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Residence Time Residence Time • Mean Water Residence Time (aka: turnover time, age of water leaving a system, exit age, mean transit time, travel time, hydraulic age, flushing time, or kinematic age) – T = V / Q = turnover time or age of water leaving a system – For a 10 L capped bucket with a steady state flow through of 2 L/hr, T = 5 hours • • Assumes all water is mobile Assumes complete mixing – For watersheds, we don’t know V or Q • Mean Tracer Residence Time (MRT) considers variations in flow path length and mobile and immobile flow Residence and Geomorphology • Geomorphology controls fait of water molecule • Soils – Type – Depth – Permeability – Fracturing • Slope • Elevation Mean Residence time (days) • Bedrock 160 MRT = 1.9(Distance) + 19.0 r^2 = 0.88 120 80 40 0 0 10 20 30 40 50 Distance from divide (m) 60 70 80 MRT estimated using Transfer Function Models Transfer Function Models • Signal processing technique common in – Electronics – Seismology – Anything with waves – Hydrology Transfer Function Models • Brief reminder of transfer function HYDROGRAPH model before returning to Hydrograph Modeling flow Precipitation • Goal: Simulate the shape of a hydrograph given a known or designed water input (rain or snowmelt) time Hydrologic Model time Hydrograph Modeling: The input signal • Hyetograph can be – A future “design” event • What happens in response to a rainstorm of a hypothetical magnitude and duration – See http://hdsc.nws.noaa.gov/hdsc/pfds/ – A past storm Hydrologic Model time flow Precipitation • Simulate what happened in the past • Can serve as a calibration data set time Hydrograph Modeling: The Model • What do we do with the input signal? – We mathematically manipulate the signal in a way that represents how the watershed actually manipulates the water Hydrologic Model time flow Precipitation • Q = f(P, landscape properties) time Hydrograph Modeling • What is a model? • What is the purpose of a model? • Types of Models – Physical • http://uwrl.usu.edu/facilities/hydraulics/projects/projects.html – Analog • Ohm’s law analogous to Darcy’s law – Mathematical • Equations to represent hydrologic process Types of Mathematical Models • Process representation – Physically Based • Derived from equations representing actual physics of process • i.e. energy balance snowmelt models – Conceptual • Short cuts full physics to capture essential processes – Linear reservoir model – Empirical/Regression • i.e temperature index snowmelt model – Stochastic • Evaluates historical time series, based on probability • Spatial representation – Lumped – Distributed Integrated Hydrologic Models Are Used to Understand and Predict (Quantify) the Movement of Water How ? Formalization of hydrologic process equations Lumped Model Semi-Distributed Model REW 2 p t pq REW 3 t .( U ) .( ) Q ss REW 4 REW 1 Distributed Model REW 5 REW 7 REW 6 q e.g: Stanford Watershed Model Process Representation: e.g: HSPF, LASCAM Parametric Predicted States Resolution: Coarser Data Requirement: Small e.g: ModHMS, PIHM, FIHM, InHM Physics-Based Fine Large Computational Requirement: 12 Hydrograph Modeling • Physically Based, distributed Physics-based equations for each process in each grid cell See dhsvm.pdf Kelleners et al., 2009 Pros and cons? Hydrologic Similarity Models • Motivation: How can we retain the theory behind the physically based model while avoiding the computational difficulty? Identify the most important driving features and shortcut the rest. TOPMODEL • • Beven, K., R. Lamb, P. Quinn, R. Romanowicz and J. Freer, (1995), "TOPMODEL," Chapter 18 in Computer Models of Watershed Hydrology, Edited by V. P. Singh, Water Resources Publications, Highlands Ranch, Colorado, p.627-668. “TOPMODEL is not a hydrological modeling package. It is rather a set of conceptual tools that can be used to reproduce the hydrological behaviour of catchments in a distributed or semi-distributed way, in particular the dynamics of surface or subsurface contributing areas.” TOPMODEL • Surface saturation and soil moisture deficits based on topography – Slope – Specific Catchment Area – Topographic Convergence • Partial contributing area concept • Saturation from below (Dunne) runoff generation mechanism Saturation in zones of convergent topography TOPMODEL • Recognizes that topography is the dominant control on water flow • Predicts watershed streamflow by identifying areas that are topographically similar, computing the average subsurface and overland flow for those regions, then adding it all up. It is therefore a quasi-distributed model. Key Assumptions from Beven, Rainfall-Runoff Modeling • There is a saturated zone in equilibrium with a steady recharge rate over an upslope contributing area a • The water table is almost parallel to the surface such that the effective hydraulic gradient is equal to the local surface slope, tanβ • The Transmissivity profile may be described by and exponential function of storage deficit, with a value of To whe the soil is just staurated to the surface (zero deficit Hillslope Element P a c asat qoverland β qsubsurface qtotal = qsub + q overland We need equations based on topography to calculate qsub (9.6) and qoverland (9.5) Subsurface Flow in TOPMODEL • qsub = Tctanβ – What is the origin of this equation? – What are the assumptions? – How do we obtain tanβ – How do we obtain T? c a asat qoverland β qsubsurface • • • Recall that one goal of TOPMODEL is to simplify the data required to run a watershed model. We know that subsurface flow is highly dependent on the vertical distribution of K. We can not easily measure K at depth, but we can measure or estimate K at the surface. We can then incorporate some assumption about how K varies with depth (equation 9.7). From equation 9.7 we can derive an expression for T based on surface K (9.9). Note that z is now the depth to the water table. a z c asat qoverland β qsubsurface Transmissivity of Saturated Zone • K at any depth • Transmissivity of a saturated thickness z-D a z c qoverland asat D β qsubsurface Equations Subsurface Surface Assume Subsurface flow = recharge rate Saturation deficit for similar topography regions Topographic Index Saturation Deficit • Element as a function of local TI • Catchment Average • Element as a function of average Hydrologic Modeling Systems Approach A transfer function represents the lumped processes operating in a watershed -Transforms numerical inputs through simplified paramters that “lump” processes to numerical outputs -Modeled is calibrated to obtain proper parameters -Predictions at outlet only -Read 9.5.1 P Mathematical Transfer Function t Q t Transfer Functions • 2 Basic steps to rainfall-runoff transfer functions 1. Estimate “losses”. • W minus losses = effective precipitation (Weff) (eqns 9-43, 9-44) • Determines the volume of streamflow response 2. Distribute Weff in time • Gives shape to the hydrograph Recall that Qef = Weff Event flow (Weff) Q Base Flow t Transfer Functions • General Concept Task Draw a line through the hyetograph separating loss and Weff volumes (Figure 9-40) W Weff = Qef W ? Losses t Loss Methods • Methods to estimate effective precipitation – You have already done it one way…how? • However, … Q t Loss Methods • Physically-based infiltration equations • Chapter 6 – Green-ampt, Richards equation, Darcy… • Kinematic approximations of infiltration and storage Exponential: Weff(t) = W0e-ct c is unique to each site W Uniform: Werr(t) = W(t) - constant Examples of Transfer Function Models • Rational Method (p443) – qpk=urCrieffAd • • • • No loss method Duration of rainfall is the time of concentration Flood peak only Used for urban watersheds (see table 9-10) • SCS Curve Number – Estimates losses by surface properties – Routes to stream with empirical equations SCS Loss Method • SCS curve # (page 445-447) • Calculates the VOLUME of effective precipitation based on watershed properties (soils) • Assumes that this volume is “lost” SCS Concepts • • Precipitation (W) is partitioned into 3 fates – Vi = initial abstraction = storage that must be satisfied before event flow can begin – Vr = retention = W that falls after initial abstraction is satisfied but that does not contribute to event flow – Qef = Weff = event flow Method is based on an assumption that there is a relationship between the runoff ratio and the amount of storage that is filled: – Vr/ Vmax. = Weff/(W-Vi) • • where Vmax is the maximum storage capacity of the watershed If Vr = W-Vi-Weff, W eff (W V i ) 2 W V i V max SCS Concept • Assuming Vi = 0.2Vmax (??) • Vmax is determined by a Curve Number Curve Number The SCS classified 8500 soils into four hydrologic groups according to their infiltration characteristics Curve Number • Related to Land Use Transfer Function 1. Estimate effective precipitation – SCS method gives us Weff 2. Estimate temporal distribution Volume of effective Precipitation or event flow Q Base flow t -What actually gives shape to the hydrograph? Transfer Function 2. Estimate temporal distribution of effective precipitation – Various methods “route” water to stream channel • Many are based on a “time of concentration” and many other “rules” – SCS method • Assumes that the runoff hydrograph is a triangle On top of base flow Tw = duration of effective P Tc= time concentration Q Tb=2.67Tr t How were these equations developed? Transfer Functions • Time of concentration equations attempt to relate residence time of water to watershed properties – – The time it takes water to travel from the hydraulically most distant part of the watershed to the outlet Empically derived, based on watershed properties Once again, consider the assumptions… Transfer Functions 2. Temporal distribution of effective precipitation – Unit Hydrograph – An X (1,2,3,…) hour unit hydrograph is the characteristic response (hydrograph) of a watershed to a unit volume of effective water input applied at a constant rate for x hours. • 1 inch of effective rain in 6 hours produces a 6 hour unit hydrograph Unit Hydrograph • The event hydrograph that would result from 1 unit (cm, in,…) of effective precipitation (Weff=1) – A watershed has a “characteristic” response – This characteristic response is the model – Many methods to construct the shape 1 Qef 1 t Unit Hydrograph 1. How do we Develop the “characteristic response” for the duration of interest – the transfer function ? • • Empirical – page 451 Synthetic – page 453 2. How do we Apply the UH?: • For a storm of an appropriate duration, simply multiply the y-axis of the unit hydrograph by the depth of the actual storm (this is based convolution integral theory) Unit Hydrograph • Apply: For a storm of an appropriate duration, simply multiply the y-axis of the unit hydrograph by the depth of the actual storm. – See spreadsheet example – Assumes one burst of precipitation during the duration of the storm In this picture, what duration is 2.5 hours Referring to? Where does 2.4 come from? • What if storm comes in multiple bursts? • Application of the Convolution Integral – Convolves an input time series with a transfer function to produce an output time series t Q ( t ) W eff U t d 0 U(t-) = time distributed Unit Hydrograph Weff()= effective precipitation =time lag between beginning time series of rainfall excess and the UH Convolution • Convolution is a mathematical operation – Addition, subtraction, multiplication, convolution… • Whereas addition takes two numbers to make a third number, convolution takes two functions to make a third function ∞ x(t) ∗ = () ≝ − −∞ ∞ U(t) ∗ = () ≝ − −∞ y(t) x(t) = input function U(t) = system response function τ = dummy variable of integration Convolution • Watch these: http://www.youtube.com/watch?v=SNdNf3m prrU • http://www.youtube.com/watch?v=SNdNf3m prrU • http://www.youtube.com/watch?v=PV93ueRg iXE&feature=related • http://en.wikipedia.org/wiki/Convolution Convolution • Convolution is a mathematical operation – Addition, subtraction, multiplication, convolution… • Whereas addition takes two numbers to make a third number, convolution takes two functions to make a third function ∞ x(t) ∗ = () ≝ − −∞ ∞ U(t) ∗ = () ≝ − −∞ y(t) x(t) = input function U(t) = system response function τ = dummy variable of integration • Unit Hydrograph Convolution integral in discrete form ∞ ∗ = () ≝ − −∞ () ≝ ∞ − () =−∞ For Unit Hydrograph (see pdf notes) Q (t ) t i 1 W ( i )U ( t i 1) Q ( t ) W tU 1 W t 1U 2 W t 2U 3 ... W 1U j J=n-i+1 Catchment Scale Mean Residence Time: An Example from Wimbachtal, Germany Wimbach Watershed Streamflow Gaging Station Major Spring Discharge Precipitation Station Maloszewski et. al. (1992) • Drainage area = 33.4 km2 • Mean annual precipitation = 250 cm • Absent of streams in most areas • Mean annual runoff (subsurface discharge to the topographic low) = 167 cm Geology of Wimbach Many springs discharge at the base of the Limestone unit Maloszewski, Rauert, Trimborn, Herrmann, Rau (1992) 3 aquifer types – Porous, Karstic, Fractured 300 meter thick Pleistocene glacial deposits with Holocene alluvial fans above Fractured Triassic Limestone and Karstic Triassic Dolomite d18O in Precipitation and Springflow • • • Seasonal variation of 18O in precipitation and springflow Variation becomes progressively more muted as residence time increases These variations generally fit a model that incorporates assumptions about subsurface water flow Modeling Approach • Lumped-parameter models (black-box models): – Origanilly adopted from linear systems and signal processing theory and involves a convolution or filtering – System is treated as a whole & flow pattern is assumed constant over the modeling period (can have many system too) Watershed/Aquifer Processes Filter/ Transfer Function 1 Weight 0 Normalized Time Modeling by Convolution C (t ) t 0 C in ( t ) g ( t ) d • A convolution is an integral which expresses the amount of overlap of one function g as it is shifted over another function Cin. It therefore "blends" one function with another where C(t) = output signature Cin(t) = input signature t = exit time from system = integration variable that describes the entry time into the system g(t-) = travel time probability distribution for tracer molecules in the system • It’s a frequency filter, i.e., it attenuates specific frequencies of the input to produce the result Convolution Illustration C (t ) 0 g() = e -a Cin() t C in ( t ) g ( t ) d Step e g(-) -(-a Folding 1 g(t-) e -a(t- 2 t Displacement Cin()g(t-) Multiplication 3 Integration 4 t C(t) t Shaded area t Transfer Functions - Piston Flow (PFM) • Assumes all flow paths have same residence time – All water moves with advection (no dispersion or diffusion) Represented by a delta function – This means the output signal at a given time is equal to the input concentration at the mean residence time T earlier. Maloszewski and Zuber 1 0.8 PFM 0.6 g(t) • 0.4 0.2 0 0 1 2 t/T 3 4 PFM Transfer Functions - Exponential (EM) • Assumes contribution from all flow paths lengths and heavy weighting of young portion. • Similar to the concept of a “well-mixed” system in a linear reservoir model 0.16 DM 0.14 0.12 g(t) 0.1 0.08 EM EM EPM 0.06 0.04 0.02 0 0 2 4 6 8 10 12 t/T Maloszewski and Zuber EM Exponential-piston Flow (EPM) • Combination of exponential and piston flow to allow for a delay of shortest flow paths • This model is somewhat more realistic than the exponential model because it allows for the existence of a delay 0.2 DM g(t) 0.15 0.1 0.05 0 0 2 4 6 t/T 8 10 12 Maloszewski and Zuber Dispersion (DM) • Assumes that flow paths are effected by hydrodynamic dispersion or geomorphological dispersion – Geomorphological dispersion is a measure of the dispersion of a disturbance by the drainage network structure 0.01 DM 0.008 g(t) 0.006 0.004 0.002 0 0 2 4 6 8 10 t/T Maloszewski and Zuber (White et al. 2004) C (t ) Input Function t 0 g ( t ) C in ( t ) d • We must represent precipitation tracer flux to what actually goes into the soil and groundwater – Weighting functions are used to “amount-weight” the tracer values according recharge: mass balance C in ( t i ) N Pi N P C i C out C out i i 1 where Pi = the monthly depth of precipitation N = number of months with observations = summer/winter infiltration coefficient Cout = mean output 18O composition (mean infiltration composition) Infiltration Coefficient was calculated using 18O data from precipitation and springflow following Grabczak et al., 1984 [ ( P i C i ) C out ( P i ) ] /[ C out ( Pi ) ( P i C i ) ] w w s s where Cout (1988-1990) = -12.82o/oo (spring water) Mean Weighted Precipitation (1978-1990) = -8.90o/oo and -13.30o/oo, for summer and winter, respectively Application of this equation yielded an value of 0.2, which means that winter infiltration exceeds summer infiltration by five times Grabczak, J., Maloszewski, P., Rozanski, K. ans Zuber, A., 1984. Estimation of the tritium input function with the aid of stable isotopes. Catena, 11: 105-114 Input Function C in ( t i ) N Pi N P C i C out C out i i 1 Convolution using FLOWPC Application of FLOWPC to estimate MRT for the Wimbach Spring Maloszewski, P., and Zuber, A., 1996. Lumped parameter models for interpretation of environmental tracer data. Manual on Mathematical Models in Isotope Hydrogeology, IAEA:9-58 Convolution Summation in EXcel • Work in progress • Your Task: – Evaluate my spreadsheet. Figure out if I’m doing it right – Get FlowPC to work • Reproduce Wimbachtal results – Run FlowPC or Excel for Dry Creek.