Dispersion of motile bacteria in a porous medium

Understanding flow and transport of bacteria in porous media is crucial to technologies such as bioremediation, biomineralization or enhanced oil recovery. While physicochemical bacteria filtration is well-documented, recent studies showed that bacterial motility plays a key role in the transport process. Flow and transport experiments performed in microfluidic chips containing randomly placed obstacles confirmed that the distributions of non-motile particles stays compact, whereas for the motile strains, the distributions are characterized by both significant retention as well as fast downstream motion. For motile bacteria, the detailed microscopic study of individual bacteria trajectories reveals two salient features: (i) the emergence of an active retention process triggered by motility, (ii) enhancement of dispersion due to the exchange between fast flow channels and low flow regions in the vicinity of the solid grains. We propose a physical model based on a continuous time random walk approach. This approach accounts for bacteria dispersion via variable pore-scale flow velocities through a Markov model for equidistant particle speeds. Motility of bacteria is modeled by a two-rate trapping process that accounts for the motion towards and active trapping at the obstacles. This approach captures the forward tails observed for the distribution of bacteria displacements, and quantifies an enhanced hydrodynamic dispersion effect that originates in the interaction between flow at the pore-scale and bacterial motility. The model reproduces the experimental observations, and predicts bacteria dispersion and transport at the macroscale.


Introduction
Bacteria are the cause of many diseases and some of them, such as cholera, are spread by contaminated water. In the 19th century, this problem led to the development of drinking water systems separated from wastewater and motivated Darcy to formulate the basic equations describing the flow of a fluid in a porous medium (Darcy 1856). Since then, bacteria transport and filtration through porous media has remained a field of intense research. However, still many practical challenges are concerned with difficulties for macroscopic standard models to provide a reliable and quantitative picture of the dispersion of bacteria transported by flow in porous media. For instance, Hornberger et al. (1992) published a study comparing the bacterial effluent curves with those of a classical filtration model including fluid convection and absorption-desorption kinetics. The model allows for a good adjustment of the long time tail of the bacteria concentration curves whereas the model gives disappointing predictions for the breakthrough curves at short times. Subsequent studies have sought to identify the influence of flow or physicochemical conditions on the model parameters. Although little consideration was given to bacterial motility, it came out that this parameter could be crucial to better understand dispersion and retention processes (McCaulou et al. 1994;Hendry et al. 1999;Camesano & Logan 1998;Jiang et al. 2005;Walker et al. 2005;Liu et al. 2011;Stumpp et al. 2011;Zhang et al. 2021). Recent studies support the idea that the swimming capacity of the bacteria allows them to explore more of the porosity (Becker et al. 2003;Liu et al. 2011). For instance, by performing flow experiments with motile and non-motile bacteria in a fracture, Becker et al. (2003) recovered at the outlet about 3% of the non-motile bacteria and only 0.6% of similar but motile bacteria. The mass loss of motile bacteria was explained by the fact that motility eases the diffusion into stagnant fluid resulting in a greater residence time in the porosity and close to grain surfaces. As a consequence motile bacteria are more likely to be filtered. This conclusion seems however inconsistent and in contradiction with earlier observations Hornberger et al. (1992) and Camesano & Logan (1998) reporting less adhesion to soil grains at low fluid velocity.
Microfluidic technology offers a unique experimental method to directly visualize the behavior of bacteria inside pores. Even when using simple geometries such as channels with rectangular cross sections researchers observed non trivial behavior of bacteria in a flow like upstream motions (Kaya & Koser 2012), back-flow low along corners (Figueroa-Morales et al. 2015) eventually leading to large scale "super-contamination" (Figueroa- Morales et al. 2020a), transverse motions due to chirality-induced rheotaxis (Marcos et al. 2012;Jing et al. 2020) and oscillations along the surfaces (Mathijssen et al. 2019). Those observations revealed that the dependence of the bacteria orientations on fluid shear adds new elements that further complicate the transport description. Some studies also point out that this dependence might affect the macroscopic transport of motile bacteria suspensions. This was revealed by the experimental study of Rusconi et al. (2014). In this work, the bacterial concentration profile across the width of a microfluidic channels was recorded as function of flow velocity. When flow was increased and concomitantly the shear rate, they observed a depletion of the central part of the profile that they attributed to a transverse flux of bacteria from low shear to high shear regions located near the surfaces (Rusconi et al. 2014). Motility was also observed to lead to bacteria accumulation at the rear of a constriction (Altshuler et al. 2013) or downstream circular obstacles (Miño et al. 2018;Secchi et al. 2020;Lee et al. 2021). Addition of pillars to microfluidic rectangular channels offers the possibility to design model bi-dimensional heterogeneous porous system suited to explore the influence of flow heterogenities and pore structures on the transport and retention of bacteria (Creppy et al. 2019;Dehkharghani et al. 2019;Scheidweiler et al. 2020;Secchi et al. 2020;de Anna et al. 2020). This approach allows for tracking of individual bacteria trajectories and the measurement of statistical quantities leading to significant progresses towards the understanding and modeling of bacteria transport and dispersion at a macroscopic scale. They all point out that motility has two major impacts, it increases the residence time close to the grains and in regions of low velocity and favors the adhesion (Scheidweiler et al. 2020). The increase of probability to be close to the grains was recently observed in periodic porous media (Dehkharghani et al. 2019). The effect on the macroscopic longitudinal dispersion was then investigated numerically using Langevin simulations. Their study revealed a strong enhancement of the dispersion coefficient particularly when the flow is aligned along the crystallographic axis of the porous medium. In this case, the dispersion coefficient is found to increase like the flow velocity to the power 4 instead of a power 2 as classically obtained for Taylor dispersion. Those examples also show that an accurate macroscopic transport model based on the pore scale observations suited to predict the fate of motile bacteria transported in a porous flow is still missing.
Current approaches to quantify the impact of motility on bacteria dispersion use the generalized Taylor dispersion approach developed by (Brenner & Edwards 1993), which is based on volume averaging of the pore-scale Fokker-Planck equation that describes the distribution of bacteria position and orientation (Alonso-Matilla et al. 2019). This approach lumps the combined effect of pore-scale flow variability and motility into an asymptotic hydrodynamic dispersion coefficient. Therefore, it has the same limitations as macrodispersion theory in that it is not able to account for non-Fickian transport features such as forward tails in the distribution of bacteria displacements and non-linear evolution of the displacement variance. The data-driven approach of Liang et al. (2018) mimics the run and tumble motion of the bacteria by a mesoscopic stochastic model that represents the motile velocity as a Markov process characterized by an empirical transition matrix, but does not provide an upscaled model equation for bacteria dispersion.
In this paper, our aim is to develop a physics-based mesoscale model for bacteria motion, and derive the upscaled transport equations, by explicitly representing porescale flow variability and motility, and their combined impact on bacteria dispersion. In order to understand and quantify the role of motility, we used the experimental data obtained by Creppy et al. (2019). Because these experiments were performed at various flow rates and with motile and non-motile bacteria, this data set offers the possibility to investigate the effect of the flow velocity on bacterial motion. We use a continuous time random walk (CTRW) approach (Morales et al. 2017;Dentz et al. 2018) to model the advective displacements of bacteria along streamlines at variable flow velocities, while the impact of motility is represented as a two-rate trapping process. A similar travel time based approach was used by de Josselin de Jong (1958) and Saffman (1959) to quantify hydrodynamic dispersion coefficients in porous media.
The paper is organized as follows. Section 2 reports on the experimental data for the displacement and velocity statistics of motile and non-motile bacteria. Section 3.1 analyzes transport of non-motile bacteria, which can be considered as passive particles. Thus, we use a CTRW approach, which is suited to quantify the impact of hydrodynamic variability on dispersion. This approach forms the basis for the derivation of a CTRWbased model for the transport of motile bacteria in Section 3.2, which accounts for both hydrodynamic transport and motility. A central element here is to consider and quantify the motility based motion of bacteria toward the solid as an effective trapping mechanism.

Experimental data
We use the extensive data set of Creppy et al. (2019) for the displacements of nonmotile and motile bacteria in a model porous medium consisting of vertical cylindrical pillars placed randomly in a Hele-Shaw cell of height h = 100µm, also termed the grains in the following. The pillar diameters were chosen randomly from a discrete distribution (20, 30, 40 and 50 µm) with mean 0 = 35µm, which is about 1/3 of the cell height. The grains filled the space with a volume fraction of 33%. This idealized model porous medium shares some characteristics with natural media in channel height and grain size (Bear 1972). A fluorescent Escherichia coli RP437 strain is used to facilitate optical tracking. Details on the microfluidic experiments are given in Creppy et al. (2019). The raw trajectory data were reanalyzed for this study. We consider data from 7 experiments that are characterized by the mean streamwise velocities of the non-motile bacteria, which are u m = 18, 43, 66, 98, 113, 139 and 197µm/s. In each experiment the motion of both motile and non-motile bacteria are considered. In the following, we refer to the experiments as 18, 43, 66 etc. according to the respective mean velocity. We choose the average grain diameter and the average absolute value of the particle velocity along the flow direction u m to define the characteristic advection time τ v = 0 /u m .

Displacement moments and propagators
Particle trajectories x(t) = [x(t), y(t)] of different lengths and duration are recorded, along which velocities are sampled, and from which the displacement moments and propagators are determined. Figure 1 illustrates trajectories of non-motile and motile bacteria from the microfluidic experiments. We focus on displacements along the mean flow direction, which is aligned with the x-direction of the coordinate system. Particle displacements are calculated by where x(t 0 ) is the starting position of the trajectory at time t 0 and t n = n∆t are subsequent sampling times. The time increment ∆t is given by the inverse framerate of the camera. The displacement moments are determined by averaging over all particle where N t denotes the number of tracks, and subscript k denotes the kth trajectory. The displacement variance is defined in terms of the first and second displacement moments by The propagators or displacement distribution is defined by where I(·) is the indicator function, which is 1 if the argument is true and 0 else, ∆x is the size of the sampling bin. Note that the number of tracks decreases with track length and sampling time t n , see the discussion in Appendix A.

Velocity statistics
Particle velocities u(t) = [u x (t), u y (t)] are obtained from the particle displacements between subsequent images, (2.5) The particle speed is defined by v(t) = u x (t) 2 + u y (t) 2 . The mean particle velocity in the following is denoted by u(t) = (u m , 0). The mean speed is denoted by v(t) = v m . Averages are taken over all tracks and sampling times. The speed PDFs are obtained by sampling over all trajectories and sampling times. Figure 2 shows the probability density functions (PDFs) of particle speeds for the nonmotile and motile bacteria, denoted by p nm (v) and p m (v), respectively, rescaled by the mean speed v m of the non-motile bacteria. Non-motile bacteria can be considered passive tracer particles. Thus, the speed distributions of non-motile bacteria serves as a proxy for the Eulerian flow speed distribution, that is, p nm (v) ≡ p e (v) which is supported by the fact that the rescaled data collapse on the same curve. The non-dimensional speed data is well represented by the Gamma distribution for α = 2.25. Speed distributions in porous media are often characterized by exponential or stretched exponential decay for v > v m and power-law behaviors at low flow speeds. Similar speed distributions have been reported in experimental particle tracking data (Holzner et al. 2015;Morales et al. 2017;Alim et al. 2017;Carrel et al. 2018;Souzy et al. 2020) and from numerical simulations of pore-scale flow (Siena et al. 2014;Matyka et al. 2016;De Anna et al. 2017;Aramideh et al. 2018;Dentz et al. 2018). The right panel of Figure 2 shows the speed PDFs for the motile bacteria rescaled by the mean speed of the respective non-motile bacteria, together with the Gamma distribution given in Eq. (2.6), which models the non-motile speed PDFs. The global shapes of the rescaled speed PDFs for the motile bacteria are very similar to the speed PDF for the non-motile bacteria represented by the Gamma distribution. However, they are shifted towards smaller values when compared to the non-motile bacteria, with a small peak at low values, which can be related to bacteria motion along the grains. The speed PDFs with u m 98µm/s scale with the mean speed v m and group together above all at intermediate and small speeds. The speed PDF of the motile bacteria measures the combined speed of the flow field and bacteria motility. The fact that the speed PDFs collapse when rescaled by the respective mean flow speeds indicates that bacteria motion scales with the flow speed. This seems to be different for the speed PDFs for u m 66µm/s. The PDFs are more scattered and shifted towards smaller values compared to the speed PDFs for the high flow rates.
Particle trajectories are tortuous due to pore and velocity structures, and thus are longer than the corresponding linear distance. The ratio between the average trajectory length of the non-motile bacteria and the linear length in mean flow direction defines the tortuosity χ. It can be quantified by the ratio between the mean flow speed v m and the mean flow velocity u m as (Koponen et al. 1996;Ghanbarian et al. 2013;Puyguiraud et al. 2019b) We obtain from the velocity data at all flow rates, tortuosity values between χ = 1.17 and 1.23.

Theoretical approach
We present here the theoretical approach to model the dispersion of non-motile and motile bacteria. We use the CTRW framework to model the stochastic motion of bacteria due to pore-scale flow variability and motility, based on a spatial Markov model for subsequent particle velocities, and a compound Poisson process for motility. This type of approach was used to upscale and predict hydrodynamic transport in porous and fractured media at the pore and continuum scales (Berkowitz & Scher 1997;Noetinger et al. 2016;Dentz et al. 2018;Hyman et al. 2019). It naturally accounts for the organization of the flow field along characteristics length scales that are imprinted in the host medium. We focus here on the quantification of the streamwise motion and 0 grain size c characteristic persistence length of particle speeds c coarse-graining length v0 magnitude of the swimming velocity of the bacteria u velocity of non-motile bacteria v = |u|, speed of non-motile bacteria vm = v , average speed um = ux , average streamwise velocity τv = 0/um, advection time χ = vm/um, tortuosity τc characteristic trapping time γ trapping rate Dnm dispersion coefficient of the non-motile bacteria Dm dispersion coefficient of the motile bacteria ρ fraction of bacteria at the grains β partition coefficient R retardation factor associated to the convection at the macroscopic scale of the motile bacteria Table 1. Notation large scale dispersion of bacteria, which play a key role for the prediction of the length of bacteria plumes and the distributions of residence times in a porous medium.

Non-motile bacteria
Non-motile bacteria are considered as passive tracer particles that are transported by advection only. Non-motile bacteria move along streamlines of the porescale flow field, and thus explore the porescale velocity spectrum, except for the lowest velocities close to the grains, due to volume exclusion or molecular diffusion. Typical trajectories are shown in Figure 1. In the following, we model the motion of non-motile bacteria using a spatial Markov model for particle speeds Morales et al. 2017;Puyguiraud et al. 2019b).

Spatial Markov model
Particle motion is characterized by the spatial persistence of particle velocities over a characteristic length scale, which is imprinted in the spatial structure of the porous medium ). This provides a natural parameterization of bacteria motion in terms of travel distance. That is, motion is modeled by constant space and variable time increments along streamlines. Thus, the equations of streamwise motion of non-motile bacteria can be written as (Puyguiraud et al. 2019b) where ∆s is the transition length along the tortuous particle path. The advective tortuosity χ accounts for streamline meandering in the pore space between the grains. It quantifies the ratio of the average streamline length to streamwise distance. Note that this meandering is different for each streamline and may be correlated to the particle speed. However, under ergodic flow conditions, the streamline lengths converge toward the average value, and thus, at scales larger than 0 tortuosity provides a good estimate for the longitudinal displacement. The point distribution p v (v) of particle speeds is given in terms of the Eulerian flow speed distribution p e (v) This speed-weighting relation is due to the fact that in this framework particles make transitions over constant distance, while the distribution of flow speeds p e (v) is obtained by measuring speeds at constant framerate, this means isochronically Morales et al. 2017;Puyguiraud et al. 2019b). Equations (3.1) constitute a CTRW because bacteria are propagated over constant (discrete) distances while time is a continuous variable. In this framework, the position x(t) of a particle at time t is given by The series {v n } of particle speeds is modeled as a stationary Markov process whose steady state distribution is given by Eq. (3.2). Specifically, we model {v n } through an Ornstein-Uhlenbeck process for the unit normal random variable w n which is obtained from v n through the transformation (Puyguiraud et al. 2019a) where P v is the cumulative speed distribution and Φ −1 (u) the inverse of the cumulative unit Gaussian distribution. The w n satisfies the Langevin equation where ξ n is a unit Gaussian random variable. The length scale c denotes the characteristic correlation scale of particle speed. It is typically of the order of the characteristic grain size 0 (Puyguiraud et al. 2021). However, its exact value needs to be adjusted from the data for the displacement variance. The increment ∆s is chosen such that ∆s c . The phase-space particle density p(x, v, t) in this framework is given by the Boltzmann-type equation (Comolli et al. 2019) ∂p (x, v, t) ∂t is the distribution of initial particle velocities. The propagator, that is, the distribution of particle displacements, is given by (3.5)

Asymptotic theory
The behavior of the upscaled model at travel distances much larger than the correlation length c , can be obtained by coarse-graining particle motion on a length scale c c , such that The transition times τ n = c /v n are independent random variables whose distribution ψ(t) is given in terms of p v (v) as where τ 0 = c /v 0 . ψ(t) is given here by an inverse Gamma distribution because the particle speed is Gamma-distributed, see Eq. (2.6). For the velocity distribution (2.6) with α = 2.25, the CTRW predicts asymptotically a Fickian dispersion. That is, for times t τ v , transport can be quantified by the advection-dispersion equation  ∂p(x, t) ∂t with the average velocity u m = v m /χ and the dispersion coefficient (Puyguiraud et al. 2021) (3.9) The mean and mean squared transition times are defined by for k = 1, 2. Γ (α) denotes the Gamma function. We find by comparison of the dispersion coefficients from the full spatial Markov model and the CTRW model (3.6) that c ≈ 1.57 c .

Motile bacteria
We provide here the theoretical framework to interpret the trajectory data and motion of motile bacteria. The motion of motile bacteria is due to advection in the flow field and their own motility as illustrated in Figure 1. At zero flow rate, bacteria fluctuate in a random walk-like manner characterized by a zero mean displacement with a characteristic 2D projected swimming velocity v 0 ≈ 12µm/s Creppy et al. (2019). At finite flow rate, bacteria tend to swim along the streamlines, and make excursions perpendicular to them in order to move toward the solid grains. Based on the observations of Creppy et al. (2019) for bacteria motility, we couple the CTRW model for hydrodynamic transport with a trapping approach. These authors found that bacteria move towards the grains at a flow dependent rate γ and dwell on the grain surface for random times θ, which are distributed according to the trapping time distribution ψ f (t).

Spatial Markov model and trapping
Within the CTRW approach outlined in the previous section, the trapping of bacteria is represented by a compound Poisson process for the time t n of the bacteria after n CTRW steps. Thus, the equations of motion are given by for n > 1. The initial displacement is x 0 = 0 for all bacteria. The initial time is set to t 0 = 0. The particle speeds v n evolve according to the process (3.3). The compound trapping time τ (r) is given by where θ i is the trapping time associated to an individual trapping event, and n r is the number of trapping events during time r. The number of trapping events n r follows a Poisson process characterized by the rate γ, that is, the mean number of trapping events per CTRW step is γ∆s/v n . The trapping rate is constant and counts the average number of trapping events per mobile time. While the trapping properties could depend for example on the local flow speeds, we use the Poisson process with constant rate as a robust and simple way of describing the average trapping properties, which is fully defined by the average number of trapping events per mobile time. The distribution of compound trapping times τ (r), denoted by ψ c (t|r), can be expressed in Laplace space by (Feller 1968;Margolin et al. 2003) ψ c (t|r) denotes the probability that the trapping time is t given that a trapping event occurred at time r. For n = 1, we distinguish the proportion ρ of bacteria that are initially trapped, and 1 − ρ of initially mobile bacteria. For the trapped bacteria, x 1 = 0 and t 1 = η 0 , where the initial trapping time η 0 is distributed according to ψ 0 (t). For the mobile bacteria, x 1 and t 1 are given by Eq. (3.11) for n = 0. We consider here steady state conditions at time t = 0. As experimental trajectories and their starting points are recorded continuously, it is reasonable to assume that steady state between mobile and immobile bacteria is attained. Under steady state conditions, the joint probability of the bacterium to be trapped and the initial trapping time to be (3.14) see Appendix C. The trapping times are assumed to be exponentially distributed, that is, with τ c the characteristic trapping time. This means, we use Poissonian statistics to account for the effective retention of motile bacteria in the vicinity of grain surfaces. This picture is classically based on the idea that the run to tumble process promoting surface detachment is itself a memory-less Poisson process (Berg 2018). However, there has been recent evidence that the run-time distribution for bacterial motion in a free fluid is a long-tail non-Poissonian process (Figueroa- Morales et al. 2020b), which is also at the origin of a long-tailed distributions of bacteria sojourn times at flat surfaces (Junot et al. 2021). For porous media, there are currently no direct measurements that offer a quantitative microscopic description of the complex exchange processes taking place between the surface regions and the flowing regions. Thus, we adopt Poissonian statistics characterized by the mean retention time τ c as a model of minimal assumptions. We hope that our conceptual approach, which provides a model of the emerging transport process, will motivate more detailed experimental investigations on this central question.
Using Eq. (3.15) in Eq. (3.14), we obtain where we define the partition coefficient β = γτ c . Thus, the fraction of trapped bacteria is ρ = β/(1 + β), and the initial trapping time distribution is ψ 0 (t) = ψ f (t). Thus, the steady state partitioning of bacteria is directly related to their motility through the trapping rate γ and mean dwelling time τ c on the grain surface. Note that this picture does not account for the tortuous particle path on the grain surfaces, which is represented as a localization event at fixed positions. Grain-scale bacteria motility could eventually be modeled by an additional process. However, here we focus on large scale bacteria dispersion and only account for tortuosity due to the flow path geometry. As above the bacteria position x(t) at time t is given by x(t) = x nt . The expressions for the displacement moments and variance are analogous.
The density p s (x, v, t) of mobile bacteria in the stream is quantified by the non-local Boltzmann equation the probability that the trapping time is larger than t. Equation (3.17) reads as follows. The evolution of the particle density in the stream is given by (second term on the left side) particle exchange between the stream and grain surface, (third term on the left) advection by the local velocity, (first term on the right side) release of bacteria that were initially on the grains, (second and third terms on the right) velocity transitions along the trajectory. The total bacteria density is given by The density p g (x, v, t) of bacteria on the grains is given by This first term on the right side reads as follows. The density of bacteria on the grains is given by the probability per time γp s (x, t ) that bacteria are trapped at time t times the probability φ(t − t ) that the trapping time is longer than t − t . The second term denotes the bacteria that are initially trapped and whose trapping time is larger than t. The speed v associated with a bacterium on the grain should be understood as the bacteria speed before the trapping events.

Asymptotic theory
Similar to the discussion in the previous section for the non-motile bacteria, for distances much larger than c , particle motion can be coarse-grained such that where the advective transition times τ n = c /v n are distributed according to Eq. (3.7). τ (r) describes the compound Poisson process defined above. The propagator p s (x, t) of bacteria in the stream for this equation of motion is quantified by the non-local advectiondispersion equation while the distribution p g (x, t) of bacteria at the grains is given by (3.23) Asymptotically this means for times t τ c , the transport of the bacteria concentration p(x, t) can be described by the advection-dispersion equation see Appendix D. The retardation coefficient R and the asymptotic dispersion coefficient D m are given by the explicit expressions By definition, R compares the average velocity of motile bacteria with the average flow velocity. In absence of trapping, ρ = 0 and R = 1, the bacteria are transported in the porous media with an average velocity equal to the average fluid velocity. If trapping is present, retardation increases, indicating a decrease of the average bacteria velocity compared to the fluid velocity. The retardation coefficient is directly related to bacterial motility, which in our modeling framework is expressed by the trapping rate γ and the mean retention time τ c for which a bacterium dwells at the grain surface. The asymptotic dispersion coefficient in Eq. (3.26) contains two terms. The first term D nm (1−ρ) corresponds to the so-called dispersion coefficient at steady state (Yates et al. 1988;Tufenkji 2007). It predicts a reduction of the dispersion coefficient of the motile bacteria compared to the non-motile concomitant with the reduction of the average velocity of the bacteria population. It accounts for the dispersion of the motile proportion 1 − ρ only. The second term quantifies a mechanism similar to the Taylor dispersion. It originates from the spread of the bacteria plume due to fast transport in the pores and localization at the grains. The resulting dispersion effect can be rationalized as follows. The typical separation distance between localized and mobile bacteria, that is, the dispersion length is u m τ c , while the dispersion time is τ c . The corresponding dispersion coefficient is dispersion length squared divided by dispersion time, which gives exactly the scaling u 2 m τ c of Eq. (3.26). As we will see in the next section, this interaction can lead to a significant increase of bacteria dispersion compared to non-motile bacteria.
Asymptotic bacteria transport is predicted to obey the advection-dispersion equation with constant parameters for two reasons. First, the distribution of particle velocities does not tail towards low values, that is, mean and mean squared transition times are finite. Second, the distribution of retention times is exponential. Thus, for times large compared to the characteristic mass transfer times, the support scale can be considered as well-mixed, and, similar to Taylor dispersion (Taylor 1953), and generalized Taylor dispersion (Brenner & Edwards 1993), transport can be described by an advectiondispersion equation.

Results
We discuss the experimental results for the displacement means and variances, as well as the displacement distributions, in the light of the theory presented in the previous section. As discussed in Appendix A, the number of experimentally observed tracks decreases with the travel time, which introduces a bias toward slower bacteria. Thus, in the following, we consider travel times shorter than 5τ v in order to avoid a too strong bias toward slow bacteria. Even so, as we will see below, there is a slowing down of the mean displacement with increasing travel time, specifically for the motile bacteria.
The proposed theoretical approach for the non-motile bacteria has one parameter that needs to be adjusted, the correlation scale c , which typically is of the order of the grain size. It is adjusted here from the data for the displacement variance for the nonmotile bacteria. The approach for the motile bacteria has two additional parameters, the trapping rate γ and the mean trapping time τ c . The partition coefficient β = γτ c is adjusted from the mean displacement data for the motile bacteria, while the trapping time τ c is adjusted from the data for the displacement variance of the motile bacteria. Thus, the non-motile CTRW model needs to adjust one parameter, which is of the order of the grain size. The motile CTRW model needs to adjust two parameters, which are related to the partitioning of bacteria between flowing and stagnant regions close to the grains. non-dimensionalized by the mean advection time over the size of a grain, which implies that the propagators are reported for the same mean travel distances. The CTRW model uses the velocity distribution (2.6) with α = 2.25, the correlation length c ≈ 2 0 and the advective tortuosity χ = 1.2. The mean displacement is linear with a slightly higher slope at short than at large times. It starts deviating from the expected behavior m 1 (t) = u m t at around t = 2τ v . We relate this behavior to a bias due to the decrease in the number of tracks as discussed in Appendix A. The displacement variance shows a ballistic behavior at t < τ v , this means it increases as t 2 . Then for t > τ v it increases superlinearly, which can be seen as a long cross-over to normal behavior. These behavior are accounted for by the CTRW model. For flow velocities u m 66µm/s, we observe a larger variance than for the higher flow rates. This, and the slightly smaller mean displacements compared to higher flow rates, can be attributed to the localization of some bacteria at the origin (see Figure 4), which causes a chromatographic dispersion effect, which is discussed in more detail for the motile bacteria.

Dispersion of non-motile bacteria
Figures 4 compares the experimental data for the propagators with the results of the CTRW model. The propagators are asymmetric but compact, meaning that there is no significant forward or backward tails in the distribution. For comparison, we plot a Gaussian shaped propagator characterized by the mean displacement and displacement variance shown in Figure 3. The asymmetry decreases with increasing travel time and the propagators become closer to the corresponding Gaussian. The CTRW model captures the initial asymmetry and the transition to symmetric Gaussian behavior for all flow rates.

Dispersion of motile bacteria
Figures 5, 6 and 7 show the displacement mean and variance, and the propagators for the motile bacteria at different flow rates. As in the previous section, time is measured in units of τ v , that is, it measures the mean number of grains the bacteria have passed. The propagators are measured at the same non-dimensional times, that is, at the same mean distance. The motile CTRW model is parameterized by the same correlation length and tortuosity as the non-motile model. The partition coefficient β = γτ c is adjusted from the early time behavior of the mean displacement, which is predicted to behave as because we consider the system to be initially in a steady state. The characteristic trapping time is adjusted from the displacement variance by keeping β fixed. We adjust τ c = 2.5τ v and β = γτ c = 0.4 for u m 98µm/s, and τ c = 2τ v and β = 1 for u m = 66µm/s. As shown in Figure 5, the mean displacement is consistently lower for the motile than for the non-motile bacteria, which is due to migration toward the grain surfaces and localization at the grains. The means displacement initially evolves linearly until a time of about 2τ v and from there, the evolution slows down. We relate this to the decrease of the number of experimentally observed tracks, which induces a bias toward slow tracks as discussed in Appendix A. In contrast to the mean displacement, the displacement variance can be larger than its non-motile counterpart for u m 98µm/s and lower for u m 66µm/s. The data seem to fall into two groups for high and low flow rates, except for u m = 18µm/s. In this case, the flow velocity is of the order of the swimming velocity v 0 ≈ 12µm/s. The data indicates that that the density of trapped particles is higher at high than at low flow rates. The possible mechanisms for these behaviors are discussed in Section 5.
These behaviors are also reflected in the propagators shown in Figure 6 for high flow rates with u m 98µm/s and in Figure 7 for u m 66µm/s. The green symbols in Figure 6 denote the experimental data rescaled by the mean grain size 0 , the solid green lines, the corresponding solution from the CTRW model for the parameters τ c = 2.5τ v and β = γτ c = 0.4. Analogously, the red symbols in Figure 7 denote the experimental data rescaled by the mean grain size 0 . The solid red lines show the corresponding solution from the CTRW model for the parameters τ c = 2τ v and β = 1. For comparison, we also plot the corresponding CTRW solution for the non-motile bacteria, marked by the blue solid lines. The motile propagators are delayed compared to the non-motile bacteria. They are characterized by a localized peak around zero and a pronounced forward tail, which can be attributed (i) to slow motion towards and around grains and (ii) to fast motion in the main pore channels. Figure 6 shows that the propagators at high flow rates (u m 98µm/s) overlap, which indicates that bacteria motion scales with the mean flow. Similarly, for the low flow rates (u m 66µm/s) shown in Figure 7, we observe overlap in the forward tails, which are advection-dominated due to transport in the pore-channels. However, the upstream tails that develop starting from the localized peak do not group together. They can be attributed to bacteria motility, which is independent of the flow rate. This is most pronounced for u m = 18µm/s, which is characterized by strong localization and an almost symmetric propagator. The features of peak localization and forward tailing show that steady state in the macroscopic transport behavior has not been attained at the largest observation time. At asymptotic times, that is, for t τ c , the theoretical model given by Eq. (3.24) predicts Fickian transport characterized by symmetric propagators.
The data for the displacement moments and propagators seems to indicate that the data are grouped in two families, which we have highlighted by using two different colors. These observations are in agreement with the behaviors of the speed PDFs shown in Figure 2. We therefore fit each family separately. From the early time evolution of the mean displacements, we adjust the partition coefficient β = 0.4 for u m 98µm/s, β = 1 for u m 66µm/s. For u m 98µm/s, we adjust from the displacement data τ c = 2.9τ v and for u m = 66µm/s, we adjust τ c = 2.3τ v .
With these parameter sets, the CTRW model is able to describe the propagators and displacement moments as shown in Figures 5 and 6. For the lowest flow velocity, bacteria are able to swim upstream over relatively long distances. The subsequent backward tail that develops because of the upstream motion is clearly visible in Figure 6 (bottom row), and also, to a smaller extent at the higher flow rates (top row). This effect is not accounted for in the model that assumes that the trapping is localized and that trapped bacteria do not move once trapped. Since τ v ∝ 1/u m , our results indicate that the trapping rate increases linearly with the average flow velocity u m while the characteristic trapping time decreases linearly with u m . We used different values for β = γτ v and τ c to adjust the two sets. Recall that the fraction of trapped bacteria ρ is β/(1 + β). Each set thus corresponds to a different value of the fraction of trapped bacteria. The fraction of trapped bacteria is high at low velocities (ρ 0.5) and decreases towards an asymptotic value of about ρ = 0.3 as the flow velocity is increased. The fraction of trapped bacteria is also related to the retardation coefficient R through Eq. (3.25), which is estimated from the experimental data for the mean bacteria displacement according to relation (4.1). The dependence of R and thus ρ on the flow rate is further discussed in the next section.

Asymptotic dispersion and retardation
The CTRW model allows to extrapolate the transport behaviors to times that cannot be reached in the experiment. The top panels of Figure 8 show the displacement mean and variance up to times of 1000τ v . We see that both observables evolve linearly at asymptotic times. The mean displacement indicates a lower average velocity for the motile than for the non-motile bacteria, which is due to trapping. The displacement variance on the other hand is larger for the motile than for the non-motile at high flow rates, which indicates stronger motile dispersion. This effect can be quantified by Eqs. (3.25) and (3.26) for the retardation coefficient and asymptotic dispersion coefficient. The retardation coefficient R = 1/(1 − ρ) = 1 + β can be estimated directly from the experimental data for the mean displacement according to Eq. (4.1). The left panel at the bottom of Figure 8 shows that the retardation coefficient decreases with increasing flow rate, which is consistent with the values adjusted for β in the previous section. Thus the data shows also that the fraction ρ of trapped particles decreases with increasing flow rate.
The behavior of D m as a function of the proportion ρ of trapped bacteria is shown in the bottom panel of Figure 8. The solid line shows the theoretical behavior of D nm for τ c = 2.5τ v and τ c = 2τ v , which corresponds to the value used in the CTRW model. The green and red symbols denote the values obtained from the CTRW models at high and low flow rates. We see that at low fractions of immobile bacteria, the Taylor term in Eq. (3.26) dominates and motile bacteria disperse more than non-motile. At high proportions of trapped bacteria, localization dominates over the Taylor mechanism, and motile dispersion is lower than non-motile. Figure (8) illustrates the competition between the trapping time τ c and the proportion ρ of trapped bacteria. For increasing τ c , motile dispersion can be significantly larger than non-motile dispersion.

Discussion
We study the interaction between bacteria motility and flow variability, and its impact on the dispersion of bacteria. To do so we use data obtained in a microfluidic chip containing randomly placed obstacles, in which thousands of non-motile and motile bacteria were tracked at different flow rates. This geometry reproduces the structure of a porous medium on the scale of a few pores, and is thus ideal to study transport phenomena at the pore scale. Because bacteria do not adhere to the surface of the flow cell, this setup allows to study the first step of filtration which consists of the transport of bacteria from the flowing fluid to regions of low flow in the vicinity of solid grains. Bacteria motion is quantified by a CTRW approach that is based on a Markov model for equidistant particle speeds. The experimental data for the displacement of non-motile bacteria is used to constrain the velocity correlation length, which is of the order of the grain size. Bacteria motility is modeled in this framework by a trapping process, which accounts for the rheotactic motion toward and along the grain surfaces by a trapping rate γ and characteristic dwelling time τ c . The ratio between trapped and mobile bacteria at steady state is measured by the partition coefficient β = γτ c . Adjustment of the model to the experimental data reveals two main features. Firstly, we observe that γ ∝ u m and τ c ∝ 1/u m . The increase of the trapping rate with the flow rate can be explained by the constant reorientation of the bacteria by the flow. The frequency by which bacteria point toward the grains increases with the flow rate, which may explain the increase of the trapping rate. Similarly, for increasing flow rate, shear increases on the grains and thus the area for motion around the grains decreases and the bacteria are more easily blown off by the flow. This can explain why the residence time decreases with flow rate. A model that supports this idea is proposed in Appendix E.
Secondly, we observe that the ratio β between trapped and mobile bacteria is different at high and low flow rates. This observation indicates a transition between a regime at low flow rates, where motility favors trapping with a high density of trapped bacteria (about 50% of trapped bacteria), to a regime at high flow rates, where the flow hinders trapping (about 30% of trapped bacteria). Two phenomena may contribute to this change. The first comes from the volume of fluid in which the bacteria can be considered as trapped. This fraction can be separated in two: a part where the velocity is very small (this part corresponds to the dark blue regions that can be seen in Figure 1 and is always present for all the flow rates used) and a second contribution which comes from the regions of flowing fluid where the average flow velocity is less than the swimming velocity. In those volumes, which are located close to the grain surface the bacteria trajectories are little influenced by the flow and they swim much like in a quiescent fluid. Bacteria can be considered trapped when they swim along the grain surface. This contribution however decreases with the flow rate reducing in turn the density of trapped bacteria as observed. The second contribution comes from the diffusion due to the constant reorientation of the bacteria. In a fluid at rest, the trajectories of the bacteria can be decomposed as a succession of runs followed by tumbles that reorient the bacteria. At large scale, the reorientation is diffusive and can be characterized by the translational diffusion coefficient D b . For E.coli we have here D b ≈ 243µm 2 /s (Creppy et al. 2019). In a shear flow, bacteria constantly tumble and are reoriented at a frequency set be the shear rateγ (Jeffery 1922). When the Péclet number defined as P e = u m 0 /2D b is of the order of 1. For a grain size of 0 = 30µm, we have P e u m /(16µm/s). Random orientation will thus dominate shear alignment for the lowest flow rate with little or no influence at high flow velocity.

Conclusions
In conclusion, to understand the dispersion of bacteria in porous media, our study focuses on the central importance of hydrodynamic flow fluctuations and the active exploration process into high shear regions around the solid grains. The rheotactic coupling between flow and bacteria motility manifests itself at small scales through non-Fickian behavior, and at large scales through a motility-dependent hydrodynamic dispersion effect. Noticeably, the interplay between fast transport in the flow and motile motion toward grain surfaces is the first necessary step before possible adhesion (Yates et al. 1988). To date, it had been assumed that the transfer between regions of high fluid flow and low flow regions in the vicinity of the grain surfaces was diffusive, like for passive solutes, and had been modeled as a kinetic single-rate mass transfer process (Yates et al. 1988;Bai et al. 2016). Our study suggests that both motility and flow play a central role in the trapping and release processes, which are characterized by two different rates. Both trapping and release rates are proportional to the average flow velocity, while the ratio between mobile and trapped particle increases with increasing flow velocity. The trapping and release mechanisms explain apparently contradictory observations of the concomitant enhancement of retention and dispersion. They are quantified in a theoretical approach that captures the salient features of the experimental displacement data, and allows for predicting the dispersion of motile bacteria at large scales. These findings shed light on the strategies microorganisms may use to maximize their survival and proliferation abilities under natural conditions, and can give new insights into bacteria filtration and biofilm growth, for which the contact with grain surfaces is determinant.

Appendix A. Track length statistics
The number of observed tracks decreases with time because tracks leave the observation window according to their average velocity. Figure 9 shows the number of tracks of nonmotile and motile bacteria for the experiments at different flow rates as a function of time measured in units of the characteristic advection time τ v , which here is the time to move over the characteristic grain length 0 by mean advection u m . We see that the number of tracks decreases to around 90% of the initial number of tracks after around 2τ v for the non-motile and motile bacteria. After around 4τ v the number of tracks decreases to approximately 35% for the non-motile and to around 40% for the motile bacteria. This means, that the number of tracks of lengths larger than 4 0 is 35% and 40% of the total number of tracks. The long tracks are tortuous low velocity tracks that can be observed for a longer time. This is supported by the observation that the mean velocity starts decreasing after about 2τ v , as shown in Figures 3 and 5 below. In the following, we consider travel times shorter than 5τ v in order to avoid too strong a bias toward slow bacteria. Even so, as we will see below, there is a significant slowing down of the mean displacement with increasing travel time, specifically for the motile bacteria.

Appendix B. Continuous time random walk model
The streamwise motion of a non-motile bacterium in the CTRW is described by Eq. (3.1). Unlike classical random walk strategies for the modeling of particle motion in heterogeneous flow fields, the CTRW approach models particle motion based on stochastic series of equidistant instead of isochronic particle speeds (e.g., Dentz et al. 2016;Morales et al. 2017), that is, particle speeds that change at equidistant points along a streamline. The streamwise displacement in the CTRW model represents the projection of the tortuous streamline onto the mean flow direction using advective tortuosity χ. This is illustrated schematically in Figure 10.

B.1. Non-motile bacteria
In the following, we provide a derivation of the Boltzmann-type equation (c) for the joint distribution p(x, v, t) of particle displacement and speed. For more details, see Comolli et al. (2019). The distribution p(x, v, t) can be written as where ψ(t|v) = δ(t − ∆s/v). The probability per time R(x, v, t) for the particle to just arrive at (x, v) at t satisfies where r(v|v ) is the transition probability from v to v, and ψ(x, t|v) = δ(x − ∆s/χ)δ(t − ∆s/v).

(B 2)
The initial condition is encoded in R 0 (x, v, t), which is defined by where p 0 (x, v) is the distribution of initial particle positions and speeds. Equations (B 1a) and (B 1b) can be combined in Laplace space to the generalized master equation Using the explicit form (B 2) for ψ(x, t|v), it can be written as In the limit of ∆s c , we can write λp * (x, v, λ) = where we localized r(v|v ) = δ(v − v ) in the advection term. By transformation back to time, we obtain the Boltzmann equation

Appendix C. Initial trapping time distribution
In order to derive the initial trapping time distribution, we employ the concept of the backward recurrence time B t0 = t 0 − t N , this means the time that has passed between a target time t 0 and the time t N of the last trapping event before t 0 . For a Poissonian trapping process, this means for an exponential inter-event time distribution, the distribution of B t0 in the steady state limit, that is, for N → ∞, is given by (Godrèche & Luck 2001) ψ B (t) = γ exp(−γt).
(C 1) It is independent from t 0 , B t0 ≡ B. The initial trapping time η 0 can be expressed in terms of B as η 0 = τ f − B. Thus, the joint distribution for a bacterium to be trapped and have the trapping time η 0 < t is It can be written as Using expression (C 1) for ψ B (t) and shifting t → t − t , we obtain Thus we obtain for the joint probabiility of being trapped and η 0 in [t, t+dt] by derivation of (C 4) with respect to t P 0 (t) = m 2 (t) = 2D nm t 1 + β + u 2 m t 2 (1 + β) 2 + 2u 2 m βτ c (1 + β) 3 (D 13) We define the retardation coefficient by comparing m 1 (t) with the mean displacement for the non-motile bacteria. This gives The displacement variance is given by Thus, we obtain for the dispersion coefficient We consider now the asymptotic equation for the total bacteria concentration. Thus, we consider the Fourier-Laplace transform of the total bacteria distribution p(x, t) = p s (x, t) + p g (x, t). From the Fourier-Laplace transform of (3.23), we obtaiñ p * s (k, λ) =p * (k, λ) − ρφ * (λ) 1 + φ * (λ)γ .