Fractal and multifractal analysis of the hydraulic property variations of karst aquifers

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1 148 Calibration and Reliability in Groundwater Modelling: A Few Steps Closer to Reality (Proceedings of ModclCARE"2002. Prague. Czech Republic. June 2002). IAHS Publ. no Fractal and multifractal analysis of the hydraulic property variations of karst aquifers B. MAJONE, A. BELLIN Dipartimettto di Ingegneria Civile ed Ambientale, Université di Trento, via Mesiano 77, Povo-Trento, Italy bruno,maione@ine.uriitn.it A. BORSATO Mttseo Tridentitto di Scienze Natural!, via Calepina 14, Trento, Italy Abstract We present the multifractal analysis of several signals recorded at the valclusian karst spring of Prese Val, located in the Dolomities area northwest of Trento (Italy). The data analysed include water discharge, temperature and electrical conductivity. Both electrical conductivity and temperature, which mimic the spatial variability of hydraulic conductivity, show the signature of multifractality, while the water discharge shows a much more complex structure. We conclude that accurately recorded signals of electrical conductivity and temperature of spring water can be used for characterization of karst systems. Key words fractal noise; karst aquifers; Levy distribution; multi-fractal analysis INTRODUCTION Hydrogeological processes in karst aquifers are controlled by hydraulic property variations at several continuous and discrete scales ranging from a few micrometres, in elements such as microfractures, to tens of kilometres, in geological fault zones and karst channels. Most of the existing studies describe the karst system by using either lumped- or distributed-parameter models. The latter rely on a suitable model of hydraulic property variations. In practice, because of the limited information available on relevant scales of variability, single or dual continuum porous equivalent models are often used, which assume uniform hydraulic properties for the medium (e.g. Teutsch & Sauter, 1998). The need for geostatistical models of hydraulic property variations of karst systems has been shown by Bauer et al. (2000) in a study that identified the importance of conduit-matrix flow exchange in the process of karstification. In this work we explore the potential for using time series analysis of discharge, temperature and electrical conductivity at a spring for inference of hydraulic property variations of the underlying karst system composed of carbonate rocks. DATA COLLECTION AND POWER SPECTRA ANALYSIS The detailed field survey, conducted by the Museo Tridentino di Scienze Naturali of Trento (Italy) at the valclusian karst spring of Prese Val, located in the Dolomites area

2 Fractal and multifractal analysis of the hydraulic property variations of karst aquifers u 250 CO SL v o Time [hours] Fig. 1 Time series of electrical conductivity E (dashed line) and water discharge Q (solid line). northwest of Trento, represents the basis for our work. Data sets include water discharge (Q), temperature (7) and electrical conductivity of water (E). All data are collected with a time step of 2 h for 1 year. Figure 1 shows the time series of electrical conductivity E and water discharge Q. We consider the electrical conductivity since it is linearly correlated with the concentration of ionic species (Ca 2+, Mg 2+ and HCO3 2 ") originating from the chemical dissolution of calcite and dolomite (White, 1988). Figures 2(a) and 2(b) shows the power spectrum of E and T, respectively. The E- spectmm shows a fractal power-law scaling which resembles the \/f noise, where /is the frequency (Hz). This result confirms the finding of Kirchner et al. (2000), who found a sharp contrast between rainfall and chloride signals in streamflow at the Plymlimon catchment, Wales, UK. While the former has a white noise spectrum, the latter exhibit fractal l/f scaling over three orders of magnitude.

3 150 B. Majone et al. Extensive experimental studies on calcite dissolution showed that the dissolution rate is affected by a sharp drop as saturation is approached, resulting in undersaturated water over long flow paths and large travel times (Hanna & Rajaram, 1998). The persistence of undersaturated water over long distances is widely recognized as one of the most important mechanisms controlling the conduit growth, and at a much smaller time-scale it introduces a strong dependence of E from the residence time of water. Thus, the fractal power-law scaling of E can be recognized as the imprint of hydraulic property variations acting over a wide range of scales. However, the sharp drop in the dissolution rate may complicate the extraction of the formation's structure from the assignai, in particular for large travel times associated with diffusive mechanisms in the rock matrix. Additional insights on the underlying hydrogeological structure can be obtained from temperature analysis. Since the spring emerges 430 m below the mean elevation of the catchment, only the water with a high residence time is in thermal equilibrium with the aquifer. As a consequence, water discharging from the slow diffusive part of the reservoir shows small to negligible fluctuations of T with the seasons or the discharge. Water moving through the conduit system, in contrast, shows a residence time of the order of hours or days, which is insufficient to reach the thermal equilibrium. As a result, T is expected to mimic the hydraulic property variations of the reservoir with a possible lack of resolution at large travel times, as for the is-signal. Figure 2(a) and 2(b) shows a significant disparity in the exponents of the power spectra of E and T, which can be explained with the different time scales characterizing thermal and chemical processes in the subsurface. The power spectrum of the discharge is in sharp contrast with the others. The most striking feature of the (9- s P e c i r u m is a progressive increase of the slope with the frequency, which suggests the co-existence of several mechanisms controlling spring flow production. At low flow conditions the spring is fed by water stored in the fractured matrix. In absence of recharge, because of the relatively small hydraulic conductivity of the matrix and the low hydraulic head gradient, the water spends much J CD T3 a. E < E-8 1E-7 1E-6 1E-5 Frequency (Hz) Fig. 3 Power spectrum of water discharge. The straight line indicates the slope of the power spectrum of E. 1E-4

4 Fractal and midtifractal analysis of the hydraulic property variations of karst aquifers 151 of its residence time in the fractured matrix. Thus, Q can be modelled by using the classical approach of the unit hydrograph with the transfer function depending on the travel time distribution within the fractured matrix (Rodriguez-Iturbe & Rinaldo, 1997). When the system is forced by rainfall or snowmelt, the spring is fed by both the channels in direct communication with the surface and the plug-flow resulting from the increased hydraulic head gradient in the fractured matrix. The water initially stored within the matrix is forced to quickly reach the closest channel and, as a consequence, the time spent to travel the channel network becomes a significant portion of the total residence time. This is particularly evident for high flow conditions. In other words, hydraulic property variations cannot be clearly detected from the hydrological signal because of the nonlinearity of the mechanisms controlling spring flow production. The overall effect observed in Fig. 3 is a steeper power spectrum at high frequency, where the geometry of the channel network contributes significantly to the shape of the signal. This reasoning is supported by the fact that at high frequency the slope of the (3-spectrum approaches that of E. MULTIFRACTAL ANALYSIS The contemporaneous presence of elements such a fractures, karst network and faults, where the geometry is characterized by a wide range of fractal dimensions, suggests the possibility of observing the signature of multifractality in the hydrological signals recorded at the spring. Multifractal analysis is a powerful method for detecting scaleinvariance properties of signals. Central to the study of multifractality is the generalized variogram of the property Y (in our case E, T and Q): Y, ( / i ) = ( r ( r + A ) - r ( o r ) ( l ) where t is the time, h is the lag, q is the order of the variogram and brackets indicate statistical average. The moments (equation (1)) are used as a tool to extract the intertwined arrangement of small and large fluctuations from the signal (Rodriguez- Iturbe & Rinaldo, 1998). For multifractal processes, equation (1) assumes the following expression y q (h) = A with the scaling function \ assuming the following form (Schmitt et al., 1995): l&q) = qk- (q a -q) (2) a-1 which identifies an important class of conservative processes (Labat et ai, 2002). In equation (2) K is a multifractal parameter, which in the mono-fractal case coincides with the Hurst coefficient, C is the co-dimension characterizing the sparsenessinhomogeneity of the mean, and a characterizes the degree of multifractality: a = 0 is representative of the mono-fractal case, while a = 2 represents the maximum or lognormal multifractal case. It is important to notice that the multifractal behaviour depends only on the values assumed by a and C. Figure 4(a) and 4(b) shows the function cj for E and 7 7, respectively. In both cases the fitting of equation (2) with the experimental data is successful, and the signals

5 152 B. Majone et al. q q Fig. 4 Experimental i; as a function of variogram-order q (open circles) compared with the monofractal curve (dashed line): (a) electrical conductivity; (b) temperature. The best fitting of equation (2) is also shown with the solid line. Best-fit parameters are C = , a = 1.46, and K = for the ^-signal and C = , a = 1.57, and K = 0.410, for the 77-signal. Open triangles indicate the experimental, for the g-signal. show a clear multifractal behaviour. Furthermore, the two signals show slight differences in the multifractal parameters. We then repeated the same analysis for the (2-signal, and the results are shown in Fig. 4(b) with open triangles. It can be observed that contrary to E and T, Q shows a more complex multi-scale structure, since the scaling function does not follow (2). This result confirms the analysis presented in the previous Section suggesting that the formation's structure can hardly be detected from the hydrological signal. FRACTIONAL LEVY MOTION In an effort to introduce realism in modelling abrupt changes of rock properties associated with strata boundary, Painter & Paterson (1994) proposed the Lévy-stable probability distribution: f(p) = - fexpf-\bkf ) cos (kp) àk (3) as an alternative to the Gaussian model. In equation (3) 5 is the Levy index and B is the width parameter, which scales as B{1) = B 0 \l\", where / is the lag, H is a scaling parameter, which quantifies the degree of interdependence of the signal, and BQ is a parameter. For independent increments H = 1/5, while the condition H > I/O is typical of persistent signals showing a positive correlation between successive increments. Furthermore, H < 1/8 indicates antipersistence in the signal. It should be noted that equation (3) does not have a closed analytical form, except for a few particular values of 6 (8 = 2, 3/2 and 1). Furthermore, 5 varies within the range 0 < 8 < 2, with the pdf converging to the Gaussian distribution as 8 converges to the upper limit of 2.

6 Fractal and multifractal analysis of the hydraulic property variations of karst aquifers 153 The analysis presented in this work is based on the assumption that the increments of the travel time follow a Lévy-stable probability distribution with tails decreasing as P(\x\ > m) = u~ h. This assumption is supported by the analysis of Painter (1996, 2001), who showed that in most cases the Lèvy-stable distribution is superior to the Gaussian distribution in describing the statistical properties of the hydraulic logconductivity increments in both fractured and unconsolidated formations. The algebraic decay of the probability distribution ensures that extreme events, which assume here the form of exceedingly large increments in the recorded signal associated with a large disparity in travel times, are more likely than in processes controlled by distributions with exponential decay. Following the approach suggested by Painter & Paterson (1994) we attempted to reproduce the signals as a fractional Levy motion (flm) with the increments distributed according to equation (3). The parameters of the Levy distribution are obtained by best fitting of equation (3) with the frequency distribution of the increments, with lag /, extracted from the original signal. As can be seen from Fig. 5(a) B(l) is described by a power law over the range h. The deviation at small scales can be attributed to the lower cut-off introduced by the detection limit of the sampling device, and the deviation at large scales is the finite-length effect: a larger lag results in a smaller number of independent samplers increasing the uncertainty in the inferred parameters. The best-fit value of the scaling parameter H is 0.922, indicating persistence in the signal. Furthermore, it is a good match with the value obtained by the rescaled-range analysis of the signal (Mandelbrot, 1983). Figure 5(b) shows ô as a function of the lag. It can be observed that for / > 20 h, 8 consistently increases, with the lag approaching the value of 2 for / = 2000 h. What we observe here with reference to the Ji-signal confirms the findings of Painter (1996) for vertical fluctuations of permeability in sedimentary formations (see also the comment by Liu & Molz, 1997) C C Lag time [hours] Lag time [h] Fig. 5 (a) The width parameter B(l), and (b) the Levy index 8 as a function of the time lag for the electrical conductivity.

7 154 B. Majone et al. CONCLUSIONS Signals in the form of electrical conductivity and temperature of the spring water have been used to infer the statistical properties of the karstic system feeding the spring. The statistical properties of the signals are shown to be consistent with recent findings on hydraulic property variation of sedimentary formations, opening interesting possibilities for characterization of complex karst systems. Furthermore, we showed that runoff production is controlled by non-uniform travel time distributions which depend on the hydrological input. REFERENCES Bauer, S., Liedl, R. & Sauter, M. (2000) Modelling of karst development considering conduit-matrix exchange flow. In: Calibration and Reliability in Groundwater Modelling (ed. by F. Stauffer, W. Kinzelbach, K. Kovar & E. Hoehn), (Proc. Int. Conf. ModeICARE'99, September 1999, Zurich, Switzerland), IAHS Publ Hanna, R. B. & Rajaram, H. (1998) Influence of aperture variability on dissolutional growth of fissures in karst formations. Water Resour. Res. 34(11), Kirchner, J. W., Feng, X. & Neal, C. (2000) Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, Labat, D., Mangin, A. & Ababou, R. (2002) Rainfall-runoff relations for karstic springs: multifractal analysis. J. 256, Liu, H. H. & Molz, F. J. (1997) Comment on "Evidence for non-gaussian scaling behaviour in heterogeneous sedimentary formations" by Scott Painter. Water Resour. Res. 33(4), Mandelbrot, B. B. (1983) The Fractal Geometry of Nature. W. H. Freeman, New York, USA. Painter, S. (1996) Evidence for non-gaussian scaling behaviour in heterogeneous sedimentary formations. Water Res. 32(5), Painter, S. (2001) Flexible scaling model for use in random field simulations of hydraulic conductivity. Water Resour. 37(5), Painter, S. & Paterson, L. (1994) Fractional Levy motion as a model for spatial variability in sedimentary rock. Res. Lett. 21, Hydro/. Resour. Res. Geophys. Rodriguez-lturbe, 1. & Rinaldo, A. (1997) Fractal River Basins: Chance and Self-Organizations. Cambridge University Press, Cambridge, UK. Schmitt, F., Lovejoy, S. & Schertzer, D. (1995) Multifractal analysis of the Greenland ice-core project climate data. Geophys. Res. Lett. 22, Teutsch, G. & Sauter, M. (1998) Distributed parameter modelling approaches in karst-hydrological investigations. In: Bulletin d'hydrogéologie. Special Issue no. 16, (ed. by Peter Lang,Centre d'hydrogéologie, Université de Neuchatel, Switzerland). White, W. B. ( 1988) Geomorphology and Hydrology of Karst Terrains. Oxford University Press, New York, USA.