1
1
2
Supporting Information for
3
“Derivation Approaches for the Theis (1935) Equation”
4
5
6
7
8
by Hugo A. Loáiciga
Equation numbers in the following appendices that are not preceded by a capital letter
9
correspond to equation numbers in the technical commentary.
10
11
In the case of a well of negligible diameter, the boundary-value problem for drawdown
being solved is embodied by Equations 1 through 4 of the technical commentary:
12
13
 2 s( r ,t )
r 2

1 s( r ,t ) S s( r ,t )

r r
T t
(1)
14
Appended to the pde (Equation 1) is one initial condition for drawdown:
15
s( r ,0 )  0
16
There are two boundary conditions:
17
s( r   ,t )  0
18
Q  2 T r
19
Modifications to Equations 1, 2, and 4 are needed when the well’s diameter is finite (that is,
20
nonzero), as shown in Equations B15, B16, and B17 of Appendix B.
21
Appendix A. The Derivation of an Analog of Equation 5 for Drawdown
for r  0
s( r ,t )
r r 0
(2)
for 0  t  
(3)
for 0  t  
(4)
22
This method was transmitted to the author during the review of the technical
23
commentary. The author has added the material here related to the delta function . The
24
method starts with the two-dimensional equation for drawdown [ s ( x , y ,t ) ] caused by the
25
removal of a volume ∆V of ground water at x = 0, y = 0, at time 0:
2
1
 2 s
x 2

 2 s
y 2

S s 
T t
   x, y   ; 0  t  
(A1)
2
The initial condition for drawdown created by the removal of a volume of ground water ∆V at
3
the origin (x = 0, y = 0) at time t = 0 is given by:
4
s ( x , y ,0 ) 
5
The delta function is defined such that:
6

V
S
 ( x, y )
s ( x , y ,0 ) dx dy 
(A2)
V
S
x y

 ( x , y ) dx dy 
V
S
(A3)
x y
7
In other words, there is a finite drawdown over an area of aquifer centered on x = 0, y = 0
8
equal to V/S and everywhere else the drawdown at time zero equals zero. The solution of
9
Equations A1 and A2 aims at finding the drawdown for t > 0 at all x, y caused by the initial
10
spike of drawdown at the origin. To this end, introduce the 2D Fourier transform for
11
drawdown:
12
1
G(  , ,t ) 
2


  s( x, y ,t ) e

i (  x  y ) dx dy
(A4)

13
Applying the Fourier transform on both sides of Equation A1 produces the subsidiary
14
equation:
15
 (  2   2 ) G(  , ,t ) 
16
which is readily integrated with respect to time to yield:
17
T
 (  2  2 ) t
G(  , ,t )  G(  , ,0 ) e S
18
Now, using Equation A2 in A4 and taking advantage of the property (Equation A3) of the
19
delta function:
S G(  , ,t )
T
t
(A5)
(A6)
3
1
1
G(  , ,0 ) 
2


  s( x, y ,0 ) e
i (  x  y ) dx dy  1  V
 
2 S
(A7)
2
Substituting Equation A7 in A6:
3
2
2
 V  S (   ) t
G(  , ,t ) 
e
2 S
4
By applying the inverse Fourier transform to Equation A8, one obtains the drawdown for all
5
x, y, t:
6
1
s ( x , y ,t ) 
2
7
Integrating the right-hand side of Equation A9 leads to the solution of Equations A1 and A2
8
(after letting r 2  x 2  y 2 ):
T
9
10
s ( r ,t ) 
 

 
V
4 t T
r 2S
e 4T t
(A9)

(A10)
The well-known probability integral was used in Equation A9 to obtain Equation A10:

11
T 2 2 
  V  (   ) t   i(  x  y )
e S
d d

e
2

S


  

(A8)

2 2

e   dv 


(A11)
12
Note that Equation A10 is analogous to Equation 5 in the technical commentary, and one
13
needs only equate in the latter equation d with s  , k with T / S , and (  / S ) dt with V.
14
The solution (Equation A10) is extended to the case of constant withdrawal Q at the well
15
by convolution. The steps that follow were the same used by Theis (1935) in his derivation.
16
Note that V = Q dt; therefore, the convolution of Equation A10 with respect to time yields
17
the desired drawdown for constant pumping rate Q:
4
r2S
t
e 4T ( t  t  )

1
Q
s( r ,t ) 
4 T

t  t
Q
dt  =
4 T
0
 v
e

2
v
dv
(A12)
r S
4T t
2
which is the Theis equation.
3
Appendix B. The Laplace-Transform Method
4
Basic Application of the Laplace Transform
5
This section shows how to use the Laplace transform to solve boundary-value problems
6
such as given in Equations 1 through 4. The key three steps of the Laplace method are (1)
7
conversion of the pde and its initial and boundary conditions to an ordinary differential
8
equation (ODE) with boundary conditions (see Equations 15 through 17 of the technical
9
commentary), (2) solution of the ODE (see Equation 18), and (3) Laplace inversion of the
10
solution of the ODE (see Equation 20). As an example, recall the boundary condition
11
(Equation 4), which is:
12
Q  2 T r
13
Applying the Laplace transform on both sides of Equation 4:

14

s( r ,t )
r r 0

e  t Q dt
=
(B1)
0

e  t Q dt =
Q

0
17
s( r ,t )
dt
r
(4)
The left-hand side of Equation B1 equals Q/ :

16

 2 T r e  t
0
15
for 0  t  
Its right-hand side equals:
(B2)
5


  t

d e
s( r ,t ) dt 


0

  2 T r dF ( r , )
 2 T r
dr
dr

1
(B3)
2
Equating Equations B2 and B3, and recalling that r  0, this produces:
3
r
4
which is identical to Equation 17. The Laplace transform is applied to Equations 1 through 3
5
in a similar fashion to produce the transformed boundary-value problem stated in Equations
6
15 through 17. The reader is referred to the literature dealing with the Laplace transform (see,
7
e.g., chapter 17 of Gradshteyn and Ryzhik [1994]) for a complete exposition of its properties.
8
9
10
11
dF ( r , )
Q

dr
2 T 
r 0
(B4)
Proof of Equation 18, the Solution to the Laplace-Transformed ODE
This proof starts with Equations 15 through 17:
d 2 F ( r , )
dr 2

1 dF ( r , ) S
  F ( r , )
r
dr
T
(15)
12
The Laplace-transformed boundary conditions (Equations 3 and 4) are:
13
F ( r   , )  0
14
r
15
The solution to the problem (Equations 15 through 17) is derivable using the theory of
16
solutions of ordinary differential equations (see equation 2.162.Ia in Kamke [1959), and
17
recalling the definition for z( r , )  i r 2 S  / T , where i 2  1 ):
18
F ( r , )  J 0 z( r , ) C 1  Y0  z( r , ) C2
19
in which C1 and C2 are constants to be determined from the boundary conditions Equations
20
16 and 17. J 0 z( r , ) and Y0 [ z( r )] are Bessel functions of the first and second kind,
21
respectively:
dF ( r , )
Q

dr
2 T 
r 0
(16)
(17)
(B5)
6
n

r 2 S /( 4T ) n
J 0 z( r ,   


1
n 0
(B6)
n! 2






3
n
n
  2

 2
2
r S /( 4T )
r S /( 4T )
 



n
2
n
Y0 [ z( r , )]  
  ln  i r S /( 4T )   C  
 an 
2
2


 

( n! )
( n! )

n 1
n  0

(B7)
4
Solve for C1 based on Equation B5:
5
C1 
6
Now the limiting operation r   is effected on Equation B8 to take advantage of the
7
boundary condition (16), that is, of:
8
F ( r   , )  0
9
so that the first term on the right-hand side of Equation B8 vanishes and C1 becomes:
2
Y [ z( r , )]
F ( r , )
 C2 0
J 0 [ z( r , )]
J 0 [ z( r , )]
(B8)
(16)
10
 Y [ z( r , )] 
 C2   i C2
C1   lim  0
r  J 0 [ z( r , )] 
11
The boundary condition Equation 17 leads to (noting that r dz / dr  z ):
12
(B9)
lim  z( r , )J 1 [ z( r , ) C1  lim  z( r , )Y1 [ z( r , ]C 2  
r 0
r 0
Q
2 T 
(B10)
13
in which the Bessel functions J1 and N1 are obtained by differentiating the Bessel functions J0
14
and
15
Y1 [ z( r , )]  dY0 [ z( r , )] / dz . The first limit on the left-hand side of Equation B10
16
equals zero. The second limit in the same equation equals 2/. The value for the constant C2
17
is then:
18
C2  
Y 0,
Q
4 T
respectively,
or,
J 1 [ z( r , )]  dJ 0 [ z( r , )] / dz
(B11)
and
7
1
By virtue of Equation B9, it follows that C1  i Q /( 4T ) . The solution F ( r , ) in Equation
2
B5 can then be written as follows:
3
F ( r , )  
Q
Y0 [ z( r , )]  i J 0 [ z( r , )] 
4T 
(B12)
4
Equation B12 can be expressed in terms of the modified Bessel function K0 [ z( r , ] by
5
noting that Y0 [ z( r , )]  i J 0 [ z( r , )]  i J 0 [ z( r , )]  i Y0 [ z( r , )]  and then using the
6
following expression (Gradshteyn and Ryzhik 1994, equations 8.405.1 and 8.407.1):
7
 i  J 0 [ z( r , )]  i Y0 [ z( r , )]   
8
where:
9
z( r , )  r 2
2

K 0  z( r , ) 
S

T
(B13)
(19)
10
Substitution of Equation B13 into Equation B12 yields Equation 18:
11
F ( r , ) 
12
Equation 18 is inverse Laplace-transformed as shown in Equation 20 to yield the desired
13
solution.
14
The Case of Finite-Diameter Well
17
(18)
The governing equation for drawdown when the radius of the well rw is finite, rw > 0
15
16
Q
K 0  z( r , ) 
2 T 
is:
 2 s( r ,t )
r 2

1 s( r ,t ) S s( r ,t )

r  rw , 0  t  
r r
T t
18
The initial condition is:
19
s( r ,0 )  0
20
There is a boundary condition at infinity:
21
s( r   ,t )  0
for r  rw
for 0  t  
(B15)
(B16)
(3)
8
1
The boundary condition at the well is a water-balance equation stating that the pumping rate
2
Q equals the ground water flow into the well plus the volume of water removed from within
3
the well’s casing per unit time:
4
Q  2 T r
s( r ,t )
r r  r
w
s( r ,t )
  rw2
 t r r
for 0  t  
(B17)
w
5
Equation B17 assumes that there is no well loss, so that the drawdown at the radius rw equals
6
the drawdown within the well. The Laplace-transformed Equation B15 is:
7
d 2 F ( r , )
dr 2

1 dF ( r , ) S
  F ( r , )
r
dr
T
(15)
8
The Laplace-transformed boundary conditions become:
9
F ( r   , )  0
10
11
Q

 2 rT
(16)
dF ( r , )
  rw2 F ( rw , )
dr
r r
(B18)
w
12
The solution of the ODE (Equation 15) is (see equation 2.162.Ia in Kamke [1959]):
13
F ( r , )  J 0 z( r , ) C 1  Y0  z( r , ) C2
14
in which:
15
z( r , )  i
16
and C1 , C2 are constants that are evaluated by substituting Equation B19 in the boundary
17
conditions Equation 16 and Equation B18 and solving for the constants. Upon evaluation of
18
C1 , C2 the solution (Equation B19) becomes:
19
F [ r , ] 
(B19)
r 2S

T
(B20)
K 0  z( r , ) 
1

Q
2 T z( rw , ) K 1  z( rw , )    rw2  K 0  z( rw , ) 
(B21)
9
1
in which z( r ,t )  r  S T and z( rw ,t )  rw  S T ; K 0 and K1 are modified Bessel
2
functions (see, for example, Gradshteyn and Ryzhik, 1994). The inverse Laplace transform of
3
Equation B21 yields the drawdown s(r,t). The inverse of equation B21 is found in equation
4
13.7.II of Carslaw and Jaeger (1959) (after changing the meaning of variables, say Q is
5
replaced by a heat source, T by a type of thermal conductivity, and so on). Papadopulos and
6
Cooper (1967) adapted result 13.7.II in Carslaw and Jaeger (1959) to obtain the drawdown
7
when the pumping well has a finite radius rw . The Laplace inversion of Equation B21 yields
8
the drawdown:
9

10
s( r ,t ) 
2Q S
2T

  1  e
0

11

v2
4u w


  J  v r v Y ( v )  2 S Y ( v )  Y  v r v J ( v )  2 S J ( v ) dv
 2
1
0
0
1
  0  rw  0
rw 


 v g( v )


(B22)
12
in which u w  rw2 S / 4Tt ; J 0 , J 1 , Y0 , Y1 are Bessel functions introduced above when
13
presenting the Laplace method applied to the case of radial flow to a well of negligible
14
diameter
15
g( v )  v J 0 ( v )  2 S J 1 ( v )2  v Y0 ( v )  2 S Y1 ( v )2 . The integral in Equation B22 must be
16
evaluated numerically.
17
Appendix C. A Hybrid Method of Separation of Variables
(see
Equations
B6,
B7,
and
B10);
18
The boundary-value problem for drawdown being solved is embodied by Equations 1
19
through 4. In the method of separation of variables (see Mathews and Walker [1970] for a
20
review), the drawdown s(r,t) is expressed as a linear combination of the product of functions
21
of r times functions of t, namely:
22
s( r ,t ) 

 Gn X n ( r ) Z n ( t )
n 0
(C1)
10
1
in which G n , n = 0, 1, 2, 3, …, are constants of integration. Substitution of a typical product
2
X ( r ) Z ( t ) into Equation 1 leads to two ODEs as follows:
3
4
5
6
7
8
9
10
11
12
13
14
d2X
dr 2

1 dX
  2 X
r dr
(C2)
where  is an unknown positive constant, and (with  = T/S)
dZ
  2  Z
dt
(C3)
The latter equation is integrated to yield that:
2
Z ( t )  M e   t
(C4)
in which M is an integration constant.
The solution to Equation C2 is (see equation 2.162.Ia in Kamke [1959]):
15
X ( r )  A J 0 (  r )  B Y0 (  r )
16
in which A and B are integration constants, and J 0 , Y0 are Bessel functions of the first and
17
second kind, respectively. Generalizing the product of X(r) Z(t) according to Equation C1,
18
one obtains the drawdown equation:
(C5)


 n2  t
 n2  t 
A
J
(

r
)
e

B
Y
(

r
)
e
n 0 n
 n 0 n



n 0

19
s( r ,t ) 
20
where An , Bn ,  n , n= 0, 1, 2, 3,…, are coefficients to be determined from the initial and
21
boundary conditions Equations 2 through 4.
22
23
24
25
26
27
28
(21)
References
Carslaw, H.S., and J.C. Jaeger. 1959. Conduction of Heat in Solids. 2nd ed. London: Oxford
University Press.
Gradshteyn, I.S., and I.M. Ryzhik. 1994. Tables of Integrals, Series, and Products. 5th ed.
San Diego, California: Academic Press.
11
1
Hermance, J.F. 1999. A Mathematical Primer on Ground Water Flow: An Introduction to the
2
Mathematical and Physical Concepts of Saturated Flow in the Subsurface. Englewood
3
Cliffs, New Jersey: Prentice Hall.
4
5
6
7
8
9
Kamke, E. 1959. Methods of Solution and Solutions of Differential Equations. (in German).
New York: Chelsea Publishing.
Mathews, J., and R.L. Walker. 1970. Mathematical Methods of Physics. Redwood City,
California: Addison-Wesley.
Papadopulos, I.S., and H.H. Cooper. 1967. Drawdown in a well of large diameter. Water
Resources Research 3, no. 1, 241–244.
10
Theis, C.V. 1935. The relation between the lowering of the piezometric surface and rate and
11
duration of discharge of a well using ground water storage. Transactions of the American
12
Geophysical Union 16, 519–524.
13
14