a conceptual framework for incorporating surface–groundwater interactions into a river...

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Environmental Modelling & Software 26 (2011) 1554e1567

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Environmental Modelling & Software

journal homepage: www.elsevier .com/locate/envsoft

A conceptual framework for incorporating surfaceegroundwater interactions intoa river operationeplanning model

David W. Rassam*

Hydrological Modeller,1 CSIRO Land and Water and eWater CRC, Ecosciences Precinct, 41 Boggo Rd, Dutton Park, QLD 4102, Australia

a r t i c l e i n f o

Article history:Received 18 May 2011Received in revised form28 July 2011Accepted 28 July 2011Available online 24 August 2011

Keywords:Groundwateresurface water interactionRiver operationeplanning modelRiver low flowGroundwater extractionRiver depletionAnalytical solutionsRivereaquifer exchange fluxConjunctive management

* Tel.: þ61 7 3833 5586, þ61 400 877 676 (mob); fE-mail address: [email protected].

1 Mail to: GPO Box 2583, Brisbane, QLD 4001, Aust

1364-8152/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.envsoft.2011.07.019

a b s t r a c t

Groundwater discharge constitutes a significant proportion of the total flow volume in most rivers. Theexchange flux between surface and groundwater greatly impacts the surface as well as the groundwaterbalance with serious implications on ecosystem health especially during low flow conditions. There isa move towards conjunctive rivereaquifer management with the integration of surfaceegroundwaterexchange fluxes into surface and groundwater models to manage water as a single resource. Ground-watereSurface water (GWeSW) exchange fluxes are seldom integrated into river operation and planningmodels. The time lags associated with the impacts of groundwater processes on nearby rivers can greatlycompromise the forecasting capacity of river models especially during low flow conditions.

This paper presents a conceptual framework for integrating GWeSW exchange fluxes into the newgeneration river operationeplanning model ‘Source Integrated Modelling System’. The proposed GWeSW Link Module adopts a simple pragmatic approach for estimating the exchange fluxes betweena river reach and the underlying aquifer using explicit analytical solutions. This flux becomes an inflow/outflow to that river reach and forms part of the routing and calibration processes. The exchange fluxcomprises four components: (1) natural exchange flux resulting from river stage fluctuations during lowflow conditions, within bank and overbank fluctuations; (2) flux due to groundwater extraction; (3) fluxdue to changes in aquifer recharge; and (4) flux due to changes in evapotranspiration. The sum of thosecomponents during every time step dictates whether the river loses water to or gains water from theaquifer.

The proposed analytical solutions were found to provide flux predictions that agree favourably withthose derived from a numerical groundwater model. Recognising that the simplifying assumptions thatunderpin the explicit analytical solution are likely to be violated in the natural world, a suite of criteriawas recommended for their use under many violating conditions related to boundary conditions, headgradients, and aquifer heterogeneity. Low flow indices were adopted to demonstrate the critical role ofGWeSW exchange flux when predicting river low flows. Explicit accounting of GWeSW interactions intoriver operation and planning models greatly enhances their forecasting capacity during low flowconditions.

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1. Introduction

1.1. Significance of surfaceegroundwater interactions

Groundwater discharge from shallow aquifers into catchmentsurface waters represents the major part of the total flow volume inmost rivers (Wittenberg, 2003). Themagnitude and direction of theexchange flux between surface and groundwater is mainly

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determined by the hydraulic gradient between a river and theunderlying aquifer. It can greatly impact the surface water andgroundwater balance with serious implications on ecosystemhealth especially during low flow conditions. Krause et al. (2007)reported that although groundwater contributions from a riverstretch in the northeast German lowlands represent only 1% of theannual total discharge within the river, its impact is much higherduring low flow conditions in summer where 30% of the riverrunoff which is generated in the catchment is originated bygroundwater discharge from the riparian zone along this river.During extreme low flow conditions, the groundwateresurfacewater(GWeSW) exchange fluxes are crucial in determining thehydro-chemical conditions and resulting ecological stress during

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D.W. Rassam / Environmental Modelling & Software 26 (2011) 1554e1567 1555

a timewhichmay coincide with the main vegetation growth period(Krause et al., 2007).

The critical issues of water resource availability and ecologicalsustainability have highlighted the need to integrate surfacee-groundwater interactions in both groundwater and surface watermodels thus leading to a conjunctive approach that manages wateras a single resource. Recent initiatives by the Australian govern-ment such as the Murray-Darling Basin Sustainable Yields Project(MDBSY; see http://www.csiro.au/partnerships/MDBSY.html) haveexplicitly required the incorporation of groundwater fluxes whenestimating surface water resources in the basin. In Kansas in theUSA, aquifer management regulations now include baseflow whenevaluating a groundwater permit application (Sophocleous, 2000).It is now recognised that in order to maintain healthy rivers andwetlands, only a small fraction of aquifer recharge can be exploited.As such, there is a move towards conjunctive rivereaquifermanagement by amending safe yield regulation to include base-flow, which is represented as a groundwater withdrawal that hasalready been appropriated (Sophocleous, 2010).

1.2. Processes contributing to the groundwateresurface waterexchange flux

A number of processes contribute to the exchange flux betweensurface and groundwater; most importantly they include:groundwater extraction, aquifer recharge (including diffuserecharge, recharge from irrigation return, and recharge fromoverbank flow), bank storage, and evapotranspiration. Aquiferrecharge represents a gain to the GW system (which may enhancedischarge to the river), while groundwater extraction and evapo-transpiration represent a loss to the GW system (which maydeplete the river). Bank storage is a dynamic phenomenonwherebya river recharges the aquifer during a flood event and then waterdischarges back to the river after the flood wave recedes. The netresult of those processes at any point is space and time can eitherlead to a gaining or a losing river. Some or all of those processesmight contribute to the exchange flux with the extent of thecontribution varying significantly in space and time depending onthe hydrogeological configuration as well as human and/or envi-ronmental drivers.

Drought conditions that result in limiting a surface waterresource can place enormous stress on a groundwater resource viaincreased groundwater extraction. Groundwater extraction, whichinitially depletes the aquifer, eventually depletes nearby rivers byeither reducing aquifer discharge to rivers or by inducing riverrecharge to aquifers. Long-term sustained extraction can lead tosignificant reductions in river flow; Mair and Fares (2010) investi-gated river flow in the Makaha Valley (O’ahu, Hawai’i) and reportedreductions in river flow of up to 36% since 1971 as a direct result ofgroundwater extraction. River depletion can lead to increasedintermittency of river flow, which has adverse ecological impacts.Stream fed aquifer recharge may be a naturally occurringphenomenon but it is enhanced by the increased downwardgradients that are developed due to extensive groundwaterabstraction. Andersen and Acworth (2009) analysed the annualflow difference between two gauging stations on the Namoi Riverin eastern Australia, which indicated that losses from the NamoiRiver are significantly larger than the combined surface waterdiversion and groundwater abstraction. Large overbank events,although not very frequent, can lead to significant aquifer recharge.Evapotranspiration is a significant discharge mechanism forgroundwater in shallow aquifers (Rassam et al., 2002; Cook andRassam, 2002), in closed hydrologic basins (Abdalla, 2008), andalong riparian buffers. Bank storage can significantly reduce storm-inflow peaks and contributes partially to baseflow, the natural

groundwater discharge to a river (Hantush et al., 2002). Exchangefluxes during bank storage can significantly affect water andnitrogen budgets in perennial, as well ephemeral streams withperched water tables (Rassam et al., 2008a).

The temporal and spatial scales at which these processescontribute to the exchange flux vary significantly. For example,river depletion resulting from groundwater extraction is delayed bytime lags that range from days to hundreds of years; the extent ofthe extraction activity may vary along a river reach thus leading togaining and losing sub-reaches. Because of the intensive spatial andtemporal variabilities there is a need for dynamicmodelling of theirimpacts on river flow.

1.3. Rationale for current work

Near-rivereaquifer systems are complex due to the difficultiesin estimating flows into and out of the aquifer, the complicatednature of the GWeSW interaction processes, and the uncertainty ofaquifer properties (Sophocleous, 2010). Because of this complexity,computer models are used to model groundwater systems andestimate the exchange flux between surface water and ground-water. Many of the large river systems around the world are highlyregulated they provide resources for a range of water needs such asirrigation, urban use, and the environment. River operationeplan-ning models are becoming increasingly complex due to the rapidgrowth in urban and agricultural sectors, environmental require-ments, over allocation, and changes to land use and climate change.The interaction between the surface and groundwater systems asrepresented by the GWeSW exchange flux is seldom integratedwithin river operation-management models (Valerio et al., 2010).Traditionally, the interaction between surface and groundwater isimplicitly accounted for during the routing calibration of rivermanagement models. The slow time-variant nature of thegroundwater processes leads to unrealised impacts that are outsidethe calibration period of the river model, which compromises theforecasting capacity when used outside its calibration period. Fullycoupled models such as MODHMS (Hydrogeologic Inc., 1996) andGSFLOW (Markstrom et al., 2008) have the capacity to simulta-neously simulate the flow of surface water, groundwater, and theirinteraction. However, they do not take into account the complexoperational aspects of river management.

The most commonly used approach to account for GWeSWexchange in river operation and planning models is linking them togroundwater models such as MODFLOW (McDonald and Harbaugh,1988). This can be achieved either via a dynamic link where themodels are run simultaneously (Valerio et al., 2010), or via anexternal link whereby fluxes estimated by the groundwater modelare imported as known inputs into the river model. The latterapproach has been adopted by the MDBSY project in Australia. Dueto the very strict time constraints that prevented dynamic couplingof the surface and groundwater models, the GWeSW interactionswere evaluated from groundwater models using a new ‘dynamicequilibrium’ approach. A number of shortcomings were identifiedin this approach (Rassam et al., 2008b). Lessons learnt from theMDBSY project have emphasized the need and demonstrated thelack of tools that can pragmatically model the GWeSW interactionson a large scale.

The world-wide increasing demand for water has led to theadoption of integrated water resource management approaches. Itis recognised now that water management strategies must bemulti-disciplinary evaluating not only the technical and scientificdimensions of a water system, but also the economic, political,legislative and organizational aspects, which are equally important(Molina et al., 2010). In addition, this approach should be accom-panied by stakeholder’s participation (Martínez-Santos et al., 2010).

http://www.csiro.au/partnerships/MDBSY.html

D.W. Rassam / Environmental Modelling & Software 26 (2011) 1554e15671556

One of the key physical developments that helps manage water asa single resource and requires serious attention is a conceptualframework for surfaceegroundwater interaction that fits intoexisting river operationeplanning models. This paper presents theGWeSW Link Module, which accounts for the exchange fluxesbetween surface and groundwater within the new generation riveroperationeplanning model ‘Source Integrated Modelling System(IMS)’ developed by the eWater Cooperative Research Centre(www.ewater.com.au/uploads/files/FL110218_SourceRivers_web.pdf).

2. Objectives of current work

In this paper, a conceptual framework for incorporating theexchange fluxes between surface and groundwater into a riveroperationeplanningmodel is presented. The GWeSW LinkModule,which has been integrated into the river operationeplanningmodel ‘SourceIMS’, explicitly accounts for the interaction betweensurface and groundwater. The GWeSW Link Module is anintermediate-complexity model that adopts analytical solutions toestimate the exchange fluxes between surface and groundwaterwith minimal data requirements. The model incorporates theeffects of individual stresses and then assumes linearity to estimatethe cumulative exchange flux, which controls low river flows(Fleckenstein et al., 2006). This concept of linearity has beenimplemented successfully in the analytical elements method (e.g.,Bakker and Strack, 2003) thus providing the correct trade-off

Fig. 1. Schematic showing; (A) 1-dimensional river link model, (B) processes for unsaturatedschematic for analytical solutions.

between model complexity and the correct representation of thegroundwater system. It is crucial that one arrives at a sensiblecompromise between data availability and model complexity(Fleckenstein et al., 2006).

This paper presents the conceptualisation of the proposedmodel, describes the basic processes that contribute to theexchange fluxes between surface and groundwater, presents a suiteof analytical solutions for evaluating the flux for various riv-ereaquifer configurations, tests the capacity of the model to eval-uate cumulative impacts by comparing its predictions to numericalsimulations and then explores the applicability limits of theanalytical solutions and discusses calibration issues. This study isstrictly a stand-alone application of the GWeSW Link Module thatdemonstrates a proof of concept. A subsequent paper will presentan implementation of the GWeSW Link Module within ‘SourceIMS’ in an actual river reach and will demonstrate how theexchange fluxes estimated by the GWeSW Link Module affect thecalibration and predictions of the river model. The complex oper-ational functionalities of Source IMS is outside the scope of thepaper.

3. Methods

3.1. Conceptualisation of GWeSW Link Module

Source IMS is designed to manage water resources across rural and urbancatchments, for both human and environmental uses of water, taking into accountan unlimited range of future water, land use and climate scenarios. Source IMSdiscretises the river system into a number of 1-dimensional links, which is usually in

connection, (C) processes for saturated connection, and (D) conceptual cross-sectional

http://www.ewater.com.au/uploads/files/FL110218_SourceRivers_web.pdf

http://www.ewater.com.au/uploads/files/FL110218_SourceRivers_web.pdf


the order of tens of kilometres, with upstream and downstream nodes (see Fig. 1A).Along any river system, there are inflows (such as rainfall-generated inputs or irri-gation returns) and losses (such as seepage under a dam or evapotranspiration).Knowing the inflow at a particular upstream gauge, the river model takes intoconsideration the inflows and losses along a particular reach, routes the flow withthe aim of predicting outflow at the downstream node. The GWeSW Link Moduleadds an explicit daily term representing GWeSW exchange fluxes, which is a time-variant gain/loss that is accounted for during calibration of the river model.

The GWeSW Link Module conceptualises a groundwater aquifer that underliesthe river, with either a saturated or an unsaturated connection (see cross-sectionalview of the link, Fig. 1B and C). The connection type greatly influences the activeprocesses that contribute to the exchange fluxes between surface and groundwater;this aspect will be discussed in detail in the following sub-section. A range of riv-ereaquifer configurations are available in the GWeSW LinkModule. They are strictlyaligned with those for which explicit analytical solutions for flux are available in thepeer reviewed literature (e.g., Glover and Balmer, 1954: Hunt, 2003). Each config-uration is associated with a set of different boundary conditions, aquifer layering,and river connection with the aquifer.

The exchange flux comprises the following components: (1) low flow flux ata pre-development condition; (2) flux due to river bank fluctuations (within andoverbank); (3) flux due to changed aquifer recharge; (4) flux due to groundwaterextraction; and (5) flux due to changed evapotranspiration. The GWeSW exchangeflux is tracked historically and projected in a time-varying manner into the futurethus accounting for the significant time lags associated with the delayed impacts ofgroundwater processes. This enhances the calibration of the river model where theunrealised impacts of existing stresses and any future groundwater developmentsare taken into consideration when running the river model in a forecasting mode.

3.2. Underpinning assumptions

B The river, which is represented with a 1-dimensional link, always interactswith the underlying aquifer. This interaction results in an exchange flux, whichis estimated (by default) on a daily time basis.

B The interaction between the river and the underlying aquifer comprises lateralexchange that is orthogonal to the river for a saturated connection, anddownward flow (river recharge) for an unsaturated connection. That is, there isno aquifer flow component parallel to the river, which implicitly means thatadjacent river links do not interact with each other.

B Flow and the associated time lags in the unsaturated zone are not considered.B The type of connection between a river and the underlying aquifer is assessed

a priori for each link based on the GWeSW connectivity mapping of the region;it remains constant during the simulation period.

B If the rivereaquifer connectivity varies spatially within a long river reach, thenit should be divided into links with a uniform connection type. Similarly, if theuser believes that the connection type is likely to change during a simulation(e.g., due a large flood event or long-term wet/dry conditions), then thesimulation should be terminated at the time at which the connection changes,then the model re-run with the new connection type.

B A linear system is assumed, which means that a head change resulting fromapplying a stress to the aquifer is not large enough to alter its transmissivity.

B The assumption of linearity means that the overall GWeSW exchange fluxalong each link at any time step can be estimated by summing the individualfluxes arising from the active GWeSW processes within that link.

B Rivereaquifer configuration is constant along a link but can vary alongdifferent links. A suite of rivereaquifer configurations are available; eachconfiguration is associated with a set of different boundary conditions, aquiferlayering, and river connection to the aquifer.

B The hydraulic parameters of the aquifer and the rivereaquifer interconnectionare constant along a link (homogeneity is assumed). Heterogeneity along a longriver reach can be modelled by dividing it into sub-links with differentconfigurations and/or hydraulic parameters.

B The analytical solutions for discharge response assume that the rivereaquifersystem is initially at equilibrium, i.e., water level in river and nearby aquifer isrepresented by a straight line.

B The exchange flux resulting from within-bank river stage fluctuations areassumed to be constant along the entire link.

B The interaction between evapotranspiration and groundwater extraction isneglected.

3.3. Hydraulic connection between river and aquifer

The interaction between surface and groundwater water is largely impacted bythe type of connection, which dictates the direction and magnitude of the exchangeflux between the two systems. A fully saturated hydraulic connection betweena river and the nearby aquifer occurs when the watertable intersects the river(Fig. 1C). Under such conditions, the flux is a linear function of the head gradientbetween the river and the aquifer. When the groundwater table in a losing stream

system drops slightly below the streambed (Fig. 1B), seepage flux continues toincrease linearly with small declines in the groundwater level but further declineseventually leads to the formation of an unsaturated zone below the river bed wherethe fluxehead relation becomes non-linear. The relationship between seepage fluxand depth to water table ultimately becomes asymptotic and approaches themaximum flux condition when the groundwater table becomes deep. Here, a linearrelationship between flux and head gradient is assumed up to some critical depth(specified by the user) beyond which the flux become independent of depth togroundwater table. This approach is similar to that adopted in MODFLOWwhere theriver bottom variable ‘RBOT’ represents the critical depth at which flux behaviourchanges (McDonald and Harbaugh, 1988).

One still needs to differentiate between full hydraulic connection and discon-nectionwhere the latter condition prevents return flows from the aquifer (Vázquez-Suñé et al., 2007). Under such conditions aquifer stresses have minimal impacts onriver flow, for example, groundwater evapotranspiration becomes irrelevant andstream depletion becomes less significant as the depth to groundwater tableapproaches the critical depth. The criteria proposed by Brunner et al. (2009) can beused to identify the formation of unsaturated conditions. This criteria needs to beestablished by the user and the decision on the type of connection be madeaccordingly.

For an unsaturated connection, the exchange fluxes between surface andgroundwater is merely the river recharge as influenced by changes in stage height,as the depth to the groundwater table is considered to remain at the maximumcritical depth. For a saturated connection, the exchange fluxes between surface andgroundwater will include the fluxes originating from pre-development conditions inaddition to those arising from changes in aquifer recharge, groundwater extraction,within and overbank river fluctuations, and evapotranspiration. Those processeswill be discussed in detail in Section 3.4.

3.4. Processes contributing to the GWeSW exchange flux

3.4.1. Natural rivereaquifer interaction driven by river stage fluctuationA river is continuously interacting with the underlying aquifer. This interaction

can be discretised into three components depending on the state of flow in the river:(1) interaction during baseflow (non-event, low flow) conditions; (2) interactionduring within-bank flood events; and (3) interaction during overbank flood events.This is mathematically represented as follows:

QnðtÞ ¼ �QlðtÞ � QbðtÞ þ QoðtÞ (1)

whereQn is the natural GWeSW interaction flux (L3/T, where L represents the lengthunits and T represents the time units) at any time step resulting from river-stagefluctuation, Ql is the interaction during low flow conditions (L3/T), Qb is the interac-tion duringwithin-bankfluctuations (L3/T), andQo is the interaction during overbankfluctuations (L3/T). Those three components are described in more detail herein.

3.4.1.1. Interaction during low (base) flow conditions. A river may continuously losewater to, or gain water from the underlying aquifer (as shown in Fig. 1C); neutralcases are also possible with no head gradient and zero exchange. Consideringa regional rivereaquifer system that is at equilibrium (i.e., recharge into the aquiferis equal to discharge to the river system), this exchange flux would remain constantwith time. It is given by:

Ql ¼ ðh� hwtÞ ��K � Y �W

M

�¼ Dh� C (2)

where Ql is the GWeSW exchange flux at low flow conditions (L3/T), hwt is thegroundwater table level (L), h is the river stage level (L), Dh is the head differencebetween the river stage and the groundwater table (L), K is the hydraulic conduc-tivity of the riverbed sediments (L/T), Y is the length of the river link (L), W is thewidth of the river link (L), M is the thickness of the riverbed sediments (L), and C isthe hydraulic conductance of the rivereaquifer interconnection (L2/T). At steadystate conditions with no climate change, no land use change, and no groundwaterextraction, the long-term average Dh should be at a state of dynamic equilibriumthus resulting in a constant average baseflow. Ideally, Dh can be obtained fromanalysing long-term river flow and groundwater records at pre-developmentconditions. It is important to note that adding this term to the flux equationprecludes the need to model recharge and ET at pre-development conditions, oneonly needs to model the change that has occur past that period. Realistically, thisterm is implicitly accounted for during calibration of the river model (since it isconstant and very difficult to evaluate).

3.4.1.2. Interaction during within-bank flow events (bank storage). River stage risesas the river flow increases following rainfall events. This triggers a change or reversalin head gradient between the river and the aquifer thus resulting in water infil-trating into the aquifer. When the flood wave recedes, this water subsequentlyreturns to the river. The significance of bank storage varies with the size of the riverfloodplain and its hydraulic properties (Knight and Rassam, 2007).

The bank storage flux formulation, derived by Moench et al. (1974) and Hall andMoench (1972) as implemented by Birkhead and James (2002), is used:


QbðtÞ ¼ 2TYb

Xti¼2

ðhi � hi�1Þ �

XNn¼1

exp

(��ð2n� 1Þp

2b

�2TDtS

ðt � iþ 0:5Þ)!

(3)

where Qb is the bank storage flow rate (L3/T) where discharge from the bank to theriver is positive (consistent with the sign convention for a gaining river), T is theaquifer transmissivity (L2/T; T¼ Kh*, where K is aquifer hydraulic conductivity and h*

is average saturated aquifer thickness), S is the aquifer specific yield, Y is the lengthof the link (L), h is the river stage height (L) at any step i,Dt is the time step (T), t is thetime level, n is the number of times the exponential term needs to be summed (pilotruns have indicated diminishing benefits in accuracy for n > 25), and b is the lateralextent of the floodplain (L). Note that this solution is for a finite boundary so even ifone uses a semi-infinite solution for the recharge/extraction effects, the effectivewidth within which bank storage effects are active need to be specified. As bankstorage is effective on both sides of the river bank, Equation (3) needs to beimplemented independently for each side of the river (with a possibility for eachside having a different width). Ideally, the simulation should start after a dry periodwhere return flow from previous bank storage processes have ceased, meaning thatthe rivereaquifer is at the low flow equilibrium condition.

3.4.1.3. Interaction during overbank flow events. Overbank flow during major floodevents results in large quantities of water that travels as sheet flow across thefloodplain. After the flood-wave subsides, the overbank water returns back to theriver minus the portion that had evaporated and/or infiltrated through the flood-plain surface. Some of the overbank water may be trapped in depressions thusleading to the formation of floodplain wetlands; some of this water will evaporateand the remainder will continue to infiltrate long after the flood-wave subsides. Allthe infiltrated water travels through an unsaturated zone and eventually rechargesthe aquifer in a time-variant manner. This recharge eventually discharges to theriver after a time lag that depends on the orthogonal distance to the river and aquiferdiffusivity. It is worthwhile noting that very large flood events may significantlyraise the water table with the potential to change the connection type between theriver and the aquifer. This condition should be independently assessed and in theevent of it happening, then a new simulation that reflects the new connection typeshould be conducted.

An algorithm is proposed to calculate the evaporative flux from a floodplainwetland in addition to the infiltrative flux that passes through its bed while main-taining water mass balance. The evaporative flow rate is given by:

fev ¼ Aw � PE (4)

where fev is the evaporative flux (L3/T), Aw is the inundated floodplain area (L2), andPE is the open water surface evaporation rate (L/T).

The infiltrative flow rate is given by:

finf ðtÞ ¼ Aw � Ks � Dhwðt � 1Þ (5)

where finf(t) is the infiltrative flux (L3/T), Ks is the hydraulic conductivity of thewetland bed (L/T), and Dhw(t� 1) is the head gradient in the wetland at the previoustime step.

The wetland volume (Vw) and water level is calculated at the end of every timestep as follows:

VwðtÞ ¼ Vwðt � 1Þ � dt�fev þ finf

�(6)

hwðtÞ ¼ VwðtÞAw

(7)

where Vw(t) is the wetland volume (L3), Vw(t � 1) is the wetland volume at theprevious time step (L3), hw(t) is the water head in the wetland (L), and dt is the timestep (T).

The time series for finf (t) becomes a recharge time series (the time lag in theunsaturated zone is neglected in this formulation). The length of the recharge timeseries depends on the initial wetland volume and the rates of evaporation andinfiltration, that is, the time series ends when Vw becomes zero (the wetlandempties). Note that evenwhen the wetland empties, the discharge impact continuesto be estimated along the entire simulation (to account for the delayed dischargeresponse). The discharge resulting from an instantaneous recharge source to a semi-finite aquifer finf (t) is given by (Knight et al., 2005):

QoðtÞ ¼�finf � dt

� a

2tffiffiffiffiffiffiffiffiffipDt

p exp��a2

4Dt

�(8)

where Qo(t) is the instantaneous river discharge resulting from aquifer rechargesourced from overbank flow (L3/T), finf (t) is the instantaneous recharge sourced fromoverbank flow during one time step (L3/T),D is aquifer diffusivity which is equal toT/S (L2/T), a is the orthogonal distance from the centre of the floodplainwetland to theriver (L), and t is time (T). The overall discharge is estimated by summing up theindividual discharge response for every recharge value pulse using a convolutionapproach similar to that used in Equation (3). It is worthwhile noting that

integrating Equation (8) yields the response for a step change as given by Glover andBalmer (1954) solution for stream depletion, which will be discussed in the nextsection as given by Equation (9).

3.4.2. River depletion due to groundwater extractionGroundwater extraction is one of the most important processes that impact the

exchange flux between surface and groundwater water. Extraction-induced riverdepletion is defined as the reduction of river flow due to induced infiltration ofstream water into the aquifer or the capture of aquifer discharge to the river. Aftera long period of uniform extraction, the cone of depression takes its final shape atsteady state. The time required to reach steady state varies linearly with aquiferdiffusivity and non-linearly with the square of the orthogonal distance between theextraction-well and the river. Other important factors that may significantly affectriver depletion include riverbed clogging, degree of stream partial penetration, andaquifer heterogeneity.

There are numerous analytical solutions for river depletion derived for a varietyof rivereaquifer configurations. Those solutions integrate the discharge responsealong an infinite river reach; the spatial distribution of fluxes along a river reach willbe discussed in the next section. Note that the extraction rates are linear multipliersin the analytical solutions; hence, they do not affect the time scales during which theimpacts of extraction occur. Modelling a variable extraction rate including decom-missioning of a extraction source is simply modelled by applying a complementary(new) source with the adequate magnitude and sign notation required for repre-senting the new extraction rate. For example, if the extraction rate is to be increasedfrom 500 m3/day to 750 m3/day then, using superposition, one needs to add a newsource with a extraction rate of 250 m3/day; similarly, if this source is beingdecommissioned, one needs to add a new source with a rate of �500 m3/day.

The following analytical solutions representing a range of rivereaquiferconfigurations are available for implementation within the GWeSW Link Module.

3.4.2.1. Glover and Balmer (1954). This solution estimates cumulative dischargeresponse resulting from a step change in extraction rate (Fig.1D; extraction source atdistance ¼ a) in an unconfined, single-layer homogenous aquifer (Fig. 1D; f ¼ 0;H1]H2), with a fully penetrating river, an infinite boundary opposite to the river(Fig. 1D; c ¼ N), where the river/aquifer systems are at initial equilibrium condition(Fig. 1D; q ¼ 0).

QpðtÞ ¼ P � ResðtÞ ¼ P � erfc

"a

2ffiffiffiffiffiffiffiffiffiðDtÞp

#(9)

where Qp is the cumulative depletion flux resulting from groundwater extraction(the fraction of pumped water being sourced from the river (L3/T), P is the extractionrate (L3/T), Res(t) is the non-dimensional discharge response as a function ofdimensional time, D is the aquifer diffusivity (L2/T), a is the orthogonal distancebetween the river and the pump (L), and t is the time (T).

3.4.2.2. Hall and Moench (1972). This solution estimates cumulative dischargeresponse Qp in an unconfined, single-layer homogenous aquifer, with semi-infiniteboundary, and a river that fully penetrates the aquifer and has a semi-pervious bed.

QpðtÞ ¼ erfc�

a

2ffiffiffiffiffiffiDt

p�� exp

�aaþ Dt

a2

�erfc

a


p þffiffiffiffiffiffiDt

p

a

!(10)

where a is by a retardation factor that represents the effect of a low-conductivitybarrier, which was defined by Hantush (1965) as the effective thickness of aquiferrequired to cause the same head loss as the semi-pervious barrier; a¼mK/K*whereK is the hydraulic conductivity of the aquifer, m is the width of the semi-perviousbarrier, and K* is its hydraulic conductivity.

3.4.2.3. Knight et al. (2005). A suite of solutions estimate cumulative dischargeresponse Qp in an unconfined, single-layer, homogenous aquifer, with a river thatfully penetrates the aquifer. Two of those solutions proposed by Knight et al. (2005)are implemented in the GWeSW Link module, the first accounts for the presence ofa no-flow boundary located at a distance ‘c’ opposite to the river (Fig. 1D):

QoðtÞ ¼ P

(a


p exp��a2

4Dt

�þXNn¼1

ð�1Þnþ1

"2nc�a


p �exp

�ð2nc�aÞ2

4Dt

!

� 2ncþa


p �exp � ð2ncþaÞ2

4Dt

!#)(11)

The second solution accounts for the presence of a head gradient in the aquifer(Fig. 1D; qs0; H1]H2):

QpðtÞ ¼ 12erfc

�a� kt


p�

� 12exp

�akD

�erfc

�aþ kt


p�

(12)

where k ¼ K tan(q)/S, which is the physical velocity of a water particle down theslope, and q is the angle of the water table (Fig. 1D).

3.4.2.4. Hunt (2003). This solution estimates cumulative discharge response Qp inan semi-confined, homogenous aquifer, with semi-infinite boundary and a river that


partially penetrates the aquifer. The formulation requires knowledge of the aquitardparameters including its thickness beneath the river, conductivity, porosity, andspecific storage, in addition to the semi-confined aquifer transmissivity and stor-ativity. Those parameters collapse into three non-dimensional parameters that areactually required for the solution.

QpðtÞ ¼ erfc�

12ffiffit

p�� exp

l

2þ tl2

4

!erfc

�1

2ffiffit

p þ lffiffit

p

2

�� l

Z10

Fða; tÞGða; tÞda (13)

Where a is a variable related to aquitard thickness, permeability, and porosity, and l

is a variable related to aquitard permeability, its thickness below the river, and riverwidth. The definitions of the functions F and G are lengthy and can be found in Hunt(2003).

3.4.3. Change in rechargeVarious land use practices such as irrigation and farm dams lead to changed

drainage below the root zone. This alters the recharge to the underlying aquifer thusleading to changing discharge to the nearby river. Recharge and groundwaterextraction are two opposite processes, the former leading to increased discharge tothe river and latter reducing it. However, the mathematical response functiondescribing them are identical, the only difference is the recharge/extraction ratehaving opposite signs.

The recharge can either be as a step change or variable in time. For the formercase, the algorithms for estimating river depletion (Glover and Balmer, 1954; Halland Moench, 1972; Knight et al., 2005) may be used to estimate the dischargeresponse resulting from a change of recharge in an unconfined aquifer. Theextraction rate P is replaced by the magnitude of change in recharge rate DRwith anopposite sign hence resulting in positive discharge flux to the river rather thana negative depletion flux. For the latter case where recharge is highly time-variant,the algorithmused for overbank return is usedwith finf(t) of Equation (8) replaced byDR(t); the convolution approach demonstrated in Equation (3) is then used toestimate the total flux resulting from any number of recharge pulses.

3.4.4. Change in evapotranspirationEvapotranspiration (ET) is a major component of the water budget in vegetated

areas that have relatively shallow water tables. In such areas, transpiration directlyfrom groundwater by near-shore vegetation can intercept baseflow that wouldotherwise discharge to a stream. Depending on the positioning of the root zonewith respect to the water table, the plant can extract water directly fromgroundwater, from the unsaturated zone, or from both. Actual evapotranspirationremains equal to the potential (PET) value as dictated by the climate, down to somedepth called the ‘transition depth’ where ET starts to shift from atmospheric-control to soil-moisture control (Rassam and Williams, 1999). This implies thatbelow this depth, soil properties and vegetation type would dictate the magnitudeof actual ET. At some depth called the ‘extinction depth’, evapotranspirationbecomes very low as the vegetation can no longer extract any groundwater. TheET-decline function proposed by Shah et al. (2007) is adopted in this work and isgiven by:

ETa ¼ ETPET

¼ 1 for d � d0 (14a)

ETa ¼ ETPET

¼ e�gðd�d0 Þ for d � d0 (14b)

where ETa is the actual evapotranspiration rate (L/T), PET is the potential evapo-transpiration rate (L/T), d is the depth to groundwater table (L), g is a decaycoefficient (T�1), and d0 is the transition depth (L) where ET shifts from atmo-spheric control to soil-moisture control. Shah et al. (2007) has recommendedestimates of the fitting parameters for various land uses and soil types which havebeen incorporated in the GWeSW Link Module. The total flow rate due to ET isgiven by:

QET ¼ Y � b� ETa (15)

where QET is the ET flux (L3/T), Y is the length of the link (L), b is the lateral extent ofthe floodplain (L), and ETa is the actual evapotranspiration rate defined by Equation(14). Extreme care should be taken to ensure that there is no double accounting forthe impacts of any process. It is emphasized that the pre-development flux baseflowimplicitly accounts for pre-development ET fluxes. Therefore, one only needs toaccount for changes in actual ET that have occurred since that time due to landclearing or re-vegetation during the post-development stage.

3.5. Spatial and temporal aspects of GWeSW interactions

3.5.1. Spatial and temporal distribution of fluxes along river reachThe spatial distribution of GWeSW exchange fluxes vary with every process.

GWeSWexchange driven by within-bank changes of river stage is assumed to occurinstantaneously along an entire link. However, all other contributors to the flux(recharge, extraction, and ET) are point-source aquifer stresses that are spatially

defined by their xey coordinates. The x-coordinate represents the distance ‘a’, whichappears in all the analytical solutions (e.g., Equations (8) and (9)). The analyticalsolutions for discharge response integrate fluxes along an infinite river reach. Sincethe link used in this model has a finite length, one needs to apportion fluxes alongthis finite length. This issue becomes important for aquifer stresses that are close toa node separating two adjacent links whereby a stress located within one link maycontribute to the flux in the adjacent link. The flux can be apportioned as follows(Rassam et al., 2004):

f ðy; tÞ ¼ ap�y2 þ a2

�exp �� y2 þ a2

4Dt

�(16)

where f is the instantaneous flux response at any lateral distance from a node perunit length of the river (L3/T) at time (t), a and y are the x- and y-coordinates of therecharge/extraction source relative to the river link (L) as shown in Fig. 1A, D isaquifer diffusivity (L2/T), and t is time (T). To obtain the total flux, Equation 16 needsto be integrated along the entire length of the link.

3.5.2. Lumping of similar aquifer stressesThe individual impact of each aquifer stress (recharge/extraction) is added to

yield the cumulative impact of a number of stresses. Hence, model run time becomesa function of the number of active stresses in the aquifer, which can be in the order ofhundreds for long river reaches. The discharge response for recharge/extraction isa function of the hydrological response time a2/D (Knight et al., 2005). One caneffectively lump the impacts of ‘hydrologically similar’ aquifer stresses thus reducingthe run time. Any number of aquifer stresses located anywhere within a river reachbut having identical schedules and similar hydrologic response times can be lumpedinto one entity having a single response function. In this case, the recharge orextraction rates for all sources are added and then multiplied by the responsefunction to yield the overall contribution to the flux at each time step.

3.5.3. Smart time steppingSource IMS by default operates on a daily time step. The GWeSW Link Module

also estimates the exchange fluxes between a river and the nearby aquifer ona daily basis. Aquifer stresses (extraction/recharge) having a very slow dischargeresponse (due to low D or large a) may lead to very marginal variations in the dailyfluxes. For such slow-response processes, the user has the option to extend the timestep of the GWeSW Link Module into weekly or monthly to reduce model runtimes by reducing the number of mathematical operations. Non-dimensionalanalysis can be adopted to smartly arrive at an optimum time step that results ina minimal acceptable error. When the dimensional time scale is normalised withrespect to the hydrological response time, one obtains a dimensionless time scales ¼ t/(a2/D). When the response Res(t) given by Equation (9) is plotted versusdimensionless time s, a unique characteristic response function is obtained. Sincethis response is a function of aquifer diffusivity and the distance between the stresslocation and the river (‘a’ and ‘D’ both known quantities), one can predict a priorithe change in discharge response during a groundwater time step and hencerecommend a longer time step during which a noticeable response occurs (e.g., 1%).The slope of the steepest section of the non-dimensional discharge response curveis evaluated then Ds corresponding to one percentile change in response is iden-tified. Substituting the known values of ‘a’ and ‘D’, the actual time (time step, Dt)required to arrive at the 1% change in discharge response is determined. One canreduce the response during a time step to increase accuracy, for example halve theresponse to 0.5%, since the relationship is linear Dt will be halved too. The modelthen uses linear interpolation between successive groundwater time steps tocalculate the daily response. Note that this is a pre-processing procedure where themodel only recommends a groundwater time step (based on a 1% response change)and the user has the option to implement it or adhere to the daily time step.

3.6. Calculating total exchange fluxes

3.6.1. Unsaturated connectionFor an unsaturated rivereaquifer connection, the exchange flux for groundwater

depths equal to or deeper than the critical depth is given by:

QunðtÞ ¼ ½hðt � 1Þ � hcr� � C (17)

where Qun is the GWeSW exchange flux for an unsaturated connection (L3/T), hcr isthe critical groundwater level below which the fluxehead relationship becomesindependent of depth to groundwater table (L), h is the river stage level (L), and C isthe hydraulic conductance of the rivereaquifer interconnection (L2/T) given byEquation (2).

3.6.2. Saturated connectionThe linearity of the governing equations that underpins the analytical solutions

means that the principle of superposition is applicable, which allows the summationof individual impacts to obtain their overall accumulation. For example, the totalstreamdepletion flux resulting from a number of extraction points across the aquiferis estimated as follows:

Table 1Details of modelling experiment (refer to Figs. 1 and 2 for conceptual models and parameter symbols).

Simulation Type Detailsa Parametersa

1. Flux response due tomultiple sourcesand processes

Analytical Knight et al. (2005) for finite domainEquation (3) for bank storage

D1 ¼ 7840 m2/day; D2 ¼ D3 ¼ 4000 m2/day;a1 ¼ 350 m; a2 ¼ 250 m; a3 ¼ 1000 m; c ¼ 3000 m;extraction rates shown below

Numerical X ¼ 3 km; Y ¼ 10 km; Y1 ¼ 5 km;XP1 ¼ 350 m; YP1 ¼ 2500 m;XP2 ¼ 250 m; YP2 ¼ 7500 m;XP3 ¼ 1000 m; YP3 ¼ 8000 m

K1 ¼ 10 m/day; K2 ¼ 19.6 m/day; S ¼ 0.05; hi ¼ 20 m;extraction rates P1 ¼ 500 m3/day; P2 ¼ 300 m3/day;P3 ¼ 500 m3/day then reduced to 200 m3/day after 200 days

2. Spatial and temporaldistribution of fluxesalong the river

Analytical Equation (16) D ¼ 1000 m2/day; a ¼ 1000mNumerical X ¼ 3 km; Y ¼ 10 km; Y2 ¼ 0 K1 ¼ 5 m/day; hi ¼ 10 m; S ¼ 0.05; a ¼ 1000 m;

Zone Budgets at reach widths a, 2a, 4a, 8a, 12a, 16a, and 20a3. GWeSW interactions

during low flowsAnalytical Knight et al. (2005) for finite domain D ¼ 6000 m2/day; a ¼ 1, 2, 4 km; Daily flux response

deducted from river flows (stage heights derived from flows (Fig. 3)4. Effect of no-flow

boundaryAnalytical Compare Glover and Balmer (1954)

to Knight et al. (2005) for finite domainnon-dimensional analysis with c ¼ 2a, 4a, 6a, and 9a;upper and lower bound envelops with 0.5 D and 2 D

5. Effect of headgradient

Analytical Compare Glover and Balmer (1954)to Knight et al. (2005) with head gradient

a ¼ 100 m, S ¼ 0.05; variable K and h to arriveat D ¼ 9000 and 18,000 m2/day; head gradienttanq and h varied to arrive h/tanq ¼ 100;250; 600; 3000; and 10,000.

6. Effect of aquiferheterogeneity

6.a Cases 1e2: variablesaturated aquiferthickness due to agroundwater mound

6.b Case 3: variablesaturated aquiferthickness due tosloping base aquifer

6.c Case 4: presence ofsemi-pervious riverbank

Analytical Cases 1e3: use Knight et al. (2005)with an effective aquifer thicknessh* based on knowledge of prevailing headsCase 4: Hall and Moench, 1972

Case 1: K ¼ 2 m/day; h* ¼ 20 m & 28 m; S ¼ 0.1; a ¼ 300 mCase 2: K ¼ 2 m/day; h* ¼ 20 m & 24 m; S ¼ 0.1; a ¼ 3,500 mCase 3: H1 ¼ 38 m; H2 ¼ 90 m; h* ¼ 51.5 m; a ¼ 500 m;K ¼ 0.5 m/day; S ¼ 0.05; 4 ¼ �3�

Case 4: D ¼ 2,000 m2/day; a ¼ 2000; a ¼ 100; 1000; 5000; 10,000Numerical Cases 1e2: X ¼ Y ¼ 25 km; Y2 ¼ 0;

Case 3: X ¼ 1,000 m; Y ¼ 50 m;Y2 ¼ 0; H1 ¼ 38 m; H2 ¼ 90 m; 4 ¼ �3�

Case 4: X ¼ 25 km; Y ¼ 50 m; Y2 ¼ 0

Case 1: K1 ¼ 2 m/day; hi ¼ variable; S ¼ 0.1; a ¼ 300 mCase 2: K1 ¼ 2 m/day; hi ¼ variable; S ¼ 0.1; a ¼ 3500 mCase 3: K1 ¼ 0.5 m/day; hi ¼ 90 m; S ¼ 0.05; a ¼ 500 mCase 4: K1 ¼ 10 m/day; hi ¼ 10 m; S ¼ 0.05; a ¼ 2,000 m;a ¼ 100; 1000; 5000; 10,000

a Unless otherwise indicated, H1]H2; q ¼ 4 ¼ 0; Equilibrium initial conditions exist between river and aquifer; hi is initial head in numerical model.


QPðtÞtotal ¼Xn1

QPða; tÞ (18)

where QP(t)total is the total depletion flux for a number of extraction sources (L3/T),QP(t) is the cumulative depletion flux (L3/T) up to time t for an individual extractionsource resulting from a step increase in extraction rate (e.g., given by Equation (9)),and n is the number of active sources along a link. A formulation similar to Equation(18) is implemented for multiple recharge sources to yield their total impactsQR(t)total.

The overall rivereaquifer exchange flux at any time is conceptualised as thesummation of the fluxes resulting from all the active GWeSW interaction processesat that time, which is given by:

Qsw�gwðtÞ ¼ �QnðtÞ � QETðtÞ � QPðtÞtotalþQRðtÞtotal (19)

where Qsw-gw is the overall rivereaquifer exchange flux (L3/T) at time t, Qn(t) is thenatural interaction flux (L3/T) due to river stage fluctuations given by Equation (1),QP(t)total is the total interaction flux (L3/T) due to groundwater extraction, QR(t)total is

River; Constant head boundary

3 No-Flow boundaries

X

Y

(P refers to recharge/extraction

source; n refers to its number)

Fig. 2. Conceptual model for MODFLOW simulations.

the total interaction flux (L3/T) due to changes in recharge, and QET(t) is the inter-action flux (L3/T) due to changes in evapotranspiration.

3.7. Modelling experiment

A comprehensive modelling experiment was carried out to test the validity ofthe proposed conceptualisation to model GWeSW interactions within the riveroperationeplanning model ‘Source IMS’. As mentioned earlier, this experiment isa stand-alone application of the GWeSW Link Module, which involves testing theproposed analytical approach and comparing its results to numerical predictionsderived from MODFLOW. The modelling experiment covers four aspects: (1) suit-ability of the analytical solutions to model multiple stresses and processes, (2)spatial and temporal distribution of fluxes along the river, and (3) significance ofGWeSW interactions during low flow conditions, and (4) applicability of theanalytical solutions. Details of the modelling experiment are listed in Table 1 withreference to the conceptual models shown in Figs. 1 and 2.

The suitabilityof theanalytical solutions tomodelmultiple stresses andprocesseswas tested by applying three extraction sources and a time-variable constant headboundary condition representing river stage fluctuations using stage heights derivedfrom the flow time series for the Namoi River shown in Fig. 3. The flow domaincontains two hydraulic conductivity zones, which serve two purposes: firstly, to

Fig. 3. Flow time series and flow duration curve for Namoi River at Boggabri.

Fig. 4. The Namoi River catchment.

-100000

0

100000

200000

300000

400000

500000

600000

700000

0 100 200 300 400 500 600 700 800 900 1000

Time (days)

m

( e

m

u l o v e g n a h c x e

W

G

- W

S

e v i t a l u m

u

C

3 )

Modflow

Analytical

Knight et al. (2005); finite domain Equation 3; bank storage

Fig. 6. Exchange flux due to two processes; extraction and bank storage.


represent heterogeneity along the river, and secondly, to demonstrate the concept oflumping sources that have similar hydrological response time. Referring to Table 1(Simulation 1), the hydrological response time (a2/D) for both P1 and P2 is equal(15.625). Hence, those two sources that have extraction rates of 500 and 300m3/day,respectively, are lumped into one source with an extraction rate of 800m3/day. Avariable extraction rate was used for P3 to demonstrate the concept of implementinga negative source to model a reduction in extraction rate. The analytical results arethen compared to results from theMODFLOWsimulations. This part of themodellingexperiment also aims at validating the concept of linearity and superposition.

The spatial and temporal distribution of fluxes along the river was analyticallymodelled using Equation (16). The flux was apportioned along the river reach atsteady state and then the results were compared to those obtained from the ZoneBudget package of MODFLOW (Table 1, Simulation 2).

The significance of GWeSW interactions during lowflowconditions is highlightedby using flow data from the Namoi catchment in eastern Australia. The Namoi Rivercatchment is a tributary of the Murray-Darling River Basin (see Fig. 4), which hasexperienced a consistent increase in the level of groundwater extraction in recenttimes. The connectivity of the Namoi River with the underlying aquifer system variesalong the river (Ivkovic, 2009). The exchange fluxes along the reach from Keepit Damto Narrabri are in the order of�(10e15) ML/day where � refers to gains and losses toand from the river (McNeilage, 2006). River flows from the Boggabri gauge on theNamoi River for the period April/1992 to December/1994 were used (see Fig. 3). Itincludes a drought period during which groundwater extraction significantlyincreased by about 10,000ML/annum for themanagement zone upstream of Boggabri(McNeilage, 2006). A hypothetical groundwater extraction zone is activated fromApril/1992 until July/1994 (just prior to the drought period). An extraction rate of27.4 ML/day was implemented, which is equivalent to the 10,000 ML/annum rise ingroundwater extraction during the drought period. Three scenarios were investigatedwhere the distance between the hypothetical source and the riverwas 1 km, 2 km, and4 km (Table 1, Simulation 3). The flux response for each case was deducted from thedaily flows, then two low flow indices were derived, namely, the 90th percentile flow(Q90)and the number of zero-flow days.

-200000

0

200000

400000

600000

800000

1000000

0 100 200 300 400 500 600 700 800 900 1000

Time (days)

Cu

mu

lati

ve S

W-G

W e

xcha

nge

vol

um

e (m

3 )

Modflow

Sum=(P1+P2)+(P3+P4)

Combined P1 & P2 @ 800 m 3 /day

P3 @ 500 m 3 /day

P4 @ -300 m 3 /day

Positive is flow into aquifer Knight et al. (2005) finite domain

Fig. 5. Exchange flux due to several extraction sources (one process).

A series of modelling simulations were carried out to test the applicability of thevarious analytical solutions implemented in theGWeSWLinkModule. The experimentaims at assessing the accuracy of the analytical solutions as their underpinningassumptionsmay be violated in the natural world. Those violationsmay arise from thefollowing effects: (a) presence of a no-flow boundary; (b) presence of head gradients;and (c) aquifer heterogeneity thatmay arise froma number of factors such as a variablesaturated aquifer thickness (due to the formation of groundwater mound or due toa sloping base aquifer), the presence of a semi-pervious river bank, or a heterogeneousriver reach. Details of this part of themodelling experiment are covered in Simulations4e6,Table1. Thegroundwatermound(numerical simulation6a,Table1) is simulated inMODFLOW with an aquifer having a uniform initial saturated aquifer thickness ofhi ¼ 20 m. Subsequently, an irrigation development is applied 500 m away from theriver covering an area 300 � 300 mwith a recharge rate of 120 mm/yr. The output ofa 33-year simulation is then used as an initial condition for two new simulations withtwo recharge sources applied individually at the mound peak and behind it, respec-tively. The semi-pervious river bank was modelled by including a vertical layer havinga lower hydraulic conductivity next to the river (see Fig. 1D; Simulation 6c, Table 1).

4. Results

4.1. Flux response due to multiple sources and processes

Fig. 5 shows that summing the individual impact of each sourceyields a cumulative impact on the river, which is in excellentagreement with estimates obtained from the numerical MODFLOWmodel. In this simulation, the concept of lumping multiple sourceshaving similar hydrological responses has also been demonstrated(where P1 and P2 were lumped into on source with a combinedextraction rate of 800m3/day). It has been shown that the reductionin extraction rate can be modelled by adding a new source witha negative rate (P4 with 300 m3/day was added to P3 to yield a netrate of 200 m3/day).

Fig. 6 shows that the GWeSW exchange fluxes are due to thecombined impacts of groundwater extraction and bank storage.

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000

Distance along river y (m)

) t , y ( F

e s n o p s e r

x u l f s u o e n a t n a t s n I

Time=infinity

Time=5 years

Time=1 year

Time=0.5 year

Fig. 7. Flux distribution along a river at various times.


Note that the extraction layout for this example is similar to theprevious one with the exception of using a homogeneous aquiferhaving a hydraulic conductivity of 10 m/day. The simulation resultsconfirms that the analytical solution provide adequate estimates forthe flux with an error margin of up to 4%, which tends to stabiliseafter 800 days. This error margin is consistent with the findings ofBirkhead and James (2002) who reported that Equation (3) mayunderestimate the bank storage flux by up to 5%.

4.2. Spatial and temporal distribution of fluxes along the river

Fig. 7 shows that at early times, most of the flux is sourced froma narrow band of the river reach opposite to the source. Theinfluence length increases with time and is function of aquiferdiffusivity; it asymptotically approaches infinity at steady state. Thetotal GWeSW exchange flux along a finite length along the river(e.g., link length) is estimated by integrating the area under thecurve. At steady state, the shape of the curve becomes independentof aquifer diffusivity as the exponential term of Equation (16)equates to unity at infinite time. Plotting the product of instanta-neous response and distance ‘a’ versus the normalised distance ‘y/a’results in a unique relationship that describes the flux distributionalong the river at infinite time (Fig. 8). For example, integrating theflux over a distance y ¼ 2a (1a on either side of the source) showsthat exactly 50% of the extraction/recharge is sourced froma narrow 2a-wide stretch of the river reach (see insert Fig. 8).Performing several integrations at various distances shows how theflux response asymptotically reaches unity with excellent agree-ment to numerical simulations using MODFLOW and the ZoneBudget package (see insert Fig. 8).

An example application of the concept of apportioning flux isdemonstrated here at a steady condition with reference to Fig. 8.Assume two equal river reaches [a] and [b] of length L¼ 35 km thatare defined by Nodes 1, 2, and 3 where Node 3 is the downstreamnode. A recharge source has coordinates a ¼ 5 km and y1 ¼ 25 km(following the convention used in Fig. 1; note that y2 ¼ L � y1). Bydefinition, the source is located at the origin y/a ¼ 0. In non-dimensional terms (y/a), Nodes 3, 2 and 1 are located relative tothe source at y1¼5, y2¼�2, and y3¼�9, respectively. The flux thatis apportioned to Reach [1] is represented by Area 1 whereas theflux apportioned to Reach [2] is represented by the sum of Areas 2and 3. The areas can be derived from insert Fig. 8, which provides

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-10 -8 -6 -4 -2

F(y

,t)

. avv

At steady state

- 'a' is orthogonal distance from source to river; source located at y/a=0 - Area 1 represnts flux apportioned to Reach [1]- Areas 2 & 3 represnts flux apportioned to Reach [2]

Reach [1]

Node 1; upstream Node 2

Fig. 8. Non-dimensional characteristic flux distribution along a river

integrals for the area on both sides of the origin (e.g., y/a ¼ 2includes an area that starts from y/a¼�1 and ends at y/a ¼þ1 andaccounts for 0.5 of the flux; from symmetry, the area on either sideof the origin is half the total area).

Therefore, Area 3 ¼ 0.88/2 ¼ 0.44 (where 0.88 is derived frominsert Fig. 8 for y/a ¼ 10; as area extends from origin to y1 ¼ 5, it ishalved to yield the area on one side of the source). Area 2 ¼ 0.72/2 ¼ 0.36 (where 0.72 is derived from insert Fig. 8 for y/a ¼ 4 asy2 ¼ 2). The flux ratio apportioned to Reach [2] is equal to (Area2 þ Area 3) ¼ 0.44 þ 0.36 ¼ 0.80. The flux apportioned to Reach 1stretches from y/a ¼ �9 to �2; it can be estimated by finding thetotal area from y/a ¼ 0 to y/a ¼ �9 then subtracting from it Area 2.Therefore, area from y/a ¼ 0 to �9 ¼ 0.92/2 ¼ 0.46 (where 0.92 isderived from insert Fig. 8 for y/a ¼ 18 as y3 ¼ 9). The flux ratioapportioned to Reach [1] is equal to 0.46e0.36¼ 0.10. Note that thetotal flux ratio for the two reaches adds up to 0.90, which leaves0.10most of which belongs to the reach downstream of Node 3. Theflux ratio apportioned during every time step of the model run isestimated by implementing Equation (16) and performinga numerical integration to estimate the area under the curve, whichincrease with time (as shown in Fig. 7).

4.3. Significance of GWeSW interactions during low flowconditions

It is very important to emphasize that the impacts of theinteractions between surface and groundwater become paramountduring low flow conditions. The flow time series for the NamoiRiver at Boggabri (see Fig. 4) indicates that under normal climateconditions (period prior to 30/6/1994) the 50th and 90th percentileflow is 265 ML/day and 35.5 ML/day, respectively (see insert Fig. 3).The GWeSW exchange flux constitutes about 3.8% of the medianflow, which is an insignificant amount that could well be within theuncertainty limits of the gauge data. However, the GWeSWexchange flux constitutes about 30% of flow during low flowconditions. This contribution would be much more significantunder drought conditions (see Fig. 3 encircled data 1995); this datashow a significant drop of more than two orders of magnitude inlow flows, which is the direct result of the increased groundwaterextraction. The groundwater extraction during 1995was double theaverage rates during the period between 1987 and 1993(McNeilage, 2006). The GWeSW Link module is used to conduct

0 2 4 6 8 10

y/a

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20Distance on both sides of source

Tot

al f

lux

rati

o

AnalyticalModlfow

0.5 at 2a

Reach [2]

Node 3; downstream

at steady state showing apportioning of flux along a river each.

0

5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Additional annual groundwater extraction (ML)

syad

wolforez

fore

bmu

N

15

20

25

30

35

Q09

)yad/L

M(

Q90

Zero flow days

1 km from river

2 km from river

4 km from river

Glover and Balmer, 1954

Fig. 9. Low flow indices versus time due to groundwater extraction.

-1

-0.8

-0.6

-0.4

-0.2

0

0 20 40 60 80 100

Dimensionless time

e s n o p s e r

x u l

F

Double the diffusivity

True diffusivity

Half the diffusivity

c=2a

c=4a

c=6a

c=9a Markers refer to the Knight et al. (2005) solution Solid lines refer to the Glover and Balmer (1954)

Fig. 10. Effect of distance to no-flow boundary.


a modelling exercise to highlight this phenomenon using theNamoi river flow data. Note that the intention of this hypotheticalexercise is not to replicate observed flows but merely to highlightthe impact of groundwater extractions on river low flows.

Results in Fig. 9 show a significant decline in Q90 and an expo-nential rise in the number of zero-flow days due to increasedgroundwater extraction. The results highlight the critical role of thedistance between the extraction source and the river. The fluxresponse to groundwater extraction varies with the square of thedistance between the extraction point and the river. During the825-day simulation, the impacts of the extraction seem to diminishwhen the extraction source is 4 km from the river. Hence, knowingthe time scales within which the impacts of groundwater extrac-tions occur, one can allow additional extractions that are distantenough to cause minimal impact on the river during a specificplanning period.

4.4. Applicability limits for analytical solutions

A set of assumptions are usually required to idealize (simplify)the flow problem in order to arrive at an explicit analytical solutionfor every rivereaquifer configuration. In order to obtain accurateresults from the analytical solutions, their underpinning assump-tions should not be violated. Since the most basic solution forestimating discharge response was introduced by Glover andBalmer (1954), a number of advancements has been made toincrementally add versatility (and hence complexity) to the solu-tion. More complexity leads to more parameters and higheruncertainty, which eventually defies the purpose of having a simplemodel. Therefore, one always endeavours to use the simplest modelthat can produce acceptable results knowing that the simplifyingassumptions may be violated to some extent under natural flowconditions thus resulting in some errors.

It is reasonable to postulate that the magnitude of the error isdirectly related to the extent to which the assumptions have beenviolated. Here, the applicability limits of various analytical solutionsused in the GWeSW Link Module are individually tested andcompared to the basic Glover and Balmer (1954) solution as well asresults obtained fromMODFLOW numerical models. Subsequently,criteria for the application of the solutions are identified for three

classes of violations relating to: (1) boundary conditions, (2) headgradients, and (3) aquifer heterogeneity.

4.4.1. Effect of no flow boundaryMost analytical solutions for discharge response assume a semi-

infinite flow domain meaning that the flow boundary opposite tothe river is infinite. One would expect that if the distance ‘a’ fromthe stress source to the river is much smaller than the distance tothe no-flow boundary ‘c’, the infinite boundary solution shouldproduce adequate results. On the other hand, if the two distancesare comparable there would be a significant discrepancy. Knightet al. (2005) proposed a solution for a finite flow domain, whichaccounts for the effect of a no-flow boundary. However, the othersolutions (e.g., Hall and Moench (1972) that accounts for a low-conductivity layer) still assume an infinite boundary opposite tothe river. Hence, the effect of a no-flow boundary needs to beinvestigated.

Fig. 10 compares results from Glover and Balmer (1954) withthose from Knight et al. (2005). The y-axis in Fig. 10 represents theflux response (Res(t) in Equation (9)) whereas the x-axis representsnon-dimensional time (time normalised by a2/D). It shows that forc > 9a, the two solutions produce identical results within the fluxresponse up to 90%. On the other hand, the effect of the no-flow

Fig. 11. Effect of head gradient.

-1

-0.8

-0.6

-0.4

-0.2

0

0 5 10 15 20 25 30

Time (years)

esnopserxul

F

Case-1 Modflow

Case-1 Analytical; h=20 m

Case-2 Modflow



Case-3 Modflow

Case-3 Analytical; h=51.5 m

Case-1

Case-2

Case-3

Knight et al., 2005; finite domain

Fig. 12. Effect of variable saturated aquifer thickness (Cases 1e3).

1

0.75

0.5

0.25

0

0 10 20 30 40 50 60 70 80 90 100

Time (years)

esno

pserx

ulF

Analytical;α = 0

Modflow; α = 100

Modflow; α = 1,000

Analytical; α = 1,000

Modflow; α = 5,000

Analytical;α = 5,000

Case 4

α = 0; Glover and Balmer 1954α > 0;Hall and Moench, 1972

Fig. 13. Effect of semi-pervious river bank (Case-4).


boundary becomes pronounced when c ¼ 2a. To put matters intoperspective, the magnitude of the error resulting fromviolating theboundary condition is compared to that resulting from a �100%uncertainty in knowledge of aquifer diffusivity, this is a conserva-tive uncertainty envelope as this parameter is usually varied by anorder of magnitude (de Marsily et al., 2005). Fig. 11 shows thatwhen c > 6a, the effects of the no-flow boundary can be neglected.

4.4.2. Effect of head gradientsThe analytical solution for discharge response assumes initial

hydrologic equilibriumbetween the river and the aquiferwhereby theaquifer water table is horizontally aligned with the river water level.Under natural conditions, most rivers are either gaining or losing andhence have head gradients to or away from the river, respectively. Adownward (positive) head gradient towards the river speeds thedischarge response whereas a reverse gradient slows it down. Themathematical formulation of Knight et al. (2005) for sloping baseaquiferswithaheadgradient (Equation12) applies tobothgainingandlosing rivers where only the sign notation of the angle needs to bereversed.Here, thesensitivityof thedischarge responsewithrespect toapositiveheadgradient is testedbycomparing results fromGloverandBalmer (1954) toKnightet al. (2005). In thismodellingexperiment, theaquifer thickness and head gradient are varied but the diffusivity keptconstant. Another trial was carried out with half the diffusivity to testthe model sensitivity to it.

The sensitivity analysis has shown that head gradient (repre-sented by the slope of the water table tanq) and aquifer thickness(h) are the two critical parameters contributing to the discrepancybetween predictions obtained from the two solutions with the ratioh/tanq as an indicator. Fig. 11 shows that the discrepancy varieswith time and reaches a maximum at a response of about 70%. TheGlover and Balmer (1954) solution exhibits a maximum error ofabout 10% when h/tanq z 600. It produces excellent results whenthis ratio approaches 3000, which is equivalent to a very commonlyoccurring combination of a head gradient of 1% and an aquiferthickness of 30 m. Results have demonstrated that the magnitudeof the diffusivity per se neither affects the errors pattern nor itsmagnitude (Fig. 11, see equal-length dotted arrows at 60% responsefor D ¼ 9000 and 18000 m2/year). However, Fig. 11 also shows thatGlover and Balmer (1954) solution grossly underestimatesdischarge response for shallow aquifers with steep head gradients(e.g., h/tanq z 100).

4.4.3. Effect of aquifer heterogeneity4.4.3.1. Variable saturated aquifer thickness. The saturated thick-ness of an aquifer may vary spatially either due to a sloping base ordue to the formation of a groundwater mound or a depression coneresulting from a recharge or an extraction source, respectively.

Firstly, the effect of a groundwater mound is investigated. Thenew irrigation development (Simulation 3a, Table 1) has resulted ina groundwatermoundwith a peak of 30m,which is 50% higher thanthe initial aquifer thickness of 20 m thus significantly adding toaquifer transmissivity. Fig. 12 shows that using the initial saturatedthickness of 20m in the analytical solution (at both locations in frontand behind the groundwater mound, Case-1 and Case-2, respec-tively), results in underestimating the response by about 5%. If onecan independently estimate the head distribution after the build-upof the mound, which can be done analytically, a modified saturatedaquifer thickness can be derived to improve the predictions of theanalytical solution. Fig. 12 shows that using such an effectivethickness that accounts for increased transmissivity resultingfrom the presence of the mound results in excellent agreementwith MODFLOW predictions (h ¼ 24 m for Case-2).

Secondly, the effect of a sloping base aquifer is investigated.Fig. 12 (Case-3) shows that using the same approach that accountsfor a non-uniform aquifer thickness improves the performance ofthe analytical solution and yields excellent agreement with thenumerical results.

4.4.3.2. Semi-pervious river bank. Fig. 13 shows that when theretardation factor a (which accounts for the presence of the semi-pervious layer) is as low as 100, the semi-pervious layer hasvirtually no effect and the response is identical to that provided bythe basic Glover and Balmer (1954) formulation (marked noretardation in Fig. 13). Note that a ¼ 100 means a 1-m semi-pervious layer having a hydraulic conductivity that is two ordersof magnitude lower than that of the aquifer. For higher retardationfactors, the response starts to be significantly affected by the semi-pervious layer where Hall and Moench (1972) formulation

Fig. 14. Schematic showing relation of GWeSW exchange fluxes to calibration and forecasting periods for river model.


continues to provide adequate predictions for a-values of up to1000. However, as a approaches 5000, Hall and Moench (1972)solution underestimates the response by a maximum of 10% atearly times but this discrepancy diminishes at large times(compared to MODFLOW predictions).

4.4.3.3. Aquifer heterogeneity along river reach. As outlined in theunderpinning assumptions, a river link is assumed to be homoge-neous with constant hydraulic parameters. Hantush et al. (2002)indicated that heterogeneity of a rivereaquifer system can beaccounted for efficiently by dividing the river reach into homoge-nous segments (sub-reaches), the outflow from one segmentbecomes the inflow to the next segment. This approach is adoptedhere to represent heterogeneity. The validity of this approach wasdemonstrated in Section 4.1 whereby the impacts of two groups ofextraction sources located in layers of different hydraulic conduc-tivities were added and their cumulative response agreeing favor-ably to MODFLOW predictions.

5. Model calibration

Models achieve their best predictive capacity via a calibrationprocess whereby their parameters are optimised to providepredictions that in best agreement with field observations.Groundwater models are usually calibrated against observedpressure head observations and the calibrated model is then usedto provide predictions for GWeSW exchange fluxes (e.g., ZoneBudget at the river boundary for a MODFLOW model). Assumingsound aquifer conceptualisation and good knowledge of variousstresses to the system, aquifer diffusivity (D) and riverbedconductance (C) are the main parameters that are optimised duringthe calibration of a groundwater model. A river model on the otherhand, is calibrated against observed flow data at a downstreamgauge. The traditional calibration of a river model such as IQQM(Simons et al., 1996) includes: (1) flow routing parameters (varieswith routing scheme); (2) a loss function (losses as a function ofriver flow that includes overbank losses and river recharge to theunderlying aquifer for losing river systems); and (3) a residual flowtime series (inflows including ungauged inputs and gains from thegroundwater for gaining river systems).

As the new generation ‘Source IMS’model includes both surfaceand groundwater processes, its calibration would entail conceptsfrom both the surface water and groundwater systems. The cali-bration would still be against observed flow data at a downstream

gauge with two additional groundwater parameters, namely,aquifer diffusivity and riverbed conductance. However, one canindependently calibrate the groundwater parameters and keepthem constant (or implement bounds to their variability) duringcalibration of the river model to control the non-uniquenessproblem. This can be done outside of ‘Source IMS’ using analyticalsolutions for the application of a flood wave response to estimateaquifer diffusivity and river conductance using the groundwaterlevels of a floodplain observation well (Ha et al., 2007).

An important issue thatneeds tobehighlightedhere is the timingof the delayed impacts of the groundwater processes and how theyrelate to the calibration period of the river systemmodel. Referringto Fig. 14, one can assume a pre-development erawhen the coupledsurfaceegroundwater system was at equilibrium thus resulting ina steady-state exchange flux, Qpd. Groundwater extractions wouldupset this state of equilibrium resulting in a new exchange flux thatis time-variant. With a high level of groundwater development, onewould expect this to be the status quo almost everywhere. There-fore, the calibration period of a river model would most likelycoincidewithagroundwater systemthat is at a transient statewherethe flux, during part or the entire calibration period, is time-variant.The traditional calibration of river models such as IQQM implicitlyaccounts for this time-variant GWeSW exchange flux. However,when the model is operated in a forecasting mode, it would disre-gard the effect of the unrealised impacts of existing developments aswell as the impacts of future developments.

The newgeneration ‘Source IMS’model overcomes this problemas it explicitly accounts for the interaction between surface andgroundwater via the GWeSW Link Module. To achieve a realisticcalibration that results in a model with a strong forecasting capa-bility, one must historically track all the changes in aquifer stressesand model their impacts prior to, and during the calibration periodof the river model (t0et1et2 in Fig. 14) and continue to account fortheir unrealised impacts during a subsequent forecasting simula-tion (t2et3 in Fig. 14). Note that the pre-development flux will mostlikely be an unknown quantity that is implicitly accounted for in thecalibration of the river model (which maintains mass balance).

6. Conclusions

The exchange flux between surface and groundwater can greatlyimpact the surface water and groundwater balance with seriousimplications on ecosystem health especially during low flowconditions. River models implicitly account for the interaction


between surface and groundwater where the exchange fluxesbecome part an unaccounted gains/losses during a black-box cali-bration process. In this paper, a conceptual framework for incor-porating surfaceegroundwater interactions into the riveroperationeplanning model ‘Source IMS’ was presented. TheGroundwateresurface water (GWeSW) Link Module explicitlyaccounts for the interaction between surface and groundwater. Itadopts explicit analytical solutions to evaluate the exchange fluxdue to individual processes, and then, uses the concept of linearityand superposition to estimate the overall exchange flux betweenthe surface and groundwater systems. The proposed approachstrikes the right balance between the complexity required to modelGWeSW interactions and the minimal data required to drive themodel. The following may be concluded:

B The total exchange flux between a river and the underlyingaquifer comprises the following components: (1) naturalexchange flux due to river stage fluctuations, which is the sumof three components, exchange during baseflow conditions,exchange during within-bank fluctuations, and exchangeduring overbank fluctuations; this component assumes a riv-ereaquifer system at dynamic equilibrium, with any stressesthat disrupt this state accounted for in the next three compo-nents; (2) exchange flux due to groundwater extraction; (3)exchange flux due to change in recharge rate; and (4) exchangeflux due to changes in evapotranspiration. The total exchangeflux estimated by the GWeSW Link Module is incorporatedinto the routing and calibration of the river operationeplan-ning model ‘Source IMS’ during every time step of a simulation.

B The adopted analytical solutions are capable of predicting theoverall exchange flux resulting from multiple sources andarising from various processes with errors not exceeding 4%compared to predictions obtained from numerical MODFLOWmodels. The results further validate the applicability of thecommon concept of linearity and superposition.

B The flux distribution resulting from a recharge/extractionsource located at a distance ‘a’ follows a normal distributionpattern where 50% of the flux is sourced from a reach-lengthequal to 2a. Results of the analytical solution were found tobe in very close agreement with predictions derived fromnumerical MODFLOW simulations. This functionality is espe-cially useful for apportioning fluxes to two adjacent links whena recharge/extraction source is located close to the node thatseparates the two links.

B Low flow indices demonstrated the significant influence ofGWeSW exchange flux on river low flows. Explicit accountingof GWeSW interactions in river models greatly enhances theirforecasting capacity during low flow conditions.

B A suite of analytical solutions are available in the GWeSW LinkModule to estimate the flux under different rivereaquiferconfigurations. One must always endeavour to select thesimplest solution that has the least number of parameters. Theapplicability limits for the analytical solutions were tested astheir simplifying underpinning assumptions are violated undernatural rivereaquifer conditions. The following criteria fortheir application were recommended: (1) the effect of a no-flow boundary on the flux response becomes marginal as thedistance from the river to a no-flow boundary becomes largerthan six times the distance from the recharge/extraction sourceto the river; (2) the effect of a head gradient becomes marginalas the ratio of aquifer thickness to head gradient approaches3000 but it can lead to serious errors for steep gradients andshallow aquifers as the ratio becomes as low as 100; (3) theeffect of aquifer heterogeneity resulting from a variable satu-rated aquifer thickness may be accounted for by estimating

aquifer diffusivity based on an effective aquifer thickness thataccounts for the non-uniform saturated aquifer thickness; (4)the effect of a semi-pervious layer can be adequately modelledusing an analytical solution that implements a retardationfactor (a) with excellent results for a < 1000 and an error ofabout 10% as a approaches 5000; and (5) aquifer heterogeneityalong the river can be modelled by discretising the river reachinto homogenous sub-reaches.

B To achieve a realistic calibration that results in a river modelwith a credible forecasting capability, one must historicallytrack all the changes in aquifer stresses and model theirimpacts prior to, and during the calibration period. When themodel is used in a forecasting mode, the remaining unrealisedimpacts of existing stresses, and any new stresses should beincluded in the future forecasting simulation.

Acknowledgement

The author wishes to acknowledge the following: the eWaterCooperative Research Centre and the National Water Commissionfor funding the project; the CSIRO team members Ian Jolly andTrevor Pickett, and the Department of Water, NSW.

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