### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Breakdown Coefficients
- 3. Solute Transport Experimental Data
- 4. Results and Discussion
- 5. Summary and Conclusions
- Acknowledgments
- References
- 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

- Top of page
- Abstract
- 1. Introduction
- 2. Breakdown Coefficients
- 3. Solute Transport Experimental Data
- 4. Results and Discussion
- 5. Summary and Conclusions
- Acknowledgments
- References
- 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

- Top of page
- Abstract
- 1. Introduction
- 2. Breakdown Coefficients
- 3. Solute Transport Experimental Data
- 4. Results and Discussion
- 5. Summary and Conclusions
- Acknowledgments
- References
- 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 *x*–*z* plane (Figure 1a) and by *m*_{t}(*l*, *x*)denote the vertically integrated mass of infiltrated dye in a column of width *l* (*l*_{0} ≤ *l* ≤ *L*, 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

where *l*_{1} < *l*_{2} and the larger-scale column covers the smaller-scale column, i.e., (*x*_{1} − *l*_{1}/2, *x*_{1} + *l*_{1}/2) ⊂ (*x*_{2} − *l*_{2}/2, *x*_{2} + *l*_{2}/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].

[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., *M*_{t}(*L*, *X*/2), is redistributed into

where *w*_{1} and *w*_{2} 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 *l*_{0}. The cascade step may be identified by an integer *k*, 0 ≤ *k* ≤ log_{2}(*L*/*l*_{0}), with *n* = 2^{k} 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., *l*_{2} = 2*l*_{1}. This produces two complementary BDCs associated with the left and right smaller-scale column, respectively, according to

where *c*_{1} + *c*_{2} = 1. If we consider an image of horizontal resolution *l*_{0} and total width *L* = *nl*_{0}, applying (3) will generate *n*_{0} BDCs at resolution *l*_{0}. The extraction procedure is then repeated for successively halved resolutions 2*l*_{0}, 4*l*_{0}, …, producing *n*_{0}/2, *n*_{0}/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 (*w*_{1} + *w*_{2} = 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

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

where *a*_{0} is the shape parameter related to the highest resolution *l*_{0}, 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*) = *a*_{0}, 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 *M*_{t}(*L*,*X*/2) into two halves according to (2), with *w*_{1} + *w*_{2} = 1, and proceeds until desired resolution *l*_{0}. 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 *w*_{1} + *w*_{2} = 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

where *Γ* is the gamma function, is beta distributed with the same parameter *a*. To obtain the weights *w*, two gamma-distributed numbers *x*_{1} and *x*_{2} are generated from (6), with *a* estimated from (5), then *w*_{1} = *x*_{1}/(*x*_{1} + *x*_{2}) and *w*_{2} = *x*_{2}/(*x*_{1} + *x*_{2}). By this procedure, the weights become beta distributed and satisfy *w*_{1} + *w*_{2} = 1. For further details on the modeling approach, see *Menabde and Sivapalan* [2000].

### 5. Summary and Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Breakdown Coefficients
- 3. Solute Transport Experimental Data
- 4. Results and Discussion
- 5. Summary and Conclusions
- Acknowledgments
- References
- 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 *a*_{0} 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.