EVALUATION OF DISPERSIVITY USING A LABORATORY MODEL*
Giuseppe Passarella - Costantino Masciopinto C.N.R. - Istituto di Ricerca Sulle Acque
Reparto Sperimentale di Chimica e Tecnologia delle Acque Bari
Riassunto
Un modello di laboratorio e' stato messo a punto per valutare una metodologia di misura per la determinazione dei valori della dispersivita' di alcuni inquinanti in sistemi di acque sotterranee. Il fenomeno simulato e' stato quello della dispersione idrodinamica in un acquifero freatico a matrice porosa non reagente situato tra due bacini aventi capacita' infinita e differenti livelli della superficie libera. Il valore della dispersivita', ottenuto per una sabbia silicea della valle del Ticino, con diametro d pari, in media, ad un millimetro, e' stato di circa 2.0 mm. Questo valore e' stato detrminato attraverso il "fit" delle curve sperimentali dei dati di concentrazione utilizzando la soluzione analitica dell'equazione di advezione-dispersione unidimensionale.
Abstract
A laboratory model has been carried out to evaluate a measurement methodology to determine dispersivity values of several pollutants in groundwater systems. The simulated phenomenon has been hydrodynamic dispersion in a phreatic, porous no-sorbing aquifer situated between two basins of infinite capacity and different piezometric heads. The value of dispersivity, obtained for a natural siliceous sand from the Ticino Valley, having diameter d=1 mm, in average, has been about 2.0 mm. This value has been evaluated by calibration of breakthrough curves of experimental data of concentration using the analytical solution of the one-dimensional advection-dispersion equation.
*
Introduction.
The model generally used to represent solute transport in homogeneous, porous, saturated media is (Parker et al., 1984)
where Dh [L2T-1] is the hydrodynamic dispersion coefficient and U [LT-1] is the average pore-water velocity.
The dispersivity, since has been considered as a characteristic single-valued of the homogeneous porous medium, should be a constant. (Bear, 1972).
In field measurements of dispersivity, several researchers found that it cannot be considered as a constant but rather depends on the mean travel distance and scale of the system (Fried, 1972, 1975; Pickens et al, 1981).
Consequently, the nonuniqueness of dispersivity value poses a difficulty in the use of classical advection-dispersion equation.
In laboratory tests, some discrepancies can be found due to the presence of immobile water and the difficulty in reproducing the real boundary conditions.
In this work, the value of dispersivity for natural siliceous sand from the Ticino Valley has been evaluated, simulating the behaviour of a phreatic granular aquifer. The values for dispersivity have been obtained by calibration of breakthrough curves of experimental data of concentration using the analytical solution of the one-dimensional advection-dispersion equation. This is a first necessary step to calibrate the laboratory equipments to determine experimental values of dispersivity.
Laboratory Tests.
Laboratory tests have been carried out to evaluate the dispersivity values of natural siliceous sand.
The physical model has been designed to simulate the hydrodynamic behaviour
of a phreatic granular aquifer, situated between two reservoirs of infinite capacity and with different piezometric heads. The difference of the piezometric heads on the boundary was chosen to be small (0.012 m), to simulate the hydrodynamic behaviour of a real aquifer and, at the same time, to limit the variation of the specific discharge in the x direction.
The length of the porous matrix, was 2.0 m (fig.1).
The sand was left to consolidate for about six months, with variable watering conditions. The box was equipped with level gauges and electronic devices able to control all the system parameters involved. On the upper part of the permeameter a drop spillway was placed to eliminate all the fluctuations of hydraulic pressure and to assure a constant hydraulic head. The data acquisition system was made up of 24 temperature and electric conductivity probes, connected to a scanner able to perform data collecting cycles almost instantaneously, at programmed times. The collecting cycles were controlled by a computer running "in-house" software that permitted also the storing of the data. The probes were placed in vertical cross-section arrays at appropriate distances.
The porosity has been 43%. This value was carefully evaluated on unconsolidated sand cores by standard methods of soil analysis of the Italian Society of Soil Science (S.I.S.S., 1985). We believe that this parameter can be considered accurate with an error ranging within ± 2-3%.
The average hydraulic conductivity was K=4.66⋅10-3 m/s. The hydraulic regime was assumed to be steady water flow through a homogeneous porous medium and, assuming that the Depuit-Forchheimer approximation is valid, the discharge formula is:
where B and L are, respectively, the width and the length of the sandbox. This formula permits the evaluation of K when the total discharge Q is known.
Each tracer test was conducted after a period of time long enough to assure that the water level in the reservoirs to be constant. The water level was continuously monitored by 7 piezometers located along the sandbox. The difference in piezometric head was h0-h1=0.012 m.
Due to the imposed boundary conditions, the value of hydraulic conductivity and, consequently, the low velocity of the flow along the longitudinal profile of water table, being of a parabolic nature, was very similar to a straight line (fig.2).
A 1.5 inch diameter PVC tube has been used for injecting the tracer. On the lateral surface of the tube we have made holes, into the saturated zone of the model, to achieve a porosity greater than 40%. It simulated a well drilled into a phreatic aquifer. The tracer has been injected into this "well", using a pump and a small tube.
The non-reactive tracer used during the tests was chloride ion (NaCl) (Butow et al., 1989). Both the pulse (t=t0) and the long term tracer (t0=∞) inputs were done directly into the centre of the cross-section in the saturated zone. The concentration of total salts in solution was C0=0.240 g/l (drinkable water) and the tracer discharge was constant and equal to 0.00167 l/s during the pulse injection (with concentration of injection C1=1.3 g/l) and 0.00033 l/s (with C1=1.3 g/l) during the long term injection.
2 2
0
-
1h
h
Q = KB
Analytical Solutions
To obtain the dispersivity value, the experimental results were interpreted with the following simplifications:
[1] the tensor of dispersion Dh has its principal direction in the direction of the water flux;
[2] the change of specific discharge in direction x is assumed to be zero and then
δq/δx=nδU/δx=0;
[3] the mathematical model adequately simulates the hydrodynamic dispersion along the centre line.
where Dd= 1.296 cm2/d at 25°C (Weast, 1968) and α [L] is the dispersivity.
The second and third simplification allows for the use of a one-dimensional model instead of a three-dimensional one. The linear equation, which represent, the hydraulic head, is:
and then δh/δx≈ -0.006 = constant, leads to:
and
Due to the third simplification the actual (theoretical) value of concentration of the injection can be calculated from the tracer tests (Heqing, 1991) by:
where C0 is the initial concentration before injection, C1 is the concentration of the tracer and Γs is an experimentally determined uniformity coefficient. The values for this
coefficient depend on the type of tracer and on the position of the probes in the cross section.
Under the foregoing conditions, the solutions of equation (1) in the case of both a Crenel pulse injection (Fried, 1975) and a long term input are well known in the literature. For a Crenel pulse injection, the solution of the differential equation (1), for t≥t0, is (Taylor, 1987, Fried, 1975):
where:
while using the boundary and initial conditions for a continuous injection, the solution is:
and C0s is the zero initial value for each probe.
Results and Discussion.
By calibration of the breakthrough curves of experimental data of concentration (fig.3÷4) it was possible to evaluate the corresponding value of α (tab.1). These parameters have been evaluated by a "best-fit" procedure.
The fitting of the experimental data was performed using a mathematical software running on personal computers (386-Matlab, 1990, the MathWorks, Inc.). The unknown parameters were Γs and the dispersivity α. The known parameters were C0, C0s, C1, U and x. The Γs values mainly influence the area beneath the BTC of C while the values of α influence the slopes of the BTC of concentration.
( )
( )
1 0.5
h
2 0.5
h
x -Ut
=
X
2 t
D
x +Ut
=
X
2 t
D
Table 1. Values of α, Dh and P
Test type Distance from injection
The Γs value was variable for each probe. For Crenel type injection the mean value of this coefficient was 0.61.
The value of α is constant for the first 0.90 m of the sand-box. This value varies in the last cross section (tab.1). Therefore, under these laboratory condition, we could suppose α constant until 0.90 m.
Other researchers found similar values of dispersivity for sand both in the field and laboratory experiments (Taylor, 1987). A similar value of dispersivity (α=1.1 mm) can be obtained using the formula (Bear, 1979):
Figure 4: Breakthrough curves for continuous injection.
h 1.2
d
D
= 0.5P
D
The following formula (Saffman, 1960), obtained by means of a statistical analysis for a Peclet number Pe=U⋅d/Dd (U=mean interstitial velocity) greater than 1,
yields α equal to 0.8 mm.
The effective value of Dheff, constant up to 90 cm from injection, was equal to 116 cm2/d. This value increases in the last section according with the dispersivity values.
The shapes of the BTC were very similar for each probe of the same cross section having the same slopes (that means same dispersivity) but smaller values of the submitted areas and smaller peaks (influencing the uniformity coefficient).
Conclusions
The estimated value of dispersivity is very similar to that evaluated in several laboratory tests reported in literature.
Values of velocity, specific discharge and Peclet number show that the system is in a transition zone (zone III) (Bear, 1979), where the spreading of the tracer is mainly due to mechanical dispersion. The dispersivity increase in the last cross section and this could be explained by local microscopic heterogeneities of the sand.
The electron microscope (fig.5) shows the presence of both sub-spherical particles having mean diameter equal to 200 μm and dead-end pores (3-4%).
These microscopical heterogeneities could have influenced the measurement of dispersivity in the last cross section; in fact the thickening of the finest particles produces a heterogeneity of the sand not negligible in macroscopic scale. Other tracer tests are in project to evaluate values of dispersivity for different types of soils.
The evaluation of dispersivity is very difficult, above all when the applicability of the theoretical model is extended to real field situations.
Several researchers have studied the well known scale-effect (Silliman et al., 1987) to explain the growth of dispersivity value with increasing distances. In fact this phenomenon influences the applicability of the classical advection-dispersion equation to real aquifers.
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