Demonstrating fractal scaling of baseflow residence time distributions using a fully-coupled groundwater and land surface model



[1] The influence of the vadose zone, land surface processes, and macrodispersion on the shape and scaling behavior of residence time distributions of baseflow is studied using a fully coupled watershed model in conjunction with a Lagrangian, particle-tracking approach. Numerical experiments are used to simulate groundwater flow paths from recharge locations along the hillslope to the streambed. These experiments are designed to isolate the influences of topography, vadose zone/land surface processes, and macrodispersion on subsurface transport of tagged parcels of water. The results of these simulations agree with previous observations that such distributions exhibit a power law form and fractal behavior, which can be identified from plots of the residence time distribution and the power spectra. It is shown that vadose zone/land surface processes significantly affect both the residence time distributions and their spectra.

1. Introduction

[2] The observation by Kirchner et al. [2000] that long-term time series of stream chemistry exhibit fractal behavior has prompted increased interest in residence time distributions of groundwater from the streambed scale (100 m) to the continental scale (106 m) [e.g., Wörman et al., 2007]. The main focus of previous work has been on the role of subsurface heterogeneity in solute transport [Haggerty et al., 2000; LaBolle et al., 2006; Maxwell et al., 2003; Tompson et al., 1999], the patterns of vegetation [Scanlon et al., 2007] and the influence of the topography of the upper boundary [Cardenas, 2007; Haggerty et al., 2002; Kirchner et al., 2001; Wörman et al., 2007]. All of these processes are shown to be scale dependent, for example the influence of a topographical upper boundary to groundwater flow may range in scale from streambed ripples to the continental scale. The work presented here illustrates scale dependant effects determined from a fully-integrated, numerical watershed model that incorporates aspects of all these systems, and points to the relative importance of these various component processes.

[3] For steady state conditions and based on the assumption that the water table closely follows the topography of the upper boundary, one can show that topography induces groundwater flow with power law or fractal behavior even if the subsurface is homogeneous. This stems from the presence of stagnation points in the flow field, i.e. locations where the flow velocities are zero. These stagnation points generate velocity distributions over a wide range of scales that lead to a wide range in residence time distributions of groundwater [Cardenas, 2007]. In addition, subsurface heterogeneity can enhance power law behavior of residence time distributions by additionally producing a range of groundwater velocities [Haggerty et al., 2000].

[4] The aforementioned studies are based on a suite of assumptions such as steady-state conditions of the potential field and that processes in the vadose zone and at the land surface do not exert any influence on subsurface flow, i.e. only flow in the saturated zone is considered. However, even if there is a free water table that closely follows the topography, it may be dynamic due to diurnal and seasonal variations in atmospheric forcing and vegetation dynamics. In this case, a steady state solution of the potential field does not suffice. Thus, in order to examine the influence of the land surface and the shallow subsurface on the residence time distributions of groundwater, one requires a more advanced analysis that takes into account the pertinent physical processes, such as three-dimensional variably saturated groundwater flow, root water uptake by plants, evaporation from bare soil, infiltration, and overland flow. This has been discussed previously by Reed et al. [2006], who pointed to the need for field measurements on the watershed scale and the development of simulation tools treating the subsurface-land surface-atmosphere system in an integrated fashion.

[5] In this study, a novel watershed simulation platform is applied to a catchment with an area on the order of 103 km2. The simulation platform consists of a parallel, three-dimensional, variably saturated groundwater/surface water flow code, coupled to a land surface model. This fully-coupled model accounts for pertinent processes at, across and below the land surface and is forced by an atmospheric time series, thus, relaxing many of the assumptions made in previous studies. Transient particle tracking of a conservative tracer is used to derive residence time distributions of parcels of water recharged either at the water table or at the land surface (thus being influenced by vadose and root zone processes) for a range of macrodispersion values. In the ensuing analysis, the power spectra of the distributions of residence time and their slope are computed. The different slopes of the power spectra are related to the simulated processes and reflect distinct changes in the statistical properties of the breakthrough curves of the simulated tracer.

2. Methods

[6] In order to arrive at residence time distributions that were used in the spectral analysis, a two-step numerical experiment was performed. First, an integrated watershed simulation platform was applied to a watershed in central Oklahoma (USA), which resulted in a time series of three-dimensional pressure fields in the subsurface. Second, a Lagrangian, particle-tracking method was applied in conjunction with these transient pressure results to develop residence distributions of subsurface water for the spectral analysis. Both steps are outlined in more detail below.

2.1. Integrated Watershed Simulations

[7] The pressure fields for the particle tracking experiment were obtained from simulations using an integrated watershed numerical code. The methodology and simulation are described in detail by Kollet and Maxwell [2008]. The numerical code consists of ParFlow, a parallel, three-dimensional variably saturated groundwater/surface water flow code with an integrated land surface model. The land surface model is the Common Land Model (CLM) [Dai et al., 2003] and calculates the mass and energy balance at the land surface. ParFlow calculates the moisture redistribution in the shallow subsurface that is influenced by evapotranspiration and infiltration as well as deep groundwater flow. For technical details we refer the reader to Ashby and Falgout [1996], Jones and Woodward [2001], Kollet and Maxwell [2006, 2008], and Maxwell and Miller [2005].

[8] As described in detail by Kollet and Maxwell [2008] the simulation platform was applied to the Little Washita watershed, Oklahoma, USA for the water-year 1999. This model of the Little Washita watershed consisted of a deep, homogeneous aquifer (∼102 m), topography, spatially distributed land and soil cover, and overland flow parameters. A one year time series of spatially uniform atmospheric forcing was applied from September 1998 until August 1999 in spinup mode until a dynamic equilibrium was obtained. The spinup procedure resulted in the development of the Little Washita River in the model domain that is the locus of particle injection in the particle tracking experiment.

2.2. Calculation of Mass Flux to the River and Spectral Analysis

[9] A Lagrangian, particle-tracking approach, described in detail in previous work [Maxwell and Kastenberg, 1999; Maxwell and Tompson, 2006; Maxwell et al., 2003, 2007; Tompson et al., 1998], was used to simulate the evolution of age of tagged parcels of water. Particle-tracking methods have been widely applied in subsurface transport problems [e.g., LaBolle et al., 1996; Tompson and Gelhar, 1990]. This particular particle model has been previously applied to simulate water age and to interpret isotopic observations [e.g., Maxwell et al., 2003; Tompson et al., 1999].

[10] Particles were placed at the bottom of fully saturated riverbed cells, located within the primary watershed, at a density of 5,000 particles per grid cell for a total number of particles, Np = 765,000. Pressure fields from Kollet and Maxwell [2008] were advanced daily for 500 years for a total of 182,500 timesteps, i.e. the pressure field time series from the one year spinup was repeated 500 times. As in the work by Tompson et al. [1999] and Maxwell et al. [2003], cell velocities are reversed and particles are transported backwards to the source location. As only the primary watershed was used, particle transport occurred away from most of the major model boundaries and care was taken to ensure accurate calculation of particle movement near the bottom of the domain. For the water table cases (case WT), to mimic an isotopic tracer such as 3H that will re-equilibrate when exposed to the atmosphere, particles were stopped when the relative saturation dropped below 0.95 and their travel times were recorded. For the vadose zone cases (case LS) particles were not stopped until they reached the land surface. Macrodispersion was added to these transport simulations to represent the dispersive effect of subsurface heterogeneity on the distribution of residence times. While a very approximate representation, this approach allowed for varying the Peclet number, Pe [−], (∼L/αl) by varying the longitudinal dispersivity, αl [L], over four orders of magnitude and using a hillslope scale of L = 7.5 km from Kollet and Maxwell [2008]. The four values of Pe = 75,000; 7,500; 750 and 75 were applied in the WT case and the LS case, which resulted in eight simulation cases in total. For all simulations the transverse dispersivity was set to αl/10. An example of the spatial distribution of travel time generated using this approach may be seen in Figure 1a.

Figure 1.

(a) Three dimensional plot of backwards in time streamlines for the Little Washita watershed. Particle ages are depicted along each pathline in years. The watershed outline is plotted as the solid black line. The time scale is in years of traveltime. Also shown are scatterplots of relative mass fraction as a function of time for particles that arrive in the river for decreasing Peclet numbers for (b) case WT and (c) case LS. Distributions binned at daily intervals shown as symbols and smoothed distributions created by using lognormally-increasing bins shown as lines.

[11] Particle travel times from recharge locations to the river bed were binned into one-day increments to create a distribution function of mass fraction. These distributions, shown in Figures 1b and 1c, may be interpreted in two ways: 1) as the distribution of age of water that enters the river as baseflow; or 2) as an arrival with time, much like a breakthrough curve, of a conservative tracer initiated at the land surface or the water table. Due to the discrete nature of particle tracking, some of the bins contained zero values, which were omitted creating an unevenly sampled distribution. These distributions where then transformed into the spectral domain using the Lomb-Scargle technique for uneven data [Lomb, 1976; Scargle, 1982] as implemented by Press et al. [1996]. The resultant spectral power-wavelength plots are shown in Figure 2.

Figure 2.

Logarithmic plot of spectral power as a function of wavelength for different values of dispersivity (noted by the Peclet number in each plot) for simulations ending at the a) water table (case WT) and b) the ground surface, thus, including vadose zone processes (case LS). Note that raw spectra are shown in gray, smoothed spectra in black and a linear fit of the spectra shown with the dashed line with the slope (m) given for each case. Note also that the power spectra are represented in terms of the Lomb periodogram [Lomb, 1976].

3. Results and Discussion

[12] Figures 1b1c shows the fraction of mass arriving in the river as a function of time for the two cases for all Peclet numbers simulated. The curves in Figure 1 use two time averaging windows, one with uniform, one day averaging (colored symbols) and one with a lognormally increasing average (black lines, solid and dashed). These curves all show a characteristic power law shape and indicate baseflow is comprised of water over a wide range of ages. Increases in macrodispersion (shown as decreasing Pe in Figure 1) change the shape and the slope of the mass breakthrough curves. Figure 1b shows a shape and trend that agree with the results shown by Cardenas [2007, Figure 4a], which shows simulated concentration breakthrough for a steady-state solution of the Toth problem. While the addition of vadose and land surface processes (Figure 1c) do not change the overall power-law behavior shown by the WT case, these processes do have an effect on the shape of the mass fraction curves and the influence of dispersion. The different curves with uniform, daily-averaging of mass fraction display fluctuations at a number of points in time, which is due the interplay of the transient forcing used in this numerical experiment, the presence of hydrodynamic stagnation points in the domain, and vadose zone processes. This behavior is further interpreted using spectral analysis below.

[13] Figure 2 shows the power spectra of the mass fraction curves shown in Figures 1b and 1c. In all simulations, the spectra demonstrate that the power law behavior shown in Figures 1b and 1c exhibits fractal scaling that has been observed previously in experimental and theoretical studies [e.g., Kirchner et al., 2000]. Increases in macrodispersion (i.e. a decrease in Pe due to an increase in dispersivity) act as a low-pass filter and smooths the curves similar to the moving average that is also shown in Figure 2. The power spectra show slopes that decrease after one year. This could be because the analysis is based on a pressure distribution forced by a one year time series repeated 500 times.

[14] In case WT (Figure 2a), the best-fit slopes of the power spectra, m, range between 1.16 and 0.74. The range of m agrees with reported values in the literature from experimental and theoretical studies. For example, Kirchner et al. [2000] obtained similar values from the analysis of chloride concentrations from a number of small catchments (∼100 km2) spanning a variety of climate and hydrologic conditions. In theoretical studies, Kirchner et al. [2001] only obtained model results that produced fractal scaling by using very large macrodispersivities, i.e. Pe < 1. Thus, the range of m obtained from the measured data was explained by the degree of heterogeneity. The results presented here in Figure 2, however, show fractal scaling for Pe values up to 75,000, many orders of magnitude larger than previous studies. This is a confirmation of previous results that an undulating topography, which is directly accounted for in the current work, plays a significant role in the apparent dispersion of residence times of parcels of water in the subsurface [e.g., Wörman et al., 2007]. For the first time, this study demonstrates that the influence of topography persists under transient conditions.

[15] The current work also includes vadose and root zone processes. Some tracers such as chloride, used in the work of Kirchner et al. [2000], as opposed to some isotopic tracers, do not re-equilibrate when exposed to the atmosphere. Chloride residence times will then reflect the time history of pathways between the ground surface and water table. Thus, vadose zone processes, not included in other studies, need to be considered in the interpretation of such data. Figure 2b shows the power spectra for case LS, where the vadose and root zones are included in the simulations of travel times. Inspection of the spectra for different Pe values shows that m of the power spectra increases with increasing macrodispersion. The slopes now range from 1.04 to 1.38. In case of Pe = 75, the power law behavior appears to weaken for larger time scales and approaches a more exponential behavior.

[16] Juxtaposition of Figures 1b and 1c with Figures 2a and 2b show directly the influence of the vadose zone and root zone on the mass flux distribution and associated spectra for varying degrees of macrodispersion. These zones add additional processes acting on particles as they follow parcels of water, which change the shapes of both the mass flux distributions and their power spectra. Vadose zone processes also introduce more noise in the power spectra (as shown in Figure 2b) and result in a lower fraction of early arrival (Figure 1c). This might be because particles (and thus parcels of water) become “locked” in the vadose zone through continuous redistribution due to evapotranspiration and infiltration. Vadose and root zone processes may, in a sense, introduce additional stagnation points, similar to those found in the classical Toth problem analysis, and introduction of macrodispersion may smooth the effects of those stagnation points on the power spectra of the age distribution. This argument is reinforced by the increase in early arrival and steepening breakthrough curve resulting from the addition of macrodispersion (Figures 1b and 1c).

[17] The slope of the power spectra, m, may be used as an indicator or measure of stationarity of a time series [Davis et al., 1994]. By this, it is meant that the statistics of the fluid parcel travel time series are invariant to temporal translation. For m < 1, the time series is generally stationary in this sense, while for m > 1, it is generally not. In the current simulations inclusion of the vadose zone appears to increase the spectral slopes from less than one in case GW to greater than one in case LS (particularly for the Pe = 75 cases with the largest dispersivity values). Although there is significant scatter in Figure 2, this suggests that nonstationary behavior in the mass fraction time series may be generated by the influence or contributions of vadose and root zone effects. Despite being a relatively thin interface, the vadose zone has a significant influence on residence time distributions, because of the non-linearity of variably saturated flow and evapotranspiration, which depends on the moisture state of the shallow subsurface. The additional locations where water may be in residence for a significant portion of time might be responsible for this shift in behavior of the power spectra.

4. Summary

[18] Using a fully-coupled, groundwater, vadose zone and land surface model in conjunction with a Lagrangian particle tracking model, a series of residence time distributions were developed for recharge from the land surface and water table to the riverbed. These residence time distributions may be expressed as a mass breakthrough in the river over time, plots of which demonstrate a power-law type behavior and indicate that baseflow in the simulated system represents water comprised of a very wide range of age. A spectral analysis of these mass fraction distributions demonstrated power spectra that exhibit fractal behavior over a wide range of Pe numbers previously documented in observational studies. Incorporation of vadose and root zone processes changes the shape of the mass flux distributions and their power spectra. Thus, the vadose zone interface of the shallow subsurface has a profound influence on the scaling behavior of the residence time distribution that needs to be considered in the interpretation of a conservative tracer introduced at the land surface. Following the notion of the slope of the power spectra, m, being an indicator of stationarity, the processes included in the vadose zone case may result in a nonstationary mass fraction time series. That fractal scaling is demonstrated for much larger Pe numbers than in previous studies confirms the importance of topography under transient conditions. However, this study also shows the influence of land surface and vadose zone processes on the apparent macrodispersion of solutes in groundwater. This indicates that the observed fractal scaling behavior in watersheds might be explained through a combination of these physical processes and not aquifer heterogeneity alone. The model of dispersion used in this work is a surrogate for the real effects subsurface heterogeneity has on mixing and transport and is quite approximate. A much more realistic and explicit depiction of subsurface heterogeneity in this system and the resultant mass fraction and power spectra will be studied in the future.

[19] The study provides a picture of overall fractal scaling of solutes in watersheds in an integrated fashion by including topography, macrodispersion, the vadose zone, and land surface processes that have previously been discussed separately in the literature. The results also suggest that the simulation platform is useful in analyzing residence time distributions and mass transport in real-world system by appropriately capturing the behavior over a wide range of scales.


[20] Authors share equal co-authorship. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. We are grateful to Andrew F. B. Tompson for discussions and comments on this work. We are also grateful to the comments of Rina Schumer and one anonymous reviewer that greatly improved the quality of this manuscript. Portions of this work were funded by LLNL under the Climate Change Initiative. Portions of this work were funded by LLNL under the Climate Change Initiative.