Pore-Scale Flow Mechanisms and the Hydrodynamic Porosity of Porous Media in Surface Water Treatment and Groundwater Remediation

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2025-09-14

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2023

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Abstract

As climate change and growing demand exacerbate water scarcity, it will become more imperative than ever to remediate our natural resources and treat our waste streams. This is especially true if we are to successfully provide clean water for all and ensure the future of endangered species and habitats. Thus, we look to surface water treatment technologies (e.g., granular media and filtration membranes) and groundwater remediation strategies (e.g., the vertical circulation well, rapidly pulsed pump and treat, and bioremediation) to add to our freshwater stores and reduce environmental pollution.

Complicating the matter is the fact that both surface water treatment and groundwater remediation are reliant, to varying degrees, on flow through porous media. Even without the added complexities of multiphase flows, immiscible fluids, and the time-dependent processes associated with chemical reactions and biofouling, characterizing flow through porous media, properly, is a cumbersome and arduous task. Heterogeneities in the morphology of the medium range from the pore scale, to, in the case of groundwater flows, meters. Resulting is a random distribution of the shape, size, and connectivity of the pore space. To quantify flow through porous media, researchers are forced to either make a set of simplifying assumptions, some more appropriate than others, or more recently, use black-box machine learning models that have little basis in the physicality of the flow. In this work, we choose to focus on one of the standard assumptions researchers make when calculating the pore-scale velocity (i.e., the supposed “static” nature of flow porosity). In relaxing this assumption, we provide a paradigm shift in the modeling of flows through porous media. We apply our theory to flow through and along the walls of microporous membranes, granular media, and aquifer substrates.

We choose to study pore-scale flow velocity because it is an essential parameter in determining transport through porous media, but it is often miscalculated. Researchers use a static porosity value to relate volumetric or superficial velocities to pore-scale flow velocities. We know this modeling assumption to be an oversimplification. The porosity conducive to flow, what we define as hydrodynamic porosity, exhibits a quantifiable dependence on Reynolds number (i.e., pore-scale flow velocity) in the laminar flow regime. This fact remains largely unacknowledged in the literature. In this work, we quantify the dependence of hydrodynamic porosity on Reynolds number via numerical flow simulation at the pore scale. We demonstrate that, for the tested flow geometries, hydrodynamic porosity decreases by as much as 42% over the laminar flow regime. Moreover, hydrodynamic porosity exhibits an exponential dependence on Reynolds number. The fit quality is effectively perfect, with a coefficient of determination of approximately 1 for each set of simulation data. We then demonstrate the applicability of this model by validating a high fit quality for a range of rectangular and non-rectangular cavity geometries. Finally, we show that this exponential dependence can be easily solved for pore-scale flow velocity using only a few Picard iterations, even with an initial guess that is over 10 orders of magnitude off. Not only is this relationship a more accurate definition of pore-scale flow velocity, but it is also a necessary modeling improvement that can be easily implemented.

In the chapters that follow our introduction of hydrodynamic porosity, we apply the concept to subsurface flow modeling for groundwater remediation via the vertical circulation well and flows over patterned membrane surfaces for surface water treatment – supposing that a hydrodynamic porosity parameter could be defined for the surface pattern of a membrane and then correlated to the rate of particle deposition (and therefore fouling) at the membrane surface.

In the future, we aim to explore the applicability of the hydrodynamic porosity model to microporous membrane wall flows. Although the characteristic length scale of the membrane wall is admittedly much smaller than the characteristic length scale of granular media, microporous membranes, like granular media, have dead-end pores. Thus, it remains necessary to determine the effect of these dead-end pore volumes on membrane wall flows. Preliminary experimental data previously collected from a hollow-fiber ultrafiltration membrane will be used to verify our numerical results.

Following our study of steady flows, we pivot to the analysis of rapidly pulsed flows and the mixing mechanisms these flows induce at the pore scale (i.e., the deep sweep and vortex ejection) in cavities and other effectively immobile zones. These mechanisms have been shown to significantly reduce contaminant recovery time in media with significant immobile zone volume. This finding suggests substantial cost-savings for treatment and remediation methods that utilize rapidly pulsed flows.

Regarding groundwater remediation, we estimate that the cost savings from utilizing rapidly pulsed flows could be on the order of magnitude of 100 billion USD. But this calculation assumes that we can remediate the entirety of a contaminated groundwater matrix with the mixing mechanisms induced by rapidly pulsed pump-and-treat. In application, induced oscillations will only reach a small volume of the flow field before dissipating to a negligible amplitude. Equally important, these oscillations will only induce a deep sweep or vortex ejection if the mean pore-scale flow velocity is above a Reynolds number of 0.1. Following, we use our model of hydrodynamic porosity to determine the magnitude of the volume we expect to benefit from rapidly pulsed pumping in a vertical circulation well.

Finally, given the similarity in characteristic length scale, we liken flow in the dead-end pore space of groundwater matrices, to flow past the channels in patterned membrane surfaces. We find that for the studied surface pattern, the vortex ejection and deep sweep are still present in highly laminar flows (i.e., a Reynolds number of 1600 for pipe flows). We hypothesize that these mechanisms can prevent particle deposition at the membrane surface, and when used as a cleaning mechanism, can remove loose deposits that would otherwise adhere to the membrane surface. It is also likely that these mechanisms would speed up the regeneration of fouled granular media used to remove suspended solids, microorganisms, and organics (i.e., sand and granulated activated carbon) from wastewater.

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Frechette, August (2023). Pore-Scale Flow Mechanisms and the Hydrodynamic Porosity of Porous Media in Surface Water Treatment and Groundwater Remediation. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/29207.

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