Management, Nakhon Pathom Rajabhat University, 85 Malaiman Road, Muang, Nakhon Pathom 73000, Kingdom of Thailand

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1 The Importance o Denity Dependent Flow and Solute Tranport Modeling to imulate Seawater Intruion into a Coatal Aquier Sytem Phatcharaak Arlai 1 and Manred Koch 2 1) Dr.-Ing., Head, Reearch Unit or Sutainable Water and Environmental Reource Management, Nakhon Pathom Rajabhat Univerity, 85 Malaiman Road, Muang, Nakhon Pathom 73000, Kingdom o Thailand 2) Full Proeor, Director, Intitute or Geotechnology and Geohydraulic, Univerity o Kael, Kael D-10400, Republic o Germany Abtract: Thi tudy dicloe the premier acet why the ue o a denity-dependent low and olute tranport model i very important to model the eawater encroaching into a coatal aquier ytem. In order to clariy thi, comparion o modeling reult rom both a contant- and a variable denity low and olute tranport model in variou point o view are done. The ummary reult proo evidence that the variable denity model i better able to ynchronize the computed eawater- intruion-band with the band reulting rom the analytical olution (the baed cae) than the contant-denity tranport model Keyword: Coatal Aquier, Seawater Intruion, Importance o Variable-Denity Modeling 1. Introduction Seawater intruion i continually becoming a eriou problem in Thailand, a the eawater ha urther invaded into many coatal aquier becaue o negative hydraulic gradient rom heavy groundwater pumping. Such i, or example, the cae or the Bangkok Aquier Sytem and the Samui Iland-coatal aquier. Subequently, the reh groundwater in thoe aquier ha been polluted by eawater water. Thu there i a need or the Thai authoritie to comprehenively undertand the eawater intruion mechanim in order to be able to utainably manage thi pollution problem. However, a evidenced in the pat groundwater modeling tudie, thi i unortunately yet not the cae ince, up-to-date, only contantdenity groundwater low and olute tranport model have been ued to imulate the eawater intruion phenomena. In a detailed eawater intruion tudy Arlai (2007) and Arlai and Koch (2007a;b) have hown that a contant- denity model cannot well imulate the high-contrat-denity groundwater regime, a in eawater intruion. Indeed, thi peciic groundwater regime need to be modeled by a variable denity groundwater and olute tranport model. A a matter o act, the article will how the importance o uing a variabledenity low and olute tranport model to imulate the eawater intruion mechanim. 2. Mathematical Theory 2.1 Contant denity groundwater low equation The three dimenional movement o groundwater o contant denity through porou earth media i decribed by the ollowing partial dierential equation (e.g. McDonald and Harbaugh, 1988):

2 Kxx + K yy + Kzz W = S x x y y z z t (1) where K xx, K yy and K zz [LT -1 ] are the hydraulic conductivity along the x, y and z coordinate axe, which are aumed to be parallel to the major axe o hydraulic conductivity; h [L], the potentiometric head; W [T -1 ], a volumetric lux per unit volume and repreent ource and/or ink o water; S [L -1 ], the peciic torage o the porou media; and t [T], the time. 2.2 Variable denity groundwater low equation The variable-denity groundwater low mechanim in a porou media i decribed by the ollowing mathematical equation. ρ ρ Z ρ K x + ρ K y + ρ K z + x x y y z z ρ z ρ C = ρ S + θ ρ q t C t (3) where h [L] i the equivalent reh water head (ee below); K x, K y, K z, [LT -1 ] the equivalent rehwater hydraulic conductivitie in the three coordinate direction; ρ [M/L 3 ], the denity o native aquier water; ρ [M/L 3 ], the denity o rehwater; S [L -1 ], the peciic torage in term o equivalent reh water head; C [M/M], the olute concentration; θ, the eective poroity (dimenionle); ρ [M/L 3 ], the denity o water entering rom a ource or leaving through a ink; and q [T -1 ], the volumetric low rate o ource and ink per unit volume o aquier. 2.3 Variable denity olute tranport equation For groundwater low with large denity variation in the olute tranported, the reditribution o the olute concentration change the denity ield itel which, in return, aect the groundwater low. Thereore, groundwater low and tranport o olute in the aquier are coupled procee in uch a ituation, o that the low equation (3) and the tranport equation C v q = ( D C) ( νc) C + R t N θ k= 1 mut be olved jointly. Here D [L 2 /T] i the hydrodynamic diperion tenor deined a D = D m + D*, where D m and D* are the coeicient o mechanical and molecular diperion, repectively, the ormer being related to the linear luid velocity v [L/T] through D m = ( v, A L, A T ), where A L [L] and A T [L] are the longitudinal and tranveral diperivity, repectively; C [M/M], the olute concentration o water entering rom ource or ink and R k [M/(L 3 *T)] i the rate o olute production or decay in reaction k o N dierent reaction, which in the preent cae o pure altwater tranport i et to zero. k (4)

3 2.4 Equation o tate between denity and concentration For iothermal condition, the luid denity o the groundwater i predominantly aected by the olute concentration. The empirical equation o tate between the denity o altwater and concentration can be written a: ρ ρ = ρ + * C (5) C ρ where ~ [M/L 3 ] i the empirically determined lope o the linear relationhip C between denity and alt water concentration. Equation (5) etablihe the coupling between the low equation (3) and the olute concentration equation (4). 3. Study area and model implementation The modeling o a cae-tudy o eawater intruion in a coatal aquier in Irael by Bear (1979) by mean o a 3D inite element model will be ued a an example to tet the importance o denity eect in a real coatal groundwater ytem aected by by eawater intruion. Here both the MODFLOW-2000&MT3DS (a contant denity code) and the SEAWAT D model (Langevin et al., 2003) are applied to the cae-tudy o Bear et al. (2001), uing the ame model-etup and the ame low and tranport parameter and attempt to undertand the importance o denity-dependent eect in thi problem. The modeling approach i auming no groundwater withdrawal (pumping). A imulation over 30 year (until a teady tate i reached) i carried out and modeled alinitie o both contant- and variable denity model are compared. The computational domain i a rectangular prim 1000 m long, 500 m wide and 105 m thick and dicretized by 20 m x 20 m cell x 15 and 10 m or layer 1 and the other (in the z direction), repectively (Figure 1.). Thu the model i compried o 10 modeled layer. The aquier i recharged horizontally by a contant rehwater inlux at a rate o q 0 = m/day on the inland boundary, while on the eawater boundary head and concentration are et to eawater condition. Note that or the contant-denity code MODFLOW- 2000&MT3DMS the eawater boundary head mut be adjuted to comply with the higher eawater- than reh water preure. Thi i done by calculating the o-called equivalent reh water head, uing the ollowing ormula ρ ρ ρ h = h Z (6) ρ ρ where h i the equivalent reh water head (m); ρ, the denity o aline aquier water (~ kg/m3); ρ, the denity o reh water (~1 kg/m3); h, the head (0 m at ea boundary); and Z, the elevation, taken or each layer o the model in the middle. For the variable-denity code SEAWAT-2000, on the other hand, the tranormation (1) i not needed explicitly, a all equation are internally already written in term o the equivalent reh-water head. To wrap up the boundary condition, the bottom boundary i aumed to be imperviou to low and

4 Figure 1: Domain and boundary condition in the coatal aquier (Bear et al, 2001) tranport. The aquier i recharged uricially through natural replenihment by precipitation with a rate o 0.15 m/year. The hydraulic- and olute tranport parameter ued are a ollow: poroity = 0.25, horizontal conductivity = 20 m/day, vertical conductivity = 2 m/day, denity o pure rehwater = 1000 kg/m 3, denity o pure eawater = 1025 kg/m 3, longitudinal diperivity = 10 m, tranveral diperivity = 1 m and the molecular diuivity = 0 m 2 /day. 4. Analytical theory or the reh- eawater interace location 4.1 The Ghyben-Herzberg approximation Ghyben and Herzberg (c. Bear, 1979) have derived a well-known ormula or the approximate location o a tationary (harp) interace between the reh- and the eawater (Figure 2), known a the Ghyben-Herzberg equation h = ( ρ /( ρ ρ ) * h = δ * h (6) with δ = ρ /( ρ ρ ) which, uing tandard value or the denity o reh water 3 3 ( ρ 1000kg / m ) and eawater ( ρ 1025kg / m ) reult in a value o δ =40, o that = = h 40 * h = (7) where h i the depth o the tationary interace below ea level, and h, the rehwater table above ea level.

5 Figure 2: Ghyben-Herzberg tationary interace 4.2 Analytical expreion baed on Dupuit aumption For a hallow phreatic coatal aquier, under Dupuit aumption, the groundwater low i eentially in the horizontal direction. Auming urther that the aquier i uniormly recharged rom above by an amount N [L/T] and dicharge laterally by the amount Q 0, the application o continuity equation reult in (Bear, 1979): Q0 + N x = K( h + h ) * = K(1 + δ )* h * (8) x x where and h( x) = δh ( x) and δ = ρ /( ρ ρ ), Integrating Eq. 8. rom x = 0 to x, h = φ0, h = B (depth rom the ea level to the bottom o the aquier) reult in φ h = (2Q x + Nx ) / K(1 + ) (9) 0 0 δ At x = L (horizontal extenion o the eawater intruion) h = 0, which give: Q 0 = KB 2L 2 (1 + δ ) NL 2 δ 2 (10) B where φ0 =. Thu the ize o L o the eawater intruion length can be determined rom δ Eq. 10 which, uing the value o the preent model etup, dicued in the previou ection, reult in L = m. Auming a parabolic hape o the reh-altwater interace in x- direction and the named L value, the poition y(x) o thi interace can be written a a parabolic equation o the orm (Bear, 1979) 2 ( y 0) = 4 * ( ) * ( x 1000) (11)

6 Figure 3: Stationary interace with the Dupuit-Ghyben-Herzberg approximation (Bear, 1979) 5. Steady-tate imulation Reult o the teady-tate imulation with no pumping, uing both the contant-denity (MODFLOW-2000&MT3DMS) and the variable-denity model SEAWAT-2000 are plotted together with the Ghyben-Herzberg interace location and the interace location baed on Dupuit aumption in Figure 2. The igure illutrate unequivocally the importance o the incorporation o denity-dependent low and tranport into the eawater intruion modeling problem, a the denity-driven intruion plume move urther landward and alo conorm better with the two analytical approximation olution than the contant-denity model. Figure 4: Steady-tate aline concentration (kg/m 3 ) at Y = 250 m o contant-denity- (dahed line), variable-denity model (olid line), Ghyben-Herzberg interace location (circle), and interace location baed on Dupuit aumption (diamond). Alo hown i the phreatic water table (triangle).

7 6. Dicuion The mechanim illutrated in Figure 4 can be explained through Darcy law: K x P K y P K z P qx =, qy =, qz = + ρg µ x µ y µ z (12) where the denity ρ o the luid i taken at the point and time or which the peciic dicharge q i to be determined. Even though Eq. (12) predict at irt ight that denity eect act only in vertical direction, one mut conider that the whole preure ditribution in a porou media i partly inluenced by the overall luid denity ditribution and, hence, alo the horizontal peciic dicharge i aected by the variable denitie in the groundwater ytem. Thi i clearly manieted alo by the reult or the low vector hown in Figure 5. Figure 5. Flow vector or teady-tate eawater intruion at Y = 250 m. o denityindependent model (top panel) and denity-dependent model (bottom panel). The vertical blue line envelop approximately the low vector area aected by the ea boundary and/or by denity driven low.

8 7. Concluion The teady-tate imulation how that a denity-dependent low and olute tranport model mimic eawater intruion (analytical olution) in a coatal aquier more realitically than a contant-denity model. In act, the intruion band o the variable-denity model conorm better to the two analytical olution or the location o the alt reh water interace (Bear, 1979) than the contant-denity low and tranport model. 8. Reerence Arlai, P. (2007), Numerical Modeling o poible Saltwater Intruion Mechanim in the Multiple-Layer Coatal Aquier Sytem o the Gul o Thailand, Dr.-Ing. Diertation, Univerity o Kael, Kael, Germany, 147p. Arlai, P., and M. Koch (2007a), Numerical Invetigation o the original Source o aline Pollution and eicient Remediation Scenario in the Bangkok multilayer-aquier Sytem, In: Proceeding o 12 th National Convention on Civil Engineering,, Phitanulok, Thailand, May 2-4, Arlai, P., and M. Koch (2007b), Need or denity-dependent Flow and Tranport Modeling o horizontal Seawater and vertical Saltwater Intruion in the Bangkok multilayer-aquier Sytem, In: Proceeding o 12th National Convention on Civil Engineering, Phitanulok, Thailand, May 2-4, Bear, J. (1979), Hydraulic o Groundwater, McGraw-Hill, New York. Bear, J., Q, Zhou, and J. Benabat (2001), Three Dimenional Simulation o Seawater Intruion in Heterogeneou Aquier, with Application to the Coatal Aquier o Irael, preented at 1t International Conerence on Saltwater Intruion and Coatal Aquier Monitoring, Modeling, and Management, Eaouira, Morocco, April McDonald, M.G. and A.W. Harbaugh (1988), A Modular Three-Dimenional Finite Dierence Ground-Water Flow Model, U.S. Geological Survey, Technique o Water Reource Invetigation, Book 6, Chapter A1, 586p. Langevin, C.D., W.B. Shoemaker, and Guo, Weixing (2003), MODFLOW-2000, the U.S. Geological Survey Modular Ground-Water Model Documentation o the SEAWAT-2000 Verion with the Variable-Denity Flow Proce (VDF) and the Integrated MT3DMS Tranport Proce (IMT): U.S. Geological Survey Open-File Report , 43 p.