### Dispersivity

The dominant mechanism of subsurface mass transport is groundwater advection. Geologic heterogeneity contributes significantly to advective transport because it creates variations in velocity that influence patterns of groundwater flow. As a result, advective transport is extremely sensitive to how subsurface geologic heterogeneities are distributed in space. When heterogeneities are distributed uniformly in space, and when the scale of prediction is much larger than the scale of the heterogeneities, dissolved mass will migrate through and around heterogeneities in much the same way as molecules diffuse through liquid under the influence of a concentration gradient. This spreading phenomenon is referred to as mechanical mixing. While it arises from an entirely different mechanism than that governing molecular diffusion, at practical scales the two processes produce the same outcome, and can be described using Fick's law:

M = - D dC/dx

where M is mass flux per unit length squared per unit time [M/L2T], D is the diffusion coefficient in units of L2/T, C is concentration [M/L3], and x is coordinate location. Extension to two and three dimensions is fairly straightforward.

Field scale transport described by this model of molecular diffusion is often referred to as Fickian transport, and its corresponding diffusion coefficient is called dispersivity (a) to reflect the fact that dispersion, rather than molecular diffusion, is the actual mechanism responsible for spreading. Dispersitivities are generally orders of magnitude larger than diffusion coefficients because, while diffusion is caused by random molecular motion due to thermal kinetic energy, mechanical mixing is a consequence of pore-scale variations in fluid velocity. Instantaneous concentration breakthrough curves for dissolved mass that spreads according to Fick's law exhibit bell-shaped, or gaussian, behavior. The rate of spreading can be related to the variance of the gaussian curve via the equations:

aL = v*s2tL/2t
aT = v*s2tT/2t

where s2tL and s2tT are variances from instantaneous breakthrough curves (C vs. time) in the longitudinal and transverse directions, v is velocity (constant over time), t is time since displacement, and x is plume displacement distance in the longitudinal direction. Note that, if breakthrough curves in the transverse direction are not available, empirical evidence suggests that aT is roughly one-tenth of aL.

 Classical Fickian Dispersivity From Breakthrough Curve Data
 a(L, T) = v*s2t (L,T)/2t a(L, T) = longitudinal or transverse dispersivity [L] s2t (L, T) = variance of C vs. t curve in either long. or trans. direction [T2] v = groundwater velocity [L/T] t = time since plume displacement began [T]
Velocity (v)
Instantaneous Breakthrough Curve Variance [s2t (L, T)]
Displacement Time (t)
Dispersivity [a (L, T)] Calculated Result

Alternatively, if concentration is known as a function of space, dispersivities can be estimated according to the relations:

aL = s2L/2x
aT = s2T/2x

where s2L and s2T are variances from C vs. x and C vs. y plots through the plume centroid along longitudinal and transverse directions, and x is plume centroid displacement distance in the longitudinal direction. If the concentration profile in the transverse direction is not available, aT can again be estimated as roughly one-tenth of aL.

 Classical Fickian Dispersivity From Concentration Profile Data
 a(L, T) = s2(L, T)/2x a(L, T) = longitudinal or transverse dispersivity [L] s2(L, T) = variance of C profile in either long. or trans. direction [L2] x = plume centroid longitudinal displacement [L]
Concentration Profile Variance [s2(L, T)]
Displacement Distance (x)
Dispersivity [a (L, T)] Calculated Result

While the classical Fickian techniques of estimating dispersivities discussed above account for spatial variability via the variance terms, they do not explicitly consider the effects of spatial structure that can come into play at macroscopic scales of variation. Geologic variables commonly exhibit spatial correlation, in that values immediately adjacent to one another are similar and those further apart are dissimilar. Spatial correlation can strongly influence the rate of plume spreading by encouraging dispersion along directions where correlation scales are large and inhibiting it in directions where spatial correlation is small.

From theoretical considerations, Gelhar and Axness (1983) derived the following equation describing Fickian macrodispersivity in the longitudinal direction:

AL = s2ln K* lL/g2

where s2ln K is the variance of the natural log of hydraulic conductivity, lL is the correlation scale of ln K in the longitudinal or mean-flow direction, as obtained from an exponential variogram, and g is a flow factor, generally assumed to equal 1.

For the case of transverse macrodispersivity, the absence of advective forces produces a theoretical transverse macrodispersivity of zero:

AT = 0

This result can be intuitively understood by considering the fact that molecular diffusion and mechanical mixing, in the absence of any driving advective forces, will tend to "cancel out" along transverse directions. That is, just as many molecules will diffuse or disperse outward, away from the plume centroid, as will diffuse or disperse inward, toward the plume centroid.

 Macrodispersivity (Gelhar and Axness, 1983)
 AL =s2ln K lln K AL= longitudinal macrodispersivity [L] s2ln K = variance of ln K lln K = correlation scale of ln K (from exponential variogram fit) [L]
Ln K Variance (s2ln K)
Ln K Correlation Scale(lln K)
Longitudinal Macrodispersivity (AL) Calculated Result

In general, scales required to attain limiting Fickian behavior, particularly in spatially correlated geologic media, are quite large and may never be attained when the scale of the plume is smaller than the scales of all heterogeneities influencing plume migration. Prior to attaining this asymptotic Fickian rate of spread, spreading will occur at smaller, non-Fickian rates. It has been estimated from theoretical considerations that, in geologic media characterized by a single scale of geologic heterogeneity, a plume must be displaced over several correlation lengths before it will undergo Fickian rates of spread. [Dagan, 1988; Neuman and Zhang, 1990; Zhang and Neuman, 1990].

In practice, most geologic media possess a hierarchy of natural scales. Plumes moving through such media are in a constant state of transition, attaining intermittent episodes of Fickian behavior while moving within statistically homogeneous geologic units, and pre-asymptotic or non-Fickian behavior when the next larger scale of variation is spanned and new sources of geologic heterogeneity encountered. In such media, macrodispersivity increases without bound in a stepwise manner with increasing displacement, as the effects of the geologic heterogeneities "average out" over successively larger representative elementary volumes (REVs).

Neuman (1990) derived a relationship between Fickian macrodispersivity and plume displacement using 130 values of dispersivity estimated from a wide variety of geologic media and over a broad range of scales by many different researchers. He assumed that geologic media are characterized by stepwise changes in variability with increasing scale which, in the limit, approach fractal behavior as the number of scales increases to infinity. After accounting for varying amounts of uncertainty in the data, he concluded that macrodispersivity (AL) depends on plume displacement distance (Ls) according to the relation:

AL = 0.017 * Ls1.5

where data for Ls > 3500 m were excluded from consideration because such data violated the theoretical constraint that AL< Ls used to derive the relation. Note that, again, according to theory, transverse macrodispersivity AT = 0.

The "universal scaling rule" created a great deal of controvery in the hydrologic community, largely because it is based on the assumption that all geologic media are characterized by similar fractal architectures. However, Neuman (1990) was careful to point out that the scaling rule does not necessarily describe conditions at any given location. Rather, it represents a measure of dispersivity over a wide range of length scales, in a variety of geologic media, and under diverse flow and transport conditions, accurate only in a global, or mean-square, sense. He qualifies the rule by stating that local deviations of ln K from a power-law variogram, caused by natural scales of geologic variation that do not approximate fractal behavior, will likely contribute significant error to macrodispersivity estimates calculated from the universal scaling relation.

 'Universal Scaling Rule' Macrodispersivity (Neuman, 1990)
 AL = 0.017 * Ls1.5 AL= longitudinal dispersivity [L] Ls = plume displacement distance [L]
Plume Displacement Distance (Ls) (must be less than or equal to 3500 m)
Longitudinal Dispersivity (AL) Calculated Result

References

Dagan, G., 1988, 'Time-dependent macrodispersion for slute transport in anisotropic heterogeneous aquifers', Water Resources Research, v. 24, n 9, 1491-1500.

Gelhar, L.W. and C.L. Axness, 1983, 'Three dimensional stochastic analysis of macrodispersion in aquifers', Water Resources Research, v. 28, n. 7, 1955-1974.

Neuman, S.P., 1990, 'Universal scaling of hydraulic conductivities and dispersivities in geologic media', Water Resources Research, v. 26, n. 8, 1749-1758.

Neuman, S.P. and Y.-K. Zhang, 1990, 'A quasi-linear theory of non-Fickian and Fickian subsurface dispersion, 1. Theoretical analysis with application to isotropic media', Water Resources Research, v. 26, n. 5, 887-902.

Zhang, Y.-K. and Neuman, S.P., 1990, 'A quasi-linear theory of non-Fickian and Fickian subsurface dispersion, 1. Application to anisotropic media and the Borden site', Water Resources Research, v. 26, n. 5, 903-913.