Determination of the Radius of Influence of Oil Exploration Wells During Steady-State Flow Testing
Abstract Generated abstract
The paper addresses the problem of estimating the radius of influence of a single oil exploration well during steady-state flow testing, where methods based on interference between wells are generally inapplicable. It develops a calculation procedure from oil compressibility, Dupuit radial-flow relations, perforation geometry, and geometric characteristics of flow near perforations, including cases of complete and incomplete perforation. By integrating the volume of oil released through elastic expansion within the drawdown region, the author derives a practical expression for the radius of influence in terms of produced oil volume, formation thickness, porosity, oil compressibility, and drawdown at the boundary of the perforation-influence zone.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV, HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
Vol. 236 1976
DETERMINATION OF THE RADIUS OF INFLUENCE OF OIL EXPLORATION WELLS DURING TESTING UNDER STEADY-STATE CONDITIONS
L. A. Pukhlyakov
(Presented by Professor A. V. Aksarin)
In investigations of wells under steady-state conditions, the radius of influence of the wells plays an important role. A method is known for determining this parameter, based on the influence of wells upon one another [1]. However, in the exploration of oil fields, as a rule, no more than one well operates within a single pool; therefore, it is not possible here to use the above-mentioned method. Below, a new method is proposed for determining the radius of influence of wells, which can be used in processing data on exploration wells during their testing under steady-state conditions. It consists in the following.
If a small quantity of oil is withdrawn from a well that has just penetrated an oil pool, in such a way that the pressure in the near-wellbore zone does not fall below the saturation pressure, then the oil will move toward it only by virtue of its own compressibility. V. N. Mamuna et al. [2] recommend determining this parameter by the formula
\[ \alpha=\frac{1}{V_0}\frac{\Delta V}{\Delta P}, \tag{1} \]
where \(\alpha\) is the compressibility, or coefficient of compressibility, of the oil in \(1/\text{atm}\), \(V_0\) is the initial volume of the oil sample in \(\text{cm}^3\), \(\Delta V\) is the change in the volume of this sample in \(\text{cm}^3\) when the pressure changes by an amount \(\Delta P\), expressed in atmospheres.
Most wells are perforated in such a way that the openings in the casing are distributed uniformly within the productive part of the formation. In such cases, two zones are distinguished in the drainage regions of wells: the zone of influence of the openings and the zone of plane-radial flow, the outer boundary of which will be determined by the radius of influence of the well \(R\). The drawdown on the formation within this zone can be expressed from Dupuit’s formula
\[ P_x=P_s\ln\frac{R}{x}, \tag{2} \]
where \(P_x\) is the drawdown on the formation at a given point, expressed in atm, \(R\) is the radius of influence of the well in cm, \(x\) is the distance from the well axis to the given point in cm, \(P_s\) is the drawdown on the formation at the boundary of the zone of plane-radial flow with the zone of influence of the openings, that is, at such a distance from the well axis which is determined by the quantity \(x_0\), expressed by the relation
\[ x_0=r+y+s, \tag{3} \]
where \(r\) is the radius of the well before running the casing string, in cm; \(y\) is the depth (length) of the channels at the openings, in cm; and \(s\) is the radius of influence of the openings.
The last of these parameters represents one half of the average distance between openings. At different perforation densities it is determined in different ways. Namely, at high perforation density, when the formation thickness per one opening, \(h/n\), does not exceed \(2.55r\), i.e., in most cases, it is calculated by the formula
\[ s=0.5\sqrt{2\pi r\,\frac{h}{n}}, \tag{4} \]
where \(h\) is the length of the perforation interval (and in the present case also the formation thickness), in cm; \(n\) is the number of openings in the casing of the given well within the limits of the formation under study.
At low well perforation density, when the formation thickness per one opening, \(h/n\), exceeds \(2.55r\), the radius of influence of the openings is calculated by the formula
\[ s^{*}=0.5\sqrt{\frac{h^{2}}{n^{2}}+0.96\pi^{2}r^{2}} . \tag{5} \]
As for the quantity \(P_s\)—the drawdown on the formation at the surface bounding the zone of influence of the openings—then in the case considered (with complete perforation of the formation) it may be determined from the relation
\[ \frac{P_s}{P_{\mathrm{pl}}-P_z}=\frac{G_R}{G_s+G_R}, \tag{6} \]
where \(P_{\mathrm{pl}}\) and \(P_z\) are the formation and bottomhole pressures in atm, and their difference is the total drawdown on the formation under the given regime; \(G_R\) is the geometric characteristic of the plane-radial-flow zone, and \(G_s\) is the geometric characteristic of the zone of influence of the openings. The first of these is calculated by the formula
\[ G_R=\frac{n}{h}\ln\frac{R}{r+y+s+\delta_{\max}}, \tag{7} \]
where \(\delta_{\max}\) is the largest component of the well perforation incompleteness (in the case of one-component incompleteness, the distance from the bottom or top of the formation to the nearest opening).
The formula for calculating the second of these characteristics depends on the perforation density. At high perforation density it is calculated by the formula
\[ G_s=\frac{1}{y}\ln\frac{(y+\lambda)s}{(y+s)\lambda} -\frac{0.25}{r+y}\ln\frac{s+y}{\lambda+y} +0.0625\,\frac{s-\lambda}{(r+y)^2}-\ldots . \tag{8} \]
At low perforation density, by the formula
\[ G_s^{*}=\frac{1}{y}\ln\frac{(y+\lambda)(y+4r)s}{(y+2r)(y+2s)\lambda} -\frac{0.25}{r+y}\ln\frac{2r+y}{\lambda+y} +\frac{2r-\lambda}{16(r+y)^2}+\ldots . \tag{9} \]
With incomplete perforation of wells, in addition to the zone of influence of the openings and the zone of plane-radial flow, within the drainage region of the well there is a zone of flow convergence caused by the incompleteness of the perforation. To determine the geometric characteristic of this zone, \(G_{\sigma}\), let us assume that some well penetrated the formation perpendicular to its bottom and top over the full thickness \(H\), but in the perforation process only some part of the given formation, \(h\), was opened. In this case, between the upper opening of the filter and the top of the formation, as well as between the lower opening of the filter and the bottom of the formation, unperforated sec-
$\delta_1$ and $\delta_2$, of which $\delta_2$ is less than $\delta_1$. Below, these quantities will be called the components of perforation incompleteness.
Let us isolate, around the perforated part of the borehole, a thin-walled body in the form of a cylinder, ending at both ends in hemispheres of radius $\rho$ and with the length of the cylindrical part equal to the length of the perforated part of the borehole $h$. Obviously, the surface of this body is expressed by the relation
\[ F=2\pi\rho h+4\pi\rho^2 . \tag{10} \]
The thickness of its walls will be equal to $d\rho$, and the initial expression for determining the geometric characteristic of the first part of the flow-convergence zone will have the form
\[ dG=\frac{n}{\rho(h+2\rho)}\,d\rho . \tag{11} \]
Integrating the left-hand side of expression (11) within the limits $0<G<G^*$, and the right-hand side within the limits $x_0<\rho<\delta_2$, we obtain
\[ G^*=-\frac{n}{h}\ln\frac{h+2\rho}{\rho}\bigg|_{x_0}^{\delta_2} \tag{12} \]
whence
\[ G^*=-\frac{n}{h}\left(\ln\frac{h+2\delta_2}{\delta_2}-\ln\frac{h+2x_0}{x_0}\right) \tag{13} \]
or, after the corresponding transformations,
\[ G^*=\frac{n}{h}\ln\frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0}. \tag{14} \]
To account for the remaining part of the space characterizing the hydrodynamic imperfection of the well with respect to the degree of penetration of the formation, let us isolate a second thin-walled cylinder-like body, differing from the first in that it ends not in two hemispheres but in one hemisphere. Its surface is expressed by the relation
\[ F=2\pi\rho(h+\delta_2)+2\pi\rho^2, \tag{15} \]
and the initial expression for determining the geometric characteristic of the considered part of the flow-convergence zone will have the form
\[ dG=\frac{n}{\rho(h+\delta_2+\rho)}\,d\rho . \tag{16} \]
Integrating the left-hand side of this expression within the limits $0<G<G^{**}$, and the right-hand side within the limits $\delta_2<\rho<\delta_1$, we obtain
\[ G^{**}=-\frac{n}{h+\delta_2}\ln\frac{h+\delta_2+\rho}{\rho}\bigg|_{\delta_2}^{\delta_1}. \tag{17} \]
or, after the corresponding transformations,
\[ G_\delta^{**}=\frac{n}{h+\delta_2}\ln\frac{(h+2\delta_2)\delta_1}{(h+\delta_1+\delta_2)\delta_2}, \tag{18} \]
Finally, adding expression (14) to expression (18), we find the complete geometric characteristic of the flow-convergence zone for two-component perforation incompleteness
\[ G_\delta^{\mathrm{II}}=\frac{n}{h}\ln\frac{(h+2x_0)\delta_2}{(h+2\delta_2)x_0} +\frac{n}{h+\delta_2}\ln\frac{(h+2\delta_2)\delta_1}{(h+\delta_1+\delta_2)\delta_2}. \tag{19} \]
The drawdown on the formation at the outer boundary of the zone of influence of the openings $P_s$ in a well hydrodynamically imperfect with respect to the degree of penetration of the formation is expressed by the relation
\[ \frac{P_s}{P_{\mathrm{pl}}-P_z}=\frac{G_\delta+G_R}{G_S+G_\delta+G_R}, \tag{20} \]
To now construct the initial expression for determining the radius of influence of wells, let us isolate, within the body of the producing formation in the zone of plane-radial flow, a thin-walled cylinder whose axis will coincide with the axis of the well, and whose radius is equal to some variable \(x\). The thickness of its walls will be equal to \(dx\), the height \(H\) (formation thickness), and the volume of the walls will be expressed by the relation
\[ V^1 = 2\pi Hx\,dx. \tag{21} \]
Obviously, the amount of oil that will be released from the walls of this cylinder will be expressed by the following relation
\[ dV = 2\pi Hm\alpha P_s\, x \ln \frac{R}{x}\, dx, \tag{22} \]
where \(m\) is the porosity, or the porosity coefficient, of the formation in fractions of unity.
Integrating the left-hand side of this expression over the limits \(0<V<V_p\), and the right-hand side over the limits \(x_0<x<R\), we obtain
\[ V_p = 2\pi Hm\alpha P_s \int_{x_0}^{R} x(\ln R-\ln x)\,dx, \tag{23} \]
whence
\[ V_p = 2\pi Hm\alpha P_s \left[ \frac{x^2}{2}\ln R - x^2\left(\frac{\ln x}{2}-\frac{1}{4}\right) \right]_{x_0}^{R}. \tag{24} \]
\[ V_p = \pi Hm\alpha P_s \left( R^2\ln R - x_0^2\ln R - R^2\ln R + x_0^2\ln x_0 + \frac{R^2}{2} - \frac{x_0^2}{2} \right) \tag{25} \]
or, after the corresponding transformations,
\[ V_p = \pi Hm\alpha P_s \left( \frac{R^2}{2} - \frac{x_0^2}{2} - x_0^2\ln \frac{R}{x_0} \right). \tag{26} \]
Here it is necessary to note that the quantity \(x_0\) is always two to three orders of magnitude smaller than the quantity \(R\), and their squares differ from one another by four to six orders of magnitude. Therefore, in practical calculations the value \(x_0\) may be neglected. As a result, the formula of interest to us assumes the simpler form
\[ R^2 = \frac{2V_p}{\pi Hm\alpha P_s}, \tag{27} \]
where \(V_p\) is the amount of oil withdrawn from the well during its testing and measured under reservoir conditions in \(\text{cm}^3\).
Expression (27), without any substantial detriment to accuracy, may also be used to determine the radius of influence of those wells in whose drainage areas there are zones of flow constriction due to incomplete perforation. This is connected with the fact that the shape of the body encompassed by the drawdown funnel will be sufficiently close to a cylinder.
REFERENCES
- V. A. Dolitskii, E. F. Frolov, N. A. Eremenko. Methodology of exploratory work for oil and gas. “Nedra,” 1964.
- V. N. Mamuna, G. F. Trebin, V. V. Ulyaninskii. Experimental study of reservoir oils. GOSINTI, 1960.