Tank Experiments, Numerical Investigations and Stochastic Approaches of Density-Dependent Flow and Transport in Heterogeneous Media

Transcription

1 Tank Experiments, Numerical Investigations and Stochastic Approaches of Density-Dependent Flow and Transport in Heterogeneous Media [1] Robert BRAATZ, [1] Manfred KOCH [1] Department of Geohydraulics and Engineering Hydrology, University of Kassel, Germany, Abstract At the hydraulic laboratory of the University of Kassel density-dependent flow experiments in a Plexiglas tank have been performed over recent years. The objective of this study is to analyze macrodispersive effects across a fresh-/saltwater interface within different heterogeneously packed sand structures of given stochastic properties. Numerous predetermined concentrations and velocities are analyzed in each sand pack for the hydrodynamically stable case once the interface zone has attuned to steady-state conditions over most of the tank. All experimental results are compared with numerical simulations. The results of both the numerical models and the experiments are complemented by Monte Carlo simulations, using predefined stochastic realizations of the permeability field. These simulations are representative for the stochastic packing of the tank. For calibration and validation of the density-dependent vertical macrodispersion in stochastically heterogeneous porous media additional investigations are intended. Keywords: Tank experiments, density-dependent transport, macrodispersion, stochastic heterogeneous media 1. Introduction Density-dependent flow and transport in porous media is a phenomenon that occurs in a multitude of scenarios in groundwater hydrogeology like, for example, seawater intrusion in coastal aquifers (Bear et al., 1999). In all respects is it important to understand the underlying physical processes, i.e. for the purpose of quantification of the migration of contaminants in the subsurface. As such a survey of the spread behaviour of groundwater contaminants may serve to prevent or to control contaminations. When groundwater is contaminated in a way similar to the here-studied hydrodynamically stable flow pattern (denser fluid below), depending on the density stratification, a solute plume may develop. Strong density effects with the parting of the plume were observed in this stable case for already moderate solute concentrations which indicates the need of taking density effects into account in the practical modeling of such flow and transport situations. For heterogenous media extra special computational efforts are required for full density-dependent transport simulations, when compared with those for the uncoupled tracer problem. In many practical situations this may be computationally prohibitive (Koch and Starke, 2003), particularly, for heterogeneous aquifer structures (Dagan and Zeitoun, 1998), thus the need for setting up engineering-kind scaling laws for the plume behaviour (Koch and Zhang, 1992). Finally, there is a need for a more accurate physical description and the development of computationally more effective modeling approaches of density-dependent flow and transport processes. One way to approach this complex problem is to find a way to relate the apparent dispersion coefficients DL and DT to the density contrast across the fluid interface between the solute plume and the freshwater and, additionally, to the stochastic properties of the porous medium. For a hydrodynamically stable configuration one expects a decrease of the dispersion with increasing density. In tank experiments a reduction of the width of the mixing zone, i.e. a decrease of the lateral dispersion (DT) with increasing density contrast is obtained only for a sand packing with a nonuniform grain distribution (Kobus and Spitz, 1985). This indicates that the density-dependent dispersion processes are more significant on a larger scale than a pore-sized one, i.e. scale-dependent macrodispersion prevails. This well-known scale-effect of macrodispersion has been fairly well investigated for density-independent (tracer) flow and transport (cf. Gelhar, 1993), but less so for density-dependent transport (Welty and Gelhar, 1991; Koch and Starke, 2003; Starke and Koch, 2006). To this avail, it is reasonable to take a full stochastic approach in variable density transport for the description of macrodispersion, using the methods of stochastic theory by Gelhar (1993). This is 1

2 the objective of this study, namely experimental investigation, numerical modeling and the use of stochastic approaches for the characterisation of longitudinal and, specifically vertical (transversal) macrodispersion in stochastic heterogeneous porous media. Ultimately, the aim is an experimental verification of some of the theoretical predictions for the effective macrodispersion that may occur in a stochastically packed heterogeneous sand structure in a laboratory tank, and the development of effective modelling strategies to that avail. In the present contribution, we describe the setup and some salient results of such tank experiments and numerical models for the dynamically stable case of saltwater injection of various concentrations and velocities underneath a layer of fresh water, statistic validations and finally the comparison with the stochastic theory. For a more detailed description of the study we refer to Starke (2005). 2. Experimental setup and numerical model At the hydraulic laboratory of the University of Kassel density-dependent flow experiments in a Plexiglas tank are being performed over the recent years. In these, basically, 2D x-z models, macrodispersive effects on the longitudinal and vertical dispersion across a fresh-/saltwater interface are analyzed in the major x-z flow plane. For this purpose, different experimental and numerical setups for each stochastic realization of the porous sand-pack structure with given stochastic properties are being investigated Experimental program Fig. 1 shows the tank with internal dimensions of 9.8 m in x-direction, 0.1 m in y-direction and m in z-direction. Two inlet and one outlet chambers are separated from the sand-package by a permeable fleece. To avoid premature mixing of fresh and saline water at the two inlet sections of the tank, a plastic plate protrudes the tank in the middle over a length of 0.25 m. Overflow chambers at the inlet and outlet of the tank adjusted in height to set up a constant head difference h across the tank that ensures steady-state flow conditions. The anisotropic sand structures are packed as 2401 blocks (each 0.2 m in x-direction by m in z- direction) with eight different pre-sieved classes of chemically pure industrial quartz-sands per stochastic realization. Depending on the latter, the hydraulic conductivities of the sand range from K = 10-2 to 10-5 m/s (i.e. a permeability of k = 10-9 to m 2 ). Experiments of three representative realizations of stochastic heterogeneity, with a logarithmic mean of Y = ln k = -13,25, predefined variances σ 2 ranging from 0.25 to 1.5 and correlation lengths in x-direction, λ X of 0.2 to 0.4 m and in z-direction, λ Z of 0.05 to 0.2 have been carried out so far. Each of the three stochastically packed heterogenous sand structures are representative of a natural aquifer system (Dagan 1989). Fig. 1. Schematic setup of tank experiment. 2

3 For each of the three sand packs numerous experiments to investigate the hydrodynamically stable case of a layer of saltwater of different concentrations underneath a layer of de-ionized water, have been carried out, whereby the concentrations of the saltwater (chemically pure NaCl-solutions) range from C 0 = 250 (tracer experiment) to ppm (brine) and the flow velocities from u = 1 to 8 m/d. Depending on the latter, steady-state conditions for the solute transport are reached after one to three weeks. By then, the interface zone has attuned to steady-state conditions in the observed section of the tank. To probe the concentrations, 150 control ports with needles that intrude 0.05 m into the tank are vertically distributed along nine observation columns at one side of the tank. This allows an almost complete in-situ probing by taking a small amount of water per sample. The electric conductivity of the samples is then measured and, using previously established calibration curves between concentration and conductivity, the corresponding solute concentration is estimated Numerical models The experimental setups and have been implemented in numerical density-dependent flow and transport models. The model grid was set up to simulate the x-z cross-sectional geometry and the experimental stochastic realization of the permeability field of the tank. Exemplarily, the latter is shown for the third sand pack in Fig. 2. Each sand block is represented numerically by 8 horizontally and 4 vertically directed elements, resulting in a total nodes, respective quadrilateral elements. The numerical simulations are compared with the experiments using initially the SUTRA and, more recently, the FEFLOW density-dependent flow and transport modes. The models are run in time until steady-state conditions for the transport, likewise to the experiments, are reached. Fig. 2. Stochastic realization and model implementation of hydraulic conductivity of the tank packing. The tank experimental conditions are simulated in the numerical models using the following boundary conditions (BC). A pressure difference p IN - p OUT between the in- and the outflow boundaries of the tank and of C = C 0 at the lower half of inflow boundary for the concentration are specified. The choice of the proper outflow BC for C poses somewhat a problem, as this value is usually varying through the course of the experiment. In the SUTRA model a first-type BC C=C MIX is taken, where C MIX is half of the input concentration C 0 at the saltwater inlet. With this choice of BC, the numerical model appears to better simulate the tank experiments (Koch and Starke 2003). Instead of using these C MIX BC, where concentration varies with the depth, a natural (free) BC is already implemented in the FEFLOW model and used at the outflow boundary. In the FEFLOW model the mass boundary conditions at the two inflow sections are similar to those used in the SUTRA model above. To mimic the separating plastic plate in the tank, as mentioned above, a no-flux mass condition (second kind) is chosen in the FEFLOW model. Also, unlike in the SUTRA model, flux viscosity dependencies and the extended Boussinesq-approximation is used in the FEFLOW model. To implement the densitydependency of the fluid in the latter a global density ratio, defined as the ratio between the density of the fresh water and that of the saltwater has to be implemented in the FEFLOW as shown in Table 1. Because of the large size of the problem, with more than nodes in the FEFLOW model, likewise to the SUTRA model, an iterative solver is used. 3

4 y [m] C = 5000 ppm C = ppm C = ppm u = 4 m/d sand pack x [m] Fig. 3. SUTRA model runs in steady-state conditions for the transport with different concentrations shown as percentage isolines of C = C S (after Starke 2005) Fig. 4. Steady-state conditions for the transport in the FEFLOW Table 1. Density ratio α of given salt concentrations C S and density ρ CS by density of water ρ 0 ρ 0 [kg/m 3 ] C S [kg/l] ρ (CS) = ρ 0 + 0,7*C S α (0,7*C S )/ρ 0 α = (ρ (CS) - ρ 0 ) / ρ 0 α [10-4 ] , Typical results obtained with the two model approaches for sand pack 3, different saline concentrations and a flow velocity of u = 4 m/d are shown in Figs. 3 and 4. 4

5 3. Experimental and numerical results All experimental results are compared with numerical simulations using the SUTRA densitydependent flow and transport model. The validation of the experiments by a rerun of some selected models and the validation of the SUTRA runs by using FEFLOW is still in progress. The results of both the numerical models and the experiments have been complemented by Monte Carlo (MC) simulations Results of deterministic experiments Spatial variances or second moments σ 2 C of the concentration distribution are calculated from the measured (or numerically simulated) mixing widths B(x) taken along the displacement distance x for each sand pack as σ 2 C = (B/2) 2. The results of the experiments, as well as of the simulations, show often (cf. Starke, 2005) that the boundaries of the concentration plumes are varying in an undulatory way across the horizontal extension of the tank, attributable possibly to non-ergodic effects. The wave lengths λ C are proportional to the correlation length λ X, but independent of the concentration differences and of the flow velocities. Using linear regression, the transversal macrodispersivity A T is computed from the spatial variances σ 2 C (x) as A T = 0.5*d σ 2 C /dx. Fig. 5 indicates that there is a nonlinear decreasing tendency of A T with increasing flow velocity u reaching a velocity-independent value for A T. The transversal macrodispersivity A T is increasing with the variance σ 2 ln k and the correlation length λ of the log-normal permeability distribution (note there is a higher A T for pack 2 and 3) and decreasing with the density contrast c between the two fluids. AT [m] u [m/d] pack 1; c = 250 ppm pack 1; c = 5000 ppm pack 1; c = ppm pack 1; c = ppm pack 2; c = 250 ppm pack 2; c = 5000 ppm pack 2; c = ppm pack 2; c = ppm pack 3; c = 250 ppm pack 3; c = 5000 ppm pack 3; c = ppm Fig. 5. Experimental results for the transversal macrodispersivity A T for various flow rates u and concentration differences c for the three different sand packs (after Starke 2005) Validation using Monte Carlo simulations The results of both the deterministic numerical models and of the experiments are complemented by Monte Carlo (MC) simulations, using stochastic realizations with predefined variance σ 2 and correlation lengths λ X, λ Z of the Y = ln k permeability field that are representative for the stochastic packing of the tank. For each experiment, a large number of Monte Carlo simulations with stochastic realizations (realized by the Turning Band Generator, Tompson et al. 1989), taken from the corresponding statistical family, are simulated in the numerical models. From the moment-analyses of the widths of the simulated fresh- saltwater interfaces, variances of the transversal dispersion are calculated as a function of the horizontal distance from the tank inlet. Using simple square root 5

6 σ² [m²] σ² ln k = 1.50; λ x = 0.3 m; λ y = m; u = 1 m/d Monte-Carlo simulations tank and numerical simulations expected value x [m] Fig. 6. Ensemble of Monte Carlo results for a tracer of the spatial variances σ 2 c (x) with expected value and of all experimental and numerical results for sand pack 3 and at a flow rate u = 1 m/d regression analysis of these variances, representative expectation values for the apparently vertical (transversal macro)-dispersivity A T are computed. Exemplarily, this is shown in Fig. 6 for a realization of the third sand pack at one flow rate u = 1 m/d for a tracer C 0 = 250ppm. For more details we refer to Starke (2005). For most of the MC-random field families, the numerical models were able to provide an asymptotically stable value for A T already after approx. 100 realizations for density-independent, and approx. 30 realizations for density-dependent simulations. The expected value for the transversal macrodispersivity of the ensemble of results is calculated using linear regression like above. In addition, the sensitivities of the computed A T to the various parameters, describing flow and transport (i.e. u and C 0 ) and the porous media (i.e. σ 2, λ X and λ Z ), are investigated and used in a multiple linear regression to establish a functional relationship between these and A T. In detail, the different sensitivities of the computed longitudinal and tranversal macrodispersive coefficients are adjusted to the various parameters describing the flow, transport and the porous media Comparison with analytical stochastic theory A theoretical formula proposed by Welty et al for the transversal macrodipersivity of densitydependent flow and transport in a stochastic porous medium could be refined and further developed. The salient features of it are outlined in Fig. 7 (see Starke, 2005, for a derivation and discussion) σ _ ln kλ1a1 AT = 1 a 2 2Γ3 x G3 γ3 a a 2 ( ξ + 1) ( ξ )( ξ + ) ( 2 1) ( ξ + ) = ξ ξ + = σ 2 ln k ( ) 3 3 ij i 3 variance of log-normal permeability distribution λ, λ correlation length in direction x respectivly x γ flow factor γ = v / K J ξ factor of anisotropy ξ = λ / λ Γ G _ x ( p / x3 0,65 Lg) /( p / x3 Lg) 3 Γ = ρ ρ γ concentration gradient dc/dx 3 3 mean displacement distance Fig. 7. The refined formula proposed by Welty et al for the transversal macrodispersivity A T 6

7 Applying multiple linear regression analysis of the experimental and numerical results for A T using various factors in this analytical formula as regressor variables as mathematically described in Fig. 7, as well, was performed. A very good agreement of the results of density-independent Monte Carlo simulations is found as indicated by a negligible p-values. For the density-dependent Monte Carlo simulations the results are not so good, as the probability error is about 5% which can still be deemed satisfactory, so that the formula of Fig. 7 may still apply with the predictions of stochastic theory (Starke and Koch, 2006). 4. Summary and outlook Tank experiments, numerical investigations and stochastic approaches are needed to understand density-dependent flow and transport processes in heterogeneous porous media. For calibration and validation purposes, the experiments and numerical models from SUTRA have recently been supplemented by numerical simulations using the FEFLOW flow and transport model which appears to allow a better representation of the intricate concentration outflow boundary condition whose effects on the plume concentrations close to the exit boundary of the tank are not yet fully understood. Together with additional tank experiments of different stochastic porous-medium representations, the present analysis will complement the earlier work of Starke (2005) with regard to this scientific topic, namely, an exhaustive characterisation of density-dependent vertical macrodispersion in stochastically heterogeneous porous media. As further steps, the results have to fit with those from different upscaling procedures, such as the ones based on Monte Carlo simulations which provide statistically relevant data for validation of stochastic theory of macrodispersion (Gelhar, 1993). The final conclusions should lead to statistical ensemble-prognoses based on adequate data of the experimental investigations, numerical models and stochastic approaches for the characterisation of lateral macrodispersion in heterogeneous porous media. For more automatization and specific investigations of future tank experiments new methods of measurements of the solute, respectively, the denser fluid are needed. Specific investigations could be, for example, the unstable hydrodynamic case where the denser fluid is above which results in a manifold of intrinsic physical phenomena, namely the development of fingers (cf. Koch, 1994). As this requires a better spatial resolution of the plumes as achieved so far with the present tank design, colour electronic page setting in connection with a solute colouring tracer could be helpful. Acknowledgements The authors gratefully acknowledge Dr. B. Starke for the complete transfer of her researched data, R. Feldner and Dr. Hassinger for the help for re-setup the tank and finally N. Dockendorf, T. Münch and A. Jaroch for a preview of the manuscript. References Bear J, Cheng AH-D, Sorek S, Ouazar D, Herrera I (1999) Seawater intrusion in coastal aquifers Concepts, methods and practices. Kluwer Academic Publishers 13-17: Dagan G (1989) Flow and transport in porous formations. Springer-Verlag:465pp. Dagan G, Zeitoun DG (1998) Seawater-freshwater interface in a stratified aquifer of random permeability distribution. Journal of Contaminant Hydrology Vol. 29 No. 3: Gelhar LW (1993) Stochastic subsurface hydrology. Prentice-Hall:385pp. Kobus HE, Spitz KH (1985) Transverse mixing of stratified flows in porous media. Paper presented at the 21st Congress of the International Association for Hydraulic Research, Melbourne, Australia, August. Koch, M. (1994), The dynamics of density driven finger instabilities in stochastically heterogeneous porous media, In: Computational Methods in Water Resources X, A. Peters et al. (eds.), Vol. 1, p , Kluwer Academic Publishers, Dordrecht. 7

8 Koch, M., G. Zhang (1992) Numerical simulation of the migration of density dependent contaminant plumes, Ground Water, 5, Koch M, Starke B (2003) Experimental and numerical investigation of macrodispersion of density-dependent transport in stochastically heterogeneous media: Effects of boundary conditions and high concentrations. Paper presented at the 2nd International Conference on Saltwater Intrusion and Coastal Aquifers, Mérida, Yucatán, México, 30 March 2 April Koch M, Starke B (2002) Experimental and numerical investigation of macrodispersion of density-dependent flow and transport in stochastically heterogeneous media. Paper presented at the International Groundwater Symposium 2002, Lawrence Berkeley National Laboratory, Berkeley, CA, March Koch M, Starke B (2001) Experimental and numerical investigation of macrodispersion of density-dependent flow and transport in stochastically heterogeneous media. Paper presented at the 1st International Conference on Saltwater Intrusion and Coastal Aquifers, Essaouira, Morocco, April Starke B, Koch M (2006) Laboratory experiments and Monte Carlo simulations to validate a stochastic Theory of tracer- and density-dependent macrodispersion. Paper presented at the 16th Computational Methods in Water Resources Conference, Copenhagen, Denmark, June Starke B (2005) Experimentelle und numerische Untersuchungen zur Dispersion von Dichteströmungen in einem stochastischen Modellaquifer [in German]. PhD, University of Kassel, Germany 187 pp. Tompson AFB, Ababou TR, Gelhar LW (1989): Implementation of the three-dimensional turning bands random field generator. Water Resources Research Vol. 25 No. 10: Welty C, Kane AC, Kauffmann LJ (2003) Stochastic analysis of transverse dispersion in density-coupled transport in aquifers. Water Resources Research Vol. 39 No. 6: SBH Welty C, Gelhar LW (1991) Stochastic analysis of the effects of fluid density and viscosity variability on macrodispersion in heterogenous porous media. Water Resources Research Vol. 27 No. 8: