[ G_1 = -\frac{n}{h}\left(\ln\frac{h+\sigma_1}{\sigma_1} - \ln\frac{h+x_0}{x_0}\right), \tag{15} \]

or, after the corresponding transformations,

\[ G_1 = \frac{n}{h}\ln \frac{(h+x_0)\sigma_1}{(h+\sigma_1)x_0}. \tag{16} \]

The formula for the geometric characteristic of the flow-constriction zone also has a comparatively simple form in the case of two-component incomple-

but under the condition that both components are equal to each other. To derive this formula, let us consider a well in which two equal sections, each of length $\delta_2$, located one at the top and the other at the bottom of the formation, have turned out to be unperforated. Around the perforated part of this well we single out a thin-walled cylindrical body of radius $\rho$, ending on both sides with hemispheres of the same radius $\rho$. Obviously, the surface of this body is expressed by the relation

\[ F = 2\pi \rho h + 4\pi \rho^2, \tag{17} \]

and the initial expression, by analogy with expression (12), will take the form

\[ dG = \frac{n}{\rho(h+2\rho)}\,d\rho . \tag{18} \]

Integrating the left-hand side of this expression over the limits $0 < G < G_2$, and the right-hand side over the limits $x_0 < \rho < \delta_2$, we obtain

\[ G_2 = -\frac{n}{h}\ln \frac{h+2\rho}{\rho}\bigg|_{x_0}^{\delta_2} \tag{19} \]

or, after the corresponding transformations,

\[ G_2 = \frac{n}{h}\ln \frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0}. \tag{20} \]

For a three-component perforation incompleteness, when all the components are equal to one another, by analogy with expression (20) we obtain the following formula for the geometric characteristic of the zone of flow constriction due to perforation incompleteness:

\[ G_3 = \frac{n}{h}\ln \frac{(h+3x_0)\delta_3}{(h+3\delta_3)x_0}. \tag{21} \]

Under analogous conditions, for a well with four-component incompleteness we obtain

\[ G_4 = \frac{n}{h}\ln \frac{(h+4x_0)\delta_4}{(h+4\delta_4)x_0}. \tag{22} \]

The same procedure may be followed for a larger number of components of perforation incompleteness, provided that their lengths are the same.

If the components of perforation incompleteness are not the same in length, then the zone of flow constriction should be divided into several subzones (according to the number of components) and, using formulas (16, 20, 21, and 22), new formulas should be composed, which will be something like derivatives of them.

Suppose that we have a well with two-component perforation incompleteness, and one of the components $\delta_1$ exceeds the other $\delta_2$ in length. We divide the zone of flow constriction of this well into two subzones: an inner and an outer one. The inner subzone is bounded externally by the component $\delta_2$ and internally by the quantity $x_0$—see formula (13). The outer zone is bounded externally by the component $\delta_1$, and internally by the component $\delta_2$.

To characterize the resistances in the inner subzone, one may use formula (20); thus, it automatically becomes the first term of the new formula. As for its second term, formula (16) must underlie it. In that formula, however, instead of the former $h$ it is necessary to substitute the sum $h+\delta_2$, and instead of the quantity $x_0$ again use $\delta_2$. As a result, the formula for calculating the geometric-

…the characteristic of the flow-constriction zone for two-component incomplete perforation and different component lengths takes the form

\[ G_{2-1}=\frac{n}{h}\ln \frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0} +\frac{n}{h+\delta_2}\ln \frac{(h+2\delta_2)\delta_1}{(h+\delta_2+\delta_1)\delta_2}. \tag{23} \]

With three-component incompleteness and different component lengths, two cases are possible. The first case is when one of the components exceeds the other two. The calculation formula has the form

\[ G_{3-1}=\frac{n}{h}\ln \frac{(h+3x_0)\delta_3}{(h+3\delta_3)x_0} +\frac{n}{h+2\delta_3}\ln \frac{(h+3\delta_3)\delta_1}{(h+2\delta_3+\delta_1)\delta_3}. \tag{24} \]

The second case is when two components of the incompleteness exceed the third. The calculation formula takes the form

\[ G_{3-2}=\frac{n}{h}\ln \frac{(h+3x_0)\delta_3}{(h+3\delta_3)x_0} +\frac{n}{h+\delta_3}\ln \frac{(h+3\delta_3)\delta_2}{(h+\delta_3+2\delta_2)\delta_3}. \tag{25} \]

With four-component incompleteness, the number of cases increases; however, there is no need to consider them here, since wells with four-component incompleteness occur extremely rarely. In any case, of the wells under consideration, three have one-component incomplete perforation and one is perforated completely. Otherwise, none of them has four-component incompleteness.

Let us examine well 260 in more detail. Its perforation interval is 2442–2453 m, and the reservoir interval is 2442–2472 m. Thus, there is one component of incompleteness here, equal to 1900 cm. The value \(x_0\) is expressed by the relation

\[ x_0=9.5+2.0+9.55=21.05\ \text{cm} \tag{26} \]

or approximately 21 cm. Accordingly, the geometric characteristic of the flow-constriction zone here is expressed by the relation

\[ G_{\delta}=\frac{180}{1100}\ln \frac{(1100+21)\cdot1900}{(1100+1900)\cdot21} =\frac{1}{6.111}\ln 33.808=0.5761\ \text{1/cm}. \tag{27} \]

In well 261 the entire reservoir is perforated; consequently, the flow-constriction zone is absent here, or, in other words, the geometric characteristic of this zone is equal to zero. For well 262, the geometric characteristic of the flow-constriction zone is equal to \(0.5859\ \text{1/cm}\), and for well 264, \(0.6814\ \text{1/cm}\) (Table 2).

The geometric characteristic of the plane-radial flow zone in the absence of a flow-constriction zone due to incomplete perforation is expressed by the relation

\[ G_R=\frac{n}{H}\ln \frac{R}{r+y+s}. \tag{28} \]

where \(R\) is the radius of influence of the well in cm, \(H\) is the total reservoir thickness in cm (the remaining parameters were explained above). If there is a flow-constriction zone due to incomplete perforation, the geometric characteristic of the plane-radial flow zone is calculated by the formula

\[ G_R^{*}=\frac{n}{H}\ln \frac{R}{\delta_{\max}}. \tag{29} \]

All parameters entering formulas (28) and (29), except for the radius of influence of the well \(R\), were either given above in final form or determined. As for the radius of influence of the well, it must be determined. For this purpose one may use the circumstance that,

that the amount of oil withdrawn from the reservoir during the testing of a well and the time of its operation remain small, owing to which the reservoir operates under an elastic regime. This means that oil advances toward the well bottoms as a result of its own compressibility.

V. N. Mamuna et al. [2] recommend determining this parameter by the formula

\[ \alpha=\frac{1}{V_0}\cdot \frac{\Delta V}{\Delta P}, \tag{30} \]

where \(V_0\) is the initial volume of the oil sample in \(\text{cm}^3\), \(\Delta V\) is the change in the volume of the sample in \(\text{cm}^3\) when the pressure changes by the amount \(\Delta P\), expressed in atmospheres. As noted above, for the oil of the Pervomayskoye field the compressibility proved to be \(17.72\cdot 10^{-5}\ 1/\text{atm}\).

To derive the formula for the radius of influence of a well, let us isolate in the zone of plane-radial flow 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 thickness of the walls of this cylinder will be equal to \(dx\), the height to \(H\) (the reservoir thickness), and the volume of the wall will be expressed by the relation

\[ V=2\pi Hx\,dx. \tag{31} \]

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

\[ dV_p=2\pi Hm\alpha xP_s\ln\frac{R}{x}\,dx, \tag{32} \]

where \(m\) is the reservoir porosity, expressed as a fraction of unity, and \(P_s\) is the depression on the reservoir at the outer boundary of the zone of influence of the openings, in ati.

In the absence of a flow-constriction zone, this quantity can be calculated by the formula

\[ \frac{P_s}{P_{\text{pl}}-P_3}=\frac{G_R}{G_s+G_R}, \tag{33} \]

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

\[ \frac{P_s}{P_{\text{pl}}-P_3}=\frac{G_\delta+G_R}{G_s+G_\delta+G_R}. \tag{34} \]

Expression (32) is the starting point for determining the radius of influence of the well. Integrating its right-hand side over the limits \(x_0<x<R\), and the left-hand side over the limits \(0<V<V_p\), we obtain

\[ V_p=2\pi Hm\alpha P_s\int_{x_0}^{R}x(\ln R-\ln x)\,dx, \tag{35} \]

whence

\[ V_p=2\pi Hm\alpha P_s \left[ \frac{x^2}{2}\ln R - x^2\left(\frac{\ln x}{2}-\frac{1}{4}\right) \right]_{x_0}^{R}; \tag{36} \]

\[ V_p=\pi Hm\alpha P_s \left( R^2\ln R - x_0^2\ln R - R^2\ln R + x_0^2\ln x_0 + \frac{R^2}{2} - \frac{x_0^2}{2} \right), \tag{37} \]

or, after the corresponding transformations,

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

Here, however, it is necessary to note that the value \(R\) always differs by two or three orders of magnitude from the value \(x_0\), and their squares differ from one another ...

Table 2

Results of determining the permeability of the Yu-1 reservoir in wells 260, 261, 262, and 264 of the Pervomayskoye oil field

from one another already by four to six orders of magnitude. Therefore, in practical calculations the quantity \(x_0\) may be neglected. In that case the formula of interest to us takes the simpler form

\[ R^2=\frac{2V_p}{\pi Hm\tau P_s}. \tag{39} \]

Within the zone of flow contraction, the relationship between the amount of oil withdrawn and the radius of influence of the well will be still more [[unclear: distorted word]], but here too expression (39) can 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. Let us consider this using well 260 as an example. This well was tested under five regimes. Under the first regime (3-mm choke) the well operated for 34.5 hours, or 124200 sec. Its average rate in this case was 358.13 \(\text{cm}^3/\text{sec}\), and the withdrawal of fluid under this regime amounted to 44480000 \(\text{cm}^3\) (Table 2). Under the second regime (5-mm choke) the well operated for 24.5 hours, or 88200 sec. Its average rate in this case was 516.69 \(\text{cm}^3/\text{sec}\); the withdrawal of fluid under this regime amounted to 45571 thousand \(\text{cm}^3\), and in all, by the end of operation under this regime, 90051 thousand \(\text{cm}^3\) of oil had been withdrawn. Data on the cumulative withdrawals under the other regimes are given in Table 2.

Substituting the corresponding quantities into expression (39) and carrying out the necessary operations, we obtain for the first regime

\[ R_1^2=295930000:P_{s-1}; \tag{40} \]

for the second regime

\[ R_2^2=599110000:P_{s-2}; \tag{41} \]

for the third regime

\[ R_3^2=1189000000:P_{s-3}; \tag{42} \]

for the fourth regime

\[ R_4^2=1414500000:P_{s-4}; \tag{43} \]

and for the fifth regime

\[ R_5^2=1664620000:P_{s-5}; \tag{44} \]

Further, the problem is solved by the method of successive approximations, namely.

1. In the first approximation, let us assume that the radius of influence of the well is equal to 5000 cm. Then the geometric characteristic of the plane-radial-flow zone, in accordance with formula (29), is expressed by the relation

\[ G_R=\frac{180}{3000}\ln\frac{5000}{1900} =\frac{1}{16.667}\ln 2.6316 =\frac{0.9759}{16.667} =0.0581\ 1/\text{cm}. \tag{45} \]

1. In accordance with formula (34), the pressure at the boundary of the zone of influence of the holes is expressed by the relation

\[ P_{s-1}=37\frac{0.5761+0.0581}{1.0732+0.5761+0.0581}=13.743\ \text{atm gage}. \tag{46} \]

1. Using expression (40), we find the radius of influence of the well in the second approximation. Namely:

\[ R_1^2=295930000:13.743=21553300\ \text{cm}^2, \tag{47} \]

whence

\[ R_1=4642\ \text{cm}. \]

1. In an analogous manner, in the second approximation we obtain: the geometric characteristic of the plane-radial-flow zone is equal to

0.0536 \(1/\text{cm}\), the pressure at the outer boundary of the zone of influence of the perforations is 13.682 atm.

1. In the third approximation we obtain: the radius of influence of the well is 4651 cm, the geometric characteristic of the zone of plane-radial flow is 0.0537 \(1/\text{cm}\), and the pressure at the outer boundary of the zone of influence of the perforations is 13.683 atm.

2. In the fourth approximation the radius of influence of the well proves to be the same as in the third approximation, 4651 cm. There is no need to repeat the remaining calculations.

Let us now substitute the data obtained, as well as the viscosity of the oil of the Pervomaiskoye field under reservoir conditions, which proved to be 0.71 centipoise, into expression (1). As a result of this substitution we obtain

\[ 251.8 - 214.8 = \frac{358.13 \cdot 0.71}{6.2832 \cdot 180 \cdot k_1} \cdot (1.0732 + 0.5761 + 0.0537), \tag{48} \]

whence

\[ k_1 = \frac{358.13 \cdot 0.71 \cdot 1.7030}{6.2832 \cdot 180 \cdot 37} = 0.010348 \ \text{darcy}, \tag{49} \]

or 10.35 millidarcies.

For the second operating regime we obtain a well radius of influence of 5547 cm, the geometric characteristic of the zone of plane-radial flow is 0.0643 \(1/\text{cm}\), and the reservoir permeability is 10.67 millidarcies. The results of calculations for the other regimes of this well and for other wells are given in Table 2.

The same table also gives the average permeability values for individual wells: in well 260—10.83 md, in well 261—4.82 md, in well 262—10.59 md, and in well 264—13.87 md.

The permeabilities of the reservoir under consideration, established from cores by G. N. Voloshchuk et al., proved to be almost the same. Specifically, in well 260—9.5 md (discrepancy 1.14), for well 261—11.7 md (discrepancy 0.412), in well 262—13.4 md (discrepancy 0.790), and for well 264—33.2 md (discrepancy 0.418). The average discrepancy proved to be 0.628.

The reason for the discrepancy apparently lies in the fact that, in determinations from cores, permeability along the bedding is taken into account, whereas in inflows into wells permeability across the bedding also plays a definite role. In addition, deterioration of permeability due to mudding had some effect here.

Indeed, the movement of clay mud through so-called through-flow pores does occur. In any case, the blowout at the Gubkinskoye gas field was stopped only after 400 m\(^3\) of clay mud had been injected into the reservoir. The 35 thousand m\(^3\) of water injected before that produced no effect. However, drilling mud penetrates into the bottomhole zone of the reservoir more readily than into the core, since in the core, inside the pores, the oil present there creates sufficient back pressure. In the bottomhole zone of the reservoir no such back pressure arises, since the oil is displaced by the drilling mud into parts of the reservoir remote from the bottomhole. As a result, the bottomhole zones of the reservoir prove to be contaminated to a greater degree than the cores.

There can be no strictly established well operating regimes, since theoretically, owing to the continuous expansion of the depression funnel, the sum of the geometric characteristics must constantly in-

increase, and permeability should increase. Indeed, in the zone covered by the depression funnel, an ever larger proportion should be made up of non-clayized rocks remote from the well bottom. In three of the wells considered, an increase in effective permeability is observed; in the last it is not observed.

It follows from the foregoing that, in individual cases, this phenomenon can be used to determine the reservoir permeability in its non-clayized parts remote from the wells; however, the results of such a determination will be less reliable than the results of determining the effective permeability of reservoirs, which are given in Table 2.

To derive the formula for the permeability of the non-clayized parts of the reservoir, we proceed as follows.

1. Let us represent the sum of the geometric characteristics of the well drainage region at the earlier regime, \(\sum G_1\), in the form of two parts: the geometric characteristic of the clayized zone, \(G_{\mathrm{cl}}\), and the geometric characteristic of the non-clayized zone, \(G_{\mathrm{nz}}\).

2. Let us represent the sum of the geometric characteristics of the drainage region at the later regime, \(\sum G_2\), in the form of three parts, of which the first two are exactly equal to the corresponding characteristics of the earlier regime, and the third is the difference of the sums under consideration

\[ G_{\mathrm{add}}=\sum G_2-\sum G_1 . \tag{50} \]

1. We compose the system of equations

\[ \begin{cases} k_2\sum G_2=G_{\mathrm{cl}}\cdot k_{\mathrm{cl}}+G_{\mathrm{nz}}\cdot k' + G_{\mathrm{add}}\cdot k',\\ k_1\sum G_1=G_{\mathrm{cl}}\cdot k_{\mathrm{cl}}+G_{\mathrm{nz}}k', \end{cases} \tag{51} \]

where \(k_1\) and \(k_2\) are the effective reservoir permeabilities at the first and second well-test regimes, \(k_{\mathrm{cl}}\) is the reservoir permeability in the clayized part, and \(k'\) is the reservoir permeability in the non-clayized part.

1. Subtracting the second equation of system (51) from the first, we obtain

\[ k_2\sum G_2-k_1\sum G_1=G_{\mathrm{add}}\cdot k', \tag{52} \]

whence the reservoir permeability in its non-clayized part is expressed by the relation

\[ k'=\frac{k_2\sum G_2-k_1\sum G_1}{\sum G_2-\sum G_1}. \tag{53} \]

In expression (53) one may substitute data relating directly to one or another operating regime of the well, and then find the average permeability values of the non-clayized zones. One may also proceed differently, namely: first find the average values of the effective permeability and the sums of the geometric characteristics for the first regimes, and then these same parameters for the last regimes. For well 260 of the Pervomayskoye field, for the first two regimes we obtain: an average effective permeability of 10.51 millidarcies, and an average sum of geometric characteristics of \(1.7083\ \mathrm{1/cm}\); for the last two regimes, 11.07 millidarcies and \(1.74315\ \mathrm{1/cm}\), respectively. Substituting these data into expression (53), we obtain

\[ k'_{260}=\frac{11.07\cdot 1.74315-10.51\cdot 1.7083}{1.74315-1.7083}=38.52\ \text{md}. \tag{54} \]

For well 261 we find the averages first for the first and second operating conditions, and then for the second and third. Substituting these data into expression (53), we obtain

\[ k'_{261}= \frac{4.80\cdot 1.67805-4.775\cdot 1.6698} {1.67805-1.6698} =9.86\ \text{md}. \tag{55} \]

For well 262 we find the averages first for the first and second, and then for the third and fourth operating conditions. Substituting the obtained data into expression (53), we obtain

\[ k'_{262}= \frac{11.04\cdot 1.79845-10.145\cdot 1.7411} {1.79845-1.7411} =38.21\ \text{md}. \tag{56} \]

Comparing the permeability of the unclayed parts of the reservoirs with the permeability of the clayed parts, we obtain a quantitative expression for the skin effect of the well. We do not, however, have the permeability of the clayed sections of the reservoirs, but rather their average effective permeabilities. Therefore, by comparing them, we obtain not the true but an approximate skin effect. The calculation formula for this parameter is

\[ \Gamma=\frac{k'}{k_{\mathrm{eff}}}. \tag{57} \]

Substituting into this formula the data relating to well 260, we obtain 3.56; in the same way, for well 261 we obtain 2.05, and for well 262, 3.61.

Thus, processing data on oil inflows into wells under stabilized operating conditions makes it possible to obtain an idea not only of the effective permeabilities of reservoirs, but also of the approximate value of the skin effect in these wells.

REFERENCES

1. E. M. Vytseni. Cumulative perforators used in oil and gas wells. 2nd ed. Moscow, Nedra, 1971.

2. V. N. Mamuna, G. F. Trebin, B. V. Ulyaninsky. Experimental study of reservoir oils. Moscow, GOSINTI, 1960.

3. L. A. Pukhlyakov. Methodology for determining the permeability of reservoirs in unclayed zones from the results of well studies under stabilized operating conditions. Information Sheet, No. 100–74. Tomsk, Tomsk TsNTI, 1974.