Grantee Research Project Results
Final Report: Diffusional Rate Limitations in Heterogeneous Porous Media: Model Structure, Scale, and Geologic Characterization
EPA Grant Number: R824768Title: Diffusional Rate Limitations in Heterogeneous Porous Media: Model Structure, Scale, and Geologic Characterization
Investigators: Freyberg, David L. , Roberts, Paul V.
Institution: Stanford University
EPA Project Officer: Packard, Benjamin H
Project Period: November 1, 1995 through October 1, 1998
Project Amount: $198,000
RFA: Water and Watersheds (1995) RFA Text | Recipients Lists
Research Category: Watersheds , Water
Objective:
The goal of our project has been to develop an improved understanding of the impact of diffusional rate 1imitations on solute transport in saturated, heterogeneous porous environments. Our context is the need for better tools to predict the transport and fate of contaminants during remediation, although a better understanding of diffusional rate limitations is of importance in many contexts beyond remediation. Our scope has ranged from the grain scale, at which diffusion within intraparticle pores may control solute interchange with interparticle pore water and access to sorption sites, to the scale of geologic units, at which spatial heterogeneity may mask or enhance the effects of grain- scale diffusional rate limitations, and at which regions of low velocity may interact with higher-velocity regions in a manner which may be represented as a large-scale diffusional rate limitation. Our tools have included mathematical models of flow and transport, the data base from the Borden transport experiment, laboratory data, and a detailed data base describing the variability in grain-size characteristics of a point-bar deposit over scales ranging from millimeters to meters.We have approached our goal of a better understanding of the impact of diffusion on transport from two different perspectives. First, we have examined the nature and effects of diffusion at the scale of individual grains, analyzing data from a set of laboratory column experiments and developing and applying models characterizing the behavior of the temporal moments of breakthrough curves under the influence of advection, dispersion, diffusion through intraparticle pore space, and sorption. Second, we have used mathematical transport models to investigate the role and importance of diffusion in low- velocity zones of heterogeneous velocity fields and the conditions under which velocity heterogeneity leads to behavior which can be modeled as apparent large-scale diffusional rate limitations during transport. The following sections of this report examine each of these approaches in turn.
Summary/Accomplishments (Outputs/Outcomes):
Our examination of the nature and effects of diffusion at the scale of individual grains has allowed us to elucidate and quantify the processes controlling diffusional rate limitations at the grain scale, to evaluate the effects of a distribution of grain-scale diffusional time scales on contaminant plume spreading and tailing, and to quantify the relative importance of grain-scale diffusional rate limitations and spatial heterogeneity in hydraulic conductivity on plume behavior. Based on an analysis of laboratory column elution data, we have developed a conceptual and mathematical model of contaminant transport which recognizes two distinguishable regimes for contaminant elution, a "fast" regime in which the majority of the contaminant can be removed quickly, sometimes in a matter of minutes, and a "slow" regime, in which the remainder of the contaminant is removed very slowly regardless of the experimental conditions. We attribute both the fast and slow contaminant removal to diffusion processes to sorption sites occurring inside individual grains or aggregates of the media. The fast diffusion appears to occur in the relatively large pores, while the slow diffusion appears to occur in very small pores branching off from the larger pores. Diffusion is slowed considerably in the very small pores because the sorbate molecule is never free of the attractive effects of the pore walls. For natural materials, the slow diffusion process is not well-modeled by a single diffusion coefficient and length scale. A gamma distribution of diffusion rate constants provides very satisfactory reproduction of experimental data. The fast diffusion parameters may be estimated a priori, but the parameters describing the distribution of slow diffusion rate constants must be determined empirically.We have studied the effects of a distribution of diffusion rate constants (sorption time scales) on transport by examining theoretically the temporal moments of breakthrough curves. We have shown analytically that a distribution of sorption time scales does not affect the first two temporal moments, but can affect strongly and even dominate the third moment (the skew or "tailing" of the breakthrough curve). Mathematically, the third temporal moment is increased by a term that is proportional to the variance of the distribution. Thus, the effect of a distribution of sorption time scales is to create long tails on breakthrough curves, a phenomenon commonly observed in a wide variety of situations.
We have extended this analysis to heterogeneous aquifers in order to evaluate the relative importance of grain-scale diffusion processes and heterogeneity of aquifer conductivity in determining the nature of contaminant breakthrough curves. We have shown analytically that the temporal moments for the arrival of a sorbing contaminant at a control plane are related to the temporal moments of a conservative contaminant in the same flowfield and to the rate of the intragranular diffusion process. Aquifer heterogeneity and intragranular diffusion have additive effects towards the spread of a contaminant plume (as measured by the second temporal moment). We have formulated a parameter that can be used to distinguish between situations where plume spread is controlled by heterogeneity of aquifer conductivity, by slow intragranular diffusion, or by a combination of both. We have demonstrated that when intragranular diffusion is fast enough, the local equilibrium assumption might be a valid and useful approximation; when intragranular diffusion is slow enough, it might be possible to treat the conductivity of the aquifer as spatially homogeneous. Either of these simplifications could greatly facilitate transport simulation for the purposes of remediation design.
While grain-scale diffusional mass transfer produces nonequilibrium sorption behavior yielding skewed breakthrough curves, pure advection alone through spatially heterogeneous groundwater velocity fields can also lead to skewed breakthrough curves. In addition, velocities in some regions of a heterogeneous velocity field may be so low that solute transport there is dominated by diffusion, even though the mean velocity is relatively high. This, of course, severely complicates the analysis of field-scale transport in the presence of grain-scale mass transfer. The temporal moment analysis described above has yielded considerable insight into this challenge. However, we have used several other approaches to attempt to better understand the interplay between grain-scale mass transfer, velocity heterogeneity, spatial and temporal scales, and transport prediction.
Using a three-dimensional model of point-bar geometry we have explored systematically the nature of low-velocity zones in which diffusion may be the dominant transport process and/or in which advective effects are manifested as a field-scale rate limitation. We have learned that the size, shape, and location of low-velocity regions depend in a quite complex manner on the dimensionality of the modeled domain, the presence of boundaries, the magnitude and direction of the hydraulic gradient, the strength of the conductivity heterogeneity, and the spatial structure of the permeability and porosity fields.
Importantly, simple statistical summaries may be inadequate to capture important characteristics of the structure of the velocity field. In order to find nonstatistical approaches for dealing with this complexity, our initial focus has been on understanding the effects on transport of a single, low-permeability, elliptical inclusion embedded in a homogeneous higher conductivity material.
Using an analytical solution for the flow field, we have simulated the transport of a line source of a non-sorbing solute through and around ellipses of various size, shape, orientation, and permeability contrast. The degree of distortion of the uniform flowfield is clearly a function of the size, shape, and orientation of the inclusion, along with permeability contrast between it and the background material. We have used a Peclet number to characterize the transition from advection to diffusion dominance within the ellipse. The primary factors affecting the Peclet number are the free-stream velocity, the diffusion coefficient, the semi-minor axis of the ellipse, and the conductivity contrast between the inclusion and the surrounding media. The effects of the orientation and eccentricity of the ellipse are secondary, although they can be significant when the semi-major axis is nearly aligned with the free-stream velocity.
For low inclusion Peclet numbers, when diffusion governs transport in the inclusion, the temporal moments for simulated breakthrough curves from a line source upstream from the ellipse vary with the time scale for diffusion through the ellipse. For large inclusion Peclet numbers, when transport within the inclusion is dominated by advection, the temporal moments of the breakthrough curve depend on the advective time scale. For the cases when advection controls transport within the inclusion, we have developed and validated approximate expressions for the first three temporal moments as functions of the system variables. In terms of the primary physical parameters, for large inclusion Peclet numbers (advection dominance), the temporal moments of a solute plume traveling through and around the inclusion depend on the semi-major and semi-minor axes, the length of the line source, the freestream velocity, the angle between the semi-major axis and the freestream velocity, and the permeability ratio. For small inclusion Peclet numbers (diffusion dominance), the moments depend on the semi-major and semi-minor axes, the length of the line source, the freestream velocity, and the diffusion coefficient within the inclusion.
Journal Articles on this Report : 1 Displayed | Download in RIS Format
Other project views: | All 1 publications | 1 publications in selected types | All 1 journal articles |
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Cunningham JA, Goltz MN, Roberts PV. Simplified expressions for spatial moments of groundwater contaminant plumes. Journal of Hydrologic Engineering 1999 (in press). |
R824768 (Final) |
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Supplemental Keywords:
RFA, Scientific Discipline, Water, Hydrology, Ecology, Water & Watershed, Physics, Geology, Watersheds, fate and transport, model structure, contaminant transport, Borden transport experiment, aquatic ecosystems, diffusion rate limitations, heterogeneous porus media, groundwaterProgress 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.