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The Analytical Solution of the Boussinesq Equation for Flow Induced by a Step Change of the Water Table Elevation Revisited

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Abstract

The classical problem of flow induced by a sudden change of the piezometic head in a semi-infinite aquifer is re-examined. A new analytical solution is derived, by combining an expression describing the water table elevation upstream, obtained by the Adomian’s decomposition approach, to an existing polynomial expression (Tolikas et al. in Water Resour Res 20:24–28, 1984), adequate for the downstream region; the parameters of both approximations are computed by matching the two solutions at the inflection point of the water table. Although several analytical solutions are available in the literature, we demonstrate that the expression we have developed in this issue is the most accurate for strong or moderate non-linear flows, where the degree of non-linearity is defined as the ratio of the piezometric head elevation at the origin to the initial water table elevation. For this type of flows the perturbation-series solution of Polubarinova-Kochina, characterized by previous studies as the best available analytical solution provides physically unacceptable results, while the analytical solution of Lockington (J Irrig Drain Eng 123:24–27, 1997), used to check the accuracy of numerical schemes, underestimates the penetration distance of the recharging front.

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Abbreviations

A :

Parameter for the description of the water table elevation by a polynomial approximation (Eq. 41)

A k :

Auxiliary function used for the determination of the Adomian’s series solution

B k :

Auxiliary function used for the determination of the Adomian’s series solution

c k :

Parameter of the Adomian’s series (Eq. 47)

D1, D2:

Variables used for the computation of the parameters of the new analytical solution

\({\tilde{D}_{\rm H}}\) :

Equivalent hydraulic diffusivity

h :

Water table elevation

h 0 :

Initial water table elevation

h 1 :

Water table elevation at the origin

h a :

Approximate value of the water table elevation

h e :

Exact value of the water table elevation

\({\tilde {h}}\) :

Representative saturation thickness

H :

Non-dimensional water table elevation, defined by Eq. 9b

K :

Hydraulic conductivity

L :

Non-dimensional penetration distance

L η :

Linear operator introduced in Sect.3.2

S :

Specific yield

t :

The time variable

u :

Dependent variable used to obtain a perturbation solution of the Boussinesq equation

v :

Dependent variable used to obtain a perturbation solution of the Boussinesq equation

V s :

Stored water volume in the aquifer during the recharge process

\({\bar{{w}}}\) :

Aquifer width

w :

Dependent variable used to obtain a perturbation solution of the Boussinesq equation

x :

Distance from the origin of the aquifer

Z :

Auxiliary function used for the determination of the Adomian’s series solution

η :

The Boltzmann similarity variable defined by Eq. 9a

η k :

Position of the inflection point of the non-dimensional variable H

\({\hat{{\eta }}}\) :

Similarity variable defined by Eq.12a

\({\tilde {\eta }_k }\) :

Variable defined by Eq.75

λ :

Parameter for the description of the water table elevation by a polynomial approximation (Eq.41)

Λ :

Auxiliary function used for the determination of the Adomian’s series solution

μ :

Parameter defined by Eq.11c

ν :

Value of the non-dimensional water-table elevation H at the inflection point

χ1, χ2:

Variables used for the computation of the parameters of the new analytical solution

Ψ :

Variable defined by Eq. 43

Ω :

Auxiliary function used for the determination of the Adomian’s series solution

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Moutsopoulos, K.N. The Analytical Solution of the Boussinesq Equation for Flow Induced by a Step Change of the Water Table Elevation Revisited. Transp Porous Med 85, 919–940 (2010). https://doi.org/10.1007/s11242-010-9599-3

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