Math Geosci (2012) 44:133–145DOI 10.1007/s11004-011-9365-2

S P E C I A L I S S U E

On Preferential Flow, Channeling and Connectivityin Heterogeneous Porous Formations

A. Fiori · I. Jankovic

Received: 19 January 2011 / Accepted: 31 August 2011 / Published online: 27 October 2011© International Association for Mathematical Geosciences 2011

Abstract This paper analyzes the emergence of channeling and preferential flow inheterogeneous porous media. Connectivity is studied through the statistical character-ization of the length L of connected, high velocity patterns in both two-dimensionaland three-dimensional media. A simple, physically based, fully analytic expressionfor the probability of L has been derived. It is found that the length L of connected,high velocity channels is flow-related and can be much larger than the conductivityintegral scale I . Heterogeneity has a considerable impact on emergence of channel-ing patterns; connectivity is considerably enhanced in three-dimensional structuresas compared to two-dimensional ones. The strong dependence on space dimension-ality is a warning against the use of two-dimensional numerical models for assessingconnectivity and preferential flow in heterogeneous media. The probability p(L) isemployed in order to determine the early arrivals of the breakthrough curve at a givencontrol plane; the simple model can be used for a preliminary assessment of preferen-tial flow. Comparison with numerical simulations confirms that the main connectivityfeatures were adequately captured by the model.

Keywords Connectivity · Preferential flow · Channeling · Heterogeneous media ·Random conductivity · Porous media · Stochastic processes

1 Introduction

Solute transport in heterogeneous porous formations has been a subject of intensiveresearch in the last three decades. The fundamental role played by the spatial het-

A. Fiori (�)Università di Roma Tre, Rome, Italye-mail: aldo@uniroma3.it

I. JankovicState University of New York at Buffalo, Buffalo, USA

134 Math Geosci (2012) 44:133–145

erogeneity of the hydraulic properties, and in particular the hydraulic conductivityK , and its impact on transport is by now generally accepted. The classic first-orderanalysis in the logconductivity variance of flow and transport has led to understand-ing of the basic mechanisms which govern transport in heterogeneous media (see,e.g., Dagan 1989; Rubin 2003). However, the first-order analysis is formally validfor low to moderate heterogeneity and, therefore, it cannot model the unique featuresoccurring in strongly heterogeneous aquifers. Among them, we are interested here inthe emergence of connected high velocity channels, which deliver solute in relativelyshort travel times; this feature is sometimes denoted as channeling or connectivity,although the latter often refers to conductivity rather than velocity. The limitations offirst order analysis can be overcame by the use of numerical models, which in turnhave other limitations, e.g., lack of generality, numerical errors for strong hetero-geneities (which generally require a very dense computational grid, with 8 or morepoints per integral scale for large heterogeneity). Also, the majority of studies con-cerning connectivity and its effect on solute transport are performed through two-dimensional numerical simulations, rather than the more realistic three-dimensionalconfigurations. As shown in the sequel, a two-dimensional analysis may seriouslyunderestimate the flow connectivity in heterogeneous formations.

The emergence of preferential flow channels (or channeling) and connectivityis increasingly recognized as an important feature of flow and transport in hetero-geneous systems. Such channels can deliver solute at velocities much larger thanthe mean, with important implications for risk assessment studies and the impacton several transport measures, like e.g. the breakthrough curve at a given controlplane (CP). However, the matter is quite difficult and it still lacks a clear cut defini-tion in quantitative terms. One of the crucial problems is the relation between flowand the underlying K structure, which is difficult to solve without resorting to nu-merical simulations. Several parameters have been proposed in the literature (see,e.g., the comprehensive review by Knudby and Carrera 2005), either related to theconductivity structure or to flow/transport features. The adequacy of such measuresand the relationship between connectivity and the main features of the conductivitystructure has not been fully clarified yet. In most of the previous works channeling isstudied as a consequence of connectivity of the conductivity field, and in particularthe highly conductive areas in two-dimensions (Wen and Gomez-Hernandez 1998;Zinn and Harvey 2003). However, preferential channels may also emerge in me-dia with no identifiable, significant channels of high K , especially when hetero-geneity is strong (Moreno and Tsang 1994; Park et al. 2008; Gotovac et al. 2009;Bianchi et al. 2011). In order to illustrate this behavior, we depict in Fig. 1a ve-locity field and a plume snapshot calculated by accurate numerical simulations of ahighly heterogeneous, two-dimensional stationary K field, with finite integral scaleand without imposing connectivity of particular classes of high K . The presence ofvariously distributed high velocity channels is clearly visible; the flow-related pref-erential channels emerge naturally from the system. Clearly, the presence of spatialconnections of high-K areas may further enhance the emergence of preferential chan-nels (see e.g. Bellin et al. 1996) which however would anyway self-organize in theporous system when heterogeneity is strong.

The scope of the present work is to derive a simple mathematical description ofchanneling by studying the distribution of the length of preferential channels in two-

Math Geosci (2012) 44:133–145 135

Fig. 1 Snapshot of a plume attime tU/I = 30; thetwo-dimensional flow field isgenerated numerically by theanalytic element method;log-conductivity is normallydistributed, with varianceσ 2Y

= 4. Lines representflowlines; convergence offlowlines show the preferentialchannels; the rectangle near theleft border is the plume locationat t = 0

and three-dimensional heterogeneous porous systems. The focus of the work is inflow-related channeling for a simple conductivity structure of stationary K of finiteintegral scale, without imposing a-priori connectivity patterns. One of the objectivesis to show that there is no need to impose particular patterns of K connectivity to getstrong preferential flow; the latter emerges naturally in highly heterogeneous forma-tions. As previously stated, the presence of strongly connected K areas will likelyincrease preferential flow, but this issue shall not be explored in the present work.

We seek for the simplest possible formulation of flow connectivity, with a clearlink to the conductivity structure, in order to gain understanding of the main phys-ical processes which determine channeling. A more comprehensive numerical in-vestigation of some of the transport features analyzed here can be made by usingthe semi-analytical model that we developed in the last years (Dagan et al. 2003;Fiori et al. 2003; Jankovic et al. 2006; Fiori et al. 2010), which is formally valid forany degree of heterogeneity and was tested against accurate numerical simulations;the model is, however, more complex than the present analysis.

The plan of the paper is as follows. The mathematical formulation is presentedfirst, with the conceptual model for the conductivity and the approximated flowmodel. The distribution of the length of preferential channels and its relation withthe breakthrough curve are derived next and results are compared with numericalsimulations. A set of conclusions closes the paper.

2 Mathematical Formulation

The proposed method is based on a simplified formulation for flow in heterogeneousformations. Such formulation builds on previous work by the same authors, as dis-cussed in the sequel. Although simplified, the flow solution is physically based and

136 Math Geosci (2012) 44:133–145

it derives from the heterogeneous K field. In particular, it is able to handle any de-gree of heterogeneity (epitomized by the log-conductivity variance σ 2

Y ), well beyondthe validity of the standard first order analysis; this point is crucial when dealing withconnectivity and preferential flow, which are highly nonlinear processes. All the stepsof the analytical procedure have been carefully tested in the past, as discussed in thesequel.

2.1 The Medium Structure and the Approximate Flow Solution

The porous medium is a stationary random log-conductivity (Y = lnK) field, of mean〈Y 〉 = lnKG (the geometric mean), variance σ 2

Y , and isotropic two-point covarianceCY (x1,x2) = σ 2

Y ρ(|x1 − x2|). Although the procedure can be formally extended toanisotropic structures, we focus here on the simpler isotropic case, as most of theconclusions of this work do not depend on anisotropy. The autocorrelation ρ has a fi-nite linear integral scale I = ∫ ∞

0 ρ(x′,0,0) dx′, which is assumed to be much smallerthan the characteristic length scale of the flow domain and of the solute plume.

The log-conductivity field is usually represented in the numerical models by a reg-ular arrangement of voxels, i.e., small cubes or squares of size much smaller than I .Along the approach pursued by Fiori et al. (2007), we simplify the above schemeby replacing the voxels by large blocks (cubes for three-dimensional, squares fortwo-dimensional) of side 2 I of independent hydraulic conductivity K . This leads toconsiderable simplifications, since the structure is completely defined by I and by theunivariate pdf f (K). The above setup implicitly assumes a linear ρ. However, fol-lowing the first-order analysis (Dagan 1989), the exact shape of ρ does not influenceresults by a significant amount. The issue will be revisited later in this manuscriptby comparing the analytic results with those based on a numerical simulation witha different shape of ρ. While this simplified structure is idealized, it can match anyisotropic random medium of given univariate f (Y ) and integral scale I in statisti-cal sense up to second order (Dagan et al. 2003). The model definitely serves ourmain purpose which is to gain understanding of connectivity and preferential flow inheterogeneous porous formations.

In order to compute a complete solution of flow driven by mean uniform ve-locity U , the total velocity is expressed as a sum of perturbations associated witheach block, each contribution being expanded in turn in an infinite series. Instead,we adopt here the self-consistent approximation, which was extensively analyzedand tested against accurate, large-scale numerical simulations, for both Gaussian andnon-gaussian f (Y ) (Jankovic et al. 2003b; Fiori et al. 2006, 2007). Furthermore, weretain the dominant term, namely the interior velocity component V , aligned withmean U . Adopting the simple expressions pertinent to isolated inclusions of circu-lar (two-dimensions) or spherical (three-dimensions) shape (see Fiori et al. 2007) toblocks, the following expressions apply

V

U= 2κ

1 + κ(2D)

(1)V

U= 3κ

2 + κ(3D)

Math Geosci (2012) 44:133–145 137

where κ = K/Kef. The effective conductivity Kef is equal to Kef = KG for two-dimensional fields; for three-dimensional flows it can be derived through the self-consistent approach, as a solution of the integral equation

∫(Kef − K)/(2Kef +

K)f (K)dK = 0, where f (K) is the pdf of K (Dagan 1989). The solution (1) corre-sponds to the interior velocity of an isolated circular (two-dimensional) or spherical(three-dimensional) inclusion of conductivity K submerged in a homogeneous matrixof conductivity Kef. Jankovic et al. (2003a) showed by detailed numerical simulationsthat the solution (1) approximates accurately the leading order term of velocity insideeach inclusion. Hence, the velocity pertaining to each block of logconductivity Y willbe described here by expressions (1). This formulation greatly simplifies the deriva-tions by providing a unique relation between hydraulic conductivity and velocity,which is the prerequisite for a simple analysis of flow/transport connectivity.

2.2 Analysis of Connected Elements with Given Velocity Threshold

We study channeling by analyzing the occurrence of strings of length L of connectedblocks which are characterized by a velocity larger than or equal to a given thresholdV ∗. The results depend therefore on the assigned threshold V ∗, and the main interesthere is in fast, preferential flow, i.e., for V ∗ > U . For the ease of referencing theblocks with V ≥ V ∗ shall be referred to as “fast” blocks.

We denote Π as the probability that a fast block has at least one adjacent fast blockalong the direction of flow. Since the conductivities of the blocks are uncorrelated,the probability pn of having a string (not necessarily straight) of n subsequent fastblocks is equal to p(n) = Πn−1(1−Π). Since each block has a size 2I , the horizontaldistance L of the connected path, with velocity larger than the threshold, is equalto L = 2In. Thus, the probability p(L) of length of connected fast blocks is givenby

p(L) = ΠL/(2I )−1(1 − Π) (2)

The mean and variance of L result in

〈L〉I

=∞∑

L=2I

Lp(L) = 2

1 − Π

(3)σ 2

L

I 2=

∞∑

L=2I

L2p(L) − 〈n〉2 = 4Π

(1 − Π)2

Hence, the crux of the matter is the calculation of the probability Π , which is per-formed along the following lines. Each block is adjacent to a downstream number N

of elements along the flow direction, and both water and solute can move from thisblock into any of N adjacent blocks. In a regular arrangement of blocks, the num-ber N of adjacent elements depends on space dimensionality, being N = 9 for cubes(three-dimensional) and N = 3 for squares (two-dimensional); we neglect possibleminor contribution of lateral blocks.

With the above definitions, the probability of presence of at least one adjacent fastblock is the total probability Π = prob(E1 ∪E1 ∪· · ·EN), where Ei is the probability

138 Math Geosci (2012) 44:133–145

of finding an adjacent fast block. Because of the relation (1), the probability Ei iseasily determined by

Ei = E = prob(V ≥ V ∗) = 1 − PK

(V ∗KG

2U − V ∗

)

(2D)

(4)

Ei = E = prob(V ≥ V ∗) = 1 − PK

(2V ∗Kef

3U − V ∗

)

(3D)

with PK the CDF of K .The total probability prob(E1 ∪E1 ∪· · ·EN) is calculated as follows. Starting with

two adjacent blocks, the probability q2 that at least one is “fast” (i.e., V ≥ V ∗) is

q2 = prob(E1 ∪ E2) = prob(E1) + prob(E2) − prob(E1 ∩ E2) = 2E − E2 (5)

The procedure is extended to an arbitrary number i of adjacent blocks, with the fol-lowing iterative procedure

q1 = E(6)

qi+1 = qi + E − qiE

and Π is simply given by

Π = qN (7)

Application of the above iterative procedure for two- and three-dimensional for-mations leads to the following expressions for Π

Π = 3E − 3E2 + E3 (2D)(8)

Π = 9E − 36E2 + 84E3 − 126E4 + 126E5 − 84E6 + 36E7 − 9E8 + E9 (3D)

The behavior of the above expression is illustrated in Fig. 2; the effect of the differentΠ on channeling shall be discussed later.

Summarizing, expressions (2), (8) allow us to compute the probability of thelength L of strings of connected, “fast” elements, characterized by velocities larger

Fig. 2 The function Π (see (8))as function of the probability E

(4), for both two- andthree-dimensional flows

Math Geosci (2012) 44:133–145 139

than a given threshold velocity V ∗. The needed prerequisites for the connectivityanalysis are therefore the conductivity distribution f (K), the conductivity integralscale I and the mean velocity U .

3 Application to Lognormal Hydraulic Conductivity

We illustrate here a few result for the case of lognormal f (K), defined by KG and σ 2Y

(the logconductivity variance). Figures 3a,b depict the probability p(L) as functionof dimensionless distance L/I for a few values of the log-conductivity variance σ 2

Y ,for both two-dimensional (Fig. 3a) and three-dimensional (Fig. 3b) flows. The resultsare obtained for threshold velocity V ∗ = 1.5U ; hence, we focus on the distribution ofconnected patterns conveying water and solute with velocity at least 50% faster thanthe mean.

The first feature which is observed is the strong dependence of p(L) on σ 2Y , which

reflects the degree of heterogeneity, for both two-dimensional and three-dimensionalflows. It is seen that the probability of the occurrence of connected, high velocitychannels is greatly enhanced by heterogeneity, which generally increases the prob-ability of finding neighboring high velocity zones. Thus, p(L) tends to be broader

Fig. 3 The probability p(L)

(see (2)) of occurrence of achannel of length L withvelocity larger than threshold(V ≥ V ∗), for a few values ofthe log-conductivity varianceσ 2Y

; (a) two-dimensional and(b) three-dimensional flows

(a)

(b)

140 Math Geosci (2012) 44:133–145

with increasing σ 2Y , for both two- and three-dimensional flows. For low or moder-

ate heterogeneity the degree of connectivity is very weak, as visible in the narrowerprobability p(L) for small σ 2

Y .The degree of connectivity is also strongly influenced by the space dimensional-

ity. In particular, the occurrence of long, high velocity channels is much stronger inthree-dimensions than in two-dimensions, as visible in the Figs. 3a,b. The increase ofconnectivity in three-dimensional flows is caused by two different mechanisms. Thefirst is the larger probability to find an adjacent fast block in three-dimensions for agiven PK(V ∗), because of the increased number of adjacent blocks N . The features isobserved in Fig. 2 which depicts Π (8) as function of E (4), for both two- and three-dimensions. That is, for a given exceedence probability E it is more likely to find anadjacent block of high velocity in three-dimensional than in two-dimensional flow.The second mechanism is the general increase of E with V ∗ in three-dimensionalflow, for a given conductivity distribution fK. In fact, velocities are generally largerin three-dimensions for high values of the hydraulic conductivity, as shown by (1).The combination of both mechanisms leads to a much stronger connectivity of hetero-geneous three-dimensional flows, as compared to two-dimensional fields with samedistribution of hydraulic conductivity. This conclusion, which is similar to that ofFogg (2010) obtained by a different analysis, acts as a warning against analysis onconnectivity based on two-dimensional models, e.g., using two-dimensional numeri-cal simulations.

The above features are recovered in synthetic results presented in Fig. 4, whichdepicts the mean 〈L〉 and coefficient of variation CV (L) of the length L of connectedhigh velocity channels, as function of σ 2

Y for different space dimensions (two- andthree-dimensions). V ∗ = 1.5U is again assumed; although the results and conclusionsare similar for other velocity thresholds. The striking impact of space dimensionalityis visible; the mean connected length for three-dimensional flows can be one order ofmagnitude larger than in two dimensions. The CV , which is typically close to unity,indicates that the length of preferential channels in the system is highly variable, ofthe same order as the mean length.

In conclusion, stationary random conductivity fields of finite integral scale maylead to occurrence of very long high velocity channels, which generally depend onthe degree of heterogeneity and space dimensionality of the problem. The spatial con-nectivity patterns that emerge in heterogeneous systems are mainly related to flow,

Fig. 4 The mean 〈L〉 andcoefficient of variation CV (L)

of the channel length L withvelocity larger than threshold(V ≥ V ∗) as function of thelog-conductivity variance σ 2

Y,

for both two- andthree-dimensional flows

Math Geosci (2012) 44:133–145 141

since hydraulic conductivity displays a finite correlation length. The latter alone can-not predict connected channels that are tens of conductivity integral scales long, asthose displayed on Fig. 4. Solute transported in such systems may be conveyed atrelatively fast velocities, leading to strong preferential flow in certain situations. Thisissue is further discussed in the next section.

4 Implications on Solute Transport: Early Arrivals, Relation with the BTC andComparison with Numerical Simulations

The emergence of channels characterized by velocities larger than the mean mayhave an important impact on solute transport, and in particular on the early arrivalsat a given control plane. We consider mean flow in the x direction and a large so-lute plume injected at x = 0, t = 0. Thus, with solute transport characterized by thebreakthrough curve (BTC) at a control plane at distance x from the source, the lead-ing edge of the curve is determined by the early arrivals, which in turn depend onthe connectivity patterns in the porous medium. We seek in the following a relationbetween the connectivity structure previously derived and the early arrivals.

With M0 the total mass of solute injected in the system, part of it (M∗0 ) is injected

in areas where V ≥ V ∗. The solute is then transported along the connected channels,if any. The fraction of mass transported by such channels to the control plane at x isequal to the proportion of channels characterized by a length L ≥ x. The travel time τ

in the generic channel to distance x is roughly τ ≤ 〈τ 〉(U/V ∗), where 〈τ 〉 is the meantravel time; we assume here that solute is injected proportional to the local flux, suchthat 〈τ 〉 = x/U (see Jankovic and Fiori 2010, on the influence of injection mode ontransport). Thus, the mass which arrives at the control plane at time τ ≤ 〈τ 〉(U/V ∗)is calculated as follows

M = M∗0

[1 − P(x)

](9)

where P(x) is the cumulated probability of p(L), i.e. P(x) = ∑xL=2I p(L). The

mass M∗0 depends on the injection mode, either resident or flux-proportional

(Jankovic and Fiori 2010). For the latter case, which is the one employed in thenumerical simulations, M∗

0 is easily calculated along the lines of Fiori et al. (2006),from the pdf of κ = K/Kef, as follows

M∗0 = M0

∫ ∞

κ∗V (κ)

Ufκ(κ) dκ

(V

(κ∗) = V ∗) (10)

The latter expresses the injected mass M0 weighted over the local velocity at theinjection plane (flux proportional condition is considered), and only the conductivitycontrasts κ such that V ≥ V ∗ are considered. For resident concentration injectionmode (not employed here), it is M∗

0 = M0∫ ∞κ∗ fκ(κ) dκ .

Summarizing, expression (9) provides the mass of solute arriving at the controlplane before t = 〈τ 〉(U/V ∗), or equivalently the integral of the BTC between t = 0and t = 〈τ 〉(U/V ∗). A similar quantity was employed in previous studies for assess-ing connectivity (e.g., Knudby and Carrera 2005). The analytical relations given here

142 Math Geosci (2012) 44:133–145

provide an unique link between the early arrivals of solute and the distribution of theconnected, high-velocity channels.

For the sake of illustration we depict and compare the above formulation to the re-sults of detailed large-scale, three-dimensional numerical simulations. The numericalmethodology is the same as the one employed in our previous work, based on the ana-lytic element method, and explained in detail in Jankovic et al. (2006). The numericalsimulations are performed with an ensemble of 100,000 densely packed spherical in-clusions of volume fraction φ = 0.7, for injection of 40,000 equally-spaced soluteparticles. The selected background conductivity was the effective one. It is remindedhere that the numerical procedure is characterized by a very high precision, whichenables handling flow and transport problems in heterogeneous media of arbitrarilyhigh heterogeneity.

Comparison between the present, analytical model and the numerical simulationsis not straightforward, as there are a few differences between the two setups. First, thenumerical simulations consider spherical inclusions at near-maximal packing, withvolume fraction φ = 0.7, while the analytical setup was based on space-filling (φ = 1)cubical blocks. The conductivity autocorrelation ρ is also different in the two mod-els, semispherical for the numerical simulations and linear in the present theoreticalanalysis. Hence, a perfect match between results cannot be expected, the main aimof the comparison being nevertheless to check the consistency and efficiency of theanalytical model. Although it is physically based, the proposed model is underlain bya few approximations which have to be tested.

In order to make the comparison possible, we need to reproduce the volume frac-tion φ = 0.7 which was employed in the numerical simulations. This is achieved in asimple way by expanding the spatial coordinates of the analytical model by the factorφ−1/3, keeping the same integral scale I , such to recover the volume fraction φ ofinclusions of the numerical model.

Figures 5a,b show the relative mass (i.e. divided by M0) which arrives at the con-trol plane at times t ≤ 〈τ 〉/1.5 (Fig. 5a) and t ≤ 〈τ 〉/2 (Fig. 5b) as function of thecontrol plane distance x/I , for a few values of the log-conductivity variance σ 2

Y . It isseen that the numerical results are reproduced reasonably by the analytical solution,despite its extreme simplicity and the structural differences between the two setups.The crucial role played by heterogeneity is visible; more and longer connected chan-nels emerge for increasing heterogeneity, leading to stronger preferential flow andmore mass arriving at the control plane, ahead of mean arrival time 〈τ 〉. The proba-bility of occurrence of channels larger than x decreases with increasing control planedistance x, i.e., a lesser portion of the total mass conveyed in such preferential chan-nels.

5 Summary and Conclusions

The present study deals with the emergence of channeling and preferential flow inheterogeneous porous media. The analysis takes advantage of the self-consistentmodel of flow and transport in highly heterogeneous formations that we developedrecently (Dagan et al. 2003; Fiori et al. 2003; Jankovic et al. 2006). The hydraulicconductivity is modeled as a stationary random function of finite integral scale I .

Math Geosci (2012) 44:133–145 143

(a)

(b)

Fig. 5 Relative mass arriving at the control plane before t = x/V ∗ as function of the control plane dis-tance x, for a few values of the log-conductivity variance σ 2

Y. Theoretical model (see (9)): solid lines;

Numerical results: dots. Threshold values are equal to V ∗/U = 1.5 (a) and V ∗/U = 2 (b)

The study focused on the statistical characterization of the length L of connected,high velocity patterns in both two- and three-dimensional media. A connected struc-ture is defined as the one in which the velocity is larger or equal than a threshold

144 Math Geosci (2012) 44:133–145

velocity V ∗, typically larger than the mean velocity U . A simple, fully analytic ex-pression for the probability of L has been derived. The probability depends on basicflow and structural parameters, namely the mean velocity U , the hydraulic conduc-tivity pdf and its integral scale I . Although the conceptual setup is simplified, themodel is physically based, with a clear and unique relation between hydraulic con-ductivity and flow. The analytical solution can serve as a tool for a first assessment ofconnectivity and preferential flow.

The length L of connected, high velocity channels is flow-related and can be muchlarger than I . It depends to a large extent on the degree of heterogeneity, i.e. onthe log-conductivity variance σ 2

Y . When the latter is small (σ 2Y < 1) the probability

p(L) is rather narrow and connectivity is poor; conversely, for large σ 2Y the degree of

connectivity increases, with broader p(L) and larger mean 〈L〉. Hence, heterogeneityhas a considerable impact on emergence of channeling patterns.

Another important factor which strongly impacts connectivity is space dimen-sionality. Our analysis shows that connectivity is considerably enhanced in three-dimensional structures as compared to two-dimensional ones. The difference is at-tributed to two factors: (1) the increased degrees of freedom and (2) the larger ve-locities pertaining to three-dimensional flows. The strong dependence on space di-mensionality is a warning against the use of two-dimensional models for assessingconnectivity and preferential flow in heterogeneous media; although they are moreconvenient than three-dimensional models, they can seriously underestimate the de-gree of connectivity in a system.

The probability p(L) is employed in order to determine the early arrivals of theBreakthrough Curve (BTC) at a given control plane. In particular, the solute arrivingat the control plane at time t ≤ 〈τ 〉/V ∗, where 〈τ 〉 is the mean travel time, can beeasily predicted by the knowledge of p(L). The simple model can be used for apreliminary assessment of preferential flow.

We have compared the analytical model developed here with numerical simula-tions, which are free from most of the assumed approximations. The comparison isreasonably good, approximations and differences notwithstanding. The main featureswere adequately captured by the model, which is physically based and does not re-quire any fitting procedure.

It is believed that the connectivity analysis proposed here is quite useful in captur-ing the main features of channeling and connectivity and their emergence in hetero-geneous porous media. Although the model is extremely simple, it is able to link theconductivity structure and the velocity field in a consistent and simple way. While amore accurate analysis is possible, e.g., through the complete self-consistent model,the results presented here are of significance because of their simplicity and abilitythe capture main effects with a parsimonious model.

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