Fractional-RWHet: An enhanced solver for solute transport with both spatiotemporal memory and conditioning on local aquifer properties

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1 Fractional-RWHet: An enhanced soler for solute transport with both spatiotemporal memory and conditioning on local aquifer properties Zhang Yong 1, Daid A. Benson 2, Eric M. LaBolle 3, Donald M. Reees 4 1 Desert Research Institute, yong.zhang@dri.edu, Las Vegas, NV, USA 2 Colorado School of Mines, yzhang@mines.edu, dbenson@mines.edu, Golden, CO, USA 3 Uniersity of California, emlabolle@ucdais.edu, Dais, CA, USA 4 Desert Research Institute, mreees@dri.edu, Reno, NV, USA ABSRAC racer transport through natural geologic formations is affected by physical and/or chemical heterogeneity, which is present across multiple scales. Local random walk transport models based on the 2 nd -order adection-dispersion equation, such as RWHet, can (at least theoretically) incorporate the site characterization information as fine as the measurement scale. Emerging nonlocal transport models, such as the fractional-order adection-dispersion equation, are demonstrated to capture coneniently the influence of fine scale heterogeneity (under the measurement scale) on solute transport. A combination of local and nonlocal models seems ery attractie and has drawn serious attention recently in the hydrology community. In this study, we combine fractional dynamics and random walks to derie an enhanced soler, named Fractional-RWHet, based on the nonlinear Langein approach. Fractional- RWHet can capture the sub-grid, non-fickian dispersion, including the solute retention process due to relatiely low-permeable sediments and the fast-moing plume front due to the fine-scale preferential flow paths. It also maintains the adantage of local approach, where the obserable heterogeneity (such as hydrofacies from driller s logs) and the measurement of aquifer properties (i.e., elocity from obseration wells) can be incorporated. he Fractional-RWHet therefore enables us to efficiently simulate solute transport with both spatiotemporal memory and conditioning on local aquifer properties. In addition, Fractional-RWHet is built on the solid footing of a fully Lagrangian framework, and can interface directly with the popular USGS MODFLOW code. INRODUCION Increasing concerns of the fate and longeity of contaminant in groundwater necessitates modeling of complex small-scale transport process oer ery large space and time. Many, if not most, natural geologic formations exhibit fractal structure [Cushman, 199] with multiscale physical/chemical heterogeneities. Recent studies [such as Berkowitz et al., 26] further show that the solute transport through heterogeneous media at arious scales can be non-fickan, which cannot be captured efficiently (if at all) by the classical simulator based on the 2 nd -order adection-dispersion equation (ADE). New computational theories/models are therefore needed to extend the 2 nd order ADE based simulator to 1) simulate non-fickian dispersion at all scales, 2) propagate transport information between arious scales, and 3) incorporate measurements of aquifer properties (i.e., elocity) at any practical resolutions. his study proposes an enhanced soler by combining the local simulator deeloped by LaBolle [26] and the nonlocal transport theory proposed by Benson [1998] to capture contaminant transport through aquifer/aquitard systems with physical/chemical heterogeneities, where the influences of sub-grid heterogeneity, local aquifer properties, and spatiotemporal nonlocal dependence on solute dynamics can be characterized efficiently. A preious effort to capture the sub-grid elocity can be found in Feehley et al. [2], where a dual-domain mass transfer (i.e., single-rate mass transfer) methodology is embedded into M3D [Zheng, 199]. See also the oeriew of multiscaling techniques for the study of microfluidics and nanofluidics by Hu and Li [27]. Here the local and nonlocal theories follow the definitions gien by Cushman [199, page 3]: If the constitutie ariable depends on what is happening at a point in spacetime or a ery small neighborhood of the point, the ariable is said to be local and deried from a local theory. On the other hand, if information is needed to define the constitutie ariables from regions of space and time distinct from a neighborhood of the space-time point where ealuation of the ariables is to be made, then the theories and constitutie ariables are said to be non-local in character. he

2 classical soler based on the 2 nd -order ADE is a local approach, where the ariation of solute concentration depends on a ery small neighborhood (i.e., a REV-scale grid) in space and time. he fractional-order adection-dispersion equation (fade) model deeloped by Benson [1998] is a nonlocal transport theory, where the local ariation of solute mass flux is affected by a wide range of upstream domains (i.e., multiple grids) and a long historical release of contaminants at the local point. he influence of sub-grid heterogeneity (i.e., below the REV or measurement scale) on solute transport can also be captured by the fade model [Benson, 1998; Benson et al., 28]. Also note that ariable nonlocal approaches hae been deeloped, such as the multi-rate mass transfer theory [Haggerty et al., 2] and the continuous time random walk (CRW) framework [Berkowitz et al., 26]. Cushman and Moroni [21] showed that the arious nonlocal theories could find a home in the nonlocal dispersie constitutie theory, where the space and time memory kernels define the nonlocality. Our soler can incorporate other forms of memory functions (such as the tame Léy flights) coneniently, so it is actually a generalized soler for arious memory kernels with physical/hydrological definitions. We name this enhanced soler the fractional random walk and the resulting code Fractional-RWHet. he random walk is a fully Lagrangian scheme, with encouraging adantages including minimal numerical dispersion, sub-flow-grid scale resolution of concentrations, easy implementation and near linear scaling on parallel computers due to computational expense that depends on independent particle numbers rather than grid dimensions [Green, 22]. It is also a promising tool to obiate the grid-aeraging problem when used for the local scale reaction with sharp concentration fronts. In addition, Fractional- RWHet can interface directly with the popular USGS MODFLOW code (note that other flow solutions can also be incorporated). his soler allows us to simulate non-fickian dispersion without the need to explicitly define local-scale heterogeneity in a numerical model. In the following we introduce the methodology and examples, including simulations of solute transport through arious heterogeneous media generated by a Marko Chain based geostatistical tool [Carle, 1998]. MEHODOLOGY DEVELOPMEN We proposed the following noel multiscaling fade to capture the local ariation of transport strength, the possibly heay-tailed particle retention processes, and the possibly preferential leading fronts β C ( x t) Θ +, = ( x t) M( dθ ) D( x t) C ( x t) + f ( x) δ ( t) t t, H 1 I,,, (1) where is the elocity ector, D is a scalar dispersion coefficient, ( < 2 ) is the order of the time fractional deriatie (Caputo type), β is the capacity coefficient for < 1 (note 1+ β is the retardation factor for = 1 ) and the parameter defining the waiting time distribution for 1 < 2, the sign function Θ = 1 for < 1 and -1 otherwise, I is the identity matrix, H -1 is the inerse of the scaling matrix is a function uniquely determined by the requirement of (1) haing a mild solution (see Baeumer et al. [25] for the definition of mild solution). C is the solute concentration in the total (mobile+immobile) phase. proiding the order and direction of the fractional deriaties, and M(dθ) is the mixing measure. f ( x ) When β = and and D are constant, (1) reduces to the multiscaling fade proposed by Meerschaert et al. [21]. When deriing (1), we selected the Langein equation (see eq. (3)), where the net flux due to nonlocal dispersion is the same form as the one in the Fokker-Planck equation (FPE). Similar treatment (to the ADE) was proposed by Cortis et al. [24]. his option allows the ariable dispersion coefficient, and it can also reduce significantly the computational burden and thus make the following particle tracking much more conenient. Additional experiments are needed to ealuate the FPE type of fractional dispersion, such as a detailed analysis of the influence of the gradient of D on particle moement. We deeloped a non-markoian particle tracking scheme to approximate the complex model (1). his scheme includes two steps. First, the indiidual jump ector of each particle at time step i, which is essentially a multiscaling compound Poisson process, can be calculated by [ tm/ dt] [ tm/ dt ] H Z() t = x i = R r i θ i, (2) i= 1 i= 1

3 where Z(t ) denotes the particle location at time t, n = [t / dt ] is the number of random jumps by time t using time step size dt, R i is the random length of the i-th jump, the jump direction θ i is a random unit ector drawn from the distribution of the mixing measure M(dθ), and t m is the operational time that particles spend in motion by the clock time t. he jump length of the particle along the eigenector belonging to the k-th eigenalue 1 /α k of H can be calculated by the Langein approach R1 /α k = (x k, t ) dt + D(x k, t )1 /α k dl α k (t ), k = 1, L d,, (3) where dl α is the random noise underlying an α -order Léy motion, k represents the direction of the k-th eigenector of H, and d is the dimension of H. (1) and (3) are also alid for the transient flow. -1 (a) ime nonlocal (α=2).16 (b) Space nonlocal (a=) C/C =.1 =.5 = α = 1.3 α = 1.7 α = (c) Conergence (=.5, α=2) FF RW with nt = 5 RW with nt = 2 RW with nt = RW with nt = Figure 1. Snapshots using the random walk approach (symbols) ersus the FF solution (lines) for the time fade (a) and the space fade (b). he dashed line is the ADE result. In (a), =1, D=.1, =3, α =2, =.1~.9, β=.1, and the initial instantaneous point source is at location. In (b), =1, D=1, =4, α=1.3~1.9, β=, and the source is at location 3. (c) Conergence of the random walk approximation. nt denotes the number of time steps used in the numerical experiments. In the second step, the operational time can be simulated as the number of renewals by time Streamline t > for a gien waiting time distribution with power-law probability tails. Here the particle motion is not instantaneous, and thus we can distinguish the status (mobile or immobile) for 4 each particle at any gien time. his distinction is critical in modeling field-measured plumes, since the sampling process tends to collect mobile solutes preferentially. Also note that the 2 indiidual operational time is an exponentially distributed random ariable, which can be drawn by generating a uniform ( 1) number corresponding to the CDF of the exponential function (with a mean corresponding to the mean motion time). In addition, when 1 < 2, scaling Source location Particle plume X (Horizontal) limit of the operational time process is split into two parts, the mean wait and the deiation from Figure 2. RWHet plume (left) s. Fractionalthe mean, to presere the near and intermediate RWHet plume (right). Here α=1.8, =5, β=. he time behaior. he number of jumps is streamline is exaggerated for ision purpose. associated with the max-process [Baeumer et al., 25; Benson et al., 28], since deiations from the mean waiting time may be negatie. Physically this means that the particle pauses until the next time the accumulated deiations become positie, so that the max increases. 2nd-order ADE Y (Longitudinal) 6 Space fade

4 We erified the aboe random walk scheme extensiely. A few examples are shown in Figure 1. he random walk solutions match well the fast Fourier transform (FF) solutions (Figures 1a, b). We also checked the conergence speed. he solution of (1) is equal to the density of the stochastic process obtained as a scaling limit of the CRW. herefore, the probability density of the CRW will conerge to the solution of (1) after long time or large time steps. Numerical experiments (Figure 1c) reeal that the conergence can be achieed after affordable time steps, such as. SIMULAION RESULS We then ealuated the particle plumes goerned by the time fade, underlying the fine-resolution elocity shown in Figure 2. Results (Figure 3) show that the index controls the slope of the late-time tail of BC, while β affects the mass decay rate. We also analyzed the particle plume distribution. In Figure 4, the background image shows the conductiity distribution. he longitudinal and horizontal dispersiity is.1 and.1 m, respectiely, and the molecular dispersion coefficient is m2/day. An instantaneous line source was used along x=185~2m and y=5m. he plume exhibits both leading and trailing edges, just as expected. We then tested the influence of mixing measure on particle plumes. Results (Figure 5) show clearly the moement of particles along the two preferential directions defined by the mixing measure. o drawn a conclusion, the aboe numerical tests show the applicability of Fractional-RWHet. In the next study, we will apply this enhanced soler to simulate contaminant transport through real aquifers, including the non-fickian, 3-d, plumes obsered at the well-studied MADE test site. Longitudinal Distance (m) C/C C/C We first applied the Fractional-RWHet to simulating solute transport through a heterogeneous porous medium, where the goerning equation of solute transport is the space fade. A 2-d heterogeneity model was generated first (see RWHet result =1.3, β= the background image (adction only) shown in Figure 2) [Carle, =1.5, β= =.3, β= ], and the steady-2 =.5, β=.1 state flow field was =1.7, β= -4 calculated by USGS MODFLOW code. We -4 =.7, β=.1 then released particles at -6 an upstream point, to track the moement of particles =1.5, β=3 along the streamline =1.5, β= -2 =.5, β=.2 (adection only). A new -2 subroutine, called =.5, β=.1 =1.5, β= MOVEP5, was added into -4 Fraction-RWHet, to -4 =.5, β=.1 calculate the fractional -6 dispersion with local ime ariation of transport strength. he fractional Figure 3. Breakthrough cures simulated by Fractional-RWHet dispersion was projected ersus RWHet (with adection only, so the BC is a ertical line). along the streamlines. he resultant particle plume has a heaier leading edge comparing to the plume goerned by the 2nd-order ADE (Figure 2) X (m) 2 22 Figure 4. Particle plumes goerned by the ADE (light dots) and the time fade (dark dots), with =1.5, β=2, and =5.

5 (b) 2nd-order ADE (a) Operator stable parameters (c) Space fade 35 α 1 =1.8 M 1 =.7 3 M 2 =.3 3o 2o Y (m) α 2 = Source Source Figure 5. Particle plume captured the 2nd-order ADE (b) and the space (multiscaling) fade (c). he multiscaling fade used for (c) has pre-defined operator stable parameters shown in (a). he direction of in (b) and (c) shows the mean flow direction., particles were released at the point showing with a diamond in (b) and (c). Plumes in (b) and (c) were simulated by RWHet [LaBolle, 26], and FractionalRWHet, respectiely X (m) ACKNOWLEDGMEN his work was supported by the National Science Foundation (NSF) under EAR , EAR , and EAR his paper does not necessary reflect the iew of the NSF. REFERENCES Baeumer, B., D. A. Benson, and M. M. Meerschaert, 25. Adection and dispersion in time and space, Phys. A, 35(2-4), Benson, D. A., he fractional adection-dispersion equation: Deelopment and application. Ph.D. hesis, Uniersity of Neada, Reno, 157 pp. Benson, D. A., M. M. Meerschaert, Y. Zhang, B. Baeumer, and R. Schumer, 28. Stochastic model for multi-rate mobile/immobile contaminant transport, Water Resour. Res., in reiew. Berkowitz, B., A. Cortis, M. Dentz, and H. Scher, 26. Modeling non-fickian transport on geological formations as a continuous time random walk, Re. Geophy., 44(2), RG23. Carle, S., PROGS: ransition Probability Geostatistical Software, 2.1. Uniersity of California, Dais, 84 pp. Cortis, A., C. Gallo, H. Scher, and B. Berkowitz, 24. Numerical simulation of non-fickian transport in geological formations with multiple-scale heterogeneities, Water Resour. Res., 4, W429. Cushman, J. H., 199. Dynamics of Fluids in Hierarchical Porous Media, Academic Press, San Diego, CA, 55 pp. Cushman, J. H., and M. M. Moroni, 21. Statistical mechanics with three-dimensional particle tracking elocimetry experiments in the study of anomalous dispersion. I. heory, Phys. Fluids, 13, Feehley, C. E., C. Zheng, and F. J. Molz, 2. A dual-domain mass transfer approach for modeling solute transport in heterogeneous porous media, application to the MADE site, Water Resour. Res., 36(9), Green, C.., 22. Effects of heterogeneity on reactie transport in geologic media. Ph.D. hesis, Uniersity of California, Dais, 42 pp. Haggerty, R., S. A. McKenna, and L. C. Meigs, 2. On the late-time behaiour of tracer test breakthrough cures, Water Resour. Res., 36(12), Hu, G. P., and D. Q. Li, 27. Multiscale phenomena in microfluidics and nanofluidics, Chem. Eng. Sci., 62, LaBolle, E. M., 26. RWHet: Random Walk Particle Model for Simulating ransport n Heterogeneous Permeable Media,.3.2. User s Manual and Program Documentation, 27 pp. Meerschaert, M. M., D. A. Benson, and B. Baeumer, 21. Operator Léy motion and multiscaling anomalous diffusion, Phys. Re. E, 63, Zheng, C., 199. M3D: A modular three-dimensional transport model for simulation of adection, dispersion and chemical reactions of contaminants in groundwater systems, S.S. Papadoulos and Associates, Inc., Rockile, MD, 163 pp.