Appendix B
Derivation of a solution for steady-state aquifer head conditions in radial
coordinates with a constant transmissivity and recharge rate
The solution for steady-state head conditions for an aquifer with a constant transmissivity and
recharge rate can be derived using the radial flow equation (1) and the following set of boundary
conditions,
s=0
, r = r0
(B1a)
,z=0
(B1b)
,z=b
(B1c)
where E is the recharge rate. The derivation of the solution is based on an approach and equation
outlined in “Conduction of Heat in Solids” (Carslaw and Jaeger, 1959). The equation as
presented (8.3.III, eq. 14, pg. 219) is for “The finite cylinder 0 < r < a, 0 < z < l. z = 0 kept at
prescribed temperature f(r), the other surfaces kept at zero”. The derivation below differs from
the one described by Carslaw and Jaeger in that the boundary conditions are no-flow at z = 0,
and a specified flow at z = l, rather than the constant-value conditions as specified in Carslaw
and Jaeger. It should be noted that at the outer radial boundary (r0 = a) is a constant-value
boundary and that the inner boundary at r = 0 is implicitly a no-flow condition as groundwater
can neither leave nor enter along this boundary. The hydraulic conductivity (K) is assumed to be
isotropic; this is modified at the end of the appendix to consider the anisotropic case.
Note that the following function satisfies boundary conditions B1b and B1c for all values of 
J0(r)cosh(z)
(B2)
in which J0 is the Bessel function of the first kind and cosh is the familiar exponential cosine
function; this function also satisfies B1a if  is a root of
J0(nr0) = 0
(B3)
where n is an index that implies multiple roots and therefore a series solution,
s(r,z) =
(B4)
Note that the first 20 zeros for equation B3 can be found in Abramowitz and Stegun (1972), table
9.5, pg. 409. The next step is to evaluate An in a fashion similar to that for the Fourier series
coefficient by applying equation B4 to the boundary condition B1c. The steps required for the
calculation of the An coefficients are outlined in Carslaw and Jaeger (1959; 7.5-6, pg. 196-8), in
particular noting eq. 7.5(1). We first note that boundary condition B1c requires a description of
the recharge flux at the upper boundary:
(B5)
where sinh is the exponential sine function. Then, in a fashion similar to the development in
equation 7.6(3) of Carslaw and Jaeger (1959), the parameter An is evaluated:
An =
(B6)
The integral in equation B6 is integrated using the identity 7.5(5) in Carslaw and Jaeger (1959),
(B7)
With the result
An =
(B8)
This result can now be used in the general solution to produce the specific solution
s(r,z) =
(B9)
With the substitution of r* and r0* for the anisotropic case, the result can now be stated as:
s(r*,z) =
where
(B10)
, and
References
Abramowitz, M., and I.A. Stegun, (editors). 1972. Handbook of Mathematical Functions. Dover
Publications, New York.
Carslaw, H.S., and J.C. Jaeger. 1959. Conduction of Heat in Solids. 2nd ed. Oxford University
Press, Oxford, UK. 510p.