Analytical solution for solute transport from a pulse point source along a medium having concave/convex spatial dispersivity within fractal and Euclidean framework

Abstract

In the present study, analytical solutions of the advection dispersion equation (ADE) with spatially dependent concave and convex dispersivity are obtained within the fractal and the Euclidean frameworks by using the extended Fourier series method. The dispersion coefficient is considered to be proportional to the nth power of a non-homogeneous quadratic spatial function, where the index n is considered to vary between 0 and 1.5 so that the spatial dependence of dispersivity remains within the limit to describe the heterogeneity in the fractal framework. Real values like $n =$ 0.5 and 1.5 are considered to delineate heterogeneity of the aquifer in the fractal framework, whereas integral values like n = 1 represent the same in the Euclidean sense. A concave or convex variation is free from demanding a limiting value as in the case of linear variation, hence it is more appropriate in the ambience of many disciplines in which ADE is used. In this study, concentration at the source site remains uniform until the source is present and becomes zero once it is annihilated forever. The analytical solutions, validated through the respective numerical solutions, are obtained in the form of an extended Fourier series with only first five terms. They are convergent to the desired concentration pattern and are stable with the Peclet number. It has been possible because of the formulation of a new Sturm–Liouville problem with advective information. The analytical solutions obtained in this paper are novel.

Appendix

Here the analytical solutions are provided with seven and eight terms of the EFS in equation (26) using the concave variation of dispersivity for the index n = 1.5 in equation (3). Peclet number is ${\text{Pe}} =$ 1.4 in the domain of length $\ell =$ 1.0 km. Other input values are $u_{0} =$ 0.14 km/yr and $D_{0} =$ 0.1 km²/yr. The solution with seven terms in the presence of the source is as below:

$$ C^{*} (X,T) = 1 + \exp ( - X/2)\sum\limits_{m = 1}^{7} {\beta_{m} (T)} \;\frac{1}{{\left\| {\psi_{m} (X)} \right\|}}\sin \left( {\alpha_{m} X} \right), $$

(A1)

where $\alpha_{m}$ are the roots of the transcendental equation associated with (21), and the seven time-dependent coefficients may be obtained from equation (31) as

$$ \left( {\begin{array}{*{20}r} {\beta_{1} (T)} \\ {\beta_{2} (T)} \\ {\beta_{3} (T)} \\ {\beta_{4} (T)} \\ {\beta_{5} (T)} \\ {\beta_{6} (T)} \\ {\beta_{7} (T)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}r} { - 0.0015} & {0.0038} & { - 0.0077} & { - 1.4525} & {0.2523} & {0.0193} & { - 0.0480} \\ {0.0017} & { - 0.0030} & {0.0088} & {0.1252} & { - 0.5417} & { - 0.0133} & {0.1152} \\ { - 0.0011} & {0.0041} & { - 0.0019} & { - 0.0143} & {0.0922} & {0.0720} & { - 0.3326} \\ {0.0017} & {0.0031} & {0.0511} & {0.0089} & { - 0.0252} & { - 0.2367} & {0.0679} \\ {0.0051} & {0.0309} & { - 0.1742} & { - 0.0017} & {0.0114} & {0.0547} & { - 0.0265} \\ {0.0304} & { - 0.1355} & {0.0392} & {0.0019} & { - 0.0042} & { - 0.0246} & {0.0107} \\ { - 0.0933} & {0.0400} & { - 0.0235} & { - 0.0004} & {0.0029} & {0.0107} & { - 0.0053} \\ \end{array} } \right)\left( {\begin{array}{*{20}r} {\exp ( - \upsilon_{1} T)} \\ {\exp ( - \upsilon_{2} T)} \\ {\exp ( - \upsilon_{3} T)} \\ {\exp ( - \upsilon_{4} T)} \\ {\exp ( - \upsilon_{1} T)} \\ {\exp ( - \upsilon_{6} T)} \\ {\exp ( - \upsilon_{7} T)} \\ \end{array} } \right), $$

(A2)

where $\upsilon_{1} = 359.9983,$ $\upsilon_{2} = 267.6400,$ $\upsilon_{3} = 174.8626,$ $\upsilon_{4} = 3.8192,$ $\upsilon_{5} = 21.4275,$ $\upsilon_{6} = 107.4537,$ $\upsilon_{7} = 55.7066$ are the seven eigenvalues of the matrix R in equation (28).

Similarly, the solution with eight terms in equation (26) may be written in the presence of the source as

$$ C^{*} (X,T) = 1 + \exp ( - X/2)\sum\limits_{m = 1}^{8} {\beta_{m} (T)} \;\frac{1}{{\left\| {\psi_{m} (X)} \right\|}}\sin \left( {\alpha_{m} X} \right), $$

(A3)

where the eigenvalues of the matrix R in (28) are obtained as $\upsilon_{1} = 481.1673,$ $\upsilon_{2} = 373.6767,$ $\upsilon_{3} = 260.3778,$ $\upsilon_{4} = 176.3123,$ $\upsilon_{5} = 3.8211,$ $\upsilon_{6} = 21.4225,$ $\upsilon_{7} = 55.8164,$ $\upsilon_{8} = 107.1349$, in terms of which the time-dependent coefficients of (A3) are obtained from equation (31) as

$$ \left( {\begin{array}{*{20}r} {\beta_{1} (T)} \\ {\beta_{2} (T)} \\ {\beta_{3} (T)} \\ {\beta_{4} (T)} \\ {\beta_{5} (T)} \\ {\beta_{6} (T)} \\ {\beta_{7} (T)} \\ {\beta_{8} (T)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}r} {0.0010} & { - 0.0020} & {0.0044} & { - 0.0083} & { - 1.4534} & {0.2527} & { - 0.0486} & {0.0198} \\ { - 0.0008} & {0.0021} & { - 0.0035} & {0.0093} & {0.1253} & { - 0.5419} & {0.1158} & { - 0.0136} \\ {0.0011} & { - 0.0016} & {0.0049} & { - 0.0024} & { - 0.0143} & {0.0919} & { - 0.3351} & {0.0738} \\ { - 0.0005} & {0.0023} & {0.0032} & {0.0514} & {0.0089} & { - 0.0253} & {0.0690} & { - 0.2381} \\ {0.0007} & {0.0056} & {0.0393} & { - 0.1826} & { - 0.0018} & {0.0112} & { - 0.0263} & {0.0534} \\ {0.0062} & {0.0234} & { - 0.1397} & {0.0450} & {0.0019} & { - 0.0043} & {0.0112} & { - 0.0257} \\ {0.0259} & { - 0.1115} & {0.0325} & { - 0.0234} & { - 0.0003} & {0.0028} & { - 0.0050} & {0.0100} \\ { - 0.0792} & {0.0342} & { - 0.0220} & {0.0100} & {0.0006} & { - 0.0013} & {0.0031} & { - 0.0056} \\ \end{array} } \right)\left( {\begin{array}{*{20}r} {\exp ( - \upsilon_{1} T)} \\ {\exp ( - \upsilon_{2} T)} \\ {\exp ( - \upsilon_{3} T)} \\ {\exp ( - \upsilon_{4} T)} \\ {\exp ( - \upsilon_{5} T)} \\ {\exp ( - \upsilon_{6} T)} \\ {\exp ( - \upsilon_{7} T)} \\ {\exp ( - \upsilon_{8} T)} \\ \end{array} } \right). $$

(A4)