Germer, Kai

Hinkelmann, R.

Martinez-Noguez, Isaac

Stadler, Leopold

English

Hydraulic Engineering Repository

Conference Paper, Published VersionStadler, Leopold; Martinez-Noguez, Isaac; Hinkelmann, R.; Germer, KaiSimulating Water Flow into a Soil Matrix with a CylindricalMacroporeZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99443Vorgeschlagene Zitierweise/Suggested citation:Stadler, Leopold; Martinez-Noguez, Isaac; Hinkelmann, R.; Germer, Kai (2014): SimulatingWater Flow into a Soil Matrix with a Cylindrical Macropore. In: Lehfeldt, Rainer; Kopmann,Rebekka (Hg.): ICHE 2014. Proceedings of the 11th International Conference onHydroscience & Engineering. Karlsruhe: Bundesanstalt für Wasserbau. S. 287-294.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding. 1 INTRODUCTION  Macropore flow is a typical phenomenon in field soils where soil fauna, flora and genesis can form a net-work of large soil pores. Water and solutes can shortly reach deeper soil regions via macropores while bypassing lower permeable soil layers (Beven and German, 1982). For this reason, it is challenging to predict, flow and transport processes in macroporous soils with numerical models. Today a wide range of scale and problem dependent models and concepts for simulating flow in macroporous soils exist (Simunek et al. 2003, Gerke 2006, Köhne et al. 2009). Macropore flow can be taken into account with a discrete model concept (Stadler et al. 2009) or an equivalent model concept such as a dual-permeability model (Stadler et al. 2012), both based on the Richards or two-phase flow equations. Several approaches have been developed to formulate the mass transfer between macropores and matrix domain (Simunek et al. 2003, Gerke 2006, Stadler et al. 2012). It is obvious that the mass transfer is the most crucial part for the mentioned model concepts. While controlled laboratory experiments with single artificial macropores allow the study of macropore flow and water exchange for specific soils (Edwards et al. 1979, Buttle and Leigh 1997, Ghodrati et al. 1999, Allaire et al. 2002, Castiglione et al. 2003, Köhne and Mohanty 2005, Akay et al. 2008), studies with undistributed soil samples (Bouma et al. 1978) are considerably more complex so that the validation of transfer formulations or the determination of soil specific transfer parameters becomes very complex. Another aspect of verification is that macropore flow and water infiltration on the field scale depend on soil formation, initial conditions and precipitation rates, soil-layers and the coating and lining on the macropore walls (Jarvis 2007).   In this paper we present the model CASCADE which simulates the flow processes in a single vertical macropore and the water exchange between macropore and matrix. The intention was to first investigate and understand macropore infiltration in well-controlled laboratory experiments before later going to large scales. CASCADE was implemented in two existing models for flow in porous media, MUFTE-UG (Helmig et al. 1998) and DuMuX (Flemisch et al. 2011). As the gas phase should be taken into account for complex multi-layered systems (Stadler et al., 2009, 2012) we coupled CASCADE with the two-phase Simulating Water Flow into a Soil Matrix with a Cylindrical Macropore L. Stadler, I. Martinez-Noguez & R. Hinkelmann  TU Berlin, Chair of Water Resources and Modeling of Hydrosystems, Germany  K. Germer University of Stuttgart, Institute for Modelling Hydraulic and Environmental Systems, VEGAS, Germany  ABSTRACT: The simulation of water transfer between the soil matrix and macropores is crucial for modeling flow in macropores soils. We present a numerical model (CASCADE) that allows to simulate the flow in a single macroporous and the water transfer between the macropore and its surrounding soil matrix. CASCADE is coupled with existing modeling software for porous media which models the flow in the soil matrix (MUFTE-UG, DuMuX). We show experimental and numerical results of two laboratory experiments. The exchange parameter is determined together with other sensitive parameters (van Genuchten parameters, permeability, residual saturation) through parameter optimization. Overall a good agreement between computations and measurements is obtained. Keywords: Macropore flow, Infiltration, Discrete model concept, Two-phase flow in porous media, Numerical model ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-82872.2 Numerical model  The basic equations for modeling immiscible two-phase flow in porous media are the mass balance equa-tions for both regarded phases α (w = water, n = air) which can be written in a general form as: ( ) 0)( =−+∂∂ααααααα ρρρφ qvdivtS  (1) where φα (LL-3) is the porosity, Sα [-] the saturation of the phase α, ρα (ML-3) the density, vα (LT-1) the Darcy velocity and qα (L3T-1) a sink/source term. The extended Darcy law is applied to compute the Dar-cy velocity and it is inserted into the mass balance equations: )( gpgradkv r αααα ρµ −−= K  (2) where K (L2) is the intrinsic permeability tensor, krα (-) the relative permeability, μα (ML-1T-1) the deter-mined dynamic viscosity and g (MT-2) the vector of gravity. The ratio of relative permeability and dy-namic viscosity is called mobility (λα= krα / μα). The relative permeability functions after van Genuchten in combination with the Mualem (1976) model are applied.  These mass balance equations build a set of partial differential equations (PDEs). Two primary varia-bles are necessary to solve the non-linear system of equations. Therefore, two supplementary constraints, one for the saturations for the wetting and non-wetting phase (Sn+Sw=1) and one for the capillary pres-sures pc (ML-1T-2) and the phase pressures (pc=pn+pw) are required to close the system. The capillary pressure pc can be described by van Genuchten (1980) as a function of the saturation and soil properties.  The mass transfer between the matrix and macropore domain can be described with different formula-tions (see also Simunek et al., 2003, Gerke 2006). The formulation chosen here is similar to Darcy’s law: )( macroporematrixr ppku −−= βµααα (3) where β (L) is a resistance factor or exchange parameter and controls the flow over the macropore sur-face. The relative permeability in Eq. (3) is computed with a fully up-winding scheme to prevent unphys-ical fluxes, assuming a mobility equal to 1 if water infiltrates from the macropore. pmatrix (ML-1T-2) is the water pressure in the matrix and pmacropore (ML-1T-2) is the water pressure in the macropore which is set to atmospheric pressure. The mass fluxes ṁα (MT-1) between macropore and matrix domain are given as: ααααα ρρ qAum ==• (4) where A (L2) is the surface between matrix and macropore and ραqα the sink/source term (see Eq. 1). CASCADE is implemented as a simple computational scheme to simulate vertical non-capillary macropore flow. Similar model concepts are widely used in hydrology. CASCADE generates for all ma-trix cells which border to the macropore corresponding CASCADE cells to calculate the cell fluxes with Eq. (4) (Fig. 1, middle). The water exchange between matrix and macropore and vice versa is computed over the whole macropore length and is a source/sink term in the two-phase model of matrix (Eq. 1). Therefore the inflow into the macropore and its value on the top must be known. Before each time step is carried out with DuMuX or MUFTE-UG, the scheme shifts the infiltrated wa-ter cell-wise downwards the macropore; starting at the top and ending at the bottom. Water transfer and downward migration are computed as follows: (i) The inflow rate into the macropore at the top is set as boundary condition for the first cell. The water flux based on the pressure difference in both domains (macropore/matrix) is calculated (Eq. 4) for each cell i of CASCADE.  (ii) Starting from the first cell, lateral water transfer qw between macropore and matrix is added to (inflow to the macropore) or subtracted from the macropore cell (outflow to the matrix). The vertical outflow qout of a cell serves as input for the next cell (Fig. 1, middle). iwiiniout qqq ,,, −=  (5) Transfer to the matrix is only calculated as long as enough water is available in the macropore, while water transfer to the macropore can always occur.   289(iii) Remaining water in the macropore at the last cell of CASCADE represents outflow through the macropore bottom.  (iv) All calculated transfer fluxes are transferred as source/sink terms to the two-phase model of the matrix (Eq. 1). CASCADE approximates the macropore flow without resolving the flow characteristics inside which are very complex (film flow, plug flow) and cannot be determined in detail. The concept behind CASCADE was chosen to keep the model and the numerical solutions as simple as possible and as accurate as neces-sary.  The model CASCADE was designed as a flexible extension for existing simulators for flow in porous media. It was written in Python and can be easily coupled with existing models via XML-RPC, embedded Python or exchange files. We tested CASCADE with the numerical toolbox MUFTE-UG (coupling via XML-RPC and embedded Python) and DuMuX (file based coupling).  Both toolboxes apply a local mass conservative box-method for the spatial discretization. The mobility at the integration points is therefore computed with a fully upwinding scheme. A fully implicit adaptive time stepping is used for the discretization in time. The resulting non-linear systems are linearized with the Newton-Raphson method and solved with a BICGstab solver (Helmig et al. 1998, Hinkelmann 2005). Both toolboxes computed the same result, however DuMuX was considerably faster.  3 RESULTS AND DISCUSSION 3.1 Model set-up We applied a three-dimensional model domain, using the setup symmetry to reduce the domain size and calculation time (see Fig. 1). A mesh with overall 2484 nodes and 1170 elements with a resolution of 46 nodes in the vertical direction is used for simulation. For the given setup it is convenient to use a pn/Sw formulation. For both phases, we defined Neumann no-flow boundary conditions (BC) on the cylinder surface. At the top we set atmospheric pressure for the gas phase (Dirichlet BC) and a Neumann no-flow BC for the water phase. However, it is also necessary to define a Dirichlet BC for the second primary var-iable, the water saturation. To minimize the influence of this BC the water saturation was close to the ini-tial condition at the outer edge at the top. The surface between matrix and CASCADE is a crucial part. Here, we chose a Neumann BC for the water phase since the CASCADE model serves as a sink/source term and a Neumann no-flow BC for the gas phase.  3.2 Parameter estimation A main problem during the study was the determination of the soil hydraulic parameters. The only known parameter for the set-up was the soil porosity (φ=0.32); all other soil hydraulic parameters were fitted during the calibration. In a first step, the capillary pressure was adjusted so that the measured pressure drops (tensiometer) after infiltration was reproduced as good as possible. Then soil parameters (VGn, VGα van Genuchten n, α; K: permeability; β: resistance factor; Swr, Swn: residual water/air saturation) were op-timized with the downhill-simplex method (Nocedal and Wright 1999) of Scipy (www.scipy.org). The re-sulting parameters are given in Tab. 1.  Table 1. Resulting soil parameters experiment VGn [-] VGα [Pa-1] K [m2] β [m] swr [-] snr [-] Q166 3.75 0.000195 8.19E-12 4.5E-10 0.060 0.020 Q375 5.50 0.000200 7.58E-12 4.5E-10 0.120 0.010  The results of the two different experiments were relatively close together except VGn. The obtained re-sidual saturations are very small. This is based on the fact that the porosity was very small and the water must be hold by the matrix to reproduce the speed of the infiltration front. The quality of the calibration was determined by comparing the measured tensiometer heads with simulated pressures applying the least square method (Fig. 4).   290REFERENCES Akay O., Fox , G.A., and Simunek, J. (2008). Numerical simulation of flow dynamics during macropore-subsurface drain in-teractions using HYDRUS. Vadose Zone J. 7:909-918. Allaire, S.E., Gupta S.C., Nieber, J., and Moncrief, J.F. (2002). Role of macropore continuity and tortuosity on solute transport in soils: 1. 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Simulating Water Flow into a Soil Matrix with a Cylindrical Macropore

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Simulating Water Flow into a Soil Matrix with a Cylindrical Macropore

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