In this article we will discuss about interference among wells and flow in a well.
Interference among Wells:
If two or more wells are located so close to each other that when they are discharging their drawdown curves intersect then one well is said to interfere with the other. Observations indicate that due to the interference of wells the discharge of each individual well is decreased, though the total discharge from the wells is increased.
In Confined Aquifers (i.e., Artesian Wells or Pressure Wells):
(i) Two Wells Spaced a Distance B Apart:
ADVERTISEMENTS:
Figure 4.34 shows interference between two wells penetrating a confined aquifer. From the principle of superposition, the drawdown at any point in the area of influence caused by the discharge of several wells is equal to the sum of the drawdown caused by each well individually. Thus for the two wells having same discharge i.e., Q1 = Q2 the composite drawdown curve may be obtained from the individual drawdown curves as shown in Fig. 4.34.
Further if the two wells are spaced a distance B apart and have same radius r and drawdown (H-h) and discharge over the same period of time then with the help of the method of complex variables it can be shown that the discharge of each of the two wells is given by the expression
where R is the radius of influence of each well (R > B).
However, if there was only one well, then under the same drawdown the discharge would be given by Eq. 4.11 as
Comparing the discharge of one well of the two well system with the discharge of a single well we get
Since R > B; (R2/rB) > (R/r) and hence Q > Q1, i.e., the discharge of each well of a two well system is less than that of a single well.
ADVERTISEMENTS:
It can however be shown that 2Q1 > Q, thereby indicating that the total discharge of both wells of a two well system is more than that of a single well.
(ii) Three Wells in a Straight Line Spaced a Distance B Apart:
Let there be three wells penetrating a confined aquifer, each well being of radius r and the wells being spaced a distance B apart in a straight line. If all the three wells are discharging over the same period of time then for the same drawdown (H-h) the discharges Q1 and Q3 of the two outer wells are given by the expression-
(iii) Three Wells Placed in a Pattern of Equilateral Triangle and Spaced a Distance B Apart:
Let there be three wells penetrating a confined aquifer, each well being of radius r and the wells being placed in a pattern of equilateral triangle and placed a distance B apart. If all the three wells are discharging over the same period of time then for the same drawdown (H-h) the discharge of each of the three wells is given by the expression-
It can be readily shown that the discharge of one well of the three well systems described above is less than that of a single well.
ADVERTISEMENTS:
In Unconfined Aquifers (i.e., Gravity Wells or Water Table Wells):
(i) Two Wells Paced A Distance B Apart:
Let there be two wells penetrating an unconfined aquifer, each well being of radius r and the wells being spaced a distance B apart. If both the wells are discharging over the same period of time then for the same drawdown (H – h) the discharge of each of the two wells is given by the expression-
Again by comparing the discharge of one well of the two well system as given by Eq. 4.44 with the discharge of a single well as given by Eq. 4.18, it can be shown that for this case also the discharge of each well of a two well system is less than that of a single well.
(ii) Three Wells in a Straight Line Spaced a Distance B Apart:
Let there be three wells penetrating an unconfined aquifer, each well being of radius r and the wells being spaced a distance B apart in a straight line. If all the three wells are discharging over the same period of time then for the same drawdown (H-h) the discharges Q1 and Q3 of the two outer wells are given by the expression
(iii) Three Wells Placed in A Pattern of Equilateral Triangle and Spaced A Distance B Apart.
Let there be three wells penetrating an unconfined aquifer, each well being of radius r and the wells being placed in a pattern of equilateral triangle and spaced a distance B apart. If all the three wells are discharging over the same period of time then for the same drawdown (H- h) the discharge of each of the three wells is given by the expression-
In this case also it can be readily shown that the discharge of one well of the three well systems described above is less than that of a single well.
Artesian Gravity Well:
Sometimes in an artesian well (i.e., a well in a confined aquifer) due to high rate of pumping the water level in and around the well may fall below the top of the confined aquifer as shown in Fig. 4.35. In such a case, the flow pattern close to the well is similar to that for a gravity well (i.e., a well in an unconfined aquifer) whereas at distance farther from the well the flow is similar to that for an artesian well. This type of well is therefore known as artesian gravity well.
The discharge for such a well can be determined from the following expression developed by Muskat:
Flow in a Well:
A particular case of a partially penetrating well is the one in which a well just penetrates the top surface of a semi-infinite porous media. In this case the flow towards the well is spherical as shown in Fig. 4.37. As such the earlier derived equations which are based on the radial flow towards the well do not apply for this case.
The discharge Qs for such a well can be determined from the following expression:
where k is coefficient of permeability and the other terms are as shown in Fig. 4.37.
For the case of radial flow towards a well fully penetrating a confined aquifer the discharge Q is given by Eq. 4.11 as-
i.e., in this case the discharge of a spherical flow well is only about 3% of that of a radial flow well, and hence a spherical flow well is highly inefficient as compared to a radial flow well.
Unsteady Flow towards Wells— Non-Equilibrium Equation:
When a well penetrating an extensive aquifer is pumped at a constant rate then if the cone of depression does not vary with time, it may be assumed that steady- state flow exists within the aquifer.
The assumption that steady-state flow exists within the aquifer. However, actually the steady-state flow does not exist because as water is pumped from the well at a constant rate then since the storage within the aquifer is reduced, the head continues to decline as long as the aquifer is effectively infinite.
Although the rate of decline of head decreases continuously as the area of influence expands, but this results in the variation of the cone of depression with time and therefore unsteady or transient flow exists within the aquifer. In order to analyse the unsteady flow towards wells the following differential equation has been derived.
in which
h = height of the drawdown curve above the impervious strata at the bottom at a radial distance r from the centre of the well
S = storage coefficient; and
T = coefficient of transmissibility
Theis (1935) obtained a solution of this equation based on an analogy between groundwater flow towards a well and flow of heat towards a sink or point where heat is removed at a uniform rate just as water is removed from the well at a constant rate. Thus by assuming that the well is replaced by a mathematical sink of constant strength and considering the boundary conditions h = H for t = 0 i.e., before pumping; and h → H as r → ∞ for t > 0, the following solution is obtained-
Wenzel tabulated the values of W(u) for a wide range of u from 10-15 to 9.9.
Using Eqs. 4.57 and 4.58, the aquifer (or formation) constants S and T can be determined by measuring the change in drawdown with time in one or more observation wells when the test well is pumped at a constant rate Q which is also measured.
However, on account of mathematical difficulties it is impossible to obtain an exact analytical solution. As such several investigators have developed simpler approximate methods of solution which can be readily used to determine S and T for field purposes. One of these methods which is commonly used is Jacob method which is described below.
Jacob Method:
Jacob indicated that for small values of r and large values of t, u is small and hence after the first two terms all other terms of the series in Eq. 4.59 may be neglected. As such the drawdown s can be expressed by the following equation-
Equation 4.63 shows that s varies linearly with log t since all other terms are constant. By using Eq. 4.63 the values of S and T may be obtained as indicated below.
For determining the values of S and T by this method, besides a test or pumping well, one observation well is required. The test well is pumped at a constant rate Q which is measured. During the pumping of the test well the drawdown s is measured in the observation well at different instants of time t. The radial distance r of the observation well from the test well is also measured. The observed values of drawdown s are plotted against log t and a straight line passing through most of the plotted points is drawn as shown in Fig. 4.38.
Further in the observation well if s1 and s2 are the respective drawdowns at times t1 and t2 since pumping was started, then from Eq. 4.63 we have
Extrapolating the straight line of s versus log t plot to intersect the zero – drawdown axis permits the determination of S as indicated below:
Let t = t0 when s = 0 as shown in Fig. 4.38, i.e., t0 is the time since pumping started upto which the drawdown in the observation well is equal to zero. Then from Eq. 4.63, we have
Introducing the value of T obtained from Eq. 4.66 in Eq. 4.67, the value of S may be determined.
However, the straight line approximation for this method should be restricted to small values of u i.e., u < 0.01 to avoid large errors.