A numerical assessment of the random walk particle tracking method for heterogeneous aquifers

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1 288 Calibration and Reliability in Groundwater Modelling: From Uncertainty to Decision Making (Proceedings of ModelCARE 2005, The Hague, The Netherlands, June 2005). IAHS Publ. 304, A numerical assessment of the random walk article tracking method for heterogeneous aquifers P. SALAMON, D. FERNÀNDEZ-GARCIA & J. J. GÓMEZ-HERNÁNDEZ Universidad Politécnica de Valencia, Deartamento de Ingeniería Hidráulica y Medio Ambiente, Camino de Vera, s/n., E Valencia, Sain salamon@hma.uv.es Abstract The random walk article tracking method has become an imortant tool for the uncertainty assessment of solute transort models due to its comutational efficiency and the non-existence of numerical disersion. Yet, in highly heterogeneous aquifers the smoothness assumtion of the velocity field, requisite for a correct solution of the random walk equations, may not hold anymore. Three different numerical methods to overcome this roblem are numerically evaluated. The results demonstrate that in an aquifer with an isotroic satial correlation, and following a multi-gaussian random function with a small connectivity of extreme values, all techniques are able to model solute transort correctly, even for a hydraulic conductivity variance of σ 2 lnk = 4. Keywords heterogeneity; local mass conservation; numerical imlementation; random walk article tracking; solute transort INTRODUCTION Eulerian transort models are often lagued by numerical disersion or artificial oscillations, esecially for advection-dominated roblems. To reduce these roblems, a higher grid resolution and smaller time stes may be alied, resulting in large comutational times. One alternative to solve transort in heterogeneous orous media is the Lagrangian aroach. In articular, the random walk article tracking (RWPT) method, treats the transort of a solute by artitioning the solute mass into a large number of articles. It moves each article through the orous medium using the velocity field obtained from the solution of the flow equation to simulate advection, and adds a Brownian random dislacement to simulate disersion. This aroach avoids solving the transort equation directly and therefore is virtually free of numerical disersion and artificial oscillations. Furthermore, comutational times, even for models with a high grid resolution and characterized by strong heterogeneities, are significantly smaller than using the traditional Eulerian, mixed Eulerian-Lagrangian, or total variation diminishing (TVD) schemes (e.g. Tomson & Gelhar, 1990; Zheng & Bennett, 2002). These advantages have made the RWPT method a oular choice for comlex, highresolution transort roblems, inverse modelling, and uncertainty assessment of contaminant transort.

2 Numerical assessment of the random walk article tracking method for heterogeneous aquifers 289 BASIC PRINCIPLES Random walk article tracking is a method from statistical hysics which has been used in the analysis of disersion and diffusion rocesses in orous media. It is based on a similarity between the Fokker-Planck equation and the advection disersion equation. Using the similarity between these equations, solute concentrations are reresented by the density function of articles that are moving following the Itô- Taylor integration scheme (Gardiner, 1990): ( t + Δt) = ( t) + A(, t) Δt + B(, t) Δw Δt (1) T with A = u + D and 2 D = B B where (t) is the osition of a article at time t; A is a drift vector; B is a tensor defining the strength of disersion which is related to the disersion tensor D as shown above; u is the groundwater velocity; and Δw, an uncorrelated stochastic force, is a vector of indeendent normally distributed random variables with zero mean and unit variance. However, using Stratonovich s interretation of a stochastic integral leads to the following equation, by which the density function of articles also fulfils the advection disersion equation: Δt Δt ( t + Δt) = ( t) + A, t + Δt + B, t + Δw Δt (2) T T with: A = u + B ( B ) and 2 D = B B 2 It can be seen that the drift vector and the disersion tensor are evaluated at a moment halfway along the time ste to the next article osition. Equation (2) is rarely used for the RWPT method, as this rocedure requires an additional iterative scheme within each time ste. Nevertheless, the modified velocity contains a derivative term reduced by a factor of 1/2. THE PROBLEM OF LOCAL SOLUTE MASS CONSERVATION Numerical imlementation of the random walk equations is relatively simle, with one excetion. When solving the flow equation using numerical methods the resulting hydraulic heads and the associated velocity field are usually comuted at discrete oints. Yet, simulation of solute transort by the random walk methodology requires continuous information about the velocity field. Therefore, a ma of velocities from this discrete information has to be generated. This velocity ma should fulfil the local fluid mass balance at any location and the local solute mass conservation at any gridcell interface. In general, there is not a simle solution to this roblem, but several aroaches have been roosed in the literature. The interolation method The interolation method is certainly the aroach most commonly used to address this roblem. It uses linear interolation to obtain the groundwater velocity at any

3 290 P. Salamon et al. oint and bilinear interolation to calculate the gradient of the disersion term and the disersion tensor. By means of this so-called hybrid scheme, the interolation method fulfils both the fluid mass balance and the local solute mass balance. Nevertheless, using bilinear interolation introduces a greater smoothing, which can lead to errors esecially for highly heterogeneous aquifers. LaBolle et al. (1996) resented a detailed analysis of this method. The reflection rincile The reflection rincile was first resented by Uffink (1985) and is based on the idea of diffusion across a boundary with different diffusion coefficients. He suggested that art of a set of articles crossing from one domain into another with different diffusion coefficients must be reflected in order to obtain the correct concentration distribution across the boundary between the domains. This is done in RWPT by assigning a certain robability for a article of crossing the domain interface, which deends on the diffusion coefficients on either side. The local flow mass balance is satisfied by using linear interolation to calculate groundwater velocity. Various authors (Ackerer, 1987; Cordes & Rouvé, 1991; Semra et al., 1993; Hoteit et al., 2002) have used this aroach and suggested slight modifications. Generalized stochastic differential equations Strictly seaking, stochastic theory only alies when transort roerties are smooth functions in sace. LaBolle et al. (2000) resented generalized stochastic differential equations (GSDE) for the case of discontinuous transort roerties and their numerical integration for RWPT. In ractical terms this means that, for the case of isotroic disersion and homogeneous orosity, a two-ste rocedure is used in order to account for the local solute mass balance. First, the velocity at the articles osition, shifted only by the disersive art, is evaluated. Then, this velocity is used to calculate the final article ste. The local fluid mass balance is fulfilled by using linear interolation as in the other two aroaches. NUMERICAL ANALYSIS The aroaches were numerically imlemented into the random walk transort model RW3D (Fernàndez-Garcia et al., 2005). Solute transort in the following synthetic cases is conservative, two-dimensional and ore-scale disersion is assumed to be isotroic (α = 0.01 m). A constant dislacement scheme is used as it is comutationally more efficient than the constant time ste scheme (Wen & Gómez-Hernández, 1996). MODFLOW (McDonald & Harbaugh, 1988) was used to solve the flow equation and to comute the cell-interface velocities. Two heterogeneous hydraulic conductivity fields with a discretization of Δx = Δy = 0.3 m and a total domain size of x = 60 m and y = 21 m were generated using stochastic

4 Numerical assessment of the random walk article tracking method for heterogeneous aquifers 291 simulation techniques as shown in Fig. 1. Whereas for model A a Gaussian random function is used, model B was created using indicator simulation. Satial correlation is isotroic for model A (λ x = λ y = 1.2 m) and strongly anisotroic for model B (λ x = 20 m, λ y = 1.2 m). The satial correlation for both fields is modelled using an exonential semivariogram and the indicator simulation was erformed using a mosaic model with nine thresholds. The variance is increased from σ 2 lnk = 0.5 to 4 in order to simulate increasing heterogeneity and the mean hydraulic conductivity is ket constant at lnk = m s -1. In this article, the interolation method, using the Itô or the Stratonovich interretation of a stochastic integral, and the GSDE method in highly heterogeneous aquifers are evaluated. The third order TVD (ULTIMATE) scheme, included in the MT3DMS model (Zheng & Wang, 1999), was used for comarison with these methods. This scheme offers an attractive alternative to the traditional mixed Eulerian-Lagrangian schemes (e.g. method of characteristics, MOC) due to its mass conservation roerty and the ability of reducing numerical disersion (Zheng & Wang, 1999; Zheng & Bennett, 2002). Nevertheless, as the resent roblem is strongly advection-dominated, the domain was refined to a discretization of Δx = Δy = 0.1 m to minimize any ossible numerical disersion. One model run for model B with σ 2 lnk = 2 was reeated with a discretization of Δx = Δy = m and the two results obtained with the TVD scheme were comared. Neither artificial oscillations nor numercial disersion were observed. The differences between the different numerical schemes are investigated by comaring the time-deendence of corresonding solute lume satial moments. The satial moments are calculated following Tomson & Gelhar (1990) Model A Model B Y Y Fig. 1 Models generated using sequential Gaussian simulation (Model A) and indicator simulation (Model B) with different satial correlation. Model dimensions are given in metres; hydraulic conductivity is as lnk RESULTS All methods erformed well for model A, even when resenting an extreme heterogeneity of σ 2 lnk = 4 (Fig. 2). For model B, all the techniques illustrate good results u to a hydraulic conductivity variance of σ 2 lnk = 2. Increasing the heterogeneity for this model, only the interolation method in combination with the Itô- or Stratonovich- Fokker-Planck equation is able to reroduce the average velocity and the macrodisersivity correctly (Fig. 3). The GSDE method underestimates the average velocity and overestimates the macrodisersivity as deicted in Fig. 3. With resect to the numerical imlementation, the interolation method in combination with the Itô-Fokker-Planck equation is the easiest method. All other

5 292 P. Salamon et al Average Velocity U(t) [m/s] Longitudinal Macrodisersivity A L (t) [m] Transient Variation Diminishing (TVD) Scheme Interolation Method (Ito-Fokker-Planck) Interolation Method (Stratonovich-Fokker-Planck) Generalized Stochastic Differential Equations Fig. 2 Average velocity and longitudinal macrodisersivity using model A with σ 2 lnk = 4 (where G (t) denotes the location of the centre of mass at time t). Average Velocity U(t) [m/s] Longitudinal Macrodisersivity A L (t) [m] Transient Variation Diminishing (TVD) Scheme Interolation Method (Ito-Fokker-Planck) Interolation Method (Stratonovich-Fokker-Planck) Generalized Stochastic Differential Equations Fig. 3 Average velocity and longitudinal macrodisersivity using model B with σ 2 lnk = 2 (where G (t) denotes the location of the centre of mass at time t). aroaches require either an additional iterative scheme (Stratonovich interretation), or are comlex to imlement for three-dimensional, heterogeneous systems (reflection rincile, GSDE). Finally, it should be noted that comutation times, esecially for the strongly heterogeneous cases, are significantly lower for the RWPT method than for the TVD scheme. Whereas the TVD scheme required from aroximately 30 minutes (for model A with σ 2 lnk = 0.5) to several hours (for model B with σ 2 lnk = 4) to solve the transort equation, RWPT, using 2000 articles, required aroximately 15 minutes, clearly demonstrating the comutational efficiency.

6 Numerical assessment of the random walk article tracking method for heterogeneous aquifers 293 Acknowledgements This work was funded by the Sanish Ministry of Science & Technology, Project ESCOPA REN REFERENCES Ackerer, Ph. (1987) Random-walk method to simulate ollutant transort in alluvial aquifers or fractured rocks. In: Advances in Analytical and Numerical Groundwater Flow and Quality Modelling (ed. by E. Custodio, A. Gurgui & J. P. Lobo Ferreira), NATO ASI Series C, vol Cordes, C. H. D. & Rouvé,G. (1991) A new very efficient algorithm for article tracking in layered aquifers. In: Comuter Methods in Water Resources II, vol. 1, Groundwater Modelling and Pressure Flow (ed. by D. B. Sari et al.), Sringer Verlag, New York, USA. Fernàndez-Garcia, D., Illangasekare, T. H. & Rajaram, H. (2005) Differences in the scale deendence of disersivity estimated from temoral and satial moments in chemically and hysically heterogeneous orous media. Adv. Water Resour. 28(7), Gardiner, C. W. (1990) Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Sringer Verlag, New York, USA. Hoteit, H., Mosé, R., Younes, A., Lehmann, F. & Ackerer, Ph. (2002) Three-dimensional modeling of mass transfer in orous media using the mixed hybrid finite elements and the random walk methods. Math. Geology (34)4, LaBolle, E. M., Fogg, G. E. & Tomson, A. F. B. (1996) Random-walk simulation of transort in heterogeneous orous media: Local mass-conservation roblem and imlementation methods. Water Resour. Res. 32(3), LaBolle, E. M., Quastel, J., Fogg, G. E. & Gravner, J. (2000) Diffusion rocesses in comosite orous media and their numerical integration by random walks: generalized stochastic differential equations with discontinuous coefficients. Water Resour. Res. 36(3), McDonald, M. G. & Harbaugh, A. W. (1988) A Modular Three-dimensional Finite-difference Ground-water Flow Model. US Geol. Survey Techniques of Water Resources Investigations, Book 6. Semra, K., Ackerer, P. & Mosé, R. (1993) Three-dimensional groundwater quality modelling in heterogeneous media. In: Water Pollution II: Modeling, Measuring and Predicition, Comut. Mech., Billerica, Massachusetts, USA. Tomson, A. F. B. & Gelhar, L. W. (1990) Numerical simulation of solute transort in three-dimensional, randomly heterogeneous orous media. Water Resour. Res. 26(10), Uffink, G. J. M. (1985) A random walk method for the simulation of macrodisersion in a stratified aquifer. In: Relation of Groundwater Quality and Quantity, IAHS Publ IAHS Press, Wallingford, UK. Wen,. H. & Gómez-Hernández, J. J. (1996) The constant dislacement scheme for tracking articles in heterogeneous aquifers. Groundwater 34(1), Zheng, C. & Wang, P. P. (1999) MT3DMS, a modular three-dimensional multisecies transort model for simulation of advection, disersion, and chemical reactions of contaminants in groundwater systems. Reort to the US Army Cors of Engineers, Washington, DC, USA. Zheng, C. & Bennett, G. D. (2002) Alied Contaminant Transort Modeling, 2nd edn. John Wiley and Sons, Inc., New York, USA.

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