PUBLICATIONS Water Resources Research TECHNICAL REPORTS: METHODS 10.1002/2014WR015283

Delineation of connectivity structures in 2-D heterogeneous hydraulic conductivity fields Alina R. Tyukhova1, Wolfgang Kinzelbach1, and Matthias Willmann1

Key Points: Numerical algorithm to delineate connected channels without ﬂow and transport simulation Channels provide main ﬂux and early solute arrival

Correspondence to: A. Tyukhova, [email protected]

Citation: Tyukhova, A. R., W. Kinzelbach, and M. Willmann (2015), Delineation of connectivity structures in 2-D heterogeneous hydraulic conductivity ﬁelds, Water Resour. Res., 51, doi:10.1002/2014WR015283.

1

Institute of Environmental Engineering, ETH Z€ urich, Switzerland

Abstract Connectivity is a critical aquifer property controlling anomalous transport behavior at large scales. But connectivity cannot be easily deﬁned in a continuous ﬁeld based on information of the hydraulic conductivity alone. We conceptualize it as a connecting structure—a connected subset of a continuous hydraulic conductivity ﬁeld that consists of paths of least hydraulic resistance. We develop a simple and robust numerical method to delineate the connectivity structure using information of the hydraulic conductivity ﬁeld only. First, the topology of the connectivity structure is determined by ﬁnding the path(s) of least resistance between two opposite boundaries. And second, a series of connectivity structures are created by inﬂating and shrinking the individual channels. Finally, we apply this methodology to different heterogeneous ﬁelds. We show that our method captures the main ﬂow channels as well as the pathways of early time solute arrivals. We ﬁnd our method informative to study connectivity in 2-D heterogeneous hydraulic conductivity ﬁelds.

Received 14 JAN 2014 Accepted 6 JUN 2015 Accepted article online 11 JUN 2015

1. Introduction Spatial heterogeneity of hydraulic conductivity is ubiquitous in nature [Neuman, 1990]. It causes heterogeneity of ﬂow as well as non-Fickian behavior of solute transport. Connectivity crucially inﬂuences anomalous transport but its inﬂuence could not yet been quantiﬁed [Zinn and Harvey, 2003; Knudby and Carrera, 2006; Willmann et al., 2008; Bianchi et al., 2011; Pedretti, 2013, 2014; Zhang et al., 2013]. Many deﬁnitions of connectivity exist [e.g., Renard and Allard, 2013; Knudby and Carrera, 2006; Knudby et al., 2006; Fernandez-Garcia et al., 2010]. They are separated into two groups: static and dynamic connectivity measures. Static measures are functions of the spatial distribution of hydraulic conductivity alone, while dynamic measures represent a hydrodynamic process such as ﬂuid ﬂow or solute transport. Static measures are ideal to relate material properties and effective transport parameters directly. An ad hoc deﬁnition of connectivity for continuous ﬁelds is based on one or more paths of the smallest resistance between two boundaries. Such paths would form a channel or a system of channels providing the main and the fastest ﬂux [Ronayne and Gorelick, 2006]. These channels arise in various types of ﬁelds [Le Goc et al., 2010] and together with the lowconductivity areas outside the channels cause ﬂux ﬁeld heterogeneity and variability in arrival times, i.e., early solute arrivals [e.g., Zinn and Harvey, 2003; Knudby and Carrera, 2006]. Prediction of such channels prior to ﬂow simulation appears crucial to better understand and ﬁnally to quantify connectivity. We call such connected channels the connectivity structure. The connectivity structure varies in topology (number of ‘‘holes’’) and geometry (thickness and cohesion of channels) depending on the spatial structure of the conductivity ﬁeld. As a hydraulic conductivity set, it can internally be homogeneous as well as very heterogeneous.

C 2015. American Geophysical Union. V

All Rights Reserved.

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Percolation theory is a powerful mathematical framework to study such type of connected structures in porous media [Stauffer and Aharony, 1992; Hunt, 2000; Hunt et al., 2014; Harter, 2005]. It deals with identifying connected structures, the backbone, and critical paths in heterogeneous ﬁelds. As a function of an a priori deﬁned percolation threshold, a connectivity structure, the backbone, can be delineated. It was also used for transport upscaling, for example based on distribution of resistance of the hydraulic conductivity ﬁeld [Hunt, 1998]. Another way to characterize connectivity is to deﬁne some cut-off value of hydraulic conductivity. Edery et al. [2014], however, showed that the cut-off methodology is often insufﬁcient. Both resistivity and thickness of the low-conductive segments affect the entire resistance of the path. The cut-off

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recognizes the resistivity (resistance per unit of volume) but not the thickness, i.e., it does not identify the resistance. This also means the cut-off method may discern a conductive path as nonconductive because the paths include a small part with high resistivity, a bottleneck. This might lead to crucial misconceptions particularly for weakly connected ﬁelds. A better way to deﬁne connectivity would be to deﬁne a path only based on resistivity. For simple geometries such path of smallest resistance can be determined analytically. For more complex geometries, a different approach is required. The main objective of this work is to develop a simple numerical method to delineate highly conductive connectivity structures in heterogeneous hydraulic conductivity ﬁelds. The method takes resistance directly into account. We demonstrate its use by applying it to a strongly connected ﬁeld with a distinct connectivity structure and to a more complex weakly connected ﬁeld.

2. Delineation of the Connectivity Structure We propose a simple numerical method to delineate the connectivity structure of a given hydraulic conductivity ﬁeld by ﬁnding the path(s) of smallest resistance. The procedure is based on two fundamental operations of mathematical morphology: erosion and dilation of a spatial set [Serra, 1982], with some adaptation for continuous ﬁelds. We delineate the path of the smallest resistance that connects two boundaries. Most of the ﬂux is expected to follow this path. Connectivity is an anisotropic material property therefore we deﬁne our connectivity structure in a given direction between two opposite boundaries. The procedure consists of two steps: (1) deﬁnition of the topology of the connectivity structure, i.e., number of holes and connections of the manifold, and (2) rectiﬁcation of its geometry: varying the thickness of channels. 2.1. Extracting the Topological Structure of the Connected Channels The main idea is to bend a line—placed initially at one boundary—according to its underlying resistivity until one or more part of the line reach the opposite boundary. Extracting the topological structure is an iterative process between two boundaries. It starts with a straight line along one boundary. The individual points of this line are then moving toward the other boundary, depending on the inverse of hydraulic conductivity (resistivity) value at that point (Figure 1a). We regard the line as a polygon, i.e., a set of ordered spatial points fðxi ; yi Þgi51;...; N , with initially an equal distance between two neighboring points ﬁxed as value D. Here we use D5cell side=4 to sufﬁciently resolve the curvature of the line. Further reﬁnement increases computation time, but does not improve the result. The points of the line move iteratively in a direction normal to the line. The direction is given at the beginning toward the outlet boundary (Figures 1b and 1c). The length of each individual step is proportional to the hydraulic conductivity of the accommodating cell. Points inside high-conductivity cells make larger steps than points inside low-conductivity cells. In a heterogeneous ﬁeld, the line becomes more and more irregular. We call its movement a ‘‘ front movement.’’ The front develops into the shape of the connectivity structure of the ﬁeld (Figure 1a). At the ﬁrst step, the polygon is a straight line through points (xi, yi). Then it transforms as follows: each point ~i of the bisector of angle i formed by unit normal to the line segments (xi, yi) is moving along the direction n connecting the points i-1 and i and the points i and i11. 0 1 ðyi 2yi21 Þ ðyi 2yi11 Þ ﬃ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C B qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B ðxi 2xi21 Þ2 1ðyi 2yi21 Þ2 ðxi 2xi11 Þ2 1ðyi 2yi11 Þ2 C C 1 B B C ~i 5 (1a) n B C Norm B 2ðxi 2xi21 Þ ðxi 2xi11 Þ C @ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A ðxi 2xi21 Þ2 1ðyi 2yi21 Þ2 ðxi 2xi11 Þ2 1ðyi 2yi11 Þ2 Norm is a normalizing constant to make ni a unit normal vector. Each edge center point (point between two neighboring points of the polygonal line) is moving in direction normal to the edge

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Figure 1. Delineation of the connectivity structure. (a) Front movement in a moderately heterogeneous hydraulic conductivity ﬁeld at three different iterations to delineate the topology. (b) Movement of polygonal points along the bisection and normal direction. (c) Line transformation during one iteration. (d) Self-crossing of the polygonal line and low-conductivity zone extraction. (e) Red polygon represents low-conductivity inclusion. Blue line represents the curve to move further. (f) Illustration of the channels erosion: polygons, bounding the low-conductivity inclusions are growing with respect to the others. The topological structure does not change.

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1 ! n’i 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðxi11 2xi Þ2 1ðyi11 2yi Þ2

2ðyi11 2yi Þ xi11 2xi

! (1b)

The length of the spatial shift of each point is proportional to the hydraulic conductivity Ki of the accommodating grid cell and is limited by a ﬁxed maximum step e to assure that not an entire cell is crossed in one step (Figures 1b and 1c). The maximum value of hydraulic conductivity for the entire line is determined as Kmax5maxKi. As a result, each point is moved by a vector D~ ri 5n~i

Ki e Kmax

(1c)

The line represented by the polygon becomes more irregular and, thus, longer with increasing iterations. This increase in length takes place where the channels form and Dri is large. A constant number of points would lead to a decrease in resolution at the channel boundaries due to it. To avoid this at each iteration, new auxiliary points are placed between two existing points. Using the Ramer-Douglas-Peucker algorithm [Douglas and Peucker, 1973], it is checked whether this auxiliary point is needed for the resolution and the curvature. Only those points are additionally used in subsequent iterations to ensure numerical efﬁciency. The resulting curve is used as the initial curve for the next iteration step. If the curve forms selfintersections, the inner loop contains a low-conductivity zone and is cut off from the curve (Figures 1d and 1e). We indicate self-intersection as intersections of pairs of segments constructing the polygon. The topology and geometry of the low-conductivity inclusions remain ﬁxed. The rest of the curve becomes a new initial curve for the next iteration. Finding low-conductivity inclusions is a key to deﬁning the topology of the channels. Iterations are repeated until the outer curve reaches the opposite boundary. As the procedure is iterative, the front-line moves further into intermediately conductive parts close to the starting boundary. This might lead to dead-ends within our connectivity structure. A dead-end contradicts the deﬁnition of the connectivity structure. Thus, we remove them by performing the connectivity structure delineation twice: from left to right and from right to left, and only the overlap the two is used for as the ﬁnal connectivity structure. For highly connected hydraulic conductivity ﬁelds, this step can be omitted as few iterations are needed to reach the other boundary. Now, we have delineated a topological structure of the connectivity structure of the hydraulic conductivity ﬁeld. The polygon represents the channel network and its counterpart a series of polygons which represent the low-conductive inclusions.

2.2. Erosion and Dilation of the Channels The topological structure of the connectivity structure is now deﬁned but the thickness of the individual channels depends on the least resistance and, thus, the number of iterations. Therefore, we extend our method to allow to increase and shrink the channels by erosion and dilation. Now, we operate with polygons conﬁning the low-conductivity inclusions. We either erode or inﬂate the inclusions. The inﬂation of inclusions should consider other inclusions, to prevent polygons from intersecting and separating channels from occlusion; it conserves the topological structure of the connectivity network. For numerical reasons, the channel thickness cannot be smaller than the grid cell-size. Our algorithm allows setting a minimum thickness for the channels and thus, varying this parameter, one can draw different connectivity structures that occupy a different spatial fraction of the conductivity ﬁeld. The procedure is again iterative. At each iteration, every polygon (line conﬁning a low-conductivity zone) transforms according to a rule similar to the one used above. The fundamental difference is that at each step we check that the distance between the new point and all other polygons is larger than the minimal thickness. The routine is as follows: for all points of all polygons, we ﬁnd the minimum conductivity value Kmin5minKi. All points are moving along the outer normal or bisector of the corresponding angle (1a, 1b). The difference

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is that now points inside the low-conductivity cells are moving a larger distance than those inside highconductivity cells. The size of the step is deﬁned by: Di 5e

Kmin Ki

(2)

Thus, the new point is: ! ~i x! new 5xold 1Di n

(3)

! If x! new is closer to other polygons than the minimum distance, the point xold remains ﬁxed. If not, it moves ! to xnew (Figure 1f). After each movement, we cut off the excess points with Ramer-Douglas-Peucker algorithm. The routine repeats until all the points reach their limits, i.e., remain ﬁxed. An alternative criterion for ﬁxing a point is the conductivity threshold, i.e., a point falling inside a high-conductivity cell does not move any more. Both of these criteria can be applied together. We delineate a series of connectivity structures with different thicknesses of channels or number of ﬁnger (if they reach the opposite boundary at signiﬁcantly different iteration steps) for each individual ﬁeld. These structures occupy different volume fractions of the domain nc.

3. Application to Heterogeneous Hydraulic Conductivity Fields To test our methodology, we apply it to two ﬁelds of different type of connectivity structures: a highly connected ﬁeld where the more permeable parts of the domain are connected; and an intermediately connected ﬁeld where the intermediate values of hydraulic conductivity are best connected. To show that our methodology is able to delineate the clearly distinguishable connectivity structure, we create a strongly connected hydraulic conductivity ﬁeld (SCF) [Zinn and Harvey, 2003]. Due to the almost binary nature of the ﬁeld, the channels can directly and intuitively be observed by looking at the hydraulic conductivity ﬁeld (Figure 2a). After applying the ﬁrst step of our procedure the delineation of the topology, a uniform dense network of equally wide channels is shown (Figure 2b). A qualitative comparison between the original hydraulic conductivity ﬁeld and the delineated channel network shows that the methodology is able to capture all the main features: the number of connections between highly permeable areas (channels) and the number and location of holes, the parts of the domain that are not part of the channels network. The frontline of the individual channels moves almost uniformly and reaches the opposite boundary at about the same iteration. This means they all represent paths of equal resistance. The sizes of the low-conductivity inclusions are of the same scale and represent an almost uniform distribution. After performing the erosion and dilation procedures, we obtained a series of seven connectivity structures with different channel width. The thickest occupies 58%, the thinnest 27% of the domain. The variability in hydraulic conductivity inside the channels is relatively small and decreases with channel width (rY252.34 for the entire domain, rY250.22 for the thickest channel structures, and r250.06 for the thinnest one). The variability stems from lower hydraulic conductivity values at the border of the channel. For the weakly connected ﬁeld (WCF), the connectivity structure is not directly visible from the hydraulic conductivity ﬁeld alone (Figure 3a). The connectivity structure is heterogeneous, different individual channels or ‘‘ﬁngers’’ reach the opposite boundary at different iterations. In contrast to SCF, these channels are of different resistance. Thus, we construct three connectivity structures, consisting of one, two, and four channels based on the series of polygons at the different iteration steps (Figures 3d–3f). To cut off the dead ends, we perform the polygon growth twice: from the right to the left boundary and from the left to the right. The ﬁnal connectivity structure is only the intersection of the two previously obtained structures. The channels consist of both highly conductive segments and the bottlenecks of small conductivity that plug the channels. These bottlenecks have a signiﬁcant inﬂuence on overall resistance of an individual channel. The thickness of the channels also is inﬂuenced by their resistance. To conﬁrm that our connectivity structure really represents the connectivity network, we solve for ﬂow [McDonald and Harbaugh, 1988]. The agreement between the connectivity structure and paths of strongest ﬂux is very good (Figure 3b). Channel 1 which has the least resistance provides the largest ﬂux. As pointed out by Edery et al. [2014], the cut-off approach is not sufﬁcient to characterize connectivity in such ﬁelds due to bottlenecks formed along

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Figure 2. Strongly connected ﬁeld, SCF, and a series of connectivity structures (red) with different width. (a) Original strongly connected hydraulic conductivity ﬁeld (SCF). The connectivity network can be observed directly. (b) Connectivity structures occupying 68% of the domain (nc 5 68%). (c) nc 5 42% (d) nc 5 22%. Note, ﬁelds in Figures 2b–2d represent the main feature of connectedness observed in Figure 2a.

the paths of smallest resistance. In Figure 3c, we see that the cut-off appraoch renders for WCF a channel network that does not reproduce the connectivity pattern reﬂected in the ﬂow ﬁeld. Contrarily the cut-off would render an excllent result for SCF. To illustrate whether the connectivity structure captures the trajectories of the particles with the earliest arrivals, we perform particle tracking [Pollock, 1988]. We compute particle traces within the original ﬁeld and compare the distribution of the arrival times for all particles with the particles whose trajectories are entirely within the connectivity structure. For the SCF, early arrival times for all distributions are about the same, i.e., connectivity structures provide the space for trajectories with the fastest transport pathways (Figure 4a). For WCF, the earliest arrival time is also similar. But the arrival time distributions are wider within the connected structures for WCF than for SCF. Connectivity structures with more than one channel for WCF also provide particles which arrive at intermediate times. The connectivity structure with four channels contains trajectories with arrival times spanning over 1.5 orders of magnitude (Figure 4b). This can be explained by the larger resistance in the fourth channel which leads to a lower advective velocity. Connectivity structures do not contain trajectories with large arrival times which contribute to tailing of the BTC. This was expected as the tail is determined by the low-conductive inclusions. The inclusions are not studied here, but they can also be studied as a result of this method because they are spatial complements to the connectivity structures.

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Figure 3. (a) Weakly connected hydraulic conductivity ﬁeld (WCF). Connectivity is not apparent here. (b) Flow ﬁeld from numerical simulation to illustrate the existing set of connected channels: one major and three minor that generate a network. (c) connectivity pattern derived using a cut-off value of hydraulic conductivity (d–f) series of the connectivity structures (red) Figure 3d one major channel nc57%, Figure 3e two channels, and Figure 3f four main paths.

4. Discussion and Conclusions We present a new numerical method to delineate the connectivity structure of a hydraulic conductivity ﬁeld based on ﬁnding the path of least resistance. The method consists of two major steps: ﬁrst, the topology of the connectivity structure is determined. The connected channels are delineated by moving a line of equal resistance between two opposite boundaries. And second, the individual channels are inﬂated or shrunk based on dilation and erosion. The method is applied to two highly heterogeneous aquifers with different types of connectedness. First, we apply it to a highly connected ﬁeld (SCF). Here, the connectivity structure can be observed directly from the hydraulic conductivity ﬁeld. Comparison with the delineated connectivity structure shows that our

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Figure 4. Histogram of arrival times for all particles and only counting particles with entire trajectories within the connectivity structure. The connectivity structures contain the pathways of the trajectories with the earliest arrival times. (a) SCF: the smaller the thickness of the channels (nc) the fewer are particles with large arrival times. (b) WCF: the more channels are included the larger is the difference in advective times.

method is able to capture the connected channels. For SCF, the channels are fairly homogeneous and all channels arrive at more or less the same iteration. The channels become more heterogeneous with increasing width, but no bottlenecks occur. The channel network for the weakly connected ﬁeld (WCF) is different. The individual channels reach the opposite boundary at different iterations. Therefore, the resistances of the individual channels are different as resistance increases with number of iterations. All the individual channels of the WCF show bottlenecks and are internally variable. Both ﬁelds are highly heterogeneous but their connected channel networks are qualitatively different. For illustration, we simulate ﬂow and transport. We show that ﬂow channels are well preserved using our approach as well as the pathways for particles with earliest arrival times. This indicates that our methodology captures features of dynamic connectivity using only information of hydraulic conductivity ﬁeld. The method presented here is two-dimensional. As connectivity in 3-D behaves fundamentally different, our ﬁndings cannot be directly extrapolated. The algorithm itself could be extended to account for 3-D ﬁelds. In this case, a front-surface of least resistance would move through the medium.

Acknowledgments Financial support by the Swiss National Science Foundation (SNF) through grant 200021 132304 and by Nagra, Wettingen, is gratefully acknowledged. The authors also thank Denis Allard for useful hints, remarks, and inspiration. There is no observation data in the paper.

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References Bianchi, M., C. Zheng, C. Wilson, G. R. Tick, G. Liu, and S. M. Gorelick (2011), Spatial connectivity in a highly heterogeneous aquifer: From cores to preferential ﬂow paths, Water Resour. Res., 47, W05524, doi:10.1029/2009WR008966. Douglas, D. H., and T. K. Peucker (1973), Algorithms for the reduction of the number of points required to represent a digitized line or its caricature, Can. Cartogr., 10(2), 112–122, doi:10.3138/FM57-6770-U75U-7727. Edery, Y., A. Guadagnini, H. Scher, and B. Berkowitz (2014), Origins of anomalous transport in heterogeneous media: Structural and dynamic controls, Water Resour. Res., 50, 1490–1505, doi:10.1002/2013WR015111. Fernandez-Garcia, D., P. Trinchero, and X. Sanchez-Vila (2010), Conditional stochastic mapping of transport connectivity, Water Resour. Res., 46, W10515, doi:10.1029/2009WR008533. Harter, T. (2005), Finite-size analysis of percolation in three-dimensional correlated binary Markov chain random ﬁelds, Phys. Rev. E, 72, 026120. Hunt, A., R. Ewing, and B. Ghanbarian (2014), Percolation Theory for Flow in Porous Media, 3rd ed. [electronic], Cham: Springer International Publishing, doi:10.1007/978-3-319-03771-4. Hunt, A. G. (1998) Upscaling in subsurface transport using cluster statistics of percolation, Transp. Porous Media, 30(2), 177–198. Hunt, A. G. (2000), Percolation cluster statistics and conductivity semi-variograms, Transp. Porous Media, 39, 131–141. Knudby, C., and J. Carrera (2006), On the use of apparent hydraulic diffusivity as an indicator of connectivity, J. Hydrol., 329(3-4), 377–389. Knudby, C., J. Carrera, J. D. Bumgerdner, and G. E. Fogg (2006), Binary upscaling: The role of connectivity and new formula, Adv. Water Resour., 29(4), 590–604. Le Goc, R., J.-R. de Dreuzy, and P. Davy (2010), Statistical characteristics of ﬂow as indicators of channeling in heterogeneous porous and fractured media, Adv. Water Resour., 33(3), 257–269, doi:10.1016/j.advwatres.2009.12.002. McDonald, M. G., and A. W. Harbaugh (1988), A modular three-dimensional ﬁnite-difference ground-water ﬂow model, Department of the Interior, Reston, Va. Neuman, S. P. (1990), Universal scaling of hydraulic conductivities and dispersivities in geologic media, Water Resour. Res., 26(8), 1749–1758. Pedretti, D., D. Fernandez-Garcia, D. Bolster, and X. Sanchez-Vila (2013), On the formation of breakthrough curves tailing during convergent ﬂow tracer tests in three-dimensional heterogeneous aquifers, Water Resour. Res., 49, 4157–4173, doi:10.1002/wrcr.20330. Pedretti, D., D. Fernandez-Garcia, X. Sanchez-Vila, D. Bolster, and D. A. Benson (2014), Apparent directional mass-transfer capacity coefﬁcients in three-dimensional anisotropic heterogeneous aquifers under radial convergent transport, Water Resour. Res., 50, 1205–1224, doi:10.1002/2013WR014578. Pollock, D. W. (1988), Semianalytical computation of path lines for ﬁnite-difference models, Ground Water, 26(6), 743–750.

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Renard, P., and D. Allard (2013), Connectivity metrics for subsurface ﬂow and transport, Adv. Water Resour., 51, 168–196, doi:10.1016/ j.advwatres.2011.12.001. Ronayne, M. J., and S. M. Gorelick (2006), Effective permeability of porous media containing branching channel networks, Phys. Rev. E, 73, 026305, doi:10.1103/PhysRevE.73.026305. Serra, J. (1982), Image Analysis and Mathematical Morphology, Academic, London, U. K. Stauffer, D., and A. Aharony (1992), Introduction to Percolation Theory, 2 ed., Taylor and Francis, London, U. K. Willmann, M., J. Carrera, and X. Sanchez-Vila (2008), Transport upscaling in heterogeneous aquifers: What physical parameters control memory functions?, Water Resour. Res., 44, W12437, doi:10.1029/2007WR006531. Zinn, B., and C. F. Harvey (2003), When good statistical models of aquifer heterogeneity go bad: A comparison of ﬂow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity ﬁelds, Water Resour. Res., 39(3), 1051, doi:10.1029/ 2001WR001146. Zhang, Y., C. T. Green, and G. E. Fogg (2013), The impact of medium architecture of alluvial settings on non-Fickian transport, Adv. Water Resour., 54, 78–99, doi:10.1016/j.advwatres.2013.01.004.

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Delineation of connectivity structures in 2-D heterogeneous hydraulic conductivity fields Alina R. Tyukhova1, Wolfgang Kinzelbach1, and Matthias Willmann1

Key Points: Numerical algorithm to delineate connected channels without ﬂow and transport simulation Channels provide main ﬂux and early solute arrival

Correspondence to: A. Tyukhova, [email protected]

Citation: Tyukhova, A. R., W. Kinzelbach, and M. Willmann (2015), Delineation of connectivity structures in 2-D heterogeneous hydraulic conductivity ﬁelds, Water Resour. Res., 51, doi:10.1002/2014WR015283.

1

Institute of Environmental Engineering, ETH Z€ urich, Switzerland

Abstract Connectivity is a critical aquifer property controlling anomalous transport behavior at large scales. But connectivity cannot be easily deﬁned in a continuous ﬁeld based on information of the hydraulic conductivity alone. We conceptualize it as a connecting structure—a connected subset of a continuous hydraulic conductivity ﬁeld that consists of paths of least hydraulic resistance. We develop a simple and robust numerical method to delineate the connectivity structure using information of the hydraulic conductivity ﬁeld only. First, the topology of the connectivity structure is determined by ﬁnding the path(s) of least resistance between two opposite boundaries. And second, a series of connectivity structures are created by inﬂating and shrinking the individual channels. Finally, we apply this methodology to different heterogeneous ﬁelds. We show that our method captures the main ﬂow channels as well as the pathways of early time solute arrivals. We ﬁnd our method informative to study connectivity in 2-D heterogeneous hydraulic conductivity ﬁelds.

Received 14 JAN 2014 Accepted 6 JUN 2015 Accepted article online 11 JUN 2015

1. Introduction Spatial heterogeneity of hydraulic conductivity is ubiquitous in nature [Neuman, 1990]. It causes heterogeneity of ﬂow as well as non-Fickian behavior of solute transport. Connectivity crucially inﬂuences anomalous transport but its inﬂuence could not yet been quantiﬁed [Zinn and Harvey, 2003; Knudby and Carrera, 2006; Willmann et al., 2008; Bianchi et al., 2011; Pedretti, 2013, 2014; Zhang et al., 2013]. Many deﬁnitions of connectivity exist [e.g., Renard and Allard, 2013; Knudby and Carrera, 2006; Knudby et al., 2006; Fernandez-Garcia et al., 2010]. They are separated into two groups: static and dynamic connectivity measures. Static measures are functions of the spatial distribution of hydraulic conductivity alone, while dynamic measures represent a hydrodynamic process such as ﬂuid ﬂow or solute transport. Static measures are ideal to relate material properties and effective transport parameters directly. An ad hoc deﬁnition of connectivity for continuous ﬁelds is based on one or more paths of the smallest resistance between two boundaries. Such paths would form a channel or a system of channels providing the main and the fastest ﬂux [Ronayne and Gorelick, 2006]. These channels arise in various types of ﬁelds [Le Goc et al., 2010] and together with the lowconductivity areas outside the channels cause ﬂux ﬁeld heterogeneity and variability in arrival times, i.e., early solute arrivals [e.g., Zinn and Harvey, 2003; Knudby and Carrera, 2006]. Prediction of such channels prior to ﬂow simulation appears crucial to better understand and ﬁnally to quantify connectivity. We call such connected channels the connectivity structure. The connectivity structure varies in topology (number of ‘‘holes’’) and geometry (thickness and cohesion of channels) depending on the spatial structure of the conductivity ﬁeld. As a hydraulic conductivity set, it can internally be homogeneous as well as very heterogeneous.

C 2015. American Geophysical Union. V

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Percolation theory is a powerful mathematical framework to study such type of connected structures in porous media [Stauffer and Aharony, 1992; Hunt, 2000; Hunt et al., 2014; Harter, 2005]. It deals with identifying connected structures, the backbone, and critical paths in heterogeneous ﬁelds. As a function of an a priori deﬁned percolation threshold, a connectivity structure, the backbone, can be delineated. It was also used for transport upscaling, for example based on distribution of resistance of the hydraulic conductivity ﬁeld [Hunt, 1998]. Another way to characterize connectivity is to deﬁne some cut-off value of hydraulic conductivity. Edery et al. [2014], however, showed that the cut-off methodology is often insufﬁcient. Both resistivity and thickness of the low-conductive segments affect the entire resistance of the path. The cut-off

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recognizes the resistivity (resistance per unit of volume) but not the thickness, i.e., it does not identify the resistance. This also means the cut-off method may discern a conductive path as nonconductive because the paths include a small part with high resistivity, a bottleneck. This might lead to crucial misconceptions particularly for weakly connected ﬁelds. A better way to deﬁne connectivity would be to deﬁne a path only based on resistivity. For simple geometries such path of smallest resistance can be determined analytically. For more complex geometries, a different approach is required. The main objective of this work is to develop a simple numerical method to delineate highly conductive connectivity structures in heterogeneous hydraulic conductivity ﬁelds. The method takes resistance directly into account. We demonstrate its use by applying it to a strongly connected ﬁeld with a distinct connectivity structure and to a more complex weakly connected ﬁeld.

2. Delineation of the Connectivity Structure We propose a simple numerical method to delineate the connectivity structure of a given hydraulic conductivity ﬁeld by ﬁnding the path(s) of smallest resistance. The procedure is based on two fundamental operations of mathematical morphology: erosion and dilation of a spatial set [Serra, 1982], with some adaptation for continuous ﬁelds. We delineate the path of the smallest resistance that connects two boundaries. Most of the ﬂux is expected to follow this path. Connectivity is an anisotropic material property therefore we deﬁne our connectivity structure in a given direction between two opposite boundaries. The procedure consists of two steps: (1) deﬁnition of the topology of the connectivity structure, i.e., number of holes and connections of the manifold, and (2) rectiﬁcation of its geometry: varying the thickness of channels. 2.1. Extracting the Topological Structure of the Connected Channels The main idea is to bend a line—placed initially at one boundary—according to its underlying resistivity until one or more part of the line reach the opposite boundary. Extracting the topological structure is an iterative process between two boundaries. It starts with a straight line along one boundary. The individual points of this line are then moving toward the other boundary, depending on the inverse of hydraulic conductivity (resistivity) value at that point (Figure 1a). We regard the line as a polygon, i.e., a set of ordered spatial points fðxi ; yi Þgi51;...; N , with initially an equal distance between two neighboring points ﬁxed as value D. Here we use D5cell side=4 to sufﬁciently resolve the curvature of the line. Further reﬁnement increases computation time, but does not improve the result. The points of the line move iteratively in a direction normal to the line. The direction is given at the beginning toward the outlet boundary (Figures 1b and 1c). The length of each individual step is proportional to the hydraulic conductivity of the accommodating cell. Points inside high-conductivity cells make larger steps than points inside low-conductivity cells. In a heterogeneous ﬁeld, the line becomes more and more irregular. We call its movement a ‘‘ front movement.’’ The front develops into the shape of the connectivity structure of the ﬁeld (Figure 1a). At the ﬁrst step, the polygon is a straight line through points (xi, yi). Then it transforms as follows: each point ~i of the bisector of angle i formed by unit normal to the line segments (xi, yi) is moving along the direction n connecting the points i-1 and i and the points i and i11. 0 1 ðyi 2yi21 Þ ðyi 2yi11 Þ ﬃ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C B qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B ðxi 2xi21 Þ2 1ðyi 2yi21 Þ2 ðxi 2xi11 Þ2 1ðyi 2yi11 Þ2 C C 1 B B C ~i 5 (1a) n B C Norm B 2ðxi 2xi21 Þ ðxi 2xi11 Þ C @ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A ðxi 2xi21 Þ2 1ðyi 2yi21 Þ2 ðxi 2xi11 Þ2 1ðyi 2yi11 Þ2 Norm is a normalizing constant to make ni a unit normal vector. Each edge center point (point between two neighboring points of the polygonal line) is moving in direction normal to the edge

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Figure 1. Delineation of the connectivity structure. (a) Front movement in a moderately heterogeneous hydraulic conductivity ﬁeld at three different iterations to delineate the topology. (b) Movement of polygonal points along the bisection and normal direction. (c) Line transformation during one iteration. (d) Self-crossing of the polygonal line and low-conductivity zone extraction. (e) Red polygon represents low-conductivity inclusion. Blue line represents the curve to move further. (f) Illustration of the channels erosion: polygons, bounding the low-conductivity inclusions are growing with respect to the others. The topological structure does not change.

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1 ! n’i 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðxi11 2xi Þ2 1ðyi11 2yi Þ2

2ðyi11 2yi Þ xi11 2xi

! (1b)

The length of the spatial shift of each point is proportional to the hydraulic conductivity Ki of the accommodating grid cell and is limited by a ﬁxed maximum step e to assure that not an entire cell is crossed in one step (Figures 1b and 1c). The maximum value of hydraulic conductivity for the entire line is determined as Kmax5maxKi. As a result, each point is moved by a vector D~ ri 5n~i

Ki e Kmax

(1c)

The line represented by the polygon becomes more irregular and, thus, longer with increasing iterations. This increase in length takes place where the channels form and Dri is large. A constant number of points would lead to a decrease in resolution at the channel boundaries due to it. To avoid this at each iteration, new auxiliary points are placed between two existing points. Using the Ramer-Douglas-Peucker algorithm [Douglas and Peucker, 1973], it is checked whether this auxiliary point is needed for the resolution and the curvature. Only those points are additionally used in subsequent iterations to ensure numerical efﬁciency. The resulting curve is used as the initial curve for the next iteration step. If the curve forms selfintersections, the inner loop contains a low-conductivity zone and is cut off from the curve (Figures 1d and 1e). We indicate self-intersection as intersections of pairs of segments constructing the polygon. The topology and geometry of the low-conductivity inclusions remain ﬁxed. The rest of the curve becomes a new initial curve for the next iteration. Finding low-conductivity inclusions is a key to deﬁning the topology of the channels. Iterations are repeated until the outer curve reaches the opposite boundary. As the procedure is iterative, the front-line moves further into intermediately conductive parts close to the starting boundary. This might lead to dead-ends within our connectivity structure. A dead-end contradicts the deﬁnition of the connectivity structure. Thus, we remove them by performing the connectivity structure delineation twice: from left to right and from right to left, and only the overlap the two is used for as the ﬁnal connectivity structure. For highly connected hydraulic conductivity ﬁelds, this step can be omitted as few iterations are needed to reach the other boundary. Now, we have delineated a topological structure of the connectivity structure of the hydraulic conductivity ﬁeld. The polygon represents the channel network and its counterpart a series of polygons which represent the low-conductive inclusions.

2.2. Erosion and Dilation of the Channels The topological structure of the connectivity structure is now deﬁned but the thickness of the individual channels depends on the least resistance and, thus, the number of iterations. Therefore, we extend our method to allow to increase and shrink the channels by erosion and dilation. Now, we operate with polygons conﬁning the low-conductivity inclusions. We either erode or inﬂate the inclusions. The inﬂation of inclusions should consider other inclusions, to prevent polygons from intersecting and separating channels from occlusion; it conserves the topological structure of the connectivity network. For numerical reasons, the channel thickness cannot be smaller than the grid cell-size. Our algorithm allows setting a minimum thickness for the channels and thus, varying this parameter, one can draw different connectivity structures that occupy a different spatial fraction of the conductivity ﬁeld. The procedure is again iterative. At each iteration, every polygon (line conﬁning a low-conductivity zone) transforms according to a rule similar to the one used above. The fundamental difference is that at each step we check that the distance between the new point and all other polygons is larger than the minimal thickness. The routine is as follows: for all points of all polygons, we ﬁnd the minimum conductivity value Kmin5minKi. All points are moving along the outer normal or bisector of the corresponding angle (1a, 1b). The difference

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is that now points inside the low-conductivity cells are moving a larger distance than those inside highconductivity cells. The size of the step is deﬁned by: Di 5e

Kmin Ki

(2)

Thus, the new point is: ! ~i x! new 5xold 1Di n

(3)

! If x! new is closer to other polygons than the minimum distance, the point xold remains ﬁxed. If not, it moves ! to xnew (Figure 1f). After each movement, we cut off the excess points with Ramer-Douglas-Peucker algorithm. The routine repeats until all the points reach their limits, i.e., remain ﬁxed. An alternative criterion for ﬁxing a point is the conductivity threshold, i.e., a point falling inside a high-conductivity cell does not move any more. Both of these criteria can be applied together. We delineate a series of connectivity structures with different thicknesses of channels or number of ﬁnger (if they reach the opposite boundary at signiﬁcantly different iteration steps) for each individual ﬁeld. These structures occupy different volume fractions of the domain nc.

3. Application to Heterogeneous Hydraulic Conductivity Fields To test our methodology, we apply it to two ﬁelds of different type of connectivity structures: a highly connected ﬁeld where the more permeable parts of the domain are connected; and an intermediately connected ﬁeld where the intermediate values of hydraulic conductivity are best connected. To show that our methodology is able to delineate the clearly distinguishable connectivity structure, we create a strongly connected hydraulic conductivity ﬁeld (SCF) [Zinn and Harvey, 2003]. Due to the almost binary nature of the ﬁeld, the channels can directly and intuitively be observed by looking at the hydraulic conductivity ﬁeld (Figure 2a). After applying the ﬁrst step of our procedure the delineation of the topology, a uniform dense network of equally wide channels is shown (Figure 2b). A qualitative comparison between the original hydraulic conductivity ﬁeld and the delineated channel network shows that the methodology is able to capture all the main features: the number of connections between highly permeable areas (channels) and the number and location of holes, the parts of the domain that are not part of the channels network. The frontline of the individual channels moves almost uniformly and reaches the opposite boundary at about the same iteration. This means they all represent paths of equal resistance. The sizes of the low-conductivity inclusions are of the same scale and represent an almost uniform distribution. After performing the erosion and dilation procedures, we obtained a series of seven connectivity structures with different channel width. The thickest occupies 58%, the thinnest 27% of the domain. The variability in hydraulic conductivity inside the channels is relatively small and decreases with channel width (rY252.34 for the entire domain, rY250.22 for the thickest channel structures, and r250.06 for the thinnest one). The variability stems from lower hydraulic conductivity values at the border of the channel. For the weakly connected ﬁeld (WCF), the connectivity structure is not directly visible from the hydraulic conductivity ﬁeld alone (Figure 3a). The connectivity structure is heterogeneous, different individual channels or ‘‘ﬁngers’’ reach the opposite boundary at different iterations. In contrast to SCF, these channels are of different resistance. Thus, we construct three connectivity structures, consisting of one, two, and four channels based on the series of polygons at the different iteration steps (Figures 3d–3f). To cut off the dead ends, we perform the polygon growth twice: from the right to the left boundary and from the left to the right. The ﬁnal connectivity structure is only the intersection of the two previously obtained structures. The channels consist of both highly conductive segments and the bottlenecks of small conductivity that plug the channels. These bottlenecks have a signiﬁcant inﬂuence on overall resistance of an individual channel. The thickness of the channels also is inﬂuenced by their resistance. To conﬁrm that our connectivity structure really represents the connectivity network, we solve for ﬂow [McDonald and Harbaugh, 1988]. The agreement between the connectivity structure and paths of strongest ﬂux is very good (Figure 3b). Channel 1 which has the least resistance provides the largest ﬂux. As pointed out by Edery et al. [2014], the cut-off approach is not sufﬁcient to characterize connectivity in such ﬁelds due to bottlenecks formed along

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Figure 2. Strongly connected ﬁeld, SCF, and a series of connectivity structures (red) with different width. (a) Original strongly connected hydraulic conductivity ﬁeld (SCF). The connectivity network can be observed directly. (b) Connectivity structures occupying 68% of the domain (nc 5 68%). (c) nc 5 42% (d) nc 5 22%. Note, ﬁelds in Figures 2b–2d represent the main feature of connectedness observed in Figure 2a.

the paths of smallest resistance. In Figure 3c, we see that the cut-off appraoch renders for WCF a channel network that does not reproduce the connectivity pattern reﬂected in the ﬂow ﬁeld. Contrarily the cut-off would render an excllent result for SCF. To illustrate whether the connectivity structure captures the trajectories of the particles with the earliest arrivals, we perform particle tracking [Pollock, 1988]. We compute particle traces within the original ﬁeld and compare the distribution of the arrival times for all particles with the particles whose trajectories are entirely within the connectivity structure. For the SCF, early arrival times for all distributions are about the same, i.e., connectivity structures provide the space for trajectories with the fastest transport pathways (Figure 4a). For WCF, the earliest arrival time is also similar. But the arrival time distributions are wider within the connected structures for WCF than for SCF. Connectivity structures with more than one channel for WCF also provide particles which arrive at intermediate times. The connectivity structure with four channels contains trajectories with arrival times spanning over 1.5 orders of magnitude (Figure 4b). This can be explained by the larger resistance in the fourth channel which leads to a lower advective velocity. Connectivity structures do not contain trajectories with large arrival times which contribute to tailing of the BTC. This was expected as the tail is determined by the low-conductive inclusions. The inclusions are not studied here, but they can also be studied as a result of this method because they are spatial complements to the connectivity structures.

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Figure 3. (a) Weakly connected hydraulic conductivity ﬁeld (WCF). Connectivity is not apparent here. (b) Flow ﬁeld from numerical simulation to illustrate the existing set of connected channels: one major and three minor that generate a network. (c) connectivity pattern derived using a cut-off value of hydraulic conductivity (d–f) series of the connectivity structures (red) Figure 3d one major channel nc57%, Figure 3e two channels, and Figure 3f four main paths.

4. Discussion and Conclusions We present a new numerical method to delineate the connectivity structure of a hydraulic conductivity ﬁeld based on ﬁnding the path of least resistance. The method consists of two major steps: ﬁrst, the topology of the connectivity structure is determined. The connected channels are delineated by moving a line of equal resistance between two opposite boundaries. And second, the individual channels are inﬂated or shrunk based on dilation and erosion. The method is applied to two highly heterogeneous aquifers with different types of connectedness. First, we apply it to a highly connected ﬁeld (SCF). Here, the connectivity structure can be observed directly from the hydraulic conductivity ﬁeld. Comparison with the delineated connectivity structure shows that our

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Figure 4. Histogram of arrival times for all particles and only counting particles with entire trajectories within the connectivity structure. The connectivity structures contain the pathways of the trajectories with the earliest arrival times. (a) SCF: the smaller the thickness of the channels (nc) the fewer are particles with large arrival times. (b) WCF: the more channels are included the larger is the difference in advective times.

method is able to capture the connected channels. For SCF, the channels are fairly homogeneous and all channels arrive at more or less the same iteration. The channels become more heterogeneous with increasing width, but no bottlenecks occur. The channel network for the weakly connected ﬁeld (WCF) is different. The individual channels reach the opposite boundary at different iterations. Therefore, the resistances of the individual channels are different as resistance increases with number of iterations. All the individual channels of the WCF show bottlenecks and are internally variable. Both ﬁelds are highly heterogeneous but their connected channel networks are qualitatively different. For illustration, we simulate ﬂow and transport. We show that ﬂow channels are well preserved using our approach as well as the pathways for particles with earliest arrival times. This indicates that our methodology captures features of dynamic connectivity using only information of hydraulic conductivity ﬁeld. The method presented here is two-dimensional. As connectivity in 3-D behaves fundamentally different, our ﬁndings cannot be directly extrapolated. The algorithm itself could be extended to account for 3-D ﬁelds. In this case, a front-surface of least resistance would move through the medium.

Acknowledgments Financial support by the Swiss National Science Foundation (SNF) through grant 200021 132304 and by Nagra, Wettingen, is gratefully acknowledged. The authors also thank Denis Allard for useful hints, remarks, and inspiration. There is no observation data in the paper.

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