Fast Approach for Unsteady Flow Routing in ComplexRiver Networks Based on Performance Graphs

Arturo S. Leon, M.ASCE1; Elizabeth A. Kanashiro2; and Juan A. González-Castro, M.ASCE3

Abstract: This paper presents a new model for unsteady flow routing through dendritic and looped river networks based on performancegraphs. The model builds on the application of a hydraulic performance graph (HPG) to unsteady flow routing introduced in a previous studyand adopts the volume performance graph (VPG) introduced in another study. The HPG of a channel reach graphically summarizes thedynamic relation between the flow through and the stages at the ends of the reach under gradually varied flow (GVF) conditions, andthe VPG summarizes the corresponding storage. Both the HPG and VPG are unique to a channel reach with a given geometry and roughnessand can be computed decoupled from unsteady boundary conditions by solving the GVF equation for all feasible conditions in the reach.Hence, in the proposed approach, the performance graphs can be used for different boundary conditions without the need to recompute them.Previous models based on the performance graph concept were formulated for routing through single channels or channels in series. The newapproach expands on the use of HPG/VPGs and adds the use of rating performance graphs for unsteady flow routing in dentritic and loopednetworks. The applicability of the proposed model to subcritical unsteady flow routing is exemplified through a looped network, and itssimulation results are contrasted with those from the well-known unsteady Hydrologic Engineering Centers River Analysis System model.The results show that the present extension of application of the HPG/VPGs appears to inherit the robustness of the HPG routing approach in aprevious study. The unsteady flow model based on performance graphs presented in this paper can be extended to include supercritical flows.DOI: 10.1061/(ASCE)HY.1943-7900.0000678. © 2013 American Society of Civil Engineers.

CE Database subject headings: Flood routing; Hydraulics; Unsteady flow; River flow.

Author keywords: Dendritic network; Flooding; Flow routing; Hydraulic routing; Looped network; Modeling; River hydraulics;Simulation; Unsteady flow.

Introduction

Most free surface flows (also called open-channel flows) are un-steady and nonuniform. Hence, in many applications, the spatialand temporal variation of water stages and flow discharges needto be determined. Unsteady flows in river systems are typicallysimulated using one-dimensional models, although two- and three-dimensional models are now being used more frequently. In a one-dimensional framework, unsteady flows in rivers are typicallysimulated by the Saint-Venant equations, the pair of partial differ-ential equations representing conservation of mass and momentumfor a control volume. These equations are as follows:

∂A∂t þ

∂Q∂x ¼ 0 ð1Þ

1

g∂V∂t þ

∂∂x

�V2

2g

�þ cos θ

∂y∂xþ Sf − So ¼ 0 ð2Þ

where x = distance along the channel in the longitudinal direction;t = time; Q = discharge; A = cross-sectional area; y = flow depthnormal to x; θ = angle between the longitudinal bed slope and ahorizontal plane; g = cceleration of gravity; So = bed slope; andSf = friction slope. In Eq. (2) (momentum equation), the first,second, third, fourth, and fifth terms represent the local accelera-tion, convective acceleration, pressure gradient, friction, and grav-ity terms, respectively.

The Saint-Venant equations are typically solved for appropriateinitial and boundary conditions to simulate the spatial and temporalvariation of water stages and flow discharges resulting from floodrouting. At present, no analytical solution for the Saint-Venantequations is known, except for special conditions (e.g., dam breakflow over a dry bed in a frictionless and horizontal channel). Hence,solutions of general open-channel flow conditions such as thosefound in practical applications are sought numerically. Solutionsto the full dynamic, one-dimensional Saint-Venant equations andtheir quasi-steady, noninertia (or diffusion), and kinematic waveapproximations [details on these approximations can be found,for example, in Yen (1986)] have been sought based on severalnumerical schemes and methods [e.g., Abbot and Basco (1989)].As emphasized in González-Castro (2000) and González-Castroand Yen (2012), despite the wide array of methods available forthe solution of the Saint-Venant equations, the lack of robustnessand accuracy issues still poses a problem.

Another important issue with unsteady flow models is computa-tional burden, especially when an unsteady model is used for opti-mization problems such as real-time operation of regulated river

1Assistant Professor, School of Civil and Construction Engineering,Oregon State Univ., 220 Owen Hall, Corvallis, OR 97331 (correspondingauthor). E-mail: arturo.leon@oregonstate.edu

2Hydraulic Engineer, Ausenco-Vector, Calle Esquilache 371,San Isidro, Lima, Perú. E-mail: Akemi.Kanashiro@ausenco.com

3Principal Engineer, Hydro-Data Management, South Florida WaterManagement District, 3301 Gun Club Rd., West Palm Beach, FL33406. E-mail: jgonzal@sfwmd.gov

Note. This manuscript was submitted on December 10, 2011; approvedon September 19, 2012; published online on September 22, 2012. Discus-sion period open until August 1, 2013; separate discussions must be sub-mitted for individual papers. This paper is part of the Journal of HydraulicEngineering, Vol. 139, No. 3, March 1, 2013. © ASCE, ISSN 0733-9429/2013/3-284-295/$25.00.

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systems [e.g., Leon et al. (2012)]. In this case, hundreds or eventhousands of runs need to be performed for each operational deci-sion (30 min), which would require numerically-efficient modelsfor unsteady flow routing or a large number of computer processors(clusters). Even if the simulations are run on computer clusters,there is no guarantee that hydraulic routing models will workfor all ranges of conditions (e.g., low stage flows up to flows inthe floodplains). In the writers’ experience, under some simulationconditions, most of the existing routing models fail to converge toa solution. In particular, the widely known unsteady HEC-RASmodel (Hydrologic Engineering Center 2010b), which has beenfound to converge for a range of conditions, fails to converge undersome conditions. When HEC-RAS fails to converge, it proceedswith the simulation based on assumed prespecified conditions(e.g., critical flow), which may yield questionable results.

In the last three decades, Ben Chie Yen’s research group at theUniversity of Illinois at Urbana-Champaign extended the conceptof delivery curves introduced by Bakhmeteff (1932). Yen’s groupproposed a general approach to summarize the dynamic relationbetween the water surface elevations (stages) or depths at the endsof a channel reach (e.g., rivers or canals) for different constantdischarges under gradually varied flow (GVF) conditions (e.g., Yenand González-Castro 2000; Schmidt 2002). This approach wascalled the hydraulic performance graph (HPG). The HPG is aset of curves of constant discharge known as hydraulic performancecurves (HPC). Each HPC defines the locus of the upstream anddownstream water depths in a channel reach for a given constantflow discharge. An example of an HPG for a mild-sloped channel isdepicted in Fig. 1. The HPG shown in Fig. 1 has few HPCs; inactual applications, the number of HPCs must be set based on aprecision goal. This must be decided based on a convergence analy-sis, which consists of successive refinement of the resolution of theset of HPCs to a resolution such that the solution for the conditionsof interest (e.g., stage and flow hydrographs at a given station)becomes nearly independent of the number of HPCs used. The pro-cedure to determine the optimal resolution of HPCs to ensure that aprescribed accuracy is afforded is outside the scope of this paper.

Hydraulic performance graphs can be used to summarizegradually varied subcritical and supercritical flows. However,they have been mostly applied to summarize subcritical GVF inchannel reaches with steep, mild, adverse, and horizontal slopes

[see methodology in Yen and González-Castro (2000) and Schmidt(2002)]. The construction of the HPG for each reach may involvehundreds of GVF simulations, each simulation corresponding toone discrete point on the HPG. When using a one-dimensionalmodel for constructing the performance graphs, any GVF modelcan be used. In the present application, the steady HEC-RAS modelwas used to generate the HPG/volume performance graphs (VPG).

Hydraulic performance graphs have been applied to solveproblems in open-channel flows including the (1) evaluation ofhydraulic performance of floodplain channels under pre andpostbreached levee conditions (González-Castro and Yen 1996);(2) assessment of the carrying capacity of channel systems in series(Yen and González-Castro 2000); and (3) theoretical developmentof discharge ratings based on the hydrodynamics of unsteady andnonuniform flows (Schmidt 2002). González-Castro (2000) as-sessed the applicability of HPGs for unsteady flow routing in singleprismatic channels and channel systems in series with successfulresults. The unsteady approach of González-Castro (2000) assumesthat the flow is steady at the different time steps of the simulation.More recently, Hoy and Schmidt (2006) relied on the volumeperformance graph instead of a finite-difference scheme like thefour-point implicit finite-difference scheme used by González-Castro (2000) to satisfy the reach-wise mass conservation duringrouting. The VPG approach is equivalent to enforcing Eq. (1) (con-servation of mass) in a reach [see details in Hoy and Schmidt(2006)]. An example of a VPG for a mild-sloped channel isdepicted in Fig. 2. The HPG and VPG are unique to a channel reachwith a given geometry and roughness, and can be computeddecoupled from unsteady boundary conditions by solving theGVF equation for all feasible conditions in the reach. They areessentially a fingerprint of all gradually varied flow conditionsin a channel reach. Consequently, HPG/VPGs need to be revisedonly when geomorphic changes modify the geometry or roughnesscharacteristics of the channel (Yen and González-Castro 2000).A significant advantage of the HPG approach with respect to otherrouting models is that the results are a little sensitive to space andtime discretization (González-Castro 2000). Even though the per-formance graph approach is robust, previous models based on theHPG/VPG engine were formulated only for a single channel, orchannels in series that are not suitable for routing in dentriticand looped networks.

Fig. 2. Example of a VPGFig. 1. Example of an HPG

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The present work extends the application of the performancegraph approach for unsteady flow routing in river networks. In asimilar fashion to the unsteady approach of González-Castro(2000), the model introduced in this paper, referred to as theOregon State University unsteady routing (OSU UR) model, as-sumes that the flow is timewise steady at the different time stepsof the routing. The application presented in this paper is limited tosubcritical flows; however, it can be extended to supercritical flows.In addition to relying on the HPG/VPG concept, the OSU unsteadyrouting model also makes use of what is referred to as the ratingperformance graph (RPG). Rating performance graphs graphicallysummarize the dynamic relation between the flow through and thestages upstream and downstream of an in-line structure. An exam-ple of an RPG is depicted in Fig. 3. Rating performance graphsare conceptually similar to look-up tables such as those utilizedto characterize the dynamics of hydraulic structures in the FullEquations model of Franz and Melching (1997b). However, RPGsare described with an adaptive spacing so as to capture changessmoothly, which leads to better interpolation estimates. Furtherdetails on RPGs are presented in the next section. The paper isorganized as follows: (1) formulation of the OSU unsteady routingmodel; (2) the OSU unsteady routing model and the unsteady HEC-RAS model are applied to inflow hydrograph routing for a test casein the Applications Guide of the HEC-RAS model (HydrologicEngineering Center 2010a); and (3) the results of these modelsare compared and discussed.

Formulation of the OSU Unsteady Routing Model

As mentioned previously, the OSU unsteady routing model is builton the performance graphs approach. Hydraulic performancegraphs and volume performance graphs are obtained for each chan-nel reach for as many flows and downstream boundary conditionsas necessary to cover the region of possible pairs of upstream anddownstream stages in the reach. The maximum water stages aregiven by the elevations of the channel banks, floodplain levees,or other topographic features. The HPG/VPGs of a system can alsoinclude the inundation of urban areas (see Fig. 4).

Overall, the limitations of the OSU UR model are the same asthose of unsteady flow routing models based on the Saint-Venantequations. In addition, the present version of the OSU UR modeldoes not have the capability to model dry-bed flows. Furthermore,the performance graphs approach applies to flow routing when the

contribution of the local acceleration to the momentum balance isnegligible. This condition is met by flows in many natural andartificial systems. As noted by Henderson (1966), even for a steepriver with a very fast-rising flood the local acceleration term issmall compared to the gravity and friction terms. Actually, therelative contribution of the local acceleration with respect to thepressure gradient is in the order of the Froude number squared.Typically, the maximum Froude number of mild-slope unregulatedriver systems and regulated river systems is much smaller than one.A flow chart of the OSU unsteady routing model is presented inFig. 5, which comprises six main modules. A brief descriptionof these modules is presented in the next section.

Module I: Definition of River Network

In this module, data of nodes and river reaches that define the riversystem network are read and stored for later use. In the OSU un-steady routing model, a river system is represented as a network inwhich all components of the river system are defined by riverreaches and nodes. A river reach is defined by its upstream anddownstream nodes and should have similar geometric propertiesalong the reach (e.g., prismatic channel) to ensure GVF conditions.Whether a channel reach is changing gradually enough so that flowthrough it can be treated as flow in a prismatic channel must beassessed based on a more general form of the ordinary differential

Fig. 3. Example of an RPG

Fig. 4. Schematic of a river cross section

Computation of HPGs and VPGs for all river reaches and RPGs for hydraulic structures

Boundary conditions (BCs)

River system routing: solve a system of non-linear equations assembled based on systems’ HPGs, VPGs and RPGs, flow continuity at nodes, compatibility conditions of water stages and system BCs.

End

t = t +∆tHas

simulation been completed?

No

Yes

Definition of river network: nodes and river reaches

Initial conditions

Module I

Module II

Module IV

Module III

Evaluation of time step (∆t)Module V

Module VI

Start of OSU Routing Model

Fig. 5. Flowchart of the OSU unsteady routing model

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equation (ODE) for GVF conditions. González-Castro (2000)discussed this issue based on the following more general ODEfor GVF in nonprismatic channels:

dydx

¼ So − Sf þ ðF2=cos2θÞðD=TÞðdT=dxÞcos θ − F2

ð3Þ

The third term in the numerator of Eq. (3) accounts for changesin nonprismatic channels. From this equation, it is clear that for acanal to behave as prismatic,

So − Sf ≫F2

cos2θDTdTdx

ð4Þ

where T = free surface width; D = hydraulic depth (¼ A=T); andF = Froude number. The criterion in Eq. (4) is met by subcriticalflows in canals with mild bed slopes for which F2=cos2θ ¼ O (0.1)and D=T ¼ O (0.1), even when dT=dx ¼ O (So - Sf).

The flow direction in a river reach is assumed to be from itsupstream node to its downstream node, as shown in Fig. 6. In Fig. 6,the subscript j and superscript n represent the river reach index andthe discrete-time index, respectively; and y and Q with the sub-scripts u and d denote the water depths and discharges at theupstream and downstream ends of the river reach, respectively.A negative flow discharge in a river reach indicates that reverseflow occurs in that river reach. A node may have inflowing and/or outflowing river reaches. A river reach is denoted as inflowing(or outflowing) when it conveys to (or from) the node. A node inthe proposed model refers to point nodes that have no storage.A node is used at the location of hydraulic structures, connectionof reaches, and boundary conditions. In the OSU routing model, areservoir is not represented by a node, but by one or a series ofreaches. The treatment of boundary conditions is presented inthe “Module IV” section.

Module II: Computation of the Performances Graphs

In this module, HPGs and VPGs for all the reaches of the riversystem and RPGs for the hydraulic structures are computed andstored for later use. In the present application, the system’s perfor-mance graphs (PG) were constructed using the HEC-RAS steadymodule. The subcritical flow mode of the steady HEC-RAS modelallows multiple steady subcritical flow simulations (e.g., usingmultiple discharges) with a fixed downstream water depth as initialcondition. This allows a rapid construction of the performancegraphs.

Once the PGs are constructed, they are plotted to ensure thatthey are free of numerically induced errors. Numerical errorsmay result in the superposition of PCs or in PCs that display os-cillatory patterns (see Fig. 7). For the discrete points of an HPC thatpresents apparent problems, the simulations must be repeated withmore stringent criteria. This requires decreasing Δx (interpolationbetween cross sections) and adjusting convergence parameters forthe GVF simulations. This initial screening of the PGs results in theelimination of the aforementioned oscillatory patterns and superpo-sition of HPCs. For a detailed description on the construction of

HPGs, the reader is referred to Yen and González-Castro (2000)and Schmidt (2002).

Module III: Initial Conditions

The initial conditions in the OSU unsteady routing model foreach river reach in the steady case can be specified by two of thevariables describing the reach’s HPG, i.e., the water stages at thereach’s end, or one of the stages and the flow discharge in the reach.In the case that the simulation starts from nonsteady conditions, theinitial conditions must be defined as the combination of the stagesat the reach’s end and the flow at one of its ends, or the flows at thereach’s ends and the stage at one of its ends.

Module IV: Boundary Conditions

The following types of boundaries are supported by the OSU un-steady routing model:1. External boundary condition (EBC), which is defined at the

most upstream and downstream ends of the river system.An EBC can have either an inflowing or outflowing river reachconnected to the node. An external boundary includes aninflow hydrograph, a stage hydrograph, or a rating curve.

2. Internal boundary condition (IBC), which is defined at internalnodes whenever two or more reaches meet. Three types ofIBCs are supported in the OSU unsteady routing model. Theseare:• A fixed in-line structure BC [e.g., weirs or dams with fixed

position of gates; see Fig. 8(a)]. A single RPG is built forthis BC. The water depth immediately upstream of the in-line structure is computed using the RPG built for thisstructure. For building the RPG of an in-line structure(fixed and mobile), the most upstream and downstreamcross sections of the simulation must be kept close tothe in-line structure to avoid large errors of conservationof mass. The criteria for the selection of cross sectionsfor the construction of look-up tables can be found in Franzand Melching (1997a). This criteria also applies to theconstruction of RPGs.

Under simple conditions, good rating equations areprobably easier to manage than both RPGs and look-uptables. However, for the operation of multitype hydraulicstructures under complex flow conditions (e.g., downstreamFig. 6. Schematic of a river reach

Fig. 7. Typical instability problems during construction of HPGs

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flow conditions ranging from low to high water stagesand/or vice versa), relying on RPGs (look-up tables) mayexpedite convergence during the simulation of unsteadyflow. Both RPGs and look-up tables can be constructedbased on measurements of flow discharge and water stagesand/or results from computational fluid dynamics simula-tions. As pointed out by Franz and Melching (1997a, b), theuse of look-up tables (or RPGs) can be used to model thedynamics of a variety of hydraulic structures such weirs,culverts, spillways, bridges, and canal expansions and con-tractions. The analyst only needs an adequate descriptionof the relation between the flow through the structure andthe water-surface elevation downstream and upstream ofthe structure. For an application of RPGs to the openingand closing of gates, the reader is referred to Leon et al.(2012).

• A mobile in-line structure BC (e.g., spillways and culvertswith control gates and pumping stations with pumps of dif-ferent capacities or operated at variable speeds). Dams witha combination of tainter and roller gates are often found incanal networks and dams. In some cases, the roller gates atdams can be raised (underflow gate) or lowered (overflowgate). These gates are often operated using predefined op-erating rules that specify the number of gates to open andpercentage of openings while addressing issues such asscour, safety, and outdraft. Complex arrays and operationscan be handled by using a group of RPGs (e.g., one for eachgate opening) for each gate or prespecified combinations ofgates, depending on how they will be operated. The RPGswill need to encompass the entire range of gate openingsand all possible ranges of downstream and upstream waterlevels. Naturally, a given flow discharge can be passedthrough the dam with more than one combination ofoperational settings. If there is any prespecified order forthe operation of gates, this should be linked to the orderof use of the RPGs. For the application of RPGs to mobilein-line structures, the reader is referred to Leon et al.(2012).

• Another IBC is a node without a hydraulic structure thatis used to connect two or more river reaches with differentroughness or bed slopes, or where an abrupt bed drop

or canal expansion occurs. This node is denoted as a junc-tion BC, and its schematic is depicted in Fig. 8(b).

Module V: Evaluation of Time Step (Δt)This module estimates the time step for advancing the solution tothe next time level. In the PG approach, the reach-wise averagedflow discharge [1= � ðQu þQdÞ] and yd are used to determine yu.Hence, the time step must be long enough to ensure that disturb-ances generated at the ends of the reaches have enough time toarrive to the opposite end. This can be expressed as

Δt > Max

�Δxj

jjunj jj þ cnj

�∀ j ð5Þ

where j = reach index; and un and cn = average reach velocity andgravity wave celerity at time level n, respectively. The averagereach-wise gravity wave celerity and velocity are defined as c ¼ðcu þ cdÞ=2 and u ¼ ðuu þ udÞ=2, respectively. A Δt smaller thanthat of Eq. (5) would mean that disturbances in some of the reachesdo not have enough time to travel from one end of the reach to theother, which would violate one of the HPG/VPG assumptions(HPG/VPG use reach-wise averaged flow variables). The time steppresented in Eq. (5) can be expressed as

Δt ¼ kMax

�Δxj

jjunj jj þ cnj

�∀ j ð6Þ

where k must be set larger than 1. Numerical tests performed toevaluate the sensitivity of k for both simulations in the section“Application to a Looped River Network” showed that the resultsare nearly insensitive to k. The simulation results presented in thispaper were generated using k ¼ 3.

Module VI: River System Hydraulic Routing

This module assembles and solves a nonlinear system of equationsto perform the hydraulic routing of the river system. These equa-tions are assembled based on information summarized in thereaches HPGs and VPGs, RPGs at nodes with hydraulic structures,continuity and compatibility of water stages at junctions, and thesystem’s initial and boundary conditions. The compatibility ofwater stages at a junction is a simplification of the energy equationignoring losses and assuming that the differences in velocity heads[u2=ð2gÞ] upstream and downstream of the junction are negligible.

In general, for a river network consisting of N reaches, there is atotal of 3 N unknowns at each time level, namely, the flow dis-charge at the upstream and downstream end of each reach andthe water depth at the downstream end of each reach; hence,3 N equations are required. The water depth at the upstream end(yu) of each reach is estimated from the reach HPG using the waterdepth at its downstream end (yd) and the spatially averaged flowdischarge ½Q ¼ ðQu þQdÞ=2�. This can be represented as

ynu ¼ HPG

�ynd;

1

2ðQn

u þQndÞ�; ∀ j ð7Þ

The application of Eq. (7) requires an interpolation process fordetermining yu. Two cases of interpolation are possible. Theseare depicted in Figs. 9 and 10. As illustrated in these figures, thelocus of the upstream and downstream water depth for a given con-stant flow discharge is denoted as hydraulic performance curve(HPC).

(a)

(b)

Fig. 8. Schematic of (a) in-line structure; (b) junction node

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The first case of interpolation is used whenever ydx is locatedbetween two critical downstream water depths (ydc1 and ydc2 ), asshown in Fig. 9. In this interpolation case (Fig. 9), the discretepoints [yda; yuðQ1; ydaÞ], [ydb; yuðQ1; ydbÞ], (ydc1 ; yuc1 ), and(ydc2 ; yuc2 ) are known, and yux for a given yux and Qx is sought.

The second interpolation case is used for all conditions otherthan the first case (Fig. 10). In the second case interpolation case(Fig. 10), the discrete points [yda; yuðQ1; ydaÞ], [yda; yuðQ2; ydaÞ],[ydb; yuðQ1; ydbÞ], and [ydb; yuðQ2; ydbÞ] are known, and yux for agiven ydx and Qx is sought.

The interpolation procedure to determine the upstream waterdepth yux for a given Qx and ydx for the first case (Fig. 9) canbe summarized as follows:1. Determine coefficient c1 as

c1 ¼Qx −Q1

Q2 −Q1

2. Determine slope s1 of HPC for Q1 as

s1 ¼yuðQ1; ydaÞ − yuc1

yda − ydc1

3. Determine slope s2 of HPC for Q2 as

s2 ¼yuðQ2; ydaÞ − yuc2

yda − ydc2

4. Determine slope sx of HPC for Qx as

sx ¼ s1ð1 − c1Þ þ s2c1

5. Determine yuðQx; ydaÞ for Qx and yda as

yuðQx; ydaÞ ¼ ð1 − c1Þ½yuðQ1; ydaÞ� þ c1½yuðQ2; ydaÞ�

6. Determine yux for Qx and ydx as

yux ¼ yuðQx; ydaÞ − ðyda − ydxÞsxThe interpolation procedure to determine the upstream water

depth yux for a given Qx and ydx for the second case (Fig. 10)can be summarized as follows:1. Determine coefficient c1 as

c1 ¼Qx −Q1

Q2 −Q1

2. Determine coefficient c2 as

c2 ¼ydx − ydaydb − yda

3. Determine yuðQx; ydaÞ for Qx and yda as

yuðQx; ydaÞ ¼ yuðQ1; ydaÞ þ c1½yuðQ2; ydaÞ − yuðQ1; ydaÞ�

4. Determine yuðQx; ydaÞ for Qx and yda as

yuðQx; ydbÞ ¼ yuðQ1; ydbÞ þ c1½yuðQ2; ydbÞ − yuðQ1; ydbÞ�

5. Determine yux for Qx and ydx as

yux ¼ yuðQx; ydaÞ þ c1½yuðQx; ydbÞ − yuðQx; ydaÞ�As mentioned previously, 3 N equations are required for a river

system of N reaches. For illustration purposes without losinggenerality, these equations are formulated for the simple networksystem depicted in Fig. 11. The network system presented in Fig. 11has eight reaches and therefore has 24 unknowns (3 × 8).

The reach-wise conservation of mass provides one equation foreach reach. This equation is typically discretized as follows:

In þ Inþ1

2−On þOnþ1

2¼ Snþ1 − Sn

Δt; ∀ j ð8Þ

where I = inflow;O = outflow; S = storage; n = value at the currenttime level, t; and nþ 1 = value at the next time level, tþΔt.The storage S is not considered an unknown because it can berelated to yd and Q through the VPG [VPG relates S, yd,and QðQu þQdÞ=2].

For this study’s simple network (Fig. 11), the application ofEq. (8) provides a total of eight equations. For an inflow hydro-graph (external boundary), ðIn þ Inþ1Þ=2 is the average flowdischarge computed from the hydrograph (integrated volumedivided by Δt). The water storage (volume of water) at any timein a river reach is determined from the reach VPG using its down-stream water depth and the spatially averaged flow dischargeQðQu þQdÞ=2 as input values as

Sn ¼ VPG

�ynd;

1

2ðQn

u þQndÞ�; ∀ j ð9Þ

For more details on the VPG approach, the reader is referred toHoy and Schmidt (2006). The application of Eq. (9) requires an

Fig. 9. Schematic of first interpolation case

Fig. 10. Schematic of second interpolation case

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interpolation process similar to that of the HPG presented previ-ously [see Eq. (7)]. Also, for the network in Fig. 11, five continuityequations are available (nodes B, C, D, E, and G). For instance,at node C, the continuity equation is given by

Qd2 ¼ Qu3 þQu6 ð10Þ

The system under study has three external boundary conditions(A, F, and H). These EBCs could be, for instance, inflow hydro-graphs [QðtÞ], stage hydrographs [yðtÞ], overflow structures withhydraulic controls for which there is a flow-stage relation, orany other boundary prespecified by the user. For each ofthese EBCs, a flow variable [e.g., QðtÞ, yðtÞ] or an equation isavailable.

For this example, so far 16 equations are available, and eightmore equations are needed. Six of the remaining eight equationscan be obtained by enforcing water stage compatibility conditionsat nodes that connect two or more river reaches and that do nothave any hydraulic structure associated to the node. In thiscase, if a node is connected to k river reaches, k − 1 water stagecompatibility conditions are available for the junction node. Theseconditions enforce the same elevation for the water stages immedi-ately upstream and downstream of the node [Fig. 8(b)]. Forinstance, at node C, two water stage compatibility conditionsare available as

zd2 þ yd2 ¼ zu3 þ yu3 zd2 þ yd2 ¼ zu6 þ yu6 ð11Þ

In Eq. (11), zd and zu are the reach bottom elevations immedi-ately downstream and upstream of a junction node, respectively.In the case of an abrupt change in channel geometry or abruptbed drop at the junction node, the energy equation instead ofthe water stage compatibility condition should be used. If a hy-draulic structure is associated to the node, the equation is obtainedfrom the RPG of the hydraulic structure. In this example, the lasttwo equations are obtained from RPGs at in-line structures (nodesB and E in Fig. 11). The treatment of fixed and mobile in-linestructures in the OSU unsteady routing model are the same. Theonly difference is that a single RPG is used for a fixed in-line struc-ture, whereas as a group of RPGs (depending on discrete gatepositions) are required for a mobile in-line structure. For instance,at node B, the water stage upstream of the structure is obtained fromthe RPG of the structure, as follows (see Fig. 11)

yd1 ¼ RPG

�yu2 ;

1

2ðQd1 þQu2Þ

�ð12Þ

The application of Eq. (12) requires an interpolation processsimilar to that of the HPG for a river reach presented previously(Eq. 7). For solving the resulting nonlinear system of equations,the OSU UR model uses the Open Source C/C++ MINPACKcode. MINPACK solves systems of nonlinear equations, or carriesout the least-squares minimization of the residual of a set of linearor nonlinear equations. For more details on this library, the reader isreferred to Moré et al. (1984).

Fig. 11. Schematic of a simple network system

Fig. 12. Plan view of HEC-RAS looped river system

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Application to a Looped River Network

To illustrate the use of the OSU unsteady routing model, this modelhas been applied to a looped river system adapted from an examplein the Applications Guide of the HEC-RAS model (HydrologicEngineering Center 2010a). The plan view of this system is de-picted in Fig. 12, and the geometric characteristics of the 26 reachesthat compose the system are presented in Table 1. To assess theOSU unsteady routing model, two test cases were simulated.For each case, the results of the OSU unsteady routing model werecompared with the results from the unsteady HEC-RAS modelversion 4.0. For this comparison, an inflow hydrograph and a ratingcurve boundary condition were specified at the most upstream(node 1) and most downstream (node 26) ends of the system, re-spectively (Figs. 13 and 14, respectively). The inflow hydrograph(Fig. 13) for the first case represents a slow flood-wave, whereasthe one for the second test case represents a fast flood-wave. Allperformance graphs used in the applications (HPGs and VPGs)were generated using the steady HEC-RAS model.

To determine the initial conditions in the two subcritical flowtest cases, a steady-flow discharge of 1.395 m3=s with a waterdepth of 1.65 m at the downstream end of reach 26 (see Fig. 12)was used. These values of water depth and flow discharge wereused for determining the water depths and flow discharges at theends of every reach. The water depth and flow discharge at the endsof each reach are the necessary initial conditions.

Case 1: Slow Flood-Wave

The simulated flow and stage hydrographs at two locationsobtained with the OSU unsteady routing model are compared with

Table 1. Geometric Characteristics of River Reaches

Reach ID Length (m) zu (m) zd (m) Slope (m=m)

1 35.05 6.2332 6.1631 0.0020002 35.05 6.1631 6.0930 0.0020003 38.10 6.0930 6.0198 0.0019204 32.00 6.0198 5.9497 0.0021905 24.38 5.9497 5.8887 0.0025006 39.62 5.8887 5.8339 0.0013857 39.62 5.8339 5.7790 0.0013858 45.72 5.7790 5.7241 0.0012009 44.20 5.7241 5.6693 0.00124110 32.00 5.6693 5.6144 0.00171411 36.58 5.6144 5.5596 0.00150012 39.62 5.5596 5.5047 0.00138513 25.91 5.5047 5.4712 0.00129414 21.34 5.9497 5.8979 0.00242915 51.82 5.8979 5.8491 0.00094116 54.86 5.8491 5.7760 0.00133317 48.16 5.7760 5.7638 0.00025318 56.39 5.7638 5.7028 0.00108119 56.39 5.7028 5.6510 0.00091920 54.86 5.6510 5.5839 0.00122221 57.00 5.5839 5.5535 0.00053522 57.91 5.5535 5.5047 0.00084223 21.34 5.5047 5.4712 0.00157124 47.24 5.4712 5.4315 0.00083925 42.67 5.4315 5.4132 0.00042926 51.82 5.4132 5.3950 0.000353

Fig. 13. In-flow hydrograph at node 1 for the slow and fast flood-wavecases

Fig. 14. Rating curve boundary at node 26

0 25 50 75 1000

5

10

15

20

Time (h)

Flo

w d

isch

arge

(m

3 /s)

HEC-RAS

Reach 4(downstream)

Reach 18(downstream)

Fig. 15. Flow hydrograph at downstream end of reaches 4 and 18 forcase 1

0 25 50 75 1007.0

7.2

7.4

7.6

7.8

Time (h)

Wat

er s

tage

(m

)

HEC-RAS

Reach 4(downstream)

Reach 18(downstream)

Fig. 16. Stage hydrographs at downstream end of reaches 4 and 18 forcase 1

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the results obtained with HEC-RAS in Figs. 15 and 16, respec-tively. As can be observed in these figures, the difference betweenthe OSU unsteady routing model results and the HEC-RAS resultsis rather small.

To evaluate the discrepancies in flow discharge, water stage, andvolume between the OSU unsteady routing and the HEC-RASmodel results, the following relative differences were defined:

EQð%Þ ¼ 100

�QOSUUnsteady −QHEC−RAS

QInflowHidromax −QInflowHidromin

�

EWSð%Þ ¼ 100

�WSOSUUnsteady −WSHEC−RAS

WSHEC−RAS max −WSHEC−RAS min

�

EVð%Þ ¼ 100

�VOSUUnsteady − VHEC−RAS

VHEC−RAS

�ð13Þ

where EQ, EWS, and EV = relative differences in flow discharge,water stage, and cumulative outflow volume, respectively. Notein Eq. (13) that the discrepancies in flow discharge and water stageare defined as the difference of results between the two modelsnormalized by the range of flow discharges or water stages.

The relative differences defined by Eq. (13) are shown inFigs. 17–19 for the flow discharge, water stage, and cumulativeoutflow volume (V) at the downstream end of reaches 9 and 18,respectively. The relative differences in flow discharge ranged from−0.6 to 0.6%, in water stage it varied from −0.25 to 0.30%, and incumulative outflow volumes varied from −1.0 to 1.5%. Althoughthe discrepancies between the results of HEC-RAS and theOSU routing model are negligible, the maximum discrepancies

in discharge and stage occur near the time of the peak discharge(see Figs. 17 and 18). The local acceleration term may be respon-sible for the discrepancies, because the local acceleration term ismore important in a sudden rising and falling of the flow (i.e., nearpeak flow). The unsteady HEC-RAS model accounts for thelocal acceleration term, whereas the OSU routing model neglectsthis term.

The relative differences in flow discharge and cumulativeoutflow volume for reaches 9 and 18 shown in Figs. 17 and 19nearly complement one another. Complementary results are tobe expected when the model of reference (HEC-RAS in this case)is exact and when the two branches of the loop are identical (samenumber of reaches, all with identical geometry and roughness).In this application example, the two branches of the loop are notidentical. The relative differences in flow discharge and cumulativeoutflow volume for the case of slow flood-wave (Figs. 17 and 19)were near complimentary, likely because the model of reference(HEC-RAS) was highly accurate in this case.

Case 2: Fast Flood-Wave

The flow and stage hydrographs at two locations simulated with theOSU unsteady routing and the HEC-RAS models are shown inFigs. 20 and 21, respectively. Simulation results obtained with bothmodels appear to be similar. However, the differences are notice-ably more significant for this case than for those with the slow

0 25 50 75 100-0,8

-0,4

0.0

0,4

0,8

Time (h)

Flo

w d

isch

arge

dif

fere

nce

(%)

Reach 18(downstream)

Reach 9(downstream)

Fig. 17. Difference in flow discharges at downstream end of reaches 9and 18 for case 1

0 25 50 75 100

-0,2

0.0

0,2

0,4

Time (h)

Wat

er s

tage

dif

fere

nce

(%)

Reach 9(downstream)

Reach 18(downstream)

Fig. 18. Difference in water stages at downstream end of reaches 9 and18 for case 1

0 25 40 75 100-2

-1

0

1

2

Time (h)

Cum

ulat

ive

volu

me

diff

eren

ce (

%)

Reach 18(downstream)

Reach 9(downstream)

Fig. 19. Difference in conservation of volume at downstream end ofreaches 9 and 18 for case 1

0 50 100 150 2000

5

10

15

20

Time (min)

Flo

w d

isch

arge

(m

3 /s)

HEC-RAS

Reach 18(downstream)

Reach 4(downstream)

Fig. 20. Flow hydrographs at downstream end of reaches 4 and 18 forcase 2

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flood-wave conditions of case 1. Figs. 22–24 show the differencesin flow discharge, water stage, and cumulative outflow volumebetween OSU unsteady routing and HEC-RAS [according withEq. (13)] at the downstream end of reaches 9 and 18. As shownin Figs. 22–24, the relative difference in flow discharge rangedfrom −1.5 to 0.7%, in water stage from −0.7 to 0.5%, and incumulative outflow volume from −1.0 to 5.5%.

Fig. 25 shows the plot of the flow discharge versus water stage(i.e., rating curve) at the downstream end of reach 18 for the fastflood-wave case (case 2). This figure shows a typical looped ratingcurve having greater flows at lower stages in the rising limb and

smaller flows at higher stages in the receding limb. The rating curvefor the slow flood-wave case is similar to that of the fast flood-wavecase; however, it is not shown because of space limitations.

To assess the robustness of the OSU unsteady routing modelwith respect to time discretization, the simulations were carriedout with three different time resolutions (t ¼ 10, 30, and 60 s).Fig. 26 shows the results of numerical accuracy of the OSUunsteady routing model attributable to time discretization for theupstream end of reach 14 and downstream end of reach 23. Thisfigure appears to show that time discretization does not signifi-cantly affect the results. With regard to space and water depth dis-cretization (Δx andΔy, respectively), the maximum value ofΔx orΔy used to obtain a good accuracy is system dependent and shouldbe obtained for each system by iteration. For instance, for Δx,two values of Δx can be used to check if the simulated results ofdischarge and water stage are similar for both discretizations. Theprocess can be repeated to find the largest Δx that produces thedesired accuracy, and in turn the minimum central processing unit(CPU) time. The reader is referred to González-Castro (2000) foran in-depth discussion on spatial discretization.

The results for the CPU times obtained using a Dell PrecisionT3500 2.67 GHz, 1.00 GB of RAM for the OSU unsteady routingand the HEC-RAS models are presented in Table 2. The CPU timein Table 2 included the time of preprocessing and computationalengine, but not that of postprocessing. The preprocessing in theOSU routing model involves the reading of performance graphs(from. dat files into matrices), whereas in HEC-RAS, it involvespreprocessing geometric and hydraulic data for import into

0 50 100 150 2007.0

7.2

7.4

7.6

7.8

Time (min)

Wat

er s

tage

(m

)

HEC-RAS

Reach 4 (downstream)

Reach 18 (downstream)

Fig. 21. Stage hydrographs at downstream end of reaches 4 and 18 forcase 2

0 50 100 150 200-3.0

-1.5

0.0

1.5

3.0

Time (min)

Flo

w d

isch

arge

dif

fere

nce

(%)

Reach 9(downstream)

Reach 18(downstream)

Fig. 22. Difference in flow discharges at downstream end of reaches 9and 18 for case 2

0 50 100 150 200-1.0

-0,5

0.0

0,5

1.0

Time (min)

Wat

er s

tage

dif

fere

nce

(%) Reach 18

(downstream)

Reach 9(downstream)

Fig. 23. Difference in water stages at downstream end of reaches 9 and18 for case 2

0 50 100 150 200-2,5

0.0

2.5

5.0

7.5

Time (min)

Cum

ulat

ive

volu

me

diff

eren

ce (

%)

Reach 18(downstream) Reach 9

(downstream)

Fig. 24. Difference in conservation of volume at downstream end ofreaches 9 and 18 for case 2

0.0 2.5 5.0 7.5 10.07.0

7.2

7.4

7.6

Flow discharge (m3/s)

Wat

er s

tage

(m

)

HEC-RAS

Start

End

Fig. 25. Flow discharge versus water stage at downstream end of reach18 for case 2

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HEC-RAS. The postprocessing typically demands more time, andit depends on the user-specified outputs. As can be observed inTable 2, the results obtained with the OSU unsteady routing modelare approximately 700% faster than that of the HEC-RAS model. Inthe OSU unsteady routing model, for the slow flood-wave, the timestep (Δt) ranged from 8.16–12.62 s, whereas for the fast flood-wave, the time step ranged from 8.31–12.62 s. For both flood-waves, the HEC-RAS model was simulated using a time stepof 10 s.

Conclusions

This paper presented a computationally efficient model for un-steady flow routing through river networks with dendritic, looped,or a combination of dendritic and looped topologies. The applica-tion of the OSU UR model focused on routing of subcritical flows;however, the model can be extended to include reaches suscep-tible to transition from subcritical to supercritical flow (and viceversa) during routing. The model builds upon the application ofhydraulic performance graph to unsteady flow routing introduced

by González-Castro (2000) and adopts the volume performancegraph introduced by Hoy and Schmidt (2006). Moreover, in theOSU UR model, the concept of performance graphs was extendedto ratings, and the rating performance graphs, which graphicallysummarize the dynamic relation between the flow through andthe stages upstream and downstream of in-line structures, wereintroduced. The OSU unsteady routing model solves a system ofnonlinear equations assembled based on information summarizedin the system’s HPGs, VPGs, and RPGs, continuity in junctions,water stage compatibility at junctions of reaches, and the system’sinitial and boundary conditions. The applicability of the OSU un-steady routing model was exemplified to a looped network, and itssimulation results were contrasted with those from the well-knownunsteady HEC-RAS model. The key findings are as follows:1. Results show that agreement between the OSU unsteady

routing and HEC-RAS models is very good for slow and fastflood-wave conditions, with better agreement for slow flood-wave conditions.

2. The use of HPGs, VPGs, and RPGs for unsteady flow routingin a river system as proposed in this paper is a relatively robustand numerically efficient approach, because the momentumand storage for all river reaches are computed prior to thesystem routing based on the momentum and mass conserva-tion principles of GVF, and most of the computations for thesystem routing only involve interpolation steps to satisfy theprescribed BCs. The OSU UR model provided an accuratesolution for the looped system right the first time, whereasthe unsteady HEC-RAS model required few adjustments toget the model to run properly. In addition, when using theperformance graphs approach, instabilities or other problems

Fig. 26. Numerical accuracy of OSU unsteady routing model attributable to time discretization for the upstream end of reach 14 and downstream endof reach 23

Table 2. Comparison of CPU Times for the Simulation of Cases 1 and 2

DescriptionOSU unsteadyrouting time (s)

HEC-RAStime (s)

Case 1 (slow flood-wave) 218.8 (Δt ¼ 9.77 s) 752.7 (Δt ¼ 10 s)Case 2 (fast flood-wave) 7.3 (Δt ¼ 10.27 s) 52.6 (Δt ¼ 10 s)

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attributable to discretization and numerical inaccuracies areremoved as the system’s HPGs, VPGs, and RPGs are beingconstructed.

3. The application examples presented in this paper suggest that,overall, the proposed model is computationally more efficientthan and affords a numerical accuracy comparable to the un-steady HEC-RAS model for unsteady flow routing throughriver networks. It is clear that the CPU time for precomputingthe PGs can be computationally demanding, but this is doneonly once. The advantage of the OSU UR model may besignificant when it is used for optimization problems suchas real-time operation of regulated river systems. In this case,hundreds or even thousands of runs would be needed for eachoperational decision that may be as short as 30 min.

4. Further assessment of the proposed approach includes evalu-ating the sensitivity of results to the algorithms for interpola-tion from the HPGs, VPGs, and RPGs, and to discretizationover space prescribed for the routing.

5. The writers are considering assessing the effect of BCs on theaccuracy and robustness of the OSU UR model as future work.

Acknowledgments

The writers gratefully acknowledge the financial support of theNSF Idaho EPSCoR Program and the National Science Foundationunder award number EPS-0814387. The first writer also would liketo thank the financial support of the School of Civil and Construc-tion Engineering at Oregon State University. The writers gratefullyacknowledge the collegiality of the anonymous reviewers; theirconstructive and insightful comments certainly helped us improvethe manuscript.

Notation

The following symbols are used in this paper:c = average gravity wave celerity;

HPG = hydraulic performance graph;I = inflow;

∀i = for all i;N = set of nodes;

NB = set of downstream boundary nodes;NS = set of source nodes;O = outflow;

Qdj = flow discharge at downstream end of reach j;Quj = flow discharge at upstream end of reach j;S = storage;u = average reach velocity;

VPG = volume performance graph;ydj = water depth at downstream end of reach j;yuj = water depth at upstream end of reach j;zdj = channel bottom elevation at downstream end of reach j;zuj = channel bottom elevation at upstream end of reach j;Δt = time step; andΔx = length of river reach.

Subscripts

d = downstream end index;i = node index;j = river reach index; andu = upstream end index.

Superscripts

n = discrete-time index.

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