2D modelling of clogging in landfill leachate collection systems

Transcription

1 1393 2D modelling of clogging in landfill leachate collection systems A.J. Cooke and R. Kerry Rowe Abstract: A 2D model for predicting clogging of a landfill leachate collection system and subsequent leachate surface position (mounding) is described. A transient finite element fluid flow model is combined with a reactive, multiple-species finite element transport model. The transport model considers biological growth and biodegradation, precipitation, and particle attachment and detachment. It uses a geometrical relationship to establish porosity from the computed thickness of the accumulated clog matter and a relationship between the porosity and hydraulic conductivity of elements in the system. The model represents the flow path within the drainage layer in profile. An iterative method is used to solve for the new hydraulic heads, surface and internal nodal positions, and redistributed clog properties (clog quantity, porosity, hydraulic conductivity) for each element and for each time step. The porosity (and consequently hydraulic conductivity) of the media can therefore change spatially and temporally. The mesh is regenerated automatically each time step (including the addition or subtraction of nodes) taking into account allowable element aspect ratios, the interfaces between differing hydrostratigraphic layers, and static point sources and openings. An integrated alternate solution for very thin mounds is included. The application of the model is demonstrated using a hypothetical field case. Key words: clogging, landfills, leachate collection systems, modelling, biofilms, mineral precipitation. Résumé : On décrit un modèle à deux dimensions pour prédire le colmatage de systèmes de collecte du lixiviat d un enfouissement sanitaire et la position subséquente du lixiviat en surface (formation de monticules). On a combiné un écoulement de fluide transitoire en éléments finis avec un modèle réactif de transport de multiples espèces en éléments finis. Le modèle de transport prend en considération la croissance biologique et la biodégradation, la précipitation, et l attachement et le détachement des particules. Il utilise une relation géométrique pour établir la porosité en partant de l épaisseur calculée de la matière colmatée accumulée, et une relation entre la porosité et la conductivité hydraulique des éléments du système. Le modèle représente le cheminement de l écoulement à l intérieur de la couche de drainage dans le profil. On utilise une méthode d itération pour résoudre pour les nouvelles charges hydrauliques, surface et positions nodales internes, et les propriétés de colmatage redistribuées (quantité de colmatage, porosité, conductivité hydraulique) pour chaque élément, et chaque étape dans le temps. La porosité (et conséquemment la conductivité hydraulique) des média peut en conséquence changer dans l espace et dans le temps. Le grillage est régénéré automatiquement à chaque étape dans le temps (incluant l addition la ou la soustraction de noeuds) prenant en compte les rapports d éléments d aspect admissibles, les interfaces entre différentes couches hydrostratigraphiques, et le point statique des sources et des ouvertures. On inclut une solution intégrée alternée pour chacun des monticules très minces. On démontre l application du modèle au moyen d un cas hypothétique sur le terrain. Mots-clés :colmatage, sites d enfouissements, systèmes de collecte de lixiviat, modélisation, biofilms, précipitation minérale. [Traduit par la Rédaction] Introduction The leakage through landfill barrier systems has received considerable interest with most of the focus having been on the performance of the composite liner (Rowe and Brachman 2004; Rowe 2005; Barroso et al. 2006; Dickinson and Brachman 2006; Touze Foltz et al. 2006; Rowe et al. 2007; Received 9 September Accepted 30 May Published on the NRC Research Press Web site at cgj.nrc.ca on 26 September A.J. Cooke. Gartner Lee Limited, 300 Town Centre Boulevard, Suite 300 Markham, ON L3R 5Z6, Canada. R.K. Rowe. 1 GeoEngineering Centre at Queen s RMC, Department of Civil Engineering, Queen s University, Kingston, ON K7L 3N6, Canada. 1 Corresponding author ( kerry@civil.queensu.ca). Saidi et al. 2008). However, the key driver for leakage through these liners is the leachate head on the liner. Thus the primary function of the leachate collection system (LCS) in modern municipal solid waste landfill is to control the leachate head on the underlying liner, thereby minimizing advective flow (leakage) through the liner to the environment. This is commonly achieved using a continuous high permeability drainage layer (e.g., sand, gravel) in conjunction with regularly spaced pipes (e.g., Fig. 1) that lead to sumps from which the leachate is removed for treatment. The leachate mound in the LCS is a function of the rate of infiltration, pipe spacing, bottom slope, and the hydraulic conductivity of the drainage layer. In some countries the maximum design head is stipulated as a design criterion (typically it is not to exceed 0.3 m except for very short periods). For some time it was common design practice to use Moore s Equations (Moore 1980, Can. Geotech. J. 45: (2008) doi: /t08-062

2 1394 Can. Geotech. J. Vol. 45, 2008 Fig. 1. Profile of typical landfill drainage system with hypothetical mound surface. 1983; USEPA 1989) to estimate the maximum depth of leachate. However, with time, the validity of the Moore equations was questioned, and since no derivation was published, it has became more common to use Giroud s equation (J.P. Giroud, Height of leachate in a drainage layer, unpublished project notes 1985) and the subsequent improved version by Giroud et al. (1992), or the equations of McEnroe (1989, 1993). These solutions are analytical and are based on a number of simplifying assumptions, most significantly: (i) flow is essentially horizontal (thus permitting use of the Dupuit approximation), (ii) hydraulic conductivity is constant and homogeneous, and (iii) infiltration is constant and uniform. Of these solutions, only McEnroe (1989) predicted the entire steady-state profile of the phreatic surface. More rigorous methods (intended for unconfined flow in general) often employ finite element routines wherein the mound surface is a boundary whose position is initially unknown and must be obtained iteratively. This type of analysis was pioneered by Taylor and Brown (1967) for steady-state conditions and by Neuman and Witherspoon (1971) for nonsteady flow through homogeneous soils. More recently, Crowe et al. (1999) presented a variable mesh technique that allowed for the surface to rise and fall within heterogeneous soils. Of the factors responsible for leachate depth in the LCS, it is the hydraulic conductivity that is the hardest to control or predict after landfill operation has commenced. Field observations (see Bass 1986; Brune et al. 1991; McBean et al. 1993; Rowe et al. 1995; Rowe 1998; Craven et al. 1999; Fleming et al. 1999; Maliva et al. 2000; Rowe et al. 2004) have concluded that the transmission of leachate through the drainage layer leads to the build-up of clog material and a reduction in hydraulic conductivity. Prediction of clog build-up (clogging) within the drainage layer and the subsequent profile of the mound, as they change temporally and spatially, would aid in the design of new landfill LCSs. At the time of writing the authors were unaware of any published model with this objective. As a first step towards modelling the clogging of the LCS, a numerical model (called BioClog) was developed to apply to column experiments in which glass beads were permeated with synthetic leachate (Cooke et al. 1999, 2001). This one-dimensional (1D) flow model was updated significantly with additional mechanisms to better predict clogging caused by real leachate which contained significant suspended solids (Cooke et al. 2005a). Using the model, clogging of experimental columns packed with glass beads and permeated with synthetic and real leachate were successfully modelled by VanGulck et al. (2003), and columns packed with gravel and permeated with real leachate were successfully modelled by Cooke et al. (2005b). The objective of this paper is to describe the incorporation of the previously developed and tested clogging processes into a 2D system for the purposes of predicting the change in the leachate mound in the LCS with time and hence to allow estimates to be made of the likely performance of different LCS designs. The following sections describe the problem domains that can be represented by the BioClog model and the technique used to model flow in a variable mesh system. Subsequently, the methods by which species transport, clogging, and species fate are modelled is discussed. This is followed by an elaboration on the issues relevant to the problem. Lastly, the application of the model is illustrated for several hypothetical field cases. Problem definition Consider a representative 2D cross section (e.g., the modelled zone in Fig. 1) through a LCS with a perforated leachate collection pipe perpendicular to the cross section being the means by which leachate is removed from the cross section. This will be modelled using a 2D finite element formulation. The initial and subsequent versions of the 2D mesh are produced by the model based on the calculated location of the phreatic surface, the location of the bottom of the mesh, and the positions of layer interfaces and boundary conditions that affect nodal placement (described later). The inclusion of layers allows the model to consider a sand protection layer beneath a gravel drainage layer and (or) a sand filter above a gravel drainage layer as well as the waste. The current implementation uses three node linear triangle elements although the concepts could be readily used in conjunction with higher order elements. Typical flow boundary conditions The percolation of leachate out of the waste above the drainage layer was modelled by specified (uniform) vertical

3 Cooke and Rowe 1395 infiltration into the upper row of elements representing the top of the leachate mound (i.e., at the upper boundary). The base below the drainage layer was assigned no flow conditions. To allow the modelling of field LCS as shown by the modelled zone in Fig. 1 or 2D laboratory mesocosm tests, such as those reported by Fleming and Rowe (2004) and McIsaac and Rowe (2007), which simulated the last 0.5 m adjacent to a leachate collection pipe, one of four arrangements may be chosen for the left and right boundaries. For the field LCS ( modelled zone in Fig. 1), symmetry implies a no flow boundary at the left boundary. The presence of a perforated leachate collection pipe at the right boundary was modelled as an open zone with the zones above and below the pipe being no flow zones due to symmetry (Figs. 1 and 2a). Within the pipe (open zone), hydrostatic conditions were assigned below the specified head corresponding to the assumed level of leachate flow in the pipe, and atmospheric conditions were specified above this level (this zone was a potential seepage face). For modelling of the left boundary in a 2D mesocosm test (Fleming and Rowe 2004; McIsaac and Rowe 2007), a static point source (Fig. 2b) or distributed specified fluid flux (Fig. 2c) can be used to simulate the lateral inflow of leachate into the system with no flow conditions above and below the lateral fluid source. Typical solute transport boundary conditions For solute transport, the model was programmed to include specified concentration (Dirichlet), specified flux (Neumann), specified total flux (Cauchy), and free exit (also called open) boundary conditions. The top boundary (the surface) may be assigned any one of the first three transport boundary conditions; although for the modelling reported herein a Cauchy condition was considered most realistic (see Padilla et al. 1997) and was therefore adopted. On the side boundaries all four of the boundary conditions may be assigned, although for the field case examined herein a Neumann condition (specifically, zero specified flux) was applied at the left no flow boundary and on the right boundary the open zone was given the free exit condition, with the remaining no flow zones given zero specified flux conditions. Fluid flow modelling with variable mesh Defining equations The basis for the methods outlined here was first published by Neuman and Witherspoon (1971) for nonsteady modelling assuming constant hydraulic conductivity. It was necessary to modify the Neuman and Witherspoon approach for the present work to allow consideration of the nonsteady porosity and hydraulic conductivity (as discussed later). Unconfined, transient saturated fluid flow is modelled ½1Š k k z þ q where h is hydraulic head measured from the x-axis, k x and k z are components of saturated hydraulic conductivity in the x and z directions, respectively, and q is specified flow rate. For confined transient modelling, the term q would normally be a specific storage term (the rate of change in storage), which is a function of media and fluid compressibility and provides transient effects. For unconfined flow the effects of compressibility are negligible, and the transient behaviour is established in the boundary conditions of the free surface (see below). The initial conditions for hydraulic head at each node, h, and the elevation of the free surface nodes, Z s, at time t =0 are given by ½2Š ½3Š hðx; z; 0Þ ¼h 0 ðx; zþ Z s ðx; 0Þ ¼Z s;0 ðxþ where h 0 and Z s,0 are the specified initial conditions. The boundary conditions are defined as described below. Where hydrostatic conditions occur, specified head boundary conditions are applied using ½4Š hðx; z; tþ ¼Hðx; z; tþ where H is equal to the specified elevation of the external fluid level. Where specified flux boundary conditions are used (including zero flux) ½5Š n x þ n z ¼ Fðx; z; tþ where n x and n z are the x and z components, respectively, of the unit normal vector, and F is the specified fluid flux. At the seepage face (Fig. 2a) the head is equal to the elevation of the node, Z ½6Š hðx; z; tþ ¼Z The free surface must satisfy two boundary conditions as both the heads and the node elevations are unknown; first, the atmospheric pressure condition, where the elevation of the free surface nodes must equal the hydraulic head, and second, the continuity condition whereupon the normal flux must correspond to the net effect of infiltration, I, and specific yield, S y (Neuman and Witherspoon 1971) ½7Š ½8Š Z s ðx; z; tþ ¼hðx; z; Z s ; tþ n þ k s z ¼ I S y n The system of equations used to obtain values of hydraulic head at each node in the mesh, in matrix form, is ½9Š ð!½cš tþt þð1!þ½cš t þ!t½kš tþt Þfhg tþ t ¼ð!½CŠ tþt þð1!þ½cš t ð1!þt½kš t Þfhg t þ tðð1!þffg t þ!ffg tþt Þ where [C] is the specific yield matrix, [k] is the global conductance matrix, {h} is a vector of hydraulic heads, {F} is the specified flow vector, u is the relaxation factor, and Dt is the time step length. Note that [C] is zero everywhere ex-

4 1396 Can. Geotech. J. Vol. 45, 2008 Fig. 2. Typical flow boundary condition arrangements for representation of (a) field and laboratory conditions at a leachate collection pipe (with fluid flowing in the pipe in the specified head zone), and lateral inflow conditions using (b) a point source (as in the experiments by McIsaac and Rowe 2007), or (c) distributed specified fluid flux. The zones are defined by elevation, Z. cept at the free surface, where the boundary condition defined by eq. [8] is employed using 2 3 ½10Š ½CŠ e ¼ S yx e and Dx e is the horizontal distance between the free surface nodes of the element (Neuman and Witherspoon 1971). The global conductance matrix, [k], is the same as is commonly employed for three node triangles (see Istok 1989). In this work, though, [C] t and [k] t are calculated using the nodal coordinates and porosity (which controls S y, k x, and k z )at the previous time step, and [C] t+dt and [k] t+dt are calculated based on current properties. The specific yield, S y, is equal to the element porosity, n, taking clogging into account and hence varies with time. The relaxation factor u controls the finite difference formulation, with u = 0 being forward difference, u = 0.5 central difference, and u = 1 backward difference. The backward difference formulation was used for the modelling herein. Solution method The system of equations (eq. [9]) cannot be used to solve h t+dt directly because the nodal elevations at time t + Dt are unknown. Additionally, n, S y, k x, and k z are considered to be unknown at time t + Dt because these parameters change as the clog material is redistributed according to changes in element location. To solve for all of these parameters each time step, the model uses two nested iterative routines; an inner routine to solve for h t+dt and Z s and an outer routine to solve for n, S y, k x, and k z. Initial element properties (n, S y, k x, and k z ) are calculated based on the elevation of the nodes from the previous time step. The element properties are kept constant while the surface heads, h s,t+dt,i, and elevations, Z s,t+dt,i, are adjusted until convergence is achieved (satisfying eq. [7]). The new nodal elevations are then used to recalculate the element properties, and the convergence routine is reiterated. The recalculation of the element properties is repeated until the surface elevation before convergence, Z s,t+dt,i=1, and after convergence, Z s,t+dt,i=ic, differ by less than a specified tolerance. Full details of the approach are described by Cooke (2007). The following highlights some important considerations for the modelling of a system where clogging is resulting in a change in element properties with time and position throughout the analysis. Additional details regarding techniques for improving computational efficiency and accuracy are provided by Cooke (2007). Element properties (i.e., k, n, or specific surface A s ) are controlled by the thickness of five films (discussed subsequently) associated with each element. When node elevations are changed, element position and area is also changed. To maintain mass conservation for clog matter, the film thicknesses are redistributed. This procedure is also used when nodes are added or subtracted (as discussed later). The three step approach originally proposed by Neuman and Witherspoon (1971) to solve for the head h t+dt method was found to result in relatively poor convergence for situations such as those examined here where the element properties change with time. Thus the following alternative approach was adopted. The specified head boundary condition was initially adopted for the seepage face and specified flux boundary condition for the free surface. All of the fluid fluxes along the free surface are functions of the infiltration rate alone, except at nodes, which are both part of the seepage face and free surface, where an unknown contribution to the flux passes through the seepage face. The fluid flux at these nodes is calculated from head conditions of the previous iteration (or previous time step, if this is the first iteration of a time step). The values of h t+dt are computed assuming the free surface boundary condition is that of eq. [8], where the normal fluid flux is equated to the net infiltration and specific yield. Using this boundary condition on the free surface produces new values of h t+dt for all nodes including the free surface, h s,t+dt. The solution is taken to have converged when the calculated head h s,t+dt and the estimated node elevation Z s,t+dt,i are suitably close for all free surface nodes (and thus the boundary conditions of eqs. [7] and [8] are satisfied). Surface node elevations are repositioned, and heads are recalculated iteratively until con-

5 Cooke and Rowe 1397 vergence occurs. To achieve ultimate convergence for the time step, after each convergence of h t+dt and Z s,t+dt,i, film thicknesses are redistributed to account for the new element positions and the solution is repeated until all surface nodes meet the error criteria applied to the difference between the elevation of the surface at the beginning, Z s,t+dt,i=1 and end Z s,t+dt,i=ic of the iteration. The error is calculated relative to the change in elevation for the time step: ½11Š Err ZZ ¼ Z s;tþt;i¼1 Z s;tþt;i¼ic Z s;tþt;i¼1 Z s;t and this error must be less than a specified tolerance. Solute transport The partial differential equation for saturated, nonuniform solute transport in two dimensions ¼ nd 2 2 C þ nd þ 2 C þ nd 2 2 ðv a;zcþ þs R where C is solute (species) concentration, D xx, D xz, D zx, and D zz are dispersion coefficients, v a,x and v a,z are the Darcy velocities in the x and z coordinate directions, and S R is a source or sink term for addition or removal of species due to biogeochemical reaction processes. Transport is solved using the finite element method and the same mesh as that used for fluid flow. Similar to the fluid flow model, the time step length can vary over time. At any given time, the time step length of the transport model can differ from that of the fluid flow routine by selecting one to be the base time step, and the other be an integer multiple of the base time step. The transport of nine leachate constituents deemed most responsible for the majority of clog accumulation (Cooke et al. 2005a) is included: three volatile fatty acids (VFAs) (acetate, propionate, and butyrate), the active biomass associated with these acids (acetate degraders, propionate degraders, and butyrate degraders), inert biomass, calcium carbonate, and inorganic particles. Clogging and species fate Film thicknesses and species reactions Details regarding the calculation of film thicknesses due to clogging processes, subsequent changes in porosity, and the calculation of species reaction terms are given in Cooke et al. (2005a) for 1D systems. The species reaction terms are used here with only modification of the b term for 2D conditions as ½13Š ¼ na e ðthree noded triangular elementþ 3 where A e is the area of an element. The majority of the calculations of film thickness and species reaction terms are the same for both 1D and 2D conditions. A summary of the mechanisms modelled is outlined next. The volume of clog matter in each element is quantified by the thicknesses of five separate films: one biofilm for each of the three active substrate degraders, the inert biofilm, and the inorganic solids film. The rate at which active biomass, inert biomass, and inorganic particles attach to the porous media is computed using one of two methods: (i) the Rajagopalan and Tien (1976) filtration model, or, (ii) the Reddi and Bonala (1997) network model. The calculated rate of attachment is used to compute the quantity of suspended matter removed from suspension (appearing in the reaction term for the species in the transport model) and the increase in thickness of the associated film (each film thickness is recalculated every time step). The utilization of substrate (VFA), and consequent substrate flux into the active biofilms, is modelled assuming Monod kinetics and molecular diffusion using the method of Rittmann and McCarty (1981). Based on the substrate flux and density of the active film, the additional thickness of biofilm due to growth is calculated using the Rittmann and Brunner (1984) approach. The rate at which mass is detached from the films and is added to the suspended concentration is calculated based on both shearing (Rittman 1982) and growth rate (Peyton and Characklis 1993). For the active biofilms (and active biomass in suspension), the rate at which mass is lost to decay contributes to the accumulation of inert film (or suspended inert biomass). The inert biomass and inorganic solids films do not lose mass to decay. While the active biofilm gains matter due to growth and attachment, and the inert biofilm gains matter by attachment and decay of the active biofilm, the inorganic solids film gains matter from attachment and precipitation of calcium carbonate. The rate of calcium carbonate precipitation is based on the rate of carbonic acid production in the system (determined from calculated VFA degradation rates) and an empirically derived yield coefficient (VanGulck et al. 2003). The resulting rate of calcium removal is used in the reaction term for calcium and in the calculation of the flux of calcium carbonate onto the porous media. Similar to the biofilms, by knowing the flux onto the surface and an estimate of density, the thickness added can be found. Porosity and hydraulic conductivity BioClog represents the porous media by assuming each element contains packed spheres of uniform diameter, onto which the films are attached. The density of the packing determines the clean media porosity. Given the total film thickness in the element, the porosity and specific surface are calculated geometrically based on Cooke and Rowe (1999). For each time step, the calculation of new film thicknesses then gives rise to new porosity and specific surface values. Published correlations of clog matter quantity (i.e., mass, volume, porosity decrease) to hydraulic conductivity are scarce. For this reason, the hydraulic conductivity (k) of each element is dependent on porosity (n) alone, using an exponential relationship ½14Š k ¼ A k e b kn where coefficients A k and b k are estimated by regression analysis of measured data (see VanGulck and Rowe 2004;

6 1398 Can. Geotech. J. Vol. 45, 2008 Fig. 3. Relationship between total porosity, n Tot, and hydraulic conductivity, k. Regression line is for measured k and calculated n Tot (based on measured drainable porosity) for 6 mm beads (series TK, S, VS, VK) and similarly sized pea gravel (series PG-1 and PG-2). Estimated modelling relationship for hypothetical sands and waste for field case demonstration (with minimum k = m/s) also shown. Cooke et al. 2005b), where data is available. In the absence of data to the contrary, isotropic conditions are assumed, thus k x = k z = k. Figure 3 shows the relationship between hydraulic conductivity and total porosity obtained from experiments examining the effects of clogging of columns permeated with synthetic or real landfill leachate. Also shown is the linear regression line (eq. [14]) and the associated parameters for this data as well as the relationships assumed for three hypothetical sands and the waste examined in the field demonstration case to be discussed later. It is assumed here that severely clogged porous media maintains some secondary porosity, and a hydraulic conductivity of m/s is adopted for the demonstration case as minimum k. This relationship between porosity and hydraulic conductivity could be readily replaced with other suitable equations when there is sufficient data to justify an alternative relationship. Issues relevant to variable mesh technique and clogging Remapping element and node properties The mesh configuration changes when the surface elevation of the leachate mound changes (e.g., rises because of a reduction in hydraulic conductivity due to clogging) or when nodes are added or subtracted from the mesh. The consequent adjustment in nodal positions causes a change in element location and shape, but does not represent movement of the clog properties that were associated with the previous location of the element (since the clog matter is attached to the porous media, which remains static). Thus the element properties associated with the old configuration are remapped onto the elements in the new configuration. Since all other clog properties (n, A s, k, etc.) can be derived from the five film thicknesses, only the thicknesses must be remapped from previous values as described by Cooke (2007). The porosity, specific surface, and hydraulic conductivity can then be computed for the elements based on the new film thicknesses. Details regarding the effect of layer interfaces (such as between a sand and gravel layer) on the movement, addition, and subtraction of nodes is explained in more detail in Cooke (2007). Maintaining adequate element aspect ratios To minimize numerical errors, the aspect ratio of the elements (the ratio of maximum to minimum dimension) should approach unity, and experience has shown that aspect ratios greater than 5 should be avoided (Anderson and Woessner 1991). To keep the number of nodes to a practical limit, the finite element mesh of a typical field case often consists of elements with horizontal dimensions larger than the vertical dimensions, but the vertical dimensions are variable over time. For this reason, after the model solves for the surface node elevations at time t+dt, the element dimensions are subjected to criteria checks controlled by input parameters. If elements are deemed to have a vertical dimension that is too large relative to the horizontal dimension, the mesh may be regenerated with an additional row of

7 Cooke and Rowe 1399 Fig. 4. Field case setup. A F is the base. D E is the layer interface between the drainage layer and the waste. Flow and solute transport boundary conditions: A B is m standing water with specified head and free-exit solute boundary conditions; B C is m collection pipe with potential seepage face (h = Z) and free-exit solute boundary condition; Z >C,Z > F, and A F are zero flow and zero solute flux. The moving top boundary (not shown) has specified fluid flux and total solute flux boundary conditions. Table 1. Model parameters varied for separate cases. Case 1: medium sand Case 2: fine sand Case 3: coarse sand Drainage media size, d g (mm) Initial hydraulic conductivity, k 0 (m/s) Hydraulic conductivity coefficient (eq. [14]), A k (m/s) Hydraulic conductivity coefficient, (eq. [14]), b k elements. Conversely, if the vertical dimension is deemed to be too small relative to the horizontal dimension, a row of elements may be removed. Nonsteady drainage model for early times If the drainage layer is coarse granular material (e.g., coarse gravel with k = 10 1 m/s), the saturated leachate mound thickness is very small (e.g., a maximum of 0.2 mm for a 30 m drainage path at a slope of 1% and infiltration rate of 0.2 m/year) at early times before there is significant accumulation of clog matter. Under these conditions, it was deemed impractical to model the mound using a 2D mesh because too many elements would be required to maintain adequate element aspect ratios. Considerable time may be required for clogging to cause the mound to rise to levels that can be adequately modelled in two dimensions. For these reasons, an approximate nonsteady mounding model was implemented that could be used to model early operation times until the time when the mound has risen to heights at which the 2D model can take over. For this method, the shape of the surface is calculated based on the governing equation and boundary conditions described by McEnroe (1989) for nonsteady drainage of leachate on sloped barriers in landfill covers and liners. A single hydraulic head, and thus mound height, is calculated at each specified x-direction nodal spacing along the length of the flow path. The entire saturated zone is considered a single homogeneous unit with respect to species concentrations and clogging processes. Clogging calculations are performed only once per time step. Similar to the full 2D method, an iterative method is used each time step to maintain a balance between mound volume change, distribution of clog volume, and hydraulic conductivity. This solution method allows accurate calculation of Table 2. Source leachate species concentrations. Species Concentration Propionate, Pr (mg/l) Acetate, Ac (mg/l) Butyrate, Bu (mg/l) 2000 Calcium, Ca (mg/l) 1500 Volatile suspended solids, VSS (mg/l) 1000 Fixed suspended solids, FSS (mg/l) 1000 VSS % active 70 VSS ratio Pr:Ac:Bu 1:1:1 mound heights and flow velocities and reasonable computational speed while sacrificing spatial variability of clog properties when the mound is very small. Once the mound height is sufficient, as determined by a user-supplied maximum mound thickness, the model will switch to the full 2D method. Details regarding the implementation of this solution are given by Cooke (2007). Model demonstration: Problem definition To illustrate the use of the model, the results of a reasonably simple hypothetical field case will be presented. For brevity, not all input parameters will be discussed (all parameters not discussed in detail are well described by Cooke et al. (2005a) and the interested reader is referred to that paper for additional details). The system is illustrated in Fig. 4. The hypothetical field case being modelled is a sand drainage layer with a uniform thickness of 0.3 m, sloped at 1% to collection pipes spaced 40 m apart in a sawtooth arrangement (thus a drainage length of 20 m is modelled). Three cases using different sand properties, as listed in Table 1, are modelled. In case 1 the sand has an initial hydraulic conductivity, k 0,of m/s and grain size, d g, of 1.0 mm;

8 1400 Can. Geotech. J. Vol. 45, 2008 Table 3. Kinetics constants and diffusion parameters for the volatile fatty acids (parameters defined in detail by Cooke et al. 2005a). Parameters Propionate Acetate Butyrate Kinetic constants Half maximum rate of concentration, Ks (mgcod L 1 ) Maximum specific substrate utilization rate, bq (mgcod mgvs 1 d 1 ) True yield coefficient, Y (mg VS mgcod 1 ) Biofilm decay coefficient, b d (d 1 ) Diffusion parameters Substrate in fluid, D o (cm 2 d 1 ) Substrate in film, D f (cm 2 d 1 ) Table 4. Clog and suspended solids parameters (parameters defined in detail by Cooke et al. 2005a). Parameters Value Clog matter parameters Calcium removal rate coefficient, Y H 0.05 (mg Ca 2+ removed per mg H 2 CO 3 produced) Variable biofilm density parameter, A X 247 Variable biofilm density parameter, B X 72 (mg VS cm 3 ) Inorganic film density, X f,is (mg NVS cm 3 ) 2750 Other precipitate ratio, f OP 0.06 Fraction degradable by decay, f d 0.8 Suspended solids parameters Active and inert diameter (cm) Active and inert density (mg VS cm 3 ) 1030 Inorganic particles diameter (cm) Inorganic particles density (mg NVS cm 3 ) 1065 Attachment and detachment Attachment model (Tien 1989) Shear detachment modifier 1.0 Growth detachment modifier 1.0 Note: Film density X f,is is mass of nonvolatile solids (NVS) per volume of inorganic solids. in case 2, k 0 = m/s and d g = mm; and in case 3, k 0 = m/s and d g = 2.0 mm (approximately representing medium, fine, and coarse sands, respectively). Each of the drainage materials has a different relationship between n and k, as defined by A k and b k in Table 1 and shown in Fig. 3. At the downstream end, the perforated collection pipe is assumed to have a diameter of 0.2 m and 5 mm of standing water, thus this boundary is modelled as a 0.2 m opening with m specified head (A B) and m potential seepage face (B C). The solute transport boundary condition is free for the open zone (A C). The remainder of the boundary (Z > C) is zero flow and zero solute flux. A constant infiltration rate of 0.2 m/year is applied to the surface boundary of the system (specified flux). For solute transport, this boundary is specified total solute flux. The upstream side boundary (Z > F) and bottom boundary (F A) are both zero flow and zero solute flux. The leachate species concentrations entering the drainage layer from above are as given in Table 2. The initial mesh used 5 rows and 245 columns of nodes (1952 triangular elements). The node spacing in the x- Table 5. Numerical and 2D specific parameters. Parameters Value Numerical settings Time step, Dt (d) Variable Relaxation factor, u 1.0 Substrate convergence tolerance, Surface convergence tolerances, 3 hz, 3 ZZ 0.05 Surface extrapolation multiplier, E f 1.0 Limit surface node movement No Element orientation All right oriented Limits Minimum hydraulic conductivity, k min (m/s) Minimum saturated thickness, g Min (cm) 0.3 Dispersivity Longitudinal dispersivity, a L (cm) b n L ¼ L;0 n 0 Initial longitudinal dispersivity, a L,0 (cm) 10 Equation parameter, b a 1.74 Transverse dispersivity, a T (cm) a T = c a a L Equation parameter, c a 1.0 direction was m at each end and gradually increased to 0.10 m for the majority of the mesh (from x = 1.8 to x = 18.2 m). In preliminary testing to assess the effect of mesh refinement, this node spacing was shown to provide adequate modelling of flow and transport without further refinement. As the mesh rises an aspect ratio of 1.2 was used to guide the addition of nodes, as required. To aid the Cauchy source boundary condition, thin elements were employed along the phreatic surface with a thickness of m. The choice of time step lengths is primarily controlled by surface convergence, and smaller time steps are generally required as time elapses. In case 1 the time step lengths were d(0 to 7.5 years) and d (7.5 to 10 years). In case 2 the time step length was d from start to finish, and in case 3 multiple time step lengths were used, beginning at d and ending at d. Tables 3 and 4 list additional model parameters selected for these cases that are not specific to the granular drainage material (sands) and 2D application. These parameters have been discussed in detail in Cooke et al. (2005a, 2005b). A relaxation factor of 1.0 is employed for both the fluid flow and solute transport models, implying backward difference interpolation of the time derivative. Other numerical parameters are listed in Table 5. Based on the findings of Taylor et al. (1990), dispersivity increased with decreasing porosity

9 Cooke and Rowe 1401 Fig. 5. Development of a leachate mound with time for three sands with different initial hydraulic conductivities, k 0 (m/s), with and without clogging modelled. Arrows illustrate the steady-state value calculated using the equation by Giroud (Giroud et al. 1992). Note: * indicates model run with waste layer omitted. according to the equation and parameters listed in Table 5. The transverse dispersivity is assumed to be equal to the longitudinal dispersivity. Model demonstration: Results The modelling was performed until the maximum thickness of the mound exceeded 0.30 m, as this is often regarded as a maximum allowable height for design purposes and also because at this point the mound will begin to be affected by the properties of the waste, which are often more uncertain than those of the drainage media. This thickness was reached just before 10, 0.75, and 32 years for cases 1, 2, and 3, respectively, demonstrating the significant effect that particle size can have on mound height and on the time the head may stay below 0.3 m. A comparison will be made to the Giroud equation (Giroud et al. 1992), and then the results for case 1 (base case) will be presented in detail, followed by the mound progression and porosity data for cases 2 and 3. The cases examined next are intended to illustrate the application of the model. They only consider one set of parameters (other than the difference in initial particle size and hydraulic conductivity of the sand). Clogging will vary depending not only on these parameters but also on the influent concentrations, kinetic parameters etc. Space does not permit a discussion of these issues in this paper. Rather, the objective here is to present a model that can examine these factors. The effect of these parameters will be discussed in a subsequent paper. Comparison to the Giroud equation To compare the model results for each case with results from the Giroud equation (Giroud et al. 1992), the three runs were first performed without clogging, and for case 2, without a waste layer. The calculated maximum mound thicknesses developed over the first 2 years are shown in Fig. 5. Without clogging, the mound rises rapidly and reaches the steady-state thickness predicted by the Giroud

10 1402 Can. Geotech. J. Vol. 45, 2008 Fig. 6. Case 1 (medium sand, k o =10 4 m/s) mound surface profiles at intervals of 2 years. Fig. 7. Case 1 porosity within the mound after 5 and 10 years. equation (0.077, 0.38, and m for cases 1, 2, and 3, respectively) within a few months for cases 1 and 3 and within about 2 years for case 2. Also shown in Fig. 5 is the calculated mound development when clogging is considered. Case 1 with clogging For case 1 (medium sand), the mound shape at 2 year intervals is shown in Fig. 6. After an initial rapid rise (see Fig. 5) it can be seen from Fig. 6 that the mound rises at a fairly constant rate for subsequent times. The typical maximum design head of 0.3 m is reached after a little less than 10 years. The calculated porosities within the mound are shown at 5 and 10 years in Fig. 7. After 10 years, porosity has decreased from the initial 0.35 to as low as 0.25 at the downstream end, with the average being While clogging was most extensive at the downstream end, which is to be expected because of the relatively large mass loading, there was also a clogged region at the upstream end of the slope. This was likely caused by the low flow and thus long residency time and decreased shear detachment at this end. Also, there was a slight increase in porosity near the base, very likely due to the removal of substrate (VFAs) and suspended solids near the surface and hence there was less mass loading near the base (except near the collection pipe). Contours of the calculated hydraulic conductivities within the mound at 5 and 10 years are shown in Fig. 8. After 10 years of clogging, the hydraulic conductivity dropped from an initial m/s to a low of m/s at the downstream end near the collection pipe. The average within the mound was m/s. Assuming a homogeneous drainage layer with this average hydraulic conductivity, the Giroud equation predicts a maximum mound thickness of 0.25 m compared to the 0.31 m calculated by BioClog assuming a heterogeneous system.

11 Cooke and Rowe 1403 Fig. 8. Case 1 hydraulic conductivity (m/s) within the mound after 5 and 10 years. Fig. 9. Case 1 thickness of active films (mm) within the mound after 5 and 10 years. Figures 9, 10, and 11 show the calculated thicknesses of active, inert organic, and inorganic films, respectively, in the mound at 5 and 10 years. The active films (Fig. 9) were the thinnest of the films and generally decreased in thickness with distance from the surface. According to the thickness of the three types of films after 5 and 10 years, the inert organic film dominated the clog matter, followed by the inorganic film. The distributions of the inert, inorganic, and total film thicknesses (Figs. 10, 11, and 12, respectively) were all similar. The largest films were located at the down-

12 1404 Can. Geotech. J. Vol. 45, 2008 Fig. 10. Case 1 thickness of inert organic films (mm) within the mound after 5 and 10 years. Fig. 11. Case 1 thickness of inorganic films (predominately calcium carbonate, in mm) within the mound after 5 and 10 years. stream end where film thicknesses increased within a relatively short 1 m region likely caused by the relatively high mass loading at this end. In the remainder of the mound, there was a slight increase in film thickness near the surface and upstream end, likely controlled by the proximity to the source of growth substrate (VFAs) and suspended solids for attachment. The latter influences, coupled with the rising mound and thus greater distance to the surface, can be seen to cause a general decrease in active film thickness between 5 and 10 years. The reduced active film

13 Cooke and Rowe 1405 Fig. 12. Case 1 total thickness of clog films (mm) within the mound after 5 and 10 years. Fig. 13. Case 1 acetate concentrations (mg/l) within the mound after 5 and 10 years. thickness near the base, over time, leads to reduced rates of calcium carbonate precipitation. Understandably, the distributions of the dominating inorganic and inert films (and thus also total film thickness) follow similar patterns to that of porosity and hydraulic conductivity. The calculated concentrations of acetate (Fig. 13) and cal-

14 1406 Can. Geotech. J. Vol. 45, 2008 Fig. 14. Case 1 calcium concentrations (mg/l) within the mound after 5 and 10 years. Fig. 15. Case 2 (fine sand, k o =10 5 m/s) mound surface profiles at intervals of 0.25 years for fine sand. Fig. 16. Case 3 (coarse sand, k o =10 3 m/s) mound surface profiles at intervals of 8 years for coarse sand.

15 Cooke and Rowe 1407 Fig. 17. Case 3 porosity within the mound after 16 and 32 years for coarse sand. cium (Fig. 14) in the mound are shown for 5 and 10 years. In the majority of the mound, the acetate has been depleted to concentrations of 800 to 1200 mg/l and calcium to concentrations of 650 to 800 mg/l. Low concentration zones appear upstream. It should be noted that significantly greater calcium depletion (and inorganic clog accumulation) would result if a larger value for the calcium removal rate, Y H, was employed (such as Y H = as reported by VanGulck et al. 2003). Results for cases 2 and 3 with clogging The mound progression for case 2 (approximating a fine sand, k 0 = m/s and d g = mm) is shown in Fig. 15 at intervals of 0.25 years. Because of the low initial hydraulic conductivity the leachate mound would exceed 0.3 m even if there was no clogging. With clogging, the top of the drainage layer was reached before 0.75 years. The rate of rise in this case was gradually decreasing, but is impacted to a greater extent by the properties of the waste. Starting at an initial porosity of 0.33, the average was 0.32 and the low (occurring at the downstream end) was 0.28 after 1 year. With the mound reaching the waste quickly, relatively little clogging occurred during the time modelled. Due to the considerably slower clogging rate associated with a drainage material of larger grain size and hydraulic conductivity of case 3 (approximating a coarse sand with k 0 = m/s and d g = 2.0 mm), it required 32 years (Fig. 16) for the top of the drainage layer to be reached compared to 10 years for the fine and medium sand. The rate of rise was relatively constant after the initial period. Figure 17 shows the predicted porosities in the mound after 16 and 32 years. The initial porosity was After 32 years, the average porosity in the mound was 0.24 and the lowest porosity, at the downstream end near the pipe, was Conclusions A 2D model for predicting the change in porosity and hydraulic conductivity of drainage layers and the subsequent increase in leachate head on the liner as a result of clogging by biological and physical processes in landfill leachate collection systems has been described. The model includes flow modelling with a dynamic surface, solute transport of the nine species most responsible for clogging, the fate of these species, and the corresponding increase in clog material volume. Porosity and hydraulic conductivity both change spatially and temporally within the leachate mound. The demonstration of the model using typical field parameters shows that, depending on the nature of the material used in the drainage layer, clogging can cause a decrease in hydraulic conductivity that can cause the leachate mound to exceed the design values in periods ranging from about 1 to 32 years for sand with initial hydraulic conductivities of 10 5 to 10 3 m/s. The model provides an opportunity to explore the effect of different drain spacing, drainage materials, and leachate characteristics. Acknowledgements This research was funded by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are very grateful for the valuable discussions with Dr. B.E. Rittmann.

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