SOME REMARKS ON THE STATISTICAL METHOD FOR CALCULATING OIL RESERVES

L. A. PUKHLYAKOV, M. V. SAMOILOVA

(Presented by the scientific seminar of the departments of combustible minerals)

As is known, the statistical method for calculating oil reserves is applied only in those cases when no measures are being carried out to intensify production, as a result of which the average daily flow rate for each well is constantly decreasing. Calculation by this method begins with determining the coefficient of change in production, \(K_{\mathrm{i}}\), understood as the ratio of the average daily production for a certain group of wells in the subsequent month \(q''\), for example February, to the average daily production in the preceding month \(q'\), for example January, of one and the same year:

\[ K_{\mathrm{i}}=\frac{q''}{q'} . \tag{1} \]

The calculation uses average daily flow rates for all months for all wells. In order to cover such a large amount of factual material, correlation tables are compiled. Correlation tables are divided into ordinary and logarithmic ones. At present only the latter are used, and on this basis the entire methodology for determining the coefficient \(K_{\mathrm{i}}\) is founded.

Logarithmic correlation tables are constructed as follows. On the left-hand side of the page are entered the mean logarithms of the average daily well flow rates for the preceding month; to the right, next to them, the limits of the intervals of these flow rates in logarithms; still farther to the right, the limits in natural numbers (Table 1). In an analogous way, at the very top of the table are given the mean logarithms of the average daily flow rates for the subsequent month; next to them, the limits of the intervals (groups) of these flow rates in logarithms; and still lower, the limits in natural numbers. It is the determination of the latter that must first of all be emphasized.

The point is that the mean logarithms for the intervals of the table and the limits in logarithms are chosen so that they are convenient to work with. For example, as mean logarithms one may choose 2.1; 2.0; 1.9; 1.8, etc., and as limits in logarithms 2.15; 2.05; 1.85; 1.75, etc.

It would seem that there is nothing simpler than to take antilogarithms for these quantities and enter them in the table. However, the authors who set forth this method [1, 3, 4] become too carried away with rounding. Such an approach proves justified if the gaugers will round the obtained

Table 1

Correlation table

results obtained by them to whole numbers. But what will happen if, for the purpose of a more accurate calculation, more accurate measurements of well flow rates are organized in the field? Distortions of the results will occur. Therefore, in Table 1 appended to this work, the values of the interval limits in numbers are given on the basis of G. Vega’s tables of seven-place logarithms [2], with accuracy to the fifth decimal place. This will make it possible to enter into this table flow rates determined with accuracy to tenths of a ton.

The correlation table should be filled in as usual. Namely, on the basis of the magnitude of the preceding flow rate one should choose the row whose interval limits are given on the left-hand side, and on the basis of the magnitude of the following flow rate—the column of that interval whose limits are given in the upper part of the table. Suppose that the average daily flow rate of a certain well in a certain year was, in January, 46 t/day, in February—45 t/day, in March—44 t/day, in April—43 t/day, and in May 42 t/day. For the first pair of flow rates (46—45) we choose the row with mean logarithm 1.7 (limits in numbers 56.234—44.668) and the vertical column with the same mean logarithm. At their intersection (in the cell belonging to both of them) we put a point. For the second pair of flow rates (45—44) the horizontal row remains the same, while the vertical column is different (the value 44 falls within the limits 44.668—35.481). The following pairs of flow rates, 44—43 and 43—42, fall into one cell—the cell with mean logarithms 1.6.

After the distribution is completed, the number of points in all cells is counted, and on the basis of this count, for each preceding flow rate, the mean logarithms of the following flow rates are determined. The calculation is performed by the formula

\[ x=\frac{x_1\cdot n_1+x_2 n_2}{n_1+n_2}. \tag{2} \]

For example, for the preceding flow rate with mean logarithm 1.7, it is expressed by the relation

\[ x=\frac{1.7\cdot72+1.6\cdot8}{72+8}=1.6900. \tag{2,a} \]

Finally, on the basis of the data obtained, the coefficient of change in production is determined. Different authors propose carrying out this operation in different ways; however, not all the proposed methods are sufficiently perfect. For example, A. M. Agadzhanov and M. I. Maksimov [1] propose first determining, from the logarithms, the values of the preceding and following flow rates in natural numbers, then plotting the former along the vertical axis and the latter along the horizontal axis, and after that connecting the points obtained with a smoothed curve. The tangent of the angle of inclination of this curve at different sections will give the coefficient of change in production at different flow rates.

However, here it is necessary to correct the third and fourth digits after the decimal point, whereas the graph makes it possible to correct only the first and second digits after the decimal point. In this connection, it is recommended here to construct a graph not of the logarithm of the mean following flow rate, but of its shortfall to the logarithm of the preceding flow rate, or of the logarithm of the coefficient of change in production. This is done in the following manner.

First of all, one more vertical column is introduced into the table—the difference of the logarithms of the mean following and preceding flow rates, which represents the common logarithm of the coefficient of change in production. Then a graph is constructed, along whose horizontal axis...

...of which the logarithms of the preceding flow rate are plotted, and along the vertical axis—its difference with the logarithm of the subsequent flow rate (Fig. 1).

\[ \tau = \lg q'' - \lg q'. \]

Fig. 1. Graph of the dependence of the difference between the logarithms of the mean subsequent and preceding flow rates (the decimal logarithm of the production-change coefficient) on the logarithm of the preceding flow rate

If relation (1) were constant or nearly constant, then all points on the graph \(\tau = f(\lg q')\) would lie on one straight line, and this straight line would be parallel to the abscissa axis of this graph. In that case the relation

\[ q = q_0 \cdot K_i^t, \tag{3} \]

would hold, where \(q_0\) and \(q\) are the average daily flow rates, respectively, for the initial month and for some month in the future; \(t\) is the time between the midpoints of these months (in months). In deriving the final formula of the statistical method, one proceeds from the validity of this relation. Meanwhile, as is evident from the graph presented (Fig. 1), the points of the differences of the logarithms of the flow rates do not lie on a straight line parallel to the abscissa axis. Therefore the data obtained are smoothed by a moving curve, and relation (3) is regarded as valid only within individual groups of wells, determined by the mean flow rates of the correlation table. In the present case such flow rates will be 126, 100, 79.5, 63.1, 50.1, 39.8, 25.1, 20.0, 15.8, 12.6, 10.0, and 2.0 t/day (as the lower limit of the last interval, the minimum flow rate at which operation is possible is always taken). For these groups, the values of the production-change coefficient are determined on the basis of the position of the moving curve (Fig. 1). It is recommended that this be done in the following manner.

First of all, the limits of the intervals over which the calculation will be made are determined, as are the average production rates for these intervals and their logarithms (Table 2). Then, referring to the graph of the dependence of the dif-

Table 2

Calculation of reserves by individual groups of wells

ferences between the logarithms of the production rates and the logarithms of the preceding production rates (Fig. 1), the average values of these differences are determined; as noted above, they represent the decimal logarithms of the coefficients of change in production. To determine the latter from them, it is sufficient to reduce these logarithms to a positive mantissa and then, using tables of logarithms, determine this coefficient. For example, if the value of the obtained difference is equal to −0,0103, then after reduction to a positive mantissa the logarithm of the coefficient of change in production will be equal to \(\overline{1},9897\), and its natural value \(0,97656\). Finally, proceeding according to the method of A. M. Agadzhanov and M. I. Maksimov [1], M. A. Zhdanov [3, 4], and other authors, it is necessary to determine the deficiency of the coefficient of change in production to unity, in this case

\[ 1 - K_{\mathrm{i}} = 1 - 0,97656 = 0,02344 \tag{4} \]

and substitute the obtained value into the calculation formula

\[ Q = \frac{q_{0} - q_{n}}{1 - K_{\mathrm{i}}} - q_{0}, \tag{5} \]

where \(Q\) is the possible production from the given well (total reserves);
\(q_{0}\) and \(q_{n}\) are the average daily production rates for the initial and final months;
\(K_{\mathrm{i}}\) is the coefficient of change in production.

The proposed method provides for dispensing with the conversion of the decimal logarithms of the production-change coefficients to a positive mantissa and the determination of the natural values of this coefficient. Instead, it is proposed to determine the values of the natural logarithms of the production-change coefficient directly from the averaged difference of the decimal logarithms $\tau$ and to carry out the calculation by a new formula, which is derived as follows.

Let us express the average daily output for a given well for some month in the future as

\[ q=\frac{1}{30.4}\cdot\frac{dQ}{dt}, \tag{6} \]

where $dQ$ is the increment of total production in tons over the time $dt$ in months; and $30.4$ is the coefficient expressing the average number of days in a month. Then, substituting into it, instead of $q$, the right-hand side of expression (3), and moving the increment of monthly production to the left-hand side of the equation, we obtain

\[ dQ=30.4\cdot q_0\cdot K_i^t dt. \tag{7} \]

To bring this expression to a form convenient for integration, let us assume that

\[ \ln\frac{1}{K_i}=a, \tag{8} \]

in which case

\[ K_i=e^{-a}, \tag{8,a} \]

and expression (7) takes the form

\[ dQ=30.4\cdot q_0\cdot e^{-at}\cdot dt, \tag{9} \]

and, integrating it within the limits from $t_1=0$ to $t_2=t$, we obtain

\[ Q=30.4q_0\int_{t_1}^{t_2} e^{-at}\,dt, \]

\[ Q=30.4\frac{q_0}{-a}\left(e^{-at}-e^{-at_1}\right) \]

\[ Q=30.4\frac{q_0}{a}\left(e^{-at_1}-e^{-at_2}\right), \]

\[ Q=30.4\frac{q_0}{a}\left(1-e^{-at}\right). \tag{10} \]

Finally, introducing the value of the initial rate $q_0$ into the brackets and taking into account expressions (8,a) and (3), we obtain the total inflow for one well:

\[ Q=30.4\frac{1}{a}\left(q_0-q_0\cdot e^{-at}\right). \]

\[ Q=30.4\frac{1}{a}\left(q_0-q_0\cdot K_i\right), \]

\[ Q=30.4\frac{1}{a}(q_0-q). \tag{11} \]

Assuming that in the given group there is not a single well, but sev-

how many there are, and the number of them is equal to \(n\), we find the total oil reserves for the given group of wells:

\[ Q = 30.4 \frac{n}{\alpha} (q_0 - q), \tag{12} \]

The calculation by this formula is recommended to be carried out in the following order. First of all, the number of wells is determined whose initial flow rates fall within the limits of the first column of the correlation table (in the present case 141.2—112.2). Suppose that there is one such well. Then the initial and final flow rates are determined, within the limits of which the coefficient of change in production may be regarded as constant. Above, it was recommended to take as such flow rates the mean flow rates of the separate columns of the correlation table; in the present case these will be 126 and 100.

From these flow rates their mean value and its logarithm are determined (in the present case 2.05), and then, on the basis of the graph (Fig. 1), the corresponding common logarithm of the coefficient of change in production (in the present case \(-0.01035\)). From the common logarithm the natural logarithm of the reciprocal of the coefficient of change in production is determined. For this purpose the sign of the common logarithm is changed to the opposite one and it is multiplied by 2.3026 (or divided by 0.4343).

\[ \alpha = -2.3026 \tau. \tag{13} \]

For the interval under consideration this quantity will be equal to 0.0238. Finally, substituting the values obtained into expression (12), we obtain

\[ Q_1 = 30.4 \frac{1}{0.0238} (126 - 100) = 33200. \]

All these data are entered in the table (Table 2).

After determining the total production for the given group of wells in the first interval, one proceeds to the second (in the present case to the interval 100.0—79.5 t/day), that is, one determines how much oil these wells will yield within this interval. Together with them, wells are considered whose initial flow rate falls within the limits of the second column of the correlation table (in the present case 112.2—89.12). Suppose that in this group there is also one well. Thus, the calculated number of wells for the second interval is equal to two. The value of the natural logarithm \((\alpha)\) is determined in an analogous manner; in the present case it will be equal to 0.0226. As a result, the calculation for this interval will take the form:

\[ Q_2 = 30.4 \frac{2}{0.0226} (100 - 79.5) = 55200. \]

For the remaining intervals the calculation is carried out in an analogous manner, and in each new interval new wells are added to those already being calculated, their initial flow rates falling within the lower intervals of the correlation table (Table 2).

Calculation by formula (12), as compared with calculation by formula (5), first of all reduces the amount of computational work, since here there is no need to bring the common logarithm to a positive mantissa, to determine from it the natural value of the coefficient of change in production and its deficiency to unity. These three operations are replaced by one—determining the natural logarithm of the reciprocal of the coefficient of change in production from the common logarithm, that is, by multiplying it by 2.3026. And this, besides reducing labor costs, also reduces...

and the probability of error, which is no less important. Finally, of some significance is also the fact that the need to consult logarithm tables one more time is eliminated.

REFERENCES

1. A. M. Agadzhanov, M. I. Maksimov. Petroleum Geology. Gostoptekhizdat, 1958.
2. G. Vega. Tables of Seven-Digit Logarithms. Geodezizdat, 1955.
3. M. A. Zhdanov, V. R. Lisunov, A. V. Velichko, D. A. Grishin. Calculation of Oil and Gas Reserves. Gostoptekhizdat, 1959.
4. M. A. Zhdanov. Oil-and-Gas Field Geology. Gostoptekhizdat, 1962.