Integrated modeling of flow and residence times at the catchment scale with multiple interacting pathways



[1] There is still a need for catchment hydrological and transport models that properly integrate the effects of preferential flows while accounting for differences in velocities and celerities. A modeling methodology is presented here which uses particle tracking methods to simulate both flow and transport in multiple pathways in a single consistent solution. Water fluxes and storages are determined by the volume and density of particles and transport is attained by labeling the particles with information that may be tracked throughout the lifetime of that particle in the catchment. The methodology allows representation of preferential flows through the use of particle velocity distributions, and mixing between pathways can be achieved with pathway transition probabilities. A transferable 3-D modeling methodology is presented for the first time and applied to a unique step-shift isotope experiment that was carried out at the 0.63 ha G1 catchment in Gårdsjön, Sweden. This application highlights the importance of combining flow and transport in hydrological representations, and the importance of pathway velocity distributions and interactions in obtaining a satisfactory representation of the observations.

1. Introduction

[2] One of the major challenges in hydrological science is the realistic and consistent representation of both hydrographs and water residence times in catchment systems [Bachmair and Weiler, 2013; Beven, 2010; Kirchner et al., 2000; McDonnell et al., 2010; McGuire and McDonnell, 2006; Rinaldo et al., 2011]. It remains a challenge because, while residence times are intrinsically linked to flow pathways and hydrograph responses at hillslope and catchment scales, the flow of water is affected by small-scale heterogeneities in complex, nonstationary and nonlinear ways as a catchment wets and dries. This complexity is compounded by the fact that the hydrograph is controlled by the changing distribution of celerities (pressure wave velocities) in the system, and the residence times by the changing distribution of velocities (water transport velocities). As noted by Beven [Beven, 2010, 2012a; Beven et al., 1989] for the simplest kinematic case, the celerity or wave speed is an inverse function of local storage deficits above the water table, which will be small when the catchment is wet. On the other hand, the distribution of pore water velocities will be a function of local storage beneath the water table. Celerities will, therefore, be faster than pore water velocities [see also Beven, 2012b, p. 145; Davies and Beven, 2012].

[3] Flow pathways also contribute to the complexity and nonstationarity of water residence times. It has been extensively shown throughout the literature of the last 30 years that preferential flow features have an important influence on flow and transport processes [Abou Najm et al., 2010; Anderson et al., 2008; Bertolino et al., 2010; Beven and Germann, 1982, 2013; Bogner et al., 2012; Chappell, 2010; Frey and Rudolph, 2011; Greve et al., 2010; Hangen et al., 2005; Henderson et al., 1996; Holden, 2004; Lin, 2010; Morales et al., 2010; Nobles et al., 2010; Sade et al., 2010; Weiler and Naef, 2003; Weiler and Flühler, 2004]. Preferential pathway features such as root channels, soil pipes, fissures, and soil fauna burrows allow water to travel at a range of velocities well above those of water traveling through tight matrix pores (note that this was also recognized by Robert Horton in the 1930s, who perceived macropore pathways as important pathways for both water and air during infiltration, see Beven [2004]). Dynamic mixing or displacement between preferential features and the soil matrix will also influence residence times.

[4] It is perhaps the complexity of this challenge that has resulted in a lack of progress toward developing integrated representations of water flow and transport that directly incorporate the influence of preferential flows [Beven and Germann, 2013]. Where models of flow and transport have been developed, they have tended to be of two types: the first is to rely on the implicit mixing characteristics of conceptual storage elements [e.g., Christophersen et al., 1990; Genereux, 1998; Hooper et al., 1990; Ogunkoya and Jenkins, 1993]; and the second is to relate transport to the local mean velocities predicted by a Darcy-Richards continuum approach under the assumptions of local continuity of gradients and homogeneous soil hydraulic characteristics at the calculation element scale [e.g., Kollet and Maxwell, 2006; Qu and Duffy, 2007; Ebel et al., 2008]. In both cases, extension to dual-domain representations have been made [Beven and Germann, 1981; Frey et al., 2012; Gerke and Van Genuchten, 1993; Haws et al., 2005; Hoogmoed and Bouma, 1980; Köhne et al., 2009; Larsson and Jarvis, 1999; Simunek and van Genuchten, 2008; Simunek et al., 2003; Weiler and McDonnell, 2007], but have not generally reflected the distributions of potential pathways for water flows.

[5] Typically, transport models are superimposed over these flow solutions in a way that is consistent with local predictions of mean velocities (e.g., the particle transport model SLIM-fast is transposed over ParFlow [Maxwell and Kastenberg, 1999], or ModPath over ModFlow [McDonald and Harbaugh, 1984] or MOCDENS3D where an advective solution that considers the density of saline waters is imposed over ModFlow [Oude Essink, 2001]). Solutions have used analytical or numerical solutions of the Advective-Dispersion equations (ADE) [e.g., Ahlstrom et al., 1977; Prickett et al., 1981; Hassan and Mohamed, 2003; Frey et al., 2012; Ramasomanana et al., 2013], which in its simplest form implies a symmetric Gaussian distribution of velocities around the local mean velocity. One solution technique is to use particle tracking. Particle tracking methods are flexible and have been used to represent “anomalous” transport [Haggerty and Gorelick, 1995], fractional ADEs [Benson et al., 2000; Schumer et al., 2001], and nonGaussian probability distributions [Berkowitz et al., 2006; Dentz and Berkowitz, 2003; Scheibe and Cole, 1994]. While these transport modeling approaches allow some representation of local heterogeneities, they are dependent on the estimation of local mean velocities from flow models that do not explicitly allow for heterogeneity at the subdiscretization scale.

[6] Thus, there is still a need for a consistent modeling approach. The multiple interacting pathways (MIPs) model is a recently developed methodology that represents flow and transport in an integrated solution, directly acknowledging preferential flow features and their possible interaction. Figure 1 summarizes the MIPs concept. The MIPs model is particle based: water entering the catchment is discretized into water particles/packets (here, the term particles and packets are interchangeable as they refer to a discretized volume of water with specified velocity characteristics, not a single water molecule or contiguous storage volume). The approach is innovative in that, unlike classical implementations of random particle tracking transport solutions, MIPs uses the particles to simulate both flow and transport in a completely consistent way, i.e., the movement of water particles according to mechanistic step equations provides the flow, and simultaneously these can be tracked throughout the system to provide water residence times and conservative solute transport. Each particle represents water moving at a certain velocity, regardless of where it might be in the pore space. In this way local heterogeneity, including preferential flow pathways can be simulated using generalized velocity distributions rather than needing to know the detailed soil structure. Water table levels are derived dynamically based on particle densities within discrete sections or grid cells of the catchment. No other approach to the authors' knowledge uses particle tracking at hillslope and catchment scales to represent the water flow directly without a prior solution for the local mean velocities (but see the SAMP model of Ewen [1996], for a similar approach including capillarity effects at the profile scale).

Figure 1.

Summary of the MIPs modeling concept.

[7] In this way, it is possible to represent a whole continuum of flow pathway velocities, which need not be symmetrical, since the distribution will include both relatively immobile water in the soil matrix and water in functioning preferential flow pathways. The distribution will integrate to a flux per unit area analogous to a Darcian velocity. Exchanges between the different velocity pathways are then implemented using transition probabilities. These may be made dependent on current pathway or hydrological conditions.

[8] This combination of particle tracking, velocity distributions, and transition probabilities makes MIPs a highly flexible framework for exploring flow and transport processes, allowing the modeler to think directly in terms of flow pathway velocities and interactions. The methodology has been previously verified against an analytical kinematic wave solution, under shared assumptions [Davies and Beven, 2012], and successfully applied to a plot scale tracing experiment [Davies et al., 2011] where a 2-D model was used in a hypothesis testing framework to model both discharge and tracer breakthrough. An interesting feature of the presented particle tracking approach is that by making direct use of particle velocities, the length scales of the flow domain are intrinsic to the model predictions. There is no reliance on assumptions of local homogeneity, implicit mixing assumptions, or numerical dispersion.

[9] This paper extends the previous work with MIPs in a number of ways:

[10] 1. A method of formulating 3-D models is detailed that allows MIPs to be applied up to the catchment scale.

[11] 2. The applicability of this 3-D formulation is demonstrated through application to a catchment-scale isotope experiment carried out at Gårdsjön, Sweden (details of which are given in section 'Gårdsjön G1 Catchment Isotope Experiment').

[12] 3. To explore the influence of flow pathways and storage exchanges in determining catchment flow and transport characteristics, several modeling hypotheses regarding flow pathways and pathway exchanges are made and compared to observations. The results demonstrate the importance of preferential flows and exchanges in determining catchment scale residence time distributions, and illustrate that these distributions are nonstationary as the catchment wets and dries.

2. Gårdsjön G1 Catchment Isotope Experiment

[13] The decisions made by the modeler in creating a mathematical representation of catchment hydrology are to some degree grounded in: (a) what is known (or supposed) regarding the characteristics of a catchment and its soils, flow pathways, and so on and (b) what the model is intended to describe, i.e., the experimental conditions it must represent and the variables it must produce for comparison with available measurements. Hence, before detailing the MIPs modeling methodology in section 'Catchment Scale 3-D MIPs Methodology', the site and experiment to which the MIPs has been applied is described here.

[14] Gårdsjön is an area on the Swedish West Coast, where experiments were started in 1991 to investigate the effects of acidification [Bishop and Hultberg, 1995]. In order to achieve this, a roof was built over the G1 catchment to intercept the natural rainfall. Precipitation in the catchment was then simulated by sprinkling higher pH water underneath the roof, pumped from the nearby lake. This also created a catchment-wide step change of δ18O in the input waters, as water from the lake is fractionated and hence enriched in δ 18O. This, therefore, brought about a unique opportunity to observe the exchange of old and new water at a catchment scale, and δ 18O concentrations were monitored in the inputs, run-off and groundwater for the subsequent 4 years (details of which are given in Rodhe et al. [1996]).

[15] Figure 2 provides an illustration of the G1 catchment topography. It is a small catchment with an area of 6300 m2, dominated by a central valley flanked by steeper slopes. It is vegetated by well-established Norway Spruce, providing many lateral rooting pathways through the topsoil. The podzol soils are developed on sandy-silty glacial till, with a clay content less than 10% deposited on a bedrock of gneissic granodiorite. Depth of soil cover varies between 0 and 1.4 m and is closely related to the topography, with deeper soils in the valley bottom (with a mean depth of 56 cm) and shallower soil on the steep slopes (mean depth of 35 cm). Mean thickness of the soil horizons were 9 cm for the O horizon, 5 cm for the Ae horizon, and 21 cm for the B horizon. The organic content was high with a mean of 15% mass and maximum of 30% mass in the valley region, correlating with areas with thick B horizons. The soil profile has a strongly nonlinear change in saturated hydraulic conductivity with depth [Nyberg, 1995].

Figure 2.

G1 catchment topography (a) at a 5 × 5 m grid scale and soil depth and (b) at the same grid scale, interpolated from measurements.

3. Catchment Scale 3-D MIPs Methodology

[16] In what follows, the elements necessary to define a MIPs model for a particular catchment are discussed. The formulation is generally transferable but is discussed in terms of application to Gårdsjön.

[17] The simplest possible assumptions are made as a starting point, after which changes may be made as new model hypotheses are formed. Hence, the model described first is the nominal model, formed on the basis of measured quantities and the simplest assumptions.

3.1. Catchment Geometry

[18] As the MIPs model is based on a kinematic representation of velocities, the physical geometries of the catchment as local slope gradients must be well represented. A 5 × 5 m digital elevation map of the catchment was used in defining an elevation grid within the model at the same scale (Figure 2). By taking a numerical gradient of these elevation data, a dominant flow direction, as defined by an Euclidean vector a with longitudinal and latitudinal vector components ax and ay, and slope angle α can be derived for each grid cell. This definition of dominant direction of flow is sufficient for application at relatively steep sites such as the catchment in question. For application to flat terrains or strongly undulating bedrock surfaces [e.g., Freer et al., 1997], then consideration of the relative water table height may be used rather than topographical derivatives. A soil depth survey was undertaken at the site to a similar scale [Seibert et al., 2011], and this was used to interpolate soil depths for each grid cell within the model (see Figure 2b). It would of course be possible to interpolate the elevation and depth data to provide a finer grid. However, the 5 m scale was used as a starting point.

3.2. Hydraulic Properties

[19] It is necessary to define the hydraulic properties of the soil profile within each cell. The hydraulic properties for the G1 site were defined and tested in the slope scale MIPs model [Davies et al., 2011] and are assumed here to hold for the whole catchment, truncated at the local soil depth. A derivation of these properties is included here for completeness.

[20] It is important that the hydraulic conductivity, porosity and assumed water velocity distribution are consistently defined. Here, assumptions are made regarding the hydraulic conductivity and water velocity distribution, from which the porosity is derived, ensuring that these three properties are consistently linked. However, it is worth noting that these properties do not need to be defined in this order. For example, in the presence of porosity and hydraulic conductivity data, a pore water velocity may be derived.

3.2.1. Hydraulic Conductivity

[21] At this site, the saturated hydraulic conductivity Ks significantly decreases with depth below the surface z, which is associated with a declining macroporosity and matrix porosity with depth. Within the model, Ks is assumed to be an exponential function of z as follows:

display math(1)

where Ko is the hydraulic conductivity at the surface and the parameter f determines the rate of decline in conductivity with depth. An exponential function is chosen as this corresponds with field measurements [Nyberg, 1995]. The local transmissivity may be defined by integrating equation (1) over the water table depth d:

display math(2)

where D is the depth to bedrock for a grid cell.

3.2.2. Velocity Distribution

[22] The water velocity v is assumed to be an exponential function of the filled pore space θ:

display math(3)

where vo is the minimum velocity for a unit slope and b defines how velocity exponentially increases with porosity. An exponential distribution is chosen as it agrees with our perceptual model of the processes, i.e., that the bulk of the soil water travels slowly within tightly bound matrix pores, with exponentially less soil volume occupied by preferential flow features that conduct water at higher velocities. In addition, it is a simple distribution requiring only one parameter to define. If data are available regarding the structure of the pore space then other distributions or multimodal expressions could be applied, at the expense of introducing further parameters.

3.2.3. Deriving Porosity

[23] Integrating equation (3) to the local porosity gives the saturated hydraulic conductivity, Ks, at depth, z:

display math(4)

[24] This can be rearranged to provide an expression for the saturated porosity θs that is consistent with equations (1) and (3):

display math(5)

[25] As θs is derived analytically from the assumptions of equations (3) and (4), it can be used to verify the choice of parameters, i.e., the saturated porosity profile produced should be consistent with expected values. The conductivity profile, porosity profile, and velocity distribution utilized here are plotted in Figure 3 and the parameters are given in Table 1. It can be seen from this figure that reasonable magnitudes of porosity are produced, and that the porosity decreases with depth, in line with the observed increase in macroporosity in the near surface.

Figure 3.

(a) Hydraulic conductivity and (b) porosity profiles for soil within the catchment. (c) An example of a random set of particle velocities across the profile depth. Particles are distributed with an exponential distribution so that at any depth, the majority of particles are slow moving with exponentially fewer faster particles [Davies et al., 2011].

Table 1. Parameter Summary With Simulation Values Where Appropriate
Parameter DescriptionSymbolUnitsValues
Grid cell longitudinal and latitudinal lengthΔxym5
Elevation above outletzm0–21
Soil depthdm0.025–1.4
Particle volumeVpl5
Simulation time steptsdVariable
Hydraulic conductivity at soil surfaceKsm/d50
Exponential parameter for hydraulic conductivity with depthfm−1−3.5
Exponential parameter for the increase of velocity with saturationbm−125
Minimum mobile velocityv0m/d0.0001
Root zone depthDrzm0.3
Fraction of local water table entering bedrock pathwayFBd−10.001–0.05
Fractional rate of exchange between pathwaysFed−10.05, 50
Overland flow velocity rangem d−10.001–15000

[26] Lacking complete spatially distributed information on how conductivity and porosity profiles vary across the catchment, it is assumed that these functions are valid for the whole catchment, regardless of changes in depth to bedrock, i.e., variations in soil depth will not change the conductivity function, but the transmissivity (equation (2)) will vary grid cell to grid cell.

3.3. Boundary Conditions

[27] It is assumed that all water entering the catchment arrives via the surface within the catchment boundary. No water enters or exits via the subsurface or overland from outside of this boundary. Precipitation in the catchment is relatively well defined as it is sprinkled across the catchment, but no explicit account is taken of the local effects of stemflows resulting from the interaction of the sprinklers and the trees on the catchment. Instead, the simplest assumption is made that the input waters are spread evenly over the catchment surface. Water particles are randomly allocated velocities from an exponential distribution (in line with equation (3)) on entry to the catchment.

[28] The particles are labeled with their time of entry to the catchment so that the percentage of old/new water in the discharge can be determined and residence time distributions calculated. Particles are also labeled with 18O concentrations. A particle volume of 5 L (equivalent to 8 × 10−4 mm over the catchment) is used within the simulations presented here. Each particle represents an equivalent number of water molecules traveling with the same velocity within the local pore space. In choosing the particle volume, a trade-off exists between decreasing the particle size to provide good resolution of input volumes, local storages, and discharge; and decreasing the total number of particles to reduce simulation times. The particle volume chosen here adequately represents the input levels and provides good repeatability in the discharge outputs (i.e., low random noise between different realizations of the same model run).

[29] It is also assumed that catchment water only exits via the outlet and evapotranspiration (ET), i.e., no water is lost through the bedrock. These are simplistic assumptions, and groundwater losses can also be incorporated in the MIPs framework through use of transition probabilities (as demonstrated in hypothesis H2 in what follows). Nonetheless, without information to inform the parameterization of these pathways and their associated storages, the simplest possible set of assumptions is made as a starting point.

[30] The trees at the site extend beyond the roof. It was found through comparison with prior conditions and a paired catchment that the roof had a minimal influence on ET. The level of actual ET during the experiment was estimated by Rodhe et al. [1996] using the SOIL model [Jansson, 1998]. Details of this model can be found in the cited sources; however, a brief summary of the process is given here. The SOIL model is a vertical multilayer heat and water flow model driven by air temperatures, precipitation, wind speed, radiation, and humidity. It uses the Richards equation and Penman equation along with these drivers to estimate the evapotranspiration rate. The root zone was assumed to be 30 cm deep. Within the model, the water particles can be taken from the soil to satisfy the transpiration rate estimated by the SOIL model by randomly choosing particles within the rooting zone across the catchment. In this version of the MIPs model, a probability characteristic was added to this process that stated that water nearer the surface is most likely to be lost via ET and that likelihood decreases linearly with depth into the rooting zone. This can be defined as:

display math(6)

where PETi is the probability of particle i to be available for ET, zi is the depth of particle i, and Drz is the rooting zone depth. Other probability characteristics could be used, but this linear option was favored as it is a simple no parameter assumption. It was assumed that uptake by roots from water stored in the soil did not involve fractionation and had no effect on δ18O concentrations (but see Brooks et al. [2009]).

[31] Using an estimate of actual ET from a model that incorporates a Richards equation is not entirely satisfactory but will not have a significant impact on the simulations in that it will be consistent with the catchment water balance during the simulated period [Rodhe et al., 1996]. In future applications, estimates of ET could be made within the MIPs model if the forcings were available.

3.4. Step Equations

[32] Particles traveling in the saturated zone are assumed to travel in parallel with the bedrock in the dominant flow direction as defined in section 'Catchment Geometry'. Therefore, at time t, for time step length ts the longitudinal and latitudinal plan position (xi (t), yi(t)) for a particle i is modified as follows:

display math(7)

where j is the grid cell which the particle occupies, α is the slope angle, and v is the particle velocity defined for a unit hydraulic gradient. The time step length is variable within the simulation and is always defined sufficiently small to ensure that particles do not cross more than one grid cell during the time step. On moving from one cell to another, the z location of the particle is adjusted such that it remains proportional to the local water table height.

[33] In the unsaturated zone, particles are assumed to travel vertically under a unit hydraulic gradient. This is consistent with viscosity dominated Stokes flow during infiltration and initial drainage [see Beven and Germann, 2013], but assumes that capillarity has a negligible effect on the mobile water flow velocities (which may be considered valid for the wet soils considered here). The step equation for unsaturated particles is:

display math(8)

[34] Following the movement of particles over the time step, the level of saturation in each grid cell needs to be updated to account for particles that have entered/exited. Within the model algorithm, the volume of water in the saturated region V in each grid cell is obtained by counting the number of particles below the current water table depth. This is then used to find the new water table depth by equating it to the filled porosity profile, which is approximated as a straight line function of depth (which is a reasonable assumption in this case, given equations (1) and (3) as shown in Figure 3, where the estimated porosities in any case show only a small range):

display math(9)

[35] This allows a simple numerical integral to be used to define an expression for V as a function of water table depth d as below:

display math(10)

[36] The new water table depth may then be found by rearranging equations (9) and (10) and solving for d:

display math(11)

[37] On redefining the level of saturation in the grid cell, the particles within that grid cell must be checked to determine whether they lie within the saturated or unsaturated zone, and relabeled accordingly such that the correct step equations are applied within the next time step.

[38] If the grid cell is saturated, the above-surface porosity is assumed to be unity and the water table calculation is adjusted accordingly. A uniform distribution of velocities is applied to particles in the overland flow pathway, the range of velocities of which is consistent with typical overland flow observations [Emmett, 1978] (see Table 1). Particles in overland flow pathways on entering a grid cell that is unsaturated are assumed to infiltrate. No limit is applied to the rate of infiltration at present; however, this may be implemented in future versions.

3.5. Transition Probabilities

[39] A transition probability expression has already been used for determining availability of particles for evapotranspiration (equation (6)). However, transition probability matrices can also be included to describe how water moves from one pathway to another in the subsurface.

[40] The simplest possible case is used as initial starting point, where all transition probabilities are set to zero. This means that once a velocity has been assigned to a particle, it will travel with that velocity unless it crosses the surface boundary or is lost by evapotranspiration. Under these assumptions, the velocities may be considered to be a Lagrangian representation of the velocities over the journey of a particle from input to output.

[41] An alternative set of transition probability assumptions is described and implemented in section 'H3: Pathway Exchange'.

3.6. Initialization

[42] Initial conditions in the catchment can be simulated in the model in one of two ways. If information is available regarding initial fluxes at the simulation starting date, then the slope can be prepopulated with particles assuming that the catchment is initially at steady state with a homogeneous recharge rate equivalent to a steady initial discharge. This is the method that was used in the slope scale simulations, and details of such an approach can be found there [Davies et al., 2011]. However, to avoid the assumptions of initial steady state conditions, an alternative is to presimulate a long run-in period starting with a dry catchment. This of course requires that monitoring data are available for a long period before the experiment. This also provides the advantage that isotopic content can be associated with all input particles entering the catchment prior to the main simulation period, allowing direct analysis of isotope content of the modeled outputs. Four hundred days of precipitation and discharge data prior to the commencement of the isotope experiment were available in this case. Potentially, waters older than 400 days are contributing to catchment outputs; however, it is assumed that this volume is small when compared to the total discharge volume and has a marginal effect on 18O concentrations. Given that the mean residence time as predicted by Rodhe et al. [1996] is 65 days, then this is considered to be a reasonable assumption.

4. Modeling Results

[43] The MIPs model can be driven with natural precipitation volume and 18O contents of rainfall measured at the site for 400 days prior to roof construction as an initialization period. 18O is based on bulk samples of daily precipitation and fortnightly pumped lake water, and each particle is labeled with the measured 18O for the period of precipitation which the bulk sample represents. After roof construction, inputs are then determined by sprinkling volumes and 18O measurements of the lake water used for irrigation. Due to the storage volume and residence time within the lake, the 18O concentrations of the irrigated water are much less variable than that of rain water. As such a lower sampling frequency is sufficient to capture changes in the 18O content. The roof was constructed on the 3 April 1991 and by 10 December 1991 18O values monitored at the outlet reached values equivalent to post-roof-construction waters. Hence, this is the main period studied with the model.

[44] Model discharge volumes are determined by counting the number of particles exiting at the outlet per day, and concentrations determined from the 18O values of those exiting particles. 18O contents can also be derived for waters exiting as evapotranspiration, including any assumptions to allow for fractionation in uptake and transpiration [Brooks et al., 2009]. However, as these were not measured in the field at the time of this experiment, they are not addressed here.

4.1. H1: Nominal Model

[45] Modeling results for the nominal hypothesis model H1, as described within the previous section, are discussed here. Figure 5 gives the measured and modeled hydrograph for the studied period, alongside measured and modeled 18O values. The discharge results produced by the model correspond well to the measured data, particularly considering that the parameters are not optimized, but based on first approximations from the measured properties of the site: time constants or gain terms have not been tuned in order to achieve peak timings or magnitudes.

[46] Conversely, the performance of this nominal hypothesis is not satisfactory when examining the 18O data. The model produces 18O concentrations that rise too quickly in comparison with the measurements. These high 18O values suggest that new (post-roof construction) waters are being transmitted too quickly to the outlet in comparison with the measured scenario, with old (preroof) waters remaining in storage within the catchment. This conclusion is supported when comparing the model performance in terms of old and new water against a mixing model that was determined by Rodhe et al. [1996], as shown in Figure 7 (mixing model details are omitted here for brevity, please see original paper for details). The H1 MIPs model releases too much new water too quickly. Care must be taken when comparing to the outputs from the mixing model however, as the mixing model contains assumptions regarding the end member values. For example, the new water forms one end member and as such it is assumed that the 18O concentration of the post-roof-construction water does not vary significantly through time. Labeling particles directly with 18O values circumvents this assumption (please refer to Rodhe et al. [1996] for a detailed discourse on the assumptions made).

[47] This poor performance in replicating transport data coinciding with good performance in flow illustrates the importance of creating models that represent multiple physical states which can be verified against multiple criteria. If evaluation of this model is made solely with respect to discharge data, then it would be considered behavioral. However, on examination of the transport results this hypothesis can be rejected. This process of rejection is vital if models are to be improved, as it forces us to consider why the models do not behave as expected and form new hypotheses which may be more valid [Beven, 2010, 2012b].

4.2. H2: Bedrock Pathways

[48] In the plot-scale simulations described in Davies et al. [2011], the hypothesis that produced the best results involved upward water movement in the lower valley region. As these simulations used a plot-scale 2-D model, this was achieved by feeding the input waters from the lateral flanks in at bedrock level in the slope and forcing preexisting waters upwards. This can be considered equivalent to water entering bedrock pathways in the steeper parts of the catchment, and displacing water in fractures that connect the flanks and the lower valley region. This is one potential mechanism of old water mobilization, as preexisting waters are forced into the highly transmissive shallow areas, and more old water enters the valley region from the bedrock pathways. Bedrock pathways have also been identified to play an important role in other experimental catchments, for example, in Coos Bay [Ebel et al., 2008]. Hence, a hypothesis that includes bedrock pathways is examined here within the 3-D model as hypothesis 2 (referred to as H2 herein).

[49] Simulation of such a process is more involved in the 3-D model in comparison to the 2-D case, where input waters could be directly rerouted. It is assumed here that:

[50] 1. water entering bedrock pathways displaces water into the lower valley region (as defined in Figure 4) via a piston-like mechanism;

[51] 2. the residence time of water in the bedrock fractures is longer than the simulated period, i.e., all the water displaced from bedrock pathways is “old”;

[52] 3. the volume of water entering the bedrock pathways each day is proportional to the water table level in the nonvalley regions, such that FB is the fraction of the local water table height entering the pathway;

[53] 4. water entering the lower valley region from bedrock fractures arrives at bedrock level, displacing existing waters upwards in the soil profile;

[54] 5. old water entering the lower valley region from bedrock pathways is given 18O concentration values set as −9% based on the 18O output on roof construction, which is in line with the analysis made in Rodhe et al. [1996].

[55] Simulation results with FB set as 0.001, 0.01, and 0.05 are displayed in Figure 5, denoted as H2a to H2c, respectively. It can be seen from this figure that inclusion of bedrock pathways within the model increases old water output. However, significant volumes need to be diverted via this pathway to bring the 18O values down to those measured during the experiment in the first few months, with the highest FB producing the best result in the first half of the simulation. This is also confirmed by examining the old water fractions in Figure 7, with increased old water being produced by H2. Too much old water contributes to the output during the second half of the simulated period, however. This may be remedied by defining a shorter residence time in the bedrock pathways (relaxing assumption 2 described above) such that increased volumes of new water contribute to outputs from September onward.

[56] However, this modification to H2 does not improve the discharge result produced by representing the bedrock flux (particularly when setting FB at the higher values required to match old water outputs in the first few months), which is not satisfactory. Diverting water into bedrock pathways alters the flow dynamics, producing higher recessions and lower peaks, which is particularly noticeable in the events after October (Figure 5).

[57] Hence, the bedrock pathway hypothesis, whilst increasing mobilization of old waters, is not a satisfactory hypothesis when both flow and transport data are evaluated.

Figure 4.

Definition of the lower valley region (denoted with dark gray shading) within the bedrock pathways hypothesis H2.

Figure 5.

Measured and MIPs simulated hydrographs for the G1 catchment 3 April to 10 December 1991. (top) The precipitation as measured at the site, and the measured 18O concentration (per mil) of the input waters (plus markers). (middle) The discharge as measured (black line) and modeled using H1 (blue line) and H2 (green, red, and cyan lines for hypotheses H2a–H2c, respectively). (bottom) The measured (plus marker) and modeled 18O concentrations (per mil) for discharge (using the same line notation as the middle figure).

4.3. H3: Pathway Exchange

[58] Another process that may mobilize old waters is pathway exchange. On arrival of new waters into the substrate, old waters can be displaced from slow matrix pathways into preferential features. This should be expected since preferential flow pathways will often have a limited length scale. At the end of a pathway, there may be a local buildup of saturation and displacement of stored old water into another fast pathway extending further downslope. This could be repeated a number of times at the length scale of the slope. This type of exchange can be implemented in MIPs using transition probabilities. In the nominal model hypothesis, no pathway exchanges were included and as such water existing in the catchment after a dry period would have a high frequency of low velocities, and new waters entering this system would have a full range of velocities as defined by the velocity distribution. As no exchange occurs between these volumes, old water is not displaced into fast pathways (although old water is mobilized by new water to an extent, as new water may increase the water table height in turn mobilizing old water in the unsaturated region).

[59] Hence, a new hypothesis (H3) can be formed that includes pathway exchanges through definition of transition probability terms, and tested against the flow and transport data. To form the simplest hypothesis that includes transitions between pathways, it is assumed that:

[60] 1. particles exchange only with other particles in their locality (in this case defined as within the saturated region of one grid cell);

[61] 2. all particles within a locality have an equal likelihood of pathway exchange with each other that is not dependant on current velocity (i.e., pathway);

[62] 3. exchange occurs at a constant rate such that a defined proportion of the saturated pore space exchanges per time step.

[63] These assumptions also introduce one additional parameter, Fe which is the fraction of the saturated pore space that exchanges per time step (in this case defined per day). Two hypotheses (H3a and H3b) are formed using these assumptions with slow and fast rates of exchange, respectively, where Fe is set at 0.005 and 0.5 day−1. The results of the simulations based on these hypotheses are shown in Figure 6.

Figure 6.

Measured and MIPs simulated hydrographs for the G1 catchment 3 April to 10 December 1991. (top) The precipitation as measured at the site and the measured 18O concentrations (per mil) of the input waters (plus markers). (middle) The corresponding discharge as measured (black line) and modeled using H1 (blue line) and H3 (magenta and yellow lines for hypotheses H3a–H3b, respectively). (bottom) The measured (plus marker) and modeled 18O concentrations (per mil) for discharge (using the same line notation as the middle figure). Insets in the bottom two figures give an enlarged view of H3b from September 1991 to October 1991 as denoted by the gray area in the main figures, illustrating the area of noncommensurability discussed in section 'Uncertainties'.

Figure 7.

Old water fractions as determined by a traditional mixing model (black line) [Rodhe et al., 1996] and the MIPs models H1 (blue line), H2a (green line), H2c (cyan line), H3a (magenta broken line), and H3b (yellow broken line).

[64] Discharge results are unchanged in comparison to the nominal model. However, exchanges improve the modeled 18O values, with the faster rate showing a good performance (also see Figure 7). The 18O concentrations produced by the model are now more event related, in comparison to the nominal modeling hypothesis. Thus, H3b may be considered behavioral.

5. Discussion

[65] The results show that whilst good discharge results can be produced by the MIPs model without including pathway interactions, pathway exchanges are potentially an important process in determining transport characteristics in the catchment. A fast rate pathway exchange hypothesis produced results which correspond well to measured data. It was also shown that whilst bedrock pathways increase old water mobilization, they change the flow dynamics of the system, producing discharge results that do not correspond with measured discharges. Therefore, the pathway exchange hypothesis is the best performing hypothesis from those tested.

5.1. Uncertainties

[66] It would be possible to “optimize” the pathway exchange rate parameter further to improve the fit of the model to the data. However, the usefulness of this exercise must be questioned. Due to uncertainties associated with the data, it is undesirable to over-fit this data. Some of the uncertainties surrounding the data and the model are discussed here.

[67] Whilst this experiment is unusual in that input to the whole catchment is controlled, making input volumes less uncertain than natural precipitation, there is still a small degree of uncertainty surrounding the sprinkled volume due to pumping rate accuracy errors. The measured 18O in precipitation, lake water and discharge have very small (0.02%) analytical errors associated with them; however, the sampling rate may not be sufficient to capture all of the 18O dynamics. This is less likely to be a problem for the sprinkled input, as the 18O concentration of the lake water varies only at low frequencies. However, it is evident within the discharge measurements that where events have been sampled at greater frequency (for example, during the mid-September event, and onward), rapid changes in 18O in the discharge are observed. It is likely that such peaks are also occurring in earlier events (such as the events in May) but are unobserved in the data. The model consistently produces spikes in 18O value in response to the events in May and June, which is particularly clear when examining the old water fractions in Figure 7. This might be mistaken for numerical variation resulting from particle-based model calculations, but all hypotheses consistently produce event driven peaks, and given that peaks are observed in later events with higher frequency sampling, then it is possible that these spikes in 18O concentrations are real and not numerical errors. Of course, it does not follow that all the spikes on the modeled 18O are nonnumerical errors, as there will be random realization variability errors when the discharge is represented by relatively small numbers of particles.

[68] One interesting area of error in the simulation is during the highly sampled event in mid-September (enlarged in the insets of Figure 6), when the H3b modeled 18O response fits the change characteristic well, apart from the fact that it is approximately 3 days in advance of the measured peak in 18O. The modeled discharge response is also in advance of that measured. This might be attributable to a soil moisture deficit error in the modeled representation after the dry summer period, producing a faster event response than that observed. Work to address this issue is planned considering soil moisture measurements and localized model storages.

[69] It is possible that there are other hypotheses that include different process mechanisms that would lead to good results for both discharge and isotope data within the limits of uncertainty of the available data. All that can be concluded from the hypotheses examined is that the nominal hypothesis is not valid, as it does not represent the observed transport characteristics sufficiently, the bedrock hypothesis is not valid as it does not satisfy the discharge requirements and that the exchange hypothesis performs well when tested against both flow and transport data. A full uncertainty analysis using generalized likelihood uncertainty estimation (GLUE) [Beven and Freer, 2001] and limits of acceptability procedures would be desirable here to better understand the influence of data uncertainties and the model space. However, at present the model has high computation demands that restrict such an analysis. Simulation times of course depend on the resolution of the discretization and the level of saturation influences time step length in this case. However, to give an indication of the simulation times at present, the model presented here takes approximately 24 h to complete one simulation year. Clearly, this is prohibitive to carrying out a meaningful uncertainty analysis given the large number of Monte Carlo runs that would be required. However, with the inexorable rise of computing capabilities and the potential for parallel processing with this method, full uncertainty analyze should be achievable in future work.

5.2. Bedrock Pathways

[70] A bedrock pathways hypothesis (H2) was examined here as a possible mechanism of old water mobilization as upward water movement in the lower valley region was found to be the most behavioral hypothesis when an artificial tracer experiment in the same catchment was modeled using a 2-D plot-scale model [Davies et al., 2011]. The upward displacement of waters, induced by water entering the valley at bedrock level, was required to move tracer into the highly transmissive near-surface zone, where most of it was observed to flow. Inclusion of upward water movement was also favorable in mobilizing old water in the 3-D catchment scale model. However, the diversion of water into bedrock fractures from upslope regions in the 3-D model changed the flow dynamics, resulting in a reduced discharge performance. Discharge volumes were unchanged in the 2-D model by the bedrock flow mechanism, as input volume timings that were determined by a linear store, were assumed to be unaffected by the inclusion of this mechanism.

[71] This does not necessarily invalidate the results of the 2-D model. Upward water forcing may be highly localized in the area of artificial tracer injection, resulting in lower volumes of water being diverted into bedrock pathways. However, this mechanism does not appear to be the dominant control of residence times at the catchment scale.

5.3. Further Analysis

[72] There has been a lot of recent discussion in the literature about the nonstationarity of residence times of water in catchments with wetting and drying over time [e.g., Hrachowitz et al., 2010, 2013; McDonnell et al., 2010; Rinaldo et al., 2011]. The MIPs modeling framework allows full analysis of residence time distributions, whether for increments of an input being transported to an output boundary (input residence time), increments of discharge at the outlet (output residence time), or storage within the catchment at any time (storage residence time). An example of the distribution of output residence times for a volume of output at a time t calculated from the model is given in Figure 8. The output distribution is analyzed for hypotheses H1, H3a, and H3b for all events during the simulation period. Two events of similar magnitude are highlighted: the first on the 13 September 1991 and the second on the 15 November 1991. The first event follows a relatively dry period and the second event is during a wet period. It can be seen from this figure that the output residence time distribution is nonstationary, with higher fractions of lower residence times produced during the wet period than in the dry period. The three plots of Figure 8 show that pathway exchange rates influence the degree to which residence times differ, with the fast exchange rate scenario producing the largest difference in the output residence time between the two highlighted events, and a wider range of residence times for all events is evident.

Figure 8.

Output residence times for the H1, H3a, and H3b (top to bottom figures, respectively) during all events in the simulation period. Two similar sized events are highlighted by the colored lines: An event with dry antecedent conditions (13 September 1991) is denoted by the solid line; and an event after a wet period (15 November 1991) is denoted by the broken line.

6. Conclusions

[73] The MIPs modeling methodology has been applied to a catchment scale isotope experiment, using a process-based hypothesis testing framework. Discharge and environmental tracer data in the form of 18O concentrations were reproduced by the model with reasonable success using a hypothesis testing approach. The nominal model (which included preferential pathways, but no dynamic exchange between pathways) reproduced discharge measurements well; however, the transport processes as indicated by the tracer data were not well represented by this hypothesis, with too much new water being present in the output volumes. Hence, new hypotheses were formed and tested. A hypothesis where bedrock fracture pathways were included was examined, resulting in increased old water volumes in discharge but reduced performance when comparing the measured and modeled discharge results. A final hypothesis was tested where pathway exchange terms were introduced to mobilize old waters. This produced both discharge and tracer results that correspond well to the measured values, and consequently this model hypothesis is considered behavioral.

[74] This flexibility to directly consider flow pathway velocities and interactions, and form process-based hypotheses which can be tested against multiple data sets, is a key strength of the MIPs methodology. The flexibility of the MIPs concept also permits many potential extensions to this framework including water chemistry, plant-water interactions, and sediment transport processes, making the MIPs concept a potentially useful tool for integrated process exploration.


[75] This work was funded by the UK's Natural Environment Research Council (NERC) under grant reference NE/G017123/1. The comments of several referees are acknowledged as leading to significantly improved presentation of the introduction and methodology.