Three-dimensional temporally dependent dispersion through porous media: analytical solution

Abstract

A three-dimensional model for non-reactive solute transport in physically homogeneous subsurface porous media is presented. The model involves solution of the advection-dispersion equation, which additionally considered temporally dependent dispersion. The model also account for a uniform flow field, first-order decay which is inversely proportional to the dispersion coefficient and retardation factor. Porous media with semi-infinite domain is considered. Initially, the space domain is not solute free. Analytical solutions are obtained for uniform and varying pulse-type input source conditions. The governing solute transport equation is solved analytically by employing Laplace transformation technique (LTT). The solutions are illustrated and the behavior of solute transport may be observed for different values of retardation factor, for which simpler models that account for solute adsorption through a retardation factor may yield a misleading assessment of solute transport in ‘‘hydrologically sensitive’’ subsurface environments.

Appendix

The conditions (12–13) in terms of new space (η) and time (T) variables may be written as

$$ c\left( {\eta ,T} \right) = C_{i} ,\,\eta \ge 0,\,T = 0, $$

(19)

$$ c\left( {\eta ,T} \right) = \left\{ {\begin{array}{*{20}c} {C_{0} ,\,0 < T \le T_{0} } \\ {0, \,T > T_{0} } \\ \end{array} } \right., \eta = 0 $$

(20)

$$ \frac{\partial c}{\partial \eta } = 0,\,\eta \to \infty ,T \ge 0 $$

(21)

Now, introducing a new dependent variable by the following transformation

$$ c\left( {\eta ,T} \right) = K\left( {\eta ,T} \right)exp\left[ {\frac{{U_{0} }}{{2D_{0} }}\eta - \left\{ {\frac{{U_{0}^{2} }}{{4D_{0} }} + \gamma_{0} } \right\}T/R} \right] $$

(22)

The set of Eqs. 19–21 and 11 reduced to

$$ R\frac{\partial K}{\partial T} = D_{0} \frac{{\partial^{2} K}}{{\partial \eta^{2} }} $$

(23)

$$ K\left( {\eta ,T} \right) = C_{i} exp\left( { - \frac{{U_{0} }}{{2D_{0} }}\eta } \right),\,\eta \ge 0,\,T = 0, $$

(24)

$$ K\left( {\eta ,T} \right) = \left\{ {\begin{array}{*{20}c} {C_{0 } \;exp\left( {\alpha^{2} T} \right), \quad 0 < T \le T_{0} } \\ {0,\quad T > T_{0} } \\ \end{array} } \right., \eta = 0;\,\alpha^{2} = \left\{ {\frac{{u_{0}^{2} }}{{4D_{0} }} + \gamma_{0} } \right\} $$

(25)

$$ \frac{\partial K}{\partial \eta } = 0,\,\eta \to \infty ,\,T \ge 0 $$

(26)

Applying Laplace transformation on Eqs. 23–26, we have

$$ Rp\overline{K} - RC_{i} exp\left( { - \frac{{U_{0} }}{{2D_{0} }}\eta } \right) = D_{0} \frac{{d^{2} \overline{K} }}{{d\eta^{2} }} $$

(27)

$$ \overline{K} \left( {\eta ,p} \right) = \frac{{c_{0 } }}{{\left( {p - \alpha^{2} } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right], \quad\eta = 0 $$

(28)

$$ \frac{{\partial \overline{K} }}{\partial \eta } = 0,\quad\eta \to \infty $$

(29)

Thus, the general solution of Eq. 27 may be written as

$$ \overline{K} \left( {\eta ,p} \right) = C_{1} exp \left( {- \eta \sqrt {\frac{Rp}{{D_{0} }}} } \right) + C_{2} exp\left( {\eta \sqrt {\frac{Rp}{{D_{0} }}} } \right) + \frac{{C_{i} exp\left( { - \frac{{U_{0} }}{{2D_{0} }}\eta } \right)}}{{\left( {p - \beta^{2} /R} \right)}},\,\beta^{2} = \frac{{U_{0}^{2} }}{{4D_{0} }} $$

(30)

Using condition 28–29 on the above solution, we get

$$ C_{1} = \frac{{C_{0 } }}{{\left( {p - \alpha^{2} } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right] - \frac{{C_{i} }}{{\left( {p - \beta^{2} /R} \right)}} \,{\text{and}}\,C_{2} = 0. $$

Thus, the particular solution in the Laplacian domain may be written as

$$ \begin{aligned} \overline{K} \left( {\eta ,p} \right) & = \frac{{C_{0 } }}{{\left( {p - \alpha^{2} } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right]exp( - \eta \sqrt {Rp/D_{0} } ) \\ & - \frac{{C_{i} }}{{\left( {p - \frac{{\beta^{2} }}{R}} \right)}}exp( - \eta \sqrt {Rp/D_{0} } ) + \frac{{C_{i} exp\left( { - \frac{{U_{0} }}{{2D_{0} }}\eta } \right)}}{{\left( {p - \beta^{2} /R} \right)}} \\ \end{aligned} $$

(31)

Taking inverse Laplace transform of 31, the solution of advection-dispersion solute transport for uniform pulse-type input condition may be written in terms of $c(x,y,z,T)$ by using Eqs. 22, 10, 8 and 6.

Similarly, Eq. 15 reduces by applying the transformations 6, 8, 10 and 22

$$ - D_{0} \frac{\partial K}{\partial \eta } + \frac{U_0 }{2}K = \left\{ \begin{aligned} U_{0} C_{0 }\exp( {\alpha^{2} T} ), & 0 < T \le T_{0} \\ 0, & T > T_{0} \\\end{aligned} \right.,\,\eta = 0 $$

(32)

Applying Laplace transformation on Eq. 32, we may get

$$ - D_{0} \frac{{d\overline{K} }}{d\eta } + \frac{{U_{0 } }}{2}\overline{K} = \frac{{U_{0 } C_{0 } }}{{\left( {p - \alpha^{2} } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right] $$

(33)

Now using input condition 33 in place of 28 in the general solution 30, we get

$$ C_{1} = \frac{{U_{0 } C_{0 } }}{{\sqrt {D_{0} } \left( {p - \alpha^{2} } \right)\left( {\sqrt p + \alpha } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right] + \frac{{C_{i} }}{{\sqrt {D_{0} } \left( {p - \beta^{2} /R} \right)\left( {\sqrt p + \alpha } \right)}} $$

(34)

Thus, the particular solution in the Laplacian domain may be written as

$$ \begin{aligned} \overline{K} \left( {\eta ,p} \right) & = \frac{{U_{0 } C_{0 } }}{{\sqrt {D_{0} } \left( {p - \alpha^{2} } \right)\left( {\sqrt p + \alpha } \right)}}\left[ {1 - exp\left\{ { - \left( {p - \alpha^{2} } \right)T_{0} } \right\}} \right]exp\left( { - \eta \sqrt {\frac{Rp}{{D_{0} }}} } \right) \\ & - \frac{{C_{i } }}{{\sqrt {D_{0} } \left( {p - \beta^{2} /R} \right)\left( {\sqrt p + \alpha } \right)}}exp\left( { - \eta \sqrt {\frac{Rp}{{D_{0} }}} } \right) + \frac{{C_{i} exp\left( { - \frac{{U_{0} }}{{2D_{0} }}\eta } \right)}}{{\sqrt {D_{0} } \left( {p - \beta^{2} /R} \right)\left( {\sqrt p + \alpha } \right)}} \\ \end{aligned} $$

(35)

Taking inverse Laplace transform of 35, the solution of advection-dispersion solute transport for varying input conditions may be written in terms of $c(x,y,z,T)$ by using Eqs. 22, 10, 8 and 6.