1. Introduction
Reactive transport in porous media is important for a number of natural and engineering processes, including geochemical cycling, in situ mining, and groundwater remediation. In any of these applications, a plume of the reagent is introduced—either naturally or deliberately—into the resident groundwater. The reaction depends on mixing the reagent with the groundwater, which fundamentally depends on molecular diffusion, but practically depends on a process called plume spreading. Plume spreading transforms the reagent plume into a fractal-like network of lamella that is thin enough for molecular diffusion to bring reagents together. Because flows in porous media are typically laminar, which precludes the turbulence that provides mixing in other engineered reactors, reactions in porous media are transport-limited. Accordingly, the transport of reagents in porous media is governed by the process of plume spreading.
Plume spreading can be classified as passive or active. Passive spreading results from the heterogeneity that is inherent in essentially any natural porous media. Finding the paths of least resistance, the fluid establishes channels of preferential flow, and the resulting velocity contrasts enhanced plume spreading compared to a hypothetical baseline of homogeneous media. In this context, mass transport by transverse dispersion is known to be an important process [
1]. By contrast, active spreading results from the deliberate manipulation of the velocity field through an approach called engineered injection and extraction, for example, through vertically separated segments of the well screen [
2] through a manifold of wells [
3,
4] confirmed by laboratory testing [
5,
6], or through a rotated dipole mixer [
7] confirmed by field testing [
8]. The present study proposes a new approach to active plume spreading by heating the injected water. Rather than imposing an engineered velocity field, this approach seeks to enhance plume spreading through the fundamental physics of fluid displacement.
Fluid displacement is the process by which a certain fluid, called the defending fluid, is replaced by a different fluid, called the invading fluid. This process can be classified as stable or unstable. Stable displacement causes the complete replacement of the defending fluid by the invading fluid, for example, when a more viscous fluid displaces a less viscous fluid of equal density or when a dense fluid displaces a light fluid from below. Neglecting hydrodynamic dispersion, stable displacement manifests itself as the plug flow, which is the default conceptual model for many environmental treatment unit operations and for groundwater remediation hydraulics, including pump-and-treat and engineered injection and extraction. By contrast, unstable displacement causes an unstable interface between the defending and invading fluids, causing incomplete replacement (
Figure 1). In the context of engineered reactive transport, this unstable interface generates the additional plume spreading of the invading fluid into the defending fluid, called
fingering, which provides an opportunity for enhanced mixing by molecular diffusion, and, consequently, a more complete reaction.
Unstable displacement results from various combinations of interfacial tension, density difference, or viscosity difference. When the interfacial tension is nonzero, the displacement is called immiscible, examples of which include enhanced oil recovery and the removal of non-aqueous phase liquids (NAPLs). When the interfacial tension is zero, the displacement is called miscible, for example, when an aqueous chemical or biological amendment is injected into a contaminated groundwater aquifer, which is the focus of the present study. When miscible fluids have an unequal density, a lighter invading fluid fingers into a denser defending fluid during the upward flow, and a denser invading fluid fingers into a lighter defending fluid during the downward flow through a process called gravity fingering [
10]. In contrast, for constant-density fluids, gravity fingering is prevented. While it is certainly possible to imagine a groundwater remediation application where the injected aqueous amendment has a significant density difference from the defending groundwater, the focus of the present study is on the viscous fingering of miscible fluids with a constant density.
Viscous fingering results when a less viscous fluid breaks through the miscible interface and creates a new pathway into the more viscous fluid [
11,
12]; viscous fingering always occurs when a less viscous fluid displaces a more viscous one, regardless of miscibility [
13]. Viscous fingering can result from the native viscosity difference of the fluids [
14,
15] or from varying the injection rate over several orders of magnitude to create new flow regimes [
16]. Here, we considered the viscous fingering caused by an imposed temperature that renders a viscosity difference between otherwise identical defending and invading fluids.
Viscous fingering between miscible fluids (
Figure 1) depends on two dimensionless numbers: the mobility ratio
R, and the Péclet number,
Pe [
15,
17]. The mobility ratio, also called the log-viscosity ratio [
17], quantified the viscosity difference between two miscible fluids, where
R > 0 was required for viscous fingering, and larger values of
R indicated that viscous fingering was more likely. The mobility ratio can be defined as:
where
μ1 is the dynamic viscosity of the defending fluid and
μ2 is the dynamic viscosity of the invading fluid. The Péclet number is a dimensionless ratio of advection to diffusion, which quantifies the general pattern of advection, imposing fine structure on plumes, while diffusion smooths out fine structure. The Péclet number is defined as:
where
v is the fluid velocity,
L is a characteristic length, and
D is the fluid’s self-diffusion constant.
There have been several studies involving temperature as a factor in groundwater remediation. Kaslusky and Udell [
18] injected steam to enhance the removal of volatile organic compounds (VOCs), especially dense non-aqueous phase liquids (DNAPLs), from groundwater but did not specifically address mixing, spreading, or fingering. Kosegi et al. [
19] found that increasing the temperature across the entire system in an aquifer remediation simulation resulted in faster cleanup times, but their study only noted changes in viscosity without considering changes in plume morphology. Similarly, Payne et al. [
20] identified thermal effects in groundwater remediation hydraulics but did not consider plume morphology. Jackson et al. [
21] modeled temperature difference between two immiscible fluids that already had a difference in viscosity, and they found that increasing the temperature difference increased the interfacial area, echoing similar results achieved by a model by Islam and Azaiez [
22] that assumed the fluids were miscible. Both of these studies graphically analyzed the interfacial length between the invading plume and the defending fluid. However, a review of the literature has yet to identify research in terms of which enhanced plume spreading can be achieved by heating the invading fluid, thus lowering its viscosity. Accordingly, the novel aspect of the present study was to explore plume spreading by injecting a hot invading fluid into a cold defending fluid.
4. Discussion
The proof-of-principle experiments presented here show the potential for the thermally enhanced spreading of injected plumes of miscible fluids in porous media. Elongated plume perimeters occur with a mobility ratio of
R = 1.2 and a Péclet number (as defined in
Table 1) of
Pe = 40, and this observation is independent of the image analysis method chosen, since both binary and morphological analysis result in similar results. The binary image analysis (
Figure 4 and
Figure 5) renders a fractal-like plume geometry with increases in perimeter compared to the isothermal control. The complementary morphological image analysis (
Figure 6 and
Figure 7) renders a solid-like plume geometry also with increases in the perimeter. The combination of these methods provides a more in-depth understanding of the thermally enhanced plume spreading of miscible plumes in porous media.
It is notable that elongated plume perimeters can be generated even within the limited temperature range of liquid water that constrains the maximum possible mobility ratio. Using Equations (1) and (3), the maximum temperature range of 0 °C to 100 °C corresponds to a maximum viscosity range of
μ1 = 1.8 × 10
−3 kg m
−1 s
−1 to
μ2 = 2.8 × 10
−4 kg m
−1 s
−1, which corresponds to a maximum theoretical mobility ratio of
R = 1.9. Practically, the lower temperature is more or less fixed, perhaps reflecting some seasonal variation, but seldom comes close to freezing. The temperature
T1 = 11 °C used here is probably a reasonable figure for temperate climates. Similarly, although the injection fluid can be heated to boiling, its temperature upon injection is limited by heat loss during fluid handling. If the tubes used in a field application are larger than the 6.4 mm (1/4 in) diameter tubes used here, the smaller area-to-volume ratio limits heat loss; if the delivery time
t =
V/
Q in a field application is smaller, this could also limit heat loss. Practically, the higher temperature may be higher than the
T2 = 73 °C used here but seldom comes close to boiling. This limitation contrasts the present study with prior work by others, where higher temperatures generated steam and, consequently, introduced the immiscible displacement of water by steam. Such higher temperatures have been used in the remediation of NAPLs [
18]. In contrast, the present study demonstrates the ability to elongate plume interfaces within a temperature range that one might expect in real aquifers and allows strictly miscible displacement.
The experiments reported here show more plume spreading with a decreasing Péclet number opposite the expectation for miscible plume spreading by viscous fingering [
29,
30], but is at least qualitatively consistent with the results of Videbæk and Nagel [
9], as shown in
Figure 1, where the left panels show the suppression of 3D fingers and the right panels show the development of 2D fingers with a decreasing Péclet number. The present study differs from these three in at least two respects. First, the present study measures plume spreading in porous media rather than Hele–Shaw cells. Second, the present study generated plume spreading thermally, so the viscosity difference, and therefore, mobility ratio, depends on both fluid mixing and thermodynamics. That is, given enough time, the two fluids would reach thermal equilibrium with an equal viscosity and mobility ratio
R = 0. Accordingly, the results presented here may be somewhat counterintuitive because lower Péclet numbers imply lower velocities and correspondingly more time for the two fluids to reach thermal equilibrium, which drives the mobility ratio back toward zero. The observation of increased plume perimeter suggests that the time scale for elongating plume interfaces is shorter than the time scale for the thermal equilibrium, at least in the experiments reported here.
Another manifestation of the Péclet number effects could be observed in both the binary image analysis (
Figure 5) and the morphological image analysis (
Figure 7). In both figures, the perimeter of the thermally enhanced invading plume began to exceed that of the isothermal control after 40 mL of the cumulative injection volume. This transition was observed only in experiment 4 with the smallest Péclet number
Pe = 40; it was not observed in other experiments with larger Péclet numbers (
Table 1). This observation suggests that there is a critical Péclet number above which little thermally enhanced plume spreading occurs. In experiment 4, when the cumulative injection volume was 30 mL or less, the Péclet number was too high; when the cumulative injection volume was 40 mL or more, the Péclet number was low enough. In experiment 3, when the cumulative injection volume was 60 mL or less, the Péclet number was too high, which was similar to experiments 2 and 1, which had even higher injection rates. Accordingly, experiment 4 suggests a critical Péclet number in the range of 40–46, while experiment 3 suggests that the critical Péclet number is less than 65. Taken together, these results suggest that thermally enhanced plume spreading might have been expected in experiment 3 at a cumulative injection volume of 160 mL (although this larger volume would correspond to a longer injection time which could allow thermodynamics to eliminate the mobility ratio as discussed above).
Further experiments are required to address the limitations of this proof-of-principle study. First, the assumption of the equal density of invading and defending fluids should be tested. Second, a modified apparatus could prevent the inlet fitting and supply tubing from appearing in the plume images and using a deeper chamber could determine whether a fully 3D apparatus might reveal experimental artifacts in our quasi-2D apparatus. Third, additional experiments are required to further constrain the critical Péclet number and to determine whether enhanced plume spreading at lower injection rates (i.e., lower Péclet number) could be suppressed by the thermal equilibrium resulting from the additional injection time. Fourth, additional experiments are required to extend these results to 3D flows and reactive transport. For example, delivering hot amendments could accelerate reactions not only by improving plume spreading but also by hastening reaction kinetics. On the other hand, boiling (or nearly boiling) the injection fluid could preclude injecting amendments that are volatile, thermally unstable, or biologically active. Having stated these limitations, the observation of elongated plume interfaces in this experiment suggests that heating the injection fluid may increase the size and extent of the reactive interface between the injected plume and the native groundwater, which, in turn, may result in a larger volume of remediated groundwater. The results of the present study are the first steps toward quantifying the effectiveness of thermally enhanced plume spreading as a tool for in situ groundwater remediation.