Estimation of vertical water fluxes (FLUX-BOT)1 Investigation of horizontal flux direction

The application of heat as a hydrological tracer has become a standard method for quantifying water fluxes between groundwater and surface water. Here, we present a new computer code (FLUX-BOT) that was developed to estimate vertical water fluxes from measured temperature profiles. Flexibility in the temperature boundary conditions allows for better representation of measured temperatures and enables the direct use of all natural temperature condition without the need of any data pre-processing (e.g. curve fitting, frequency analysis).

Method:

FLUX-BOT uses a numerical solver of the one dimensional conduction equation. The program automates the entire workflow to calculate vertical water fluxes from raw temperature time series and, thus, provides a handy and flexible tool to allow analysis of transient vertical exchange fluxes in saturated porous media.The automated time-varying functionality of FLUX-BOT (24 hours windows or greater) and the automated uncertainty assessment are features not available in established one-dimensional numerical models [1].

Code availability:

FLUX-BOT is distributed as open source code. The most up to date version can be downloaded from the following web site: https://bitbucket.org/flux-bot/flux-bot.

Temperature probe setup and data structure:

FLUX-BOT requires at least three temperature timeseries measured in a vertical profile and estimations of the thermal diffussivity of the saturated sediment

Application and verification (sand box experiment under controlled hydrological conditions [2]):

- Accurate estimates of vertical water fluxes (a, b)- Time variable vertical flux estimates for long time series and a wide range of temperature boundary condition

- The goodness of fit can be derived immediately by comparing measured and modeled temperatures (c)

Characterizing the flow conditions in the streambed can be challenging because of the complex, multi-dimensional flow patterns driven by time variable vertical hydraulic gradients and groundwater discharge. The identification of horizontal flow paths is essential to identify hyporheic exchange processes at the interface between river and groundwater Here, we present a methodology to analyse the geometry of subsurface water flux using vertical temperature profiles. The approach relies on changes in daily temperature amplitude between subsurface temperature sensors.Beyond the one-dimensional vertical exchange flow, major flow patterns could be identified and systematically categorised in purely vertical and horizontal (hyporheic, parafluvial) components.

Method:

Stationary and vertical (1D) heat transport:

Non stationary two dimensional heat transport [3]:

Results:

- Deviation from stationary 1D heat transport result in non-linearity in ln(A /A )z 0

- Non-linearity in ln(A /A ) can be described by higher order polynomialsz 0

- Polynomial parameter a is independent of the magnitude of the exchange flux and the sediment thermal 2

properties, and captures the normalised difference between horizontal and vertical flux

Application:

- Polynomials (p(z)) of 1st to 3rd order are clearly identified as best supported model concerning model residuals and model complexity

- River groundwater exchange flux at the head, crest and tail of the geomorphological structure is characterized by a strong horizontal advective component

Contact: Matthias MunzUniversity of PotsdamKarl-Liebknecht-Str. 24-2514476 [email protected]

2

-2.0

-1.5

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-0.5

0

ln(A

/A)

z0

Depth (m)

a = 02a < 02a > 02 a < 0; a > 02 3

1 3

X (m)0 5 10 15 20

0

2

4

z (

m)

1 2

2

a)

b)

0 0.60.3

Depth (m)0.60.3

Depth (m)0.60.3

Depth (m)0.60.30 0 0 -0.8 -0.4 0 0.4 0.8

-4

-2

0

2

4

(v - v ) / max(v - v )x z x z

ln(a

)2

Position at x (m):

0.5 2.0 5.0 6.0 6.5-10

19.5 18 15 14 11.5-13.5

c)

3 -1q = 1.0 m s3 -1q = 2.0 m s

black: outflux area

3 -1q = 0.5 m s

red: influx area

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0 0.2 0.4 0.6Depth (m)

0 0.2 0.4 0.6Depth (m)

st1 order; AICc = -26nd2 order; AICc = -22rd3 order; AICc = -17

0 0.2 0.4 0.6Depth (m)

0 0.2 0.4 0.6Depth (m)

ln(A

/A)

z0

a)

b)

+

+

PaleochannelSubmerged Riffle

RiverRiver Flow Direction

1 order model

2nd order model

3rd order model

>3rd order model

datast1 order; AICc = -11

nd2 order; AICc = -38rd3 order; AICc = -34

datast1 order; AICc = -14

nd2 order; AICc = -26rd3 order; AICc = -24

datast1 order; AICc = - 9

nd2 order; AICc = - 8rd3 order; AICc = -22

data

c)

A

B

CD

A

B

CD

st

Figure: (a) Flow vectors idealising hyporheic exchange. Exchange flux range from vertically (v ) dominated subsurface flow at the left hand z

side to horizontally (v ) dominated subsurface x

flow at the centre of the domain; e.g. the horizontal flow component increases from the left to the centre of the domain;

(b) Simulated amplitude ratio profiles for flow field and positions shown in (a);

(c) Relation between dominant flow direction and polynomial coefficient a2.

Figure: (a) Experimental amplitude ratio profiles, fitted model set and corres-ponding performance measures;

(b) Schematic showing relative freq-uency of best supported model for the entire observation period from June 2011 until July 2013 in a losing reach of a gravel bed river (Selke). Position of the pie diagrams correspond to the measurement location;

(c) Photographs of the point bar (top) and the in-stream gravel bar (bottom) during low

3 1 3 1(q = 0.25 m s , left) and high (q = 3.60 m s , right) river discharges.

References: 1] Munz, M., Oswald, S. E., and Schmidt, C. (2011): Sand box experiments to evaluate the influence of subsurface temperature probe design on temperature based water flux calculation, Hydrol Earth Syst Sc, 15, 3495-3510, 2011.

2] Munz, M., and Schmidt, C. (2017): Estimation of vertical water fluxes from temperature time series by the inverse numerical computer program FLUX-BOT. Hydrol. Process., doi: 10.1002/hyp.11198.

3] Munz, M., Oswald, S. E., and Schmidt, C. (2016): Analysis of riverbed temperatures to determine the geometry of subsurface water flow around in-stream geomorphological structures, J Hydrol, 539, 74-87, 2016.

0111001 M;λ;P;qfaz;atAtAn;etAtA zza

z

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Ōnz zazazazaatAtAnzp � ...3

32

2100l

A : Amplitude of daily temperature oscillation at 0

the river bottom (°C)A : Amplitude of the corresponding Temperature z

oscillation at depth z (°C)-2 -1q: Flow velocity (l m d )

2 -1 : Thermal diffusivity (m s )P: Period of the temperature oscillation

Analysis of temperature time series to estimate direction and magnitude of water fluxes in near-surface sediments

1 1 2Matthias Munz , Sascha Oswald and Christian Schmidt1University of Potsdam, Institute of Earth and Environmental Science, Karl-Liebknecht-Str. 24-25,14476 Potsdam 2Helmholtz-Centre for Environmental Research - UFZ, Permoserstraße 15, 04318 Leipzig, Germany

-1000 -500 0 500 1000 1500-1500

-1000

-500

0

500

1000

1500

2000

-2-1

Estim

ate

d flo

w v

elo

city [l m

d]

-2 -1Measured flow velocity [l m d ]

FLUX-BOT (RMSE = 0.16)Hatch and Keery Amplitude (mean of all sensor pairs) (RMSE = 0.27)Hatch and Keery Amplitude (shallow sensor pair) (RMSE = 0.24)Hatch and Keery Phase (mean) (RMSE = 1.08)Hatch and Keery Phase (shallow sensor pair) (RMSE = 0.64)Luce and McCallum (combinde Amplitude and Phase) (RMSE = 0.24)

Time [d]0 10 20 30 40 50 60 70 80 90

-2000

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0

500

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measuredmean qz

mean q ± 2 z σ

-2-1

Estim

ate

d flo

w v

elo

city [l m

d]

Upw

ard

flux

Do

wnw

ard

flux

Measured Temperature [°C]5 10 15 20 25 30 35

Estim

ate

d T

em

pera

ture

[°C

]

5

10

15

20

25

30

35

a) c)

Depth [m]:

Root Mean Square Error:

Corellation Coefficient:

0.015

0.26

1.00

1.00

0.065

0.24

0.99

1.00

0.165

0.30

0.99

1.00

0.165 m

0.065 m

0.015 m

Depth [z]

Time 0 z1 z2 z3 z4 …. zn

t1 TSFW(t1) T(z1,t1) T(z2,t1) T(z3,t1) T(z4,t1) …. T(zn,t1)

t2 TSFW(t2) T(z1,t2) T(z2,t2) T(z3,t2) T(z4,t2) …. T(zn,t2)

…. …. …. …. …. …. …. ….

tm TSFW(tm) T(z1,tm) T(z2,tm) T(z3,tm) T(z4,tm) …. T(zn,tm)

e.g.:

Time 0 0.15 0.20 0.25 0.35 0.55

01.01.2011 00:00:00 15.71 16.22 16.05 16.05 15.95 15.88

01.01.2011 00:10:00 … … … … … …

Date28/10/12 02/11/12 07/11/12 12/11/12 17/11/12 22/11/12 27/11/12 02/12/12

Tem

pera

ture

[°C

]

1

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3

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5

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9

0.35 m0.25 m0.15 m0 m

b)

0.55 m