On the hydrodynamic imperfection of wells associated with incomplete perforation

ON THE HYDRODYNAMIC IMPERFECTION OF WELLS ASSOCIATED WITH INCOMPLETE PERFORATION

L. A. Pukhlyakov

(Presented by the scientific seminar of the Department of Combustible Minerals)

In one of the works devoted to hydrodynamically imperfect wells [4], the drainage region of a well is divided into three zones: the zone of influence of the holes, the zone of flow contraction due to incomplete perforation, and the zone of plane-radial flow.

The zone of influence of the holes is understood as that part of the drainage region of the well where the flow of reservoir fluid is divided into a large number of streams (according to the number of holes), and within each stream the motion of the fluid takes place in the direction of the center of the given hole or in the direction of the axis of the channel at it. The inner boundary of this zone is the outer surface of the cement sheath of the well; the outer surface is a cylinder described around the well axis with radius \(x_0\), expressed by the relation

\[ x_0 = r + y + s, \tag{1} \]

where \(r\) is the radius of the well before the casing string is run, in cm; \(y\) is the depth (length) of the channel at the hole, in cm; and \(s\) is the radius of influence of the holes, or one half of the average distance between the centers of adjacent holes, measured along the surface of the cement sheath, in cm. At high perforation density of wells, when the filter length per one hole does not reach \(2.55r\), this parameter is calculated from the formula

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

where \(h\) is the length of the well perforation interval in cm and \(n\) is the number of holes in the well filter. At low perforation density of the well, when the filter length per one hole exceeds \(2.55r\), the radius of influence of the holes is calculated from the formula

\[ s = 0.5 \sqrt{0.96\pi^2 r^2 + \frac{h^2}{n^2}}. \tag{3} \]

The zone of flow contraction due to incomplete perforation of the well is understood as that part of its drainage region where the flow of reservoir fluid still preserves its unity (i.e., is not yet divided into a multitude of streams according to the number of holes), but is no longer plane-radial. The inner boundary of this zone corresponds to the outer boundary of the zone of influence of the holes. Its outer boundary is the surface—

…the surface of a cylinder described around the axis of the well with a radius equal to the length of the largest component of the incomplete perforation of the well, \(\delta_{\max}\).

By components of incomplete perforation of wells are meant the unperforated intervals within the productive formation, if they adjoin the floor or the roof, or halves of these intervals if they are bounded on both sides by perforated intervals; otherwise, if they adjoin neither the floor nor the roof of the formation.

The following nomenclature for the components of incomplete perforation of wells has been adopted.

If there is one unperforated interval in the well, adjoining either its roof or its floor, then it will be called the first component of incomplete perforation and denoted by \(\delta_1\). A well having only such an unperforated interval will be called a well with one-component incomplete perforation.

If there is one unperforated interval in the well, but it adjoins neither the roof nor the floor of the formation, that is, perforated intervals are located on both sides of it, then such a well will be called a well with two-component incomplete perforation. In this case each of the components will be considered equal to half of the unperforated interval and denoted by \(\delta_2\). Two-component incomplete perforation of a well will also be considered to occur in the case where there are two unperforated intervals in the well, one of which adjoins the floor of the formation, and the other the roof. In this case, if they have the same length, both will be denoted by \(\delta_2\); if they have different lengths, the larger will be denoted by \(\delta_1\) and called the first component, while the second will be denoted by \(\delta_2\) and called the second component.

Three-component incomplete perforation will be considered to occur when, within the productive formation, two intervals remain unperforated, one of which adjoins either the roof or the floor, while the other is located somewhere in the middle. In this case, the interval located at the roof or floor will correspond to a full component, and the one located in the middle of the formation—to two.

If all three components are equal to one another, they will be denoted by \(\delta_3\) and called third components. If one of the components exceeds the other two, then it will be called the first and denoted by \(\delta_1\), while the other two will be called third components and denoted by \(\delta_3\). If, in the case under consideration, two components exceed the third, then these two will be called second components and denoted by \(\delta_2\), while the remaining one will be called the third and denoted by \(\delta_3\).

The number of a component must express its repeatability. In the latter case the smallest component may be set aside within the limits of the larger ones twice and within its own limits a third time. In the preceding case it could be set aside once within the limits of the largest component and plus twice within its own limits. Finally, in the first case, when all three components were equal to one another, the given component was likewise repeated three times.

It follows from what has been said that the largest component of incomplete perforation corresponds to its smallest number.

Four-component incomplete perforation will be considered in two cases. The first case is when, within the productive part of the formation, two intervals remain unperforated, neither of which…

adjoins neither the bottom nor the top of the formation. As the components of incompleteness here, one-half of these intervals will be taken. In the second case, within the productive part of the formation three intervals remained unperforated, one of which is located somewhere in the middle of the perforation interval, while the other two adjoin either the bottom or the top of the formation. In this case the fourth will be regarded as the smallest component, while the remaining numbers are assigned in accordance with the rule stated above.

The incompleteness of well perforation may also be characterized by a larger number of components; however, there is no need to consider all possible variants in this direction. In the present case we are interested in the question of what should be regarded as the largest component of perforation incompleteness and how, accordingly, to determine the outer boundary of the flow-constriction zone.

The plane-radial flow zone is characterized by the following: if the formation is cut by a plane containing the axis of the well, then the projections of the individual elements of the flow of reservoir fluid will be parallel to one another. But if this system is viewed from above (along the well axis), then the individual elements of the flow will be directed toward the well axis along radii.

The inner boundary of the plane-radial flow zone is the outer boundary of the flow-constriction zone due to the incompleteness of perforation (see above), and its outer boundary is the surface of a cylinder described by the radius of influence of the well.

In accordance with the foregoing, the formula for the pressure drop during inflow into a hydrodynamically imperfect well will have the form

\[ P_{\text{pl}} - P_{3} = \frac{Q_{\text{pl}}\mu}{2\pi k_{\text{eff}} n} \left(G_s + G_{\delta} + G_R\right), \tag{4} \]

where \(P_{\text{pl}}\) and \(P_3\) are the reservoir and bottom-hole pressures in ati, \(Q_{\text{pl}}\) is the inflow into the well under reservoir conditions in \(\mathrm{cm^3/s}\), \(\mu\) is the oil viscosity under reservoir conditions in cP, \(k_{\text{eff}}\) is the effective permeability of the formation in darcies, \(n\) is the number of openings in the filter (dimensionless), and \(G_s\), \(G_{\delta}\), and \(G_R\) are the geometric characteristics of the zone of influence of the openings, of the flow-constriction zone due to perforation incompleteness, and of the plane-radial flow zone, respectively, in \(1/\mathrm{cm}\).

The inflow under reservoir conditions is calculated from the formula

\[ Q_{\text{pl}} = 11.574\, Q_{\text{pv}} \cdot b, \tag{5} \]

where \(11.574\) is a coefficient expressing the number of \(\mathrm{cm^3/s}\) in \(\mathrm{m^3/day}\), \(Q_{\text{pv}}\) is the inflow into the well measured under surface conditions in \(\mathrm{m^3/day}\), and \(b\) is the oil formation volume factor, a dimensionless quantity.

The geometric characteristic of the zone of influence of the openings at high perforation density and in the absence of channels at the openings (unconsolidated formation) is calculated from the formula

\[ G_s = \frac{1}{\lambda} -\frac{1}{s} -\frac{0.25}{r}\ln\frac{s}{\lambda} +0.0625\,\frac{s-\lambda}{r^2} -0.0117\,\frac{s^2-\lambda^2}{r^3} +\ldots, \tag{6} \]

where \(\lambda\) is the radius of the openings in cm, and \(r\) is the radius of the well before the casing string is run, in cm. Low perforation density is now encountered extremely rarely, and therefore the corresponding formula for determining the geometric characteristic of the zone of influence of the openings is not given in this work. When channels are present at the openings, the parameter under consideration is calculated from the formula

\[ G_s^*=\frac{1}{y}\ln\frac{(y+\lambda)s}{(y+s)\lambda} -\frac{0.25}{r+y}\ln\frac{s+y}{\lambda+y} +0.0625\,\frac{s-\lambda}{(r+y)^2}, \tag{7} \]

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

The geometric characteristic of the plane-radial flow zone in the absence of a flow-constriction zone is calculated by the formula

\[ G_R=\frac{n}{H}\ln\frac{R}{r+y+s}, \tag{8} \]

and when such a zone is present, by the formula

\[ G_R^*=\frac{n}{H}\ln\frac{R}{\delta_{\max}}, \tag{9} \]

where \(H\) is the total thickness of the formation, in cm, and \(R\) is the radius of influence of the well, in cm. In turn, the radius of influence of the well is calculated by the formula

\[ R^2=\frac{2V_{\Sigma}}{\pi m\alpha H P_s}, \tag{10} \]

where \(V_{\Sigma}\) is the total volume of oil or other fluid withdrawn from the well during its testing, in cm\(^3\); \(m\) is the formation porosity in fractions of unity; \(\alpha\) is the coefficient of compressibility of the oil or other fluid, in \(1/\text{atm}\); and \(P_s\) is the drawdown on the formation at the boundary of the zone of influence of the holes, in atm, calculated by the formula

\[ P_s=(P_{\text{pl}}-P_z)\frac{G_\delta+G_R}{G_s+G_\delta+G_R}. \tag{11} \]

Thus, it remains here to consider only the formula for the geometric characteristic of the flow-constriction zone due to incomplete perforation of the well, \(G_\delta\), to which the present article is devoted.

Other authors dealing with the problem of oil inflow into hydrodynamically imperfect wells [1, 2, 5, 6, 7] considered only wells with one-component incomplete perforation. This was due to the fact that the theory of hydrodynamically imperfect wells developed on the basis of wells whose bottomholes did not reach the base of the formation and which are now called hydrodynamically imperfect with respect to the degree of formation penetration. It was assumed in this case that the entire interval of the formation penetrated by the well is perforated.

The study of this question by the author of the present article showed that the interval of the formation penetrated by the well is perforated incompletely, and that two unperforated intervals are not equivalent to one whose length is equal to the two original ones. Accordingly, a formula was proposed for determining the geometric characteristic of the zone under consideration, which was suitable for solving the problem posed both for one- and for two-component incomplete perforation of a well [4].

However, further study of this question showed that wells occur with more than two components of incomplete perforation. In this connection it became necessary to compile the corresponding formulas for determining the geometric characteristic under consideration. At the same time it was established that, for the most commonly encountered wells with one-component incomplete perforation, it is expedient to use a formula simpler than that for two-component incompleteness; its derivation is given below.

To derive the formula for the geometric characteristic in the case of a single-component perforation incompleteness, we proceed as follows.

1. Within the flow-constriction zone, let us isolate a thin-walled cylinder-like body ending in a hemisphere of radius \(\rho\), with cylindrical-part length \(h\) and wall thickness \(d\rho\). Obviously, the surface of this body will be expressed by the relation

\[ F = 2\pi \rho h + 2\rho^2, \tag{12} \]

and the initial expression for obtaining the geometric characteristic of the zone under consideration will have the form

\[ dG^1 = \frac{n}{2\pi\rho(h+\rho)}\,d\rho. \tag{13} \]

1. From the expression obtained, \(2\pi\) is transferred into the basic formula for the pressure drop (4) without change. As a result, expression (13) takes the form

\[ dG = \frac{n}{\rho(h+\rho)}\,d\rho. \tag{14} \]

1. We integrate the left-hand side of the expression obtained within the limits \(0 < G < G_{\delta}^{\mathrm{I}}\), and the right-hand side within the limits \(x_0 < \rho < \delta_1\). As a result, we obtain

\[ G_{\delta}^{\mathrm{I}} = -\frac{n}{h}\ln \frac{h+\rho}{\rho}\Bigg|_{x_0}^{\delta_1}, \tag{15} \]

\[ G_{\delta}^{\mathrm{I}} = -\frac{n}{h} \left( \ln \frac{h+\delta_1}{\delta_1} - \ln \frac{h+x_0}{x_0} \right) \tag{16} \]

or, after the corresponding transformations,

\[ G_{\delta}^{\mathrm{I}} = \frac{n}{h} \ln \frac{(h+x_0)\delta_1}{(h+\delta_1)x_0}. \tag{17} \]

In the case of a two-component perforation incompleteness of the well and equality between the two components, that is, in the absence of the first component, the initial expression for deriving the formula of the geometric characteristic will have the form

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

After the corresponding transformations, analogous to those carried out in finding expression (17), we obtain

\[ G_{\delta}^{\mathrm{II}} = \frac{n}{h} \ln \frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0}. \tag{19} \]

If, however, in a well with two-component perforation incompleteness the individual components differ from one another in size, that is, if the first component is also present here, then the formula for the geometric characteristic for it can be obtained as follows.

We divide the constriction zone into two subzones: an inner and an outer one, in such a way that the boundary between them is the surface of a cylinder described around the well axis by the component \(\delta_2\). Obviously, for the inner subzone the geometric characteristic will be expressed by relation (19).

For the outer subzone this characteristic will be expressed by a relation analogous to relation (17),

\[ G_1'=\frac{n}{h'}\ln \frac{(h'+x_0')\delta_1}{(h'+\delta_1)x_0'}, \tag{20} \]

where

\[ x_0'=\delta_2, \tag{21} \]

\[ h'=h+\delta_2. \tag{22} \]

After the corresponding transformations, expression (20) takes the form

\[ G_1'=\frac{n}{h+\delta_2}\ln \frac{(h+2\delta_2)\delta_1}{(h+\delta_2+\delta_1)\delta_2}, \tag{23} \]

and the formula for the geometric characteristic of the flow-constriction zone for a two-component imperfection in general form will be expressed by a relation that is the sum of relations (19) and (23):

\[ G_{\delta}^{\mathrm{II-I}} = \frac{n}{h}\ln \frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0} + \frac{n}{h+\delta_2}\ln \frac{(h+2\delta_2)\delta_1}{(h+\delta_1+\delta_2)\delta_2}. \tag{24} \]

The formula for the geometric characteristic of the flow-constriction zone for a well with a three-component perforation imperfection can be obtained in the same manner as the analogous formula for a well with a two-component imperfection; namely, it is first necessary to derive the formula for the case in which all three components of the perforation imperfection are equal to one another. By analogy with expression (14), the initial expression for deriving this formula will have the form

\[ dG=\frac{n}{\rho(h+3\rho)}\,d\rho, \tag{25} \]

and the geometric characteristic itself will finally be expressed by the relation

\[ G_{\delta}^{\mathrm{III}} = \frac{n}{h}\ln \frac{(h+3x_0)\delta_3}{(h+3\delta_3)x_0}. \tag{26} \]

With a three-component imperfection of well perforation, two more cases are possible. The first (the most common) is that one component of the imperfection, \(\delta_1\), exceeds the other two, which in turn are equal to each other and are denoted by \(\delta_3\). The formula for the geometric characteristic is two-term; in this case the first term is expressed by relation (26). To derive the second term of this formula, expression (20) is taken as the basis, but in it \(h+2\delta_3\) is substituted for \(h'\), and \(\delta_3\) for \(x_0'\). As a result the formula takes the form

\[ G_{\delta}^{\mathrm{III-I}} = G_{\delta}^{\mathrm{III}} + \frac{n}{h+2\delta_3}\ln \frac{(h+3\delta_3)\delta_1}{(h+2\delta_3+\delta_1)\delta_3}, \tag{27} \]

where \(G_{\delta}^{\mathrm{III}}\) is expressed by relation (26).

Another case with a three-component imperfection of well perforation is that two mutually equal components \(\delta_2\) exceed the third, \(\delta_3\). The formula is derived in the same manner as the preceding one; however, in deriving the second term, expression (19) is taken as the basis, which is reduced to the form

\[ G_2'=\frac{n}{h'}\ln \frac{(h'+2x_0')\delta_2}{(h'+2\delta_2)x_0'}. \tag{28} \]

Substituting into it, however, for \(h'\) the sum \(h+\delta_3\), and for \(x_0'\) — \(\delta_3\), we obtain

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

where \(G_{\delta}^{\mathrm{III}}\) is still expressed by relation (26).

With four-component incompleteness of well perforation, several cases are also possible.

First case. All components are equal to one another and are denoted by \(\delta_4\). The geometrical characteristic of the flow-constriction zone here is expressed by the relation

\[ G_{\delta}^{\mathrm{IV}} = \frac{n}{h} \ln \frac{(h+4x_0)\delta_4}{(h+4\delta_4)x_0}. \tag{30} \]

Second case. One component has a smaller length in comparison with the others, and the other three are equal to one another, \(x_0<\delta_4<\delta_3\). The geometrical characteristic of the flow-constriction zone here is expressed by the relation

\[ G_{\delta}^{\mathrm{IV-III}} = G_{\delta}^{\mathrm{IV}} + \frac{n}{h+\delta_4} \ln \frac{(h+4\delta_4)\delta_3}{(h+\delta_4+3\delta_3)\delta_4}, \tag{31} \]

where \(G_{\delta}^{\mathrm{IV}}\) is expressed by relation (30).

Third case. Two components have one length, and two another, \(x_0<\delta_4<\delta_2\). The geometrical characteristic of the flow-constriction zone is expressed by the relation

\[ G_{\delta}^{\mathrm{IV-II}} = G_{\delta}^{\mathrm{IV}} + \frac{n}{h+2\delta_4} \ln \frac{(h+4\delta_4)\delta_2}{(h+2\delta_4+2\delta_2)\delta_4}, \tag{32} \]

where \(G_{\delta}^{\mathrm{IV}}\) is expressed by formula (30).

Fourth case. Three components are equal to one another and are inferior in length to the fourth, \(x_0<\delta_4<\delta_1\). The geometrical characteristic of the flow-constriction zone is expressed by the relation

\[ G_{\delta}^{\mathrm{IV-I}} = G_{\delta}^{\mathrm{IV}} + \frac{n}{h+3\delta_4} \ln \frac{(h+4\delta_4)\delta_1}{(h+3\delta_4+\delta_1)\delta_4}, \tag{33} \]

where \(G_{\delta}^{\mathrm{IV}}\) is expressed by formula (30).

Fifth case. The two smaller components have different lengths, and the two larger ones have the same length \((x_0<\delta_4<\delta_3<\delta_2)\). The formula for the geometrical characteristic of the flow-constriction zone here has the form

\[ G_{\delta}^{\mathrm{IV-III-II}} = G_{\delta}^{\mathrm{IV-III}} + \frac{n}{h+\delta_4+\delta_3} \ln \frac{(h+\delta_4+3\delta_3)\delta_2}{(h+\delta_4+\delta_3+2\delta_2)\delta_3}, \tag{34} \]

where \(G_{\delta}^{\mathrm{IV-III}}\) is expressed by relations (31) and (30).

Sixth case. The smallest and largest components are unpaired; the components of intermediate length are paired, \(x_0<\delta_4<\delta_3<\delta_1\). The formula for the geometrical characteristic of the flow-constriction zone will have the form

\[ G_{\delta}^{\mathrm{IV-III-I}} = G_{\delta}^{\mathrm{IV-III}} + \frac{n}{h+\delta_4+2\delta_3} \ln \frac{(h+\delta_4+3\delta_3)\delta_1}{(h+\delta_4+2\delta_3+\delta_1)\delta_3}, \tag{35} \]

where \(G_{\delta}^{\mathrm{IV-III}}\) is expressed by relations (31) and (30).

The seventh case (the most common in wells with four-component incompleteness). The two smallest components are even, the largest are odd, that is, they differ in length: \(x_0<\delta_4<\delta_2<\delta_1\). The formula for the geometric characteristic of the flow-constriction zone in this case will have the form

\[ G_{\delta}^{\mathrm{IV-II-I}} = G_{\delta}^{\mathrm{IV-II}} + \frac{n}{h+2\delta_4+\delta_2} \ln \frac{(h+2\delta_4+2\delta_2)\delta_1}{(h+2\delta_4+\delta_2+\delta_1)\delta_2}, \tag{36} \]

where \(G_{\delta}^{\mathrm{IV-II}}\) is expressed by relations (32) and (30).

Incomplete perforation of wells with a number of components greater than four is possible, but unlikely. Therefore there is no need, within the scope of the present article, to consider the geometric characteristics of the flow-constriction zone for wells with incompleteness of this kind.

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