Mathematical Modeling of Laboratory-Scale Solute Transport in Porous Media

Transport processes in porous media are ubiquitous, and their manifestation depends on the scale of observation. The effect of pore scale processes is typically upscaled by considering full mixing and assuming full mixing within pores and Fickian transport fully defined by a mean pore velocity and a constant Darcy’s scale dispersivity. Theoretical development, supported by experimental results, evidenced the emergence of incomplete mixing accompanied by non-Fickian transport, which impacts the reactions upon mixing. We performed laboratory scale experiments in a porous media created by hydrogel spheres with a refraction index matching that of the water. Rhodamine B was injected at the inlet of a 9 cm x 9 cm x 17 cm sample, and its spreading was monitored by using Planar Laser-Induced Fluorescence (PLIF),  which provided the distribution of the concentration within a control plane normal to the mean flow direction at a high resolution. The images were obtained at constant time intervals of two seconds.

The PLIF images were processed by calibrating pixel intensity values against Rhodamine B concentrations using standard PLIF calibration procedure. This calibration enabled the determination of spatial concentration distributions within the imaging plane.  Breakthrough curves (BTCs) were obtained from these image data, and variance was computed at each time step. The breakthrough curve (BTC) provides a macroscopic representation of solute transport, capturing the temporal evolution of solute concentrations at a downstream control plane. In a Lagrangian framework, the BTC is determined by displacement moments, which describe the key characteristics of the transport process.

Three models for displacement moments were analyzed. The classical Fickian model considers dispersivity values in the longitudinal and transverse directions. The stochastic macrodispersion model (Dagan, 1989) uses medium variance and its integral scale. The extended Saffman model incorporates sphere diameters and interstitial flow speed as its parameters.The Fickian model provided dispersivity values consistent with those reported by Eames and Bush (1999) for a medium composed of impermeable spheres. However, it struggled to capture the early and late parts of the BTC, indicating that incomplete mixing and pre-Fickian transport behavior can occur even at the laboratory scale.In contrast, the stochastic macrodispersion model and the extended Saffman model yielded more accurate results. Both models successfully reproduced BTCs across the entire observation period, including early arrivals and tails. Additionally, the Saffman model effectively represented physical properties of the medium, such as interstitial velocity and pore size, which aligned with the measured values.

 

References 

1. G. Dagan. Flow and transport in porous formations. Springer-Verlag, New York, 1989

2. Eames, I. and Bush, J.W.M., 1999. Longitudinal dispersion by bodies fixed in a potential flow. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455(1990), pp.3665-3686.

3. P. G. Saffman. A theory of dispersion in a porous medium. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 251(1264):313–328, 1959.