Formulas for Determining the Reduced Well Radius and the Formation Thickness Reduced to a Given Well
Abstract Generated abstract
The paper derives practical formulas for characterizing hydrodynamically imperfect wells by comparing pressure drop expressions for imperfect and ideal wells. It first obtains an expression for the reduced well radius, defined as the radius of a perfect well with the same inflow, and illustrates that this value can become inconveniently small in field calculations. To avoid this difficulty, the author introduces and derives a formula for the formation thickness reduced to a given well, using the real well radius while preserving equivalent production behavior. An example from well 123 of the Olenye oil field yields a reduced thickness of 562.7 cm and an imperfection coefficient of 0.1521, supporting the use of reduced thickness in transmissibility estimates to account for well imperfection.
Full Text
Formulas for Determining the Reduced Well Radius and the Formation Thickness Reduced to a Given Well
L. A. Pukhlyakov
(Submitted by Professor A. V. Aksarin)
The reduced radius of a hydrodynamically imperfect well is understood as the radius of such a perfect well for which, all other conditions being equal, the inflow is equal to the inflow into the well under consideration. To derive a formula for this parameter, it is necessary to take the geometric parts of the pressure-drop formulas for inflow into these wells and equate them, and then determine the required quantity from the resulting relation. The formula for the pressure drop during inflow into a hydrodynamically imperfect well has the form:
\[ P_{\text{pl}}-P_{3}=\frac{Q_{\text{pl}}\mu}{2\pi kh}\left(G_s+G_{\delta}+G_R\right), \tag{1} \]
where \(P_{\text{pl}}\) and \(P_3\) are the reservoir and bottom-hole pressures, in atm; \(Q_{\text{pl}}\) is the inflow into the well of oil or another liquid under reservoir conditions, in \(\text{cm}^3/\text{s}\); \(\mu\) is the viscosity of the oil or other liquid under reservoir conditions, in cP; \(k\) is the formation permeability, in darcies; \(h\) is the number of holes in the casing string (dimensionless); \(G_s\), \(G_{\delta}\), 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, in \(1/\text{cm}\). Formulas for determining these quantities are given in [1]. However, when working with them it must be borne in mind that, to determine the radius of influence of the well, one should use the formula
\[ \frac{R^2}{\ln R-\ln x_0}=\frac{2V}{\pi m\alpha H\Delta P_s}, \tag{2} \]
where \(V\) is the volume of oil or other liquid withdrawn from the well during its test in the first operating regime, in \(\text{cm}^3\); \(m\) is the formation porosity as a fraction of unity; \(\alpha\) is the compressibility of the oil or other liquid, in \(1/\text{atm}\); \(\Delta P_s\) is the drawdown on the formation at the outer boundary of the zone of influence of the holes, in atm; \(x_0\) is the radius of influence of the filter, in cm; \(H\) is the formation thickness, in cm.
The formula for the pressure drop during inflow into a conventional perfect well, whose radius we consider as the reduced radius of the real well, will be as follows:
\[ P_{\text{pl}}-P_3=\frac{Q_{\text{pl}}\cdot\mu}{2\pi kH}\ln\frac{R}{r_{\text{pr}}}, \tag{3} \]
where \(r_{\text{pr}}\) is the reduced radius.
Comparing the geometric parts of formulas (1) and (3), we obtain
\[ \frac{1}{H}\ln\frac{R}{r_{\mathrm{pr}}} = \frac{1}{n}\left(G_s+G_\partial+G_R\right), \tag{4} \]
whence, after the corresponding transformations,
\[ \lg r_{\mathrm{pr}}=\lg R-\frac{H}{2.3026n}\left(G_s+G_\partial+G_R\right). \tag{5} \]
The values of the parameter under consideration may be very small. It is measured in millionths and billionths of a micron. As an example, let us consider the reduced radius of well 123 of the Olenye oil field in Tomsk Oblast, the initial data for which are given in [1]. These parameters are as follows: formation thickness 3700 cm, well radius \(r_{\mathrm{sk}}\)—9.5 cm, radius of influence of the well by the end of operation in the first regime 6978.6 cm, number of holes in the string 120, geometric characteristic of the zone of influence of the holes 1.0882 1/cm, geometric characteristic of the flow-constriction zone due to incomplete perforation 0.2752 1/cm, and geometric characteristic of the plane-radial-flow zone 0.0439 1/cm. Substituting all this into expression (5), we obtain
\[ \lg r_{\mathrm{pr}}=\lg 6978.6-\frac{3700}{2.3026\cdot 120}\cdot(1.0882+0.2752+0.0439), \tag{6} \]
whence the desired parameter proves to be equal to \(9.9791\cdot 10^{-16}\) cm, or about one billionth of a micron.
Naturally, it is very inconvenient to deal with such a quantity, and therefore it is recommended instead to use the formation thickness reduced to the given well. This term means the thickness of a formation penetrated by a hydrodynamically perfect well, which in diameter is equal to the hydrodynamically imperfect well under consideration and which, all other conditions being equal, has the same production rate as the well under consideration.
To derive the formula for this parameter, one may use expression (4), which was the starting point in deriving the formula for the reduced radius; however, in it, instead of the reduced radius \(r_{\mathrm{pr}}\), the real well radius \(r_{\mathrm{sk}}\) should be substituted, and instead of the penetrated formation thickness \(H\), the reduced one should be substituted. After this, by means of simple transformations, we obtain
\[ h_{\mathrm{pr}}= \frac{n\cdot \ln \dfrac{R}{r_{\mathrm{sk}}}} {G_s+G_\partial+G_R}. \tag{7} \]
This is the formula for determining the formation thickness reduced to the given well.
As an example, let us consider the determination of the formation thickness reduced to well 123 of the Olenye oil field. For this purpose, the data pertaining to this well are substituted into formula (7). As a result we obtain
\[ h_{\mathrm{pr}}= \frac{120\cdot \ln \dfrac{6978.6}{9.5}} {1.0882+0.2752+0.0439}, \tag{8} \]
whence, by the corresponding transformations, we obtain 562.7 cm.
The ratio of the reduced formation thickness to the true thickness is called the coefficient of well imperfection
\[ \hat{\eta}=\frac{h_{\mathrm{pr}}}{H}. \tag{9} \]
In the case under consideration this coefficient is 0.1521.
At present, the solution of many problems of subsurface hydrodynamics is carried through to the determination of the transmissibility, which is expressed by the ratio
\[ \frac{kh}{\mu}, \tag{10} \]
where \(h\) is the formation thickness. As this thickness, practicing researchers substitute the true formation thickness and obtain data that are not at all comparable with the results of core-material studies in laboratories.
The reason for this phenomenon lies in the fact that, when processing data on inflows into wells, as a result of which the transmissibility is obtained, the fact that the well is hydrodynamically imperfect is ignored. To account for well imperfection, it is sufficient to use the reduced thickness as the calculated formation thickness in the transmissibility.
The foregoing, as well as the fact that the reduced formation thickness is always commensurable with the true one, make this parameter a convenient means for expressing the hydrodynamic imperfection of wells.
REFERENCES
- L. A. Pukhlyakov. Determination of the permeability of formation Yu-1 of the Olenye oil field from inflows under steady-state conditions. Izvestiya TPI, vol. 288, 1976.