Mixed analytical and numerical modelling of an oceanic peninsula using the Dupuit-Ghyben-Herzberg approach

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1 Calibration ami Reliability in Groundwater Modelling: A Few Steps Closer to Reality (Proceedings of ModclCARLV Prague. Czech Republic. June 2002). 1A11S Publ. no Mixed analytical and numerical modelling of an oceanic peninsula using the Dupuit-Ghyben-Herzberg approach L. M. NUNES CVRM/IST-FCMA. Universidada do Algctrve, Campus de Gambe/as, P Faro, Portugal lnunes@ualg.pl A. CARVALHO DILL Department of hlydrogeology IGM, Estrada da Porte/a, Alfragide, P Amadora, Portugal L. RIBEIRO CVRM, Institute) Superior Téenieo, Av. Rovisco Pais I, P-I049-00I Lisboa, Portugal J. VIEIRA CVRM/IST-FCMA, Universidade do Algarve, Campus de Gambelas, P Faro, Portugal Abstract Analytical and numerical models were applied to an oceanic peninsula to estimate the volume of available freshwater and the amount of water that could be pumped without depleting the resource. Analytical models were used at regional scale, whereas numerical models were applied in a small domain to obtain aquifer parameters. The results showed that even small pumping rates of 10% of the annual net recharge would have important impacts on the thickness of the water lens, which can be even higher during dry years, when the annual recharge may be half of the multiple-year long term average value. Key words analytical modelling; freshwater-saltwater interface; island; numerical modelling INTRODUCTION In this paper an application of mixed analytical and numerical models of the oceanic Troia peninsula is presented. The objectives of the study were to estimate the volume of freshwater available, and the amount of water that could be pumped without depleting the resource. This was of major importance for the promoter of a tourist project to be developed in the peninsula, mainly to irrigate two golf courses and gardens. This study is part of the Environmental Impact Assessment report for the Troia resort (IMOAREIA, 2001). The peninsula is located 30 km south of Lisbon (Portugal), bordered westward by the Atlantic Ocean, and eastward by the Sado River estuary. It is an emergent portion, 2000 m wide and m long, of a Quaternary sandy formation of variable granulometry (from fine to coarse), with a thickness varying from 50 to 90 m. This

2 240 L. M. Nîmes et al. porous formation is limited below by Pliocene clayish formations with very low vertical hydraulic conductivity that constitute a natural flow barrier. Fieldwork included geophysical surveys (VLF-EM and RMT-R), tide tests, and tracer tests with marine bacteriophages (Imoareia, 2001). Tide tests were needed to evaluate the tidal effect on the surface of the freshwater lens. The tidal effect is defined as the variation in the head due to tidal variations. Analytical models were applied at the peninsula scale, whereas inverse numerical modelling in a small domain was used to obtain estimates of the average hydrogeological parameters of the porous formation that were scarce or not reliable. The paper is organized as follows: in the next section analytical approximations to the thickness of the freshwater lens are revised and the finite-difference solution to the small-scale domain is presented; in the third section, the main results are shown and discussed; the most relevant conclusions are drawn in the fourth section. COUPLED ANALYTICAL AND NUMERICAL MODELLING Complementary field-work with a limited number of vertical electrical conductivity profiles and geophysics showed that the freshwater lens is limited below by saltwater. The first approximation to the thickness of the lens indicated that it should be around 30 m in the axes of the peninsula; therefore well above the bottom of the Quaternary porous formation. The modelling domain was then considered as an island of infinite length and 2000 m wide. The analytical models rely on the approximation of a static and narrow saltwaterfreshwater interface with respect to the thickness of the aquifer. This is not a limitation since the aquifer has low rates of abstraction, and therefore small non-natural temporal piezometric oscillations. Analytical modelling The Ghyben-Fferzberg relation states that in a unconfmed aquifer: z(x) = P/ h(x) = och(x);x = (x,y) (1) for a particular location X, the depth z(x) (m) of saltwater-freshwater interface below mean sea level is proportional to the height h(x) (m) of the freshwater lens above sea level. Variables p s and p/(kg m" J ) are the saltwater and freshwater density. The estimate of z(x) and h(x) in any location X, is achieved with the application of the former equation and the one proposed by Fetter (1972) for oceanic islands with infinite length: K(l + l/a) (2)

3 Mixed analytical and numerical modelling of an oceanic peninsula 241 where / (m day" 1 ) is the recharge rate, a (m) corresponds to half the island's width, x (m) is the perpendicular distance of X to the coast and K (m day" 1 ) is the hydraulic conductivity. Equations (1) and (2) are only reliable in situations where horizontal flux can be assumed, i.e. where the equipotential lines are vertical (for Dupuit assumption see e.g. Bear, 1979). Examples of application of the Dupuit-Ghyben-Pterzberg approach in oceanic island case studies can be found in Fetter (1972), Vacher (1988) and Vacher & Wallis (1992). This approximation is used for distances perpendicular to the coast, x, where the tidal effect in the piezometric head of the aquifer is imperceptible. The tidal effect extends inland in the peninsula to a distance calculated with the expression of Jacob (1950) for confined aquifers: (3) where H x (m) is the variation in the piezometric head observed at the perpendicular distance x off the coast due to the tide, Ho (m) is the tidal range, to (day) is the tidal period (12:25/24), T (m 2 day" 1 ) and S (dimensionless) are hydrogeological parameters characterizing the aquifer, namely transmissivity and storavity. This relation is valid because H x is small when compared to the saturated thickness of the aquifer, b (m). In this case, T is a constant (T = bk) and S is the effective porosity. For the regions where the Dupuit assumption was no longer valid, the expressions proposed by Glover (1964) are used: ak ^ K 2açA V K where q' (m 2 day" 1 ) is the freshwater outflow rate per unit length of coastline, calculated by: q'=fa (6) The decrease in the thickness of the freshwater lens may be estimated for different pumping rates. Obviously, the water amount pumped would otherwise have recharged the aquifer. The approximation of Van Dam (1999) allows the calculation of: (a) the volumes (m 3 ) extracted per unit of length (m) of coastline of the aquifer, indicated by AV(m 2 ), for each abstraction, and (b) the time T r (day) to recover from one situation of dynamic equilibrium (recharge rate f\ (m day" 1 )) to another (recharge rate fi (m day" 1 )):

4 242 L. M. Nîmes et a!. where (j) (dimensionless) is the porosity, and f\ and fi represent the two situations of recharge rate (without any abstraction and with abstractions, respectively). For the majority of the situations, T r has values in the order of several years. As a consequence, the calculation of the reserves and of the natural equilibrium can be made, assuming an annual mean recharge rate distributed over the most important precipitation months. Numerical modelling In the numerical approach, an area of 300 x 300 m was defined as the domain, corresponding to a region in contact with the estuary where the tracer and the tide tests were made, which in turn permitted the calibration and validation of the model. The modelling problem was solved with a finite-difference approach, using the well-known code MODFLOW (McDonald & Harbaugh, 1988, 1996). The distance between the estuarine border and the first three piezometers used in the tide and tracer tests, Hi, H 2 and H3 was just 30 m. Between these three and the other two inland piezometers, H 5 and HO, the distance was 100 m (Fig. 1). In this area the water levels are highly influenced by tide fluctuations, and therefore the modelling is more complex and prone to error. The landside and the estuarine boundaries were defined as general head boundaries (McDonald & Harbaugh, 1988): as a constant-level reservoir in the interior; and as a variable water-level reservoir at the saltwater-freshwater interface. The landside boundary was defined as a constant-level reservoir in agreement with the water-level data along the axes of the peninsula. The tidal effect was just perceptible in the 200 m closest to the coast. The piezometric head in the middle of the peninsula should not be higher than 1 m above sea level according to the geophysical surveys. A general head boundary was used to make the bridge between the landside border of the model and the axes of the peninsula, 700 m inland. The estuarine boundary should represent the contact between the freshwater of the aquifer and the saltwater of the estuary. The filtering of the tide (range and time gap) was made by calibrating the conductance. This approximation to the phenomena of oscillation of the interface was possible considering that the soil inside the domain is saturated with freshwater (0-10 J mg total dissolved solids (TDS) per litre, Freeze & Cherry, 1979). TDS were obtained indirectly using electrical conductivity (EC) measurements: TDS = EC (Grohman, 1987) or TDS= xec, with 0.55 < % < 0.75 (Hem, 1970), in p_s cm" 1. The EC measurements made inside the domain indicated that the approximation was valid (TDS always <10 J ). Mercer et al. (1980) and Reilly (2001) state that the problem of saltwater intrusion can be studied with constant density models for systems where the interface is abrupt, and that the mass balance errors are tolerable in terms of their impact on computed heads and interface elevations. Inverse numerical modelling, with PEST (Hill, 1992), then approximated the hydrogeological parameters of the porous formation until the simulated head series closely approximated the observed series collected during the tide tests.

5 Mixed analytical and numerical modelling of an oceanic peninsula 243 RESULTS AND DISCUSSION Numerical model The model calibrated well, with a fine adjustment of the simulated values against the observed values. The mean error of the estimation was 8 x 10 m, the standard deviation of the estimation was 1 x 10" 3 m and the square root of the mean square error was 3.6 x 10"" m. Five observation points were used (Hi, H 2, H3, H5 and H 6 ), and their location can be seen in Fig. 1. The hydrogeological parameters obtained by inverse numerical modelling, T and S, are presented in Table 1. Transmissivity, T, was then used in the analytical models. From the results of the numerical model there is clearly an alteration in the direction of flux between the flood and the ebb tide. During the ebb tide, the flux is in the direction of the estuary, as expected in a coastal area (Fig. 1(a)), and for tide heights greater than 0.4 m, the flux reverses near the coast, propagating inland until the flood tide peak (Fig. 1(b)). During this latter period, a depression zone is formed about 100 m off the coast, as a consequence of the higher piezometric heads inland and the ascent of freshwater pushed by seawater rise. These results are in agreement with the tracer test results, and help to explain why tracers were first detected in the further inland piezometers, H 5 and H 6 (Imoareia, 2001). In reality the tracers have a very complex migration path due to the rapid variations in the flow field. This aspect is not considered in this article. Analytical model The analytical models were applied to the peninsula scale and the necessary parameters are presented in Table 1. It was also assumed that the peninsula might be split into two distinct regions: region A, an outer region 200 m wide where the tidal (b) i t / / t Seawater 1:2 ;.0 2.A 3.0 x100 (meters) x100 (meters) Fig. 1 Simulated piezometric heads (metres above sea level) and flux direction (a) during flood tide situation; and (b) during ebb tide situation.

6 244 L. M. Nunes et al. Table 1 Model parameters used in the analytical models at regional scale to estimate the freshwater lens in different situations of the recharge rate. Parameter Value Recharge,/(m day" 1 ) 2.33 x 10" 4 Hydraulic conductivity, K (m day" 1 ) 10 Storavity, S (dimensionless) 0.25 Saturated thickness, b (m) 60 Transmissivity, T(m~ day" 1 ) 600 Maximum width (m) 2000 Tidal period, t 0 (day) 0.53 Tidal range, H a (m) 0.98 Porosity, a? (dimensionless) 0.30 Freshwater density, p r (kg m ) 1000 Saltwater density, p s (kg m" 3 ) 1025 effect is noticed; and region B, an inland region where the tidal effect is imperceptible. The outer region was determined with equation (3) and validated with the tide test results. In region B, the Dupuit-Ghyben-Herzberg approximation was considered valid, and h and z were estimated with equations (1) and (2). For region A, h and z were calculated with the alteration proposed by Glover equation (4) and (5). In a transversal section across the peninsula, the results for the depth of the interface, z, vary between 7 m in the region near the contact with the sea and more than 30 m far from the coast (Fig. 2). These results are in agreement with the geophysical surveys (Imoareia, 2001) Fig. 2 Thickness of the freshwater lens across a transversal cut in the Troia peninsula. Location z of freshwater-lens bottom for the actual recharge, and with 90%, 80%, 70% and 60% of the actual recharge, 0.9/ 0.8/ 0.7/and 0.6/ A is the region under the tidal effect; B is the region with no tidal effect.

7 Mixed analytical and numerical modelling of an oceanic peninsula 245 The decrease in the thickness of the freshwater lens was estimated for pumping rates of 10, 20, 30 and 40% of the recharge rate. The annual available volumes with these abstractions from the aquifer (per unit length perpendicular to the cross-section) were estimated as 750, 1540, 2380 and 3290 m 2, respectively (equation (7)), with a decrease of 5.1, 10.6, 16.3 and 22.5% in the thickness of the freshwater lens (Fig. 2), when compared with the location z of the freshwater-lens bottom for the situation without groundwater abstraction. The period for the recovery of the dynamic equilibrium was estimated (equation (8)) as 50 years. CONCLUSIONS In this paper a coupled analytical numerical approach to the modelling of an oceanic peninsula was presented. A numerical solution was looked for inside a small domain with 300 x 300 m in contact with estuarine water (saltwater). This model was developed in an area where tide and tracer tests had been made. It should allow determination of the hydrogeological parameters necessary to the analytical estimates of the freshwater lens thickness, the volume of water that could be pumped during one year with different pumping rates, the effect it would have on the resource, and the time necessary to attain equilibrium after pumping. The results showed that even small pumping rates of 10% of the annual net recharge would have important impacts in the thickness of the water lens. Due to the normal climatic variation, the average annual recharge may be significantly lower than the value used in the calculations in dry years the annual recharge may be half of the long-term multiple-year average value (examples are the years , ). If such dry periods extend for several years, the pumping of those volumes would result in a greater reduction in the thickness of the freshwater lens. The exploration of the freshwater lens should be carefully planned in order to guarantee that the roots of plants near the coast are never soaked in saltwater, and that the natural wetlands in the interior of the peninsula are preserved. Acknowledgements The authors would like to acknowledge the project developer IMOAREIA SA for the authorization to publish the results. REFERENCES Bear, J. (1979) Hydraulics of'groundwater: McGraw-Hill, New York, USA. Fetter, C. W. (1972) Position of the saline water interface between oceanic islands. Water Resour. Res. 8(5), Freeze, R. A. & Cherry, J. A. (1979) Groundwater. Prentice-Hall, Englewood Cliffs, New Jersey, USA. Glover, R. E. (1964) The pattern of fresh-water flow in a coastal aquifer. US Geol. Survey Paper Grohman, A. (1987) Die Trinkwasserverordnung (The drinking water regulation), Erich Schmidt, Berlin, Germany. Hem, J. D. (1970) Study and interpretation of the chemical characteristics of natural waters. US Geol. Survey Paper Hill, M. C. (1992) MODFLOW/P- A computer program for estimating parameters of a transient three-dimensional, groundwater flow model using non-linear regression. US Geol. Survey, Open-file Report IMOAREIA (2001) Environmental Impact Study of the Troia's Resort marina and new feme's quayage, Vol. II. IMOAREIA, SA, Lisbon, Portugal.

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