Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov
Vol. 236, 1976

On the Influence of Cracks in the Cement Sheath on Inflows into Oil Wells

L. A. Pukhlyakov

(Presented by Professor A. V. Aksarin)

Suppose that a certain well has penetrated an oil-bearing formation of small thickness \(h\), in which there is one opening in the casing string, and that through this opening, parallel to the axis of the well, there passes a crack in the cement sheath of length \(h\) and width \(2u\). Obviously, the resistance to the motion of oil inside the crack itself will be many times less than the resistance along the path of approach to it. On this basis, the first of these may be neglected and only the second considered. For this it is first necessary to determine the filtration area at various distances from the axis of the crack.

It is not difficult to see that beyond two radii of the well this area will be expressed by the relation

\[ F_1=2\pi \rho h, \tag{1} \]

where \(\rho\) is the distance from the axis of the crack. Thus, to express the pressure drop in this zone, Dupuit’s formula will be valid, which is written in the following form:

\[ P_0-P_1=\frac{Q\mu}{2\pi kh}\ln\frac{R}{2r} \tag{2} \]

where \(Q\) is the inflow into the well under reservoir conditions, in \(\text{cm}^3/\text{s}\); \(k\) is the permeability of the formation, in darcies; \(h\) is the thickness of the formation (and the length of the crack), in cm; \(R\) is the radius of influence of the well, in cm; \(r\) is the radius of the well before the casing string is run into it (half the bit diameter), in cm.

In the zone immediately adjoining the crack, the pressure drop will have a more complex form. To establish its mathematical expression, it is first necessary to determine the area cut out by the body of the cement sheath in a cylinder described by radius \(\rho\) about the axis of the crack. For this purpose let us turn to Fig. 1. In this figure it is seen that half of the arc cut off by the cement sheath in the circumference of the described radius can be determined through the radius \(\rho\) and the angle \(\angle O_1O_2C\), expressed in radians,

\[ \frac{1}{2}x=\rho \angle O_1O_2C. \tag{3} \]

But this angle is not constant; therefore, in order to carry out various transformations with it, it must be expressed through the arccosine. In that case expression (3) takes the form

\[ \frac{1}{2}x=\rho \arccos\frac{\rho}{2r}. \tag{4} \]

Fig. 1. Elements determining the area of the cut made in a cylinder when the axis of one of them lies on the surface of the other.

Hence the area of the cut can be written in the following form:

\[ F' = 2\rho h \cdot \arccos \frac{\rho}{2r} \tag{5} \]

and the filtration area

\[ F_2 = 2\pi\rho h - 2\rho h \cdot \arccos \frac{\rho}{2r}. \tag{6} \]

Expanding the arccosine in a series, we obtain

\[ F_2 = 2\pi\rho h - 2\rho h \left[ \frac{\pi}{2} - \left( \frac{\rho}{2r} + \frac{\rho^3}{6\cdot 8 \cdot r^3} + \frac{3\rho^5}{40\cdot 32 \cdot r^5} + \frac{15\rho^7}{336\cdot 128 \cdot r^7} +\ldots \right) \right], \tag{7} \]

or, after the corresponding transformations,

\[ F_2 = h\left( \pi\rho + \frac{\rho^2}{r} + 0.0416667\,\frac{\rho^4}{r^3} + 0.0046875\,\frac{\rho^6}{r^5} + 0.0006975\,\frac{\rho^8}{r^7} +\ldots \right). \tag{8} \]

Obviously, the original expression for determining the pressure drop will have the form

\[ dP = \frac{Q\mu}{ kh\left( \pi\rho + \frac{\rho^2}{r} + 0.0416667\,\frac{\rho^4}{r^3} + 0.0046875\,\frac{\rho^6}{r^5} + 0.0006975\,\frac{\rho^8}{r^7} +\ldots \right) } \,d\rho . \tag{9} \]

or, after reducing it to a form convenient for integration,

\[ dP = \frac{Q\mu}{2\pi kh} \left( \frac{2}{\rho} - 0.636620\,\frac{1}{r} + 0.202640\,\frac{\rho}{r^2} - 0.091028\,\frac{\rho^2}{r^3} + \right. \]

\[ \left. + 0.037420\,\frac{\rho^3}{r^4} - 0.017564\,\frac{\rho^4}{r^5} + 0.007752\,\frac{\rho^5}{r^6} - 0.003712\,\frac{\rho^6}{r^7} + \right. \]

\[ \left. + 0.001692\,\frac{\rho^7}{r^8} - 0.000744\,\frac{\rho^8}{r^9} +\ldots \right) d\rho . \tag{10} \]

Integrating expression (10) over the limits \(u < \rho < 2r\), where \(u\) is half the width of the crack in cm, we obtain

\[ P_1 - P_2 = \frac{Q\mu}{2\pi kh} \left( 2\ln\rho - 0.636620\,\frac{\rho}{r} + \frac{0.202640}{2}\,\frac{\rho^2}{r^2} - \frac{0.091028}{3}\,\frac{\rho^3}{r^3} + \right. \]

\[ +\frac{0.037420}{4}\frac{\rho^4}{r^4} -\frac{0.017564}{5}\frac{\rho^5}{r^5} +\frac{0.007752}{6}\frac{\rho^6}{r^6} -\frac{0.003712}{7}\frac{\rho^7}{r^7} + \]
\[ +\frac{0.001692}{8}\frac{\rho^8}{r^8} -\frac{0.000744}{9}\frac{\rho^9}{r^9} +\ldots\Bigg)\Bigg|_{u}^{2r}. \tag{11} \]

Substituting the specified limits into the expression obtained and taking into account that the radius of the well is many times greater than the width of the crack, with a known degree of approximation we obtain

\[ P_1-P_2=\frac{Q\mu}{2\pi kh} \left( 2\ln\frac{2r}{u} -1.273240+0.405284-0.242740+ \right. \]
\[ \left. +0.149680-0.112536+0.082688-0.067867+0.054144-0.042324+\ldots \right) \tag{12} \]

\[ P_1-P_2=\frac{Q\mu}{2\pi kh} \left( 2\ln\frac{2r}{u}-1.04692 \right), \tag{13} \]

If the calculation by the method described is continued to higher powers of \(\rho\), then the second term in the parentheses of expression (13) proves to be equal to 1.024. Moreover, if it is assumed that the crack length \(l\) is not equal to the formation thickness, then expression (13) must be written in the following form

\[ P_1-P_2=\frac{Q\mu}{2\pi kl} \left( 2\ln\frac{2r}{u}-1.024 \right). \tag{14} \]

Finally, adding expression (14) to expression (2), we obtain

\[ P_1-P_2=\frac{Q\mu}{2\pi k} \left( \frac{2}{l}\ln\frac{2r}{u} -\frac{1.024}{l} +\frac{1}{h}\ln\frac{R}{2r} \right). \tag{15} \]

Now let us assume that we have a well whose radius before running the casing string was equal to 10 cm, its radius of influence \((R)\) 210 m, the oil viscosity 0.52 cP, the permeability of the formation 91.6 millidarcies, the formation thickness per one opening 14.286 cm, and the inflow under reservoir conditions 18.3 cm\(^3\)/sec per opening; and let us see what pressure drop we obtain if the perforator bullet made an opening only in the casing and created in the cement sheath a crack 14.286 cm long and 0.4 mm wide \((u=0.02\ \text{cm})\). In the presence of an opening and in the absence of a crack in the cement sheath, under these conditions the pressure drop proved to be equal to 33 atm [1].

As a result of substituting the listed data into formula (15), we obtain

\[ P_0-P_2= \frac{18.3\cdot 0.52}{2\pi 0.0916\cdot 14.286} \left( 2\ln\frac{20}{0.02} -1.024+\ln\frac{21000}{20} \right) \]

or, after the corresponding transformations,

\[ P_0-P_2=\frac{9.51652}{8.22215} \left( 2\ln 1000-1.024+\ln 1050 \right), \]

\[ P_0-P_2=1.157424\,(2\cdot 6.90775-1.024+6.9566)=22.857\ \text{atm}. \tag{16} \]

Thus, in comparison with inflow through an opening, inflow through a crack requires a somewhat smaller pressure drop; however, the difference between them is not very large. It barely exceeds 30%. Moreover, comparing with one another the separate terms of the part of expression (16) enclosed in parentheses, it is not difficult to see that, even when cracks are present, the principal resistance to fluid motion is created by the zone immediately adjacent to the wellbore. It is almost twice the resistance of the zone of plane-radial flow. Thus, longitudinal cracks in the cement sheath, so long as their width remains small, have little effect on inflow into the well.

Let us consider how matters stand with annular cracks. For this purpose, we choose a crack of the same width \(2u\) and a formation of thickness \(h\), within which, exactly opposite its middle, there is a single opening in the casing. It is understood that the crack under consideration is connected with the opening.

To express the filtration area in the zone of influence of the crack, we select in it an elementary strip. The length of this strip is expressed by the relation

\[ x = 2\pi (r+\rho\sin\alpha), \tag{17} \]

where \(\rho\) is the distance from the crack along a straight line passing through the axis of the well, and \(\alpha\) is the angle formed by this straight line and the axis of the well (Fig. 2). The width of the strip is expressed by the relation

\[ y = \rho d\alpha. \tag{18} \]

Hence its area is

\[ dF = 2\pi (r+\rho\sin\alpha)\rho d\alpha . \tag{19} \]

Fig. 2. Elements determining the filtration area at a distance \(\rho\) from an annular crack during inflow through this crack. The casing body is hatched with oblique hatching; the cement sheath is cross-hatched. \(2u\) is the crack width; \(\rho d\alpha\) is the width of the elementary strip of the surface under consideration.

and the area of the upper half of the filtration surface, bounded by the limits \(0<\alpha<\frac{\pi}{2}\), can be expressed through the integral of expression (19)

\[ \frac{1}{2}F = 2\pi\rho r \int_{0}^{\pi/2} d\alpha + 2\pi\rho^{2}\int_{0}^{\pi/2}\sin\alpha d\alpha, \tag{20} \]

\[ \frac{1}{2}F = 2\pi\rho r\alpha\bigg|_{0}^{\pi/2} - 2\pi\rho^{2}\cos\alpha\bigg|_{0}^{\pi/2}, \tag{21} \]

\[ \frac{1}{2}F=2\pi\rho r\frac{\pi}{2}+2\pi\rho^2. \tag{22} \]

Thus, the total area of the filtration surface is expressed by the relation

\[ F=2\pi^2\rho r+4\pi\rho^2 \tag{23} \]

and the initial expression for determining the pressure drop in the case under consideration will have the form

\[ dP=\frac{Q\mu}{k\,2\pi(\pi\rho r+2\rho^2)}\,d\rho. \tag{24} \]

Reducing expression (24) to a form convenient for integration, we obtain

\[ dP=\frac{Q\mu}{2\pi k\cdot 2\rho\left(\frac{1}{2}\pi r+\rho\right)}\,d\rho \tag{25} \]

and after integration

\[ P_1-P_2= \frac{Q\mu}{2\pi k\cdot 2} \left( -\frac{1}{\frac{1}{2}\pi r} \ln \frac{\frac{1}{2}\pi r+\rho}{\rho} \right)\Bigg|_{u}^{h/2} \tag{26} \]

\[ P_1-P_2= \frac{Q\mu}{2\pi k}\cdot\frac{1}{\pi r} \left( \ln\frac{\pi r+2u}{2u} -\ln\frac{\pi r+h}{h} \right), \tag{27} \]

\[ P_1-P_2= \frac{Q\mu}{2\pi k}\cdot\frac{1}{\pi r} \ln \frac{(\pi r+2u)h}{2u(\pi r+h)}. \tag{28} \]

Finally, excluding from the numerator of expression (28) \(2u\), since this quantity (the crack width) is always incommensurably smaller than \(\pi r\) (half the circumference of the well along the outside diameter), and adding to the expression obtained the geometrical characteristic of the zone of plane-radial flow, we obtain

\[ P_0-P_2= \frac{Q\mu}{2\pi k} \left[ \frac{1}{\pi r}\ln\frac{\pi r h}{2u(\pi r+h)} + \frac{1}{h}\ln\frac{R}{r+0.5h} \right]. \tag{29} \]

As is known, the fluid experiences the main resistance to its motion in the immediate vicinity of the crack. Therefore, if the crack in the cement ring occupies not the entire circumference of the well but only part of it, and if its length is denoted by \(l\), then without great error the expression for the inflow into the well under consideration may be written in the following form:

\[ P_0-P_2= \frac{Q\mu}{2\pi k} \left[ \frac{2}{l}\ln\frac{lh}{2u(l+2h)} + \frac{1}{h}\ln\frac{R}{r+0.5h} \right]. \tag{30} \]

Now let us see how much the pressure drop in longitudinal cracks differs from the pressure drop in transverse cracks. For this purpose we perform a calculation by formula (30), substituting into it \(l=14.286\) cm, and for the remaining quantities the same values that were used in the calculation by formula (15). Incidentally, the value \(14.286\) cm in the calculation by formula (15) represented not only the reservoir thickness, but also the crack length. As a result of substituting these quantities into formula (30), we obtain

\[ P_0-P_2= \frac{18.3\cdot0.52}{6.28318\cdot0.0916} \left[ \frac{2}{14.286}\ln \frac{14.286\cdot14.286}{2\cdot0.02(14.286+28.372)} + \right. \]

\[ +\frac{1}{14{,}286}\ln\frac{21000}{10+7{,}143}\right], \]

\[ P_0-P_2=\frac{9{,}516}{0{,}57554}\,(0{,}141\ln119{,}05+0{,}071\ln1224{,}98), \]

\[ P_0-P_2=16{,}476(0{,}66913+0{,}49777)=19{,}226\ \text{atm}. \tag{31} \]

When calculating by formula (15), we obtained 22.857 atm. The discrepancy is comparatively small and can be fully explained by the fact that, in the zone of influence of the crack, when calculating by formula (15) we proceeded from the assumption that the length of the crack corresponded to the width of the flow, whereas when calculating by formula (30) the length of the crack was always less than the width of the flow. In view of the foregoing, to determine the pressure drop for inflow into a well in which one crack in the cement sheath passes through each opening, one may use the following formula, based on formula (15):

\[ P_0-P_2=\frac{Q\mu}{2\pi kh}\left(\frac{2n}{l}\ln\frac{2r}{u}-\frac{1{,}024n}{l}+\frac{n}{h}\ln\frac{R}{2r}\right). \tag{32} \]

Here \(Q\) is the inflow into the well under reservoir conditions in cm\(^3\)/s, \(\mu\) is the viscosity of the oil under reservoir conditions in centipoises, \(k\) is the permeability of the formation in darcies, \(n\) is the number of openings in the casing, \(h\) is the formation thickness in cm, and \(l\) is the average length of the cracks in cm.

For large crack dimensions this formula will also be valid in cases where the opening in the casing is simultaneously also an opening in the cement sheath. In cases where, under the same conditions, the crack dimensions are small, the pressure drop and other parameters must be calculated as follows.

1. The first two terms in parentheses in expression (32) are represented in the form of the geometric characteristic of the zone of influence of the cracks

\[ G_u=\frac{2n}{l}\ln\frac{2r}{u}-\frac{1{,}024n}{l} \tag{33} \]

1. In accordance with the theory of inflow into wells that have no cracks in the cement sheath [1, 2], the geometric characteristic of the zone of influence of the openings is determined. At a high perforation density, when the formation thickness per one opening does not reach 2.55 well radii, or 1.27 bit diameters,

\[ \frac{h}{n}\leq 2{,}55r; \tag{34} \]

the calculation is carried out by the formula

\[ G_\lambda=\frac{1}{\lambda}-\frac{1}{s} -\frac{0{,}25}{r}\ln\frac{s}{\lambda} +0{,}0625\,\frac{s-\lambda}{r^2} -0{,}0117\,\frac{s^2-\lambda^2}{r^3} + \]

\[ +0{,}0026\,\frac{s^3-\lambda^3}{r^4} -0{,}0008\,\frac{s^4-\lambda^4}{r^5} +0{,}0002\,\frac{s^5-\lambda^5}{r^6} -0{,}0001\,\frac{s^6-\lambda^6}{r^7} +\ldots \tag{35} \]

where \(\lambda\) is the radius of the opening on the outer section of the cement sheath in cm, \(s\) is the radius of influence of the openings, which at a high perforation density is calculated by the formula

\[ S=0{,}5\sqrt{2\pi r\,\frac{h}{n}}, \tag{36} \]

and \(r\) is the radius of the well before running the casing, in cm. At a low perforation density, when the left-hand side of expression (34) exceeds the right-hand side, other formulas [2] are used instead of formulas (35) and (36).

1. After determining the geometric characteristic of the zone of influence of the cracks \(G_{\mathrm{u}}\) and the zone of influence of the holes \(G_{\lambda}\), their overall geometric characteristic \(G_{\mathrm{ob}}\) is determined. The calculation is carried out by the formula

\[ \frac{1}{G_{\mathrm{ob}}}=\frac{1}{G_{\mathrm{u}}}+\frac{1}{G_{\lambda}} \tag{37} \]

If the number of cracks passing through each hole is expressed by the number \(i\), and all of them are approximately identical, then formula (37) takes the form

\[ \frac{1}{G_{\mathrm{ob}}}=\frac{i}{G_{\mathrm{u}}}+\frac{1}{G_{\lambda}} \tag{38} \]

The total pressure drop in this case will be expressed by the relation

\[ P_0-P_2=\frac{Q\mu}{2\pi kh}\left(G_{\mathrm{ob}}+\frac{n}{h}\ln\frac{R}{s+r}\right). \tag{39} \]

Suppose that, in a certain case, each hole in the filter has two cracks, the geometric characteristic of each of which is equal to 0.66913, while the geometric characteristic of the zone of influence of the holes without taking the cracks into account, \(G_{\lambda}\), is expressed by the number 1.5264. Let us further suppose that the geometric characteristic of the plane-radial-flow zone, expressed by the relation

\[ G_R=\frac{n}{h}\ln\frac{R}{r+s} \tag{40} \]

is equal to 0.49777, and the other flow parameters are the same as in expressions (16) and (31). In this case, calculation by formula (38) gives

\[ \frac{1}{G_{\mathrm{ob}}}=\frac{2.0}{0.66913}+\frac{1.0}{1.5264}=2.989+0.655=3.644, \tag{41} \]

whence \(G_{\mathrm{ob}}\) proves to be equal to 0.274. Substituting this value into expression (31) instead of 0.66913, we obtain:

\[ P_0-P_2=16.476(0.274+0.49777)=12.7\ \text{atm}. \tag{42} \]

With one crack present and the hole ignored, we obtained in one case 22.9 and in another 19.2 atm (see expressions 16 and 31), while when the cracks were ignored, 33 atm [1].

LITERATURE

1. L. A. Pukhlyakov. Determination of reservoir permeability from inflows into a well under steady-state conditions. Izvestiya TPI, vol. 177, 1971.

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