Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

Abstract

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a d-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.

Appendices

Appendix 1: Trapping Time Scales

The mass exchange process exhibits two distinct time scales. The first one is given by the inverse of the trapping rate $\tau _\gamma = \gamma ^{-1}$. This time scale represents the time at which, on average, particles undergo the first trapping event. Moreover, it is also the average time that the particles spend in the mobile phase. For times larger than $\tau _\gamma$, advection ceases to be the only process that governs transport.

The second time scale represents the moment at which particles start spending on average more time trapped than in the mobile phase. In order to determine this scale, we need to compare the average time spent mobile, which is given by $\tau _\gamma$ to the average time spent immobile at a given time t. Note that the average trapping of the stable distribution (37) does not exist. Thus, in order to determine the characteristic trapping time after n trapping event, we first consider the mean number $\nu (t_f) \equiv \langle n_{t_f} \rangle$ of trapping times needed to arrive at a given total trapping time $t_{f,n} = \sum _{i=1}^{n} \tau _{f,i}$, which is given by the renewal theorem as

$$\begin{aligned} \nu (t_f) = 1 + \int \limits _0^{t_f} \hbox {d} t^\prime \nu (t_f - t^\prime ) p_f(t^\prime ). \end{aligned}$$

(55)

This equation is solved for the Laplace transform of $\nu (t_f)$ as

$$\begin{aligned} \nu ^*(\lambda ) = \frac{1}{\lambda } \frac{1}{1 - p_f^*(\lambda )}. \end{aligned}$$

(56)

Using the stable distributions (37) for small $\lambda \tau _g$ gives $\nu ^*(\lambda ) \approx \tau _g (\lambda \tau _g)^{-1-\delta }$, from which we obtain in time

$$\begin{aligned} \nu (t_f) \approx \left( \frac{t}{\tau _g}\right) ^{\delta }. \end{aligned}$$

(57)

The latter gives us a relation between the total trapping time and number of trapping events. We can use this relation to define a total mean trapping time as a function of step number by setting $\nu (\langle t_f \rangle ) = n$, which gives $\langle t_f(n) \rangle = \tau _g n^{1/\delta }$. Thus an average trapping time after n steps is simply $\langle \tau _f(n) \rangle = \langle t_f(n) \rangle /n$, which gives

$$\begin{aligned} \langle \tau _f(n) \rangle = \tau _g n^{\frac{1-\delta }{\delta }}. \end{aligned}$$

(58)

Notice that the latter is not strictly an average trapping time because for the stable distributions (37) the mean does not exist. It is rather a characteristic trapping time after n trapping events.

Appendix 2: Moments of the Spatial Distribution

1.1 Weak Advective Heterogeneity

In the following, we will derive the expressions for the scaling of the moments and the variance of particle displacements in both the longitudinal and transverse directions in the pre-asymptotic and in the asymptotic time regimes. In particular, we will consider all the possible cases corresponding to different choices of the distribution of trapping times. For the derivation of the moments, we will make use of the inverse Gamma distribution for $\delta$ in (1, 2) and of stable distribution for $\delta$ in (0, 1). This choice has as a consequence the fact that the Laplace transform of the distribution of immobile times in the asymptotic limit $(\lambda \rightarrow 0)$ can be approximated for $\lambda \tau _g \ll 1$ as

$$\begin{aligned} p_f^*(\lambda ) \approx 1 - (\lambda \tau _g)^\delta . \end{aligned}$$

(59)

By inserting Eq. (59) into Eq. (18), we get the expression for the distribution of transit times

$$\begin{aligned} \psi ^*(\lambda ) \approx \frac{1}{1+\lambda \tau _0 +\gamma \tau _0(\lambda \tau _g)^\delta } \end{aligned}$$

(60)

From Eq. (60), we identify the trapping time scale $\tau _e$ as follows. The second term in the denominator dominates for $\lambda \ll \gamma (\gamma \tau _g)^{\frac{\delta }{1 - \delta }}$ and analogously for times $t \gg \tau _\gamma (\gamma \tau _g)^{\frac{\delta }{\delta - 1}} \equiv \tau _e$.

We first derive the scalings of the moments in the pre-asymptotic time regime. Under this conditions, the PDF of transition times can be expanded and approximated with

$$\begin{aligned} \psi ^*(\lambda ) \approx 1-\lambda \tau _0. \end{aligned}$$

(61)

By inserting Eq. (61) into the equations for the first moment (43) and for the second moment (44) in this time regime and by considering the leading term in $\lambda$, we obtain

$$\begin{aligned} {m}^*_1(\lambda ) \propto \lambda ^{-2}, \quad {m}^*_{11} (\lambda ) \propto \lambda ^{-3}+\lambda ^{-2}, \quad m^*_{22}(\lambda ) \propto \lambda ^{-2}. \end{aligned}$$

(62)

We recall that the first moment in the transverse direction is always null. By applying the Tauberian theorems, we calculate the scaling in time from the expressions in the Laplace domain of Eq. (62). Thus, we find

$$\begin{aligned} m_1(t) \propto t, \quad m_{11}(t) \propto t^2+t, \quad m_{22}(t) \propto t. \end{aligned}$$

(63)

These results are valid for $\delta \in (0,2)$. The variance is computed using Eq. (42). Note that the term proportional to $t^2$ in Eq. (63) will cancel out with the square of the first moment. Therefore, we get for the variance

$$\begin{aligned} \kappa _{11}(t) \propto t, \quad \kappa _{22}(t) \propto t. \end{aligned}$$

(64)

The procedure here described will be adopted to calculate all the scalings of the first moment and the variance in the following.

At long times, i.e., in the asymptotic time regime, the PDF of transition times can be approximated with

$$\begin{aligned} \hat{\psi }(\lambda ) \approx 1 - \tau _0\gamma (\tau _g\lambda )^\delta . \end{aligned}$$

(65)

By substituting this expression into Eqs. (43) and (44), we get the scalings of the moments in the Laplace space. The mean value in the longitudinal direction behaves asymptotically as

$$\begin{aligned} m^*_1(\lambda ) \propto \lambda ^{-1-\delta } \end{aligned}$$

(66)

By making use of the Tauberian theorems, we derive the asymptotic behavior of the variance in the temporal domain and we get

$$\begin{aligned} m_1(t) \propto t^\delta \end{aligned}$$

(67)

This procedure is repeated to calculate all the scalings of the moments. In the transverse direction, as we have already pointed out before, the first moment in the transverse directions is always null $m_2 = 0$. For the spatial variances we obtain

$$\begin{aligned} \kappa _{11}(t) \propto t^{2\delta }, \quad \kappa _{22}(t) \propto t^\delta . \end{aligned}$$

(68)

1.2 Strong Advective Heterogeneity

We derive here the scalings of the moments for the case of strong advective heterogeneity. The latter is mapped onto heavy-tailed distributions of the mobile transition times. In particular, we will refer to inverse Gamma distributions for $\beta$ in (1, 2). Therefore, the Laplace transform of $\psi _m$ at long times can be approximated with

$$\begin{aligned} \psi _m(\lambda ) \approx 1 - \alpha _1 \lambda \tau _v + \alpha _2 (\lambda \tau _v)^\beta . \end{aligned}$$

(69)

where $\alpha _1$ and $\alpha _2$ are constants. By inserting this expression into Eq. (18), we get the Laplace transform of the PDF of the compound process

$$\begin{aligned} \psi ^*(\lambda ) \approx 1 - \alpha _1 \tau _v(\lambda + \gamma [1-p_f^*(\lambda )]) + \alpha _2 (\tau _v\lambda + \tau _v\gamma [1- p_f^*(\lambda )])^\beta . \end{aligned}$$

(70)

The Laplace transform of the distribution of trapping times at long times is approximated by Eq. (59). By substituting the latter into Eq. (70), we get the distribution of total transition times in the Laplace space

$$\begin{aligned} \psi ^*(\lambda ) \approx 1 - \alpha _1\left[ \lambda \tau _v + \gamma \tau _v(\lambda \tau _g)^\delta \right] + \alpha _2\left[ \lambda \tau _v + \gamma \tau _v(\lambda \tau _g)^\delta \right] ^\beta . \end{aligned}$$

(71)

We now follow the same procedure described in “Weak Advective Heterogeneity” of Appendix to derive the scaling of the moments in the pre-asymptotic and asymptotic limit for different choices of the parameters $\beta$ and $\delta$. In the pre-asymptotic limit, the mean value of particle displacements scales linearly with time

$$\begin{aligned} m_1(t)\propto t \end{aligned}$$

(72)

in the longitudinal direction, while in the transverse directions the mean is zero $m_2(t) = 0$.

For the mean squared displacement, we find

$$\begin{aligned} \kappa _{11}(t) \propto t^{3-\beta } \end{aligned}$$

(73)

in the direction of advection, while in the transverse direction the variance scales linearly

$$\begin{aligned} \kappa _{22} (t) \propto t. \end{aligned}$$

(74)

Unlike the previous case, late behavior is strongly conditioned by the trapping properties of the medium and, as a consequence, the scalings of the moments will in general depend on the distribution of times that the particles spend in the immobile phase. In particular, the first moment in the longitudinal direction scales as

$$\begin{aligned} m_1(t) \propto t^{\delta }, \end{aligned}$$

(75)

while in the transverse directions the mean value is always null $m_2(t) = 0$.

The scaling of the mean squared displacement along the direction in which advection occurs is given by

$$\begin{aligned} \kappa _{11}(t) \propto t^{2\delta }, \end{aligned}$$

(76)

while in the transverse direction we get

$$\begin{aligned} \kappa _{22} (t) \propto t^{\delta }. \end{aligned}$$

(77)

Appendix 3: First Passage Time Distribution

Here we will derive the scaling of the FPTD in the asymptotic and pre-asymptotic regimes for both weak and strong advective heterogeneity. The derivation will be performed for the same scenarios discussed in the previous section.

1.1 Weak Advective Heterogeneity

We will derive the expressions for the asymptotic regime. It has been previously shown that the distribution of transition times is given by Eq. (60). For $\lambda \tau _e \ll 1$, we obtain

$$\begin{aligned} \psi ^*(\lambda ) \approx 1 - \gamma \tau _0 (\lambda \tau _g)^\delta . \end{aligned}$$

(78)

Inserting the latter into (50) and using (48) gives for the Laplace transform of the FPTD

$$\begin{aligned} f^*(\lambda ,x_c) = \exp [-\langle n_c \rangle \gamma \tau _0 (\lambda \tau _g)^\delta ], \end{aligned}$$

(79)

which is again a stable distribution characterized by the exponent $\delta$. This gives directly the scaling $f(t,x_c) \propto t^{-1-\delta }$ for $t \gg \tau _e$.

1.2 Strong Advective Heterogeneity

Inserting (71) in (50) gives for $f^*(\lambda ,x_c)$

$$\begin{aligned} f^*(\lambda ,x_c) \approx \exp \left[ - \langle n_c \rangle \left( \alpha _1\left[ \lambda \tau _v + \gamma \tau _v(\lambda \tau _g)^\delta \right] + \alpha _2\left[ \lambda \tau _v + \gamma \tau _v(\lambda \tau _g)^\delta \right] ^\beta \right) \right] . \end{aligned}$$

(80)

For $\lambda \ll \gamma$, we approximate

$$\begin{aligned} f^*(\lambda ,x_c) \approx \exp \left[ - \langle n_c \rangle \alpha _1 \gamma \tau _v(\lambda \tau _g)^\delta \right] , \end{aligned}$$

(81)

which gives $f(t,x_c) \propto t^{-1-\delta }$ for $t \gg \tau _\gamma$. For $\lambda \gg \gamma$, we have

$$\begin{aligned} f^*(\lambda ,x_c) \approx \exp \left( - \langle n_c \rangle \left[ \alpha _1\lambda \tau _v + \alpha _2 (\lambda \tau _v)^\beta \right] \right) , \end{aligned}$$

(82)

which gives the pre-asymptotic scaling $f(t,x_c) \propto t^{-1-\beta }$ for $t \ll \tau _\gamma$.