Evaluation of Dispersivity using a Labor (1)

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 nosorbing 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.

*

IRSA (1994): “Migration and Fate of Pollutants in Soils and Subsoils", Quad. Ist.
Ric. Acque, 96, Roma, 1994

Introduction.
The model generally used to represent solute transport in homogeneous, porous,
saturated media is (Parker et al., 1984)
2
∂C
∂C
∂C
= D h 2 -U
∂t
∂x
∂x

(1)

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).

Figure 1: Experimental equipment.

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:

2

Q = KB

2

h0 - h1
2L

(2)

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.

Figure 2: Piezometric heads in deformed scale.

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.
Using assumption [1], the following equation can be written:

D h = αU + D d

(3)

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:

h(x) ≈ 0.317 - 0.006x

(4)

and then δh/δx≈ -0.006 = constant, leads to:

q(x)= _K

∂h
≈ 0.03 • 10 -3m/s = constant
∂x

(5)

and

∂q(x)
≈0
∂t

(6)

Due to the third simplification the actual (theoretical) value of concentration of
the injection can be calculated from the tracer tests (Heqing, 1991) by:

C inj = Γ s( C 1 - C 0 )

(7)

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):

C(x,t)= C 0s +0.5Γ s( C 1 - C 0 )[erf( X 2 ) - erf( X 1 )]

(8)

where:

U
t
X 1=
0.5
2 [ D h]
x-

(9)

xX 2=

U
(t - t 0 )

⎡
⎤
2 ⎢ Dh ⎥
⎣ (t - t 0 ) ⎦

0.5

while using the boundary and initial conditions for a continuous injection, the solution is:

⎡
⎤
⎛ Ux ⎞
C(x,t)= C 0s +0.5Γ s( C 1 - C 0 ) ⎢erfc( X 1 )+ exp ⎜
⎟ erfc( X 2 )⎥ (10)
⎝ Dh ⎠
⎣

⎦
where:

X 1=

2 ( D ht )
x -Ut

0.5

(11)

X 2=

2 ( D ht )

x +Ut
0.5

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.

Table 1. Values of α, Dh and P
Test type

Distance from
injection
x (cm)

Dispersivity
α (cm)

Hydrodyn.
dispersion
coefficient
Dh (cm2d-1)

Peclet number
of column
P

Crenel
injection

10
40
90
140

0.19
0.19
0.19
0.20

116
116
116
122

53
211
474
700

Continuous
injection

10
40
90
140

0.19
0.19
0.19
0.45

116
116
116
279

53
211
474
311

Figure 3: Breakthrough curves for Crenel pulse type injection.

Figure 4: Breakthrough curves for continuous 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):

D h = 0.5 1.2
P
Dd

(12)

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,

Dh = U

d
ln ( 1.5Pe - 0.25 )
6

(13)

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.

Figure 5: A picture of the sand made by the electron microscope.

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.

References.
- Bear, J., Dynamics of Fluids in Porous Media, 764 pp., Elsevier, new York, 1972.
- Bear J., Hydraulics of Groundwater, Mc Graw-Hill, New York, 1979.

- Butow, E. and E. Holzbeker, Approach to Model the Transport of Leachates from
Landfill Site Including Geochemical Process. Proceeding of the International
Symposium on Contaminant Transport in Groundwater, Stuttgart, 4-6 april 1989.
- Fried, J.J., Miscible Pollution of Groundwater: A Study of Methodology, in Proceedings
of the International Symposium on Modelling Techniques in Water Resources
Systems, vol.2, edited by A.K. Biswas, pp.362-371, Environment Canada,
Ottawa, 1972.
- Fried, J.J., Groundwater Pollution, 330 pp., Elsevier, Amsterdam, 1975.
- Heqing, H., On a One-dimensional Tracer Model, Ground Water, vol. 29, No.1,
January-February 1991.
- Parker, J.C. and M.Th. van Genuchten: Boundary Conditions for Displacement
Experiments Through Short Laboratory Soil Columns, Soil Sci. Am. J. 48,
703-708, 1984.
- Pickens, J.F. and G.E. Grisak, Scale-Dependent Dispersion in a Stratified Granular
Aquifer, Water Resources Research, vol.17, pp.1191-1211, 1981.
- Pickens, J.F. and G.E. Grisak, Modelling of Scale-Dependent Dispersion in
Hydrogeologic Systems, Water Resources Research, vol.17, n.6, pp.1701-1711,
1981.
- Saffman P.G., Dispersion Due to Molecular Diffusion and Macroscopic Mixing in Flow
Through a Network of Capillaries, Journal of Fluid Mechanics, London (UK),
n°7, pages 194,208, 1960.
- Silliman S.E. and E.S. Simpson, Laboratory Evidence of the Scale Effect in Dispersion of
Solutes in Porous Media, Water Resources Research, 1987.
- Societa' Italiana della Scienza del Suolo (S.I.S.S.), Metodi Normalizzati di Analisi del

Suolo, Edagricole, Via Emilia Levante, 31, Bologna, Italy, 1985.
- Taylor S.R., G.L. Moltyaner, K.W.F. Howard and R.W.D. Killey, A Comparison of
Field and Laboratory Methods for Determining Contaminant Flow Parameters,
Ground Water, n° 3, vol. 25, pages, 321, 330, 1987.
- Weast R.C., Handbook of Chemistry and Physics, The Chemical Rubber CO. 18901
Cranwood Parkway, Cleveland, Ohio, 1968.