ESTIMATION OF TRANSVERSE DISPERSTVITY IN THE MIXING ZONE OF FRESH-SALT GROUNDWATER

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ModelCARE 90: Calibration and Reliability in Groundwater Modelling (Proceedings of the conference held in The Hague, September 1990). IAHS Publ. no. 195, 1990.

1 ModelCARE 90: Calibration and Reliability in Groundwater Modelling (Proceedings of the conference held in The Hague, September 1990). IAHS Publ. no. 195, ESTIMATION OF TRANSVERSE DISPERSTVITY IN THE MIXING ZONE OF FRESH-SALT GROUNDWATER T. HOSOKAWA Department of Civil Engineering, Kyushu Sangyo University, Matukadai, Higashi-ku, Fukuoka 813, Japan K. JINNO & K. MOMII Department of Civil Engineering (SUIKO), Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812, Japan ABSTRACT This paper presents a method for the estimation of transverse dispersivity in a homogeneous isotropic coastal aquifer where salt water intrusion takes place and the mixing zone of fresh-salt water is observed. An approximate solution of the concentration profile in the mixing zone is derived from the simplified convection-dispersion equation in steady-state flow with respect to the curvilinear coordinate system, in which an axis parallel to the fresh-salt water interface is taken as the stream direction. The validity of the proposed method is confirmed through both a laboratory and a field experiment. INTRODUCTION Numerical simulations of the salt water intrusion into a coastal aquifer require the information on both longitudinal and transverse dispersivities which characterize the extent of the mixing zone. Since a theoretical method using the vertically measured profile of salt water concentration in a borehole has not yet been developed for the estimation of these dispersivities inherent in the coastal aquifer, one may have to deterraine these dispersivities by trial and error until a reasonable solution is obtained. Harleman & Rumer (1963) gave the experimental formulae between dispersion coefficients and the Reynolds number for isotropic and homogeneous porous media. However, it is difficult in a practical problem to apply the formulae to the dispersivity estim ation because an actual salt water profile may be influenced by the non-uniform distri bution of grain size or locally anisotropic matrix structure. Moreover, the flow pattern is affected by the density variation in salt water because of gravitational effect and tjien an analytical solution of mass transport with the uniform flow can not be apprised any more. It is therefore expected that an alternative method for the estimation of dispersivity will be developed based on a reasonable and physical implementation from the field data measured in boreholes. In this paper, a simple method which takes into consideration the above mentioned requirements is proposed for the estimation of transverse dispersivity. An analytical solution obtained by the similarity approximation is adapted to the vertical profile of the salt water concentration. To confirm the validity of the proposed method, a laboratory experiment has been performed and the method has been checked through a field experiment as well. BASIC FLOW EQUATION AND TRANSPORT EQUATION IN UNSATURATED- SATURATEDZONE The numerical simulation used in the field experiment is performed to check the validity 149

2 150 T.Hosokawa et al. of the proposed method for the estimation of transverse dispersivity. Fig. 1 shows the cross-sectional view of the actual unconfined aquifer. In order to solve the flow filed and concentration distribution for the coordinate system O r XY (Oj is the origine) in Fig. 1, the following basic equations are used: (a) equation of continuity: (Cw + ccos)-^- - (1) (b) equation of motion: i 3h Vx= - k ax (2).. 3h p. Vy = -k( ) (3) 9Y p f - where d8/dh : unsaturated zone \ 0 : saturated zone and oc 0 : 0 : unsaturated zone 1 : saturated zone (c) mass transport equation: where! l = i- (6C) + V-(v'eC) = V-(0D-VC) (4) dc a t v' = ï- (5) and 6D = 6Dxx 9DxY. 9DYX 6DYY. (6)

3 Transverse dispersivity in fresh-salt mixing zone edxx = a L Ix+a T^+ 0Dj Vx w=a T v +a L^+ 6D M V 2 (7) D XY =9D YX =(a L -a T )- v xv Y v=(^ + v?) 1/2 (8) Symbols used herein are as follows: h : equivalent fresh-salt water hydraulic pressure head (cm) k(6) : hydraulic conductivity (cm s" 1 ) v(v x,v Y ) : Darcy velocity (cm s" 1 ) v'(v x,v Y ) : actual velocity (cm s" 1 ) p : density distribution (g cm" 3 ) S : specific storage coefficient (cm -1 ) Cy, : specific water capacity (cm" 1 ) OCQ : dummy number C : normalized mass fraction given by C= P Pf x 100 (%) Ps-Pf (9) M Hean sea water level\ w Salt water depth ê o, ' \/>..Fresh-salt ;- ^ O v interface 26.0 m 8.0 n 18.0 o ^-Ground surface»? /www Groundwater table-* :. ; k s =0.02 cm/s Fresh water p f =1.000g/cm 3 Salt water ^^l-~<; : p s = l-025g/cu 3 Impervious boundary S Q? /W/W ^ ( [ i k s = 1.6x10-6 ci/s -J LEGEND g : surface soil (U : fine-coarse si ind g : sandy silt g: silt //W/W WW V Fresh water depth > e *& CD C rn I C o FIG. 1 Cross-sectional view of the unconfined aquifer (field set-up).

4 152 T.Hosokawa et al. pf : fresh water density (g cnr 3 ) p s : salt water density (g cm" 3 ) : density difference = (p s -p f )/p f 9 : soil water content(= porosity when saturated) D M : molecular diffusion coefficient (10~ 5 cm 2 s" 1 ) OL Oj. : longitudinal dispersivity (cm) : transverse dispersivity (cm) SIMILARITY SOLUTION ALONG THE CURVILINEAR AXIS According to the previous works of Ueda et al (1977), Jinno et al (1989) and Momii et al. (1989), when one axis is taken parallel to the flow direction and the other perpendicular to the flow direction, equation (4) is reduced to a simple form Ux Ç =CCT _ (Ux Ç ) (10) ox oy ay where U x is the actual velocity along the x-axis, i.e. interface, and the x-axis (denoted by lower case x) runs along the interface, while the y-axis (lower case y) is perpendicular to the x-axis. The disappearance of the transverse convection and the longitudinal dispersion terms is a consequence of choosing our coordinate system parallel to the flow direction. The assumption of neglecting the transverse convection and the longitudinal dispersion terms is confirmed in the numerical simulation. In the present analysis, such an axis is taken parallel to the fresh-salt water interface as shown in Fig. 1. The distance between the immiscible interface, which is given by equation (11) (JSCE, 1985), and the iso-concentration line which is determined by the similarity solution, is depicted by the symbol a. The analytical solution of the immiscible interface for the unconfined aquifer is: h(x) = (ek)???-x /2 (ID where q (cm 2 s _1 ) is the fresh water discharge per unit width toward the sea. The similarity solutions are given as: U x = U co f(ti) (12) c = 100(1 - fen)) (%) (13) where U<» denotes the velocity at a point distant from the height of h(x)+a and U x is the actual velocity along the x-axis (along the interface). By substituting equation (13) into equation (9), the density distribution is expressed as: P = Ps-(Ps-pf)f(Tl) (14) The similarity variable T] is defined by:

5 Transverse dispersivity in fresh-salt mixing zone 153 ^RW and R(x) is a function of x. The similarity function f(r]) and R(x) are obtained by substituting equations. (12) and (13) into equation (10): (15) f(tl) = 1- e-^dç 1/2 (16) R(x) = Y2a T x (17) ESTIMATION PROCEDURE OF TRANSVERSE DISPERSIVITY The concentration profile is commonly measured in a borehole drilled vertically in an aquifer. On the other hand, the above mentioned similarity solution is given along the curvilinear coordinate. It is therefore necessary to transform the measured data in a borehole into the data set in the curvilinear coordinate. The transformation of the coordinates is done by: f X ' JO l + (dh(x)/dx) 2 ] 1/2 dx (18) r? 211/2 91 y=[(x-x') +(Y-h(X')-arj (19) where the point on the x-axis parallel to the immiscible interface is expressed by (X',h(X')+a). The point (x,y) in the 0 2 -xy coordinate is equivalent to the point (X,Y) in the O r XY coordinate. The transverse dispersivity 04 appearing in equation (17) controls the shape of concentration profiles. When the sum of the squares of fitting error in the similarity solution (equation (13)) for the measured profile in a borehole has a minimum value, the best estimate of the transverse dispersivity is obtained. Then the following criterion is adapted: N J(«T) = 2 j=l r C M(X 0,Yj) - C(Xj,Yj) (20) where C M (X 0,Yj) denotes the concentration at the depth Yj of the measurement point at X=X 0 and C(x;,yj) the similarity solution. N is the number of the measurements in a borehole. Changing the magnitude of 04. the best estimate of 04 is obtained.

6 154 T.Hosokawa et al. VERIFICATION OF PROPOSED METHOD BY LABORATORY EXPERIMENT The similarity and the numerical solutions are compared with the measured profiles obtained in the laboratory experiments. The conditions of the experiment and numerical calculation are given in the previous study (Momii eial., 1989). The method of characteristics is adapted in the numerical calculation for the mass transport equation. As shown in Fig. 2, the similarity solution is in good agreement with both the measured profile and the numerical solution. The validity of the similarity assumption for the velocity and concentration profiles with respect to the curvilinear axes can be found in Fig. 3. :Measured : Similarity solution (a T =0.01cm) :Numerical ra:t=0.01cni -I >-a L =0.32cm J Concentration (%) FIG. 2 Concentration profile for the laboratory experiment. APPLICATION TO ACTUAL UNCONFINED AQUIFER The location of the observation In Fig. 1, the cross-sectional view of the observation area is shown. The origin of the X coordinate is moved from the actual beach line toward the fresh water region so that the measured isoline of 90% of salt water concentration at the mean sea water level matches the immiscible interface given by equation (11). One may, alternatively, be able to utilize the numerically obtained immiscible interface instead of equation (11). The distance a is determined as 74.9 cm at this point The sea water level oscillates from time to time and therefore the vertical rise and fall of the concentration profile in the borehole is observed. Strictly speaking, the steady-state assumption may not be valid. However, the similarity solution which assumes the steady-state is adapted to the initial guess of the transverse dispersivity. The bold solid line in Fig. 1 shows the immiscible interface when the sea water level is equal to the mean sea water level.

7 Transverse dispersivity in fresh-salt mixing zone y io-$ 5H- (cm) Similarity solution x Numerical U»=0.14cm/s y(cm) I * Concentration(%) (a) Concentration distribution. -Similarity solution x Numerical i U/Uc (b) Velocity distribution. FIG. 3 Validity of similarity assumption. a x O :Measured :Similarity solution (a T =0.36cm)."Numerical (a T =al=0.36cm) J i i i i. i i i i Concentration (%) FIG. 4 Concentration profiles. Comparisons of the measured concentration and similarity solution In Fig. 4, the measured profile of salt water concentration is indicated by cross marks. Unlike the similarity solution, the concentration in the range of 6.5 to 7.0 m depth does

8 156 T.Hosokawa et al. not gradually increase with depth. This is perhaps affected by the locally inhomogeneous permeability. However, the present method is applied in order to obtain an over-all transverse dispersivity. In Fig. 5, the dependency of J calculated by equation (20) on 04. is shown. The best fit of 04. has been determined as 0.36 cm. Using this value, the similarity solution is plotted by the solid line as shown in Fig. 4. The numerical solution using the best fitting is indicated by circles. The same value of 04 is used for ot L because of no information on oc L. However, it should be remarked that the dependency of the transverse concentration profile in a steady-state on the longitudinal dispersivity is essentially small. The similarity and numerical solutions represent the measured profile CD a 3 T3 as <«13.OH o -" J T Transverse dispersivity a T (cm) FIG. 5 Sensitivity of J against Oj-. Concentration distribution andflowpattern In Figs 6 and 7, the numerical solutions are shown. The calculated fresh water discharge q is cm 2 s" 1. The cross marks in Fig. 6 indicate the numerically obtained immiscible interface. The results obtained by the formula for the immiscible interface (equation (11)) matched this interface very well. In Fig. 7, a very slow movement of salt water toward the fresh water region is obtained below the interface. This flow transports salt water from the sea toward the fresh water region through the mixing zone. CONCLUSIONS A simple method for the estimation of transverse dispersivity is discussed. Unlike the guess of the dispersivities by trial and error, the present method uses the measured data from a theoretical and physical point of view. Even though the present method does not consider the relation between the dispersivities, and a matrix structure (anisotropy and inhomogeneity), initial information on the value of transverse dispersivity will be obtained.

9 - * - *. *. - * - < -< < * «* ^ < Transverse dispersivity in fresh-salt mixing zone >^_- Ground surface /!W/W ^ ^Groundwater table "=" : Numerical ^. x : Imœiscible interface ^ l ^ r 1 0 % ^J?\ è$i>^ Salt water region ^Impervious boundary Fresh water region ^^^SSSsSjs^ ^~^"^ ^ 5^ X(m) FIG. 6 Concentration distribution by numerical solution * : Velocity ca/s i n- < < < < < < * -_ «* ^ -f_ (- <e n.ii- ^fctl ss"ç^> tj>- "* * * *-*- +- «* 1 0-i J ^%^S H t t t k ^ ÎN ^ *-*. X^ *«^ * * - c *. l'^oîsï^''^'**'*- *- "*" > < - R ^ ^ ^ - ^ ^ ^ - (. > > v*^'f^-'*.-*.-*,-*. > > * w Y ^» "*- *«* *. > > > i. - i.ç - * *. - *. *. > > > > * > v Y * * > > > > > > A v T -v. > > > XT T =» > > A > > > > > > > > > A w > > > > > > > > > > < < < < < < i i J t î < - - «- * - * - «- - * - * - «- * - * - * ^ - *. - *. *. <«. «.<«.-«. -«. «.«.-«. *. - «. *. *. *. - «. - «. - *. >ç. - *. - *. *. *. f. -*.». C T * * A V v v -v > * V T > > > < > > > > > > > > < < * - * «- * - *. * * * v * * - * - < - < - <-«.<- «- *- -e- * * s "* ï >* J < ^ < r < < «* X Î j J J <- ; <- i * <- <- <- «. Î ; «- * ; ;» -* ^ * t -s» «* < < < < X * * t t % t t t <- < < I ;! * * < * * * < < <- î -f * Km) FIG. 7 Calculated flow pattern. REFERENCES Harleman,D.R.F. & Rumer,R.R. (1963) Longitudinal and lateral dispersion in an isotropic porous medium. J. Fluid Mechanics 16, Japan Society of Civil Engineers (1985) Handbook of Hydraulics 384, in Japanese. Jinno,K., Berndtsson,R., Momii,K. & Hosokawa,T. (1989) Estimation of the lateral

158 T.Hosokawa et al. dispersivity in a homogeneous isotropic coastal aquifer. Proc. 10th SWIM. Ghent Belgium, 300-307. Momii,K., Hosokawa,T., Jinno.K. & Ito.T. (1989) Estimation method of transverse dispersivity based on vertical salt concentration distribution in coastal aquifer.

10 158 T.Hosokawa et al. dispersivity in a homogeneous isotropic coastal aquifer. Proc. 10th SWIM. Ghent Belgium, Momii,K., Hosokawa,T., Jinno.K. & Ito.T. (1989) Estimation method of transverse dispersivity based on vertical salt concentration distribution in coastal aquifer. Proc.of Japan Society of Civil Engineers. 411(II)-12,45-53, in Japanese. Ueda,T., Jinno,K. & Fujino,K. (1977) On the mixing zone of fresh-salt water interface of groundwater flow. Technical Report of Kyushu University Vol.50, No.3, , in Japanese.

In all of the following equations, is the coefficient of permeability in the x direction, and is the hydraulic head.

In all of the following equations, is the coefficient of permeability in the x direction, and is the hydraulic head. Groundwater Seepage 1 Groundwater Seepage Simplified Steady State Fluid Flow The finite element method can be used to model both steady state and transient groundwater flow, and it has been used to incorporate