An Euler-Lagrange approach to transport modelling

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2 Faculty of Forestry, Geosciences and Hydrosciences, Dresden Technical University, Helmholtzstrasse 10, .... porous media, which is given by (Warrick, 2003): sf ..... Springer, Berlin / Heidelberg / New York. Hoteit, H., Mose, R., Younes, A., Lehmann, F. & Ackerer, P. (2002) Three-dimensional modeling of mass transfer in ...
Managing Groundwater and the Environment (Proceedings of ModelCARE 2009, Wuhan, China, September 2009). IAHS Publ. 341, 2011.

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An Euler-Lagrange approach to transport modelling in coupled hydrosystems J.-O. DELFS1, C.-H. PARK1 & O. KOLDITZ1,2 1 Department of Environmental Informatics, UFZ-Helmholtz Centre for Environmental Research, Permoserstrasse 15, D-04318 Leipzig, Germany [email protected] 2 Faculty of Forestry, Geosciences and Hydrosciences, Dresden Technical University, Helmholtzstrasse 10, D-01069 Dresden, Germany

Abstract Hydrosystems are very complex systems with numerous processes occurring simultaneously at different spatial and temporal scales. We present an Euler-Lagrange approach for the analysis of flow and transport processes in coupled hydrosystems. Hydrological processes are described by diffusion equations (the diffusive wave approximation for overland flow and Richards equation for flow in the unsaturated zone) and coupled with exchange fluxes. The governing equations of the flow processes are spatially discretized with Euler methods (finite elements, finite volumes). A new feature of this study is the use of Lagrangian stochastic particles (random walk particle tracking methods) for numerical investigation in coupled hydrosystems. This new model is applied to a laboratory experiment on the effects of the capillary fringe during streamflow generation. The numerical results agree well with the experimental data for a steady flow field. Key words coupled hydrosystem analysis; random walk particle tracking; Abdul & Gillham experiment; streamflow generation

1

INTRODUCTION

Studies on transport between surface and subsurface compartments include numerical investigations such as the Euler methods (e.g. Jones et al., 2006; Sudicky et al., 2008). Randomwalk particle tracking methods have been used extensively to simulate transport of solutes in groundwater systems, and are particularly advantageous in cases of heterogeneous flow and advection-dominated transport (e.g. Kinzelbach, 1988; Park et al., 2008). One issue in catchment hydrology is the “rapid mobilization of old water” (Kirchner 2003), indicated by isotope hydrograph separation techniques (Sklash & Farvolden, 1979). A variety of conceptual models have been proposed to describe the flashy hydrographs, which were found in combination with a strongly damped response in passive tracer tests, including: groundwater ridging (e.g. Sklash & Farvolden 1979), macropores (e.g. Weiler & McDonnell, 2007), hydrodynamic mixing by mechanical dispersion and molecular diffusion (Jones et al., 2006), translatory flow (Renshaw et al., 2003). In this context, the laboratory experiments by Abdul & Gillham (1984) were designed to examine the role of capillary fringe groundwater ridging on runoff generation processes. Cloke et al. (2006) concluded in a numerical investigation on this setup that the rapid water table rise does not necessarily lead to pre-event water contribution to discharge. VanderKwaak (1999), Kollet & Maxwell (2006), Delfs et al. (2009a), Kumar et al. (2009) used this data set when they introduced a coupled surface–subsurface flow model. In this work we extend a compartment approach (Kolditz et al., 2008; Delfs et al., 2009ab) in OpenGeoSys (www.opengeosys.net) to simulate transport processes in coupled hydrosystems. We present the coupled model for flow processes (diffusive wave overland flow and Richards flow in the variably saturated zone) and mass transport (random walk particle tracking). Subsequently, we present results of a benchmark test which is based on the experiments by Abdul & Gillham (1984). 2

CONCEPTUAL MODEL

We calculate flow processes in a fixed reference system with finite element methods (Kolditz et al., 2008). For mass transport, the velocity field is calculated with a node based finite element method (Park et al., 2008) and then pathlines of stochastic particles with the random walk particle Copyright  2011 IAHS Press

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J.-O. Delfs et al.

tracking method (Hoteit et al., 2002; Delfs et al., 2009a). For this combination—finite element methods for flow and random walk particle tracking for mass transport—in this paper we use the term “Euler-Lagrange approach”. 2.1 Hydrology Flow in the overland compartment is described by the diffusive wave approximation to the SaintVenant equations reading (VanderKwaak, 1999):

hof     Hq of  qsof t



of



 hof S h    x

C Hl q   r1k  h of Sh of

2

  hof      y

  

2 1/ 2

  

(1)

where hof, hydraulic head, is used as a primary variable for overland flow, H = max(hof – a – b, 0) is the mobile water depth (Fig. 1), a is the immobile water depth, b is the bottom elevation,  is the two-dimensional nabla operator, qsof is a source/sink term, 0  of(Ha)  1,is the surface porosity which is unity for flow over a flat plane and varies between zero and unity for flow over an uneven surface, Ha = hof – b is the surface water depth. The flux qof depends on bottom friction parameters Cr, k, l and the absolute value of head slope Sh. Flow in the soil compartment is described by the Richards equation for variably saturated porous media, which is given by (Warrick, 2003): S (2)    q sf  q ssf q sf  k r Kh sf t where S(hsf), the soil saturation, depends on the hydraulic soil water pressure head hsf, which is used as a primary variable,  is the soil porosity,  is the three-dimensional nabla operator, and qssf is a source/sink term. The flux qsf depends on the relative permeability kr, and the saturated soil hydraulic conductivity K. The van Genuchten-Mualem soil-water characteristic curves for saturation-capillary pressure S(), where z –g hsf, z is the vertical coordinate (positive upwards), is the liquid bulk density, g the gravitational acceleration, and relative permeability kr(S) are given by:



1   S ()  1   1 m   



m



kr S   Se1 / 2 1  1  Se1 / m



m 2

 S  Sr   Se   0,  1  Sr 

(3)

where Sr is the residual saturation, Se the effective saturation,  parameterizes pore size and m grain distribution.

z Ha H

Cr

a

K qsfsc

of

h

sf

h

b

c

x

K

Fig. 1 Hydrological coupling between overland and soil compartments with flux qssfc via an interface with conductivity Kc and thickness a = a.

An Euler-Lagrange approach to transport modelling in coupled hydrosystems

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Fluxes between the overland and soil compartments are given by (VanderKwaak, 1999):





Kc (4) qsofc  qssfc a´ where  is the leakance, a´ the thickness of a hydraulic interface, and Kc the hydraulic interface conductivity. The interface thickness is set in the following according to the immobile depth a´ = a and the hydraulic interface conductivity according to the saturated soil hydraulic conductivity Kc = K (Delfs et al., 2009b). The scaling factor ka´(Ha) varies between zero for dry and unity for fully saturated interface (Ha  a´) to ensure that infiltration does not exceed the available surface liquid. qssfc  ka ' hof  h sf



2.2 Solute transport Stochastic particles represent the solute concentrations in the liquid of the surface compartment cof and in the soil compartment csf. Particle positions x for a new time step are determined for each component i  d (d = 1, 2) according to the stochastic equation (Kinzelbach, 1988; Park et al., 2008): d   d  ij*  t   2 ij* t Z j xi t  t   xi t    qi* x      j 1 x j  j 1 

(5)

where t is the time step length, d the dimension and Zj are random variables with the mean value of zero and the variance of one. The fluxes q* are given in equations (1) and (2) for advection in the liquid on the surface and in the soil, respectively. The hydrodynamic dispersion tensors of,sf are given by:



of ij

D

of ij

 ijsf   ij  Dijsf

of ij

D

  ij  q t t

of

qiof q ofj     q of



t l

t t



sf i

q q Dijsf   ij  t q sf   l   t   sf q

(6)

sf j

where  is the diffusion coefficient in water, is the tortuosity tensor, Dof, Dsf are dispersion coefficient tensors, tt, lt represent turbulent mixing, and t, l are the longitudinal and transverse dispersivity, respectively. Particles switch from the soil to the surface compartment with the flow field in the variably saturated zone qsf at the common boundary. The particles in the overland flow compartment switch to the soil compartment with a probability P, which is given by:   a' '   P  min  max  0,  , 1  Ha   

(7)

where a is again understood as an interface thickness. In the following we will set a = a. Particles which switch from the overland to the soil compartment at a seepage face (qssfc > 0) are basically pressed at the common boundary by the flow field in the soil qsf for the present time. Particles which switch at an infiltration zone (qssfc < 0) experience the transport processes of the soil such as advection and diffusion/dispersion (equation (6)). Alternative approaches were proposed by VanderKwaak (1995) for solely diffusive/dispersive exchange and by Therrien et al. (2004) and Delfs et al. (2009a) for solely advective exchange between the overland-soil compartments. 3

LABORATORY EXPERIMENTS BY ABDUL & GILLHAM (1984)

The laboratory experiments by Abdul & Gillham (1984) showed that if the zone of tension saturation extends to, or near, ground surface, a small amount of precipitation can cause an immediate rise in the water table. Tracer results with chloride indicated that at early times preevent water is the predominant component in streamflow. The event/pre-event ratio after steady

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J.-O. Delfs et al. 1.2

Infiltration Infiltr ation

1

100

Precipitation

& ov erlan d flow

S eep age f ace Initial water table

98

0.8

Outflow 78

z (m)

96

80

0.6

82 94 84

0.4

92

0.2

86

90

0

0

0.2

0.4

88

0.6

0.8

1

1.2

1.4

x (m)

Fig. 2 Benchmark set-up based on the laboratory experiments by Abdul & Gillham (1984). 2-D soildiscretization and pressure field in steady state (parameters in Table 1). Overland flow on the soil surface is simulated with a corresponding 1-D mesh.

Table 1 Parameters of the benchmark test.

Cr

a





m

Sr

K





l

l

lt

130 s-1 mm-1 0.1 mm 0.34 2.4 1/m 0.8 0.05 19 cm/h 10-5 cm2/s 10-4 0.1 mm 0.01 mm 1 mm

conditions were reached was found to be determined mainly by the extent of the seepage face. Simulated and measured event and pre-event components were compared for the steady flow field. Several experimental runs were performed for different precipitation and initial water table locations on a soil flume with a length of 1.4 m, a width B = 8 cm, and a slope of 12%. Figure 2 shows the simulated pressure field after steady state was reached for the experimental run where the sand flume was initially saturated below the outlet height and precipitation with a rate of qsofp = 4.2 cm/min contained chloride with a concentration of cofp = 60.6 mg/L. The accumulated discharge from the flume was measured at the toe of the slope over 30-second intervals. The parameter set used in the present study for the flow and transport simulations can be found in Table 1. The saturated soil hydraulic conductivity and water content–pressure relationships were determined for a relatively homogeneous packing.

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RESULTS AND DISCUSSION

The flow dynamics represented by the coupled model are ruled by capillarity and friction processes, transport of chloride by advection and hydrodynamic dispersion. Figure 3 compares experimental hydrographs and chlorographs with critical depth outflow in the simulations Qctof = (gH3)1/2 B, where g = 9.81 m/s2, and the chloride concentration in its outflow cctof, respectively. The influence of the saturated soil hydraulic conductivity K and the interface parameter a on a reference solution, obtained with the parameters in Table 1, is shown as well as results for solely advective transport. Surface bottom, which is described by the laminar Darcy-Weisbach relationship by setting k = 1, l = 2 in equation (1), was found as of low importance for this setting. Chloride in the precipitation is represented in the simulations by 143 particles which are distributed along the surface for each time step t = 1 s. The reference solution for chloride

An Euler-Lagrange approach to transport modelling in coupled hydrosystems

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cof p

80

60

q of sp

50 60

Reference K = 18 cm/h a = 0.5 mm Reference Solely advective K = 18 cm/h a = 0.5 mm Measured hydrograph Measured chlorograph

20

0

0

5

t (min)

10

30

(mg/l) cof ct

40

of

Q ct(cm3/min)

40

20

10

0 15

Fig. 3 Comparison of experimental with simulated hydrographs (lines) and chlorographs (lines with square and triangle symbols). Influence of saturated soil hydraulic conductivity K, hydrodynamic dispersion, and interface parameters a = a´ = a´´. Parameters of the reference solution are given in Table 1. Calculated chloride concentrations in outflow cctof are averaged for 30-second intervals.

prediction in the outflow cctof agrees well with the measured chlorograph after about 10 min when the latter reached a maximum value. The simulated chloride concentrations in outflow are highly sensitive to the saturated soil hydraulic conductivity K, which is attributed to its large influence on the seepage face extent in accordance to the findings by Abdul & Gillham (1984). An increase in the interface parameters a = a = a by a factor of five (Ha < 0.9 mm throughout the simulations) decreases the chloride concentration prediction in outflow by about 3 mg/L compared to the reference solution. Hydrodynamic dispersion increases the predicted pre-event/event ratio (cf. Jones et al., 2006). However, the behaviour of the experimental chlorograph for the early unsteady flow field could not be reproduced with the coupled model presented here. 5 SUMMARY AND CONCLUSIONS The compartmental approach of the OpenGeoSys software environment has been extended for the numerical investigation of transport processes in coupled hydrosystems. Euler methods (finite elements and finite volumes) for the solution of the diffusive wave overland flow equation and the Richards equation for flow in variably saturated porous media are combined with a Lagrangian method (random walk particle tracking) for advective–diffusive/dispersive mass transport. The laboratory experiments by Abdul & Gillham (1984) are an excellent test case to validate numerical models for their use in streamflow generation studies. In particular, the “rapid mobilization of old water” problem (Kirchner, 2003) in catchment hydrology is a laboratory scale benchmark-suited example. Our numerical results of the selected experimental run agree well with experimental hydro- and chlorographs for the steady flow field. The modelling approach (diffusive wave overland flow and Richards flow) is assessed to represent an incomplete description of the flow mechanisms in the soil flume during the early fast rising hydrograph of the selected experimental run. This study indicates that processes other than hydrodynamic mixing by mechanical dispersion and molecular diffusion, preferential flow through macropores, and capillarity driven groundwater ridging, as described by the presented coupled model, operate in this particular setting, such as water displacement by gas phase movement (Helmig, 1997). Understanding the functioning of environmental systems is one of the grandest scientific challenges. The holistic systems approach is the ultimate prerequisite to understand and maybe

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influence the evolution of our environment. Concerning climate change there is no doubt about the importance of the hydrosphere. Numerical investigations of large-scale hydrosystems with coupled surface–subsurface flow and transport models are possible in principle. Recent studies with physically-based approaches include Li et al., 2008; Sudicky et al., 2008; Delfs et al. (2010). Simplified benchmarks such as the Abdul & Gillham (1984) problem are essential to follow the holistic concept. The Euler-Lagrange approach presented here is shown to be quite suitable for the numerical investigation of laboratory experiments. For large-scale applications (catchment, mesoscale) the inclusion of further stochastic and conceptual approaches (e.g. Samianego & Bárdossy, 2007; O’Connor & Rossi, 2009) for surface runoff modelling appears promising.

Acknowledgements This work was funded by the German Ministry of Education and Research (BMBF) Project “IWAS – International Water Research Alliance Saxony” (no. 02WM1027).

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