# An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions

## Abstract

**:**

## 1. Introduction

**x**) = ln K(

**x**), where K is the local hydraulic conductivity and

**x**is the vector of spatial coordinates, is a homogeneous random space function, normally distributed and completely characterized by constant mean and variance and by a fast-decaying correlation function. The above characteristics imply the existence of a single representative scale of heterogeneity, i.e., the so-called integral scale.

_{Y}(r) = ar

^{b}, where a is a dimensional constant and one-half of the scaling exponent H = b/2 is known as the Hurst coefficient [4].

**x**) = ln T(

**x**), where the transmissivity T indicates the vertical average of K, are critically influenced by the depositional process. In such context, although the existence of several scales of heterogeneity seems reasonable, there is no direct experimental evidence supporting the power-law model. Neuman [3] provided an indirect justification of it, analyzing the scaling behavior exhibited by the longitudinal dispersivity of tracer plumes.

## 2. Formulation

**x**) = lnK(

**x**) = < Y > + Y’(

**x**), where the angle brackets indicate the (assumed constant) ensemble mean and the prime indicates the deviation about that mean, is given by:

^{th}order of the hierarchy. The generic heterogeneous sub-unit is therefore represented by the superposition of a given finite number of stationary fields of increasing order:

^{th}heterogeneous sub-unit and

^{th}component of velocity fluctuation and

**U**= (U

_{1},U

_{2},U

_{3}) is the constant ensemble mean velocity. From Equations (24) and (25) one can see that, at the first-order in the log-conductivity variance, each Fourier component of the log-conductivity field corresponds to a single component of hydraulic head and velocity. Therefore, the same properties of linear superposition holding for Y (see Equation (2)) apply to h and

**v**as well. Consider now an initial solute injection at a uniform concentration C

_{0}, confined within a volume

**a**represents its initial position within V

_{0}, t is the time, and

**X**

_{B}the zero-mean Brownian component. Given Equations (16) and (25), it is also:

**a**+

**U**τ), disturbed only by the local-dispersive contribution represented by

**X**

_{B}. Such an assumption is common to all first-order (linearized) analytical formulations of subsurface flow and transport. Its physical meaning is that one neglects the self-feeding mechanism of advective dispersion that would emerge from the solution of the exact integro-differential equation:

**x**,t) indicates the concentration in

**x**at time t. For a single-particle injection in

**x**=

**a**(e.g., [1]):

_{0}is the associated initial concentration, n is the generic volume porosity, and n

_{0}is the volume porosity at injection site. Integrating over the whole initial volume V

_{0}for n ≅ n

_{0}gives:

_{0}= const and M = n

_{0}C

_{0}V

_{0}yields:

^{th}component of initial centroid vector position. Ensemble averaging will be performed on Equation (39) recalling that, by definition, the ensemble mean of a random fluctuation is zero and that Equation (16) and flow linear treatment allow it to be assumed that:

_{ij}is Kronecker’s Delta. Based on Equation (31), for negligible local dispersion and provided that the integral scales of all fields of the $\overline{Y}$-hierarchy are larger than the initial plume size, the corresponding components of particle positions can be viewed as almost fully correlated (i.e., as if the particles were concentrated in a single point) at any time:

_{Mij}for log-conductivity fields characterized by evolving scales of heterogeneity and power-law semi-variograms can be pursued by computing the corresponding coefficient for any stationary field of the continuous hierarchy (exponential covariance for 0 < b <1 and Gaussian covariance for 1 ≤ b < 2) and by integrating the result over the truncated frequency domain. See Appendix A for the derivation of D

_{Mij}($\infty ,\mathrm{\Lambda}$) in the case of 3-D stationary exponential and Gaussian log-K covariance. To obtain the global asymptotic macro-dispersion coefficient, given by the linear combination of the macro-dispersion coefficients characterizing the heterogeneous single-scale fields of the hierarchy, one has to compute:

_{0}= 1/l

_{0}. From Equation (A7), respectively for exponential and Gaussian hierarchy:

_{0}, particles’ displacement includes a local-dispersive component that makes the original distances increase as ~$\sqrt{2Dt}$. Thus, the threshold sub-unit corresponding to the subdivision into $\tilde{X}$- and $\overline{X}$-displacement hierarchy changes in time, and the boundary wave-number is now ${\mathrm{\Lambda}}_{0}^{\prime}={\left({l}_{0}+\chi \sqrt{2Dt}\right)}^{-1}$, with $\chi $ indicating a suitable constant related to the assumed width of the Brownian-Gaussian bell. Equations (56) and (57) transform into:

_{M}

_{11}increases in time).

## 3. Results

_{ij}> in order to cut-off the long tail of the log-conductivity spectrum, obtaining an asymptotically constant value. As a matter of fact, the centroid covariance ${\tilde{R}}_{ij}$ is affected by a restricted range of heterogeneity scales and tends to zero at large times. As a consequence, the time derivative of <S

_{ij}> tends to coincide with the time-derivative of ${\tilde{X}}_{ij}$, which envisions an asymptotically ergodic transport process. Additionally, the assumed linearity of the problem and the integration of the asymptotic macro-dispersion coefficient obtained for a generic single-scale log-conductivity field over the truncated hierarchy domain lead to an invariably constant asymptotic macro-dispersion coefficient and, therefore, to Fickian transport conditions. It should be in any case emphasized that the never-decaying dependence of this coefficient on the initial plume size l

_{0}in Equation (64) which means that the system is characterized by persistent memory.

_{0}. As one can see, the effect produced by high Péclet (Pe

_{0}) numbers is opposite to the effect produced by high scaling exponents (b). Specifically, the higher the Péclet number, the closer the transport process to asymptotic Fickian conditions, represented by a constant longitudinal macro-dispersion coefficient (although, for τ → 0, the dispersion coefficient is higher for higher Péclet numbers due to the larger number of heterogeneity scales initially sampled). Conversely, the higher the scaling exponent, the faster the macro-dispersion coefficient increases. Notice that, in Figure 2, the red dotted line represents the Fickian, linear behavior corresponding to a single-scale log-conductivity field (exponential or Gaussian log-K covariance) characterized by the truncated-hierarchy variance and by an integral scale equal to the initial plume size. Thus, unlike the case of fast-decaying correlation functions, the coexistence of evolving-scale advective heterogeneity and intensive diffusive mixing acts as an antagonist mechanism in the process of solute dilution and Fickian regime achievement. Such a behavior, which is in definite contrast with what was previously found for stationary porous media, was detected here for the first time.

## 4. Discussion and Conclusions

_{11}, and detected invariably super-diffusive regimes.

## Conflicts of Interest

## Appendix A

_{1}is:

_{r}= k

_{r}I

_{Y}and S

_{uij}is the velocity spectrum. If the axes of the Cartesian reference frame (x

_{1}, x

_{2}, x

_{3}) coincide with the principal axes of the dispersing plume, the velocity spectrum and log-conductivity spectrum S

_{Y}are related by the following expressions (e.g., [1]):

_{M}

_{11}for both exponential and Gaussian covariance is therefore:

_{1}= λ

_{1}Pe/2π, where Pe = UI

_{Y}/D >> 1, from [28]:

_{2}and λ

_{3}:

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**Figure 1.**Longitudinal macro-dispersion coefficient for variable Péclet numbers and scaling exponents.

**Figure 2.**Longitudinal particle trajectory covariance for variable Péclet numbers and scaling exponents. The red-dotted line refers to Fickian linear behavior corresponding to a single-scale log-conductivity field (exponential or Gaussian log-K covariance) characterized by the truncated-hierarchy variance and by an integral scale equal to the initial plume size.

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Pannone, M.
An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions. *Water* **2017**, *9*, 751.
https://doi.org/10.3390/w9100751

**AMA Style**

Pannone M.
An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions. *Water*. 2017; 9(10):751.
https://doi.org/10.3390/w9100751

**Chicago/Turabian Style**

Pannone, Marilena.
2017. "An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions" *Water* 9, no. 10: 751.
https://doi.org/10.3390/w9100751