Grantee Research Project Results
Final Report: Effect of Heterogeneities on Instability and Fingering of DNAPLS Below the Water Table
EPA Grant Number: R826157Title: Effect of Heterogeneities on Instability and Fingering of DNAPLS Below the Water Table
Investigators: Neuman, Shlomo P. , Smith, James E.
Institution: University of Arizona , McMaster University
EPA Project Officer: Aja, Hayley
Project Period: March 1, 1998 through February 28, 2001
Project Amount: $330,000
RFA: Exploratory Research - Physics (1997) RFA Text | Recipients Lists
Research Category: Water , Land and Waste Management , Air , Safer Chemicals
Objective:
The objective of this research project is to analytically, experimentally, and computationally investigate the effect of soil heterogeneities on instability and fingering of dense non-aqueous phase liquid (DNAPL) contaminants below the water table in aquifers. Specific issues to be investigated include: (1) the conditions that may lead to instability when fronts of DNAPL propagate through water-saturated, randomly heterogeneous soils in multidimensional flow fields below the water table of aquifers; (2) the manner in which multidimensional fingers evolve in finite time intervals following the onset of DNAPL front instability; and (3) the effect of systematic trends in permeability, and high-permeability preferential flow paths, on DNAPL front instability and finger development.
Summary/Accomplishments (Outputs/Outcomes):
Chlorinated organic solvents such as trichloroethylene (TCE) and perchloroethylene (PCE) are among the most ubiquitous and problematic groundwater contaminants. They usually enter the subsurface in the form of organic liquids, which exhibit low miscibility with water. Though such solvents are nonwetting on mineral surfaces relative to water, their interfacial tension with water is low enough to allow penetration into narrow pores and fractures with relative ease. These solvents are heavier and often less viscous than water. DNAPLs exhibit enhanced downward mobility, which allows them to penetrate deep below the water table. Their downward movement tends to be unstable and to occur in the form of rapidly advancing narrow fingers. Though instability and fingering may occur in uniform porous media due to differences in density and viscosity between DNAPL and water across their interface, in nonuniform porous and fractured media they tend to develop preferentially along channels, or fractures, with elevated permeability. A recent review by Chen, et al., (1995) shows that little is known about the manner in which medium heterogeneity affects the development and propagation of unstable fingers between immiscible fluids. The goal of this project is to quantify and understand the relative importance of unstable DNAPL fingering, and preferential flow due to medium heterogeneity, in porous media below the water table.
Theory and Computation
In the theoretical and computational arenas, we successfully analyzed the movement of DNAPL within a three-dimensional randomly heterogeneous porous medium saturated with water (i.e., initially pooled above a water table or flows at a constant flux). We considered the front to form a sharp boundary at which the capillary pressure head, assumed equal to the entry pressure head of DNAPL, is prescribed either deterministically or randomly. We treated log conductivity as a statistically homogeneous random field with a given mean, variance, and covariance. We cast the corresponding boundary-value problem in the form of an integro-differential equation, in which the parameters and domain of integration are random. We expanded this equation in a Taylor series about the mean position of the front, and took the ensemble mean. To quantify the predictive uncertainty associated with this mean solution, we developed a set of integro-differential equations for the corresponding second ensemble moments. We solved the resulting moment equations analytically and numerically in one and two dimensions to second order in the standard deviation of log conductivity. A comparison of our one-dimensional solutions with the results of Monte Carlo simulations has verified its accuracy. We also have shown that a probabilistic analysis of wetting front instability due to Chen and Neuman (1996) application to a DNAPL front.
Our theoretical and computational work has led to the following conclusions:
1. Using a sharp front approximation, we derived stochastic equations and boundary conditions that control DNAPL propagation in randomly heterogeneous, saturated porous media. We found that these equations have a similar form to those describing the dynamics of a wetting front in randomly heterogeneous, unsaturated soils.
2. In line with Tartakovsky and Winter (2001), we proposed a new moment approach to the analysis of the stochastic equations by converting them into integro-differential equations. Tartakovsky and Winter had approximated the domain and boundary of integration by their mean counterparts, resulting in what they termed a "leaner solution." Their approximation introduces a systematic error in the prediction of mean front position. Upon treating the domain and boundary of integration as being random, we obtained an additional second-order term in our equations for mean head, which eliminated the error.
3. Due to our choice of deterministic boundary conditions, our moment expressions are formally the same for flow under prescribed head and flow under prescribed flux. The difference between the two solutions stems from a difference in the Green's functions.
4. The zero-order approximation of the mean hydraulic head satisfies a standard boundary-value problem with moving boundaries for a medium with known properties, driven by a deterministic forcing term. Randomness, and nonlocality of the mean flow problem manifest themselves solely in second-order (and higher) terms.
5. Our instability analysis implies that a DNAPL front is stable when the gradient of pressure head immediately above it is negative (the vertical coordinate pointing downward and pressure head increasing upward) and unstable when this gradient is positive. We showed that a probabilistic analysis of wetting front instability, in Chen and Neuman (1996), applies to a DNAPL front.
6. We solved our moment equations analytically in one dimension and compared the results with Monte Carlo simulations. We found the result to be accurate for both mildly and moderately heterogeneous soils. Our analytical solution allows one to investigate the effect that the variance and spatial correlation scale of log hydraulic conductivity have on front behavior under the action of various forcing terms.
7. Our one-dimensional analysis demonstrated the advantages of our moment solution over Monte Carlo simulation. The number of Monte Carlo simulations necessary for convergence of the two leading moments to a stable result increases with heterogeneity (the variance of log conductivity). There generally is no assurance that the Monte Carlo solution would stabilize at the theoretical ensemble solution. We nevertheless have taken the Monte Carlo solution to represent the theoretical solution.
8. Our second-order front position is much closer to the one obtained from Monte Carlo simulation than the zero-order front position, which constantly overestimates the Monte Carlo solution.
9. We found that a one-dimensional front driven by a deterministic flux Q propagates at a fixed deterministic velocity Q | θ, θ , θ being DNAPL content, so that front variance σ2ε, velocity variance σ2ν, second-order mean front position ξ(2)(t) , second-order mean front velocity , and related cross-covariances are zero. While trivial (this result can be obtained directly from the mass-conservation principle), it indicates that our moment solution is free of internal contradictions. While the front moves through a random porous medium at a deterministic velocity, the actual head and its gradient remains random. We found that when one-dimensional flow is driven by constant head, heterogeneity reduces the mean front propagation rate regardless of gravity. Mean front depth and its rate of advance increase with (H-a) / lY , (i.e., increase with the driving head, and decrease with the correlation length lY). The latter effect is amplified by the fact that dimensionless time also decreases as lY increases. The larger the b (i.e., the density difference between driving and driven fluids), the faster the one-dimensional gravity front propagates. In the absence of gravity, the front prorogation rate is independent of density difference between the two fluids.
10. The one-dimensional front variance increases with the log conductivity variance σ2y, the driving head, and the correlation length lY, regardless of gravity. In the presence of gravity, the front variance also increases with b. The variance of a gravity-free front does not depend on b.
11. We found that gravity-free DNAPL fronts (µD >> µw ) driven by either constant flux or constant head remain stable. This is in full agreement with Saffman and Taylor (1958), who stated that such displacements are stable to small deviations if motion is directed from the more viscous to the less viscous fluid, whatever are their relative densities.
12. A gravity DNAPL front driven by constant head is initially stable but becomes unstable with time regardless of σ2y.
13. A gravity wetting front (b = 1) can be stable or unstable depending on σ2y and (H - a) / lY. For a mean wetting front to be stable, (H - a) / lY must be sufficiently large. Increasing σ2y has a destabilizing effect on the front under gravity. Changes in correlation length do not have significant effect on the front.
14. A gravity DNAPL front driven by constant flux is stable for and unstable otherwise. In other words, heterogeneity (increasing σ2y) has a stabilizing effect on gravity flow driven by constant flux. The one-dimensional moment solutions for DNAPL flow driven by constant flux do not depend on correlation length.
15. We solved our moment equations numerically in two dimensions. Our numerical solution reproduces known phenomena, such as finger splitting and shielding in a homogeneous porous medium. It shows that: (1) random layering reduces the mean finger propagation rate; and (2) the front velocity variance is highest at the fingertip, increases with σ2y, and decreases with spatial correlation.
16. Our results are of both theoretical and practical importance, because in most soils, conductivity behaves as a correlated random field.
Laboratory Experiments
In the experimental arena, our objective was to accurately and precisely measure conditions for the onset of instability, the development of DNAPL fingers, their morphology, number, spatial distribution, size, and rate of advance in homogeneous and heterogeneous saturated porous media. This was achieved through a series of laboratory-based experiments in transparent two-dimensional flow cells recorded with analog to digital video imaging. A set of replicated experiments investigated vertical penetration of PCE under constant head inlet conditions in homogeneous porous media. Fifteen experiments were completed using five textures of glass beads, each having three replicates. A second set of replicated experiments investigated vertical penetration of PCE under constant head inlet conditions in heterogeneous porous media. Twelve experiments were completed in layered heterogeneous, systems and a thirteenth was completed in a system with block shaped heterogeneity. The behavior of the heterogeneous system was compared and contrasted with the homogeneous systems.
Our experimental work has led to the following conclusions:
1. The migration of DNAPL fingers in porous media with systematic heterogeneity exhibited the primary mechanisms of flow instability-induced fingering, pooling at the interface between layers of differing displacement pressure, and channeling induced by permeability and displacement pressure heterogeneity. The fingering processes of a DNAPL in the top layer of a water-saturated layered porous medium were not affected by the layers below it. When the DNAPL fingers encountered a layer boundary between a coarser layer and an underlying finer layer, vertical PCE movement temporarily ceased. Consequently, the DNAPL moved laterally above the finer layer, and a DNAPL pool formed. When the pool thickness reached the displacement value, the DNAPL started to penetrate into the finer layer as fingers. When the DNAPL fingers encountered a layer interface where a finer layer was lying above a coarser layer, the fingers entered the lower coarser layer immediately because the coarser layer has a smaller displacement pressure.
2. All the values of finger spacing (λ) for the finer layers were similar regardless of the position of the layer. The values of λ in the coarser top-layers were larger than those in the coarser sub-layers, which had values of λ similar to those in the overlying finer layer.
3. The displacement pool thickness (Hd) was found to be proportional to the difference between the displacement pressures of the two layers and inversely proportional to the density difference between DNAPL and water. Predicted values of Hd fit our observation. The steady-state pool thickness was slightly larger (e.g., about 0.2 cm for our experiments) than the displacement pool thickness. DNAPL saturations in the fingered region were between 3.7 percent and 14.0 percent, while those in the DNAPL pools above the finer layers were between 60 percent and 88 percent.
4. Finger velocities in the finer layers and in the top coarser layers were similar to those in corresponding homogeneous media. Average finger velocity in the coarser sub-layer was larger than that in the finer layer and smaller than that in the coarser top-layer.
5. In the experiment with discrete blocks of different porous media having only slightly different textures, three mechanisms of DNAPL movement were observed: fingering, channeling, and pooling. The results suggest that a small threshold driving force is needed to overcome resistance to lateral movement. Our results indicate that there is a required minimum difference in permeability and displacement pressures before permeability-caused channeling would dominate fingering induced by flow instability. This is important because the common assumption is that the only mechanism that is important under field conditions is channeling along a "path of least resistance" due to variations in permeability. The magnitude of the permeability and displacement pressure differences required to cause "channeling" to dominate over "fingering" is fundamentally important to our basic understanding of the behavior of these systems. It is also critical for the further development of accurate multiphase flow simulation models that incorporate heterogeneity.
Journal Articles on this Report : 3 Displayed | Download in RIS Format
Other project views: | All 10 publications | 3 publications in selected types | All 3 journal articles |
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Type | Citation | ||
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Smith JE, Zhang ZF. Determining effective interfacial tension and predicting finger spacing for DNAPL penetration into water-saturated porous media. Journal of Contaminant Hydrology 2001;48(1-2):167-183. |
R826157 (1999) R826157 (Final) |
not available |
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Zhang ZF, Smith JE. The velocity of DNAPL fingering in water-saturated porous media: Laboratory experiments and a mobile-immobile-zone model. Journal of Contaminant Hydrology 2001;49(3-4):335-353. |
R826157 (Final) |
not available |
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Zhang ZF, Smith JE. Visualization of DNAPL fingering processes and mechanisms in water-saturated porous media. Transport in Porous Media 2002;48(1):41-59. |
R826157 (Final) |
not available |
Supplemental Keywords:
water, groundwater, soil, sediments, dense non-aqueous phase liquid, DNAPL, physics, engineering, hydrology, geology, mathematics, modeling., Scientific Discipline, Water, Waste, Environmental Chemistry, Physics, Remediation, Engineering, Chemistry, & Physics, Groundwater remediation, environmental monitoring, DNAPL, instability, soil screening, aquifer remediation design, chemical kinetics, soil contaminants, multiscale permeability fields, groundwater contamination, hydrologic site characteristics, stochiometry, time domain reflectrometry, contaminated aquifersProgress and Final Reports:
Original AbstractThe perspectives, information and conclusions conveyed in research project abstracts, progress reports, final reports, journal abstracts and journal publications convey the viewpoints of the principal investigator and may not represent the views and policies of ORD and EPA. Conclusions drawn by the principal investigators have not been reviewed by the Agency.