Contents preface (VII) introduction 1—37

4.5.2. Non-Equilibrium Equations
For an unsteady flow in confined aquifer, Eq. (4.5) can be written in polar cylindrical coordinates (r, θ, z) as

1 F ∂

∂hI

∂^2h

∂^2h

S ∂h

G

r

J

+

+

=

(4.23)

r^2 ∂θ^2

∂z^2

T ∂t

r H ∂r

∂r K

which reduces to

∂ ^2 h

+

1

∂h

=

S

∂h

(4.24)

∂r^2

r

∂r

T

∂t

if one assumes radial symmetry and the aquifer to be horizontal and of constant thickness. The general form of solution of Eq. (4.24) is h(r, t). For unsteady flow towards a well penetrating the entire thickness of a confined aquifer, Eq. (4.24) needs to be solved for the following boundary conditions:
(i) h (∞, t) = h₀

	F
(ii)	2 πrw B HG ^K
∴	F r
	G w
	H

∂hI	= – Q for t > 0 (flux condition)
		J
	^∂^r ^K r = rw
	∂hI			Q
		J	= –
		2		πT
	^∂^r ^K r = rw

132					IRRIGATION AND WATER RESOURCES ENGINEERING
which can be approximated as
F	∂hI			Q
G r	J	= –
2	πT
H			∂r K r → 0

(iii) h(r, 0) = h₀ (initial condition).
Theis (7) obtained a solution of Eq. (4.24) by assuming that the well is replaced by a mathematical sink of constant strength. The solution is expressed as

				Q ^∞	e− u
	s = h0 – h = –		zu		du	(4.25)
	4 πT	u
in which	u =	r ^2S
4Tt

where, t is the time since the beginning of pumping. Equation (4.25) is also written as

s = –	Q	W(u)	(4.26)
4πT

in which, W(u) is known as the well function (Table 4.2) and is expressed as a function of u in the form of the following convergent series:

W(u) = – 0.5772 – ln u + u –	u 2	+		u 3	−		u4	+ ...	(4.27)
2 × 2 !	3	× 3 !	4	× 4 !

An approximate form of the Theis equation (i.e., Eq. (4.25)) was obtained by Cooper and Jacob (8) dropping the third and higher order terms of the series of Eq. (4.27). Thus,

s = –

Q

[– 0.5772 – ln u]

4

πT

L

0.25 Tt O

Q

or

s = –

Mln

2

P

4

r

S

πT N

Q

∴	s = –	0.183 Q L	2.25 Tt O	(4.28)
T	Mlog	r	2		S	P
		N				Q

For values of u less than 0.05, Eq. (4.28) gives practically the same results as obtained by Eq. (4.26). Note that Q is to be substituted as a negative quantity for a pumping well.
Because of the non-linear form of Eq. (4.6), its solution is difficult. Boulton (9) has presented a solution for fully penetrating wells in an unconfined aquifer. The solution is valid if the water depth in the well exceeds 0.5 H₀. The solution is

s =		Q	(1 + Ck) V (t′, r′)	(4.29)
2	πKH0

in which, C_k is a correction factor which can be taken as zero for t′ less than 5, and according to Table 4.3 for t′ greater than 5 (when C_k depends only on r′). V (t′, r′) is Boulton’s well function dependent on r′ and t′ defined as

	t′ =		Kt
			SH0
and	r′ =		r
		^H0

GROUND WATER AND WELLS 133

	9.0	0.000012	0.26	1.92	4.14	6.44	8.74	11.04	13.34	15.65	17.95	20.25	22.55	24.86	27.16	29.46	31.76
	8.0	0.000038	0.31	2.03	4.26	6.55	8.86	11.16	13.46	15.76	18.07	20.37	22.67	24.97	27.28	29.58	31.88
u	7.0	0.00012	0.37	2.15	4.39	6.69	8.99	11.29	13.60	15.90	18.20	20.50	22.81	25.11	27.41	29.71	32.02