Bifurcating -Paths the relation between preferential flow bifurcations void and tortuosity on the Darcy scale . Avioz Dagan1 Yaniv Edery1

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Bifurcating-Paths: the relation between preferential flow bifurcations, void,

and tortuosity on the Darcy scale.

Avioz Dagan1, Yaniv Edery1

1 Faculty of Civil and Environmental Engineering, Technion, Haifa, Israel

Abstract

In recent years, Darcy scale transport in porous media was characterized to be Fickian or non-Fickian due

to the homogeneity or heterogeneity of the porous medium conductivity layout. Yet, evidence shows that

preferential flows that funnel the transport occur in heterogeneous and homogenous cases. We model the

Darcy scale transport using a 2D conductivity field ranging from homogenous to heterogeneous

and find that these preferential flows bifurcate, leaving voids where particles do not invade while

forming a tortuous path. The fraction of bifurcations decreases downflow and reaches an

asymptotical value, which scales as a power-law with the heterogeneity level. We show that the

same power-law scaling of the bifurcations to heterogeneity level appears for the void fraction,

tortuosity, and fractal dimension analysis with the same heterogeneity level. We conclude that the

scaling with the heterogeneity is the dominant feature in the preferential flow geometry, which

will lead to variations in weighting times for the transport and eventually to anomalous transport.

1. Introduction

Transport in saturated porous medium has been extensively investigated over the last few decades,

with implications in many disciplines like chemical transport in soils and fractured rocks [Bear,

2013; Haggerty et al., 2001; Raveh‐Rubin et al., 2015], oil recovery [Edery et al., 2018;

Lenormand et al., 1983], filtration [Tufenkji and Elimelech, 2004], fuel cells [Pharoah et al.,

2006], and even percolation of coffee [Fasano and Talamucci, 2000]. The general advective-

diffusion equation describes Fickian transport in a homogenous porous material, using:

1. 

   ,

where c is the tracer concentration, D is the diffusion constant, v is solvent velocity, and S is the

dissolved source that includes the concentration injected into the field [Koch and Brady, 1987].

However, a natural porous environment is characterized by a heterogeneous structure of the

medium, which leads to non-Fickian transport with a heavy tail for the tracer migration that can

be captured by various models [Berkowitz and Scher, 1997; Cirpka and Kitanidis, 2000; Cushman

and Ginn, 1993; Dullien, 2012; Haggerty et al., 2000; Kang et al., 2011; Le Borgne et al., 2008;

Morales-Casique et al., 2006a; b; Sánchez-Vila and Carrera, 2004; Willmann et al., 2008]. This

transport in a heterogeneous porous media can be regarded as transport within locally homogenous

areas, distributed in a lognormal way, and spatially varying following a correlation length [Gómez-

Hernández and Journel, 1993], while the variance of the distribution marks the field heterogeneity

[Sanchez‐Vila et al., 2006]. This heterogeneity formation allows solving the transport as Fickian

on the local scale while giving rise to a non-Fickian transport in the field scale [Edery, 2021; Edery

et al., 2014]. A persistence outcome of the heterogeneity, both experimentally and numerically, is

the emergent preferential flow paths defined as the tracer's movement in unequal parts through the

porous medium [Cirpka and Kitanidis, 2000; Willmann et al., 2008]. In these preferential flow

paths, the fluid funnels into narrower flow paths with the lowest resistance to flow, and

increasingly forms stagnation areas, or voids, in regions with high resistance [Webb and Anderson,

1996]. This relation between preferential flow and hydraulic conductivity distribution is observed

at the field scale [Bianchi et al., 2011; Edery et al., 2016a; Riva et al., 2010; Riva et al., 2008],

numerically [Fiori and Jankovic, 2012], and even at the pore-scale. Moreover, studies have shown

the importance of preferential flows to the reaction pattern in reactive transport [Edery et al., 2011;

Edery et al., 2015; Edery et al., 2016b; Raveh‐Rubin et al., 2015]; and specifically for the vadose

zone they are related to contaminant distribution in soil [Hagedorn and Bundt, 2002], microbial

communities in soil [Bundt et al., 2001; Morales et al., 2010], and even landslides [Hencher, 2010;

Shao et al., 2015]. Preferential flows pattern starts as a uniform tracer front, which funnels and

splits into distinct flowing branches, sometimes referred to as channel branching [Fiori and

Jankovic, 2012; Liao and Scheidegger, 1969; Moreno and Tsang, 1994; Torelli and Scheidegger,

1972]. As the channel branches, the tracer concentrations vary and sample the conductivities non-

uniformly. This branching is similar to single and multiphase flow at the pore scale [Ferrari et al.,

2015; Yeates et al., 2020], and with a topology analogous to graph theory [Kanavas et al., 2021;

Liao and Scheidegger, 1969; Torelli and Scheidegger, 1972].

We identify the channels as the continuous transport of at least one particle in our particle tracking

(PT) model, while the point at which a channel branch is a bifurcation of the transport. This

bifurcation phenomenon was studied in similar fields, such as small-scale heat transfer bifurcation

in porous media [Yang and Vafai, 2011a; b], and bifurcations in braided rivers [Amooie et al.,

2017; Bolla Pittaluga et al., 2003; Zolezzi et al., 2006]. However, to date, there is no study

characterizing the bifurcation of flow in porous media at the Darcy scale and in the context of

preferential flow paths. This work characterizes the preferential flow patterns and bifurcation on

this Darcy scale transport. We identify the bifurcation points and the stagnant zones (voids) in a

numerical conductivity field ranging from homogenous to heterogeneous. We further show that

the bifurcations decrease from inlet to outlet, reaching an asymptotical value that scales with the

heterogeneity level. We identify a power-law scaling that correlates the bifurcation and void

fractions with the heterogeneity level. Surprisingly, the same power-law scaling with heterogeneity

holds for the transport tortuosity and characteristic fractal dimension.

2. Methodology

We characterize the bifurcation of preferential flows using a set of 2D numerical simulations,

where a second-order stationary random field of conductivities is distributed by a lognormal

distribution with mean  and a variance of  , established by a sequential

Gaussian simulator (GCOSIM3D) [Gómez-Hernández and Journel, 1993]. This conductivity

field is made of 120×300 conductivity bins (each with a size of   ). Each field is

produced by a statistical homogenous and isotropic Gaussian field in the ln(k), with a normalized

correlation length  0.016, where  is the domain length along the main flow direction and

=1 is produced by an exponential covariance. This correlation length leads to   , which

provides an accurate description for the small-scale fluctuations generated by the  field and

advective transport [Ababou et al., 1989; Riva et al., 2009]. One hundred realizations are

produced for each variance with a stability test to verify the mean conductivity for all the results

presented here; see supplementary for details. Each realization had a deterministic pressure drop

translated to the total head drop ( ) imposed from the inlet (left) to the

outlet (right), and a finite element numerical model with Galerkin weighting function calculated

the local head drop in 2D per bin [Guadagnini and Neuman, 1999]. Thus, the streamline for each

realization is retrieved, and from it, the local velocities, given that the porosity θ=0.3. Each PT

realization tracks a pulse of 105 particles. These particles are flux-weighted at the inlet according

to the inlet permeability distribution, and at t=0, the particle pulse advances according to the

local advection and diffusion term, following the Langevin equation:

2. ,

where  is the displacement,  is the particle known location at time  ,  is the fluid

velocity at that location,  

 is the temporal displacement magnitude ( is the modulus of v)

and is the diffusive displacement. The displacement size δs is selected to be an order of

magnitude less than Δ so to interpolate the velocity within each bin correctly. This diffusive

displacement is randomly generated from a normal distribution between 0 to 1 (,

multiplied by the square root of the diffusion coefficient ( 

) representing the

diffusion of ions in water [Domenico and Schwartz] and the displacement magnitude as

illustrated from the following equation:

3. .

The simulations were verified using 106 particles and δs <Δ/10 with no significant numerical

dispersion. Using the same GCOSIM3D model was proved valuable in reproducing and

analyzing field data [Eze et al., 2019; Obi et al., 2020], uncertainty[Ciriello et al., 2013;

Franssen et al., 2004; Riva et al., 2005], and upscaling [Li et al., 2011], while the PT method

proved to be very robust and appropriate in this modeling configuration [Peter Salamon et al.,

2006; P Salamon et al., 2006].

Figure 1. a. Example of particles visitations per bin, where a

heat map represents the particle visitations throughout the

simulation, on a logarithmic scale, white areas mark locations

with no particle’s visitation. i. Example of a single bifurcation

point where there is no visitation of particles in the bifurcation

point and downstream from it, and there are visitations above

and below the bifurcation points. ii. An example of a

bifurcation front, where multiple cells have no visitations. iii.

An example for a single stagnant point which is not defined as

a bifurcation point. b. Combined description of the particle

visitation overlay on the conductivity field of a single

realization with variance of ln(k)= 5, and a color bar for ln(k).

The darker hue describes the flow paths, while the complete

circle marks a bifurcation around a low conductivity, the small,

dashed circles marks a higher-than-average conductivity in

which there is no transport, and the bigger dashed circle marks

a higher-than-average conductivity that funnels the transport.

While defining the bifurcation as the point where a channel branches into two is straightforward,

searching for the bifurcations within the preferential flows is more challenging. We, therefore,

distinguish a set of points where the channel first splits while changing the flow direction

tangentially. Defining the bifurcation point as a cell with transport up-flow from it and no transport

at down-flow from it, yet there is transport transversely to the down-flow direction (see example

in figure 1. a.i). However, in some cases, the bifurcation does not occur at a single cell but rather

at several vertically consecutive cells (= 'bifurcation front') as shown in figure 1.a.ii. The

bifurcation front is typically around 1% of the total bifurcations and a maximum of 8% of the

bifurcations per realization. This work does not consider a bifurcation front since they are

negligible and uncertain in conductivity value. In addition, as shown in figure 1.a.iii, individual

voids points are not defined in this work as bifurcation points, although locally, these are points

that split the flow.

3. Results

In the following, we identify and characterize the bifurcation from inlet to outlet for fields with

varying heterogeneity levels. While many parameters may affect transport on the Darcy scale,

e.g., isotropy, correlation length[Edery, 2021], and heterogeneity [Edery et al., 2014], we focus

on the transition from a homogenous to a heterogenous case; and show how this transition affects

the topology of preferential flows. We demonstrate that a power-law correlates the bifurcation to

heterogeneity. The same power-law correlates the void, the tortuosity, and the fractal dimension

of the preferential flows to the heterogeneity of the field.

While the definition of bifurcations is established for this study, the mechanism leading to these

bifurcations is not. In the simplified case where there is a non-conductive area or a big difference

between conductivities, it is evident that a bifurcation will occur around the obstacle, which makes

it a local event. However, bifurcations mainly occur upstream to the location with the low

conductivity in a way that maximizes the path of least resistance globally. Evidence for this kind

of bifurcation around a low conductive area can be found in Levy and Berkowitz [Levy and

Berkowitz, 2003], where the tracer is branching around low permeable zones, yet some tracer does

enter these low conductive zone, as it serves

the overall minimization of flow through the

global solution of transport as shown in the

recent study by [Zehe et al., 2021]. This

branching of transport is presented in the

supplementary material (figure S1 modified

permission from Elsevier 2003), and the

temporal and spatial evolution of the pulse

tracer exhibits similar behavior as in our

simulations, as shown by [Berkowitz, 2021].

The preferential flows are marked by a

substantial spatial change of the tracer

concentration from inlet to outlet. We

encounter these preferential flows in all

scales, from the pore scale to the field. Yet,

Figure 2. Bifurcation fraction from all the cells along the y-axis, and their

change downflow on the x-axis. Each color represents a different

heterogeneity level, yet all reach an asymptotical value, shown in the inset.

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Bifurcating-Paths:therelationbetweenpreferentialflowbifurcations,void,andtortuosityontheDarcyscale.AviozDagan1,YanivEdery11FacultyofCivilandEnvironmentalEngineering,Technion,Haifa,IsraelAbstractInrecentyears,DarcyscaletransportinporousmediawascharacterizedtobeFickianornon-Fickianduetothehomogeneityo...

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Bifurcating -Paths the relation between preferential flow bifurcations void and tortuosity on the Darcy scale . Avioz Dagan1 Yaniv Edery1

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