Computational modelling of reactive transport in hydrogeological systems

Water Resources Management III 239 Computational modelling of reactive transport in hydrogeological systems N. J. Kiani, M. K. Patel & C.-H. Lai School of Computing and Mathematical Sciences, University of Greenwich, U.K. Abstract This study presents numerical simulations of solute transport through homogeneous and layered heterogeneous porous media taking the effects of solute partitioning and particle deposition processes into account. The model is based around PHYSICA s multi-physics simulation environment and utilizes finite volume discretisation techniques to investigate the attenuation of reactive contaminants subject to equilibrium and kinetic adsorption isotherms. A number of test cases have been conducted for simulating one and two-dimensional advective-dispersive transport in homogeneous and layered soils. The accuracy of the model has been examined by means of analytical/numerical solution comparisons available in the literature. The model predicts that the shape and spread of contaminant plume can be strongly influenced by the governing adsorption mechanism for solute migration. Breakthrough curves of solute transport subject to nonlinear adsorption (advection-dominant problems) exhibit concentration shocks and sharp fronts with a retarded plume velocity as opposed to the linear adsorption. The contaminant concentration in layered physical and geochemically heterogeneous systems is observed to be affected by contrasting physical/geochemical properties and velocity variations across the layers. Keywords: adsorption, finite volume, layered soils, nonlinear isotherm, colloid transport, geochemically heterogeneous. 1 Introduction Adsorption is one of the mass transfer processes that control the attenuation of organic/inorganic compounds in hydrogeological systems. It involves the

240 Water Resources Management III partitioning of dissolved contaminants from aqueous phase (groundwater) onto the solid phase (aquifer surfaces) thereby retarding the entire process of contaminant migration. Adsorption processes may be classified either as fast/instantaneous or equilibrium adsorption where the chemical reactions are reversible and locally in thermodynamic equilibrium or slow/kinetic or nonequilibrium adsorption where the reactions are irreversible and require a kinetic rate law to determine the reaction rate. The use of mathematical models to describe the reactive transport of solutes in hydrogeological systems is quite common and has been studied theoretically in the literature. Many theories based around analytical or finite difference/finite element methods have been proposed to predict the solute transport in porous media coupled with competitive adsorption mechanisms (Lapidus and Admundson [1], Istok [2], Manoranjan and Stauffer [3], Wu et. al [4], Sheng and Smith [5]). The purpose of the current study is to provide a brief description of transport of reactive contaminants subject to equilibrium and kinetic adsorption isotherms by employing finite volume (FV) discretisation techniques. The choice of FV method is based on its conservation principle of underlying physical properties together with the ease with which complex domains can be represented. 2 Adsorption model The governing reactive transport equation for a homogeneous saturated medium taking the effects of advection, dispersion and adsorption into account can be represented as C ρ S b =. ( D C).( v C) + (1) η where C = concentration of solute [mg/l], S = adsorbed solute concentration [mg/kg], D = dispersion coefficient [m 2 /s], v (= q/η) = pore water velocity [m/s], q = Darcy flux [m/s], ρ b = bulk density of the medium [kg/m 3 ] and η = porosity of medium [dimensionless]. The dispersion tensor D is defined as vv D ( ) i j ij = αt v δij + αl αt + Ddτδij (2) v where α L = longitudinal dispersivity [m], α T = transverse dispersivity [m], D d = molecular diffusion coefficient [m 2 /s], δ ij = Kronecker delta and τ = tortuosity of porous medium. 2.1 Instantaneous (fast) or equilibrium adsorption An equilibrium adsorption reaction is fast in relation to contaminant transport processes at groundwater velocity thus resulting in an instantaneous mass transfer in the porous medium. Most transport models use three types of instantaneous adsorption isotherms: linear, Langmuir and Freundlich.

Water Resources Management III 241 2.1.1 Linear isotherm Linear isotherm is the simplest adsorption isotherm. It describes the linear relationship between S and C and is valid for dissolved species present at low concentrations. It can be expressed mathematically as S = KdC (3) where K d = distribution coefficient which describes the partitioning between liquid and solids for the linear isotherm. A dimensionless retardation factor R which takes into account the effect of sorption on contaminants transport by retarding the actual contaminant velocity in groundwater system can be defined by using eqn (3) in (1) as ρbkd R = 1+ (4) η 2.1.2 Langmuir isotherm Langmuir isotherm assumes that there are a finite number of adsorption sites available on the solid surface. At low solute concentration adsorbed concentration increases linearly with increase in solute concentration whereas it becomes constant at higher concentrations. The equation describing singlecomponent Langmuir isotherm is given by K1KC 2 S = (5) 1 + K1C where K 1 = constant which is the measure of the adsorbate bond strength [m 3 /g] and K 2 = maximum adsorption capacity. 2.1.3 Freundlich isotherm Freundlich isotherm describes a nonlinear relationship between the adsorbed and the solute concentrations. It assumes that the number of available adsorption sites is unlimited. It is expressed as n S = K f C (6) where K ƒ = coefficient of Freundlich isotherm [m 3n /g n ] and n = power law coefficient for the adsorption isotherm. The value of n varies between 0.4 (heavy metals) to near 1.0 (organic solutes). In case n = 1, Freundlich isotherm reduces to linear isotherm. 2.2 Kinetic (slow) or non-equilibrium adsorption For cases where the equilibrium condition is not satisfied kinetic or slow adsorption models are incorporated more commonly. 2.2.1 First-order kinetically-controlled adsorption The rate of adsorption for a first-order kinetically-controlled model can be described as

242 Water Resources Management III S = kc 1 k2s (7) where k 1 = finite rate constant, sorption [m/h] and k 2 = finite rate constant, desorption [h -1 ]. 2.2.2 Colloid transport in geochemically heterogeneous medium Micro-organisms such as viruses and bacteria with a range in size from 0.001 to 1 micron are termed colloids. Colloids are ubiquitous in subsurface systems and can play a crucial role in transport of strongly sorbing solute species, heavy metals and pesticides. The general advection-dispersion equation describing transfer of colloidal particles from liquid suspension onto stationary surfaces through colloid deposition and release (Sun et al. [6]) can be described as N f θ =. ( D N). ( v N) (8) 2 π ap where N = colloid number concentration [m -3 ], θ = fractional surface coverage of deposited colloids, ƒ = specific surface area [m 2 /m 3 ] and a p = colloid particle radius [µm]. The particle surface coverage rate of the porous medium for patchwise geochemically heterogeneous model is given by (Johnson et al. [7]) θ θ f θu = λ + ( 1 λ) (9) where λ and (1-λ) represent the favourable and unfavourable representative elementary volume surface fractions respectively. The favourable and unfavourable surface coverage rates while taking the dynamic aspects of colloid deposition and first-order kinetic release into account can be represented as θ f 2 = π apkdep, f B( θ f) N kdet, fθ f (10) θu 2 = π a k B θ N k θ ( ) p dep, u u det, f f where k dep = colloid deposition rate constant [m/s] and k det = colloid release rate constant [s -1 ]. The dynamic blocking function B characterizes the probability of colloid deposition by quantifying the fraction of collector surface still available for deposition of colloids (N. Sun et al [6]). Two types of dynamic blocking functions are generally recognized. The Langmuirian blocking function derived from the molecular adsorption model of Langmuir is presented as B ( θ ) s θs = 1 (11) θ where subscript s denotes favourable (f) or unfavourable (u) surface fractions and θ max = maximum attainable surface coverage. According to recent research studies, the dynamics of particle deposition in hydrogeological systems can be better described by the random sequential adsorption (RSA) dynamic blocking function expressed as max

Water Resources Management III 243 2 3 s s s B( s ) 1 a θ 1 a θ θ 2 a θ = + + 3 (12) θmax θmax θmax where the coefficients a 1, a 2 and a 3 are employed as those given by Johnson and Elimelech [8]. 3 Numerical simulation method The use of computational fluid dynamics (CFD) techniques is increasingly becoming popular in predicting fluid flow and transport problems in general. The technique is very powerful and spans a variety of industrial and non-industrial applications. The FV method serves as the numerical algorithm around which most CFD codes are constructed. The transport phenomenon which describes the conservation of a general flow variable φ within a finite control volume is expressed as a balance between various processes and is represented in mathematical form by Ct + div ( Ccvφ) = div ( Γ φgradφ) + Sφ (13) Transient Convection Diffusion Source where C t and C c are the coefficients of transient and convection terms respectively and Γ φ = diffusion coefficient. 4 Model verification and examples The adsorption isotherms described above have been implemented within PHYSICA [9] and validated with analytical solutions and numerical results available in the literature. The influence of different processes on the transport of contaminants through homogeneous and layered heterogeneous porous media is illustrated by means of examples described below. 4.1 Example 1 A set of cases is considered in this example by accounting for linear and Langmuir isotherms given by eqns (3) and (5) respectively. The material properties for theses cases are shown in table 1. The initial and boundary conditions are given as follows Case I & II: Cx (,0) =0,0 x 10 C( x,0) =0,0 x 10 C( 0, t) = 1 ; Case III: C( 0, t) = 1 C,0 < t tmax,0< t t = 0 C( x= 10, t) = 0 D x x= 10 max

244 Water Resources Management III Table 1: Material properties for example 1. Case η ρ b v D Transport type I 0.5 1.5 1.157 10-5 0.0 Pure advection II 0.5 1.5 1.157 10-5 1.157 10-6 Advection-diffusion III 0.5 1.5 0.0 1.157 10-6 Pure diffusion For case I the simulated concentration in Physica for three differencing schemes is compared with the analytical solution presented by Sheng and Smith [5]. The effects of numerical dissipation and spurious oscillations are observed to be suppressed by the use of TVD schemes (Van Leer). A retardation in solute migration under the influence of the equilibrium adsorption isotherm being considered is observed as shown in figure 1b.The concentration profiles for advection-dispersion and pure dispersion cases are compared with the reference solution presented by Sheng and Smith [5] in figure (1 c and d) respectively. (a) (b) (c) (d) Figure 1: Concentration profiles at t = 2 days. (a) case I without adsorption, (b) case I with adsorption, (c) case II, (d) case III.

Water Resources Management III 245 4.2 Example 2 Nonlinear adsorptive transport through a homogeneous medium (case I) and a layered heterogeneous medium (case II) using Freundlich isotherm is presented in this example. Case I considers a soil column 100 cm in depth subjected to a steady infiltration of 0.2 cm/day. The material properties are given as: 3 3 ρ 1.5 /, 0.45, 1.5, 1 ( / )( / ) 0.5 b = g cm η = n = K f = cm g L mg, α = 0.01 cm, Dd = 0 The initial and boundary conditions are: 1, t Tp C( z, 0) = 0, 0 < z 100; C( 0, t) = ; C( 100, t) = 0 0, t > Tp where T p = 10 days is the time period during which a constant concentration source is applied at the surface boundary. Shown in figure 2 is the comparison between Physica s simulation results and analytical and numerical solutions presented by Wu et al. [4] at t = 10, 50 and 100 days. Case II involves onedimensional solute transport through a layered heterogeneous medium. Two model layers are considered: first being 100 cm in depth and the second being semi-infinite. The medium is subjected to a steady infiltration of 0.4 cm/day with material properties for the two layers given as 3 3 ρ = 1.5 g / cm, η = 0.4, n = 1.5, K = 0.64 cm / g l/ mg Layer 1: ( )( ) 0.5 b 3 3 Layer 2: ρ = 1.6 g / cm, η = 0.25, n = 5, K = 0.25 ( cm / g)( l / mg) 4 b The initial and boundary conditions for case II are: 10, t Tp C( z, 0) = 0; C( 0, t) = ; C( z = outflow boundary,t) = 0 qt, > Tp The comparison results for simulated concentration profiles for case II are shown in figure 3. An abrupt change in the slope of concentration is observed at the interface of two layers which is due to contrasting material properties and the velocity variation across the interface. f f Figure 2: Concentration profiles for case I at three different times.

246 Water Resources Management III Figure 3: Concentration profiles for case II at three different times. Figure 4: Concentration profiles comparison at 0.25, 0.5, 1, 2 and 4 days. 4.3 Example 3 This case computes the solution of the one-dimensional pure diffusion problem coupled with eqn (7) by taking the following material properties into account: 2 3 1 D = 0.5 m / d, v = 0.0, η = 1.0, ρb = 1.0 kg / m, k1 = 0.1 m/ d, k2 = 0.5d The initial and boundary conditions chosen are: C( 0, t) = 1 C( x,0) = 0,0 x 2;,0< t tmax C( 2, t) = 0 ( ) ( ) S x,0 = 0; S 0, t = 0 Shown in figure 4 are comparison results of Physica s simulation with the analytical solution presented by McGrail [10]. The two results for kinetically controlled linear adsorption are in good agreement.

Water Resources Management III 247 4.4 Example 4 2-dimensional colloid transport in geochemically heterogeneous media is considered in this example. The transport model is validated with the physically homogeneous and layered heterogeneous test cases presented by Sun et al. [6]. The physically homogeneous case involves a rectangular domain 3 m in length along the x-axis and 1 m thick along the z-axis. The basic model parameters, listed in table (2) are close representative of colloid transport in sandy aquifer. The initial and boundary conditions for the colloid concentration are: N x, z, t = 0 = 0; θ = θ = 0, ( ) f 14 ( = 0,, ) = 2.8 10 N x z t N D = 0 x x= 3 u,0< t T where T p = 0.5 days. Figure (5) shows the comparison results of Physica simulations with the numerical solution presented by N. Sun et al. [6]. As can be seen in Figure (5a), an increase in geochemical heterogeneity increases colloid deposition rate but decreases colloid concentration in the bulk solution. It is revealed from Figure (5b) that at a constant hydraulic head gradient, an increase in hydraulic conductivity increases colloid concentration spreading which is due to an increase in colloid advection velocity. The layered heterogeneous case involves the distribution of the model domain into three horizontal layers along the x-axis. Layer I (0-0.3 m) and III (0.7-1.0 m) are assigned the same material properties and parametric values. A different set of parameter values is assigned to layer II (0.3-0.7 m) in order to incorporate the heterogeneity. The boundary conditions are changed to as: 14 N N( x = 0, z, t) = 2.8 10 ; D = 0 x = x 3 p (a) (b) Figure 5: Effect of (a) geochemical heterogeneity, (b) hydraulic conductivity values variation on colloid transport for a period of 0.75 days.

248 Water Resources Management III Figure 6: Concentration contours showing the effect of (a) physical layered heterogeneity, (b) geochemical layered heterogeneity on colloid transport at an observation time of 0.75 day. Table 2: Parameters and basic values for example 5. Source: Sun et al. [6]. Parameter Basic value Hydraulic gradient, h 0.01 Hydraulic conductivity, K (m/day) 100 Longitudinal dispersivity, α L (m) 0.05 Dispersivity ratio, R = α L /α T 5:1 Porosity, ε 0.4 Specific surface area, ƒ(m 2 /m 3 ) 3 10 4 Particle radius, a p (µm) 0.15 Patchwise heterogeneity parameter, λ 0.001 Unfavourable particle deposition rate, k dep,u (m/day) 6.5 10-6 Favourable particle deposition rate, k dep, ƒ (m/day) 6.5 10-3 Detachment rate from unfavourable surface, k det,u (h -1 ) 0.0 Detachment rate for favourable surface, k det, ƒ (h -1 ) 0.0 Maximum attainable surface coverage, θ max 0.2 Figure 6 illustrates the effect of layered physical heterogeneity on colloid transport under different conditions. (a) represents the case with contrasting hydraulic conductivity values. Layer I and III have the conductivity of 50 m/day and the layer II is assigned a conductivity value of 100 m/day. A higher conductivity value causes faster flow through the central layer and faster accompanying colloid transport. In (b) layer I and III are assigned a geochemical heterogeneity value of 0.025 and layer II with a much smaller value of 0.001 in a physically homogeneous medium with conductivity value of 100 m/day throughout the domain. The results show that the increased deposition rate of colloid particles onto the favourable surface fractions of layer I and III can result in preferential flow of colloid particles through the central layer. 5 Conclusions A CFD model based on finite volume discretisation techniques has been presented in this study. The performance of the model as a reliable tool has been

Water Resources Management III 249 validated by means of several test case comparisons. Issues relating to accuracy of predictions in advection-dominated and diffusion problems have been addressed and simulations have been performed to establish more accurate and stable differencing scheme for concentration distributions. The effects of medium properties have been determined on colloid transport through geochemically heterogeneous systems. The spread and shape of particle breakthrough curves are observed to be strongly influenced by alteration in conductivity and elementary volume surface fractions. In layered cases, the system heterogeneities can result in preferential colloid particle transport. Owing to different transport processes, the extent to which the migration of dissolved solute species entering the subsurface flow systems is retarded can be investigated using adsorption isotherms. References [1] Lapidus, L. & Amundson, N.R., Mathematics of adsorption in Beds. VI. The effect of longitudinal diffusion in ion exchange and chromatographic columns. J. Phys. Chem., 56, pp. 984-988, 1952. [2] Istok, J., Groundwater modeling by the finite element method. Water Resour. Monogr., AGU, Washington, D. C., 13, pp. 495, 1989. [3] Manoranjan, V. S. & Stauffer, T. B., Exact solution for contaminant transport with kinetic Langmuir sorption. Water Resour. Res., 32, pp. 749-752, 1996. [4] Wu, Y. S., Kool, J. B., Huyakorn, P. S. & Saleem, Z. A., An analytical model for nonlinear adsorptive transport through layered soils. Water Resour. Res., 33, pp. 21-29, 1997. [5] Sheng, D. & Smith, D. W., Numerical modelling of competitive components transport with nonlinear adsorption. Int. J. Numer. Anal. Meth. Geomech., 24, pp. 47-71, 2000. [6] Sun, N., Elimelech, M., Sun, N. Z. & Ryan, J. N., A novel twodimensional model for colloid transport in physically and geochemically heterogeneous porous media. J. Contam. Hydrol., 49, pp. 173-199, 2001. [7] Johnson, P. R., Sun, N. & Elimelech, M., Colloid Transport in geochemically heterogeneous porous media: modeling and measurements. Environ. Sci. Technol., 30, pp. 3284-3293, 1996. [8] Johnson, P. R. & Elimelech, M., Dynamics of colloid deposition in porous media: blocking based on random sequential adsorption. Langmuir, 11, pp. 801-812, 1995. [9] PHYSICA+, http://www.multi-physics.com [10] McGrail, B. P., Inverse reactive transport simulator (INVERTS): an inverse model for contaminant transport with nonlinear adsorption and source terms. Env. Model. Software, 16, pp. 711-723, 2001.