A Compartmental–Spatial System Dynamics Approach to Ground Water Modeling

A Compartmental–Spatial System Dynamics Approachto Ground Water Modelingby Jesse Roach1 and Vince Tidwell2

AbstractHigh-resolution, spatially distributed ground water flow models can prove unsuitable for the rapid, interactive

analysis that is increasingly demanded to support a participatory decision environment. To address this shortcom-ing, we extend the idea of multiple cell (Bear 1979) and compartmental (Campana and Simpson 1984) groundwater models developed within the context of spatial system dynamics (Ahmad and Simonovic 2004) for rapidscenario analysis. We term this approach compartmental–spatial system dynamics (CSSD). The goal is to balancespatial aggregation necessary to achieve a real-time integrative and interactive decision environment while main-taining sufficient model complexity to yield a meaningful representation of the regional ground water system.As a test case, a 51-compartment CSSD model was built and calibrated from a 100,0001 cell MODFLOW(McDonald and Harbaugh 1988) model of the Albuquerque Basin in central New Mexico (McAda and Barroll2002). Seventy-seven percent of historical drawdowns predicted by the MODFLOW model were within 1 m of thecorresponding CSSD estimates, and in 80% of the historical model run years the CSSD model estimates of riverleakage, reservoir leakage, ground water flow to agricultural drains, and riparian evapotranspiration were within30% of the corresponding estimates from McAda and Barroll (2002), with improved model agreement during thescenario period. Comparisons of model results demonstrate both advantages and limitations of the CCSD modelapproach.

IntroductionIn 2000, an estimated 83,500 million gallons per day

of fresh water were extracted from U.S. aquifers to sup-port agricultural, domestic, and industrial uses, represent-ing a 9% increase over 1995 levels and a 145% increaseover 1950 levels (USGS 2004). As ground water utiliza-tion has grown, so too has the need for long-term resourcemanagement planning. To support this need, long-rangeground water planning efforts have increasingly

employed numerical ground water modeling to providea scientifically defensible basis for resource allocationand policy decisions. There has also been a recognition ofthe importance of an integrated, basin-scale approach towater management (e.g., Jury and Vaux 2005; USBoR2005) and active stakeholder engagement during allphases of the planning process, including model develop-ment and analysis (e.g., Tidwell and van den Brink 2008;van den Belt 2004). To connect stakeholders to integratednumerical modeling, simulation tools are needed that pro-vide a real-time and interactive environment for educationand enable lay participation in ground water decisionmaking. Such models provide a scientifically informedbasis for exploring alternative resource utilization scenar-ios within the context of interrelated surface water andhuman demand systems that are not endogenous to mostground water model conceptualizations. Here we outlinea single ground water modeling approach able to providescientifically defensible results in a structure that is easily

1Corresponding author: Earth Systems Department, Sandia Na-tional Laboratories, P.O. Box 5800, MS 0735, Albuquerque, NM 87185-0735; (505) 284-9367; fax (505) 844-7354; [email protected]

2Earth Systems Department, Sandia National Laboratories,P.O. Box 5800, MS 0735, Albuquerque, NM 87185-0735; (505)844-6025; [email protected]

Received September 2008, accepted March 2009.Journal compilationª 2009 National Ground Water Association.No claim to original US government works.doi: 10.1111/j.1745-6584.2009.00580.x

686 Vol. 47, No. 5–GROUND WATER–September-October 2009 (pages 686–698) NGWA.org

integrated across systems and computationally fastenough to be accessible in real time to stakeholders.

Numerical modeling of ground water behavior forpolicy analysis is dominated by models with high spatialresolution. Such models typically incorporate the bestavailable spatial data and conceptual understanding ofa ground water system and are calibrated to point data atobservation wells. When done well, they represent thebest available numerical representation of complexground water systems. However, for real-time, interactivescenario analysis, such models can be limiting for threemain reasons. First, it is difficult to dynamically link suchtools to models of other influencing systems such as thesurface water and human demand systems. Second, thetext-based input files cannot be easily manipulated forreal-time interactive scenario testing. Finally, the minutesor several minutes required for spatially complex groundwater model run times can be too long for effective,group-based, real-time model interaction. Although soft-ware solutions can help alleviate input-output issues, andcurrent computing power allows variable grid finite dif-ference (e.g., Afshari et al. 2008) and variable mesh finiteelement algorithms to be solved in real time even forbasin-scale problems, solving all three problems togetherremains a challenge.

Here we explore a compartmental (Campana andSimpson 1984) model approach to ground water model-ing within a context of spatial system dynamics (Ahmadand Simonovic 2004). The approach combines runtimeadvantages conferred by compartmental ground watermodeling with integrated modeling and user interface ad-vantages conferred by spatial system dynamics. We termthis approach compartmental spatial system dynamics(CSSD).

Compartmental models can be thought of as aspatially distributed and communicating set of lumped-parameter models (models that do not use spatial coordi-nates [Gelhar and Wilson 1974]). Compartmental modelswith explicit development of hydrologic parameters arereferred to as distributed parameter models (e.g., Adarand Sorek 1989) or as multiple cell models (Bear 1979)when described with a spatial coordinate system. Com-partmental ground water models have been used to modelsteady-state ground water flow as a function of hydro-chemical data when hydrologic parameter data are sparse(e.g., Campana and Simpson 1984; Adar et al. 1988), butnot to model ground water flow as a function of head. Aclose exception to this trend appears in Adar and Sorek(1989) where hydrochemical data are used to solve forquasi-steady-state flows between compartments, whichare then combined with available head data to estimatetransmissivities based on Darcy’s law.

The name spatial system dynamics (SSD) was coinedby Ahmad and Simonovic (2004) to describe a grid-basedinteraction of spatially distributed system dynamics mod-ules. SSD methodology has been used extensively inecological modeling (e.g., Costanza et al. 2002; Dealet al. 2000), and combines the power of temporal analysisthat is a hallmark of systems dynamics with the power of

spatial analysis characteristic of geographic informationsystems (GIS). Here we extend this body of work toaddress issues related to ground water management whileusing numerical compartments of irregular shape and sizeto increase model speed.

The CSSD model can be part of a larger systemsmodel and also preferably developed as a screeningmodel to complement higher resolution physical models.The challenge is to balance spatial aggregation necessaryto achieve a real-time interactive decision environmentwhile maintaining sufficient model complexity to yielda meaningful representation of the regional ground watersystem. We begin by describing the development ofa mathematical model to simulate transient ground waterflow between ground water compartments of irregularshape as a function of average head in the compartments.After development of the mathematical model, choice ofcompartments is considered and one approach to theparameterization of compartment properties is suggested.Finally results are presented from application of thisapproach to a ground water basin in New Mexico.

Model Formulation

Mathematical ModelHere we develop a CSSD approach to ground water

modeling for purposes of rapid, interactive scenario eval-uation. This development is analogous to the develop-ment of a multiple cell ground water model described byBear (1979) without the reliance on an underlying coordi-nate system. Consider an aquifer divided into the irregularspatial compartments shown in Figure 1. Applying con-servation of mass, the change in storage in compartmenta is equal to the sum of flows into a less the sum of flowsout of a. Mathematically,

dSadt

¼ Qab 1 Qac 1 Qad 1 QaB ð1Þ

where

Figure 1. Example aquifer (plan view) delineated into fourirregular compartments.

NGWA.org J. Roach, V. Tidwell GROUND WATER 47, no. 5: 686–698 687

dsadt

¼ change in storage through time in compartment

a [L3/T]Qab;Qac;Qad ¼ flows into a from b, c, and d, respec-

tively [L3/T]QaB ¼ sum of boundary flows for compartment

a [L3/T].All flow terms in Equation 1 are positive for flow

into compartment a and negative for flow out. Boundaryflows are flows entering or leaving the saturated groundwater system, including terms such as evapotranspiration(ET), well extraction or injection, recharge, stream leak-age, and drain capture. Applying Darcy’s law, flowbetween ground water compartments can be described asa function of average compartmental ground water head,

Qab ffi�TabCab

Labðha � hbÞ ð2Þ

whereTab ¼ effective transmissivity between compartments

a and b [L2/T].Cab ¼ length of horizontal contact between compart-

ments a and b [L].Lab ¼ effective distance between compartments

a and b [L].ha; hb ¼ representative heads in compartment a and

b, respectively [L].In an unconfined aquifer, effective transmissivity

between compartments Tab is a function of ground waterhead. When spatial scale is large compared to headchanges, Tab may be assumed to be independent ofground water head. Although not necessary for modeldevelopment, this assumption allows Equation 2 to besimplified to:

Qab ffi aabðhb � haÞ ð3Þ

where

aab[TabLab

Cab ð4Þ

Equation 3 estimates flows between ground watercompartments as a linear function of heads in the com-partments. A key assumption in CSSD modeling implicitin Equation 3 is that a single head value is representativeof head throughout the compartment, an approximationthat improves as compartment size and heterogeneitydecrease.

Substituting Equation 3, and analogous terms forcompartments c and d, Equation 1 can be rewritten as

dSadt

¼ aabðhb � haÞ 1 aacðhc � haÞ

1 aadðhd � haÞ 1 QaB ð5Þ

Using a finite timestep approximation for storagechange, adding superscript notation to specify time, andconverting to matrix form for all possible generic ground

water compartments, Equation 5 can be rewritten to solvefor storage in aquifer compartment i at time t11 as a func-tion of storage and head values at time t:

St11i ¼ Sti 1 �t

"Xnj¼1

ðQtijÞ 1 Qt

iB

#ð6Þ

wheren ¼ total number of ground water compartments.St11i Sti ¼ storage vectors for n ground water compart-

ments at times t11 and t [L3].�t ¼ timestep duration [T].Qt

iB ¼ boundary flow vector for n ground water com-partments at time t [L3/T].

Qtij ¼ n by n matrix representing flows to i from j at

time t for all i and j [L3/T].The flow matrix Qt

ij is developed from Equation 3expressed in matrix form:

Qtij ¼ aij �

Xni¼1

Xnj¼1

�htij ð7Þ

whereaij ¼ symmetric conductance (and connectivity)

matrix for all i and j [L2/T].�htij ¼ hj � hi at time t [L].The conductance value for compartments that are not

hydrologically connected is zero, so the conductancematrix aij also expresses the connectivity betweencompartments.

Note that an initial condition storage vector Sti is nec-essary to evaluate Equation 6 at the first timestep, andthat in Equation 6, the flow matrix Qt

ij is summed acrossall j to result in a vector of total flows to each of ncompartments.

If all boundary flows (QtiB) are either known or

a function of aquifer heads, the aij matrix is known, andstorage and head conditions at the beginning of the time-step are known, we end up with a system of n equationsand n unknowns. In effect we have formulated a forwarddifference explicit solution for calculating ground waterheads at one timestep from head values at the previoustime step. Aquifer storage is related to aquifer head usingthe following relationship between storage and head in anunconfined aquifer:

Si ¼ ðhi � zbotiÞ � Fi � syi ð8Þ

whereFi ¼ the horizontal area (footprint) of compartment

i [L2]syi ¼ specific yield of unconfined compartment i [–].zboti ¼ bottom elevation of compartment i with

respect to common datum [L].Equation 8 is used to update the heads from changes

in compartment storage so that Equations 7 and 6 can besolved at the next timestep.

Because the forward difference explicit formulationpredicts the future state based on the present state, the

688 J. Roach, V. Tidwell GROUND WATER 47, no. 5: 686–698 NGWA.org

system of equations can be unstable if the timestep is toolong relative to the spatial scale and rate of movement ofwater between compartments. Conditional stability forEquation 6 (and by analogy the set of equations repre-sented by Equation 7) is satisfied if the following stabilitycriterion is met:

�tXnj¼1

aij , Fisyi ð9Þ

For a given distribution of heads, Equation 9 statesthat the amount of water that can move between groundwater compartments (left side of inequality) cannot begreater than the available storage in those compartments(right side of inequality). This is analogous to the well-known unconditional stability criteria for a forward dif-ference explicit two-dimensional square grid solution:�t

�2Tx 1 2Ty

�, F � sy where Tx; Ty are transmissivities

in the x and y direction, and are doubled because there aretwo faces in each direction through which water canreach the square cell of interest (modified from Bear andVerruijt [1987], equation 9.3.5).

Spatial Compartmentalization and ParameterizationThe objective of a CSSD ground water model is to

efficiently capture salient ground water dynamics forreal-time systems level analysis. The choice of physicallymeaningful compartments is critical, and typically in-volves trial and error as compartments are refined toallow for model behavior to more closely match systembehavior. Compartments should represent areas of rela-tively homogenous aquifer properties, ground waterbehavior, and system stresses. Campana et al. (2001,p. 37) suggest that compartment differentiation dependson ‘‘hydrogeological uniformity, the availability of data,the degree of resolution desired, and constraints imposedby numerical solutions’’ (e.g., stability criteria). The reso-lution desired will depend on the questions being ad-dressed by the model, the model speed desired, and thespatial resolution required to meaningfully couple toother modeled systems.

Once initial ground water compartments have beendefined, the model can be calibrated to observed groundwater elevation data, or ground water flows predicted by anaccepted high resolution, spatially distributed ground watermodel (high resolution model). In the latter case, averagehead and flow associated with chosen CSSD compartmentscan be analyzed in the high resolution model. By rearrang-ing Equation 3, the effective conductance between all com-partment pairs can be calculated in the high resolutionmodel at each timestep by dividing ground water flowsfrom one compartment to another by the average compart-mental head difference, as shown in Equation 10.

Qij ¼ aijðhj � hiÞ0aij ¼Qij

hj � hið10Þ

Note that Qij values are calculated as a spatial sum ofthe high resolution model flow between compartments i

and j, and hi is calculated as the average of all head valuesfrom the high resolution model that fall within compart-ment i.

A positive conductance value (aij) suggests that flowcalculated by the high resolution model for the selectedcompartment configuration is, on average, from a com-partment of higher average head to a compartment oflower average head. The compartmentalization is refinedin a process of trial and error, in search of a compartmentgeometry that will result in positive average conductancevalues over all timesteps for all compartment pairs. Thetime averaged conductance values are then used to createthe conductance matrix (aij ) used in Equation 7 todescribe ground water movement in the CSSD model.Finally, parameters associated with boundary flows areselected such that long-term average boundary flows arethe same between the CSSD and higher spatial resolutionmodels. Without a high resolution model, CSSD com-partmentalization and parameterization proceed in a simi-lar fashion until average ground water level observationsare acceptably matched through time. The remainder ofthis paper considers a specific application of the CSSDapproach to ground water modeling.

Model ApplicationThis section describes the implementation of a CSSD

model using Equations 1 through 10 to describe groundwater dynamics in the Albuquerque ground water basin incentral New Mexico. Although there are well-developedground water models of the Albuquerque ground waterbasin, the CSSD model was developed to address the lackof a rapid ground water model that could be coupleddirectly with other resource models of the region. Themodel described here was built as part of a basin-scalesurface water, ground water, and human water demandmodel (Roach 2007), and can run 40-year scenarios ona standard laptop computer in tens of seconds. The modelwas built in Powersim Studio Expert 2007 (http://www.powersim.com), a commercial, object oriented systemdynamics software platform with powerful graphical userinterface capabilities to facilitate user interaction.

The Albuquerque Ground Water BasinLocated in central New Mexico (see Figure 2), and

underlying the Rio Grande from Cochiti Reservoir in thenorth to the San Acacia Narrows in the south, the Albu-querque ground water basin covers an area of approxi-mately 8000 km2 (McAda and Barroll 2002). Overdraft ofthe aquifer to support growing municipal populations andresulting effects on Rio Grande leakage into the groundwater system have resulted in a significant amount ofeffort toward understanding and characterizing the groundwater system. A legacy of distributed ground water mod-els has led to a widely accepted and mature ground waterflow model of the Albuquerque Basin most recently up-dated by Douglas P. McAda and Peggy Barroll in 2002.The McAda and Barroll (2002) model is highly spatiallydistributed, with more than 100,000 MODFLOW cells


arranged in 80 columns, 156 rows, and 9 layers. Cell sizein the x and y directions is 1 km. The MODFLOW modelruns from 1900 to 1990 at an annual timestep and from1990 to 2000 on a semiannual timestep (McAda andBarroll 2002).

A 51-compartment CSSD model, developed as a spa-tially simplified representation of the McAda and Barroll

(2002) model, is shown in Figure 2. ZONEBUDGET(Harbaugh 1990) was used to determine the mass balancein a given compartment at each model timestep from1975 through 1999 in the McAda and Barroll (2002)model. After initial compartmentalization based onspatial distribution of geohydrological characteristics,surface water features, and locations of surface water flow

Figure 2. Albuquerque Basin ground water model extent. Shaded compartments representing the alluvial aquifer are associ-ated with the top two MODFLOW layers only as shown in the typical cross section A to A’.


observations, delineation was refined and compartmentswere added through trial and error evaluation of compart-ment behavior compared to behavior of the McAda andBarroll (2002) model. The 51-compartment configurationshown in Figure 2 satisfied the non-negative conductancerequirement (Equation 10) for all compartment pairsexcept one. Additional compartments could have beenadded to address this problem; however, in the balancebetween resolution and CSSD run time, it was decidedthat 51 compartments captured sufficient behavior of theMcAda and Barroll model for purposes of rapid, interac-tive, high level system analysis. The one negative averageconductance value was for flow between shallow aquifersin north and south Albuquerque (near a significant pump-ing-induced cone of depression), and was set to a positivevalue based on average conductance values and contactareas for similar compartmental junctions.

An important factor used in initial compartment demar-cation in the Albuquerque ground water basin was the pres-ence of high-conductivity alluvial sediments located closeto the river. These alluvial sediments are relatively dynamicfrom a ground water perspective, with a strong seasonal sig-nal as water is gained from river, canal, and crop seepage,and lost to agricultural drain capture and riparian vegetationET. These hydrologically active alluvial sediments act dif-ferently enough from the rest of the aquifer that they areconceptualized as a shallow alluvial aquifer on top ofa more stable regional aquifer. The first two layers of theMODFLOW model near the river represent these high-conductivity sediments (McAda and Barroll 2002, p. 20),and compartmentalization efforts defined a shallow aquiferassociated with the top two MODFLOW layers, making theCSSD three dimensional, as shown in cross section A to A’in Figure 2. A specific yield value of 0.2 was assumed forall compartments, consistent with the McAda and Barroll(2002) model.

A key purpose of the CSSD approach described hereis to create a rapid ground water module that is part ofa larger, monthly timestep, interactive decision supportmodel. For this reason, the CSSD model was set up torun on a monthly timestep, and fluxes between the sur-face water and ground water system were set up to takeadvantage of monthly surface water information.

Albuquerque basin boundary flows treated as inde-pendent of ground water head by McAda and Barrollinclude well extraction, specified ground water inflowsalong model margins, and recharge from surface sourcesthat are not connected hydrologically to the aquifer,including recharge from the mountain front, ephemeraland tributary channels, disconnected streams, irrigationcanals, irrigated crops, and septic tanks. These terms arealso treated as independent of ground water head in theCSSD model. Boundary fluxes modeled as ground waterhead dependent by McAda and Barroll include aquiferinteraction with hydrologically connected surface water,including the Rio Jemez and Rio Grande and the JemezCanyon and Cochiti Reservoirs, agricultural drains, andET. These fluxes are also modeled as ground water headdependent in the CSSD model as described subsequently.

In the 51-compartment model, Rio Jemez and RioGrande river-aquifer interactions and reservoir-aquifer in-teractions, Qi�SWGW [L3/T], were modeled as follows:

Qi�SWGW ¼ Ki�bedFi�bed

bi�bedðzi�sw � bÞ

3

�b ¼ zi�bed if zi�bed � bi�bed � hib ¼ hi if zi�bed � bi�bed , hi

ð11Þ

wherezi�sw is the surface water elevation [L].zi�bed is the elevation of the top of the bed sediments

[L].bi�bed is thickness of the flow limiting bed sediments

[L].Ki�bed is the hydraulic conductivity of the flow limit-

ing bed sediments [L/T].Fi-bed is the horizontal area of the river channel [L2].hi is the ground water head [L].All terms are specific to compartment i; and all head

and elevation terms are defined based on a commondatum. Equation 11 describes hydrologically separateflow when ground water head is below the flow-limitingsediments, and head-dependent flow to or from the sur-face water system otherwise, and is consistent with theconceptual approach used by MODFLOW in the riverpackage (McDonald and Harbaugh 1988). For RioGrande leakage, bed thickness (bi�bed) and bed conductiv-ity (Ki�bed) were set initially to 5 feet and 0.5 feet/day,respectively, consistent with a value of 0.1 day-1 for Ki�bed

bi�bed

used by McAda and Barroll (2002). Powersim forces unitconsistency and automatically converts all time-based pa-rameters to their monthly equivalent. River leakage wascalibrated to McAda and Barroll (2002) average values bymodifying the river bed elevation of each shallow aquifercompartment within an acceptable range. River bed con-ductivity Ki�bed was also adjusted during calibration ofthe shallow aquifer north of Albuquerque, where spatialaggregation seems to result in larger leakage near thecones of depression than is predicted by the MODFLOWmodel. For the Rio Jemez, bed thickness was set to 1 footconsistent with McAda and Barroll, and both bed eleva-tion and river bed conductivity were adjusted during cali-bration. For reservoir leakage, values of reservoir bedthickness and conductivity were adjusted during calibra-tion. Jemez Reservoir was assumed hydrologically sepa-rate from the ground water system. Fi–bed was found bysumming the number of MODFLOW cells in the toplayer of each compartment and multiplying by the cellsize of 1 km2.

In unconfined aquifers where ground water flow toa surface water sink is predominantly horizontal and thereis no significant seepage face, a Dupuit–Forchheimer-based approach can be used to model flux (e.g., Fetter1980, equation 5-59). This approach was used to modelground water flow to the agricultural drains, QDUP [L3/t],in the Albuquerque Basin:

QDUP ¼ Ki�aLixi

ðh2i � zi�sw2Þ ð12Þ


whereKi�a is the hydraulic conductivity of the aquifer com-

partment [L/T].Li is the length of the drain [L].xi is a characteristic distance beyond which the drainhas negligible effect on ground water head [L].

All terms are specific to compartment i. All otherterms are as defined previously. Equation 12 is expressedas double the typical Dupuit-Forchheimer equation torepresent flow to a drain from two sides. Drain elevationswere set to 5 feet below the corresponding river bed ele-vation, and flow to the drains was calibrated by adjustingaquifer conductivity (Ki�a) and characteristic length val-ues (xi).

Average monthly surface water stage (zsw) for riversand drains was found at each timestep by iterative solu-tion of Manning’s equation for open channel flow (e.g.,Grant and Dawson 1997, p. 130):

QMAN ¼ AR2=3S1=2

nð13aÞ

whereQMAN is discharge in liters per second [L3/T].S is the dimensionless drain slope [–].n is the dimensionless Manning coefficient of rough-ness [–].

A is the cross-sectional area of flow in square meters[L2].

R is the hydraulic radius in meters [L].For a channel with vertical sides,

A ¼ ðzsw � zbedÞ �W ð13bÞ

R ¼ A

2ðzsw � zbedÞ 1 Wð13cÞ

where W is the channel width [L].Ground water fluxes from the river are small com-

pared to river discharge and thus have negligible effecton river stage. In the case of drains, however, groundwater movement is the primary source of any surfacewater flow in the drains. The amount of water that will

move to the drain depends on the stage in the drain,which itself determines how much water will movethrough the drain as surface flow. In the Albuquerquebasin model, an iterative solution was used to finda drain stage (zsw) that would result in a Dupuit-Forchheimer predicted ground water flow to the drain(Equation 12) equal to the Manning-based channel flow(Equation 13a). This iterative solution was implementedin Powersim using an embedded Visual Basic loop. Seethe equations in the online Supporting Information foradditional details. A Manning coefficient of 0.028 corre-sponding to a clean straight natural channel (Grand andDawson 1997) was used for the river and drain channels.Average drain width was assumed to be 3 m. Rio Grandeaverage channel width between major surface watergauges was calculated by dividing flow-dependent esti-mates of river area developed for the Upper Rio GrandeWater Operations Model (USACE et al. 2002) by reachlengths (see Table 1). The Rio Jemez was assumed to havean average width of 7.6 m independent of flow rate. Riverand drain slopes were assumed equal (drains only work byhaving a slope slightly less than the river slope; however,this difference was assumed negligible). Table 1 showsreach lengths, slopes, and area parameters for the surfacewater river reaches overlying the Albuquerque groundwater basin.

Ground water ET is modeled in the CSSD as a head-dependent flux. Consistent with the McAda and Barroll(2002) approach, ET is modeled as 1.5 m/yr when waterlevel is at or above the land surface, then decreases line-arly to 0.6 m/yr when depth to ground water is 2.7 m,then decreases linearly to 0.23 m/yr when depth to groundwater is 4.9 m, then decreases linearly to 0 m/yr whendepth to ground water is 9.1 m, and is 0 m/yr for allground water depths greater than 9.1 m below the surface(McAda and Barroll 2002, p. 38). ET predicted by theCSSD model was calibrated to McAda and Barroll esti-mated ET fluxes by adjusting the representative surfaceelevation of shallow aquifer compartments containingriparian vegetation.

The CSSD equations described in this section wereimplemented in a system dynamics framework and can

Table 1Surface Water Parameters used in the CSSD Model to Predict Surface Water–Ground Water Interactions

River Reach

River Area(ha) as a Function

of Flow Rate (Q) in m3/sBank FullArea (ha) Reach Length (km) Reach Slope (–)

Cochiti to San Felipe 91.103Q0.1988 253 24 0.0014San Felipe

to Albuquerque146.97Q0.4099 1100 53 0.001

Albuquerque to Bernardo 238.33Q0.4375 2094 85 0.0008Bernardo to San Acacia 34.209Q0.5291 427 23 0.0009Jemez River 37 37 48 0.0032

Note: All data except the reach slopes were taken from values and relationships used in the Upper Rio Grande Water Operations Model (USACE et al. 2002).


be seen along with associated diagrams in the SupportingInformation online materials.

ResultsThis section summarizes the behavior of the 51-

compartment CSSD ground water model of the Albuquer-que Basin described previously, during a historicalcalibration period (1975 to 2000) and a scenario period(2001 to 2040). In both cases, model outputs includingaquifer heads and head-dependent ground water fluxesare compared to analogous output from the McAda andBarroll (2002) MODFLOW model of the Albuquerqueground water basin.

Calibration Period 1975 to 2000

Spatial aggregation leads to a loss of spatial head dis-tribution information, which affects the ability to predictboth intercompartmental ground water flows and head-dependent boundary flows. Figure 3 shows predicteddrawdowns between 1975 and 2000 for each model, aswell as the distribution of differences between them. Itcan be seen that the driving changes to the Albuquerqueground water system between 1975 and 2000 are moundingunder Cochiti Reservoir in the northeast, and pumping-induced drawdown under Albuquerque and Rio Rancho inthe center of the basin. The 51-compartment model

captures the major head changes predicted by the MOD-FLOW model. Drawdown predictions in 77% and 91% ofMODFLOW cells are within 1 and 3 m, respectively, ofthe drawdown predicted in the corresponding compart-ment in the CSSD model.

Specified boundary flux terms in the CSSD modelare the same as those used by McAda and Barroll.Head-dependent boundary flux terms include river andreservoir leakage, drain capture, and ET, and are mod-eled in the CSSD model with Equations 11 and 12,respectively, with drain stage from Equation 13. Fig-ure 4 shows river leakage, Cochiti and Jemez Reservoirleakage, ground water flow to drains, and riparian ETvalues calculated by each model from 1975 to 2000,both as time series and annual averages compared.Cumulative fluxes for the 25-year period served as cali-bration targets. The difference in magnitude of fluctua-tions seen in the river and reservoir leakage and drainflow plots is in part a result of temporal resolution dif-ferences between the models. The CSSD model usesmonthly data and the MODFLOW model uses annualdata until 1990 and biannual data from 1990 to 2000.River and reservoir leakage are driven by surface waterstage, which varies significantly at a monthly timestep,to a lesser degree in a biannual timestep, and negligiblywhen averaged across an entire year. The drains capturea significant fraction of river leakage and thus respond to

Figure 3. Cumulative drawdown in the Albuquerque Basin from 1975 to 2000 as modeled by McAda and Barroll (2002), andthe 51-compartment CSSD ground water model. The histogram bins are 2 m wide centered on zero. Seventy-seven percentof drawdowns predicted by McAda and Barroll (2002) are within 1 m of the corresponding compartment drawdown; 91% ofdifferences are within 3 m.


river leakage variations, resulting in greater variation forthe monthly CSSD model than the annual/semiannualMODFLOW model. ET drops slightly in both modelsbeginning in 1984 because total riparian area before 1984is estimated with a 1975 United States Bureau of Reclama-tion (USBoR) spatial dataset (~145 km2), and after 1984with a 1992 USBoR spatial dataset (~126 km2) (McAdaand Barroll 2002, p. 38). The visible match between mod-els is easiest to see after 1990 when the MODFLOWmodel begins using seasonal data. The scatter plot showsannual flow values calculated by each model and the dif-ference between models that is not exceeded in 80% of his-torical model years. For example, annual river leakagecomputed by the compartment model was within 19% ofthe McAda and Barroll (2002) estimates in 80% (20 of25) of historical model years.

Robustness Analysis 2001 to 2040

The 51-compartment model is a standalone represen-tation of the Albuquerque Basin ground water system, isable to capture salient ground water system behavior dur-ing the calibration period, and thus provides a reasonableapproximation to ground water system behavior whensystem forcings are within a historical range. We nowconsider the robustness of the model by analyzing modelbehavior for system forcings outside of historical rangeswhile maintaining parameter values at the settings deter-mined from the calibration analysis.

To compare model behavior for conditions differentfrom those observed during the 1975 to 2000 calibrationperiod, an existing analysis of the McAda and Barroll(2002) model for three different ground water use scenar-ios for the city of Albuquerque from 2001 to 2040 done

Figure 4. Time series and annual averages compared for river leakage, Cochiti and Jemez Reservoir leakage, ground waterflow to agricultural drains, and riparian ET in the Albuquerque ground water basin, as simulated by McAda and Barroll(2002) and the CSSD model for the 1975 to 2000 historical period.


by Bexfield and McAda (2003) was used. A high pump-ing scenario assumed all projected Albuquerque demandbetween 2001 and 2040 would be satisfied by groundwater pumping. A medium pumping scenario froze Albu-querque ground water pumping at 2000 levels for the2001 to 2040 period. The low pumping scenario assumedthat Albuquerque would use direct diversion of surfacewater from the river beginning in 2006 to reduce relianceon ground water pumping. Projected demand and pro-jected available surface water supply were based on cal-culations and a synthetic climate sequence used by theCity of Albuquerque (Bexfield and McAda 2003, p. 9).For simplicity, only the low and high pumping scenariosare considered here.

As was the case in the calibration period, to matchspecified model forcings, the spatially aggregated modelwas forced with MODFLOW specified recharge and wellextraction terms for the robustness analysis. Jemez andCochiti Reservoir stage and area were specified based onBexfield and McAda (2003) values. Bexfield and McAda(2003) specified each year from 2001 to 2040 as dry, wet,or normal, but did not report actual river flows used. For

the robustness analysis, surface water conditions for dry,wet, or normal years were set to average conditions forthe 6 lowest, 6 wettest, and 13 remaining years from the1975 to 1999 historical record, respectively, as defined byflow below Cochiti Reservoir. Riparian ET, drain flow,and river and reservoir leakage were calculated within theCSSD model.

Figure 5 shows total pumping assumed for each sce-nario, basin ground water storage change, net surfacewater to ground water flux from the combined river anddrain system, and riparian ET values simulated by theCSSD model and the Bexfield and McAda model. The51-component model is able to track overall ground watermovement patterns predicted by the Bexfield and McAdamodel for both pumping scenarios; however, the year toyear variability is significantly dampened. Despite greaterintra-annual variability in river leakage and drain flow(Figure 4), the spatial aggregation of the CSSD modelleads to a dampening of response in the ground water sys-tem to interannual changes in overall magnitude of riverleakage and ET flux. The MODFLOW lines in Figure 5show climate variability from year to year that is

Figure 5. Model behavior for two extreme ground water use scenarios in the Albuquerque Basin simulated by the CSSD andMODFLOW (Bexfield and McAda 2003) models.


dampened in the CSSD model. Both models suggest thatunder the high pumping scenario, around 2015, the netcontribution of the river leakage less drain capture goesfrom net gains to the surface water system to net gains tothe ground water system. This type of consistencybetween models suggests that the CSSD model can beused with relative confidence for first order scenario anal-ysis. Note that the range of ET fluxes shown in Figure 5(107 m3/yr) is an order of magnitude less than the rangeof net surface water to ground water fluxes (108 m3/yr),and that the Bexfield-McAda ET flux values increaseslightly to start the run and remain above the CSSD esti-mates. This initial divergence may be a result of slightdifferences between the Bexfield-McAda MODFLOWmodel used for scenario analysis and the McAda-BarrollMODFLOW model to which the CSSD model wascalibrated.

Both models suggest that river leakage and draincapture are more sensitive to changes in pumping regimethan is riparian ET. In both models, the majority of theET response to reduced pumping occurs between 2010and 2020 after the dramatic drop in pumping that beginsin 2006 in the reduced pumping scenario. Because stor-age change in the entire aquifer is essentially a functionof specified and head-dependent fluxes, and the head-dependent fluxes match reasonably, it is not surprising tosee that storage change predicted by the compartmentmodel follows that predicted by the Bexfield and McAda(2003) model to a reasonable degree. Overall, the spa-tially aggregated 51-compartment model appears to bea reasonable proxy of the spatially resolved MODFLOWmodel, even outside the range of calibration conditions.

The ability of the spatially aggregated model to cap-ture the behavior of the MODFLOW model during therobustness analysis period can be compared to the sameability during the calibration period by using the rootmean squared error (RMSE) of model outputs as thecomparison metric. The comparison metrics are derivedon an annual basis for the ground water compartments inand around Albuquerque. In general, the 51-componentmodel would be expected to match the behavior of theMODFLOW model more closely during calibration than

during robustness analysis; however, as seen in Table 2,the opposite is true. The same comparisons made ata monthly basis also yield a better match to the MOD-FLOW models during the robustness analysis than thecalibration period. This unexpected improved perfor-mance during future conditions is likely to be a result ofthe relative simplification of the future system for sce-nario analysis. During the scenario analysis period, allmodel drivers are held constant, with the exception ofthree hydrologic year types and Albuquerque pumping.The historical period on the other hand is driven by con-stantly changing pumping regimes and river and reservoirconditions. Though unexpected, this result suggests thatour ability to predict future ground water conditions islimited more by our inability to predict future modelstresses than by the errors associated with spatial aggre-gation of the ground water model. Given that the futureis inherently unknown, the spatially aggregated modelprovides a legitimate proxy to the more complexMODFLOW models for rapid scenario evaluation of awide range of alternative futures.

ConclusionsThe CSSD approach was demonstrated for the Albu-

querque ground water basin in central New Mexico. Spe-cifically, a highly distributed regional ground watermodel (McAda and Barroll 2002) was used to calibratea spatially aggregated 51-unit CSSD model, which in turnwas able to capture the dominant system behavior pre-dicted by the more distributed model in both the calibra-tion and scenario evaluation periods. Seventy-sevenpercent of 1975 to 2000 drawdowns predicted by theregional model are within 1 m of the correspondingCSSD estimates; 91% of those differences are within 3 m.In 80% of the model run years between 1975 and 2000,the CSSD model estimates of river leakage, reservoirleakage, ground water flow to agricultural drains, andriparian ET were within 30% of the correspondingregional model flux estimates. For 2000 to 2040, theCSSD estimated heads and flux values were even closerto the corresponding regional model estimates than

Table 2Evaluation Table for Spatially Aggregated 51 Compartment Model Annual Fluxes Compared to Those from

McAda and Barroll (2002) and Bexfield and McAda (2003) MODFLOW Models

Ground WaterFlux [106 m3/yr]

Calibration (1975 to 2000) Robustness Analysis (2000 to 2040)

Mean RMSE vs. MODFLOW Mean RMSE vs. MODFLOW

Aquifer storage change -63 45 –50 8.7River-drain flux to ground water -29 40 –13 8.3ET flux from ground water 111 6.3 97 4.8

Note: Fluxes are summed across the entire model extent, and robustness analysis metrics are an average of high and low pumping scenario results.


during the calibration period. Thus, in the Rio Grande,a CSSD approach was used successfully to build a rapid,systems level ground water model without sacrificing theability to capture salient system behaviors.

The CSSD model is not appropriate for problemsrequiring a detailed analysis of ground water dynamics; itis instead a supplement to the more highly distributedmodels that exist for the basin. The Rio Grande CSSD isappropriate for real-time, integrated, systems level analy-sis of a preliminary nature. The ground water modeldescribed here is part of a basin-scale surface watermodel and human water demand model (Roach 2007)that can run 40-year scenarios in tens of seconds. Aninteractive user interface with easily modified inputs fa-cilitates basin-scale scenario analysis in open and partici-patory processes aimed at public education and preliminarypolicy analysis. The user interface can be viewed in theSupporting Information online materials.

Supporting InformationAdditional Supporting Information may be found in

the online version of this article:Table S1. RoachTidwell2009EquationsAndDiagrams.

xls.Presentation. RoachTidwell2009InterfaceCaptures.

ppt.Please note: Wiley-Blackwell are not responsible for

the content or functionality of any supporting materialssupplied by the authors. Any queries (other than missingmaterial) should be directed to the corresponding authorfor the article.

AcknowledgmentsThe authors would like to thank Michael Barden and

two anonymous reviewers for valuable feedback and sug-gestions for improvement of this paper. In addition, wewould like to acknowledge the assistance provided duringmodel development and review by the Upper Rio GrandeWater Operations Model technical team. The first authorreceived support from Sandia National Laboratoriesthrough a Campus Executive Fellowship while attendingthe University of Arizona. Funding for this project wasprovided through Sandia National Laboratories’ Labora-tory Directed Research and Development program.Sandia is a multiprogram laboratory operated by SandiaCorporation, a Lockheed Martin Company, for the U.S.Department of Energy’s National Nuclear SecurityAdministration under contract DE-AC04-94AL85000.

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A Compartmental–Spatial System Dynamics Approach to Ground Water Modeling