A Semi-Analytical Solution for the Reactive Henry Saltwater Intrusion Problem

Abstract

Because of the existence of a semi-analytical solution, the Henry saltwater intrusion problem has been widely used for benchmarking non-reactive density-driven flow models. In this work, we extend the semi-analytical solution of Henry to reactive transport in variable-density fluid flow. Accurate semi-analytical solutions are provided for three test cases dealing with saltwater transport including dissolution and degradation reactions. About 6,195 terms are required in the Fourier series to obtain a stable solution for these test cases instead of the 78 initially used by Henry (Sea Water in Coastal Aquifers 1613-C:70–84, 1964) for the non-reactive problem. The resolution of the highly non-linear system is made possible due to the modified Powell hybrid algorithm with an analytical evaluation of the Jacobian. Numerical simulations are performed using different numerical methods and grid sizes to evaluate the benefits of these new test cases for benchmarking reactive density-driven flow models.

Appendix

The non-linear algebraic equations are as follows:

$$ \begin{array}{c}\hfill {\varepsilon}_2a{\pi}^2{A}_{g,h}\left({g}^2+{h}^2\right)\xi ={\displaystyle \sum_{r=0}^{\infty }{B}_{r,h} hN\left(g,r\right)+\frac{4}{\pi }W\left(g,h\right)}\hfill \\ {}\hfill {\varepsilon}_1b{\pi}^2{B}_{g,h}\left({g}^2+\frac{h^2}{\xi^2}\right)\xi ={\displaystyle \sum_{n=0}^{\infty }{A}_{g,n} gN\left(h,n\right)+{\varepsilon}_1{\displaystyle \sum_{s=1}^{\infty }{B}_{g,s} sN\left(h,s\right)}+\mathrm{Quad}+\frac{4}{\pi }W\left(h,g\right)+{R}_{g,h}}\hfill \end{array} $$

where

$$ \begin{array}{l}\begin{array}{ccc}\hfill {\varepsilon}_1=\left\{\begin{array}{l}2,\mathrm{if}:g=0\hfill \\ {}1,\mathrm{if}:g\ne 0\hfill \end{array}\right.\hfill & \hfill \mathrm{and}\hfill & \hfill {\varepsilon}_2=\left\{\begin{array}{l}2,\mathrm{if}:h=0\hfill \\ {}1,\mathrm{if}:h\ne 0\hfill \end{array}\right.\hfill \end{array}\hfill \\ {}N\left(h,n\right)=\frac{{\left(-1\right)}^{h+n}-1}{h+n}+\frac{{\left(-1\right)}^{h-n}-1}{h-n}\hfill \\ {}W\left(h,g\right)=\left\{\begin{array}{l}\frac{{\left(-1\right)}^h-1}{h}, if:g=0\hfill \\ {}\begin{array}{cc}\hfill 0,\hfill & \hfill if:g\ne 0\hfill \end{array}\hfill \end{array}\right.\hfill \\ {}\mathrm{Quad}=\frac{\pi }{4}{\displaystyle \sum_{m=1}^{\infty }{\displaystyle \sum_{n=0}^{\infty }{\displaystyle \sum_{r=0}^{\infty }{\displaystyle \sum_{s=1}^{\infty }{A}_{m,n}{B}_{r,s}\left(\mathrm{msLR}-\mathrm{nrFG}\right)}}}}\hfill \\ {}{R}_{g,h}=\left\{\begin{array}{ll}\left(2\xi {B}_{g,h}-4\xi \frac{{\left(-1\right)}^h}{\pi h}\right)\left(e+f\right)-4e{C}_s\xi \frac{1-{\left(-1\right)}^h}{\pi h}\hfill & \mathrm{if}\kern0.5em g=0\hfill \\ {}\xi {B}_{g,h}\left(e+f\right)\hfill & \mathrm{elsewhere}\hfill \end{array}\right.\hfill \end{array} $$

with

$$ \begin{array}{l}F={\delta}_{\left(m-r\right),g}+{\delta}_{\left(r-m\right),g}-{\delta}_{\left(m+r\right),g}\hfill \\ {}L={\delta}_{\left(m-r\right),g}+{\delta}_{\left(r-m\right),g}+{\delta}_{\left(m+r\right),g}\hfill \\ {}G=\frac{{\left(-1\right)}^{h+n-s}-1}{h+n-s}+\frac{{\left(-1\right)}^{h-n+s}-1}{h-n+s}-\frac{{\left(-1\right)}^{h+n+s}-1}{h+n+s}-\frac{{\left(-1\right)}^{h-n-s}-1}{h-n-s}\hfill \\ {}R=\frac{{\left(-1\right)}^{h+n-s}-1}{h+n-s}+\frac{{\left(-1\right)}^{h-n+s}-1}{h-n+s}+\frac{{\left(-1\right)}^{h+n+s}-1}{h+n+s}+\frac{{\left(-1\right)}^{h-n-s}-1}{h-n-s}\hfill \end{array} $$

and δ _i,j is the Kronecker delta such that ${\delta}_{i,j}=\left\{\begin{array}{c}\hfill 1,\mathrm{if}:i=j\hfill \\ {}\hfill 0,\mathrm{if}:i\ne j\hfill \end{array}\right.$.