Experimental Investigation and Pore-Scale Modeling Interpretation of Compound-Specific Transverse Dispersion in Porous Media

Transcription

1 Transp Porous Med (2012) 93: DOI /s Experimental Investigation and Pore-Scale Modeling Interpretation of Compound-Specific Transverse Dispersion in Porous Media Massimo Rolle David Hochstetler Gabriele Chiogna Peter K. Kitanidis Peter Grathwohl Received: 28 July 2011 / Accepted: 30 January 2012 / Published online: 15 February 2012 Springer Science+Business Media B.V Abstract In this study, we performed multitracer laboratory bench-scale experiments and pore-scale simulations in different homogeneous saturated porous media (i.e., different grain sizes) with the objective of (i) obtaining a generalized parameterization of transverse hydrodynamic dispersion at the continuum Darcy scale; (ii) gaining an improved understanding of the role of basic transport processes (i.e., advection and molecular diffusion) at the subcontinuum scale and their effect on the macroscopic description of transverse mixing in porous media; (iii) quantifying the importance of compound-specific properties such as aqueous diffusivities for transport of different solutes. The results show that a non-linear compounddependent parameterization of transverse hydrodynamic dispersion is required to capture the observed lateral displacement over a wide range of seepage velocities ( m/day). With pore-scale simulations, we can prove the hypothesis that the interplay between advective and diffusive mass transfer results in vertical concentration gradients leading to incomplete mixing in the pore channels. We quantify mixing in the pore throats using the concept of fluxrelated dilution index and show that different solutes undergoing transport in a flow-through system with a given average velocity can show different degrees of incomplete mixing. Furthermore, it is this compound-specific incomplete mixing within pores that causes different local transverse (mechanical) dispersion to result at the Darcy scale for high flow velocities. We conclude that physical processes at the microscopic level significantly determine the observed macroscopic behavior and, therefore, should be properly reflected in up-scaled parameterizations of transport processes such as local hydrodynamic dispersion coefficients. Keywords Transverse dispersion Multitracer experiments Pore-scale modeling Incomplete mixing Dilution index M. Rolle (B) G. Chiogna P. Grathwohl Center for Applied Geosciences, University of Tübingen, Sigwartstrasse 10, Tübingen 72076, Germany massimo.rolle@uni-tuebingen.de D. Hochstetler P. K. Kitanidis Department of Civil and Environmental Engineering, Stanford University, 473 Via Ortega, Stanford, CA 94305, USA

2 348 M. Rolle et al. 1 Introduction In groundwater systems, the complex interplay between physical and (bio)chemical processes determines the transport and degradation of contaminant plumes. Mixing of reactants often limits (bio)chemical reactions thus controlling natural attenuation and engineered remediation strategies of groundwater pollution. For reactions requiring two or more substrates, mixing of these species undergoing transport in a low-reynolds number (non-turbulent) flow regime typically occurs at the narrow fringes of contaminant plumes. In particular, when contaminant plumes approach steady-state conditions, transverse hydrodynamic dispersion controls the mixing of reaction partners (e.g., electron donors and acceptors for microbially mediated redox reactions). Despite decades of extensive research on dispersion (e.g., Bear 1972; Dentz et al. 2011), the correct quantification of mixing processes in subsurface environments remains a challenging task. Mixing processes have been studied at different scales using experimental and modeling approaches. At the field scale, the importance of transverse mixing was shown for organic contaminant plumes in high-resolution site investigations (e.g., Prommer et al. 2009) and in a number of modeling studies (e.g., Cirpka et al. 1999; Chu et al. 2005). The distinction between plume spreading and true mixing processes (Kitanidis 1994) exposed the need for detailed laboratory investigations. Flow-through bench-scale experiments were performed to study transverse mixing of conservative and (bio)reactive solutes in homogeneous (e.g., Delgado and Carvalho 2001; Klenk and Grathwohl 2002; Huang et al. 2003; Gaganis et al. 2005; Tartakovsky et al. 2008; Rolle et al. 2010) and heterogeneous (e.g., Bauer et al. 2009a, Bauer et al. 2009b; Rolle et al. 2009) porous media. At the smaller pore scale, recent advances include numerical modeling studies (Cao and Kitanidis 1998; Knutson et al. 2007; Acharya et al. 2007; Bijeljic and Blunt 2007), theoretical investigation of solute dispersion (Porter etal. 2010) and microfluidic experiments (Willingham et al. 2008; Zhangetal. 2010). These detailed studies at the pore scale have contributed to improve the understanding of transverse mixing and mixing-controlled reactions by directly taking into account the physical mechanisms governing solute transport. Often the outcomes of Darcy- and pore-scale studies pointed out the limitations of traditional continuum scale description based on the spatial average of pore-scale processes over a representative elementary volume (REV). For instance, the numerical investigation of Tartakovsky et al. (2009)compared pore-and Darcyscale simulations, and highlighted that the accuracy of continuum Darcy-scale modeling deteriorates with increasing grain Péclet numbers (Pe). Nonetheless, a continuum averaged description is still required for practical applications of solute transport and this formulation should be based on up-scaled parameters which still capture the important effects of physical processes at smaller scales. Focusing on transverse dispersion, we recently carried out multitracer bench-scale experiments (Chiogna et al. 2010) which showed a compound-specific behavior over a wide range ofpe and resulted in a non-linear parameterization of local transverse dispersion retaining a significant dependence on aqueous diffusion even at high flow velocity. In this study, we extend the experimental investigation based on conservative multitracer flow-through experiments to different porous media, i.e., different grain sizes, in order to test the validity of the non-linear compound-specific parameterization of transverse dispersion. Moreover, with the aim of bridging pore and Darcy scales, we interpret the experimental results using pore-scale modeling. The purpose of the modeling exercise is not to exactly reproduce the experimental setup, but to gain a better understanding of the macroscopic observations based on the simulation of the physical processes taking place at the microscopic level. The detailed description of basic transport processes such as advection and

3 Experimental Investigation and Pore-Scale Modeling Interpretation 349 Fig. 1 Laboratory experimental setup with steady-state fluorescein plume and oxygen-sensitive stripes to measure oxygen concentrations (upper half). Pore-scale model domain with the tracer injection width (w/2 = 0.6 cm) and details of the pore channels (lower half) diffusion in a liquid solid medium (i.e., pore-scale model domain) allows us to explain the compound-specific lateral displacement observed at the larger scale (i.e., Darcy-scale experimental setup). 2 Multitracer Bench-Scale Experiments The experimental setup was described in detail in Chiogna et al. (2010); here, we summarize the principal features (Fig. 1). Tracer experiments were carried out in a quasi two-dimensional (2D) flow-through chamber with inner dimensions cm 3 (L H W). Steady-state flow conditions were established with three high-precision peristaltic pumps (IPC-N, Ismatec, Glattburg, Switzerland) connected to ten inlet and outlet ports with 1.2 cm spacing. Glass beads (Fisher Scientific GmbH, Germany) with diameters in the range and mm were used as porous media resulting in average packed bed porosity (n) of 41.6 and 40.3%, respectively. These grain sizes are, respectively, smaller and larger than the ones used in our previous study, mm (Chiogna et al. 2010). The experiments were performed in a thermostatic room at a temperature of 22 C. We selected three different conservative tracers: fluorescein characterized by a low aqueous diffusion coefficient ( m 2 /s at 22 C, Worch 1993) and bromide and oxygen with considerably higher diffusion coefficients ( and m 2 /s, respectively, at 22 C, Worch 1993). The tracer solutions of fluorescein and bromide (20 mg/l) were depleted in oxygen and injected through the central inlet port (port number 6 from the bottom). In this way, each experimental run was characterized by the simultaneous displacement of two conservative solutes: fluorescein and oxygen, or bromide and oxygen, respectively. After the establishment of steady-state plumes, fluorescein and bromide were conventionally sampled at the outlet ports and analyzed with a fluorescence spectrometer (Perkin Elmer LS-3B)

4 350 M. Rolle et al. and an ion chromatograph (Dionex DX-120). A non-invasive optode technique, described in detail in Haberer et al. (2011), was used to measure oxygen concentrations at high-spatial resolution (2.5 mm) along three oxygen-sensitive polymer stripes (PreSens, Germany), located at 40.5, 57, and 74 cm from the inlet. A range of typical groundwater flow velocities was investigated (v = m/day), corresponding to grain Péclet numbers (Pe = vd/d aq ) from 1.5 to 975. These values of flow velocity are representative of typical groundwater flow in natural porous media and cover both diffusion- and advection-dominated transport regimes. 3 Pore-Scale Modeling Pore-scale simulations were carried out in a 2D domain (6 2.4cm 2 ) corresponding to the upper fringe of the conservative tracer plumes close to the inlet of the flow-through chamber (Fig. 1). Given the symmetry of this problem across the x-axis, for computational efficiency only the top half of the fringe (i.e., upper half of the plume) was modeled. The geometry of the porous medium consists of a staggered array of circles with diameter equivalent to the average grain sizes used in the experiments. The pore networks were constructed using a periodic arrangement of unit cells maintaining the same average porosity determined for the experimental packed beds. The flow and transport problems were solved using the commercial finite element software COMSOL Multiphysics 4.0a. A triangular finite element mesh was created in COMSOL 4.0a for each of the three different pore-scale models: small grain, intermediate grain, and large grain corresponding to circular grains with diameters of 0.25, 0.625, and 1.25 mm, respectively. Stokes equation describes the flow in the liquid solid pore-scale model domain: ρg ϕ + μ 2 u = 0 (1) where ρ is the fluid density [ML 3 ], g is the constant of acceleration due to gravity [LT 2 ], φ is the hydraulic head (p/(ρg) + z) [L],μ is the dynamic viscosity [ML 1 T 1 ], and u is the velocity vector [LT 1 ]. The solid liquid interfaces along the grain boundaries are no-slip (u = 0); the top and bottom boundaries are no-flux boundaries (u n = 0); the left boundary is a Dirichlet boundary condition for the pressure (p = dp); and the right boundary is also a Dirichlet boundary condition for the pressure (p = 0). The constant pressure on the left boundary, dp, is changed from simulation to simulation in order to yield the correct average velocity for the porous media. Figure 2 shows a schematic of the large grain (d = 1.25 mm) pore-scale domain and flow boundary conditions as well as the velocity field resulting from the solution of the flow problem. The color range is for the magnitude of the velocity within the saturated fluid in [m/day]. The insert focuses on details of the velocity field within an arbitrary pore channel; the black arrows indicate the direction of flow and their size scales with the velocity magnitude. At the pore scale, solute transport is described by the advection diffusion equation, which at steady state reads as: u C i D aq,i 2 C i = 0 (2) where u [LT 1 ] is the 2D velocity field determined by the solution of the Stoke s flow problem (Eq. 1), and C i [ML 3 ]andd aq,i [L 2 T 1 ] are the concentration and the aqueous diffusion coefficients of the tracer i, respectively. Two tracers were considered in the pore-scale simulations: fluorescein (D aq = m 2 /s) and oxygen (D aq = m 2 /s).

5 Experimental Investigation and Pore-Scale Modeling Interpretation 351 Fig. 2 Pore-scale model boundary conditions and computed velocity field for the large grain size (1.25 mm) and an average velocity of 1 m/day. The color code refers to the velocities (m/day) in the pore channels obtained by solving Stokes flow equation (Eq. 1) The latter was selected as representative solute with high diffusive properties. In fact, bromide has practically the same D aq as oxygen and numerical simulations (not shown here) did not result in remarkable differences between the mixing behavior of the two tracers. Dirichlet boundary conditions for the transport problem were imposed at the inlet boundary, where the tracers were injected with a constant concentration through a line source corresponding to half the width of the source in the experimental setup (Fig. 1). Zero concentration gradient boundary conditions were assigned at the outlet boundary. The domain was discretized with a fine mesh of triangular elements (692, ,000), with extra refinement in the highvelocity regions of the pore space and close to the plume fringe to enhance the accuracy in the areas of mixing. Multitracer numerical experiments were carried out in a range of groundwater flow velocities ( m/day) similar to those considered in the flow-through experiments (Table 1). The maximum simulated average velocity was limited to 16.5 m/day to avoid an excessive increase of the grid Péclet number. The plots of steady-state concentration distribution for all three of the different grain size models for each of the two tracers, oxygen and fluorescein, illustrate the effects of both geometry and compound-specific diffusivity of a solute on the magnitude of transverse dispersion and thus the amount of mixing (Fig. 3). 4 Data Evaluation For each laboratory and numerical experiment, we evaluated the steady-state vertical concentration profiles measured/observed at the outlet of the flow-through domain using the 2D analytical solution of the transport equation for a line source (adapted from Domenico and Palciauskas 1982): C(x, z) = C 1 + C 0 C 1 er f z + w 2 er f z w 2 (3) where C 0 [M/L 3 ] is the continuously injected tracer concentration, C 1 [M/L 3 ] is the ambient concentration (different from zero only for oxygen in the laboratory experiments), D t x v D t x v

6 352 M. Rolle et al. Table 1 Flow and transport parameters for the Darcy-scale experiments and the pore-scale simulations Parameter Darcy-scale experiment Pore-scale model Domain dimension (m) Grain diameter Small grain size (mm) Intermediate grain size (mm) a Large grain size (mm) Porosity of packed bed Small grain size (%) Intermediate grain size (%) 39.5 a 39.5 Large grain size (%) Range of seepage velocity (m/day) Source width (m) Aqueous diffusion coefficient at 22 C Fluorescein (m 2 /s) Bromide (m 2 /s) Oxygen (m 2 /s) a Data from Chiogna et al. (2010) Fig. 3 Normalized concentration (C/C 0 ) plots of oxygen (a, c, e) and fluorescein (b, d, f) for the large, intermediate and small grains at an average velocity of 1 m/day

7 Experimental Investigation and Pore-Scale Modeling Interpretation 353 w [L] is the source width, D t [L 2 /T] is the hydrodynamic transverse dispersion coefficient, v [L/T] is the average groundwater velocity, and x [L] and z [L] are the longitudinal and transverse coordinates, respectively. This simplified analytical solution was used to evaluate the laboratory data under the assumption of homogeneous system and parallel flow. As done in previous works (e.g., Acharya et al. 2007), the analytical solution, rigorously valid for a continuum description of solute transport, was also applied to the spatially averaged results of the multitracer pore-scale simulations. A trust-region-reflective method for the minimization of non-linear least squares problems was adopted to fit the experimental and pore-scale model results with Eq. 3,usingD t as the fitting parameter. Thus, values of the hydrodynamic transverse dispersion coefficient were estimated at each velocity and for each tracer in both the Darcy-scale laboratory setup and the pore-scale simulations. Hydrodynamic dispersion in porous media depends on the basic transport mechanisms of advection and diffusion. Despite the fact that dispersion results from these two mechanisms acting together (Bear 1972), the dispersion coefficient is typically parameterized as an additive contribution of a velocity-independent pore diffusion term and a diffusion-independent mechanical dispersion term. To interpret our experimental and pore-scale modeling results with the aim of obtaining a dispersion parameterization that correctly reflects the fundamental transport processes occurring at the pore scale, we adopt an empirical formulation of D t (Chiogna et al. 2010), inspired by the statistical model of Bear and Bachmat (1967). This formulation retains an explicit dependence of the mechanical dispersion term on the aqueous diffusion coefficient of the transported tracer and a (possibly) non-linear relationship with the average seepage velocity: ( Pe 2 ) β D t = D p + D aq Pe δ 2 (4) where D p [L 2 /T] is the pore diffusion coefficient given by the ratio of the aqueous diffusion coefficient and the tortuosity (τ) of the porous medium, Pe = vd/d aq is the dimensionless grain Péclet number, δ[ ] is the ratio between the length of a pore channel and its hydraulic radius, and β is an empirical exponent. The concept of tortuosity is important to describe transport in porous media (e.g., Bear 1972; Valdés-Parada et al. 2011). In the pore-scale systems considered in this study, the tortuosity was determined by computing the flux from aqueous diffusion across a unit cell (i.e., REV), J D,l s = n(d aq /τ)dc/dx, and comparing it to the flux for a completely porous cell of the same size, J D,l = D aq dc/dx (Bear 1972, Sect ). For the small, intermediate, and large grain sizes, the tortuosity was 1.59, 1.62, and 1.62, respectively. In the laboratory setups, it was not possible to perform an accurate measurement of the tortuosity of the randomly packed beds and we approximated τ as the inverse of the porosity (Boving and Grathwohl 2001). In order to obtain a general formulation of the hydrodynamic transverse dispersion coefficient for the considered tracers and each different grain size investigated in the laboratory and numerical experiments, Eq. 4 was fitted to the observed pattern of D t values over the range of applied flow velocities. The same fitting procedure described above (i.e., trust-regionreflective method for the minimization of the non-linear least squares problem) was adopted to determine the values of the empirical exponent β and the geometrical parameter δ by minimizing the sum of the relative error squared of the N simulations/experiments: ( N D obs t i D pred ) 2 t i Dt obs (5) i i=1

8 354 M. Rolle et al. The uncertainty in the fitted parameters β and δ were then evaluated using the following nonlinear least squares method in which the estimation error is dependent on the measurement error and on the sensitivity to the parameters: V = s 2 (J T J) 1 (6) where V is the covariance matrix of the estimated parameters, s 2 is the square measurement error, and J is the Jacobian matrix. The elements of the Jacobian are the sensitivity coefficients of the predicted D t to the model parameters: J ij = 1 D obs t i pred Dt i (7) θ j in which θ j is the model parameter β and δ. For the numerical simulations, the measurement error, s 2, can be estimated by the sum of the relative error squared divided by the number of degrees of freedom: ŝ 2 = 1 N 2 ( N D obs t i D pred ) 2 t i Dt obs (8) i i=1 This estimate accounts for both measurement errors and the uncertainty in the analytical model used to determine D t (Eq. 3). For the laboratory experiments, the measurement error is dominant; therefore, it was directly used in Eq. 6 for s 2. A relative error of 15% was determined according to Chiogna et al. (2010). Finally, for both the experimental and numerical results, the standard deviations for the two parameter estimates, V 1,1 and V 2,2, were used to calculate the 95% confidence intervals. 5 Experimental and Modeling Results 5.1 Physical Displacement Bell-shaped vertical concentration profiles were measured/observed at the outlet of both the homogeneously packed flow-through chamber and the 2D pore-scale model domain. Figure 4 illustrates the results for an average grain size d = 1.25 mm at different velocities representative of advection-dominated flow regimes. The plots on the left show normalized concentration profiles observed in the laboratory flow-through system for fluorescein and bromide (measured at the outlet ports) and for oxygen (measured at the third oxygen-sensitive stripe). The plots on the right show the outcomes of the pore-scale simulations carried out continuously injecting fluorescein and oxygen at the inlet boundary of the model domain. The values of the numerically simulated concentrations (symbols) are averaged over a vertical distance of a pore channel. Despite the fact that at the considered flow velocities the mechanical dispersion term is predominant, significant differences between the profiles of the considered tracers can be observed. This behavior reflects the compound-specific diffusive properties of the different tracers. In fact, the compound with a lower aqueous diffusion coefficient (i.e., fluorescein) displays less spreading and higher concentrations along the plume centerline.

9 Experimental Investigation and Pore-Scale Modeling Interpretation 355 Fig. 4 Vertical concentration profiles of the different tracers at the outlet of the flow-through chamber (left panels) and the pore-scale model domain (right panels). Observed (diamonds fluorescein, circles oxygen, crosses bromide) versus fitted values at different advection-dominated flow conditions for an average grain sizes of 1.25 mm. Data in the gray area are extended by symmetry from the values calculated in the pore-scale domain (upper half) 5.2 Darcy-Scale Results The laboratory multitracer experiments were evaluated with the procedure summarized in Sect. 4. The values of the fitting parameters of the non-linear transverse dispersion parameterization (Eq. 4) were determined for each considered grain size: the exponent β varied between 0.47 and 0.50, whereas the obtained values for the geometrical parameter δ were in a range (Table 2). In order to provide a parameterization of D t for the different investigated porous media, we can consider the average of the fitted parameters. Therefore, the transverse dispersion coefficient can be written as: D t vd 1 τ Pe + 1 (9) Pe where D t /vd represents an inverse dynamic Péclet number, depending non-linearly on the (molecular) grain Péclet number. Using such normalized representation, traditional linear parameterizations of D t (de Josselin de Jong 1958; Saffman 1959; Scheidegger 1961) used in contaminant hydrogeology read as:

10 356 M. Rolle et al. Table 2 Fitted parameters for the laboratory and numerical multitracer experiments Porous medium Darcy-scale experiment Pore-scale model β [ ] δ[ ] β [ ] δ [ ] Mean 95% CI Mean 95% CI Mean 95% CI Mean 95% CI Small grain size Intermediate grain size a Large grain size a Experimental data from Chiogna et al. (2010) Fig. 5 Results of laboratory multitracer experiments: data collected in this study (filled symbols) and in a previous study (Chiognaet al. 2010) with identical setup but different grain size (empty symbols) plotted with Eq. 9 (continuous line) and a classical linear parameterization (dash-dot line, Eq.10 with c = 3/16) D t vd 1 τ Pe + c (10) in which c is a dimensionless constant term, equal to 3/16 in the classical statistical models of de Josselin de Jong (1958) andsaffman (1959). A graphical representation of the experimental results and the linear and non-linear parameterizations of D t are shown in Fig. 5. The results of the multitracer flow-through experiments with different grain sizes lie on the non-linear parameterization curve (Eq. 9) with little scattering. Instead Eq. 10, predicting a constant value of the inverse dynamic Péclet number (D t /vd) with increasing Pe does not capture the observed behavior with the exception of the few experimental points corresponding to a clearly diffusion-dominated regime, where the two parameterizations converge. 5.3 Pore-scale Results The results of the numerical multitracer experiments show the interplay between the fundamental transport processes of advection and diffusion that determine the mixing and dilution of the tracers in the pore channels. Figure 6 illustrates the concentration distribution in the pore channels in a portion of the large grain pore-scale model domain (0.35 cm by 0.3 cm, at 3 cm from the inlet) at the fringe of the fluorescein and oxygen plumes. Notice that in a

11 Experimental Investigation and Pore-Scale Modeling Interpretation 357 Fig. 6 Concentration distribution and vertical concentration gradients in a pore channel at x = 3cm, z = 0.6 cm. Values calculated for fluorescein and oxygen at different flow velocities for the large grain size (d = 1.25 mm) diffusion-dominated flow regime (v = 0.1 m/day), mixing in the pore throats is complete as demonstrated by the absence of concentration gradients in the central vertical section of a pore channel. With increasing flow velocities, concentration gradients start to develop. In a transition flow regime (v = 2 m/day), concentration gradients are found for fluorescein, whereas the higher diffusive properties of oxygen counteract the local concentration gradients induced by the solute advection. At higher flow velocities, mixing is incomplete for both tracers which show significant concentration gradients in the pore channels. In order to quantify the amount of mixing, we use the concept of the dilution index (Kitanidis 1994). In particular, we calculate the flux-related dilution index. This property, introduced by Rolle et al. (2009) and further developed by Chiogna et al. (2011a,b)represents the volumetric water flux over which the solute flux is diluted. In an unbounded three-dimensional domain, at the Darcy scale, the flux-related dilution index, E Q (x), can be computed at different locations along the mean flow direction as: E Q (x) = exp + + p Q (x, y, z) ln(p Q (x, y, z)) q x (x, y, z) dydz (11) where q x (x, y, z) is the normal component of the specific discharge with respect to a crosssectional area in the plane y z and p Q (x, y, z) is a flux-related density function: c(x, y, z) p Q (x, y, z) = + + c(x, y, z) q (12) x(x, y, z) dydz p Q (x, y, z) represents the probability that a solute particle, at a specified cross-section at distance x, is within a certain fraction of the water flux. To quantify dilution at the fringe of the tracers plumes in the pore-scale domain used in this study, we consider a cross-section through the center (x mid ) of a single pore channel. In such a bounded domain, a normalized flux-related dilution index (M Q ) can be computed (Kitanidis 1994; Rolle et al. 2009):

12 358 M. Rolle et al. Fig. 7 Mixing in a pore channel: normalized flux-related dilution index calculated for fluorescein and oxygen in the considered velocity range for the different grain sizes M Q (x mid ) = E Q(x mid ) E max Q (x mid) = exp [ amin 0 p Q (x mid, z) ln(p Q (x mid, z)) u x (x mid, z) dz ] Q channel (13) where p Q is the flux-related density function in 2D, a min is the minimum vertical height of a channel (at the longitudinal middle point, x mid ), u x is the horizontal component of the flow velocity computed by numerically solving the Stokes flow equation in the pore-scale domain. E max Q (x mid) is the maximum possible dilution (i.e., the distribution of the solute mass across the channel is uniform) and corresponds to Q channel, which is the total flow rate in a channel. Figure 7 shows an example of the calculated values of the normalized flux-related dilution index for all investigated flow velocities in a selected pore channel (at x = 0.3cm,z = 0.6cm) in the different pore-scale domains. When solute transport is diffusion dominated, mixing in the pore throats is complete and M Q = 1. With increasing velocities, the effect of advection starts to become dominant, leading to incomplete mixing. It can be observed that the degree of dilution depends on the diffusive properties of the solute and the transition between fully and partially mixed conditions takes place at different velocities for fluorescein and oxygen. Moreover, for both solutes, the degree of dilution depends on the porous medium grain size and the effects of incomplete mixing are more significant for the intermediate and large grain sizes. The transition between complete and incomplete mixing is found to start at lower flow velocities (<1 m/day) for the larger grain size. Analogous to the laboratory Darcy-scale experiments, the numerical pore-scale experiments also were evaluated following the procedure described in Sect. 4. The non-linear transverse dispersion parameterization (Eq. 4)wasfittedtotheD t values obtained from multitracer pore-scale simulations carried out at different velocities. The fitted parameters β and δ are reported in Table 2. The fitted values of the exponent β were found to be similar to the ones obtained in the laboratory experiments. These values, averaging at 0.51, express the non-linear increase of the solute spread in the flow-through domain. In a direct comparison of the single grain sizes between Darcy- and pore-scale experiments, the most noticeable difference was found for the small grain size for which a higher value of β resulted from the simulations, thus pointing to a lesser extent of incomplete mixing in the modeling setup. The geometrical factor δ determined in the pore-scale simulations was found to be smaller than the one obtained from the laboratory experiments. These differences can be attributed to the

13 Experimental Investigation and Pore-Scale Modeling Interpretation 359 Fig. 8 Transverse dispersion coefficients as a function of flow velocity for d = 0.625mm. Symbols results of the pore-scale simulations and of the laboratory experiments for fluorescein (diamonds) and oxygen (circles). Lines Eq. 4 with best-fit parameters reported in Table 2 for the pore-scale modeling and the Darcyscale experiments, respectively different geometric characteristics of the pore channels in the quasi 2D laboratory setup and in the idealized 2D pore-scale domain. In fact, in the laboratory setup, δ is an average value taking into account the different geometrical and topological characteristics of the 3D pore space in the randomly packed porous medium, whereas in the periodic pore-scale setup δ is the same for all 2D channels, which are identical throughout the flow-through domain. Figure 8 shows the results of the pore-scale simulations and the non-linear parameterization of transverse dispersion for the grain size d 50 = mm (with the fitted β and δ reportedintable2). The observed D t values as a function of the flow velocity show a trend similar to the one observed in the Darcy-scale experiments, with a clear compounddependency and a non-linear increase in the advection-dominated flow regime. 6 Discussion and Conclusions The laboratory and the numerical multitracer experiments show a compound-dependence of the transverse dispersion coefficient over a wide range of flow velocities. The results of both setups show a similar pattern that can be divided into three zones: low flow velocities: in this zone, diffusion is the dominant process and a clear difference in lateral displacement of distinct tracers can be noticed; intermediate flow velocities: advection becomes progressively more important and determines the transition between a diffusion- and an advection-dominated regime. A minimum of the effect of the compound-specific diffusive properties on lateral displacement is observed in this velocity range; high flow velocities: in this advection-dominated regime, the tracers aqueous diffusion coefficient plays an important role since it limits the lateral diffusion of mass between adjacent streamlines at the pore scale, and, therefore, influences the mechanical dispersion term at the Darcy scale. The distinction between different zones to characterize dispersive processes is traditionally done on the basis of the grain Péclet number (e.g., Bear 1972; Delgado and Carvalho 2001). However, the transport of compounds with different diffusive properties in the same porous medium (e.g., the conservative multitracer experiment of this study or the reactive transport of an organic contaminant and a soluble electron acceptor) is characterized by a given value

14 360 M. Rolle et al. of seepage velocity to which different Pe correspond. As illustrated in Fig. 6, this compoundspecific difference influences the degree of mixing of the solutes in the pore channels. A non-linear compound-specific parameterization of transverse dispersion in which the mechanical dispersion term retains a dependence upon the tracers aqueous diffusivities is necessary to capture the behavior observed in the three velocity ranges described above (Figs. 5, 8). The results of our multitracer experiments and, in particular, the non-linear relationship with the average flow velocities are in line with the findings of previous studies (e.g., Delgado and Carvalho 2001; Klenk and Grathwohl 2002; Olsson and Grathwohl 2007; Chiogna et al. 2010). The pore-scale simulations were performed in 2D porous media using regular arrays of circles. This certainly represents a simplification of the unknown 3D geometrical structure of the randomly packed beds used in the laboratory experiments. Nonetheless, the outcomes of the modeling investigation allowed us to point out that the interplay of the basic transport processes of advection and diffusion at the subcontinuum scale determines the macroscopic behavior observed in the experiments. In fact, the relative magnitude of these processes determines the presence or absence of transverse concentration gradients in a pore channel and, therefore, the degree of mixing (Fig. 6). At low flow velocities, mixing in the pore channels is complete whereas increasing velocities result in an insufficient amount of time for diffusion to counteract the gradients induced by advection, thus resulting in incomplete mixing. This incomplete mixing and resulting concentration gradients within each pore space, which are dependent upon the specific compound aqueous diffusivity, result in compound-dependent local transverse (mechanical) dispersion even at high velocities at the macroscopic scale. The transition between complete and incomplete mixing depends not only on the flow velocities but also on the geometrical characteristics of the porous medium (e.g., grain size) and on the considered solute (Fig. 7). The interpretation of the multitracer transport experiments at the pore-scale strongly supports the hypothesis that incomplete mixing in pore throats results in non-linear and compound-dependent D t and shows that processes at the microscopic level significantly contribute to the observed macroscopic behavior. Therefore, up-scaling procedures and macroscopic parameterizations necessary to describe solute transport at larger scales should attempt to properly capture these processes. In particular, for transverse hydrodynamic dispersion, which was the focus of this study, parameterizations such as the one proposed in Eq. 8 are more appropriate to describe lateral displacement in porous media than linear relationships traditionally used in contaminant hydrology (Scheidegger 1961). Therefore, we suggest that such descriptions should be adopted when the objective is to accurately describe conservative and mixing-controlled reactive transport in porous media. Acknowledgments The authors thank Dr. Christina Eberhardt and Huixiao Wang for their help in the experimental work. The support from the DFG (Deutsche Forschungsgemeinschaft) Research Group FOR 525 Analysis and modeling of diffusion/dispersion-limited reactions in porous media (grants GR971/18-1, 18-3) and NSF grant EAR Nonequilibrium Transport and Transport-Controlled Reactions is gratefully acknowledged. Student funding from Government support and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a is also gratefully acknowledged. M.R. acknowledges the support of the Marie Curie International Outgoing Fellowship (DILREACT project) within the 7th European Community Framework Programme. References Acharya, R.C., Valocchi, A.J., Werth, C.J., Willingham, T.W.: Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media. Water Resour. Res. 43, W10435 (2007). doi: /2007wr005969

15 Experimental Investigation and Pore-Scale Modeling Interpretation 361 Bauer, R.D., Rolle, M., Bauer, S., Eberhardt, C., Grathwohl, P., Kolditz, O., Meckenstock, R.U., Griebler, C.: Enhanced biodegradation by hydraulic heterogeneities in petroleum hydrocarbon plumes. J. Contam. Hydrol. 105, (2009) Bauer, R.D., Rolle, M., Kürzinger, P., Grathwohl, P., Meckenstock, R., Griebler, C.: Two-dimensional flow-through microcosms: versatile test systems to study biodegradation processes in porous aquifers. J. Hydrol. 369, (2009) Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1972) Bear, J., Bachmat, Y.: A generalized theory on hydrodynamic dispersion in porous media. In: IASH Symposium on Artificial Recharge and Management of Aquifers, Haifa, Israel, vol. 72, pp (1967) Bijeljic, B., Blunt, M.J.: Pore-scale modelling of transverse dispersion in porous media. Water Resour. Res. 43, W12S11 (2007). doi: /2006wr Boving, T., Grathwohl, P.: Tracer diffusion coefficients in sedimentary rocks correlation to porosity and hydraulic conductivity. J. Contam. Hydrol. 53, (2001) Cao, J., Kitanidis, P.K.: Pore-scale dilution of conservative solutes: an example. Water Resour. Res. 34, (1998) Chiogna, G., Eberhardt, C., Grathwohl, P., Cirpka, O.A., Rolle, M.: Evidence of compound dependent hydrodynamic and mechanical transverse dispersion by multi-tracer laboratory experiments. Environ. Sci. Technol. 44, (2010) Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Transverse mixing of conservative and reactive tracers in porous media: quantification through the concepts of flux-related and critical dilution indices. Water Resour. Res. 47, W02505 (2011). doi: /2010wr Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Relevance of local compound-specific transverse dispersion for conservative and reactive mixing in heterogeneous porous media. Water Resour. Res. 47, W07540 (2011). doi: /2010wr Chu, M.P., Kitanidis, P.K., McCarty, P.L.: Modeling microbial reactions at the plume fringe subject to transverse mixing in porous media: when can the rates of microbial reaction be assumed to be instantaneous?. Water Resour. Res. 41, W06002 (2005). doi: /2005wr Cirpka, O.A., Frind, E.O., Helmig, R.: Numerical simulation of biodegradation controlled by transverse mixing. J. Contam. Hydrol. 40, (1999) de Josselin de Jong, G.: Longitudinal and transverse diffusion in granular deposits. Eos Trans. AGU 39, (1958) Delgado, J.M.P.Q., Carvalho, J.F.R.: Lateral dispersion in liquid flow through packed beds at Pe< Transp. Porous Med. 44, (2001) Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reactions in heterogeneous media: a brief review. J. Contam. Hydrol , 1 17 (2011) Domenico, P.A., Palciauskas, V.V.: Alternative boundaries in solid waste management. Ground Water 20, (1982) Gaganis, P., Skouras, E.D., Theodoropoulu, M.A., Tsakiroglou, C.D., Burganos, V.N.: On the evaluation of dispersion coefficients from visualization experiments in artificial porous media. J. Hydrol. 307, (2005) Haberer, C.M., Rolle, M., Liu, S., Cirpka, O.A., Grathwohl, P.: A high-resolution non-invasive approach to quantify oxygen transport across the capillary fringe and within the underlying groundwater. J. Contam. Hydrol. 122, (2011) Huang, W.E., Oswald, S.E., Lerner, D.N., Smith, C.C., Zheng, C.: Dissolved oxygen imaging in a porous medium to investigate biodegradation in a plume with limited electron acceptor supply. Environ. Sci. Technol. 37, (2003) Kitanidis, P.K.: The concept of dilution index. Water Resour. Res. 30, (1994) Klenk, I.D., Grathwohl, P.: Transverse vertical dispersion in groundwater and the capillary fringe. J. Contam. Hydrol. 58, (2002) Knutson, C., Valocchi, A., Werth, C.: Comparison of continuum and pore-scale models of nutrient biodegradation under transverse mixing conditions. Adv. Water Resour. 30, (2007) Olsson, A., Grathwohl, P.: Transverse dispersion of non reactive tracers in porous media: a new nonlinear relationship to predict dispersion coefficients. J. Contam. Hydrol. 92, (2007) Porter, M.L., Valdés-Parada, F.J., Wood, B.W.: Comparison of theory and experiments for dispersion in homogeneous porous media. Adv. Water Resour. 33, (2010) Prommer, H., Anneser, B., Rolle, M., Einsiedl, F., Griebler, C.: Biogeochemical and isotopic gradients in a BTEX/PAH contaminant plume: model-based interpretation of a high-resolution field data set. Environ. Sci. Technol. 43, (2009)

16 362 M. Rolle et al. Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O.A., Grathwohl, P.: Enhancement of dilution and transverse reactive mixing in porous media: experiments and model-based interpretation. J. Contam. Hydrol. 110, (2009) Rolle, M., Chiogna, G., Bauer, R., Griebler, C., Grathwohl, P.: Isotopic fractionation by transverse dispersion: flow-through microcosms and reactive transport modeling study. Environ. Sci. Technol. 44, (2010) Saffman, P.G.: A theory of dispersion in porous media. J. Fluid Mech. 6, (1959) Scheidegger, A.E.: General theory of dispersion in porous media. J. Geophys. Res. 66, (1961) Tartakovsky, A.M., Redden, G.D., Lichtner, P.C., Scheibe, T.C., Meakin, P.: Mixing-induced precipitation: experimental study and multi-scale numerical analysis. Water Resour. Res. 44, W06S04 (2008). doi: /2006WR Tartakovsky, A.M., Tartakovsky, G.D., Scheibe, T.D.: Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32, (2009) Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Med. 88, 1 30 (2011) Willingham, T.W., Werth, C.J., Valocchi, A.J.: Evaluation of the effects of porous media structure on mixing-controlled reactions using pore-scale modeling and micromodel experiments. Environ. Sci. Technol. 42, (2008) Worch, E.: A new equation for the calculation of diffusion coefficients for dissolved substances. Vom Wasser 81, (1993) Zhang, C., Dehoff, K., Hess, N., Oostrom, M., Wietsma, T.W., Valocchi, A.J., Fouke, B.W., Werth, C.: Porescale study of transverse mixing induced CaCO3 precipitation and permeability reduction in a model subsurface sedimentary system. Environ. Sci. Technol. 44, (2010)