MODELING SURFACE WATER AND GROUNDWATER INTERACTIONSAT MESOSCALE USING A PSEUDO 3-D NUMERICAL SCHEME

Diego Rivera & José Luis ArumíDepartment of Water Resources, University of Concepcion

Alexander FernaldNew Mexico State University

November 26, 2008

Abstract

Water and solute ?uxes are generally related with agricultural practices and di-rectly controlled by local hydrological processes. The spatial and temporal patternof water flow (irrigation water, surface water, groundwater) establishes restrictions toagricultural productive potential. Understanding of transport characteristics relatedto hydrological processes and agricultural management practices could help to under-stand and predict the temporal patterns of diffuse pollution sources, and also improvemanagement practices. However, water management strategies must be adapted orevaluated before field implementation, because the application of these strategies isconstrained by hydrological, economical, environmental, technological and field-specificissues. For this task, numerical simulation is a fast and inexpensive approach to study-ing optimal management practices, but there is a scale issue wich must be included toget operational and useful tool across spatio-temporal scales. Given that the spreadof pollutants in a system is a function of the velocity field, the knowledge of the flowpatterns spatial-temporal variations is an useful tool to explore and understand trans-port phenomena, transient storage of pollutants and how boundary conditions imposedby anthropogenic activity affect these patterns in both contaminants and groundwaterrecharge, but also crop quality and productivity levels.

Based on field data and recent literature, the objetive of this paper is to presenta review about the development and implementation of an advanced numerical model(finite volumes on unstructured triangular meshes and quasi-3-dimensional modeling)that included irrigation-related local processes and the interaction between surface andgroundwater system at a meso-scale. This conceptual framework allows the inclusion ofrelevant eco-components such as orchards, irrigation systems, irrigation canal networks,rivers and aquifers.

The aim of this work is to develop a numerical model which will be used as avirtual laboratory, for the design of monitoring systems as well as the evaluation of theeffects of channel networks and irrigations systems on the flow and velocity fields andwater fluxes. The use of collective knowledge is also considered, in order to develop an

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advanced generic model. The proposed sheme must still be verified with field data andalgorithms solution of the equation of Richards should be contrasted with experimentaldata or published data. Keywords:

Contents

1 INTRODUCTION 3

2 WHY ANOTHER MODEL? 4

3 THEORETICAL BACKGROUND & STATE OF THE ART 53.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Agriculture and groundwater/surface water interactions . . . . . . . . . . . . 63.3 Advection-Diffusion-Reaction Equation & Richards Equation . . . . . . . . . 73.4 Conservation Laws Discretization using the Finite Volume Method . . . . . 9

4 NOVEL ASPECTS & CURRENT APPROACHES 124.1 Quasi 3-dimensional modeling scheme . . . . . . . . . . . . . . . . . . . . . . 134.2 Proposed steps for model implementation . . . . . . . . . . . . . . . . . . . . 15

5 CONCLUDING REMARKS 20

List of Figures1 Conceptual scheme of the Quasi 3-dimensional Modeling Scheme application. 162 (A) Digital elevation model (DEM) from NASA Shuttle Radar Topography

Mission(SRTM Farr et al., 2007) for was used to contruct vertical cuts us-ing Global Mapper 7. (B) After that, unstrutured triangular meshes weregenerated using EASYMESH . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 (A) Zoom for the grid showed in 2B. (B) Zoom for the grid showed in 2Bwhere black dots indicate mesh control points. . . . . . . . . . . . . . . . . . 18

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1 INTRODUCTIONThe preservation of the water quality, adequate irrigation management and suitable plan-nification and monitoring strategies are essential components of the economical, technicaland social sustainability of agriculture. With increasing demands on water resources, im-proved decision-making, requires improved models (Beven, 2001a). Water and solute fluxesare generally related with agricultural practices and directly controlled by local hydrologi-cal processes, and the spatial and temporal pattern of water flow (irrigation water, surfacewater, groundwater) establishes restrictions to agricultural productive potential.

The analysis of the influence of agricultural activity should consider not only the defi-nition of loads and concentrations, but should also include the search for spatial-temporalpatterns in the flow field to develop a better understanding of the system and to contributeto monitoring system design and hypothesis testing (Kang and Lin, 2007). Understandingof transport characteristics related to hydrological processes and agricultural managementpractices could help to understand and predict the temporal patterns of diffuse pollutionsources, and also improve management practices.

To model hydrological processes and the effect of agricultural activity on water resourcesquality and quantity, the hydrological cycle should be considered in its widest scale rangeand spatio-temporal variability. The conceptualizations used to generate functional modelsshould consider that the concept of scale is not arbitrary not even in continuous ranges(Klemes, 1983). Interactions dentro del agro-hydrological system are appreciable at interme-diate scales (meso-scales), especially in the transient stage of the groundwater/surface wa-ter systems interaction (Wroblicky et al., 1998; Winter, 1999; Addiscott and Mirza, 1998).Examples of these meso-scale phenomena are: the effect of filtrations of irrigation waterdistribution channel networks on the groundwater flow, the water and pollutants storage invadose, riparian and boundary areas (Sophocleous, 2002), i.e. transitory pollutant storagezones, the nitrate movement below irrigated areas (Böhlke, 2002) and the riparian zonesprocesses, both biogeochemical and hydrological, (Gribovszki et al., 2008; Szilágyi et al.,2008; Rivera et al., 2005; Molenat et al., 2008).

The analysis of interaction between surface water, groundwater and irrigation watersystems, their relation with variations on spatio-temporal patterns of both contaminants andgroundwater recharge,and also crop quality and productivity levels (Böhlke, 2002) requires anumerical tool to simultaneously solve Richards equation, the Advection-Diffusion-Reactionequation that includes in its conceptualization the non-homogeneous, non-linear, and 3-dimensional nature of agro-hydrological systems as well as boundaries conditions variable intime and space (e.g crops water requeriments).

Based on the available evidence and recent literature, the objetive of this paper is topresent a review of some issues about the development an implementation of an advancednumerical model (finite volumes on unstructured triangular meshes and quasi-3-dimensionalmodeling) including the main features of agro-hydrological systems. The numerical model isconsidered as a tool (a virtual lab), wich will facilitate the design of monitoring systems aswell as the evaluation of the effects of channel networks and irrigations systems on the flowand velocity fields and water fluxes in the. The use of collective knowledge is also considered,

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in order to develop an advanced generic model.

2 WHY ANOTHER MODEL?Dispersion of solutes in groundwater is caused mainly by spatial variations in aquifer prop-erties but additional dispersion can be induced by temporal fluctuations in the flow field orchanges on the spatial configuration of the groundwater flow systems (Vissers and van derPerk, 2008; Kim et al., 2000). Temporal fluctuations in recharge, discharge or boundaryconditions will also increase velocity variance and thus might also be expected to contributeto plume spreading (Goode and Konikow, 1990). The boundaries of flow systems may beunstable or move due to climatologically or anthropogenically induced changes in boundaryconditions (Vissers and van der Perk, 2008). However, the interactions and effects betweenagricultural systems (e.g. irrigation and channel networks) and groundwater systems is oftenunknow (Rivera, 2006). For example, off season water inputs and the water managementactivities (both agricultural and urban) can lead to stable water table level, but this impliesthat, in general, contaminants transported by groundwater flow remain within the flow sys-tem (Vissers and van der Perk, 2008), allowing the generation of transitory storage zones forcontaminants (Rivera et al., 2007b) and limiting dispersion. Variations in groundwater levelcould allow mixing processes, spreading contamiants in the systems, buffering or controllingconcentration fluctuations (Rivera, 2006; Molenat et al., 2008; Grimaldi et al., 2004).

Agro-hydrological systems have features wich limit the use of regional and/or local models(Addiscott and Mirza, 1998), because water flux patterns are affected not only by the spatialcrop pattern and crop extension, but also by water management practices and crop’s phe-nology. Indeed, Nakayama et al. (2006) reproduced the observed groundwater fluctuationson the whole system using the Integrated Catchment-based Ecohydrology (NICE) model,but it could not reproduce some fluctuations that were due to local water withdrawal. Forthese reasons, coupling groundwater, surface water and agricultural models is and useful toolfor environmental assessment and management in agrohydrological systems (see Nakayamaet al., 2006; Young et al., 2007; Ocampo et al., 2006b; Singh et al., 2006a,b).

In Chile, climate is strongly influenced by global climatological patterns (e.g. El NiñoSouthern Oscillation), soils are highly stratified and heterogeneous, and agricultural zonesare mainly in fluvial valleys. For these reasons,the applications of regional model have to dealwith changing boundary conditions and heterogenous hydraulics and hydrological systems.Indeed, the results of the Water Management Technologies for Intensive Sustainable Agri-culture Project (FONDEF grant D02I-1146) indicated that in Peumo Valley the productivelevel was dependent on the actual hydrological and hydrogeological situation. Besides, filtra-tions from the irrigation channels in Peumo Valley have implications of in the assignment ofground water rights and changes in groundwater flow and recharge patterns (Rivera, 2006;Rivera et al., 2007b).Based on the data obtained from FONDEF project and the availableliterature, it could be established that:

(i) Filtrations from the irrigation channel network interact with the groundwater system

(ii) Agricultural zones of intensive irrigation interact with groundwater system

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(iii) Water table rising in summer( Once the irrigation period has begun) is an expressionof the coupling of the surface water (rivers, springs), groundwater and irrigation water(channel networks and irrigated areas) systems

(iv) The effect of the irrigation systems is not only local (meso-scale)

(v) Irrigation systems are an integral part of the agro-hydrological system and are not onlyboundary conditions

Therefore, the main characteristics of an agro-hydrological system in the Central ChileanValley are: the presence of unlined irrigation channels networks, the presence of intensiveirrigation areas, and the off seasons input of water into the system. Therefore, irrigationchannel networks and irrigation systems are mobile and dynamic boundary conditions in timeand space. Based on collected observations and the bibliographic review, working questionsare established to define the problem to be studied:

(i) What is the effect of rising water table, produced by the interaction between irriga-tion water, distribution system and groundwater, on the velocity and field flow at theregional level?

(ii) How should the coupling between the irrigation water and groundwater systems bemodeled?

(iii) How should the interaction between the irrigation water and groundwater systems beincluded in a numerical model?

3 THEORETICAL BACKGROUND & STATE OF THE ART3.1 Theoretical framework

Groundwater is a geological agent that interact with the environment through chemical,physical, and kinetic processes that become manifest in the hydrology and hydraulics, thevegetation, the geomorphology and the transport and accumulation processes (Tóth, 1999).These interactions with the environment, and especially with surface waters, are complexgiven the ecological significance and the impacts of human activity on these relations (Sopho-cleous, 2002).

Equations and constitutive relationships can represent the movement of water in the hy-drological continuum. These equations are used to estimate the velocity field and water flowfield in the radicular, non-saturated, and saturated areas. The non-saturated or vadose areais the link between surface water and groundwater and is a determining factor in groundwa-ter dynamics. The velocity and flow field can be estimated solving the Richards equation.To study transport processes, the Advection-Diffusion-Reaction (ADR) equation is applica-ble to passive and reactive pollutant transport in variably saturated porous media (Ewinget al., 2000). The transport equation is function of the velocity field, medium properties andpollutant characteristics. Since, Richards equations estimates the velocity and flow field, is

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used to solve the ADR equation for the pollutant concentration field (Cariaga et al., 2005;Sepúlveda and Vera, 2006).

The pollutant transport models actually available are applicable at regional and localscales. For example, at regional scale, MODFLOW (McDonald and Harbaugh, 1984) presentsproblems when simulating processes near the domain boundaries and interfaces, where gen-erally there is exchange between the agricultural system and the environment (Sophocleous,2002; Wroblicky et al., 1998). It is possible to model smaller domains decreasing mesh size,but in a local-basis modelling (see Lautz and Siegel, 2006). At local scale, HYDRUS 2D(Simunek et al., 1999) simulates water and pollutants in movement using finite elements,which are applicable, for example, to the design of drip irrigation equipment and nitratemovement in the soil column, but do not include interaction processes with surface waters.However, the meso-scale modeling does not implies mesh refinement or the local applicationof models in distinct locations, but rather a conceptualization that integrates the differentspatio-temporal scales in the equations and models used in such a way that the responsesare integrated and the influences between systems can be appreciated.

Modeling couls also considers the integration of collective knowledge (Beven, 2001b) inorder to test the model with data available in studies and literature, to study its structureand to generate a tool that can be used in the monitoring system design in agro-hydrologicalsystems. In this context, the models are virtual hydrological laboratories where both aspecific study and a conceptual generalization are developed to explore and test hypotheses(Sivapalan, 2005). Because almost any model with sufficient free parameters can yield goodresults when applied to a short sample from a single catchment, effective testing requiresthat models be tried on many catchments of widely differing characteristics, and that eachtrial cover a period of many years (Linsley, 1982). Virtual intercomparisons can be extremelyinstructive and they can definitely help to improve models and assess their generality (Perrinet al., 2003).Therefore, data should be used to corroborate hypotheses and not only tocalibrate models developed a priori, i. e. the application of the Equifinality Principle (Beven,2006, 2001a).

3.2 Agriculture and groundwater/surface water interactions

In an extensive review of groundwater/surface water (GW-SW) interaction, Sophocleous(2002) reviewed the interaction mechanisms, scale influence, quantitative analysis methods,and ecological implications. This theoretical and conceptual framework establishes that themass and energy transference processes determine the recharge-discharge areas within thesystem and determine the conditions for ecosystem and crop maintenance (Wroblicky et al.,1998; Winter, 1999). New conceptual frameworks for the study of agro-hydrological systemshould include a new continuum, irrigation water (IW), which interacts with GW and SW.Examples of interaction are groundwater recharge by filtration from channels (Khepar et al.,2000) or the infiltration from pressurized irrigation equipments (Taghavi et al., 1984). Irri-gation acts not only as a recharge source but also as a new condition that locally determinesflow patterns (Wroblicky et al., 1998).

In the interface between the surface water and groundwater continuums, i.e hyporheiczone, biogeochemical processes play an important role in the transfered water quality. For

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example, denitrification is one of the principal processes of nitrogen removal together withthe dilution with groundwater, and consequently the study of these processes and theirrelation with basin hydrology and groundwater flow patterns will provide knowledge on flowdynamics (Cey et al., 1999; Cirmo and Mcdonnell, 1997). Hantush and Mariño (2001) presenta denitrification model in the hyporheic zone where the riparian zone acts as a natural filterand storage for nitrates produced in high cultivation areas. Additionally, these areas act asflow buffers, increasing the residence time of these pollutants in the vadose zone.

Two different numerical approximations are defined for the GW-SW coupled models.Fully coupled models (e.g. Vanderkwaak, 1999) consider coupled variably saturated flowequations (Richards Equation) and open channel flow equations (Chèzy-Manning Equation,Kinematic Wave Approximation Equation, Saint Venant Equations) in a single global matrixthat is simultaneously solved for each time step. This is a robust approximation that isless susceptible to numerical oscillations and errors in the mass balance, but it does notconsider the differences in time scales, and consequently the transient study implies smalltime steps. The second one combines and solve a discrete system of balance equations,where the water and solute flows are part of the solution, achieve the coupling of surfaceand subsurface flow. The transport and transference equations are defined as a function ofthe surface interaction scales, fluid or solute properties and system parameters (e.g. Husseinand Schwartz, 2003; Perkins and Koussis, 1996; Niazi, 2000), and they consider solvingindependently the equations of both continuums, continually updating the flows betweencontinuums to maintain the mass balance.

In most available models, the link between surface water and groundwater is the flowfrom rivers or unlined channels. In general, the interaction models consider that the flowfrom channel bottom is a boundary condition (Dirichlet, Neumann or Robin) that is includedin the flow model for variably saturated medium (e.g. Swain and Wexler, 1996), or numericalmethods that consider the flow from the channel as unknown in the system to be solved (e.g.Vanderkwaak, 1999). In these formulations, functions in which the flow is proportional tothe difference in hydraulic head (discrete formulation) or the potential gradient in a SW-GWcontinuum (continuous formulation) are considered. The flow’s magnitude depends on thehydraulic characteristics of the vadose zone and the channel bottom, and the relative depthof water table respect to channel bed. The flow exchange equations can be manipulated torepresent free drainage, evaporation, evapotranspiration, rain, and irrigation conditions.

3.3 Advection-Diffusion-Reaction Equation & Richards Equation

The macroscopic models of solute transport in porous medium are usually described by theAdvection-Diffusion-Reaction equation (Bear, 2007):

∂c

∂t+∇ (v · c−D · ∇c) = S on Ω×RN (1)

where Ω corresponds to the spatial domain, c (x, t) corresponds to the spatio-temporal dis-tribution of a pollutant due to advective transport in the velocity field v (x, t), D (x, t) isthe diffusion tensor and S (x, t) corresponds to the source term. The problem is defined

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if the initial conditions c (x, t) = c0 (x) ,x ∈ Ω are included, and Dirichlet-type boundaryconditions c = gD (x, t) and Neumann boundary conditions D∇c · n = gN (x, t) are speci-fied in the domain boundary ∂Ω. This equation is highly non-linear due to scale problemsof the diffusion tensor (Eymard et al., 1997; Ewing et al., 2000; Bertolazzi and Manzini,2004b,a; Michev, 1996, 1998; Eymard et al., 2004). Generally, this equation is discretizedapplying separation techniques for advective and diffusive components.The velocity field re-quired in (1) to study soil-water dynamic (Gong et al., 2006; Short et al., 1995) and thepollutants that are both, transported and transformed within the system, can be estimatedsolving Richards equation (Vanderkwaak, 1999; Farthing et al., 2002; Chu, 2002; Bertolazziand Manzini, 2004b), which describes the water flow in a variably saturated porous medium(Richards, 1931; Clement et al., 1994) :

∂θ

∂t−∇ [K (ϕ)] · ∇ (ϕ+ z) = s, on Ω×RN (2)

where Ω corresponds to the spatial domain, ϕ (x, t) corresponds to the pressure head, θ (x, t)is the soil water content, z is the vertical coordinate defined positive upward, K (x, t) isthe hydraulic conductivity tensor, and s (x, t) corresponds to the source term. The problemis defined if the initial conditions ϕ (x, t) = ϕ0 (x) ,x ∈ Ω are included and the Dirichletboundary conditions ϕ = gD (x, t) and Neumann boundary conditions K∇ (ϕ+ z) · n =gN (x, t) are specified in the domain boundary ∂Ω.

The main difficulty in the numerical solution of Richards equation for flow in a variablysaturated medium is related with the instabilities that are generated when solving non-linearequation systems, given the non-linear characteristics of the constitutive relationships K (ϕ)and θ (ϕ). Another difficulty is the definition of the adequate position of the moving boundaryconditions, such as water table position by vertical water flow towards a non-confined aquifer(Eralingga, 2004; Simpson and Clement, 2003). Celia et al. (1990) demonstrated that themixed form of Richards Equation, where the temporal derivative is a function of the soil watercontent and the spatial derivatives are functions of the pressure head, offers the advantagesof pressure-based and water content-based formulations.

The numerical solution schemes of the temporal derivative are principally based on ex-plicit and implicit Euler schemes. From Celia et al. (1990), the use of the Taylor series in theexpansion of the temporal derivative in an iterative Picard scheme has become a standardgiven its mass conservation properties. Some alternative schemes or modifications can befound in Binning (1994), Vanderkwaak (1999), Mazzia et al. (2000), Farthing et al. (2002),Farthing and Miller (2001), Eymard et al. (1997) and Barth and Ohlberger (2004). An ex-tensive review on numerical methods for time-dependent advection-diffusion problems canbe found in Ewing and Wang (2001).

Currently, numerical methods, using the finite element method (FEM) or the FiniteDifferences Method (FDM), can obtain solutions for the Richards equation with relativeefficiency for the non-linear equation systems coupled in time and space. These methods havebeen replaced by mixed solution frameworks that take advantage of the different formulationsfor the Richards equation and the boundary conditions within the hydrological system. Celia

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et al. (1990) showed the advantages of using a mixed formulation of Richards equation anda Picard iterative approach for the solution of the temporal derivative, which diminishes themass balance error. This work identifies two principal problems in the solution of Richardsequation: the equation’s form that determines the discretization and the mass balance, andthe treatment of the temporal derivative. In spatial discretizations, the structured meshesare generally associated to FDM, while unstructured domains and meshes are associated toFEM. The election of one or other method depends principally on the modeler’s capabilitiesand choice and on the spatio-temporal domain as well as the variability of parameters andboundary conditions (Simpson and Clement, 2003).

3.4 Conservation Laws Discretization using the Finite Volume Method

The Finite Volume Method (FVM) is an adequate discretization method for the numericalsimulation of several types of conservation laws, which has been widely used in engineer-ing sciences such as fluid mechanics, mass and heat transfer (Antilén et al., 2006; Moragaand Salinas, 1999, 2000; Moraga and Vega, 2004; Moraga and López, 2004), or petroleumengineering. Its principal advantages lie in that it can be applied to structure and unstruc-tured meshes in arbitrary geometries, and that the schemes obtained are robust and possessthe property of local conservation (Eymard et al., 1997; Barth and Ohlberger, 2004). Thismethod combines FEM’s advantages to work in domains with complex geometries and com-plicated boundary conditions and FDM’s implementation simplicity (Ewing et al., 2002).The conservation property makes FVM attractive in problems where the fluxes, and conse-quently conservation, are important (Morton et al., 1997). FEM preserves mass conservationin an aproximate way with an asymptotic limit when the mesh size tends to zero, althoughthis is a disadvantage when the objective is to model complex geometries and small time in-tegration steps. However, FVM is local and exactly conservative (Ewing et al., 2000; Michev,1996) since it is based in the balance for each control volume. About the mathematical baseof the finite volume approximation in conservation problems. Eymard et al. (1997), Barthand Ohlberger (2004), Michev (1996) and Cai et al. (1991) are fundamental references andmathematically rigorous

FVM is a suitable tool for the numerical solution of equations based on conservationlaws, such as Richards equation (for a rigorous mathematical formulation see Eymard et al.,1999)and the ADR Equation. It is applicable in problems with high gradients, such as thehyporheic zone, or discontinuities in ,for example, stratified soils. Bertolazzi and Manzini(2004b,a,c,d), Ewing et al. (2000), Eymard et al. (2004, 2006), Morton et al. (1997) and Caiet al. (1991) are examples of the application of FVM to pollutant transport problems withina porous medium, and Loose et al. (2006, 2005) and Versteeg and Malalasekera (2007) forhydrodynamic problems. Finite Volume Methos has been also applied to flow problems inporous medium applying Richards equations in two phases (air and water) can be found inMichev (1996), Binning (1994), Eymard et al. (1997) and Cariaga et al. (2005, 2007a). Otherexamples are Eymard et al. (1997) with a detailed mathematical approach, Vanderkwaak(1999) applying the control volume finite element method, Eralingga (2004) and Manziniand Ferraris (2004) on unstructured meshes. However, the studies on the application ofthese methods in Richards equations and their testing with field data are recent. Field

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testing (Loague et al., 2005; Loague and Vanderkwaak, 2004; Vanderkwaak and Loague,2001; Vanderkwaak and Sudicky, 2000) for coupled models and field explorations can befound in Ocampo et al. (2006a), Bonilla et al. (1999) and Molenat et al. (2008).

Equation 3 presents the formulation in the form of a Cauchy initial value problem for aConservation Law, conceptually equivalent to the equation 1:

∂u∂t

+∇F (u)− ε∇2u = 0, on RN ×R+

u (x, 0) = u0 (x) , on RN (3)

where u (x, t) : RN × R+ → R+ is the solution of (3), F (u) ∈ C1 (R) is the flow function,and ε ≥ 0 is the diffusion coefficient or tensor. When applying the Divergence Theorem orGauss-Green Theorem in a region T, the integral form (equation 4)is obtained for problem3 This equation establishes that the change rate for a substance with density u in a fixedcontrol volume T is equal to the total advective flow minus the flows due to diffusive processesand the net substance flow in the borders TΩ:

d

dt

∫Tudx+

∫∂TF (u) dν −

∫∂Tε · ∇udν = 0 (4)

FVM requires that the domain Ω ⊂ RN be discretized in a set of control volumes (CV)= = Tj, so that

⋃T∈= T = Ω. The integral form for the flows on the left side of the equation

4 can be approximated for the CVs Tj and Tk that share a edge ejk, so that ejk = Tj ∩ Tkby: ∫

∂Tj

F (u) dν ≈∑

∀ejk∈∂Tj

gjk (uj, uk) (5)

∫∂Tj

ε · ∇u dν ≈∑

∀ejk∈∂Tj

djk (uj, uk) (6)

These numerical flows are assumed to satisfy the conservation property that assures thatthe flows through ejk cancel each other, i.e. gjk (u, v) = −gkj (v, u), and the consistencyproperty that assures that the value of the flow function for uj = uk is exactly reduced tothe flow integral, so that gjk (u, u) =

∫ejkF (u) dν.

From the equations 4, 5 and 6 and a flow function that satisfies the conservation andconsistency properties, the totally discrete finite volume approximation is defined for 3 inthe time interval [tn, tn + ∆t] using a explicit Euler scheme with advective numerical flowfunction gjk

(unj , u

nkj

)and diffusive numerical flow djk

(unj , u

nkj

)in the following discrete sys-

tem:un+1j = unj −

∆t

|Tj|∑

∀ejk∈∂Tj

[gjk

(un+1j , un+1

k

)− djk

(un+1j , un+1

k

)], ∀Tj ∈ = (7)

Therefore, the conservation equation is represented as a discrete mass balance equation foreach control volume Tj, where σj corresponds to the set of control volume indexes sharingan edge of Tj:

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|Tj|dcjdt

+∑k∈σj

(Gjk +Hjk) +∑k∈σj′

Fjk′ = Sj (8)

where Gjk corresponds to the advective flow function, Hjk corresponds to the diffusive flowfunction, and Fjk′ corresponds to the boundary flow function.

Finite Volumes schemes based on higher-order reconstructions are based in the "recon-struction" of values within the cell, allowing better estimation of gradients and flows in theinterfaces between CVs. Gradient estimation can be done using least squares or by applyingthe Gauss-Green Theorem. The use of least squares in a 2-dimensional simplex (triangle)considers gradient estimation from point values for a given stencil.

The FVM is highly flexible in CV definition. In the case of cell-centered FVM, thetriangle is the CV and the unknowns are stored, for example, in the baricenter. In the caseof vertex-centered FVM, the CV is build through a dual mesh, for example Voronoï diagrams(known as Thiessen Polygons in hydrology) where the unknowns are stored in each trianglevertex.

Use of unstructured triangular meshes allows work with complex geometries, either onthe domain boundaries or within the domain, through geometric restrictions that allow, forexample, meshes densification in the interfaces (e.g. hyporheic zone). There is a consider-able quantity of meshers, generally used in finite elements. Du and Wang (2005) present anextensive review of the available methods and algorithms. The mesh generation algorithmsand software are varied, and include the free available TRIANGLE algorithm, which gener-ates a Delaunay triangular mesh (Shewchuk, 1996, 2002) and the P2MESH solver for partialdifferential equations solving (Bertolazzi and Manzini, 2002).

The literature on error estimators, convergence and stability is extensive and is based onthe search for a priori or a posteriori measures that compare the solution functions approxi-mated with the problem’s exact solution in a time step. The existence and uniqueness of thesolution is investigated through the generation of entropy functions for different polynomialspaces (L2, L , Hilbert spaces H1, H2). For this point, the references are only indicative: therigorousness and generalization of Eymard et al. (1997), Eymard et al. (2006) for diffusionproblems, the seminal works of Cai et al. (1991) and Kröner et al. (1995), and the studiesof Coudière et al. (1999), Morton et al. (1997), Ewing et al. (2000) and Ewing et al. (2002)for diffusive and convective-diffusive problems.

For gradient reconstruction analysis, the most important studies are Farthing and Miller(2001) and Ohlberger (2004) on the use of meshes and adaptive stencils for gradient recon-structionand Eymard et al. (2006) on piecewise constant functions. Bertolazzi and Manzini(2004b,a) use linear and polynomial reconstruction applying limiting strategies (Bertolazziand Manzini, 2004d) and vertex value reconstruction utilizing least squares (Bertolazzi andManzini, 2004c; Manzini and Ferraris, 2004; Bertolazzi and Manzini, 2005), based principallyon Coudière et al. (1999). Rigorous reviews respect to gradient reconstruction can be foundin Eymard et al. (1997), Eymard et al. (2006), Mavriplis (2003) and Barth and Ohlberger(2004).

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4 NOVEL ASPECTS & CURRENT APPROACHESKeith Beven (Beven, 2001a), summarized the modeling problems as the non-linearity prob-lem, the scale problem, equifinality problem, the uniqueness problem, and the uncertaintyproblem. In all the cases, the models should be analyzed first in their physical significanceand in the truth-value of field data. Kirchner (2006) indicates that hydrology is at a crucialpoint for the advance, lead by new hydrological measurements, new methods to analyzehydrological data, and new approaches to modeling hydrological systems. This author in-dicates five important points in order to advance in hydrology, and this vision is shared byBeven (2001a) and Klemes (1983), and formalized in Sivapalan (2005):

(i) Design of new monitoring networks, field observations, and field experiments that ex-plicitly consider spatio-temporal heterogeneity of hydrological processes

(ii) Replacement of linear and additive black box models by gray box approaches thatbetter capture the non-additive and non-linear character of the hydrological systems

(iii) Development of physically based equations for hydrological behavior at the basin orhillslope level recognizing that they can seem different from the equations that describesmall-scale physical processes

(iv) Development of minimally parameterized models that consequently can fail and shouldbe tested, and

(v) Development of methods to comprehensively and incisively test models.

This ongoing research is oriented to the problems indicated in points (ii), (iii) and (iv) inorder to develop in the future point (i). For example, Kirchner et al. (2001) define a middlepath, which directly captures the essentially extended character of hillslopes and basins in thegoverning equations without explicitly requiring spatial disaggregation and the subsequentproliferation of free parameters. Other examples are: Fox and Durnford (2003) propose aformulation that defines three flow conditions, where the border condition in the flow passesfrom the Dirichlet to the Neumann type; Osman and Bruen (2002) simplify and developsimple and functional equations that depend on field measurable parameters applicable tosystems in which the irrigation water distribution network has notable variations since itconsiders the water table position over time. Other studies are presented in Thoms (2003),Vanderkwaak (1999), Swain and Wexler (1996) and Hussein and Schwartz (2003).

We seek to address the coupling of conservation law systems in agro-hydrological systemsand the development of context-dependent mass balance estimators. Current approachesabout finite volumes are oriented to the development of advanced mathematics in the appli-cation of finite volume approximations for partial differential equations (e.g. Eymard et al.,2006). Their application to water flow in variably saturated porous medium and pollutanttransport in porous media are academic, principally 2-dimensional in specific problems toverify their numerical performance (e.g. Bertolazzi and Manzini, 2004b; Manzini and Ferraris,2004). The models are applied in problems where the velocity field is known, generally in

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saturated flow in porous medium, and consequently is needed to develop models that incor-porate the flow in the vadose zone. The advanced topics in the Finite Volume Method refer tothe construction of discretizing frameworks for diffusive problems (e.g. Eymard et al., 2006;Bertolazzi and Manzini, 2004b), the extension to non-linear conservation law systems, e.g.coupling of the Richards equation and the Advection-Diffusion-Reaction equation (Cariagaet al., 2005, 2007a,b).

Regional scale groundwater models generally represent surface water as boundary con-dition (e.g. McDonald and Harbaugh, 1984; Sophocleous, 2002). In the case of groundwa-ter/surface water interaction problems, the surface water is considered as distributing sourceswith known pollutant flows and loads (e.g. Russo et al., 2001; Gardener, 1999), but the de-velopment of integrated hydrological models has been limited by the non-linearity problemsof the constitutive equations, scaling, and parameter estimation (Moench, 2003).

There is extensive literature on groundwater (GW) and surface water (SW) interactionmodels at watershed scale, but none of the cited models include in their conceptualization themeso-scale problems associated with agricultural activity. The GW model MODFLOW (Mc-Donald and Harbaugh, 1984) was extended for surface waters with BRANCH (Schaffranek,1987) and to the interface with MODBRANCH (Swain and Wexler, 1996); the MODFLOW-SURFANCT 3D model (HYDROGEOLOGIC, 1998) and the MODFLOW-WhaT (Water-shed Hydrology and Transport) developed by Thoms (2003). The work of Osman and Bruen(2002) improved the mathematical formulation of aquifer recharge sources due to losses fromwater flows. The models that explicitly consider the GW-SW interaction are the model Sys-tem Hydrologique Europèen (SHE; Abbott et al. (1986)), the coupled GW-SW interactionanalysis model SW Adaptative Hydrology Model ((ADH; Schimdt and Roig, 1997, citedby Hussein and Schwartz, 2003), the InHM model (Integrated Hydrology Model) devel-oped by Vanderkwaak (1999), the FTSTREAM Model (Hussein and Schwartz, 2003) andphysically-based, spatially-distributed model for simulation of surface/subsurface flow andthe interactions between these domains (Panday and Huyakorn, 2004).

4.1 Quasi 3-dimensional modeling scheme

The new aspects to develop are related with the application of the quasi 3-dimensional mod-eling approach, the inclusion of meso-scale processes and agro-hydrological system’s char-acteristics. The quasi 3-dimensional framework, which integrates knowledge and modelingtechniques that have not been applied together and essentially include from its concep-tualization: agricultural systems characteristics, the 3-dimensional nature of the modeledsystem using 2-dimensional equations and the possibility to solve the flow problem in non-homogenous systems (constitutive and boundary functions are independently applied in eachcontrol volume).

For the solution of Richards equation by finite volumes, the Finite Volumes Methodwas applied in unstructured triangular meshes following Bertolazzi and Manzini (2005),Manzini and Ferraris (2004) and Coudière et al. (1999). This formulation allows solvingthe flow field on variably saturated and heterogeneous soils (different conductivity functionsand boundary condition, e.g. Dirichlet, Neumann, Robin). Algorithms were codified inMATLAB environment and Delaunay mesh was generated using EASYMESH. A numerical

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model was implemented that solves Richards Equation for variable saturated flux, includingvariable spatial and temporal boundary conditions. A cell-centered formulation was usedon 2-dimensional non-structured triangular meshes. A linear reconstruction was used toobtain and second-order approximation. Vertex values are reconstructed from the cell valuesusing the minimal square method Celia et al. (1990). This formulation allows the use oftemporal and spatial functions for hydraulic conductivity and soil water content. We usedthe following expressions for Richards Equation and boundary conditions:

∂θ

∂t−∇ [K (ϕ) +∇ (ϕ+ z)] = S (9)

K (x, t) = gk (x, t) , in Ω, t > 0 (10)

ϕ (x, t) = gD (x, t) , in ΓD, t > 0 (11)

n ·K · ∇ (ϕ+ z) = gN (x, t) , in ΓN , t > 0 (12)

For the spatial discretization,the original 3-dimensional domain is divided into orthogo-nal 2-dimensional domains, where the Z-axis corresponds to the vertical distance (positiveupwards). Each cell may have different function for hydraulic conductivity and boundaryconditions. A set T = Ti, i = 1, . . . , NT of quasi-Delaunay triangles with centroid coordi-

nates xi forms a two-dimensional gridNT⋃i=1

Ti ∼= Ω, where eij is the segment shared by triangles

Ti and Tj. The set of vertices of the triangular grids is V = Vi and let be the set of pointsP = Pi = P (xpl) where P ⊆ V . The elements of P are defined as control points con-straining the Delaunay mesh, because the control points must correspond to centroids of theelements. Let be Tα = Tαm , α = X, Y, Z the 2-dimensional meshes defined for the planesY-Z and X-Z, that must satisfy the condition V X ∩ V Y = P . If in each Tαm a finite volumeapproximation of Richards Equation is applied, then an approximation of the 3-dimensionaldomain solution is obtained.

The discretized expression of the balance equation for pressure head and water contentfor each control volume Ti, given by the finite volumes approximation of Richards Equation(Equations 9 to 12) are:

|Ti|dθidt

+∑j∈σi

Gij (ϕ) +∑j∈σi

Hij (ϕ) +∑j′∈σ′i

Bij′ (ϕ) = Si (13)

where Gij (ϕ) is the gravitational component, Hij (ϕ) is the diffusion component andBij′ (ϕ) is the boundary conditions component. The expresions for each term are (for furtherdetails and test problems see Manzini and Ferraris, 2004):

∑j

Gij (ϕ) = −∑∫

eij

n ·K · ∇(ϕ) d` (14)

14

∑j

Hij (ϕ) = −∑∫

eij

n ·K · ∇(z) d` (15)

∑j′

Bij′ (ϕ) = −∑∫

eij

gN d`−∑∫

eij

n ·K · ∇(gD + z) d` (16)

In the Quasi 3-dimensional Modeling Scheme, also called 2.5-dimensional (Rivera, 2006;Rivera et al., 2007b,a), the boundary conditions for the numerical model are defined apply-ing an approach that uses measurable field variables and smooth and continuous piecewisefunctions in ∂Ω, like K (x, t) = gk (x, t) which represents the spatio-temporal variation ofhydraulic conductivity, ϕ (x, t) = gD (x, t) which represents the spatio-temporal variationof Dirichlet boundary conditions, and n · K · ∇ (ϕ+ z) = gN (x, t) which represents thespatio-temporal variation of the Neumann boundary conditions.

In this scheme (Figure 1), each orthogonal triangular mesh m Tα = Tαm , α = X, Y, Z,defines the 2-dimensional triangulations in the planes Y-Z, X-Z and X-Y, where the finitevolumes formulation for Richards equation is solved, resulting in the vector ϕα. In thevertical planes, the condition V X ∩ V Y = P , which assures that there is a link betweenmeshes, should be satisfied.The P points or mesh control points restrict the Delaunay mesh,which means that the points xpl must correspond to the element’s centroids. For meshes Tαmresulting the pressure head vector ϕX . Then, the values in the control points are appliedas Dirichlet boundary conditions and each one of the M-meshes are solved in plane Y. Inthis way, the values obtained can be considered as interpolators between each one of the Xplanes. The process is repeated considering the Y axis as the principal flow axis, with whichthe vector ϕY is built. As an example of orthogonal grid for Peumo Valley in Chile (≈ 16000ha) is showed in

Given that the orientation of the field flow vector is not defined by only one axis, thevector ϕ

(ϕX, ϕY

)must be reconstructed using some method or algorithm. If the considered

vectors ϕX,Y are oriented in a orthogonal system, it can be assumed that they correspond withthe spatial base, with which the total vector total can be calculated as a linear combinationof the vectors ϕX,Y , where wi are weighting factors such that ϕ =

∑iwi · ϕi, i = X, Y .Once

the values of the pressure heads ϕiβ , β = X, Y resulting from the consideration of plane βas the principal flow plane have been obtained. Considering the set of pressure head valuesin control points of the mesh Pk, the weights W1 and W2 can be chosen so that the functionF is minimum:

F =∑k∈P

(ϕPk −W1 · ϕXk −W2 · ϕYk

)2(17)

4.2 Proposed steps for model implementation

To analyze agricultural pollutant transport in an agro-hydrological system, a numericalmodel will be implemented applying finite volumes and quasi 3-dimensional modeling basedon conceptual model for the specific system. Then, this model will be applied to temporaland spatial scaling, generating optimal spatio-temporal scale ranges, i.e. meso-scale def-

15

Figure 1: Conceptual scheme of the Quasi 3-dimensional Modeling Scheme application.

inition in the proposed problem. These ranges will allow visualization of the interactionbetween agriculture and the hydrological continuum.

Programming architecture is modular since one of the advantages of the approximationusing FVM is that the discretization is independent of the constitutive relationships. Conse-quently, it is possible, for example, to study the adjustment of different hydraulic conductivityfunctions in the velocity field without needing to change the balance equations. Visualizationalgorithms, such as isoline concentration maps, flow vectors, plane visualization and borderconditions, could also be developed.

Performance indicators must be be defined or adapted. The mass balance could beanalyzed applying the mass balance relation (additional mass in the domain/total flow inthe domain) given in Celia et al. (1990). The discrete version of the relative error εrel is(Bertolazzi and Manzini, 2004c; Manzini and Ferraris, 2004):

εrel =

√∑i |Ti| |ci − c (xi)|2√∑

i |Ti| |c (xi)|2(18)

where ci is the approximation using finite volumes of the concentration (or water content)and c(x, t) is the analytical (or mesasured) solution.

Domain definition is a fundamental activity in model development since it begins concep-tualizing the system to be represented, defining the relations and processes to be considered,and the auxiliary models to be used. Then, spatial and temporal domains will defined, i.e.

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rrrrrrrrrrrrrrrrrrrrrrrrrrrrr(A)

(B)

Figure 2: (A) Digital elevation model (DEM) from NASA Shuttle Radar Topography Mis-sion(SRTM Farr et al., 2007) for was used to contruct vertical cuts using Global Mapper 7.(B) After that, unstrutured triangular meshes were generated using EASYMESH

17

(A)

(B)

Figure 3: (A) Zoom for the grid showed in 2B. (B) Zoom for the grid showed in 2B whereblack dots indicate mesh control points.

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maximum and minimum mesh size, simulation time steps, processes to be modeled, and theparameters to be considered (hydraulic characteristics of the porous medium, pollutant char-acteristics). Once the domains are defined, the balance equations (Richards Equation, ADREquation), the process equations (e.g nitrification, retardation, sorption) and constitutiverelations (hydraulic conductivity, diffusion tensor, reaction constants) is defined.

To analize the effect of agricultural activities, is needed to estimate the natural condition.For example, (Vissers and van der Perk, 2008) to estimate the effect of the artificial drainagepattern simulated the natural historical, situation by removing the artificial streams andditches in the model area. Also, is possible to test models using qualitative scenarios (Nguyenet al., 2007).Then, a benchmark problem will be designed using available information andliterature including the characteristics of Chilean agro-hydrological systems.

For the evaluation of the irrigation network effect, technical literature and available in-formation will be collected and classified appliying collective knowledge techniques, so thatthe model’s parameter variation ranges can be defined for parameters such as freatic levelvariation, hydraulic conductivity, diffusion coefficient estimations, reaction rates, stratifica-tion. Additionally, the boundary conditions, such as irrigation rates, fertilizer applicationrates, crop evapo-transpiration demand, flow and distribution channel geometries, will bedefined.This problem will be used to simulate different scenarios that allow analysis of theeffect of the interaction between surface water, groundwater and irrigation water in pollutanttransport and the effect of channel networks in velocity and flow fields. At the end of thisstep a qualitative analysis of the effect of the channel network in an agro-hydrological systemon the transport of agricultural pollutants will be crried out.

As an example, for Chilean condiction, to evaluate the effect of the interaction betweensurface, ground, and irrigation waters in pollutant transport (differences between the veloc-ity and flow field for the situations with and without irrigation), at least four scenarios mustbe defined: (1) Base Condition: Valley without agricultural activity and without irrigationchannels, (2) Crop Condition: Valley with intensive agricultural activity and un-lined chan-nel network; (3) Management Condition I: Changes in the irrigation infrastructure such aschannel lining and increases in irrigation system efficiency; (4) Management Condition II:Changes in crop patterns, species and location. The analyzed time step will be daily for ahydrological year (12 months from April to April). The system’s transient must be analyzedin detail, being careful with changes in distribution network operating conditions.

To evaluate numerically the transport of contaminants and the changes produced in thewater table depth by the interaction between surface water, groundwater and irrigation watersystems a reference problem is used. Performance indicator, both qualitative and quantita-tive must consider the Equifinality Principle theorical framework (Beven, 2006; Beven andFreer, 2001).This reference problem should be characterized by its similarity with respectto an average Chilean agro-hydrological system: an irrigation water distribution network,intensive crop areas close to the water flow and pressurized and gravity irrigation systems.The application to a real case is considered as the next step since the monitoring systemneeds to be carefully designed so that the simulated and measured data are comparable. Wehave advanced work in this area with respect to the analysis of monitoring systems and the

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similarity between the real system and the numerical model.For spatial and temporal scaling, proposed model will be applied to problems where the

condition ranges have been fixed for at least the following cases: mesh size, time step andparameter values. In this way, at least scenarioa will be defined for a maximum, average andminimum value for each parameter is defined.

5 CONCLUDING REMARKSAt present, the available technology makes possible to obtain data sets at small time steps forhydrological variables such as water table depth, channel flow, soil water content and waterquality parameters. The time-density of these series allows to study phenomena within ahydrological system at different temporal states: transient and steady-state.

Given that the spread of pollutants in a system is a function of the velocity field, theknowledge of the flow patterns spatial-temporal variations is an useful tool to explore and un-derstand transport phenomena, transient storage of pollutants and how boundary conditionsimposed by anthropogenic activity affect these patterns.

The proposed modeling approach seeks from its initial conceptualization, to integratemeasured data from monitoring systems with simulation models in a continuous and dy-namical interaction. The question that may occur then is: what is the purpose of measuringand simulating in parallel?. Often measuring-based approaches and simulation-based ap-proaches, confront its truth-values, validating simulated data (truth-value near zero) againstmeasured data (truth-value near 1), but often we do not take account about representativelyand quality of measured data, given us incomplete information about both model perfor-mance and model’s reality representation capabilities. In this way, numerical models arenot only a useful tool in the simulation of certain events or phenomena, but it also allowsfeedback between water resources management constructed by knowledge-based approaches,"real" data from monitoring systems and "simulated" data from numerical models.

There must be a compromise between the simulation and monitoring systems, in orderto improve management strategies, including water quality issues, water efficiency at basinscale and singular scenarios forecasting. In agricultural ecosystems, water fluxes and contam-inants transport phenomena are in both spatial and temporal, meso-scale. As an examplethe Peumo Valley, the interaction between the regional groundwater system, the local char-acteristics of the distribution systems of irrigation water and on-farm irrigation systems areclear, the evidence comes from water table stability. Also, climatological variables affectday-to-day management decisions; therefore small time discretization is hended.

However, the objectives set forth in this discussion are high. The optimal design andoperation of a system of monitoring is a difficult task in itself, as well as and engagementbetween simulated and measured data. Therefore, the proposed wording must still be verifiedwith field data and algorithms solution of the equation of Richards should be contrasted withexperimental data or published data.

The driving-idea for this ongoing research is the conceptualization of processes and in-teraction between surface water systems and ground water systems in an agricultural valley.Data from the Peumo Valley show and verified the main hypothesis of this work: the inter-action of irrigation systems with groundwater, and the dependence of the productive system,

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but have yet to verify the proposed model with these data.

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