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1 in Water Resources Elsevier Editorial System(tm) for Advances Manuscript Draft Manuscript Number: Title: From rainfall to spring discharge: Coupling conduit flow, subsurface matrix flow and surface flow in karst systems with a discretecontinuum model Article Type: Research Paper Keywords: Karst; modeling; flow coupling Corresponding Author: Dr. Rob De Rooij, Ph.D. Corresponding Author's Institution: University of Florida First Author: Rob De Rooij, Ph.D. Order of Authors: Rob De Rooij, Ph.D.; Pierre Perrochet, Professor; Wendy Graham, PhD

2 Highlights (for review) A discrete-continuum model for simulating flow in karst systems is presented The model couples flow between matrix, overland, conduit and channel domains The model accounts for variably saturated conditions in the subsurface Conduit-matrix coupling is based on a projection well-index

3 *Manuscript Click here to download Manuscript: MS.doc Click here to view linked References From rainfall to spring discharge: Coupling conduit flow, subsurface matrix flow and surface flow in karst systems with a discrete-continuum model Rob de Rooij a, Pierre Perrochet b, Wendy Graham c a,c Water Institute, University of Florida, 0 Weil Hall, PO Box, Gainesville, FL- -01, USA b Centre of Hydrogeology and Geothermics, University of Neuchâtel, Emile Argand, 000 Neuchâtel, Switzerland a r.derooij@ufl.edu b pierre.perrochet@unine.ch c wgraham@ufl.edu Corresponding author: Rob de Rooij Water Institute University of Florida 0 Weil Hall PO Box Gainesville FL--01 USA Telephone: 1--- Fax: 1---

4 Abstract Physics-based distributive models for simulating flow in karst systems are generally based on the discrete-continuum approach in which the flow in the three-dimensional fractured limestone matrix continuum is coupled with the flow in discrete one-dimensional conduits. In this study we present a newly designed discrete-continuum model for simulating flow in karst systems. We use a flexible spatial discretization such that complicated conduit networks can be incorporated. Turbulent conduit flow and turbulent surface flow are described by the diffusive wave equation whereas laminar variably saturated flow in the matrix is described by the Richards equation. Transients between free-surface and pressurized conduit flow are handled by changing the capacity term of the conduit flow equation. This new approach has the advantage that the transients in mixed conduit flow regimes can be handled without the Preissmann slot approach. Conduit-matrix coupling is based on the Peaceman s well-index such that simulated exchange fluxes across the conduit-matrix interface are less sensitive to the spatial discretization. Coupling with the surface flow domain is based on numerical techniques commonly used in surfacesubsurface models and storm water drainage models. Robust, accurate and efficient algorithms are used to simulate the non-linear flow processes in a coupled fashion. The model is verified and illustrated with simulation examples. Keywords: Karst; modeling; flow coupling 1. Introduction

5 Physics-based distributive numerical models or process-based numerical models have been used successfully to gain insights into the complex hydrodynamic functioning of hydrogeological systems. Over the years a variety of process-based models have been developed that deal with coupled flows in complex hydrogeological system such as coupled conduit-matrix flow in karst aquifers [1-] and coupled surface-subsurface flows in watersheds [-1]. Not surprisingly, there are similarities between the numerical modeling approaches used to simulate coupled conduit-matrix flow and those used to simulate coupled surfacesubsurface flow. Typically, a discrete-continuum approach is used in which the coupled flow domains have different spatial dimensions such that representative flow equations can be applied to each flow domain. Coupled flows may be solved separately using a conjunctive approach [, -]. This approach requires an iteration scheme to match the conditions along the interfaces of the flow domains. An advantage of the conjunctive approach is that that each flow domain may be solved with a different scheme and with a different time stepping procedure. An alternative approach is the fully implicit approach in which all the discretized equations are combined into a single matrix system such that the equations can be solved simultaneously [,, -1]. This approach is generally more robust [1]. However, this method requires a certain degree of similarity between the flow equations. Coupling of the flow domains may be either based on the assumption of continuous pressure heads on the interface [,, 1] or on flux relations defined by firstorder exchange parameters [,,,, 1, 1]. Over the years, following the pioneering work of Kiraly [1, ] discrete-continuum models for coupled conduit-matrix flow have been applied successfully to gain insights into

6 the factors that influence the exchange of water between the conduits and the surrounding matrix and its implications for spring hydrograph analysis [, 1-1]. Discrete-continuum models have also been used for simulating karst genesis [, -1]. Typically, discretecontinuum models for karst aquifers are often based on simplifications such as laminar conduit flow and / or saturated flow conditions. De Rooij developed a discrete-continuum finite element model for coupling turbulent conduit flow with laminar matrix flow which accounted for variable saturation in both flow domains []. The matrix is represented by a tetrahedral mesh. Conduits elements can be added along the edges of the tetrahedra. Although tetrahedral grids provide flexibility to represent conduit networks with a complex geometry, generating good quality tetrahedral grids is non-trivial. Turbulent conduit is described by the diffusive wave equation in conjunction with the Manning s equation. Laminar matrix flow is governed by the Richards equation. The transition between free-surface and pressurized flows in the conduits is handled similarly as the transition between unsaturated and saturated flows in the matrix. Namely, the transition is handled by a modifying the capacity term in the flow equation. The coupled model is based on the fully implicit approach. Coupling of the flow domains is based on imposing head continuity on the conduit-matrix interface. As pointed out by De Rooij [], imposing head continuity may result in some serious problems. Firstly, simulated exchange fluxes between the matrix and the conduit are sensitive to the spatial discretization of the matrix in the vicinity of the conduits. Secondly, under variably saturated conditions where the pressure head in the matrix can be negative, the head continuity breaks down since the diffusive wave equation used for the conduits is restrained to positive water depths. In the finite element model of De Rooij, this problem is handled

7 by deactivating dry conduits from the computation, where dry conduits are defined if the water depth drops below a certain threshold. However, the head continuity than implies that a deactivated conduit can only be activated once the matrix at the conduit-matrix interface has become saturated. This means that deactivated conduits may act as unphysical barriers for conduit flooding, particularly for relative coarse matrix discretizations. The USGS has developed MODFLOW-00 Conduit Flow Process MODE 1 (CFPM1), a discrete-continuum model for simulating saturated karst aquifers []. The packages that deal with simulating the flow in the conduits and the matrix-conduit coupling are largely based on the Carbonate Aquifer Void Evolution (CAVE) code []. In the CFPM1 model the matrix is represented by a three-dimensional rectilinear grid and each vertex of a one- dimensional conduit cell is required to coincide with a cell center of a matrix cell. As such the geometry of the conduit network is restrained to the underlying matrix grid. Turbulent pipe flow is based on the Darcy-Weisbach equation whereby the friction factor is given by the Colebrook-White formula. Partially filled conduits are simulated as fully saturated pipes with a corrected diameter. This corrected diameter is defined such that the hydraulic radius of the saturated pipe corresponds to the hydraulic radius of the partially filled conduit. The major drawback of this approach is that storage associated with partially filled conduits is not accounted for []. CFPM1 uses the conjunctive approach to simulate coupled conduit-matrix flow. The exchange fluxes between the matrix and the conduit are simulated by specifying a lumped first-order exchange parameter. Several studies have found that simulation results obtained by CAVE and CFPM1 are quite sensitive to the value of this parameter [, -]. Furthermore, the CFPM1 model is sensitive to the space discretization around the conduits []. Finally, in

8 the majority of simulation studies based on CFPM1 the matrix is represented by a single layer of brick cells []. Representing the matrix by an essentially two-dimensional grid, however, implies that the conduits drain or recharge the matrix as if they were trenches cutting through the entire thickness of the model domain [1]. In this study we recognize that the problem of conduit-matrix coupling is analogue to the problem of coupling of one-dimensional flow in a well with the three-dimensional flow in a surrounding aquifer. Well-aquifer coupling is commonly based on the Peaceman wellindex [, ]. This well-index takes into account that the computed pressure in the aquifer at the location of the well depends on the spatial discretization of the aquifer. The fact that existing discrete-continuum models for coupled conduit-matrix do not include an approach similar to the well-index explains why the exchange fluxes computed with these models are usually found to be sensitive to the spatial discretization of the matrix. Whereas De Rooij simulated conduit flow using the diffusion wave equation [], storm water drainage models simulate conduit flow with equations that include inertia effects. Models that simulate free-surface and pressurized flows in conduits with the Saint- Venant equations and the waterhammer equations, respectively, solve the two flow regimes separately [-]. In order to solve the two flow regimes these models need to track the moving interface between the two flow regimes. This shock-fitting approach typically relies on solving additional mass and momentum equation across the bore. Models based on the Preissmann slot approach apply the Saint-Venant equations for both free-surface as well as pressurized flows [0-]. Following this approach pressurized flow is approximated by a flow with a very narrow free-surface. To accomplish this, the geometry of the conduit cross-section is changed by adding a narrow slot to the top of the conduit. The advantage of

9 the Preissmann slot approach is that models capable of simulating a free-surface model can be easily extended to simulate pressurized flows as well. However, the Preissmann slot also has some disadvantages. Firstly, the slot provides additional and artificial storage of water for pressurized flows []. Secondly, the wave speed in pressurized flows will depend on the width of an artificial slot instead of the compressibility of the water []. Storm water drainage models may include a coupling of conduit-overland flow and this coupling can be defined by using weir flow equations []. The Storm Water Management Model of the EPA [] has been applied to applied to karstic conduits []. The drawback of applying storm water drainage models to simulate flow in karst aquifers, however, is that the flow in the surrounding limestone matrix is not accounted for. Recently, Reimann et al. [] modified the MODBRANCH code to include the transient simulation of variably saturated pipe flow. In this new model, MODBRAC, conduit flow is described by the Saint-Venant equations and the transition between freesurface and pressurized conduit flow is handled by a Preismann slot approach. Like CFPM1 the model solves the flow in the conduits and the matrix separately. In model applications the matrix is represented by an essentially two-dimensional grid, such that the conduits drain or recharge the matrix as if they were trenches. Although MODBRAC seems to overcome some of the limitations of CFPM1, especially with respect to the simulation of storage in partially filled conduits, the model does not account for variably saturated conditions in the matrix. Well-known models for coupling surface-subsurface flows include MODHMS [], InHM [1], HydroGeoSphere [], CATHY [], WASH1D [], ParFlow [1] and OpenGeoSys []. An important requirement for simulating coupled surface-surface flows

10 is that saturated groundwater flow and surface flows can be coupled across the unsaturated zone. In general coupled surface-surface models accomplish this by simulating subsurface flow with the Richards equation. Coupling of the flow domains may be based on imposing continuous heads [1]. In general, however, most surface-subsurface models are based on first-order exchange coefficients [, 1, 1]. In MODHMS a special coupling technique based on weir equations is used for coupled channel-overland flow []. MODHMS and HydroGeoSphere can also simulate subsurface flow in pipes and drains. As such these two models offer interesting possibilities to simulate karst systems. However, to the authors knowledge these two models have not yet been applied to karst systems. In MODHMS transitions between free-surface and pressurized conduit flow are handled with the Preissmann slot approach []. Recently, a similar approach has also been implemented in HydroGeoSphere [0]. In HydroGeoSphere the coupling between the drains and the surrounding porous medium is based on an embedded node approach and resistance adjustment factors. These factors, like the Peaceman s well index seeks to eliminate mesh size effects on the computed exchange fluxes. In this study we present our Discrete-Continuum model (DisCO) for coupling conduit flow, channel flow, overland flow and matrix flow in karst systems. This newly designed model permits simulation of spring discharges in response to rainfall events while accounting for a wide range of hydrodynamic processes. Compared to existing modeling approaches for karst systems which only consider coupled conduit-matrix flow, a new feature of our model is that it includes a surface flow domain. We argue that such an advanced numerical tool is important since many karst systems are characterized by the

11 presence of surface water flows. For example, sinkholes may be recharged by surface flow [1, ], surface streams may recharge water into the aquifer [], and flood flows that exceed the carrying capacity of the conduits may spill over into the surface water domain [1-]. Moreover, surface water flow may also simply occur due to Hortonian and Dunne runoff processes. We also emphasize that the surface flow domain represents a natural boundary condition for variably saturated subsurface flow. Without this natural boundary condition, a model for variably saturated subsurface flow can generate physically incorrect results, unless it can be assumed that all the water can be stored within the subsurface domain. We also implement the well index approach to couple conduit-matrix flow. We illustrate that this is a more accurate methodology to simulate coupled conduit-matrix flows in karst systems. A distinct difference from MODHMS and HydroGeoSphere is that our model does not rely on a Preissmann slot approach to handle the transition between freesurface and pressurized flows. Instead our model uses the approach from De Rooij [] in which the transition is handled via a modification of the capacity term in the conduit flow equation. Our model is based on a cell-centered finite difference scheme. A rectilinear mesh is used to discretize the matrix domain. The faces of the matrix cells that coincide with the land surface define the overland cells. Channel and conduit networks of arbitrary geometry can be accommodated. Variably saturated flow in the matrix is described by the Richard s equation. Conduit flow, channel flow and overland flow are described by the diffusive wave equation such that a fully implicit coupling scheme can be implemented. Coupling of overland-matrix flow is based on the classical conductance concept. The methodology to couple channel-matrix flow follows from similar considerations as those

12 that apply to coupled conduit-matrix flow and is similar to the methodology proposed Herbert []. Finally, conduit-overland flow and overland-channel flow are governed by weir flow equations. The distributive modeling approach adopted here has some limitations, particularly in applications to karst systems. A general limitation of distributed models is that spatially distributed information about the hydraulic parameters is required. In the case of karst systems this includes the geometry of the conduit network, which is generally not wellknown []. Therefore, many modeling studies in karst hydrogeology have been restricted to simple hypothetical karst systems [, ]. Nonetheless, from a scientific point of view models on hypothetical systems have been useful for gaining insights into processes not yet fully understood, such as the hydrodynamic functioning of the epikarst []. Additionally, these models have been used to test classical methods for spring hydrograph analysis [, ] and to test the sensitivity of different modeling approaches with respect to the conduit network [, 1]. A relatively new area of research is the stochastic generation of karst conduit network []. This line of research offers interesting new possibilities to generate more realistic karst conduit networks.. The conceptual model In our conceptual model the karst system is subdivided into four distinct flow domains: a three-dimensional porous medium representing the fractured limestone volumes, a network of interconnected one-dimensional conduits, a network of interconnected one-dimensional channels and a two-dimensional overland domain. Within the matrix the flow is assumed to be laminar. In the other domains flow is assumed to be turbulent. Conduits and channels are

13 assumed to have a circular and rectangular cross section, respectively. The four distinct flow domains are connected via matrix-conduit, matrix-channel, matrix-overland, overlandconduit, overland-channel and conduit-channel interfaces.. The mathematical model.1. Matrix flow Variable saturated subsurface flow in a three-dimensional porous medium is governed by the well-known Richards equation: p s Sss krk h qms 0 (1) t t where p is the pressure head [L], h the hydraulic head [L], s the saturation [-], S s the specific storage coefficient [L -1 ], φ the porosity [-], S s the specific storage coefficient [L -1 ] k r is the relative permeability [-], K the saturated hydraulic conductivity tensor [LT -1 ] and q Ms a sink term [T -1 ]. To solve the Richards equation constitutive relationships are required for the saturation as well as the relative permeability. In this study we assume the Van Genuchten- Mualem model []: s s e e s sr s s s r n m 1 p if p0 () 1 if p 0 k =s 1-1-s 1 1 m m r e e

14 where s r, s s and s e are respectively the residual, maximum and effective saturation [-], α the inverse of the air entry pressure head [L -1 ], n the pore size distribution index [-] (n>1) and m1 1 n... Channel and conduit flow One-dimensional channel flow and sub-horizontal one-dimensional conduit flow are governed by following mass conservation equation []: p Q C qcs 0 t s C where C is a capacity term [L], Q the volumetric flux [L T -1 ], s C the spatial coordinate in the direction parallel to the conduit or channel [L], q Cs a source term [L T -1 and p the pressure head subjected to the restriction p 0. For free-surface flows the pressure head p equals the water depth ψ. The capacity term is given by []: W for free surface flows C () ga for pressurized flows w c w where W is the top width of the flow [L], ρ w the density of water [ML - ], g the acceleration due to gravity [LT - ], A c is the cross-sectional area of the conduit [L ] and κ w the compressibility of water [LT M -1 ]. For pressurized conduit flow, it is assumed that storage of water can only take place by the compression of water. Within the framework of the diffusive wave approximation of the Saint Venant equations the momentum equation is simplified by neglecting the inertia terms. The resulting momentum equation then reads: ()

15 S f h () s The volumetric flux in equation () can be expressed in term of the friction slope by using empirical relationships such as the Manning-Strickler, Chezy or Darcy-Weisbach equations. In general form the empirical relationship can be written as: Q K C c Sf () where K c is the conveyance factor [L T -1 ]. Combining equation (), () and () results in the diffusive wave approximation for channel and conduit flow: S p K h C t s s c qs C S f C Adopting the Manning-Strickler equation the conveyance factor is given by: K c f 0 () ARh () where η is the Manning s roughness coefficient [L -1/ T], A is the cross-sectional area of flow [L ] and R h is the hydraulic radius [L]. The hydraulic radius is defined by R h A p w with p w the wetted perimeter [L]. In general, the variables W, A and p w depend on the pressure head p and the geometry of the channel / conduit cross-sections. For free-surface flow in a circular conduit with a radius r c the width of the flow is given by: W r p c p 1 ()

16 For pressurized flow in a circular conduit the cross-sectional area of flow and the wetted perimeter are given by: w A r p w c r For free-surface flow in a sub-horizontal circular conduit: p r A r c c arccos 1 p p rc 1 rc p p p rcarccos 1 rc For free-surface flow in rectangular channel with width B: A Bp c p B p w () () (1) As explained later on, from a computational perspective it is desirable to define a height above the bottom slope, a rill storage height, below which no flow occurs. In this work we account for the effects of the rill storage height on the flow by redefining the crosssectional area of flow: rs A( p) max A p A h,0 (1) It is noted that we assume here that the storage capacity in channels and conduits is not affected by the presence of rills that block the flow. In MODHMS [] and HydroGeoSphere [], channel and conduit flow are also described with a diffusive wave approximation. A crucial difference in our work, however, is that the transition between free-surface and pressurized conduit flow is not handled with the Preissmann slot approach. Instead mixed conduit flow regimes are handled by

17 modifying the capacity term C in equation () according to equation (). We argue that this approach is more accurate and also easier to implement. It is worth mentioning that the Preissmann slot approach only provides a distinct advantage in terms of computational efficiency, if it is used to overcome the need of solving additional mass and momentum equations across the interface between the two conduit flow regimes. However, if conduit flow is modeled by the diffusive wave equation, then there is no need for additional computations across these bores. Consequently, there is no advantage in using the Preissmann slot approach. In fact an accurate implementation of the Preissmann slot approach would define a slot width equal to w ga c w. A more straightforward approach in our opinion is to use equation () directly without using the Preissmann slot... Overland flow Overland flow is governed by the following mass conservation equation: f t os mob qos 0 v (1) where ψ= p is the water depth [L] subjected to the restriction p 0, ψ mob the mobile water depth [L], v the flow velocity vector [LT -1 ] and q Os a sink term per unit surface [LT -1 ]. The mobile water depth equals max h,0 mob where h rs is the rill storage height [L], a rs height above the surface bottom below which no lateral flow occurs. The dimensionless parameter f os is the fraction of the surface available for surface water storage [-] which may also be thought of as the ponded fraction of the overland domain. This fraction is unity for a flat plane but varies between zero and unity in the presence of rills. Without defining an

18 explicit function, Panday and Huyakorn argued that a parabolic function for f os is favorable from a computational perspective [1]. In this study we define the following parabolic function for f os : f os 1 fos,min 1 h rs if h hs 1 if h where f os,min defines a minimum value for f os. Figure 1 illustrates how the dimensionless parameter f os varies with water depth. It can be observed that the function has a continous derivative at ψ= h rs. Similar to channel and conduit flow we adopt the diffusive wave approximation. and therefore the momentum equations are given by: fi i rs rs (1) S h x (1) where S fi is friction slopes in the x i -direction. The velocity vector in equation (1) is expressed in term of the friction slopes. Adopting Manning-Strickler this vector is expressed as: v () v mob i i Sf i where η i is the Manning s roughness coefficient in the x i -direction. Using v follows that: 1 v v 1 v mob Sf1 1 Sf (1) Combining equation (1), (1), () and (1) results in the diffusive wave approximation for overland flow: it

19 T K ovh qs 0 () t where K ov is second order tensor: 0 K ov F mob (0) 0 1 with: 1 f f (1) F S S Numerical implementation.1. Spatial discretization The spatial discretization of the matrix domain is based on a three-dimensional rectilinear grid of a box-shaped domain. Around the outer boundaries of this grid, cells may be deleted to accommodate for non-rectangular domains and a lateral varying topography. The twodimensional overland cells are directly linked to the matrix grid in the sense that each overland cell corresponds to the top face of a matrix cell. The conduits and the channels are represented by a network of interconnected one-dimensional cells. Within this network the channel and the conduit cells may be connected. The geometry of the network is not constrained to any geometrical conditions arising from the two-dimensional and threedimensional grids. Multiple one-dimensional cells may be located within a single matrix cell and these cells do not need to be aligned with the gridlines of the matrix grid. There is also no requirement that the vertices of the one-dimensional cells must coincide with the cell centers of a matrix cells as in CFPM1. We only require that each one-dimensional cell

20 is located within a single matrix cell. In other words at locations where conduits and channels intersect with a matrix face, we need to define a vertex in the one-dimensional grid. This requirement facilitates the computation of exchange fluxes with other flow domains. Moreover at junctions where more than two one-dimensional segments are connected we also require a vertex. After finding all the required vertices, extra vertices may be defined to obtain a finer discretization of the one-dimensional network. To define the finite difference stencils we let η I denote the set of cells and Dirichlet cell boundaries which are connected to cell I. Each finite difference connection I, J is then associated with a cell I and a cell J I or a Dirichlet cell boundary J I. In a threedimensional grid, the finite difference connections define a classical -point finite difference stencil and each finite difference connection is associated with the midpoint of a cell face. The overland grid may be projected on a horizontal plane and within this projected grid the finite difference connections define a classical -point stencil and are associated with midpoints of cell edges. In the one-dimensional grid the finite difference connections are associated with the cell vertices. If a vertex defines a junction between more than two cells, multiple connections are defined. For example, on a junction connecting the one-dimensional cells I, J and K we define the connectionsi, J,IK, and J, K. As described later on, the coupling between the flow domains results in additional finite difference connections. Let I denote the vector between the connection I, J and the cell center of cell I. If this connection is associated with a cell J then let J denote the vector between I, J I

21 and the cell center of cell J. If the connection is associated with a Dirichlet cell boundary J then J 0. In the two-dimensional overland grids the vectors I and J are I measured within the horizontal projected grid. In all the other flow domains these vectors are measured in three-dimensional space. We now define the distances L IJ, between the grid cells used to approximate the spatial derivatives with a finite difference approach: I J () L IJ,. Cell-centered finite difference formulation Our model is based on cell-centered finite differences and on an implicit time marching scheme. Using single subscripts to denote the cell indices and superscripts to denote the time level, the discretization of the flow equations (1), (), and () can be written in the following general form: 0 R B p p O s s T h h Q () tt tt t tt t tt tt tt tt I I I I I I I IJ I J I J I where T IJ is a transmissibility term, B I and O I capacity terms and R I a residual term. The transmissibility terms T IJ are obtained using a suitable averaging of material parameters and variables at the connection I, J. In the following the subscripts H, A and U denote weighted harmonic means, weighted arithmetic means and upstream weighted values, respectively. For weighted harmonic means and weighted arithmetic means of cell values we use the lengths I and J. Upstream weighting between two cell values β I and β J is defined as:

22 if U tt tt I I J if h h h h tt tt J I J For matrix flow the capacity and transmissibility terms are given by: SsI sivi BI t IVI OI t 1 T k K A IJ r U ( I, J ) H ( I, J ) L( I, J) where V I is the volume of matrix cell I, A( IJ, ) () () the area of the face associated with the connection I, J, K( IJ, ) the saturated hydraulic conductivity at the face in the direction perpendicular to the face. Harmonic weighting is generally the preferred method to obtain an average hydraulic conductivity, since it gives an accurate equivalent conductivity if cells with a different conductivity are put in series. Moreover, within a layer of cells with a constant conductivity, the harmonic mean will simply return the actual conductivity. However, it may be emphasized that there exists patterns of heterogeneity for which the harmonic mean is not accurate. For example for a checker board pattern the average is given by the geometric mean [0]. Upstream-weighting to average the relative permeability is necessary to avoid spurious oscillations in simulating variably saturated subsurface flows in porous media as pointed out by Forsyth and Kropinski [1]. For channel and conduit flow the capacity and transmissibility terms are given by:

23 CL I I BI t O 0 where L I is the length of cell I, ( IJ, ) T I IJ AR h L U S ( I, J ) ( I, J ) H f ( I, J ) associated with connection I, J and Sf ( I, J ) hj hi L IJ,. () the Manning s roughness coefficient at vertex For overland flow the capacity and transmissibility terms are given by: fos AI BI t O 0 T I IJ 1 F ( IJ, ) A mob E U ( I, J ) L( IJ, ) ( IJ, ) H where A I is the area of the overland cell I, E ( IJ, ) the length of the cell edge associated with the connection I, J, ( IJ, ) the Manning s roughness coefficient at the cell edge in the direction perpendicular to the edge and F ( IJ, ) is the value of F as defined by equation (1) at the connection I, J. The values F ( IJ, ) are constructed by taking a weighted arithmetic mean of the cell values F I and F J which in turn are derived from an arithmetic mean of the values of Sf ( IJ, ) at the cell edges. As pointed out by Panday and Huyakorn, the standard procedure to evolve upstream weighted terms for free-surface flows may hinder convergence []. Namely, if the upstream water depth is used, then a sudden jump in the transmissibility value occurs once a cell with a lower elevation becomes the upstream node. Instead it is preferred to construct the ()

24 upstream water depth by subtracting the maximum of the two cell elevations from the upstream water elevation. Taking into account the rill storage heights, the upstream water depth for overland flow is then expressed as: h max( z h, z h ) () U U I rsj J rsj In our model the upstream weighted terms for channel and conduit flow are not constructed from the upstream water depth, but from the upstream pressure head. Moreover, in the channel and conduit flow domains we assume that the rill storage height is constant. Therefore the upstream pressure head for channel and conduit flow are constructed using: pu hu max( zi, zj) () In a turbulent flow domain it is necessary to ensure non-zero positive water depths as well as non-zero friction slopes. This is done by defining a minimum water depth p min as well as a minimum friction slope (typically - m and -, respectively). The minimum friction slope is necessary to avoid division by zero in calculating the transmissibility terms. The role of the minimum water depth is to avoid zero rows in the matrix system as defined by equation (), where we note that zero rows would result in singular matrix system. However, to avoid zero rows it is sufficient that the capacity term for each cell is greater than zero. By ensuring non-zero capacity terms with a minimum water depth, transmissibility terms are permitted to be zero for water depths below the rill storage height. This means that the essential behavior of dry turbulent flow domains, namely no flow, can be simulated provided that hrs p min. It is important to observe that the rill storage height also plays an important role in conserving mass. Namely, once the water depth drops below the rill storage, no flow implies that the pressure head remains constant such that the

25 generation of water depths smaller than the minimum water depth can be avoided. Avoiding the generation of water depths smaller than the minimum water depth is important because resetting simulations with the minimum water depth results in mass balance errors... Coupling techniques The general discretization defined by equation () permits a fully implicit coupling of the flow domains. Namely, equation () then represents a single matrix system in which all the discretized flow equations are combined. Exchange fluxes between the flow domains can be incorporated into this matrix system in two distinct ways. Firstly, the exchange flux between two cells I and J may be defined in terms of a first-order exchange parameter, which simply equals the transmissibility term T IJ in equation (). Secondly the exchange flux may be incorporated using two source term Q I and Q J where Q Q Coupling conduit-matrix flow and channel-matrix flow Coupling of conduit-matrix flow is based on the Peaceman well-index [, ], a special first-order exchange parameter originally developed for coupling well-aquifer flow. The Peaceman well-index is based on an analytical solution for radial flow in a homogeneous aquifer around a well and may be derived from the Thiem equation []. Moreover, the Peaceman well-index also assumes a regular grid for the aquifer in the vicinity of the well. In our model we use the projection well-index []. This well-index is based on the same assumptions as the Peaceman well index but can be applied to wells that are not aligned with the gridlines. I J

26 Consider a conduit cell I and a matrix cell J. Let L be the vector along the conduit cell Furthermore as mentioned in section.1, we require that the entire conduit cell is located within the matrix cell. The projections of the vector L along the principal gridlines are defined as: l Le (0) i where e i is the unit vector in direction x i -direction. The well-indices WI i are now defined by: WI i l i i ji r ln rc where r eq,i is the equivalent conduit radius, θ the angle open to radial flow and K jj the principal saturated hydraulic conductivity of the surrounding matrix cell in the x j -direction. For a circular conduit θ = π. The equivalent radius is defined by: r 0. eq, i K K x jj k, k i, k j ji Kkk, k i, k j ei, 1 K K jj jj ji kk, k i, k j where Δx k defines the spatial discretization of the matrix cell in the x k -direction. Finally the well index WI is expressed by: (1) () WI WI WI WI () The well-index WI is only applicable to saturated flow. To include the effects of unsaturated flow conditions, we define the exchange parameter as follows:

27 IJ ex T f WI f U * * pw ( p ) ex kr max pw where f ex denotes the fraction of the interface area available for exchange fluxes [-], * p max( hi, hj ) zi, () max p w the maximum value for the wetted perimeter given by πr c and * kr a dimensionless factor [-] which permits zero values for the exchange parameter at water depths close to the minimum water depth. If * kr is unity for all pressure heads then the fraction f ex is always greater than zero, since the pressure head in the conduits is constrained to a minimum positive pressure head. To ensure that conduits with a water depth close to the minimum depth do not recharge a surrounding unsaturated matrix and to avoid the generation of water depths in the conduits smaller than the minimum depth, defined to be zero if * kr is * p h min and unity otherwise. The height h min with pmin hmin hrs is a height above the conduit bottom and is usually defined to be an order of magnitude larger than p min. It is observed that the factor f ex for conduit-matrix flow varies rather smoothly between zero and non-zero values even if * k r jumps between zero and unity. This is because the wetted perimeter of a circular conduit can be assumed to be close to zero if * p h min. In our model the coupling of channel-matrix flow is handled by the same methodology as used for conduit-matrix coupling. This is done by defining an equivalent channel radius that can be used in equation (1) to calculate the well index: c c r B d () where d c is the depth of the channel [L]. Furthermore, for channel flow the angle open to radial flow in equation (1) is approximated by θ = π. The fraction f ex is evaluated by using

28 the relations for the wetted perimeter as established for rectangular channels and therefore p B d. It is observed that for a rectangular channel, the minimum value of for the max w c wetted perimeter is close to the value B. To ensure that the fraction f ex for channel-matrix flow can vary smoothly between zero and non-zero values, the dimensionless factor * k r in equation () does not jump from zero to unity at a certain water depth, but varies smoothly according to: * 0 if p hmin 1 h 1 k sin p h if h p h * 1 if p hrs * * rs * r min min rs hrs Figure illustrates the function continuous derivatives. () * kr as defined by equation () and that this function has Our methodology of coupling channel-matrix flow is similar to the one proposed by Herbert [] in the sense that the exchange fluxes are based on an analytical solution for radial flow towards a channel. Our methodology is thus very different from the one used in MODHMS where the exchange parameter for channel-subsurface flow is a function of a channel bed with a given thickness and a given vertical hydraulic conductivity... Coupling matrix-overland flow The exchange coefficient between a overland cell I and an matrix cell J is based on the classical conductance concept []: T IJ 1 fexkai () L ( IJ, )

29 where K is the vertical saturated hydraulic conductivity in the matrix cell, AI the area of the overland cell and f ex k. The parameter evaluated with equation () where p * max( h, h ) z. * r * k r for coupled matrix-overland flow is I J I.. Coupling channel-overland flow and conduit-overland flow Channel cells are always coupled with the overland cell. If a channel cell I is located in matrix cell K and if this matrix cell K is coupled with an overland cell J, then the channel cell is coupled with the overland cell J. For conduit cells the same approach is used except that conduit cells are only coupled with the overland domain if one of its vertices is located on the overland cell. The exchange fluxes between the channel domain and the overland domain are expressed by weir flow equations following the approach of Panday and Huyakorn []. Weir equations are also used to approximate the exchange fluxes between the conduit domain and the overland domain similar as in the work of Leandro et al. []. For simplicity we assume that the height of the weir is above the rill storage heights of the connected flow domains such that the exchange fluxes can be zero for small water depths. Let h w, L w and C w denote the height of the weir [L], the length of the weir [L] and the weir discharge coefficient [-], respectively. Let h U and h D denote the upstream and downstream hydraulic head at the weir, respectively. These heads are simply the heads in the two connected cells. If hu hwand hd hw then the sink terms for the cells are given by the free weir equation []: Q U QD LwCw g hu hw ()

30 If hu hwand hd hw then the sink terms for the cells are given by the submerged weir equation []: 1 QU QD LwCw g hu hd hu hw () For a coupled channel-overland flow the length of the weir is given by: L w L (0) where L c is the length of the channel cell. For coupled conduit-overland flow the length of the weir is given by: L w c r (1) We note that the average hydraulic head in the overland cells will depend on the spatial discretization of the overland domain and may not reflect the actual value at the location of the weir. Simulated exchange fluxes at the weirs may thus be sensitive to the spatial discretization of the overland domain. Although, we have not resolved this problem, it is interesting to note that the well-index for coupled well-aquifer flow is designed to solve a similar problem. Namely in the case of coupled well-aquifer flow the simulated head in the aquifer cell containing the well depends on the spatial discretization of the aquifer and may not reflect the actual value at the well-aquifer interface... Boundary conditions For subsurface flow in the porous medium the boundary conditions are given in terms of classical Dirichlet and Neumann conditions. For the domains with turbulent free-surface flows zero-gradient conditions can also be defined. These zero-gradient conditions may be c

31 specified in terms of either Dirichlet or Neumann conditions. In general a zero-gradient condition is specified by setting the friction slope equal to the bottom slope. For conduit flow, however, this condition is always implemented as a Dirichlet condition, such that the pressure head at the outlet can be constrained to a maximum value which equals the diameter of the conduit. This is a necessary constraint because flow at the conduit outlet is in equilibrium with atmospheric pressure. If an effective rainfall rate q R [LT -1 ] is applied then this rate needs to be distributed among the matrix domain and the overland domain. Factors that determine this distribution are the ponded fraction of the overland cell and the maximum infiltration capacity of the matrix. We assume that the fraction f os as defined in equation (1) may be used to define the ponded fraction of the overland cell. Consider an overland cell I and a matrix cell J. We define the maximum infiltration rate q R,max as: zi h J qr,max min max 1 G K,0, q L ( IJ, ) R () where G = 0 for a flat overland domain with h rs = 0 and G fos if h rs > 0. With A I the area of the overland cell, the sink terms for the cells are then given by:. Solution strategies..1 Newton-Raphson scheme Q q A J R,max I Q q q A I R,max R I ()

32 To solve the non-linear flow equations the Newton-Raphson procedure is used. Using the residual R I as defined in equation (), the matrix system to be solved becomes: [ J]X R () where [J] is the Jacobian matrix defined by JIJ RI X J and X the primary variable. For turbulent flows the primary variable is the pressure head. For matrix flow, we use the primary variable switching technique [, ] such that the primary variable is either the pressure head or the saturation. Two tolerances tol f and tol b with tol tol 1 are used to switch between the saturation and the pressure head as the primary variable. In cells where the saturation is the current primary variable switching occurs if the saturation becomes greater than tol f. Likewise, in cells where the pressure head is the current primary variable switching occurs if pressure head becomes smaller than tol b. In our model the Jacobian matrix in equation () is computed analytically. Within the Newton-Raphson procedure the non-linear terms F ( IJ, ) and Sf ( IJ, ) b f are only updated with the hydraulic heads from a previous iteration while the derivatives of these terms with respect to the water depth are set to zero. This is similar as the methodology used by Panday and Huyakorn []... Convergence criteria and time-stepping procedure The efficiency and accuracy of numerical models dealing with non-linear and transient problems depend on the convergence criteria as well as the time-stepping procedure. A complete overview of implemented numerical techniques to measure convergence errors and to handle adaptive time stepping, is beyond the scope of the present study. Instead, we

33 only present a single set of procedures to measure convergence errors and to adapt the time steps and refer to other studies for alternative procedures. The convergence criterion for the Newton-Raphson procedure in our model can be defined in terms of an absolute error measure e A,i and a relative error measure e R,i : e X X tt, m1 tt, m AI I I e e X tt, m R I A, i J where the second subscript denotes the iteration level. The convergence criterion is then defined by: max AI RI () e e () where ε is the convergence tolerance. This convergence criterion is a stringent criterion in the sense that it measures local convergence at the cell level. More lenient criteria may be used to achieve faster convergence. We refer to work of Diersch [] for alternative convergence criteria. The standard time-stepping procedure in our model is based on a target-based timestepping approach [] in which the time step is changed based on the convergence behavior of the Newton-Raphson procedure as well as on a target-based change in the primary variables. Let it and it max be the number of non-linear iterations and the maximal number of permitted non-linear iterations respectively. If it it divided by two and the iteration procedure is repeated. If it it multiplier M 1 is defined: w max max then the time step then a time step it M1 () it

34 where it w is the number of wished iterations. Based on the changes in the primary variables we define a second time step multiplier M : where X M min tt t XI XI X is the wished absolute change in the primary variable. The new time step is now defined by: max 1 () t min M,max M, M t () where M max is a maximum permitted time-step multiplier. Finally the time step may also be adjusted to write simulation results at certain times or to account for changes in boundary conditions at certain times. From a mathematical point of view, a more rigorous approach is to adapt the time step according to truncation errors in the numerical solution. In general such schemes rely on a one-step Newton scheme which solves the non-linear matrix system by performing a single iteration [, ].. Verification examples.1 Matrix flow: Forsyth et al. s [] problem This problem considers the infiltration intro an initially dry caisson consisting of heterogeneous materials. The problem originates from a study by Forsyth et al. [] where it was used to test the performance of simulators for very dry initial conditions. The problem has been used as a verification example in several other studies [, ]. The caisson consists of four different zones as depicted in figure. Table 1 lists the material

35 properties. Along the surface between x = 0 m and x =. m, a constant recharge of 0.0 m/day is applied. The spatial domain is discretized into a uniform two-dimensional grid consisting of 0 x cells. Figure illustrates that our model compares very well to the HydroGeoSphere model (reference). It can be observed that the saturation fronts produced by DisCo are sharper than those generated by HydroGeoSphere and Forsyth et al.. This is a logical result of different spatial and temporal discretizations. We used a relatively dense grid and compared to Forsyth et al. we also used smaller time steps.. Conduit flow: Rossman s [] test cases To verify the capability of our model to handle free-surface and pressurized conduits flows, we compare our model with two test cases as performed by Rossman [] using the Storm Water Management Model version.0 (SWMM.0) of the Environmental Protection Agency. The first test case corresponding to the second challenge test case of Rossman consists of five conduit sections in series, each being 00 foot long. The first, third and fifth section have a diameter of 1 foot. The second and third have a diameter of foot. The bottom slope of the conduits is 0.0. Manning s roughness coefficient is given by 0.0 m - 1/ s. Initially the sections are dry. During the first 1 minutes of the simulation the recharge at the upstream end is linearly increased from 0 to 0 ft /s. This recharge remains constant for hours and is then linearly decreased to zero is 1 minutes. The simulation time is hours. In our model each conduit section is discretized into 0 cells. Figure shows the flow rate in the middle of the first section. It can be observed that the middle of the first conduit section becomes surcharged after approximately 1. hours. The simulation result