Mixing and Chaotic Microstructure

Yuefan Deng, James Glimm, Mixing and and David H. Sharp Chaotic Microstructure

ixing is a process in which performance of pellets in laser-fu- nonlinear, and they typically do not distinct fluids intermingle sion experiments. admit a small parameter for useful Min a complex way. Chaotic Without doubt the problem of expansions. Together these features microstructure refers, more broadly, chaotic microstructure, in its many lead to an extremely complex phe- to all small-scale stochastic or ramifications, is the most fundamen- nomenology. chaotic behavior that affects dynam- tal in classical continuum physics. We propose here a systematic ics at large length scales. Problems Why is this? First, because the program for addressing this class of that involve mixing and chaotic mi- problem is pervasive, and second problems. This program follows a crostructure are of fundamental im- because the only systematic solution line of development that will be fa- portance in both basic science and occurs at the engineering level miliar to many-body theorists. The engineering: They are the central through the method of conservative first step is the identification and issue in turbulence, pipeline flow, overdesign, or experimental trial and analysis of elementary modes, or co- and the dynamics of supernovae. error. The classical methods of sci- herent structures. For example, in The microstructure of porous rocks, ence—theory, computation, and ex- turbulence the elementary modes are which is stochastic on all length periment—have so far delivered vortices; in phase transitions, den- scales, is a dominant aspect of the much less than is needed either for drites; in Hele-Shaw cells, viscous geology of aquifers and oil reser- engineering purposes or for scientif- fingers; in solid materials, lattice voirs. Chaotic microstructure also ic understanding. dislocations. The second step is to plays a key role in determining the These problems are difficult for understand the pairwise interaction properties of common materials such three reasons. They possess an ex- of elementary modes. This step is as metals and plastics. Mixing is a ceedingly large number of active followed, not by an analysis of dominant phenomenon limiting the degrees of freedom, they are usually multi-mode interactions, but by the

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Mixing fig1 3/27/93

(a) statistical analysis of an ensemble of regularities are inevitably present at Ceiling modes governed by pairwise interac- the interface. Those portions of the tions. From this analysis we want to fluid at the interface that lie higher Water derive continuum-level constitutive than the average experience more laws (such as equations of state), pressure than is necessary for their which are then used (fifth step) in support. They begin to rise, pushing Air macroscopically averaged flow aside water as they do so. Neigh- equations such as the Navier-Stokes boring portions of the fluid, where equation. Thus we have a five-step the surface hangs a little lower than Floor procedure relating microscopic dy- average, require more than average namics to macroscopic observables. pressure for their support. They Although each of the steps may begin to fall. The air cannot supply (b) seem rather obvious, their integra- the specific variations in pressure tion is less commonly discussed. A from place to place necessary to pre- large fraction of work in classical vent the interface irregularities from physics is related to this program in growing. The initial irregularities Ripples that it addresses one or another of therefore increase in magnitude, ex- its steps. ponentially in time at the beginning. In this paper we will illustrate this The water falls to the floor. program by discussing its implemen- The same layer of water lying on tation in examples drawn from our the floor would have been perfectly work and that of our collaborators. stable. Irregularities die out. Thus This work as well as collateral work we can infer a simple criterion for (c) Bubbles of others can be traced from the fur- the onset of Rayleigh-Taylor insta- ther reading given at the end of the bility at the interface between two article. fluids of different densities: If the heavy fluid pushes the light fluid, the interface is stable. If the light Rayleigh-Taylor Instability fluid pushes the heavy fluid, the in- Spikes terface is unstable. Our first example concerns Rayleigh-Taylor instability occurs Rayleigh-Taylor instability. The oc- in diverse situations. Let us take a currence of this instability can be quick look at one example from re- Figure 1. An Example of Rayleigh- understood in the following simple search into inertial-confinement fu- Taylor Instability way. Imagine the ceiling of a room sion. Figure 2 shows a highly (a) The pressure of the air is quite suffi- plastered uniformly with water to a schematic picture of the implosion cient to support a perfectly uniform layer depth of 1 meter (Figure 1). The of a deuterium-tritium (DT) pellet. of water 1 meter thick against the ceiling. layer of water will fall. However, it A spherical glass or metal tamper is (b) But the air pressure cannot constrain is not through lack of support from filled with DT gas. The tamper is the air-water interface to flatness. Rip- the air that the water falls. The irradiated with intense laser light, ples or irregularites will inevitably be pre- pressure of the atmosphere is equiv- which causes it to accelerate inward, sent at the interface. (c) The irregulari- alent to that of a layer of water 10 compressing the DT gas inside the ties grow, forming “bubbles” and “spikes.” meters thick, quite sufficient to hold cavity in order to bring about nu- The water falls to the floor. the water against the ceiling. But in clear fusion. During irradiation the one respect the atmosphere fails as a outer surface of the tamper is the in- sion, when the light fluid pushes the supporting medium. It fails to con- terface between a heavy fluid (glass heavy fluid. As the pellet is com- strain the air-water interface to flat- or metal) and a light fluid (vapor- pressed to perhaps 1000 times its ness. No matter how carefully the ized glass or metal) and is unstable normal density, the pressure in the water layer was prepared, small ir- during the initial phase of the implo- cavity increases until it is sufficient

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Mixing and Chaotic Microstructure

to slow the inward motion of the interactions among bubbles and connecting interior region (mixing tamper. This phase of the implosion spikes lead to the formation of a layer). The bubble regime has been is also Rayleigh-Taylor unstable; chaotic mixing layer at the interface. the most carefully studied and has here the high-pressure light gas in Understanding the growth rate and the simplest properties. It is found the cavity is pushing the tamper. structure of this mixing layer is the in the Read and Youngs experi- Although this picture of the implo- central problem in the study of ments, as well as in numerical simu- sion of a DT pellet is oversimplified Rayleigh-Taylor instability. lations (Figure 3), that the edge of in a number of ways, it nevertheless This question has been investigat- the mixing layer in the bubble suggests quite clearly that Rayleigh- ed experimentally by Read and regime is dominated by a collection Taylor instability can have an im- Youngs. Their important findings of bubbles. portant effect on the compression of bring the questions to be understood The average height of the bubbles the pellet. into sharp focus and can be summa- relative to the mean position of the A closer look at the time evolu- rized as follows. The mixing layer interface, h(t), grows with time ac- tion of a Rayleigh-Taylor unstable has three principal regions: the edge cording to the scaling law interface reveals a complex phenom- where bubbles of light fluid are pen- enology. As the instabilities devel- etrating into the heavy fluid (bubble h(t) = Æ Agt2, op, bubbles of light fluid and spikes regime),Sharp fig2 the edge where spikes of 4/9/93 of heavy fluid form, each penetrat- heavy fluid are penetrating into the where A, the Atwood number, is a ing into the other phase. Complex light fluid (spike regime), and the modification due to buoyancy of the acceleration force (gravity) g and is Laser light defined as A = ( Ω 2 ° Ω 1)/( Ω 2 + Ω 1), in which Ω i represents the density of fluid i. It is a remarkable experimental finding of Read and Tamper Youngs that for incompressible flu- Vapor (vapor) ids the acceleration constant Æ in Tamper this equation is an approximately Solid universal constant, in that it is nearly independent of initial conditions DT gas DT gas and of fluid properties such as density, viscosity, and surface tension. P1 P1 Finally, it is observed that the average number of bubbles at the inter- P2 P2 face decreases with time, and that

P1 > P2 their average radius increases.

P 3 The main objective of our work

P3 > P2 has been to understand the physical properties of the Rayleigh-Taylor mixing layer. The fluid in the mixing layer is in a chaotic state, and it is necessary to have a strategy to guide Figure 2. Rayleigh-Taylor Instabilities in Laser Fusion one through the complexities of this On the left appears an extremely simplified view of a laser-fusion pellet at an early problem. The fundamental challenge stage of a fusion experiment. Laser irradiation of the solid tamper vaporizes the outer in modeling fluid chaos is to provide layer of the tamper and pushes the tamper inward. Because the light vapor is push- a simple macroscopic description of ing the heavier solid, this stage of the process is Rayleigh-Taylor unstable. On the a chaotic fluid state that expresses a right is sketched the same experiment at a later stage. Here the DT gas, now at high statistical average of information de- pressure (P1), slows the inward motion of the tamper. Since the DT gas is less dense scribing its chaotic microstructure. than the tamper, this stage of the process is also Rayleigh-Taylor unstable. For the case of chaotic microstruc-

126 Los Alamos Science Number 21 1993 Mixing and Chaotic Microstructure

ture at the molecular level, the solution to this problem is highly developed and has given rise to the sub- jects of thermodynamics and statistical physics to describe the macro- and microphysics respectively. Adapting the statistical-physics viewpoint to the analysis of fluid chaos,we proceed to study the one- and two-body problems and follow this with an analysis of a statistical ensemble. A renormalization-group fixed point emerges as a simple description of this ensemble. In molecular physics the elementary modes, or units of analysis, are atoms or molecules. We identify bubbles as the fundamental modes governing the microphysics at one edge of the Rayleigh-Taylor mixing zone. The one-body problem concerns the dynamics of a single bub- Figure 3. A Simulation of Rayleigh-Taylor Mixing ble. A relatively complete theory of The figure shows the late-time behavior of Rayleigh-Taylor instability. The dimension- the motion of a single bubble has less compressibility M2 = 0.5. The simulation was performed on a 32-node parallel been developed, which is valid for iPSC/860 computer. both compressible and incompressible fluids and includes a formula height and radius. We draw an en- the single-bubble theory, and ve is for the bubble velocity as a function velope through the tips of adjacent the velocity of a bubble in the enve- of mode amplitude. This formula bubbles. The envelope defines a lope. The resulting formula agrees contains a small number of parame- long-wavelength collective excita- very well with experiment; note that ters, which have been fixed on the tion of the fluid and has the appear- it contains no free parameters. basis of analytic formulae, numeri- ance of a set of broader bubbles and The superposition hypothesis cap- cal computations, and experiments spikes. These broader bubbles can tures only one aspect of bubble in- on periodic arrays of bubbles. be viewed as additional elementary teractions. The other is bubble When more than one bubble is modes, which themselves obey the merger. Merger refers to the behav- present, the bubbles interact. The single-bubble theory. Both experi- ior of a pair of neighboring bubbles: interactions have a pronounced ef- ment and simulation led us to the as- a larger, advanced, faster-moving fect on the behavior of bubbles. The sumption that, although each funda- bubble and a smaller, less advanced, first effect, observed in both experimental bubble is in a deeply nonlin- slower-moving bubble. (Since larg- ments and simulations, is that the ear regime, the interaction of such a er bubbles move faster, eventually velocity of a single bubble in a bubble with the collective mode is they are always more advanced than chaotic array of bubbles is typically linear, so that the net velocity of a smaller bubbles.) It is observed that a factor of 2 greater than that pre- bubble in a chaotic array is given by the smaller bubble is rapidly washed dicted by the single-bubble theory. a simple superposition: downstream, while the larger bubble This is a rather surprising result, increases in size to occupy the por- which can be understood as follows. v = vb + ve, tion of the interface vacated by the In a chaotic array of bubbles, a smaller bubble. Thus, the dynamics given bubble has left and right where vb is the velocity of a bubble of the fundamental modes comprises neighbors which generally differ in of the same amplitude as given by the dynamics of a single nonlinear

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mode (single bubble) plus an ap- length scale of the instability. The Sharp fig4 4/8/93 proximately linear mode-mode ve- next step is to introduce rescaled locity coupling plus a highly nonlin- variables, which subtract out these ear mode-merger process. changing length scales. Once this is 0.14 We next consider a statistical en- done, the result is a renormalization- semble of interacting modes (bub- group equation. bles). We suppose there is a proba- Analysis of this equation revealed 0.12 (dimensionless)

bility distribution for an infinite col- the existence of a non-trivial renor- α lection of bubbles, which defines the malization-group fixed point. This single-particle distribution (the means that the equations have the 0.1 probability of picking a bubble of property that, no matter what the height h from the ensemble), the initial values of the variables are, 0.08 pair correlations, and so forth. The they approach the same asymptotic bubble dynamics outlined above de- values, or fixed point. At this fixed fines an evolution in this statistical point, the width of the mixing layer Acceleration parameter 0.06 ensemble of modes. Although the grows at a constant acceleration, Æ . 0 0.2 0.4 0.6 0.8 1.0 full statistical model could be solved The theory provides a zero-parame- FigureFluid 4. compressibility Dependence M2 (dimensionless)of the numerically, we instead introduce ter determination of Æ , which turns Mixing-Layer Growth Rate on simplifying assumptions to arrive at out to be in remarkable agreement Compressibility a more tractable form of the model. with experiment. Since Æ is deter- Plot of the growth rate Æ versus the As when the Boltzmann equation is mined by a fixed point, its insensi- compressibility, M2. The vertical bars in- derived in statistical mechanics, we tivity to initial conditions is also ex- dicate the variance associated with the neglect pair correlations; then the plained. Thus our analysis has choice of random-number seed. probability measure defining the sta- shown that the main properties of tistical ensemble is determined by the bubble-dominated part of the approximate validity of the model- the single-particle distributions. chaotic mixing layer are direct con- ing assumptions employed to under- Those distributions in turn are ap- sequences of scaling and the exis- stand bubble interactions and to proximated in terms of just three tence of a renormalization-group identify conditions where those as- time-dependent parameters: the av- fixed point. sumptions were not valid. erage height of the bubbles, h(t), the We emphasize that our program Our numerical simulations of variance in bubble height, æ (t), and has depended in an absolutely essen- Rayleigh-Taylor mixing employed a the bubble radius, r(t). The bubble tial way on numerical simulation of front-tracking method. Front-track- dynamics leads to a set of ordinary the solutions of the full two-dimen- ing works by “hard-wiring” into the differential equations for these sional, two-fluid Euler equations. code maximal information about the quantities. These calculations were used in a analytically known behavior of the The bubble dynamics can be un- variety of ways. Calculations of the solution near a discontinuity or front derstood from the perspective of the evolution of Rayleigh-Taylor insta- (such as the jump in density at the renomalization-group method. This bility at a random interface were heavy/light interface in the method has two key steps: “coarse- used as a model-independent way to Rayleigh-Taylor problem or at a graining” and “rescaling.” The bub- calculate the magnitude of Æ , as shock front in gas-dynamics prob- ble-merger process accomplishes a well as to establish the insensitivity lems). This approach leads to two dynamical coarse-graining of the dy- of Æ to a wider range of initial con- benefits as well as a cost. The first namics, because sets of smaller bub- ditions than were explored experi- benefit is the scientific understand- bles are replaced by larger bubbles, mentally and in a way that was not ing about the behavior of the solu- without changing the physics. The tied to possible idiosyncrasies of the tion near a discontinuity, generated differential equations for the bubble experimental apparatus. Numerical in the course of implementing the dynamics thus reflect the coarse- simulations were used to define pa- method. The second benefit is that graining of the merger process, as rameters in our single-bubble model. highly accurate solutions can be well as a dynamic increase in the They were also used to establish the achieved without recourse to very

128 Los Alamos Science Number 21 1993 Mixing and Chaotic Microstructure

fine computational grids. In fact, flow variables Ω = hΩ i+ ±Ω , v = (solutes) in groundwater. There is a the front-tracking method typically hvi+ ±v, etc., are expressed as sums nontrivial linear version of this prob- allows an increase in computational of mean quantities hΩ i, hvi, etc., lem, and in this sense it is simpler resolution by a factor of about 3.5 in and fluctuating quantities ±Ω , ±v, than the Rayleigh-Taylor problem. each spatial dimension and in time, etc. An effort to write governing The fundamental methodology again that is, by a factor of about 40 in a equations directly for mean quanti- consists of the integration of field two-dimensional, time-dependent ties such as hΩ ifails due to the observations, theory, and computa- problem and 150 in a three-dimen- nonlinearity of the governing equation, but the linearity permits the sional, time-dependent problem. tions and the fact that the average of specialization of the full arsenal of This dramatic increase in efficiency a product is not equal to the product techniques described above and a allowed a corresponding increase in of the averages. For this reason, the more complete analysis of the prob- the detail and scope of the computa- averaged conservation laws contain lem. In particular, since the modes tions attempted and was instrumen- new quantities, such as the Reynolds in a linear problem do not interact tal to the success of the program. with each other, the study of mode- h½v ih½v i The cost is that code development R = h½v v i¡ i j : mode interactions is unnecessary. ij i j h½i for front-tracking is more demand- ½ As Einstein recognized, the phe- ing because more complex algo- stress: 1 ¯ nomenon of dispersion (the spread = max ;1 ¡ : rithms are required to express the in- The continuum 2equations,2 even of one medium through another) is formation about solution discontinu- after quantities such as R are intro- the macroscopic consequence of ran- 0 0 0 0 0ij 0 ities. This is basically a duceddh as=dt new= d unknowns=dt = dr with=dt = their 0 dom microscopic events. The clas- software-complexity, issue, which own dynamical equations, require sical theory of dispersion, based on has been dealt with by the construc- the introduction of further new the assumption that the random tion of highly modular code, the use quantities (do not close). This events occur on a much shorter scale of more powerful programming lan- means that further modeling as- than the macroscopic observations, guages, and other modern code- sumptions must be made in order to leads to the familiar relation development methods. arrive at a complete system of equa- The Read-Youngs experiments tions. We view these modeling rela- l(t) = O(t1/2), were carried out with nearly incom- tions as an extension of thermody- pressible fluids. Both our simula- namics. Our work in progress in the where l is the position of the disper- tion methods and our model apply area is to evaluate the correctness of sion front and t is time. Field data also to more compressible fluids. It different modeling approaches. One for transport of fluids through is thus possible to explore, and in- step in this program is to understand porous media do not satisfy this sim- deed for the first time to predict, the the structure of the fluctuating flow ple relationship. Porous media of properties of a chaotic mixing layer moments, such as Rij, in a Rayleigh- practical interest are heterogeneous in a parameter regime outside the Taylor unstable flow. on all length scales owing to random range of existing experiments. Our variations in the geology. Thus the analysis shows that the mixing-layer scale on which the randomness oper- growth rate in a compressible fluid Flow in Porous Media ates is not small relative to the ob- can be a factor of 2 larger than in in- servations, and the assumptions un- compressible fluids (see Figure 4), In the example of Rayleigh-Taylor derlying the classical theory of dis- and that Æ develops a dependence instability discussed above, multiple- persion are not valid. This creates a on initial conditions. The impor- length-scale chaos arises sponta- serious problem for practical engi- tance of these results in various ap- neously in a nonlinear dynamical neering in hydrology and in environ- plications seems clear. problem. We now turn to an exam- mental studies of contaminants dis- The final step in our program for ple in which chaos is forced, through persing in groundwater, for example. the study of high-dimensional chaos the influence of multiple-length- In fact there is no known body of is to study continuum-level constitu- scale data in the definition of the knowledge to connect laboratory and tive laws and equations. Here the problem. This example concerns the field-scale values of dispersivity. flow is regarded as stochastic and transport and dispersion of pollutants Therefore in practice, dispersivity

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Mixing fig5 3/30/93

that the entire statistical variation of Shape of tracer front the flow is determined by the two- point correlation function hª(x)ª(y)i, which is then a function t = 8 of the difference variable x°y; that is, hª(x)ª(y)i= f(x°y). Any choice of f defines a model of the stochastic structure of the rock. A common t = 6 choice, f = b exp(°|x°y|/l), has a single correlation length scale, l, and thus does not represent variability on all length scales. The simplest form of f that does allow variability t = 4 on all length scales is the scale-in-

Direction of flow variant function

°Ø t = 2 f = b|x°y|

with Ø > 0. Such a scale-invariant, Tracer front at or self-similar, function is known as t = 0 a fractal. Somewhat greater gener- t = 0 ality is obtained by allowing Ø to be a slowly varying function of |x°y|. Randomly varying porous structure (that is, the specification of a ª ξ ξ ξ ξ 1 2 3 4 field) introduces a statistical aspect to the flow. In Figure 5 we plot suc- cessive positions of a tracer front Figure 5. Simulation of Flow through a Porous Medium (an interface between one part of the Depicted are positions of a tracer front in a fluid flowing through a porous medium. fluid, which has been marked with a The front is shown at five times (t = 0, 2, 4, 6, 8) for each of four independent simula- dye or in some other way, and the tions. Each simulation uses a different realization of the ª field drawn from a statisti- rest of the fluid) in four simulations. cal ensemble of geologies. The simulations used the same initial configuration of the tracer front must be estimated conservatively ogy to electrical conductivity, it is but different realizations of the ª from carefully instrumented field an inverse resistivity for the flow in field, each drawn from the same sta- studies conducted at some other site the sense that the flow rate equals ∏ tistical ensemble of geologies. The but at comparable length scales. times the negative pressure gradient. statistical variation between the sim- Flow through porous media is We make the common assumption ulations arises, not from a lack of controlled by unknown (and, at nec- that ∏ = exp ª, where ª is a normal- determinism of the flow, but from essary fine scales of detail, unknow- ly distributed, random field that de- unpredictability of the flow due to able) features of the geological pends on the position x in space and unknown details of the geology. To medium such as its porosity. In our represents the random variations in illustrate this idea, we consider the studies, we treat those features as the structure of the porous medium. simplest flow problem: transport of random fields, that is, random vari- Assuming stationary statistics for a passive concentration sample of an ables defined at every point of the simplicity and denoting the ensem- impurity by a background flow media. The transmissibility ∏ is the ble average with angle brackets, we through porous rock. The flow un- most important variable relating the note that hª(x)iis independent of x certainty would be observable partly geology to flow properties. In anal- and can be normalized to be zero, so as an uncertainty of arrival times of

130 Los Alamos Science Number 21 1993 Mixing and Chaotic Microstructure

Sharp fig6 4/2/93 a concentration sample and partly in 102 the spreading of the concentration Perturbation theory gradient due to differing arrival Computer simulation times of the individual particles that b = 0.76 Pure fractal theory compose it. Apart from the rather small effect of molecular mixing on 101 length scales typically of interest, b = 0.18 these two aspects of flow uncertainty are identical because the structure of the rock is heterogeneous on all length scales and consequently affects the flow on all scales. The dis- tinction between the two types of 100 b = 0.01 uncertainty lies in the resolving ability of the measuring instrument, Mixing length the size or volume of the initial concentration profile, its initial concentration gradient, and the distance of travel before measurement occurs. 10-1 We use dispersion to represent both aspects of uncertainty described above. We can study this dispersion mathematically in terms of the expected values hc(x,t)iof a concentration c. The fluctuations, 10-2 0 1 2 ±c = c°hci, and higher moments 10 10 10 Travel distance such as the covariance, h±c±ci, are of interest as well. Similarly, the Figure 6. Three Calculations of the Mixing Length in a Porous Medium flow velocity v can be expressed as The figure is a log-log plot of mixing length, l, as a function of travel distance in a the sum of its mean and its fluctuat- medium having Ø = 0.5, where Ø is the fractal exponent characterizing the stochastic ing parts: v = hvi+ ±v. At this structure of the rock. Each triple of curves shows a comparison between pure fractal point, the flow problem is similar to theory, second-order transport perturbation theory including transients, and a numeri- the transport of a concentration by a cal simulation. Each triple of curves corresponds to a different value of the permeabil- turbulent velocity field, such as a ity field coefficient variation, as labeled. pollutant in the atmosphere. The following methods are avail- means that the mixing length l(t) exponent Ø of the geology: able for this class of problems: nu- associated with the concentration for Ø > 0. However, l(t) at short and merical simulation of the transport profile hc(x,t)igrows in time more intermediate times deviates from process, followed by an ensemble rapidly than t1/2. When the geology this law and in fact at short times is average, to determine hcidirectly; is fractal, l(t) has a fractal behavior fractal with a different exponent. perturbative solutions of the trans- as well in the limit of long time, In contrast to many problems of port equations in powers of ±v, in- namely, turbulent mixing, the different meth- cluding resummation methods such ods give consistent results. As Fig- as renormalized perturbation theory; l(t) ª O(t∞ ), ure 6 shows, perturbative and nu- analysis of field data; and exact so- h½v ih½v i merical solutions agree quantitative- R = h½v v i¡ i j : lution of simplified model problems. and ∞ ij can ibej relatedh½i to the fractal ly, simple soluble models agree These methods lead to consistent re- ½ asymptotically at long times, and 1 ¯ sults and show anomalous (that is, = max ;1 ¡ : each is consistent with geological nonclassical) dispersion. This 2 2 field data. However, each of these dh0=dt0 = d 0=dt0 = dr0=dt0 = 0 1993 Number 21 Los Alamos Science 131 ,

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methods is at variance with the results of classical diffusion theory, with a fixed diffusion constant. Our conclusion is that for geology with unknown fine-scale heterogene- ity, dispersion of contaminants in groundwater should depend on time or travel distance, with transient cor- rections to fractal asymptotics. The understanding of dispersivity and its relation to multiscale geological het- erogeneities is important because dispersivity is an essential input into large-scale computational modeling of groundwater-remediation projects.