(I-356) A two-phase porous media flow model based on the incompressible Navier-Stokes equation

Introduction:: This paper presents a new two-phase porous media flow model based on the incompressible Navier-Stokes equation. This model aims to address the limitations of existing methods by indicating partial saturation distribution in porous media. The model's basic parameters are the mean pore size, standard deviation, and contact angle. To validate the model, we solved five analytical solutions and implemented experiments. The experiment measured penetration length data agree with the model predictions, demonstrating its accuracy. In addition, the model correctly solves the droplet spreading problem that the Richards equation fails to address. Our findings suggest that the new two-phase porous media flow model can provide a valuable tool for researchers in fields such as lateral flow tests and paper chromatography.

Materials and Methods:: We used nitrocellulose membrane (AE99, 13401787, Whatman) as the experimental porous media. To achieve the desired geometries, a laser cutter (VSL350, Universal Laser Systems) was used to cut the nitrocellulose sheets. In the two-dimensional scenarios, we employed PEEK tubing (1569, IDEX) and a syringe (1 mL, BD) to ensure a continuous water supply by maintaining a small droplet ($\sim$ 50 $\mu$L) around the point of contact between the tubing and the membrane, making the water absorbed and flow through the membranes. To obtain data on the relationship between penetration length and time, we took videos of the liquid flow process and analyzed them using ImageJ. To maintain data consistency, the experiments were always implemented in the same conditions.

We assume the pore size in the media obeys Gaussian distribution. The liquid flows in the pores with different velocities due to the pore size variation. The velocity difference causes partial saturation at the liquid front. Integrating the saturated pores and dividing by the total pores at arbitrary locations result in the water content.

Results, Conclusions, and Discussions::

In the 2D experiments, as the total water source volume diminished, the circular wetting region's migration eventually stopped upon depletion. Surface tension-induced capillary pressure served as the driving force for flow, with the pressure gradient arising from the difference between capillary pressure at the water/air interface and the atmospheric pressure of the water source. Depletion of water led to a balance in surface tension and a zero pressure gradient, halting the wetting region's expansion. The validation experiment using water droplets on a nitrocellulose membrane confirmed this behavior, where wetting area expansion ceased within approximately 2 seconds. Our model accurately reproduced these observations, unlike the Richards equation, which assumed capillary pressure as a function of water content and failed to capture the experimental outcome.

 

We want to highlight several key points in our study. Firstly, it is essential to ensure that the pores within the porous media are randomly distributed to maintain consistent macroscopic properties such as porosity and permeability throughout the entire domain of interest. Secondly, our analytical solutions only consider capillary pressure without other pressures, and the influence of gravity was negligible. Our model is based on the Navier-Stokes equation, allowing for the inclusion of additional pressures in the pressure gradient term.

 

We have demonstrated the limitations of the Richards equation in two specific scenarios, highlighting the incorrectness of Richards' assumption regarding capillary pressure as a function of water content. To address this issue, we have developed a novel two-phase porous media flow model based on the Navier-Stokes equation and solved five analytical solutions. Experimental data agreed closely with the model solutions, confirming the validity and accuracy. Furthermore, the applicability of this model extends beyond our specific study, as it can be readily employed in other research areas where two-phase porous media flow is of significant interest.

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