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The following article will guide you about how does water flow into a well.

**Flow into a Well****: **

The principle objective of ground water studies is to determine how much ground water can be safely withdrawn perennially from the aquifers in the area under study.

**This determination involves:**

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(i) Transmissibility and storage coefficients of the aquifers.

(ii) The lateral extent of each aquifer and the hydraulic nature of its boundaries.

(iii) Vertical leakage if any, i.e., vertical seepage of water from either above or below the aquifer being tested, for which the hydraulic characteristics of overlying and underlying beds should be known.

(iv) The effect of proposed developments on recharges and discharge conditions. Most of the above information can be obtained by conducting aquifer or pumping tests.

**In all these studies two different flow conditions are assumed: **

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(a) A steady state condition, i.e., when the flow is steady and the water levels have ceased to decline.

(b) A non-steady state condition, i.e., when the rate of flow through the aquifer is changing and the water levels are declining; water is taken from storage within the aquifer and the water level or piezometric head (in confined aquifer) is gradually reduced.

**For steady flow of water in homogeneous and isotropic media, Eq. (4.31): **

**Steady Radial Flow into a Well****: **

After prolonged pumping from the well, the drawdowns stabilise, when the cone of depression spreads to natural discharge and recharge areas, i.e., the steady state and steady shape conditions have been developed, Fig. 5.1 (a), (b).

R: Radius of influence

P.W.L: Pumping Water Level

1, 2: Observation Wells

dw: Dewatered Portion

Sc: Screened Portion

**(a) Water Table Conditions (Unconfined Aquifer), Fig. 5.1 (a):**

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The yield from the well is-

Applying Eq. 5.3 between the points of zero-drawdown (radius of influence R) and face of the well (r_{w}) on the drawdown curve-

Where T = transmissibility of the water table aquifer (= KH); Q = yield from the well; H = saturated thickness of the aquifer; h_{w} – depth of water in the well during pumping; r_{w} = radius of the well; R = radius of influence (usually assumed as 300 m) and H – h_{w} = s_{w} the resulting drawdown in the pumped well.

**(b) Confined Aquifer (Artesian Conditions), Fig. 5.1 (b): **

The yield from the well is-

If the drawdowns in the observation wells are s_{1} and s_{2}, then h_{2 }– h_{1} = s_{1} – s_{2}; also Kb = T, the transmissibility of the confined aquifer. Applying Eq. (5.6) to the points of zero- drawdown (radius of influence R) and the face of the well (r_{w}) on the drawdown curve-

Which is similar to Eq. (5.5). From Eq. (5.5), it follows that for the same drawdown in the pumping well, the yield is inversely proportional to log_{10 }R/r_{w}.

**The assumptions in the Thiem’s equations are: **

(i) Stabilised drawdown.

(ii) Constant thickness of the aquifer with constant permeability (isotropic).

(iii) Complete penetration of the well with 100% efficiency.

(iv) Radial flow into the well.

Equations (5.3) and (5.6) can be used with reasonable accuracy even when the water table or piezometric surface has an initial slope, provided that h_{1} and h_{2} are taken from wells lying in a straight line through the well being tested, and in the direction of the initial slope of the water table or piezometric surface, and that h_{1} and h_{2} are taken as the average values at a distance r_{1} and r_{2} on the upslope and downslope of the test well respectively. From Fig. 5.2 it is apparent that the circular area of influence associated with a radial flow pattern becomes distorted. The distance r_{1} should be great enough to extend beyond the immediate distortion of streamlines near the test well.

**Unsteady Radial Flow into a Well****: **

**Confined Aquifer: **

A well pumping at a constant rate from an extensive confined aquifer produces an area of influence which expands with time. Water is taken from storage within the aquifer as the piezometric head is reduced. In 1935, C.V. This developed the non-equilibrium equation some-times referred to as the nonleaky-artesian formula-

Where s = drawdown in the observation well at radius r; Q = pumping rate; T = transmissibility coefficient; t = time after pumping started; r = distance of the observation well from the pumping well; S = storage coefficient, dimensionless (5 × 10^{-5} to 5 × 10^{-3} for confined aquifers, 0.05 to 0.30 for water table aquifers); u= argument and W(u) = well function, dimensionless.

The plot of u verses W(u) on standard log paper is called the type curve, Fig. 5.3.

The aquifer constant S and T can be determined by observing the drawdowns at several observation wells at a particular instant or the drawdown in a single well over a long period of time, for a constant pumping rate from the main well, by drawing time-drawdown and distance- drawdown curves or from the match point of the type curve.

The pumping rate during the test is held constant by changing the setting of a valve in the discharge line and not by changing the pump speed. Fairly long time prediction drawdown can be made from S and T by the methods of Theis, modified Thesia (Jacob, Chow) and other investigators.

**The assumptions in the Theis equations are: **

(i) Radial flow into the well with no entrance losses (horizontal stream lines).

(ii) Instantaneous release of water from storage upon a lowering of the drawdown curve (no lag) when pumping, i.e., proportional to the rate of change of head.

(iii) All water comes from storage (no influent seepage or aquifer leakage).

(iv) Constant coefficient of storage (independent of time and pressure).

(v) Non-pumping piezometric surface horizontal.

(vi) Complete penetration of the aquifer.

(vii) Homogeneous, isotropic aquifer of uniform saturated thickness and infinite aerial extent.

(viii) Discharge from well is constant and radius of well is infinitesimal.

**Water Table Aquifer: **

In confined aquifers, water comes mainly by gravity drainage, which is not instantaneous. The storage coefficient reduces with time and ultimately becomes equivalent to the specific yield.

**Non-equilibrium equation can be applied to unconfined aquifers with the following limitations: **

(i) The drawdown should be small in relation to the saturated thickness.

(ii) The observation well should be at a distance of 0.2 to 0.6 times the saturated thickness of the aquifer.

(iii) The minimum time interval after start of pumping for the non-equilibrium equation to be applicable is-

t > 5 S_{y}H/K …(5.10)

For example, if S_{y} = 0.15, H = 30 m and K = 10^{5} lpd/m^{2} or 100 m^{3}/day/m^{2} or 100 m/day,

t > 5 (0.15 × 30)/100

> 0.23 day, or 5.5 hr

Limitations- (ii) and (iii) are suggested by Boulton.

(iv) Jacob’s correction for very thin aquifers under water table conditions. When water is withdrawn from unconfined aquifer the transmissibility decreases as the aquifer is dewatered and the drawdown is more than that in confined aquifers of equivalent initial transmissibility. Jacob (1944) has shown that adjustment can be made for the effect of the dewatering with the following equation-

s’ = s – s^{2}/2H …(5.11)

Where s’ = drawdown in an equivalent confined aquifer; s = drawdown observed in water table aquifer and H = saturated thickness of the water table aquifer prior to start of pumping.

If s^{2}/2H < 0.003 m, the correction need not be applied.

Chow (1952) developed a method of solution of Theis equation by avoiding the curve matching technique and not being restricted to large values of t and small values of r’ as in the Jacob’s method.

Chow introduced the function-

F(u) = W(u)e^{u}/2.3 …(5.12)

The relation between (Fu), (Wu) and u is shown in Fig. 5.8 and in Table 5.9. F(u) is calculated from the time-drawdown data on an observation well. In Chow’s method the time- drawdown data on an observation well is plotted on a semi-logarithmic paper, an arbitrary point P is chosen on the plotted curve and a tangent to the curve is drawn at P. The drawdown (s_{p}) and the slope of the tangent to the curve at P (Δs_{p}), i.e., the drawdown difference per log cycle of time is read on the graph.

The value of F(u) is then calculated as-

F(u)= s_{p}/Δs_{p} …(5.13)

**Example 1:**

For u = 3 × 10^{-2}, F(u) = 1.33

The values of u and W(u) corresponding to F(u) are determined from Fig. 5.8, and then T and S are calculated from Eqs. (5.8) and (5.9). For the data given in Example 5.5, from Fig. 5.9, s_{p} = 0.49 m and Δs_{p} = 0.3 m. Then F(u) = 0.49/0.3 = 1.63, and from Fig. 5.8, W(u) = 3.8 and u = 0.011.

Hence, T = Q/4πs_{p} W(u) = 1150/4π (0.49) × 3.8 = 710 lpm/m or 1022 m^{2}/day

S = 4T tu/r^{2} = 4 × 0.71 × 14 × 0.011/(12.3)^{2} = 0.0029

As compared to T = 700 lpm/m and S = 0.0037 obtained by Jacob’s method.

**Law of Times****: **

**From the Eqs. (4.81), (4.83) and (4.85) it follows that: **

t_{1}/r_{1}^{2} = t_{2}/r_{2}^{2} = t_{3}/r_{3}^{2} = … = t_{n}/r_{n}^{2} …(5.14)

That is, the time of occurrence of zero-drawdown or equal drawdown vary directly as the square of the distances of the observation wells from the discharge well and are independent of the rate of pumping. It may also be noted from Eqs. (4.80) and (4.82) that Δs per log cycle of distance is twice Δs per log cycle of time. If from a time-drawdown curve on a particular well Δs per log-cycle of time is 1.5 m, Δs per log cycle of distance is 3 m; if the drawdown at a distance of 30 m from the pumping well is 1 m, the drawdown at 3 m from the pumping well is 4 m and at 0.3 m from the pumping well is 7 m.

Also from the time-drawdown curve, interpretation of aquifer boundaries can be made. Recharge boundaries cause the plot of observed drawdowns to diverge below the Theis-type curve, Fig. 5.3, and above the time-drawdown straight line plot of the Jacob method, Fig. 5.6. Barrier (impermeable) boundaries cause the plot of observed drawdowns to diverge above the Theis-type curve and below the time-drawdown straight line plot of the Jacob’s method. If a change in slope (divergence), depending on whether it is a recharge or barrier boundary is indicated at a time t on the plot, Fig. 5.10, the distance to the boundary from the discharging well is taken as nearly equal to x/2, where x is given by (from the theory of image wells, Fig. 5.24).

R^{2}/t_{0} = x^{2}/t …(5.15)

Where,

r = distance of the observation well from the discharging well;

T_{0} – time for zero drawdown taken from the time-drawdown curve (Jacob), and

t = time at which a change of slope is indicated.

**Hydraulic Boundary Condition: **

At t – 450 min, the plot of observed drawdowns diverges above the Jacob’s time-drawdown straight line plot (so that the drawdowns are reduced with continued pumping) indicating the existence of a recharge boundary and the distance to the recharge boundary is given by Eq. (5.15).

r^{2}/t_{0} = x^{2}/t

(0.3)^{2}/9.9 × 10^{-6} = x^{2}/450

x = 2100 m

The distance to the recharging image well x = 2100 m and the distance to the recharging boundary from the pumping well is x/2 = 1050 m (Fig. 5.10).

**Note:**

If time-drawdown data on an observation well is given, then x ≈ r, (Eq. 5.41), x/2 ≈ r_{1}/2. Jacob’s method gives fair predictions only when steady shape and steady state conditions are developed. This method is specially applicable to artesian conditions since it takes a long period to reach steady shape conditions in water table aquifers.

The method is extremely useful in making long term predictions of withdrawals from aquifers whose boundary conditions and storage and transmissibility coefficients are known.

When the pumping well penetrates only a part of the aquifer (partial penetration) the transmissibility coefficient obtained from the distance drawdown plot is not reliable, while the time-drawdown plot gives a reliable value of the transmissibility coefficient if boundary effects or leakage do not develop before the data yields a straight line plot. For observation wells affected by partial penetration, the storage coefficient S cannot be determined from the time-drawdown plot.

**Theis Recovery, Fig. 4.26****: **

**Example 2:**

A 30 cm well was pumped at the rate of 1080 1pm for one day and the drawdown in the well was 1.85 m. The pumping was stopped and the residual drawdowns during recovery for one hour are given in Table 5.11 below. Determine the aquifer constants S and T.

**Solution:**

The data processed for plotting the recovery curve is given in Table 5.12.

The values of s’ versus t/t’ are plotted on a semi-log paper, Fig. 5.12. The residual drawdown Δs’ per log cycle of time tit’ is measured and T is determined as-

The storage coefficient can be determined from the observed drawdown in the well (s_{t1}) when pumping stopped.

**Confirmation of the Boundary from the Recovery Curve****: **

If the recovery curve intercepts the time axis, i.e., s’ = 0 at some positive values of t/t, Fig. 5.13, it may be interpreted that a recharge boundary is encountered. On the other hand, if the recovery curve intercepts the axis of drawdown giving some positive value of s’ at t/t’ → 1, it may be interpreted that an impervious (barrier) boundary is encountered. The existence of the boundary can be confirmed by drawing a time-drawdown curve (during pumping) and also the distance to the boundary can be calculated.