3.1. Fluctuation scales

The distribution of heterogeneity scales $l$ is encoded in the correlation spectrum $\tilde{\mathscr{E}}(\boldsymbol{k})$ , which quantifies the amplitude associated with the wavenumber $\boldsymbol{k}$ . Note that $|\boldsymbol{k}|$ is related to the fluctuation length scales $l$ by $l=1/|\boldsymbol{k}|$ . Thus we may obtain the probability density function (p.d.f.) $p_{l}(l)$ of fluctuation length scale from the p.d.f. of wave numbers $p_{k}(\boldsymbol{k})$ through the mapping $l=1/|\boldsymbol{k}|$ . The p.d.f. of wave numbers is given in terms of the correlation spectrum $\tilde{\mathscr{E}}(\boldsymbol{k})$ as

Specifically, for the Gaussian spectrum (2.10), $p_{k}(k)$ is given by

Thus we obtain from the mapping $l=1/|\boldsymbol{k}|$ the explicit p.d.f. of fluctuation scales

where ${\it\Omega}_{d}$ is the surface area of the $d$ -dimensional unit sphere. The scale distribution of different correlation spectra can be determined in the same way. The scale distribution (3.3) is illustrated in figure 2. It is cut off exponentially at scales $l\ll l_{c}$ and falls off algebraically for $l\gg l_{c}$ . It has a maximum at $l_{max}=l_{c}/\sqrt{d+1}$ . Note that a scalar can be assumed to have sampled a representative part of the spatial heterogeneity when it has swept about 10 heterogeneity length scales. Thus we now define $l_{m}$ , the length $l\leqslant l_{max}$ at which $p_{l}(l_{m})=p_{l}(10l_{c})$ , as a minimum heterogeneity scale. For $d=3$ , we obtain $l_{m}=0.18l_{c}$ , for $d=2$ , $l_{m}=0.2l_{c}$ and for $d=1$ , $l_{m}=0.26l_{c}$ . In the same way in which heterogeneity scales $l>10l_{c}$ do not play a role in scalar transport, scales below $l_{m}$ have only limited impact.

Figure 2.

Distribution $p_{l}(l)$ of fluctuation length scales given by (3.3) for $d=3$ (solid line), $d=2$ (dashed line) and $d=1$ (dotted line) spatial dimensions.

3.2. Mixing scale

The fluctuation scales $l$ of the heterogeneous flow field are contrasted with the mixing scale ${\it\lambda}(t)$ , which can be defined as the characteristic gradient scale of a heterogeneous mixture (Villermaux & Duplat Reference Villermaux and Duplat2006; Le Borgne et al. Reference Le Borgne, Dentz, Davy, Bolster, Carrera, De Dreuzy and Bour2011). Thus it marks the scale below which the scalar mixture can be assumed homogeneous. The temporal evolution of the mixing scale is governed by the competition of dispersion, which tends to increase the mixing scale, and compression due to deformation of material segments, which tends to decrease it. This can be illustrated by considering solute transport in the coordinate system attached to the material fluid element that deforms as it is transported in the flow field. The concentration ${\hat{c}}(z,t)$ across a material fluid element evolves according to the advection dispersion equation (Ranz Reference Ranz1979; Meunier & Villermaux Reference Meunier and Villermaux2010)

The spatial variance

is a measure for the square width of the solute distribution across the deformed material element. It satisfies the evolution equation (Tennekes & Lumley Reference Tennekes and Lumley1972; de Barros et al. Reference de Barros, Dentz, Koch and Nowak2012)

We now set the mixing scale ${\it\lambda}(t)$ equal to the width of the scalar concentration in a deformed material strip, which captures the competition between dispersive expansion and advective compression, ${\it\lambda}(t)=\sqrt{{\it\kappa}(t)}$ . It satisfies the equation (Villermaux & Duplat Reference Villermaux and Duplat2006; Villermaux Reference Villermaux2012)

which in the Introduction was phenomenologically motivated. For the steady flow fields under consideration here, stretching and compression of a material element is algebraic (Villermaux Reference Villermaux2012) such that ${\it\gamma}(t)\sim {\it\beta}t^{-1}$ . For the choice of ${\it\gamma}(t)={\it\beta}/[{\it\tau}_{v}({\it\beta}+t/{\it\tau}_{v})]$ , the evolution of the mixing scale is determined by solving (3.7) subject to the initial condition ${\it\lambda}(t=0)={\it\lambda}_{0}$ . We obtain after a quick calculation

with $f_{{\it\beta}}=1/(1+2{\it\beta})$ . Figure 3 illustrates the evolution of ${\it\lambda}(t)$ with time. At short times, the mixing scale decreases as a result of compression until it assumes a minimum at the mixing time ${\it\tau}_{m}={\it\tau}_{v}({\it\lambda}_{0}^{2}/D{\it\tau}_{v})^{1/(2{\it\beta}+1)}$ (Villermaux Reference Villermaux2012), at which advective compression and dispersive expansion equilibrate. For times larger than the mixing time, $t>{\it\tau}_{m}$ , dispersive expansion prevails and ${\it\lambda}(t)$ evolves dispersively, that is, ${\it\lambda}(t)\sim \sqrt{2Dtf_{{\it\beta}}}$ . Note that the dispersive long-time behaviour is affected by compression through a reduction in the dispersion coefficient by the factor of $0<f_{{\it\beta}}\leqslant 1$ .

Figure 3.

Evolution of the mixing scale ${\it\lambda}(t)$ for $\mathit{Pe}=10^{3}$ and ${\it\lambda}(t=0)/l_{c}=10^{-1}$ . The dashed lines show the dispersive $\sqrt{2Dtf_{{\it\beta}}}$ behaviour.

5.1. Random shear flow

We first consider the problem of dispersion in a random shear flow, as occurs in a randomly stratified porous medium (Matheron & de Marsily Reference Matheron and de Marsily1980). It is characterized by the flow velocity $\boldsymbol{v}(\boldsymbol{x})=\boldsymbol{e}_{1}v(x_{2})$ with $\boldsymbol{e}_{1}$ the unit vector in the 1-direction. We assume that the hydraulic gradient is aligned with the direction of stratification and points in the 1-direction of the coordinate system. For this case, the projectors ${\it\psi}_{ij}(\boldsymbol{k})={\it\delta}_{1i}{\it\delta}_{1j}$ , and thus the velocity covariance function (2.4) reduces to

As outlined in § 3, for the Gaussian spectrum (2.10), the fluctuation spectrum is given by (3.3) for $d=1$ . Transport in random shear flows such as the one under consideration here (Matheron & de Marsily Reference Matheron and de Marsily1980) has been studied in the literature as an exactly solvable model for heterogeneous aquifers (Zavala-Sanchez, Dentz & Sanchez-Vila Reference Zavala-Sanchez, Dentz and Sanchez-Vila2009, and references therein) characterized by stratification in the horizontal plane. Both steady and unsteady random shear flows have been studied in the physics literature as simplified models for heterogeneous and turbulent flows (Bouchaud & Georges Reference Bouchaud and Georges1990).

The stretching of a material fluid element in random shear flow is a non-sequential process (Duplat, Innocenti & Villermaux Reference Duplat, Innocenti and Villermaux2010), in the sense that the shear properties are constant in a given stratum but vary between strata. The elongation of a material fluid element due to the random shear ${\it\sigma}=\partial v(x_{2})/\partial x_{2}$ relative to its initial length is given by

Note that the shear rate depends on the vertical position, and is a random space function as the stratified flow velocity $v(x_{2})$ . The stretching rate ${\it\gamma}(t)=\text{d}\ln {\it\rho}(t)/\text{d}t$ is now given by

At times $t\gg {\it\sigma}^{-1}$ it is ${\it\gamma}(t)\approx 1/t$ . Inserting (5.3) into the evolution equation (3.7), we obtain the mixing scale

For times $t>({\it\lambda}_{0}^{2}/D{\it\sigma}^{2})^{1/3}$ , it behaves like ${\it\lambda}(t)\approx \sqrt{2Dt/3}$ . It is independent of the random shear rate ${\it\sigma}$ .

In order to derive explicit expressions for the filtered dispersion coefficients, we substitute $\tilde{v}_{i}(\boldsymbol{k})={\it\delta}_{i1}\tilde{v}(k_{2})$ into (4.6) and (4.8). For the noise mean $\langle {\it\zeta}_{i}(t)\rangle$ and the cross-moment $\langle {\it\zeta}_{i}(t){\it\zeta}_{j}(t^{\prime })\rangle$ this gives

where $\tilde{c}(k_{2},t\mid a)=\langle \exp [\text{i}k_{2}x_{2}(t)]\rangle$ is the Fourier transform of the scalar distribution for the random walk

with the initial position $x_{2}(t=0)=a$ . It is given by

Using (5.6) together with (5.8) and (5.1) in (4.14) gives

for the longitudinal ensemble dispersion coefficient. The trajectory fluctuations (4.15) are obtained by using (5.5) together with (5.8) and (5.1):

Further, using the definition (4.7) of $\mathscr{A}(k_{2},t)$ and the Gaussian filter (4.3) with the mixing scale ${\it\lambda}(t)=\sqrt{2Dt/3}$ , the integrals in (5.9) and (5.10) can be evaluated explicitly. Thus we obtain for $D_{11}^{ens}(t)$ the explicit expression

For the trajectory fluctuations, we obtain

The effective dispersion coefficient $D_{11}^{eff}(t)$ is obtained from (5.11) and (5.12) according to (4.13). For comparison, the unfiltered effective dispersion coefficient, i.e. for ${\it\lambda}\rightarrow \infty$ , is given by (Clincy & Kinzelbach Reference Clincy and Kinzelbach2001)

Note that the ensemble and effective dispersion coefficients are independent of the distribution $c_{0}(\boldsymbol{x})$ of initial particle positions.

The behaviours for the filtered effective and ensemble dispersion coefficients in the early-time regime $(6D^{2}/l_{c}^{2}\overline{v}^{2}{\it\sigma}_{\mathscr{E}}^{2})^{1/3}{\it\tau}_{D}\ll t\ll {\it\tau}_{D}$ are $D_{11}^{ens}(t)\sim D_{11}^{eff}(t)\sim (t/{\it\tau}_{D})^{3}$ . In the long-time regime $t\gg {\it\tau}_{D}$ , they show the characteristic scaling as $(t/{\it\tau}_{D})^{1/2}$ (Matheron & de Marsily Reference Matheron and de Marsily1980). The unfiltered effective dispersion coefficient (5.13) behaves at $t\ll {\it\tau}_{D}$ as $(t/{\it\tau}_{D})^{2}$ , which is characteristic of dispersion in a linear shear flow (Bolster, Dentz & Le Borgne Reference Bolster, Dentz and Le Borgne2011). Thus the unfiltered effective coefficient incorporates dispersion effects due to shear deformation, while its filtered counterparts are at the same time of the order of the local dispersion coefficient, i.e. unaffected by advective heterogeneity. Figure 4 illustrates the evolution of the filtered effective and ensemble dispersion coefficients as well as the unfiltered effective dispersion coefficient (5.13). The effective and ensemble dispersion coefficients are almost indistinguishable, i.e. the trajectory fluctuations are small compared to the ensemble dispersion coefficient. The evolution of the mixing-scale dependent dispersion coefficients reflects the sampling of the pertinent heterogeneity length scales. The dispersion coefficients are of the order of the local-scale dispersion coefficient until the mixing scale ${\it\lambda}(t={\it\tau}_{c})=l_{m}$ is of the order of the minimum heterogeneity scale defined in § 3. For the activation time scale this gives ${\it\tau}_{c}=l_{m}^{2}/(2Df_{{\it\beta}})$ , where ${\it\beta}=1$ . The dispersion coefficients increase for $t>{\it\tau}_{c}$ as the solute scans through the distribution of heterogeneity scales as indicated in the inset in figure 4. For times $t>(10l_{c})^{2}/(2Df_{{\it\beta}})$ the mixing scale comprises 10 characteristic heterogeneity lengths $l_{c}$ and the filtered dispersion coefficients assume the characteristic correlation-induced $t^{1/2}$ scaling (Matheron & de Marsily Reference Matheron and de Marsily1980; Bouchaud & Georges Reference Bouchaud and Georges1990).

Figure 4.

Mixing-scale dependent effective (solid lines) and longitudinal ensemble (dashed lines) dispersion coefficients for $\mathit{Pe}=10^{3}$ in a $d=2$ dimensional stratified flow. The dash-dotted line represents the unfiltered effective dispersion coefficient. The dotted vertical line indicates the time scale ${\it\tau}_{c}=l_{m}^{2}/6D$ , and the solid vertical line the time scale $\hat{{\it\tau}}_{D}=(10l_{c})^{2}/2D$ . The thin solid lines indicate the early-time $t^{3}$ and late-time $t^{1/2}$ behaviours of the filtered effective dispersion coefficient, respectively, as well as the $t^{2}$ behaviour of the unfiltered effective dispersion coefficient. The dispersion coefficients are normalized by ${\it\sigma}_{\mathscr{E}}^{2}\overline{v}^{2}l_{c}^{2}/D$ . The inset illustrates the scale distribution $p_{l}(l)$ , and the horizontal axis denotes $l/l_{c}$ . The vertical lines indicate $l_{m}=0.26l_{c}$ and $10l_{c}$ .

5.2. Heterogeneous porous media flow

We now consider dispersion in the flow through heterogeneous media as described in § 2. The transport problem does not possess a closed-form solution for the dispersion coefficients, so we study the dispersion behaviour in the frame of a second-order perturbation expansion in the fluctuations of the random velocity field $\tilde{\boldsymbol{v}}^{\prime }(\boldsymbol{k})$ .

This implies the substitution of $\tilde{c}(\boldsymbol{k},t\mid \boldsymbol{x}^{\prime })$ in (4.6) and (4.8) by $\tilde{c}_{0}(\boldsymbol{k},t\mid \boldsymbol{x}^{\prime })=\langle \exp [\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{0}(t)]\rangle$ , where the ‘unperturbed’ particle trajectory is given by

Executing the noise average gives, for the propagator $\tilde{c}_{0}(\boldsymbol{k},t\mid \boldsymbol{x}^{\prime })$ ,

Using this approximation in (4.8), we obtain for the ensemble dispersion coefficients (4.14) the perturbation expression

which is similar to the one derived by Rubin et al. (Reference Rubin, Sun, Maxwell and Bellin1999) for a constant ideal filter. For the trajectory fluctuations, we derive by using (5.15) in (4.6) and (4.15) the expression

The mixing scale ${\it\lambda}(t)$ is set to ${\it\lambda}(t)=\sqrt{2Dtf_{{\it\beta}}}$ in the filter (4.3).

Notice that the scalar may start sampling the flow heterogeneity earliest when it has been transported advectively over a disorder correlation length. Thus we focus our analysis on times larger than the advection time scale ${\it\tau}_{v}$ . Furthermore, in most practical applications transport is advection-dominated, that is, the Péclet number is $\mathit{Pe}\gg 1$ . Under these conditions, we obtain for the longitudinal ensemble dispersion coefficient the compact expression

The approximation method leading to this expression is outlined in Dentz et al. (Reference Dentz, Kinzelbach, Attinger and Kinzelbach2000). The first term in the square brackets quantifies the unfiltered ensemble dispersion coefficient. Note that lowest-order perturbation theory renders the transverse dispersion coefficients of the order of the inverse Péclet number. Thus we focus here on the longitudinal coefficients only. Note that at times $t\ll {\it\tau}_{D}$ , the mixing-scale dependent ensemble dispersion coefficient is significantly reduced compared to its unfiltered counterpart, which evolves to its asymptotic value on the advection time scale ${\it\tau}_{v}$ (Dagan Reference Dagan1991).

The evolution of the trajectory fluctuations, which quantify the difference between them (see (4.15)), is obtained as

The expression on the first line quantifies the trajectory fluctuations with respect to the ensemble average flow velocity $\overline{v}$ (Dentz et al. Reference Dentz, Kinzelbach, Attinger and Kinzelbach2000). As in the previous section, both the effective and ensemble dispersion coefficients are independent of the distribution of initial particle positions $c_{0}(\boldsymbol{x})$ .

Figure 5 illustrates the evolution of the mixing-scale dependent effective and ensemble dispersion coefficients. The mixing-scale dependent effective and ensemble dispersion coefficients evolve much more slowly than their unfiltered effective counterpart. The unfiltered effective dispersion coefficient picks up for times smaller than the advection time scale ${\it\tau}_{v}$ . This behaviour can be traced back to advective spreading of the scalar distribution as its size increases due to dispersion. The filtered dispersion coefficients, in contrast, remain essentially at the value of the local dispersion coefficient until the time ${\it\tau}_{c}$ , which depends on the compression factor $f_{{\it\beta}}$ and the smallest heterogeneity scales. As above, it is set by the heterogeneity scale $l_{m}$ through ${\it\lambda}({\it\tau}_{c})=l_{m}$ , and equivalently ${\it\tau}_{c}=l_{m}^{2}/(2Df_{{\it\beta}})$ . For $t<{\it\tau}_{c}$ , i.e. for scales smaller than $l_{m}$ , the mixing scale is smaller than the minimum heterogeneity scale. Thus the flow field is homogeneous, and therefore dispersion is equal to local dispersion. For $t>{\it\tau}_{c}$ , heterogeneity is activated as a dispersion mechanism. As the mixing scale increases, it scans through the distribution of heterogeneity scales, illustrated in the inset in figure 5, and incorporates fluctuations of increasing scale into the dispersion process. For times $t\gg {\it\tau}_{c}$ , the dispersion coefficients evolve towards their asymptotic long-time value. For times $t\gg {\it\tau}_{D}$ , that is, as the mixing scale ${\it\lambda}(t)\gg l_{c}$ , the full spectrum of velocity values is available and the velocity fluctuations that cause dispersion are statistically equal in all realizations.

Figure 5.

Mixing-scale dependent effective (solid lines) and longitudinal ensemble (dashed lines) dispersion coefficients for $\mathit{Pe}=10^{3}$ and $d=3$ spatial dimensions. The dash-dotted line represents the unfiltered effective dispersion coefficient. The dotted vertical lines indicate the time scale ${\it\tau}_{c}=l_{m}^{2}/2Df_{{\it\beta}}$ for ${\it\beta}=1/2$ and ${\it\beta}=4$ . The solid vertical lines denote the time scale $\hat{{\it\tau}}_{D}=(10l_{c})^{2}/2Df_{{\it\beta}}$ . The dispersion coefficients are rescaled by ${\it\sigma}_{\mathscr{E}}^{2}\overline{v}l_{c}$ . The inset illustrates the scale distribution $p_{l}(l)$ , and the horizontal axis denotes $l/l_{c}$ . The vertical lines indicate $l_{m}=0.18l_{c}$ and $10l_{c}$ .