Before pumping, the water level in the well stands up to the same elevation as the water table or piezometric surface depending on the type of aquifer. When pumping starts, the water is removed from the aquifer surrounding the well, and in and around; the well the water table or piezometric surface is lowered and assumes the shape of an inverted cone which is known as cone of depression. The area of the base of this cone is known as the area of influence, because it is this area which gets affected by the pumping of the well.

The boundary of the area of influence is known as the circle of influence. The radius of the circle of influence is known as the radius of influence. Further at any point the difference in elevation of the water table or piezometric surface before and after pumping is known as drawdown. The maximum drawdown occurs at the well and it decreases with increase in the distance from the well. The variation in drawdown with distance from the well is shown by a drawdown curve. (see Figs. 4.28 and 4.29).

The analysis of radial flow of ground water towards a well was first proposed by Dupuit (1863) and later modified by Thiem (1906).

The Dupuit-Theim theory is based on the following assumptions:

(i) The aquifer is homogeneous, isotropic, of uniform thickness and of infinite areal extent.

(ii) The well penetrates and receives water from the entire thickness of the aquifer.

(iii)The pumping has been continued for a sufficiently long time at a uniform rate so that an equilibrium stage or a steady flow condition has been reached.

(iv) The coefficient of transmissibility is constant at all places and at all times.

(v) The flow lines are radial and the flow of groundwater is horizontal.

(vi) Flow is laminar and Darcy’s law is applicable. However, the hydraulic gradient may be represented by tan 0 instead of sin 0 where 0 is the angle between the hydraulic grade line and the horizontal.

(vi) The well is infinitely small with negligible storage and all the pumped water comes from the aquifer.

On the basis of these assumptions the radial flow equations which relate the well discharge to drawdown for steady flow condition have been derived for wells completely penetrating a confined aquifer and an unconfined aquifer as indicated below.

#### Steady State Flow to Wells in Confined Aquifer (i.e., Artesian Wells or Pressure Wells):

Figure 4.28 shows a well of radius r fully penetrating a confined aquifer. Let b be the thickness of the aquifer measured between the top and bottom impervious strata, and H be the height of the initial piezometric surface measured above the impermeable strata at the bottom.

When the well is pumped at a constant rate Q for a long time so that the water level in the well has been stabilised then the drawdown curve as shown in Fig. 4.28 is developed. At this stage let h be the depth of water in the well measured above the impermeable strata at the bottom. Further let R be the radius of influence.

Let (x, y) be the coordinates of any point P on the drawdown curve with respect to origin O at the centre of the well at its bottom, if a vertical cylindrical surface passing through point P and surrounding the well located at its centre is considered then the area of the portion of the cylindrical surface which is lying within the aquifer is equal to (2πxb). Further if (dy/dx) is the hydraulic gradient at P, then from Darcy’s law the rate of flow of water through this portion of the cylindrical surface is equal to [k(dy/dx)2πxb] which by continuity is also equal to the well discharge and hence-

Further for a confined aquifer since the coefficient of transmissibility T = kb, Eqs. 4.11 and 4.12 become-

Again as indicated below the use of R can be avoided if the observation wells are available.

As shown in Fig. 4.28 let there be two observation wells at radial distances r1 and r2 and the depth of water in them be h1 and h2 respectively. Integrating Eq. (i) between the limits, at x = r1, y = h1 at the observation well No. 1 and at x = r2, y = h2 at the observation well No. 2, the following equation may be obtained which does not involve R.

#### Steady State Flow to Wells in Unconfined Aquifer (i.e., Gravity Wells or Water Table Wells):

Figure 4.29 shows a well of radius r completely penetrating an unconfined aquifer. Let H be the thickness of the aquifer measured from the impermeable strata to the initial level of the water table.

When the well is pumped at a constant rate Q for a long time so that the water level in the well has been stabilised, i.e., an equilibrium stage or a steady flow condition has been reached, then the drawdown curve as shown in Fig. 4.29 is developed. At this stage let h be the depth of water in the well measured above the impermeable strata. Further let R be the radius of influence (or the radius of inappreciable or zero drawdown) measured from the centre of the well to a point where the drawdown is inappreciable.

Considering the origin at a point O at the centre of the well at its bottom, let the coordinates of any point P on the drawdown curve be (x, y). If a vertical cylindrical surface passing through point P and surrounding the well located at its center is considered then the area of the portion of cylindrical surface which is lying within the aquifer below point P is equal to (2πxy). Further if (dy/dx) is the hydraulic gradient at P then from Darcy’s law the rate of flow of water (or discharge)

Equation 4.21 is similar to the one derived for a confined aquifer.

These equations can however be used only if the radius of influence R is known. In practice, the selection of the radius of influence R is approximate and arbitrary, but the variation in Q is small for a wide range of R. The values of R in general fall in the range of 150 to 300 metres.

Alternatively, R may be computed from the following approximate expression given by Sichardt:

R = 3000 s √k …(4.22)

in which

R = radius of influence in metres;

s = drawdown at the well in metres; and

k = coefficient of permeability in metres per second

However, as indicated below the use of R can be avoided if the observation wells are available.

As shown in Fig. 4.29 let there be two observation wells at radial distances r1 and r2 and the depths of water in them be h1 and h2 respectively. Integrating both sides of Eq. (ii) between the limits at x = r1 y = h1 at the observation well No. 1 and at x = r2, y = h2 at the observation well No. 2, the following equation may be obtained which does not involve R.

#### Seepage Face (Or Surface of Seepage) and Free Surface Curve:

(a) Seepage Face (Or Surface of Seepage):

When a gravity or water table well is being pumped the drawdown curve (or lowered water table profile or free surface curve) that is actually developed lies above the theoretical drawdown curve obtained on the basis of Dupuit-Thiem theory as shown in Fig. 4.30.

This is so because in the Dupuit-Thiem theory the flow of water in the aquifer is assumed to be horizontal, whereas the actual velocities of the same magnitude (given by Darcy’s law) have a downward vertical component so that a greater saturated thickness is required for the same discharge. Further at the well boundary a discontinuity in flow forms because no consistent flow pattern can connect a water table directly to the free water surface in the well.

The water table actually approaches the well boundary tangentially above the water surface in the well and forms a seepage face as shown in Fig. 4.30. The seepage face (or surface of seepage) is that boundary where the seepage leaving the flow region enters a zone free of both liquid and soil. The pressure on this surface is both constant and atmospheric, and the surface is neither an equipotential line nor a streamline.

The length of the seepage face m is approximately given by the following empirical formula:

(b) Free Surface Curve:

When a gravity or water table well is being pumped the free surface curve (or lowered water table profile) that is actually developed does not coincide with the theoretical drawdown curve obtained on the basis of Dupuit-Thiem theory.

The free surface curve that is actually developed has been formulated empirically by Babbitt and Caldwell as:

where

k = coefficient of permeability;

H = thickness of aquifer (or the height of the initial water table) measured above the impermeable strata;

hx = height above the impermeable strata of any point on the actual free surface at a radial distance r from the centre of the well (see Fig. 4.30);

R = radius of influence; and

Cx = a correction factor.

The value of the correction factor Cx to be used in Eq. 4.27 depends on the ratio(rx/R), and it may be obtained from Fig. 4.31 which is a plot of Cxv/s(rx/R).

It has, however, been found that the value of Cx may be approximated as-

Cx = 0.3 log 10(R/rx) …(4.28)