# Euler-Euler simulations of colloidal transport and retention at the pore scale

Nassim Cheikh, Emmanuel Le Trong, Cyprien Soulaine

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Nassim Cheikh, Emmanuel Le Trong, Cyprien Soulaine. Euler-Euler simulations of colloidal transport and retention at the pore scale. 17èmes Journées d'Étude des Milieux Poreux, Nov 2025, Orléans, France., 2025. hal-05358624

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Submitted on 11 Nov 2025

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# Euler-Euler simulations of colloidal transport and retention at the pore scale

Nassim Cheikhª, Emmanuel Le Trongª, Cyprien Soulaineª

ªInstitut des Sciences de la Terre d'Orléans, Orléans, France

**Keywords:** Euler-Euler simulation, Clogging, NAPL

Non-aqueous liquids (NAPLs) are water-immiscible pollutants that can disperse and release significant amounts of toxic compounds into underground water resources. Numerous methods have been developed to remove these pollutants from aquifers, but NAPL droplets remain trapped by capillary forces in the porous matrix even after depollution. The aim of our research is to explore the potential of colloids (e.g. nanoparticles, bacteria) to remobilize immiscible pollutant droplets. Our final objective is to develop a multiphase model at the pore scale that includes colloidal interactions with the solid phase, the aqueous phase, and the NAPL. Euler-Lagrange models allow an accurate description of colloidal particle interactions (e.g. transport and retention) but remains very time- and resource-intensive. In this work, we are developing Euler-Euler approaches based on OpenFOAM's multiphaseEuler module to represent particles as a dispersed phase in water. The use of this model in the context of the remediation of polluted porous media is currently under development. Our package includes new viscosity models and plastic behavior, as well as poromechanic properties to account for the compressibility threshold of clogged particles. Aggregation forces in between particles and between particles and grains are also considered. To this end, we are using DLVO theory, which characterizes the forces exerted between two charged surfaces separated by a liquid. We present preliminary results such as: i) comparisons with Euler-Lagrange simulations for very diluted suspension, ii) response to fluid stresses and its compressibility, and iii) study of the impact of particle aggregation in the context of porous media.
## INTRODUCTION

* **Context:** Non-aqueous phase liquids (NAPLs) are water-immiscible pollutants that can contaminate groundwater by releasing toxic compounds. Despite various remediation methods, NAPL droplets often remain trapped in porous media due to capillary forces.

* **Objectives:** Our research investigates the potential of colloids—such as nanoparticles or bacteria—to mobilize these trapped pollutants.

* **Challenge:** While Euler-Lagrange models offer detailed insight into colloid transport and retention, they are computationally demanding. To address this, we are developing a more efficient Euler-Euler approach using OpenFOAM®'s multiphase Euler module, treating colloids as a dispersed phase in water.

## PORE-SCALE MODEL

<figure><img src="image_2.png" /></figure>

$$ \left\{ \begin{aligned} 0 &= -\phi_f \nabla \hat{p}_f + \phi_f \nabla \cdot \mu_f (\nabla \hat{u}_f + (\nabla \hat{u}_f)^T) - \phi_f^2 \mu_f K^{-1} (\mathbf{u}_f - \mathbf{u}_s) \\ 0 &= -\phi_s \nabla \hat{p}_s - \nabla \hat{p}_s + \phi_s \nabla \cdot \mu_s^{eff} \mu_f (\nabla \hat{u}_s + (\nabla \hat{u}_s)^T) + \phi_f^2 \mu_f K^{-1} (\mathbf{u}_f - \mathbf{u}_s) \end{aligned} \right. $$

<table><tr><td>+ Mass conservation</td></tr><tr><td>+ Boundary Conditions</td></tr></table>

* **Drag forces :**

* Stokes spherical particles, $Re \ll 1$

* Schiller-Naumann spherical particles, $Re < 800$

* Wen Yu for low $\phi_s$ and Ergun for high $\phi_s$. Considers particle-particle interaction

* **Solid pressure :**

* Gidaspow and Ettehadieh (1983) exponential dependence on $\phi_s - \phi_{s,max}$

* Boyer et al. (2011) shear depending solid pressure

## COMPRESSION IN A SINGLE PORE

* Column half-filled with a **solid phase** of a volume fraction of 0.45

* The liquid phase can pass through the lower boundary condition, **but not the solid phase**

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Compression as a function of fluid pressure</figcaption></figure>

<figure><img src="image_5.png" /></figure>

## REFERENCES

1. Maya Fogouang et al., 2025, "Particulate transport in porous media at pore-scale. Part 1: Unresolved-resolved four-way coupling CFD-DEM"

2. Soulaine, 2024, "Micro-Continuum Modeling: An Hybrid-Scale Approach for Solving Coupled Processes in Porous Media"

3. Hewitt et al., 2016, "Flow-induced compaction of a deformable porous medium"

4. Boyer et al., 2011, "Unifying Suspension and Granular Rheology"

## CLOGGING IN A SINGLE PORE

<figure><img src="image_6.png" /><figcaption>Ratio of solid volume fraction to maximum solid volume fraction averaged on the radius along the channel</figcaption></figure>

## PERMEABILITY REDUCTION TEST

* Injection of a **dilute suspension** during 4 Pore Volumes

* Followed by injection of clear water to **flush out** the solid phase

<figure><img src="image_7.png" /><figcaption>Simulation of permeability damage evolution during injection of a particulate phase (0.05%)</figcaption></figure>

<figure><img src="image_8.png" /></figure>

## CONCLUSION

* Validation of the model with **Lagrangian** approach for permeability damage and **analytical solution** for compression test

* Adding a **solid pressure** and **effective viscosity** model according Boyer et al.

* Ability of simulating **clogging by sieving** in constricted channel

* **Perspectives:** sensitivity analysis of parameters for clogging in a single pore, simulation of bridging in complex geometry

## ACKNOWLEDGEMENTS

We would like to acknowledge European Union for founding this project (ERC COCONUT n°101043288)