
Research Article


Estimation of Two Dimensional Solute Transport Parameters Using Tracing Experiment Data 

R. Derakhshani
and
A. Bazregar



ABSTRACT

The effective solute transport parameters, longitudinal dispersivity (α_{x}), transversal dispersivity (α_{y}) and retardation factor (R) for a twodimensional plume are estimated using timeconcentration data of a sampling borehole through which the plume passes. The plume was produced by injection of 2 kg Uranine (Sodiumfluorescien; Color Index: 45350) in a borehole in uniform groundwater flow. Twodimensional analytic equation of solute transport in a uniform groundwater flow was used to estimate the parameters. The results were compared with one well method of dispersivity estimation. A timeconcentration curve was drawn using estimated parameters and compared with the observed timeconcentration curve. Comparison shows the accuracy of the estimated parameters obtained by the former method. Based on the estimated parameters and the twodimensional analytic equation the movement of the plume is predictable at any arbitrary time after injection for the study area. Based on the estimated parameters and the twodimensional analytic equation the movement of the plume is predicted at 85 days after injection for the study area, where as it is shown in this study the predictable and observed concentration curve are exactly equal.







INTRODUCTION
Dispersivity parameters are needed for prediction of movement of a contaminant
and can be estimated in the field by two general methods. The first method involves
measurement of a real contaminant plume that exists in the field and the second
method involves injection of a tracer in the aquifer. The latter is a much more
common approach for determining dispersivities than using an existing contaminant
(Zou and Parr, 1994). Local transverse dispersion was lionized as an important
factor in the controlling the rate of dilution of conservative solutes and smoothing
of concentration fluctuations (Fiori and Dagan, 2000).
Local transverse dispersion transfers longitudinal spreading of solute plumes
to effective mixing in heterogeneous media (Dentz et
al., 2000; Cirpka and Kitanidis, 2000; Cirpka,
2002; Benekos et al., 2006). Chen
et al. (2005), Cirpka and Kitanidis (2001)
and Benekos et al. (2006) proposed some methods
for determination transverse dispersivity. Benekos et
al. (2006) and Cirpka and Kitanidis (2001) introduced
a method to estimate the transverse dispersion coefficient by best fitting breakthrough
curves with numerical simulations. Transverse dispersion coefficient was determined
from net rate of nonaqueous phase liquid pool dissolution by Eberhardt
and Grathwohl (2002). Cirpka et al. (2006)
estimated transverse dispersion coefficient by measuring plume length of alkaline
plume in acidic ambient while Klenk and Grathwohl (2002)
have studied transverse vertical dispersion in groundwater, by using the results
of experiments on mass transfer of volatile compound across the capillary fringe.
More recently, Massabo et al. (2007) introduced
a quick method for the laboratoryscale estimation of the transverse dispersion
coefficient which was based on the analytical solution of the advectiondispersion
equation where a pulselike injection of a nonreactive solute was conducted
in a column packed with a homogeneous porous medium.
MATERIALS AND METHODS
This study which was conducted in 2008 is concentrated on estimation of two
dimensional solute transport parameters using tracing experiment data in Korbal
plain, situated in South of Iran. Two estimation methods are used in the present
paper in which transport parameters can be estimated for a twodimensional plume
generated by a slug tracer injection in a uniform ground water flow field. In
the first method, the twodimensional advectiondispersion equation of Kinzelbach
(1986) is employed by which the longitudinal and transversal dispersivity
parameters were determined using an observed time concentration curve in a
sampling well.
In the second method, one well method of dispersivity estimation of Zou and Parr (1994) is used to estimate these parameters.
RESULTS
The field tracing experiment was conducted in the Korbal plain, situated in
South of Iran, 55 km NorthWest of the city of Shiraz (Lat.: 29°, 30. Lon.:
52°, 45). Nearest mountain to the Korbal plain that highly has affected
the geology and hydrogeology of the plain, is Rahmat anticline with the NW
to SE trend corresponds to the Zagros ranges trend. The Zagros foldthrust
belt, which extends for about 2000 km from Southeastern Turkey through Northern
Syria and Iraq to Western and Southern Iran, with its numerous supergiant hydrocarbon
fields, is the most resourceprolific fold thrust belt of the world (Derakhshani
and Farhoudi, 2005; Sepehr and Cosgrove, 2004). This
fold and thrust belt is a result of the structural deformation of the Zagros
(peripheral) proforeland system, whose presentday expression is the marine
Persian Gulf and continental Mesopotamia basins and underlying preproforeland,
mostly platformal and continental shelf deposits (Alavi,
2004). The belt has structurally evolved as a prism of stacked thrust sheets,
composed of uppermost Neoproterozoic and Phanerozoic sedimentary strata approximately
7 to 12 km thick, in the external part of the southwestmigrating Zagros orogenic
wedge (Alavi, 1994). More than a hundred stratigraphic
columns have been studied from both subcrop (well) and outcrop (measured) sections
in various parts of the Zagros belt (Alavi, 2004; Rahnama
et al., 2008; Bahroudi and Koyi, 2004; Adabi
et al., 2008; Ghavidelsyooki and Vecoli, 2008;
Fakhari et al., 2008; Shirazi,
2008).
The dominant formation of Rahmat anticline is Sarvak (L. Cretaceous, Limestone) and in some area along the anticline axes at the crest, the Kazhdumi Formation (AlbianCenomanian, shale and marly limestone) is exposed (Fig. 1). Eroded material of this Formation has produced fine texture sediment of the study area. The soil texture in the study area is reported to be siltyclay with 30% prosity and thickness of that is about 30 m. The groundwater flow velocity is very slow. The general direction of flow is down dip from the  Fig. 1: 
Geologic map and borehole positions in study area 
NorthWest towards the SouthEast (along the Rahmat anticline foot). However, the groundwater flow direction in the tracing site in the time of experiment was exactly determined using several borehole data dug for this purpose. The location of the boreholes and direction of local groundwater is shown in Fig. 1. It is assumed that a constant linear velocity for a uniform saturated groundwater flow field exists and that the tracer is conservative and the diffusion component of hydrodynamic dispersion is ignored. The advectiondispersion equation for a conservative solute in these conditions is:
where, c is mass of solute per unit volume of solution, x and y is Cartesian
coordinates, t is time, is
liner velocity of field flow in the x direction, α_{x} and α_{y}
are dispersivities in the x and y directions.
The instantaneous input of a mass ΔM of solute as a vertical line source
at x and y in a uniform infinite flow field results in a two dimensional plume
which is derived from Eq. 1 and described by Kinzelbach
(1986) as follows:
Where:
m 
= 
Is aquifer thickness 
n_{e} 
= 
Effective porosity 
R 
= 
Retardation factor 
λ 
= 
Is a decay constant 
DISCUSSION Eleven boreholes with depth of about 20 m and diameter of 10 cm were dug in the experiment site, as it is shown in Fig. 1. The bore hole I is used as injecting point which 2 kg of Uranine mixed with 50 L of water was injected into it (Fig. 1). Other 10 surrounding bore holes (W1, S1, S2 …) were used as sampling point which they were sampled for a period of 162 days. The Uranine was detected and measured in the samples by the means of a Shimadzu RF 5000 spectrofluorophotometer with a sensitivity of 0.001 ppb. Due to the fine texture of the soil in the experiment site, the flow velocity was very low and dye was detected in only two sampling boreholes, E1 and S1 (Fig. 1). The timeconcentration curves for these two bore holes are shown in Fig. 2. Due to some limitations and the low velocity of groundwater, the timeconcentration curve for sampling bore hole E1 was not completed. Therefore, the observed timeconcentration curve of S1 was used to estimate the solute transport parameters.  Fig. 2: 
Observed timeconcentration curve of boreholes S1 and E1 
The twodimensional solute transport equation of Kinzelbach
(1986) was used to estimate the effective parameters based on the minimum
differences between observed data and analytic data produced by Eq.
2. The results are given in Table 1 using observed data
of sampling bore hole (S1) with Cartesian coordinates of x = 2 m and y = 4.5
m. Analytic timeconcentration curve using estimated parameters is shown and
compared with the observed curve in Fig. 3. These curves are
close enough to prove that the estimated values are greatly close to the real
values of the field characteristics. In Fig. 3, the observed
and analytic curves cross each other at the point of 85 days after injection.
It shows that at this time, the estimated concentration have the most agreement
with the field measured data.
Dispersivity for a two dimensional plume can be determined by several methods
such as one well method, two well method, areal method and inverse method. In
this experiment, because there is available only one sampling well that, timeconcentration
data is complete, therefore, one sampling well method of dispersivity estimation
(Zou and Parr, 1994), is used to compare with the first
method explained above. One sampling well method requires concentration versus
time data from one sampling well through which the plum passes. In order to
determine the transversal dispersivity, the observation well should be off the
plume centerline. This method is an extension of the twowell method proposed
by Zou and Parr (1993). Equations that used for calculation
of longitudinal and transversal dispersivity, given by Zou
and Parr (1994) are as bellow:
where, R_{1} = c_{max}t_{max}/ c_{l}t_{l} and c_{max} and t_{max} are concentration and related time of maximum concentration in timeconcentration curve. C_{1} is the half of the maximum concentration c_{max}/2 and t_{1} is the time of c_{1}.
These equations are used to calculate α_{x} and α_{y}
when and
the plume direction are known. According to above equations and observed timeconcentration
of borehole S1, dispersivities are estimated as: α_{x} = 6.7 m
and α_{y} = 0.76 m. Table 1 shows the estimated
parameters from these two methods for comparison. The known parameters are presented
also in Table 1 too.
It is obvious that the Zou and Parr (1994) method only uses two pairs of timeconcentration
data to estimate the dispersivitis coefficient. Also, by this method other parameters
like λ and R cannot be estimated. In order to examine the accuracy of the
dispersivity coefficients observation and analytic timeconcentration curve
based on parameters given by Zou and Parr (1994) method is shown in Fig.
4.
 Fig. 3: 
Observation and estimated timeconcentration curve of borehole
S1 using parameters derived from Kinzelbach (1986) 
 Fig. 4: 
Observation and estimated timeconcentration curve of borehole
S1 using parameters derived from Zou and Parr (1994) 
Comparison of Fig. 3 and 4 show the accuracy
of dispersivity estimation using the Kinzelbach (1986)
equation. Therefore, parameters that can be given by this method are more reliable
and were used to predict the plume concentration in groundwater system in the
study area.
CONCLUSIONS The accuracy of the estimated parameters obtained by the twodimensional analytic equation of solute transport in a uniform groundwater flow method is demonstrated in the study area. Based on the estimated parameters and the twodimensional analytic equation, the movement of the plume is predictable at any arbitrary time after injection for the study area. Movement of the plume is predicted at 85 days after injection for the study area and as it is shown in this paper the predicted and the observed concentration curves are exactly equal. ACKNOWLEDGMENTS The authors would like to extend their thanks to Dr. Zare for his useful suggestions. The authors greatly appreciate the scientific comments of Dr. Raeisi. We appreciate the critical reading by the arbitration committee and we welcome any enlightening suggestions and insightful comments.

REFERENCES 
1: Adabi, M.H., A. Zohdi, A. Ghabeishavi and H. AmiriBakhtiyar, 2008. Applications of nummulitids and other larger benthic foraminifera in depositional environment and sequence stratigraphy: An example from the Eocene deposits in Zagros Basin, SW Iran. Facies, 54: 499512. CrossRef 
2: Alavi, M., 2004. Regional stratigraphy of the Zagros foldthrust belt of Iran and its proforeland evolution. Am. J. Sci., 304: 120. CrossRef  Direct Link 
3: Alavi, M., 1994. Tectonics of the Zagros Orogenic belt of Iran: New data and interpretations. Tectonophysics, 229: 211238. CrossRef  Direct Link 
4: Bahroudi, A. and H.A. Koyi, 2004. Tectonosedimentary framework of the Gachsaran Formation in the Zagros foreland basin. Marin Petroleum Geol., 21: 12951310. CrossRef  Direct Link 
5: Benekos, D., O.A. Cirpka and P.K. Kitanidis, 2006. Experimental determination of transverse dispersivity in a helix and a cochlea. Water Resour. Res., 42: 74067407. CrossRef 
6: Chen, J.S., C.W. Liu and C.P. Liang, 2005. Evaluation of longitudinal and transverse dispersivities distance ratios for tracer test in a radially convergent flow field with scaledependent dispersion. Adv. Water Resour., 27: 887898. CrossRef 
7: Cirpka, O.A. and P.K. Kitanidis, 2001. Theoretical basis for the measurement of local transverse dispersion in isotropic porous media. Water Resour. Res., 37: 243252. Direct Link 
8: Cirpka, O.A., and P. K. Kitanidis, 2000. Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments. Water Resour. Res., 36: 12211236. Direct Link 
9: Cirpka, O.A., 2002. Choice of dispersion coefficients in reactive transport calculations on smoothed fields. J. Contam. Hydrol., 58: 261282. CrossRef 
10: Cirpka, O.A., A. Olsson, Q. Ju, M.A. Rahman and P. Grathwohl, 2006. Determination of transverse dispersion coefficients from reactive plume lengths. Ground Water, 44: 212221. CrossRef 
11: Dentz, M., H. Kinzelbach, S. Attinger and W. Kinzelbach, 2000. Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. Pointlike injection. Water Resour. Res., 36: 35913604. CrossRef  Direct Link 
12: Derakhshani, R. and G. Farhoudi, 2005. Existence of the oman line in the empty quarter of Saudi Arabia and its continuation in the red sea. J. Applied Sci., 5: 745752. CrossRef  Direct Link 
13: Eberhardt, C. and P. Grathwohl, 2002. Time scales of organic contaminant dissolution from complex source zones: Coal tar pools vs. blobs. J. Contam. Hydrol., 59: 4566. PubMed 
14: Fakhari M.D., G.J. Axen, B.K. Horton, J. Hassanzadeh and A. Amini, 2008. Revised age of proximal deposits in the Zagros foreland basin and implications for Cenozoic evolution of the High Zagros. Tectonophysics, 451: 170185. CrossRef 
15: Fiori, A. and G. Dagan, 2000. Concentration fluctuations in aquifer transport: A rigorous firstorder solution and applications. J. Contam. Hydrol., 45: 139163. CrossRef 
16: Ghavidelsyooki, M. and M. Vecoli, 2008. Palynostratigraphy of Middle Cambrian to lowermost Ordovician stratal sequences in the High Zagros Mountains, southern Iran: Regional stratigraphic implications, and palaeobiogeographic significance. Rev. Palaeobot. Palynol., 150: 97114. CrossRef 
17: Kinzelbach, W., 1986. Groundwater Modeling, An Introduction with Sample Programs in Basis. Elsevier, New York, ISBN: 0444425829, pp: 331
18: Klenk, I.D. and P. Grathwohl 2002. Transverse vertical dispersion in groundwater and the capillary fringe. J. Contam. Hydrol., 58: 111128. CrossRef 
19: Massabo, M., F. Catania and O. Paladino, 2007. A new method for laboratory estimation of the transverse dispersion coefficient. Ground Water, 45: 339347. CrossRef 
20: Rahnama, R.J., R. Derakhshani, G. Farhoudi and H. Ghorbani, 2008. Basement faults and their relationships to salt plugs in the arabian platform in Southern Iran. J. Applied Sci., 8: 32353241. Direct Link 
21: Sepehr, M. and J.W. Cosgrove, 2004. Structural framework of the zagros foldthrust belt, Iran. Marine Petroleum Geol., 21: 829843. CrossRef  Direct Link 
22: Shirazi, M., 2008. Calcareous Algae from the Cretaceous of Zagros Mountains (SW Iran). World Applied Sci. J., 4: 803807.
23: Zou, S. and A. Parr, 1993. Estimation of Dispersion Parameters for TowDimensional Plumes. Groundwater, 31: 389392. CrossRef 
24: Zou, S. and A. Parr, 1994. TwoDimensional Dispersivity Estimation Using Tracer Experiment Data. Groundwater, 32: 367373. CrossRef 



