A flume experiment is carried out to explore jamming of Large Woody Debris (LWD) in streams with complex morphology, occurring in mountain streams with in channel boulders or vegetation, in braided rivers or in floodplains during flood events. Non rooted, defoliated LWD is modeled using wood dowels and obstacles to motion are represented by vertical wood rods. Congested transport of LWD is simulated by insertion of a number (100) of dowels. The final position of the dowels is mapped and the observed jams are classified according to their size and position. The key member of each jam is identified and its trapping mechanism evaluated, either by leaning against a single obstacle or by bridging two obstacles. To mimic uncongested transport, the experiment is repeated for single pieces of wood, with subsequent removal. Longer dowels and shallower water result in shorter traveled distance. Wood pieces travel farther when congested transport is observed. The traveled distance of the wood pieces can be modeled using a Gamma distribution, for both congested and uncongested transport. Jams instead display Uniform traveled distance. The number of pieces displays an Exponential distribution. The degree of uniformity in space of jams and wood pieces is evaluated using a neighbor K statistic. Wood pieces show considerable clustering, while jams show sparse distribution. Eventually, the relationship between jams magnitude and position is explored, showing negative correlation. Model application is then discussed and some conclusions and future developments are outlined.
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 In rivers congested transport of wood pieces is likely to be observed [Braudrick et al., 1997; Abbe and Montgomery, 2003], where LWD pieces interact to provide complex aggregation patterns, resulting in formation of wood jams. This is particularly true during floods, when a number of wood pieces is likely to be transported.
 Here, a flume experiment is carried out to explore the interaction of multiple pieces of LWD with complex channel geometry. It is assumed that formation of jams results from feeding of wood pieces from the upstream part of a reach, mimicked here by insertion of pieces of wood of regular size. A number of 100 dowels is sequentially placed into the flume and their final setting mapped. The wood jams resulting from multiple stops at the same place are classified according to the number of elements (stationary single pieces are also considered) and their position, i.e., distance from the flume inlet. Also, the position of the single pieces of LWD is described. Furthermore, the spatial distribution of the jams and of the wood pieces is evaluated using a neighbor K statistic.
 In some cases, a few pieces of LWD are transported contemporarily, so resulting into negligible interaction. Here, also the situation of uncongested load is studied by experimentally observing the passage of single logs inside the gauntlet and the results therein compared with those for the wood jams.
 Of great interest is the attitude of wood pieces to either cluster (i.e., when they are closer to each other than in the case of uniform distribution) or segregate (i.e., when they are more regularly spaced than in the uniform case). Kraft and Warren  evaluated the distribution in space of debris dams (i.e., jams) and wood pieces for a number of streams in the Adirondack Mountains (NY), after the occurrence of a major storm in 1998, providing a relevant source of woody debris from the hillslopes. They speculate that after a disturbance event such as a considerable storm LWD is uniformly distributed in space. Therefore the distribution of LWD after some time (therein, two years) is driven by channel flow. Using a one dimensional neighbor K statistics, they found a considerable degree of clustering for single pieces for a range of small scales (about 1 to 40 m), while considerable segregation was observed at larger scales (about 80 to 100 m). Concerning wood dams, they found more uncertain results, with segregation observed in some cases at a scale of about 100 to 300 m. Also, they discuss the existence of a relationship between the spatial pattern of the wood pieces and the morphology of streams, including bends, narrow sections and the presence of in channel obstacles such as large boulders.
Bocchiola et al. [2006a] carried out a flume experiment on the transport of LWD pieces in the presence of obstacles. The distance traveled by a single piece of LWD is shown to be a random variable, with expectation and variance depending on the length of the piece, the inter-obstacle spacing and the force exerted by the current. The probability that a single piece of LWD is stopped and also its stopping mode, by leaning against one obstacle or by bridging two obstacles depend on the length of the LWD and on the flow conditions.
 Albeit the present experiment partially builds on those by Bocchiola et al., 2006a, it extends the results therein for a number of reasons. First, the exploration of LWD accumulation processes is carried out in the case of congested transport, i.e., considering interaction of more LWD pieces. Further, here an approach based on maximum likelihood (hereon, ML) for censored series is introduced to evaluate the statistics of traveled distance. The traveled distance is then modeled using standard distributions, so providing a statistical interpretation of the accumulation processes. The approach suggested by Kraft and Warren  is also applied, to evaluate the degree of either clustering or segregation of LWD.
2.2. Motion of LWD in Streams
 Two major modes occur in the motion of LWD in rivers [Braudrick et al., 1997; Braudrick and Grant, 2001; Haga et al., 2002; Bocchiola et al., 2006a]. First, LWD can move in contact with the bed, by rolling or sliding. Second, floating occurs when water is deep enough to achieve LWD buoyancy. One needs a criterion to discriminate these two different situations. Following Braudrick and Grant  and Haga et al. , if a piece of wood with diameter DLog is immersed in a water flow with depth dw, the dimensionless variable h* = dw/DLog is the factor discriminating between floating and rolling or sliding. Bocchiola et al., 2006b accounted explicitly for wood and water mass density, ρLog and ρw by taking h* = ρwdw/ρLogDLog. The “flotation threshold” should be h* = 1, associated to the expected buoyancy force in hydrostatic conditions. However, Bocchiola et al. [2006b] showed that the local perturbation of the flow results into the log being immersed to an average depth lower than dw. The buoyancy force is then less than that occurring under hydrostatic conditions and flotation occurs for a larger flow depth [see Bocchiola et al., 2006b, equation (13) to (15)]. A floating threshold is well approximated by setting h* = 1.26. Because transport of wood by flotation is of more practical interest, particularly during floods, the case of floating LWD is considered here.
3. Flume Experiment
3.1. Flume Settings and Obstacles
 The experimental setup is shown in Figure 1, where some wood jams are observed, formed into a straight channel with known bed slope and roughness, where obstacles to motion are present. The adopted flume has width WFl = 1 m and length LFl = 30 m, with a bed slope if = 0.006. With respect to the paper by Bocchiola et al. [2006a], the authors designed here a new gauntlet, with a longer panel and closer obstacles, to increase the number of dowels captured inside the gauntlet. The gauntlet length was fixed to LP = 4 m and the average nearest neighbor distance between the obstacles was set to L0 = 0.16 m, resulting into a number of 156 obstacles. The panel was fixed to the flume bed and covered with sand with known specific weight and grain size distribution (γs = 25.7 KNm−3, D16 = 1.5 mm, D50 = 2 mm, D84 = 2.5 mm). The authors decided to fix the channel bed, neglecting the effect of bed load. The in channel obstacles were simulated using wood dowels of 0.1 m length and 1 cm in diameter. They were randomly distributed in space (i.e., their coordinates are extracted independently from uniform distributions) and glued to the bed, to prevent motion.
3.2. Adopted Dowels
 Handmade cylindrical dowels were used. Although this geometry is not representative of woody debris with complex shape, it can provide a reasonable picture of non rooted, defoliated and cylindrical logs that often occur in rivers [see, e.g., Braudrick and Grant, 2001]. For example, the biomass deriving from wood harvesting and maintenance often occurs in the form of cylindrical logs. Cylindrical logs also resemble the relics of forest fires, that are an important source of woody debris in rivers [Zelt and Wohl, 2004; Rulli et al., 2006; Rosso et al., 2007]. The dowels are made of Beech (Fagus Sylvatica). For the experiment, 200 hand made dowels are used, 100 with diameter DLog = 0.014 m and length LLog = 0.1 m and another 100 with the same diameter and LLog = 0.15 m. The size and mass density of the dowels used in the experiment are indicated in Table 1.
Table 1. Properties of the Dowels
Length LLog, m
Dimensionless Length L*Log [.]
Diameter DLog, m
Average Dry Density ρLog,d, Kgm−3
Average Wet Density ρLog,w, Kgm−3
3.3. Methods of Observation
 First, the flow conditions in the flume were considered. Using HEC-RAS® software, a model of the flume was created, including the sand bed, the plastic bed and the obstacles. Roughness, quantified using Manning's n, was evaluated for both the sand bed (n = 0.029) and the plastic bed (n = 0.016), by iteratively fitting the modeled water profile to the measured one. This was done for the four considered discharges (see Table 2), showing values of n constant in practice. The obstacles were observed to only slightly increase the average water depth, as compared with flume flow without obstacles. More importantly, due to the smaller roughness of the plastic flume bed before and after the sand covered panel, a considerable decrease of the water depth was observed therein (not shown here for shortness), resulting into noticeably changing water depth along the sand bed. For the proposed experiment, reasonably uniform flow conditions needed to be set into the flume, to reproduce approximately uniform flow inside a stream. To obtain as much as possible uniform motion the authors chose to insert a 1 meter wide, 2 cm high step 1.5 m downstream of the sand bed, to force tailwater effect and increase the water depth on the sand bed. This resulted in flow conditions on the sand bed closer to uniform motion.
Table 2. Summary of the Experimental Flow Conditions
Water Depth Dowel
1.0E + 04
2.3E + 03
2.0E + 04
2.9E + 03
3.6E + 04
3.6E + 03
6.0E + 04
4.4E + 03
 To perform the set of experiments, steady values of depth and velocity were established using the flume circulation facility. Water depth was manually measured at 50 sparse points and than Kriged (using Surfer-8®, linear correlation) to obtain distributed depth maps. Flow mass rate (discharge Q) was measured by a standard rectangular (Bazin) weir and eye-ball piezometric glass. Flow velocity was evaluated using continuity equation based on water depth and discharge. Average surface flow velocity along the panel was also measured using a float and stop watch and correction for depth-averaged velocity [e.g., Braudrick and Grant, 2000]. This was compared with velocity from continuity equation, indicating substantial agreement. Here, the velocity values are shown evaluated by continuity equations, due to their more accurate assessment. Four different sets of experiments have been carried out, featuring different values of (average) water depth and velocity, indicated in Table 2. A comprehensive description of the flow conditions is shown in Figure 2. Therein, the average (cross section) flow conditions along the sand bed are shown. These are the water depth dw in cm obtained by averaging of the Kriged maps, its average value along the panel dwav and the corresponding depth for uniform flow dw0. Notice that a horizontal line for dw indicates uniform flow. Also, the energy line is shown H (in cm, with level zero as referred to the last cross section, L = 4 m), together with the energy line for uniform flow H0. Here, uniform flow is no more represented by a horizontal line, but instead by a line with constant slope (i.e., the friction slope). This is because the kinetic head Uw2/2g amounts to few millimeters, due to low velocity (see Table 2), thus making difficult to visualize water surface (dw) and energy line (dw + U2/2g) in the same scale. Also water velocity (secondary axis) is reported.
 Although the flow conditions are variable along the plate, the deviation from uniform flow seems acceptable. The greatest relative deviation of dw from dw0 amounts to 21% for the inlet section in case H2, while the average depth dwav never differs from dw0 for more than 2%, in case H3. Therefore the authors feel confident in considering as the representative depth and velocity, the average value dwav and Uwav = Q/dwav.
 Each set of experiments was composed of 2 trials, corresponding to two different lengths of the dowels, namely L1 = 0.10 m and L2 = 0.15 m. Each trial was repeated four times, for more statistical robustness. Each repetition was carried out by sequentially inserting a number of 100 dowels perpendicularly to main flow direction. To mimic a random effect with respect to the position of the obstacles, 5 starting positions inside the flume were considered (i.e., 5 different values of the abscissa x in Figure 3, identified as A, B, C, D and E) far 15 cm from each other and 20 cm away from the walls. Because the dowels would move by floating with their axes parallel to flow, they would not touch the flume walls easily and a distance of 20 cm was observed to be sufficient to avoid side effects. When a dowel stops and reaches a visually steady position, another dowel is inserted. When all the 100 dowels have been inserted and have reached their final setting, their positions is mapped and another repetition is carried out. After four repetitions, another trial is carried out, by changing dowels' length. Eventually, a number of 32 repetitions was carried out (i.e., considering four flow depths, two log lengths and four repetitions), for a grand total of 3200 mapped dowels.
 A jam is formed when a number of pieces are grouped together so that the all pieces are in touch. For each jam, the forming key log is found and classified according to the different stopping mode, i.e., by leaning, Le, or bridging, Br. Although in some cases this operation is somewhat subjective, particularly when a high number of logs is observed, in most of the cases the key log is clearly identified. Also, the experiments were filmed using a digital camera, so aiding in the identification of the key logs in case of doubts.
 The choice of using 100 dowels is arbitrary. For instance, one could consider a number of pieces necessary to reach some equilibrium conditions in the stream, e.g., when the number of dowels incoming over a reference period is equal to the number of those running out of the flume. In real rivers the reference period would probably be linked to the average rate of supply of wood pieces (e.g., pieces/day). For rivers with a high supply rate, this means a short period (e.g., few days), while for low supply rate, this could mean a long period (e.g., months). In turn, the supply rate is related to the occurrence of relevant storms, increasing abruptly the amount of wood conveyed into the channel [e.g., Kraft and Warren, 2003], but also sweeping away the accumulated wood pieces. When relatively long term accumulation is considered, wood decay would also affect the accumulation of wood. Therefore a quantitative assessment of the dynamic equilibrium of a stream and the timescale therein, that should be then reproduced at the flume scale seems complicate and the authors are not aware of any such attempt in practice.
 The present results simply mimic the behavior of a stream that is initially empty, after a number of 100 dowel is inserted, that seems to describe in a first approximation the dynamics of jamming into a river. In the future, more experiment could be carried out to investigate the issue of dynamic equilibrium.
 The experiment of uncongested transport was carried out in the same hydraulic conditions. For each velocity and dowels' length a trial was carried out. For each trial, at the starting positions (A, B, C, D, E), ten dowels were sequentially inserted in the flow and subsequently removed. Eventually, 8 trials were carried out, for a total of 400 mapped dowels.
3.4. Scaling Issues
 Dimensional analysis indicates that the experiment mimics transport of wood pieces in mountain streams with in channel boulders or vegetation, in braided rivers or in floodplains during flood events. First, one has to consider the Froude number Fr = Uw/(gdw)0.5, with g gravity acceleration and the Reynolds number, Re = 4Uwdw/ν, with ν kinematic viscosity (see Table 2). The former ranges from 0.46 to 0.51, while the latter ranges from 1E4 to 6E4. These provide conditions similar to those observed for low water depths, when interaction with wood is of interest [see, e.g., Braudrick and Grant, 2000, Table 3]. To provide similarity at the field scale, both Re and Fr should be maintained. This cannot be done when working with the same fluid (i.e., water). According to, e.g., Wallerstein et al.  resistive forces scale logarithmically with Reynolds number and become in practice constant for turbulent flows, albeit not fully developed as here. Therefore Reynolds scaling can be relaxed in a first approximation. As far as the LWD is concerned, the scaling was described by Bocchiola et al. [2006a] and requires some changes here. In plane, similarity holds provided the ratio between LWD length and obstacles spacing L*Log = LLog/L0 is maintained. The scaling in the vertical direction depends on LWD motion conditions. When LWD is in touch with bed the scaling is dictated by the (dimensionless) excess of force acting on the wood piece, with respect to the threshold for motion. When LWD floats, a null threshold occurs, because the wood is no more in touch with the bed. In the simplified hypothesis that LWD moves at flow velocity (see, e.g., Braudrick and Grant, 2001), dimensional analysis (Biesuz and Zanetti, 2005, not shown here for shortness) show that the scaling parameters are the Froude and Reynolds number of the log, FrLog = Uw/(gDlog)0.5 and ReLog = UwDLog/ν [see also Wallerstein et al., 2001]. Particularly, FrLog and ReLog affect the value of drag coefficient, so influencing the interaction of wood and flowing water [Wallerstein et al., 2002]. Here, ReLog ranges from 2.3E + 03 to 4.4 E + 03. The drag coefficient is practically constant against ReLog for values of the latter up 1E6 [e.g., Wallerstein et al., 2001; Alonso, 2004] and Reynolds scaling can therefore be relaxed in a first approximation. The Froude number of the logs FrLog ranges here from 0.21 to 0.86, showing instead more influence [e.g., Alonso, 2004]. Also, considerable values of the blockage of single wood pieces with respect to the flow area, B = DLogLLog/WFldw affect the drag coefficient [e.g., Shields and Gippel, 1995]. However, B is quite low here, as it ranges from 0.03 to 0.13, thus providing no practical influence on the results [e.g., Hygelund and Manga, 2003]. Therefore the parameters dictating the scaling is here FrLog.
4. Experimental Results
4.1. Motion Patterns and Quantitative Description of the Jams
 The dowels quickly self-adjust their direction to match the flow direction and converge toward the centerline. Two mechanisms of motion are observed. The first occurs when the average water depth is only slightly greater than the flotation threshold (case H1, with * = 1.33). In such case, the dowel occasionally touches the channel's bed, due to local decrease of water depth. Bocchiola et al. [2006a] termed this condition “Just Floating” (JF). The second condition is “Fully Floating” (FF), i.e., when the water depth exceeds notably the floating threshold (case H2, H3 and H4, with * = 2.07, 2.97 and 3.96, respectively), so that the dowels always floats. Here, no significant difference has been observed in the JF case, practically equivalent to a FF condition, and no difference is made in the forthcoming. In some cases, stopping of a first “key log” would lead to further stopping of dowels, so forming a jam. Conversely, the “key log” could be swept away after collision with another incoming dowel. The formed jams feature a different number of pieces, ranging from one (i.e., stationary single pieces) to a greatest number (as observed here) of Maxp = 33, for H2 = 2.45 cm and L2 = 0.15 m. In the literature concerning wood transport in rivers, different types of jams are described [see. e.g., Abbe and Montgomery, 2003] and some of these were also observed during the experiment. However, jams type classification is strictly linked to river morphology, including lateral constriction, bends, vertical structures (steps, pool), floodplains and the presence of bed load. Because here such phenomena are not represented and therefore comparison would be difficult, the authors make here no distinctions between different types of jams and focus is directed on the number of pieces. In Figures 3 and 4 visual summary of the experiments is given. Therein, for each couple of H and L, the position and size of the formed jams are given. The number of dowels in a jam is reported for each obstacle. This is the average value of the number of dowels that was observed in each of the four repetitions for given flow depth and log length, rounded to the nearest integer.
 In Table 3, a quantitative description of the results is given. First, the number of jams NJ is reported, averaged over the four repetition and rounded to the nearest integer. Then, the greatest number of dowels in a jam is reported for one single repetition, Maxp, together with its average on the four repetitions, Maxpav. NJ is shown in Figure 5 against FrLog. The number of jams NJ slightly decreases with FrLog, particularly for L1.
Table 3. Summary of the Experimental Results
 The average E[Np] and the coefficient of variation CV[Np], i.e., the ratio between standard deviation and average value, of the number of pieces into the jams are also reported in Table 3. E[Np], in Figure 5, is higher for L2. The fraction of stopping dowels inside the gauntlet is given, StopJ, together with that of the jams formed by a bridging key log, BrJ (the fraction of leaning key logs is LeJ = 1 − BrJ). These values can be compared with those for uncongested transport, Stops and Brs also reported in Table 3. The comparison is shown in Figures 6 and 7. Generally speaking, the probability of a log to be trapped increases with its length and decreases with flow velocity (scaled to log diameter, as given by FrLog). Here, it is shown that the probability of stopping is higher for single logs than for jams, as evaluated in steady state (i.e., after each repetition) condition.
 One can define a removal rate, Rem = 1 − StopJ/Stops, measuring the degree of removal of trapped (key) logs due to flow conditions. This is reported in Table 3 and increases with FrLog. This is due to removal of lodged logs after collision with moving logs. As a final result, the fraction of retained logs inside the channel is smaller than that of retained logs (i.e., the probability of stopping) when uncongested transport is observed. Also, one can define an efficiency of trapping
with Type either Br or Le. Eff can be either positive or negative, the former case indicating a higher fraction of wood pieces in jams retained in mode Type than in the case of single pieces, and the latter case the vice versa. Eff is given in Table 3. As expected, this is positive (and slightly decreasing with FrLog) for Br and negative (quite constant with FrLog) for Le. The efficiency of bridging, EffBr is high for L1 when low values of FrLog are observed (EffBr = 2 or so), but drops noticeably for increasing FrLog (EffBr = 0.08 or so). Accordingly, logs tend to stick for low velocity, but are swept away for high velocity. For L2, EffBr depends more weakly on FrLog, ranging from 0.54 to 0.21. Changes in velocity affect less the longer bridging logs than the shorter ones. The efficiency of leaning, EffLe is negative for both L1 and L2, meaning that leaning logs are more likely to stand in uncongested than in congested transport. For L2 lower values are observed (EffLe from −0.84 to −0.60, not depending on FrLog), than for L1 (EffLe from −0.59 to −0.43, again not depending on FrLog).
4.2. Statistics of the Traveled Distance
 The average and variance of the distance traveled by the dowels are evaluated here. Because a number of woods travels out of the panel, the evaluation of these statistics by sample calculation is not accurate. Therefore the authors have used here the method proposed, e.g., by Kendall and Stuart  for evaluation of population statistics from censored series using a maximum likelihood (ML) approach. This evaluates the moments of a given distribution using a sample of values that are censored (i.e., truncated) either above o below a given value. Here, the censored variable is the traveled length LT and the considered sample is censored above LT = 4 m. If a number nout of dowels is observed to travel out of the panel, the ML approach is carried out by maximizing the Likelihood function
where F and f represent the hypothesized distribution (cdf and pdf, respectively) and θ is the appropriate set of parameters, here mean and variance, or coefficient of variation, CV. The ML function was here maximized as a function of the unknown mean and CV, using EXCELS® solver (Generalized Reduced Gradient, GRG2). The initial conditions (first estimate) were set according to the moments of the observed sample. Convergence of the solution to the ML value was set to 1E−4 of the LF function. The choice of the starting point is critical and the final value can represent a local maximum. However, this is true with any maximization method. ML estimation was carried out for each of the four repetitions and the values so obtained were compared to each other for consistency. In all cases, consistent results were found (not shown for shortness). The values of mean and variance obtained from averaging of the four repetitions are reported in Table 3. To carry out the ML function, a preliminary assessment of the F function is required. For each of the variables considered here, preliminary testing was carried out using five standard distributions, namely Uniform (UN), Exponential (EXP), Normal (NR), Lognormal (LN) and Gamma (GA). The distribution was chosen giving the best (eyeball) fitting to the observed data. The so obtained distributions are shown in the next section. In the following, the traveled distance is considered made dimensionless with respect to L0, or L*T = LT/L0. This is consistent with scaling LLog to L0 and allows to use geometric similarity in field studies or flume experiments. First, the average distance of the jams from the channel inlet E[L*T]J is calculated, together with its coefficient of variation CV[L*T]J. Notice that the ML estimation showed a Uniform distribution of L*T of jams, so giving CV[L*T]J = 0.58. Then, the same statistics are calculated considering the single pieces of wood into the jams, namely E[L*T]w and CV[L*T]w. Also, these statistics are calculated in the case of uncongested transport (i.e., single woods), E[L*T]s and CV[L*T]s. The values of E[L*T] are shown in Figure 8. E[L*T] is smaller for single pieces than for the jams, due to wood redistribution during congested transport. Also, E[L*T] is higher for single pieces in jams, than their counterpart for uncongested transport. Notice also that E[L*T] increases with FrLog, while CV[L*T] seems only slightly variable against FrLog (see Table 3).
4.3. Frequency Distributions of the Traveled Distance and the Jams Size
 The frequency distributions resulting from the ML approach are reported here. Also the distribution of Np is studied. For the latter only sample statistics are used, in the reasonable hypothesis that the wood pieces running outside the gauntlet do not change substantially the size of the jams. Here, the sample values obtained for different values of FrLog were made dimensionless with respect to their average, as follows
This allows distribution fitting by using the sample obtained by grouping together the values of L′T and N′p for the different values of FrLog. This is tantamount to assume that different values of FrLog can influence the average traveled length and its standard deviation, but the ratio between the latter and the former, i.e., the CV remains more or less constant. Strictly speaking, this procedure is allowed if the distribution of the dimensionless values is equivalent for each value of FrLog, i.e., if it shows the same coefficient of variation CV. Analysis of the estimated CV values, in Table 3, shows only slight dependence of the latter upon the value of FrLog. More influence is given by the dowels length, because the CV value for L1 is lower than for L2. On a principle line, some statistical tests should be carried out to assess homogeneity of the CV coefficients. Here, due to the very preliminary nature of the proposed attempt, equivalence of the CV values with respect to FrLog is tentatively assumed. Also, plotting positions of the observed frequencies was preliminarily carried out (not shown here for shortness), showing indeed no relevant visual difference. Distribution fitting is instead carried out separately for the two different log lengths. The observed frequency distributions so obtained are shown in Figures 9 (a for L1, b for L2), 10 (a for L1, b for L2), 11 (a for L1, b for L2), and 12 (a for L1, b for L2) for single logs, jams, single pieces in jams and jams size, respectively.
 Therein it is shown the probability (evaluated using Weibull plotting position, F = n/(ntot + 1), with n rank of the sample in the series ordered in decreasing order and ntot whole sample size) that a piece (or a jam) is positioned within a given traveled distance L′T, or that a jam is made of a given number of elements smaller or equal to N′p. For the traveled length, the plotting position is compared to the standard distribution used for the ML approach. No test for distribution fitting is carried out here and only confidence limits are given to aid goodness of fit evaluation (Kolmogorov Smirnov, α = 5%). Best fitting to the observed traveled distance L′T of the single pieces is given by GA, with more scatter for L2. Notice that the scatterplot shows frequencies lower than one, as a result of the censoring for high values of L′T (i.e., for LT > 4m). The traveled distance of jams is distributed according to an UN. The single pieces in jams are well represented by a GA, again with more scatter for length L2, apparently consistent with the GA distribution for single pieces. Eventually, the frequency of N′p is reasonably well accommodated by an EXP distribution. Worse fitting is observed for L2, but the EXP seems to capture the core of the scatterplot.
4.4. Neighbor K Statistics for the Jams and the Wood Pieces
 To emphasize the attitude, if any, of the wood pieces to either cluster or segregate at given spatial scales, neighbor K statistics are here calculated. The same approach proposed by Kraft and Warren  is used. In short, the K statistic evaluates the number of wood pieces (or jams) within a given distance from another wood piece (or jam), averaged on the whole sample
with ns sample size, xij distance between samples i and j, and ID = 0 if xij > D; ID = 1 if xij ≤ D. Distance D represents the scale at which the degree of sparseness is tested. Calculation of K is here carried out for each repetition of the experiment. Then, the Rank (i.e., frequency of exceedance) of K is calculated. This indicates the chance that the value of K obtained for a given distance is representative of a uniform distribution. To evaluate the Rank, a reference statistic is needed, based on Monte Carlo approach. For each repetition, a number of 1000 equivalent synthetic simulations is carried out, considering the same number of jams, or wood pieces as observed. Each simulation provides a possible distribution in space that would be obtained if the coordinates of the jams or the wood pieces were extracted from uniform random variables (i.e., if they were uniformly distributed in space). For each of the synthetic distributions, the value of K(D) is estimated. The values of K(D) so obtained is Normally distributed [e.g., Kraft and Warren, 2003]. Using this Normal distribution, the Rank of the observed value of K(D) is evaluated. Taking a confidence level α = 5%, the Rank can indicate either clustering (Rank > 0.975) or segregation (Rank < 0.025). Here, the distance is considered made dimensionless with respect to L0, or D′ = D/L0, so that the proposed results can be scaled to obstacles spacing. The physical extent tested is from D = 0.05 m, to D = LP = 4 m, that is the greatest possible distance between the jams, with steps of 0.05 m. Considering 32 repetitions for both wood pieces and jams one has 64E3 simulated values of K, for each considered distance. The Rank is here averaged on the four repetitions, to obtain more robust evaluation, for given values of LLog and dwav. In Figures 13 and 14, the Rank is reported for the four investigated cases (L1 and L2, jams and wood pieces), as a function of water depth. For comparison, it is also reported the Rank of the obstacles, Uniformly distributed. Jams show substantial uniformity at all scales. This seems consistent with the findings by Kraft and Warren , showing apparently sparse distribution of jams for short scales below 100 m or so (see Figures 4 to 6 therein).
 Somewhat surprisingly, the distribution of the jams seems not to fully reflect that of the obstacles, as the Rank of the former is higher than that of the latter for a considerable range of scales. As far as the wood pieces are considered, clustering is clearly evident for the smallest scales (up to 5–10 times the value of L0). Again, this seems to match with the findings by Kraft and Warren , showing clustering of LWD for small scales (up to 40 m or so, Figure 3 therein). For values of about D′ = 15 the Rank decreases until values indicating segregation, particularly for L2, then increasing again to indicate clustering. This pattern seems not reflected in the findings from Kraft and Warren . Also water depth (i.e., velocity) influences the spatial patterns of wood pieces, because for H1 clustering is practically always observed and generally shallower depths result in more clustering.
4.5. Dependence Between Jams Position and Size
 From Figures 3 and 4 one notices that the size (Np) of a jam is related to its position, because the greatest jams are formed close to the inlet. This agrees with the intuitive statement that more pieces of wood tend to gather at the inlet, while fewer pieces will be able to travel farther.
 Here the (linear) correlation coefficient r is evaluated between LT and Np for each trial (i.e., for each couple of FrLog and LLog), reported in Table 3. It is different from zero (significance tests for r, α = 5% reliability level) and always negative. Therefore longer distances imply smaller jams. Also, r (absolute value) is always greater for L2 than for L1, i.e., longer logs are remain more correlated in space than shorter ones. Further r seems weakly dependent on FrLog, meaning that the dowels length influences their correlated distribution in space more than flow velocity. Because LT and Np are likely to be significantly correlated, their joint distribution is different from the product of the two marginal distributions and should be carried out according to a bivariate approach [e.g., Kottegoda and Rosso, 1997, Ch. 3]. The latter is possibly a research issue in itself and goes beyond the scopes of the present paper. A more refined approach including for instance bivariate distribution fitting or copulas could be used in the future to model the accumulation process of jams, in turn increasing the required experimental effort, because bivariate approaches claim for considerable sample dimensionality for robust model estimation.
5. Discussion of Model Application in Real Rivers
 A number of field campaigns have been carried out in the near past to evaluate the patterns of deposition of wood in streams (among others, it is worth quoting here the works by Prof. Angela Gurnell and her team, concerning the Tagliamento river, in Italy, reported in the references). Accumulation patterns of wood are qualitatively described, in some cases with specific reference to stream geometry [as, e.g., by Jackson and Sturm, 2002; Abbe and Montgomery, 2003], but usually with no modeling purposes. In this sense, the present experiment sketches a simple model to assess the statistical distribution and degree of aggregation of wood pieces in streams for predictive purposes. Model application is not straightforward, in view of the utmost complexity of the interaction between wood accumulation patterns and flow in streams. However, it is possible to draw here some guidelines for a qualitative comparison of the proposed flume scale results against findings of equivalent field experiments, based on similarity of geometric and hydraulic conditions. First, one has to consider relatively low flow depths in respect to wood size, or h* ≈ 1–5 and control provided by bed geometry, in presence, e.g., of boulders in mountain streams [as, e.g., by Kraft and Warren, 2003] or bars in braided rivers [as, e.g., by Gurnell et al., 2000a]. Relative low blockage should be considered as here, or B ≈ 0.01–0.2, i.e., small wood pieces as compared to flow area. Uncongested and congested transport can be discriminated according to the observed degree of interaction between wood pieces. A small number of jams (evaluated as, e.g., NJ/WFlLP) as compared to those reported here suggests use of uncongested hypothesis, while more jams suggests vice versa. Also, an average wood feeding rate (e.g., wood pieces per day) can be assessed, providing a first guess of the required period for the equilibrium conditions as here given (i.e., for 100 pieces or so). Then, one must check that wood geometry is appropriate, i.e., simple cylindrical logs can represent the wood pieces. This is reasonable if rootwads, canopy and branches are negligible as compared with the size of the pieces. If average flow depth dw and velocity Uw are known one can calculate Re and Fr. Given the size of the wood pieces, on can calculate FrLog and ReLog. Then, one needs to estimate the average nearest neighbor distance between the in channel obstacles L0. Also one can evaluate their distribution in space, e.g., according to the neighbor K statistic. For illustrative purposes, one can imagine that the length of the shortest dowels here (LLog = 0.1 m) mimic the shortest length for a LWD, i.e.1 m. This results in a geometric scaling ratio 1:10. The average nearest neighbor distance between the obstacles L0 = 0.16 m corresponds to 1.6 m and the extent of the investigated gauntlet LP = 4 m to a stream reach with length 40 m. Flume width WFl = 1 results in stream width of 10 m. These reasonably approximate geometric conditions in mountain streams. If the same scaling is applied in the vertical direction, i.e., to the wood diameter DLog = 0.014 m, this results into a diameter of 0.14 m, close to the lowest bound for the definition of a LWD, i.e., 0.1 m. Scaling of the water depth dwav, with an average value here of 0.03 cm, gives a water depth of 0.30 m. In turn, the scaled bottom roughness D50 = 0.002 m results in 0.02 m, i.e., a gravel bed. Considering the same slope as here used if = 0.006 and a reasonable value of the Manning coefficient in pool-bar forested streams of n = 0.02–0.05 [as, e.g., by Braudrick and Grant, 2000, Table 3, against n = 0.03 here], this gives (for uniform flow conditions), Uw = 0.69–1.74 ms−1 (here, an average Uw = 0.26 ms−1). Accordingly, the Reynolds and Froude number range from Re = 2E5–5E5 and Fr = 0.40 − 1.01 (here, Re = 3.15E4 and Fr = 0.48). The Reynolds and Froude number of the wood pieces for the field conditions range from ReLog = 1E5–2E5 and FrLog = 0.58–1.49 (here, ReLog = 3.3 E3 and FrLog = 0.70). If scaling of Re and ReLog is relaxed as suggested in section 3, scaling based on Fr and FrLog drives model application.
 In trying to apply the model to real rivers, it might happen that scaling in the vertical direction (i.e., for DLog, dwav, D50, Uw) is different from scaling in plane (i.e., for LLog, L0, LP, WFl).
 For instance, in lowland braided rivers with low flow depths, one could have a scaling of 1:100 or so in plane, i.e., LLog = 10 m, L0 = 16 m, LP = 400 m and WFl = 100 m, while in the vertical direction a scaling of 1:10 or so still applies.
 In the assumption of steady and uniform flow, i.e., if complexity of the flow field due to obstacles is neglected in a first approximation, the horizontal and vertical scales are unrelated and one can apply different scaling. Comparison based on geometric and hydraulic similarity might still be applied, at least qualitatively.
 To deal with the neighbor K statistic, notice that distance D is here scaled on L0. Therefore one can interpret results at the field scale considering the Rank associated to the dimensionless ratio D′.
 According to the guidelines her given, qualitative comparison of either field or laboratory scale results against the model here proposed can be carried out. In doing so the simplifying hypothesis here introduced should be reasonably fulfilled and scaling respected. This results in relatively restrictive conditions as compared to the complexity observed in real rivers, thus requiring care in model application.
 The formation of wood jams in streams is a multifaceted process, involving a combination of deterministic mechanisms and random factors. The present approach provides a simplified sketch of LWD and stream flows geometry as a first attempt to represent such complexity, to be developed henceforth. Some preliminary findings can be stressed. The probability that LWD is entrained in jams increases with its length and decreases with its Froude number. Trapping is more efficient if bridging against two obstacles is likely, i.e., when obstacles are close compared to LWD length. Congested transport leads LWD pieces to occupy more extensively the available space than they would do singularly, i.e., with little congestion. The observed frequency of the distance traveled by LWD and its quantitative aggregation (i.e., the number of pieces) can be explained fairly well using proper statistical distributions. Analysis using neighbor K statistic shows considerable aggregation of the pieces of LWD, while sparse occupation is attained by wood jams. Not surprisingly, there is significant statistical linkage between jams size and traveled distance, that is worth of more investigation maybe using bivariate approach. The accumulation of LWD and the formation of wood structures influence river geomorphology and riverine environment quality, but also river conveyance and hydraulic hazard, especially where man-made structures yield supplementary obstacles to LWD. The proposed statistical approach could be adopted as a basis to predict wood accumulation patterns for river management purposes, wood insertion for habitat restoration and evaluation of hazard for man-made structures, such as dams or bridges.
 Further developments need to include the study of more complex LWD geometry, including roots and foliage. Further, the accumulation process needs to be studied in more complex (in field and flume scaled) stream geometry, including bends and braiding and vertical structure. Also, the presence of sediment load must be introduced, particularly relevant during flood events. Eventually, the proposed results and the way they pave for future investigation seem of interest for researchers and river managers and can contribute to the understanding of the interaction between vegetation and river geomorphology.
 The authors kindly acknowledge Eng. S. Biesuz and Eng. A. Zanetti, for their contribution to the presented research, in partial fulfillment of their master's thesis. Giuseppe Passoni, at Politecnico di Milano, is kindly acknowledged for fruitful discussion concerning scaling issues in stream flows. Jochen Aberle and three other anonymous reviewers are acknowledged for providing precious suggestions that guided the authors in making the paper more comprehensive and readable.