Baseflow Analysis. Objectives. Baseflow definition and significance

Transcription

1 Objectives Baseflow Analysis. Understand the conceptual basis of baseflow analysis.. Estimate watershed-average hydraulic parameters and groundwater recharge rates. Baseflow definition and significance Portion of (stream) flow that comes from groundwater or other delayed sources (Tallaksen, 995. J. Hydrol., 65: 49). Understanding of low-flow condition is important for water resource management and environmental protection. Why? We will review: () Baseflow recession analysis () Baseflow separation technique 0.6 discharge (m s - ) baseflow discharge (m s - ) 0. straight-line recession / 5/ 6/0 7/0 7/0 8/9 5/ 5/ 6/0 7/0 7/0 8/9 Stream discharge gradually decreases after storm events. Various baseflow separation techniques have been proposed. What purpose? Regardless of sophisticated algorithms, they are all arbitrary. Recession hydrographs commonly plot as straight lines on a semi-long graph. Q(t) = Q 0 exp(-at) Q 0 : discharge at t = 0 a : constant (s - ) What causes the exponential behaviour?

2 Reservoir model for recession analysis Exponential function is the solution of: ds Q = as and = Q (linear reservoir) S: volume of water stored (m ) S Q A more general reservoir model is given by: ds Q = as p and = Q p: dimensionless constant Non-linear (p > ) reservoir represents the effects of complex processes such as the transmissivity feedback. Q /Q The solution of the non-linear reservoir equation is: Q(t) = Q 0 ( + at) -p / (p - ) p = p = 0 4 at See Tallaksen (995) for an excellent review. Physically-based aquifer model Baseflow from a homogeneous, confined aquifer is described by: h Kb x x T h h = S x c t = S s h = h s h 0 piezometric surface, not WT h q b b t x = 0 x = B T = Kb : transmissivity (m s - ) S c = S s b : storage coefficient or storativity Boundary conditions and initial condition h(x 0 ) = h s for all t > 0 No flow at the divide (x = B) for all t > 0 h = h s + h 0 at t = 0 for all 0 x B Rorabaugh (964. Int. Assoc. Scientific Hydrol. Pub. 6: 4-44, Eq.) reported the Fourier-series solution for flow per shore line, q (m s - ): q = (Th 0 /B)[exp(-at) + exp (-9at) + exp(-5at) + ] where a = π T / (4B S c ) 4

3 High-order terms in the series are negligible for tt / (B S c ) > 0.. t c = 0. B S c / T is called critical time. q q 0 exp{-at} for t >t c where q 0 = Th 0 /B Note the similarity between this and the exponential decay equation of hydrograph in Page. What does this mean? q / (Th 0 /B) 0 0. all terms first term only t T/(B S c ) Remember the recession coefficient in this model: a = π T / (4B S c ). The recession coefficient in Page represents the average properties of aquifer over the entire watershed. 5 Models for unconfined aquifer Streams are usually connected to unconfined, not confined, aquifers. The rigorous analysis of unconfined aquifers would require the solutions of the Richards equation. The Dupuit-Forchheimer (D-F) approach offers a reasonable approximation of complex problems (e.g. Paniconi et al., 00. Water Resour. Res., 9: 7). The transient flow equation based on the D-F approximation is called the Boussinesq equation: h Kh x x = S y h t K: aquifer conductivity (m s - ) S y : drainable porosity D c x = 0 h(x,t) D x = B Exact solution of the non-linear Boussinesq equation is available only for special cases. Brutsaert (005, Hydrology an introduction. Ch. 0, Cambridge Univ. Press) presented a summary of various solutions for the cross section shown above. 6

4 Early-time drainage The first solution considers a riparian aquifer with a thickness D, which is fully saturated at t = 0. The depth of the stream (D c ) is assumed much smaller than D. D c D q x = 0 x = B q: flow per shore length (m s - ) Using a technique known as Boltzmann s transform, early-time drainage flux from the near-shore zone is (Brutsaert, 005, Eq.0.64): q = 0. KS y D / t Multiplying q by the total channel length L in the watershed and (from both sides of the channel), baseflow Q (m s - ) measured at the watershed outlet is (Brutsaert, 005, Eq.0.60): Q Note: = L KS D / t = KS D L t Eq. [] dq y y / = KS yd L / t Q 7 Long-term drainage The next solution considers gradual drainage of the entire hillslope, assuming no flow at x = B. D c x = 0 q D x = B Approximate analytical solution is obtained by linearizing the Boussinesq equation: h h Kh m = S h h y Kh m = Sy x x t xx t where h m is the average saturated thickness. Solving the linearized equation, drainage flux per shoreline is: q = (Kh m D /B)[exp(-at) + exp (-9at) + exp(-5at) + ] where a = π Kh m / (4B S c ) This is almost identical to the Rorabaugh (964) equation. Therefore, for t > 0.B S c / Kh m q = (Kh m D /B) exp{- π Kh m t / (4B S c )} 8

5 Brutsaert (005, p.400) proposed expressing h m as a fraction of D: h m = pd where p 0.5 for Dc << D p (D + D c )/(D) for other cases Using p, the flux is written as (Brutseart, 005, Eq.0.6): q = (KpD /B) exp{- π KpDt / (4B S c )} Eq. [] We note that the average distance from the channel to drainage divide, B, can be estimated by: A B = A / (L) Why? B L Total baseflow Q (m s - ) at the watershed outlet is (Brutseart, 005, Eq.0.64): Q = L (KpD L/A) exp{- π KpDt 4L / (4A S c )} = (8KpD L /A) exp{- π KpDL t/(a S c )} Eq. [] Note: dq exp{( π KpDL t /( A Sc )} Q 9 Watershed-scale flow parameterization Brutsaert & Lopez (998. Water Resour. Res., 4: ) used Eqs. [] and [] to estimate average K and S y for watersheds. This method examines the relation between Q and its time derivative in a form: dq b = aq a and b are constants. It can be shown that: b = and a =.6/(KS y D L ) for Eq. [] (early time) b = and a = π KpDL / (S y B ) for Eq. [] (late time) When daily values of Q and -dq/ are plotted on a log-log graph, log[-dq/] = loga + blogq The cluster of the points is enveloped by lines of b = and b =. The envelope lines represents the lowest recession rate (-dq/) for a given Q, which is considered the baseflow condition. -dq/ (m s - d - ) Q (m s - ) 0

6 K or S y may be estimated from the intercepts (loga) of the envelope lines, if other parameters in the equations are known. One can also determine K and S y simultaneously from a and a (Brutsaert & Lopez, 998, p. 7): K =. 6 π p a a ( A LD ). 6 p S y = π a a DA In practical computation, daily values of Q and dq/ are given by: Q av Q i + Q dq Q + i i + Q i = = av t Eq. [4] Sample data from the Marmot Creek watershed in Canada will be used to demonstrate the techniques in the computer exercise. Baseflow separation Given a hydrograph, quick flow and baseflow can be separated by a number of different methods. - Connecting local minima - Variation of local-minima method - Using inflection points discharge time All methods use arbitrary criteria for baseflow, and are time consuming for manual operation. Automated techniques are at least objective, and are efficient for processing many data sets. We will use a digital-filter algorithm of Arnold et al. (995. Ground Water, : 00) to demonstrate the usefulness and limitation of automated baseflow separation.

7 Recursive digital filter The algorithm, originally described by Nathan & McMahon (990), calculates the quick flow component q i at time step i from q i- at previous time step and total flow Q i and Q i- : + β q i = βq i + ( Qi Qi ) where β is a filter constant ranging between 0.9 and Baseflow b i is calculated as: b i = Q i q i In this example from the Marmot Creek watershed in 005, the filter was successively applied three times with β = The first pass appears to have produced reasonable results. Q (m s - ) 0 raw data st pass nd pass rd pass 5/0 6/9 6/9 6/9 7/9 Baseflow index By applying the digital filter to the entire 005 summer discharge data set (May - September 0) for Marmot Creek, it was found that: Total discharge = m Total baseflow = m The ratio of total baseflow to discharge is base flow index (BFI). In this example, BFI =.7 /.7 = 0.6. Automated baseflow separation offers a convenient tool to calculate BFI for multiple watersheds having different size and geology, or for a single watershed in multiple years having different meteorological forcing or landuse practice. We will use a computer program Baseflow to sample data from the Marmot Creek watershed in the computer exercise to calculate BFI. 4

8 Marmot Creek watershed (4 km ), Alberta, Canada Elevation m, the eastern edge of the Rocky Mountains. Mean annual precipitation = 640 mm at the base. N gauging station km 5 Baseflow Computer Exercise Watershed parameter estimation In this exercise, we will apply the Brutsaert & Lopez (998) method to estimate watershed-average values of K and S y for the Marmot Creek watershed. () Open Marmot_data.xls file, which contains daily stream discharge and precipitation values for May - September 0 in 009, 00, and 0. () Open dq_template.xls file, and paste the discharge and precipitation data from the raw data file. () Compute Q av = (Q i+ + Q i ) / (4) -dq/ is calculated for the days without significant rain, and with a sufficiently high difference in discharge to minimize the influence of data noises. Columns G and H in the template file sets the flags (0 or ) depending on the critical values at the top. Copy the cell formula down and compute -dq/. 6

9 (5) Plot -dq/ vs Q for all points, using different symbols for different years. (6) Plot straight lines of -dq/ = aq b with b = and b = for a suitable range of Q. Use a as a fitting parameter to adjust the position of the envelope lines. (7) Once the values of a are determined for both lines, calculate K and S y from Eq. [4]. Assume that A = 4 km, L = 5 km, D = m and p = (8) Discuss the magnitude of values. 7 Automated baseflow separation In this exercise, we will use the Baseflow program to separate baseflow and compute base flow index (BFI). () Open file.lst file using Notepad or Wordpad, and change the input and output file names to mrmt009.prn and mrmt009.out, respectively. The input file mrmt009.prn has already been prepared in the required format. () Double click on bflow.exe on the file folder to run the program. () Open the output file and covert the date from YYYY, MM, DD to actual dates by a cell formula =Date(YYYY,MM,DD). (4) Plot stream flow, pass, pass, and pass on the same chart. Observe the quality of separation. (5) Calculate the total stream flow and total baseflow (for appropriate pass), and compute the BFI. (6) Repeat ()-(5) for 00 and 0, and compare the results. 8