Stochastic pore-scale growth models of DNAPL migration in porous
media
q
Robert P. Ewing
a, Brian Berkowitz
b,*a
Department of Agronomy, Iowa State University, Ames, IA 50011, USA b
Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot, Israel
Received 10 December 1999; received in revised form 21 June 2000; accepted 31 August 2000
Abstract
Stochastic models that account for a wide range of pore-scale eects are discussed in the context of two-phase, immiscible dis-placement problems in natural porous media. We focus on migration of dense, nonaqueous phase liquids (DNAPLs) through water-saturated geological materials. DNAPL movement is governed by buoyancy, capillary, and viscous forces, as well as by the details of the porous medium. We examine key issues relevant to development of stochastic growth models. We then present a particular stochastic growth model, based on a generalization of invasion percolation and Eden growth approaches, which realistically simulates two-phase ¯ows in a computationally ecient manner. Fingering patterns are shown to depend critically on the competing buoyancy, capillary, and viscous forces between the DNAPL and the water, and their interactions with the porous medium at the local scale. We conclude with recommendations for future research. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction
Characterizing and quantifying the migration of im-miscible liquids in geological formations has been the subject of intense investigation over many decades. Ini-tially, attention was focused on secondary recovery methods in the petroleum industry, where injected water and/or other liquids are used to displace oil. More recent study of two-phase immiscible ¯ows has been motivated by the need to treat spills and leaks of so-called non-aqueous phase liquids (NAPLs), such as trichloroethane (TCA), trichloroethylene (TCE), and benzene, which can severely reduce the quality of subsurface water supplies. Although NAPL solubilities in water are generally low, their toxicity is such that NAPLs that leak into an aquifer and dissolve slowly over long periods of time will con-taminate large volumes of water. Clearly, there is a strong need for realistic simulation models with predictive ca-pabilities, both for application to problems of pollution remediation and prevention as well as for oil recovery.
In the case of secondary recovery in petroleum res-ervoirs, water displacing oil is generally an imbibition process, if we assume that the rock is predominantly water-wet. In contrast, the advance of a dense NAPL (DNAPL) into a previously water-saturated porous medium is a drainage process, in that a nonwetting ¯uid displaces a wetting ¯uid. Generally speaking, the initial downward migration of a DNAPL is due mainly to gravity, whereas the lateral spreading usually results from capillary resistance to entering layers of lower pore-size material, such as clay lenses. The interactions among the capillary, buoyancy, and viscous forces and the details of the porous medium can result in a complex ¯ow pattern characterized by rami®ed features that range in size from the pore scale to tens of meters or more.
The physics of two-phase, immiscible ¯uid movement in porous media are well understood under simple con-ditions e.g. [1,2,11,20]. Realistic simulation in well-de-®ned, continuum-scale media has been achieved using macroscopic multi-phase ¯ow equations (e.g. [1,2,29± 31]). The simulations generally succeed in reproducing the ®nger-and-pool con®guration observed in some DNAPL spills (e.g. [9]). Despite being 2-D, however, these continuum-based simulation models are generally cpu-intensive. More signi®cantly, DNAPL ¯ow patterns are often highly rami®ed and sensitive to small-scale
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q
Journal Paper No. J-18735 of the Iowa Agricultural and Home Economics Experiment Station, Ames, Iowa, Project No. 3262, and supported by the Hatch Act and State of Iowa.
*
Corresponding author.
E-mail address:brian.berkowitz@weizmann.ac.il (B. Berkowitz).
details. As a consequence, these models are severely limited because use of a coarse grid will likely yield meaningless results, but the memory requirements and computation times required for ®ner grids are prohibi-tive. These computational diculties become more marked in full 3-D simulations, which are necessary to properly capture realistic ¯ow patterns.
More recent alternatives to these macroscopic ap-proaches to describing multi-phase ¯ows have been de-veloped on the basis of stochastic growth models (e.g. [14,18]). These models seem particularly promising be-cause of their inherent ability to account for the dy-namic, and often unstable, interaction among capillary, buoyancy, and viscous forces, along with the high sen-sitivity to pore-scale structure. Stochastic, discrete growth models are based upon frameworks such as in-vasion percolation (IP) and diusion-limited aggrega-tion (DLA). Such models capture the essential physics of a process, and migration patterns can be simulated without the need to solve large systems of equations. Simple growth models that mimic physical growth processes are known for speci®c instances, such as vis-cous ®ngering [34,40], capillary ®ngering [34,50], and gravity ®ngering [21,37]. A broad review of stochastic and continuum two-phase ¯ow models is given by Sahimi [41]. Note that in some physical situations (e.g., slow capillary-driven processes), large-scale behavior is controlled by small-scale mechanisms, whereas in other cases (e.g., high pressure injection of an oily phase), small-scale behavior is controlled by large-scale mech-anisms. In general, models that use local rules (as do most stochastic growth models) will work better when local mechanisms dominate, so conditions that are controlled by small-scale mechanisms will play to the growth model's strength, whereas the latter one will not. In this contribution, we ®rst discuss the application of various stochastic growth models to modeling two-phase, immiscible displacements in geological porous media. We focus on drainage-type processes relevant to the migration of DNAPLs. Our purpose is to examine current growth models to identify places where they fail to realize their potential as useful predictive tools. We then present a particular growth model, based on a generalization of invasion percolation and Eden growth approaches, which realistically simulates DNAPL mi-gration in a computationally ecient manner. We con-clude with a discussion of future research directions.
2. Stochastic growth models for DNAPL movement
Invasion percolation was developed as an analog to quasi-static capillary displacement [50], and drying (or drainage) processes in particular. DNAPL migration is certainly a drainage process at the advancing front, but the contributions of viscous and buoyancy forces
de-mand that a growth model more complex than simple IP be used. In addition to considerations as fundamental as which growth model and conceptual framework to use, other complications are encountered in bringing the model from an abstract concept to a real-world appli-cation. In this section, we discuss some of the challenges of developing percolation-based DNAPL models and examine how these challenges may be addressed.
2.1. Stochastic growth models: background
The patterns formed by immiscible displacement of a wetting ¯uid by a nonwetting ¯uid in the absence of buoyancy forces fall into three regimes [34], with an intermediate transition zone that links them. These regimes are de®ned as regions in a 2-D ``phase diagram'' (Fig. 1(a)), with one axis being log M (M viscosity ratiold=lr, where l is viscosity, r the resident ¯uid,
and d is the displacing ¯uid), and the other log Ca
(Cacapillary numberVld=c, whereV is velocity of
the displacing ¯uid, andcis interfacial tension). At high viscosity ratios and high capillary numbers, stable dis-placement is found (Fig. 1(a)). At high M and low Ca,
capillary-driven ®ngering occurs, whereas at low M,
viscous ®ngering is seen. Signi®cantly, these three ¯ow regimes each correspond to a speci®c stochastic growth model: IP is an analog of capillary ®ngering [50], stable growth is mimicked by anti-DLA [36,40], and DLA is similar to viscous ®ngering [40,52]. Some researchers (e.g. [14]) have used Eden growth [13] rather than anti-DLA to represent stable growth; we shall address this issue later.
As discussed in the following section, in order to be applicable to DNAPL migration, growth models must account for density dierences between liquids in ad-dition to dealing with viscous and capillary forces. The experiments of Lenormand et al. [34] were conducted in ¯at, horizontal model porous media, so gravity eects were negligible. But, for DNAPLs invading a porous medium that is not 2-D and horizontal, buoyancy forces can aect the stability of the interface (e.g. [17,21,25,51]). For example, a light NAPL (LNAPL) injected at the top of the medium would tend to have its interface with the aqueous phase stabilized by buoyancy [27], whereas a DNAPL injected at the top would have its interface destabilized [17]. As such, one can consider that Lenormand et al.'s [34] phase diagram could be extended to include a third dimension describing buoy-ancy forces (Fig. 1(b)). Position in this dimension is given by the bond number (BogrlDq=c, wheregis the gravitational acceleration, r the characteristic pore ra-dius,lthe characteristic distance between pores, andDq
shrinking with negative ones. The viscous ®ngering re-gime will likewise change size according to the bond number and the critical velocity, as given in Chuoke et al. [11]. Yortsos et al. [54] have developed a stable/ unstable ¯ow phase diagram for immiscible displace-ment which also includes the in¯uence of the bond number on stabilization and destabilization. Combining this work with that of Lenormand et al. [34] may better de®ne the 3-D stable and unstable displacement regions. The literature on the stochastic growth models is large and well developed, and so here we will not dwell on their descriptions. For more detailed discussions of these dierent growth models, we refer the interested reader to the original articles referenced in the preceding paragraph in addition to, e.g., Stauer and Aharony [47] for percolation in general and Barabasi and Stanley [4] for Eden and DLA. Brie¯y, the IP algorithm involves de®ning a medium as a regular or irregular grid, with each site being randomly assigned an ``invasion prob-ability'' between 0 and 1. For each invasion step, the interface along the invading boundary is scanned for the site with the highest probability; then this site is invaded (Fig. 2(a)). The DLA algorithm, on the other hand, begins with a ``seed'', which can be a point, line, or other shape. ``Sticky'' particles are then released from random locations at a distance from the interface and diuse freely until they touch the seed, whereupon they stick to it and become part of it (Fig. 2(b)).
Eden and anti-DLA algorithms lead to patterns that initially look similar, but over time anti-DLA remains stable while Eden develops features. Historically, Eden was developed as an analog for bacterial growth [13],
and anti-DLA for stable ¯ow [40]. In terms of imple-mentation, Eden scans the interface and chooses one of the interface sites at random; in contrast to the IP al-gorithm, all sites are equally likely (Fig. 2(c)). Anti-DLA, on the other hand, starts a random walker at the DNAPL source and allows it to diuse to the interface, where it sticks and becomes part of the DNAPL (Fig. 2(d)). The result is that Eden is more unstable because parts of the interface that protrude slightly have extra interfacial area and so a slightly higher chance of growth. Conversely, with anti-DLA, interface sites along a protuberance are at a greater distance from the source, so random walkers that start at the source are more likely to encounter closer sites than those at the end of the protuberance, thus dissipating instabilities or perturbations.
Generalizations and combinations of these and other growth processes have been attempted for several years. In physics, Martin et al. [35] noted that the Eden stable growth model and IP are essentially extremes of a continuum, and the transition between them can be accomplished by using an exponent to bias a pseudo-random selector toward zero (for IP) or unity (for Eden). Similarly, Smith and Collins [44] hypothesized that the Eden and DLA models could be generalized to a single model using a method based on variable diu-sion lengths, with large steps corresponding to DLA and in®nitely short to Eden. The next year, Siddiqui and Sahimi [43] developed a model that interpolated smoothly from Eden to DLA. In petroleum engineering, Leclerc and Neale [33] used a Monte Carlo method that transitioned among three models (IP, DLA, and
Log(M)
Log(Ca)
Bond Number
DLA
IP
Log(M)
-8 -4 0 4
Log(Ca)
-8 -4 0 4
Transition
Stable (Eden)
Capillary (IP) Viscous
(DLA)
(a) (b)
anti-DLA) using probabilities, based on values ofM and Ca, to select a random walker type and a sticking pro-bability. Their model produced excessively stable be-havior at very lowCa and did not properly predict the stable to capillary transition, but otherwise showed good
agreement with experiments. The lowCaproblems were
partially corrected by Kiriakidis et al. [28], but results were still not close enough to experimental behavior. In geophysics, Stark [46] modeled patterns of plateau erosion by interpolating among three models (IP, Eden, DLA). Stark's [46] algorithm assigned an original strength to each point (as in IP), then used random walkers to weaken perimeter sites by a factorg, as well as Eden-like selection to weaken perimeter sites by a
factor h. The choice of internal or external walkers
(stable ¯ow or viscous ®ngering) and the balance
be-tween g and h shifted the process among the dierent
growth models. As an aside, we note that in DLA, anti-DLA, and Eden as originally formulated, the medium is implicitly assumed to have no eect, so medium eects are essentially ignored.
Some researchers (e.g. [14,46]) have used the Eden growth model [13] for simulating stable ¯ow, despite Paterson's [40] and Lenormand et al.'s [34] identi®ca-tion of stable ¯ow with anti-DLA. The Eden model is likely chosen because generalizing across Eden and IP is much more straightforward than generalizing across anti-DLA and IP. As discussed previously, however, Eden and DLA do not behave identically: anti-DLA resists perturbations in the front just as viscous-stabilized ¯ow does, whereas the Eden model tends to propagate them. Other researchers (e.g. [18,49]) have used DLA to model ¯ow that is primarily capillary-driven. Although the fractal dimensions of the resulting patterns are similar, which may lead the human eye to initially perceive the ¯uid patterns as similar, the driv-ing physics are dierent, as are the resultdriv-ing patterns and often their topologies. The key issue here is that the model should closely follow the relevant physics of the process of interest: it is not sucient to simply repro-duce the essential features of the process being simu-lated.
2.2. Complicating issues in stochastic growth model development and application
In this section, we identify and discuss six factors that contribute to the complexity of modeling DNAPL movement in the real world: buoyancy, viscosity, sen-sitivity to local detail, geological uncertainty, partial saturation, and dynamics. These factors complicate both continuum scale and stochastic growth models, but our focus here is on the stochastic growth models. A seventh factor, capillarity, poses diculties for continuum models but is built into invasion percolation-based models at a fundamental level, so we will not discuss it here. Our purpose in this section is to summarize areas of needed future work.
2.2.1. Buoyancy
Most NAPLs have a density dierent from that of water, so that in addition to capillary forces, buoyancy forces are exerted on NAPLs and aect their move-ment. At ®rst glance this is not a problem: percolation concepts can be adapted by assigning an invadability value to each site that accounts for buoyancy as well as capillarity (see, e.g. [21,37]), such that the eective in-vadability IeffI0Dqgh, where I0 is the intrinsic
(capillary) invadability and h is the vertical distance from the source. But this approach assumes no viscosity (a basic assumption of invasion percolation), which can cause notable deviations from the equation. For ex-ample, a DNAPL ®nger migrating downward at non-zero velocity does not exert the full force given by that simple equation, because some of the pressure dierence is lost by viscous drag within the ®nger. Similarly, where the DNAPL encounters and spreads along the top of a con®ning layer, viscous forces cause the DNAPL to accumulate into an inverted bowl shape with nonzero depth [19]. In other words, the assump-tion of no viscosity implies that simple hydrostatic forces dominate everywhere, unaected by viscosity, which is clearly an extreme. We propose an alternative extreme: by allowing for viscosity eects, we assume that DNAPL height eects do not propagate unaltered through the entire DNAPL-invaded region. Speci®cally, we can assume that the hydrostatic force at the bottom of a pool feeding a ®nger is that given from the top of the pool (viscous drag through the pool itself is negli-gible). For this alternative, buoyancy force calculations should be updated dynamically as the con®guration changes. The two extremes will serve to constrain the solution and point toward a more reasonable middle ground.
A further complication is that buoyancy forces have a dierent sensitivity to length than do capillary forces [27]. To put it simply, if the grid is made too coarse, then the buoyancy forces driving invasion of a site one level lower will overwhelm any capillary considerations.
Speci®cally, the vertical grid size l at which buoyancy forces would equal capillary forces is approximately lPc=gDq, wherePc is the capillary pressure at
break-through [27]. Because l should be much smaller than
this, the vertical grid spacing must be relatively small, which may greatly increase memory requirements for a realistic 3-D simulation. An alternative approach would be to use the calculated invadability when comparing sites on the same level but to moderate local buoyancy eects by a much smallerlwhen invading either upward or downward. To our knowledge this approach has never been used, but it oers a potential way to avoid being forced into impractically ®ne discretization.
2.2.2. Viscosity
Viscous forces aect the con®guration of an ad-vancing front [34], but because invasion percolation assumes zero viscosity, viscous forces are not explicitly accounted for by a pure invasion percolation model. As discussed in Section 2.1, invasion percolation can be blended with one or more other growth models to ac-count for viscous forces. For high viscosity NAPLs like coal tars, viscous ®ngering eects may be safely ignored. However, low viscosity NAPLs such as carbon tetra-chloride or chloroform require a model that includes DLA behavior. A general growth model must be able to handle both high and low viscosity ratios.
2.2.3. Sensitivity to local detail and upscaling consider-ations
Invasion percolation was developed to mimic pore-scale properties, and the capillary forces that dominate DNAPL movement in medium- and ®ne-grained ma-terials operate at the pore scale. Nonetheless, it is of course impractical to model an aquifer at the pore scale. Several authors (e.g. [15,21,32,53]) have used a form of upscaling to move percolation models to larger scales than a few pores, and aside from the concerns about grid size raised by Ioannidis et al. [27], mentioned earlier, upscaling percolation theory appears to work well. We note in passing that upscaling a pore-scale model raises some question as to what exactly constitutes pore-scale modeling: if scale alone is the determining factor, then a ®nite-element model could presumably be made pore-scale by simply reducing the size of the elements. However, we believe the key point is that what are called pore-scale models treat the medium as being composed of discrete pores rather than being a continuum; when they are also at or near the pore scale, it is convenient to call them pore-scale models.
Features smaller than the grid size, e.g., a thin clay layer, may not show up in a coarsely discretized model but will stop DNAPL migration in actuality [19]. Con-versely, a small fracture that crosses an otherwise im-permeable layer will provide an ample pathway for migration [48]. Sensitivity to detail implies that, given absolute knowledge of the medium, ®ner resolution will always lead to more accurate simulations, whether one is using a continuum-based or a stochastic growth model. Because accuracy is always desirable, modelers will likely be tempted to work near the memory limitations of their computers, and so memory limitations will fre-quently be a limiting factor for both kinds of models. One approach that has not yet been taken by any per-colation modelers (to the best of our knowledge) is to use an adaptive memory scheme with ®ner grids at the interface and coarser grids where detail is not needed. The implementation is dicult, but it would greatly reduce the memory overhead and may thereby speed execution. In the end, however, cpu time is more likely than memory to be the limiting computer resource, be-cause inversion of the large sparse matrices that result from high resolution 3-D simulations requires far more cpu time than do stochastic growth processes.
2.2.4. Geological uncertainty
All models of DNAPL movement, be they continuum or percolation based, are limited by geological uncer-tainty. At best, the model is only as good as the data it is given, and geological information is known only sket-chily. Worse, many kinds of relevant data are not known at all. For example, DNAPL movement is locally sensitive not only to porosity and conductivity, but also to contact angle [8], pore shape, pore roughness, and so
on. These parameters are in turn functions of position in the pro®le (which in¯uences weathering and organic matter content) and lithology, among other factors. Because the data requirements are so high and what is known so sparse, many of these parameters are simply assumed to be homogeneous. This means that the model is run within a virtual porous medium created by a combination of data, inference, assumptions, and sto-chastic generation. Because both uncertainty and sensi-tivity to detail are so great, a single simulation has a very low chance of correctly predicting the DNAPL dispo-sition.
As suggested by Borchers et al. [7], a more useful approach is to generate many realizations of the porous medium and run the DNAPL movement simulation on each to obtain probabilistic estimates of the DNAPL location. This Monte Carlo approach is a very powerful way of handling geological uncertainty. But, the price of that power is the need for many realizations, which requires a model that executes rapidly. This can be partially dealt with by a coarse-grained parallelism: one can run simulations in parallel on many dierent ma-chines. But a phenomenological growth model such as modi®ed invasion percolation has the advantage of being much more abstract than either a continuum model with its multiple linked partial dierential equations, or a pore-scale model that handles each of the many possible pore-scale events explicitly. Sto-chastic growth models are fast to the extent that they remain abstract, but accurate to the extent that they remain true to all the relevant physics; this tension is part of the challenge of modeling.
2.2.5. Partial saturation
As discussed in Section 2.2.4 this is an area that can quickly become too complex to model with reasonable speed, and percolation-based NAPL models aimed at practical application will need to ®nd ways to abstract these complexities into simpler patterns if they are to succeed.
2.2.6. Dynamics
The aquifer into which the DNAPL moves is not a static system. Even aside from the DNAPL's in¯uence, the water table may rise and fall, and the water may ¯ow. The medium is likely to have hysteretic water retention properties, so the history as well as the cur-rent height of the water table will need to be known. Schwille [42] performed experiments in which DNAPL was injected with a high water table, which was sub-sequently lowered. He also indicated that water ¯ow had little eect on DNAPL movement, but at the least it will in¯uence ganglion dissolution rates. To our knowledge, no one has yet replicated these experiments either quantitatively or in a simulation model, sug-gesting that changes in the water table height and ¯ow in an aquifer represent areas of poor current under-standing.
Once the DNAPL comes into contact with water, further changes will take place. The DNAPL will slowly dissolve into water, changing the interfacial tension as well as the density and viscosity of water. Likewise, water will slowly dissolve into the DNAPL, changing its properties. Because these changes are partially a func-tion of the time of contact between the two phases, ¯uid properties, such as interfacial tension, density, and vis-cosity of both ¯uids, and contact angle, may change locally over time. In fact, they will likely even dier at dierent points along a single ®nger. Finally, the oscil-lations mentioned previously (Section 2.2.2) may also occur, further complicating the movement.
When the DNAPL source is removed, the pressure at the injection point changes and the DNAPL will retreat downward, generating ganglia by snap-os as buoy-ancy-induced pressure is reduced. A complete DNAPL movement model will need to inform us where we will ®nd DNAPL in pools, where in ganglia, and where in intact ®ngers. Then, after the initial DNAPL migration has occurred and the ganglia are apparently stable, slow dissolution of DNAPL into the water may eventually remobilize them [3]. Experimentally, these dynamic ef-fects are generally avoided by holding the water table constant (e.g. [26]), pre-equilibrating the water and DNAPL (e.g. [23]), and collecting data only up to the point of breakthrough. Real spills are not so convenient, however, and eventually experiments will be needed to assess the magnitude of the eect of these dynamics on the eventual DNAPL disposition so that models can be altered as needed.
3. An in-depth example of a generalized growth model
The following model expose is not given with the
claim that the model fully addresses all issues raised above. We examine this model because it is one of the few stochastic growth models being developed, and be-cause we are quite familiar with it. Both new develop-ments and de®ciencies in the model are examined in this discussion.
Berkowitz and Ewing [5] noted that the phase dia-gram relating viscous, capillary, and stable ¯ow regimes should have a third dimension to account for buoyancy forces (Fig. 1). This implies that the growth models must allow modi®cation or biasing to account for gravity stabilizing and destabilizing. Onody et al. [39] accom-plished this through the use of four adjustable parame-ters that controlled the bond number, the downward bias due to gravity, the local change in capillary forces due to curvature of the interface, and the number of simultaneous invasions (which allowed multiple ®ngers to form). Their model required a data structure that identi®ed ®ngers as distinct entities, but ®nger dynamics such as splitting and sensitivity to local details appeared to be lacking.
We [14] blended the IP and Eden models similarly to Martin et al.'s [35] model. The IP portion of the model involves assigning an invadabilityI to each cell, using a function of capillary forces with buoyancy-biasing (e.g. [21])
I g qDphDq ÿ2ccos h=r; 1
where g is the gravitational acceleration, q the NAPL
density, Dp the depth of NAPL ponding above the
water table, h the vertical position (positive downward withh0 at the water table),hthe local contact angle,
and r is the local mean pore size. The Eden portion
involves making a stochastic selection of what interface cell to invade, rather than always invading the most invadable as in normal IP. As with Martin et al. [35], the model can be biased toward the IP and/or Eden ends of the continuum by generating a pseudo-random
number R, raising it to an exponent W, and choosing
the cell in the interface list whose invadability-weighted position in the sorted list corresponds toW. To obtain multiple ®ngers with merging and splitting behavior, as often seen in laboratory and ®eld experiments, the model used an intermediate stochastic step that ®rst stochastically selected among the continually changing number of individual ®ngers, then chose a cell within that ®nger. Eq. (1), combined with a bi-scale (®nger and cell) adaptation of Martin et al.'s [35] biased se-lection procedure, is the core of the Ewing and Berkowitz [14] model.
The capillary number's eect on the ¯ow regime was controlled through stochastic weighting of the exponent
cells at each time step. Time was explicitly accounted for by assuming that the ¯ow rate at the source was constant, and calculating the time required to NAPL-saturate one cell given the capillary number at the in-jection point. A more detailed model might include partial saturation capabilities, but our model currently assumes a given location is either water- or NAPL-saturated. The resulting model is the only percolation-based NAPL migration model to date that can span a range of capillary numbers and produce dynamic ®n-gers. Given the improvement in model behavior when ®ngers were explicitly included in the model's data structure, we speculate that even better simulations would be obtained if the data had a hierarchical, re-cursive pool and ®nger data structure instead of the current ¯at model.
Since the publication of Ewing and Berkowitz's [14] model, we have implemented a trapping rule [12], based on a path-®nding algorithm that determines whether the resident ¯uid has a possible escape route. Trapping oc-curs when the displacing ¯uid surrounds an ``island'' of the resident ¯uid. If the resident ¯uid is incompressible and the invasion process is assumed to proceed suf-®ciently quickly that ®lm ¯ow is negligible, then the is-land is considered trapped: it cannot be invaded. The decision of whether to implement the trapping rule in a given instance is therefore partially dependent on the time scale, and the two extremes ± with and without trapping ± should serve as constraints to intermediate cases. Additionally, if a NAPL invades an aquifer quickly but then stays in place for an extended time, ®lm ¯ow may be important. In such a case, examining the dierence between invasion with trapping and without may give some hints as to where slow changes should appear. In 2-D, trapping causes holes to appear in the invaded region (Fig. 3), but does not by itself cause ¯ow to have, for example, more or fewer ®ngers. However, given the history-dependent progression of IP, a small change in the invasion sequence may result in large dierences in the ®nal con®guration.
We have also extended the model to the third di-mension. This is important for several reasons. First, 2-D and 3-2-D ¯ow patterns are dierent, partially because connected regions across the domain boundaries can co-exist for each of the two ¯uid phases in 3-D, but not in 2-D. Additionally, real-world problems are 3-D, and modeling a single 2-D slice of a 3-D site is not as useful, or potentially as accurate, as modeling the complete problem domain. Interestingly, trapping is less extensive in 3-D than in 2-D [53], but it can still be useful to be able to examine how important it is in any given situation. When modeling a quasi-2-D experi-ment, such as those of Hofstee et al. [26] or Glass et al. [19], it may actually be advantageous to simulate the process in 3-D, with the third dimension (thickness) being a few layers thick, because observations indicate
that saturation is often not uniform from one face to the other [19]. Because these physical quasi-2-D ex-periments are, strictly speaking, 3-D, modeling them as 2-D is likely to introduce model artifacts that could be avoided in 3-D.
Finally, we have developed a procedure to readjust the capillary number in time and space based on local changes in the area of the interface (see Section 2.2.2). Capillary readjustment is performed because if the DNAPL/water interfacial area increases while the spill rate and the porosity remain constant, the mean velocity of the interface ± and therefore of the invading ¯uid ± must decrease. Because a decrease in velocity corre-sponds to a decrease in the capillary number, the capillary number should be continually updated to ac-count for this change in velocity. Simply multiplying the injection capillary number by the proportional increase in interfacial area will not produce accurate results, however, because the velocity of the invading ¯uid is not uniform along the entire interface. A viscous-stabilized (high capillary number) invasion may expand fairly uniformly, but a slowly advancing DNAPL ®nger moves at the tip but changes little along the rest of the ®nger. In other words, the degree to which increases in interfacial area should be used in updating the capillary number, itself depends on the capillary number.
Our capillary readjustment algorithm therefore pro-ceeds as follows. First, the capillary number from the previous time step tÿ1 is used to calculate the
sto-chastic ®nger weighting parameterW
W 3=41= 4P; 2a
where
P mlog Catÿ1 n
n : 2b
The intermediate value P is simply a log-transformed
capillary number normalized to the range 0±1 from the original range Ca10m±10ÿn, for m0 to ÿ1 and n5±9. This original range corresponds to the range over which transition occurs from viscous-stabilized ¯ow to capillary ®ngering [34], itself a function of lattice size and, likely, other variables such as dimension of the
medium. In our simulations, we used m0 andn6,
but we show the exponents as being variables to em-phasize that their actual values may vary somewhat depending on the speci®cs of the system. Eq. (2a) is a strictly ad hoc formulation for relating the normalized
capillary number to the behavior produced by Martin
et al.'s biasing scheme. WhenP60,W is set to in®nity, resulting in pure IP.
Once a ®ngerFis stochastically selected usingW(as detailed in [14]), the ``eective'' interfacial area AF of
that ®nger is calculated by summing the ®rstIF=W worth
of entries in the sorted interface list, whereIFis the sum
It is evident that at low capillary numbers, W will be large, and so AF will only be a small fraction of the
number of entries in the list. In other words, at low ¯ow
rates, only a small fraction of the interface is actually moving. The eective area forFis then used to calculate the eective area of the entire interface Ai
AiWAFIi=IF; 3
whereIiis the sum of the invadabilties of the entire
in-terface. Finally, Ai is used to calculate a new capillary
number
CatCainjAinj=Ai; 4
where the subscript ``inj'' indicates initial values at the injection point. The end result is an algorithm in which the capillary number changes according to velocity of the moving portions of the interface, allowing ®ngering to develop even when ¯ow is viscous stabilized at the injection point (Fig. 3). The method is oset in time: the
Ca calculated from one ®nger at time t is used to
ap-proximate Ca for the entire system at timet1, given knowledge of the interfacial area of each ®nger. The assumption is that even though the individual ®ngers' Ca's span a wide range, all ®ngers together will average to approximately the systemCa.
The discussions so far have implied that the ¯ow patterns are largely functions of the two ¯uids. How-ever, the structure of the porous medium also exerts considerable in¯uence in developing ¯ow patterns. As an illustration, Fig. 4 shows invasion patterns for media similar to those in the previous ®gures, but diering in spatial arrangement. The imprint of the porous medium structure is evident in the DNAPL migration patterns, showing that the ®nal pattern is the result of interaction of the porous medium with the capillary, buoyancy, and viscous forces.
A series of detailed experiments reported in [22,23] aords us an opportunity to compare our model results to experimental data. The experiments are a series of controlled DNAPL spills performed in a water-satu-rated, uniformly packed sand tank. Following DNAPL detection at the lower tank outlet, the DNAPL source was removed and the tanks were excavated in serial horizontal sections. The addition of red dye to the
(a)
(c) (d)
(b)
DNAPL allowed discrimination of the DNAPL in photographs. In the current paper, we compare our re-sults to their experiment IV, injection of TCE into me-dium sand. Held's [22] results are statistical, in the sense that duplicate experiments may produce similar statis-tics with respect to (for example) saturation with depth, but would yield dierent speci®c ®ngering patterns. The photographic results are not strictly quantitative, how-ever, because the DNAPL spill volume calculated from image analysis is approximately 200% of that measured probably due to their methodology: they [22] used a red ®lter, which masked much of the contrast between the sand and the red dye, resulting in a pixel brightness histogram approximately seven brightness values wide. Partially NAPL-saturated pores would frequently be interpreted as NAPL-saturated during thresholding, over-estimating the amount of NAPL in the image.
In Fig. 5, we present two photographs from Held [22], showing DNAPL con®guration at 16.5 and 73.5 cm
below the injection port, and results of one of our sim-ulations of that experiment. Simulation inputs corre-sponded to experimental parameters in terms of ¯uid properties (DNAPL density, viscosity, contact angle, and interfacial tension), porous medium properties (porosity, pore size distribution, and a [assumed] ran-dom pore structure), size of the tank, size and location of the DNAPL injection port, and experimental pa-rameters such as DNAPL head and injection rate. A listing of numerical values of simulation and exper-imental parameters, taken from the input ®le, is given in Table 1. There are no ®tting parameters as such: num-bers are either derived directly from the experiment (e.g., the NAPL density), or are arbitrary (e.g., the random
number seed). Values for nandmdo not appear in the
input ®le; as mentioned earlier, we assume here that
m0 andn6. Held [22] used a constant head upper
boundary with an initially rising water table. The re-sulting DNAPL injection rates were nearly constant
(a)
(c) (d)
(b)
over time, so for convenience we used a constant rate injection, also with an initially rising water table. Sim-ulations used trapping and capillary readjustment and
were performed on a 556144144 grid with unit cell
size of 1.8 mm3. Simulations required approximately 5
min on a 800 MHz Athlon processor.
The qualitative comparison between horizontal sec-tions is quite good, indicating that the model captured the essentials of the DNAPL behavior. Examination of a side view of the simulation results (Fig. 6) shows that, as Held [22] stated, ``A long stretched plume reached 35 cm down (below the injection point), where multiple ®ngers developed''. In agreement with Held's [22] data, the region just below the injection point showed high saturation and stable behavior. Additionally, below about 35 cm under the injection point the number of ®ngers started to decline, with only a small number of ®ngers still advancing near the bottom. Quantitative comparison between experiment and simulation can be made using (inferred) saturation with depth, area/ perimeter ratio, or fractal dimension [22], but the
pattern itself is more dicult to quantify. A more quantitative comparison between Held's [22] experi-mental results and our simulations will be presented in a future publication.
Strictly speaking, our model is not directly applicable to Held's [22] experiment. Their experimental viscosity ratio,M 0:57, is slightly less than unity, whereas our model is intended for M >10:0 (as noted in [14]). Moreover, there is no DLA component to our model. However, their viscosity ratio places the experiment in the transition region, but closer to the stable ¯ow regime than to the viscous ®ngering regime, so little DLA-like behavior is to be expected. It is encouraging that the model captures the observed DNAPL movement, without adjusting an array of ®tting parameters; all needed input parameters are speci®ed in [22]. Examining the range of behaviors and robustness of the model, aorded by changing the random number seed, or the changes induced by (for example) changing the input capillary number, is beyond the scope of the current study. In this context, we stress that we are not
adver-Table 1
Parameter values for the Ewing and Berkowitz model simulating Held and Illangasekare's [23] experiment IV, TCE movement into medium sand
Parameter Value Comment
Porous medium parameters:
Mean pore radius, m 0.00011 From [22]
Standard deviation of pore size distribution, m 0.00003033 From [22]
Porosity 0.4 From [22]
Medium type Random Random decoration (no spatial structure)
Fluid parameters:
Fluid density dierence waterÿNAPL, kg mÿ3 460.0 Positivegravity-unstable Initial depth of water table, m 0.05 Must be P0 from top of grid Initial ponding depth of NAPL at surface, m 0.02 Must be P0 from top of grid Water/NAPL interfacial tension, J mÿ2 0.0164 From [22]
NAPL wetting angle, radians 2.2176 From [22] Absolute viscosity of NAPL, poise 0.0057 From [22] Capillary number at injection point 1:6410ÿ4 From [22]
Size parameters:
Grid size,X (vertical) 556 Grid height, units
Grid size,Y 144 Grid width, units
Grid size,Z 144 Grid depth, units
Injection point inX, units 56 Start injection at this grid depth Injection pointYmin, units 72 Start injection here (or center if circular) Injection pointYmax, units 11 Stop injection here (or radius if circular) Injection pointZmin, units 72 Start injection here (or center if circular) Injection pointZmax, units )1 Stop injection here.)1 denotes circular injection.
Grid unit size, m 0.0018
Experimental parameters:
Total volume of NAPL injected, m3 9:410ÿ4 From [22]
Horizontal area of actual problem domain, m2 0.20885 From [22]. For water table rise, when problem domain is smaller than physical experiment.
Use trapping? Yes
Use capillary readjustment? Yes
Use wrapping? No
Run-speci®c parameter:
tising our model as addressing all the issues raised earlier in the paper. Rather, the model represents one of the current state of the art, and is used here as a foil to raise and illustrate issues we believe to be important.
4. Future directions and perspectives
A consistent diculty in evaluating both continuum-based and stochastic growth DNAPL ¯ow models is that there is no straightforward way to validate a model. Proper validation requires a data set that is suciently detailed in both space and time that comparisons can be made at all stages of the simulation, not just at the end. In fact, comparison with many experimental results spanning a wide range of experimental parameter space is needed before we can have con®dence that the model performs both correctly and generally. Currently avail-able data fall into three general classes: micromodel studies (e.g. [24]), macroscopic experiments using an arti®cially structured porous medium (e.g. [26]), and macroscopic experiments using an unstructured porous medium (e.g. [23]). For the purposes of this article, micromodel studies are less relevant, so we will discuss only the macroscopic experiments. Laboratory studies are few in number (see, e.g. [10]) and fewer yet when one requires quantitative descriptions of the porous medi-um, the NAPL and the conditions under which it was injected, and the migration pattern. Structured medium experiments are useful for verifying that a model per-forms in a generally correct manner, but unless the medium is fairly intricate and very well characterized, a single dataset is of limited value because it tests so few aspects of the model. Unstructured media, meanwhile, will produce somewhat random patterns due to sensi-tivity to unknown local details. It is dicult to ®nd meaningful ways to compare experimental results to simulations, eectively comparing a single physical Monte Carlo realization to many virtual ones. Com-parisons between experimental and model results, across many depths, of the number of ®ngers, NAPL-saturated area, and NAPL area/perimeter ratios give some indi-cation of how well a model performs, but because of the random nature of the medium it is not useful to be too strict in the comparisons.
The most useful experiments will be those that use intricately constructed, well-characterized media and quantitatively captured saturation data over time. To our knowledge, the only data that meet these criteria are from the recent experiments of Glass et al. [19]. Their medium was designed to test several hypotheses con-cerning capillary barriers, and data were collected at frequent intervals and analyzed to yield NAPL satura-tion as a funcsatura-tion of posisatura-tion and time. It is our hope that more experiments of this caliber will be performed and published, as they oer a more stringent and quantitative avenue for model validation than has heretofore been available.
To summarize, percolation-based models of DNAPL movement have much to recommend them over models based on partial dierential equations, in terms of con-ceptual foundation, speed, and potential for accuracy.
However, we identify ®ve principal ways in which per-colation-based NAPL movement models need to ad-vance. First, the overly simplistic invadability equation, by which invadability increases linearly with depth, needs to be modi®ed dynamically to account for the NAPL's common pool/®nger structure and the viscosity of the NAPL. Second, ways must be found to blend the anti-DLA and IP models so the correct process model is used for stable ¯ow. Third, models should be expanded to function properly in the unsaturated zone and where the water table is ¯uctuating, even incorporating three-phase ¯ow dynamics as needed. Fourth, the number, quality, and availability of data sets for model valida-tion must be increased. And ®nally, modelers should work with computer scientists to overcome some of the speed and memory issues that are already evident, and which could eventually delay or prevent practical ap-plication of the models.
Acknowledgements
The authors thank Rudi Held for permission to re-produce photographs from his M.S. thesis, and several anonymous reviewers whose comments contributed signi®cantly to the clarity and completeness of this paper. This research was partially supported by a grant from the Raymond and Mary Baker trust.
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