DETERMINATION OF THE PERMEABILITY OF RESERVOIR Yu-1 OF THE OLENYE OIL FIELD FROM INFLOWS UNDER STEADY-STATE REGIMES

L. A. Pukhlyakov

(Submitted by the Department of Combustible Minerals)

The method for determining the permeability of reservoirs from inflows under steady-state regimes was based on the formula for the pressure drop during inflow into a hydrodynamically imperfect well,

\[ P_{\text{pl}} - P_{\text{z}} = \frac{Q_{\text{pl}}\cdot \mu}{2\pi k_{\text{ef}} h} \left(G_s + G_\delta + G_R\right), \tag{1} \]

where \(P_{\text{pl}}\) and \(P_{\text{z}}\) are the reservoir and bottom-hole pressures in atm, \(Q_{\text{pl}}\) is the oil inflow into the well, measured under reservoir conditions in \(\text{cm}^3/\text{sec}\), \(\mu\) is the viscosity of the oil under reservoir conditions in centipoises, \(K_{\text{ef}}\) is the effective permeability of the reservoir in darcies, \(n\) is the number of holes in the filter (dimensionless quantity), \(G_s\), \(G_\delta\), and \(G_R\) are the geometric characteristics of the zone of influence of the holes, the zone of flow constriction due to incomplete perforation, and the zone of plane-radial flow, respectively, in \(1/\text{cm}\).

The inflow under reservoir conditions is calculated by the formula

\[ Q_{\text{pl}} = 11.547 Q_{\text{pv}}\cdot b, \tag{2} \]

where 11.547 is the coefficient expressing the number of \(\text{cm}^3/\text{sec}\) in \(\text{m}^3/\text{day}\), \(Q_{\text{pv}}\) is the inflow into the well, measured under surface conditions in \(\text{m}^3/\text{day}\), and \(b\) is the volumetric coefficient of oil, that is, the ratio of its volume under reservoir conditions to the volume under surface conditions (a dimensionless quantity always greater than unity).

Data on oil inflows into various wells under various regimes are given in Table 1. The volumetric coefficient and other physical properties of the oil of the Olenye field, obtained as a result of its investigation on the UIPN-2M unit, are given in Table 2.

The geometric characteristic of the zone of influence of holes under different conditions is expressed differently. First of all, it depends on the presence or absence of channels at the holes. It is affected in a definite way by the perforation density.

The perforation density is taken into account through the so-called radius of influence of the holes \(s\), which is equal to half the average distance between two neighboring holes. At high perforation density, when the filter length per one hole does not reach \(2.55 r\), where \(r\) is the radius of the well before running the casing string in cm, this parameter is calculated by the formula

\[ s = 0.5 \sqrt{\frac{2\pi r h}{n}} \tag{3} \]

Table 1

Results of tests of seams 123, 125, and 129 of the Olenegorsk nepheline deposit

Table 2

Physical properties of the oil of the Olenye oil field and data on reservoir porosity

where \(h\) is the length of the perforated part of the wellbore within the given productive reservoir, in cm; the remaining notation is as before.

At a low perforation density, when the filter length per one opening exceeds \(2.55r\), the radius of influence of the openings is calculated by the formula

\[ s = 0{,}5 \sqrt{\frac{h^2}{n^2} + 0{,}96\pi^2 r^2} \tag{4} \]

The radii of all wells considered in the present article are equal to \(9.5\ \text{cm}\); hence the quantity \(2.55r\) is equal to \(24.2\ \text{cm}\). The greatest filter length per one opening is \(20.0\ \text{cm}\) (well 123). Thus, in all the wells under consideration the perforation density is high, and, consequently, for calculating the radii of influence of the openings one should use formula (3). For well 123 this parameter is expressed by the relation

\[ s = 0{,}5 \sqrt{6{,}283 \cdot 9{,}5 \cdot \frac{1200}{120}} = 12{,}21\ \text{cm} \tag{5} \]

The results of determining this parameter for other wells are given in Table 3.

The geometrical characteristic of the zone of influence of the openings in the absence of channels at the openings (a loose reservoir) and with a high perforation density is calculated by the formula

\[ G'_s = \frac{1}{\lambda} - \frac{1}{s} - \frac{1}{4r}\ln\frac{s}{\lambda} + \frac{s-\lambda}{16r^2} - 0{,}0117\,\frac{s^2-\lambda^2}{r^3} + 0{,}0026\,\frac{s^3-\lambda^3}{r^4} - \ldots \tag{6} \]

where \(\lambda\) is the radius of the opening in cm.

Under the same conditions (absence of channels), but with a low perforation density, this parameter is calculated by the formula

\[ G''_s = \frac{1}{\lambda} - \frac{0{,}14362}{r} - \frac{1}{4r}\ln\frac{2r}{\lambda} - \frac{1}{2s} - \frac{r^2}{8s^3} - \frac{r^4}{32s^5} - \ldots \tag{7} \]

In the case where channels are present at the openings (a competent reservoir) and with high

for a high perforation density the geometric characteristic of the zones of influence of the holes is calculated by the formula

\[ G_s^{\mathrm{III}}=\frac{1}{y}\ln\frac{(y+\lambda)s}{(y+s)\lambda} -\frac{0.25}{r+y}\ln\frac{s+y}{\lambda+y} +0.0625\,\frac{s-\lambda}{(r+y)^2}, \tag{8} \]

where \(y\) is the depth (length) of the channels at the holes, in cm, beyond the cement sheath, and \(\lambda\) is the mean radius of these channels, also in cm.

In the case of a low perforation density and the same other conditions (the presence of channels at the holes), the geometric characteristic of the zone of influence of the holes is calculated by the formula

\[ G_s^{\mathrm{IV}}=\frac{1}{y}\ln\frac{(y+\lambda)2r}{(y+2r)\lambda} -\frac{0.25}{r+y}\ln\frac{2r+y}{\lambda+y} +0.0625\,\frac{2r-\lambda}{(r+y)^2} +\frac{1}{y}\ln\frac{(y+4r)s}{(y+2s)2r}. \tag{9} \]

In the case under consideration, the perforation density in all wells is high (see above), the formation is competent, the perforation method is shaped-charge, and the perforator is PK-80; therefore formula (8) is adopted as the calculation formula.

The values of two of the parameters used in formula (8) (the well radius and the radius of influence of the holes) were given above. To substantiate the channel depths and their mean radii, let us recall that the PK-80 perforator creates channels up to \(5\ \text{cm}\) deep and with an initial radius up to \(0.5\ \text{cm}\) [1]. However, most of the channel (about \(3\ \text{cm}\)) falls within the metal of the casing and the cement sheath. Thus, the calculated length of the channel may be taken as \(2\ \text{cm}\). The channel at the hole in the casing, as is known, narrows almost conically with distance from its beginning. Therefore, for that part of it which falls within the rock, the mean radius may be taken as \(0.2\ \text{cm}\). Substituting these values into formula (8), we obtain

\[ \begin{aligned} G_s&=\frac{1}{2}\ln\frac{(2+0.2)\cdot 12.21}{(2+12.21)\cdot 0.2} -\frac{0.25}{9.5+2}\ln\frac{12.21+2}{0.2+2} +\frac{0.0625\cdot(12.21-0.2)}{(9.5+2.0)^2} =\\ &=1.1231-0.0406+0.0057=1.0882\ \text{1/cm}. \end{aligned} \]

For well 125 this parameter turned out to be \(1.0569\ \text{1/cm}\), and for well 129, \(1.0753\ \text{1/cm}\) (Table 3).

The geometric characteristic of the zone of flow contraction due to incomplete perforation \(G_{\delta}\) also has different expressions for different types of incompleteness. It depends primarily on the number of incompleteness components. Therefore, consideration of this question must begin with the definition of a component of perforation incompleteness.

By a component of perforation incompleteness of a well is meant either half of an unperforated interval, if this interval is adjacent neither to the bottom nor to the top of it, or its full length, if it is adjacent to the top or to the bottom of the formation. In this case, only those components that exceed the radius of influence of the filter \(x_0\), expressed by the relation, are taken into account

\[ x_0=r+y+s. \tag{11} \]

A component of incompleteness is called first and is denoted by \(\delta_1\) if it exceeds all the others and occurs only once. The first component always corresponds to an unperforated interval adjacent either to the top or to the bottom of the formation.

A component of incompleteness is called second and is denoted by \(\delta_2\) if it is repeated twice. For example, a well has one unperforated interval adjacent neither to the top nor to the bottom—

the reservoir top. According to the definition given above, in this interval two components of incompleteness are distinguished, each of which is equal to half its length. The second example is that the well has two unperforated intervals, one of which adjoins the reservoir top and the other the reservoir bottom. Each of them corresponds to a component of incompleteness. The smaller of these components is repeated in the larger; consequently, it is the second and is denoted by $\delta_2$. The larger component of incompleteness in such cases is called the first and is denoted by $\delta_1$. Wells with two-component incomplete perforation may or may not have the first component of incompleteness.

A component of incomplete perforation is called the third and is denoted by $\delta_3$ if it is repeated three times. With three-component incomplete perforation, within the reservoir there are two unperforated intervals, one of which adjoins either the top or the bottom of the reservoir, while the second is located somewhere in the middle. The first and second components of incompleteness in wells of this kind may be present or may be absent.

A component of incomplete perforation is called the fourth and is denoted by $\delta_4$ if it is repeated four times. Here two cases are possible. First: the well has three unperforated intervals, two of which adjoin the bottom and top of the reservoir, while the third is located somewhere in the middle. Second case: the well has two unperforated intervals located somewhere in the middle of the reservoir. The first, second, and third components of incompleteness in wells with four-component incomplete perforation may be present or may be absent.

Of the wells under consideration, two (Nos. 125 and 129) have two-component incomplete perforation, and the third, well No. 123, has four-component incomplete perforation. Wells with a number of components of incomplete perforation greater than four are not encountered within the Olen’e oil field and therefore are not considered in the present article.

With one-component incomplete perforation, the geometric characteristic of the flow-constriction zone is calculated by the formula

\[ G_{\delta}^{\mathrm{I}}=\frac{n}{h}\ln\frac{(h+x_0)\delta_1}{(h+\delta_1)x_0}, \tag{12} \]

where $h$ is the length of the well-perforation interval in cm, $n$ is the number of holes in the casing (a dimensionless quantity), and $\delta_1$ is the first (and in this case the only) component of incomplete perforation in cm.

With two-component incomplete perforation of wells, when both components $\delta_2$ are equal to each other, the geometric characteristic of the flow-constriction zone is calculated by the formula

\[ G_{\delta}^{\mathrm{II}}=\frac{n}{h}\ln\frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0}. \tag{13} \]

Under analogous conditions (two-component incompleteness), but in the case when one of the components of incompleteness exceeds the other,

\[ x_0<\delta_2<\delta_1, \tag{14} \]

the geometric characteristic of the flow-constriction zone is calculated by the formula

\[ G_{\delta}^{\mathrm{II+I}}=G_{\delta}^{\mathrm{II}}+\frac{n}{h+\delta_2}\ln\frac{(h+2\delta_2)\delta_1}{(h+\delta_2+\delta_1)\delta_2}, \tag{15} \]

where $G_{\delta}^{\mathrm{II}}$ is calculated by formula (13).

With three-component incomplete perforation of wells, three cases are possible. The first case is when all components of incompleteness are equal to one another.

with one another, or, in other words, only the third components \(\delta_3\) participate in the calculation. The formula for determining the geometric characteristic of the flow-constriction zone has the form

\[ G_{\delta}^{\mathrm{III}}=\frac{n}{h}\ln \frac{(h+3x_0)\delta_3}{(h+3\delta_3)x_0}. \tag{16} \]

Second case. One component exceeds the other two. The relation between the components has the form

\[ x_0<\delta_3<\delta_1. \tag{17} \]

The geometric characteristic of the flow-constriction zone is expressed by the formula

\[ G_{\delta}^{\mathrm{III-I}}=G_{\delta}^{\mathrm{III}}+\frac{n}{h+2\delta_3}\ln \frac{(h+3\delta_3)\delta_1}{(h+2\delta_3+\delta_1)\delta_3}, \tag{18} \]

where \(G_{\delta}^{\mathrm{III}}\) is expressed by relation (16).

Third case. Two components exceed the third, or, in other words,

\[ x_0<\delta_3<\delta_2. \tag{19} \]

The geometric characteristic of the flow-constriction zone is expressed by the formula

\[ G_{\delta}^{\mathrm{III-II}}=G_{\delta}^{\mathrm{III}}+\frac{n}{h+\delta_3}\ln \frac{(h+3\delta_3)\delta_2}{(h+\delta_3+2\delta_2)\delta_3}, \tag{20} \]

where \(G_{\delta}^{\mathrm{III}}\) is determined from formula (16).

With a four-component imperfection of well perforation, the following seven cases are possible. The first case—all components are equal to one another, or, in other words, only the fourth components \(\delta_4\) participate in the calculation. The formula for determining the geometric characteristic of the flow-constriction zone here has the form

\[ G_{\delta}^{\mathrm{IV}}=\frac{n}{h}\ln \frac{(h+4x_0)\delta_4}{(h+4\delta_4)x_0}, \tag{21} \]

Second case. One component has a smaller length compared with the others, while the other three are equal to one another. This means that only the fourth and third components participate in the calculation

\[ x_0<\delta_4<\delta_3. \tag{22} \]

The geometric characteristic of the flow-constriction zone here is expressed by the formula

\[ G_{\delta}^{\mathrm{IV-III}}=G_{\delta}^{\mathrm{IV}}+\frac{n}{h+\delta_4}\ln \frac{(h+4\delta_4)\delta_3}{(h+\delta_4+3\delta_3)\delta_4}. \tag{23} \]

where \(G_{\delta}^{\mathrm{IV}}\) is determined from relation (21).

Third case. Two components have one length, and two another. In other words, the fourth and second components are present here

\[ x_0<\delta_4<\delta_2, \tag{24} \]

and the first and third components are absent. The geometric characteristic of the flow-constriction zone here is expressed by the formula

\[ G_{\delta}^{\mathrm{IV-II}}=G_{\delta}^{\mathrm{IV}}+\frac{n}{h+2\delta_4}\ln \frac{(h+4\delta_4)\delta_2}{(h+2\delta_4+2\delta_2)\delta_4}, \tag{25} \]

where \(G_{\delta}^{\mathrm{IV}}\) is determined from relation (21).

Fourth case. Three components are equal to one another and are shorter than the fourth, or, in other words, the first and fourth components participate in the calculation

\[ x_0<\delta_4<\delta_1, \tag{26} \]

and the second and third are absent. The geometric characteristic of the flow-constriction zone here may be expressed by the formula

\[ G_{\delta}^{IV-I}=G_{\delta}^{IV}+\frac{n}{h+3\delta_4}\ln\frac{(h+4\delta_4)\delta_1}{(h+3\delta_4+\delta_1)\delta_4}, \tag{27} \]

where \(G_{\delta}^{IV}\) is calculated by formula (21).

The fifth case. The two smaller components have different lengths, and the two larger ones are identical. The fourth, third, and second components participate in the calculation,

\[ x_0<\delta_4<\delta_3<\delta_2, \tag{28} \]

the first being absent. The formula for the geometric characteristic of the flow-constriction zone in this case has the form

\[ G_{\delta}^{IV-III-II}=G_{\delta}^{IV-III}+\frac{n}{h+\delta_4+\delta_3}\ln\frac{(h+\delta_4+3\delta_3)\delta_2}{(h+\delta_4+\delta_3+2\delta_2)\delta_3}, \tag{29} \]

where \(G_{\delta}^{IV-III}\) is determined by formulas (21) and (23).

The sixth case. The smallest and largest components are unpaired, the components of intermediate length are paired; otherwise, the fourth, third, and first components participate in the calculation,

\[ x_0<\delta_4<\delta_3<\delta_1, \tag{30} \]

the second component is absent. The formula for the geometric characteristic of the flow-constriction zone here has the following form:

\[ G_{\delta}^{IV-III-I}=G_{\delta}^{IV-III}+\frac{n}{h+\delta_4+2\delta_3}\ln\frac{(h+\delta_4+2\delta_3)\delta_1}{(h+\delta_4+2\delta_3+\delta_1)\delta_3}, \tag{31} \]

where \(G_{\delta}^{IV-III}\) is determined from relations (21) and (23).

The seventh case is the most probable. The two smallest components are paired (identical in length), while the largest are unpaired (not identical in length); otherwise, the fourth, second, and first components participate in the calculation,

\[ x_0<\delta_4<\delta_2<\delta_1, \tag{32} \]

the third is absent. The geometric characteristic of the flow-constriction zone in such a case is determined by the relation

\[ G_{\delta}^{IV-II-I}=G_{\delta}^{IV-II}+\frac{n}{h+2\delta_4+\delta_2}\ln\frac{(h+2\delta_4+2\delta_2)\delta_1}{(h+2\delta_4+\delta_2+\delta_1)\delta_2}, \tag{33} \]

where \(G_{\delta}^{IV-III}\) is calculated by formulas (21) and (25).

The imperfection of perforation of well 123 is four-component; the smallest component is unpaired, its length is 100 cm. It is repeated in all the others; hence it may be regarded as the fourth, \(\delta_4\). The next component is paired, its length is 300 cm. In addition, it is repeated in the largest. Thus it is the third, \(\delta_3\). The largest component of perforation imperfection has a length of 1800 cm. It is repeated only in itself, that is, once, and therefore is regarded as the first, \(\delta_1\). This relation between the components corresponds to the sixth case and to formulas (21), (23), and (31).

Substituting into formula (21) the values given above, and also the radius of influence of the filter, which in this case, in accordance with formula (11), is expressed by the relation

\[ \chi_0=9.5+2.0+17.3=28.8\ \text{cm}, \tag{34} \]

we obtain

\[ G_{\delta}^{IV}=\frac{120}{1200}\ln\frac{(1200+4\cdot23.7)\cdot100}{(1200+4\cdot100)\cdot23.7}=0.1228\ 1/\text{cm}. \tag{35} \]

Performing analogous substitutions in formula (23), we obtain

\[ G_{\delta}^{\mathrm{IV-III}}=0.1228+\frac{120}{1200+100}\ln \frac{(1200+4\cdot100)\cdot300}{(1200+100+3\cdot300)\cdot100} = \]

\[ =0.1228+0.0720=0.1948\;1/\mathrm{cm}. \tag{36} \]

Finally, substituting the corresponding data into formula (31), we obtain

\[ G_{\delta}^{\mathrm{IV-III-I}}=0.1948+\frac{120}{1200+100+600}\ln \frac{(1200+100+3\cdot300)\cdot1800}{(1200+100+2\cdot300+1800)\cdot300} = \]

\[ =0.1948+0.0804=0.2752\;1/\mathrm{cm}. \tag{37} \]

The imperfection of the perforation of wells 125 and 129 is two-component; the components of imperfection in both cases are different. Thus, the geometric characteristic of the zone of flow constriction here must be calculated from formulas (13) and (15). For well 125 this parameter proved to be equal to \(0.7913\;1/\mathrm{cm}\), and for well 129, \(0.4755\;1/\mathrm{cm}\) (Table 3).

The geometric characteristic of the zone of planar-radial flow in the wells of the first type is expressed, by analogy with the case in which the effect of flow constriction is absent, by the formula

\[ G'=\frac{1}{h}\ln\frac{R}{r_{\mathrm{s}}}. \tag{38} \]

where \(h\) is the total thickness of the reservoir and \(R\) is the radius of the drainage area of the well.

In this case the geometric characteristic of the zone of planar-radial flow, or of simple radial flow, is defined by the formula

\[ G_{R}''=\frac{1}{H}\ln\frac{R}{r_{\mathrm{w}}}. \tag{39} \]

The value \(R\) participating in formulas (38) and (39), as a rule, requires a special calculation. Some authors recommend determining it by the formula

\[ R^{2}=2.25\chi t, \tag{40} \]

where \(\chi\) is the piezoconductivity of the reservoir and \(t\) is the operating time of the well in the process of testing it. Here, however, it should be noted that the piezoconductivity of the reservoir is a parameter that requires preliminary determination. In this connection, at the first stage of calculating a well, the radius of influence is proposed to be determined not from the piezoconductivity, but from the volume of withdrawn liquid; for this purpose it is recommended to use the following formula:

\[ R^{2}=\frac{2V}{\pi m a H P_{\mathrm{s}}}, \tag{41} \]

where \(V\) is the volume of oil or other liquid withdrawn from the well in the first regime, in \(\mathrm{cm}^{3}\) (for well 123 this quantity is equal to \(9\,474\,000\;\mathrm{cm}^{3}\)); \(m\) is the reservoir porosity in fractions of unity (for well 123 this quantity is equal to 0.187); \(a\) is the compressibility coefficient of the oil or other liquid in \(1/\mathrm{atm}\) (for the oil of the Olenye field this quantity is equal to \(15.37\cdot10^{-5}\;1/\mathrm{atm}\)); \(H\) is the reservoir thickness in \(\mathrm{cm}\) (in the case under consideration this quantity is equal to \(3700\;\mathrm{cm}\)); and \(P_{\mathrm{s}}\) is the drawdown on the reservoir at the boundary of the zone of influence of the openings, determined by the formula

\[ P_{\mathrm{s}}=(P_{\mathrm{pl}}-P_{\mathrm{c}})\frac{G_{\delta}+G_{\mathrm{R}}}{G_{\mathrm{s}}+G_{\delta}+G_{\mathrm{R}}}. \tag{42} \]

This determination is carried out by the method of successive approximations. Let us consider this for the example of the first regime of well 123. In the first approximation we take the radius of influence of the well to be equal to \(4000\;\mathrm{cm}\). Then ge-

the geometric characteristic of the plane-radial flow zone, in accordance with formula (39), is expressed by the relation

\[ G_R=\frac{120}{3700}\ln\frac{4000}{1800}=0.0258\ 1/\mathrm{cm}. \tag{43} \]

The drawdown on the formation at the outer boundary of the zone of influence of the perforations under these conditions, in accordance with formula (42), is expressed by the relation

\[ P_s=29.2\frac{0.2752+0.0258}{1.0882+0.2752+0.0258}=6.3268\ \text{atm}. \tag{44} \]

In accordance with this, from formula (41) we obtain

\[ R^2=\frac{2\cdot 9474000}{3.1416\cdot 0.187\cdot 15.37\cdot 10^{-5}\cdot 3700\cdot 6.3268} =8{,}964{,}720\ \mathrm{cm}^2, \tag{45} \]

whence the radius of influence of the well in the second approximation proves to be equal to 2980 m.

The geometric characteristic of the plane-radial flow zone in the second approximation proved to be equal to \(0.0163\ 1/\mathrm{cm}\), and the drawdown at the outer boundary of the zone of influence of the perforations to \(6.16931\ \text{atm}\).

In an analogous manner, in the third approximation we obtain: the radius of influence of the well \(3032\ \mathrm{cm}\), the geometric characteristic of the plane-radial flow zone \(0.0169\ 1/\mathrm{cm}\), and the drawdown at the outer boundary of the zone of influence of the perforations \(6.17932\ \text{atm}\).

In the fourth approximation the radius of influence of the well proves to be equal to \(3030\ \mathrm{cm}\); the remaining characteristics remain the same as in the third approximation. Therefore we take them as final (Table 3).

To determine the radius of influence of the well in the second and subsequent operating conditions, formula (40) may be used; for this purpose the data relating to the first operating condition are first substituted into it (the radius of influence of the well \(3030\ \mathrm{cm}\) and the operating time of the well \(50400\ \mathrm{s}\)). As a result we obtain the piezoconductivity of the formation, which proved to be equal to \(80.96\ \mathrm{cm}^2/\mathrm{s}\).

Next, into the same formula (40) we substitute the piezoconductivity obtained and the total time of liquid outflow from the well by the end of operation under the given condition. For the second condition this time is expressed by the relation

\[ t_2=50400+133200=183600\ \mathrm{s}, \tag{46} \]

for the third condition

\[ t_3=50400+133200+75600=259200\ \mathrm{s}. \tag{47} \]

For the fourth condition this time proved to be equal to \(349200\ \mathrm{s}\).

In accordance with this, the radius of influence of the well for the second condition was expressed by the relation

\[ R=\sqrt{2.25\cdot 80.96\cdot 183600}=5783\ \mathrm{cm}. \tag{48} \]

For the third condition this parameter proved to be equal to \(6871\ \mathrm{cm}\), and for the fourth to \(7976\ \mathrm{cm}\) (Table 3).

The geometric characteristics of the plane-radial flow zone proved to be as follows: for the second condition \(0.0379\ 1/\mathrm{cm}\), for the third \(0.0434\ 1/\mathrm{cm}\), and for the fourth \(0.0483\ 1/\mathrm{cm}\) (Table 3).

Substituting the obtained values into formula (1) and carrying out the corresponding transformations, we find the effective permeability of the formation during operation of the well under the first condition

\[ k_1=\frac{137.979\cdot 0.64}{6.2832\cdot 120\cdot 29.2}\cdot(1.0882+0.2752+0.0169)=0.007534\ \text{darcy}, \tag{49} \]

or \(7.534\) millidarcies.

Table 3

Results of determining the effective permeability of the Yu-1 reservoir from wells 123, 125, and 129 of the Olenye oil field

For the other regimes this parameter was found to be 9.803, 11.434, and 11.200 millidarcies. The average value for the four regimes was 9.995 millidarcies (Table 3). For well 125 the average value of the effective permeability is 10.646 millidarcies, and for well 129 it is 49.0 millidarcies (Tables 3 and 4).

Table 4

Comparison of the results of determining effective permeability from inflows under steady-state regimes with the results of determining this parameter from cores

The question arises as to how far the data obtained correspond to those permeabilities that are obtained when this parameter is determined from cores. According to an oral communication by G. N. Voloshchuk, the average permeabilities of the formation under consideration, when determined from cores, were as follows: for well 123, 6.5 millidarcies; for well 125, 32.0 millidarcies; and for well 129, 60 millidarcies (Table 4).

To compare the data under consideration, it is proposed to use the convergence coefficient, by which is meant the ratio of the permeability determined from inflows into the well to the permeability determined from cores. This coefficient, as shown in Table 4, deviates from unity now in one direction, now in the other, and at times to a considerable degree. However, the average value of this coefficient for all the wells considered, which in the present case is the cube root of the product of its three initial values, is 0.746, which is fairly close to unity.

The following detail is also of interest. In the determination from cores, the largest permeability value (60 millidarcies) was found in well 129. In this same well it was also the largest (49 millidarcies) when determined from inflows under steady-state regimes. Thus, we have the right to say that the permeabilities obtained from oil inflows under steady-state regimes are sufficiently close to those obtained when the parameter under consideration is determined from cores.

It is evident from Table 3 that in most cases, as the radius of influence of the wells increases, the permeability of the formation also increases. The reason for this phenomenon apparently lies in the fact that the effective permeability is something intermediate between the permeability of the near-wellbore zone, which was clogged during the opening of the formation, and the permeability of the unclogged zone remote from the well. And the greater the share of unclogged intervals in this sum, the greater the effective permeability.

Proceeding from the foregoing, in order to determine the permeability of unclogged zones it is proposed to use the formula

\[ K_{\mathrm{nz}}=\frac{K''\Sigma G''-K'\Sigma G'}{\Sigma G''-\Sigma G'} \tag{50} \]

where \(k'\) and \(\sum G'\) are the effective permeability and the sum of the geometric characteristics under the regime with the smaller radius of influence of the well, and \(k''\) and \(\sum G''\) are the effective permeability and the sum of the geometric characteristics under the regime with the larger radius of influence of the well.

It should be noted, however, that, in order to determine the permeability of the uncolmatized zones, it is impractical to take the data directly from Table 3. This is due primarily to the fact that each well is tested, as a rule, not under two but under three, four, or more regimes, as a result of which not two effective permeabilities are obtained, but many more. Formula (50) is adapted for two of them.

In hydrodynamic methods of investigation in such situations, a large amount of data is reduced to a smaller amount either by graphical constructions or by finding averages. In the case under consideration, when the number of effective permeabilities and sums of geometric characteristics is four, it is expedient to find the arithmetic means from the first two and the last two measurements for each well.

For well 123, the average effective permeability over the first two regimes proved to be \(8.673\) millidarcies, and the corresponding average sum of the geometric characteristics was \(1.3908\ \text{cm}^{-1}\). For the last two regimes, these parameters proved to be \(11.317\) millidarcies and \(1.40925\ \text{cm}^{-1}\), respectively. As a result, the calculation of the permeability of the uncolmatized zones for this well, in accordance with formula (50), takes the form

\[ k_{\text{n.z.}} = \frac{11.317 \cdot 1.40925 - 8.673 \cdot 1.3908}{1.40925 - 1.3908} = \]

\[ = \frac{3.8861}{0.01845} = 210.6\ \text{millidarcies}. \tag{51} \]

Continuing the calculation in the same manner, we find that in well 125 the permeability is \(61.7\) millidarcies, and in well 129 it is \(1854\) millidarcies (Table 5).

In order to take into account how much the permeability changed during the transition from uncolmatized zones remote from the bottomhole to near-bottomhole colmatized zones, it is proposed to introduce the term colmatization coefficient, understood as the ratio of these permeabilities. Within the range of the wells under consideration, this coefficient varies over very wide limits, from \(37.8\) to \(5.8\), but it is not difficult to see that it increases with an increase in effective permeability. It follows from this that colmatization of beds depends to some extent on permeability. This is also natural, since clayey solution should penetrate more easily into wider pores.

There exists an opinion according to which clayey solution does not penetrate into the pores of a bed, or penetrates no more than \(2\)–\(3\ \text{cm}\) (variant \(2\)–\(3\ \text{mm}\)), and the reservoir properties of the bed supposedly do not deteriorate because of this. In the extreme case, supporters of this theory acknowledge penetration into the bed of filtrate, that is, water separated from the clayey solution.

As an objection to such an opinion, the following fact may be cited. During the killing of one large gas blowout on the Pur River, a second well was drilled near the flowing well; through it, \(35\) thousand \(\text{m}^3\) of water was initially pumped, but this produced no effect. When clayey solution began to be pumped into it, \(400\ \text{m}^3\) of this solution proved sufficient to colmatize the bed and stop the flowing of the emergency well. Thus,

Table 5

Calculation of the permeability of reservoir Yu-1 in the non-clayey parts for wells 123, 125, and 129 of the Olen’e field

that is, at sufficiently low pressures the clay solution rather intensively fills the large pores that determine the permeability of the formation and, consequently, makes it impermeable. At high formation pressures, penetration of the solution into the formation proceeds less intensively, but this too leads to some deterioration of the formation permeability in the bottom-hole zone.

It is precisely in connection with the clayization of formations in the process of opening them that the concept of the “skin effect” was introduced; to some extent it corresponds to the concept of the “clayization coefficient” used in the present article.

The method set forth for determining the effective permeability of formations from inflows under steady-state conditions is not the only one. A similar method was set forth by N. M. Bazlov, who recommends determining the additional resistances in the zone of influence of the perforations and in the zone of flow constriction by means of the charts of V. I. Shchurov [5]. A shortcoming of these charts is that they are suitable only for manual processing of data on oil inflows into wells. To use these charts in machine processing of data on inflows into wells, they must first be converted into tabular form, and this entails such expenditures of manual labor that machine processing of inflow data is not justified. Carrying out analogous calculations by the formulas given in the present article makes it possible to use electronic computers for them.

The second shortcoming of V. I. Shchurov’s charts is that they were compiled on the basis of the possibility of only a single-component incomplete perforation of wells. In any case, the data obtained from the charts intended for determining the coefficient \(C_2\) correspond to such values of the geometric characteristic of the flow-constriction zone as are obtained from formula (12). Practice has shown that wells with a number of components of incomplete perforation greater than one occur quite often.

This was the reason why the above-described processing of data on inflows into wells 123, 125, and 129 of the Olen’e field was carried out not according to the method set forth by M. N. Bazlov and other authors, but in accordance with the developments made in works [2, 3, 4].

LITERATURE

1. E. M. Videny. Cumulative perforators used in oil and gas wells. Izd-vo “Nedra,” 1971.

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

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

4. L. A. Pukhlyakov. A method for determining formation permeability in unclayized zones from the results of well studies under steady-state conditions. Information sheet No. 100-74 of the Tomsk TsNTI, 1974.

5. B. S. Chernov, M. N. Bazlov, A. I. Zhukov. Hydrodynamic methods for studying wells and formations. Gostoptekhizdat, 1960.