On the Question of the Specific Capacities of Wells

V. A. Nudner

Taking part in the work of the “Kuzbassuglegeologiya” trust connected with the use of groundwater for solving problems of water supply to the industrial centers of Kuzbass, the author of the present paper has more than once had occasion to encounter the concept of the specific capacity of wells and its use for practical purposes. It turned out that the very concept of the specific capacity of a groundwater well is not sufficiently definite, and that its use in solving practical problems is difficult. These circumstances compelled us to undertake an investigation of the concept of “specific capacity of a well.” The results of this investigation are set forth in the present article.

As is known, the specific capacity of a well is understood to mean the capacity of the well per unit lowering of the water level in it caused by pumping.

Let us find an expression for the specific capacity of a perfect groundwater well under conditions of the linear law of filtration. For this purpose we shall use the well-known formula for the capacity of a well in the form:

\[ Q=\frac{\pi K}{\ln \frac{R}{r_c}}(2H-S)S \tag{1} \]

Here \(K\)—coefficient of filtration,

\(R\)—radius of influence,

\(r_c\)—radius of the well,

\(H\)—thickness of the aquifer,

\(S\)—lowering of the level in the well caused by pumping.

Let us denote the aggregate of the terms on the right-hand side of formula (1) that are constant for the given well and the given aquifer by \(A\)

\[ \frac{\pi K}{\ln \frac{R}{r_c}}=A, \tag{2} \]

then

\[ Q=A(2H-S)S . \tag{3} \]

Hence

\[ q_c=\frac{Q}{S}=A(2H-S). \tag{4} \]

Here \(q_c\) is the specific capacity of the well.

In order to understand the essence of this quantity, let us find the gradient of the discharge, i.e., take the first derivative of \(Q\) with respect to \(S\) in equation (3)

\[ q=\frac{dQ}{dS}=2A(H-S). \tag{5} \]

Let us denote

\[ H-S=h_c, \tag{6} \]

where \(h_c\) is the thickness of the water layer in the well when the level is lowered by \(S\).

Then

\[ q=2Ah_c. \tag{7} \]

It follows from this that the discharge gradient of a water-table well is directly proportional to the thickness of the water layer in it.

Let us find the discharge gradient \(\delta_c\) per unit thickness of the water layer in the well:

\[ \delta_c=\frac{q}{h_c}=2A=2\frac{\pi K}{\ln \dfrac{R}{r_c}} . \tag{8} \]

We shall call the quantity \(\delta_c\) the unit capacity of the well. Formula (8) shows that the unit capacity of a perfect water-table well under conditions of the linear filtration law is, for a given well and a given aquifer, a constant quantity, directly proportional to the filtration coefficient and inversely proportional to the logarithm of the ratio of the radius of influence to the radius of the well.

Let us now find an expression for the specific capacity and the discharge gradient of an artesian well, also under conditions of the linear filtration law. For this we shall use the well-known formula.

\[ Q=2\pi Km\,\frac{S}{\ln \dfrac{R}{r_c}} . \tag{9} \]

Here \(m\) is the thickness of the aquifer. The remaining notation is as before.

As is easy to see, in this case the discharge gradient is the specific capacity

\[ q=q_c=2\frac{\pi K}{\ln \dfrac{R}{r_c}}\,m . \tag{10} \]

Keeping formula (2) in mind, we may write

\[ q = q_c = 2Am. \tag{11} \]

The specific capacity of an artesian well will be

\[ \delta_a = \frac{q}{m} = 2A. \tag{12} \]

Comparing formulas (8) and (12), we find that

\[ \delta_c = \delta_a = \delta, \tag{13} \]

i.e., the specific capacity of a well does not depend on which horizon—confined or with a free surface—is being exploited by the well.

Let us consider formulas (7) and (11).

It follows from (11) that the discharge gradient and the specific capacity of an artesian well are constant because the thickness of the water layer in the well, which in this case is the thickness of the aquifer, remains constant.

It follows from (7) that the discharge gradient of a groundwater well is not constant and changes as the drawdown changes, since in this case the thickness of the water layer in the well changes.

Thus, formula (7) is a general formula both for an artesian well and for a groundwater well; but in the first case \(h_c\) is a constant quantity (the thickness of the aquifer), independent of the drawdown, whereas in the second case \(h_c\) depends directly on \(S\).

It is obvious that formula (7) is also valid for the case when a confined flow moving toward the well, at some distance from it, as a result of a large drawdown of the water level in the well, changes into a flow with a free surface.

Let us now determine the maximum, minimum, and mean discharge gradients of a groundwater well. From formula (7) it follows that for \(h_c = H\)

\[ q_{\max} = 2AH, \tag{14} \]

for \(h_c = 0\)

\[ q_{\min} = 0, \tag{15} \]

As for the mean discharge gradient, it obviously depends on the magnitude of the drawdown \(S\) at which it is determined. Thus, for \(S = H\), bearing in mind that the discharge gradient changes with change in drawdown according to the law of a straight line,

\[ q_{\mathrm{avg},\,n} = \frac{q_{\max} + q_{\min}}{2} \tag{16} \]

\[ q_{\mathrm{avg},\,n} = \frac{q_{\max}}{2}. \tag{17} \]

Or, taking formula (14) into account,

\[ q_{\mathrm{avg},\,n} = AH, \tag{18} \]

For an arbitrary drawdown \(S\), the mean discharge gradient is

\[ q_{cp_s}=\frac{q_{\max}+q}{2} \tag{19} \]

Using formulas (7) and (14), formula (19) can be represented in the following form:

\[ q_{cp_s}=\frac{2AH+2Ah_c}{2} \quad \text{or} \]

\[ q_{cp_s}=2A\frac{H+h_c}{2}. \tag{20} \]

Denote

\[ \frac{H+h_c}{2}=h_{cp}, \]

then

\[ q_{cp_s}=2Ah_{cp}. \tag{21} \]

Recalling formulas (8) and (13), we obtain

\[ q_{cp_s}=\delta h_{cp}. \tag{22} \]

Let us now return to the specific discharge of an unconfined well. Formula (4) can be represented, replacing in it the drawdown \(S\) by the difference \(H-h_c\), in the following form:

\[ q_c=A(H+h_c). \tag{23} \]

Comparing formulas (20) and (23), we find that

\[ q_c=q_{cp_s}, \tag{24} \]

i.e., the specific discharge of an unconfined well at a drawdown of level \(S\) is equal to the mean discharge gradient at the same drawdown. Such, consequently, is the content of the concept of the “specific discharge” of an unconfined well.

Using formulas (4), (22), and (24), one can write the following formula for the discharge of a well:

\[ Q=\delta h_{cp}S. \tag{25} \]

Formula (25) can be used both for an artesian well and for an unconfined well, and also for the case when the confined flow in the vicinity of the well changes over (at large drawdowns of the level) into flow with a free surface.

In the first case,

\[ h_{cp}=m, \tag{26} \]

in the second—

\[ h_{cp}=\frac{H+h_c}{2}, \tag{27} \]

in the third—

\[ h_{cp}=\frac{m+h_c}{2}. \tag{28} \]

Using formula (25), one can obtain the equation of the well-yield curve.

For this, let us find the maximum yield \(Q_{\max}\) of the well. In this case, evidently,

\[ S=H-S_{\max}. \]

\(S_{\max}\) is here the maximum possible drawdown,

\[ h_{cp}=\frac{S_{\max}}{2}, \]

\[ Q_{\max}=\delta \frac{S_{\max}^2}{2}. \tag{29} \]

Let us take the ratio \(\dfrac{Q}{Q_{\max}}\)

\[ \frac{Q}{Q_{\max}}= \frac{\delta h_{cp} S}{\delta \dfrac{S_{\max}^2}{2}} \]

Keeping in mind that

\[ h_{cp}=\frac{H+h_c}{2}=\frac{S_{\max}+h_c}{2}, \]

after simple transformations we obtain

\[ Q=Q_{\max}\frac{S_{\max}+h_c}{S_{\max}^2}S. \tag{30} \]

The obtained equation of the yield curve of a groundwater well is more convenient for practical use than the well-known Thiem equation

\[ (S_{\max}-S)^2=\frac{S_{\max}^2}{Q_{\max}}(Q_{\max}-Q). \tag{31} \]

Tomsk Polytechnic Institute
named after S. M. Kirov