Continuous Time Random Walks for the Evolution of Lagrangian Velocities

# Continuous Time Random Walks for the Evolution of Lagrangian Velocities

Marco Dentz Spanish National Research Council (IDAEA-CSIC), Barcelona, Spain    Peter K. Kang Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea    Alessandro Comolli Spanish National Research Council (IDAEA-CSIC), Barcelona, Spain    Tanguy Le Borgne Université de Rennes 1, CNRS, Geosciences Rennes, UMR 6118, Rennes, France    Daniel R. Lester School of Civil, Environmental and Chemical Engineering, RMIT University, 3000 Melbourne, Victoria, Australia
###### Abstract

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

## I Introduction

The dynamics of Lagrangian velocities in fluid flows are fundamental for the understanding of tracer dispersion, anomalous transport behaviors, but also pair-dispersion and intermittent particle velocity and acceleration time series, as well as fluid stretching and mixing. A classical stochastic view-point on particle velocities in heterogeneous flows is their representation in terms of Langevin models for the particle velocities Pope (2000), which accounts for temporal persistence, and the random nature of velocity through a Gaussian white noise. Such approaches assume that velocity time series form a Markov process when measured isochronically along a particle trajectory Meyer and Saggini (2016).

The observation of intermittency in Lagrangian velocity and acceleration time series in steady heterogeneous flow De Anna et al. (2013); Kang et al. (2014); Holzner et al. (2015) questions the assumptions that underly the representation of Lagrangian velocity in terms of a classical random walk. Observed intermittency patterns manifest in long episodes of low velocities and relatively short episodes of high velocity. This indicates an organizational principle of Lagrangian velocities that is different from the one implied in a temporal Markov processes, which assumes that velocities are persistent for a constant time interval of characteristic duration . Observed intermittency for flow through disordered media De Anna et al. (2013); Kang et al. (2014); Holzner et al. (2015) suggests that particle velocities are persistent along a characteristic length scale along streamlines. Approaches that model particle velocities as Markov processes in space, assign to particle transitions a random transition time, which is given kinematically by the transition distance divided by the transition velocity. Thus, such approaches are termed continuous time random walks (CTRW) Montroll and Weiss (1965); Scher and Lax (1973); Metzler and Klafter (2000); Berkowitz et al. (2006). They are different from classical random walk approaches, which employ a constant discrete transition time.

Particle motion and particle dispersion have been shown to follow CTRW dynamics for flow through pore and Darcy-scale heterogeneous porous and fractured media Berkowitz and Scher (1997); Le Borgne et al. (2008a, b); Kang et al. (2011); Bijeljic et al. (2011); Edery et al. (2014), as well as turbulent flows Shlesinger et al. (1987); Thalabard et al. (2014). While the CTRW provides an efficient framework for the quantification of anomalous dispersion and intermittency in heterogeneous flows, some key questions remain open regarding the relation of particle velocities and Eulerian flow statistics, and the stationarity of Lagrangian velocity statistics.

In classical CTRW formulations, particle velocities are non-stationary. This means, for example that the velocity mean and covariance evolve in time. This property is termed aging Sokolov (2012). However, for steady divergence-free random flows, such as flow through porous media, it has been found that particle velocities may in fact be stationary Dagan (1989); specifically the Lagrangian mean velocity may be independent of time. Furthermore, it has been found for flow through random fracture networks that the Lagrangian velocity statistics depends on the initial particle distribution Hyman et al. (2015); Frampton and Cvetkovic (2009); Kang et al. (). Hence, in general, Lagrangian velocities are expected to evolve from an arbitrary initial distribution toward an asymptotic stationary distribution. Quantifying this property, which is not described in current CTRW frameworks, is critical for upscaling transport dynamics through disordered media, whose transport properties are sensitive to the initial velocity distribution.

In this paper, we study the evolution of Lagrangian velocities and their relation with the Eulerian velocity statistics. To this end, we discuss in the following section the concepts of Lagrangian velocities determined isochronically and equidistantly along streamlines and their relation to the Eulerian velocity. Furthermore, we recall some fundamental properties that elucidate the conditions under which they are transient or stationary. Section III derives the Lagrangian velocity statistics in the classical CTRW and develops a Markov-chain CTRW approach that models the evolution of equidistant streamwise Lagrangian velocities as a stochastic relaxation process. In this framework, we derive explicit expressions for the one and two-point statistics of Lagrangian velocities, and analyze the evolution of the mean particle velocity, its covariance as well as particle dispersion in Section IV.

## Ii Lagrangian Velocities

We consider purely advective transport in a heterogeneous velocity field . Particle trajectories are described by the advection equation

 dx(t,a)dt=v(t,a), (1)

where denotes the Lagrangian particle velocity. The initial particle position is given by . The particle motion can be described in terms of the distance traveled along a trajectory, which is given by

 ds(t,a)dt=vt(t,a), dt(s,a)ds=1vs(s,a), (2)

We define the t-Lagrangian particle velocity as , the s-Lagrangian velocity . The initial velocities are denoted by .

The absolute Eulerian velocities are defined by . Their probability density function (PDF) is defined through spatial sampling as

 pe(v)=limV→∞1V∫Ωdxδ[v−ve(x)], (3)

where is the sampling domain and its volume. We assume here Eulerian ergodicity, this means that spatial sampling is equal to ensemble sampling such that

 pe(v)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯δ[v−ve(x)], (4)

where the overbar denotes the ensemble average. In the following, we discuss the t-Lagrangian velocities , which are sampled isochronally along particle trajectories, and the s-Lagrangian velocities , which are sampled equidistantly along particle trajectories. Here and in the following, we assume both Eulerian and Lagrangian ergodicity.

### ii.1 Steady Lagrangian Velocity Distributions

The PDF of the t-Lagrangian velocity is defined by isochrone sampling along a particle trajectory as

 pt(v,a)=limT→∞1TT∫0dtδ[v−vt(t,a)], (5)

Under Lagrangian ergodic conditions, it is independent of the initial particle position and equal to the average over an ensemble of particles

 pt(v)=limV0→∞1V0∫Ω0daδ[v−vt(t,a)]. (6)

The latter is equal to the Eulerian velocity PDF due to volume conservation,

 pt(v)=limV0→∞1V0∫Ω(t)dxδ[v−ve(x)]≡pe(v), (7)

which can be seen by performing a change of variables according to the flow map and recalling that the Jacobian is one due to the incompressibility of the flow field.

The PDF of the s-Lagrangian velocity is defined in analogy to (5) by equidistant sampling along a particle trajectory as

 ps(v,a)=limL→∞1LL∫0dsδ[v−vs(s,a)]. (8)

Changing variables under the integral according to the kinematic relationship (2) between and gives immediately

 ps(v,a)=vpt(v,a)⟨vt⟩, (9)

this means the s-Lagrangian velocity PDF is equal to the flux weighted t-Lagrangian velocity PDF. This can also be understood intuitively by the fact that isochrone sampling as expressed through gives a higher weight to low velocities because particles spend more time at low velocities, while equidistant sampling assigns the same weight to high and low velocities.

Under conditions of Lagrangian ergodicity, we thus have that (i) is independent of the particle trajectory and equal to the average over an ensemble of particles and (ii) that the s-Lagrangian velocity PDF is related to the Eulerian velocity PDF through flux weighting as

 ps(v)=vpe(v)⟨ve⟩. (10)

The latter establishes the relation between s-Lagrangian and Eulerian velocity distributions.

### ii.2 Transient Lagrangian Velocity Distributions

In the previous sections, we considered the PDFs of t- and s-Lagrangian velocities under stationary conditions. Here we focus on their transient counterparts, which are defined through a spatial average over an arbitrary normalized initial particle distribution .

The PDF of t-Lagrangian velocities then is defined by

 ^pt(v,t)=∫daρ(a)δ[v−vt(t,a)]. (11)

Its temporal average is given by

 limT→∞1TT∫0dt^pt(v,t)=pt(v)=pe(v), (12)

and thus its steady state PDF is of course given by the Eulerian velocity PDF. In analogy, we consider the PDF of s-Lagrangian velocities for an arbitrary initial PDF

 ^ps(v,s)=∫daρ(a)δ[v−vs(s,a)]. (13)

Its average along a streamline is given by

 limL→∞1LL∫0ds^pt(v,s)=ps(v)=vpe(v)⟨ve⟩. (14)

The initial conditions for both the t-Lagrangian and s-Lagrangian velocity PDFs are identical,

 ^p(v,s=0)=^p(v,t=0)=p0(v) (15)

Thus, as their respective steady state PDFs are different, either one or both of them need to evolve, depending on whether the initial PDF is the flux weighted Eulerian PDF, (the steady state PDF for ), the Eulerian PDF (the steady state PDF for ), or neither of the two.

The initial velocity PDF depends on the particle injection mode. For example, a uniform in space particle injection corresponds here to an initial velocity PDF equal to the Eulerian PDF,

 p0(v)=limV0→∞1V0∫Ω0daδ[v−v0(a)]≡pe(v) (16)

because of Eulerian ergodicity. As this initial distribution is equal to the asymptotic steady t-Lagrangian velocity distribution, the is independent of time for this initial injection condition, while the evolves with distance from the injection.

A flux weighted particle injection mode corresponds to an initial velocity PDF equal to the flux weighted Eulerian PDF

 p0(v)=limV0→∞1V0∫Ω0dav0(a)⟨ve⟩δ[v−v0(a)]≡vpe(v)⟨ve⟩ (17)

again because of Eulerian ergodicity. As this initial distribution is equal to the asymptotic steady s-Lagrangian velocity distribution, is independent of for this initial injection condition, while evolves with time.

A point-like injection at the initial position corresponds to the delta initial PDF

 p0(v)=δ[v−v0(a)]. (18)

For this initial condition, both the t-Lagrangian and s-Lagrangian velocities are unsteady.

The evolution of Lagrangian velocities may be very slow and thus have a strong impact on the transport dynamics. This is the case in particular for heavy-tailed (towards low velocities) velocity distributions that induce long-range temporal correlations of particle velocities. In the following, we study the quantification of the evolution of the Lagrangian velocity PDFs in a Markov model in , this means distance along streamline.

### ii.3 Lagrangian Velocity Series

We have established that the Lagrangian velocity PDFs evolve with travel time or travel distance along a streamline, unless the initial velocity distribution coincides with the respective steady state PDF. In order to quantify this evolution, we need to model the Lagrangian velocity series. As mentioned in the Introduction, a classical approach is to model the t-Lagrangian velocity as a Markov process, based on the assumption, or observation that velocities decorrelate on a characteristic time scale . Thus, the equations of motion (2) may be discretized isochronically as

 tn+1=tn+Δt, s(tn+1)=s(tn)+vt(tn)Δt. (19)

Velocity time series have been modeled by Langevin equations of the type Pope (2000)

 ~vt(tn+1) =~vt(tn)−Δtτc~vt(tn)+√2σ2vΔtτcξ(tn), (20)

which describes an Ornstein-Uhlenbeck process for the velocity fluctuation . The noise is Gaussian distributed with zero mean and unit variance. The steady state distribution here is Gaussian with mean variance . Under stationary conditions, the velocity correlation is exponential with correlation time . Evidently, this modeling framework is limited to Gaussian statistics and short range correlation in time.

Here, we consider a different modeling approach. As pointed out in the Introduction, there has been ample evidence that particle motion in the flow through random porous and fractured media may be quantified by a CTRW Berkowitz et al. (2006). In fact, as a consequence of the existence of a spatial correlation length scale for, e.g., the hydraulic conductivity or pore-structure, flow velocities are expected to vary over a characteristic length scale . This implies for t-Lagrangian velocities that a given velocity persists for a duration of , and specifically that small velocities are stronger correlated in time than high velocities. This characteristic can explain intermittency in velocity and acceleration time series De Anna et al. (2013); Kang et al. (2014); Holzner et al. (2015). The existence of a characteristic length scale suggests to discretize the equations of motion (2) along a particle trajectory equidistantly such that

 sn+1=sn+Δs, t(sn)=t(sn)+Δsvs(sn). (21)

Here, the s-Lagrangian velocity series is modeled as Markov process, which renders the equations of motion (21) a CTRW. In the following, we analyze the evolution of the Lagrangian velocity statistics in the setup of a classical CTRW characterized by independent s-Lagrangian velocities, and a novel CTRW in which the velocity series is modeled as a Markov process through a stochastic relaxation.

## Iii Continuous Time Random Walk

We study now the evolution of space and time Lagrangian velocities in the CTRW framework. The classical approach assigns to each particle transition a transit time that is sampled at each step from its PDF . The transition times are related to the characteristic transition length and s-Lagrangian velocities as . Thus, independence of subsequent transit times implies indepence of subsequent s-Lagrangian velocities. In the following, we first consider the evolution of t-Lagrangian velocities in this classical CTRW formulation. The velocity statistics turn out to be non-stationary at finite times. We then study a CTRW formulation that is based on a Markov process for the s-Lagrangian velocities that allows both for an evolution of the s- and t-Lagrangian velocities.

### iii.1 Independent s-Lagrangian Velocities

Particle motion along a particle trajectory is quantified in the framework of a classical CTRW by the recursion relations

 sn+1=sn+ℓc, tn+1=tn+τn, (22)

where the transition length denotes a characteristic length scale on which streamwise velocities decorrelate. In this framework, the particle velocity is constant between turning points. Thus, the transition times are independent identically distributed random variables. Their PDF is given by . It is related to the distributions of s-Lagrangian and Eulerian velocities by

 ψ(τ)=ℓcτ2ps(ℓc/τ)=ℓcτvτ3pe(ℓc/τ), (23)

where we defined the advection time scale . Note that the mean transit time is equal to the characteristic advection time.

In this framework, the t-Lagragian velocity is given by

 vt(t)=vnt, (24)

where the renewal process denotes the number of steps needed to arrive at time . The PDF of the t-Lagrangian velocity is given by

 ^pt(v,t)=⟨δ[v−vnt]⟩. (25)

This expression can be expanded as

 ^pt(v,t)=ps(v)ℓc/v∫0dzR(t−z), (26)

for and for ; is the probability per time that a particle arrives at a turning point at time , see Appendix A. Thus, the t-Lagrangian velocity PDF is determined by the sampling of the steady s-Lagrangian PDF between turning points. The right side of (26) expresses the probability of encountering velocity at a turning point times the probability that the particle has arrived within an interval of length before the observation time. The arrival time frequency at a turning point satisfies the Kolmogorov-type equation

 R(t)=δ(t)+t∫0dt′R(t′)ψ(t−t′). (27)

The probability per time to just arrive at a turning point is equal to the probability to be at a turning point at any time times the probability to make a transition of duration to arrive at the next turning point. The t-Lagrangian velocity PDF (26) is non-stationary.

From (27), the Laplace space solution for is

 R∗(λ)=11−ψ∗(λ). (28)

In the limit , it can be approximated by and therefore for , we approximate . Thus, in the limit in the limit of , we obtain from (26)

 ^pt(v,t)=pe(v)+…. (29)

Thus asymptotically, converges toward the Eulerian velocity PDF .

Similarly, we obtain for the two-point PDF of the t-Lagrangian velocity the equation

 ^pt(v,t;v′,t′)=ps(v′)× ℓc/v′∫0dz′^p(v,t−t′+z′)R(t′−z′), (30)

where , see Appendix A. It is non-stationary as indicated by its explicit dependence on . Again, in the limit , we approximate

 ^pt(v,t;v′,t′)=pe(v′)^p(v,t−t′). (31)

It is therefore asymptotically stationary.

In summary, the classical CTRW describes the evolution of the t-Lagrangian velocity PDF from the flux weighted Eulerian to the Eulerian velocity PDF. The t-Lagrangian velocities are non-stationary Baule and Friedrich (2005). This property is also called aging in the literature Sokolov (2012). In the following, we analyze a CTRW formulation that allows for stationary t-Lagrangian statistics and accounts for the evolutions of the t- and s-Lagrangian velocity PDFs from any initial distribution.

### iii.2 Markov Process of s-Lagrangian Velocities

In order to introduce correlations between subsequent particle velocities, and thus quantify the evolution of Lagrangian velocity statistics, we describe the velocity series measured equidistantly along a streamline as a Markov process Le Borgne et al. (2008a); Kang et al. (2011, 2015); Meyer and Saggini (2016). The evolution of the s-Lagrangian velocity PDF is now given by the Chapman-Kolmogorov equation

 ^ps(v,s+Δs)=∞∫0dv′r(v,Δs|v′)^ps(v′,s), (32)

where we assume that the transition PDF is stationary in . The evolution of particle time in this CTRW is given by

 t(s+Δs)=t(s)+Δsvs(s). (33a) The joint Markov process [vs(s),t(s)] of streamwise velocity and time is characterized by the joint transition density ψ(v,t−t′,Δs|v′)=r(v,s|v′)δ(t−t′−Δs/v′). (33b) Note that a Markov-chain may be characterized by the convergence rate of the transition PDF r(v,nΔs|v′) toward its steady state, which here is given by limn→∞r(v,nΔs|v′)=ps(v). (33c) The (spatial) convergence rate is given by the inverse of the correlation distance ℓc along the streamline. We consider now a process that is uniquely characterized by the steady state PDF ps(v) and the streamwise correlation distance ℓc, and model the s-Lagrangian velocity series by the stochastic relaxation process vs(s+Δs) =[1−ξ(s)]v(s)+ξ(s)ν(s). (33d) The random velocities ν(s) are identical independently distributed according to the steady s-Lagrangian velocity PDF ps(ν). The ξ(s) are identical independently distributed Bernoulli variables that take the value 1 with probability 1−exp(−Δs/ℓc) and 0 with probability exp(−Δs/ℓc). Thus, its PDF is pξ(ξ) =exp(−Δs/ℓc)δ(ξ) +[1−exp(−Δs/ℓc)]δ(ξ−1). (33e) The initial velocity distribution is given by p0(v). The transition probability r(v,s|v′) for the process (33d) is given by r(v,s|v′) =exp(−s/ℓc)δ(v−v′) +[1−exp(−s/ℓc)]ps(v). (33f)

The velocity process is fully defined by the transition PDF (33f) and the PDF of initial velocities.

#### iii.2.1 Space-Lagrangian Velocity Statistics

Using the explicit expression (33f) in (32) and performing the continuum limit , we obtain the following Master equation for the streamwise evolution of ,

 ∂^ps(v,s)∂s=ℓ−1c[ps(v)−^ps(v,s)] (34)

subject to the initial condition . Its solution

 ^ps(v,s)=ps(v)+exp(−s/ℓc)[p0(v)−ps(v)] (35)

converges exponentially from toward the steady state distribution , and for it is stationary. The mean s-Lagrangian velocity is defined by

 ⟨vs(s)⟩=∞∫0dvv^ps(v,s), (36)

and from (35) we obtain the explicit expression

 ⟨vs(s)⟩=⟨vs⟩+exp(−s/ℓc)[⟨v0⟩−⟨vs⟩], (37)

Under stationary conditions, this means for , it is constant equal to .

The velocity covariance is then defined by

 Cs(s,s′)=⟨vs(s)vs(s′)⟩−⟨vs(s)⟩⟨vs(s′)⟩, (38)

where the velocity cross-moment is

 ⟨vs(s)vs(s′)⟩= ∞∫0dv∞∫0dv′vv′r(v,s−s′|v′)ps(v′,s′), (39)

for . Using (34) and (33f), we obtain for the explicit expression

 Cs(s,s′)=(⟨v0⟩−⟨vs⟩)2exp(−s/ℓc)[1−exp(−s′/ℓc)] +σ2vsexp[−(s−s′)/ℓc]+(σ2v0−σ2vs)exp(−s/ℓc). (40)

For stationary initial velocities , it reduces to .

#### iii.2.2 Time-Lagrangian Velocity Statistics

Here we quantify the temporal evolution of the Lagrangian velocity distribution. The existence of a spatial correlation length entails short range correlation in space and long range correlation in time for the Lagrangian velocities, which we quantify in the following.

In the continuum limit of , the time process (33a) becomes

 dt(s)ds=1vs(s). (41)

The conjugate process , which is the distance traveled along the streamline until time is defined by . The t-Lagragian velocities are now given in terms of as

 vt(t)=vs[s(t)], (42)
##### One-Point Statistics

Thus, the t-Lagrangian velocity PDF reads now as

 ^pt(v,t)=⟨δ(v−vs[s(t)])⟩. (43)

Using the properties of the Dirac-delta, we can expand this equation into

 ^p(v,t)=∞∫0dsv−1R(v,t,s), (44)

where we defined the probability density that a particle has the velocity and the time at a distance along the trajectory as

 R(v,t,s)=⟨δ[v−v(s)]δ[t−t(s)]⟩. (45)

Note that is the density of the joint Markov process (33) for . Thus, it satisfies the Chapman-Kolmogorov equation

 R(v,t,s+Δs)= ∞∫0dv′t∫0dzψ(v,t−z,Δs|v′)R(v′,z,s). (46)

Inserting (33b) and (33f) into the right side of (46) and taking the limit gives the Master equation (see Appendix B)

 ∂R(v,t,s)∂s =−ℓ−1cR(v,t,s)−v−1∂R(v,t,s)∂t +ℓ−1cps(v)∞∫0dv′R(v′,t,s), (47)

with the initial condition . Integrating this equation over according to (44) gives for the t-Lagrangian velocity PDF the integro-differential equation

 ∂^pt(v,t)∂t =−vℓc^pt(v,t)+ps(v)∞∫0dv′v′ℓc^pt(v′,t) (48)

with the initial condition . Its solution in Laplace space is given by (see Appendix B)

 ^p∗t(v,λ) =p0(v)g∗0(v,λ) +v⟨ve⟩pe(v)g∗0(v,λ)ψ∗0(λ)1−ψ∗s(λ), (49)

where we defined the propagator

 g0(v,t)=exp(−tv/ℓc), (50)

whose Laplace transform is given by . We define the transit time distributions , , and through

 ψi(t)=τ−1v∞∫0dvg0(v,t)vpi(v)⟨ve⟩ (51)

with . Note that its initial values is . Its Laplace transform is given by

 ψ∗i(λ)=τ−1v∞∫0dvvpi(v)(λ+v/ℓc)⟨ve⟩. (52)

It can be seen from (III.2.2) that is steady for the initial condition and is unsteady for any other initial condition by noting that .

Expression  (III.2.2) quantifies the evolution of the t-Lagrangian velocity distribution through potentially long-range temporal correlations reflected by the transit time distributions (51). Note that the transition time PDFs (51) are different from definition (23) for the classical –discrete CTRW framework discussed in Section III.1.

##### Two-Point Statistics

The two-point velocity density is defined here by

 ^p(v,t;v′,t′)=⟨δ(v−v[s(t)])δ(v′−v[s(t′)])⟩. (53)

Along the same lines as above, we derive by using the properties of the Dirac-delta

 ^p(v,t;v′,t′) =∞∫0ds∞∫0ds′v−1R(v,t−t′,s−s′|v′) ×v′−1R(v′,t′,s′). (54)

The conditional PDF describes the joint distribution of conditional to and . It satisfies the Master equation (III.2.2) with the initial condition . Note that is stationary in and due to the stationarity of the velocity and time processes as expressed by the transition PDF (33b). Using definition (44), we can now write (54) as

 ^p(v,t;v′,t′) =^p(v,t−t′|v′)^pt(v′,t′). (55)

where we defined

 ^p(v,t|v′)=v−1∞∫0dsR(v,t,s|v′). (56)

It satisfies the integro-differential equation (48) for the initial condition . Its Laplace space solution is obtained from (III.2.2) by setting as

 ^p∗t(v,λ|v′) =g∗0(v,λ)δ(v−v′) +vv′⟨ve⟩2τcpe(v)g∗0(v,λ)g∗0(v′,λ)1−ψ∗s(λ), (57)

where we note that here . Recall that the one-point PDF is stationary and equal to for the initial condition . Under these conditions, the two-point density (55) is then

 ^p(v,t;v′,t′)≡^p(v,t−t′,v′)=^p(v,t−t′|v′)pe(v′), (58)

and so is stationary. In the following, we determine the mean and covariance of the t-Lagrangian velocities as well as the corresponding particle dispersion.

## Iv Velocity Mean, Covariance and Dispersion

We study here the t-Lagrangian mean velocity, its covariance and the particle dispersion for the CTRW model presented in Section III.2. We investigate these quantities for the following –distribution of Eulerian velocities

 pe(v)=(v/v0)α−1exp(−v/v0)v0Γ(α) (59)

for , which provides a parametric model for the low end of Eulerian velocity distributions in porous media both on the pore and on the Darcy scale Berkowitz et al. (2006); Holzner et al. (2015). As initial conditions we consider either the Eulerian (59) or steady s-Lagrangian velocity PDF (10), which is obtained from the Eulerian velocity PDF through flux weighting

 ps(v)=(v/v0)αexp(−v/v0)v0Γ(α+1). (60)

Note that the Eulerian and flux-weighted mean and mean square velocities are

 ⟨ve⟩ =αv0, ⟨v2e⟩=α(α+1)v20 (61) ⟨vs⟩ =v0(α+1), ⟨v2s⟩=v20(α+1)(α+2). (62)

Inserting (59) into (51), we obtain for the transit time distribution

 ψe(t)=ατ0(1+t/τ0)1+α. (63)

where . For the transit time distribution , we obtain analogously

 ψs(t)=α+1τ0(1+t/τ0)2+α. (64)

The Laplace transforms of and can be expanded by using Tauberian theorems. For , is

 ψ∗e(λ)=1−aα(λτ0)α, (65)

where . For , we have

 ψ∗e(λ)=1+λτ0ln(λτ0). (66)

In the range , we obtain for the expansion

 ψ∗s(λ)=1−λτv+bα(λτ0)1+α, (67)

where and . For , one obtains

 ψ∗s(λ)=1−λτ0−(λτ0)2ln(λτ0). (68)

Note that the case corresponds to an exponential distribution of Eulerian velocities.

For , both the first and second moments of exist, such that can be expanded as

 ψ∗s(λ)=1−λτv+⟨τ2s⟩2λ2. (69)

In the following, we will discuss the mean t-Lagrangian velocity, the velocity covariance and particle dispersion. We present general Laplace space expressions based on the explicit expressions for the one- and two point velocity PDFs derived in Section III.2.2, and study their temporal behavior for the Eulerian velocity PDF given by the –distribution (59). To this end, we perform random walk particle tracking simulations based on (33) and derive explicit expressions for the early and late time behaviors using the expansions (67)–(69) of the Laplace transform of the streamwise transition time PDF .

### iv.1 Mean Velocity

The mean particle velocity is equal to the one-point t-Lagrangian velocity moment

 m1(t)=∞∫0dvv^pt(v,t). (70)

Using (III.2.2), we obtain for the Laplace transform of

 m∗1(λ) =ℓcψ∗0(λ)+∞∫0dvv2⟨ve⟩pe(v)g∗0(v,λ)ψ∗0(λ)1−ψ∗s(λ). (71)

For the stationary initial conditions, , the particle velocity is constant, and equal to the mean Eulerian velocity.

For the non-stationary initial conditions we obtain at short times

 m1(t)=ℓcψs(t). (72)

This means it decreases from its initial value as . For times and , we use the expansion (67) in (71), which gives in leading order

 m∗1(λ)=⟨ve⟩λ+⟨ve⟩τ0bαb1(λτ0)α−1. (73)

For we obtain

 m∗1(λ)=⟨ve⟩λ−ℓcln(λτ0). (74)

Thus, the long-time behavior of for is

 m1(t)=⟨ve⟩+c⟨ve⟩(t/τ0)−α, (75)

where we defined for and for . This means, the mean velocity converges as a power-law toward its asymptotic value, which is given by the Eulerian mean velocity.

For , we use (69) in order to obtain in leading order for

 m∗1(λ)=⟨ve⟩λ+ℓc+⟨ve⟩⟨τ2s⟩2τv. (76)

This means, for , can be written as

 m1(t)=⟨ve⟩+(ℓc+⟨ve⟩⟨τ2s⟩2τv)δ(t). (77)

Note that the Dirac-delta indicates that the convergence toward its asymptotic value is faster than . These behaviors are illustrated in Figure 1, which shows the evolution of the t-Lagrangian mean velocity with time under Eulerian and flux-weighted Eulerian initial conditions for and .

### iv.2 Velocity Covariance

The t-Lagrangian velocity covariance is given by

 Ct(t,t′)=m2(t,t′)−m1(t)m1(t′), (78)

where we defined the two-point velocity moment by

 m2(t,t′)=∞∫0dv∞∫0dv′vv′^pt(v,t;v′,t′), (79)

which can be written in terms of (55) for the two-point velocity PDF as

 m2(t,t′)=∞∫0dv′m1(t−t′|v′)v′^pt(v′,t′), (80)

where we defined the conditional velocity moment as

 m1(t|v′)=∞∫0dvv^pt(v,t|v′). (81)

The Laplace transform of (81) is then obtained from (57) as

 m∗1(λ|v′) =v′g∗0(v′,λ) +∞∫0dvv2v′⟨ve⟩2τvpe(v)g∗0(v,λ)g∗0(