Analytical Solution of the Linearized Boussinesq Equation Considering Time-Dependent Downslope Boundary, Variable Recharge and Bedrock Seepage

Abstract

The present study proposes an analytical solution to a linearized Boussinesq equation considering unsteady downslope river stage, spatio-temporal rainfall recharge, and bedrock seepage. This analytical solution alleviates the limitations of the previous studies like varying river boundary, leaky aquifer bed, and non-uniform recharge simultaneously. The time-dependent river stage depicts the seasonal water level variation in the river, whereas the leakage from the confined to the unconfined aquifer or vice versa represents the bedrock seepage. The analytical expression is derived using the separation of variable method along with a variable transformation technique. The proposed solution is validated using nonlinear numerical and linearized analytical solution. The study highlights that the aquifers with higher bedrock conductivity tend to have a higher inflow rate to the aquifer from the river during monsoon season and a higher outflow rate to the river from the aquifer in the lean period. The analysis also illustrates that for a relatively flat catchment area, with the rise of the water level in the river, there is a reverse flow from the river to the aquifer. Further, the stream stage variation rate is found to be pivotal in deciding the duration of flow direction alteration and volume of water released from the aquifer. Furthermore, a sensitivity analysis is performed to assess the influence of flow parameters on the discharge function. The discharge is found to be positively sensitive to hydraulic conductivity, aquifer slope, and recharge. Finally, the residence time distributions corresponding to various bedrock seepage conditions are examined. Results indicate that the residence time for the various conditions tends to exhibit the heavy-tailed distributions.

Appendix: Derivation of the Initial Condition \(h_{s} (x)\) for the Transient Flow Problem

The initial condition of the flow problem considered in this study is the steady-state solution of Eq. (4) subjected to a constant surface recharge and uniform downstream river boundary. The governing equation for the steady-state solution with constant recharge, bedrock seepage, and uniform downstream boundary is given by

$$K_{s} \overline{h}_{s} \cos \theta \frac{{d^{2} h_{s} }}{{dx^{2} }} + K_{s} \sin \theta \frac{{dh_{s} }}{dx} - \frac{{K_{l} }}{b}h_{s} + \left( {\frac{{H_{0} K_{l} }}{b} + R_{s} } \right) = 0,$$

(A1)

subjected to the boundary conditions

$$h_{s} (x) = H_{s} , \quad x = 0,$$

(VI)

$$\overline{h}_{s} \frac{{dh_{s} }}{dx}\cos \theta + h_{s} \sin \theta = 0, \quad x = L,$$

(VII)

where, $R_{s}$ and $H_{s}$ represents constant recharge and uniform downstream boundary, respectively. The variable $\overline{h}_{s}$ represents average steady water table height, evaluated by the method explained in Section 3.2. Upon minor adjustment in Eq. (A1), the same can be written as

$$\frac{{d^{2} h_{s} }}{{dx^{2} }} + A\frac{{dh_{s} }}{dx} - Bh_{s} + W = 0,$$

(A2)

where A and B are constants already defined in Eqs. (6) and (7);

and

$$W = H_{0} B + \frac{{R_{s} }}{{K_{s} \overline{h}_{s}\cos \theta }}$$

(A3)

The Eq. (A2) is a second-order ODE with constant coefficients. The solution of the same is expressed as

$$h_{s} (x) = C_{1} \exp \left( {\frac{{ - A + \sqrt {A^{2} + 4B} }}{2}x} \right) + C_{2} \exp \left( {\frac{{ - A - \sqrt {A^{2} + 4B} }}{2}x} \right) + \frac{W}{B}$$

(A4)

Applying boundary conditions (VI) and (VII) in Eq. (A4) leads to the formation of two linear algebraic equations with unknown $C_{1}$ and $C_{2} .$ The coefficients $C_{1}$ and $C_{2}$ are then calculated by solving the linear equations by Cramer’s rule. The analytical expression $h_{s} (x)$ presented in Eq. (A4) is used as an initial condition for the transient problem considered in this study.