Enhanced landfill’s characterization by using an alternative analytical model for diffusion tests

Abstract

Landfills have been extensively used as waste deposits in most of the big cities in the world. Therefore, considerably large urban areas have contamination threats. Engineering solutions applied to prevent or contain soil, and water contamination often involve the application of liners, which are low permeability barriers made of materials such as compacted clay and geomembranes. In many liner applications, once reduced rates of seepage are expected, diffusion has proved to be relevant, if not dominant, to the process of contaminant transport. Normally, diffusivity parameters can be assessed by a single-reservoir pure diffusion test, where a contaminant solution is placed above a saturated soil sample and the solution’s concentration is monitored over time. Once the temporal variation of concentration is measured, the process of back-calculating diffusivity parameters is not standardized. In this paper, an analytical model of the diffusive transport of contaminants is revisited considering the initial and boundary conditions of the pure diffusion test. In this model, the contaminant solution reservoir is included in the analysis domain as an equivalent contaminated soil layer. The analytical solution relies on a series evaluation, which may be a drawback to everyday engineering situations. Therefore, we build a high-accuracy exponential approximation to the solution. Expedited evaluation procedures are proposed to provide reasonable estimates for the fitting parameters. Also, in order to illustrate the applicability of the new solution, test datasets of a soil around the Jockey Club Landfill (JCL) site, one of the major landfills in Latin America, have been modeled. We discuss possible issues of considering linear isotherms to model the sorption characteristics of soils, indicating that convex isotherms, if linearly modeled, may lead to overestimated values of the diffusivity parameter.

Appendix

New analytical solution derivation

As discussed, the no-flux conditions can be mathematically represented as:

$$\begin{aligned} \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {c_w}(x = L + b,t> 0)}}{{\partial x}} = 0}\\ {\frac{{\partial {c_w}(x = 0,t > 0)}}{{\partial x}} = 0} \end{array}} \right. \end{aligned}$$

(30)

Besides, the initial condition for the contaminant concentration can be given as ${c_w}(x,t = 0) = {c_0}u(b - x)u(x)$ , where u(x) denotes the Heaviside step function, which is 0 if $x < 0$ and 1, otherwise. 6 In order to solve the PDE in Eq. (2) subjected to the initial and boundary conditions described, the Laplace transform with respect to time and the Finite Cosine Fourier transform with respect to space shall be used. Thus, let the Laplace transform $L_t \{\}$ of the concentration function $c_w$ be given as (Boyce & Diprima, 2008):

$$\begin{aligned} {L_t}\left\{ {{c_w}(x,t)} \right\} (r) = \int \limits _0^\infty {{c_w}(x,t){e^{ - rt}}dt} = {\widetilde{c}_w}(x,r) \end{aligned}$$

(31)

where r is the transformed time variable.

According to Boyce and Diprima (2008), the Laplace transform of the derivative of a function with respect to the transforming variable can be given as:

$$\begin{aligned} {L_t}\left\{ {\frac{\partial }{{\partial t}}{c_w}(x,t)} \right\} (r) = r{\widetilde{c}_w}(x,r) - {c_w}(x,0) \end{aligned}$$

(32)

On the other hand, according to Sneddon (1951), the Finite Cosine Fourier transform $F_t\{\}$ can be applied to the function $c_w$ considering its spatial domain from 0 to $L + b$ and leads to:

$$\begin{aligned} {F_t}\left\{ {{c_w}(x,t)} \right\} (m) = \frac{2}{{L + b}}\int \limits _0^{L + b} {{c_w}(x,t)\cos \left( {\frac{{m\pi x}}{{L + b}}} \right) dx} = {\widehat{c}_w}(m,t) \end{aligned}$$

(33)

where m is the transformed space variable.

Sneddon (1951) also explores the relation between the Finite Cosine Fourier transform of derivatives of the transformed function and states that:

$$\begin{aligned} {F_t}\left\{ {\frac{{{\partial ^2}}}{{\partial {x^2}}}{c_w}(x,t)} \right\} (m) = \\ - {\left( {\frac{{m\pi }}{{L + b}}} \right) ^2}{\widehat{c}_w}(m,t) - \frac{2}{{L + b}}\left[ {\frac{\partial }{{\partial x}}{c_w}(0,t) + {{( - 1)}^{m + 1}}\frac{\partial }{{\partial x}}{c_w}(L + b,t)} \right] \end{aligned}$$

(34)

This way, by applying the Laplace transform with respect to time on Eq. (2), the following is obtained:

$$\begin{aligned} r{\widetilde{c}_w}(x,r) - {c_w}(x,0) = \frac{{{D^*}}}{R}\frac{{{\partial ^2}{{\widetilde{c}}_w}}}{{\partial {x^2}}} \end{aligned}$$

(35)

Now, by both applying the Finite Cosine Fourier transform with respect to space on Eq. (35) and considering the conditions in Eq. (30), one shall get:

$$\begin{aligned} r{\widehat{\widetilde{c}}_w} - {F_t}\left\{ {{c_w}(x,0)} \right\} (m) = - \frac{{{D^*}}}{R}{\left( {\frac{{m\pi }}{{L + b}}} \right) ^2}{\widehat{\widetilde{c}}_w} \end{aligned}$$

(36)

where for simplicity, ${\widehat{\widetilde{c}}_w} = {\widehat{\widetilde{c}}_w}(m,r)$ is the double transformed function. The Finite Cosine Fourier transform of the initial concentration distribution is:

$$\begin{aligned} {F_t}\left\{ {{c_w}(x,0)} \right\} (m) = \frac{2}{{L + b}}\int \limits _0^{L + b} {{c_0}u(x - b)u(x)\cos \left( {\frac{{m\pi x}}{{L + b}}} \right) dx} = \frac{{2{c_0}}}{{m\pi }}\sin \left( {\frac{{bm\pi }}{{L + b}}} \right) \end{aligned}$$

(37)

Thus, from Eq. (36), one may get:

$$\begin{aligned} {\widehat{\widetilde{c}}_w} = \frac{{\frac{{2{c_0}}}{{m\pi }}\sin \left( {\frac{{bm\pi }}{{L + b}}} \right) }}{{r + \frac{{{D^*}}}{R}{{\left( {\frac{{m\pi }}{{L + b}}} \right) }^2}}} \end{aligned}$$

(38)

In order to obtain the function ${c_w}(x,t)$ from Eq. (38), both Laplace and Finite Cosine Fourier transforms need to be inverted. By applying the inverse transforms presented by Boyce and Diprima (2008) and Sneddon (1951), the solution in Eq. (5) follows.

Computational code

In order to make the usage of the formulas and the methodology hereby presented easier, a Python code is provided. The code below defines the function NLFitECL, which gets the dataset, performs the nonlinear regression, and outputs a plot with the best-fit curve. If specific experimental parameters are known ($H, L, n, K_d,\rho _d$), the value of $\alpha$ is calculated and the fitting parameters are $c_0$ and $D^*/RL^2$. Otherwise, when those parameters are not available, the best-fit parameters are $\alpha$, $c_0$ and $D^*/RL^2$. The R-squared value is also presented in the legend of the plot.

Important remarks:

• The function NLFitECL takes as inputs two lists or numpy arrays. The first one is made of the pairs $(t_i, c_i)$, while the last one is related to other experimental parameters $[H, L, n, K_d, \rho _d]$. If such parameters are not available, simply input an empty list as [].

• The units of $c_i$ and $t_i$ must be consistent with the values of the experimental parameters. For simplicity we consider units as mg/L and h, respectively.

• The value of $D^*/RL^2$ is outputted in units of time raised to -1, therefore $h^{-1}$.

• As previously discussed, in order to correctly obtain the diffusivity parameters, the experiment must be carried out until near-equilibrium is reached.