A discrete particle representation of hillslope hydrology: hypothesis testing in reproducing a tracer experiment at

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1 HYDROLOGICAL PROCESSES Hydrol. Process. 25, (211) Published online 5 April 211 in Wiley Online Library (wileyonlinelibrary.com) DOI: 1.12/hyp.885 A discrete particle representation of hillslope hydrology: hypothesis testing in reproducing a tracer experiment at Gårdsjön, Sweden Jessica Davies, 1 * Keith Beven, 1,2 Lars Nyberg 3 and Allan Rodhe 2 1 Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK 2 Department of Earth Sciences, Geocentrum, Uppsala University, Uppsala, Sweden 3 Centre for Climate and Safety, Karlstad University, Karlstad, Sweden Abstract: Despite the long history of the continuum equation approach in hydrology, it is not a necessary approach to the formulation of a physically based representation of hillslope hydrology. The Multiple Interacting Pathways (MIPs) model is a discrete realization that allows hillslope response and transport to be simultaneously explored in a way that reflects the potential occurrence of preferential flows and lengths of pathways. The MIPs model uses random particle tracking methods to represent the flow of water within the subsurface alongside velocity distributions that acknowledge preferential flows and transition probability matrices, which control flow pathways. An initial realization of this model is presented here in application to a tracer experiment carried out in Gårdsjön, Sweden. The model is used as an exploratory tool, testing several hypotheses in relation to this experiment. Copyright 211 John Wiley & Sons, Ltd. KEY WORDS hillslope model; particle tracking; tracer experiment; run-off processes Received 11 November 21; Accepted 1 March 211 INTRODUCTION Classical approaches to hillslope hydrology modelling Traditionally, there have been two primary approaches to modelling hillslope hydrology. The first is based on continuum mechanics as outlined in the Freeze and Harlan (1969) blueprint (Freeze, 1972; Beven, 1977, for finite difference and finite element numerical solutions; Beven, 1981, 1982; Troch et al., 22 for analytical solutions to the simplified Boussinesq forms of the continuum equations). The second is based on simple lumped conceptual storage approaches. The two overlap when solutions to the continuum equations under simplifying assumptions lead to storage type equations (Beven and Kirkby, 1979; Sloan and Moore, 1984). There has been significant criticism the way in which continuum approaches are applied in practice (Beven, 1989, 21b, 22, 26b; Grayson et al., 1992). One particular difficulty is in defining an appropriate parameterization of the highly nonlinear unsaturated zone processes at practical model element scales in the face of field evidence of heterogeneity of soil properties and preferential flow (Beven and Germann, 1982; Beven, 1989, 21b, 26b; Henderson et al., 1996; Anderson et al., 29; Abou Najm et al., 21; Chappell, 21; Jones, 21). * Correspondence to: Jessica Davies, Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK. j.davies4@lancaster.ac.uk Efforts have been made in extending continuum-based models to address this problem, through inclusion of exchanges with immobile pore waters or dual porosity soil water characteristics (Hoogmoed and Bouma, 198; Beven and Germann, 1981; Zuidema, 1985; Germann and Beven, 1986; Bronstert and Plate, 1997; Larsson and Jarvis, 1999; Simunek et al., 23; Simunek and van Genuchten, 28; Haws et al., 25; Weiler and McDonnell, 27). However, none of these have seemed particularly satisfactory: in part because of the way in which a dual porosity concept is invoked to represent a wider potential range of preferential flows and bypassing; in part because they add additional parameters that are not easily identified. In fact, it can be argued that the continuum approach is not applicable to structured soils at any reasonable calculation scale. The continuum approach requires that there be well-defined boundary conditions, some local equilibration of pressure over some representative elementary volume (REV) of soil, and continuous gradual changes in potential and soil properties that allow the definition of gradient terms to be valid (Bear, 1972). This may be an adequate representation in homogeneous media, but not in structured or unsaturated soils that exhibit heterogeneity at the sub-calculation element scale (Beven, 1989, 21b, 22, 26a; Binley et al., 1989). A discrete domain, discrete particle conceptualization Despite the long history of the continuum approach in hydrology, it is not a necessary approach to the formulation of a physically based representation of hillslope Copyright 211 John Wiley & Sons, Ltd.

2 A DISCRETE PARTICLE REPRESENTATION OF HILLSLOPE HYDROLOGY 363 hydrology. This has been shown by the work of Reggiani et al. (1999, 2); Reggiani and Rientjes, (25) who have developed a framework for a discrete representation of the landscape that is consistent with mass and energy conservation at the scale of any calculation unit (they use representative elementary watersheds (REW) but their work applies at any scale of calculation unit). However, models based on REW concepts have not yet formulated adequate representations of the fluxes between calculation units in order to close their system of equations and complete the model (in fact, most recent implementations of the REW concepts have resorted to using small scale continuum equations at the REW scale, see Beven, 26a). The introduction of such a formally explicit discrete framework then poses the question as to whether a discrete conceptualization of hillslope hydrology might be possible that will represent the known complexity of the unsaturated and saturated zone processes at sub-rew scales in a more satisfactory way than the continuum approach. To the authors knowledge, the only previous attempts to formulate a discrete model of soil water flows have been the MIPs (Multiple Interacting Pathways) model of Beven et al. (1989) and the SAMP (Subsystems And Moving Packets) model of Ewen (1996). The MIPs model was applied to both profile and hillslope scales, but only for steady-state flows. The SAMP model was applied to dynamic cases, but only at the profile scale. The model presented here builds directly upon the MIPs model of Beven et al. (1989), extending the simple steady-state case to fully dynamic conditions, including preferential flows, on hillslopes of variable form as a possible simple methodology for representing the scale dependence of process representations at REW scales. The conceptualization of the MIPs model is outlined in more detail in the Section Multiple Interacting Pathways Conceptualization. This model is then applied to simulate a slope scale experiment that was carried out in the Gårdsjön catchment in Sweden, as described in Section Application to a Slope Scale Site. Simulation results for a number of feasible flow and transport hypotheses for this example are then discussed within the Section Tracer Transport Hypothesis Simulations. Finally, conclusions are drawn and the future directions of research addressed within the Conclusions Section. MULTIPLE INTERACTING PATHWAYS CONCEPTUALIZATION The MIPs model attempts to represent different possible flow pathways within the system in a probabilistic way through combined use of random particle tracking to represent parcels of water within different pathways defined by velocity distributions, and transition probability matrices to represent the exchange of water between these pathways. A transition probability matrix is an array that describes the probability of a particle entering a certain pathway, conditional on its current pathway i.e. element ij is the probability that a particle will move from pathway i to pathway j, under current hydraulic conditions. Antecedent storage of water within the slope and inputs during an event are represented by a very large number of discrete particles. At every time step, each particle moves within the potential flow pathways of the slope according to a step equation that reflects its current position within the flow. The velocity of the particles is assigned with a distribution that attempts to be consistent with the flow pathways of the profile. The distribution of velocities represents the Lagrangian effective velocities for each particle over its lifetime in the flow domain. This form of representation does not necessitate that exchange be made between mobile particles in order to represent the exchange that occurs between slow and fast pathways within the soil profile, as the velocities are effective velocities over the whole of the particle s transport to an outflow boundary. However, introducing transition probability matrices that effect exchange can add further dynamics into the system, and add further dependency on the dynamic slope conditions. A transition probability matrix can also be used to simulate exchange between the mobile pore space and immobile pore space, and root uptake or deep drainage loss. The use of velocity distributions and transition matrices in this form is a way of modelling the effects of preferential flow and bypassing within the hillslope soils without explicit consideration of the geometry of potential flow pathways: something that will not be knowable in nearly all real applications. Note that continuity is explicitly maintained within a model of this type, as particles are neither lost nor gained during simulation (accounting for any sink terms associated with water uptake by roots or losses to deeper layers). It is also possible to label each particle with a time of entry to the hillslope so that residence time distributions, whether for increments of an input being transported to an output boundary, increments of discharge at the outlet, or storage within the slope at any time, can be calculated directly by this type of model formulation. These residence time distributions will all be different under dynamic conditions (Botter et al., 21) and will also differ from the response time distribution for the hydrograph which will be controlled by wave celerities rather than water velocities (Beven, 21a, p. 176). Model requirements Given this conceptualization, it is pertinent to consider the requirements of such a model. Three main requirements may be identified. Input conditions. The conditions controlling the inputs to any particular pathway must be formulated. In a real hillslope, this will be controlled by the antecedent condition of the near surface soil; by the condition of the soil surface (including crusting, hydrophobicity and organic cover); the role of vegetation in distributing incoming rainfalls as throughfall and stemflow; and by Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

3 364 J. DAVIES ET AL. Figure 1. A slope section of the central valley region of the G1 catchment, Ga rdsjo n will be simulated (catchment map from Nyberg et al., 1999) the potential for local saturation in soil peds or breaks in horizon properties to generate preferential flow locally at depths below the soil surface and so on. Step equations. Step equations for the random particle tracking must be defined in a way that takes account of capillarity effects in small pores and the momentum losses for flow in secondary porosity as a function of flow rate. Transition probability matrices. Definition of the transition probability matrices that will control the exchanges between flow pathways, including any immobile pore space, must properly reflect the structural geometry of the hillslope; the changing effects of capillarity; the effects of saturating different parts of the pore space; the connectivity of that saturated pore space; and the local boundary conditions associated with some types of preferential flow (e.g. the effects of cutans, decayed root material and organic coatings). APPLICATION TO A SLOPE SCALE SITE Here, the MIPs concept is applied to a tracer experiment carried out in the lower part of the G1 covered catchment at Ga rdsjo n, Sweden (Nyberg, 1995; Nyberg et al., 1999). It is recognized that, while there have been many studies showing the importance of preferential flows on the response of a wide range of real hillslopes and soil columns, the type of data required to define the three general components of the model will not generally be available. While the properties of the soil matrix could be considered to be reasonably well known for many Copyright 211 John Wiley & Sons, Ltd. different types of soil, the nature of the structural porosity of the soil and its effect on flow pathways is very poorly known. Indeed, with the exception of small samples of soil subject to destructive experiments, the detailed nature of the secondary porosity pathways will be essentially unknowable for practical applications. Thus, it will be necessary to infer the nature of the flow pathways and input conditions from limited response data. Such inferences will not be unique but will be subject to significant uncertainty in both model structural assumptions (the changing nature of the transition probability matrices with moisture content) and parameter sets within the model. All that will be possible, therefore, will be to explore which parameterizations (as hypotheses about water velocities and exchanges) are consistent with the observed responses and which are not (see discussion of Beven, 22). It is readily apparent, however, that this is not a parsimonious parameterization if identifiability was a concern. This is not, however, the aim of this paper. The aim is rather to explore the types of minimal hypotheses that would be required to represent the perceived complexities of hillslope responses explicitly. However, this does mean that a wide range of models may need to be examined in order to test which hypotheses produce results which are in keeping with the observed and perceived behaviours of the slope. Slope set-up The simulation presented here is of a hillslope scale plot within the catchment. The G1 catchment is approximately 63 m2 with a long central valley (11 m) and steep flanks (Figure 1). A 4 m by 1 m plot is simulated which corresponds to a section of the central valley near Hydrol. Process. 25, (211)

4 A DISCRETE PARTICLE REPRESENTATION OF HILLSLOPE HYDROLOGY 365 Depth, m Assumed Hydraulic Conductivity Profile Depth, m Derived Porosity Profile Hydraulic conductivity, m/day Porosity Figure 2. Profiles of saturated hydraulic conductivity and derived porosity profile based on data from the Gårdsjön site with parameters K D 5 m/day; f D 3Ð5; v o D Ð1 m/day and b D 25 (terms defined in main text) the outlet, where discharges were monitored in a trench dug at the foot of the plot. The initial model has been implemented as a twodimensional (2D) vertical slice, the depth of which is variable (Ð3 Ð8 m). This could be extended to 3D within the MIPs framework, at the expense of greater simulation times. However, given that there is only limited data on the 3D soil properties of the slope and the hydraulic conductivity profile derived in part from drainage data naturally integrates over this variability, a 2D solution is the preferred choice for this initial study. The surface slope is approximated as constant and it is assumed that it is underlain by impermeable bedrock of variable depth, shallowing towards the outflow boundary. An important consideration is that there should be a consistent framework for linking the profiles of hydraulic conductivity, porosity and assumed velocity distribution at any level in the profile. In this site, there is a dramatic decrease in saturated hydraulic conductivity with depth associated with a decline in macroporosity and changes in the matrix porosity in the soil. This is an important factor in both the hydrological and tracer responses. To provide a simplified functional representation of the hydraulic characteristics of the soil, it is assumed (based on field measurements described in Nyberg, 1995) that the hydraulic conductivity decreases as an exponential function of depth as shown in Figure 2, and that pore water velocity is a consistent function of porosity over all depths. More formally, over the soil profile: K s D K o exp fz 1 Where K s is the saturated hydraulic conductivity at any depth below the surface z, K o is the hydraulic conductivity at surface level and f is a parameter. The local transmissivity will then be defined by the integral of Equation (1) from the local soil depth D to the depth of the water table d as: T i D D d K o exp fz D K o f [ exp fd exp fd ] 2 At any depth z it is then assumed that velocity v is an exponential function of the pore space filled with mobile water, such that: v D v o exp b 3 where v o is the minimum velocity given a unit gradient at a mobile water content D, and b is a parameter that determines the exponential increase of velocity with porosity. Integrating this to the local porosity for that depth gives an expression for the local saturated hydraulic conductivity at any depth: s K s D v D v o b [exp b s 1] D K o exp fz 4 From this, an expression for the change in porosity with increasing depth into the soil that reflects the decline in larger flow pathways inferred from the conductivity function can be derived: s D lnf1 C K ob exp fz /v o g 5 b Note that Equation (5) follows analytically from the assumptions of Equations (1) and (3) and can therefore provide a check on chosen values of parameters. Note also that other forms of distribution could be used for Equations (1) and (3) depending on the conductivity profile and availability of additional information about the structure of the pore space at a site. Finally, it is assumed that these functions hold regardless of the soil depth at a point so that changes in soil depth will not change the conductivity function, but will change the transmissivity function (Equation 2) at any point on the slope. This will affect the depth of the water table that builds up on the slope if the soil depth is variable (which is the case at this site). Defining initial conditions Initial conditions on the hillslope are represented by pre-populating the slope with particles in accordance with Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

5 366 J. DAVIES ET AL. assumptions regarding the nature of the fluxes on the slope prior to the simulation period. In all the simulations reported here, the assumption that the catchment is initially in steady state with a homogeneous recharge rate equivalent to a steady specified initial discharge is used. Initial slope saturation is then calculated in a series of finite steps along the length of the slope. The size of this length will affect the resolution at which saturation is represented in the slope, and the computational requirements. In this case, a slope length step size of 1x D 1m is used. The saturation at the top of the slope is calculated based on the initial upslope flux per unit width q u (m 2 /day), initial lateral flux per unit length along the slope q l (m 2 /day), and initial recharge from the unsaturated zone per unit area r o (m/day). This gives an expression for the volumetric flow rate across the slope: Q i D Q i 1 C 1xq l C 1xW i r o 6 where W i is the slope width at step i, 1x is the slope step length and: Q 1 D W q u C 1xq l C 1xW 1 r o 7 with W the slope width at the upslope boundary. Using the assumed hydraulic conductivity profile, and a hydraulic gradient equal to the local surface slope, the discharge may be defined as: q D K o exp f D h sin ˇ 8 where ˇ is the local surface slope angle, which is assumed constant over the 4 m slope length in this case. Integrating this downslope flux within the water table and equating this with our volumetric flow rate provides a water table depth profile along the slope length: d i D 1 f ln (exp fd i f K sin ˇ Qi ) 9 The number of particles needed to fill the pore space below the water table consistent with the saturated flux can then be calculated by integrating the porosity function below the water table. Taking the numerical integral of the porosity function between the soil depth D and the local water table depth d i gives: S i D 1 2 D i d i W i ( 1 b ln ( 1 C K b v ) exp fd i C 1 ( b ln 1 C K )) b exp fd i 1 v which defines the local water storage in the saturated zone, expressed in metre cube per unit length of slope. This may then be converted to the number of particles required, depending on the chosen volume of a single particle. Particle volume dictates the degree of accuracy with which the water storage on the slope can be represented, and as such a smaller volume is desirable. However, increased particle numbers increase computational overheads. Here, a particle volume of Ð5 l is used within this simulation, which provides adequate representation of the required input levels and good repeatability across realizations. This particle volume can be thought of as representing the equivalent number of water molecules, which might be distributed locally in the pore space, but all of which have the same velocity. The particles are placed randomly within the saturated depth of each slope length division and a velocity is assigned randomly with an exponential distribution of velocities according to Equation (3) i.e. the majority of particles are slow moving with exponentially fewer faster particles. The definition of particle velocities is further illustrated in Figure 3, where the depths and velocities for a random set of particles are given, the distribution of which is highlighted at depth a. In the unsaturated zone, the movement of all particles is assumed to be vertical and to be moving under a unit hydraulic gradient. Thus, capillarity is assumed (at least for this wet soil) to have a minimal effect on the mobile water flow velocities. The number of particles to place within the unsaturated zone is only dependant on the initial steady recharge rate: D 1 ( b ln 1 C r ) ob 11 The particles representing the initial storage in both unsaturated and saturated zones prior to an event can be labelled as old or pre-event particles (e.g. Sklash, 199). Defining the input boundary conditions Precipitation is intercepted within the Gårdsjön catchment by a plastic roof, and subsequently rainfall is simulated by irrigating the area with lake water. The model requires assumptions to be made about the way in which the surface inputs are distributed and how they then enter the soil matrix. Here the simplest (no parameter) set of assumptions is made that the surface inputs are distributed uniformly across the surface of the slope. Velocities are assigned in the manner Depth a D vo va vsmax Velocity v o Velocity definition in the MIPS model No. of particles vo Velocity Maximum velocity Particle velocities Velocity distribution at depth a Figure 3. Particles at each depth are assigned with an exponential distribution between a minimum and maximum velocity according to v o and Equation (3) respectively va Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

6 A DISCRETE PARTICLE REPRESENTATION OF HILLSLOPE HYDROLOGY 367 described in the previous section, depending on whether the area of slope at which the particle is located is saturated, unsaturated or is experiencing overland flow. The particles can be labelled by the time of entry to the slope to determine residence time distributions or simply as event water particles. Inputs from the upslope and lateral regions of the catchment are also considered within the model. Water from these areas is assumed to enter directly into the water table of the slope. The volume of input from these areas is determined based on the volume of water sprinkled on their respective surfaces. However, the time at which these inputs reach the slope is modelled using a simple time-invariant linear store. The upslope and lateral regions can be represented as two stores in parallel, with different time constants, that feed into the upper and lateral boundaries of the tracer experiment plots. In the flanking regions, a groundwater zone only existed in conjunction with irrigation events, and as such, the time constant for these areas is set to allow a fast response at 1Ð5 days. The convergent central valley region, however, maintains a groundwater zone during periods of non-irrigation, suggesting that the time constant of the store representing this region is longer. A value of 4Ð5 days is used within the simulations presented. These linear storage models are clearly simplifications of the real processes at work within the catchment but serve to get the timing of the boundary inputs to the tracer plot approximately correct. A nonlinear or dynamic storage could be used but at the expense of adding further parameters. Losses due to evapotranspiration are also neglected within this version of the model, as the level of evapotranspiration in the period of simulation is relatively low (the period simulated is November to January). However, as noted earlier, these losses are easily handled within the current model by assuming a sink state within the transition probability matrix. Defining the step equations Saturated particles are assumed to move laterally, in parallel with the slope bed. Hence, the horizontal change in particle position may be defined by: x i D x i 1 C tan ˇ vt s 12 where t s is the time step, x is measured horizontally and v is defined for the unit hydraulic gradient. If the width and depth of the soil profile changes in the downslope direction, mass balance is invoked so that the depth of the particle is adjusted to remain proportional with the changing depth of saturation. In the tracer experiment plot it is assumed that only the depth of the soil profile changes downslope. Particles within the unsaturated flow are assumed to move vertically, and their depth is updated as: y i D y i 1 C vt s 13 After moving the particles during the time step, and adding more particles according to the inputs of the system, the water table level across the slope is recalculated. The water table recalculation is achieved by: 1. Counting the number of laterally moving particles gives a first estimate of the local volume of particles below the water table, V, within the unit length increment 1x. 2. Approximating the porosity as a straight line function of depth over the depth of the soil (Figure 3) the water table height can be found by treating all laterally moving particles as if they are below the water table: V D Ð5 d C D D d W1x 14 and d D C D d 15 D so that it is necessary to solve the quadratic equation: D d 2 C 2 d C 2V D W1x D D C D After redefining the water table levels, particles that were previously unsaturated, but now lie within the saturated region are relabelled as saturated and vice versa. 4. If the slope is over-saturated, resulting in overland flow, then the porosity above the surface is assumed to be unity. Whilst Equation 3 still holds for the overland particles, the velocities are assigned with a uniform distribution within the overland region, rather than an exponential distribution which represented a majority of matrix flow (Figure 3). A multiplier is also used in order to achieve average velocities in keeping with observations of overland flow (Emmett, 1978). Water in the overland region runs-on if it enters a slope area that is unsaturated. There is no infiltration rate controlling this process at present, but this is something that may be included in the future. Defining the transition probability matrices Treating the translation of water particles between soil pathways as a transition probability matrix is the most conceptual part of this discrete model in its representation of both matrix and preferential flows, and the most difficult to parameterize as it will not normally be expected to be constant over the whole range of flows to be considered. In the general case, it would be expected that the probability of exchange will vary with the wetting of the soil due to capillarity effects, evaporative demand and local interactions between macropores and matrix. Again, the simplest possible case for the initial run is made, in which all the off-diagonal transition probabilities are set to zero i.e. once a particle has been assigned a velocity, it will travel at that velocity unless it exfiltrates onto the surface, or infiltrates from an overland flow. The Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

7 368 J. DAVIES ET AL. Table I. A summary of the parameters and boundary conditions required by the MIPs model Parameter Units Initial values Slope length Slope width Soil depth Slope angle Downslope length increment Time increment Particle volume Hydraulic conductivity at soil surface Exponential parameter for decline of hydraulic conductivity with depth, f Exponential parameter for the increase of velocity with saturation, b Minimum mobile velocity, vo Mean residence time of lateral regions Mean residence time of upslope regions (m) (m) (m) (deg) (m) 4Ð 1Ð Ð3 Ð8 2Ð87 1Ð (h) (l) (m/d) 1Ð Ð5 5Ð Figure 4. Visualisation from the MIPs simulation 1 (m ) ( ) (m/d) 3Ð5 25Ð Ð1 (d) 1Ð5 (d) 4Ð5 velocity assignments therefore represent the Lagrangian velocities over the pathway of a particle towards the outlet collection point. The effect of the various simplifying assumptions will be explored in future work. Parameter summary Table I gives a list of the parameters required by this conceptualization of the flow processes. To summarize the parameterization requirements: the model at this site requires conductivity profile parameters (K and f) to be derived from field observations (in this case made by Nyberg, 1995). Porosity parameters ( and b) are derived analytically given the assumptions of the exponential velocity distribution. The mean residence times for the lateral and upslope input storages are calibrated (by trial and error in this initial study) to obtain the appropriate hydrograph timing. TRACER TRANSPORT HYPOTHESIS SIMULATIONS The model was run to simulate the period within which a tracer experiment was performed at the Ga rdsjo n site. On 2 November 1992, Tritium (3 H) was line-injected at a depth of 6 cm, 4 m above the outlet in the central valley region. To model the tracer transport, 1 tracer particles are injected into the simulation. These particles Copyright 211 John Wiley & Sons, Ltd. are hypothetical; in that, they do not have a volume (and therefore do not contribute to the saturation of the slope), but they move in the same manner as the water particles in the slope. This set-up is useful in allowing a finer representation of the tracer in the system. An example visualization of the MIPs simulation is given in Figure 4. This diagram displays the 2D representation of the slope, underlain with an impermeable base and shows the water particles within the saturated and unsaturated zones, alongside the tracer particles. Only 2% of the simulated particles are plotted to aid clarity. The merits of a particle-based simulation are evident within this visualization. Particle tracking allows information regarding water and tracer location, trajectory, age and source to be easily obtained, monitored and visualized. Flow conditions following the injection were highly transient. A large, double peaked event occurred 27 h after injection, which produced some surface run-off. Another two large events occurred before the beginning of January. The hydrograph and precipitation during this period are shown in Figure 5. The MIPs model results are also given in this figure. The results of an ensemble of 5 simulations (each with the same parameter set and approximately 1Ð2 million particles) are plotted. An ensemble is simulated in order to average the effects of the random aspects of the model. However, the results in this case do not significantly vary, suggesting that the resolution of each of the realizations was sufficient to represent the range of velocities and particle pathways in this case. It can be seen that a good correspondence is attained in the discharge result for the second and third events. Response to the first double peaked event is less accurate. This may be attributable to the model not performing as well during the first wetting of the slope. Once the slope is wet, the model performance improves. A number of hypotheses (H1 H4) with respect to tracer injection and transport were simulated and examined (Beven, 21). In what follows these hypotheses and their results are described and discussed in turn. The first hypothesis is the simplest as suggested by the information available regarding the experiment. Hydrol. Process. 25, (211)

8 A DISCRETE PARTICLE REPRESENTATION OF HILLSLOPE HYDROLOGY 369 Precipitation, mm Discharge, mm 1 Catchment input 2 1/11/92 1/12/ Hydrographs for MIPs simulations 1/11/92 1/12/92 Date Model Data Figure 5. Modelled and measured hillslope hydrograph for a slope section of G1 catchment. The results of 5 simulations run with the same parameter set and a particle volume of Ð5 l are plotted. Multiple simulations are made in order to assess the influence of the random aspects of the model on the final result. It is important that particle numbers are high enough to represent the velocity distributions and saturation levels adequately. The high repeatability of the results here indicates the water within the system has been represented sufficiently at the simulated resolution Tracer Discharge % of input Tracer Breakthrough for Hypothesis 1 1/11/92 1/12/92 Date Model Avg. Model Measured Tracer Figure 6. Tracer breakthrough results for Hypothesis 1 where the MIPs simulation is set-up in keeping with the experimental information available. An ensemble of 1 simulations has been run to remove the effects of random variation in particle positions/velocities, and an average for this ensemble is given H1: Standard simulation In hypothesis H1 tracer is injected at 6 cm depth, in keeping with the experimental set-up, at the top of the slope. The results of this simulation are given in Figure 6. It can be seen that despite the reasonable correspondence between the measured and modelled hydrograph, the tracer results are extremely inaccurate. The tracer particles in the simulation travel through the slope far too slowly and there is a large delay (in comparison to the experimental results) before any particles arrive at the outlet. The initial velocity distribution of the tracer particles results in the majority of the particles travelling very slowly. Changes to this velocity only occur due to the variation in the depth profile of the slope. Particles may also increase in velocity if they rise in the profile due to rising of the water table. However, this has a marginal effect within this simulation as the slope remains highly saturated throughout the simulation. Hence, the tracer particles remain deep within the soil throughout the simulation. How representative is this result of reality? The concentration of tracer within the slope profile was monitored during the experiment in a piezometer nest downslope of the injection point. Concentrations were also recorded in the overland flow. It is stated in Nyberg (1995) that there was very little tracer concentration at depth below 6 cm, a medium level of concentration at 6 cm depth and highest concentrations were found in the piezometers above 32 cm depth. Concentrations at 25 cm and within surface run-off were higher than in deeper layers and as such it was concluded that most of the 3 H flowed towards the superficial soil layers after injection. The flow of tracer within H1 does not occur in this manner, as particles remain too deep within the profile. What conditions or mechanisms could lead to tracer rising rapidly within the slope profile? Two mechanisms by which tracer could rise within the soil have been identified: disturbance in the soil on installing the injection pipes could have led to tracer rising quickly along the disturbed soil around the injection pipes; and lateral inputs to the slope arrive at the base of the profile, forcing resident waters upwards, resulting in tracer rising within the profile. These mechanisms are explored in the simulation hypotheses that follow: H2: Disturbed injection depth In this hypothesis the tracer is uniformly injected at depths 6 cm representing a rapid rise of tracer in the soil profile on injection due to soil disturbances. The results of this are given in Figure 7. It is evident that the volume of tracer recovered in the simulation is still far too small in comparison to the measured data. However, there is no delay between actual tracer exit and simulated tracer exit within this scenario, suggesting that the range of velocities achieved by injecting the tracer over the depths of 6 cm has produced some tracer particles that travel with sufficient speed. The peak of outflow of the model also coincides with the real peak tracer outflow. The second release of tracer induced by the second event does not feature within the simulated result. The range of tracer depths is now more realistic within this scenario. However, no tracer rises above the surface Tracer Discharge % of input Tracer Breakthrough for Hypothesis 2 1/11/92 1/12/92 Date Model Avg. Model Measured Tracer Figure 7. Tracer breakthrough for simulation Hypothesis 2 where the tracer is injected at depths 6 cm Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

9 361 J. DAVIES ET AL. Tracer Discharge % of input Tracer Breakthrough for Hypothesis 3 1/11/92 1/12/92 Date Model Avg. Model Measured Tracer Figure 8. Tracer breakthrough and hydrograph for Hypothesis 3 where the lateral inputs arrive at the base of the slope and tracer is injected at 6 cm depth Tracer Discharge % of input Discharge, mm /11/92 1/12/92 Date Hydrographs for MIPs simulations 1 5 Tracer Breakthrough for Hypothesis 4 Model Avg. Model Measured Tracer Model Data into overland flow within the model, which is not in keeping with the observed evidence (Nyberg, 1995). The results of H2 suggest that rapid rise of tracer in the soil on injection may have taken place, but perhaps further factors such as lateral inflows from the base or a large amount of tracer entering a preferential flow path may have also occurred, influencing tracer transport considerably. H3: Lateral inputs at slope base The second suggested mechanism for upwards tracer movement was lateral inflows from the base of the slope. This possibility is explored in H3. As in H1, the tracer is injected at 6 cm depth; however, the lateral inputs are now added at the base of the profile displacing the older waters upwards. This is achieved in the model by calculating the volume of the water entering laterally, displacing the existing water upwards by a height corresponding to that volume, and placing the new lateral input particles within that height. Figure 8 gives the results of this hypothesis. The simulated tracer breakthrough, once again, does not coincide with the measured result. There is a delay in the modelled tracer transport. This is the time spent rising in the profile before the velocities are greatly increased. A substantial pulse of tracer then occurs on day 8 owing to a large percentage of the tracer particles entering the upper soil layer and overland flow region in a small time frame subsequently flowing very quickly to the outlet. Some reinfiltration after the first event and further tracer rising forced by new lateral inputs results in a second release of tracer coinciding with the second measured release. Whilst the results of H3 are far from the measured tracer breakthrough, the volume of outflow is vastly improved from H1 and H2, and the depth of flow is now much more representative of that witnessed within the slope. The second release of tracer also suggests that a mechanism similar to H3 may be a feasible suggestion for the processes occurring within the real slope. The delay to peak is the major error within this hypothesis. This may be mitigated by spreading the injection depth upwards (as suggested by H2). Hence, the situation where both lateral influx from the slope base and disturbed injection depth occurs is examined in the next hypothesis. 1/11/92 1/12/92 Date Figure 9. Tracer breakthrough and hydrograph for Hypothesis 4 where the injection depth is 6 cm and lateral inputs are introduced at the base of the slope. The hydrograph is provided to illustrate the correlation of tracer discharge to overall discharge within the model, and indicate that improvement in this area may result in improved tracer results H4: Lateral inputs at slope base and disturbed injection depth H4 is a combination of H2 and H3, where the injection depth is disturbed and lateral inputs appear at the slope base, the results of which are shown in Figure 9. The tracer discharge results of this hypothesis are much more promising. The volume of tracer released in the first and second events are comparable to the modelled volumes. Injecting the tracer over the slope depth of 6 cm reduces the delay involved in tracer entering fast pathways. The timing and peaks of the tracer release in the first event do not exactly coincide with the measured tracer. However, they match as well as may be expected given the accuracy of the simulated discharge in this first event. Improvements to the discharge results may be instrumental in enhancing the tracer breakthrough. The inaccuracy of the discharge on the first wetting of the slope after a dry period may be attributable to the influence of the upslope areas and the simplified initialization of the slope at the start of the simulation. The upslope inputs to the slope are the largest due to the area of this region. As mentioned previously, the input from this region is simulated using a linear store which acts as a low pass filter. However, approximating the behaviour of the upslope region as linear is a strong assumption. During the wetting of the catchment after a dry spell the output behaviour is likely to be changed due to storage-discharge hysteresis. A nonlinear store could be included to represent this hysteresis. Another alternative would be to simulate the whole catchment, eliminating the need to estimate upper and lateral boundary fluxes. However, this solution is computationally demanding, due to the greatly increased particle quantities and water table calculations required. Nonetheless, catchment-scale simulation for this experiment site is to be addressed in future work, in order to simulate step changes in input isotope concentrations that were performed across the whole catchment. Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

10 A DISCRETE PARTICLE REPRESENTATION OF HILLSLOPE HYDROLOGY 3611 Discussion and extensions The results of applying the MIPs model to the Gårdsjön tracing experiment are promising. Both slope discharge and tracer transport have been reproduced with some success within a unified framework consistent with the field estimated hydraulic conductivities at the site. The MIPs model has been used in this instance not as a definitive representation of the system, but as an exploratory tool in determining the active system processes. It was shown that a number of factors relating to the tracer injection and flow regime needed to coincide in order to produce both discharge and transport results that were consistent with experimental data. This might be considered a surprising result. We expected that, in keeping with the equifinality concept of Beven (26b), similar results could have been obtained by multiple sets of parameter values in the model, rather than this coincidence of conditions. However, this result is indicative of the highly state-dependant nature of flow and transport responses within a subsurface flow domain, a point that was acknowledged by Freeze when he observed that, for a Darcy-based model, significant subsurface saturation would only occur under rather stringent conditions (Freeze, 1972). It is also indicative of the additional information content of combined discharge and tracer transport data. This has been acknowledged by other studies (Güntner et al., 1999; Uhlenbrook and Leibundgut, 22; Soulsby and Dunn, 23; Iorgulescu et al., 25; McGuire et al., 27; Tetzlaff et al., 28), though not all of those have correctly differentiated between wave celerities and particle velocities in fitting both flow and tracer concentration data. In particular, information regarding the depth of tracer flow has been useful in rejecting hypotheses about the Gårdsjön slope. For example, mass transfer of tracer alone could have been replicated by assigning the tracer particles with a different, faster velocity distribution. However, this would not have been consistent with the observed levels of tracer in the near-subsurface and overland flow. It has proven very useful in this work to have the possibility of thinking about the flow processes directly in terms of velocity distributions rather than velocities inferred from a pressure field solution. Nonetheless, it is anticipated that there will be other parameterizations or conceptualizations of the slope within this MIPs framework that will provide equally valid results that are consistent with the observations. The unknowability of the details of the flows on the slope would be reflected in these multiple behavioural representations. Further, in-depth exploration of possible parameterizations or conceptualizations is required to assess the uncertainties involved. The influence of velocity distributions and soil parameters in particular need to be formally examined. Further conceptual additions should also be considered. For example, under the assumption that the assigned velocities are Lagrangian velocities, it has not been necessary to consider explicitly the role of pathway exchange within this paper. Pathway exchanges can be used to provide varying mobilization of particles, and recent work suggests that this may have a significant effect on residence times. Capillarity effects can be handled in this way (Ewen, 1996). Other extensions are possible. The effect of vegetation on the distribution of inputs to the surface is easily implemented. The effects of rooting depth distributions, preferential transpiration (like those described in Brooks et al., 21) and bedrock leakage may also be explored within this framework through use of transition probability matrices that describe the probability of particles moving from one state to another. Longer simulations will also require the effects of transpiration to be considered. Deeper soils might require the assumption of a constant lateral hydraulic gradient to be relaxed. CONCLUSIONS The first implementation of the MIPs model (Beven et al., 1989) was an attempt to acknowledge the complexity of the unsaturated and saturated zone processes that occur at the sub-element scales of continuous representations with a discrete representation of preferential flow pathways (Beven, 1989, 21). However, this model was limited to steady-state simulations only. This paper has presented progress in extending this model to represent dynamic slope conditions. The MIPs concept was discussed and its main requirements outlined. Several realizations of the MIPs model were then described in application to a tracing experiment at Gårdsjön in Sweden. Results show that the MIPs model can reproduce both flow and transport behaviour at a hillslope scale with reasonable success. However, the coincidence of a number of assumptions about conditions on the slope was required in order to produce a result that was consistent with the available discharge and tracer transport data. This may signify how informative the combination of discharge, tracer transport and flow depth data is in diagnosing the flow regime and may also indicate that the flow and transport response is highly dependant on slope conditions. However, further research is required in assessing the uncertainty and equifinality of the results. A GLUE analysis (Beven and Binley, 1992) will be performed on this model as a next step. ACKNOWLEDGEMENT This work was funded by the UK s Natural Environment Research Council (NERC) under grant reference NE/G17123/1. REFERENCES Abou Najm MR, Jabro JD, Iversen WM, Mohtar RH, Evans RG. 21. New method for the characterisation of three-dimensional preferential Copyright 211 John Wiley & Sons, Ltd. Hydrol. Process. 25, (211)

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