Upscaling of Anomalous Pore-Scale Dispersion

Abstract

We study the upscaling of advective pore-scale dispersion in terms of the Eulerian velocity distribution and advective tortuosity, both flow attributes, and of the average pore length, a medium attribute. The stochastic particle motion is modeled as a time-domain random walk, in which particles move along streamlines in equidistant spatial steps with random velocities and thus random transition times. Particle velocities describe stationary spatial Markov processes, which evolve along streamlines on the mean pore length. The streamwise motion is projected onto the mean flow direction using tortuosity. This upscaled stochastic particle model predicts accurately the (non-Fickian) transport dynamics obtained from direct numerical simulations of particle transport in a three-dimensional digitized Berea sandstone sample. It captures all aspects of transport and sheds light on the dependence of the upscaled transport behavior on the flow heterogeneity and the initial particle distribution, which are critical for the accurate modeling of dispersion from the pre-asymptotic to asymptotic regimes.

Appendices

Appendix: A Tortuosity

We derive here the average of the $\omega _1(s,\mathbf {a})$ along a streamline under ergodic conditions. To this end, we first note that the position $x_1(s,\mathbf {a})$ can be written by integration of (3) as

$$\begin{aligned} x_1(s,\mathbf {a})=s\left[ \frac{1}{s}\int _0^s \omega _1(s',\mathbf {a}) \mathrm{d}s'\right] . \end{aligned}$$

(33)

The expression in the square brackets denotes the average of $\omega _1(s,\mathbf {a})$ along a particle trajectory. At the same time, it denotes the ratio of linear to streamwise distance,

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle _\mathrm{s} = \lim _{s \rightarrow \infty } \frac{1}{s}\int _0^s \omega _1(s',\mathbf {a}) \mathrm{d}s' = \frac{x_1(s,\mathbf {a})}{s}, \end{aligned}$$

(34)

where the angular brackets with subscript s denote the streamwise average along a trajectory. The average of $\omega _1(s,\mathbf {a})$ over an ensemble of particles is defined by

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle = \lim _{V_0 \rightarrow \infty }\frac{1}{V_0}\int _{\varOmega _0} \frac{v_1[\mathbf {x}(s,\mathbf {a})]}{v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]} \rho (\mathbf {a}) \mathrm{d}\mathbf {a}. \end{aligned}$$

(35)

We consider a flux-weighted initial condition, see (4). Under ergodic conditions, this initial condition corresponds to the steady state velocity PDF $p_\mathrm{s}(v)$, which is equal to the flux-weighted Eulerian velocity PDF. This can be seen by using

$$\begin{aligned} \rho (\mathbf {a}) = \frac{1}{V_0}\frac{v_\mathrm{e}(\mathbf {a})}{\langle v_\mathrm{e}(\mathbf {x}) \rangle } \mathbb I(\mathbf {a}\in \varOmega _0), \end{aligned}$$

(36)

in the limit $V_0 \rightarrow \infty$. Also, Koponen et al. (1996) pointed out that it is natural for porous media to consider a flux-weighted average, see also Ghanbarian et al. (2013). Furthermore, under ergodic conditions, the average over a single-particle trajectory is equal to the average over the initial ensemble of particles and so

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle _\mathrm{s} = \langle \omega _1(s,\mathbf {a}) \rangle = \frac{\langle x_1(s,\mathbf {a})\rangle }{s} = \chi ^{-1}. \end{aligned}$$

(37)

Using expression (36) in (35), we obtain

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle = \lim _{V_0 \rightarrow \infty } \frac{1}{V_0} \int _{\varOmega _0} \frac{v_1[\mathbf {x}(s,\mathbf {a})]}{v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]} \frac{v_\mathrm{e}(\mathbf {a})}{\langle v_\mathrm{e}(\mathbf {x}) \rangle }\mathrm{d}\mathbf {a}. \end{aligned}$$

(38)

In order to evaluate this expression, we perform the variable change $\mathbf {a}\rightarrow \mathbf {x}(s,\mathbf {a})$,

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle = \lim _{V_0 \rightarrow \infty } \frac{1}{V_0} \int _{\varOmega (s)} \frac{v_1[\mathbf {x}(s,\mathbf {a})]}{v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]} \frac{v_\mathrm{e}(\mathbf {a})}{\langle v_\mathrm{e}(\mathbf {x}) \rangle } \mathbb J(\mathbf {a},s)^{-1} \mathrm{d}\mathbf {x}, \end{aligned}$$

(39)

where $\mathbb J(\mathbf {a}, s)$ is the Jacobian of the transformation. It can be determined by noting that (Batchelor 2000, p. 75)

$$\begin{aligned} \frac{\mathrm{d}\mathbb J(\mathbf a,s)}{\mathrm{d}s} = \mathbb J(\mathbf a,s) \nabla \cdot \frac{{\mathbf {v}}[{\mathbf {x}}(s,\mathbf {a})]}{v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]}. \end{aligned}$$

(40)

This differential equation can be integrated by noting that $\nabla \cdot {\mathbf {v}}(\mathbf {x}) = 0$ and

$$\begin{aligned} \frac{\mathrm{d}v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]}{\mathrm{d}s} = \nabla v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})] \cdot {\mathbf {v}}[\mathbf {x}(s,\mathbf {a})], \end{aligned}$$

(41)

which follows by using the chain rule and (3). Thus, we obtain for the initial condition $\mathbb J(\mathbf {a},s = 0) = 1$ that

$$\begin{aligned} \mathbb J(\mathbf {a},s)=\frac{v_\mathrm{e}(\mathbf {a})}{v_\mathrm{e}[\mathbf {x}(s,\mathbf {a})]}. \end{aligned}$$

(42)

Inserting this expression into (38) gives

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle = \lim _{V_0 \rightarrow \infty } \frac{1}{V_0}\int _{\varOmega (s)} \frac{v_1(\mathbf {x})}{\langle v_\mathrm{e}(\mathbf {a}) \rangle }\mathrm{d}\mathbf {x}= \frac{\langle v_1 \rangle }{\langle v_\mathrm{e} \rangle }. \end{aligned}$$

(43)

This result is consistent with Koponen et al. (1996). This implies that at $s \gg \ell _\mathrm{p}$, we can set

$$\begin{aligned} \langle \omega _1(s,\mathbf {a}) \rangle = \chi ^{-1} = \frac{\langle v_1 \rangle }{\langle v_\mathrm{e} \rangle }. \ \end{aligned}$$

(44)

Appendix: B Continuous-Time Random Walk

For transition length of the order of the correlation length $\ell _\mathrm{c}$, subsequent particle velocities can be considered independent and thus, the space-time particle motion (13a) may be approximated by

$$\begin{aligned} x_{n+1} = x_n + \frac{\ell _\mathrm{c}}{\chi }, \qquad t_{n+1} = t_n + \tau _n, \end{aligned}$$

(45)

where $x_n = x(s_n)$ with $s_n = n \ell _\mathrm{c}$. The random transition time $\tau _n$ is given by

$$\begin{aligned} \tau _n = \frac{\ell _\mathrm{c}}{v_\mathrm{s}(s_n)}. \end{aligned}$$

(46)

The time increments for $n > 0$ are distributed as

$$\begin{aligned} \psi (t) = \frac{\ell _\mathrm{c}}{t^2} p_\mathrm{s}(\ell _\mathrm{c}/t). \end{aligned}$$

(47)

For $n = 0$, the transition time PDF is distributed according to

$$\begin{aligned} \psi _0(t) = \frac{\ell _\mathrm{c}}{t^2} p_0(\ell _\mathrm{c}/t). \end{aligned}$$

(48)

Under steady state conditions, this means for $p_0(v) = p_\mathrm{s}(v)$ and thus $\psi _0(v) = \psi (v)$, Eq. (45) describes a continuous-time random walk as discussed in Berkowitz et al. (2006). Thus, the asymptotic behavior of the breakthrough curves and displacement moments can be predicted based on the scalings of the transition time distribution. For $\psi (t) \propto t^{-1-\beta }$ at large times, the breakthrough curves scale as $f(t,x_1) \propto t^{-1-\beta }$, the mean displacement as $m_1(t) \propto t$, and the displacement variance as $\sigma ^2(t) \propto t^{3 - \beta }$. Note that this scaling for $\psi (t)$ implies that the velocity distribution $p_\mathrm{s}(v) \propto v^{\beta -1}$ at small velocities.