Fractal and Multifractal Analysis of the Hydraulic

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Calibration and Reliability in Groundwater Modelling: A F ew Steps Closer to Reality ( P r o c e e d i n g so f

M o d c l C A R E 2 0 0 2 . P r a g u e . C z e c h R e p u b l i c . J u n e 2 0 0 2 ) . I A H S P u b l . n o . 2 7 7 . 2 0 0 2 .

Fr ac t a l an d m u l t i f r ac t a l an a ly s i s o f t h e h yd r au l i c

p r op e r t y var ia t ion s o f k ar s t aq u i f e r s

B . M A J O N E A . B E L L IN

Dipartimettto di Ingegneria Civile ed Ambientale, Universit di Trento, via Mesiano 77,

1-38050 Povo-Trento, Italy

b r u n o , m a i o n e @ i n e . u r i i t n . i t

A . B O R S A T O

Mttseo Tridentitto di Scienze Natural , via Calepina 14, 1-38100 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 f r a c t a l n o i s e ; k a r s t a q u i f e r s ; Levy d i s t r i b u t i o n ; m u l t i - f r a c t a l a n a l y s i s

INT RO DUCT IO N

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 m odels of hydrau lic 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.

DAT A CO L L E CT IO N AND PO WE R S PE CT RA ANAL YS IS

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
mailto:[email protected]:[email protected]


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

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3 0 0

u

2 5 0

CO

SL

1 0 0 0 0 0

v

1

1 0 0 0

1 0 0

.2

o

8 0 0 0

1 0 0 0 0

0 0 0 6 0 0 0

Time [ h o u r s ]

F i g . 1 Tim e series of electr ical con ductivity (dashed l ine) 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 of2h for 1 year. Figu re 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 +

, M g

2 +

and

H C O 3

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 / i s

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.


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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 (Hann a 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 ofE 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 m ay c om plicate the extraction of the form ation's structure from the as

signai, 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. A s a result, Tis 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

an d

T,

which can be explained w ith 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

1 0 0

1 0 J

CD

T 3

a.

E

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Fractal and Multifractal Analysis of the Hydraulic