Theoretical analysis of the worthiness of Henry and Elder problems as benchmarks of density-dependent groundwater flow models

Transcription

1 Advances n Water Resources (003) Theoretcal analyss of the worthness of Henry and Elder problems as benchmars of densty-dependent groundwater flow models M.J. Smpson a, T.P. Clement a,b, * a Centre for Water Research, Department of Envronmental Engneerng, The Unversty of Western Australa, Nedlands, 07 Australa b Department of Cvl Engneerng, Auburn Unversty, Auburn, AL 330, USA Receved 1 February 00; receved n revsed form 1 June 00; accepted 3 August 00 Abstract Computer models must be tested to ensure that the mathematcal statements and soluton schemes accurately represent the physcal processes of nterest. Because the avalablty of benchmar problems for testng densty-dependent groundwater models s lmted, one should be careful n usng these problems approprately. Detals of a Galern fnte-element model for the smulaton of densty-dependent, varably saturated flow processes are presented here. The model s tested usng the Henry salt-water ntruson problem and Elder salt convecton problem. The qualty of these benchmar problems s then evaluated by solvng the problems n the standard densty-coupled mode and n a new densty-uncoupled mode. The dfferences between the solutons ndcate that the Henry salt-water ntruson problem has lmted usefulness n benchmarng densty-dependent flow models because the nternal flow dynamcs are largely determned by the boundary forcng. Alternatvely, the Elder salt-convecton problem s more suted to the model testng process because the flow patterns are completely determned by the nternal balance of pressure and gravty forces. Ó 00 Elsever Scence Ltd. All rghts reserved. Keywords Groundwater-modelng; Densty-dependent flow; Unsaturated flow; Contamnant transport 1. Introducton * Correspondng author. Tel ; fax E-mal addresses smpson@cwr.uwa.edu.au (M.J. Smpson), clement@eng.auburn.edu (T.P. Clement). Benchmarng the performance of a numercal code aganst standard analytcal solutons s the necessary frst step n testng the correctness of the numercal approxmatons. The next logcal step n the benchmarng process s testng the code to ether reflect a laboratoryscale expermental data set or a feld-scale case study. Completng these two benchmarng steps for a denstydependent flow model s a dffcult tas because the avalablty of analytcal solutons or standard laboratory/feld data sets for the densty-dependent flow problem s lmted [31]. Ths contrasts wth those problems nvolvng lnear solute transport and/or densty nvarant groundwater flow, for whch there exsts several well nown analytcal and laboratory solutons to the governng equatons [1,17]. In the lterature there has been much dscusson concernng the phlosophy of the model benchmarng processes. Whle t s not the purpose of ths communcaton to enter nto ths phlosophcal dscusson, the nherent problems wth model verfcaton should be acnowledged. Konow and Bredeheoft [0] and Oreses et al. [] argue that verfcaton and valdaton of numercal models of natural systems s mpossble, as true natural systems are not closed and numercal results are always non-unque. Whle the mportance of ths argument should not be understated, the recognton of these phlosophcal ssues should not detract from the need for accurate model development and testng, as the process of mathematcal modelng s one of the useful optons avalable for ganng nsght nto complex natural systems. Konow and Bredeheoft [1] suggested that a smple acnowledgement by the groundwater modelng communty mght crcumvent some of the phlosophcal ssues assocated wth model valdaton and verfcaton /03/$ - see front matter Ó 00 Elsever Scence Ltd. All rghts reserved. PII S (0)0005-

2 1 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) They argued that the terms valdaton and verfcaton should not be used n relaton to model testng as these terms convey a false sense of accuracy to the general publc. Furthermore, they stated that groundwater modelers should acnowledge the phlosophcal problems wth the terms valdaton and verfcaton and refran from usng them, and they suggest the term hstory matchng as a substtute. We propose and use the term benchmarng as a way to descrbe and expand upon the meanng of hstory matchng. By usng the term benchmarng, we mply that the numercal algorthm can reproduce the pror hstory of a well-defned problem. The term benchmarng s also broader n scope than hstory matchng and refers to model testng aganst standard problems that have been suffcently tested and are wdely accepted by model developers. An addtonal property of benchmar problems s that they are well understood and have been tested aganst analytcal expressons and/or well-controlled feld and laboratory studes, as well as aganst several numercal solutons n publshed ntercode comparson studes. A revew of the benchmar problems avalable for testng densty-dependent groundwater flow ndcates that the Henry salt-water ntruson problem, for whch an analytcal soluton exsts, and the Elder salt convecton problem, for whch laboratory and numercal data are avalable, are the most popular ones. Note that whle ths study focuses upon the Henry salt-water ntruson problem and the Elder salt convecton problem, there are several other problems that have been used n the lterature to benchmar densty-dependent groundwater flow codes. For example, the HYDROCOIN level 1, case 5 salt-dome problem [5] has also been frequently cted n the lterature as a benchmarng problem e.g., [15,1,3]. More recently, several newer problems have been devsed and proposed as benchmarng standards e.g., [1,31]. Although we restrcted ths analyss to the two most popular benchmar problems, the methodology developed here could be used to perform a smlar analyss upon other benchmar problems. The obectve of ths study s to provde a comprehensve assessment of the worthness of the Henry and Elder problems by benchmarng a densty-dependent groundwater flow model. Detals of a densty-dependent numercal model are provded, n part, to show the complexty of the governng relatons and therefore llustrate the mportance of ensurng a thorough benchmar process. Once developed, the numercal model s used to solve the common Henry salt-water ntruson problem and the Elder salt convecton problem. The model s then used to resolve these benchmar problems n an uncoupled mode. Comparng the coupled and uncoupled results facltates a novel way to separate the effects caused purely by densty-dependent effects from those caused by the boundary condtons. By analyzng the results of these coupled and uncoupled solutons for the Henry and Elder problems, a qualtatve assessment of the worthness of each problem s accomplshed.. Modelng densty-dependent groundwater flow.1. Governng equatons and consttutve relatons The equatons governng the movement of a flud through a varably saturated porous medum subect to varable densty condtons can be derved from mass and momentum conservaton prncples [10,13]. For the present study, only two-dmensonal (vertcal) flow s consdered. The governng equatons are a set of two, coupled, non-lnear partal dfferental equatons wrtten n terms of the fresh water pressure head as the dependent varable [] obhþ þ S s b h ot / ¼ o ox ow ot bkhþ ow ox þ o oz h oc ot ¼ b o D x h oc ox ox V x h oc ox V zh oc oz bkhþ ow oz þ b o oz D z h oc oz þ o bk h Þ oz 1Þ Þ where h s the water content of the porous medum, b s the rato of the flud densty to a reference freshwater densty, K [LT 1 ] s the hydraulc conductvty of the porous medum, w [L] s the freshwater pressure head of the flud, S s [L 1 ] s the specfc storage coeffcent for the porous medum, / s the porosty of the porous medum, C [ML 3 ] s the concentraton of the solute whch contrbutes to the densty varaton, D [L T 1 ] s the total dsperson coeffcent n the th Cartesan drecton and V [LT 1 ] s the flud velocty n the th Cartesan drecton. Eqs. (1) and () are coupled through the b term whch represents the relatve dfference between the densty of the flud to a reference freshwater densty q ¼ q 0 b ¼ q 0 1 þ ec 3Þ where q [ML 3 ] s the flud densty, q 0 [ML 3 ] s the reference fresh water densty, C s a dmensonless concentraton C ¼ C=C max, where C max [ML 3 ] s the concentraton correspondng to the maxmum flud densty, and e s a dmensonless measure of the dfference between the maxmum densty and freshwater densty defned by e ¼ q max q 0 Þ q 0 Eq. (1) governs the movement of the flud under varably saturated, densty-dependent condtons. Eq. () represents the movement of the solute through the acton of

3 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) dsperson and advectve processes. In addton the equatons are coupled through the flud velocty, whch may be calculated through the applcaton of DarcyÕs law once Eq. (1) has been solved V ¼ K ow þ g b 5Þ h ox where g s the vector ndcatng the drecton of the acton of gravtatonal forces. Once the spatal dstrbuton of velocty s nown, then the coeffcents of dsperson can be evaluated usng the relatons [10] Vx D x ¼ a L V þ a T Vz D z ¼ a L V þ a T V z V þ D m V x V þ D m Þ where a L [L] and a T [L] are the longtudnal and transverse dspersvtes, respectvely, V [LT 1 ] s magntude of the flud velocty and D m [L T 1 ] s the molecular dffuson coeffcent for the solute wthn the porous medum. Eqs. (1) () are the governng equatons for denstydependent flow n varably saturated porous meda. The varably saturated form of the densty-dependent flow equatons s consdered because t s the most comprehensve way to descrbe general densty-dependent flow condtons. Although the benchmar problems consdered n ths paper are only concerned wth fully saturated condtons, there are several practcal problems that nvolve varably saturated flow phenomena. For example, Boufadel et al. [3] showed how varably saturated condtons could nfluence the groundwater flow beneath salt-laes... Numercal soluton strategy The soluton of equatons defned by the system (1) () poses an nterestng numercal challenge, partcularly because the system s non-lnear. Under fully saturated densty-dependent flow condtons the non-lnearty n the system arses from the presence of b n both the groundwater flow and solute transport equatons. For the case of varably saturated densty-dependent condtons, the governng equatons are further complcated by the more complex non-lnear relatonshps between the flud pressure and mosture content, as well as the flud pressure and the unsaturated hydraulc conductvty of the porous medum. The numercal soluton of the governng equatons was sought usng the Galern fnte-element technque. The dscretzaton of the spatal doman s acheved usng lnear trangular elements. The transent, varably saturated groundwater flow equaton s approxmated usng a lumped, bacward Euler tme steppng scheme. The modfed Pcard procedure s ncorporated to allevate the non-lnearty n the flow equaton for unsaturated condtons [,5]. Once the flow equaton s solved, then the flud velocty s evaluated usng a method proposed by Yeh [3]. Ths method for the evaluaton of flud veloctes ensures that the Darcy flux across each element boundary s contnuous and therefore guarantees conservaton of mass at each element nterface [3]. Once the spatal varaton of the flud velocty s nown, the dsperson coeffcents n the solute transport equaton can be calculated. The solute transport equaton s then solved usng a lumped Galern fnte element formulaton that ncorporates the use of a tme-weghtng scheme. Whle the equatons resultng from the applcaton of the Galern fnte-element-method have been documented for smple varably saturated flow [1] as well as for fully saturated, densty-dependent flow [10], there has been no clear documentaton of the ntegrated element equatons for varably saturated, densty-dependent groundwater flow. Therefore, the detals of the Galern element matrces derved for the varably saturated densty-dependent groundwater flow equatons are summarzed n Appendx A for reference purposes. The matrx equatons have been derved by analytcally ntegratng the governng equatons over lnear trangular elements; therefore the algorthm has the advantage of avodng the need for any numercal ntegraton, and the entres n the stffness matrx and force vector can be drectly assembled wth lttle effort. In addton, whle mplementng the Yeh [3] algorthm t should be recognzed that the global stffness matrx for the evaluaton of the x and z components of the Darcy flux are dentcal. Furthermore, the entres n these flux matrces depend only upon the area of the elements, so sgnfcant computatonal savngs can be made by assemblng the flux matrx once only durng the smulaton. The steps used n the numercal soluton algorthm are summarzed n Fg. 1. To solve a well posed problem, at the begnnng of each tme step the concentratons are assumed nown and therefore the spatal dstrbuton b s nown. The dstrbuton of b s used to solve the groundwater flow equaton to yeld the freshwater pressure head dstrbuton. The flud pressure and hydraulc conductvty are used to evaluate the flud velocty, whch then allows the evaluaton of the dsperson coeffcents. The flud veloctes and dsperson coeffcents are ntroduced nto the transport equaton, whch s lnearzed usng the value of b from the prevous couplng teraton. The concentratons at the new tme step are obtaned and used to update b, whch s substtuted bac nto the flow equaton. Ths soluton procedure s repeated teratvely untl the maxmum change n the flud pressure (or solute concentraton) falls below some specfed tolerance. The lnearzed systems of equatons generated by the procedure are solved usng a banded LU factorzaton algorthm.

4 0 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) Fg. 1. Conceptual flow dagram showng the computatonal steps used to model densty-dependent flow n varably saturated porous meda. 3. Testng coupled groundwater flow and solute transport Due to the complexty of the soluton algorthm t s mportant to use benchmar problems to test whether the soluton scheme s an accurate approxmaton of the partal dfferental equatons. The relatonshps and couplng schemes are complex, therefore the only way to chec the consstency s to use a prevously benchmared soluton and compare t wth the newly generated soluton. To test the performance of the algorthm proposed n ths study, two benchmar problems are solved and the results are compared aganst publshed solutons Henry s salt-water ntruson problem In the lterature, varable-densty models are always benchmared by solvng the well-nown HenryÕs saltwater ntruson problem [1]. HenryÕs problem s unque because an analytcal soluton exsts; however, HenryÕs analytcal soluton s rather controversal. After almost 0 years of research, there has been a lot of dscusson about the reproducblty and qualty of HenryÕs orgnal analytcal soluton [7,30]. Further complcatons have been ntroduced by the presentaton of ncorrect results; Voss and Souza [33] clamed that several analysts used an ncorrect value for the total dsperson coeffcent and that ths trend was started by the wor of Pnder and Cooper []. Several authors have also modfed the orgnal boundary condtons used by Henry to mae the problem more realstc, ths trend was started by Segol et al. []. The modfed boundary condton nvolves mang the orgnal seaward boundary a combnaton of a partly freshwater and partly seawater boundary. Whle ths may mae the problem more realstc, the use of nonhomogeneous boundary condtons precludes the use of HenryÕs analytcal soluton. Croucher and OÕSullvan [7] speculated that the orgnal evaluaton of HenryÕs analytcal soluton mght have also been n error. Segol [30] re-evaluated the analytcal soluton and showed that the resultng governng equatons dffered n three places from those reported by Henry. Segol [30] then evaluated the revsed analytcal soluton and demonstrated that t was n close agreement to a numercal soluton of HenryÕs problem [30]. Whle ths updated evaluaton of HenryÕs analytcal soluton does correspond well wth numercal solutons [30], t s the more realstc verson of the Henry problem wth the mxed seawater and freshwater boundary, as descrbed by Frnd [10] that s most frequently used as a benchmar problem. For example, several publshed models have used the solutons from Frnd [10] to valdate ther densty-dependent codes, e.g., [,1,1]. Therefore, n ths study, we used the modfed Henry problem of Frnd [10] as the standard benchmar problem. The modfed HenryÕs problem conssts of a confned aqufer, whch s subected to a horzontal fresh-

5 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) Fg.. Boundary condtons appled for solvng the Henry salt-water ntruson problem. Table 1 Aqufer propertes used for the Henry problem [10] Symbol Quantty Value Unt D m Coeffcent of molecular dffuson m s 1 g Magntude of gravtatonal.0 m s acceleraton K Hydraulc conductvty ms 1 q Recharge rate ms 1 S s Specfc storage 0.0 m 1 b max Maxmum densty rato 1.05 u Porosty 0.35 q o Reference densty 1000 g m 3 q max Brne densty 105 g m 3 water recharge. The boundary condtons mposed on the aqufer are summarzed n Fg.. The freshwater exts towards a sea boundary that s composed of a hydrostatc dstrbuton of freshwater overlyng heaver seawater. The model parameters used n the problem are shown n Table 1. The smulaton was performed upon a regularly dscretzed aqufer usng 31 (1 horzontally, 11 vertcally) nodes and 00 rght-angled trangular elements. The ntal condton for the problem was a quescent aqufer composed of fresh water. The tme step for the smulaton was gradually ncreased by a factor of 1., startng at 1 s; the maxmum tme step allowed was 00 s. Temporal weghtng of the transport equaton was chosen such that a fully mplct soluton was obtaned. The teratve couplng scheme between the flow and transport equaton was such that a mnmum of two teratons was always performed, and then subsequent teratons were employed untl the maxmum change n the pressure head over any gven tme step was less than m. Ths value of the couplng tolerance was selected because t was comparable to the couplng tolerance reported for a prevously benchmared code [33]. The model was run for 0 mn, after whch the densty feld dd not change apprecably; ths s consstent wth other analysts who have obtaned the steady soluton wthn a smlar tme perod [10]. Once the densty feld was statonary, the sochlor postons were obtaned. Fg. 3 shows the 5%, 50% and 75% sochlors along wth data ponts obtaned from Frnd [10]. The comparson of the 50% sochlor wth that prevously publshed shows that the present model s capable of reproducng the result observed by Frnd [10]. In general, the Henry salt-water ntruson problem s usually numercally reproduced usng a transent model Fg. 3. Steady-state sochlor dstrbuton for the Henry problem.

6 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) that s allowed to approach a steady profle over a long smulaton perod [,10,1,1,,33]. Whle the focus of the Henry salt-water ntruson problem s upon the generaton of the steady-state profle, t s also possble to use the transent nformaton to benchmar the model performance. However, care must be taen when usng transent nformaton snce the results are senstve to the ntal condton used for the smulaton. Normally, the ntal condton for HenryÕs salt-water ntruson problem s a quescent freshwater aqufer, although some analysts have started the smulaton usng the salt dstrbuton predcted by the Ghyben-Herzberg hydrostatc assumpton [10,]. For the freshwater aqufer ntal condton, t s possble to locate the poston of the base of the 50% sochlor as t moves along the lower boundary of the aqufer. Frnd [10] also solved the problem for the freshwater aqufer ntal condton. The model data for the movement of the 50% sochlor provded by Frnd [10] s compared aganst the present model results for the 5%, 50% and 75% sochlors n Fg.. Snce the transent data are readly avalable from a densty-dependent flow smulaton, t s envsaged that Fg.. Transent poston of the toe of the salne water for the Henry problem. ths temporal data analyss should become a common tas n the benchmarng process. 3.. The Elder salt convecton problem The orgnal problem studed by Elder [] concerned lamnar flud flow n a closed rectangular box modeled n cross-secton. The flow wthn the box was ntated by a vertcal temperature gradent, and thermally nduced densty gradents caused a complex pattern of fngerng of the denser water to mx through the box. Ths problem was studed both physcally n the laboratory wth the use of a Hele-Shaw cell as well as beng numercally reproduced. Snce the laboratory condtons restrcted the flud flow to lamnar condtons, Elder [] also developed a modfed problem wth parameters suted to porous meda flow where the denstydependence was caused by solute varatons. Ths modfed Elder problem s another standard benchmar problem used for testng several densty-dependent flow codes [3,15,1,3], ths problem shall be referred to heren as the Elder salt-convecton problem. The maxmum flud densty for the Elder salt convecton problem s 100 g/m 3, hence the problem s often classfed as a brne transport problem. Some of the dffcultes n descrbng and predctng brne flows have been dscussed by Hassanzadeh and Lense [13] and Watson and Barry [3]. The doman and boundary condtons for the Elder salt convecton problem descrbe a closed aqufer, a zero pressure head s mantaned on the two upper corners of the doman (Fg. 5). The aqufer propertes and solute transport characterstcs are summarzed n Table. In our study the doman s regularly dscretzed usng 3131 nodes (101 horzontally, 31 vertcally) and 000 rghtangled lnear trangular elements. The temporal dscretzaton was mantaned at a constant nterval of one month, and the teratve couplng between the groundwater flow and solute transport equatons was consdered complete when the maxmum change n the freshwater pressure head between teratons was smaller than m [33]. A fully mplct temporal weghtng Fg. 5. Doman and boundary condtons for the Elder problem.

7 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) Table Aqufer and transport propertes used for the Elder problem [33] Symbol Quantty Value Unt D m Coeffcent of molecular m s 1 dffuson g Magntude of gravtatonal.1 m s acceleraton Intrnsc permeablty m b max Maxmum densty rato 1. S s Specfc storage 0.0 m 1 u Porosty 0.1 l Dynamc vscosty g m 1 s 1 q o Reference densty 1000 g m 3 q max Brne densty 100 g m 3 scheme was used to approxmate the transport equaton. The dstrbuton of the coupled flow and transport characterstcs was determned after, and 10 years of smulaton. The solute concentraton profles produced by the model show the generaton of a seres of transent vortces, whch spread the solute throughout the aqufer through the acton of both advecton and molecular dffuson (Fg. ). As expected, the dstrbuton of the lobes of dense flud s symmetrc about the centerlne of the aqufer. Smlar to the Henry problem, there are several publshed profles avalable for the Elder salt convecton problem. For example, prevously publshed profles by Voss and Souza [33] and Elder [] are compared to the present profles n Fg. 7. Comparng these profles, one could draw two conclusons; frstly, the current model s capable of reproducng the solute concentraton patterns reasonably well. Secondly, the comparson shows the Fg. 7. Comparson of the 0% and 0% sochlors for the Elder problem after 10 years, by Elder [] ( ), Voss and Souza [33] () and the present model (- -). dffculty n quanttatvely benchmarng the Elder salt convecton problem. Unfortunately, benchmarng the Elder results can be a tedous tas as the solutons avalable n the lterature can dffer dramatcally dependng upon the numercal dscretzaton and level of modelng sophstcaton used to generate the results. For example, Koldtz et al. [1] showed that the Elder salt convecton profles depend upon the level of modelng sophstcaton chosen for the numercal representaton of the densty-dependent flow and transport processes. Furthermore, ther wor ncorporated a mesh-refnement procedure to show that the results can be qute senstve to the level of dscretzaton used n the numercal representaton of the problem [1]. More recently, a locally adaptve grd technque was used n conuncton wth a perturbaton study to nvestgate the non-unque nature of the statonary solutons to the Elder problem [11]. Another pont of nterest wth the Elder salt convecton problem was shown by Boufadel et al. [3], where the problem was smulated usng lnear trangular fnte elements. The results of Boufadel et al. [3] showed that the problem s senstve to mesh-nduced ansotropy when dscretzed wth regularly algned lnear trangular elements and that the mesh had to be refned to overcome these problems. Therefore, gven the range of solutons presented n the lterature, at best the analyst can only show a qualtatve comparson to demonstrate that the profles generated by the new algorthm capture the essental features of the problem. Ths s why only a qualtatve comparson between the profles n Fg. 7 was possble for the benchmarng of the current numercal algorthm. In spte of ths dffculty, the results from the present model captures the essental features of the flud flow and the solute profles are smlar to those reported n the lterature [3,1,3,33].. Evaluaton of the worthness of the modfed Henry and Elder salt-convecton problems Fg.. Evoluton of the flow pattern of the dense flud nto the aqufer for the Elder problem (0% sochlor sold, 0% sochlor dotted), showng the pattern after, and 10 years. In order to assess the worthness of both the Henry and Elder salt convecton problems for the evaluaton of the consstency of a densty-dependent algorthm, the

8 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) problems were agan solved after gnorng the couplng between the flow and transport equatons. The basc obectve of ths tas s to obtan a qualtatve assessment of the degree to whch the couplng s necessary to ensure that the problem of nterest s solved correctly. If the coupled and uncoupled solutons are smlar, then non-densty-dependent processes must cause the nteractons and therefore the problem s less worthy of beng a benchmar problem. Alternatvely, f there s a dstnct dfference between the coupled and uncoupled profles then t s clear that the correct soluton wll only be obtaned under the condtons where the numercal model accurately determnes the balance of denstydrven pressure and gravty forces..1. Uncoupled Henry s salt-water ntruson problem To nvestgate the soluton of HenryÕs problem wthout couplng, b was fxed as unty for the soluton of the flud flow, flud velocty and solute transport formulatons. Other condtons n the numercal formulaton were left the same. The smulaton was performed for the same length of tme, after whch the uncoupled concentraton feld was statonary. The poston of the 50% sochlor after ths smulaton was measured and compared wth the coupled soluton n Fg.. Comparng the poston of the 50% sochlor for the coupled and uncoupled solutons shows that the profles are qute smlar. The reason for ths smlarty could be explaned by comparng the coupled and uncoupled steady-state velocty profles for the problem. The steady-state velocty felds for the Henry saltwater ntruson problem for the coupled and uncoupled condtons are shown n Fg.. A comparson of the coupled and uncoupled velocty felds reveals some nterestng results. The coupled velocty feld shows that the horzontal veloctes assocated wth the heaver salne water ntrudes further nto the aqufer than for the uncoupled soluton. The regon of nteracton of the heavy salne flud wth the fresh recharged flud s much broader under coupled condtons. For the coupled soluton, the change n velocty n the lower area of the aqufer, where Fg.. Comparson of coupled and uncoupled solutons to the Henry problem. Fg.. Steady-state velocty feld for the Henry problem under (a) coupled condtons and (b) uncoupled condtons. the fresh and salne fluds nteract, s qute sharp where as for the uncoupled soluton the area of nteracton s more gradual and smooth. Ths explans why the coupled 50% sochlor profle s not as smooth as the uncoupled 50% sochlor n Fg.. In general, the coupled and uncoupled flow patterns are largely the same. Ths smlarty means that the densty-dependent effects of the flow are not the ey component n ths test problem. In fact, the flow pattern n ths problem s mostly determned by the mposed boundary condtons, whch force the resultng dstrbuton of pressures and concentratons. It s also of nterest to see whether ths smlarty between the coupled and uncoupled soluton to HenryÕs saltwater ntruson problem s persstent for some smple varatons of the basc problem. In order to nvestgate ths, the orgnal Henry problem was solved by ncreasng the value of the fresh water recharge rate to two tmes the orgnal value and then agan to four tmes the orgnal recharge rate. The algorthm was run under these modfed flow condtons n both coupled and uncoupled modes. In each case the algorthm was executed for 0 mn, after ths tme the densty feld became steady (Fg. 10). As the recharge nto the aqufer ncreases, the equlbrum poston of the ntruded water s pushed bac towards the sea boundary. Interestngly, the smlarty between the coupled and uncoupled profles ncreased as the recharge s ncreased. Ths serves to show that the smlarty n the coupled and uncoupled solutons observed for the orgnal case was not an solated example. The Henry problem does suffer from the problem that the true profle s largely determned by the boundary forcng and much less by the densty-dependent effects.

9 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) Fg. 10. Comparson of the coupled and uncoupled 50% sochlor profles for varyng recharge n the Henry problem. Fg. 11. Comparson of the transent poston of the 50% sochlor movng along the base of the aqufer for the Henry problem. aqufer, the only way that the solute can enter the aqufer s through pure molecular dffuson of the solute nto the porous medum. The smulaton of the uncoupled profles shows a smple dffuson pattern nto the aqufer (Fg. 1). Ths s a rather obvous result; the smple dffuson pattern s expected because the flud n the aqufer s statonary wth a fxed concentraton boundary along the upper surface. Snce there s no feedbac between the flud concentraton and the velocty, there s no advecton under the uncoupled condtons. A comparson of the results shown n Fgs. and 1 clearly demonstrates that the correct soluton to the Elder problem s completely dependent upon the accuracy of the couplng scheme. The symmetrc lobes generated n the coupled soluton are caused by the dffuson of the solute from the top boundary nto the underlyng fresh aqufer water. Ths then enters a feedbac loop whereby the heaver salne water moves down faster than the surroundng waters. Ths causes advecton of the solute nto the aqufer and sets up the complcated convectve fngerng pattern. Once ths occurs, then the spatal and temporal changes n b affect the groundwater flow equaton and the entre system becomes strongly coupled. Analyzng the velocty feld for the Elder salt-convecton problem can show the mportance of ths couplng. Fg. 13 shows the coupled velocty feld for the left-hand sde of the aqufer after four years of smulaton. Whle prevous nvestgatons have reported the velocty feld for the Elder problem [3,33], re-teratng ths dagram serves to show the sgnfcance of the problem as a benchmarng tool. It s also possble to obtan transent nformaton regardng the soluton to HenryÕs problem n the uncoupled mode. Fg. 11 shows the transent movement of the poston of the uncoupled 50% sochlor for the Henry problem as t progresses along the base of the aqufer towards the steady poston. The results from ths observaton show that there s a mared dfference between the coupled and uncoupled profles. Frstly, as observed n Fg., the uncoupled sochlor does not advance as far nto the aqufer as does the coupled sochlor; secondly, the uncoupled profle becomes steady well before the coupled soluton. Therefore, t seems that the best methodology for the observaton of the dfferences between the uncoupled and coupled solutons s n the analyss of ths transent data... Uncoupled Elder salt convecton problem The analyss of the uncoupled Elder salt convecton problem s qute straghtforward. Snce the boundary condtons for the flow equaton descrbes a closed Fg. 1. Evoluton of the solute profle nto the aqufer for the uncoupled Elder salt convecton problem (0% sochlor sold, 0% sochlor dashed) showng the pattern after, and 10 years.

10 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) Dscusson Fg. 13. The velocty feld for the Elder problem after four years 0% sochlor ( ); 0% sochlor (- - -). The dstrbuton of the solute s dependent upon the velocty feld; but the velocty feld s also dependent upon the solute concentraton and wthout the couplng between the flow and solute processes the flud velocty wthn the aqufer would be zero. Snce the uncoupled Elder salt convecton problem s a smple dffuson problem, we can use an analytcal soluton to chec the predcted locaton of the advancng concentraton front. Assumng one-dmensonal condtons along the centerlne of the front, the apparent length-scale of the advancng front can be computed usng the followng analytcal soluton [] C z ¼ erfc p C 0 ffffffffff 7Þ Dt where C=C 0 s the relatve concentraton of the solute, z [L] s the length-scale traveled by the solute, and erfc s the complmentary error functon. Table 3 compared the length-scales of the 0% and 0% relatve concentratons, after, and 10 years predcted usng the analytcal and numercal solutons. The numercal model length-scales were estmated from the results presented n Fg. 1. The Table shows a good correspondence between the analytcal and numercal results. A comparson of the coupled and uncoupled solutons to the Elder problem shows that the problem s ndeed worthy of beng used as a benchmar. Ths s because the problem can only be solved correctly when the numercal code encompasses all of the ntrcaces of the correct representatons of the equatons, ther couplng, and the assocated consttutve relatonshps. Table 3 Comparson of length-scales of the dffuson front predcted by the uncoupled Elder soluton and the analytcal soluton Tme (years) Isochlor Length numercal (m) 0% % 7 0% % % % 5 0 Length analytcal (m) The results of ths worthness analyss have several mplcatons for studes that attempt to benchmar a densty-dependent groundwater model. Tradtonally, the focus of benchmarng studes has been centered upon the Henry problem. In the past, several denstydependent groundwater models have been used to smulate feld-scale problems after only beng tested aganst the Henry problem. Segol and Pnder [] smulated a real feld-scale saltwater ntruson problem usng the models developed by Pnder and Cooper [] and Segol et al. [] after benchmarng aganst the Henry problem only. Smlarly, the wor of both Frnd [10] and Galeat et al. [1] developed densty-dependent models that were tested aganst the HenryÕs problem alone and then mmedately used to solve feld-scale problems. Snce the actual soluton to HenryÕs problem s rather obscure due to the complcatons wth the analytcal soluton as well as the publcaton of ncorrect solutons, t s clear that care must be taen when comparng the new model results to those prevously establshed. Gven that the coupled soluton n terms of the sochlor dstrbuton loos very smlar to the uncoupled soluton, t s concevable that an algorthm whch does not correctly solve and couple the governng equatons could generate a soluton that mght appear to be smlar to the true soluton. These problems wth the Henry salt-water ntruson problem have been further compounded by the way the results for the problem are commonly presented n the densty-dependent flow modelng lterature. Typcally, n the soluton of the Henry problem, the analyst presents the sochlor dstrbuton alone [,1,1]. Whle t s a necessary condton for the algorthm to be correct that the sochlors must be comparable to those from prevous wor, t may not be suffcent to clam that that the problem has been accurately solved smply because the sochlor postons are comparable. Several ponts that can ad the benchmarng of a densty-dependent model usng the Henry problem have been rased n ths dscusson. Frstly, the velocty feld should be used as a qualtatve chec of the correctness of the nternal mxng envronment wthn the aqufer before the soluton s deemed adequate. Secondly, f the densty-dependent model s used n a transent fashon to obtan the sochlor profles, then t s mportant to locate the transent poston of the ntruded toe of the salne water and use ths as an addtonal pont of reference to benchmar the code. Fnally, t s worthwhle to test the senstvty of the model to the coupled and uncoupled condtons as demonstrated n ths wor. There are some obvous fundamental dfferences between the Elder and the Henry problems. The most strng dfference s that the Henry problem nvolves a stable densty stratfcaton where the denser flud un-

11 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) derles the lghter flud, whle the Elder problem nvolves an unstable densty stratfcaton where the heaver flud overles the lghter fresh water. In addton, a comparson of the Raylegh number for the Henry and Elder problems further explans n part, the fundamental dfferences between the problems. The Raylegh number s the rato between the buoyancy-drven fluxes and those caused by dsperson processes. The Raylegh number determnes the onset of nstablty and unstable fngerng s expected when the Raylegh number exceeds some crtcal value [31]. Evaluatng the Raylegh number usng a standard expresson gven by Smmons et al. [31] shows that the Elder problem has a Raylegh number whch s two orders of magntude hgher than that of the Henry problem. Whle the consequences of ths dfference s ntutve, t s not possble to assess whether ths fundamental dfference alone shall dctate the worthness of the problem for benchmarng purposes. The magntude of the densty dfferences observed n the Henry and Elder problems s another maor dfference between these two problems. The Henry salt-water ntruson problem deals only wth seawater concentratons; therefore the maxmum value of b n the flow and transport equatons s Ths contrasts wth the Elder salt-convecton problem where the maxmum value of b s 1.. The mmedate effect of ths dfference s that the couplng between the equatons s much stronger than for HenryÕs problem. Ths was observed durng the numercal smulatons for ths study as the soluton of the Henry problem requred only two couplng teratons between the groundwater flow and transport equatons, whle the Elder salt convecton problem requred about four teratons. The addtonal teratons were requred because, when solvng the transport equaton wth the values of b from the prevous teraton, there s lely to be some dscrepancy between the concentraton values predcted and the values of b used n the transport equaton. Ths dscrepancy was always more pronounced for the Elder problem, and explans why more couplng teratons were requred than for the Henry problem. In fact, hstorcally the Henry problem has been successfully solved usng a sngle step updatng scheme where there s no teraton between the groundwater flow and transport equatons, nstead the values are smply updated [,10]. Because the larger value of e (Eq. ()) for the Elder salt-convecton problem, a smple updatng scheme s nadequate to provde suffcent couplng between the groundwater flow and solute transport equatons. Therefore, the Elder salt convecton problem has a greater worthness as a benchmar problem as t s a thorough test for evaluatng the ntegrty of the teratve couplng scheme used for the smulaton of denstydependent groundwater flow. Some lmtatons wth the use of the Elder problem for benchmarng have also been dentfed. Due to the senstvty of the Elder problem to numercal dscretzaton and numercal representaton there are several profles n the lterature, none of whch are completely unform. Therefore, at best the analyst can only use these profles as a qualtatve measure for the benchmarng process.. Summary and conclusons The worthness of both the Henry salt-water ntruson problem and the Elder salt convecton problem were studed to assess ther sutablty to be used as benchmar problems for a densty-dependent groundwater modelng code. Detals of a Galern fnte element numercal algorthm for solvng densty-dependent flow n varably saturated porous medum are provded. The mathematcal model was tested and benchmared by solvng both the Henry and Elder salt convecton problems and comparng the solutons aganst other publshed profles. Ths testng procedure was able to show that the present model could reproduce establshed solutons of denstydependent groundwater flow problems. The tested model was then used to solve the problems n an uncoupled mode. A comparson of the coupled and uncoupled solutons provded a novel way to separate the denstydependent effects from other boundary effects that may mpact the soluton. The uncoupled soluton of the Henry saltwater ntruson problem demonstrates that the patterns observed wthn the soluton are nsenstve to the densty couplng of the equatons. The soluton from the uncoupled smulaton loos smlar to the correct soluton. Therefore, the dstrbuton of the salne water n HenryÕs problem s almost completely determned by the boundary forcng mposed on the problem, and not necessarly the result of densty coupled flow and transport processes. Alternatvely, the Elder salt convecton problem has the advantage that the correct flow patterns are only acheved when the governng equatons are correctly coupled. Therefore, the worthness of the Elder problem as a benchmar standard s greater than that of the Henry problem. In summary, both the Henry and Elder problems have ther own advantages and dsadvantages. The dsadvantage of the Henry problem s that t s almost nsenstve to densty couplng effects, as llustrated n ths study; however, the advantage s that the problem can be exactly reproduced to match prevous results. On the other hand, the Elder problem s hghly senstve to densty couplng effects, but the dsadvantage of ths problem s the results can be matched only n a qualtatve sense because the problem s hghly senstve to dscretzaton and other numercal convergence effects. The results from ths study ndcate that a comparson of the coupled and uncoupled solutons to any standard densty-dependant flow problem should always be

12 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) undertaen n order to assess the worthness of the partcular problem. Once the worthness s establshed, then the benchmarng process can be focused upon those cases that show a large dfference between the coupled and uncoupled solutons, such as the Elder problem, thereby ensurng the rgor of the benchmarng process. Acnowledgements We would le to acnowledge the comments from the Advances n Water Resources revewers, n partcular, Dr Crag Smmons. We would also le to than Dr Davd Reynolds for hs thorough revew of the materal. Ths wor was made possble by the support provded by the Australan Postgraduate Award system through the Centre for Water Research and the Unversty of Western Australa. Appendx A. Fnte element matrces A.1. Flow equaton The soluton to Eq. (1) s found by applcaton of the Galern fnte-element-method. Because the method has been dscussed at length elsewhere only the most pertnent steps n the soluton procedure are descrbed to enable the reproducton of the algorthm. The dependent varable, w s approxmated by a fnte seres of the form, wx; z; tþ Xn ¼1 N x; zþw b x; z; tþ A1Þ The shape functons N ; correspond to those assocated wth a lnear trangular fnte element nterpolaton scheme, bw x; z; tþ ¼ w b tþn x; zþþw b tþn x; zþ þ b w tþn x; zþ where, N r ¼ 1 ½ A a þ b x þ c zš N s ¼ 1 A a þ b x þ c z N t ¼ 1 ½ A a þ b x þ c zš a ¼ x z x z b ¼ z z c ¼ x x a ¼ x z x z b ¼ z z c ¼ x x a ¼ x z x z b ¼ z z c ¼ x x AÞ where b w [L] s the nterpolated freshwater pressure head, w, w and w [L] are the dscrete values of the pressure head at the element vertces, N, N and N are the lnear shape functons whch depend upon the Cartesan coordnates x; zþ of the element vertces, and A [L ] s the area of the element. The bass of the Galern approach s that the tral soluton shall form a resdual when bac substtuted nto the governng equaton, the resdual expresson for the flow equaton ntegrated over a partcular element taes the form, >< > R R R >; ¼ Z X ½NŠ T " o bkhþ ow ox ox þ o oz þ o b KhÞ obhþ S s b h oz ot g bkhþ ow oz # ow dx A3Þ ot Ths ntegral s then splt nto fve separate ntegrals correspondng each to the dfferent terms n the equaton. The shape functons correspond to those used for lnear trangular elements, and then the ntegraton s performed to yeld the Galern matrces, < R R R 3 þ b K m þ b K m b b b b b b b b b b 7 5 1A b b b b b w mþ1;nþ1 >< b w mþ1;nþ1 > >; Km þ b K m þ b K m 1A w mþ1;nþ1 c c c c c >< w mþ1;nþ1 c c c c c 7 5 w mþ1;nþ1 > >; c c c c c w mþ1;nþ1 3 þ 1 c b Km þ c b Km þ c b Km c b Km þ c b Km þ c b Km 7 5 c b Km þ c b Km þ c b Km S b h m 0 0 >< w mþ1;nþ1 s 0 b 3/Dt h m 0 5 w mþ1;nþ1 0 0 b h m > >; w mþ1;nþ1 þ S b h m 0 0 >< w n s 0 b 3/Dt h m 0 5 w n 0 0 b h m > w n >; A b 0 0 >< h m;nþ1 h n 0 b 3Dt 0 5 h m;nþ1 h n 0 0 b > >; h m;nþ1 h n A b Cap m 0 0 >< w mþ1;nþ1 0 b 3Dt Cap m 0 5 w mþ1;nþ1 0 0 b Cap m > >; w mþ1;nþ1 þ A b Cap m 0 0 >< w m;nþ1 0 b 3Dt Cap m 0 5 w m;nþ1 0 0 b Cap m > >; w m;nþ1 A h m 0 0 >< b nþ1 b n 0 h m 0 5 b nþ1 b n 3Dt 0 0 h m AÞ > >; b nþ1 b n = b ; ¼ Km where the superscrpt n þ 1 refers to the value at the current tme step whle the superscrpt m þ 1 refers to the value of at the current Pcard teraton step, and Cap

13 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) [L 1 ] s the slope of the retenton functon, all other terms have been prevously defned. The matrces n expresson (A.) whch pre-multply the column vectors of the unnown pressure heads contan the terms that form the global element stffness matrx entres. The remanng vectors form the global element force vector. These expressons are evaluated ndvdually and are assembled va the drect stffness assembly procedure to form the global stffness and global force vectors at each teraton wthn the algorthm. The applcaton of Drchlet and Neumann boundary condtons n the algorthm s straghtforward, the methodology for mplementng boundary condtons has been descrbed elsewhere [7]. A.. Velocty evaluaton The applcaton of DarcyÕs law s made usng the usual Galern approach to ensure that the flud velocty s contnuous over the entre doman, the resdual expresson for the vertcal flux s, < R R R = ; ¼ Z X ½NŠ T q z þ K ow Kb dx oz A5Þ Once (A.5) s ntegrated, the element matrces become, < R = R ; ¼ A < q z = 5 q z ; R q z þ c K þ K þ K w þ c w þ c w < = K þ K þ K ; K þ K þ K þ A >< K b þ K b þ K b K b þ K b þ K b AÞ > K b þ K b þ K b >; The formulaton for the horzontal flux s very smlar, wth the excepton that there s no gravty term, < R = Z R ; ¼ ½NŠ T q x þ K ow dx A7Þ ox R X The Galern resdual expresson for the horzontal flux s, >< R R > >; ¼ A >< q x 7 5 q x > >; þ b w þ b w þ b w R q x >< K þ K þ K K þ K þ K AÞ > >; K þ K þ K Once the global force vector and global stffness matrx are assembled and solved, the soluton yelds the nodal values of the flud flux. Inspecton of the formulatons (A.) and (A.) reveal that the global stffness matrces for the evaluaton of the fluxes are dentcal and depend only upon the area of the elements. Therefore, the global stffness matrx need only be assembled once at the begnnng of the algorthm and not at each teraton step as requred by both the groundwater flow and solute transport equaton. The nodal values of flud flux obtaned are converted nto the nodal value of flud velocty by dvdng the flux value by the nodal water content. A.3. Transport equaton The soluton to equaton () s found by applyng a temporal weghtng scheme to the usual Galern fnteelement soluton procedure. The general expresson for the resdual s, >< > R R R >; ¼ Z X½NŠ T b o D x h oc þ b o ox ox oz V x h oc ox V zh oc oc h oz ot D z h oc oz dx AÞ Integratng the terms n (A.) and ntroducng a tmeweghtng factor w, yelds the element matrces, < R R R 3 h nþ1 0 0 < C nþ1 0 h nþ1 = C nþ1 3Dt 0 0 h nþ1 C nþ1 ; þ A h n 0 0 < C n 0 h n = 0 5 C n 3Dt 0 0 h n C n ; bhd xþ nþ1 ave w b b b b b < C nþ1 b b b = b b 5 C nþ1 A b b b b b C nþ1 ; bhd xþ n b ave1 wþ b b b b < C n b b b = b b 5 C n A b b b b b C n ; bhd zþ nþ1 ave w c c c c c < C nþ1 c c c = c c 5 C nþ1 A c c c c c C nþ1 ; bhd zþ n c ave1 wþ c c c c < c c c c c 5 Cn = C n A c c c c c C n ; hv xþ nþ1 ave w b b b < C nþ1 = b b b 5 C nþ1 b b b C nþ1 ; hv xþ n b ave1 wþ b b < C n = b b b 5 C n ; b b b = ; ¼ A C n

14 30 M.J. Smpson, T.P. Clement / Advances n Water Resources (003) hv x hv x c c c >< C nþ1 7 c c c 5 C nþ1 > >; c c c C nþ1 c c c 1 wþ >< C n 7 c c c 5 C n > c c c C >; n A10Þ Þ nþ1 ave w Þ n ave where the superscrpt n þ 1 refers to the unnown value at the current tme step whle the superscrpt n refers to the nown value at the prevous tme step. Note that the subscrpt ave refers to that partcular quantty arthmetcally averaged across the element from the nodal values. The tme weghtng factor w, s lmted to values between zero and one. Usng a tme weghtng factor of one gves a fully mplct soluton whle usng a weghtng factor of a half gves a Cran Ncholson soluton. Usng a tme-weghtng factor of less than a half generally result n excessve oscllatory behavor of the soluton [7]. A.. Retenton functons The model element equatons presented n ths Appendx mae reference to the varably saturated nature of the governng equatons. To solve a varably saturated problem, the functonal relatonshp between the pressure head and the water content as well as the relatonshp between the hydraulc conductvty and the pressure head must be nown. The sol propertes can be descrbed usng van GenuctenÕs [3] sol-water-retenton functon and MualemÕs [] varably saturated hydraulc conductvty functon. A detaled dscusson about the parameters requred n varably saturated flow models and ther meanng can be found n Clement et al. [5] and Wse et al. [35]. A.5. Dscusson of the fully saturated flow soluton Both the Henry salt-water ntruson problem and the Elder salt-convecton problem solved n ths wor are fully saturated flow problems. Mang the varably saturated formulaton relax to smulate a fully saturated problem s qute smple because the hydraulc conductvty of the porous medum s no longer a functon of the pressure head, and the water content of the porous medum becomes a constant. Under these condtons, the capacty (Cap) terms n the element equatons relax to zero, the hydraulc conductvty terms become uncoupled from the pressure heads and the governng flow equaton becomes lnear. Therefore, solutons to fully saturated problems may be obtaned wthout the need for the nner modfed Pcard teraton. References [1] Bear J. Hydraulcs of groundwater. New Yor McGraw-Hll; 17. [] Boufadel MC, Sudan MT, Venosa AD. A numercal model for densty-and-vscosty-dependent flows n two-dmensonal varably saturated porous meda. J Contamn Hydrol 1; [3] Boufadel MC, Sudan MT, Venosa AD. Numercal modelng of water flow below dry salt laes effect of capllarty and vscosty. J Hydrol 1; [] Cela MA, Bouloutas ET, Zarba RL. A general mass-conservatve numercal soluton of the unsaturated flow equaton. Water Resour Res 10;13. [5] Clement TP, Wse WR, Molz FJ. A physcally based, twodmensonal, fnte-dfference algorthm for modelng varably saturated flow. J Hydrol 1; [] Cran J. Mathematcs of dffuson. 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