EXPERIMENTAL STUDY OF COUPLED FLOW AND MASS TRANSPORT: A MODEL VALIDATION EXERCISE

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1 ModelCARE 90: Calibration and Reliability in Groundwater Modelling (Proceedings of the conference held in The Hague, September 1990). IAHS Publ. no. 195, EXPERIMENTAL STUDY OF COUPLED FLOW AND MASS TRANSPORT: A MODEL VALIDATION EXERCISE S.M. HASSANIZADEH National Institute of Public Health and Environmental Protection (RIVM), P.O.Box 1, 3720 BA Bilthoven, The Netherlands ABSTRACT A series of laboratory experiments are carried out to study the applicability of Darcy's and Fick's laws in high-concentration situations. In these experiments, a low-concentration salt solution (1 to 3 g I* 1 ), initially filling the column, is displaced with a higher concentration solution (3 to 280 g 1" 1 ). The salt mass fraction is monitored as a function of time using an array of 16 electrodes installed in five rows. The computer package SPRINT is employed for simulating the experiments. It appears that using classical Fick's and Darcy's laws very good agreement between calculated and experimental results for the cases of low-concentration-gradient experiments, whereas appreciable differences are found for high-concentration gradient situations. After checking a number of plausible modifications to Fick's law, it appears that a nonlinear relationship for the dispersive mass flux provides a satisfactory agreement between calculated and experimental results. INTRODUCTION Disposal of radioactive waste in geological formations is seriously being considered in a number of countries. An integral part of the safety assessment of nuclear waste disposal is the study of nuclide transport in the geosphere. This requires a profound understanding of the various processes which take place within soil formations; among them groundwater flow. Such studies are commonly made possible by means of mathematical models. Mathematical models serve as a powerful tool for prediction of release of waste into the geosphere, movement of water and contaminants through the geosphere, and calculation of doses of radioactive material reaching the biosphere. Often such predictions need to be made for thousands of years extended into the future. The question then arises how good such predictions are and how much one can trust the results. Answers to such questions are normally sought by the validation of mathematical models. One of the important aspects of validation exercises is the identification of appropriate processes related to the problem under study (Tsang, 1987) A candidate type of formation for hosting a waste repository is a salt rock. Groundwater in the aquifers surrounding and overlying salt formations often contains a high concentration of salt. This feature has to be properly accounted for in the studies of nuclide transport in the geosphere. However, in the vast literature on flow and 241

2 242 S.M.Hassanizadeh transport in porous media, there is a lack of information on the effect of high salt concentration on groundwater movement and pollutant transport. Recently, a number of works on theoretical and numerical aspects of modelling brine transport have been reported. Among these are the works by Leijnse (19J5), Herbert et al. (1988), Hassanizadeh (1986, 1988), and Hassani?.-tdeh & Leijnse (1988). Some of these works have concentrated on i lentification of appropriate processes related to brine transport in soil formation. Obviously, laboratory and field studies are also needed to assist with achieving our goals in model validation. Based on these considerations, a series of laboratory experiments have been designed and carried out in RIVM in collaboration with the Geotechnique Laboratory of Delft University of Technology, The Netherlands. The experiments have been used for two main purposes: i) investigating some of the relevant processes in brine transport in porous media, and ii) providing data sets to be used for (partial) validation of transport models. Results of the experiments have been incorporated as Test Case 13 within the international model validation study INTRAVAL (Anderson et al^, 1989). In this paper, results of a validation exercise based on this experiment is reported. First, the experimental set-up and methods are briefly explained. Then, the validation strategy is described and the governing equations for coupled flow and transport are given. Various conceptual models are proposed and results of calibration and validation of models are presented. EXPERIMENTAL SET-UP, MATERIALS AND METHODS In this section a brief description of the experimental set-up and methods is given. Details of the experiment can be found in Hassanizadeh et ajl. (1990). 3-way valve fresh water outflow res. If thermometer? i inflow recorder 1 res. inflow res. MAM" '% salt water outflow res. %J! r* fresh water reservoir salt water reservoir a electrodes «mitt distribution cylinder I flow B-T* & meter 3-way valve flow meter recorder e n FIG. 1 General configuration of the experimental set-up.

3 Coupled flow and mass transport: model validation 243 The general layout of the experimental set-up is depicted in Fig. 1. A sample of soil is placed between two parallel plates which form a two-dimensional vertical column. The plates are made of plexiglass and have a thickness of 12 mm. They are spaced 10 mm apart. Inner dimensions of the column are 60 cm (b) by 125 cm (h). Upper part of the column is open. There is one row of inlet/outlet holes at the bottom of the column and one at the top. Each of the holes at the bottom are served with a valve which may be operated independently. In principle, vertical flow both in upward and downward directions is possible. There are two separate flow circuits: fresh-water and saltwater. Each circuit consists of a main reservoir, a pump, a constantlevel inflow reservoir and a constant-level outflow reservoir. By means of the 3-way valves, one can let the fresh water or the salt water flow through the column. The fluid is distributed from the 3-way valve to the holes at the bottom through a distribution cylinder installed immediately before the holes (Fig. 1). The column is packed with glass beads, with diameters ranging from 0.40 to 0.52 mm. The volumetric flow rate has been measured at two locations: before inlet to the column where an in-line flowmeter is installed, and after outlet from the column where water could be collected in a measuring glass and the filling time is measured. Intrinsic permeability of the column was determined by means of classical Darcian Experiments. The salt mass fraction of the solution outside the column has been determined with a conductivity meter, and inside the column it could be monitored by means of 16 pairs of electrodes. Two types of displacement experiments have been carried out: lowconcentration- gradient (LCG) and high-concentration-gradient (HCG). In all experiments, the original fluid in the column had a low salt concentration (approximately 1-3 g l" 1 ). In a LCG experiment, the resident water in the column is displaced by water containing a slightly higher salt concentration (2-4 g 1" 1 ). This concentration is high enough to enable the measurement of the breakthrough curves with a high degree of accuracy. However, the effect of differences in fluid density can be neglected. In an HCG experiment, the salt concentration of the displacing fluid was much higher so as to cause the density differences to have an appreciable effect on the displacement process. VALIDATION STRATEGY Various authors have proposed different strategies for validation of mathematical models (see e.g. Tsang, 1987; Eisenberg et al. 1987). The central theme in all strategies is about the comparison of model predictions with experimental results. The aim is to obtain insight into the working of the mathematical model under its operating conditions (i.e. the conditions for which the model is supposedly designed). However, depending on the spatial and/or temporal scale of the problem and the validation experiments, one may need to follow different strategies. Also different validation strategies have to be adopted for cases where experimental data are already available and no new experiments are possible as opposed to when the experiments are going on and/or are yet to be planned. In the present case, where we have data from laboratory experiments already carried out, a simplified validation strategy,

4 244 SM.Hassanizadeh illustrated below is selected. H ZE CONCEPTUAL NUMERICAL _J VERIFICATION -~» CALIBRATION -* PREDICTION -=> ^ ^ ^ y ^ MODEL -> MODEL -> VERIFICATION T According to this strategy, first by studying experimental data, one has to decide about relevant processes. Accordingly, a conceptual model, in the form of (differential) equations and corresponding boundary and initial conditions, must be developed. The conceptual model and governing equations are given in the next section. These are then incorporated into a numerical model. We have employed the numerical model SPRINT, described shortly, to solve the set of governing equations. Verification aspects of the mathematical model are treated elsewhere (Hassanizadeh, 1990). Calibration of the model has been performed using data from one of the LCG experiments. In principle, then, one should be able to predict other LCG and HCG experiments using the calibrated model. If a satisfactory agreement between predictions and experimental results is not obtained, one has to go back and revise the conceptual and the numerical models and recalibrate. If there are effects which will be important only at high concentration gradients, then calibration has to be based on an HCG experiment as well. Finally, one should be able to simulate all LCG and HCG experiments with a single set of parameters. An important issue not addressed in this work is the measure of goodness of fit in comparisons of calculated and experimental results. In this study, we base our judgement simply on a visual comparison of results. That seems to be sufficient for purposes of this study. CONCEPTUAL AND NUMERICAL MODELS Movement of brine in a porous medium is a coupled phenomenon. It involves dispersion of salt within a moving fluid. The fluid is made up of salt and water and therefore salt dispersion affects the fluid movement and vice versa. One must also take into account the dependence of fluid density and. viscosity on the salt mass fraction. Basic equations describing fluid flow and solute transport in porous media are the equations of conservation of mass. These are (Hassanizadeh, 1986): dng at + V.(pq) - 0 (1) ry> dw + pq.vu + V.J = 0 (2) at where n is the medium porosity, p is the fluid mass density, q is the fluid velocity vector, V is the divergence vector, w is the solute mass fraction, and J is the solute dispersive mass flux. These equations must be supplemented by appropriate relationships for J and q. Equations (1) and (2) are physical laws and are considered to be

5 Coupled flow and mass transport: model validation 245 valid for the full range of thermodynamic processes and for all kinds of solutes and fluids. Relationships for J and q, on the other hand, are to be obtained from equations of motion subject to a number of conceptual assumptions. Depending on the assumptions made, one mayobtain various conceptual models. In our first attempt, we employ the classical formulation of Darcy's and Fick's laws. Therefore, the conceptual model I is written as: q - - *. (Vp - pg) (3) P J = - p D(q).Vw (4) where k is the permeability tensor, D is the dispersion tensor assumed to depend nonlinearly on q, and p is the fluid viscosity. Both p and p are known nonlinear functions of mass fraction: p = p exp (7w + p (p-p )) (5) 2 3 p-p. ( «- 4.1to to ) (6) where p and p axe the mass density and viscosity of fresh water at the reference pressure p, 7 = , and fi = 4.5 x 10 " 10 m 2 N" 1. The dispersion tensor is given by D - o T q I + (o L - a T ) q q/ q (7) where a_ and a are the transversal and longitudinal dispersivities, respectively, and I is the unit tensor. Equations (1) to (7) have been solved with the aid of the numerical model SPRINT. The SPRINT package is a general-purpose computer program for solving mixed systems of nonlinear time-dependent algebraic, ordinary and partial differential equations. The software is the result of joint research at Shell Research Limited and the School of Computer Studies at Leeds University, United Kingdom. The partial differential equations solved by SPRINT may be timedependent but may have only one spatial dimension. Otherwise, the class of equations and boundary conditions handled by SPRINT is quite general. The user should write a few subroutine to describe the format of his problem according to the general format of equations solved by SPRINT. Therefore, it is easily possible to replace any of equations (3) or (4) with a modified version, if necessary. For details about SPRINT package see Berzins et al_^ (1986) and Blom & Zegeling (1989). MODEL CALIBRATION AND VALIDATION As mentioned earlier, as our first conceptual model we use the classical formulation of Darcy's and Fick's laws given by (3) and (4). Because permeability is already known from independent experiments, we need only to evaluate the medium porosity and dispersivity values. Because we are considering a one-dimensional situation, only longitudinal dispersivity needs to be evaluated. Following our validation strategy, we use a low-concentrationgradient experiment (coded L1D01) for calibration purposes.

6 246 S.M.Hassanizadeh o en m m to j FIG. 2 Calibration fit based on an LCG experiment (L1D01). Porosity and dispersivity are evaluated such that a good agreement is obtained between the measured (solid circles) and the calculated (solid line) breakthrough curves at the center electrodes. Here, dispersivity = 0.9 mm, and porosity values in regions between inlet and/or electrodes are: 0.395, 0.463, and 0.411, respectively. o en (0 -t-> m & l.b 0.0 FIG. 3 Comparison of calculated (solid line) and measured (circles) breakthrough curves for experiment L1D02; classical theory. Calculations with various values of porosity and dispersivity were performed until a satisfactory agreement was obtained between experimental and calculated breakthrough curves (see Fig. 2). The arrival time of the middle points of breakthrough curves determines the average value of porosity between two consecutive curves. It appears that porosity is not constant along the column. Dispersivity is determined from the slope of breakthrough curves. Values of dispersivity and porosity are given in Fig. 2. Using these values, two other LCG experiments and two HCG experiments were simulated. It appears that LCG experiments can be predicted satisfactorily (Fig. 3)

7 Coupled flow and mass transport: model validation 247 whereas the agreement for HCG experiments is very poor (Figs 4 and 5). 100 i FIG. 4 Comparison of calculated (solid line) and measured (circles) breakthrough curves for experiment H1D01; classical theory. FIG. 5 Comparison of calculated (solid line) and measured (circles) breakthrough curves for experiment H1D02; classical theory. This leads us to the conclusion that there are certain effects in high-concentration-gradient situations which are not present in our conceptual model. Thus, we need to go back and modify them and come up with a more general formulation. In this exercise, based on theoretical considerations not described here, seven different conceptual models were considered. In these models, we still employ the classical Darcy's law as given by (3), but use various extended

8 245 SMMassaniiadeh versions situation. of Fick's law as listed below for a one-dimensional Model II Model III Model IV Model V Model VI Model VII J - P" L Iql y z - «K {f z + pg) J = - p(a J = - pa^ e pa L I dw 9z 2 a!«+ /9 2 w ) q 3z q Sz J = - pa exp (- /8 J = - p(a dw dz dw dz ) q 3w dz > q âz Model VIII: J P" L q ~ " P Ul 2 Model II allows for a direct coupling between Qarcy's and Fick's laws and has been suggested by Hassanizadeh (1986). Models III and IV assume a nonlinear dependence of dispersivity on salt mass fraction. Model V assumes a linear and models VI and VII assume a nonlinear dependence of dispersivity on the gradient of mass fraction. Finally, model VIII proposes a nonlinear relationship in J. It is apparent that an extended formulation typically contains one or two new parameters which need to be determined as part of the calibration procedure. Thus, one of the HCG experiments was used to evaluate the coefficient(s) fi (or /3 X and /3 2 )- However, it turned out that no satisfactory agreement between experimental results and calculated curves could be obtained for models II through VI for any value of p (or fi x and /3 2 ). Thus, the models are rejected in the calibration phase. Some success was obtained with model VII for values of fi x = 1.4 cm and j3 = 7.5 cm 2. However, using these values and the porosity and dispersivity given in Fig. 2, we could not make a satisfactory prediction of the other HCG experiment. Thus, model VII is also rejected. o 300 f _ - ^-* a if) en cu CO en g 4 1 u- v^y "FIG. 6 Calibration fit to evaluate p of Model VIII based on HCG experiment H1D02.

9 Coupled flow and mass transport: model validation 249 o en to CO CD en o !!!! -f a I' 4 / 1!! 4/ ' 1 ^ _j_l_! i J _i...!.. 1 T-+ J r ^ TT, _ * / a a.. _. _L! ^! 1! FIG. 7 Comparison of calculated (solid line) and measured (circles) breakthrough curves for experiment H1D01; nonlinear dispersion theory. Repeating the same procedure for model VIII provided strikingly satisfactory results. It appeared that for a value of fi = 4 x 10 s cm 2 s g ~ 1, all LCG and HCG experiments can be predicted very well (Figs 6 and 7). Therefore, model VIII is accepted as a suitable conceptual model for describing high-concentration-gradient transport of solutes in a porous medium. Definite conclusions on the general validity of this extended version of Fick's law, however, must be based on results of additional experimental studies. CONCLUSION Experiments on dispersion of salt water in a porous medium have been carried out. These experiments provide data for calibrating and validating coupled flow and transport models. Experiments involving displacement of a low-concentration solution by a slightly higher concentration solution can be satisfactorily simulated using conceptual models based on classical Darcy's and Fick's laws. However, if the concentration difference between resident and displacing solutions is high, appreciable deviation between calculated and experimental results are obtained. A number of variations to Fick's law, as given by models II to VIII, are employed. It appears that the nonlinear relationship of model VIII can simulate all experiments satisfactorily. Other models which do not include the proper mechanism are rejected in the calibration phase. ACKNOWLEDGEMENTS The experiments reported here have been originally designed and set up in the Geotechnique Laboratory of Delft University of Technology, The Netherlands. Expertise of Ab Mensinga and Joop van Leeuwen of this laboratory has been essential in the success of these experiments. Toon Leijnse of the Soil and Groundwater Research Laboratory of RIVM has been involved throughout the study in designing

250 SM.Hassanizadeh the experimental procedure, interpreting the results and modelling efforts. Their contributions are greatly appreciated. REFERENCES Anderson, K., Nicholson, T., Grundfelt, B.

10 250 SM.Hassanizadeh the experimental procedure, interpreting the results and modelling efforts. Their contributions are greatly appreciated. REFERENCES Anderson, K., Nicholson, T., Grundfelt, B. & Larsson, A. (1989) INTRAVAL as an integrated international effort for geosphere model validation - A status report. Int. Symp. on the Safety Assessment of Radioactive Waste Repositories, Paris, October 9-13, Berzins, M., Dew, P.M. & Furzeland, R.M. (1989) Developing software for time-dependent problems using the method of lines and differential-algebraic integrators. Appl. Numer. Math Blom, J.G. & Zegeling, P.A. (1989) A moving-grid interface for systems of one-dimensional line-dependent partial differential equations. Department of Numerical Mathematics, Center for Mathematics and Computer Sciences, Amsterdam, Report NM-R8904. Eisenberg, N.A.'; Van Luik, A.E., Plansky, L.E. & Van Vleet, R.J. (1987) A proposed validation strategy for the U.S. DOE office of Civilian Radioactive Waste Management geologic repository program. GEOVAL-87, SKI, Stockholm, Hassanizadeh, S.M. (1986) Derivation of basic equations of mass transport in porous media, 2. Generalized Darcy's and Fick's laws. Adv. Water Resour. 9, Hassanizadeh, S.M. (1988) Modelling species transport by concentrated brine in aggregated porous media. Transport in Porous Media 3., Hassanizadeh, S.M. (1990) Verification and validation of coupled flow and transport models. GEOVAL-90 Symposium (May 1990), Swedish Nuclear Waste Inspectorate (SKI), Stockholm. Hassanizadeh, S.M. & Leijnse, A. (1988) Modelling of brine transport in porous media. Water Resour. Res Hassanizadeh, S.M., Leijnse, A., De Vries, W.J. & Stapper, R.A.M. (1990) Study of brine transport in porous media: Experimental setup and modelling results. Report no , RIVM, Bilthoven, The Netherlands. Herbert, A.W., Jackson, C.P. & Lever, D.A. (1988) Coupled groundwater flow and solute transport with fluid density strongly dependent on concentration. Water Resour. Res. 24, Leijnse, A. (1985) Modelling the flow of groundwater in the vicinity of a salt dome. In: Waste Management. Vol. 3., , Ariz. Board of Regents, Tucson. Tsang, C.F. (1987) Comments on model validation, Transport in Porous Media g,

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