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Theory of single well tests in a leaky aquifer. Analytical solution for non steady flow

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The evaluation of a pump test or a ‘slug’ test in a single well that completely penetrates a leaky aquifer does not yield a unique relation between the hydraulic properties of the aquifer, independent of the testing conditions. If the flow is transient, the drawdown is characterized by a single similarity parameter that does not distinguish between the storativity and the leakage factor. If the flow is quasi stationary, the drawdown is characterized by a single similarity parameter that does not distinguish between the transmissivity and the leakage factor. The general non steady solution, which is derived in closed form, is characterized bythree similarity parameters.

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a :

e γ≈0.8905 = auxiliary parameter

b :

thickness of the aquifer

b c :

thickness of the semipervious stratum


auxiliary function


auxiliary functions in the complex plane


auxiliary functions of time

h :

undisturbed level of the phreatic surface

K :

conductivity of the aquifer

K c :

conductivity of the semipervious stratum

m 0 :

leakage factor

m :

dimensionless leakage factor


auxiliary function in the complex plane

Q′ w (t):

discharge flux

Q :

steady discharge flux

Q 0 :

constant discharge flux during limited time


dimensionless discharge flux

r 0 :

radius of the well

r′ :

radial coordinate

r :

dimensionless radial coordinate

s :

complex variable

s 0 :


S :

storativity of the aquifer

S n :

n'th part of an integration contour

t′ :


t :

dimensionless time

T :

transmissivity of the aquifer

α,β,ε,κ,λ,θ :

dimensionless parameters

γ :

Euler's number

ξ :

dummy variable

ψ 1(ξ),ψ 2(ξ):

auxiliary functions

σ′(r′, t′):



drawdown in the well

σ(r, t):

dimensionless drawdown

σ 0(t):

dimensionless drawdown in the well


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Rehbinder, G. Theory of single well tests in a leaky aquifer. Analytical solution for non steady flow. Appl. Sci. Res. 55, 211–225 (1995). https://doi.org/10.1007/BF00867512

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Key words

  • single well
  • aquifer
  • slug test