New Analytical Solutions for Unsteady Flow in a Leaky Aquifer between Two Parallel Streams

Abstract

Analytical simulation of groundwater flow is a necessary and useful technique to predict the various behavior patterns of a groundwater system. The main aim of the present study is to derive new analytical solutions to compute the unsteady flows inside an aquifer between two parallel streams of constant and varying heads. The problems are solved by means of Laplace transform method and the solution results are verified with the results of MODFLOW. It is observed that the obtained results agreed very well with the results of MODFLOW. The solutions are carried out for two cases of ascending and descending water levels and the obtained results are compared with each other. In addition, the sensitivity of hydraulic heads to aquifer parameters and how locations of water divide change by change in aquifer parameters are investigated. In sensitivity analysis of hydraulic heads to changes in recharge rate with different values of hydraulic conductivity, thickness, and length of the aquifer, it is shown that among these parameters the length of the aquifer is the most important parameter affecting the hydraulic heads. Furthermore, the sensitivity of flow rates to recharge rates and water level change rates are analyzed.

Appendix A:

The Laplace transform can be defined as:

$$ \Lambda \kern0.1em \left( X, Y, p\right)=\underset{0}{\overset{\infty }{\int }}{e}^{- p\eta} H\left( X, Y,\eta \right)\mathrm{d}\eta $$

(A1)

Where Λ denotes the Laplace transform of H and p is the Laplace variable. Application of the Laplace transform to Eqs. (6), (8) and (9), yields:

$$ \frac{d^2}{{d X}^2}\kern0.1em \Lambda \left( X, p\right)+\frac{R}{p}= p\kern0.1em \Lambda \left( X, p\right)- H\left( X,\eta =0\right) $$

(A2)

$$ \Lambda \left( X=0, P\right)=-\frac{1}{p+\lambda} $$

(A3)

$$ \Lambda \kern0.1em \left( X=1, p\right)=0 $$

(A4)

Combining Eq. (A2) with Eq. (7) becomes:

$$ \frac{d^2}{{d X}^2}\kern0.1em \Lambda \left( X, p\right)- p\kern0.1em \Lambda \left( X, P\right)=- R/ p- X+1 $$

(A5)

Equation (A5) is an ordinary differential equation, which can readily be solved as below:

$$ \kern0.1em \Lambda \left( X, p\right)=\left\{ A \sinh \left( X\sqrt{p}\right)+ B \cosh \left( X\sqrt{p}\right)\right\}+\frac{X-1}{p}+\frac{R}{p^2} $$

(A6)

where A and B are constants which can be determined by invoking Eqs. (A3) and (A4) in Eq. (A6):

$$ B=\frac{ p\lambda - R\left(\lambda + p\right)}{p^2\left(\lambda + p\right)};\kern1em A=\frac{- p\lambda + R\left(\lambda + p\right)}{p^2\left(\lambda + p\right)}\frac{ \cosh \sqrt{p}}{ \sinh \sqrt{p}}-\frac{R}{p^2 \sinh \sqrt{p}} $$

(A7)

Substituting these values in Eq. (A6) and simplifying, we get:

$$ \kern0.1em \Lambda \left( X, p\right)=\frac{-\lambda}{p\left(\lambda + p\right)}\frac{ \cosh \sqrt{p}}{ \sinh \sqrt{p}} \sinh \left( X\sqrt{p}\right)+\frac{\lambda}{p\left(\lambda + p\right)} \cosh \left( X\sqrt{p}\right)+\frac{X-1}{p}+\frac{R}{p^2}\frac{ \cosh \sqrt{p}}{ \sinh \sqrt{p}} \sinh \left( X\sqrt{p}\right)-\frac{R}{p^2 \sinh \sqrt{p}} \sinh \left( X\sqrt{p}\right)-\frac{R}{p^2} \cosh \left( X\sqrt{p}\right)+\frac{R}{p^2} $$

(A8)

The inverse Laplace transform can be defined as:

$$ H\left( X,\eta \right)=\sum_{n=1}^{\infty}\underset{p={p}_n}{\operatorname{Re} s}\left[{e}^{p\eta}\Lambda \left( X, p\right)\right] $$

(A9)

Taking the inverse Laplace transform of Eq. (A8) results in Eq. (10).