STABLE ISOTOPES AS TOOLS FOR WATERSHED HYDROLOGY MODEL DEVELOPMENT, TESTING
A. B. MAZURKIEWICZ Oregon State University,
2. METHODS
2.1. Residence time theory
Water residence times are typically determined by black-box modeling of environmental tracers (e.g. 18O and 2H), in which input (rainfall) and output (discharge) tracer concentrations are used to estimate parameters of an assumed time-invariant distribution that represents the residence time [15]. The tracer composition of precipitation that falls on a catchment will be delayed by some timescale(s) according its physical properties and current state. Explicitly,
the stream outfl ow composition at any time δout(t) consists of past inputs lagged δin(t – τ) according to their travel time distribution g(τ) [15]:
(1)
where τ are the lag times between input and output tracer composition. This model is similar to the linear system approach used in unit hydrograph models;
however, only tracer is considered here and thus, g(τ) represents the tracer transfer function. Equation 1 is only valid for systems at steady-state or when the mean fl ow pattern does not change signifi cantly in time [15].
The convolution equation (1) must also include recharge weighting w(t – τ) so that the streamfl ow composition refl ects the mass fl ux leaving the catchment [16]:
0
0
( )
( ) ( )
in out t
g w t d
∞
= ∞
− −
g( ) (w t ) (t )d
τ τ τ
δ
∫
∫
−τ τ δ τ τ
(2)
The travel time or RTD or (g(τ)) describes fractional weighting of how mass (i.e. tracer) exists the system, which is equivalent to the probability density function (pdf) or transfer function of tracer in the stream. If the tracer is conservative, then the tracer RTD is equal to the water RTD. The defi nition of residence time herein is the time elapsed since the water molecule entered the catchment as recharge to when it exits at some discharge point (i.e. catchment outlet, monitoring well, soil water sampler, etc.). RTDs used in equation 1 are time-invariant, spatially-lumped characteristics of the catchment and thus describe average catchment behavior.
It is important to note that the timescale of the runoff response is different than the residence time because fl uctuations in hydraulic head (the driving force in water fl ux) can propagate much faster than the transport of conservative tracer or individual water molecules. Thus, the timescales between the rainfall-runoff response and transport (i.e. residence time) are effectively decoupled. This partially explains why the majority of a stormfl ow hydrograph is composed of ‘old’ water even though runoff response to rainfall is often immediate.
2.2. Study site
The Maimai research catchments are a set of highly responsive, steep, wet watersheds on the west coast of the South Island of New Zealand. Maimai has a long history of hillslope hydrological research (see [14] for a complete review). Maimai M8 is the watershed examined in this study. M8 is a small 3.8 ha study catchment with short (<300 m) and steep (average 34o) slopes with local relief of 100–150 m. Stream channels are deeply incised and lower portions of the slope profi les are strongly convex. Areas that could contribute to storm response by saturation overland fl ow are small and limited to 4–7%.
Mean annual precipitation is approximately 2600 mm, producing an estimated 1550 mm of runoff. The summer months are the driest; monthly rainfall from December to February averages 165 mm and for the rest of the year between 190 to 270 mm. On average, there are 156 rain days per year and only about 2 snow days per year. The M8 watershed is a textbook headwater research catchment: it is underlain by a fi rmly compacted poorly impermeable conglomerate and seepage losses to deep groundwater are negligible (estimated at 100 mm/yr based on 25 years of water balance data). In addition to being wet, the catchments are highly responsive to storm rainfall. Quickfl ow comprises 65% of the mean annual runoff and 39% of annual total rainfall. The period of record used for model simulation in this study was September–December, 1987. There were 11 major runoff events during this period with a maximum runoff of 6 mm/h.
A major diffi culty in generalizing the Maimai perceptual model to other watersheds in the area or watersheds in other areas is the considerable heterogeneity of hillslope topography, soils, vegetation and most importantly, fl owpath diversity. While the fl ow pathways to the stream are indeed complex at Maimai, as in other experimental watersheds, the residence times computed at the catchment outlet and internal to the hillslope show clear patterns of downslope aging. Stewart and McDonnell [16] showed that between-storm matrix water varied in age from approximately one week at the catchment divide to over 100 days at the main M8 channel margin. These are some of the shortest hillslope water residence times recorded in the literature and refl ect the steep, wet, responsive nature of the catchment.
2.3. Hydrologic model structure
A simple, conceptual model structure was developed to correspond closely to the runoff generation processes that dominate in the Maimai catchments.
The model represents a balance between a highly detailed conceptual model and the data available to support the numerical model. Our goal was to develop
a model with just enough complexity to provide estimates of the available measurements and residence times. Testing of this model structure against discharge and residence time measurements was then used to indicate whether additional complexity is warranted given the available data.
One of the fi rst decisions in the development of model structure was the implementation of reservoir theory. The reservoir theory is based upon the understanding that there exists in the environment discreet units of space for which we can know volumes and fl uxes, but that understanding the dynamics internal to those volumes would require more data then currently available.
As a consequence of this lack of data, the reservoirs are often assumed to completely mix and can be combined, through fl uxes to develop quasi- distributed simulations. The Maimai catchments are steep and are perennially wet, with the degree of saturation over 80 percent most of the year. In combination with the essentially impermeable bedrock, the climatology results in lateral water fl ow occurring in the transient saturated zone and moving under a gradient that is very well approximated by the topography. Measurements which have been used to develop this understanding have been distributed throughout the catchment, including upslope areas, through the use of wells and soil lysimeters. Given this level of detail, the reservoirs outlined within the model are distributed in space based upon a 10 m DEM. The volume of water within each reservoir is accounted for using the familiar continuity equation:
in out out
dV P SS ET SS SOF
dt = + − − − (3)
where V is the specifi c volume of water in each reservoir (m), t is current time (days), P is the precipitation rate, ET is the evapotranspiration rate, Kd is the loss to groundwater (in this case set to the measured yearly average of 100mm), SSout is the rate of subsurface outfl ow, SSin the rate of subsurface infl ow, and SOFout is the output rate of saturation excess overland fl ow. An increase in water volume results in an increase in the depth of the saturated zone, and a corresponding decrease in storage of the unsaturated zones. These depths are characterized by model parameters representing soil depth (SD) and porosity (phi). Routing out of each reservoir and SS is based upon the multidimensional fl ow algorithm [17] (Equation 5). Transmissivity (T) is assumed to decline with depth as a power law, with the degree of decline modulated by the power law exponent (PLE). The decline is defi ned following from [18] as:
PLE s
SD z PLE
SD z K
T ¸
¹
¨ ·
©
§ −
= * 1
)
( (4)
where z is the depth to the water table measured from the soil surface.
Subsurface fl ows (SS) follows from this equation as
SSi,j = Ti,j * Slopei,j (5) where subscripts indicate the multidimensional aspect of fl ow. We utilize calculated water surface elevations to determine slope in each direction.
Infi ltration is assumed to occur when the soil is not saturated, but when the saturation defi cit reaches zero, infi ltration cannot occur. In these instances, excess precipitation is ponded and subsequently delivered directly to the stream network as SOF. Existing data representing temperature based hourly ET estimates were read directly into the model during simulations. Instantaneous and complete mixing within each reservoir is assumed, and the unsaturated zone is not explicitly accounted for.
2.3.1. Conservative tracer: an additional state variable
Modeled tracer simulations were used to develop estimates of the MRT for simulations. The tracer model is similar to the hydrologic model, based upon reservoir theory and continuity. In this case, the model equation is defi ned as a mass balance of some arbitrary conserved tracer:
t
e in t
dM
dt =nC + tC – tC (6)
where Mt is the tracer mass within the model unit, n is rainfall rate (m/d), Ce is the concentration of tracer in rainfall, tout and tin are the water fl ux rates out of and into the reservoir and Ctout and Ctin are the concentrations of tracer out of and into the reservoir (taken from the previous time). The incorporation of this model is designed to provide an ability to track the source of water within the model through time. Time source composition can be derived through clear assumptions about event and pre-event concentrations, and the standard 2 component mixing model. Geographic sources can be similarly tracked, but require that different tracers be ‘applied’ to different source areas.
2.3.2. Residence time
While unusual in the catchment modeling literature, the direct simulation of MRT within conceptual models, is well-established in the groundwater literature. The MRT can be derived by this concentration breakthrough and is defi ned as
³
³
∞
∞
=
0 0
cdt tcdt
MRT (7)
where c is breakthrough concentration and t is time. The numerator is the fi rst moment (concentration weighted average) of the tracer distribution and the denominator is the zeroth moment, or total mass. At this point it is worth stating clearly that strictly speaking, the MRT defi ned by equation 6 is equivalent to that defi ned through convolution only when the direct simulation incorporates the same fl owpath distribution as is incorporated by the isotope- based procedure (which is a top-down estimate of the true fl owpath complexity within the catchment). The environmental tracers behind the convolution approach access the full catchment volume and more importantly is maintained within zones of essential immobility. Clearly, as a simplifi cation, the catchment model would not be expected to incorporate that degree of heterogeneity. Our goal is evaluate the degree to which the simplifi cation affects model residence time. If we can establish that the differences are highly signifi cant, we can then successfully reject the model as a simulation of catchment chemistry due to over simplifi cation, and use that as a sound basis to iteratively incorporate additional complexity.