APPENDIX, GSA REPOSITORY

Transcription

1 APPENDIX, GSA REPOSITORY Te Crow Wing model developed for tis study ad to be able to represent all aspects of te surface and subsurface ydrologic cycle (Fig. 18). A.1 Groundwater Flow Time-dependent water-table fluctuations across te Crow Wing Watersed, MN are represented by te following tree-dimensional groundwater-flow equation of te saturated zone: were ' [ K ( x )] = S y Q (1) x + t K is te ydraulic conductivity tensor (m/s), is te equivalent fres water ead, S is te modified specific yield ( S =S y /b; m -1 ), ' y ' y is te gradient operator ( x x () () () () = + + ), x y z b t Q is te local aquifer tickness is time (s), is a source/sink term representing te rate of removal/addition of water by evapotranspiration (ET)/infiltration (I) per unit volume of aquifer (s -1 ) 1

2 Because te Crow Wing aquifer represented in te model is unconfined, we used a modified specific yield to represent unconfined conditions ( S, Harbaug et al., 2000). Te above equation was solved by specifying recarge along all land surface nodes wit te exception of te constant-ead nodes below te outlet of te watersed near te Mississippi River. Constant ead boundary conditions were also applied along te sides of te solution domain directly beneat te water-table constant-ead boundaries. No flow boundary conditions were imposed along te sides of te watersed in te uplands. Recarge to te aquifer across te land surface was calculated wit te soil-zone model described below. ' y A.2 Vadose Zone A scematic diagram illustrating te surface processes represented in our model is presented in Figure 18. For eac montly time step surface runoff was generated at eac nodal patc of te triangulated surface grid. For eac nodal patc (ie. area represented by eac node in te triangular finite element surface mes), a water balance calculation is performed using: were = Qb (2) t v R P ET I + R P is te montly runoff for a given surface node, is montly precipitation, 2

3 ET is evapotranspiration calculated as a function of montly average temperature and vadose-zone water-balance equation (discussed below), v is cange in soil-moisture content represented as te cange in water dept witin te soil zone, and Q b is baseflow for nodes were te water table is at te land surface. A.3 Runoff Snow was converted to equivalent liquid precipitation assuming a snow density of 0.6 gm/cm 3. During winter monts wen te montly temperature was below 0 o C, te snow was allowed to accumulate on te land surface. Te snow is released during te first subsequent mont tat te mean montly temperature exceeds freezing. Te surface runoff is routed across a land-surface nodes using a DEM aving a orizontal resolution of 100m. Runoff is ten added to baseflow to generate total stream flow. Baseflow is evaluated for eac stream node. Te routing equation is given by: N i Qout = Qin + Q b + R i= 1 (3) were i Q in is total upstream runoff from te i t up gradient node Q out N is total outflow from tat node is te number of up gradient nodes 3

4 Note tat runoff is independent of te unsaturated ydraulic conductivity of te soil and tus cannot represent extreme storm-flow events. Baseflow (Q b ) is calculated using a form of Darcy s Law: Q b K b = ' ' [ Z ]Lw sb (4) were, Z sb is elevation of te stream bed, b is tickness of te stream bed, K L w is te ydraulic conductivity of te stream bed, is stream segment lengt, is stream segment widt If te water-table elevation exceeds te elevation of te top of te stream bed, baseflow is computed (gaining stream). We assumed tat K equals te ydraulic conductivity of te aquifer and tat te stream bed dept is equal to te soil-zone tickness (2 m). If te ead in te aquifer falls below te base of te stream, infiltration is calculated assuming a unit ydraulic gradient (losing stream). Tis amount of infiltration is ten removed from te stream network (-Q b ). If te water table is below te stream bed and tere is no upstream runoff, ten te cell is treated as an upland cell. 4

5 Wile te approac is mass-conservative, it neglects temporal variations in stream eigt and stream-bank storage. Altoug tis would be inappropriate for quantifying event-based ydrologic processes it is a reasonable assumption for quantifying long-term annual ydrologic canges. For runoff calculations, flow direction at eac node is calculated based on te maximum gradient determined wit te elevations of te adjacent nodes. Te down-gradient node as te steepest slope among its neigboring nodes. Tis sceme prevents circular flow and produces a topograpically based partitioning of waterseds. However, tis algoritm fails to resolve situations in wic a river splits into two or more brances (Renssen and Knoop, 2000). Te basic assumption of our approac is tat water flows downill along te mes segments connecting any two nodes, called flow segments. Te surface mes was designed to follow te course of major perennial streams. A topograpically based routing algoritm is used to connect individual nodes into line segments starting from te upstream nodes to teir final downstream nodes tat constitute te stream network. Tis routing algoritm does not require te triangled surface mes to be oriented in a particular direction and tus is not biased by a regular grid (Coe, 1998; Ducarne et at., 2003). Wile tis algoritm is robust, routing problems may arise due to grid-resolution issues. For example, closed topograpic depressions or pits can arise due to te use of a relatively coarse DEM. A pit is a topograpic depression tat as not many inflows, but no outflow tat interrupts te stream network. A pit removal algoritm was implemented tat searces for a potential downstream node witin a distance of two elements and ten modifies te flow direction to adjust te new flow relation to maintain te continuity of te stream network. Tis pit removal algoritm eliminates most of te pits but some manual adjustments are required (i.e. stream 5

6 burning; Maidment, 1996; Renssen and Knoop, 2000). Tese adjustments identify te incorrectly positioned streams by matcing tem wit te actual vectorised river network and ten tey manually correct te mes wit te actual locations of te rivers. Of course, undrained topograpic depressions containing wetlands were preserved witin te model. Te routing tecnique enables all te mes segments potentially to be a part of a streamline, but some may be eliminated by a tresold flow below wic it is not regarded as a stream segment. Te model produces a dynamic flow routing network tat canges its flow depending on water balance and connectivity, wic depend on relative canges in water-table topograpy and lake levels. Tat is, stream segments are permitted to dry up and re-wet depending on temporal canges in local water-table elevation. A.4 Evapotranspiration& Infiltration Potential evapotranspiration is calculated as a function of average montly temperature wit a temperature-based equation, following Malmstrom (1969) and described in Dingman (2002): were 17.3T e sat = 6.11exp T (5) PET = e sat (6) e sat is te saturated vapor pressure (mbar) T is te montly average air temperature ( o C) at grid location i for te current time step, 6

7 PET is te montly average potential Evapotranspiration (m/mont). Actual evapotranspiration (ET) is calculated for all surface nodes tat ave te water table below te root zone by multiplying potential evapotranspiration (PET) by a weigting factor (f) wose value ranges from 0 to 1, depending on te water content of te soil orizon and elevation of te water table (Fig. 19): f = 1 ; > Z, v > fc soil f v fc = 1+ ; w < v < (7) fc fc w f = 0 ; v w ET = f PET were f v fc w ET PET Z soil weigting factor (unitless) montly soil moisture (m) montly mean water-table elevation (m) montly field capacity of soil (m) montly wilting point (m) montly actual evapotranspiration (m) montly potential evapotranspiration (m). soil base of te soil zone (i.e. land-surface minus rooting dept) 7

8 Infiltration (I) occurs wen te soil-moisture content is above field capacity ( fc ) of te soil, te point at wic te force of gravity acting on te water is greater tan te surface tension, tus resulting in gravity drainage. It sould be noted tat te total infiltration ere is assumed to recarge te aquifer, were recarge is used to represent te drainage of water from te lower portion of te unsaturated soil zone into te zone of saturation. Te upper limit of infiltration is restricted by te maximum saturated porosity ( φ ) of te soil orizon at wic te excess soil moisture contributes to runoff. Te lower limit of soil moisture is defined by te soil wilting point ( w ). If te water table () touces te base of te soil orizon (Z soil ), ten groundwater-supported evapotranspiration at its potential rate is represented as a negative infiltration. I = φ fc v φ I = v fc I = 0; I = PET; ; ; v v > > > Z fc fc soil (8) were φ soil water volume (per square meter) at saturation expressed in terms of a water level (m) Note tat I as units of lengt (cm), but since montly time steps are used its actual units are cm/mont. Actual evapotranspiration (ET) occurs at its potential value (PET) for all lake nodes and cells were te water table rises to te land surface. If te lake stage falls 8

9 below te lake bottom, te lake node is treated as a surface node and te soil-moisture balance equation is solved (eq. 5). Dry lake nodes rewet (activated) if te water table rises above te lake node bottom. At te beginning of eac mont, evapotranspiration, runoff, and infiltration are calculated. Tis approac is similar to te groundwater and surface-water algoritm implemented in MODCOU (Ledoux, et al., 1989; Ducarne et al. 2003). Here a montly forcing of climatic parameters was cosen because te aim of tis study is to assess te effects of aquifer ydrodynamics on average annual ydrologic conditions and not sort term events suc as floods. A.5 Numerical Metods Te groundwater flow equation is solved using te finite element metod (Zienkiewicz, 1977). Linear sape functions are used to approximate unknown eads across te tetraedral elements. Te resulting system of linear algebraic equations is solved wit an indirect matrix solver (Mendoza et al. 1994). Vadose zone soil-moisture levels ( v ) in equation (11) were solved wit te improved Euler's or Huen's metod (Carnaan et al. 1969). Several iterations were required to calculate accurately actual soil-moisture levels and te weigting factor "f" in equation (7). 9

10 Figure DR1. Scematic diagram illustrating ydrologic processes witin te land-surface model of te Crow Wing Watersed. Figure DR2. Scematic diagram illustrating variables used in te soil-zone model.

11 References Cited Carnaan, B., Luter, H.A., and Wilkes, J.O., 1969, Applied numerical metods: New York, Wiley, 604 p. Coe, M.T., 1998, A linked global model of terrestrial ydrologic processes; simulation of modern rivers, lakes, and wetlands: Journal of Geopysical Researc, D: Atmosperes, v. 103, p , doi: /98JD Dingman, S.L., 2002, Pysical ydrology: New York, Macmillan, 646 p. Ducarne, A., Golaz, C., Leblois, E., Laval, K., Polcer, J., Ledoux, E., and de Marsily, G., 2003, Development of a ig resolution runoff routing model, calibration and application to assess runoff from te LMD GCM: Journal of Hydrology, v. 280, p Harbaug, A.W., Banta, E.R., Hill, M.C., and McDonald, M.G., 2000, MODFLOW-2000, te U.S. Geological Survey modular ground-water models; user guide to modularization concepts and te ground-water flow process: U.S. Geological Survey Open-File Report OF , 121 p. Ledoux, E., Girard, G., de Marsily, G., Villeneuve, J.P., and Descenes, J., 1989, Spatially distributed modeling; conceptual approac, coupling surface water and groundwater, NATO ASI series: Matematical and Pysical Sciences, Ser. C, v. 275, p Maidment, D.R., 1996, Environmental modeling witin GIS, in Goodcild, M.F., et al., eds., GIS and environmental modeling; progress and researc issues: [[Q22: Please give city of publication and publiser. Q22]], p Malmstrom, V.H., 1969, A new approac to te classification of climate: Journal of Geograpy, v. 68, p Mendoza, C.A., Terrien, R., and Sudicky, E.A., 1994, ORTHOFEM user s guide, version 1.02: Waterloo, Ontario, University of Waterloo, Centre for Groundwater Researc. Olcott, P.G., 1992, Ground water atlas of te United States; segment 9, Iowa, Micigan, Minnesota, and Wisconsin: U.S. Geological Survey Hydrologic Investigations Atlas Report 0730-J, p. J1 J31. Renssen, H., and Knoop, J.M., 2000, A global river routing network for use in ydrological modeling: Journal of Hydrology, v. 230, p , doi: /S (00)

12 Rorabaug, M.I., 1964, Estimating canges in bank storage and groundwater contribution to streamflow: International Association of Scientific Hydrology Publication 63, p Rowntree, P.R., and Lean, J., 1994, Validation of ydrological scemes for climate models against catcment data: Journal of Hydrology, v. 155, p , doi: / (94) Webster, K., Soranno, P.A., Baines, S.B., Kratz, T.K., Bowser, C.J., Dillon, P.J., Campbell, P., Fee, E.J., and Hecky, R.E., 2000, Structuring features of lake districts: Landscape controls on lake cemical response to drougt: Freswater Biology, v. 43, p , doi: /j x. Zienkiewicz, O.C., 1977, Te finite element metod in engineering science: London, McGraw-Hill, 520 p.

13 Figure 18. Scematic diagram illustrating ydrologic processes witin land surface model of te Crow Wing Watersed.

14 Hydrogeology Datum = mean sea level Land Surface Soil φ Saturation fc Field Capacity Soil Moisture v w Wilting Point Z soil Soil Zone Datum = 0 Water Table