Pavel Pech

Radimir Novotny

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JOURNAL OF
ENVIRONMENTAL HYDROLOGY
The Electronic Journal of the International Association for Environmental Hydrology
On the World Wide Web at http://www.hydroweb.com
VOLUME 13       2005
Journal of Environmental Hydrology          Volume 13  Paper 28  November 20051
ANALYTICAL SOLUTIONS FOR WELL DRAWDOWN WITH
WELL LOSSES
2. REAL WELL NEAR BOUNDARY - SOLUTION BY IMAGE
WELL
Radimir Novotny1
Pavel Pech2
1Department of Constructions
2Department of Water Resources
Czech University of Agriculture
Prague, Czech Republic
The solutions in the preceding Paper 1 were based on the assumption of an infinite aquifer.
This assumption, however, is not valid in the case where a well is located near an infinite
boundary, which may be impermeable or have different permeability, and where the boundary
is within the drawdown cone of the punping well. A solution is presented for drawdown in a
pumping well that includes well skin losses and an infinite boundary. The solution is based on
the theory of mirror reflection of image wells.
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20052
Real Well Near Boundary    Novotny and Pech
INTRODUCTION
The influence of an infinite impermeable boundary is solved by the application of the mirror
reflection theory. We assume that the well drawdown during the discharge from the well is identical
with the influence resulting from the superposition of the effect of discharge, Q, from the given real
well and the effect of simultaneous discharge of the same Q from an imaginary well. This image well
is a mirror image of the real well beyond the impermeable boundary at the same distance from the
impermeable boundary as the real well.
THEORY
Real Well in the Proximity of Lateral Impermeable Boundary
With the discharge Q of a well in the proximity of a lateral impermeable boundary, which is “within
reach” of the well, the resulting ground water drawdown, s, is determined by the sum of the theoretical
drawdown, sre , due to the discharge from the real well and the drawdown, sf  , due to the discharge
from the image well placed in a mirror-reflected position.
                                   s = sre + sf (1)
for the real well
                             sre = ste + sSKIN (2)
where
           ste  = drawdown at an “ideal” well (W = 0)
          sSKIN  =  additional drawdown at a well caused by additional resistance or well loss.
If we consider the non-stationary flow regime, the water level drawdown in the real well is given
by solving the equation in dimensionless form (Agarwal et al., 1970):
∂
∂
∂
∂
∂
∂
2
2
1s
r r
s
r
s
t
D
D D
D
D
D
D
+ = (3)
The complete solution of Equation (3) may be obtained by the application of the Laplace
transform. The Laplace transform is inverted numerically using the Stehfest algorithm 368 (Wei
Chun Chu, 1980; Pech, 1986):
( ) ( )s Q
T
con j k
m
iV i
n
i
n
i
=





 −
= =
∑ ∑2 11 0π , .
( ) ( )
( ) ( ) ( )( )[ ]
( )
( )
K c W c K c
c c K c C c K c W c K c
K r c
c K cD
D0
1 2 1 2
1
1 2
3 2 1 2
1
1 2 1 2
0
1 2 1 2
1
1 2
0
1 2
2
1
1 2
/ / /
/ / / / / / /
/
/
−
+ +
+







 (4)
where
T = transmissivity; S = storativity; Q = well discharge; CD = a dimensionless well bore storage
constant (Ramey, 1970); c = the variable in Laplace space ;   rD  =  2x0 /rW  ; x0 (see Figure 1),
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20053
Real Well Near Boundary    Novotny and Pech
rw = well radius; W = skin factor; K0 (x) is a Bessel function of the second kind of order zero; K1 (x)
is a Bessel function of the second kind of order one; n= 10; and
( ) ( ) ( )( )con j k k
k
j
j m
t r
m
m m
j
k
D D
,
ln !
! !
=
− 





−
−
−1 2 2
1
1
1 (5)
where
tD = dimensionless time (Tt/rw
2s)
or, for tD < 25, i.e. in the condition when the Jacob semilogarithmic approximation of the Theis well
function is applicable, it is






′
++=
Sr
tT
W
Sr
tT
T
Q
s
W
V 22
246,2
ln
246,2
ln
2π (6)
Figure 1.  Schematic drawing of the well in the proximity of impermeable lateral boundary (1 -
drawdown due to imaginary well, 2 - drawdown due to the real well, 3 - overall drawdown.
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20054
Real Well Near Boundary    Novotny and Pech
If r = 0, then r´ = 2x0. (see Figure 1).
After modification, the drawdown in the well is expressed by the relation
( ) 




+= W
Sxr
tT
T
Qs
W
V 2
1
2
246,2ln
0π
(7)
The drawdown in any point B determined by the coordinates x, y (Figure 1) for a non-stationary
flow regime, is
( ) ( )
( ) ( )
( )s
Q
T
con j k
m
i
K r c K r c
c K cB i
n
i D D
i
m
=





 −
+ ′






= =
∑ ∑2 11
0
1 2
0
1 2
2
1
1 2
0π
,
/ /
/ (8)
where
rD = r/rW  and rD 
t = rt  /rW
The distances  r  and  r´ are determined from the relations
′ = + +r x x y0
2 2a f (9)
r x x y= − +0
2 2a f          (10)
If we consider the application of the Jacob semilogarithmic approximation, the water level
drawdown in point B is
s
Q
T
T t
r S
T t
r SB
= +
′





2
2 246 2 246
2 2π
ln
,
ln
,
         (11)
or, for the sake of simplification,
s
Q
T
T t
r r SB
=
′



π
ln
,2 246
         (12)
Well in the Proximity of Lateral Constant Head Recharge Boundary
By way of example let us use a well in the proximity of a constant head recharge boundary (surface
flow) fully penetrating the aquifer. The solution is analogous with that applied to the impermeable
boundary. The difference is that in the case of an imaginary well we consider constant injection Q
(it is an injection well). Overall drawdown produced by the discharge Q from the real well is
determined as the sum of the drawdown in the real well and the negative drawdown due to injection
Q into the imaginary well.
Analogous to the impermeable lateral boundary (Equation 1) the overall drawdown is
s = sre  -  sf          (13)
The drawdown in the discharge well is given by Equation 3.
In case of application of the Jacob semilogarithmic approximation, the drawdown in the discharge
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20055
Real Well Near Boundary    Novotny and Pech
well is






′
−+=
Sr
tTW
Sr
tT
T
Qs
W
V 22
246,2ln246,2ln
2π
         (14)
or, after modification,






+
′
= W
r
r
T
Qs
W
V ln2π
         (15)
Figure 2.  Schematic drawing of the well in the proximity of lateral feeding boundary (1 – drawdown
due to imaginary well, 2 – drawdown due to real well, 3 – overall drawdown).
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20056
Real Well Near Boundary    Novotny and Pech
The drawdown,  sV,  does not depend on time and the curve,  sV = f(log t),  proceeds horizontally,
which means that the influence of the lateral boundary will manifest itself after certain time by the
stabilization of the drawdown. The transmission capacity and storage capacity computations can be
based only on the initial part of the inflow test before the part which can be evaluated by the Jacob
semilogarithmic approximation is reached.
The drawdown in any point B (analogous to the case of impermeable boundary) is
( ) ( )
( ) ( )
( )s
Q
T
con j k
m
i
K r c K r c
c K cB i
n
i D D
i
m
=





 −
− ′






= =
∑ ∑2 11
0
1 2
0
1 2
2
1
1 2
0π
,
/ /
/          (16)
and, when considering the applicability of the Jacob semilogarithmic approximation,
s
Q
T
T t
r S
T t
r SB
= −
′





2
2 246 2 246
2 2π
ln
,
ln
,
                                (17)
or, after modification,
s
Q
T
r
rB
=






′
2π
ln                                                                    (18)
CONCLUSION
In this contribution the relations are derived for the determination of the drawdown in a well with
well losses situated in the proximity of the permeable or impermeable boundary. By means of the
derived relationships we can determine drawdown at an arbitrary place when the real well, with
additional resistance and wellbore storage taken into account, is situated near boundary.
ACKNOWLEDGMENTS
The authors would like to thank Prof. Ing. Vladimir Havlik, Ph.D., Czech Technical University,
Prague,  and Doc. Ing. Josef Buchtele, Institute of Hydrodynamics, CAV for their review of this
paper.
REFERENCES
Agarwal, R. G., R. Al-Hussainy, and H. J. Ramey Jr. 1970. An investigation of wellbore storage and skin effect
in unsteady liquid flow. 1. Analytical treatment. AIME Trans. 279-290.
Bear, J. 1979. Hydraulics of groundwater. New York; McGraw-Hill.
Pech, P. 1986. Evaluation of the hydraulic parameters from the aquifer tests. Czech Univ. Of Agriculture , Prague.
Ramey, Jr., H.J. 1970. Short-time well test data interpretation in the presence of skin effect and wellbore storage.
J. Pet. Tech.; 22:97–104.
 Stehfest, H. 1970. Algorithm 368: Numerical inversion of Laplace transforms. Comm. ACM. 13:47–49.
Todd, D.K. 1980. Groundwater Hydrology. Second Ed. New York; Wiley.
Walton, W.C., 1970. Groundwater Resource Evaluation. McGraw-Hill, New York.
Chu, W.C., J. Garcia-Rivera, and R. Raghavan. 1980. Analysis of interference test
data influenced by wellbore storage and skin at the flowing well. JPT. Trans. AIME 249 p.
Journal of Environmental Hydrology                            Volume 13  Paper 28  November 20057
Real Well Near Boundary    Novotny and Pech
ADDRESS FOR CORRESPONDENCE
Dr. Pavel Pech
Department of Water Resources
University of Agriculture Prague
Kamycka ulice
165 21 Prague 6
Czech Republic
E-mail: pech@lf.czu.cz


ANALYTICAL SOLUTIONS FOR WELL DRAWDOWN WITH WELL LOSSES 2. REAL WELL NEAR BOUNDARY -SOLUTION BY IMAGE WELL

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ANALYTICAL SOLUTIONS FOR WELL DRAWDOWN WITH WELL LOSSES 2. REAL WELL NEAR BOUNDARY -SOLUTION BY IMAGE WELL

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