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Keywords:

  • solute;
  • transport;
  • dynamics;
  • random;
  • cascade;
  • model

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] The small-scale variability and scaling properties of solute transport dynamics were investigated by laboratory experiments. Dye-stained water was applied at a constant flux in a Plexiglas Hele-Shaw cell filled with soil material, and the transport process was registered by digital photographs of the cell front, producing a very high resolution in both time (minutes) and space (millimeters). The experiments comprised different cell materials (uniform and natural sand) and different water fluxes. Scaling properties of vertically integrated dye mass distributions were analyzed and modeled using so-called breakdown coefficients (BDCs), which represent the multiplicative weights in a microcanonical random cascade process. The pdfs of BDCs varied with scale, indicating self-affine scaling, and were accurately approximated by beta distributions with a scaling parameter. Two versions of BDC-based random cascade models were used to simulate mass distributions at different time steps. The results support the applicability of random cascade models to subsurface transport processes.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Solute transport in field soils is often characterized by a pronounced spatial variability [e.g., Flury et al., 1994]. This is often attributed to the presence of macropores, through which water and solutes may travel substantially faster than through the surrounding more or less homogeneous soil matrix, so-called preferential flow [e.g., Luxmoore, 1981; Flury et al., 1994]. However, also in seemingly homogeneous soils, preferential-type flow patterns have been observed, a phenomenon known as wetting front instability or fingered flow [e.g., De Smedt and Wierenga, 1984] which is related to, e.g., small-scale textural variability or water repellency. Regardless of underlying mechanism, preferential flow may have a significant impact on transported substances.

[3] Although there is a great interest to study the small-scale variability of transport velocity, progress has been limited by the difficulties associated with obtaining high-resolution observations of the process. Small-scale measurement probes based on different techniques have been developed for measurements down to the millimeter scale [Nissen et al., 1998; Ghodrati, 1999]. These can be used for studying the high-resolution temporal variability in a specific location but are not practical for observing the spatial variability. An alternative is to use dye tracers, which can provide a detailed picture of spatial transport properties over extended segments. On the other hand, dye tracer data are typically collected at only one instant in time, by excavation in dye-stained soil volumes, preventing any assessment of transport dynamics. Persson et al. [2005], however, showed how dye tracer experiments can be performed in a Plexiglas Hele-Shaw cell to obtain solute transport observations with a high resolution in both space and time. In the experiments, the dye infiltration process was recorded by digital photographing at regular time intervals. By image analysis, the photographs were converted to concentration patterns through calibration equations. The result is a sequence of very detailed dye images separated in time by only a few minutes.

[4] Concerning solute transport modeling, several approaches have been developed over the years. Physically based equations have been developed to describe solute transport in a perfectly homogeneous medium with perfect mixing of the solutes in the horizontal plane, i.e., the convection-dispersion equation (CDE). Also other fundamental concepts have been used for solute transport modeling [e.g., Jury and Roth, 1990]. Analyses of experimental field data have shown that a certain solute transport model concept may yield perfect predictions of solute transport given macroscopic parameters like pore water velocity and dispersion for a given soil type under given conditions. For different flow conditions the fundamental transport processes may differ, leading to that the same model can fail to give satisfactory results [Persson and Berndtsson, 2002]. Even in the same soil volume, the most appropriate solute transport concept might be different for different water flow regimes [e.g., Persson and Berndtsson, 1999]. This ambiguity suggests a fundamental lack of agreement between the actual processes taking place in the subsurface and the existing theories and modeling concepts. Moreover, current models do not predict the small-scale variation in solute transport. There is a need to consider alternative approaches to detailed characterization and modeling of solute transport.

[5] Owing to the development of high-resolution measurement techniques, the possibility to analyze solute transport data for statistically scale-invariant (scaling) properties has emerged. Scaling is generally expressed as a power law relationship between some statistical property, e.g., power spectral density or raw statistical moment, and a parameter representing scale or resolution. Scaling has been found for a wide range of subsurface variables, including soil aggregate sizes [e.g., Young and Crawford, 1991] and hydraulic conductivity values [e.g., Boufadel et al., 2000]. Concerning actual solute transport as represented in dye images, scaling has been found both for single images [e.g., Hatano and Booltink, 1992; Baveye et al., 1998; Öhrström et al., 2002] and two-dimensional dye patterns obtained by combining multiple images [Olsson et al., 2002]. As a mechanism to reproduce empirical scaling properties, random cascade processes have been proposed [e.g., Mandelbrot, 1974]. In general terms, these processes operate by successively increasing the resolution of the available space, simultaneously redistributing some associated quantity according to predefined rules. The rules are typically expressed in terms of the probability distribution of multiplicative weights used in the redistribution. An example of a multiplicative random cascade model is the so-called universal multifractal model of Schertzer and Lovejoy [1987], which has been used to reproduce the spatial variability in both one-dimensional conductivity data [e.g., Boufadel et al., 2000] and two-dimensional dye patterns [Olsson et al., 2002].

[6] The results of Olsson et al. [2002] indicated that the spatial variability of infiltrated solute in soil can be accurately reproduced by a random cascade process. The overall aim of the present study is to verify this indication, by applying a new random cascade concept to a new set of solute transport data. The main feature of the study is that the temporal dynamics of the spatial variability is considered, by analyses of the time sequences of dye images obtained by Persson et al. [2005]. This makes it possible to estimate the temporal evolution of cascade model parameters for the solute transport process, something not previously attempted to our knowledge. Further, for the analysis and modeling we introduce and employ so-called breakdown coefficients, which constitute a relatively simple but efficient way of exploring and utilizing scaling properties.

2. Breakdown Coefficients

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[7] The concept of breakdown coefficients (BDCs) was introduced by Novikov in the late 1960s for characterizing turbulent intermittency, and more recently further developments and tests have been made with the same application in mind [see, e.g., Novikov, 1990; Pedrizzetti et al., 1996, and references therein]. Although the concept has obvious links with the theories of (multi)scaling processes and random cascade models developed and widely applied within different geophysical fields of research since the 1980s [e.g., Schertzer and Lovejoy, 1987; Gupta and Waymire, 1990; Young and Crawford, 1991], it was not until very recently that BDCs were explicitly applied within geophysics. Menabde et al. [1997] introduced BDCs as a tool for analyzing spatial rainfall fields and developed a BDC-based model. Harris et al. [1998] extended the analysis to temporal rainfall and further investigated the procedures for BDC extraction and model parameter estimation. Both these investigations concluded that BDC analysis meaningfully complements more standard methods of detecting scaling properties, e.g., statistical moments, and provides a parsimonious framework for scaling-based modeling. Menabde and Sivapalan [2000], in their model for continuous rainfall time series, used BDCs for disaggregating the total rainfall of an event into small time increments (6 min). In the light of previous evidence of scaling in fields representing subsurface transport [e.g., Baveye et al., 1998; Boufadel et al., 2000; Olsson et al., 2002], it is reasonable to assume that the BDC concept is suitable also for describing this process, and in the following we will investigate this hypothesis.

[8] In a general sense, BDCs express the ratio between two local mean values from a certain field, obtained at two different scales (i.e., resolutions). Consider a 2-D image of infiltrated solute (dye) concentration in the xz plane (Figure 1a) and by mt(l, x)denote the vertically integrated mass of infiltrated dye in a column of width l (l0lL, where L = X), centered at location x (0 < x < X, where X is the location of the rightmost edge of the image on an x axis with its origin at the leftmost edge), at time t (Figure 1). A breakdown coefficient c may be defined as

  • equation image

where l1 < l2 and the larger-scale column covers the smaller-scale column, i.e., (x1l1/2, x1 + l1/2) ⊂ (x2l2/2, x2 + l2/2). For more general as well as comprehensive treatments of the BDC definition, see, for example, Novikov [1990], Pedrizzetti et al. [1996], and Harris et al. [1998].

image

Figure 1. (a) Observed dye concentration images at time M and (b) the vertically integrated dye mass distributions m at times (top to bottom) B, M, and E in the three experiments. Times B, M, and E refers to the beginning, middle, and end of each experiment.

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[9] Equation (1) by its generality allow for different strategies to calculate BDCs, depending on how the values of l and x are chosen. In practice, a “reverse cascade” type of procedure is generally adopted [e.g., Harris et al., 1998], which is essentially a reconstruction of the steps in a random cascade process. When applied for a certain quantity (i.e., dye mass m), a random cascade process is based on a successive division of the available space into smaller segments, typically halved, with an associated redistribution of the quantity based on multiplicative weights w. If considering dye mass distribution, in the first cascade step the total mass, i.e., Mt(L, X/2), is redistributed into

  • equation image

where w1 and w2 are associated with a theoretical probability distribution. In subsequent steps the procedure is repeated, successively doubling the resolution of the dye mass distribution, until some final desired resolution l0. The cascade step may be identified by an integer k, 0 ≤ k ≤ log2(L/l0), with n = 2k specifying the total number of l-sized columns in the step.

[10] In the “reverse cascade” approach to calculate BDCs, when applied to 2-D dye images, the mass in two adjacent columns are compared with their total mass, i.e., l2 = 2l1. This produces two complementary BDCs associated with the left and right smaller-scale column, respectively, according to

  • equation image

where c1 + c2 = 1. If we consider an image of horizontal resolution l0 and total width L = nl0, applying (3) will generate n0 BDCs at resolution l0. The extraction procedure is then repeated for successively halved resolutions 2l0, 4l0, …, producing n0/2, n0/4, …, number of BDCs. It may be noted that this “reverse cascade” implementation of (1) results in an optimal use of the data, as compared with alternative implementations, and it further makes it possible to relate BDC analysis to scaling of statistical moments [Harris et al., 1998]. It further corresponds to a microcanonical cascade framework, where the mass of the modeled quantity is strictly conserved between cascade steps (w1 + w2 = 1), as opposed to canonical cascades in which mass conservation holds only in an average sense [e.g., Schertzer and Lovejoy, 1987].

[11] The BDCs are generally characterized by their probability distribution function (pdf) [e.g., Pedrizzetti et al., 1996], which according to (3) will be symmetrical and centered at c = 0.5. Menabde et al. [1997] employ the pdfs of BDCs from different resolutions to distinguish between two types of scaling processes, self-similar and self-affine. For a self-similar process, the pdf is independent of the resolution, to within statistical scatter. For a self-affine process, on the other hand, the pdf changes with resolution. Typically, the pdfs widen with decreasing resolution, reflecting an increase in the variance. Such self-affine scaling behavior has been modeled using so-called bounded random cascade models, characterized by an explicit dependence of the multiplicative weights on the cascade step [e.g., Marshak et al., 1994].

[12] BDC-based random cascade modeling of a self-similar process may be performed by fitting some theoretical probability distribution to the empirical pdfs associated with different resolution l, which are similar by definition, and based on this theoretical distribution estimate the multiplicative cascade weights w. For a self-affine process, different possibilities exist. One is to first make the original data self-similar, e.g., by calculating absolute gradients, then extract BDCs and model as outlined above, and finally readjust to the properties of the original data, e.g., by power law filtering [Menabde et al., 1997]. This procedure has, however, some drawbacks such as loss of information in the transformations and generation of negative values [e.g., Harris et al., 1998]. A more direct approach is to find a theoretical distribution that is able to reproduce the empirical pdfs from different resolutions, and parameterize the variation of obtained parameter(s) as f(l). This prospect has been explored by Harris et al. [1998] and Menabde and Sivapalan [2000] for spatial and temporal rainfall, respectively.

[13] As (1) it is elegant and relatively easy to implement and (2) it appears to fit very well our data, we have chosen to adopt the approach of Menabde and Sivapalan [2000] in the present study. The BDC pdfs are approximated by a single-parameter beta distribution

  • equation image

where a is a shape parameter and B the beta function. The beta distribution is defined only in the interval [0 1] with an expected value of 0.5, which agrees with the properties of the BDCs as defined here. If a > 1, an increase in a makes the pdf more narrow and peaked, reflecting a decrease in variance.

[14] By fitting (4) to BDC pdfs at different resolutions l, the variation of a with l may be estimated. For rainfall, this relationship has been found to be scaling, i.e., possible to express on the form

  • equation image

where a0 is the shape parameter related to the highest resolution l0, and H an exponent describing how rapidly a decreases with decreasing resolution. We assume that this scaling type of parameterization is valid also for our data, and this assumption is investigated below. If (5) is valid, plotting a(l) as a function of l in a log-log diagram produces a straight line, the slope of which is an estimation of H. It may be remarked that for a self-similar process a(l) = a0, i.e., H = 0.

[15] Provided that the data may be properly characterized by (4) and (5), random cascade modeling is feasible. If we consider a dye image at time t, the cascade process employed by Menabde and Sivapalan [2000] starts by redistributing the total mass Mt(L,X/2) into two halves according to (2), with w1 + w2 = 1, and proceeds until desired resolution l0. The weights w are equivalent to the BDCs c obtained at the same resolution. Thus, at a certain resolution l, w may in principle be drawn from a beta distribution with parameter a estimated from (5). A difficulty, however, concerns the criterion w1 + w2 = 1, which is not trivial to fulfill when generating random numbers. Menabde and Sivapalan [2000] solves this by using the fact that the ratio of two independent random variables from a single-parameter gamma distribution

  • equation image

where Γ is the gamma function, is beta distributed with the same parameter a. To obtain the weights w, two gamma-distributed numbers x1 and x2 are generated from (6), with a estimated from (5), then w1 = x1/(x1 + x2) and w2 = x2/(x1 + x2). By this procedure, the weights become beta distributed and satisfy w1 + w2 = 1. For further details on the modeling approach, see Menabde and Sivapalan [2000].

3. Solute Transport Experimental Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[16] Solute transport data were obtained in a series of laboratory experiments, in which dye patterns were photographed and analyzed throughout the infiltration process. In this section, the experiments and analyses are described and the selected data characterized in terms of general features and descriptive statistics.

3.1. Experimental Setup and Image Analysis

[17] Two-dimensional solute transport was studied in a Hele-Shaw cell, 0.95 × 0.975 × 0.025 m (w × h × d). The material used for constructing the cell back and front walls was 0.01 m thick Plexiglas. In the bottom of the cell, ten drainage pipes were placed, spaced 0.1 m apart. The cell material was packed using the packing procedure presented by Glass et al. [1989]. The irrigation system consisted of a Plexiglas pipe, 1 m long and 0.012 m in diameter. Holes, 0.0006 m in diameter, were drilled at 0.02 m spacing along the pipe. Each end of the pipe was connected to a 5 ml syringe, which was connected to a syringe pump (Soil Measurement System, Tucson, AZ, USA). In order to further distribute the water from the pipe over the cell surface, a 0.02 m thick layer of coarse sand was placed on top of the material. Some minor problems with the irrigation system performance may have slightly affected some experiment(s). One is that a small leakage of air into the syringes may have taken place. Another is that the weight of the dyed water may have made the irrigation pipe bend slightly downward close to the center of the cell, locally increasing the flux. Further information about the experimental setup is given by Persson et al. [2005].

[18] The water was dyed using the food-grade dye pigment Vitasyn-Blau AE 85 (Swedish Hoechst, Gothenburg, Sweden), which chemically is almost identical to the frequently used dye Brilliant Blue FCF. This tracer has been used in several field experiments due to its good visibility, low toxicity, and weak adsorption on sandy soils [e.g., Flury and Flühler, 1994].

[19] Solute transport experiments were conducted in the cell using two different materials, uniform fine sand (particle size 0.3 mm) and natural loamy sand (<2 mm), and different water fluxes [Persson et al., 2005]. For the uniform sand, water fluxes of 1.47, 0.61, 0.43, and 0.31 mm min−1 were used. In the present study, we investigate only the experiments with the highest and lowest flux, respectively, denoted eUh and eUl in the following, since the intermediate fluxes produces similar dye patterns. The loamy sand experiment, denoted eN, was conducted using a flux of 0.56 mm min−1. In Table 1 a description of the different experiments is given. The average water content in the cell presented in Table 1, equation image , was estimated from dye images according to the method described by Persson et al. [2005].

Table 1. Properties of the Different Experimentsa
ExperimentMaterial (Grain Size)Water Flux, mm min−1equation image, m3 m−3Time Step, sPore Volume pv
BME
  • a

    In the experiment designation, U denotes uniform sand, N denotes natural sand, h denotes high flux, and l denotes low flux. Times B, M, and E refer to the beginning, middle, and end of each experiment.

eUhfine sand (0.3 mm)1.470.2441470.320.921.39
eUlfine sand (0.3 mm)0.310.2073260.210.701.09
eNloamy sand (<2 mm)0.560.2561800.461.452.25

[20] Each experiment began with achieving steady state conditions (dθ/dt = 0) within the cell by applying tap water through the irrigation system at the chosen water flux. Tap water was then replaced by a dye solution containing 5 g L−1 of Vitasyn-Blau AE 85. The time t for each experiment was set to 0 when the first dye reached the surface of the cell material. The experiments were continued until the entire cell was completely filled with dye-stained water. Between the experiments the cell was washed with several pore volumes of tap water. This proved to be an efficient way of completely removing all dye from the sand. Thus the two experiments conducted in the fine sand involved the same sand structure. During the course of the experiments, photos of the dye distribution in the cell as observed through the front wall were taken at preset constant intervals, ranging from 2.5 to 5.5 min depending on the water flux (Table 1). In total, between 110 and 400 photos were taken during each experiment.

[21] The photos were converted to images of dye concentration, Cs (g dm−3 of bulk soil), using the methods described by Persson et al. [2005] and Persson [2005]. As the light source was located in front of the cell, Cs is representative of the first few pores close to the front wall. The image pixel size was ∼0.0006 × 0.0006 m. In order to minimize boundary effects, pixels within 0.09 m from the sides of the box were discarded. This way, each photograph was converted into a 1441 × 1287 matrix of concentration values Cs(x, y). The pixel to pixel noise in the concentration measurements was fairly high. In order to reduce noise the images were downscaled to 573 × 512 matrices by ordinary averaging. These final concentration matrices had a pixel size of l0 = 0.0014 m. Further noise reduction was obtained by a 3 by 3 median filter was applied. The effect of filtering on the concentration noise level is discussed in detail by Persson [2005]. Analyses were performed for both nonfiltered and filtered data.

3.2. General Characterization of Dye Mass Distributions

[22] Some typical examples of concentration images from each experiment are shown in Figure 1a. Experiments eUh and eUl generated dye patterns that were relatively homogeneous. The homogeneity seemed to increase with increasing water flux. In eUh, the dye front was sharp and uniform. However, a closer look revealed some horizontal variations including a few vertical fingers of higher concentration. In eUl this fingering was more pronounced in the 0.1 to 0.5 m zone. The fingers were separated by ∼0.05 m. The location of areas with higher transport velocities was consistent between the experiments. In eN the fingering was much more pronounced with slightly wider fingers, 0.06–0.10 m. In all experiments we believe that the variation in transport velocity is mainly linked to small changes in bulk density, and thus pore geometry, and by microlayers formed during packing of the cell.

[23] The dye infiltration process in all experiment could be divided into three phases [Persson et al., 2005]. The first phase, from the soil surface down to a depth of 0.1 m, was identified as a mixing zone. Here the dye was redistributed from the area directly under each hole of the irrigation pipe to an approximately homogeneous distribution at the 0.1 m depth. This zone was dominated by horizontal mixing and a decreasing spatial variability of the dye front with depth. Fully developed unsaturated flow was evident in the 0.1–0.5 m interval. In this zone θ was constant with depth. Below 0.5 m, the increasing water content compressed the dye front and decreased its spatial variability. In the present study, we use only data obtained during the middle phase, representing fully developed unsaturated flow. The beginning and end of the analysis time window was defined as the time when the midpoint of the horizontally averaged dye front passed the 0.1 m and 0.5 m depth, respectively. Results are generally presented for three images in each experiment, representing three different times within the selected time window: beginning (B), middle (M) and end (E). In terms of time from the start of the experiments, for eUh the time window extends from 27 min (time B) to ∼2 h (E), eUl from 70 min to ∼6 h, and eN from 105 min to 8.5 h. The time elapsed may also be defined in terms of the added amount of dye solution expressed as a fraction of the water filled pore volume of the cell (i.e., pore volume times water content), we denote this amount pv. In Table 3, the values of pv for times B, M and E in each experiment are given. The higher value of pv for eN at the end of the time window, as compared with the other experiments, is due to higher retardation because of a higher clay content [see Öhrström et al., 2004].

[24] Since the measured concentration Cs is expressed in g per dm3 of bulk soil, the dye mass of each pixel was calculated by multiplying Cs by the pixel volume (in dm3). Note that the measured concentration Cs corresponds to the dye concentration in the soil solution multiplied by the water content, thus local variations in water content are included in Cs. The mass was vertically integrated over each “pixel column”, in order to produce the vector mt(l0, x) (in the following abbreviated m) representing the spatial distribution of dye mass at time t. These vectors were used in the analyses below, and in Figure 1b the dye mass distributions obtained from the three selected images from each experiment are shown. Areas with larger and smaller amounts of dye mass developed early and then persisted over time. This corresponds to areas within and between individual fingers, discussed in connection with Figure 1a. It may further be noted that the mass distributions in eUh and eUl displayed several similarities. The consistently low values of m close to the edges of the cell may partly reflect a slightly higher flux at the center of the cell, caused by the imperfect irrigation pipe (section 3.1).

[25] Table 2 contains some descriptive statistics of the observed dye mass distributions. The mean values show that the selected images correspond to times with a similar amount of dye in the box, on average ∼4 g at time B, ∼13 g at M, and ∼19 g at E. The minimum and maximum values are also fairly similar between experiments for corresponding time steps, although eUl has a slightly narrower range in the latter part of the experiment, as compared with eUh and eN. Standard deviations are also on similar levels, in all experiments increasing with time. In relative terms, however, variability decreases with time; this is reflected in the variation of CV in Table 2. Nearly all images are characterized by a negative skewness, which generally becomes more negative with time. This indicates a process where the small values become more extreme with time. In terms of solute transport, rather than fast, preferential flow in small regions, this behavior suggests that the development of small regions with slow, delayed transport is more prominent (see, e.g., Figure 1b). Kurtosis is fairly similar in all images with no clear pattern of variation. The autocorrelation ac is represented by lags 7 and 21 pixels, corresponding to length scales 0.01 and 0.03 m, respectively. Ac(7) is consistently close to 1, ac(21) lower by 0.2–0.4 units. Both are generally increasing slightly with time, most clearly for eN. This likely reflects the gradual development of sections with faster and slower transport, respectively, from a more random pattern at earlier stages (Figure 1b). In terms of descriptive statistics, the only systematic difference between on one hand the uniform sand experiments (eUh, eUl) and on the other hand the natural soil experiment (eN), is a consistently slightly lower level of the autocorrelation in the latter.

Table 2. Descriptive Statistics of m in Observations and Average Values From the Random Cascade Models (RCMe and RCMa) for Times B, M, and E in All Experimentsa
ExperimentTimeTypeMean, mgMinimum, mgMaximum, mgSD, mgCVSkewnessKurtosisAc(7)Ac(21)
  • a

    Obs, observations; B, beginning; M, middle; E, end; Ac, autocorrelation.

eUhBObs4.481.836.571.020.23−0.652.610.940.75
eUhBRCMe4.482.677.090.920.210.452.980.720.40
eUhBRCMa4.372.596.930.900.210.402.950.700.35
eUhMObs14.08.4317.42.110.15−0.932.910.960.81
eUhMRCMe14.010.518.41.730.120.242.660.740.41
eUhMRCMa14.010.518.41.730.120.232.700.750.45
eUhEObs21.013.424.92.250.11−0.983.520.950.76
eUhERCMe21.016.726.01.990.100.192.790.740.43
eUhERCMa20.916.725.81.960.090.192.710.730.42
eUlBObs3.280.865.491.010.310.002.320.910.61
eUlBRCMe3.281.247.331.260.380.743.370.710.39
eUlBRCMa3.460.919.681.780.511.014.070.720.39
eUlMObs12.17.2914.91.650.14−0.413.130.960.80
eUlMRCMe12.18.9315.71.490.120.142.670.750.42
eUlMRCMa12.09.1315.41.360.110.142.660.730.39
eUlEObs17.712.720.71.940.11−0.502.360.970.86
eUlERCMe17.714.821.01.330.080.172.690.730.41
eUlERCMa17.614.421.31.520.090.162.650.750.44
eNBObs4.252.056.290.900.21−0.322.830.880.40
eNBRCMe4.251.938.281.380.320.613.180.760.45
eNBRCMa4.512.118.481.380.310.543.060.750.43
eNMObs11.97.6416.71.920.160.092.550.910.48
eNMRCMe11.97.3618.12.370.200.342.810.760.44
eNMRCMa12.07.6317.92.260.190.382.880.750.44
eNEObs19.412.823.42.840.15−0.702.500.940.69
eNERCMe19.414.226.32.620.140.322.810.750.43
eNERCMa19.313.626.92.920.150.352.770.740.41

4. Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[26] BDC-based analysis and modeling have been performed for all images obtained within the selected analysis time window of each experiment. The presentation of results is, however, mainly made for times B, M and E in each experiment. The 3 by 3 median filtering described in section 3.2 proved to have no qualitative influence on the results. Estimated exponents were, however, slightly more stable than for nonfiltered data, and therefore results from analyses of filtered data are reported in the following.

4.1. Analysis

[27] Before looking into the behavior of BDCs we plotted the power spectra of all dye mass distributions to verify the existence of scaling. Scaling in terms of the power spectrum E(f) is expressed as

  • equation image

where f is frequency and β an exponent. It has been demonstrated that β < D, where D is the Euclidean dimension of the data (D = 1, in our case), implies a self-similar process, and β > D a self-affine process [e.g., Menabde et al., 1997]. The value of β is further related to the smoothness (or correlation) of the data, with a higher β indicating an increased smoothness [e.g., Davis et al., 1994].

[28] Figure 2a shows a typical example of a power spectrum obtained from the present data set. The spectral energy has been averaged over logarithmically spaced frequency intervals. Despite some statistical scatter at low frequencies, it is apparent that scaling is accurately respected over the entire frequency interval (R2 = 0.91), corresponding to length scales from 2.8 mm (two times the pixel size) to 0.70 m. In total over all experiments, R2 = 0.91 ± 0.03.

image

Figure 2. (a) Empirical power spectrum (circles) with fitted regression line for eUl, time M, and (b) variation of β with pore volume pv in all experiments.

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[29] Figure 2b shows the variation of spectral exponent β over time, expressed in terms of pv, in all experiments. For both eUh and eUl, the value of β exhibits an overall decreasing trend with time, with β for eUl being constantly slightly higher (equation image is 2.35 for eUl and 2.16 for eUh). The overall higher level of β for eUl suggests a higher correlation for lower water flux, and the temporal trend implies a decreasing correlation with elapsed time. The latter is however not in perfect agreement with the autocorrelation, which generally indicated an increased correlation with time (Table 2). It may be remarked that in both cases the temporal trends are weak and not always monotonous, and an alternative interpretation is that both β and ac remain essentially constant throughout the experiments, with minor deviations. Experiment eN displays a quite different pattern with an initial decrease in β, followed by a steady and rather sharp increase during most of the experiment. This increase is most probably related to the pronounced development of vertical fingers in eN, i.e., the dye is successively concentrated into preferred sections (Figure 1b). This increases the overall dye mass variability but also the correlation, as compared with earlier stages of the transport process. This behavior is also clearly supported in the autocorrelation analysis (Table 2). This qualitative difference in the temporal evolution of β may reflect the properties of the solute transport process in uniform and natural soils, respectively. We finally note that β is constantly distinctly above 1, indicating a self-affine process.

[30] We now turn to the BDC analysis, and Figure 3a shows some typical examples of pdfs and their variation with resolution. We use an integer i to denote the degradation in resolution from 2il0 to 2i+1l0, i.e., for our data i = 0 means going from l = l0 = 1.4 to l = 2.8 mm, i = 1 from 2.8 to 5.6 mm, etc. Overall, the value of c is close to 0.5, indicating a limited difference between the dye mass in two adjacent columns of equal width. There is, however, a clearly self-affine pattern, where the pdfs change to reflect an increased variance in c with decreasing resolution.

image

Figure 3. (a) Empirical pdf of c as a function of resolution shift i for eUl, time M, and (b) the standard deviation of c as a function of i for times B, M, and E in eUh.

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[31] The pdfs from all images in all experiments are similar to Figure 3a, but one trend is found with time elapsed in each experiment, and this is shown in Figure 3b. The increase in standard deviation of c with i corresponds to the change in pdf shapes shown in Figure 3a, but the standard deviation further decreases with time for a particular i. This pattern is initially somewhat surprising, as it seems to indicate that the dye mass variability decreases with time, in contrast with the picture in Figure 1b. It is, however, a reflection of the fact that CV decreases with time in the experiments (Table 2). This behavior, i.e., that the increase in mean value over time dominates over the increase in variance, is reflected in c values approaching 0.5 with time, which makes standard deviation decrease.

[32] Beta distributions have been fitted to all empirical pdfs by the MATLAB function BETAFIT, and Figure 4 shows some examples of such fits, for the pdfs with i = 0, i = 2, and i = 4 in Figure 3a. Note that in Figure 4 smaller c intervals are used; therefore the y axis' scale is different from Figure 3a. Generally the fit is very accurate for low i values, i.e., high resolutions (e.g., Figure 4a; histogram estimated from 512 values). As i increases, fewer values become available for estimating the pdf and the statistical scatter increases; even so beta distributions generally provide at least reasonable approximations also for larger i values (e.g., Figure 4c; histogram estimated from 32 values).

image

Figure 4. Empirical pdf of c and fitted beta distribution for eUl, time M, for (a) i = 0, (b) i = 2, and (c) i = 4.

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[33] To test the accuracy of the fits, beta-distributed samples were generated using the parameters estimated from the empirical pdfs. Kolmogorov-Smirnov testing was used to test the hypothesis that the generated samples and the corresponding set of c values have the same distribution. At significance level 0.05, the hypothesis could not be rejected for any pdf in the range 0 ≤ i ≤ 4. For i > 4 the hypothesis was rejected but only in a few cases, but this is not unexpected as the number of values available for estimating the pdf is 16 or less and proper distribution fitting not possible.

[34] Figure 5 shows an example of how the beta distribution parameter a varies with resolution l, for the case used in Figures 3a and 4 (eUl, time M). It is apparent that the scaling property expressed by (5) holds very well over most of the l range. At large l values (∼0.05–0.1 m), the curve appears to level off and approach a constant a value. This tendency is to various degree present in all images and is most probably related to the very limited data at these resolutions, although it may also reflect a characteristic scale related to the observed fingering structures (section 3.2). If performing the regression in the range 0 ≤ i ≤ 5, as in Figure 5, in total over all experiments R2 = 0.99 ± 0.01.

image

Figure 5. Beta parameter a as a function of resolution l (symbols) and fitted regression line for eUl, time M. Solid circles were not used in the regression.

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[35] The parameters a0 and H in (5) may be estimated from straight regression lines such as in Figure 5; a0 as the y axis intercept and H as the (negative) slope. Figure 6 shows the variation of these parameters during the course of eUh. Whereas H remains relatively constant, a0 increases steadily in an approximately linear fashion. The picture is similar for experiments eUl and eN. This overall increase of the a parameter with time corresponds to the decrease of the standard deviation of c shown in Figure 3b, and thus reflects the temporal evolution of CV. It is difficult to discern any clear and systematic variation in the values of a0 and H between the different experiments. The value of H is ∼1.45, slightly higher for eN and slightly lower for eUh. The value of a0 varies within a wide range (Figure 6), and the limits are of a similar order of magnitude in the different experiments.

image

Figure 6. Temporal variation of parameters a0 and H during eUh (symbols) and fitted approximations (solid lines).

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4.2. Modeling

[36] In the previous section, the validity of (4) and (5) were verified for the present data. Thus random cascade modeling according to Menabde and Sivapalan [2000], as outlined in section 2 above, is possible.

[37] This is achieved by estimating parameter a from (5), defining a gamma distribution with parameter a according to (6), and from it drawing the random numbers x1 and x2 used to generate the weights w1 and w2. One complication, however, concerns the fact that a approaches a constant value at low resolutions, as shown in Figure 5. If using (5), a will thus be underestimated at these resolutions, which gives a too high probability for very uneven redistributions of mass in the first few cascade steps, in turn producing an overestimated variance in the final field. To solve this, in the model (5) is reformulated as

  • equation image

to reflect the pattern in Figure 5. This does lead to some, generally minor, overestimation of a at large l values, which does not appear to have any negative influence on model performance. Often (5) is valid also for l > 44.8, but for consistency l = 44.8 is used as the limit throughout. We emphasize that the need for this reformulation is most likely related to the limited size of our experimental box, rather than reflecting some real discontinuity in the scaling properties of solute transport.

[38] Modeling was performed by generating a number of realizations, calculating average properties, and comparing with the observations. Different numbers of realizations were tested, but 100 proved sufficient for obtaining stable estimates of the average properties. The model has two parameters, a0 and H, and a small sensitivity test illustrates their effect on the generated field. In Figure 7, the (average) pdfs of the generated fields are shown for three combinations of a0 and H, with values similar to the ones obtained for the present data. Essentially decreasing a0 or increasing H has much the same effect of widening the pdf, increasing the variance.

image

Figure 7. Principal effect of model parameter values (a0, H) on the pdf of the generated field.

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[39] In the first round of cascade model experiments, we used the exact parameter values (and the exact amount of dye mass) obtained from each image. We denote this “exact” random cascade model RCMe. This theoretically inclined type of test thus investigates the ability of the model to reproduce the observations when parameter values are exactly known.

[40] Figures 810 show some examples of results from RCMe, for the middle time step in each experiment. In Figures 810a, typical examples of single realizations are shown. Overall, both the range and the variability in the model output well mimics the observed patterns. The variability in the model realizations does, however, appear somewhat overestimated. In particular, the model outputs contain sudden “jumps” in the value of m, a typical feature of cascade processes, whereas in the observations transitions are more gradual. Also the small-scale fluctuations appear somewhat more pronounced in the model, especially for eN in which the observed data are notably smoother than in the uniform soil experiments. There are several possible explanations for the more discontinuous and erratic nature of the model generated dye mass distributions, as compared with the smoother observed ones. Conceivably, with time elapsed differences in dye mass between adjacent columns are gradually smoothed out due to local-scale mixing, which may be reflected in the increasing autocorrelation of m with time (Table 2). Such local-scale mixing is not accommodated in the RCM model, which instead redistributes mass between adjacent intervals, at all scales. Owing to the self-affine property of the model, however, redistribution becomes less pronounced with decreasing scale. Still, this may be a conceptual limitation of the RCM approach when applied in the present context, which partly explains the overestimated small-scale fluctuations. Another reason for the discontinuities is the discrete character of the RCM model, which make “jumps” from larger-scale redistributions remain unaffected as the process proceeds to smaller scales. It may be considered to apply some postfiltering procedure to adjust for this property. A third reason may be that the fitted beta distributions do not perfectly reproduce the empirical BDC distribution, but sometimes the probability of having BDCs close to 0.5 is slightly underestimated (see, e.g., Figure 4c), leading to an overestimated frequency of “jumps”.

image

Figure 8. (a) Example of random cascade model realization (RCMe) compared with observations (Obs) for eUh, time M, and (b) agreement in terms of pdfs.

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image

Figure 9. (a) Example of random cascade model realization (RCMe) compared with observations (Obs) for eUl, time M, and (b) agreement in terms of pdfs.

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image

Figure 10. (a) Example of random cascade model realization (RCMe) compared with observations (Obs) for eN, time M, and (b) agreement in terms of pdfs.

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[41] Figures 810b shows comparisons between the observed pdf and the average pdf obtained from 100 model realizations. Considering the variability in the observed pdfs, the generated pdfs overall reasonably well approximate the main features of the observed ones, although differences exist for the single cases shown. The model pdfs, however, tend to be shifted slightly to the right as compared with the observed ones, indicating somewhat overestimated minimum and maximum values.

[42] Model evaluation in terms of descriptive statistics is shown in Table 2, where the model results are averages over 100 realizations. The mean value is by default identical. As suggested from Figures 810b, both the minimum and maximum values are generally overestimated by the model, the minimum by on average 20% and the maximum by 12% of the observed value. Contrary to what could be expected from visual inspection of Figures 810a, standard deviation is generally slightly underestimated by the model, for six out of the nine images. On average, however, it is slightly overestimated (∼1.5%), mainly owing to pronounced overestimation during the early and middle stages of eN. The observed increase in standard deviation with time is reproduced by the model except for one single time step (eUl, time E). Naturally, as mean values are identical, CV follows an identical pattern with many underestimations but a slight overestimation in total, and a correct principal variation in time. A qualitative difference between observations and model outputs is found for the skewness, which is generally negative for the observations and always positive for the model output. The values are, however, small and the skewness thus not very pronounced, as also indicated in Figures 810b. The temporal trend found in the observations, toward more negative skewness with time, is qualitatively reproduced by the model. Kurtosis is on average overestimated by 7% in the model output. As suggested from the “jumpy” nature of the realizations (Figures 810a), autocorrelation is underestimated in the model output, by on average 21% for lag 7 pixels and 34% for lag 21. The general (weak) increase in autocorrelation with time found in the observations is not clearly reproduced by the model, but the values are similar for all time steps.

[43] In the second round of experiments, the temporal variation of parameter values and dye mass were approximated by simple functions; this “approximate” random cascade model is denoted RCMa. This represents a more practically inclined perspective, for example aimed at predicting future evolutions of the dye mass distribution.

[44] Concerning parameters a0 and H, in RCMa these were estimated from functions such as shown in Figure 6, i.e., H was approximated by its mean value equation image and a0 as a linear function of the fractional pore volume, i.e.,

  • equation image

where c1 and c2 are constants. As seen in Figure 6, such approximations are well applicable over the entire time interval. In Table 3 the values of the RCMa parameters equation image , c1 and c2 are given, as are the resulting average errors of equation image and a0 within the analysis time window. For eUh and eN the average error is of order 5% or below, whereas the error of a0 caused by the linear approximation in eUl reaches 16%. In RCMa, dye mass was expressed as a function of time. As the flux was constant during the experiments, this function is linear in theory. However, in practice the flux decreased slowly with time, probably owing to the small leakage of air into the syringes (section 3.1). Therefore the linear function had to be replaced by a third-order polynomial, which however closely reproduced the observed flux, the difference being on average 2% (Table 2).

[45] The results from RCMa are summarized in Table 2. Overall, the results from RCMa are very similar to the results from RCMe. In fact, out of all 72 comparisons (nine images, eight statistics (mean excluded)), in 32 cases RCMa performs better than RCMe, in 11 cases they are equal, and only in 29 cases is RCMe superior. On average, however, the difference from observations is 2% higher in RCMa, as compared with the difference for RCMe. Table 3 shows the overall relative model error for both models RCMe and RCMa in all three experiments. This error was calculated using the statistics in Table 2 except skewness which had an unreasonably large impact on the overall error. For eUh and eN, with only small errors of equation image and a0 in RCMa, the RCMa result is similar to or slightly better than RCMe. For eUl, the 6–16% error in model parameters is reflected in a 5% increase in the RCMa model error. The use of approximate parameter values does not seem to have affected any particular statistic. Instead, in cases where the approximations provide less accurate parameter values, performance deteriorates in terms of all statistics (e.g., eUh, time E, and eUl, time M). We conclude that the model is relatively insensitive to limited uncertainties in the parameter values.

Table 3. Parameter Values in the RCMa Approximations (Mean Value equation image and Coefficients c1, c2 in (9)), Corresponding Average Errors in Model Parameters (H, a0), and Comparison of Overall Model Error Between RCMa and RCMe
ExperimentEstimated RCMa ParametersModel Parameter Error, %Model Error, %
equation imagec1c2Ha0RCMaRCMe
eUh1.36521801−17392.54.618.918.4
eUl1.42248155−103746.01627.522.2
eN1.537777812701.15.814.816.1

5. Summary and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[46] Three laboratory experiments were performed to investigate the small-scale variability and temporal evolution of solute transport. A Plexiglas box was filled with soil (uniform sand or natural soil) after which dyed water was added from the top with a constant flux. The infiltration process was documented by successive photographs of the tank front (1 × 1 m). These images were processed and converted into horizontal transects, representing the spatial variability of vertically integrated dye mass.

[47] Scaling properties of the resulting data sets were analyzed in terms of power spectra and breakdown coefficients (BDCs). The spectra consistently exhibited the power law shape characteristic of scaling processes, with values of the exponent β generally in the range 2–2.5. The pdf of BDCs exhibited a clearly self-affine behavior, changing regularly in the direction of increased variance with decreasing spatial resolution. The BDC variance further decreased with time in the experiments, reflecting a decrease in CV. One-parameter beta distributions proved to well approximate the empirical pdfs, with the parameter a being related to resolution in a scaling fashion, defined by parameters a0 and H.

[48] In light of the accurate description of the data obtained in the BDC analysis, BDC-based random cascade modeling was performed using the model described by Menabde and Sivapalan [2000]. The model employs a one-parameter gamma distribution to generate the cascade weights, where the parameter is estimated from the scaling relationship of the beta parameter a. Some modification of the scaling relationship was required for low resolutions, as the limited number of values invalidated distribution fitting.

[49] In the first round of model experiments, the exact parameter values obtained from a certain image were used to reproduce the dye mass distributions. Single realizations were generally visually similar to the observed data in terms of range and variability, although sudden “jumps” were more frequent in the model output. Also the mean pdf from 100 realizations was overall reasonably similar to the observed pdf. The main differences between observed and modeled dye mass distributions were (1) a clear underestimation of the autocorrelation, caused by the “jumps” in the model output, and (2) a consistently positive skewness in the model generated pdfs, as opposed to the generally negative skewness in the observations.

[50] In the second round of experiments, approximate parameter values obtained from simple functions of time were used. The limited differences as compared with the exact parameter values proved to have a very small influence on model performance, which was similar to the “exact” model. This indicates that the approach is rather robust, and that parameters may be properly estimated from simplified relationships.

[51] We conclude that the concept of BDCs appears meaningful and suitable for describing small-scale solute transport. The pdfs of BDCs were overall accurately characterized by self-affine beta distributions; the two-parameter model overall well reproduced key features in the observations. These findings support the hypothesis that subsurface transport of water and solutes exhibits scaling properties. A novel contribution from the investigation is the assessment of how the scaling properties vary in time. The spectral analysis indicated a possible qualitative difference between uniform and natural soil in this respect. In the modeling, however, no clear differences in parameters were found between the uniform and the natural soils, which raises some concern whether model parameters can be meaningfully related to general, physical soil properties. However, the only reasonably clear difference between the soil types in terms of descriptive statistics was found in the autocorrelation, and this was one of the properties least well reproduced by the model. Besides the autocorrelation, another limitation of random cascade models concerns the ability to reproduce skewness, in particular negative skewness. These issues clearly require further investigation before the model can be developed for practical purposes, e.g., predictive simulations under field or experimental conditions. Numerical and physical experiments in a wide range of soil types and flow conditions need to be conducted and analyzed, and such extended experiments are to commence in the near future.

[52] Finally, the possibility to describe solute transport by BDCs in a qualitatively different way is currently under investigation. By the present approach it appears possible to generate a realistic solute front or mass distribution at a certain time step, or by, e.g., Monte Carlo simulation estimate statistics, but the models used at different time steps are independent. Another possibility would be to model the evolution over time as one single random cascade. In a qualitative sense, this type of RCM would produce successively more detail as the solute moves downward. The value of such an approach is not obvious in light of the sand experiments presented here, in which spatial correlation generally increases with depth. However, in other soils the temporal evolution may certainly be quite different, notably if flow through macropores and preferential pathways is pronounced. A preliminary study has supported the feasibility of the approach for experimental data but also highlighted some difficulties. For example, to reproduce extended segments with higher or lower transport, a spatial dependence of the cascade weights appears to be required. Model development is in progress and we will present these results elsewhere.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[53] This study was funded by the Swedish Research Council, the Japan Society for the Promotion of Science, and the Scandinavia-Japan Sasakawa Foundation. We thank five anonymous reviewers for helpful and constructive criticism.

References

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  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Breakdown Coefficients
  5. 3. Solute Transport Experimental Data
  6. 4. Results and Discussion
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
wrcr10650-sup-0001-t01.txtplain text document1KTab-delimited Table1.
wrcr10650-sup-0002-t02.txtplain text document2KTab-delimited Table 2.
wrcr10650-sup-0003-t03.txtplain text document0KTab-delimited Table 3.

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