1. The Concept of Block-Effective Macrodispersion for Numerical Modeling of Contaminant Transport
Yoram Rubin
University of California at Berkeley
Thanks to Alberto Bellin and Alison Lawrence

2. ModelCARE 2002, Prague, June 16-20, 2002
Background
Upscaling of permeability has been a major area of research;
Important results available for effective conductivity for various models of spatial variability, various flow regimes and space dimensionalities;
Theories are also available for upscaling to the numerical grid-block scale (length scale of the homogenized domain is comparable to the scale of heterogeneity);
Much less work has been done on the transport side, specifically:
how to assign dispersion coefficients to numerical grid blocks?
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ModelCARE 2002, Prague, June 16-20, 2002

3. Goal of this presentation:
Propose an approach toward a rational design of numerical analysis of transport which accounts for the various length scales affecting transport including: scales of heterogeneity, pore-scale dispersivity, dimensions of the solute plume, numerical grid block dimensions and travel distances, as well as space dimensionality;
On a more fundamental level: bridge between stochastic concepts and numerical applications;
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ModelCARE 2002, Prague, June 16-20, 2002

4. ModelCARE 2002, Prague, June 16-20, 2002
A common stochastic approach for modeling transport: The Concept of Macrodispersivity
Mean velocity
Macrodispersion
tensor
Local pore-scale
dispersion tensor
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5. ModelCARE 2002, Prague, June 16-20, 2002
Transport in complex geological structure can be analyzed: multi-scale, hierarchical heterogeneity
Hierarchical organization of lithofacies and corresponding permeability
Modes (Ritzi et al., Water Resources Research, 40(3), 2004)
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ModelCARE 2002, Prague, June 16-20, 2002

6. Important considerations:
Concept is limited to modeling plumes that are large with respect to the scales of heterogeneity (ergodic*), because:
When plume is ergodic, all variability is local, and its effects can be modeled deterministically through dispersion coefficients;
This concept is not useful in numerical applications, where we usually deal with non-ergodic plumes, and in that case:
It is important to capture the large scale spatial variability directly on the grid;
*(Dagan, G., JFM, 1991)
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ModelCARE 2002, Prague, June 16-20, 2002

7. ModelCARE 2002, Prague, June 16-20, 2002
Large scale variability of the hydraulic properties can be identified using GPR
Kowalsky, M., et al, Water Resources Research, 37(6), 2001
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8. ModelCARE 2002, Prague, June 16-20, 2002
Subsurface imaging using GPR at the Oyster site in Virginia:
These works suggest that we are improving our ability to capture the large scale components of the spatial variability, but are still limited in our ability wrth the small scale
Regardless of these two important results, modelers are largely focused on high resolution, detailed characterization
Hubbard et al., Water Resources Research, 37(10), 2001
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ModelCARE 2002, Prague, June 16-20, 2002

9. Detailed Site Characterization and Fine Grid Simulation
Heterogeneous
velocity field
MONTE CARLO SIMULATIONS
This is a complete departure from the previous ideas that emphasizes the use of effective parameters, towards a very detaialed description of the flow domain
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ModelCARE 2002, Prague, June 16-20, 2002

10. High computational cost
Block-scale dispersion tensor
Reproduced over the grid
ENLARGE THE
GRID BLOCK
Detailed, high resolution
Spatial distributions
A Hybrid Approach: combine detailed characterization with
Scaling concepts
Introduction of the Hybrid approach which combines the use of detailed characterization with scaling concepts.
Here the interplay between the various length scales comes into focus, because questions are raised such as: how to capture the effects of the wiped out variability? How to avoid duplicating the effects of the variability modeled directly over the grid and those modeled using dispersion coefficients? Another question is that if the dimensions of the blocks are not too much larger than the scales of wiped-out variability, then what would be the meaning of repersenting these effects using effective parameters, after all, the patterns of spatial variability at the sub-grid scale vary between blocks, so how can you represent them using uniform dispersion coefficients?
Measures should be taken
to compensate for the wiped-out
variability
High computational cost
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11. ModelCARE 2002, Prague, June 16-20, 2002
Length-scales
: plume dimensions
: spacing between
measurements
: grid dimensions
: smallest length
scale reproducible
on the grid
In Generel, Lm will determine the smallest scales of variability identifiable, but assuming this not to be a limitation, the smallest scales of variability that can be reproduced over the grid are determined by , the characteristic length of homogenized regions (which can be larger than the block dimensions 
Practical limitation
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ModelCARE 2002, Prague, June 16-20, 2002

12. ModelCARE 2002, Prague, June 16-20, 2002
Nyquist Theorem: relates between the sampling scale and the identifiable scales
LARGE PLUME
SMALL PLUME k
S(k) k
S(k)
How do all these length scales interact?
In a complex manner shown in this figure.
Here one needs to emphasize that for large plume the block D_b is deterministic, and not a function of l at all.
You definitely do not want to be in the small plume situation, and that already starts to give you an idea about grid design.
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ModelCARE 2002, Prague, June 16-20, 2002

13. Block-scale macrodispersion
Wiped-out variability
Variability reproduced
directly on the grid
Assumptions/limitations: small variance in the log-conductivity (which acts also to remove small scale – large scale correlations), uniform in the average mean flow;
The role of the filter here is admit into play only those length scales that affect the plume’s dispersion, as opposed to those which affect dispersion and which are modeled directly over the grid, with the separation between these two based on Nyquist theorem.
Alternative filters (Gaussian) were also tested, and are useful particularly in modeling the lateral macrodispersivities.
What we get from this expression is the block-scale macrodispersion for an ergodic plume, because it models the dispersion around the mean displacement and not the actual one, but ergodicity now is determined by much smaller length scales, those which characetrize the sub-grid variability
HIGH PASS FILTER
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14. longitudinal block-scale macrodispersion
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15. ModelCARE 2002, Prague, June 16-20, 2002
Longitudinal macrodispersion is a function of Pe=UIY/Dd. The ’ values denote the dimensions of the homogenized regions.
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16. ModelCARE 2002, Prague, June 16-20, 2002
Small plume case
What we found is that for wahtever lamda (the scale of the homogenizes regions), when the plume is about 50% larger, the macrodispersion no-longer depends on the plume’s scale.and this suggests that we have a design tool
The block-scale macrodispersion reaches the ergodic limit for
At this ratio, the plume becomes ergodic (=deterministic), and no-longer a function of the plume scale.
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ModelCARE 2002, Prague, June 16-20, 2002

17. First-order Instantaneous Sorption
Negative correlation between the hydraulic conductivity and the distribution coefficient is often applicable. Positive correlation is also plausible. We will consider the extremes:
(A) perfect positive correlation;
(B) perfect negative correlation and
(C) no correlation.
Finally a quick discussion on recent results. We are looking at first-order instantaneous and kinetic (rate limited) sorption. Here we will show some results for the instantaneous case. By now there is a substantial body of work on correlation models for the distribution coefficients, although not conclusive.
Macrodispersivities can be related to the distribution coefficients through correlations between the retardation coefficients and the log-conductivity. This idea was pursued in several studies, only that now for the first time we know how to use this information to develop block-scale macrodispersivity.
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18. ModelCARE 2002, Prague, June 16-20, 2002
Longitudinal Macrodispersion with Spatially variable distribution coefficient (for =1)
Results are for lamda=1 integral scale
For small blocks the large time limit is attained very fast
Different correlation models lead to different macodispersion coefficients
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19. ModelCARE 2002, Prague, June 16-20, 2002
Summary
A theory is presented for modeling the effects of sub-grid scale variability on solute mixing, using block-scale macrodispersion coefficients;
The goal is to allow flexibility in numerical grid design without discounting the effects of the sub-grid (unmodeled) variability, while at the same time:
Avoiding unnecessary high grid density;
The approach incorporates several concepts:
Rational treatment of the relationships between the various length scales involved;
Nyquist’s Theorem is used to separate between the length scales affecting mixing and those which affect advection. The outcome is a Space Random Function;
Ergodicity: The block-scale macrodispersion coefficients are defined in the ergodic limit (about 50% larger than the scale of the homogenized blocks), which allows to treat them as deterministic;
Flexibility: avoiding unnecessary high grid density
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20. ModelCARE 2002, Prague, June 16-20, 2002
References:
Rubin, Y., Applied Stochastic Hydrogeology, Oxford University Press, 2003;
Rubin, Y., A. Bellin, and A. Lawrence, Water Resources Research, 39(9), 2003;
Bellin, A., A. Lawrence and Y. Rubin, Stochastic Env. Research and Risk Analysis (SERRA), 18, 31-38, 2004.
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ModelCARE 2002, Prague, June 16-20, 2002