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There is a shortage of high quality drinking water caused by the introduction of contaminants into aquifers from various sources including industrial processes and uncontrolled sewage. Studies have shown that colloids, collections of nanoparticles, have the potential to remediate polluted groundwater. For such applications of nanoparticles, it is important to understand the movement of colloids. This study aims to enhance the previously developed MNM1D (Micro- and Nanoparticle transport Model in porous media in one-dimensional geometry) by making more realistic assumptions about physical properties of the groundwater-porous medium system by accounting for a non-constant flow velocity and the presence of electromagnetic interactions. This was accomplished by coupling the original model with the Darcy-Forchheimer fluid model, which is specific to transport in porous media, coupled with electromagnetic effects. The resulting model also accounts for attachment and detachment phenomena, both of the linear and Langmuirian type, as well as changes to hydrochemical parameters such as maximum colloidal particle concentration in the porous medium. The system of partial-differential equations that make up the model was solved using an implicit finite-difference discretization along with the iterative Newton’s method. A parameter estimation study was also conducted to quantify parameters of interest. This more realistic model of colloid transport in porous media will contribute to the production of a more efficient method to counteract contaminants in groundwater and ultimately increase availability of clean drinking water.

According to a 2013 report of the World Health Organization ([

The STANMOD suite builds off analytic models developed from 1980, including CFITM and CXTFIT, which accounts for particle degradation and the equilibrium state at a single reaction site. The numerical HYDRUS model released in 1991 includes temperature dependent kinetic coefficients and allows for two reac- tion sites [

Although the existing model encompasses the previous mentioned properties, there are still additional im- provements that can be made. In this study, the flow velocity of the fluid was assumed to be non-constant, governed by the Darcy-Forchheimer fluid model. The electromagnetic interactions that occur in the fluid were also introduced by adding an electric field term based on the Helmholtz-Smoluchowski equation to the men- tioned fluid model. In addition, a parameter estimation algorithm was implemented for a single kinetic coefficient. These parameters govern the ionic strength of the surrounding groundwater, potentially giving researchers the ability to attribute certain characteristics to the liquid.

The structure of the paper is as follows. Section 2 introduces the mathematical model and governing dif- ferential equations. Section 3 describes the implementation of the governing system via implicit finite-difference discretization. In Section 4, we present computational experiments for a benchmark application that involves step-wise injection of nanoparticles. This application, which was presented in [

Transport of colloids is governed by advection-dispersion phenomena and particle-soil interactions. Colloidal particle transport in porous systems is modeled using a modified advection-dispersion equation. It describes the dual-phase, non-equilibrium interactions between particles in the liquid and solid phase. Colloid deposition on the grain surface is generally referred to as attachment, colloid release from the surface as detachment. Earlier theories, such as the classical filtration theory (CFT) consider first-order attachment kinetics, but consider the detachment ones to be negligible. However, MNM1D includes both phenomena in the particle-soil interactions. The physical system described is summarized in

Several 1D forms of the model have been proposed, each taking into account some or all of the above characteristics. However, we can write the general form of the modified advection-dispersion equation for the colloid movement taking into account the liquid-solid phase transfer:

where:

Many types of reaction sites exist, and thus there are many models that describe the mass transfer between the solid and liquid phases. MNM1D takes into account two reaction sites. The first is assumed to have blocking phenomena, and so the colloid concentration at the site is limited to a fixed values

where:

The second site is assumed to have a linear exchange term, and so the transfer can be described as:

where:

The hydrochemical parameters, such as ionic strength, can be correlated with the concentration of some conservative tracer, such as salt. The transport of the tracer can also be described by the advection-dispersion equation as:

where:

The concentration of the tracer ultimately affects the values of the attachment/detachment kinetic coefficients. The functional relationship between the salt concentration and kinetic coefficients below was proposed in [

where:

Finally, we now introduce a fluid model to describe the flow velocity of the groundwater over time and space. Here, we will implement the Darcy-Forchheimer model [

where:

We now wish to introduce how the presence of electromagnetics affects this model. To do this, we will turn to a modified version of the above fluid model goverened by the Helmholtz-Smoluchowski (H-S) equation as discussed in [

where:

As described by Fourie, the fluid model then becomes:

where:

In this section, we will develop an implicit finite difference scheme to solve the coupled system Equations (5)- (14). To rewrite the derivatives of

where

which implies

We now see that in order to solve for pressure, we require initial values for

We can solve for

We see here that

Note, for the sake of simplicity, we will assume

We can now solve for our interested quantities,

and by a similar method, we can solve for

where

can write Equations (11)-(14) in matrix form as

where

With

Due to the non-linearity that exists in matrix

where the Jacobian

Equation (17) can be rewritten as:

The solution is now obtained iteratively by calculating the residuals

In this section, we perform the following computational experiment which will be used to validate the results from [

We will first compare the models at a point that is close to the source of the nanoparticles,

We will now compare the models at a point that is farther away from the nanoparticle source,

Variable | Value |
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To permit users to easily test the model, a graphical user interface (GUI) was created. This allows users to enter model parameters including physical characteristics of the system, affecting both the dispersion of the colloids and the electric field, and the particle-soil interaction kinetic coefficients. The GUI presents the colloidal con- centrations at both of the above positions,

with user-entered parameters.

The kinetic coefficients in the description of the attachment and detachment phenomena determine the ionic strength of the colloids. Being able to estimate the values of these given experimental data would allow for approximating this characteristic of the groundwater that would ultimately help determine properties such as the mobility of radioactive isotopes in the system [

listically represent experimental data. The accuracy and error were recorded for different magnitudes of noise, the results are shown in

In this work, a new enhanced model that incorporates non-constant flow velocity governed by the Darcy-Forch- heimer fluid model as well as influence of electromagnetic interactions using H-S equation has been studied. We were not only able to validate the results in the absence of these effects in comparsion to published work, but were also able to show that these non-constant flow velocity and electromagnetic effects impact the colloid concentrations. Two other contributions include the development of a parameter estimation strategy to extract parameters of interest by comparing the computed data against experimental data. In this paper, the experimental data was generated by adding different levels of “noise” to the computed data. Finally, a graphical user interface, that helps users to enter model parameters and obtain the colloidal concentrations in real-time, was also pre- sented. The software that has been developed through this work provides flexible infrastructure to evaluate the influence of parameters such as

The current paper does not include explicitly the interaction of colloidal particles with various contaminants that can exist in groundwater. In a forthcoming paper, we wish to enhance our model by including this interac- tion as colloidal particles are well-known to remediate contaminants because of their unique physiochemical properties. Understanding this interaction will also help us to investigate how the colloidal particles can be used in the removal of various groundwater contaminants including chlrorinated compounds, pesticides, heavy metals, nitrates, and explosives in water ([