Contents preface (VII) introduction 1—37

for different values of
6.0	0.00036	0.45	2.30	4.54	6.84	9.14	11.45	13.75	16.05	18.35	20.66	22.96	25.26	27.56	29.87	32.17
5.0	0.0011	0.56	2.47	4.73	7.02	9.33	11.63	13.93	16.23	18.54	20.84	23.14	25.44	27.75	30.05	32.35
function W(u)
	4.0	0.0038	0.70	2.68	4.95	7.25	9.55	11.85	14.15	16.46	18.76	21.06	23.36	25.67	27.97	30.27	32.58
4.2 Well	3.0	0.013	0.91	2.96	5.23	7.53	9.84	12.14	14.44	16.74	19.05	21.35	23.65	25.96	28.26	30.56	32.86
Table
	2.0	0.049	1.22	3.35	5.64	7.94	10.24	12.55	14.85	17.15	19.45	21.76	24.06	26.36	28.66	30.97	33.27
	1.0	0.219	1.82	4.04	6.33	8.63	10.94	13.24	15.54	17.84	20.15	22.45	24.75	27.05	29.36	31.66	33.96

u

× 1

–1×10

–2×10

–3×10

–4×10

–5×10

–6×10

–7×10

–8×10

–9×10

–10×10

–11×10

–12×10

–13×10

–14×10

–15×10

	4
	2
	1
	0.8
	0.6
′fort> 5	0.4
k
Values ofC	0.2
Table 4.3	0.1
	0.08
	0.06
	0.04
	0.03
	′r

C –0.27 –0.24 –0.19 –0.16 –0.13 –0.05 –0.02 0.05 0.05 0.05 0.03 0

134 IRRIGATION AND WATER RESOURCES ENGINEERING
Values of the function V have been tabulated in Table 4.4. The approximate values of V can also be calculated as follows:
For t′ < 0.01 and t′/r′ > 10.
V ≈ ln (2t′/r′)
For t′ < 0.01,

V ≈ sin h^–1

(t′/r′) –

1

1 + r′ ^2

For t′ < 0.05,

L

O

L

1 + t′ O

–1

1

–1

–1

V ≈ sin h

M

P

+ sin h

[t′/r′] – sin h

M

P

For t′ > 5.0,

N r′ Q

N

r′ Q

V ≈

1

W (u)

2

r^2n

in which,

u =

e

4Tt

Here, n_e is the effective porosity of the aquifer.
The well equations mentioned in this section are valid only for a single well of small diameter having no storage capability and fully penetrating an extensive aquifer. The equations will be modified for the effect of partial penetration, well storage, bounded aquifers, interference of adjacent wells, and multi-layer aquifer systems.
Example 4.4 A fully penetrating artesian well is pumped at a rate Q = 1500 m³/day from an aquifer whose storage coefficient and transmissivity are 4 × 10^–4 and 0.145 m²/min, respectively. Find the drawdowns at a distance 3 m from the production well after one hour of pumping and at a distance of 350 m after one day of pumping.

Solution:

At r = 3 m and t = 1 h,

r^2 s

=

3 × 3 × 4 × 10^− ^4

u =

= 1.03 × 10^–4

4 Tt

4 × 0.145 × (1 × 60)

∴

w (u) = 8.62

From Eq. (4.26)

∴

s = –

Q

W (u) = 1500 / (24 × 60)

× 8.62

4

πT

4 × 3.14 × 0.145

= 4.93 m

Similarly, at r = 350 m and t = 1 day

GROUND WATER AND WELLS																							135
																																	(Contd.)...
		0.09	0.104	0.210	0.310	0.410	0.500	0.585	0.670	0.750	0.825	0.890	1.42	1.74	1.96	2.14	2.26	2.38	2.46	2.55	2.64	3.09	3.31	3.46	3.60	3.69	3.78	3.82	3.90	3.92
			0.08	0.118	0.236	0.350	0.460	0.560	0.650	0.740	0.825	0.910	0.985	1.52	1.85	2.07	2.24	2.38	2.50	2.58	2.66	2.75	3.20	3.45	3.59	3.71	3.80	3.90	3.95		4.00	4.05
				0.07	0.135	0.270	0.400	0.525	0.630	0.735	0.835	0.925	1.02	1.09	1.65	1.98	2.20	2.38	2.50	2.64	2.71	2.80	2.90	3.34	3.57	3.70	3.85	3.92	4.00		4.10		4.15	4.19
					0.06	0.158	0.316	0.465	0.600	0.725	0.840	0.950	1.04	1.14	1.22	1.80	2.14	2.36	2.52	2.67	2.80	2.86	2.96	3.04	3.50	3.72	3.86	4.00	4.09		4.18		4.25		4.30	4.35
′andr					0.05	0.192	0.378	0.555	0.710	0.850	0.970	1.09	1.20	1.29	1.38	1.98	2.32	2.54	2.72	2.85	2.99	3.05	3.15	3.22	3.68	3.90	4.04	4.19	4.26		4.35		4.40		4.49	4.52
					0.04	0.240	0.468	0.675	0.850	1.010	1.15	1.28	1.39	1.50	1.58	2.20	2.54	2.76	2.95	3.09	3.20	3.26	3.36	3.45	3.90	4.12	4.26	4.40	4.50		4.58		4.65		4.70			4.80
′ ′ ′V(t,r)fordifferentvaluesoft
							0.03	0.322	0.610	0.860	1.07	1.24	1.40	1.54	1.65	1.75	1.85	2.46	2.84	3.05	3.24	3.37	3.50	3.55	3.65	3.74	4.18	4.42	4.55	4.68	4.78		4.85		4.92				5.00	5.02
′Valueofr						0.02	0.474	0.860	1.18	1.42	1.60	1.78	1.91	2.04	2.14	2.25	2.87	3.24	3.46	3.65	3.76	3.90	3.96	4.05	4.10	4.59	4.82	4.95	5.05	5.20	5.25		5.30		5.35				5.40
						0.01	0.875	1.42	1.79	2.06	2.28	2.45	2.60	2.72	2.84	2.94	3.56	3.90	4.15	4.30	4.45	4.55	4.65	4.75	4.83	5.25	5.50	5.70	5.80	5.85	5.95		6.05	6.10	6.14
							0.009	0.95	1.52	1.88	2.17	2.38	2.56	2.70	2.84	2.95	3.04	3.66	4.01	4.26	4.45	4.56	4.68	4.80	4.90	4.95	5.35	5.60	5.75	5.85	5.95		6.05	6.10						6.15		6.20
									0.008	1.04	1.64	2.00	2.29	2.50	2.68	2.82	2.95	3.05	3.15	3.80	4.14	4.38	4.55	4.68	4.80	4.90	5.00	5.05	5.50	5.75	5.90		6.00	6.10						6.20			6.25	6.30	6.35
								4.4 Function		0.007	1.15	1.76	2.13	2.42	2.64	2.80	2.96	3.09	3.20	3.30	3.93	4.28	4.50	4.70	4.82	4.92	5.00	5.12	5.20	5.60	5.85		6.00	6.10		6.20				6.30			6.35	6.40	6.45
										0.006	1.28	1.92	2.28	2.58	2.79	2.96	3.12	3.24	3.35	3.45	4.10	4.43	4.67	4.85	4.98	5.08	5.20	5.30	5.35	5.75	6.00		6.15	6.25		6.35		6.40		6.50			6.55	6.60
								Table				0.005	1.42	2.09	2.46	2.75	2.96	3.15	3.30	3.42	3.54	3.63	4.28	4.61	4.85	5.00	5.15	5.25	5.35	5.50	5.55		5.92	6.20		6.35		6.50		6.55			6.62				6.70	6.75	6.80
											0.004	1.64	2.30	2.70	2.98	3.20	3.36	3.51	3.65	3.75	3.85	4.50	4.85	5.08	5.25	5.40	5.50	5.60	5.70	5.75	6.15	6.45	6.58	6.65		6.75		6.85	6.90	7.00			7.05
				0.003							1.90	2.58	3.00	3.26	3.49	3.65	3.80	3.94	4.05	4.14	4.78	5.12	5.35	5.58	5.65	5.80	5.85	6.00	6.05	6.45	6.70	6.85	7.00	7.10		7.15		7.20	7.25	7.35
											0.002	2.30	2.97	3.40	3.68	3.90	4.06	4.20	4.34	4.47	4.54	5.15	5.50	5.75	6.00	6.10	6.20	6.25	6.35	6.45	6.88	7.15	7.28	7.35		7.50	7.55	7.60	7.65			7.75
												0.001	2.99	3.68	4.08	4.35	4.58	4.76	4.92	5.08	5.18	5.24	5.85	6.24	6.45	6.65	6.75	6.88	7.00	7.10	7.14	7.60	7.85	8.00		8.15	8.20	8.25	8.30				8.32					8.35
			′t									0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	2	3	4	5		6	7	8	9				10

Table 4.4 (Contd..)
	Values	of	r′
	t′
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	2	3	4	5
0.01		0.093				0.0430	0.0264		0.0180		0.0132	0.0100	0.0078	0.0062		0.0049	0.0040	0.00057		0.00015
0.02		0.187				0.0865	0.0530		0.0365		0.0268	0.0205	0.0160	0.0125		0.0100	0.0081	0.00118		0.00020
0.03		0.278				0.130		0.0800		0.0550		0.0405	0.0310	0.0240	0.0190		0.0150	0.0122	0.00184		0.00032
0.04		0.368				0.174		0.107			0.0735		0.0540	0.0415	0.0322	0.0255		0.0202	0.0165	0.00244		0.00043
0.05		0.450				0.215		0.133			0.0920		0.0675	0.0520	0.0400	0.0320		0.0255	0.0206	0.00305		0.00055
0.06		0.530				0.257		0.160			0.110		0.0810	0.0610	0.0478	0.0380		0.0305	0.0250	0.00365		0.00065
0.07		0.610				0.298		0.186			0.130		0.0950	0.0725	0.0565	0.0450		0.0360	0.0292	0.00430		0.00078
0.08		0.680				0.340		0.214			0.148		0.108	0.0825	0.0645	0.0510		0.0412	0.0336	0.00500		0.00090
0.09		0.750				0.378		0.236			0.164		0.122	0.0930	0.0730	0.0585		0.0470	0.0380	0.00570		0.00105
0.1			0.815				0.415		0.260			0.180		0.134	0.103			0.0805	0.0640		0.0515	0.0420	0.00635		0.00118
0.2			1.32						0.750		0.500			0.359		0.268	0.208			0.165		0.132		0.107		0.0880	0.0145		0.00278
0.3			1.64						1.02			0.700			0.515		0.392	0.308		0.246		0.200		0.164		0.135			0.0238		0.00490
0.4			1.86						1.22			0.870			0.650		0.510	0.405		0.328		0.268		0.220		0.182			0.0350		0.00750		0.00160	0.00038
0.5			2.03						1.37			1.00			0.770		0.610	0.490		0.400		0.330		0.275		0.230			0.0450		0.0104			0.00240	0.00056
0.6			2.16						1.49			1.12			0.875		0.700	0.570		0.468		0.390		0.325		0.276			0.0580		0.0138			0.00320	0.00080
0.7			2.28						1.60			1.22			0.965		0.775	0.640		0.525		0.445		0.375		0.320			0.0715		0.0175			0.00425	0.00108
0.8			2.36						1.69			1.30			1.04			0.850	0.715		0.600		0.500		0.425		0.364			0.0840		0.0212			0.00525	0.00140
0.9			2.45						1.75			1.38			1.11			0.920	0.775		0.650		0.550		0.475		0.404			0.0980		0.0260			0.00630	0.00165
1				2.54						1.85			1.45			1.18			0.975	0.825		0.700		0.595		0.510		0.444			0.113			0.0310			0.00840	0.00235
2				2.97						2.29			1.88			1.60			1.38			1.22			1.07		0.950		0.840		0.750			0.259			0.0950			0.0330		0.0115
3				3.20						2.50			2.10			1.82			1.60			1.42			1.28		1.15			1.05			0.960			0.388			0.165			0.0700		0.0275
4				3.36						2.66			2.25			1.97			1.75			1.58			1.42		1.30			1.20			1.10			0.495			0.235			0.112			0.0535
5				3.49						2.78			2.38			2.09			1.87			1.69			1.54		1.42			1.30			1.21			0.580			0.300			0.150			0.0715
6				3.59						2.90			2.47			2.18			1.95			1.78			1.65		1.52			1.40			1.30			0.660			0.360			0.195			0.0990
7				3.66						2.96			2.55			2.25			2.04			1.85			1.70		1.58			1.48			1.38			0.730			0.415			0.230			0.125
8				3.74						3.00			2.60			2.32			2.11			1.94			1.79		1.66			1.55			1.44			0.790			0.465			0.272			0.155
9				3.80						3.09			2.67			2.39			2.17			2.00			1.85		1.72			1.60			1.50			0.850			0.515			0.307			0.182
10				3.84						3.12			2.74			2.45			2.24			2.05			1.90		1.77			1.65			1.55			0.890			0.550			0.340			0.210

136

IRRIGATION AND WATER RESOURCES ENGINEERING

GROUND WATER AND WELLS				137
			(350)^2 × 4 × 10^− ^4
	u =			= 5.87 × 10^–2
∴		4 × 0.145 × 24 × 60
	w (u) = 2.316
	and	s =	1500/(24 × 60)	× 2.316
				4 π × 0.145
	= 1.32 m.