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Groundwater Fluctuations in Sloping Aquifers Induced by Time-Varying Replenishment and Seepage from a Uniformly Rising Stream

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Abstract

This paper analyses the classical problem of transient surface–groundwater interaction in a stream–aquifer system under rather realistic conditions. The downward sloping unconfined aquifer is in contact with a constant water level at one end, and a fully penetrating stream at the other end whose water level is rising at a uniform speed. Furthermore, the aquifer is replenished by a vertical time-varying recharge. Closed form analytical expressions for hydraulic head and flow rate in the aquifer are obtained by solving the linearized Boussinesq equation using Laplace transform method. Effects of aquifer parameters on transient water table and flow rate are illustrated with a numerical example. To assess the efficiency of the linearization method, analytical solutions are compared with numerical solutions of the corresponding non-linear equation.

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Abbreviations

tan (β):

Downward slope of the aquifer’s bed

h 0 :

Piezometric head at the left boundary [L]

h L :

Initial water level in the stream [L]

x :

Horizontal x-axis [L]

t :

Time [T]

q :

Flow rate per unit area of the aquifer [L2T−1]

h :

Water table height above impermeable bed [L]

t r :

Time in which the stream water rises from h L to h 0 [T]

s :

Laplace transform variable for dimensionless time τ

ħ :

Mean saturated depth of the aquifer [L]

S :

Specific yield [–]

K :

Hydraulic conductivity [LT−1]

L :

Length of the aquifer [L]

N 0 + N 1 :

Initial recharge rate per unit area of the aquifer [LT−1]

X :

Dimensionless spatial coordinate equal to x/L

H :

Dimensionless water head

Q :

Dimensionless flow rate

\({N_0^{\prime}, N_1^{\prime}}\) :

Dimensionless values N 0 and N 1 respectively

Q X=1 :

Dimensionless flow rate at the stream–aquifer interface

Q * :

Steady state flow rate at the stream–aquifer interface

H * :

Steady-state dimensionless water head

α :

(L tan β)/2pD

τ :

Dimensionless time

γ :

Dimensionless time corresponding to t =  t r

λ:

Positive constant [T−1]

λ1 :

Dimensionless value of parameter λ

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  1. Department of Mathematics, National Defence Academy, Khadakwasla, Pune, 411023, India

    Rajeev K. Bansal

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Bansal, R.K. Groundwater Fluctuations in Sloping Aquifers Induced by Time-Varying Replenishment and Seepage from a Uniformly Rising Stream. Transp Porous Med 94, 817–836 (2012). https://doi.org/10.1007/s11242-012-0026-9

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