Modeling solute/contaminant transport in heterogeneous aquifers

Abstract

A fissured aquifer may be considered as a dense network of fissures separated by low permeability matrix blocks. A conceptual modeling of such a system consists of an infinite number of parallel fractures separated by constant width matrix slabs. While the fissures are assumed to be main flow conduits, the fluid in the porous matrix blocks are considered to be virtually immobile. The mathematical model of the transport of a solute and/or contaminant which assumes a purely convective flow in fissures and diffusion into the matrix blocks consists of two coupled differential equations. An analytical solution of this model for the case of solute entering into the system at a constant concentration has been presented by Skopp and Warrick in Soil Sci Soc Am Proc 38:545-550, 1974. Note however, Skopp and Warrick (Soil Sci Soc Am Proc 38:545-550, 1974) have not considered the additional processes of adsorption and radioactive decay. Unfortunately, their solution had computational limitations as it involved numerical integration of a quite complex expression. Therefore, one had to turn to employing numerical Laplace transform inverters to compute the solutions. This work presents simple real space analytical solutions for the contaminant transport model described above including the adsorption and radioactive decay. The real space solutions have been developed using the method of double Laplace transform and binomial series approximation. An accurate approximate solution has also been presented which converges to the exact solution only after computing three terms in the series full solution. The developed model has been used for 1) assessment of the efficiency of numerical Laplace transform algorithms and 2) investigation of the degree and scale of contamination, and 3) designing remediation schemes for the already contaminated aquifers.

Appendix

Appendix A: novel solution with a different set of dimensionless variables

The transport equations, Eqs. 1 and 2, may be written in terms of a different set of dimensionless variables. Let’s define the following new set of dimensionless variables.

$$ \begin{array}{cc}\hfill {t}_D=\frac{\phi_m^2{R}_m{D}_mt}{R^2{b}^2}\hfill & \hfill\;\lambda =\frac{k{R}^2{b}^2}{\phi_m^2{R}_m{D}_m}\hfill \end{array} $$

(A1)

$$ \begin{array}{cc}\hfill {x}_D=\frac{R_m{\phi}_m^2{D}_mx}{Ru{b}^2}\hfill & \hfill {z}_D=\frac{R_m{\phi}_mz}{Rb}\hfill \end{array} $$

(A2)

Based on the above dimensionless variables, Eqs. 1 and 2 reduce to

$$ \frac{\partial {C}_D}{\partial {t}_D}+\frac{\partial {C}_D}{\partial {x}_D}+\lambda {C}_D-{\left.\frac{\partial {C}_{mD}}{\partial {z}_D}\right]}_{z_D=0}=0 $$

(A3)

$$ \frac{\partial {C}_{mD}}{\partial {t}_D}+\lambda {C}_{mD}-\frac{\partial^2{C}_{mD}}{\partial {z}_D^2}=0 $$

(A4)

In terms of the dimensionless variables, the related initial and boundary conditions given in Eqs. 13 and 15 remain unchanged but Eqs. 14 and 6 becomes the following:

$$ {C}_D= \exp \left(-\lambda {t}_D\right)\;\mathrm{at}\kern0.62em {x}_D=0 $$

(A5)

$$ \frac{\partial {C}_{mD}}{\partial {z}_D}=0\kern0.5em \mathrm{at}{z}_D={a}_D $$

(A6)

Closely following the same method of solution, namely Laplace transform and binomial series expansion employed to solve Eqs. 11 and 12, we obtain the following complete solution in terms of the new variables:

$$ {\overline{C}}_D=\left[\begin{array}{l}{\displaystyle \sum_{n=0}^{\infty }{\left(-1\right)}^nF(q)}+\\ {}{\displaystyle \sum_{n=1}^{\infty }{\left(-1\right)}^{n+1}n{x}_D{G}_1(q)}+\\ {}{\displaystyle \sum_{n=2}^{\infty }}{\displaystyle \sum_{m=2}^n{\left(-1\right)}^{n+m}}\left(\begin{array}{l}n\\ {}m\end{array}\right)\frac{x_D^m}{m!}G(q)\end{array}\right] $$

(A7)

where

$$ q=s+\lambda $$

(A8)

$$ F(q)= \exp \left(-q{x}_D\right)\left\{\begin{array}{l}\frac{ \exp \left(-{\alpha}_1\sqrt{q}\right)}{q}+\\ {}\frac{ \exp \left(-{\alpha}_2\sqrt{q}\right)}{q}\end{array}\right\} $$

(A9)

$$ {G}_1(q)= \exp \left(-q{x}_D\right)\frac{2}{\sqrt{q}}\left(\begin{array}{l} \exp \left(-{\alpha}_1\sqrt{q}\right)+\\ {} \exp \left(-{\alpha}_1\sqrt{q}\right)\end{array}\right) $$

(A10)

$$ G(q)=\left\{\begin{array}{l}{2}^m{q}^{\left(m-1\right)/2-1/2}\\ {} \exp \left(-q{x}_D\right)\left(\begin{array}{l} \exp \left(-{\alpha}_1\sqrt{q}\right)+\\ {} \exp \left(-{\alpha}_2\sqrt{q}\right)\end{array}\right)\end{array}\right\} $$

(A11)

$$ {\alpha}_1=2n{a}_D+{x}_D\kern0.5em \mathrm{and}\kern0.5em {\alpha}_2=2\left(n+1\right){a}_D+{x}_D $$

(A12)

Note that Eqs. A9–A11 differ from their counterparts Eqs. 32, 33, and 34, only with exp(−qx _D ) which is absent in Eqs. 32, 33, and 34. In the Laplace inversion process, this additional term introduces a Heaviside step function and shift in the time variable to yield:

$$ \begin{array}{l}F\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.36em \exp \left(-\lambda {t}_D\right)\\ {}\kern2.88em \left\{\begin{array}{l} erfc\left(\frac{\alpha_1}{2\sqrt{t_D-{x}_D}}\right)+\\ {} erfc\left(\frac{\alpha_2}{2\sqrt{t_D-{x}_D}}\right)\end{array}\right\}\end{array} $$

(A13)

$$ \begin{array}{l}{G}_1\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.36em \exp \left(-\lambda {t}_D\right)\\ {}\kern3em \left\{\begin{array}{l}\frac{2 \exp \left(-\frac{\alpha_1^2}{4\left({t}_D-{x}_D\right)}\right)}{\sqrt{\pi \left({t}_D-{x}_D\right)}}+\\ {}\frac{2 \exp \left(-\frac{\alpha_2^2}{4\left({t}_D-{x}_D\right)}\right)}{\sqrt{\pi \left({t}_D-{x}_D\right)}\kern0.24em }\end{array}\right\}\end{array} $$

(A14)

$$ \begin{array}{l}G\left({t}_D\right)=H\left({t}_D-{x}_D\right)\kern0.96em \\ {}\kern1.92em \exp \left(-\lambda {t}_D\right)\kern0.36em \frac{2^{\left(m+1\right)/2}}{\sqrt{\pi}\;{\left({t}_D-{x}_D\right)}^{m/2}\;}\\ {}\kern1.8em \left\{\begin{array}{l}{e}^{-\frac{\alpha_1^2}{4\left({t}_D-{x}_D\right)}}H{e}_{m-1}\left(\frac{\alpha_1}{\sqrt{2\left({t}_D-{x}_D\right)}}\right)+\\ {}{e}^{-\frac{\alpha_2^2}{4\left({t}_D-{x}_D\right)}}H{e}_{m-1}\left(\frac{\alpha_2}{\sqrt{2\left({t}_D-{x}_D\right)}}\right)\end{array}\right\}\end{array} $$

(A15))

Writing both Eqs. 35, 36, and 37 multiplied by the term exp(-λ₂ x _D) and Eqs. A13–A15 in terms of dimensional variables, one can see that they have the same arguments. The only difference the Heaviside step function, H(t _D  − x _D ), appearing in Eqs. A3–A15 is implicitly included in Eqs. 35, 36, and 37 as Eqs. 35, 36, and 37 are defined only for positive t _D values.

This verifies that dimensionless variables are correctly employed and Eq. 11 is indeed an unsteady state equation. Only algebra is minimized as an advantage.

Appendix B: Laplace inversion relation by Skopp and Warrick

Skopp and Warrick (1974) provide the following Laplace transform relation in their notable work.

$$ {F}_1(s)= \exp \left[-{x}_D\sqrt{q} \tanh \left(\sqrt{q}\right)\right] $$

(B1)

First, they derive the inverse transform of Eq. B1 given by Eq. B2.

$$ {F}_1=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\lambda \exp \left(-{\lambda}_R\right)\;Cos\left(\frac{\lambda^2\tau }{2}-{\lambda}_I\right)}\kern0.24em d\lambda $$

(B2)

where

$$ {\lambda}_R=\frac{x_D\omega }{2}\frac{Sinh\left(\omega \right)- Sin\left(\omega \right)}{Cosh\left(\omega \right)+Cos\left(\omega \right)} $$

(B3)

$$ {\lambda}_I=\frac{x_D\omega }{2}\frac{Sinh\left(\omega \right)+ Sin\left(\omega \right)}{Cosh\left(\omega \right)+Cos\left(\omega \right)} $$

(B4)

Let us rewrite Eq. 23 as follows:

$$ {\overline{C}}_D={e}^{-{\lambda}_2{x}_D}\frac{ \exp \left[-{x}_D\sqrt{q} \tanh \left(\sqrt{q}\right)\right]}{q} $$

(B5)

and its Laplace space variable q,

$$ q=s+{\lambda}_1 $$

(B6)

Based on the above solution given by Eq. B2 and using the shifting and convolution properties of Laplace transform (Abramowitz and Stegun 1972; Bateman and Erdelyi 1954) the inverse of Eq. B5 can be written as follows:

$$ \begin{array}{l}{C}_D={e}^{-{\lambda}_2{x}_D}\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\lambda \exp \left(-{\lambda}_R\right)d\lambda}\;\\ {}\kern1.44em {\displaystyle \underset{0}{\overset{t_D}{\int }} \exp \left(-{\lambda}_1\tau \right)Cos\left(\frac{\lambda^2\tau }{2}-{\lambda}_I\right)\;}d\tau\;\end{array} $$

(B7)

carrying out the integration with respect to τ, we obtain the following:

$$ {C}_D=\frac{2{e}^{-{\lambda}_2{x}_D}}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-{\lambda}_R}\;\frac{\left[g\left({t}_D\right)+g(0)\right]}{\lambda_1^2+\frac{\lambda^2}{4}}}d\lambda $$

(B8)

where

$$ g\left({t}_D\right)={e}^{-{\lambda}_1{t}_D}\left(\begin{array}{l}\frac{\lambda^2}{2}\mathrm{Sin}\left(\frac{\lambda^2{t}_D}{2}-{\lambda}_I\right)-\\ {}\;{\lambda}_1Cos\left(\frac{\lambda^2{t}_D}{2}-{\lambda}_I\right)\end{array}\right) $$

(B9)

$$ g(0)=\left(\frac{\lambda^2}{2} Sin\left({\lambda}_I\right)+{\lambda}_1Cos\left({\lambda}_I\right)\right) $$

(B10)